Min Max Sliding-Mode Control for Multimodel Linear Time Varying Systems

Transcription

1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 12, DECEMBER Min Max Sliding-Mode Control for Multimodel Linear Time Varying Systems Alex S Poznyak, Member, IEEE, Yuri B Shtessel, Member, IEEE, and Carlos Jiménez Gallegos Abstract An original linear time-varying system with unmatched disturbances and uncertainties is replaced by a finite set of dynamic models such that each one describes a particular uncertain case including exact realizations of possible dynamic equations as well as external bounded disturbances Such a tradeoff between an original uncertain linear time varying dynamic system and a corresponding higher order multimodel system with a complete knowledge leads to a linear multi-model system with known bounded disturbances Each model from a given finite set is characterized by a quadratic performance index The developed min max sliding-mode control strategy gives an optimal robust sliding-surface design algorithm, which is reduced to a solution of an equivalent linear quadratic problem that corresponds to the weighted performance indices with weights from a finite dimensional simplex An illustrative numerical example is presented Index Terms Min max, optimality, sliding-mode control I INTRODUCTION SLIDING-MODE control is a powerful nonlinear control technique that has been is intensively developed during last 35 years (see [16], [15], [2], and [9]) The sliding-mode controller drives the system state to a custom-built sliding (switching) surface and constraints the state to this surface thereafter A system motion in a sliding surface that is named sliding-mode is robust to disturbances and uncertainties matched by a control but sensitive to unmatched ones The sliding-mode design approach comprises of two steps First, the switching function is designed such that the system motion in sliding-mode satisfies design specifications Second, a control function is designed that makes the switching function attractive to the system state For the case of matched disturbances only the optimal sliding surface design is available [14], [15], [2], [7], and [8] In [13], a robust hyperplane computation scheme for sliding-mode control is proposed A sensitivity index for sliding eigenvalues with respect to perturbations in the system matrix, the input matrix and the hyperplane matrix is suggested to be minmized The effect of external (unmatched) perturbations has not been considered Manuscript received August 14, 2002; revised February 27, 2003 Recommended by Associate Editor P A Iglesias A S Poznyak and C J Gallegos are with the Department of Automatic Control, CINVESTAV-IPN, CP 07360, Mexico City, Mexico ( apoznyak@ctrlcinvestavmx) Y B Shtessel is from Department of Electrical and computer Engineering, The University of Alabama in Huntsville, Huntsville, AL USA ( stessel@eceuahedu) Digital Object Identifier /TAC A Motivation In the case of unmatched uncertainties the optimal sliding surface design can not be formulated, since an optimal control requires a complete knowledge of system dynamic equations Therefore, in this situation another design concept must be developed The corresponding optimization problem is usually treated as a min max control dealing with different classes of partially known models [12], [3], [6] The min max control problem can be formulated in such a way that the operation of the maximization is taken over a set of uncertainty and the operation of the minmization is taken over control strategies within a given resource set In view of this concept, the original system model is replaced by a finite set of dynamic models such that each model describes a particular uncertain case including exact realizations of possible dynamic equations as well as external bounded disturbances This is a tradeoff between the original low order dynamic system with uncertainty and the corresponding higher order multi model system with the complete knowledge Such approach improves robustness of the sliding-mode dynamics to unmatched uncertainties and disturbances To do that, a min max sliding surface design algorithm is to be developed For example, the reusable launch vehicle attitude control deals with a dynamic model containing uncertain matrix of inertia (various payloads in a cargo bay) and affecting by unknown bounded disturbances such as wind gusts (usually modeled by table look up data corresponding to different launch sites and months of a year) [11] The design of the min max sliding-mode controller that optimizes the worst flight scenarios will reduce the risk of loss of a vehicle and a loss of a crew B Basic Assumptions and Restrictions Since the original system model is uncertain, in this work we consider a finite set of dynamic models such that each model describes exactly a particular uncertain case including exact realizations of possible dynamic equations as well as external bounded disturbances; sure such approach makes sense only for reasonably not large (small) number of possible scenarios; each model from a finite set is supposed to be given by a system of linear time-varying ordinary differential equation (ODE); the performance of each model in the sliding-mode is characterized by linear quadratic (LQ)-criterion with a finite horizon; /03$ IEEE

2 2142 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 12, DECEMBER 2003 the same control action is assumed to be applied to all models simultaneously and designed based on a joint sliding function; this joint sliding function, defined in the extended multimodel state space, is suggested to be synthesized by minmization of the maximum value of the corresponding LQ-criteria C Main Contribution This study demonstrates that the designed sliding surface provides the best sliding-mode dynamics for the worst transient response to a disturbance input from a finite set of uncertainties and disturbances; the LQ problem formulation leads to the design of the min max sliding surface in a liner format with respect to system state; the corresponding optimal weighting coefficients are computed based on Riccati equation parametrized by a vector, defined on a finite-dimensional simplex; it is shown that the design of the min max optimal sliding surface is reduced to a finite-dimensional optimization problem given at the corresponding simplex set containing the weight parameters to be found D Structure of the Paper The paper starts with the system description and problem setting The extended system model and a transformation to a regular form is presented in the next section Then, the min max sliding surface design algorithm is developed The control function that stabilizes the min max sliding surface is designed in the next section An illustrative example concludes this study Several lemmas on min-max sliding surface design with proofs are given in Appendix II SYSTEM DESCRIPTION AND PROBLEM SETTING A Plant Model Consider the following collection of finite multimodel linear time varying systems : is the state vector of the system ( a given finite set), is the control vector that is common for all models from a given set, is a disturbance vector with integrable and bounded components, is a time varying-matrix is a time varying-matrix of the full rank, that is, or for any and B Control Strategy The control strategies considered hereafter will be restricted by sliding-mode control [16], [2] (1) Definition 1: A sliding-mode is said to be taking place in the multi-model system (1) for all if there exists a finite time, such that all solution satisfy for all (2) is a sliding function and (2) defines a sliding surface in C Performance Index and Problem Formulation For each and, the quality of the system (1) motion in the sliding surface (2) is characterized by the performance index [15] Here, we will show that (1) motion in the sliding surface (2) does not depend on the control function, that is why (3) is a functional of and only Now, we are ready to formulate the optimal control problem for the given multimodel system (1) in the sliding-mode in the sense of Definition 1 Problem Formulation: For the given multimodel system (1) and, define the optimal sliding function (2) providing the worst-case optimization in the sense of (3) in the sliding-mode, that is is the set of the admissible smooth (differentiable on all arguments) sliding functions So, we wish to minmize the worst scenario case varying (optimizing) the sliding surface Remark 1: For a single model system (1) that corresponds to the optimal sliding surface design problem was addressed in [2] and [15] Remark 2: The original uncertain system model is replaced in the paper by a finite number of fully known dynamic systems The question is when such a replacement is adequate One can realize that even the system contains only one constant parameter known to belong to, the number of the corresponding exact systems is infinite The solution of the corresponding min max problem is given in [5] There is shown that the min max control at each time can be represented as a vector minmizing an integral over a parametric uncertainty set of the standard Hamiltonian functions corresponding a fixed parameter value That is why for any small can be found a finite-sum approximation of this integral which guarantees the -min max solution to the initial problem given on a compact set This technique helps to avoid the questions how much approximative points should be selected (3) (4)

3 POZNYAK et al: MIN MAX SLIDING-MODE CONTROL 2143 III EXTENDED MODEL AND TRANSFORMATION TO REGULAR FORM The models collection (1) can be rewritten in the following extended model: and the sliding function becomes (12) Remark 3: The matrices, and are supposed to be symmetric Otherwise, they can be symmetrized as follows: (5) Following to the standard technique [16, Sec 56], introduce new state vector defined by, the linear nonsingular transformations are given by and represent the matrices in the form Applying (7) to (1), we obtain (6) (7) (13) Assumption A1: We will look for the sliding function (12) in the form (14) If the sliding-mode exists for (8) in the sliding surface under Assumption A1, then for all the corresponding multimodel sliding-mode dynamics, driven by the unmatched disturbance, are given by (15) and (8) with the initial conditions Defining as a virtual control, that is (15) is rewritten as (16) (9) and the performance indexes (11) become (17) Using the operator it follows new variables defined as (10) and, hence, the performance index (3) in may be rewritten as (18) In view of (17) and (18), the min max sliding surface design problem (4) is reduced to the following one: (19) For any IV MIN MAX SLIDING SURFACE (11) (20)

4 2144 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 12, DECEMBER 2003 satisfying the Ric- let us define the matrix cati differential equation and the sliding mode starts at any point the manifold is defined as (27) V SLIDING-MODE CONTROL FUNCTION DESIGN and the shifting vector generated by (21) The control function in (5) that stabilizes the sliding surface (26) is given by the following theorem Theorem 2 (Sliding-Mode Control Function Design): If for the given plant (5) and the sliding surface (26) 1) and are differentiable at ; 2) for any ; 3) ; then, for the given the sliding-mode control, stabilizing the sliding surface (26) to zero in the finite time, is as follows: (28) (22) Proof: The sliding surface dynamics are derived from (5) and (26) and are as follows: The solution of (19) is given by the next theorem Theorem 1: Under Assumption A1, for any initial conditions the min max sliding function (14), that gives a solution to the min max problem (19), is defined as (23) (29) A candidate of the Lyapunov function is introduced as and its derivative is to be enforced (30) that provides to a finite time for the origin reaching, that is, In view of (29), we derive and, taking, (31) becomes (31) (24) Proof: Based on Lemma 3 of the Appendix, the following virtual control is obtained (25) the matrix and the vector are defined by (21) and (22) Then, the sliding function (14) becomes as (23) The selection (24) follows from Lemma 4 of Appendix Corollary 1: In the original state variables (5), the min max sliding function becomes (26) that implies (30) Taking into account the assumption of the theorem, we obtain as (28) Since all predefined models are known a priori we can run them in current time and have access to making the sliding surface to be available Actually, this is a component of the proposed min max sliding surface design algorithm VI MINIMAL-TIME REACHING PHASE CONTROL In this section, we consider the plant in the format (5) The control strategies considered here will be restricted at the first part of the movement (presliding motion or reaching phase), by program strategies minmizing the reaching time of some sliding surface

5 POZNYAK et al: MIN MAX SLIDING-MODE CONTROL 2145 given in the extended space ; the control actions are supposed to be defined within a given polytope resource set (32) at the second part of movement (sliding-motion), by sliding-mode control (28) The problem discussed in this section is to design the bounded control function given at the polytope (32) that moves the trajectory from the given initial conditions to the manifold (27) in minmal-time Here, we will consider the manifold as a hyperplane given by (33) Previously, we have shown that the optimal sliding surface indeed is a hyperplane, so, the next considerations are really make sense For the given and the minmal reaching time problem is such that (34) In view of (38) and the commutation property, it follows that and, hence, (37), becomes (20) For the polytope re- given by (32), the expression (41) implies source set (40) (41) (35) Following [4] and rewriting the terminal condition (35) for some pair as (42) (36) In view of (40), it follows that the solution of the multimodel minmal-time optimization problem (34) and (35) is given by the transition function satisfies (43) (37) is the vector-function satisfying to the following differential equation: (38) with the terminal conditions (44) This implies that is a function of the distribution and the terminal point, that is, Substituting this control minmizing reaching time into (5), we obtain (45) with the transferring-matrix satisfying Taking into account that if then and for all, the following normalized adjoint variable can be introduced: So, as it follows from (45), the vector solution to the following nonlinear equation: (46) is the (39) (47)

6 2146 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 12, DECEMBER 2003 For any fixed distribution the minmal, for which there exists a solution of the nonlinear equation (47), is the corresponding reaching time of the given sliding-manifold (33) That is why the optimal reaching time may be calculated as follows: (48) Denote (49) VII SUCCESSIVE APPROXIMATION OF INITIAL SLIDING HYPERPLANE AND JOINT OPTIMAL CONTROL It is worth noting that the values of and depend on the parameters and On the other hand, for given and the hyperplane parameters and are computed uniquely using the min max solution (27) Fig 1 F ( ) dependence that is (50) (51) It means that the pair is a fixed point of the mapping (51) The existence and uniqueness conditions of a fixed point of this mapping (contraction mapping) are discussed in [1, Ch 15] and [10, Ch 2] Assuming these conditions are met, the fixed point can be obtained by the method of successive approximation starting from any arbitrary initial pair, ie, and as increases (52) (53) Computing for each and the corresponding pair a minmal reaching time control as a solution of the time optimization problem (34) and (35), we obtain and Then, the values are computed using (50) In view of (53), we design the series and that converge to their optimal values Fig 2 Trajectory behavior x(t) which yields the unique optimal sliding function (26) So, finally the min max joint optimal control that provides for robust minmal time reaching phase and robust linear quadratic optimal performance in sliding-mode is (54) both phase optimal controllers and have the structure of relay type containing SIGN-operator Fig 3 Control action u(t) VIII ILLUSTRATIVE EXAMPLES Example 1 (Two Models Two States Each): Here, we took

7 POZNYAK et al: MIN MAX SLIDING-MODE CONTROL 2147 Fig 4 Sliding surface (x; t) for x = x =0 Fig 6 Trajectory behavior x(t) Fig 5 Sliding surface (x; t) for x = x =0 Fig 7 Control action u(t) The dependence of (24) is depicted at Fig 1 The optimal and, minmizing this function, are equal to and The corresponding trajectories are given at Fig 2, the control is depicted at Fig 3 and the sliding-manifold (27) defined for is given in at Figs 4, 5 in its projections to the surfaces and, correspondingly The comparison of the functional (3) for this control with the standard LQ-control shows the following results: and So, the difference is practically negligible that means that the multimodel-sliding-mode controller provides practically a min max behavior to the given systems collection Example 2 (Three Models Two States Each): In this example and Fig 8 Sliding surface (x; t) for x = x = x = x = 0: The time optimization is The optimal weights are as follows and The corresponding trajectories are shown at Fig 6, the control at Fig 7 and the sliding-manifold (27) defined for is given in Figs 8 and 9 in its projections to the surfaces and, correspondigly

8 2148 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 12, DECEMBER 2003 For the convenience, every feasible control is assumed to be right-continuous, that is, for and, moreover, is continuous at the terminal moment, ie, The initial point is fixed Consider several cost functions containing the integral term as well as a terminal one, that is (56) The end time-point is not fixed The worst (highest) cost can be defined as follows: Fig 9 Sliding surface (x; t) for x = x = x = x =0 One can see a nice trajectory behavior for different models controlled over the same sliding surface The analogue comparison of the functional (3) for this control with the standard LQ-control gives the following results: and It means that the multi-model-sliding-mode controller works a little bit worse than the min-max optimal controller for the given systems collection From another point of view, this controller may completely make zero the influence of a, so called, match uncertainty This fact is well-known in sliding-mode control theory and now it is out of the scope of this paper IX CONCLUSION For a linear multimodel time varying system with bounded disturbances and uncertainties an optimal sliding-surface is designed based on min max approach Each model from a given finite set is characterized by a linear quadratic performance index It is shown that the min-max optimal sliding surface design is reduced to a finite-dimensional optimization problem on the simplex set containing the weight parameters to be defined The obtained robust sliding surface provides the best sliding-mode dynamics for the worst transient response to an unmatched disturbance input from a given finite set The minmal time multi model control is designed for the reaching phase completing the overall min max optimal multi model sliding-mode control problem solution (57) The function depends only on the considered admissible control In other words, we wish to construct the control feasible action which provides a good behavior for a given collection of the cost functions that may be associated with the multi criteria min max optimal control design Thus, multicriteria min max optimization problem consists of finding the feasible control action, which realizes (58) the minmum is taken over all admissible control strategies This is the Min max Bolza Problem [6] The necessary condition for the min-max optimality is given by the following lemma Lemma 1 (Min Max Optimal Control): Let be a min max optimal control and be the corresponding solution of (55) with the initial condition For the min max optimality of the control it is necessary that there exist vectors and nonnegative real values defined on such that the following conditions are satisfied i) The Maximality Condition: Denote by the solution of the (59) APPENDIX Consider a controlled plant (55) is its state vector, is the control that may run over a given control region and The usual restrictions are impose to the right-hand side, that is, the continuity with respect to the collection of the arguments and differentiability (or Lipschitz condition) with respect to A function is said to be an feasible control if it is piecewise continuous and for all with the terminal condition for all then the min max optimal control satisfies the maximality condition (59) (60) (61)

9 POZNYAK et al: MIN MAX SLIDING-MODE CONTROL 2149 ii) Complementary Slackness Conditions: For every, either the equality holds, or, that is, iii) Transversality Condition: For every, the equalities and satisfies (66) Then, there exist nonnegative real values defined on such that the min-max optimal LQ-control is (68) hold iv) Nontriviality Condition: There exists such that either, or at least one of the numbers is distinct from zero, that is Proof: It follows straightforward by applying the previous lemma using (64) (67) and the definition (69) Proof: There exists the solution to the multimodel min max optimization problem [3], [6] the criteria (56) are applied to the multimodel system (62) The next lemma represents the min max optimal LQ-control (68) in a feedback format Lemma 3: The min max optimal LQ-control (68) in a feedback format is as follows: In order to use the result (56) in obtaining the solution of the formulated multi criteria min max optimization problem (5), (56), and (58), we assume that the multi-model system (62) consists of the same models of the form (5), ie, for each (63) The direct use of the result from [6] gives the claim of this lemma Now, we apply the result of the previous lemma to LQ multicriteria min max optimization problem Consider a controlled plant (5) of the form with the cost functions (64) the matrices and the vectors for all satisfy the Riccati differential equation (70) (71) (72) Let with the terminal condition satisfies (65) (66) (67) Proof: We look for a solution to (66) and (67) in the following format: Substituting (73) into (66) and (68), we obtain (73) Remark 4: Throughout this paper, only the case is considered Lemma 2 (LQ Multicriteria Min Max Optimal Control): Let be an LQ min max optimal control and be the corresponding solution of (64) with the initial condition

10 2150 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 12, DECEMBER 2003 Then, it follows that which implies (71) and (72) The following lemma gives a tool for an optimal selection of the weighting parameters Lemma 4: Under the assumptions of the previous Lemma, the optimal vector parameter is as follows: [7] C Dorling and A Zinober, Two approaches to hyperplane design in multivariable variable structure control systems, Int J Control, vol 44, pp 65 82, 1986 [8] L Fridman, Sliding-mode control for systems with fast actuators: Singularity perturbed approach, in In Variable Stricture Systems: Toward the 21st Century, X Yu and J X Xu, Eds London, UK: Springer- Verlag, 2002, vol 274, pp Lecture Notes in Control and Information Science [9] L Fridman and A Levant, Higher order sliding-modes in sliding-mode control in engineering, in Control Engineering Series, W Perruquetti and J P Barbot, Eds New York: Marcel Dekker, 2002, vol 11, pp [10] H K Khalil, Nonlinear Systems New York: McMillan, 1996 [11] Y Shtessel, C Hall, and M Jackson, Reusable launch vehicle control in multiple time-scale sliding-modes, J Guid, Control, Dyna, vol 23, no 6, pp , 2000 [12] Y Shtessel, Principle of proportional damages in multiple criteria LQR problem, IEEE Trans Automat Contr, vol 41, pp , Mar 1996 [13] H K Tam, D W C Ho, and J Lam, Robust hyperplane synthesis for sliding-mode control systems via sensitivity minimization, Opt Control: Applicat Meth, vol 23, pp , 2002 [14] V Utkin and K K Young, Method for constructing discontinuity planes in multidimensional variable structure systems, Automat Remote Control, vol 39, pp , 1978 [15] V Utkin, Sliding-Mode: Control and Optimization Berlin, Germany: Springer-Verlag, 1992 [16] V Utkin, J Guldner, and J Shi, Sliding-Mode Control in Electromechanical Systems London, UK: Taylor & Francis, 1999 (74) Alexander S Poznyak (M 97) graduated from Moscow Physical Technical Institute (MPhTI), Moscow, Russia, in 1970 He received the PhD and Doctor degrees from the Institute of Control Sciences of the Russian Academy of Sciences, Moscow, Russia, in 1978 and 1989, respectively From 1973 to 1993, he served this institute as Researcher and Leading Researcher, and in 1993, he accepted a post of Full Professor (3-E) at CINVESTAV of IPN, Mexico City, Mexico He has published more than 100 papers in different international journals and 7 books Dr Poznyak is a Regular Member of the Mexican Academy of Sciences and System of National Investigators (SNI-3) He is an Associate Editor of Iberamerican International Journal on Computations and Systems He was also an Associate Editor of the Conference on Decision and Control, the American Control Conference, and a Member of Editorial Board of the IEEE Control Systems Society (75) Proof: The substitution of (64) and (70) into (65) implies (74) and (75) REFERENCES [1] J P Aubin, Mathematical Methods of Game and Economic Theory Amsterdan, The Netherlands: North Holland, 1979 [2] C Edwards and S Spurgeon, Sliding-Mode Control: Theory and Applications London, UK: Taylor & Francis, 1998 [3] V G Boltyanski and A S Poznyak, Robust maximum principle in min max control, Int J Control, vol 72, no 4, pp , 1999 [4], Linear multi-model time optimization, Optimal Control Applicat Meth, vol 23, pp , 2002 [5], Robust maximum principle for a measured space as uncertainty set, in Dynamic Systens & Applications Atlanta, GA: Dynamic Publishers, Inc, 2002, vol 11, pp [6] A Poznyak, T Duncan, B Pasik-Duncan, and V Boltyanski, Robust maximum principle for multi-model LQ- problem, Int J Control, vol 75, no 15, pp , 2002 Yuri B Shtessel (M 93) received the MS and PhD degrees in automatic Control from the Chelyabinsk State Technical University, Chelyabinsk, Russia, in 1971 and 1978, respectively From 1979 to 1991, he was with the Department of Applied Mathematics and Control Science at The Chelyabinsk State Technical University From 1991 to 1993, he was with the Electrical and Computer Engineering Department and the Department of Mathematics at the University of South Carolina, Columbia Currently, he is a Professor at the Electrical and Computer Engineering Department, the University of Alabama, Huntsville He conducts research in sliding-mode control with applications to reusable launch vehicle control, aircraft reconfigurable flight control systems, missile control systems, an autonomous conventional, and nuclear reactor systems of electric power supply He published over 120 technical papers Carlos Jiménez Gallegos was born in 1975 at San Luis Potosí, México He received the BcSc degree in electronic engineering and the MsSc degree from the University of San Luis Potosí, México, in 1999 and 2000, respectively He is currently working toward the DSci degree in robust and min max multimodal optimization at the same university His research interests include robust control of nonlinear dynamical systems, optimal control, and game theory