Robust Control for Networked Control Systems with Admissible Parameter Uncertainties

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1 37 International Journal of Control, Kun Ji and Automation, Won-jong and Kim Systems, vol. 5, no. 4, pp , August 7 Robust Control for Networed Control Systems with Admissible Parameter Uncertainties Kun Ji and Won-jong Kim* Abstract: his paper discusses Robust H control problems for networed control systems (NCSs) with time delays and subject to norm-bounded parameter uncertainties. ased on a new discrete-time model, two approaches of robust controller design are proposed. A numerical example and experimental verification with an NCS test bed are given to illustrate the feasibility and effectiveness of proposed design methodologies. Keywords: Linear matrix inequalities (LMIs), networed control system, uncertainty, time delays.. INRODUCION Ever-increasing computational capabilities and bandwidths in the networing technology enabled researchers to develop NCSs [-4]. Stability regions of NCSs were proposed using a hybrid-system technique by Zhang et al. []. Ji and Kim [4] proposed a real-time co-design methodology of NCSs via the Ethernet. However, controller design and the effect of the controller to the stability of the NCS were not dealt with in these papers. A technique to stabilize delay systems was proposed by Artstein [5] by transforming a delay system into a linear finite-dimensional system. he standard H control problem for linear systems with delay was solved based on Riccati equations in [6-8]. oyd et al. [9] emphasized that many problems arising in system theory could be cast into the form of LMIs. Niculescu [] introduced an approach based on LMIs to derive sufficient conditions for the stabilization of systems with uncertain input delay. Shi et al. [] discussed the problem of control of discrete time-delay linear systems with Marovian jump parameters and the time-delay case considered is state delay which is not the case of an NCS. here are some discussions about robust stability for time-delay system with parameter uncertainties and the situation of delay in the state was considered [,3]. Similar Manuscript received December, 5; revised March 7, 7; accepted May 8, 7. Recommended by Editorial oard member Lihua Xie under the direction of Editor ae- Woong Yoon. his wor was supported by the National Science Foundation under Grant No. CMS-664. Kun Ji is with Siemens echnology to usiness Center, 995 University Ave., ereley, CA 9474, U.S.A. ( unjitamu@gmail.com). Won-jong Kim is with the Department of Mechanical Engineering, exas A&M University, College Station, X , U.S.A. ( wjim@tamu.edu). * Corresponding author. h ca w( Fig.. An NCS with networ-induced time delays. approaches were presented in [4,5] they focused on continuous-time models with constant time delays. In this paper, we consider an NCS framewor as shown in Fig... PROLEM SAEMEN he NCS shown in Fig. can be described by x( + ) = ( A+ Au) x( + ww( + ( + u) h, xm ( ) = θ( m), m [ h ), z ( ) = Ex ( ), n () x( R is the state, R is the p control input, w( R is the disturbance input, h is an integer that denotes the number of the sampling periods as the length of time delay at time instant, and h = h sc + h ca for a fixed control law []. θ ( ) is the initial condition, z( is the controlled output, A,, w, and E are nown real constant matrices of appropriate dimensions, and A u and u are parameter uncertainties. he sampling period is... Assumptions. he admissible uncertainties are assumed to be A = L F ( E, = L F ( E, u A A A Plant Controller z( y( u h sc m

2 Robust Control for Networed Control Systems with Admissible Parameter Uncertainties 373 A ( ) = A+ LAFA( E ) A, ( ) = + LF( E ), () F A ( and F ( are unnown matrices with Lebesgue measurable elements satisfying F ( ; F (,, (3) A and L A, L, E A, and E are nown real constant matrices.. he upper bound of the networ-induced time delays is H sampling periods. 3. here exists a real scalar S > such that for any real symmetric positive-definite matrix P and any x(, the following inequality holds. [ ] P[ x x ] S+ x ( ) Px ( ) x ( ) x ( + ) ( ) ( + ) > (4).. Robust control problems. Robust Stabilization: Given H >, find a linear state-feedbac control law for the system (-3) such that the resulting closed-loop system is robustly stable for any time delay h satisfying h H for all.. Robust H Control: Given scalars γ > and H >, find a linear state-feedbac control law for the system (-3) such that the resulting closed-loop system is robustly stable with disturbance attenuation γ for any time delay h satisfying h H for all, i.e., the closed-loop system satisfies the inequality, zw γ, (5) zw is the H norm of the transfer function zw from w( to z(. 3. Robust Parameter Optimization: (a) given γ, find the largest H, i.e. determine an upper bound for the time delay such that the system (-3) is robustly stabilizable, and (b) given H, find the smallest γ, i.e. determine the lower bound of disturbance attenuation for the uncertain system (-3). 3. ROUS CONROLLER DESIGN AND SUFFICIEN CONDIIONS FOR ROUS SAILIY AND H CONROL Lemma : For any vectors u, v R and any real n n symmetric positive-definite matrix P R, the following inequality holds u v u P u + v Pv. Lemma [6]: Let A, L, E, and F be real matrices of appropriate dimensions with F. hen, the following inequalities hold. () For any real symmetric positive-definite matrix P n and scalar s > such that si EPE >, ( A+ LFE) P( A+ LFE) APA + APE ( si EPE ) EPA + sll. () For any real symmetric positive-definite matrix P and scalar s > such that P sll >, ( A + LFE) P ( A + LFE) A ( P sll ) A + s E E. (3) For any scalar s >, + se E. LFE + E F L s LL 3.. Memoryless controller design We design memoryless controller (6) to solve robust-control problems,, and 3 for system (-3), u ( ) = Kx ( ), (6) m n K R is a constant matrix. 3.. Robust stabilization problem Assuming w ( =, we present the following two theorems for the uncertain system (-3). heorem : Given H >, the system (-3) is robustly closed-loop stable with a control input of (6) for any time delay h satisfying h H if there exist real symmetric positive-definite matrices P, P, P, and Q, a matrix K, and scalars r i >, i =,, 5, such that following inequalities hold RPP (,, P, QKHr,,, i ) <, (7) ri 3 EQE >, P rl 4 ALA >, P r L L >, (8) 5 P P P, Q KP K, (9) RP (, P, P, Q, K, H, r) ( A + K) P + P( A + K) + P( r L L + r L L ) P i A A r EA EA HP Q r K E EK r3 LL QE r I E QE E Q P + ( 3 ) ( ) ( 4 ) ( ) A A 4 A A ( 5 ) 5. + H A I P r L L A I + r E E + K P r L L K + r K E E K () Proof: From (-) and (6), we obtain x( + ) x( = ( A( I) x( + ( Kx( h ),

3 374 Kun Ji and Won-jong Kim x( h ) = x( ( A( + i) I) x( + i) + ( + ikx ) ( h + i), x ( + ) x ( ) = ( A ( ) + K ( ) I) x ( ) ( K ( A( + i) I) x( + i) + ( + ikx ) ( h + i). Define a discrete-time Lyapunov function as () V( = x ( Px( + W( + S(, () P is a real symmetric positive-definite matri W( x ( + i) A( + i) I j= h i= j P A ( + i) I x ( + i) + x ( + i) K ( h ) j= h i= j h P ( h) K x ( + i), S( S( ) + x ( + i) Px( + i) i= ] (3) [ x ( + i) x ( )] (4) i= Px [ ( + i) x ( )], P and P are real symmetric positive-definite matrices to be chosen. From (), we obtain V(,) x = V(, x + ) V(,) x = X + W + S, X = x ( + ) Px( + ) x W = W( + ) W(, S = S( + ) S(. For X, we have ( Px(, () [ ( ) ()] [ x x ] Px X X = x P x + x + ( + ) ( ) ( ) +, X = [ x + x ] P[ x + x ] (5) (6) (7) ( ) ( ) ( ) ( ). Substituting () into (7), and applying Lemmas and, we obtain { X x ( ( A+ K I) P+ P( A+ K I) PrL ( ALA rl L ) P A A HP Q r3 LL + r E E + r K E E K + QE ( r3 I EQE ) EQ P x( + X + X + X, 3 Q KP K, P P P, and X = x ( + i) ( A( + i) I) P ( A( + i) I) x( + i), X3 = x ( h + i) K ( + i) P ( + i) ( + i) K x( h + i). For W, from (3) we obtain x ( i) ( A ( i) I) } (8) (9) W = h x ( ( A ( I) P ( A( I) x( + hx ( K ( + h) P ( + h) Kx ( ) P ( A( + i) I) x( + i) x ( h + i) K ( + i) P ( + i) ( + ik ) x ( h + i). Applying Lemma () and Assumption, we obtain { W x ( H ( A I) ( P r L L ) ( A I) 4 A A 4 A A ( 5 ) 5 ( ) 3. + r E E + K P r L L K + r K E E K x X X For S, from (4) we obtain } () S x ( ( P+ P) x( X. () Substituting (8-) into (6), we obtain V(,) x = X + W + S () x ( RPPP ) (,,, P3, QKHr,,, i ) x ( ), V( x ( R( P, P, P, P3, Q, K, H, ri ) x( (3) <.

4 Robust Control for Networed Control Systems with Admissible Parameter Uncertainties 375 he above sufficient condition is equivalent to the solvability of a system of following LMIs. heorem : Given an H >, the system (-3) is robustly closed-loop stable through a control input of (6) for any time delay h satisfying h H if there exist real symmetric positive-definite matrices Y, P, P, and Q, a matrix Z, and scalars r i >, i =,, 5, such that the following LMIs hold: M( Y, Z) M( Y, Z) M3 M4( Y, Z, H) M ( Y, Z) M, < M3 M33 M4 ( Y, Z, H) M 44 (4) Q Z, (5) Z Y Y P P Y P P, (6), M( YZ, ) = YA + AY+ Z + Z+ HQ (7) + rl L + r L L + HrL L, M ( P, K ) = YE Z E, (8) 3, A A 3 A M = HQE (9) 4 (,, ) ( ) M Y Z H = H Y A I YEA Z Z E, (3) M = diag ri, r I, (3) { } M = H( r I E QE ), (3) 33 3 { A A } M 44 = Hdiag P r4 L L, r4 I, P r5 L L, r5 I, (33) and a stabilizing controller is u ( ) = ZY x ( ). Proof: Define the new variables Y and Z in (4-6) as Y = P, Z = KY. (34) Multiplying both the sides of the inequality (3) by Y, and then by Schur complements, we obtain that the conditions in (7-9) are equivalent to the LMIs (4-6). 3.. Robust H control problem With w ( ), we present a sufficient condition as the following theorem for the uncertain system (- 3) to be robustly stable with prescribed level of disturbance attenuation. heorem 3: Given an H > and γ >, the system (-3) is robustly closed-loop stable with disturbance attenuation γ with a control input of (6) for any timedelay h satisfying h H if there exist real symmetric positive-definite matrices Y, P, P, P 3, and Q, a matrix Z, and scalars r i >, i =,, 5, such that the following LMIs hold: M M ( Y, Z) M ( Y, Z) M M Y Z M 3 (, ) M3 M33 4 ( Y, Z, H) M5 ( Y ) M4( Y, Z, H) M5( Y) <, M 44 M 55 (35) Q Z, (36) Z Y Y P P P (37) 3, M5 = YE w, (38) I M55 = γ I Hw, (39) Hw HP 3 and a guaranteed controller is u ( ) = ZY x ( ). Proof: Apply (6) to (-3), the closed-loop transfer function zw from w( to z( is given by = E( zi A Kz ). (4) zw h w From (5) and (4), and by the result of heorem, Lemma, and Schur complements, we obtain heorem Robust-parameter optimization he problem of finding the largest H for a given γ or the smallest γ for a given H, can be easily solved using standard LMI approaches by heorem 3. For instance, the largest H which ensures that the system (-3) is robustly stabilizable with disturbance attenuation γ can be determined by solving the following quasi-convex optimization problem: LMIs: (35-37), Objective: Maximize H subject to H >, Y >, P >, P >, P3 >, Q >, Z >, r >, i =,...,5. i On the other hand, the smallest γ obtainable from heorem 3 which ensures that the system (-3) with a

5 376 Kun Ji and Won-jong Kim given H is robustly stabilizable can be determined by solving the following quasi-convex optimization problem: LMIs: (35-37), Objective: Minimize γ subject to Y>, P>, P>, P3>, Q>, Z>, r >, i=,...,5. i 3.. Dynamic controller design A novel approach of using previously stored control data to compensate for time delays and pacet losses is discussed in [7]. Here we give a sufficient condition. Assuming w( =, the discrete-time analogue of Artstein transform [5] for the system (-3) is given by ( h ) x ( = x( + A( ( + i). (4) Lemma 3: Let (x(, ) be a solution (admissible pair) for (-3), defined by initial condition (x(), u (.)). hen ( ~ x (, ) with ~ x ( ) defined by (4) is a solution (admissible pair) for the system ~ ) ~ h x( + ) = A( x( + A( ( ) (4) with the initial condition ( ~ x (),.)). Conversely, let ( ~ x (, ) be a solution of (4) defined by some initial condition ( x (),.)). hen given some h and taing u (.) defined on [ ) ( h ) x() = x () A() i), (43) i= h the solution for () by the initial condition (43) is given by ( h ) x( = x ( A( ( + i). (44) Lemma 4: Let u ( = Fx ~ ( be a feedbac controller stabilizing the system (4). hen the controller = + ( h ) ( ) ( ) ( ) i u F x A ( + i) (45) i= h is stabilizing the system (-3). he proof is straightforward and relies entirely on Lemma 3. Since the admissible uncertainties are unnown, the dynamic controller is given as u ˆ( ) = Fx ( ) + A u ( + i). ( h ) ˆ i= h (46) his compensator can be constructed by steps. he corresponding nominal state equation of the transformed system is h xˆ( + ) = Axˆ( + A uˆ(, u ˆ( ) = Fx ˆ( ). hus the augmented system equation is h ( ) ( ) ( ) x + A A F x ( =. xˆ( + ) h A A F xˆ( + (47) (48) heorem 4: Given an H >, the system (-3) is robustly closed-loop stable through a dynamic control input of form (46) for any time-delay h satisfying h H if the largest singular value of H A( A( F A + A is less than. H F he proof is straightforward. 4. NUMERICAL EXAMPLE Consider NCS shown in Fig. with parameter uncertainties and time delay described by x( + ) = ( A+ Au) x( + ww( + ( + u) h, (49) z () = Ex (), sampling period is. s and.5.. A =,,,. =. w = E =. LA =, =, =,.. L EA E [ ], =.. Applying the robust control result of heorem to the above system (49) and solving LMIs (4-6) gives that the upper bound of H is Applying heorem 3 to the above system (49) and solving LMIs (35-37) with H = 8.46 and γ = 3 gives the memoryless state-feedbac control gain K = [..4 ]. 3. (a) Finding the maximum H: Given γ = 3, applying the H control result of heorem 3 to the system (49) gives that the maximum H is 8.4. If given γ =.5, then H decreases to.6. (b) Finding the minimum γ: Given H =, applying the H control result of heorem 3 to the system (49) gives that for time delay h satisfying h, the smallest value of γ is.. If given H = 8, the smallest value of γ increases to.7. he relation between the maximum H and the minimum γ is shown in Fig.. As expected, H monotonically increases as γ gets greater. However,

6 Robust Control for Networed Control Systems with Admissible Parameter Uncertainties 377 all Position(mm).5.5 Fig.. Relation between the maximum H and the minimum γ. the disturbance attenuation performance beyond a certain threshold (γ.3) maes little difference, and H approaches EXPERIMENAL VERIFICAION We constructed an NCS as shown in Fig.. A ball magnetic-levitation (maglev) setup [7] is used. he discrete-time model of this maglev system is x( + ) = x(, 4 + z ( ) = x ( ), [ ] the sampling period is 3 ms, and (5)... LA =, L, EA, E. = = = Applying the robust control result of heorem to (5), the upper bound of H was found to be. he maximum round-trip time delay induced by Ethernet in out lab is about 3.4 ms which is less than, thus the maglev system is supposed to be stabilizable with a control loop closed over the Ethernet. he response of the system is shown in Fig. 3. he ball maintained its equilibrium position and did not fall down. hen we increase the length of time delay to 3 (9 ms) after s, the system response is shown in Fig. 4. he zero value of the vertical axis denotes that the system lost its stability and that the ball could not all position (mm) ime (s) Fig. 3. NCS test bed response with a control loop closed over the Ethernet. 5 5 ime (s) Fig. 4. NCS test bed response with 3 -long time delay occurring after s. maintain its equilibrium position and fell down. hus from Figs. 3 and 4, we can conclude that the upper bound of time delay this ball maglev system can accommodate is, i.e., H =, which is consistent with that from heorem. 6. CONCLUSIONS Robust-control strategies for NCSs were proposed in this paper based on a discrete-time NCS model. Delay-dependent methods of designing linear memoryless state-feedbac controllers and dynamic state-feedbac controllers to solve robust control problems were presented. We provided new robust stability criteria in terms of LMIs. A numerical example was wored out to illustrate the presented robust control methodologies. We also used our NCS test bed to experimentally verify the feasibility and effectiveness of the proposed design methodologies. REFERENCES [] Y. ipsuwan and M.-Y. Chow, Control methodologies in networed control systems, Control Engineering Practice, vol., no., pp. 99-, Feb. 3. [] W. Zhang, M. S. ranicy, and S. M. Phillips, Stability of networed control systems, IEEE Control System Magazine, vol., no., pp , Feb.. [3] G. C. Walsh, Y. Hong, and L. G. ushnell, Stability analysis of networed control system, IEEE rans. on Control System echnology, vol., no. 3, pp , May. [4] K. Ji and W.-J. Kim, Real-time control of networed control systems via Ethernet, International Journal of Control, Automation, and Systems, vol. 3, no. 4, pp. 59-6, Dec. 5. [5] Z. Artstein, Linear systems with delayed control: A reduction, IEEE rans. on Automatic Control, vol. AC-7, no. 4, pp , 98. [6] C. Foias, A. annenbaum, and G. Zames, Weighted sensitivity minimization for delayed systems, IEEE rans. on Automatic Control,

7 378 Kun Ji and Won-jong Kim vol. AC-3, no. 8, pp , 986. [7]. Van Keulen, H -control for Distributed Parameter Systems: A State-Space Approach, asal, irhauser, 993. [8] A. Kojima, K. Uchida, E. Shimemura, and S. Ishijima, Robust stabilization of a system with delays in control, IEEE rans. on Automatic Control, vol. 39, no. 8, pp , 994. [9] S. oyd, L. E. Ghaoui, E. Feron, and V. alarishnan, Linear Matrix Inequalities in System and Control heory, SIAM, Philadelphia, PA, 994. [] S. I. Niculescu, H memoryless control with an α stability constraint for time-delays systems: An LMI approach, Proc. of the 34th Conferences on Decision and Control, New Orleans, LA, pp. 57-5, 995. [] P. Shi, E.-K. ouas, and R. K. Agarwal, Control of Marovian jump discrete-time systems with norm bounded uncertainty and unnown delay, IEEE rans. on Automatic Control, vol. 44, no., pp , Nov [].-J. Su and W.-J. Shyr, Robust D-stability for linear uncertain discrete time-delay systems, IEEE rans. on Automatic Control, vol. 39, no., pp , Feb [3] S.-J. Chen and J.-L. Lin, Robust stability of discrete time-delay uncertain singular systems, IEE Proceedings of Control heory and Applications, vol. 5, no., pp. 45-5, Jan. 4. [4]. Lee and J.-G. Lee, Robust stability and stabilization of linear delayed systems with structured uncertainty, Automatica, vol. 35, no. 6, pp , 999. [5] X. Li, M. Guay,. Huang, and D. G. Fisher, Delay-dependent robust H control of uncertain linear systems with input delay, Proc. of the American Control Conference, pp. 8-84, June 999. [6] L. Xie, Y. Wang, and C. E. de Souza, Robust control of a class of uncertain nonlinear systems, Systems & Control Letters, vol. 9, no. 3, pp , 99. [7] W.-J. Kim, K. Ji, and A. Ambie, Networed real-time control strategy dealing with stochastic time delays and pacet losses, ASME Journal of Dynamic Systems, Measurement and Control, vol. 8, no. 3, pp , Sep. 6. Kun Ji received the.s. and M.S. degrees in Precision Instrumentation and Engineering from singhua University, eijing, China, in 999 and, respectively, and the Ph.D. degree in Mechanical Engineering from exas A&M University, College Station, in 6. His research interests focus on analysis, design of networed control systems, real-time control systems, and precision mechatronics systems. Won-jong Kim received the.s. (summa cum laude) and M.S. degrees in Control and Instrumentation Engineering from Seoul National University, Seoul, Korea, in 989 and 99, respectively, and the Ph.D. degree in Electrical Engineering and Computer Science from Massachusetts Institute of echnology (MI), Cambridge, in 997. Since, he has been with the Department of Mechanical Engineering, exas A&M University, College Station, currently he is Associate Professor and the Holder of the Dietz Career Development Professorship II. His current research interests include focus on the analysis, design, and real-time control of mechatronic systems, networed control systems, and nanoscale engineering and technology.