Sliding Mode Control Part 2 pdf

24 Sliding Mode Control Ming-Chang, S & Ching-Sham, L (1993) Pneumatic servomotor drive a ball-screw with fuzzy sliding mode position control, Proceedings of the IEEE Conference on Systems, Man and Cybernetics, Vol.3, 1993, pp 50-54, Le Touquet Nguyen, V.M & Lee, C.Q (1996) Indirect implementations of sliding-mode control law in buck-type converters, Proceedings of the IEEE Applied Power Electronics Conference and Exposition, pp 111–115, Vol 1, March, 1996 Palm, R (1992) Sliding mode fuzzy control, Proceedings of the IEEE international conference on fuzzy systems, San Diego CA pp 519-526, 1992 Passino, M.K (1998) Fuzzy control, Addison-Wesley, London Qiao, F.; Zhu, Q.M.; Winfield, A & Melhuish, C (2003) Fuzzy sliding mode control for discrete nonlinear systems Transactions of China Automation Society, Vol 22, No June 2003 Sahbani, A.; Ben Saad, K & Benrejeb, M (2008) Chattering phenomenon suppression of buck boost dc-dc converter with Fuzzy Sliding Modes control, International Journal of Electrical and Electronics Engineering, 1:4, pp 258-263, 2008 Sahbani, A., Ben Saad, K & Benrejeb, M (2008), Design procedure of a distance based Fuzzy Sliding Mode Control for Buck converter, International Conference on Signals, Circuits & Systems, SCS 08, Hammemet, 2008 Slotine, J.J.E & Li, W (1991) Applied nonlinear control, Prentice Hall Englewood Cliffs, New Jersey Tan, S.C.; Lai, Y.M.; Cheung, M.K.H & Tse C.K (2005) On the practical design of a sliding mode voltage controlled buck converters, IEEE Transactions on Power Electronics, Vol 20, No 2, pp 425-437, March 2005, ISBN 0-13-040890-5 Tan, S.C.; Lai, Y.M & Tse, C.K (2006) An evaluation of the practicality of sliding mode controllers in DC-DC Converters and their general design issues, Proceedings of the IEEE Power Electronics Specialists Conference, pp.187-193, June 2006, Jeju Tse, C.K & Adams K.M (1992) Quasi-linear analysis and control of DC-DC converters, IEEE Transactions on Power Electronics, Vol No 2, pp 315-323, April 1992 Rashid, M.H (2001) Power electronics handbook, Academic Press, New York, ISBN 0849373360 Utkin, V.I (1996) Sliding mode control design principles and applications to electric drives, IEEE Transaction on Industrial Electronics, Vol 40, No 1, pp 1-14, 1996 Utkin, V.I (1977) Variable structure systems with sliding modes, IEEE Transactions on Automatic Control, Vol AC-22, No 2, April, pp 212-222, 1977 Wang, S.Y.; Hong, C.M.; Liu, C.C & Yang, W.T (1996), Design of a static reactive power compensator using fuzzy sliding mode control International Journal of control, Vol 2, No.63, 20 January 1996, pp 393-413, ISSN 0020-7179 Investigation of Single-Phase Inverter and Single-Phase Series Active Power Filter with Sliding Mode Control Mariya Petkova1, Mihail Antchev1 and Vanjo Gourgoulitsov2 2College 1Technical University - Sofia, of Energetics and Electronics - Sofia Bulgaria Introduction The effective operation of the power converters of electrical energy is generally determined from the chosen operational algorithm of their control system With the expansion of the application of the Digital Signal Processors in these control systems, gradually entering of novel operational principles such as space vector modulation, fuzzy logic, genetic algorithms, etc, is noticed The method of sliding mode control is applicable in different power electronic converters – DC/DC converters, resonant converters (Sabanovic et al., 1986) The method’ application is expanded in the quickly developing power electronic converters such as active power filters and compensators of the power factor (Cardenas et al., 1999; Hernandez et al, 1998; Lin et al., 2001; Mendalek et al., 2001) In this chart, results of the study of a single-phase inverter and single-phase active power filter both with sliding mode control are discussed A special feature is the use of control on only one output variable Single-phase inverter with sliding mode control 2.1 Schematic and operational principle Different methods to generate sinusoidal voltage, which supplies different types of consumers, are known Usually, a version of a square waveform voltage is generated in the inverter output and then using a filter the voltage first order harmonic is separated Unipolar or bi-polar pulse-width modulation, selective elimination of harmonics, several level modulation – multilevel inverters are applied to improve the harmonic spectrum of the voltage (Antchev, 2009; Mohan, 1994) Inverters with sinusoidal output voltage are applicable in the systems of reserve or uninterruptible electrical supply of critical consumers, as well as in the systems for distributed energy generation In this sub-chart an implementation of sliding mode control of a single-phase inverter using only one variable – the inverter output voltage passed to the load, is studied As it is known, two single-phase inverter circuits – half-bridge and bridge, are mainly used in practice (see Fig.1) The inverters are supplied by a DC voltage source and a LC -filter is connected in their outputs The output transformer is required at use of low DC voltage, and under 26 Sliding Mode Control certain conditions it may be missing in the circuit The voltage passed to the load is monitored through a reverse bias using a voltage transducer The use of a special measuring converter is necessitated by the need of correct and quick tracing of the changes in the waveform of the load voltage at the method used here In the power electronic converters studied in the chart, measuring converter CV 3-1000 produced by LEM is applied a) b) Fig a) half-bridge and b) bridge circuits of an inverter Fig.2 displays a block diagram of the control system of the proposed inverter The control system consists of a generator of reference sinusoid U m sin Θ (it is not shown in the figure), a comparing device, a comparator with hysteresis and drivers The control system compares Investigation of Single-Phase Inverter and Single-Phase Series Active Power Filter with Sliding Mode Control 27 the transitory values of the output voltage of the inverter to these values of the reference sinusoid and depending on the result (negative or positive difference) control signal is generated The control signal is passed to the gate of the transistor VT1 or VT2 (for halfbridge circuit) or to the gates of the transistor pairs – VT1-VT4 or VT2-VT3 (for bridge circuit) Fig Block diagram of the control system with hysteresis control The process of sliding mode control is illustrated in Fig.3 Seven time moments are discussed – from t0 to t6 In the moment t0 the transistor VT1 of the half-bridge schematic, transistors VT1 and VT4 of the bridge schematic, respectively, turns on The voltage of the inverter output capacitor C increases fed by the DC voltage source At the reach of the upper hysteresis borderline U m sin Θ + H , where in H is the hysteresis size, at the moment t1, VT1 turns off (or the pair VT1-VT4 turns off) and VT2 turns on (or the pair VT2-VT3) The voltage of the capacitor C starts to decrease till the moment t2, when it is equal to the lower hysteresis borderline Um sin Θ − H At this moment the control system switches over the transistors, etc Therefore the moments t0, t2,, t4 … and moments t1, t3, t5… are identical Fig Explanation of the sliding- mode control 28 Sliding Mode Control 2.2 Mathematical description Fig.4 displays the circuit used to make the analysis of sliding mode control of the inverter The power devices are assumed to be ideal and when they are switched over the supply voltage U d with altering polarity is passed to the LC -filter L iL + (−) id Ud − (+) C iC Fig Circuit used to make the analysis of sliding mode control of the inverter The load current and the current of the output transformer, if it is connected in the schematic, is marked as iL From the operational principle, it is obvious that one output variable is monitored – the voltage of the capacitor uC Its transient value is changed through appling the voltage U d with an altering sign The task (the model) is: uREF = U M sin ωt (1) As a control variable, the production u.U d may be examined, where in: u = sgn[( uC − uREF ) − H ] u = +1 when(uC − uREF ) < H u = −1 when(uC − uREF ) > H (2) The following relationships are valid for the circuit shown in Fig.4: U d = uC + L id = iC + iL iC = C did dt (3) duC dt Using (3) and after several transformations, it is found: uC = duC 1 i = U d dt − uC dt − L dt L.C ∫ LC ∫ C (4) In conformity with the theory of sliding mode control, the following equations are written (Edwards & Spurgeon, 1998): xd = uREF x = uC x = uC (5) Investigation of Single-Phase Inverter and Single-Phase Series Active Power Filter with Sliding Mode Control 29 The control variable ueq corresponding to the so-called “equivalent control” may be found using the following equation (Utkin, 1977): s = x − xd = (6) Using (1) and (4) and taking in consideration (5) and (6), it is found: ueq = u.U d = uC + L diL − L.C ω U M sin ω t dt (7) The value found may be considered as an average value when the switching is between the maximum U MAX and minimum U MIN values of the control variable (Utkin, 1977; Utkin, 1992) If they could change between +∞ and −∞ , in theory, there is always the probability to achieve a mode of the sliding mode control in a certain range of a change of the output variable In order to be such a mode, the following inequalities have to be fulfilled: U MIN < ueq < U MAX (8) for physically possible maximum and minimum values In this case they are: U MIN = −U d U MAX = +U d (9) Resolving (7) in respect to the variable, which is being monitored uC , and substituting in (9), the boundary values of the existence of the sliding mode control could be found: uC = ±U d − L diL + LC ω U M sin ωt dt (10) The equation (10) may be interpreted as follows: a special feature of the sliding mode control with one output variable – the capacitor voltage, is the influence of the load current changes upon the sliding mode, namely, at a sharp current change it is possible to break the sliding mode control within a certain interval From this point of view, it is more suitable to operate with a small inductance value As the load voltage has to alter regarding a sinusoid law, let (10) to be analyzed around the maximum values of the sinusoid waveform It is found: ⎛ di ⎞ uC = ±U d − L ⎜ L ⎟ ⎝ dt ⎠t =π + L.C ω U M ( ±1) (11) 2.ω Where in (11) the positive sign is for the positive half period and the negative one – for the negative half period After taking in consideration the practically used values of L and C (scores microhenrys and microfarads), the frequency of the supply source voltage ( f = 50 or 60 Hz ) and its maximum value U M ( ≈ 325 or 156V ) , it is obvious that the influence of the last term could be neglected Thus the maximum values of the sinusoidal voltage of the load are mainly limited from the value of the supply voltage U d and the speed of a change of the load current So, from the point of view of the sliding mode control, 30 Sliding Mode Control it is good the value of U d to be chosen bigger Of course, the value is limited and has to be considered with the properties of the power switches implemented in the circuit 2.3 Study through computer simulation Study of the inverter operation is made using an appropriate model for a computer simulation Software OrCad 10.5 is used to fulfill the computer simulation Fig.5 displays the schema of the computer simulation The inverter operation is simulated with the following loads – active, active-inductive (with two values of the inductance – smaller and bigger ones) and with a considerably non-linear load (single-phase bridge uncontrollable rectifier with active-capacitive load) Only the load is changed during the simulations The supply voltage of the inverter Ud is 250V, C= 120 μF and L = 10 μH Fig Schematic for the computer simulation study of sliding mode control of the inverter The simulation results are given in Fig.6, Fig.7, Fig.8 and Fig.9 The figures display the waveform of the voltage feeding the load, and the load current, which is displayed multiplied by 100 for the first three cases and by 40 for the last one Investigation of Single-Phase Inverter and Single-Phase Series Active Power Filter with Sliding Mode Control 31 Fig Computer simulation results of the inverter operation with an active load equal to 500Ω using sliding mode control Curve – the voltage feeding the load, curve – the load current Fig Computer simulation results of the inverter operation with an active-inductive load equal to 400Ω/840μН using sliding mode control Curve – the voltage feeding the load, curve – the load current Fig Computer simulation results of the inverter operation with an active-inductive load equal to 400Ω/2Н using sliding mode control Curve – the voltage feeding the load, curve – the load current 32 Sliding Mode Control Fig Computer simulation results of the inverter operation with single-phase bridge uncontrolled rectifier load using sliding mode control Curve – the voltage feeding the load, curve – the load current The results support the probability using sliding mode control on one variable – the output voltage, in the inverter, to obtain a waveform close to sinusoidal one of the inverter output voltage feeding different types of load 2.4 Experimental study Based on the above-made study, single-phase inverter with output power of 600VA is materialized The bridge schematic of the inverter is realized using IRFP450 transistors and transformless connection to the load The value of the supply voltage of the inverter is 360V Fig.10, Fig.11, Fig.12 and Fig.13 display the load voltage and load current waveforms for the load cases studied through the computer simulation Fig 10 The load voltage and load current in the case of active load Investigation of Single-Phase Inverter and Single-Phase Series Active Power Filter with Sliding Mode Control Fig 11 The load voltage and load current in the case of active-inductive load with the smaller inductance Fig 12 The load voltage and load current in the case of active-inductive load with the bigger inductance 33 44 Sliding Mode Control Conclusion The included results in the chart prove the effective operation of the single-phase inverter and single-phase active power filter studied with sliding mode control on one output variable – the voltage feeding the load The results found concerning the sliding mode control of inverters and series active power filters based on only one variable may be expanded and put into practice for three-phase inverters and three-phase series active power filters References Akagi H (2006) Modern active filters and traditional passive filters Bulletin of the Polish Academy of Sciences, Technical Sciences, vol 54, No 3, 2006 Antchev, M.H., Petkova, M.P & Gurgulitsov, V.T (2007).Sliding mode control of a series active power filter, IEEE conf EUROCON 2007, Proc., pp 1344-1349, Warshaw, Poland Antchev, M.H., Gurgulitsov, V.T & Petkova, ,M.P.(2008) Study of PWM and sliding mode controls implied to series active power filters, conf ELMAR 2008, Zadar, Chroatia, 2008, pp.419 - 422 Antchev, M (2009) Technologies for Electrical Power Conversion, Efficiency and Distribution, Methods and Processes, IGI Global, USA Cardenas V.M., Nunez, C & Vazquez, N (1999) Analysis and Evaluation of Control Techniques for Active Power Filters: Sliding Mode Control and ProportionalIntegral Control, Proceedings of APEC’99, vol.1, pp.649-654 Edwards Ch & Spurgeon, S (1998) Sliding mode control theory and applications, Taylor and Francis Hernandez, C., Varquez, N & Cardenas, V (1998) Sliding Mode Control for A Single Phase Active Power Filter, Power Electronics Congress, 1998 CIEP 98 VI IEEE International, pp.171-176 Lin, B.-R., Tsay, S.-C & Liao, M.-S (2001) Integrated power factor compensator based on sliding mode controller, Electric Power Applications, IEE Proceedings, Vol 148, No 3, May 2001 Mendalek, N., Fnaiech, F & Dessaint, L.A (2001) Sliding Mode Control of 3-Phase 3-Wire Shunt Active Filter in the dq Frame, Proceedings of Canadian Conf Electrical and Computer Engineering, vol.2 pp.765-769, 2001, Canada Mohan R (1994) Power Electronics: Converters, Applications and Design, John Wiley and Sons Sabanovic, A., Sabanovic, N & Music, O (1986) Sliding Mode Control Of DC-AC Converters, IEEE 1986, Energoinvest – Institute for Control and Computer Science Sarajevo, Yugoslavia, pp.560-566 Utkin V.I (1977) Variable structure system with sliding modes IEEE Trans On A.C., Vol.AC-22, April 1977, pp 212-222 Utkin V.I.(1978) Sliding Modes and Their Application in Variable Structure Systems, Moskow, Mir Utkin V.I.(1992) Sliding Modes in Control and Optimization, Springer Ferlag, Berlin ISBN 9780387535166 Sliding Mode Control for Industrial Controllers Khalifa Al-Hosani1, Vadim Utkin2 and Andrey Malinin3 1,2The Ohio State University, 3IKOR USA Introduction This chapter presents sliding mode approach for controlling DC-DC power converters implementing proportional integral derivative (PID) controllers commonly used in industry The core design idea implies enforcing sliding mode such that the output converter voltage contains proportional, integral and derivative components with the pre-selected coefficients Traditionally, the method of pulse width modulation (PWM) is used to obtain a desired continuous output function with a discrete control command In PWM, an external high frequency signal is used to modulate a low frequency desired function to be tracked However, it seems unjustified to ignore the binary nature of the switching device (with ON/OFF as the only possible operation mode) in these power converters Instead, sliding mode control utilizes the discrete nature of the power converters to full extent by using state feedback to set up directly the desired closed loop response in time domain The most notable attribute in using sliding mode control is the low sensitivity to disturbances and parameter variations (Utkin, Guldner, & Shi, 2009), since uncertainty conditions are common for such control systems An irritating problem when using sliding mode control however is the presence of finite amplitude and frequency oscillations called chattering (Utkin, Guldner, & Shi, 2009) In this chapter, the chattering suppression idea is based on utilizing harmonic cancellation in the so-called multiphase power converter structure Moreover, the method is demonstrated in details for the design of two main types of DC-DC converter, namely the step-down buck and step-up boost converters Control of DC-DC step-down buck converters is a conventional problem discussed in many power electronics and control textbooks (Mohan, Undeland, & Robbins, 2003; Bose, 2006) However, the difficulty of the control problem presented in this chapter stems from the fact that the parameters of the buck converter such as the inductance and capacitance are unknown and the error output voltage is the only information available to the designer The problem is approached by first designing a switching function to implement sliding mode with a desired output function Chattering is then reduced through the use of multiphase power converter structure discussed later in the chapter The proposed methodology is intended for different types of buck converters with apriory unknown parameters Therefore the method of observer design is developed for estimation of state vector components and parameters simultaneously The design is then confirmed by means of computer simulations 46 Sliding Mode Control The second type of DC-DC converter dealt with in this chapter is the step-up boost converter It is generally desired that a sliding mode control be designed such that the output voltage tracks a reference input The straightforward way of choosing the sliding surface is to use the output voltage error in what is called direct sliding mode control This methodology leads to ideal tracking should sliding mode be enforced However, as it will be shown, direct sliding mode control results in unstable zero dynamics confirming a nonminimum phase tracking nature of the formulated control problem Thus, an indirect sliding mode control is proposed such that the inductor current tracks a reference current that is calculated to yield the desired value of the output voltage Similar to the case of buck converters, chattering is reduced using multiphase structure in the sliding mode controlled boost converter The results are also confirmed by means of computer simulations Modeling of single phase DC-DC buck converter The buck converter is classified as a “chopper” circuit where the output voltage ν C is a scaled version of the source voltage E by a scalar smaller than unity The ideal switch representation of a single-phase buck converter with resistive load is shown in Fig Simple applications of Kirchhoff’s current and voltage laws for each resulting circuit topology from the two possible ideal switch’s positions allow us to get the system of differential equations governing the dynamics of the buck converter as it is done in many control and power electronics textbooks We first define a switch’s position binary function u such that u = when the ideal switch is positioned such that the end of the inductor is connected to the positive terminal of the input voltage source and u = otherwise With this, we get the following unified dynamical system: diL = ( uE − ν C ) dt L (1) dν C ⎛ ν ⎞ = ⎜ iL − C ⎟ dt C⎝ R⎠ (2) Most often, the control objective is to regulate the output voltage vC of the buck converter towards a desired average output voltage equilibrium value vsp In many applications, power converters are used as actuators for control system Also, the dynamics of the power converters is much faster than that of the system to be controlled Thus, it might be reasonable to assume that the desired output voltage vsp is constant By applying a discontinuous feedback control law u ∈ {0,1} , we command the position of the ideal switch in reference to an average value u avg Depending on the control algorithm, u avg might take a constant value as in the case of Pulse Width Modulation PWM (referred to as duty ratio) or a time varying value as in the case of Sliding Mode Control SMC (referred to as equivalent control ueq ) In both cases (constant, or time varying), the average control u avg takes values in the compact interval of the real line [0,1] Generally, it is desired to relate the average value of state variables with the corresponding average value of the control input u avg This is essential in understanding the main static features of the buck converter In steady state equilibrium, the time derivatives of the average current and voltage are set to zero and the average control input adopts a value 47 Sliding Mode Control for Industrial Controllers given by u avg With this in mind, the following steady-state equilibrium average current and voltage are obtained: (3) vC = uavg E iL = vC R = u avg E (4) R According to equation (3) and given the fact that the average control input u avg is restricted to the interval [0,1], the output voltage vC is a fraction of the input voltage E and the converter can’t amplify it L E + - u iL + v C - iC iR C R Fig Ideal switch representation of a single phase DC-DC Buck converter Sliding mode control of single phase DC-DC Buck converter Control problems related to DC-DC converters are discussed in many textbooks The control techniques used in these textbooks differ based on the problem formulation (e.g available measured state variables as well as known and unknown parameters) PWM techniques are traditionally used to approach such problems In PWM, low-power signal is amplified in average but in sliding mode, motion in some manifold with desired properties is enforced by discontinuous control Moreover, sliding mode control provides a better solution over PWM due to the binary nature of sliding mode fitting the discrete nature of the available switches in modern power converters In this section, the problem of regulating the output voltage vC of a DC-DC buck converter towards a desired average output voltage vsp is presented Consider the DC-DC buck converter shown in fig A control law u is to be designed such that the output voltage across the capacitor/resistive load vC converges to a desired unknown constant reference voltage vsp at a desired rate of convergence The control u is to be designed under the following set of assumptions: • The value of inductance L and Capacitance C are unknown, but their product m = / LC is known • The load resistance R is unknown • The input voltage E is assumed to be constant floating in the range [ Emin , Emax ] • The only measurement available is that of the error voltage e = vC − vsp • The current flowing through the resistive load i R is assumed to be constant The complexity of this problem stems from the fact that the voltage error e = vC − vsp and constant m is the only piece of information available to the controller designer 48 Sliding Mode Control The dynamics of the DC-DC buck converter shown in fig are described by the unified dynamical system model in equations (1-2) For the case of constant load current i.e iR ≈ constant , the model can be reduced to that given by equations (6-7) by introducing new variable i : i= ν 1 ( iL − i R ) = ⎛ iL − C ⎞ ⎜ ⎟ C C⎝ R⎠ (5) di = ( uE − ν C ) = m ( uE − ν C ) dt LC (6) dν C =i dt (7) Note that the model described by equations (6-7) is only valid when the load resistance is constant and thus, the load current at steady state is constant Alternatively, the load current i R might be controlled through an independent controller such that it’s always constant For the case of changing load resistance, the model given by equations (6-7) becomes inaccurate and the converter must instead be modeled by equations (1-2) or equations (8-9) using the change of variable given by equation (5) di = m ( uE − ν C ) − i dt RC (8) dν C =i dt (9) A conventional method to approach this problem ( vC to track vsp ) is to design a PID controller with the voltage error e = vC − vsp being the input to the controller Here, a sliding mode approach to implement a PID controller is presented (Al-Hosani, Malinin, & Utkin, 2009) A block diagram of the controller is shown in Fig The dynamics of the controller is described by equations (10-11) where L1 and L2 are design constants properly chosen to provide stability (as will be shown later) dν = L1 ν C − ν sp dt ( ) ( (10) ) di = m ⎡u − ν + L2 ν C − ν sp ⎤ ⎣ ⎦ dt (11) Sliding mode is to be enforced on the PID-like switching surface given by: s= ν C − ν sp c m + i =0 m (12) where c is a constant parameter that is selected by the designer to provide desired system’s characteristics when sliding mode is enforced The control law based on this surface is given by: 49 Sliding Mode Control for Industrial Controllers u= ⎧0 if s < ( − sign ( s ) ) = ⎨1 if s < ⎩ (13) Fig Sliding Mode implementation of PID Controller For sliding mode to exist, the condition ss < must be always satisfied Using the equivalent control method, the solution to equations s = and s = must be substituted in equations (6-7) to get the motion equation after sliding mode is enforced along the switching plane s = in the system’s state space, thus s=0⇒i =− ( ( m ν C − ν sp c ) ( ) (14) ) s = ⇒ ueq = −L2 ν C − ν sp + L1 ∫ ν C − ν sp dt − νC c m (15) As evident from equation (15), the equivalent control ueq is indeed equivalent to a PID controller (Fig 2) with respectively proportional, derivative, and integral gains K P = − L2 , K I = L1 , and K D = −1 c m The equivalent control u eq in (15) is then substituted into equations (6-7) and (10-11) resulting in the following 3rd order system equations: d 2ν C E dν C ⎡ ⎤ = m ⎢ Eν − − EL2 ν C − ν sp − ν C ⎥ dt c m dt ⎣ ⎦ (16) dν = L1 ν C − ν sp dt (17) ( ( ) ) To analyze stability, we write the characteristic equation of the close loop system λ3 + E m λ + m ( EL2 + ) λ − mEL1 = c (18) The constant parameters L1 , L , and c are chosen to provide stability of the system, i.e c > 0, EL2 + > 0, L1 < m ( EL2 + 1) > −L1 c (19) 50 Sliding Mode Control Satisfying the above conditions will provide stability and the system will have the following equilibrium point: ν c = ν sp , ν c = i = 0, Eν = ν sp (20) Fig shows simulation of the system with parameters C = 10 μ F , L = μ H , c = 0.0045 , L1 = −10 , and L2 = 199.92 The switching device is implemented using a hysteresis loop set such that the switching frequency is controlled to be about 100 KHz As evident from fig 3, the output voltage vC converges (with finite amplitude oscillation or chattering) to the desired reference voltage vsp at a rate set by the chosen controller’s parameters Fig Sliding mode PID control of single-phase buck converter Estimation in sliding mode PID controlled single phase DC-DC buck converter The sliding mode PID controller described by equations (10-13) assumes only the knowledge of the parameter m and the voltage error measurement e = vC − vsp However, the parameter m might be unknown or varying from one converter to another Hence, the problem is to complement the sliding mode PID controller with a parameter estimator such that the controller’s parameter can be selected automatically For this, two types of observers are presented next (Al-Hosani, Malinin, & Utkin, 2009) The first type is a sliding mode observer, which assumes that both the reference voltage vsp , and the voltage error e = vC − vsp are known On the other hand, the second type is an asymptotic observer that assumes that the voltage error e = vC − vsp measurement is the only piece of information available A Sliding Mode Observer The converter’s parameter m is assumed to be unknown in this case Thus, we initially feed the sliding mode PID controller described by equations (10-13) with a guess value of m0 In 51 Sliding Mode Control for Industrial Controllers addition, both the voltage error e = vC − vsp and the desired set point voltage vsp are assumed to known Defining the known function G = Eu − vC = Eu − e − vsp , equations (6-7) can be rewritten as: di = mG dt (21) dν C =i dt (22) The following sliding mode observer is now proposed: ˆ di ˆ = mG dt (23) ˆ dm = − M sign Gi dt ( ) (24) ˆ ˆ where m is the estimate of converter parameter m , and i is the estimate of variable i with ˆ − i Choosing the constant M > m will enforce sliding mode along the mismatch i = i ˆ sliding surface i = , and the average value (easily obtained using a low pass filter) of m will tend to m The only restriction in this observer design is that estimation should be fast enough (such that the real value of the estimated parameters is reached before G = ) To overcome this problem, we introduce the following modification to equation (24): ˆ dm dt = ⎧ ⎪ ⎨ ⎪ ⎩ − M sign ( G i ∗ ) if e ≥Δ if e vC / R since 56 Sliding Mode Control (36) sdirect = ⇒ vC = vsp ( ) sdirect = ⇒ − ueq iL = ν sp R ( ) ⇒ − ueq = ν sp (37) iL R Thus, the motion in sliding mode is governed by the following first order differential equation: ν sp ⎞ diL ⎛ ⎟ = ⎜E − dt L ⎜ RiL ⎟ ⎝ ⎠ ss (38) The system has an equilibrium point iL = vsp / ER It is not difficult to see that the system exhibits an unstable equilibrium point (Sira-Ramírez, 2006) and we shall establish this via several approaches First, using an approximate linearization approach, the local stability around the equilibrium point of the zero dynamics i ss = v / ER will be investigated Expanding the L sp ss right-hand side of equation (38) into its Taylor series about the point iL = iL , we obtain: ν sp ⎞ diL ⎛ ss ⎟ = ⎜E − + a1 iL = iL + O i dt L ⎜ RiL ⎟ ss ⎝ ⎠ iL = iL where a1 = ( ) () ν ⎞⎞ d ⎛ 1⎛ E2 R ss ⎜ ⎜ E − sp ⎟ ⎟ = , i = iL − iL diL ⎜ L ⎜ RiL ⎟ ⎟ ν sp ⎠ ⎠ i =iss ⎝ ⎝ L L di L E2 R = i L + H O.T dt ν sp (39) lim i →0 ( ) =0 O i i (40) If we restrict our attention to a sufficiently small neighborhood of the equilibrium point such that the higher order term O ( i ) is negligible, then di L E2 R = iL dt ν sp ss (41) Clearly, the equilibrium point iL = vsp / ER is unstable in view of the fact the linearized zero dynamics exhibits a characteristic polynomial with a zero in the right-half part of the complex plane Another way of showing instability of the zero dynamics of a direct controlled DC-DC boost converter is to utilize Lyapunov stability theory For example, consider the following positive definite Lyapunov candidate function: V ( iL ) = ν2 ⎞ 1⎛ ⎜ EiL − sp ⎟ R ⎟ 2⎜ ⎝ ⎠ (42) 57 Sliding Mode Control for Industrial Controllers Fig Phase Portrait diagram for the case of Direct Sliding Mode control of boost converters The time derivative of the above Lyapunov candidate function (taking into account that iL > ) is: ν2 ⎞ E ⎛ ⎜ EiL − sp ⎟ ≥ V ( iL ) = LiL ⎜ R ⎟ ⎝ ⎠ (43) Thus, the system exhibits an unstable zero dynamics The phase portrait diagram shown in Fig also confirms this The fact that the direct voltage controlled boost converters exhibits an unstable equilibrium point is addressed in the control system literature, by stating that the output voltage is a non-minimum phase output On the contrary, as we will see next, the inductor current is said to be minimum phase output and thus, current controlled boost converter doesn’t exhibit an unstable equilibrium point Fig 10 Sliding Mode Voltage controlled (direct) boost converter 58 Sliding Mode Control Simulation (left) and (right) in Fig 10 demonstrates the instability of the zero dynamics for the case of direct voltage controlled DC-DC buck converters For both of these simulations, the following converter parameters are used: L = 1H , R = 1Ω , C = 1F , E = 1V , vsp = 1.5V Both simulations assume an initial voltage of vc ( t = ) = 1.5V The ss initial inductor current value is iL ( t = ) = 2.2A < iL = 2.25A for simulation and ss iL ( t = ) = 2.3A > iL = 2.25A for simulation B Indirect sliding Mode control of DC-DC boost converter Fig 11 Sliding Mode current controlled (indirect) boost converter As demonstrated before, voltage-sliding mode controlled DC-DC boost converters exhibit unstable equilibrium points The alternative is then to use a sliding surface based on the inductor current such that when it is set to zero, leads to a desired value of the input inductor current in correspondence with the desired output equilibrium voltage Consider the following indirect current based sliding mode control: sindirect = iL − isp u= (44) ( − sign ( sindirect ) ) (45) Where isp is a desired input inductor current calculated in accordance with desired output voltage vsp To calculate the needed isp , we compute the equilibrium point of the system under ideal sliding mode conditions, thus: ( ) sindirect = ⇒ iL = isp , sdirect = ⇒ − ueq = E νC , dν C ν ν2 = ⇒ − ueq isp = C ⇒ isp = C dt R ER ( ) (46) ... ( t = ) = 2. 2A < iL = 2. 25A for simulation and ss iL ( t = ) = 2. 3A > iL = 2. 25A for simulation B Indirect sliding Mode control of DC-DC boost converter Fig 11 Sliding Mode current controlled... Institute for Control and Computer Science Sarajevo, Yugoslavia, pp.560-566 Utkin V.I (1977) Variable structure system with sliding modes IEEE Trans On A.C., Vol.AC -22 , April 1977, pp 21 2 -22 2 Utkin... chosen controller’s parameters Fig Sliding mode PID control of single-phase buck converter Estimation in sliding mode PID controlled single phase DC-DC buck converter The sliding mode PID controller