Compressors and Compressor Systems
Compressors play a crucial role in various modern applications that require gas or fluid pressurization, including jet propulsion engines, industrial gas turbines, turbochargers, and pipeline compressors in the petrochemical and mining sectors They are primarily classified into two categories based on their pressure-raising mechanisms: positive displacement compressors and dynamic compressors Positive displacement compressors, like reciprocating and rotary screw compressors, generate pressure by reducing the gas volume, in accordance with the ideal gas law.
Dynamic compressors, also known as turbocompressors, increase pressure by initially accelerating gas with a rotating impeller and subsequently decelerating it to convert kinetic energy into static pressure For comprehensive insights into the mechanisms of both positive displacement and dynamic compressors, refer to sources [24] and [61].
Dynamic compressors are divided into two types according to the direction of the flow leaving the impeller The first type is the axial compressor, where the gas
S.Y Yoon et al., Control of Surge in Centrifugal Compressors by Active Magnetic Bearings, Advances in Industrial Control,
Axial and centrifugal compressors are two primary types of dynamic compressors used in various applications In an axial compressor, gas flows parallel to the shaft, with a decreasing cross-sectional area from inlet to exhaust, where the rotating impeller accelerates the gas and the stator blades contribute to pressure rise through deceleration Conversely, a centrifugal compressor directs gas radially out of the impeller, with the flow entering axially and being accelerated towards the impeller tip Pressure increase occurs both in the impeller and the static diffuser, which decelerates the gas to maximize kinetic energy conversion into pressure Centrifugal compressors are particularly efficient for high-pressure, low-flow rate applications, while axial compressors excel in high flow rate and speed scenarios This book will primarily focus on centrifugal compressors, with the intention of extending findings to axial compressors as well.
Centrifugal compressors can feature either vaned or vaneless diffusers, each with distinct advantages Vaned diffusers excel in efficiently pressurizing gas over short distances but are prone to flow instabilities caused by separation at the vanes In contrast, vaneless diffusers, while less effective in gas pressurization, offer a more economical and straightforward manufacturing process.
Fig 1.2 A cross section of a centrifugal compressor [88]
Fig 1.3 Compressor characteristic curves for increasing compressor speed
N, and demand load curves for high and low loads at the compressor exhaust
The static performance of a compressor is portrayed by its characteristic curve.
The compressor characteristic curve illustrates the relationship between pressure rise and inlet flow rate at a constant speed, with each speed corresponding to a unique curve In contrast, the demand load curve represents the pressure drop at the compressor exhaust relative to the inlet flow rate The steady-state pressure rise and flow rate for a compressor operating under optimal design conditions are determined by the intersection of the characteristic curve and the demand load curve.
Finally, a compression system is composed of the compressor and other upstream and downstream components affecting the flow through the machine A general
A general compression system, illustrated in Fig 1.4, consists of a compressor, a plenum or collector volume, a throttle valve, and connecting piping In this system, gas enters the compressor, where it is pressurized before being directed to the plenum volume for collection The throttle valve plays a crucial role in regulating the flow rate of the pressurized gas exiting the system This configuration is commonly referenced in the modeling and active control of compressor instabilities, which will be further explored in this chapter.
Active Magnetic Bearings in Compressors
Active magnetic bearings (AMBs) have increasingly been applied in turbomachinery over the past few decades, evolving from small turbomolecular pumps to large megawatt-range compressors By utilizing magnetic forces from electric coils acting on conductive shaft materials, AMBs effectively levitate the rotor, eliminating mechanical contact between static and rotating components This technology enhances compressor efficiency and allows for higher operational speeds with minimal frictional losses, resulting in extended service intervals without maintenance These advantages are particularly critical in demanding applications like subsea oil and gas development, where accessibility for routine diagnosis and upkeep is limited.
The Active Magnetic Bearing (AMB) system operates through a closed-loop mechanism that stabilizes the rotor at the center of the bearing air gap Key components of the AMB system include magnetic actuators, a rotor, proximity sensors, a power amplifier, and a controller The proximity sensors measure the rotor's position, allowing the controller to calculate commands for the power amplifiers These commands adjust the output currents to the magnetic actuators, generating the necessary electromagnetic force to levitate the rotor effectively For more detailed information on the principles and recent advancements in AMB technology, refer to [101].
There are many benefits of employing AMBs over other more conventional bear- ings in compressors A comprehensive list of these benefits are presented in [101], and are summarized as follows.
Contact-free operation minimizes wear and eliminates the need for lubrication, resulting in reduced maintenance requirements and extended bearing lifespan This makes them particularly suitable for high-temperature applications and environments involving corrosive or pure gases.
Fig 1.5 Diagram of a rotor suspended by AMBs
• Low bearing losses: the low losses compared to traditional bearings allow the AMB supported compressors to operate more efficiently at high speeds.
Active control of rotor dynamics is achieved through active magnetic bearings (AMBs), which provide adjustable stiffness and damping tailored to various operating conditions These bearings also incorporate unbalance compensation schemes that significantly enhance their performance and capabilities.
Smart machines equipped with sensors and magnetic actuators have the capability to perform self-diagnosis, allowing them to assess bearing integrity By programming these machines to modify Active Magnetic Bearing (AMB) characteristics, optimal operational performance can be achieved.
• Enhanced compressor stability: Using the AMBs as build-in actuators, new capa- bilities can be added to compressors such as active stabilization of flow instabili- ties.
Designing and implementing active magnetic bearing (AMB) systems presents a unique challenge due to the need for expertise in electronics, control theory, and rotor dynamics These systems must meet precise rotor-dynamic requirements, often mirroring the characteristics of traditional mechanical bearings Engineers must grasp the rotor-dynamic needs specific to the rotating machines they are working on, and effectively apply bearing design and control theory to meet these requirements Furthermore, it is essential to present results in a standardized format that is comprehensible to both control and rotor-dynamic engineers.
Compressor Instability
Stall
Stall is a localized flow instability in turbocompressors, often occurring just before or overlapping with the surge line, as illustrated in the characteristic curve During stall initiation, viscous shearing forces lead to flow separation at the boundary layers, resulting in areas of reduced or stalled flow that disrupt the compressor's internal flow pattern A graphical depiction of this stall pattern in the impeller blade passage of centrifugal compressors is provided.
The illustrated stall phenomena depicted in Fig 1.7 reveals how unstable flow patterns at the passage walls create eddies that obstruct the flow path in certain areas, leading to regions of reduced flow.
Fig 1.6 Surge line and commonly observed stall region in the characteristic curve for a general centrifugal compressor [97]
Stall cells in centrifugal compressors generate a circumferentially non-uniform flow in the compressor annulus, traveling at speeds between 20% to 70% of the rotor speed These complex flow phenomena can interact, leading to varied stall behaviors reported in the literature Three primary types of stall instability have been identified: progressive rotating stall, abrupt stall, and non-rotating blade stall Progressive stall, the most prevalent type, results in a gradual decrease in compressor output pressure ratio as mass flow diminishes, with stall cells rotating at a frequency proportional to compressor speed In contrast, abrupt stall is marked by a sudden drop in output pressure rise, creating a sharp discontinuity in the characteristic curve Non-rotating stall cells typically initiate at the inducer blade and may interact with rotating stall cells in the impeller and diffuser sections.
Detecting stall cells in compressors is challenging due to their localized nature, especially during progressive stall where pressure drops between the inlet and outlet are gradual, often leaving no clear signs in flow measurements Kọmmer and Rautenberg successfully identified stall cells in centrifugal compressors by using strategically placed temperature and pressure sensors along the shroud, which allowed them to analyze pressure pulsations at various impeller regions, revealing frequency components aligned with the harmonics of the traveling stall cells.
Stall significantly impacts the performance and efficiency of axial compressors, leading to potential impeller failures due to increased load and temperature To enhance compressor performance and reliability, active stall control has been extensively researched, as detailed in various studies However, most existing literature primarily addresses multistage axial compression systems, with limited findings on centrifugal machines, as the significance of stall instability in centrifugal compressors remains a debated topic.
Surge
As flow restriction in a compression system increases, pressure in the plenum can reach a critical level where the compressor fails to match this pressure rise, leading to a reversal of flow towards the compressor inlet This disruption initiates a rapid axisymmetrical oscillation known as compressor surge, with the surge inception point marking the critical pressure threshold Identifying this surge point is crucial for safe compressor operation, as it delineates stable and unstable flow regions The surge line, connecting surge points across various compressor speeds, illustrates this relationship Unlike stall, surge represents a flow instability that impacts the entire compression system, causing significant oscillations in pressure and mass flow, particularly in centrifugal compressors Researchers have extensively studied the mechanisms behind surge initiation in both centrifugal and axial compressors.
Fig 1.8 Example of a surge limit cycle The cycle can be divided into the emptying
Surge instability in centrifugal compressors arises from the compressor impeller and diffuser's failure to generate an adequate pressure field to align with the output collector's pressure Research indicates that, during low-speed operation, surge is often triggered by flow oscillations occurring at the compressor inducer.
Surge in high-speed compressors is often triggered by flow oscillations occurring at the diffuser throat A common precursor to this surge is stall, which creates the initial flow disturbance necessary to initiate the surge oscillation.
Centrifugal compressors can exhibit multiple surge points, but their output flow behavior during surge is generally summarized by de Jager as a limit cycle in the compressor characteristic curve As the throttle valve closes, the mass flow rate decreases, moving the operating point toward the surge inception point, initiating the surge limit cycle This first phase, known as the collapse or blowdown period, is marked by a rapid pressure drop throughout the system, causing the flow characteristic to shift quickly into the negative flow region Subsequently, the system enters a recovery period, where pressure is gradually restored, and the flow characteristic moves back to the stability region Unless the compression system undergoes changes to stabilize the operating point, this surge process will continue to repeat.
Surge in compression systems can be categorized into four types: mild, classic, modified, and deep surge Mild and classic surges exhibit small amplitude oscillations, with frequencies comparable to the Helmholtz frequency, which acts as the resonance frequency of the system Typically, these surges do not result in flow reversal and are common in compressors with smaller plenum volumes In contrast, modified surge features the coexistence of surge and stall instabilities, displaying characteristics of both phenomena Deep surge is characterized by larger oscillation amplitudes and flow reversal during part of the cycle, with lower oscillation frequencies determined by the plenum's filling and emptying times Regardless of the surge type, failure to manage surge effectively can lead to significant structural damage due to the intense vibrations and elevated gas temperatures associated with this instability.
Compressor Surge Modeling
The study of compression system modeling for controlling surge instabilities in turbomachinery has gained significant attention due to the advantages of expanding the stable operating range of centrifugal and axial compressors Mathematical models for these systems are categorized into one-dimensional models, primarily used for predicting compressor surge, and two-dimensional models that address both surge and rotating stall While centrifugal compressors predominantly focus on surge instability through one-dimensional approaches, axial compressors are more associated with rotating stall, necessitating the development of specialized two-dimensional models.
Emmons et al conducted an early stability analysis of a compression system by utilizing a one-dimensional linearized model of compressor dynamics, which illustrated the onset of surge oscillations through the analogy of a self-excited Helmholtz resonator While their model effectively predicted the initiation of surge, it fell short in accurately replicating the significant perturbations associated with deep surge pulsation Following a similar approach, Greitzer expanded on this foundational work.
A nonlinear lumped-parameter model for one-dimensional incompressible flow in compression systems was introduced, initially focusing on axial compressors This model was later shown to be applicable to single-stage centrifugal compressors with low compressibility.
Since the introduction of the original Greitzer model, various enhancements in modeling compressor flows in axial and centrifugal systems have emerged Notably, Macdougal and Elder developed a one-dimensional model for compressors with compressible flow, while modular models for complex configurations were created by Badmus et al., Elder and Gill, and Morini et al Additionally, mathematical models addressing one-dimensional flow and varying rotor speed were introduced In two-dimensional compressor flow modeling, Moore and Greitzer's early model effectively predicted rotating stall for incompressible axial flow and was later adapted for compressible flow Despite the advancements, the Greitzer model remains favored for studying active surge control in centrifugal compressors due to its simplicity and low-order equations Its lumped-parameter nature simplifies the mathematical description but limits the model's ability to capture the complexities of fluid dynamics in distributed systems, such as acoustic waves and flow pulsations Research has also explored the stability of compression systems with added capacitance in piping configurations and the impact of flow pulsations on system stability.
Helvoirt and Jager observed that piping acoustics significantly influence the shape of pressure oscillations during surge conditions They suggested implementing a transmission line model, initially proposed by Krus et al., to analyze the impact of pipeline dynamics on pressure oscillations in deep surge scenarios While deep surge oscillation is crucial in turbomachinery, it represents only a portion of the overall dynamics within the compression system Understanding the system dynamics during both stable compressor operation and the onset of surge is vital for studying compressor instabilities and developing effective stabilizing controllers Consequently, a comprehensive mathematical model is essential to capture the effects of piping acoustics across stable and unstable operating conditions, as well as during transitions between these states.
Surge Avoidance and Suppression
Surge Avoidance
To protect compressors from surge, it is essential to avoid flow conditions that lead to instability, with several common surge avoidance methods available These strategies utilize flow controllers to enhance flow and alleviate pressure when conditions approach the surge line The primary method is ratio control, where flow controller set points align with the compressor's surge line, creating a surge avoidance line that maintains a separation margin, known as the surge margin, from the surge line Various actuators, such as blow-off or bleed valves, are employed to manage compressor operation safely, while recycle valves can redirect part of the compressed gas back to the inlet to keep the compressor within the safe operational region.
Most of the available industrial solutions are based on these two surge avoidance methods.
Surge avoidance options, while practical and easy to implement, often compromise the operating efficiency of compression systems due to the uncertainty surrounding the surge line's exact location To prevent damage to machinery, large surge margins are typically enforced, which restricts the compressor's operation in the high-pressure region of its characteristic curve, ultimately limiting overall system performance Conversely, a smaller surge margin increases the risk of the compressor entering the surge cycle, potentially leading to equipment damage A typical surge margin is around 10% for most compressors, though critical applications may necessitate margins as high as 25%.
Surge Suppression and Control
Compressor surge mitigation and control are increasingly important for enhancing the safety and efficiency of industrial compressors Effective surge suppression methods depend on precise compressor models to develop strategies that stabilize the system beyond the surge point Passive surge controllers utilize physical modifications, such as mass-spring-damper systems, to react to environmental changes and extend the stable flow region by inducing pressure variations in the compression system.
Active surge control has been the focus of extensive research since its introduction by Epstein et al It utilizes a sensor and actuator combination to actively adjust the compression system in response to feedback measurements.
A control circuit or algorithm computes in real-time the command signal to the ac- tuator that stabilizes the compression system based on the flow measurements from the sensors.
This article presents a comprehensive comparison between passive and active surge control systems, highlighting that the passive controller utilizes a hydraulic oscillator, while the active system employs a throttle valve Similar analyses are found in the works of Arnulfi et al., where surge controllers incorporate a movable wall in the compression system's collector to adjust plenum volume The passive approach in these studies features a spring damper system for controlling the plenum wall's motion, whereas the active method utilizes an actuator to modify the wall's position.
Selecting appropriate actuators is a key focus in surge control research, as highlighted by Simon et al in their theoretical comparison of common sensor-actuator pairs The study utilized a linearized Greitzer model alongside a proportional surge controller to assess system stability during surge conditions It emphasized that ensuring controllability of the compressor's unstable modes while maintaining sufficient bandwidth to stabilize flow dynamics is challenging The analysis also took into account bandwidth limitations and actuator constraints, concluding that bandwidth often serves as the primary limiting factor in practical surge control applications.
Jager and Gu et al conducted comprehensive surveys on passive and active surge control, highlighting the importance of selecting appropriate feedback sensors for effective implementation of active surge controllers Flow sensors in industrial compressors are prone to noise, which can negatively impact surge controller performance, potentially leading to destabilization and driving the compressor into surge limit cycles Mass flow sensors are particularly vulnerable to noise, often providing only steady-state measurements due to bandwidth limitations in larger compressors To address these challenges, Bohagen and Gravdahl, along with Chaturvedi and Bhat, proposed nonlinear mass flow observers that utilize pressure measurements in the compressor, enabling their integration with various surge controllers that depend on flow rate feedback.
Proper actuator selection is essential for effective surge control, with the throttle valve at the system exhaust being a prominent choice explored in various studies, including the Greitzer and Moore–Greitzer models Research by Banchini et al highlighted a high-gain proportional active surge controller that utilizes the throttle valve to stabilize compressor flow based on plenum pressure measurements This controller features an adaptive algorithm that adjusts the gain according to the compressor's behavior to optimize system response Experimental results showed the controller's effectiveness in damping surge limit cycles at low compressor speeds, although its performance diminished at higher speeds due to increased measurement noise and mechanical actuator limitations.
The active surge controller described in [9] utilizes a throttle valve to modify bifurcation properties within the stall/surge region of axial compressor systems, effectively eliminating hysteresis associated with stall and limit cycles caused by surge In a similar approach, Krstic [79] developed a backstepping method controller that stabilizes both stall and surge, employing pressure and mass flow rate measurements for feedback to determine the throttle valve's opening percentage A key advantage of Krstic's method is its ability to ensure global system stability with minimal information about the plant However, both studies lack experimental data to corroborate their mathematical analyses.
The closed-coupled valve is frequently discussed in literature for active surge control, typically positioned at the system inlet near the compressor Gravdahl and Egeland demonstrated through analytical and simulation results that a surge controller for centrifugal compressors can effectively stabilize flow by integrating the closed-coupled valve with compressor speed, ensuring operation at the desired equilibrium on the compressor characteristic curve By treating compressor speed as a state variable, surge instability can be managed during both acceleration and deceleration phases, which are critical moments for flow fluctuations A similar strategy was explored in another study, where the combination of compressor speed and throttle valve aimed to maintain a constant pressure rise, with the proposed controller specifically designed for fuel-cell power systems.
A bleed-valve ring, positioned between stages or at the exhaust of axial compressors, was investigated for stall and surge control Unlike the air injection method, bleed-valve rings require minimal hardware modifications for implementation in compression systems The control strategy utilizes a simple Linear Quadratic Regulator (LQR), derived from a linear approximation of an enhanced Moore–Greitzer model To improve the closed-loop system's stability, a nonlinear term was integrated into the linear feedback law Experimental findings demonstrated that the designed 2D actuator effectively mitigated stall hysteresis, although data on surge control were not provided.
Active magnetic bearings (AMBs) are emerging as innovative servo actuators aimed at mitigating flow instabilities in compression systems Research by Senoo and Ishida explored how varying the clearance between the compressor impeller tip and the shroud impacts centrifugal compressor performance Additionally, studies have investigated the potential of using AMBs to actively control stall in axial compressors by adjusting the radial impeller tip clearance While these analyses indicate a promising approach for stall stabilization, they lack experimental validation Sanadgol proposed utilizing AMBs to regulate the impeller tip clearance in single-stage centrifugal compressors, aiming to leverage pressure variations for surge control He also developed a mathematical model to illustrate the relationship between tip clearance and compression system dynamics; however, the study was confined to simulations, leaving the practical effectiveness of AMBs in industrial compressors unaddressed.
Objectives of This Book
This book serves as a comprehensive reference on integrating Active Magnetic Bearing (AMB) technology in turbocompressors, highlighting its role in stabilizing flow during surge conditions It presents a control method that utilizes AMB's capability to adjust the axial tip clearance of the impeller, effectively compensating for flow fluctuations that signal the onset of surge.
This book explores the theory behind a surge suppression method and demonstrates its feasibility through simulations and practical implementation in an experimental compressor test rig designed to replicate surge instability It focuses on a novel approach to controlling surge instability in unshrouded single-stage centrifugal compressors Early chapters introduce essential concepts related to compression systems, active magnetic bearings (AMBs), and rotor dynamics, with Chapter 2 reviewing rotor dynamics pertinent to turbocompressors and industrial compressor stability and performance requirements Chapter 3 provides a comprehensive overview of AMB operating principles and introduces the key components of the AMB system.
The article outlines the theoretical formulation and experimental validation of a surge control method, beginning with a detailed description of a compressor test rig established for this purpose in Chapter 4 Key components, including the Active Magnetic Bearing (AMB) system that levitates the compressor rotor, are highlighted The experimental characterization of compressor dynamics is crucial for validating dynamic models of both the AMBs and the compression system, which will be developed in Chapter 5 Chapter 6 introduces fundamental concepts in linear control theory, laying the groundwork for the design of AMB controllers in the subsequent chapters Finally, Chapters 7 and 8 focus on the derivation of the AMB levitation controller and the active surge controller, respectively, detailing their design, experimental implementation, and testing results.
Rotor dynamics is an engineering discipline focused on analyzing the lateral and torsional vibrations of rotating shafts, aiming to predict and control rotor vibrations within acceptable limits Key components of a rotor-dynamic system include the shaft or rotor with disk, bearings, and seals Typically, industrial applications utilize flexible rotors, designed with long and thin geometries to optimize space for components like impellers and seals, while operating at high speeds to enhance power output The first supercritical machine, a steam turbine by Gustav Delaval in 1883, set the stage for modern high-performance machines, which generally operate above the first critical speed—considered the most critical mode—while avoiding prolonged operation at or near these speeds A standard practice in the industry is to maintain a critical speed margin of 15% between the operating speed and the nearest critical speed.
The main components of rotor-dynamic systems include bearings and seals, which play crucial roles in system stability and performance Bearings support the rotating elements and provide necessary damping to control rotor vibrations, while seals prevent leakage of processing or lubricating fluids However, seals can introduce rotor-dynamic properties that may exacerbate vibrations Rotor vibrations are typically classified into two categories: synchronous and subsynchronous Synchronous vibrations occur at frequencies that match the shaft's rotational speed, often due to unbalance or other synchronous forces, whereas subsynchronous vibrations, or whirling, occur at frequencies below the operating speed and are primarily caused by fluid excitation resulting from cross-coupling stiffness.
In this chapter we present a short introduction to rotor dynamics, with the in- tention to familiarize the reader with basic concepts and terminologies that are of-
S.Y Yoon et al., Control of Surge in Centrifugal Compressors by Active Magnetic Bearings, Advances in Industrial Control,
This article discusses the use of Active Magnetic Bearings (AMBs) in rotor-dynamic systems, drawing from rotor-dynamics course notes by Allaire and various authoritative texts by Childs, Genta, Kramer, Vance, and Yamamoto and Ishida It introduces fundamental rotor-dynamic principles through a simple rotor/bearing system model, addressing key concerns such as critical speed, unbalance response, gyroscopic effects, and instability excitation Additionally, it details the American Petroleum Institute's standards for auditing rotor response in compressors, emphasizing their relevance to AMB-equipped compressors and their significance in designing the AMB levitation controller for the compressor test rig discussed in Chapter 7.
Fửppl/Jeffcott Single Mass Rotor
Undamped Free Vibration
Undamped free vibration analysis focuses on rotor vibrations when both unbalance eccentricity and damping are negligible (e u=0, c s=0) In this scenario, the equations of motion simplify to mu¨xC + k s u xC = 0 and mu¨yC + k s u yC = 0, allowing for a clearer understanding of the system's behavior.
The solution to the second-order homogeneous system is represented by the equations \( u_xC = A_x e^{st} \) and \( u_yC = A_y e^{st} \), where \( A_x \) and \( A_y \) are complex constants determined by the initial conditions of the rotor disk By substituting these solutions into the corresponding equations, we derive \( ms^2 A_x e^{st} + k s A_x e^{st} = ms^2 + k s \).
A x e st =0, (2.8a) ms 2 A x e st +k s A x e st ms 2 +k s
The above equations hold true for any value ofA x andA y if the undamped charac- teristic equation holds, ms 2 +k s =0 (2.9)
Solving the above equality for the complex constant s, we obtain the following solution: s 1,2= ±j ω n , (2.10) whereω n is the undamped natural frequency of the shaft defined as ω n k s m 48EI
The solutions to the equation of motion reveal undamped oscillatory functions with a frequency of ±ω n, where the undamped critical speed of the system is defined as ω cr = ±ω n This encompasses both the forward component, which signifies lateral vibrations in the direction of shaft rotation, and the backward component, indicating vibrations that move in the opposite direction The final solutions for undamped free vibrations are expressed as a linear combination of the solutions derived from Eqs (2.7a), (2.7b), and Eq (2.10), represented by the equation u xC = A x1 e j ω n t + A x2 e − j ω n t.
=B y1cos(ω n t )+B y2sin(ω n t ), (2.14) for some values ofA xi andA yi , orB xi andB yi , which can be found from the initial conditions of the rotor.
Damped Free Vibration
The free vibration analysis of the Fửppl/Jeffcott rotor incorporates non-zero effective shaft damping within the system The governing equations of motion are expressed as mu¨ xC + k s u xC + c s u˙ xC = 0 and mu¨yC + k s u yC + c s u˙yC = 0, highlighting the relationship between mass, stiffness, and damping forces acting on the rotor.
The solutions to the above system of homogeneous second order differential equa- tions take the same form as in Eqs (2.7a), (2.7b) Substituting these solutions into Eqs (2.15a), (2.15b), we obtain ms 2 +k s +c s
These equations hold for any initial condition if the damped characteristic equation holds: ms 2 +k s +c s =0 (2.17)
The zeros of the characteristic equation, also know as the damped eigenvalues of the system, are found to be s 1,2 = − c s
Generally, the rotor/bearing system is underdamped, which means that c s
2m 0 ensures that the energy associated with the control effort remains finite.
An analytical solution to the optimization problem involving a quadratic objective function can be derived The optimal control input for the state space system, which minimizes the quadratic cost function, is given by u(t) = -R^(-1) B^T P x(t), where P represents the unique positive definite solution to the algebraic Riccati equation (ARE).
The algorithm for solving the above Riccati equation and finding the optimal con- troller for the LQR problem is implemented in the Matlab Control Toolbox as the lqr function.
The LQR problem aims to minimize the output signal's energy as defined in Eq (6.62), yet it does not impose restrictions on the energy of the states x(t) Consequently, while the LQR solution for a general state space system ensures BIBO stability, it cannot control or stabilize states that are unobservable from the system output y(t) The following theorem presents the necessary condition for the LQR problem to ensure the internal stability of the closed-loop system.
Theorem 6.4 [123] Let Q=C T C If (A, C) is observable, then the closed-loop system ˙ x A−R − 1 BB T P x (6.65) is asymptotically stable.
In the context of the LQR problem, observing the system states through the objective function enables the synthesis of an optimal control law This law effectively limits the energy of the internal states, ensuring efficient system performance.
The LQR problem, as outlined in Eq (6.63), leads to the development of a state feedback controller; however, direct state measurements are often unavailable in practical applications To address this, state estimators or observers are utilized to derive the feedback signal from the plant's output measurements When the optimal LQR solution is integrated with an optimal Kalman filter for state estimation, the resulting feedback controller is referred to as a Linear Quadratic Gaussian (LQG) regulator or H2 controller This controller is characterized by its ability to minimize the H2 norm of the closed-loop system.
More details on LQG regulators can be found in Chap.7 A list of examples for
H 2controllers implemented on AMB systems were summarized in Sect.3.13.1of Chap.3.
TheH 2norm ofG(s)can be seen graphically in the Bode plots as the area under the magnitude curve, as illustrated in Fig.6.22 For example, if the transfer function
G(s)describes the dynamics from the input disturbance to the system output, the
The H2 controller enhances disturbance rejection by minimizing the area under the magnitude curve of the transfer function However, a limitation of this approach is its inability to prevent high peaks in the magnitude curve across narrow frequency bands, even while maintaining a reduced overall area This issue is particularly prevalent in rotor resonance frequencies within Active Magnetic Bearing (AMB) levitation systems, leading to a significant decline in closed-loop performance when disturbances occur within these critical frequency ranges.
Fig 6.22 H 2 and H ∞ norms in the Bode plots
The second optimal control synthesis method discussed here is H ∞ control design, which focuses on minimizing a cost function derived from the infinity norm of the closed-loop system The infinity norm, also known as the supremum norm, for a real rational proper transfer function G(s), is defined as the highest maximum magnitude value across all frequencies ω.
The H ∞ norm of a transfer function G(s) represents the maximum magnitude |G(jω)| across all frequencies ω This concept is visually represented in the magnitude plot shown in Fig 6.22 The objective of the H ∞ control problem is to minimize the infinity norm of the closed-loop system's transfer function.
G(s)between the input and output signals selected in the problem definition If
G(s)is the transfer function from the disturbance to the controlled output, then a smallerG(s) ∞ indicates a stronger disturbance rejection capability for all input frequencies.
Consider the interconnected system in Fig.6.23, where the state space represen- tation of a systemG(s)is given as
In H ∞ control synthesis, frequency-dependent weighting functions are integrated into the system model G(s), akin to the weighting matrices Q and R in the LQR problem The weighted input w(t) and output z(t) serve as the primary variables for different weighting functions, which help quantify the control objectives of the H ∞ controller The main goal of H ∞ control design is to determine a stabilizing controller K(s) that minimizes the H ∞ norm T zw ∞ of the closed-loop transfer function from input w to output z.
Fig 6.23 Interconnected system in H ∞ controller design
For an LTI system with the state space representation in the form of Eq (6.68), the optimalH ∞ control can be found analytically as
, (6.72) and the positive definite matricesX ∞ andY ∞ are, respectively, the unique solutions to the Riccati equations,
According to the feedback law, the norm of the closed-loop transfer function is constrained by T zw (s) ∞ < γ The hinfsyn function, implemented in the Matlab Robust Control Toolbox, utilizes an iterative method to compute the analytical solution for progressively lower values of γ This algorithm yields the minimum γ for which an analytical solution exists, along with the corresponding feedback control law K(s) Additionally, a review of the literature on the application of H ∞ controllers for Active Magnetic Bearing (AMB) systems is discussed in Section 3.13.2 of Chapter 3.
Theμsynthesis controller is commonly viewed as an extension of the traditional
The H ∞ control method focuses on optimizing the closed-loop transfer function by minimizing the infinity norm between the input and output signals This approach is essential for effectively measuring and enhancing control performance.