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Applications of Algebraic Geometry to System Theory-Part I, ZEEE A New Method of Switching Surface Design for Multivariable Variable Structure Systems

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Applications of Algebraic Geometry to System Theory-Part I, ZEEE A New Method of Switching Surface Design for Multivariable Variable Structure Systems
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  414 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 39, NO. 2. FEBRUARY 1994 specific techniques for solving this bilinear Sylvester equation, will be presented in detail elsewhere. IV. CONCLUSION In this note, a geometric theory has been presented for output stabilization and eigenstructure assignment by output feedback for linear multivariable systems. We emphasized the importance of C, A, B)-invariant subspaces, which constitute open-loop informa- tion, in revealing the possible structure of the closed-loop system under output feedback. We also proposed a computationally efficient technique for the characterization of candidate “output-stabilizable subspaces.” Necessary and sufficient conditions in terms of the C, A, B)- invariance property were presented for stabilization and eigenstruc- ture assignment by output feedback. We provided the links between the proposed geometric theory and the existent algebraic results. We proposed a bilinear generalized Sylvester equation, the solutions of which completely characterize the structure of the closed-loop system under the action of output feedback. We have also shown how differ- ent descriptions of this bilinear equation lead to different necessary and sufficient conditions for the output stabilization problem. Finally, we present a preview of possible ways to solve the proposed bilinear equation. Therefore, for the eigenstructure problem, the solutions to a bilinear matrix equations have to be studied. ACKNOWLEDOMENT The authors would like to thank the reviewers for their helpful comments, suggestions, and criticisms that contributed to the overall presentation of this note. REFERENCES F. M. Brasch and J. B. Pearson, “Pole assignment using dynamic compensators,” ZEEE Tms. Automat. Contr., vol. AC-15, pp. 348-351, June 1970. J. W. Brewer, “Kronecker products and matrix calculus in system theory,” ZEEE Trans. Circuits Sysr., pp. 772-785, Sept. 1978. R. W. Brockett and C. I. Bymes, “Multivariable Nyquist criteria, root loci, and pole placement: A geometric viewpoint,’’ ZEEE Trans. Automar. Contr., vol. AC-26, 271-284, Feb. 1981. E. I. Davison, “On pole assignment in linear systems with incomplete state feedback,” ZEEE Trans. Auromat. Contr., vol. AC-15, pp. 348-351, June 1970. “A note on pole assignment in linear systems with incomplete state feedback,” ZEEE Trans. Automar. Contr., vol. AC-16, pp. 98-99, Feb. 1971. E. J. Davison and S. G. Chow, “An algorithm for the assignment of closed-loop poles using output feedback in large linear multivariable systems,” ZEEE Trans. Auromat. Contr., vol. AC-18, pp. 74-75, Feb. 1973. E. J. Davison and S. H. Wang, “On pole assignment in linear multivari- able systems using output feedback,” ZEEE Trans. Automat. Contr., vol. AC-20, pp. 516-518, Aug. 1975. L. R. Fletcher and J. F. Magni, “Exact pole assignment by output feedback Part 1,” Znr. J. Cow., pp. 1995-2007, 1987. C. Giannakopoulos and N. Karcanias, “Pole assignment of strictly proper and proper linear systems by constant output feedback,” Znt. J. Contr., R. Herman and C. F. Martin, “Applications of Algebraic Geometry to System Theory-Part I,” ZEEE Trans. Automat. Contr., vol. AC-22, pp. 19-25, 1977. H. Kimura, “On pole assignment by gain output feedback,” ZEEE Trans. Automar. Contr., vol. AC-20, pp. 509416, Aug. 1975. H. Kimura, “On pole assignment by output feedback,” Znt. J. Contr., vol. 28, no. 1, pp. 11-22, 1978. pp. 543-565, 1985. “A further result on the problem of pole assignment by gain output feedback,” ZEEE Trans. Automat. Conrr., vol. AC-22, pp. 458-463, June 1977. B.-H. won and M.-J. Youn, “Eigenvalue-eigenvector assignment by output feedback,” ZEEE Trans. Auromar. Contr., vol. AC-32, pp. 417421, May 1987. J. F. Magni, “On exact pole placement by output feedback 3,” Znt. J. Contr., vol. 45, pp. 2021-2033, 1987. J. F. Magni and C. Champetier, “A general framework for pole place- ment assignment algorithms,” in Pm. 7rh CDC, ustin, TX, ol. 3, G. S. Miminis, “A direct algorithm for pole assignment of time invariant linear systems using output feedback,” in Proc 25rh CDC, 985, pp. P. Misra and P. V. Patel, “Numerical algorithms for eigenvalue assign- ment by constant and dynamic output feedback,” ZEEE Trans. Automat. Conrr., vol. AC-34, pp. 579-588, June 1989. J. M. Schumacher, “Compensator synthesis using C, A, B)-pairs,” ZEEE Trans. Automat, Contr., vol. AC-25, pp. 1133-1138, Dec. 1980. “Regulator synthesis using C, , B)-pairs,” ZEEE Trans. Automar. Conrr., vol. AC-27, pp. 1211-1221, Dec. 1982. S. Srinathkumar, “Eigenvaludeigenvector assignment using output feed- back,” IEEE Trans. Auromat. Conrr., vol. AC-23, pp. 79-81, Feb. 1978. A. I. Vardulskis, “A sufficient condition for n specified eigenvalues to be assigned under constant output feedback,” ZEEE Trans. Automat. Contr., vol. 20, pp. 428-429, June 1975. W. M. Wonham, Linear Multivariable Control: A Geometric Approach. New York: Springer-Verlag, 1984. 1988, pp. 2223-2229. 541-547. A New Method of Switching Surface Design for Multivariable Variable Structure Systems Ju-Jang Lee and Yangsheng Xu Abstract-In this paper, we discuss the problem of the switchjng surface for a multivariable variable structure system. We 6rst propose a new switching surface by which robustness of the controlled system to disturbances is provided. Based on the switching surface, we develop control input to drive the system to track the desired trajectory. This control input does not produce the chattering that normally occurs when a conventional control input is used. We apply the proposed method to a multivariable linear system and discuss the simulation results to show the computational procedure and its effectiveness. I. INTRODUCTION The Variable Structure Control (VSC) is a special class of nonlinear control method characterized by a discontinuous control action which changes a structure upon reaching a switching surface a(x(t)) = 0. For Variable Structure Control System (VSCS), the system is time- invariant in the sliding mode, and thus the motion is independent of parameters and disturbances. The representative point of the system is constrained to move along a predetermined switching surface [3], [5], [ 11 Therefore, design of the switching surface completely determines the performance of the system. With a conventional switching surface, the sliding mode occurs after the system reaches the switching surface. Before the system reaches the switching surface, however, the system is sensitive to Manuscript received July 21, 1991; revised September 14, 1992. The authors are with The Robotics Institute, Carnegie Mellon University, IEEE Log Number 9214019. Pittsburgh, PA 15213. 0018-9286/94 04.00 0 1994 IEEE  IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 39, NO. 2, FEBRUARY 1994 0 1 0 0 e.. 0- 0 0 1 0 ... 0 0 0 0 0 ... 1 .. .. .. 7 -a1 -a2 -a2 -a4 ... -an - 415 parameter variations and disturbances. This produces discontinuous trajectory and causes a chattering which is particularly undesirable when the actuator mechanism may be damaged by rapid switching [2]. Therefore, a new method to design control input is necessary. In this note, we propose a new time-varying switching surface for a multivariable variable structure system which ensures the robustness of the controlled system to disturbances. Based on the switching surface, we develop a control input to drive the system in tracking the desired trajectory. II. VARIABLE TRUCTURE ONTROL OR SCALAR INEAR SYSTEMS A. Switching Sudace Design It is known that if a linear system represented by a triple (A, B, ) is completely controllable, then the system can be reduced by a nonsingular transformation to an equivalent controllable form [4], and it is often advantageous to work with such an equivalent system. A scalar VSCS is usually represented in a controllable canonical form [4]]. Consider a scalar linear system i(t) = Az(t) + Bu(t) and assume the coefficient matrices A and B have the following forms: n u(z(t)) = Cz(t) = Ccizi, c; = constant, cn = 1. (2) For convenience, throughout this note, the arguments t and z are sometimes omitted when no confusion is likely to arise. In the sliding motion, a(z(t)) emains zero, and thus i=l n-1 n = - 22 i=l (3) The resultant sliding mode i = x,+l, = 1, 2,--.,n - n-1 in-1 = -~c*zt 4) i l depends only on the parameter c,. Using this switching surface, when the system does not reach the switching surface, the system is sensitive to parameter variations and disturbances and dependent on parameters of a, and bi. Only when the switching surface has been reached is the system robust to the parameter variation and disturbance. To overcome this disadvantage, we propose a new switching surface which provides robustness of control through whole control intervals. Consider the following system: 2, = zl+l, = 1, 2,-..,n - 1 n -Ca, t)z, + f(t) + u(t) (5) where u(t) s an input, f(t) s a disturbance, a,(t) is the system parameter which is time-varying, or unknown. In what follows, we denote a,(t) by az for brevity. ,=1 A model system can be described by i zz+1, = 1, 2,... ,n - n in-1 = -CC~X~, , = constant. (6) ,=1 The response of the system 6) depends only on the constant parameter cI If the controller makes the system 5) follow the model system (6), ndependently of parameter variations and disturbances, the robustness of the controller is ensured. To this end, we define a new switching surface which drives the actual system (5) to follow the model system 6). \ a(z(t)) z - lt -clzl 2z2 - . . - ,z,)dt - zn 0), t > 0. 7) If the control input guarantees the sliding mode, then the switching surface remains zero. That is, zn =~(-c1z1--~2~~----.-c~z~)dt+z~(O), >0. (8) In this way, the system response of (5) follows the model system 6) exactly. Using this new switching surface, the sliding mode occurs from the initial time, and thus the proposed switching surface guarantees the robustness of the control. B. Control Input Design Based on the switching surface designed, the control scheme to guarantee the sliding mode can be developed. When the system reaches the sliding mode from either side of the surface near the switching surface, the state remains on the switching surface a(z(t)) 0 for some finite time. The condition for the sliding motion on the switching surface is stated as follows: lim &(z(t)) 0, lim &(z(t)) 0 U-O+ U-0- or, equivalently, dz(t)P(z(t)) 0 in the neighborhood of o(z(t)) = 0. The following control input u(t) was proposed by [3]: n u(t) = -Eqzz, 6 sgn (u(z(t))) *=1 where for o(z(t))z; > 0 for a(z(t))z, 0, 9) Differentiating 2) with respect to time and substituting it into (5) yields  416 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 9, NO. , FEBRUARY 1994 From (2), (lo), (ll), we obtain a(z(t))&(z(t)) C(cI-l - ,(t) - bQs)a(z(t))zz n *=1 +[f(t) - t)b. sgn (44t)))]a(z(t)). 12) To satisfy the existence condition, we obtain the gain constants: 1) a(z(t))z, > 0 a, > maxt (l/b(t))(cl-l - ,(t)), 2) a(z(t))zZ 0 P < mint (l/b(t))(ct-l - a,(t)), 3) a 4t))[f t) - b t)S sgn(a(4t)))l < 0 > maxt (l/b(t))fmax. The control structures described so far are designed to drive the system state into an ideal sliding mode. Whether this can be achieved relies on idealizations. Because of delays in switching, hysteresis, and other factors that are not considered in dynamic modeling, the ideal sliding mode may not be easily obtained in practice. Moreover, the system state remains only within a neighborhood of a switching surface, and continuously passes back and forth across the switching surface. This behavior is known as chattering, which is particularly undesirable for the actuator mechanisms which may be damaged by the rapid switching. To overcome the chattering problem, a smoothing control scheme was proposed by [2], one of which can be represented by DP(0I Fig. 1 Proposed control scheme when b(t) = 1. where DP(t) = -xAa,z; + f t), Au*(t) = Au(t) - DPu(t). i=l In general, DP t) is an unknown time-varying function with an unknown structure. In the following, we address the problem of uncertainty cancella- tion. Using the switching surface a(z(t)), he system (5) is govemed by &(z(t)) DP t) + Au*(t) = DP t) + (Au(t) - DPu t)). (17) The auxiliary control input DPu t) is defined in the following where ueq is the continuous equivalent control input, UN is the discontinuous part of (lo), L is the feedback gain vector, and 6 is a small positive constant. M can be selected to ensure the existence condition for the maximum unknown bounded factor. Now, we present a control scheme based on the switching surface proposed previously. Theorem: Consider the system (5), (6), and the switching surface (2), and assume that a continuous function DP t) represents the plant uncertainty due to the unknown disturbance and unmodeled dynamics. We define a continuous function Au(t) as Au(t) = -Kpa(z(t)) (13) where Kp s positive. Then the proposed switching surface (7) and control input (13) guarantee the sliding mode of the control system. Proof: From (l), (2), (4), S), the derivative of the switching surface, a(z(t)), an be obtained &(z(t)) Clkl + C2kZ + . + kn = ClzZ+cZz3 + '+Cn-1Zn - alzl+ azzz + ... + anzn) + f t) + ~(t) - an - n-l)rn + f t) + u(t). = -a121 - a2 - c1)~ (14) The time-varying parameters a; can be divided into two parts: constant part a,, and time-varying part Aa,; i.e., a,=a,,+Aa;, i=l,2,...,n (Aa;)~,, a; 5 (Aar)max. = DP(t-) + (Au(t-) - DPu(t-)) - Au(t) - DPu t-)) = DP t-) + (A.(t-) - A.(t)) (18) in which t- denotes t - dt) dt is a sufficient small value). If DP t) and Au(t) are continuous functions, i.e., DP t) DP t-) and Au(t) E Au(t-), DPu t) % DP(t). (19) &(z(t)) Au(t). (20) Therefore, from (17), we have A block diagram of the proposed control scheme is shown in Fig. 1. Equation (20) results in a t))& z )) = Au( )a z )). (21) Au(~) -Kpu(z(t)), Kp > 0 (22) (23) If control input Au(t) is defined as follows: then u(z(t))&(z(t)) -Kp[a(z(t))y 0. This proves the existence condition of the sliding mode of the control system. Corollary: Consider the following system: kz = zt 1, i = 1, 2,-..,n - n in = --Casz, + f(t) + b(t)u(t> (24) 8=1 6(z(t)) = DP(t) + Au*(t) (16) where  IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 39, NO. 2, FEBRUARY 1994 ~ 417 WO Fig. 2. Proposed control scheme when b(to) = b + Ab(t). and 1 bo Au*(t) = -(Au(t) - DPu t)). If DP(t), Au(t). and Ab(t) are continuous functions, then &(z(t)) Y Au(t). Prooj? If the switching surface a(s(t)) s given by (2), the derivative of a(c(t)) s determined by n n-1 (s(t)) -Cais, + Ccizz+1 + f(t) + b(t) ~(t). 25) t=l i=l It is noted that ue,(t) can be written as 1 bo eq(t) = -[~Io~I (azo - I)ZZ + + Uno - n-i)~,] 26) thus (25) becomes n &(~(t)) -CAatI, + f(t) Ab(t) ueq(t) + b(t). Au*(t) i=l = DP t) + b(t) Au*(t) = DP t) + 1 + Au(t) - DPu(~)]. (27) [ The auxiliary control input DPu(t) s defined as DPu(t) = &(s(t-)) Au(t) - DPu(t-)) = (Au(t-) - DPu(t-) 1 + ::)I + DP(t-) - (Au(t-) - DP t-)) = (Au(t-) - Au(t)) + (Au(t-) - DPu(t-)) + DP(t-). b(t-) .- bo Using (28) and the following relation: DPu(t-) = b(z(t-*)) - Au(t-) - DPu(t-')) (29) where t-' denotes (t- - dt), b(z(t)) can be written as &(z(t)) [(Au(t) DP t-)) - Au(t-) - Au(t)) + DP(t)] + [(Au(t) DP(t-)) - Au(t-) - Au(~))]- b(t) bo - (Au(t-) - DP(t-')) - (Au(t-'j - Au(t-)] .-+... + -l)n-l [(Au(t-( -')) DP(t-( -'))) bb(t-) bo - Au(t-(n-2)) Au(t-(n-l) Ab(t) . Ab(t-(n-l)) b2t + (-l) [(AU(t- ) DP(t- -')) - Au(t- -') - Au(t- ))] This implies that (s(t)) an be approximated by Au(t), .e., b(z(t)) S Au(t). (31) This condition is the same as 20). A block diagram of the control structure is illustrated in Fig. 2.  111. VARIABLE mum CONTROL FOR MULTIVARIABLE INEAR YSTEMS A. System Model The multivariable linear system to be considered in this section is = As + Bu (32) where I E R and U E R represent the state and control vectors, respectively. To develop a variable structure control, we specify a particular canonical form for the system. This form is closely related to the controllable canonical form for a multivariable linear system, and is similar to that used by [6]. If the matrix B has full rank m, then there exists an orthogonal n x n transformation matrix T such that TB= [L] where Bz is an m x m nonsingular matrix. A set of new state variable y can be transformed from 2 .e., y = TI. Let us define the state vector, y E R as Y = [Yl yzIT. By using new state variable y. the system (32) can be written 1 = Ally1 + Aiz~z z = Aiz~i Azzyz + Bzu (33) where y1 and y~ are vectors formed by the first n - m) and last m components of the vector y, respectively, and A,, are block matrices with corresponding dimensions. B. Conventional Methods for Switching Sulface that The conventional switching surface u(z(t)) an be defined such (34) where CI and CZ are m x n - m) and m x m matrices. In the sliding mode, a(z(t)) = 0 and from (34), we obtain a(s(tj) = Cz(t) = CT-'y(t) = [Cl CZI . [Yl(tjY2(t)lT Yz(t) = -CT'Civi(t) = -Fyi(t), Vt > t, (35) where F = -CF'C1 E R and t, represents the time at which the sliding mode is reached. This indicates that the evolution of yz t) in the sliding mode is linearly related to that of yl(t). From (33) and (33, an ideal sliding mode is govemed by Yl(t) = (A11 - A12F)?/l(t) = Ae,yl(t) (36) which indicates that the design of a stable sliding mode (i.e., y + 0 as t 00, requires the determination of the gain matrix F such that A,, has n - m left-hand half-plane eigenvalues. This can be achieved by modifying a standard design method of a linear feedback control for a linear system. However, since F does not uniquely determine C,m2 degrees of freedom remain due to Ci = CzF. (37) The simplest method to determine C from F, as suggested by Utkin and Yang (1978), is to force CZ = 1,. In this way, we have C = [Ci CZ] [Flm]. (38)  418 3.00 1 2.00 1.00 0.00 0.00 609.00 :zOu.30 leOC.00 tdOO JO 33C9.jO Time( ms) (a) 2.00 , . - \ s.20 0.10 0.40 0.00 0.00 600.00 1200.00 lIIOO.09 2400.00 3000.00 Time(ms) 6 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 39, NO. 2. FEBRUARY 1994 10.00 0.00 10.00 -LO.OO 10.00 0.00 10.00 10.00 -30.00 so. 00 -70. 00 -9o.w ~ 0.00 600.00 1200.00 1EOO.00 2400.00 30C0.00 Time( ms) 00 600.00 1200.00 lSOO.00 1490.00 3000.00 Time( ms) Fig. 3. (a) State q(t) and sz t) vs. time (state feedback controller). (b) Control ul(t) and u~(t) s. time (state feedback controller). (c) State ~(t) and sz(t) vs. time (proposed controller). d) Control ~(t) nd uz(t) vs. time (proposed controller). C. Switching Surjihce Design When the sliding mode is achieved, o(y(t)) equals zero, i.e., (42) Consider the following system: where All, A 2 are constant matrices, and the others are the unknown time-varying matrices. y E R , E R , and f(t) E R represent the state variables, control inputs, and unknown disturbances, respec- tively. y1 E R - and yz E R are the substate variables. We denote the matrices formed by the nominal value of each element of the matrices Azl,Azz, and B2 by Azlo,Azzo, and Bzo. Consider the model system Jo Thus, the system (39) exactly follows the system 40). D. Control Design input for the system. From (41), we have Based on the switching surface proposed, we can derive control If the time-varying matrices A21(t), A22(t), BZ can be written in the following forms: &(t) = B20 + &(t) 44) where Azlo, Azzor Bzo are time independent, and Az~ t), &(t), &(t) epresent the time-varying part, in a manner similar to 16) or (24), then (45) where yL E R , ; E R - , and y; E R are the state and substate variables. We define the switching surface as follows: u Y ~)) [g1g2***omlT = YZ - t(-Czlyl - C22~2)dt z(0). (41) ~(t) ueq(t) Au*(t)
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