The Marine Diesel Propulsion System
Historical Note l
The majority of modern merchant ships, such as containerships and very large crude carriers (VLCCs), rely on marine Diesel engines for propulsion These vessels typically feature a single, long-stroke, slow-speed, turbocharged, two-stroke Diesel engine that is directly connected to a large-diameter, fixed-pitch propeller This design allows for significant power outputs, reaching up to 30-40 MW from a single engine, while also ensuring high operational reliability due to its straightforward configuration.
Since the mid-19th century, the mechanisation of propulsion in shipping has undergone significant evolution Initially, early motor ships utilized side wheels or screw propellers powered by reciprocating steam engines strategically placed in the hull Over time, the industry transitioned to steam turbine powerplants, a shift that was largely completed by the end of World War II.
Today, Diesel engines are the predominant choice for marine propulsion due to three key factors: their superior thermal efficiency compared to other propulsion systems, the ability to burn heavy fuel oil thanks to alkaline cylinder lubrication oils, and the direct connection of slow-speed Diesel engines to the propeller without the need for a gearbox or clutch, allowing for reversibility However, Diesel engines necessitate a larger engine room than gas turbines, their main competitor, as they have lower specific power per unit volume and weight This limitation can pose challenges for applications requiring extremely high power outputs, such as aircraft carriers or large high-speed vessels.
Marine Engine Configuration and Operation
Large merchant vessels can efficiently meet their propulsion needs with a single slow-speed, direct-drive Diesel engine, which is capable of burning low-quality fuels like HFO more effectively than medium-speed Diesel engines due to its larger combustion space and time These slow-speed engines typically feature fewer cylinders and moving parts, enhancing the reliability of the propulsion system.
Figure 1.1 Section of large marine Diesel engine
The main parts of the engine are:
The bedplate is essential for supporting the engine and maintaining the alignment of the shaft, while the crankcase serves as the housing for the crankshaft In larger engines, these components are typically integrated into a single unit, enhancing structural integrity and efficiency.
The crankshaft, a crucial and expensive component of large marine engines, plays a vital role in engine performance Paired with the flywheel, it effectively dampens vibrations caused by the individual firings in each cylinder, ensuring smoother operation and enhanced durability.
The engine body serves as the foundational structure that supports the engine cylinders, ensuring both mechanical strength and flexibility Additionally, it is designed to allow easy human access to the engine's internal components for maintenance and inspection.
• Cylinder blocks and liners: In large marine diesels each cylinder is contained in it's own separate cylinder block
Pistons, comprising the piston crown, piston rings, and piston rod, play a crucial role in transferring mechanical power to the crankshaft via connecting rods.
Cylinder heads play a crucial role in securing the combustion chamber's top and providing essential mechanical support for key engine components, including exhaust valves and fuel injectors.
• Camshaft(s): The camshaft is one of the most critical engine parts because it ensures timing of exhaust valves opening/closure, as well as fuel injection
• Fuelling system: This is comprised of the high-pressure fuel pumps, the high-pressure pipelines and the fuel injectors (there can be more than one for each cylinder)
Diesel engines designed for ship propulsion are optimized to function close to a steady-state along a specific operating curve, commonly referred to as the "fouled-hull" or propeller curve This propeller loading curve, applicable to full-bodied hulls, is characterized by a defined shape that enhances performance efficiency.
The relationship between power (P) in kilowatts, shaft RPM (N), and the propeller law constant (Kpow) is expressed as P = Kpow * N^3 ~ 10g, which can also be represented as P = log Kpow + 3 * log N Additionally, the propeller curve, which describes the torque (Q), follows a specific mathematical form that illustrates its operational characteristics.
The cubic relationship for power, represented by the equation Q = KQ ãN^2 ~ 10gQ = logKQ + 2ã}ogN (1.2), provides an approximation of propeller demands However, this approximation is influenced by various factors, including hull resistance and propulsion components, indicating a more complex functional relationship Despite this complexity, the cubic approximation remains generally valid within limited power ranges.
Engine-propeller matching is achieved by analyzing the engine's operating envelope, which illustrates the relationship between propeller power demands and shaft RPM Key operational limit lines for large marine Diesel engines are influenced by combustion efficiency, represented by the surge or smoke line at low RPM, and the strength of the shafting system bearings, indicated by the torque limit line at higher RPM.
Figure 1.2 Chart used for engine-propeller matching
The propeller demand curve intersects the Maximum Continuous Rating (MCR) of the engine, indicating the optimal engine power output and speed The MCR coordinates on the power-rpm chart signify the maximum continuous power delivery and the engine's speed threshold.
The power absorption characteristics of a ship's propeller can be influenced by various factors, including sea conditions, wind strength, hull condition, and vessel displacement As these factors become more severe, an increase in power is necessary to maintain the same speed, shifting the propeller power demand curve left towards Curve A Consequently, it is essential for the propulsion plant to be capable of generating full power under less favorable conditions, such as adverse weather, deeper draughts, or hull fouling This leads to the establishment of a performance line, represented as Curve B, which is positioned to the right of the nominal propeller demand curve for optimal engine-propeller matching This difference in performance introduces the concept of "sea margin," ensuring that the ship's propulsion system has adequate power available during service and throughout its docking cycle.
Steady-state engine load is represented by the power rating, typically as a fraction of MCR power The engine's operating point is regulated by the fuel index position at the fuel pump, usually expressed in a dimensionless format ranging from [0,1] or [0%,100%] This fuel index position correlates directly with the engine's steady-state power, assuming mechanical losses are negligible Consequently, marine Diesel engines are engineered for linear operation regarding power generation However, the relationship between engine power delivery and fuel mass injected per cycle is not linear To determine the fuel mass needed at various loading points on the propeller curve, a blend of theoretical thermodynamics, simulation data, and experimental results is utilized Subsequently, the fuel pump's mechanical design is adjusted to ensure a linear steady-state response of engine power to changes in the fuel index.
Marine engine operation relies on the relationship between engine torque delivery and index position, which is proportional at a constant engine speed (rpm) when the engine has "excess air" for optimal combustion This relationship does not contradict the linear steady-state power dependence on index, as the rpm varies across different steady-state operating points.
The engine mechanical power delivery is determined by the following design features:
• Cylinder bore and piston stroke; these parameters determine the volume
V h swept by the piston displacement during a stroke (piston displacement volume), i.e.:
4 Brake Mean Effective Pressure (BMEP, Pe), defined as follows:
Pe= 'y:r- zc ,V h %0 where P is power in watts and N is shaft rpm
Maximum BMEP is observed at MCR where P = P MCR (MCR power) and
N=N MCR' As argued in the next chapter, engine torque is directly proportional to BMEP, and therefore maximum engine torque delivery is observed at MCR as well
Indicated Mean Effective Pressure (IMEP, Pi) represents the average in-cylinder pressure per cycle, while Friction Mean Effective Pressure (FMEP, Pt) accounts for the engine's mechanical losses The relationship between IMEP, FMEP, and Brake Mean Effective Pressure (BMEP) is crucial for understanding engine performance.
The Screw Propeller
Large cargo ships utilize a single large-diameter propeller, typically fixed-pitch, with diameters exceeding 5-6 meters and consisting of 3-6 blades Introduced in the 19th century, screw propellers positioned at the aft of the ship offer significant advantages in propulsion, including superior hydrodynamic efficiency and ease of construction and operation The efficiency of a propeller is measured as the ratio of the thrust power delivered to the water to the mechanical rotational power supplied by the shaft.
The overall torque Q L developed by the water and exerted to the propeller shaft is given as follows:
The equation Q L = Kq,prop p D!rop ,N 2 (1.7) illustrates the relationship between propeller diameter (Dprop), seawater density (p), and a constant (Kq,prop) This formula can be compared to the direct-drive engine load torque equation (Eq (1.2)), highlighting the similarities in their relationships.
Significant uncertainty arises in propeller torque, which constitutes the load torque on the propulsion engine, because Kq,prop is not a constant In fact, for fixed-pitch propellers, the coefficient Kq,prop is highly dependent on the advance coefficient J,crew.
The advance speed (Vadv) of the propeller, measured in m/s², is approximately equal to the ship's advance speed This relationship is expressed by the equation J = Vadv (1.9) screw N.D prop Understanding this correlation is crucial for optimizing propeller performance and efficiency in marine engineering.
K Q, is a decreasing function of J,crew'
Engine operation is limited by an rpm cap due to its direct connection to the propeller, which affects propeller efficiency (denoted as 118) based on the coefficient J_screw Propeller efficiency peaks when J_screw is within a specific range of 0.3 to 0.7; deviations from this range lead to a rapid decline in efficiency Consequently, engine and propeller rpm should not exceed a narrow range of 60 to 250 rpm, as exceeding this threshold can reduce propeller efficiency to below 30% Additionally, the torque coefficient Kq,prop is influenced by the propeller's cavitation status, represented by the cavitation number G_cav, which generally increases with G_cav However, G_cav decreases as propeller rpm (N) and advance speed (V_adv) rise.
An important source of uncertainty in engine-propeller dynamics is propeller inertia When calculating the combined inertia of the engine and propeller, it is essential to consider the inertia of the water entrained by the propeller This entrained water's mass and moment of inertia can vary significantly, contributing an additional surplus of 5% to 30% relative to the propeller's inertia.
Contribution of this Work
Statement of the Problem
Robust control design methods for marine diesel propulsion systems are essential to ensure safe powerplant operation, particularly during near Maximum Engine Rating (MeR) conditions and significant propeller load fluctuations caused by heavy weather and rough seas Such fluctuations can lead to critical issues, including main engine overspeed, especially when the engine operates close to its upper limits When a large propeller torque demand occurs, it can accelerate the engine-propeller shaft, risking the engine speed exceeding the maximum allowable limit and resulting in emergency shutdown due to excessive loading To mitigate these risks, the operating point of the main propulsion engine is proactively reduced during heavy weather conditions, leading to a voluntary decrease in ship speed.
To ensure reliable performance and avoid emergency shutdowns due to critical overspeed, it is essential to enhance the sea margin of the main propulsion engine This adjustment accommodates inevitable hull fouling and guarantees sufficient propulsion power in rough sea conditions However, the resulting larger propulsion plant leads to higher installation costs and potentially increased running expenses, particularly if the optimal operating point is near the maximum engine rating (MeR) Even in favorable sea states, these considerations remain crucial for effective marine operations.
Under specific conditions, such as when a large containership experiences significant rolling in a beam sea (e.g., sea state 3 or 4), fluctuations in propeller torque demand can occur This situation can severely restrict the operation of the near MeR propulsion plant due to increased fuel index limiter activity, often leading to a voluntary reduction in engine speed setpoint Consequently, this can disrupt the trading schedule, representing another form of voluntary speed loss.
Limiters in marine engine electronic control units provide essential protection against critical and off-design operations by directly influencing the fuel quantity injected per cycle These limiters set upper and lower boundaries for the fuel index position, which are determined by specific measured variables such as engine speed (rpm) and boost pressure When the control action from the linear component of the controller, typically a PI control law, exceeds these established limits, it is constrained to the nearest boundary, ensuring optimal engine performance and safety.
A detailed examination reveals that considering worst-case disturbances during the synthesis of the controller's linear components could significantly minimize or eliminate limiter activity, thereby ensuring reliable operation of marine plants Furthermore, ongoing research emphasizes the need for a systematic approach to marine control systems, aiming to develop robust engine control systems capable of handling conditions beyond the typical calm sea scenarios Currently, the conventional PI speed governors are primarily calibrated for such calm conditions, highlighting a gap in adaptability for varied operating environments.
Overview of the Approach
The solution to the operational problems involves three key approaches: modeling the engine and propulsion plant for effective control, linearizing the model based on reasonable assumptions, and designing two robust controllers—one utilizing PID control law and the other employing full state feedback The following sections detail the specific methodological steps taken in this process.
Before developing a control system, it is essential to understand the dynamics of the open-loop (uncontrolled) plant This is typically achieved in control engineering by creating transfer function or state-space models that describe the transient response of the open-loop plant Due to various factors such as ship trading schedules and feasibility constraints, it is often more effective to tune the transfer function or state-space model of the marine plant using physical thermodynamic engine simulation models rather than relying on onboard measurements and experiments Consequently, this work focuses on analyzing engine operation from an energetic and thermodynamic perspective for control purposes, leading to the establishment of a transfer function and state-space model for the marine propulsion system.
This article presents a novel PI(D) speed regulator tuning method for marine plants, utilizing a reduced-order transfer function and loop-shaping to meet disturbance rejection specifications While demonstrated on marine propulsion systems, the approach is applicable to any process characterized by a transfer function with a single, stable dominant pole To address challenges with signal differentiators in marine propulsion applications, an alternative control strategy is proposed The effectiveness of the PI(D) controller design is validated through testing on a large containership's propulsion plant, showing that the gains achieved offer robust worst-case disturbance rejection compared to the existing PI governor Furthermore, the method's robustness against neglected dynamics is analyzed using a full-order transfer function derived from thermodynamic engine simulation models.
To address the necessity for a structured design of the feedback propulsion controller, a non-linear state-space model for the propulsion powerplant is developed This model integrates the advanced mapping capabilities of neural networks with comprehensive calibration datasets derived from a cycle-mean, quasi-steady thermodynamic engine simulation.
This article presents a full-state-feedback controller design methodology based on the plant state-space description, utilizing formal Hoo-synthesis and robust control theory's real parametric uncertainty analysis frameworks The method addresses propeller fluctuations as disturbance signals that must be mitigated through feedback control It incorporates robustness analysis to account for real parametric uncertainties arising from variations in thermodynamic properties and propeller-entrained water inertia Ultimately, the proposed propulsion control system integrates supervisory control for smoother engine operation with feedback control to maintain the powerplant's performance close to its nominal desired behavior.
Text Outline
A brief reference to the topics covered in the upcoming chapters is now given
Chapter 2 focuses on the thermodynamic analysis of turbocharged marine Diesel engines, examining the physical processes involved in power torque generation This analysis leads to the development of a cycle-mean, quasi-steady model that offers valuable insights and serves as a validation platform for control developments The chapter illustrates the simulation model using a typical marine engine, detailing and assessing the numerical solution procedure.
Chapter 3 deals with the problem of propulsion powerplant modelling for control purposes The modelling starts with shafting system dynamical analysis, aiming to depict the effect of the engine-propeller shaft dynamics on controller design Then the transfer function of the propulsion powerplant is formulated, using the "black-box" approach, in combination with ad hoc assumptions for the dynamics of the marine plant Finally, identification is performed by employing a detailed, filling-and-emptying thermodynamic model of the engine processes The procedure is validated using the propulsion powerplant of a large containership
Chapter 4 explores the PI and PID control of marine propulsion powerplants through the lens of modern linear robust control theory, focusing on H∞ disturbance rejection Building on the analysis from Chapter 3, the closed-loop scalar transfer function with a PI(D) controller is developed, and the PI(D) gains are determined to ensure the H∞-norm of the compensated system meets specifications The robustness of PI and PID compensated plants against neglected dynamics is evaluated, highlighting the advantages of PID regulation over PI Additionally, an alternative method for implementing the D-term in practical applications is introduced, which eliminates the need for differentiating the rpm feedback signal by calculating the rpm derivative from the shaft torque feedback signal.
In Chapter 5, the marine plant's state-space representation is derived from the thermodynamic engine model presented in Chapter 2, utilizing neural network capabilities to accurately model non-linear mappings The formulation of state equations incorporates neural torque approximators and propeller laws, while assessing the parametric uncertainty within these equations The chapter then details the linearization process of the marine power plant equations, culminating in the determination of the open-loop transfer function matrix, which is compared to the scalar transfer function obtained in Chapter 3.
Chapter 6 deals with the marine propulsion powerplant control problem using state-feedback linear robust control theoretical results, in combination with open- loop optimised schedules for operating point changes The disturbance rejection specifications are appropriately decomposed based on the analysis of Chapter 5 In effect, gains of the controller are calculated Finally, applicability of criteria for robust stability and performance, as well as the effect of integral control on steady- state error, are briefly examined
Chapter 7 concludes this work Assessment of the modelling approaches, as well as of the PI(D) and state-feedback control options investigated, is done Proposals are given for future research and investigations
Physical Engine Modelling
Large two-stroke turbocharged Diesel engines play a crucial role in the propulsion systems of modern cargo vessels, serving as the primary actuators for ship movement Effective control of these engines is essential for managing ship acceleration and deceleration, particularly in response to external factors such as weather conditions and hull fouling, which can impact speed The relationship between ship speed and engine load is complex and non-linear, necessitating advanced control strategies that leverage digital electronics and sensor technologies In control design, state-space mathematical models are typically employed; however, in the absence of such models, detailed physical simulations can provide the necessary framework for developing effective control strategies.
In the case of marine Diesel engines no state-space models are usually available This is due to a number of reasons, the most important of which are the following:
High costs hinder the construction of marine engines solely for testing purposes, unlike in the automotive industry where creating testbeds and engines is more feasible Consequently, conducting experiments in the automotive sector effectively formulates state variable maps that illustrate system dynamics, capturing the relationship between temporal derivatives and state variable values In contrast, the development of large marine engines only begins after a specific order is placed and the ship is in its early construction phases Additionally, building marine engine testbeds capable of transient loading is both challenging and expensive, often rendering it practically unachievable.
The physical and chemical processes within an engine are characterized by high complexity and nonlinearity While the differential equations governing engine and turbocharger shaft dynamics resemble those found in typical electromechanical systems, the combustion processes that generate power and torque are significantly more intricate These governing equations are akin to those used in chemical plants and the process industry, indicating that a straightforward linearized approach is inadequate without considering the inherent limitations and approximations.
Due to limited experimental data, various analytical thermodynamic and Computational Fluid Dynamics (CFD) models are utilized for predicting the performance of large marine engines during transient operations These models are based on thermodynamics, fluid dynamics, and chemical kinetics principles, allowing for the prediction of both steady-state and transient engine performance before manufacturing While their accuracy can vary, even simpler models generally offer sufficient precision for control development Consequently, employing these physical simulation models remains a practical and reliable method for gaining valuable insights into the physicochemical processes of marine engines.
Quasi-steady, cycle-mean-value thermodynamic models are crucial for analyzing physical engine systems, as they extend steady-state equations to dynamic situations These models aim to estimate the cycle-averaged temporal evolution of thermodynamic and mechanical variables, focusing on pressures and temperatures Given the highly spatially distributed nature of engines, quasi-steady models simplify this complexity by eliminating the distributed characteristics of thermodynamic variables and deriving spatial averages This involves partitioning the turbocharger and engine interconnected volumes into lumped-parameter models Ultimately, a set of intermediate variables is established, representing the cycle and plenum averages of the corresponding distributed thermodynamic variables.
The complex non-linear algebraic equations among the intermediate variables can be solved to determine the torques of the engine, turbine, and compressor as functions of the engine and turbocharger shaft RPM, along with the fuel index position as a control action Subsequently, the motion of the two shafts in the powerplant, governed by Newton's laws of motion, can be analyzed through differential equations.
This chapter provides an in-depth examination of a standard quasi-steady cycle-mean-value thermodynamic engine model, while Chapters 3 and 5 discuss techniques to effectively solve the associated algebraic system numerically.
Turbocharged Engine Model Variables
Table 2.1 outlines the key components of a large turbocharged marine Diesel engine, highlighting the associated lumped and cycle-averaged thermodynamic variables of interest for each part.
Table 2.1 Engine thermodynamic variables of interest
Turbocharger Air mass flow rate (rnA) compressor
Scavenging receiver Scavenging pressure (PI)
Engine cylinders Fuel mass flow rate (m F )
Air-to-fuel ratio (A/F) Combustion efficiency (1JJ
Brake Mean Effective Pressure (BMEP, Pe)
Indicated Mean Effective Pressure (IMEP, Pi)
Friction Mean Effective Pressure (FMEP, PI)
Fuel chemical energy proportion in exhaust gas ('a)
Exhaust receiver Exhaust pressure (PE)
Turbocharger turbine Exhaust mass flow rate (mE)
Turbine flow coefficient (aT) Turbine isentropic efficiency (1JiT)
The above intermediate variables can be calculated if the powerplant operating point is known The plant operating point is, in turn, determined if:
• Engine crankshaft rotational speed (rpm) N E'
• Turbocharger shaft rotational speed (rpm) N TC and
• Fuel index (rack) position FR are given Additionally, the following external variables are also necessary for the determination of the engine operating point:
• Ambient (atmospheric) pressure (Pa> typical value 1 bar = 10 5 N/m2)
• Ambient (atmospheric) temperature (T a, typical value 290 K = 17°C)
• Intercooler coolant (water) temperature (Tw)
In modern ship engine rooms, ambient conditions such as pressure (Pa) and temperature (Ta) are kept nearly constant through ventilation and air conditioning systems to ensure optimal machinery performance Additionally, coolant temperature is regulated within a narrow range using an external heat exchanger with adjustable heat transfer capacity, further supporting the main engine's specifications.
The modeling of a turbocharged two-stroke Diesel engine relies on the thermodynamic equivalence of air and exhaust gas flows through orifices, where the air flow is likened to that through an orifice with a constant effective area, Ayeq This parameter is influenced by the exhaust valve and inlet port configurations, valve timing, and cylinder count, and is treated as a mean value over a complete engine cycle, despite the intermittent communication between inlet and exhaust ports In two-stroke engines, one crankshaft revolution corresponds to one thermodynamic cycle, unlike four-stroke engines, which require two revolutions Additionally, the turbocharger turbine acts as an orifice for exhaust gas flow, while the turbocharger compressor functions as a pump that supplies pressurized air to the engine, with its output pressure and load torque dependent on the turbocharger's RPM The overall performance is further affected by temperature and mass increases from fuel injection and combustion, along with a drop in air temperature at the intercooler.
The physical modeling approach illustrated in Figure 2.1 features a single pump (compressor) connected in series with two orifices, representing the engine ports and valves, along with the turbine, highlighting the various gas flows involved in the system.
The primary goal of the thermodynamic analysis is to calculate the torques of the engine, turbine, and compressor, which are essential for understanding the dynamics of the propulsion system, using the differential equations outlined in the following section.
Turbocharged Engine Dynamical Equations
The dynamics of the propulsion system are determined by the equations governing the engine-propeller and turbocharger shafts In this analysis, both shafts are assumed to be fully elastic with no damping, allowing for significant simplification of the equations while maintaining accuracy.
For crankshaft rpm (N E ) or propeller rpm (N) the differential equation is as follows:
The cycle-mean torque delivery of the engine (Q E) is balanced against the cycle-mean torque demand (Q L) placed on the powerplant's shafting system, while Itotal represents the total inertia of the shafting system averaged over a complete cycle.
Under the assumption of engine-propeller direct coupling (no gearbox or clutch):
• No reduction ratio is introduced between engine and propeller rpm, i.e.:
• Propeller-law engine loading can be considered, i.e.:
The equation QL(N E) = KQ ãN; = (KQO +.t1KQ)ãN; (2.3) defines the relationship between propeller torque coefficient KQ and its nominal value KQO, with an added uncertainty kQ This uncertainty is discussed in detail in Chapter 1, highlighting its impact on the overall calculations.
• The shafting system inertia can be expressed as follows:
The total inertia (I) of the system is calculated as the sum of the engine crankshaft inertia (IE) and the propeller inertia (I prop), with an additional nominal 15% added to account for the inertia of the entrained water This relationship is expressed as I = IE + 1.15I prop It is important to note that there is some uncertainty associated with this nominal value of 10, which arises from variations in the entrained water inertia.
Engine torque delivery (QE) is determined by thermodynamic variables, with a direct proportionality to the engine's Brake Mean Effective Pressure (BMEP) BMEP is defined as the in-cylinder pressure that correlates with the engine's rotational power output (P).
However, the following holds for mechanical power:
When adjusting the fuel index position (FR'), a dead time (T) occurs before the engine generates torque This delay is because torque is produced only after fuel injection and subsequent combustion in the engine cylinders For two-stroke engines, the new torque value, influenced by the altered fuel index position, is fully realized after one complete crankshaft revolution, while for four-stroke engines, it takes two revolutions During this period, torque fluctuates according to the firing order of the engine cylinders The dead time for engine torque in two-stroke engines falls within a specific range.
However, this effect (delay) becomes significant only for low engine speeds, typically below 1 rps = 60 rpm) Furthermore, in the worst case:
In summary, the impact of engine torque delay (T) can be considered negligible in cycle-mean-value engine models, as this delay is less than the model's sampling time interval The time step corresponds to the duration of a single crankshaft revolution, measured in minutes.
The turbocharger dynamics are depicted in the following differential equation, which is analogous to the one holding for the engine-propeller shaft:
The torque delivery of the turbine (QT) and the load torque of the compressor (Qc) are critical components in turbocharger performance The combined moment of inertia (ITC) of the turbocharger encompasses the inertias of the turbocharger shaft, turbine wheel, and compressor impeller Importantly, this system parameter remains constant, as the inertias involved do not fluctuate.
Q T and Qc are calculated from thermodynamic variables with the following algebraic relations:
Constants rA and rE are the specific heat ratios for air and exhaust gas respectively: rA = 1.4, 'It = 1.34 (2.13)
The parameters mentioned are influenced by both air and exhaust gas temperatures Typically, these values are relevant for standard air temperatures of 288 K (15°C) and exhaust gas temperatures ranging from 400 to 1100 K.
CP •• ir and CP•exh are the specific heat at constant pressure of air and exhaust gas respectively In general, Cp is calculated according to the following mathematical relationship:
The equation C p = r - 1 * r Mmol (2.14) defines the relationship between specific heat capacity (C p), the ideal gas constant (R = 8.314 J/(mol·K)), and the molecular weight (Mmol) of the gas For air and exhaust gases, specific assumptions are made regarding these values to facilitate calculations in thermodynamic analyses.
Mmol.air = 28.96 gr/mol, Mmol.exh = 30.0 gr/mol (2.15)
By substituting the values of y and Mmol in the general formula one obtains:
The Constant 1J TC incorporates the mechanical efficiency 1J mTC of the turbocharger along with the isentropic efficiency 1J iC of the compressor, with both parameters assumed to be constant and approximately equal to one.
1J TC = 1J mTC '1J iC => 1J TC "" 1J iC (2.17)
Turbine and compressor torques exhibit notable similarities, as both are derived from the principles of adiabatic air compression in compressors and adiabatic exhaust gas expansion in turbines These theoretical thermodynamic processes form the foundation for establishing the relationships between turbine and compressor torques.
(2.18) This is because in the case of the compressor PI> P a and in the case of the turbine
Quasi-steady engine models primarily focus on the dynamical equations related to engine and turbocharger RPM In contrast, more analytical thermodynamic models also consider gaseous mass accumulation during the cycle, necessitating the inclusion of differential relations for air and exhaust flow rates, which increases the number of dynamical equations However, for effective engine control, a cycle-mean-value approach is often sufficient, as control actions, such as fuel index positioning, influence engine operation once per cycle, provided that modeling accuracy is maintained The subsequent sections will analyze the algebraic interdependence of thermodynamic variables essential for calculating engine, turbine, and compressor torques, following the gaseous flow through the major components of the powerplant.
Turbocharged Engine Algebraic Equations
Turbocharger Compressor
The turbocharger compressor operates under an adiabatic compression process, ensuring the necessary air mass flow rate for optimal combustion in the engine's combustion chambers This process is crucial for maximizing engine efficiency and enhancing power and torque output In quasi-steady engine models, the air mass flow rate is treated as continuous, although airflow is restricted by inlet ports and exhaust valves, leading to interruptions For analytical purposes, it is assumed that air flows from the scavenging area to the exhaust receiver through an orifice with an equivalent effective area (Ayeq), with a calculated pressure difference (PI - PE).
The coefficient Cv represents the resistance to air flow through the effective area of inlet ports and exhaust valves In two-stroke engines, which utilize inlet ports instead of valves, the coefficient Cy is typically assumed to be 0.9 The equivalent effective area is determined by calculating the cycle mean of Ayeq(ffi), where ffi denotes the crank angle.
Finally, Aye/ffi) can be calculated from the instantaneous (function of ffi as well) openings of the engine inlet ports and exhaust valves, i.e.:
A ( ffi) = A;nle,( ffi ) Aexhaust ( ffi )
A,nle, (0) and A exhaus, (0) are engine configuration data provided by the engine manufacturer.
Intercooler
Compression of air causes a temperature rise in the scavenging air, given by the thermodynamic relation concerning adiabatic processes, i.e.:
The impact of compression on density is moderated by the installation of an intercooler between the compressor outlet and the scavenging receiver This water-cooled heat exchanger effectively reduces the temperature of scavenging air by transferring heat to the coolant It is important to note that the efficiency of the intercooler decreases as the air mass flow rate increases For initial assessments, a specific relationship can be established to evaluate the intercooler efficiency, denoted as Tllc.
The intercooler efficiency, represented by the equation Tllc = 1 - /(ICã rnA (2.23), is typically considered to be around 95% according to various studies The constant /(IC can be derived from manufacturer specification data sheets While the intercooler does introduce a slight pressure drop in the scavenging air pressure, this effect is minimal and can be disregarded for accurate modeling.
Scavenging Receiver
The key thermodynamic variables influencing an engine's scavenging air receiver are the temperature and pressure of the scavenging air The air pressure is primarily determined by the turbocharger's operational status, which is closely linked to its rotational speed (turbocharger rpm, NTd) This relationship is particularly relevant for marine Diesel plants, where the engine and turbocharger are optimized to function along a specific operating line, typically represented on a compressor map chart The map's coordinates include the corrected mass flow rate (x-axis) and pressure ratio, providing essential insights into the engine's performance.
In a compressor performance map, the lines representing constant corrected speed and constant isentropic efficiency curves intersect at various operating points, which align with the engine-turbocharger matching requirements For large marine engines, this matching is optimized to maintain operation near a constant efficiency curve, allowing the isentropic efficiency to remain relatively stable across all relevant operating points This stability holds even during rapid transients, where the operating point shifts parallel to the constant efficiency curve, resulting in minimal deviations of 1-2% However, this assumption may not be accurate in the event of a malfunction.
A constant l1ic curve can be approximated using a second-order polynomial based on corrected speed This approximation arises when considering the enthalpy rise from compression as a function of turbocharger RPM and pressure ratio Consequently, a relationship for the pressure ratio can be derived, expressed as p[ = /(c [N Tc ]2 +1.
Coefficient /(c can be calculated if one point of the operating line is known, i.e a pair of values for p[ and NTC-
The scavenging air temperature (T) is a crucial thermodynamic variable, determined by the compressed air temperature (Tc), which is elevated compared to the ambient temperature (Ta) due to the compression process This calculation incorporates the influence of the intercooler, highlighting its significance in optimizing engine performance.
T[ = (l-l1Ic)' Tc +l1Ic -Tw where Tw is the cooling water temperature
Finally, scavenging air enthalpy h[ is defined as: h[ == CP,air T[
Engine Cylinders
The two most important processes of engine mechanical power/torque generation take place in the engine cylinders:
Large marine engines utilize camshaft-driven fuel pumps to facilitate fuel injection, with the timing of this injection determined by the camshaft's geometry Recently, advancements in Variable Injection Timing (VIT) electromechanical systems have been implemented on ships, alongside the development of a new generation of marine engines featuring fully electronic, hydraulically actuated injection timing However, traditional quasi-steady models fall short in accurately representing the effects of fuel injection timing, as they focus on average cycle values instead of in-cycle variations Consequently, fuel injection timing is modeled primarily as an influence on power and torque outputs, affecting specific fuel consumption (SFOC, g/kWh) based on the same fuel quantity.
The fuel mass injected into the cylinders per cycle, known as the fuel mass flow rate (m F in kg/s), is a crucial variable for engine performance This value is influenced by the position of the fuel index (fuel rack, F R) and is also dependent on the engine's RPM.
The maximum fuel injection per cylinder per cycle, denoted as mF.max (in kg), is a critical parameter in engine performance The fuel index position, FR, is dimensionless and ranges from 0 to 1, indicating its relative value within this interval.
The fuel index plays a crucial role in determining the Indicated Mean Effective Pressure (IMEP) of an engine An increase in the fuel index results in a higher IMEP, assuming optimal combustion conditions are sustained In quasi-steady engine models, the combustion regime is quantified by the air-to-fuel ratio (NF).
NF in turn determines combustion efficiency 11e according to the following relationship:
, if (NF),ow < (NF) < (NF)high (2.29)
IMEP is then calculated according to the following relationship:
The equation Pi = 11e Pi,max FR illustrates the relationship between the indicated mean effective pressure (IMEP) and fuel index, highlighting that Pi,max, specified by the manufacturer, represents the maximum achievable IMEP While IMEP shows a linear dependence on the fuel index, this relationship becomes nonlinear if combustion efficiency falls below unity A reduction in the fuel index (NF) below certain thresholds—typically 20-27 for heavy fuel oil (HFO) and 17-20 for diesel—indicates suboptimal combustion, leading to decreased power and torque This linear degradation of IMEP with lower NF values occurs because insufficient air mass for combustion results in inadequate fuel burning Furthermore, if NF drops below a critical limit (AIF) of 5-8, excessive air cooling in the cylinder can prevent combustion altogether, causing IMEP and engine power to drop to zero.
As already mentioned, BMEP, which is related proportionally to engine torque, is connected with IMEP and FMEP by the relation:
FMEP is usually calculated as a multi-linear (affine) function of IMEP and engine rpm, i.e.:
The formula for calculating the performance factor \( P_f \) is given by \( P_f = IC_{fl} \cdot P_i + IC_{fl} \cdot N_E + IC_{ro} \), where the constants \( IC_{fl} \), \( IC_{fl} \), and \( IC_{ro} \) are supplied by the engine manufacturer For large marine propulsion Diesel engines, MAN-B&W defines the friction mean effective pressure (FMEP) as a linear function of the fuel index.
The formula for expressing FMEP can be reformulated by setting ICfl to zero, leading to the transformation of the equation P f = ICo FR + IC f4 into a more simplified version through the substitution of constants ICo and IC f4.
ICo = ICfl • Pi,max' IC f4 = ICro (2.34) Also, note that combustion efficiency is assumed to be equal to unity
The MAN-B&W FMEP form is more convenient for calculating BMEP in steady-state as a linear function of fuel index position:
In steady-state conditions, the combustion efficiency is maintained at unity due to the turbocharger's optimal matching, which ensures sufficient combustion air mass is supplied to the cylinders The equation P e = Pi - P f = Pi,max FR - ICf3 • FR - ICro = (Pi,max - ICo) FR - ICro illustrates this relationship, where ICro represents the pressure-equivalent of mechanical power losses during idle or motoring engine operation.
The fuel chemical energy proportion in exhaust gas, denoted as 'a', is a key thermodynamic variable in cylinders This parameter shows a linear correlation with Brake Mean Effective Pressure (BMEP) The relationship described is applicable to all two-stroke Diesel engines.
For two-stroke engines and BMEP expressed in (N/m 2): IC Z\ = 0.0105 X 10- 5 and ICzo = 0.3120.
Exhaust Receiver
Exhaust pressure and temperature are very important because they can be used for benchmarking any zero-dimensional engine model as:
(a) they are relatively easily measured in either testbed or installed plants and engines;
(b) they comprise a measure for combustion modelling accuracy as the exhaust gas is the direct outcome of combustion;
Turbocharger RPM is influenced by the flow of exhaust gas through the turbine, which accelerates or decelerates based on its thermal properties relative to the surrounding environment.
Exhaust pressure is determined by the exhaust mass flow rate through the turbine, represented by the equation Rexh = M~,exh = 277.133 kg/K This equation closely resembles the one used for calculating air mass flow rate, as both assume equivalence of flow through an orifice The coefficient CT serves as the resistance coefficient for exhaust gas flow through the mean effective area of the turbocharger turbine Modern marine engine turbochargers feature advanced designs in the turbine nozzle and rotor blades, allowing for optimized performance.
Areq represents the effective area of a turbine, determined by its geometric configuration Due to the numerous blades on the rotor, cycle-averaging is unnecessary, as the effective area remains relatively constant throughout an engine cycle The calculation of the turbine's effective area follows a specific formula.
A,:ozzLe + A:otor (2.39) where A"ozzle is the minimum flow area of the nozzle and A,.otor is the minimum flow area of the wheel However, due to the complexity of the phenomenon calculated, A Teq does not provide adequate modelling accuracy In effect, the multiplicative flow correction parameter aT is introduced in order to compensate the modelling accuracy error
In the case of the turbine, choked flow may occur also if the value of exhaust pressure exceeds the threshold value dictated by the following inequality:
When the exhaust mass flow rate surpasses a certain threshold, the relationship between the turbine pressure ratio (Pa / PE) becomes invalid In this case, PE can be determined using a simplified equation involving mE.
Exhaust temperature (TE) is determined by the exhaust gas enthalpy (hE'), which is influenced by the fuel's specific calorific value (Hu) at 0 K, the chemical energy proportion of the fuel in the exhaust gas (parameter 'a'), and the air-fuel ratio (AIF) The relationship can be expressed as hE' = a * Hu.
Coefficient 11exh stands for the exhaust temperature correction factor Indeed, the quantity (hI + 'a Hu )V(l + _1_) stands for the exhaust gas enthalpy at the
The enthalpy increase in the exhaust receiver is calculated based on the scavenging air enthalpy (hI'), resulting from fuel injection and combustion within the engine cylinders While some enthalpy is lost during the heat exchange in the exhaust receiver, the key values for calculating the exhaust flow rate (mE) through the turbine are taken at the turbine inlet For initial estimations, it is reasonable to assume that the heat exchange effect at the exhaust receiver can be neglected, treating the exhaust pressure and temperature values as approximately equal to 1.0.
Finally, the exhaust temperature TE can be calculated directly from the exhaust gas enthalpy:
Remember that Cp , em = 1117.0 kgãK J
Turbocharger Turbine
The exhaust mass flow rate is calculated as the sum (conservation of mass) of air mass flow rate m A and fuel mass flow rate mF' i.e.: mE =mA +mp (2.44)
The performance of a turbine is typically illustrated through charts that depict its swallowing capacity (mE) in relation to the pressure ratio (Pa/PE) For modeling purposes, the previously discussed relationship is utilized, and this analysis is further enhanced by incorporating plots of isentropic turbine efficiency (ηh) Overall, these elements provide a comprehensive understanding of turbine behavior.
( Pa U T ) lh =lh PE'"C; (2.45) where U T is the velocity (in mls) of the rotor blade tip:
U T = 1[' D turb N TC (2.46) where D turb is the turbine wheel diameter and Cs is the exhaust gas velocity For isentropic processes:
The dependence of the turbine's isentropic efficiency on the velocity ratio \( U_T / C_s \) is significantly stronger than its dependence on the pressure ratio \( P_a / P_E \) For modeling purposes, this efficiency can be effectively approximated using a second-order polynomial.
The coefficients of the interpolation polynomial, /(T02' /(TOI, and /(TOO, are derived from the turbocharger manufacturer's charts for the average pressure ratio value, 1'1iT It is essential to select the appropriate 1'1iT curve and then perform polynomial interpolation based on the specified format.
Finally, the parameter aT can be approximated by a second-order polynomial of the turbine pressure ratio P a / P E , i.e.: a, = "m {;; J + IE -CK DO > IE' it holds that:
The reduction of proportional (P) and integral (I) gains in the control law is a beneficial outcome of utilizing shaft torque signals The fuel index value is constrained within the range of 0-100%, meaning that lower P and I gains lead to a greater margin before control action saturation occurs, which is a non-linear effect to be avoided Additionally, it's important to note that the feedforward gain (K FF) has an inverse relationship with the disturbance observer gain (K DO'); an increase in shaft torque causes engine deceleration, and the opposite is true as well.
Typical Case Numerical Investigation
In Section 4.4.5 the speed regulation of the propUlsion powerplant of containership
The "Shanghai Express" study revealed that only the Hinf PIO regulator satisfies the Hoo-norm requirement while ensuring robustness against neglected dynamic terms The PIO control law is replaced with the Hinf PI+FF scheme, and it is important to note that the shaft torque signal is available for feedback due to the presence of a torque meter installed in the engine-propeller shafting system.
The gains of the Hinf PID control law are given in Table 4.2 and have been translated to the gains ofthe HinfPI+FF scheme shown in Table 4.3
Table 4.2 Hinf PID regulator gains for "Shanghai Express" powerplant
(%index s/rpm) 2.53 Table 4.3 HinfPI+FF regulator gains for "Shanghai Express" powerplant
For the above calculation the assumption IE"" I p "" 112 has been used, instead of the exact value of engine inertia IE'
The Hinf PI+FF control scheme has been successfully validated through simulation, utilizing the block diagram illustrated in Figure 4.13 for the "Shanghai Express" powerplant This diagram incorporates the calculation of shaft torque based on a quasi-steady relationship.
In the simulation, the signal (2ã r(tằ is generated using the current engine torque delivery and propeller torque demand values This signal is then propagated to influence the fuel index value via a feedforward gain of K FF 12.
The simulation utilized the linearized version of the propeller law, as illustrated in Figure 4.13 Additionally, the block representing neglected dynamics mirrors the one employed for validating the Hinf PI and PID regulators, incorporating a turbocharging/combustion term with a time constant of 'fTC = 0.25 s and an actuator term with a time constant of 'fact = 0.10 s.
Figure 4.13 HinfPI+FF scheme for "Shanghai Express" plant used for simulations
Figure 4.14 illustrates the response of the Hinf PI+FF scheme alongside the Hinf PID regulator, highlighting the closed-loop plant's behavior in terms of shaft RPM and fuel index This response is evaluated following a torque coefficient (kQ) step of 0.05 kN mlrpm² at t = 5 seconds.
PI.FF full Hinf PIO full
Figure 4.14 Response of the Hinf PI+FF scheme compared to Hinf PID regulator
The Hinf PI+FF performance, when integrated with the reduced-order transfer function of the marine plant, matches that of the Hinf PID regulator However, when analyzing the full-order transfer function responses for both control schemes, slight differences in response shape emerge despite similar major features like overspeed and settling time Notably, the Hinf PI+FF response displays minor ripples and undershoot in the fuel index, which the Hinf PID does not exhibit This discrepancy arises because the rpm derivative approximation, derived from the shaft torque signal, is valid primarily in the low-frequency range, while a step excitation introduces significant high-frequency spectral content Consequently, although the plant's response contains some unwanted high-frequency components, the Hinf PI+FF scheme remains effective.
The Hinf PI+FF scheme is effective for improving propeller disturbance rejection on ships While the control law gains can be derived from the Hinf PID regulator, fine-tuning is often necessary when integrating the controller with the actual plant to enhance performance and adapt to real operating conditions.
Summary
A robust PID speed governor design method is introduced, focusing on Hoo-norm to address the challenges of severe and rapid load disturbances in marine engine control This issue often necessitates reducing engine ratings to prevent overspeed and potential overload The method employs loop-shaping with PID control to achieve effective disturbance attenuation, quantified by the Hoo-norm of the closed-loop transfer function from propeller disturbances to shaft RPM Given the need for fast closed-loop systems to reject severe load disturbances, it is crucial to consider neglected dynamics The study demonstrates that only a PID controller can fulfill the robustness requirements for marine propulsion systems using the full-order open-loop transfer function To eliminate real-time differentiation of RPM feedback signals, a disturbance feedforward technique is utilized The proposed control scheme incorporates two feedback signals—shaft RPM and torque—allowing for a regulator based on Hoo PID speed governor tuning, with the torque signal replacing the D-term.
STATE-SPACE DESCRIPTION OF THE MARINE PLANT
Introduction
Overview of the Approach
The initial step in implementing advanced control schemes for effective disturbance rejection and robustness against uncertainties is constructing a state-space model for the plant's operation This chapter focuses on developing a non-linear state-space model that captures the dynamic interaction between the marine engine and turbocharger while addressing inherent physical uncertainties and disturbances The model is derived using the non-linear mapping capabilities of artificial neural networks Unlike previous applications in the automotive industry, this approach does not seek to fill gaps in physical modeling; instead, it leverages the comprehensive cycle-mean, quasi-steady thermodynamic model from Chapter 2 to avoid the complexities of solving a non-linear algebraic system Additionally, neural networks are treated as mathematical entities rather than components of a broader intelligent powerplant modeling and control strategy, leading to the analytical manipulation of typical feedforward neural net structures to derive a linearized yet uncertain perturbation state-space model.
Unlike automotive engines, many marine propulsion engines are not turbocharged, making it challenging to experimentally determine their state equations or transfer functions This difficulty arises from several factors, including the high power output and large physical size of marine engines, which lead to costly testbed facility construction and time-consuming deployment for new engine testing Additionally, prototype marine engines are rarely available, and the performance of actual shipboard plants often differs significantly from results obtained in shore facility tests.
An alternative method for constructing and identifying the state-space equations of marine propulsion powerplants is based on physical principles The complexity of these processes necessitates detailed numerical simulation models that incorporate thermodynamic principles As discussed in Chapter 2, the relationships among various plant variables are not straightforward, appearing instead in a complex set of non-linear algebraic equations While the standard form of state-space equations cannot be easily formulated for marine propulsion systems, the numerical iterative solutions of the thermodynamic model can generate a grid of points where the mapping function is known This grid can then be used to train appropriately sized neural networks to accurately approximate the mapping function in a clear mathematical form.
An effective decomposition scheme can be applied to the plant state equations, leading to the partitioning of control actions and controllers during the controller synthesis stage This process generates two discrete open-loop plant models: the Non-linear Nominal Model (N2M) and the linear Uncertain Perturbation Model (UPM) The goal is to separate the control action, specifically the engine fuel index for marine propulsion plants, into two main components: the steady-state fuelling demand and the perturbation control action aimed at minimizing propeller disturbances The fuelling demand reflects the ship's propulsion power requirements under specific conditions, including loading, weather, and aging effects like hull fouling While short-term feedback control is unnecessary for fuelling demand, long-term adjustments can enhance ship management, based on monitoring relevant conditions and engine performance indices These adjustments can be made manually or automatically, with intelligent engine management systems optimizing fuel index offsets to reduce fuel consumption and emissions.
Feedback controls are essential for achieving closed-loop transient performance in UPM, particularly when dealing with significant propeller torque fluctuations The principles of control theory for linear systems can be effectively utilized, but it is crucial to consider the uncertainties involved A primary concern is the parametric uncertainty arising from the linearization process, as the values of partial derivatives are contingent upon the equilibrium point used for calculations Additionally, inherent uncertainties in physical parameters pose further challenges Therefore, the development of robust feedback control for the UPM state-space equations is a central focus of this study.
To achieve our goal, we must first create an open-loop state-space model and a corresponding open-loop transfer function matrix for the marine plant This process will utilize a neural state-space model based on the thermodynamic engine description outlined in Chapter 2 Ultimately, we will derive the transfer function matrix from the state-space model and compare it with the empirical results obtained.
Mathematical Formulation and Notation
Advanced control methods significantly enhance closed-loop dynamical behavior by utilizing state-feedback, which incorporates all relevant variables from the system's dynamical equations The state equations are generally expressed as x = f(x,u) and y = h(x), where x is an n-dimensional state vector representing key dynamical variables, u is an m-dimensional vector of control actions, and y is a p-dimensional vector of measured outputs The function f(x,u) maps the state and control inputs, while h(x) represents the relationship between the state variables and the system outputs.
In this article, we define a mapping on a subset of I!{n, where italic letters represent scalar variables and lower-case, non-italic letters denote vector quantities Matrices are indicated by capital, non-italic letters or by lower-case, non-italic letters with a bar above them.
Equilibrium points, denoted as I!{(n+m) (i.e (n + m)-ads), occur at x = f(x,u) = 6 (5.3) and are crucial for understanding the steady-state operating conditions of marine plants These points serve as the foundation for regulatory mechanisms necessary for disturbance rejection.
Recent advancements in controlling systems characterized by non-linear state equations have been notable A comprehensive analysis of a broad range of non-linear systems is presented in [43], which also tackles various control challenges such as feedback linearization, disturbance decoupling, and output regulation The state equations discussed in this work lead to a "control-action-affine" form represented as y = h(x), where gj(x) = gj(Xi', x2", , x.) are JR n-valued functions defined over a specific subset.
The function g(x) is defined as an (n x m)-matrix, represented as g(x) = [g1(x) g2(x) gm(x)] To address the control problem of non-linear plants, one effective method is linearisation, which involves approximating the state equations around specific equilibrium points For a given equilibrium point, such as [x~ u~ r] in JR(n+m), the linearisation of f(x, u) can be expressed as f(x, u) ≈ f(x0, u0) + L1(xj - x0) + L2(uj - u0).
,=1 dX, ( ) Xo,Uo ,-I dU, ( ) Xo,Uo
In the above, the n-dimensional partial derivative vectors are calculated by the following definitions, provided that f(x,u) is known:
~ (Xo,Uo) I (Xo,Uo) I (Xo,Uo) J (Xo,Uo)
In the sequel, the following definitions for the ( n x n) and ( n x m)
"derivative" matrices of the non-linear system are adopted: f;(xo,uo) £ [~I ~I dx dx ~I dx ]
I (Xo,Uo) 2 (xo,no) n (Xo.Uo)
Also, the perturbation (around the equilibrium point [x~ u~ r) state and control action vectors, ox and OU respectively, can be defined as following: ox=x-xo and ou=u-uo (5.11)
Taking into account the fact that f(xo'u o) = 6, the linearisation equation takes the following form, in an appropriately small vicinity of the equilibrium point
[x~ u~r : f(x,u) "" f:(xo,uoHix + f:(xo,uoHiu (5.12) Note that if the equilibrium point is not assumed to change, either deliberately or unwillingly, then the following holds: ox= ~(x-x )=x (5.13) dt °
Therefore, the non-linear state equation: x = f(x,u) is reduced to the following linear one:
The control problem of a non-linear system is significantly simplified by utilizing perturbation vectors for state and control action, represented as ox and ou, respectively This approach leverages a wealth of theoretical tools and techniques available for linear systems analysis and control It's important to note that the calculation of measured outputs y from the system states x, while omitted in this discussion, can be easily extended using the linearization method.
The simplification achieved through system linearization comes with a drawback, as both the perturbation plant model and the resulting controls exhibit a "localized" nature This limitation arises because the linearization of f(x,u) is only applicable within a narrow range around the specific equilibrium point [x~ u~r being analyzed.
To address the challenges of modeling non-linear systems, it is beneficial to utilize a family of linearized plant models rather than relying on a single approximation near a specific operating point [x~ u~ r The complexity arises from the fact that the derivative matrices, f: and f:, vary with the operating point, which prevents the system from being linear However, in many scenarios, particularly in marine plants, these derivative matrices can be bounded within a defined region of JRn,, and JRnxm for all steady-state operating points [x~ u~ r that are of practical interest This allows for a more reliable linearization approach, ensuring that fx.w ~ f: (xo,uo) ~ fX.HI and fu.w ~ f: (xo,uo) ~ fU.HI.
The nonlinearity of the original plant model translates into parametric uncertainty within the derivative matrices of the linearized perturbation models For linear systems characterized by this type of parametric uncertainty, linear robust control theory can offer a unique state or output feedback controller that ensures optimal transient performance and meets specified disturbance rejection criteria, either in the time domain or, more commonly, in the complex frequency domain Consequently, the challenge of nonlinear control is effectively transformed into the synthesis of a controller for a comprehensive set of linear systems.
The robust control of uncertain linear systems is extensively discussed in studies [44] and [45], highlighting the necessity of describing systems with non-linear state equations for effective application of their findings Following the linearization of these non-linear state equations, it becomes possible to establish bounds for the derivative matrices As detailed in Chapter 2, the dynamical equations of marine plants encompass both the engine-propeller shaft and the turbocharger shaft, which are reiterated here for clarity.
N(t):= NE(t)::: QE -QL and NTc(t)::: QT +Qc
Therefore, the state vector of the marine plant is: x::: [ ::J (5.17)
In conventional marine engines, the primary variable for feedback control is the amount of fuel injected into each cylinder per cycle, represented by the fuel index (rack) value.
Propeller load torque is primarily influenced by the rotational speed (rpm) of the propeller, with higher-order dynamics of the engine-propeller shaft being negligible.
• engine load torque coincides with propeller load torque, and
• propeller speed coincides with shaft and engine speed
Propeller law establishes the relationship between load torque and a specific state variable of the marine plant To develop the non-linear state-space equations for the marine plant, it is essential to derive direct mathematical and differentiable expressions for the torque of the engine, turbine, and compressor Essentially, we are looking for functions that exhibit this form.
The total torque applied to the turbocharger shaft is represented by QTC(t), which combines the turbine and compressor torque as QTC(t) = QT(t) + Qc(t) The expressions for QE(t) and QTC(t) are defined in relation to the number of turbines, turbine characteristics, and fuel rate, ensuring that all variables, including torque, RPM, and fuel index, reflect the thermodynamic cycle's mean values This approach aligns with the assumptions made in the engine thermodynamic model presented in Chapter 2.
In Section 5.2 a method employing artificial neural nets for approximating the above functions for the engine and turbocharger torque variables, namely
The article presents the QE(N,NTC,FR) and QTc(N,NTC,FR) metrics, utilizing steady-state and performance data from the quasi-steady, cycle-mean thermodynamic engine model discussed in Chapter 2 For ease of understanding, a vector (triad) of inputs for the neural networks is defined.
The Neural Torque Approximators
Configuration of the Approximators
In Chapter 2, the thermodynamic marine engine model is divided into a dynamical part, which governs the state differential equations, and an algebraic part that highlights the non-linear nature of engine power and torque generation The algebraic component consists of a complex set of non-linear equations that can only be solved numerically, resulting in a grid of points defined by the triad (N, N TC, FR) These points allow for the calculation of engine, turbine, and compressor torque values (QE, QT, and QC, respectively) However, the generated torque maps do not explicitly define the relationship between torque variables and state variables, such as engine and turbo RPM, as well as control actions like fuel index While these maps can aid in reducing computational effort for transient simulations of marine engines, they are not suitable for advanced controller synthesis using analytical theoretical tools.
Alternatively, curve-fitting techniques can be used for the approximation of the torque maps For example in [47] the following expression for engine torque delivery Q E is given:
The coefficients of the expression can be determined through least-squares curve fitting of available experimental data or from the engine torque map produced by a quasi-steady thermodynamic model However, a challenge arises as the specific functions for fitting experimental or model-predicted data, particularly for turbine and compressor torque (QT and QC), are not well-defined Even if an initial estimate for one or more torque functions can be derived from curve inspection, its general applicability across different marine engines remains uncertain Consequently, there is a need for an automated procedure to accurately approximate the three torque hypersurfaces, which are subsets of R4 and are parameterized by a three-dimensional input vector X generated by the thermodynamic model.
Artificial neural networks are increasingly recognized for their potential in overcoming the challenges of torque approximation The key advantage of three-layer, fully connected, feed-forward neural networks lies in their capability to represent non-linear mappings in a standardized mathematical format, which is essential for modern control theory applications Additionally, these networks eliminate the need for manual tuning, as automated calibration methods derived from optimization theory facilitate the alignment of the neural network's mathematical representation with the desired mapping Significant theoretical advancements in neural networks address approximation accuracy, training methodologies, and convergence issues Notably, under the assumption of continuity for the functions QE(N,NTC,FR) and QTc(N,NTC,FR), appropriately sized neural networks can effectively approximate the generated torque maps, as supported by the theorems of Kolmogorov (1957) and Hecht-Nielsen (1987) Kolmogorov's theorem asserts that any continuous scalar function can be represented by a three-layer feed-forward neural network with a hidden layer containing (2v + 1) neurons, ensuring that the function can be expressed in a defined mathematical form.
The function F(X) is defined as F(X) = LXj(L'I'/XJ), where X consists of continuous functions Xj(~) for j = 1, , (2v + 1) and monotonic functions 'I'ij(~) for i = 1, , y and j = 1, , (2Y + 1) Unlike Fourier series, which are linear combinations of sine and cosine functions based on frequency multiples, Kolmogorov's result presents a "non-linear" expansion using selected basis functions 'I'i/~)' for i = 1, , v and j = 1, , (2y + 1), emphasizing the unique nature of its mathematical structure.
"propagate" through the next layer's functions X/~), j = 1, ,(2Y + 1)
In effect, Kolmogorov's expansion for any non-linear mapping F implies the fully connected feed-forward layered architecture with weights shown in Figure 5.1
Figure 5.1 Typical architecture of fully connected, feed-forward three-layered neural net
In practice, the following additional architectural features are used for feedforward nets [49]:
• The outputs of the net are also normalised in the "unit" interval g = [0,1], although Kolmogorov's theoretical result requires normalisation of the inputs only
• The input layer is not a neural computing element, i.e the nodes do not have input weights and activation functions assigned to them, i.e 'i'ij(XJ = Xi'
• As indicated by Kolmogorov's result, there is only one neuron at the output layer, summing up the outcome of the hidden layer non-linear neurons
• The activation function for the hidden layer nodes X/~) can be any monotonically increasing function that is everywhere differentiable Usually it is the logistic sigmoid, i.e.:
• Biases, are added to the output of each hidden and input layer node Biases are used in order to regulate the net input to each unit (node)
Neural networks have been utilized for approximating torque hypersurfaces in marine plants, with torque maps generated from a thermodynamic model parameterized by the triad (N, Nrc, FR) Following Kolmogorov's findings, the hidden layer is configured with seven nodes Ultimately, these neural approximators effectively model engine torque.
QE (X) and turbocharger total torque Qrc (X) = Qr (X) + Qc (X) take the following standard mathematical functional form:
QE(X) = QEmax {woo + Wo h(x) = [~ ~] (5,42)
This means that the effect of the measuring devices is ignored Although this may not always be the case, the following reasons support this modelling assumption:
The construction of the state vector has overlooked sensor dynamics, incorporating only variables dictated by the physical model This analysis is intentionally kept simple, concentrating on the control challenges within the marine plant Future work will aim to expand these methods to include sensor dynamics and address non-linear features.
In Section 5,4, the state-space equations presented here are linearised and decomposed, in order for the feedback (closed-loop) and feedforward (open-loop) control problems to be formulated.
State-space Decomposition and Uncertainty
Manipulation of Equations and Variables
The standard linearization technique applied to non-linear state equations, commonly used for control purposes, involves decomposing plant signals, state variables, and control actions into a steady-state value and a superimposed perturbation signal This analysis focuses on the shaft/turbo RPM and fuel index signals.
The relationship between various parameters in a system can be expressed as N = No + n, N TC = NTCO + nTC, and FR = FRO + fR, where the subscript 0 indicates steady-state values and the lower-case letters represent perturbations The propeller torque coefficient KQ is modeled as a time-varying signal, comprising a steady-state equilibrium value and a stochastic fluctuation, represented as KQ(t) = KQO + kQ(t) The steady-state values for shaft rpm (No), turbo rpm (NTCO), fuel index (FRO), and propeller torque coefficient (KQO) reflect the equilibrium points of the system, which can be identified using specific conditions.
~ == ~E = O} {QE(NEO,NTCO,FRO) =_QL(KQO,No)
N TC -0 QTc(NEo,NTco,FRo)-O (5.44)
The calculation of equilibrium points is based on the marine plant equations, assuming that the system's uncertain parameters are fixed at their nominal values, represented as p~ = [KQO 101].
In marine plants, it is algebraically beneficial to treat the uncertainty in KQ as a form of plant disturbance Consequently, a plant disturbance vector can be represented as: d T = [ kQ (t) 0].
In effect, the state equation is rewritten as: x = f(x,u,d)
(5.46) Note that, although uncertainty AI is omitted, it is not dropped permanently, but just for the sake of simplifying the formulas
Under this assumption, the previously presented generic linearisation equation, which was derived using a first-order Taylor expansion for f, is modified as follows in order to include the disturbance vector:
The derivative matrix \( f; (x_0, u_0, O) \) is defined similarly to the other matrices \( f: \) and \( f: \), with the notation \( f; (x_a, u_a, ii) \) representing the components of the matrix This matrix is structured as \( f; (x_0, u_0, O) \sim [f_1 (x_a, u_a, D) \, | \, f_2 (x_a, u_a, D) \, | \, \ldots \, | \, f_n (x_a, u_a, D)] \), where \( M \) is equal to \( d \).
In the analysis of the marine plant, non-linear state-space equations are transformed to create a linearized perturbation component that exclusively includes the perturbation signals.
Irc Nrc = aQE I aQLI aN Xo aN No.KQo aQrcl aN E Xo
The derivation of the linearization equation from the plant state-space description is detailed here, focusing on the calculation of the relevant partial derivatives To achieve this, the torque variables associated with the engine, turbocharger, and propeller are expanded using Taylor series, retaining only the first-order term for accuracy Specifically, this process is applied to the torques of the engine and turbocharger.
QE""'QEO+[qE,l qE,2l[ n ]+qE,o'fR nrc
Qrc "'" Qrco + [ qrc,l qrc,2 J [ n ] + qrc,o fR nrc
The partial derivatives of the torque approximator functions, denoted as qE,O', qE,l', qE,2', qrc,o', qrc,l', and qrc,2, are calculated with respect to key variables such as shaft/turbo RPM and fuel index These derivatives are evaluated at an equilibrium point defined by a specific vector.
Specifically, the partial derivatives of engine torque are calculated as follows: qE'O=a~ QE(N,Nrc,FR)1 =QEmax.JWOã(a~ $(Wb+W,X)l} (5.51)
R x=x o 1 R =x o qE'l=a~QE(N'Nrc,FR)I_ =QEmax'{WO'(a~$(Wb+W'X)l_} (5.52)
X-Xo -Xo qE.2=a~ QE(N, Nrc' FR)I =QEmax.JWoã(a: $(Wb+W.X)l } (5.53) rc x=x o 1 rc =x o
The partial derivatives of turbocharger torque are computed as follows: qrr.o" a~, Qrr(N,Nrr,F,t" "Q_.{ VO -( a~, -i.~ITJI -~) y(t)
Figure 6.4 Continuous-time integrator in block form u~) ~n) z
Figure 6.5 Discrete-time integration with backwards difference
The key practical element of the marine supervisory controller lies in the forms utilized for implementing the engine and turbocharger torque functions, specifically QE(N,Nrc,F R) and Qrc(N,Nrc,F R).
2 and 5 the following three alternative ways were presented for the determination of the torque variables:
• as solutions of a non-linear, perplexed algebraic system (see Sections 2.5.1 and 2.5.2)
• as 3D numerical maps with fixed step in Section 2.5.4
• as neural nets trained to approximate the numerical maps (see Section 5.2)
The first method is not suitable for real-time implementation due to the prohibitive computational overhead of numerical iterative solutions for algebraic systems In contrast, torque functions can be effectively utilized through look-up tables or neural approximators While look-up tables offer the advantage of not needing calibration or training, there are compelling reasons to consider neural approximators, which are discussed briefly below.
Neural networks are considered more computationally efficient than traditional look-up tables, particularly in high-dimensional scenarios They require fewer computational resources, such as memory and operations, when replacing look-up tables Additionally, the rise of commercially available parallel architectures based on neural computing further enhances the advantages of using neural networks over conventional von Neumann machines.
Training neural approximators enhances the handling of sparse, incomplete, and noisy data sets, unlike traditional look-up tables that may introduce significant numerical errors In marine plants, important variables such as lubrication oil pressure, temperature, and ambient conditions, which influence steady-state performance, are often overlooked in thermodynamic modeling Although their impact is minor, it is not negligible Constructing a theoretical model to depict these effects is challenging, and experimental data can be inconsistent across different plants and time periods Consequently, the nominal plant model within the supervisory controller requires tuning during operation to adapt to specific conditions This necessity makes neural networks an appealing solution, as they can be re-trained with new data, providing a robust system of associative, cumulative memory.
The robustness of the feedback controller in marine plants is significantly influenced by the use of neural torque approximators, as highlighted in Chapter 5 The parametric uncertainty of the UPM system matrices A and B is linked to inertial uncertainty and the torque derivatives of the engine and turbocharger These derivatives are calculated using neural torque approximators, which can derive analytical expressions around each steady-state operating point defined by a triad (No, NTCO, FRo) when the weight matrices of the approximators are known By retraining these approximators within the supervisory controller, it becomes possible to automate the determination of uncertainties in the UPM system matrices using updated weight matrix values This allows for an automatic robustness assessment of the feedback controller whenever changes in the nominal plant dynamics, as represented by the parameters in the N2M equations, are detected The effectiveness of neural torque approximators in implementing torque functions within a marine plant supervisory controller is further explored in the context of the MAN-B&W 6L60MC propulsion plant discussed in Chapter 5.
Test Case Investigation
The supervisory controller module for the MAN-B&W 6L60MC plant, previously analyzed for its open-loop thermodynamic behavior and neural torque approximators, is now subject to numerical investigation.
The dynamical response of the engine thermodynamic model from Chapter 2 is analyzed in comparison to the neural torque approximators during transient conditions, specifically examining 35% to 100% and 50% to 100% steps in fuel index value Importantly, there is no low-pass setpoint filter between the power level setpoint and the fuel index offset value, and for simplicity, the power level setpoint is assumed to align with the rated fuel index value In this context, the coefficients K~rake.l and K~rake.O are set to 1.0 and 0.0, respectively The responses of shaft and turbo RPM, engine torque delivery, and turbocharger total torque are illustrated in Figures 6.6 and 6.7 for both transient scenarios Additionally, for the first case (35% to 100% step), the outputs from both the thermodynamic model and the neural state model are presented.
In the second case, where the step ranges from 50 to 100%, only the differences between thermodynamic and neural values are presented for each variable This approach helps to avoid confusion, as the modeling error is relatively small.
Discrete-time integration was utilized in both models with a time step of 0.1 seconds, enhancing the realism of the simulation This fixed integration time step significantly reduces simulation time in Matlab/Simulink It is crucial that the time step remains sufficiently small; for instance, MAN-B&W recommends that the rpm value at MCR for the 6L60MC engine should not exceed 123 rpm, necessitating a sampling interval that is less than or equal to this value.
Therefore, the value of 0.1 s is approximately five times smaller and well below the above sampling threshold required for cycle-mean modelling temporal resolution
600 ãã_-+ãããã ãããã1ããã ããããj.ããã ããã!ãããfãããã
Figure 6.6 Transient 1: 35 to 100% load - Step response predicted by both models
0,1 ~ ' ~ """"~"""""~""'''''''!'''' eO.1 -r -r -r -( -~ - gO.S + ! ããã?ãããlããã~ãããã
Figure 6.7 Transient 2: 50 to 100% load - modelling error (thermodynamic- neural)
The analysis of Figures 6.6 and 6.7 indicates that when the thermodynamic model's response is treated as the actual engine response, the neural state model, utilizing neural torque approximators, exhibits minimal modeling error It is observed that the modeling error increases with the magnitude of the step, which is expected due to the non-linear nature of both models and the significant impact of excitation magnitude Additionally, a step change in load represents a worst-case scenario for rapid engine loading; however, such load changes are infrequently encountered in real marine operations.
It is essential to acknowledge that modeling errors in the thermodynamic engine model may necessitate the tuning and training of neural torque approximators in real installations Additionally, after a certain period of service or following an overhaul, re-training of the torque approximators may be required to ensure their predictions align more closely with the actual plant's performance.
The Low-pass Setpoint Filter
A setpoint filter is integrated into the supervisory controller to mitigate rapid changes in engine loading, which is crucial for maintaining the thermo-structural integrity of plant components and preventing issues such as shafting system resonance and turbocharging lag These rapid changes can lead to significant engine and powerplant failures, necessitating a controlled rate of engine loading adjustments Implementing a low-pass filter for the setpoint is one effective method to achieve this, although other solutions like slew-rate limiters are also available.
This article focuses on the thermodynamic aspects of powerplant operation, specifically addressing the limitation of turbocharging system lag As illustrated in Figures 2.3-2.6 and 6.6-6.7, a significant fuel index step leads to a temporary failure in engine torque delivery, resulting in a shaft deceleration This phenomenon occurs due to a sudden rise in exhaust pressure, which reduces air flow through the engine, causing the air mass flow rate to drop below (AIF)low However, the increased energy in the exhaust gas eventually accelerates the turbocharger, enhancing scavenging pressure, air flow rate, and engine torque delivery.
Non-linear numerical techniques can effectively address various engineering problems, with the MAN-B&W 6L60MC engine serving as a representative example of ship propulsion powerplants This case focuses on determining the time constant l' f of a first-order low-pass filter characterized by its specific transfer function.
The goal is to filter the fuel index signal to prevent any deceleration of the engine-propeller shaft during the transient range of 35 to 100% Figures 2.3-2.6 and 6.6-6.7 illustrate a worst-case scenario of engine loading, highlighting a notable reduction in shaft RPM within the first second of the transient The design will utilize a neural state model, while validation will incorporate a thermodynamic model as well.
In both cases the discrete-time integration with the backwards difference method is used Therefore, the filter transfer function is transformed in the Z- domain, where its representation is:
(6.15) where T, = 0.1 s is the sampling period determined in Section 6.2.3 Taking into account Shannon's sampling theorem, we obtain that a time constant value l' f < 2ã T, = 0.2 s is meaningless
The impact of the filter on shaft rpm response during a 35 to 100% transient is illustrated in Figure 6.8, comparing two different filter time constants, 0.25 and 1.0 seconds A filter with a small time constant, slightly above Shannon's limit, significantly enhances rpm response, yet some shaft deceleration remains evident Conversely, a larger time constant leads to unnecessary delays in engine acceleration.
G.I : : : i : g, 80 ãããfããã -ã,j,ã -ã-+ ããã1-ã-ããã ~ ããã-ãã w : : , : :
Figure 6.S Effect of filter on response: r f = 0.25 s (left) and r f = 1.0 s (right)
A trial-and-error search was conducted to determine the optimal value of r_f within the range of [0.25 s, 1.0 s] with a 0.01 s step, aiming to achieve a condition of No ~ 0 (no shaft deceleration) throughout the transient period of 35% to 100% The optimal time constant identified was r_f = 0.47 s, which is illustrated in Figure 6.9, showcasing the plant response from both the neural state model and the thermodynamic engine model.
The neural state model closely aligns with the thermodynamic engine model, validating the use of neural approximators in filter design Additionally, the neural state model demonstrates reduced execution time on computers compared to the thermodynamic model, facilitating quicker development and broader coverage of the time constant value set.
The analysis of Figure 6.9 indicates that utilizing a setpoint filter allows for smoother acceleration of the plant, with only a slight increase in transient duration (no more than 0.5 seconds) compared to the scenario without a filter This smoother acceleration is reflected in the engine torque delivery profile, aligning with predictions from the reference thermodynamic engine model While the torque signal does experience a drop, it remains above the 35% load threshold, contrasting sharply with the significant reduction observed in engine torque without the filter Additionally, for comparative analysis, the filtered fuel index schedule is presented for both the thermodynamic engine model and the neural state model in Figure 6.9.
It OO ããããt.ããã'jããã ; ~ - ~
"E12 ã-ãããi ããã-ããã~ããã-ãrãã ããããiãããtããã
S10 ãããTããã'ããããjããã :ããã1ããã[ããã
Figure 6.9 Effect of the filter on response with r f = 0.47 s
The implementation of the setpoint filter enables smoother engine operation while maintaining a quick response to transients More sophisticated filtering techniques, such as multi-pole filters and non-minimum-phase filters, can be employed to enhance performance These advanced filtering methods can effectively shape the fuel index signal, resulting in a reduced initial increase rate during transients, which is crucial for addressing AIF mismatch and other operational challenges.
Full-state-feedback Control of the Marine Plant
Theoretical Background
Supervisory control enhances plant operation and engine performance while boosting installation reliability, as discussed in Section 6.2 By integrating non-linear optimisation theories with intelligent modelling, control, and thermodynamic principles, effective optimisation can be achieved A critical requirement for this approach is the implementation of a feedback control scheme that keeps plant operations close to their optimal state, which has undergone thorough reliability analysis This chapter focuses on full-state-feedback linear control, while PID control was addressed in Chapter 4.
The control problem for the linear Uncertain Parameter Model (UPM) is established by considering the state equations with disturbance, represented as x = Aãx + Bãu + d Additionally, the nominal perturbation model (NPM) is defined as x = Ao'x + Bo'u + d The objective is to determine the gain matrix K for the full-state-feedback control law, expressed as u = Kã x, ensuring effective control despite the presence of parametric uncertainties in the system matrices.
The proposed method aims to design a state-feedback controller with a gain matrix K to ensure effective disturbance rejection for the NPM, assuming that both disturbances and uncertainties in the system matrices are limited The robustness of the closed-loop system will be assessed against parametric uncertainties in the open-loop system matrices using standard theoretical and methodological tools While robust control of linear systems is a well-explored topic in control literature, this approach introduces unique features that are not typically found in conventional methods To highlight the distinctions, a brief overview of the standard H∞ controller synthesis procedure is provided, which relies on the H∞ norm of a transfer function matrix, derived from the singular values of a real matrix M, specifically the square roots of the positive eigenvalues of the non-negative definite matrix (MT M).
It can be proven that the number of singular values is equal to the rank of M, kRANK ~ min{p,m}; moreover, singular value indexing, i, corresponds to their descending order, i.e.:
0"1~0"2~"'~O"k"NK >0 (6.20) Therefore, the largest singular value of M, denoted 0" max (M), is equal to 0"1'
The transfer function matrix of a linear system, represented as M(s), is a matrix function of the complex frequency s The maximum singular value, denoted as O"max (M(s = jro)), is a frequency-dependent function Consequently, the Hoo-norm of the transfer function matrix M(s), indicated as ||M(s)||∞, is defined based on this relationship.
,~o i.e the supremum or least upper bound of the scalar, real-valued function O"max (M(jroằ)
In linear systems, the Hoc-norm provides a significant interpretation When considering the system setup illustrated in figure 6.10, if the linear system M(s) is stimulated by a sinusoidal input vector u, represented as uT = u~ãsin(cot) = [uolãsin(rot) uo2ãsin(cot) uomãsin(cot)], the resulting system output vector y will also exhibit sinusoidal functions at the same frequency co, with amplitudes represented as y~ = [YOI Y 02 ••• Yam].
Figure 6.10 MIMO dynamical system with input-output description
Furthermore, the Euclidean norm of the output, IIYII:
IIYII = IIYl12 = ~~ Y~i will satisfy the following relationship:
IIYII $IIM(s)ll~ '11ull where Ilull is the Euclidean norm of the input vector
The application of the above theory to Nonlinear Predictive Control (NPM) involves linear full-state feedback controls The closed-loop transfer function matrix, G(s), describes the relationship between disturbances and the state vector, represented as x = Ao'x + Bo'u + d This can be expressed in the frequency domain as x(s) = G(s)ãd(s) = (sI2 - Ao - BoãK )^-1 des(s), where u = Kãx.
To effectively address the issue of propeller load disturbance rejection, it is essential to select the controller gain matrix K such that the IIG(s) norm remains below a predetermined threshold This challenge is commonly known in the literature as the suboptimal Hoc control problem A viable solution involving a full-state-feedback controller K can be achieved if specific conditions regarding the plant's structure align with the upper design limit for the IIG(s) norm The process typically involves a trial-and-error approach, necessitating multiple resolutions of a related mini-max differential game and an algebraic Riccati equation until the desired termination criteria are satisfied.
The Structured Singular Value (SSV, J1) framework, grounded in Linear Fractional Transformation, effectively addresses disturbance rejection issues in uncertain plant scenarios.
Robust stability and performance tests, as outlined in the Main Loop Theorem, are essential for validating designs in closed-loop uncertain systems Additionally, the H∞ control problem solution is generalized through the D-K iteration method, leading to a double iterative optimization process When successfully completed, this procedure yields a controller that guarantees both disturbance rejection and robustness for the closed-loop plant.
The theoretical background related to marine plants is extensively documented in the literature; however, practical limitations necessitated a more tailored approach, as detailed in Section 6.3.2 The developed technique effectively addresses these limitations and demonstrates efficacy in both marine propulsion plants and linear systems with a single disturbance source and a state-space description.
Practical Hoo-norm Requirements
As already mentioned, control requirements for both the NPM and UPM are:
In Chapter 5, it was argued that: d T =[ -rl'(No ãk Q +No 11) 0] = [d l o]=> \\d\\ =\d l\ (6.27)
By substituting, the closed-loop transfer function matrix of the NPM linear state feedback controls becomes: with
(6.28) where the closed-loop characteristic polynomial Pees) in the above is given by:
Pe (s) = SZ - (au + a22 + blkl + b2kz)' s + (allan - a12a21 )
The equation (6.29) is expressed as (a2 A - a12b2)kl + (allb2 - a2Ibl)k2, where the index 0 denotes the nominal values of the elements within the matrices Ao = [aij,o] and Bo = [bi,o] in the UPM system.
A and B, has been temporarily dropped in order to simplify the formulae Later, however, when robustness of the closed-loop system is examined, index 0 is restored
Applying the results of Section 6.3.1 for the Hoo-norm in the scalar case, the following inequalities are obtained for Inl and l'7c I :
The Hoo-norm calculations discussed above focus exclusively on scalar (closed-loop) transfer functions, omitting transfer function matrices Additionally, the analysis in Section 5.4.2 indicates that the disturbance \( d_l \) can be constrained within a symmetrical interval around zero.
Therefore, by combining the above:
Inl::;;IIGl(s)t ãDd and InTcl::;;IIG2(s)II~ ãDd
(6.32) Finally, the requirements for Inl and InTCI are translated to the following Hoo- norm requirements (which will be used in design) for the scalar transfer functions
Gl (s) and G2 (s) from disturbance d l to shaft and turbo rpm, nand nTC respectively:
A formal requirement for IIG(s)IL is proposed to ensure individual disturbance rejection for each marine plant state variable, comparable to the disturbance rejection achieved by the previously mentioned practical Hoo-norm requirements.
Inl::;; ~lnl2 +lnTC12 ::;; G Omin ãIdll::;; GlO ãIdll InTC I ::;; ~lnl2 + InTC 12 ::;; G Omin ãIdll ::;; G20 ãIdll
However, such a specification for IIG(s)t can be expected to be harder to meet, as mentioned in literature, e.g [44]
On the other hand, if the upper bound chosen for IIG(s)t is larger than GOmin then it is not guaranteed that the two separate practical requirements for Gl (s) and
G 2 (s) are met The argumentation starts by considering a bound:
Assuming GOmin equals GlO, the specified requirement can be satisfied through a set of inequalities that apply to the Euclidean norm of each state variable independently.
InTc I2 ~ (£0 -£ )'ldJ for some £ in the interval 0 < £ < £0' By combining concluded that the requirement for IIG(s)ll~ is met; guaranteed that Inl ~ On or, equivalently, IIGI (s)t ~ GIO •
(6.38) the above, it is easily however, it cannot be
The conclusions are illustrated through the open-loop marine plant, with its transfer functions detailed in Section 5.5.1 In this scenario, no control is assumed, resulting in fR equating to zero.
Both TXd,1 (s) and T xd,2 (s) are all-pole transfer functions with real poles Thus:
In Section 5.5.1, it is noted that TXd,l(s) exhibits a single pole at s = all, while Txd,2(s) has two poles at s = all and s = a22 Importantly, for Txd,l(s), zero-pole cancellation occurs at s = a22, rendering this pole unobservable.
IITxd,l(st =_1 1 1 "" I and IITxd,2(St =I~I= I '1IqTC'III (6.41) all 2KQoNO a ll a 22 2KQoNo qTC,2
Neglecting feedback control in marine powerplants can lead to dangerous overspeed conditions, particularly near the maximum continuous rating (MCR) operating range For instance, the MAN-B&W 6L60MC propulsion powerplant exemplifies this risk, highlighting the critical need for effective control mechanisms to maintain safe shaft RPM levels.
IqTC,lIôlqTC,21=>IITxd,2(St ôIITxd,I(St and InrclôonTc (6.42)
Despite the requirement being easily fulfilled without feedback control, engine overspeed may still occur in terms of shaft RPM.
~lnl2 +lnTc l2 II II ~(on)2 + (onTC )2
Improving the values of n and nrc in relation to No and NTCO or MCR will yield only minimal enhancements, as turbocharger speed shows limited dependence on propeller torque coefficient Additionally, adjusting the contributions of n and nTC in Ilxll will not significantly impact outcomes if the upper bound contributions of On and on TC are weighted accordingly Therefore, it is essential to implement repeated reductions in the upper bound of IIG(s)1L and redesign the Hoo controller to meet the new requirements Subsequently, the requirement for Inl must be evaluated separately; if it is satisfied, the generated controller is deemed acceptable; if not, further reductions in the IIG(s)ll~ requirement are necessary until the target value (onlod) is achieved.
In most cases the above procedure is expected to work out after the bound for
IIG(s)1L has been reduced to a value closer to GOmin ' However, a simpler approach would be to design the controller so that the individual Hoo-norm requirements for
The standard Hoo synthesis procedures, which heavily depend on numerical iterative methods, allow for the "arbitrary" placement of poles and zeros in the closed-loop system, making it more challenging to ensure robustness against unmodeled plant dynamics However, a technique specifically designed for second-order systems with scalar disturbance is presented, featuring a significantly reduced numerical component This approach not only simplifies the process but also enhances understanding, as it fundamentally operates as a pole placement method.
Marine Plant Regulator Synthesis
After examining the key factors influencing marine plant dynamics and the challenges associated with the direct application of the Hoo controller synthesis method, this article proposes an alternative design procedure tailored specifically for marine plants An essential assumption relevant to marine plants was discussed in Section 5.5.1.
The turbocharging system is effectively matched to the engine, ensuring sufficient combustion air reaches the engine cylinders Consequently, it can be concluded that engine torque remains largely unaffected by variations in turbocharger RPM across a broad range Based on this premise, the previously derived closed-loop characteristic polynomial, Pees(s), is expressed as: p/s) = S² - (alJ + a22 + b1k1 + b2k2)ãs + a'la22 + a22b1k1 + (alJb2 - a21b1)k2.
(6.46) For design purposes the following standard form for the scalar transfer functions of interest, G1(s) and G2(s), will be used:
By equating the coefficients of the two forms of PeeS), the following two equations for the regulator gains kl and k2 are obtained: blkl + b2k2 = -(all + a22 ) - (PI + P2)
(6.49) a22blk l + (a ll b2 -a2Ibl )k2 = PIP2 -a ll a22
The values of system poles and zero can be determined by the requirements for
The peak value of the magnitude Bode plot for the transfer functions IIGdsl and IIGz{sl is influenced by their shape Assuming the closed-loop poles and zeros are purely real and stable, the peak value for G2(s), which has only two poles, can be determined by evaluating the transfer function at ω = 0.
The peak value and its corresponding frequency for gain interleaving (GI) are influenced by the relative position of the closed-loop zero (z) in relation to the smallest pole (P) Specifically, the relationship can be expressed as |GI(jω)| = ~2 ~ P: 2 when z is less than P.
In this analysis, we assume that PI is the minimum of pp and P2, which simplifies our manipulation of the equations Consequently, we derive specific constraint inequalities that govern the positioning of the real poles and zeros in the system.
(6.52) and la21 + b2kll ::; PIP2 G20 where GIO and G20 respectively are the required upper bounds for IIG I (s)t and IIG 2 (s)t
The comprehensive Hoo admissible value set for Ph P2 can be derived by reformulating the existing inequalities through a strategic change of variables By utilizing specific variables in place of Ph, the process becomes more straightforward and effective.
P2), the majority of the inequalities to be satisfied become linear:
Controller gains can be calculated on the basis of Sand P, from the pole- placement equations, as follows:
(6.54) where ki0 , kiS ' k,1" i = 1,2 are readily calculated using the values of the open-loop system parameters (elements of the open-loop system matrices A and B)
The inequalities for P and S are derived directly from the previously outlined Hoo-norm requirements Consequently, we can establish five linear inequalities that define the relationship between the pair (S, P).
To achieve a minimum-phase requirement for G1(s), the inequality PI ≤ z is simplified to 0 < z Omitting PI ≤ z allows the system of inequalities to be expressed in a linear form, facilitating easier analytical or numerical solutions.
To establish the values for S and P, we can subsequently determine the value set for the closed-loop poles P1 and P2 These poles are calculated as the roots of a specific second-order parametric equation for each pair of values (S, P).
X2+SãX+p=O (6.56) provided that the following constraints are satisfied:
• S > 0, so that the closed-loop system is stable,
• S2 ;::: 4P, so that Plo P2 are real, and, finally,
• PI = min {PI' P2} ::;; z = -a22 - b2k2 in order for the calculated value of IIG1(S)II~ to be valid
In conclusion, the admissible subset of R² for the dyad (P1, P2) will be identified, along with the corresponding value of the closed-loop zero z The selection of closed-loop poles and zero for the NPM, using controller gains, can be derived from this subset to address additional considerations, such as robustness against UPM modeling uncertainty, as demonstrated in the subsequent test case.
Test Case: MAN B&W 6L60MC Marine Plant
The controller synthesis procedure was tested on a propulsion plant featuring the MAN B&W 6L60MC marine Diesel engine, which is directly coupled to the propeller Key system parameters, including partial torque derivatives necessary for calculating the transition matrix A and matrix B of the state equation, are detailed in Table 6.1.
Table 6.1 Specifics of powerplant with MAN B&W 6L60MC engine qE,l = 0.46 N m1rpm qE,2 = 0.0 N m1rpm qrc,l = 4.70 N m1rpm qrc,2 = 180 N m1rpm qE,O = 9.0 kN m1% qrc,o = 700 N m1%
By substituting the values in Table 6.1, the following matrices A, B are obtained
9.2923 -355.8744 s , B = 1384 %index The linear state-space NPM is shown in Figure 6.11
Figure 6.11 Linear perturbation model of marine plant for regulator synthesis
To estimate the necessary bounds for IIGI(s)ll~ and IIGls)ll~, the maximum absolute uncertainty akQ is assumed to be 10% of the nominal value K QO, which is approximately 6 N m/rpm² Consequently, the introduced disturbance d1 is anticipated to fall within this specified range.
The design targets for the system include a shaft RPM fluctuation limited to 1.2 RPM and a turbo RPM fluctuation capped at 30 RPM Consequently, the upper bounds for the Hoo-norm of the two relevant transfer functions are established based on these specifications.
The open-loop poles (K=O) are the eigenvalues of the transient matrix A, i.e
-2.319 and -355.8744 The values for IIGJs)t and IIG2(S)II~ when no feedback is applied are:
IIGJst = -6.6 dB and IIGJs)ll~ = -38.3 dB Although the requirement for IIG2(S)II~ is met by far, the upper bound for
The open-loop plant's magnitude and phase Bode plots, illustrated in Figure 6.12, indicate that IIGJs)L exceeds the threshold by approximately 14.0 dB To effectively redistribute the impact of disturbances, an appropriate linear state-feedback control law can be employed As long as the values of IIGJs)ll~ and IIG2(s)t remain within acceptable limits, the closed-loop poles can be strategically positioned to satisfy additional criteria, including robustness against neglected dynamics.
Figure 6.12 Bode plots of open-loop transfer functions from disturbance to states
The open-loop system exhibits unobservability, particularly due to the fast pole P2 = a 22, which coincides with the open-loop zero z This phenomenon can be physically understood through the minimal impact of turbo rpm variations on shaft rpm, as long as the turbo ensures sufficient air flow for complete combustion and effective scavenging in engine cylinders This principle is generally applicable to marine engines, where typical steady-state air-to-fuel ratio values range from 30.0 to 40.0, while the ideal combustion threshold is below 20.0.
In this example the following two cases are included
(i) Double real pole at -PI = - P 2 = -19.9 This choice can be achieved by setting the state feedback gains to:
K = [kl k 2 ] = [-0.6270 0.2306] (6.57) The zero of GI (s) is then z = 36.8 > 19.9 = P!,2 (6.58) The values achieved for the Roo-norm of the two transfer functions of interest are:
IIG I (s)1L = -20.65 dB and IIG 2 (s)ll~ = 6.72 dB
Compliance with the Roo requirements is therefore concluded The magnitude and phase Bode plots of the compensated system in this case are shown in Figure 6.13 a
Figure 6.13 Bode plots of the closed-loop transfer functions - case (i)
(ii) Discrete real poles at -PI = -11.0 and -P 2 = -250.0 This choice can be achieved by setting the state feedback gains to:
K = [kl k 2 ] = [-4.3256 0.0746] (6.59) The zero of GI (s) is then z = 252.7 > 11.0 = PI (6.60) The values achieved for the Roo-norm of the two transfer functions of interest are:
IIGI(s)1L =-20.74 dB and IIG 2 (s)1L =6.74 dB
The compliance with the Roo requirements has been successfully achieved, as demonstrated by the magnitude and phase Bode plots of the compensated system, illustrated in Figure 6.14.
Figure 6.14 Bode plots of the closed-loop transfer functions - case (ii)
The transfer function from d1 to the control action y can be optimized to a minimum phase by carefully selecting controller gains The choice of controller should also consider additional requirements for the closed-loop system transfer function matrix For instance, a double real pole may be too close to neglected fast system poles, which could undermine the robustness of the closed-loop system against unmodeled fast dynamics Conversely, a slower pole PI enhances robustness against these fast dynamics, although zero-pole cancellation may occur if the fast system pole pz is situated near the zero z of G1(s) This situation can lead to unobservability in the system, similar to the open-loop case, especially when parametric uncertainty is present.
The impact of limiting functions, like the scavenging pressure limiter, is effectively replaced by feedback from the turbo shaft RPM This applies to both controllers analyzed, where k2 is greater than zero Consequently, when turbo RPM falls below the desired level, the increase in fuel index is regulated to avoid a potential deficiency in air charge.
Robustness Against Model Uncertainty
The UPM system matrices exhibit considerable uncertainty in their elements, necessitating a thorough robustness check of the resulting closed-loop system after controller synthesis Two primary forms of robustness can be distinctly identified in this context.
• robust stability of the closed-loop system, and
• robust performance of the closed-loop system
For each one of the above there exists a major theoretical result (test) for a given linear system The related results are as follows:
• Robust stability of linear systems - Theorem of Kharitonov
This finding ensures the stability of a family of characteristic polynomials with variable coefficients constrained within specific intervals, known as interval polynomials Stability can be confirmed by examining four clearly defined "edge" polynomials from the family.
• Robust performance of linear systems - Zero-Exclusion Principle
The Hoo-norm of a family of proper scalar transfer functions, with variable coefficients within specified subsets, can be ensured to remain below a defined performance level by verifying the robust stability of an appropriately derived complex polynomial To assess the robust stability of these complex polynomials with uncertain parameters, the Zero Exclusion Condition can be employed.
The detailed statements of the two theoretical results are extensively documented in the literature, with various proof versions available [61-71], so they are not included here A crucial prerequisite for applying these results is the clear definition of the subsets where the parametric uncertainty of the transfer functions resides In the context of marine plants, two nominal closed-loop transfer functions have been derived following the implementation of full-state-feedback controls to the Nonlinear Plant Model (NPM).
[ n(s)] = [G l,nom (S)] d (s) = P c, o(s) dl(s) nTC(s) G2,nom(s) I a21,O +b2,okl
(6.61) where the nominal closed-loop characteristic polynomial Pc,o(s) in the above is given by:
In the context of robustness analysis for the open-loop UPM, it is important to restore the notation for its elements Initially, the value assigned to l2 during synthesis was set to zero for specific reasons However, a complete form of the characteristic polynomial is necessary for conducting a thorough robustness assessment Additionally, Gl,nom(s) and G2,nom(s) represent the fixed-coefficient scalar transfer functions derived by substituting the nominal values of the UPM system matrices into Gl(s).
The literature identifies the uncertainty structure of linear, scalar transfer functions based on the relationship between the coefficients of the numerator and denominator polynomials and the uncertain parameters For marine plants, the standard form for the transfer functions G1(s) and G2(s) is represented as s + C/C.
The transfer functions G1(s) = 2Nl and G2(s) = 2N2(6.63)s + CID•S + COD•S + CID•S + COD illustrate that the coefficients of these polynomials are derived from the elements of the UPM state matrices A = [aij] and B = [bj] Consequently, these coefficients are not constant; rather, they are dependent on the vector of state-space parametric uncertainty.
= (all.O +a22 0 +kl ãb 1• O +k2 ãb2 0 )+(L1all +L1a22 +kl ãL1b 1 +k2 ãL1b2)
Note that p belongs to a sphere of ]R5 with radius:
In the above, as already mentioned, lower case delta denotes the uncertainty radius of elements of UPM system matrices A and B, which have been determined in detail in Section 5.4.2
To evaluate the robustness of closed-loop systems, it is essential to determine the bounds of the coefficients of the transfer functions, which are confined within specific intervals This assessment can be conducted similarly to the approach outlined in Section 5.4.2, which addresses the uncertainty of physical parameters in marine plants to derive state-space parametric uncertainty For instance, consider the case of clD.
So = (all,o + a22,O + kl bl.O + k2 b2 0 ) - (Da 1) + Da 22 - k) Db) + k2 Db2) (6.68)
So = (all,o +a22,O + k] ãb),O + k2 b 2 ,o) + (Da)] + Da 22 - k) ãDb) +k2 ãDb2)
It is assumed that k1 is less than zero and k2 is greater than zero, which is applicable to common marine plants as discussed in Section 6.3.4 In contrast, determining the bounds for Chemical Oxygen Demand (COD) is more complex due to the involvement of products between various elements of the UPM system matrices A and B; nevertheless, this can be effectively managed using the methodology outlined in Section 5.4.2.
Independent bounding of the coefficients of closed-loop transfer functions G1(s) and G2(s) often leads to conservative outcomes, including unrealistic scenarios This conservatism can result in misleading conclusions, such as the rejection of a controller due to perceived instability arising from coefficient combinations in G1(s) and G2(s) that are not attainable within the defined limits of vector p.
One way to deal with this problem is to "tighten" the coverage of the uncertainty included in the coefficients of closed-loop transfer functions Gl (s) and
G2 (s) From the expressions for these coefficients it can be easily seen that all functions are affine with respect to the elements of p; the only exception is
COD (p), which is, however, multiaffine A real-valued function of several variables
F(p), P E Jl{l, is said to be affine if it is the sum of a linear function plus a constant, i.e if it has the following form:
(6.69) where C: = [CCI CC2 CCl] and Cco are constant Moreover, a function
A function F(p), where P is an element of Jl{l}, is classified as multiaffine if each component of p satisfies a specific condition: when all but one component of p are held constant, the function exhibits affine behavior This allows for the multiplication of different elements of p, while prohibiting the use of powers.
The Zero Exclusion Condition is crucial for ensuring the robust stability of polynomials with coefficients exhibiting affine or multi-affine uncertainty structures, such as those found in marine plants This condition asserts that a polynomial p(x) is considered robustly stable if it remains stable across all polynomial uncertainty sets.
• it is stable for some Po E 5, and
The extension of polynomial stability conditions to the Zero Exclusion Principle facilitates the verification of robust performance in closed-loop plants This approach introduces a method for parameterizing uncertainty in the closed-loop transfer functions of second-order systems with scalar disturbances, based on open-loop state-space uncertainty Consequently, standard robustness assessment tools can be effectively applied to marine plants and other systems exhibiting similar dynamic characteristics.
State-feedback and Integral Control of the Marine Plant
Steady-state Error Analysis
A significant issue with feedback controllers that rely solely on error-proportional action is the persistence of steady-state error As discussed in Section 4.1, both P and PD controllers are unable to fully eliminate this steady-state error; however, it can be minimized by effectively increasing the proportional gain.
In feedback control systems, the selection of gains for linear state-feedback control laws is primarily influenced by the transient response to disturbances in propeller torque coefficients This approach often leads to a notable steady-state error By applying the Final Value Theorem of the Laplace transform, we can derive the steady-state response to a step change in disturbance, represented mathematically as x(t ~ 00) = lim(sãx(sằ)= G(O)ãdl(t ~ 00).
Pc (0) = (al\a22 - a\2a21 ) + (a22bl - al2b2 )kl + (al\b2 - a2lbl)k2
If the assumption al2 '" 0 is valid for steady-state then:
Pc(O) = Pc,o(O) = (all + blkl )a22 + (a\lb2 - a21b l )k2
Furthermore, because as explained in Section 5.4.3:
I I Q I and because in steady-state either No = 0 or AI = 0 or both:
(qE,l - 2KQONo + qE,ok1 )qTC,2 + (qE,l - 2KQONo )qTC,O - qTC,lqE,O )k2
A significant contributor to steady-state error in the context of UPM and NPM is the alteration of the propeller-law coefficient KQ, particularly any permanent deviation from its nominal value KQo This alteration can occur when propulsion power requirements rise, which may be due to factors such as hull or propeller fouling, changes in loading, or adverse operating conditions like weather and sea states For instance, prolonged service without drydocking can lead to hull fouling from pollution, necessitating increased power to maintain the same service speed due to heightened resistance against the vessel's movement Consequently, this results in greater propeller torque demands and an elevated value of KQ.
A significant contributor to steady-state error in state variables is the decreased efficiency of the engine and turbocharger, which can result from powerplant performance degradation or aging, such as piston ring wear or turbine wheel fouling.
QE (No,NTCO ' FRO) = QEO = KQo ' N; (6.75)
However, due to degradation, it can happen that at one or more N 2 M steady-state operating points:
QE (No,NTCO,FRO) = Q~o < QEO = KQo ' N; and/or
QTC (No, N TCO' FRO) = Q;co i= 0 Using linearisation analysis we obtain that:
QE (N,NTC,FR) = Q~o +qE,O' fR QTC (N,NTC ' FR) = Q;co + qTC,2 ãnTC +qTC,O fR
To simplify the expressions without losing generality, only the dominant terms are retained, considering the torque derivative values applicable to common marine plants as discussed in Section 5.4.2 The steady-state error in the state vector \( \tilde{x}_s = [ n_{ss} \, n_{Tc,ss} ] \) can be determined using the aforementioned linear approximations, ensuring that shaft equilibrium is achieved for the closed-loop plant with linear state-feedback controls represented by \( f_R = k_1 \cdot n + k_2 \cdot \dot{n} \).
{ Q~o + qE.Ok\ nss + qE,ok2 ' nTC,SS = QEO + 2KQoNo ' nss}
Q;co + qTC,2 nTC,SS + qTc,Ok\ nss + qTc,ok2 ' nTC,SS = 0 n
[ nss ] [qE,Ok\ - 2KQoNo qE,ok2 ]-\ [QEO - Q~o] xss = = ' , (6.79) nTC,ss qTc,Ok\ qTc,ok2 + qTC,2 -QTCO
In the above, no deviation from the nominal propeller law is assumed (i.e
To analyze the isolated impact of power plant degradation on steady-state error, we consider the equation KQ = KQo This approach can be readily expanded to incorporate the influence of kQ as time approaches infinity, similar to previous methodologies.
In the open-loop scenario, where k1 and k2 are both zero, the off-diagonal elements are absent, indicating that engine torque deviation solely affects shaft RPM, while turbocharger torque deviation only impacts turbo RPM, resulting in decoupled states Conversely, state-feedback control enables the redistribution of steady-state errors from one torque variable to both state variables This principle remains valid even when kQ approaches zero, as demonstrated by previous expressions; specifically, in the absence of feedback control, deviations from the nominal propeller law primarily influence shaft RPM rather than turbo RPM, due to the typically small value of the parameter qTC in two-stroke engines.
The presence of steady-state error in one or both state variables results in a persistent deviation between actual plant performance and ideal operation, represented by N 2 M Identifying this mismatch is crucial for automated monitoring systems to detect anomalies and schedule maintenance To restore zero steady-state error, recalibrating N 2 M is essential to update the model with modified system parameters This involves determining the new nominal propeller torque coefficient and potentially retraining the neural torque approximators using the latest engine time series data.
Integral Control and Steady-state Error
Incorporating an integral term in the control law effectively eliminates steady-state error, as demonstrated in Section 4.1.3 This principle is particularly evident in PID control laws, although their application to marine plants relies on scalar transfer functions Notably, when the shaft rpm integral is integrated into the control law, it increases the order of the closed-loop system by one, highlighting the complexity introduced by this approach.
In cases where a MIMO state-space description of the open-loop plant is available, it is essential to enhance the state vector with relevant integrals of the key state variables This approach is illustrated through the state equations of marine plants, where the state vector is expanded to incorporate these critical elements.
The following extended open-loop state equations are then formulated:
[ x ~lj X3 2 - [all a a 1 2l a 0 0 0 l2 22 0 OJ 0 0 [Xlj x X3 2 + [blj b2 0 ãf R + [dlj 0 0 x 4 0 1 0 0 x 4 0 0
Then, full-state-feedback control is applied to the above extended ("hat") system: fR=K.x=[kl k2 k3 k 4 }[X I x 2 X3 x 4 r (6.82)
Therefore the following closed-loop transfer function from disturbance to the state is obtained:
To effectively manage disturbances in a control system, it is necessary to design four scalar transfer functions corresponding to the four state variables, resulting in a fourth-order characteristic polynomial The integration of state variables Xl and x2 in the control law indicates that the closed-loop system will achieve zeros at s = jro = 0, effectively eliminating steady-state error Additionally, by constraining the Hoo-norm of the transfer functions from disturbance to state variables X3 and x4, one can determine the necessary controller gains However, calculating the roots of the fourth-order polynomial complicates the design process, making standard Hoo synthesis the most practical approach for implementation.
Introducing the integral of a single state variable, typically shaft RPM, into the control law reduces the characteristic polynomial's order to three While this may lead to steady-state error in the other state variable, the benefit lies in the availability of an analytical method to determine the roots of third-order polynomials in the s-plane Originally proposed by Vishnegradskii regarding the asymptotic stability of third-order polynomials, this method was further developed by Aizerman to evaluate system stability Notably, the method incorporates a damping factor, making it a natural extension of the analysis for second-order systems discussed in Chapter 4.
The propulsion control architecture, based on the powerplant model decomposition from Chapter 5, aims to establish a reference value set for optimal plant operation while compensating for deviations The N2M model serves as the foundation for a supervisory controller module that generates setpoint values for feedback compensatory control Emphasizing the significance of filtering, this approach enhances transient response and ensures smoother engine performance, especially given the non-linear characteristics of the plant model The feedback control mechanism is analyzed through the lens of modern robust control theory for linear systems, addressing the limitations of standard techniques A tailored controller design method for marine plants is proposed, focusing on individual closed-loop shaping for each state variable to meet specific H∞-norm requirements Additionally, the method considers robustness against state-space parametric uncertainty, evaluating the applicability of standard theoretical tools The analysis concludes with a steady-state error assessment combined with integral control strategies.