Observer-Based Robust-& Control Laws for Uncertain Linear Systems Yeih J. Wang * a n d Leang S. Shieh t University of Houston, Houston, Texas 77204-4793 Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2688 J . W. Sunkel $ N A S A Johnson Space Center, Houston, Texas 77058 Abstract Based on the algebraic Riccati equation approach, this paper presents a simple and flexible method for designing obseri erbased robust-H, control laws for linear systems with structured parameter uncertainty. The observer-based robust-H, outputfeedback control law, obtained by solving three augmented algebraic Riccati equations, provides both robust stability and disturbance attenuation with H,-norm bound for the closed-loop uncertain linear system. Several tuning parameters are embedded into the augmented algebraic Riccati equations so that flexibility in finding the symmetric positive-definite solutions (and hence, the robust-H, control laws) is significantly increased. A benchmark problem associated with a mass-spring system, which approximates the dynamics of a flexible structure is used to illus trate the design methodologies, and simulation results are presented. I. I n t r o d u c t i o n The problems of robust stabilization of uncertain linear SI stems have been studied by many reserchers. The algebraic Riccati equation (ARE) approach to the stabilization of systems with structured parameter uncertainty has been developed by Petersen and Hollot,' Petersen,2 S~hmitendorf,~ Jabbari and S~limitendorf,~ Khargonekar et al.,' and Wang et al.' These approaches have generally utilized the concept that a given AREbased control law guarantees the existence of a quadratic L l a punov function (and hence, stability) for a closed-loop uncertain linear system. Also, other recent research attention, e.g., P e t e r ~ e n Khargonekar ,~ et al.,' Glover and Doyle,g Bernstein and Haddad," Doyle et al.," and Scherer," has shown that AREbased robust controllers are able to not only stabilize linear S! stems with no uncertain parameters but also reduce the effect of disturbances on the controlled output to a prespecified level. Moreover, Madiwale el al.I3 and Veillette et al.I4 have proposed alternative ARE-based robust controllers which pro1 Ide both robust stability and disturbance attenuation with I I , norm bounds for systems with structured parameter uncertaint). In Ref. 13, their design method involves solving, in general, four coupled modified Riccati/Lyapunov equations, and the designed robust controllers guarantee robust stability, robust ( H z ) performance, and H , disturbance attenuation for uncertain lin- * Graduate student, Department of Electrical Engineering. Professor, Department of Electrical Engineering, Member AIAA. 1 Aerospace Engineer, Navigation Control and Aeronautics Division, Member AIAA. t Copyright 0 1991 by the American Institute of Aeronautics and Astronautics. Inc. No copyright is asserted in the United States under Title 17, U.S. Code. The U.S. Government has a royalty-free license t o exercise all rights under the copyright claimed herein for Governmental purposes. All other rights are reserved by the copyright owner. 741 ear systems. In Ref. 14, in order to achieve both robust stability and disturbance rejection with H,-norm bounds, their design method embedded the information of structured system uncertainty into the ARES which were used for nominal H , disturbance-attenuation design. In this paper, we present a simple and flexible method based on the ARE approach for determining observer-based robust-H, output-feedback dynamic control laws for systems with structured parameter uncertainty. The developed state- and outputfeedback control laws provide both robust stability and I I , disturbance attenuation for closed-loop uncertian linear systems. The design procedure is described in the following. Firstly, lye determine a robust-H, state-feedback control law by solving the first augmented ARE which accounts for both structured system uncertainty and H , disturbance attenuation. Secondly, based on a dual concept, a full-order robust-H, observer is obtained by solving the second augmented ARE which is dual to the one used in designing the robust-H, state-feedback control law. The structure of our observer is different from that developed in Ref. 11 which considers the estimation of the worst disturbance. Thirdly, when the third augmented ARE has a symmetric positive-definite (SPD) solution, the resulting observer-based dynamic control law guarantees both robust stability and H, disturbance attenuation for closed-loop uncertain lineal system. Several tuning parameters are embedded into the augmented ARES to enhance flexibility in finding the SPD solutions (and hence, the robust-Hi, control laws). A benchmark problem associated with a mass-spring system, which approximates the dynamics of a flexible structure''-'* is used to illustrate the design methodologies, and simulation results are included. 11. N o t a t i o n and Problem Formulation Throughout this paper, all matrices and vectors are considered to be real and of appropriate dimensions. Also, we denote: maximum singular value of a matrix A f i %(Ad) minimum singular value of a matrix A f ; d&I) IlAdll matrix norm, ll~dll @ ( A I ) ; I identity matrix of appropriate dimension; 0 null matrix of appropriate dimension; matrix 1\.I is symmetric positive (semi)definite; n/f > (2)0 Ad < ( 5 )0 P >(>) Q P <(<) Q matrix A4 is symmetric negative (semi)definite; means P - Q >(>) 0; means P - Q <(<) 0. Consider the uncertain linear system k(t) = ( A + A A ) z ( t )+ E u ( t ) + D w ( t ) , + y(t)= Cz(t) Sv(t), 1 where the term --DTPlz,(t) in ( s a ) is interpreted as the esti62 1 mation of the worst disturbance, i.e., w ( t ) = -DTPiz(t)." 62 In this paper, we consider an alternatwe observer-based output-feedback dynamic control law of the form where z ( t ) is the state, u ( t ) is the control input, y ( t ) is the measured output, w ( t ) and v ( t ) are disturbances, and ~ ( t is) the controlled output. The unknown-but-bounded structured uncertainty is described by .! AA= (sa.) aiAi, (7b) i=l Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2688 where a; are uncertain parameters and A; are known constant matrices. Without loss of generality, we assume that which does not assume the presence of the worst disturbance in the observer design, for both robust stabilization and H , disturbance attenuation of the uncertain linear systemin (1).Applying the dynamic control law in (7) to the uncertain system in (1) gives the following closed-loop system Applying the singular value decomposition (SVD) technique in ( A 2 ) (see Appendix) to Ai, we can decompose each Ai as [ &(t) [ z'.(t)] = i = 1, ...,e, Ai = TiUT, where Ti and Ui are weighted unitary matrices. We assume that the nominal system trio ( A ,B , C) is controUable and observable. Let a scalar 6 > 0 be the given disturbance-attenuation constant. The problem is to design an output-feedback dynamic control law of the form + &(t) = A c z E c ( t )B C y ( t ) , BK c + B K c + K,C] [ By introducing the observer error e ( t ) = z ( t ) - z c ( t ) , we can transform the system in (8) to the following augmented system: (30) u ( t ) = Cczc(t), A + AA -K,C A [f::,'] [ (36) = such that the closed-loop systemin (1)is stable and the H,-norm of the closed-loop transfer function matrix from the disturbance T the controlled output z ( t ) in ( 1 ) input C ( t ) [ w T ( t ) , v T ( t ) l to is less than 6 for all A A in (2). When A A = 0, DTDl = I , S S T = I, and the trio ( A ,D, C1) is controllable and observable, suRcient conditions for the exis- A + A A+ E K , AA -EK, A KO.] + [ e Now, the problem is reduced to designing a state-feedback gain K , and an observer gain K O in (7) such that thc wgmented system in ( 9 ) is stable and H , norm of the transfer function matrix from tL~(t)= [ w T ( t ) , v T ( t ) l Tto ~ ( t in ) ( 9 ) is less than some prespecified value for all A A in ( 2 ) . In the development below, we utilize the following matrix identity: tence of a dynamic controller which stabilizes the nominal system in (1) and achieves the H,-norm bound 6, are given by Dovle et al." as follows: There exists a matrix PI equation > 0 satisfying the following Riccati 1 ATP1 + P I A - PI (BET - -DDT)Pt 62 and the resultingmatrix A (ii) There exists a matrix Pz equation AP2 + P2AT - > + CTC1 = 0, (4) EXXT 1 is stable; 62 0 satisfying the following Riccati - + l Y Y T f ( X Y T + YAYT)2 0, (BET - -DDT)P1 1 P2(CTC - -CTC1)P2 62 + DDT = 0, where X and Y are any two appropriately dimensioned matrices and E is a positive scalar. The following lemma will be utilized in the sequel. (5) _~ Lemma 1. Let A, D , and I? be matrices of appropriate dimensions. For a given positive scalar 6, if there exist a SPD matrix P and a positive scalar E such that 1 and the resulting matrix A - PZ(CTC - -CTC1) is sta62 ble; 1 (iii) The matrix ( I - -P2P1)-'P2 is symmetric positire62 d e h i t e. ATP Suppose that there exist PI > 0 and Pz > 0 satisfying the conditions (i-iii), then an observer-based output-feedback dynamic control law is obtained" as + P A + FPDDTP 6 tL €6E T E < 0, then A is a stability matrix, and 742 R(s)= &(SI - A)-'D (11) satisfies L Proof. Suppose that P > 0 satisfies the inequality in (11). It follows immediately from Lyapunov stability theory" that d is a stability matrix. Furthermore, from (11)we have ( - ~ w I - A ) ~ P + P ( ~ w I - A? P ) -D D T P - L E T E 6 e6 > 0, + ^c, c, E6 ' Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2688 &%$T( - j w ) and post-multiply T - P,B I ( Ei - t~P~DDTP. - -1; - D T D ~ ) B T Pt. Q= = o (17) 4s6 . has a SPD solution Pc, where Ti and Ui, i = 1,.. ,.! are defined in (2c). Then, the closed-loop system in (16) is stable and the H, norm of the transfer function matrix from G ( t ) to ~ ( t in) (16) is less than 6 for all A A in ( 2 ) with (13a) for all w E R. Since A is a stability matrix, ( j w I - A) is invertible. Define f$(jw) E ( j w I - A)-'B, and pre-multiply (13a) to obtain 1 + -uiu:) A ~ P +, P A t Ci=l (E~P.T~T?P, K , = -ycBTP,, m$( j w ) to (18a) where yc satisfies either 216 1 1 216 1 l > y > or ->7,>--> 0. e6DTP$(jw)+ ~ 6 $ ~ ( - j w ) P-DE ~ $ ~ ( - ~ w ) P D D ~ P $ ( ~ wiP(D1) ) 2 - ' -2 2 c'(D1) 2 (lab) - $T(-jW)ETEl$(jW) > 0. (13b) Proof. Suppose that P, > 0 satisfies the Riccati equation in Applying the matrix identity in (10) to (13b) gives (17). In order to show that the systemin (16) with K , given by 6'1 - € 6 D T P $ ( j w )- ~ 6 $ ~ ( - j w ) P D (18) is atable and satisfies the H,-norm bound 6 for all A A in E"J( - j w ) P D D T P $ ( j w ) 2 0. ( Z ) , by Lemma 1, it suffices to show that (13c) Combining (13b) and (13c) yields Q, 2 - ( A t A A t B K , ) ~ P-. P,(A t AA B K , ) 621 > $ T ( - j W ) E T E $ ( j W ) = H T ( - j w ) H ( j w ) 1 1 - - P , D D ~ P ,- ,(cTcl 6 €6 n for all w E R, which implies llH(s)ll, < 6. + + +KTDTD~K,) is SPD for all A A in (2). From (17), it follows that R e m a r k 1. When (A,,!?) is observable, the inequality in (11) can be relaxed to ATP + P A + 56 P D D T P t +€6E T & 5 0. Accordingly, the result in (12)becomes ~ ~ ~ ( a5 )6.~ Note ~ , that an additional tuning parameter E has been introduced in (11).R Also, using the assumption in (2b) and the matrix identity in (lo), we obtain the following inequality: 111. R o b u s t S t a t e - Feedback Control If the state of the system in (1) is available for measurement and a state-feedback control law is utilized, then the disturbance u ( t ) in ( l b ) has no effect on the controlled output z ( t ) of the closed-loop system. Let the state-feedback control law be given by v(t) = KEz(t). The closed-loop system in (1) becomes i ( t )= ( A 4t)= + A A +- B K , ) z ( t ) + D w ( t ) , [;kc] 4t). (14) (15a) (15b) Aa a result, Qc 2 Note that when e ( t ) = 0, the closed-loop system in (9) reduces to & ( t ) = ( A t &A 4 t )= [& ] + B K , ) z ( t ) t [D,O]&(t), (16a) 4th (16b) > 0, which complete8 the proof. n firnark 2. The Riccati equation in (17) is constructed to account for both structured uncertainty in (2) and H, disturbance attenuation 6. If there is no system uncertainty (i.e., A A = 0 ) and disturbance attenuation is not required (Le., 6 + m), this augmented Riccati equation in (17) reduces to an ordrnary Riccati equation which arises in the linear quadratic regulator problem." which is identical to the one in (15). The following theorem is developed for finding a robust-Hi, state-feedback gain K , such that the closed-loop system in (16) is stable and the H , norm of the transfer function matrix from w(t) to z ( t ) in (16) is less than some prespecified value for all a A in (2). m The introduction of tunins puameters c i , i = 1,. . .,L and 2 in (17)enabler the propond approach more flexible in obtaining the robust-H, state-feedback gain K,. For instance, when A A = 0, DfD1 = I, and ( A , CI) is observable, the Riccati equation in (4), which is used for determining the H, r t a t c feedback gain ( - B T P 1 ) in (6b)," is a special cane of (17) with Theorem 1. Consider the closed-loop system in (16) with the structured uncertainty described in ( 2 ) . Let 6 > 0 be the given disturbance-attenuation constant. Suppose that there exist positive scalars ~i > 0, i = 1,.. . , e (.! is the number of uncertain parameters as in (2)) and 00 > 1 > such that the Riccati equation Qc 1 and Q E = 0. In this case, we have P . = 2P1. Also, 26 it should be noted that the inequality in (18b) gives an explicit bound for which K , is allowed to vary without afFecting robust stability and disturbance attenuation of the closed-loop system. 1= 2 0 80, and a matrix Q E> 0 46 743 - i Remark 4. When A A = 0, SS' = I , and ( A ,D ) is controllable, the Riccati equation in (5), which is used for determining the H, IV. Robust Output - Feedback Control When the state is not available for state-feedback design, the design problem reduces to finding an observer-based outputfeedback dynamic control law in (7) for both robust stabilization and disturbance attenuation of the system in (1) with structured uncertainty in (2). Let 6 > 0 be the desired disturbanceattenuation constant and P, > 0 be a matrix satisfying (17). With the state-feedback gain K , in (7b) obtained using Theorem 1, the closed-loop system in (16), which can be reduced to ( 9 ) when e ( t ) = 0, is stable and the H , norm of the transfer function matrix from G ( t ) to z ( t ) in (16) is less than 6 for all A A in (2). Now, the problem remaining to be solved is to find a robust-H, observer gain K O in (7a) for not only reconstructing the state but also achieving both robust stability and H , disturbance attenuation of the closed-loop system in ( 9 ) . Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2688 - ~ To determine the desired robust-H, reformulate the system in (9) as 1 observer gain ( - ( I - - P ~ P I ) - ' P ~ C T ) in (6a)," is a special 62 1 case of (21) with i = - and Q o = 0. In this case, we have 26 Po = 2P2. n Let 6 > 0 be the given disturbance-attenuation constant, and P, > 0 and Po > 0 be the matrices satisfying (17) and (21), respectively. Then, the respective If, and K O in the observerbased dynamic control law in ( 7 ) can be obtained using the respective Theorem 1 and Theorem 2 for both robust stabilization and H , disturbance attenuation of the system in (1) with the structured uncertainty in (2). However, the resulting closed-loop system in (9) may not be stable and the H , norm of the transfer function matrix from G ( t ) to z ( t ) in (9) may not be less than 6 for all A A in (2) due to the failure of the separation theorem for robust stabilization and/or H , control design of uncertain linear s y s tems . observer gain K O ,we For convenience, we rewrite the augmented system in (9) as follows: where S ( t ) When z$) = 0, the above system can be reduced to i ( t )= ( A + A A + K , C ) e ( t ) + [ D ,If,,S]CJ(t), From (2), the structured uncertainty matrix A A can be expressed as Theorem 2. Consider the closed-loop system in (20) with the structured uncertainty described in (2). Let 6 > 0 be a given disturbance-attenuation constant. Suppose that there exist pos- > where ai,Ti,and rr20, and a 46 Lemma 2. Consider the closed-loop system in (23) with the structured uncertainty described in (24). Let 6 > 0 be the given disturbance-attenuation constant. Suppose that there exist positive scalars (j > 0, i = 1 , . . and 00 > ( > 0, and a matrix > 0 such that the Riccati equation + POAT+ ~ ( 6 i P o U i U ~+P-2'iT:) 1, + iPoC,'CIP, 6 ti i=l CP, t Q o = 0 (21) f ATP 2i6 l > T 2 - O 1 >- 2 or + P A+ C((ia?iF:P i=l + -PDDTP + 6 (22Q) 1 _ETE + Le#:) Ci +Q =0 (25) (6 ,e has a SPD solution P, where f'i and fii, i = 1 , . . . are defined in (24). Then, the closed-loop system in (23) is stable and the H , norm of the transfer function matrix from W ( t ) to ~ ( t in ) (23) is less than 6 for all AA in (24). where yo satisfies either CqS) .,e' 4 has a SPD solution Po, where Ti and Ui, i = 1 , . . ., are defined in (2c). Then, the closed-loop system in (20) is stable and the H, norm of the transfer function matrix from CJ(t) to z ( t ) in (20) is less than 6 for all A A in (2) with K O= -yoP,CT Vi,i = 1,.. .,e are defined in ( 2 ) . The following lemma gives a sufficient condition for both robust stability and H , disturbance attenuation of the closedloop system in (23) with the structured uncertainty in (24). f AP, w ( t ) is as previously cirfined and (20a) which is dual to the one in (16). Hence, the following theorem, which is dual to Theorem 1, is deve1:ped to find the observer gain If, such that the closed-loop system in (20) is stable and the H , norm of the transfer function matrix from G ( t ) to z ( t ) in (20) is less than some prespecified value for all A A in (2). and 03 > itive scalars 6; > 0 , i = matrix Qo > 0 such that the Riccati equation [ ~ ' ( t )e'(t)]', , 1 2i6 1 - > T o > - - >- 0.- (22b) 2 n"S) 2 Proof. Suppose that P > 0 satisfies the Riccati equation in n (25). Using the equality in (25) and the matrix identity in ( l o ) , 744 we obtain the following matrix inequality: + - (A A A ) ~ P P(A + AA) - -( P* "D" D ~ -P 1 E T E 6 16 ?Q>O, Fig. 1 A two-mass/spring system. Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2688 for all AA in (24). Hence, by Lemma 1, we conclude that the closed-loop system in (23) is stable and satisfies the N,-norm R bound 6 for all A A in (24). 6 ( t )= (A R e m a r k 5 . Suppose that there exist matrices PI > 0 and Pz > 0 satisfying the conditions (i) and (ii) in (4) and (5), respectively, which are utilized for finding an output-feedback dynamic control law in ( 6 ) for the standard H , control prob1em.l' Let the closedloop sytem with the control law in (6) be represented by & ( t )= Ai.(t) Z(t) = + BW(t), Eqt). + ~ A ) z ( t+) B u ( t ) + D w ( t ) , + Y ( t ) = C z ( t ) Sv(t), with the nominal system matrices given by (260) (26b) The simple condition (iii) for hiding a H, control law in ( 6 ) is the resultI4 of restricting P = block diag[PI, 6'P;' - PI] such that P is SPD and satisfies the following Riccati equation D=[O 0 0 I]*, C=[O 1 0 01, and the structured parameter uncertainty given by r o However, the matrix P in (25) is not limited to be a block diagonal matrix. As a result, the solution space of the Riccati equation n in (25) is larger than that in (27). V. Design for a Benchmark Control P r o b l e m In order to highlight the concepts and methodologies presented in the previous sections, the design problems #1 and # 3 of the benchmark example16-'* associated with a mass-spring system, which approximates the dynamics of a flexible structure, are considered here. The two-mass/spring SISO system, shown in Fig. 1, is described by 0 01 0 where ~k is a scalar such that A & 2 ( A k l . Furthermore, the matrix A1 can be decomposed into Al = T1UT as that in (2c) using the SVD technique described in (A2) (see Appendix) with [o T1= 0 a -&ZIT, u1= [-aa 0 oIT Design p r o b l e m #l. This problem considers only robust stabilization but not disturbance attenuation (Le. 6 -+ co) of the uncertain system in (28). To find a suitable control law in (7) which guarantees the stability of the uncertain system in (28) for 0.5 5 k 5 2.0, we let k,, = 1.25 and A & = 0.75. Firstly, a robust state-feedback gain K , in (7) is determined using Theorem 1 as follows. With Q c = I and €1 = 0.01 (the terms :PcDDTPc 6 1 and in (17) vanish for 6 --t co),the Riccati equation in ;6CT C1 (17) has a SPD solution with a non-colocated measurement ~ and the controlled output pc= where u ( t )is an actuator input, w ( t )is a disturbance input, v ( t )is a white Gaussian noise process with unit power spectral density and S = 0.01, ml = m2 = 1, k is an unknown-but-bounded uncertain stiffness parameter with 0.5 5 k 5 2, and the weighting matrix D1 are to be chosen upon design. Let knom and A k denote the nominal value and variation of a k , respectively. Then, by the uncertain parameter k = k,,, defining z ( t ) = [zl ( t ) , z 2 ( t ) ,i1 (t), i, ( t ) l T , we can represent the uncertain linear system in (28) as [ With yC= (18) as 50.6654 -41.9673 10.3392 -41.9673 41.7742 -8.5123 10.3392 - 8.5123 4.6588 22.2718 -7.0898 2.7947 1 2 -, a robust 22.2718 -7.0898 2.7947 55.6969 1 ' (30a) state-feedback gain can be obtained from K , = -7.BTPc = [ -5.1696 4.2561 -2.3294 - 1.39731. (30b) Then, Theorem 2 is utilized to find a robust observer gain li, in (7) as follows. With Q,, = I and €1 = 0.005, the Riccati equation in (21) has a SPD solution + 745 Po= [ 86.0750 3.0272 -8.8498 28.8675 3.0272 5.3441 -11.9683 13.7695 -8.8498 -1119683 66.6080 -66.8285 28.8675 13.7695 -66.8285 76.6126 1 B , = [ 12.088 7.2389 -8.7427 16.18311 ' C , = [ -14.9991 (31a) The stability range of the robust control law (35)is found to be 0.48 5 k 5 2.03. The dynamic control law (35) is stable but non-minimum phase with poles located at -5.2672 f jl.8091, -1.8912 f j3.1209, and zeros located at -0.1688, 0.4826 f j0.9470. Root locus versus overall loop gain of this control law for the nominal evaluation stiffness k = 1.0, is shown in Fig. 2. The gain and phase margins of the loop transfer function with the robust control law for k = 1.0 are 1.679 and 30.53",respectively. 1 From (22), we choose yo = - and obtain 2 KO= -y,P,CT = [ -1.5136 -2.6720 5.9842 -6.8848IT. (31b) Combining K , in (30b) and K O in (3lb) yields the following output-feedback dynamic control law as that in (3): &(t) = A,z,(t) + B,y(t), ~ ( t=) C c z c ( t ) , 6.9698 -7.0779 -12.77841. (32) Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2688 where A, = A + B K , + KoC -1.5136 1 0 0 0 -2.6720 -6.4196 11.4903 -2.3294 1.2500 -8.1348 0 0 B, = - K O = [ 1.5136 2.6720 -5.9842 6.8848l'f n (32) guarantees the for all 0.5 5 k 5 2.0, equation in (25) has a to be stable for all 0.36 5 k 5 3.27. Note that the stability range of the stiffness k has been significantly increased. The dynamic control law (32) is stable but non-minimum phase with poles located at -0.9988 f j3.0265, -1.5019 f j1.5378, and zeros located at -7.6327, -0.0766, 0.2238. The gain and phase margins of the loop transfer function with this robust control law for k = 1.0 are 1.214 and 17.17', respectively. The robust control law (32) is somewhat conservative for this particular control problem. A less conservative control law is obtained as follows. We choose knOm= 0.88 and AB = 0.38. Following the same procedure as before, a robust state-feedback gain K , is determined as K , = [ -14.9991 6.9698 -7.0779 -12.77841, Fig. 2 Root locus versus overall loop gain with control law in eq. (35)for k = 1.0. Let w ( t ) be a unit-impulse disturbance and the initial cotiditions be zero for the following simulations. The time responses of zl(t), z Z ( t ) , and u(t) with the control law in (35) for k = 1.0 are shown in Fig. 3. It is observed from Fig.3 that the peak magnitude of the controlled variable .zZ(t) is around 1.35 and that z z ( t ) has settled down in less than 10 seconds for k = 1.0. Note that the control law (32) achieves a larger stability range (0.36 5 k 5 3.27) than the control law (35) does (0.48 5 k 5 2.03) at the expense of a larger control effort u ( t ) but less satisfactory time responses (with a peak magnitude of z z ( t ) at around 5.2 and a settling time longer than 15 seconds for k = 1.0) due to its conservativeness. (33) by solving the Riccati equation in (17) with QC = 201 and EI = 0.008 and by choosing yc = 1 in (18). Then, a robust observer gain K Ois determined as K O= [ -12.0880 -7.2389 8.7427 -16.1831IT, Design problem #3. This problem considers both robust stabilization and disturbance attenuation of the uncertain linear system in (28). In particular, we consider the case that the desired value of H, disturbance-attenuation bound 6 is one. When w ( t ) is a sinusoidal disturbance of frequency 0.5 rad/sec with unknown-but-constant amplitude and phase, determination of an observer-based robust controller, which rejects this cyclic disturbance, is also considered. (34) by solving the Riccati equation in (21) with Qo = 201 and €1 = 0.004 and by choosing yo = 1 in (22). The resulting outputfeedback dynamic control law in (7) with K , in (33) and K Oin (34) is &(t) = Aczc(t) + &Y(t), u(t) = C,zc(t), (35) where Case 1. We set the desired disturbance-attenuation constant 6 = 1 and let the weighting matrix D1 = 0.005. Again, we let k,, = 1.25 and ~k = 0.75. Firstly, a robust-Hi, state-feedback gain is determined as 0 A==[ -12.0880 1 -7.2389 0 -15.8791 16.5925 -7.0779 -12.7784 0.8800 -17.0631 0 0 746 This control law is guaranteed t o stabilize the system in (28) and achieve disturbance-attenuation H,-norm bound 6 = 1 for all 2 0.5 5 k 5 2.0, since the associated Riccati equation in (25) has a SPD solution for Q = I, (1 = 0.01, and = 0.001. The poles and zeros of this dynamic control law are -18.338 f j16.237, - 8.9966 f j12.090, and -0.2579, -0.2404 f j0.7961, respectively. Notice that the dynamic control law is minimum phase and stabilizes the uncertain system in (28) for a considerably large stability range 0.27 5 k 5 3.41. Root locus versus overall loop gain of the control law (38) for the nominal evaluation stiffness IC = 1.0, is shown in Fig. 4. The gain and phase margins of the loop transfer function with the robust control law for IC = 1.0 are 3.612 and 50.33', respectively. 5 1 0 Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2688 1 10 20 40 30 50 time (sec) 2 ........ :.................. :.................. :........ 3 I -2 20 10 0 40 30 50 time (sec) t 0 ........... + ........ .;.+ ....... +..... +... :+.+.+.....+ ..... :+.. Fig. 3 Time responses to a unit impulse disturbance with control law in eq. (35) for k = 1.0. -5 ...................................................................... -10....................... K , = [ -165.083 -252.865 -18.686 - 444.2041, (36) by solving the Riccati equation in (17) with Q. = I,E , = 5.Oe-5, 1 and i= 1.25e-5 and by choosing -yc = - in (18). Then, a robust 2 H, observer gain is determined as IC,, = [ - 376.635 -35.984 527.92 -647.013IT, (37) j............................. : ....................... ;... +. x Fig. 4 Root locus versus overall loop gain with control law in eq. (38) for k = 1.0. by solving the Fliccati equation in (21) with Q. = I, = 1 1.0e - 6, and = 5.0e - 5, and by choosing -yo = - in (22). The 2 in resulting output-feedback dynamic control law in (7) with hTC (36) and K O in (37) is + & ( t ) = A c z e ( t ) &y(t), u(t) = cczc.(t), (38) where :I -376.635 1 -35.984 0 A c = [ -166.333 276.305 -18.686 -444.204 ' 1.2500 - 648.263 0 0 0 B, = [ 376.635 35.984 -527.920 647.0131 C , = [ - 165.083 -252.865 -18.666 For the following simulations, w e l e t w(J) = sin0.5t and initial conditions be zero. The time responses of z l ( t ) , z Z ( t ) , and ~ ( twith ) the control law in (38) for k = 1.0 are shown in Fig. 5. It is seen that z z ( t ) satisfies the IT, disturbance-attenuation bound 6 = 1.0 and that z z ( t ) has settled down around 20 seconds for k = 1.0. Case 2. Let ~ ( tbe) a cyclic disturbance described by w(t) = 7 A, sin(0.5t t d), where A,,, and 4 are unknown but constant. Since the frequency of the disturbance is 0.5 rad/sec, we can differentiate (28a) and (286) twice until w ( t ) disappears in the resulting system," and -444.2041, 747 I where the nominal system matrices are given by 2.0 ' A = n i -1 0 Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2688 -2.0 I o 0 1 0 n i o o 0 0 0 0 1 0 o o o i n -0.25 0 0 d=[o o 10 I 30 40 , B= -1.25 0 1 n o 01, and the structured parameter uncertainty is given by I I 20 n o 1 - o o o -1.0 n 0.25 50 time (sec) A = G,lAl, 200 - 0 o -0.43 0 A1 = 100 0 0 0 0.43 0 0 0 0 IG,11 5 1, o n o o- 0.1075 0 0.43 0 0 o 0 0 0 0 -0.1075 0 0 0 0 0 0 ' 0 -0.43 0 - 0 -100 I -200 20 M 40 30 time 50 (SCC) Fig. 5 Time responses to a sinusoidal disturbance with control law in eq. (38) for k = 1.0. obtain z(4) 1 ( t )= -k[Zl(t) + 0.252i(t) - iiz(t) - Kc= [-7.567 -2.804 1.242 -2.512 2.395 -7.241, 0.25~z(t)] + - 0.25Z,(t) G ( t ) , (39Q) z y ' ( t ) = - k [ Z z ( t ) 0.2522(t) - & ( t ) - 0.25z1(t)] + - 0.2512(t), (39b) where G ( t ) is a new control variable defined as G(t) f C(t) K O = [ 2.61 -1.354 -2.475 -5.698 -5.917 2.057IT, + 0.25u(t). (40) The new system (39) contains uncontrollable poles at 3 = f j 0 . 5 . Hence, a new state, 6 l ( t ) 2 Z l ( t ) 0.25zl(t), is introduced to remove the uncontrollable poles from (39). Then, (39) becomes + + k [ i i z ( t )+ 0.25zz(t)] + G ( t ) , + 0.25)Zz(t) - 0.25kZz(t) + k i i ( t ) . gi(t) = - k P l ( t ) 2(4) 2 ( t ) = -(k For the above uncertain system, we assume that k,, A k = 0.43 as in Ref. 18. Define =(A (4lb) i,(t) = A,i,(t) r A, = 748 o -8.567 0 0 0 kz(t), (42fl) + ircy(t), G ( t ) = d,y(t), (46) where = 1.0 and g ( t ) = [ & ( t ) ,&(t), a&), + A A ) Z ( t ) + kG(t), (45) by solving the Riccati equation in (21) with Q o= I, €1 = 0.0n5, 1 and by choosing -yo = - in (22). The resulting output-feedback 2 dynamic control law in (43) with K, in (44) and K O in (45) becomes (41~) Z z ( t ) , 2?'(t)lT. Then (28c) and (41) can be represented as Z(t) (44) by solving the Riccati equation in (17) with Q e = I, €1 = 0.005, 1 and by choosing 7e = - in (18). Then, a robust observer gain is 2 determined as - 1 1 -2.804 0 0 0 0 2.610 0.137 -2.475 -5.698 -5.917 1.807 0 -2.512 1 0 0 0 0 3.395 0 1 0 -1.25 0 -7.24 0 0 1 0 B , = [ -2.61 1.354 2.475 kc = [ -7.567 -2.804 5.698 1.242 5.917 -2.512 -2.0571' 2.395 , -7.241. Combining (40) and (46) yields the following 8-th order dynamic output-feedback control law 4 2 -.* . ,,+-., ,' I * Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2688 0 The poles and zeros of the dynamic control law are f j 0 . 5 , -1.2990 f j0.4363, -0.9097 f j2.1877, -0.4305 f j2.4992 and -1.4071, -0.0803, 0.5169, 0.0661 f j0.4765, respectively. Notice that the dynamic control law in (47) has three noli-minimum phase zeros and stabilizes the uncertain system in (28) for 0.54 5 k 5 1.57. Root locus versus overall loop gain of the control law (47) for the nominal evaluation stiffness k = 1.0, is shown in Fig. 6. The gain and phase margins of the loop transfer function with the robust control law for k = 1.0 are 1.435 and 26.44', respectively. ............................ 3 : +\i ~.......... ...; ............. :* ............ :............. i........... ...;.. ........... 10 time ' 2 ............. .............. :............. ;.............. :.. + 30 40 50 30 40 50 (sec) 1.o -1 .o :* i&+ k-4. ++ j .................. ,..-......................... 20 10 20 + fime ( s a ) Fig. 7 Time responses to a sinusoidal disturbance with control law in eq. (47) for L = 1.0. -2 ............. ;............. .;............. .;............. .;.............. j . . ........... ;. ............ .:............. c-++++ ; -3 ..............j .............. j ............. .............. j+ i i -3 -2.5 -2 -1.5 .... i............................. Standard H , control problem. The method of Doyle et al." as well as the proposed method are utilized for finding a disturbance-attenuation control law for the nominal system in (28) (i.e., A A = 0) with D1 = 1 and S = 1. When the disturbance-attenuation constant 6 = 3.125, there exist matrices PI > 0 and PZ > 0 satisfying the conditions (iiii) for finding a H , control law using Doyle's method." And a dynamic control law which solves this standard H , control problem with 6 = 3.125 can be obtained from (6)" as follows: i-+ :.............. +& -1 -0.5 0 0.5 1 Fig. 6 Root locus versus overall loop gain with control law in eq. (47) for k = 1.0. For the following simulations, again, we let w ( t ) = sin0.5t and initial conditions be zero. The time responses of el ( t ) , x Z ( f ) , and u ( t ) with the control law in (47) for k = 1.0 are shown in Fig. 7. The peak magnitude of the controlled variable x z ( t ) for k = 1.0 is around 9.6. Also, Fig. 7 shows that for k = 1.0, x ~ ( f ) has settled down and the cyclic disturbance is rejected in z z ( t ) within 20 seconds. Furthermore, when K, in (44) is replaced by 2 K , and KO in (45) is replaced by 2K0, the associated resulting 8-th order control law greatly enlarges the stability range of the uncertain system in (28) to 0.40 5 k 5 2.75. This control laiv, however, increases the peak magnitude of z z ( t ) to 17.4 and delai-9 the settling time to 22 seconds for k = 1.0. 749 where 0 0 -2.3835 1.4086 B , = [ 59.2407 C , = [ -1.1335 -59.2407 -54.3956 -13.8439 -60.9855 54.3596 -0.0559 1 0 -1.5555 0.1249 15.0379 -1.5555 ;I -1.2199 0.2398 ' 59.78301: -1.21991. The poles and zeros of the above control law are -53.2861, -1.5432, -0.4410 f j1.9832, and -0.2786, 0.1008 5 j1.6119? respectively. Wlieii 6 = 3.115, tlierr rxist 1’1 > 0 and 1’2 0 satisfying the conditions (i) and (ii); howrvrr, t h y do not satisfy the condition (iii). As a result, 1)oyle’s rnrthotl” fails t o yield a H , control law for the standard I I , control problem sritli 6 5 3.115. o n the other harid, tbe proposed method is ;ibk to find a If, control law for 6 : 2.325, wliirli is considc~rablgsmaller tliaii 3.115. This is accomplished in the following. k?rstIy, a I I , statefeedback gain is determined as Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2688 K c = [ -1.3172 -0.0091 -1.6808 - 1.41061, mark problem associated with a mass-spring system has been used to illustrate the design methodologies. Appendix . __ Lemma A l . (Singular value decomposition.”) M = uncv,‘, (49) = block diag[xk, 01 with +Bc~(t), k n/r = A,= [ Bc = [ 0.7473 1.3197 UiUiVT (Ale) = ukxkvz, i=l Where = 1211, U z , . . . ,U k ] € R n x kand vk = [V1,V2, . . ., v k ] E n Rmxk Consider a real n x m matrix M of rank IC. Immediately from Lemma A l , the matrix &I can be decomposed as the product of two rank-k matrices as follows: ~ ( t=) C c z c ( t ) , -0.7473 1 -1.3197 0 0.9717 - 1.6808 -2.0107 0 & = diag[ul, U Z ,. . . ,uk], . (60) where 0 0 -2.5672 1.2500 (Ala) (Alb) where k 5 min(n, m ) is the rank of k2 and 61, u2, . . ,(Tk are the non-zero singular values of M. Furthermore, the matrix n/2 can be written as The resulting output-feedback dynamic control law with ICc in (49) and K Oin (50) is = Aczc(t) R”’” where .X E R”’“ is defined as by solving the Riccati equation in (21) with Q o = 0 and a tuning parameter i = 0.2, and by choosing another tuning parameter yo = 0.431 in (22). ic(t) M (uTu~ by solving the Riccati equation in (17) with Q e = 0 and a tuning parameter i = 0.2, and by choosing another tuning parameter ye = 0.431 in (18). Then, a H, observer gain is determined as K , = [ -0.7473 -1.3197 -0.2692 - 0 . 7 6 0 7 I T , Let be any real matrix. Then there exist unitary matrices V n = [ u l ,u 2 , .. . , un] E Rnxn = 6i,j) and V,, = [ V I , v2, . . .,urn] E Rmxm( v T v j = 6 i , j ) such that 0 1 -1.4106 0 hf = h&,I@; with Mu= ukz’;’’ and hIw = ~ r k x i ’ z , (’42) where uk E Rnxk,x in Lemma AI. k Itkxk, and v k E Rrnxk,are defined as 0.2692 0.7607]7 C , = [ - 1.3172 -0.0091 -1.6808 -1.4106]. Acknowledgments This work was supported in part by the U.S. Army Research Office, under contract DAAL-03-91-G0106, and NASA-Johnson Space Center, under grants NAG 9-380 and NAG 9-385. The dynamic control law in (51) indeed stabilizes the nominal system in (28) and provides a If,-norm bound 6 = 2.325, sinre the associated Riccati equation in (25) has a SPD solution for Q = I and a tuning parameter (“ = 0.01. The poles and zeros of the above control law are -1.2811 f j 0 . 8 6 3 8 , -0.2191 k j l . i 9 9 3 , and -0.2303, 0.0052fj1.7144, respectively. Hence, as the result of the introduced tuning parameters in the augmented Rirrati equations, the solution space of the proposed method has been significantly increased. References ‘Petersen, I. R. and Hollot, C. V., “ A Riccati Equation Approach to the Stabilization of Uncertain Linear Systems,” Automatica, Vol. 22, No. 4, 1986, pp. 397-411. ‘Petersen, I. R., “A Stabilization Algorithm for a Class of Uncertain Linear Systems,” Systems and Control Letters, Vol. 8, NO. 4 , 1987, pp. 351-357. 3Schmitendorf, W. E., “A Design Methodology for Robust Stabilizing Controllers,” AIAA Journal of Guidance, Control and Dynamics, Vol. 10, No. 3, May-June 1987, pp. 250-254. 4Jabbari, F. and Schmitendorf, W. E., “A Noniterative Method for Design of Linear Robust Controllers,” Proc. o f 28th IEEE CDC, Tampa, Florida, December 1989, pp. 1690-1692. ‘Khargonekar, P. P., Petersen, I. R., and Zhou, K., “Robust stabilization of uncertain linear systems: Quadratic stabilizability and H, control theory,” IEEE Transactions on Automatic Control, Vol. 35, No. 3, 1990, pp. 356-361. ‘Wang, 1’. J., Shieh, L. S., and Sunkel, J. W., “A linear quadratic regulator approach to the stabilization of uncertain linear systems,” Proc. of A l A A GN&C Conference, Portland, Oregon, August 1990, pp. 1742-1749. VI. Conclusion Based on the algebraic Riccati equation approach and Lvapunov stability theory, a new observer-based robust-Hi, outputfeedback control law has been developed for both robust stabilization and disturbance attenuation with H,-norm bound for a uncertain linear system. These obsever-based disturbanreattenuation robust-stabilizing control laws can be easily ronstructed from the symmetric positive-definite solution of a pair of augmented Riccati equations. A simple dual concept has been utilized for finding the robust-Hi, state-feedback gain If, and the robust-IT, observer gain K O . Also, the proposed approach is more flexible than some existing methods in the sense that additional tuning parameters (such as E , E, and <,etr.) have Iwen introduced in the derivations to achieve robust staldization and disturbance attenuation for uncertain linear svsterns. A beiirli 750 L Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2688 7Petersen, I. R., “Disturbance attenuation and H , optimization: A design method based on the algebraic Riccati equation,” IEEE Transactions on Automatic Control, Vol. 32, No. 5, 1987, pp. 427-429. ‘Khargonekar, P. P., Petersen, I. R., and Rotea, M. A., “H,-optimal control with state-feedback,” IEEE Transactions on Automatic Control, Vol. 33, No. 8, 1988, pp. 786-788. ‘Glover, K. and Doyle, J.C., “State-space formulae for all stabilizing controllers that satisfy an H,-norm bound and relations to risk sensitivity,” Systems and Control Letters, Vol. 11, NO. 3, 1988, pp. 167-172. “Bernstein, D. S. and Haddad, W. M., “LQG control with a n H, performance bound: A Riccati equation approach,” IEEE Transactions on Automatic Control, Vol. 34, No. 3, 1989, pp. 293-305. “Doyle, J. C., Glover, K., Khargonekar, P. P., and Francis, B. A., “State-space solutions to standard H2 and H, control problems,” IEEE Transactions on Automatic Control, Vol. 34, NO. 8, 1989, pp. 831-847. ”Scherer, C., “H,-control by state-feedback and fast algorithms for the computation of optimal H,-norms,” IEEE Transactions on Automatic Control, Vol. 35, No. 10, 1990, pp. 10901099. I3Madiwale, A. N., Haddad, W. M., and Bernstein, D. S., “Robust H, control design for systems with structured parameter uncertainty,” Systems and Control Letters, Vol. 12, No. 5, 1989, pp, 393-407. 751 I4Veillette, R. J., Medanic, J. V., and Perkins, W. R., “RObust stabilization and disturbance rejection for systems with structured uncertainty,” Proc. of 28th IEEE CDC, Tampa, Florida, December 1989, pp. 936-941. 16Byun K. W., Wie, B., and Sunkel, J. W., “Robust control synthesis for uncertain dynamical systems,” Proc. AIAA GN&C Conference, Boston, Massachusetts, August 1989, pp. 792-801. “Wie, B. and Bernstein, D. S., “A benchmark problem for robust control design,” Proc. ACC, San Diego, California, May 1990, pp. 961-962. 17Collins, E. G. and Bernstein, D. S., “Robust control design for a benchmark problem using a structured covariance approach,” Proc. ACC, San Diego, California, May 1990, pp. 970971. 18Rhee, I. and Speyer, J. L., “Application of a game theoretic controller to a benchmark problem,’’ Proc. ACC, San Diego, California, May 1990, pp. 972-973. ”Anderson, B. D. 0. and Moore, J. B., Linear Optimal Control, Prentice-Hall, Englewood Cliffs, New Jersey, 1990. 20Skelton, R. E., Dynamic Systems Control, John Wile? & Sons, New York, 1988.

1/--страниц