MATLAB LMI工具箱使用教程算例及论文原文

LMI工具箱使用教程算例工具箱使用教程算例 算 例 是 论 文 ： H state feedback control for generalized continuous/discrete time-delay system 中第三部分里面的一个例子。论 文原文附在后面。 算例程序如下： A = 2 1; 0 1; Ad = 0.2 0.1; 0 0.1; B1 = 0.1 0.1; B2 = 1 1; Bd = 0.1 0.1; C = 1, 1; Cd = 0.1, 0.1; D11 = 0.1; D12 = 1; Dd = 0.1; gammar = 1; % Initial a LMI system setlmis(); % Define Variables % Q is a symmetric matrix, has a block size of 2 and this block is symmetric Q = lmivar(1, 2 1); % S1 a symmeric matrix, size 2 S1 = lmivar(1, 2 1); % S2 is 1 by 1 matrix S2 = lmivar(1, 1 0); % Type of 2, size 1 by 2 M = lmivar(2, 1 2); % Q, S1, S2 0 QN = newlmi; lmiterm(-QN 1 1 Q, 1, 1); S1N = newlmi; lmiterm(-S1N 1 1 S1, 1, 1); S2N = newlmi; lmiterm(-S2N 1 1 S2, 1, 1); MAllN = newlmi; % pos in (1, 1) lmiterm(MAllN 1 1 Q, -1, 1); lmiterm(MAllN 1 1 S2, Bd, Bd); lmiterm(MAllN 1 1 S1, Ad, Ad); % pos (1, 2) lmiterm(MAllN 1 2 Q, A, 1); lmiterm(MAllN 1 2 M, B2, 1); % pos(1, 3) lmiterm(MAllN 1 3 0, B1); % pos(1, 4) lmiterm(MAllN 1 4 S2, Bd, Dd); lmiterm(MAllN 1 4 S1, Ad, Cd); % pos(2, 2) lmiterm(MAllN 2 2 Q, -1, 1); % pos(2, 4) lmiterm(MAllN 2 4 Q, 1, C); lmiterm(MAllN 2 4 -M, 1, D12); % pos(2, 5) lmiterm(MAllN 2 5 -M, 1, 1); % pos(2, 6) lmiterm(MAllN 2 6 Q, 1, 1); % pos(3, 3) lmiterm(MAllN 3 3 0, -(gammar2); % pos(3, 4) lmiterm(MAllN 3 4 0, D11); % pos(4, 4) lmiterm(MAllN 4 4 0, -1); lmiterm(MAllN 4 4 S1, Cd, Cd); lmiterm(MAllN 4 4 S2, Dd, Dd); lmiterm(MAllN 5 5 S2, -1, 1); lmiterm(MAllN 6 6 S1, -1, 1); lmis = getlmis; tmin, feas = feasp(lmis) 运行上述程序后，在命令窗口 （1）输入：Q = dec2mat(lmis, feas, Q) 得出： Q = 1.9253 -2.2338 -2.2338 9.1054 （2）输入：P = Q-1 P = 0.7261 0.1781 0.1781 0.1535 （3）输入：S1 = dec2mat(lmis, feas, S1) 得出： S1 = 16.0928 -14.9665 -14.9665 30.3214 输入：R1=S1-1 得出： R1 = 0.1149 0.0567 0.0567 0.0610 （）输入：S2 = dec2mat(lmis, feas, S2) 得出： S2 = 16.0495 输入：R2=S2-1 得出： R2 = 0.0623 （5）输入：M = dec2mat(lmis, feas, M) 得出： M = -0.2279 -6.2240 （6）输入：K=M*P 得出： K = -1.2741 -0.9961 与论文中结果保持一致！ Automatica 35 (1999) 14431451 Technical Communique H= state feedback control for generalized continuous/discrete time-delay system1 Jong Hae Kima,*, Hong Bae Parkb aSensor Technology Research Center, Kyungpook National University, Taegu, 702-701, South Korea bSchool of Electronic and Electrical Engineering, Kyungpook National University, Taegu, 702-701, South Korea Received 5 February 1998; revised 20 July 1998; received in nal form 3 February 1999 Abstract In this paper, we consider the problem of designing H= state feedback controller for the generalized time-delay systems with delayed states and control inputs in continuous and discrete time cases, respectively. The generalized time-delay system problems are solved on the basis of linear matrix inequality (LMI) technique considering time delays. The su$cient condition for the existence of controller and H= state feedback controller design methods are presented. Also, using some changes of variables and Schur complements, the obtained su$cient condition can be rewritten as an LMI form in terms of transformed variables. The proposed controller design method can be extended into the problem of robust H= state feedback controller design method easily. ( 1999 Elsevier Science Ltd. All rights reserved. Keywords: H= control; State feedback; Delayed system; Linear matrix inequality 1. Introduction Since the time delay is frequently a source of instability and encountered in various engineering systems such as chemical processes, long transmission lines in pneumatic systems, etc., the study of time-delay systems has received considerable attention over the past years. Because some works of analytic H= controller design method (see e.g. Doyle et al., 1989; Gahinet, 1996) and software toolbox (Gahinet et al., 1995) have been developed, many state feedback controller design methods of time-delay sys- tems were presented (Shen et al., 1991; Lee et al., 1994; Mahmoud and Al-Muthairi, 1994; Choi and Chung, 1995, 1996; Kim et al., 1996; Ge et al., 1996; Yu et al., 1996). Lee et al. (1994) presented a memoryless H= con- troller design which is a delay-independent stabilizer for * * Corresponding author. Tel.: #82-53-940-8848; fax: #82-53- 950-6827; e-mail: kimjhstrc.kyungpook.ac.kr. 1 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Editor Peter Dorato. the state delayed system. And the work proposed by Choi and Chung (1995) was extended to the problem of memoryless H= controller design for linear systems with delayed state and control using the Riccati equation approach.But not onlytheir works (Lee et al., 1994; Choi and Chung, 1995, 1996) but also other results were con- servative in pre-determination of some starting values determined whether there exists a positive-denite solu- tion, and were not considered delayed state and control input in the controlled signal output. Also Niculescu (1995) presented H= memoryless control with an a-stab- ility constraint for time-delays systems using the linear matrix inequality (LMI) approach. However, the work did not consider time-varying delay in states and control inputs. For a linear system with time-varying delay in all states and control inputs, it is more complicated to ob- tain the controller. Also Niculescu (1995) did not con- sider the controller design method for discrete time-delay system. Therefore, our results deal with controller design methods of generalized time-delay systems in continuous time case and discrete time case, respectively. Jeung et al. (1996) proposed robust controller design method for uncertain systems with time delays using the LMI approach. But the work did not treat disturbance at- tenuation H= problem. Also some starting variables 0005-1098/99/$-see front matter ( 1999 Elsevier Science Ltd. All rights reserved PII: S0 0 05 -1 0 9 8 (9 9 ) 0 0 0 38 - 2 are pre-selected in order to obtain positive-denite solutions. This is restrictive in terms of the existence of positive-denite solution. Therefore, this paper presents controller design methods without choice of some vari- ables and considers the robust H= control problem of parameter uncertain time-delay systems in Corollaries 1 and 2. The rst aim of this paper is to nd solutions at a time without the pre-selection of some variables using LMI technique. Recently, many works (Xie and Souza, 1992; Garcia et al., 1994; Yuan et al., 1996) related robust problem or robust H= problem against parameter uncer- tainties were presented. Also, robust control problem with time-delay (Choi and Chung, 1996; Kim et al., 1996; Mahmoud and Al-Muthairi, 1994; Shen et al., 1991; Jeung et al., 1996), H= control problem with time-delay (Lee et al., 1994; Choi and Chung, 1995; Niculescu, 1995), and robust H= control problem with time delay (Yu et al., 1996; Ge et al., 1996) were proposed. However, many related works treated controller design method in continuous time case only. Therefore, it is important to deal with a controller design method in discrete time case because most of engineering systems are controlled by digital computer. Garcia et al. (1994) presented a robust stabilization of discrete time linear systems with norm-bounded time-varying uncertainty, and Yuan et al. (1996) proposed a robust H= control for linear discrete time systems with norm-bounded time-varyinguncertainty. But they did not consider time- delay. Therefore, the second objective of this paper is to present an H= state feedback controller design methodofdiscretetime-delaysystems.Also,the proposed controller design method can be extended into the problem of robust H= state feedback controller design method for parameter uncertain time-delay sys- tems through some manipulations using the existing results (Xie and Souza, 1992; Gu, 1994; Garcia et al., 1994; Kokame et al., 1995; Yuan et al., 1996). To nd an H= state feedback controller, we consider the bounded real lemma for the closed-loop system as LMI problems. In this paper, we propose H= state feedback controller design methods of the generalized time-delay systems in continuous time and discrete time cases, respectively. The existence conditions and the design methods of state feedback H= controllers are given. Through some changes of variables and Schur complements, theobtainedsu$cientconditionischangedinto an LMI form in terms of each nding variable. The H= state feedback controller can be easily obtained using LMI Toolbox (Gahinet et al., 1995) because the transformed su$cient condition is an LMI form in terms of variables. The state feedback H= control- ler guarantees not only the quadratic stability of the closed-loop system but also the H= norm bound within a c. 2. Continuous time controller design Consider a continuous time linear system with time- varying delays x R (t)Ax(t)#Adx(t!d1(t)#B1w(t)#B2u(t) #Bdu(t!d2(t), z(t)Cx(t)#Cdx(t!d1(t)#D11w(t)#D12u(t) #Ddu(t!d2(t),(1) x(t)0,t(0,x(0)x0, where x(t)3Rn is the state, u(t)3Rm is the control input, w(t)3Rl is the disturbance input, which belongs to 20, R), and z(t)3Rp is the controlled signal output. And we assume that all states are measurable. In here, time-varying delays are satised with 04di(t)(R, d Q i(t)4bi(1, i1, 2.(2) As an H= controller of the time-delay system (1), we propose a continuous time state feedback law u(t)Kx(t).(3) When we apply control (3) to the time-delay system (1), the closed-loop system from w(t) to z(t) is given by x R (t)AKx(t)#Adx(t!d1(t)#B1w(t) #BdKx(t!d2(t), (4) z(t)CKx(t)#Cdx(t!d1(t)#D11w(t) #DdKx(t!d2(t), where, AKA#B2K and CKC#D12K. Lemma 1. For a given constant c0, system (1) is quad- ratically stable with an H= norm bound c by the controller (3) if there exist positive-de,nite matrices P, R1, and R2such that AT KP#PAK#R1#KTR2K PAd PBdPB1CT K AT dP !R I 1 00CT d BT dP 0!R I 2 0DT d BT 1P 00!c2I DT 11 CKCdDdD11!I (0(5) holds for time delays (2). In here, R I i(1!bi)Ri, i1, 2, are positive-de,nite matrices. 1444J.H. Kim, H.B. Park/Automatica 35 (1999) 14431451 Proof. Firstly, we dene a Lyapunov functional as (x(t): x(t)TPx(t)#P t td1(t) x(q)TR1x(q)dq #P t td2(t) x(q)TKTR2Kx(q)dq.(6) And it is noticed that condition (5) implies AT KP#PAK#R1#KTR2K PAd PBd AT dP !R I 1 0 BT dP 0!R I 2 (0. (7) Taking the derivative of the Lyapunov functional (6) along the solution of Eq. (4) yields Q (x(t)x R (t)TPx(t)#x(t)TPx R (t) #x(t)TR1x(t)#x(t)TKTR2Kx(t) !(1!d R 1(t)x(t!d1(t)TR1x(t!d1(t) !(1!d R 2(t)x(t!d2(t)TKTR2Kx(t!d2(t), (8) which is negative-denite when the matrix (Kreinder and Jameson, 1972) Q a(x(t)x R (t)TPx(t)#x(t)TPx R (t) #x(t)TR1x(t)#x(t)TKTR2Kx(t) !(1!b1)x(t!d1(t)TR1x(t!d1(t) !(1!b2)x(t!d2(t)TKTR2Kx(t!d2(t) (0.(9) When assuming the zero input, we have x(t) x(t!d1(t) Kx(t!d2(t) T AT KP#PAK#R1#KTR2K PAdPBd AT dP !R I 1 0 BT dP 0!R I 2 x(t) x(t!d1(t) Kx(t!d2(t) (0,(10) which ensures the quadratic stability of the closed-loop system (4). In the next place, assume the zero initial condition and let us introduce JP = 0 z(t)Tz(t)!c2w(t)Tw(t)dt.(11) Then for any nonzero w(t)320, R), J4P = 0 z(t)Tz(t)!c2w(t)Tw(t)# Q (x(t)dt 4P = 0 z(t)Tz(t)!c2w(t)Tw(t)# Q a(x(t)dt (12) and further substituting Eq. (9) into Eq. (12) and let f(t)x(t)T x(t!d1(t)T x(t!d2(t)TKT w(t)TT, then J4P = 0 f (t)TZf (t)dt,(13) where Z is dened Z HPAd#CT KCd AT dP#CTdCK CT dCd!(1!b1)R1 BT dP#DTdCK DT dCd BT 1P#DT11CK DT 11Cd PBd#CT KDd PB1#CT KD11 CT dDd CT dD11 DT dDd!(1!b2)R2 DT dD11 DT 11Dd !c2I#DT 11D11 ,(14) where HAT KP#PAK#CTKCK#R1#KTR2K. This Z(0 in Eq. (14) implies Ez(t)E24cEw(t)E2for any non- zero w(t)320, R). Therefore, when Z(0, t50, the system (1) is quadratically stable with an H= norm bound c by controller (3). Using Schur complements (Boyd et al., 1994), Z(0 in Eq. (14) is transformed into Eq. (5).K Theorem1. Consider the continuous time-delaysystem (1). For a given positive constant c, if there exist positive- de,nite matrices Q, S1, S2, and a matrix M such that ;1B1;2MTQ BT 1 !c2IDT 11 00 ;T 2 D11;300 M00!S20 Q000!S1 (0(15) holds for the time delays (2), then Eq. (1) is quadratically stable with an H= norm bound c by controller (3). In here, J.H. Kim, H.B. Park/Automatica 35 (1999) 144314511445 some terms are de,ned as follows: ;1QAT#AQ#MTBT 2#B2M#(1!b1)1AdS1ATd #(1!b2)1BdS2BT d , ;2MTDT 12#QCT#(1!b1)1AdS1CTd #(1!b2)1BdS2DT d, ;3!I#(1!b1)1CdS1CT d#(1!b2)1DdS2DTd, MKP1, QP1, SiR1 i ,i1, 2.(16) Proof. Using Schur complements and some changes of variables,theproof is completed.The inequalityof Eq.(5) is equivalent to AT KP#PAK#R1 PAdPBd *!R I 1 0 *!R I 2 * * * PB1CT K KT 0CT d 0 0DT d 0 !c2IDT 11 0 *!I0 *!R1 2 (0(17) 8 AT KP#PAK PAdPBdPB1 *!R I 1 00 *!R I 2 0 *!c2I * * * CT K KTI CT d 00 DT d 00 DT 11 00 !I00 *!R1 2 0 *!R1 1 (0(18) 8 AT KP#PAK#PAdR I 11 AT dP PBdPB1 *!R I 2 0 *!c2I * * * CT K#PAdR I 11 CT d KT I DT d 00 DT 11 00 !I#CdR I 1 1 CT d 00 *!R1 2 0 *!R1 1 (0(19) 8 AT KP#PAK#PAdR I 11 AT dP#PBdR I 12 BT dP PB1 *! c2I * * * CT K#PAdR I 11 CT d#PBdR I 12 DT d KTI DT 11 00 ! I#CdR I 1 1 CT d#DdR I 12 DT d 00 *! R1 2 0 *! R1 1 (0(20) 8 P1AT K#AKP1#AdR I 11 AT d#BdR I 12 BT d B1 *!c2I * * * P1CT K#AdR I 11 CT d#BdR I 12 DT d P1KTP1 DT 11 00 !I#CdR I 1 1 CT d#DdR I 12 DT d 00 *!R1 2 0 *!R1 1 (0,(21) where,*mean symmetric terms. Using some changes of variables, MKP1, QP1, and SiR1 i , i1, 2, Eq. (21) is changed to Eq. (15).K 1446J.H. Kim, H.B. Park/Automatica 35 (1999) 14431451 Eq. (15) is an LMI form in terms of Q, M, S1, and S2. Therefore, the continuous time H= state feedback con- troller K can be calculated from the MKP1 after nding the LMI solutions, Q, M, S1, and S2, from Eq. (15). Using LMI Toolbox (Gahinet et al., 1995), the solutions can be easily obtained at a time because Eq. (15) is an LMI form in terms of variables. Corollary 1. For the same controller, the generalized con- tinuous parameter uncertain system with time-varying delay in states and control inputs x R (t)A#*A(t)x(t)#Ad#*Ad(t)x(t!d1(t) #Bu#*Bu(t)u(t)#Bd#*Bd(t)u(t!d2(t) #Bw#*Bw(t)w(t), z(t)Cz#*Cz(t)x(t)#Czd#*Czd(t)x(t!d1(t) #Dzu#*Dzu(t)u(t) #Dzd#*Dzd(t)u(t!d2(t) #Dzw#*Dzw(t)w(t),(22) C *A(t)*Bu(t)*Bw(t)*Ad(t)*Bd(t) *Cz(t)*Dzu(t)*Dzw(t)*Czd(t)*Dzd(t)D CHx HzD F(t) ExEuEwEdxEdu(23) can be transformed into the system without the parameter uncertainties x R (t)Ax(t)#Adx(t!d1(t) #Buu(t)#Bdu(t!d2(t)#BwcjHxC w(t) w L (t)D , C z(t) z (t)DC Cz 1 j ExD x(t)#C Czd 1 j EdxD x(t!d1(t) #C Dzu 1 j EuD u(t)#C Dzd 1 j EduD u(t!d2(t) #C DzwcjHz 1 j Ew0 D C w(t) w (t)D (24) under preserving quadratic stability and H= norm bound through some manipulationsusing the existing results (see e.g. Xie and Souza, 1992; Gu, 1994; Kokame et al., 1995).K In here, w (t), z (t), and j are the additional disturbance input, the additional controlled signal output, and posit- ive real number. Unknown matrix is dened as F(t)3): MF(t):F(t)TF(t)4I, the elements of F(t) are Lebesgue measurableN.(25) Therefore,the robustH= state feedbackcontrollerdesign problem for parameter uncertain delay system can be solved using the proposed method. Example 1. Consider a generalized continuous time- delay system x R (t)C2 1 01D x(t)#C0.2 0.1 00.1D x(t!d1(t) #C 0.1 0.1D w(t)#C 1 1D u(t)#C 0.1 0.1D u(t!d2(t), z(t)11x(t)#0.10.1x(t!d1(t)#0.1w(t) #u(t)#0.1u(t!d2(t), c1,d1(t)2#0.2cost,d2(t)5#0.2sin(3t). From the solutions satisfying Eq. (15) and changes of variables (16), all solutions are obtained at a time as follows: PC 25.8838!8.7399 !8.73994.2486D , R1C 0.2676!0.0082 !0.00820.2200D , M!1.1255!1.4670, R20.2177. Therefore, the continuous time state feedback gain is K!16.31143.6044. The obtained controller guarantees the stability for time- varying delays and satises H= norm bound of the closed-loop system. 3. Discrete time controller design Consideradiscretetime linearsystem withtime delays x(k#1)Ax(k)#Adx(k!d1) #B1w(k)#B2u(k)#Bdu(k!d2), z(k)Cx(k)#Cdx(k!d1)#D11w(k) #D12u(k)#Ddu(k!d2),(26) x(k)0,k(0,x(0)x0, J.H. Kim, H.B. Park/Automatica 35 (1999) 144314511447 *k x(k) x(k!d1) Kx(k!d2) T AT KPAK!P#R1#KTR2K AT KPAd AT KPBd AT dPAK !R1#AT d PAdAT d PBd BT d PAKBT d PAd!R2#BT d PBd x(k) x(k!d1) Kx(k!d2) (0,(34) AT KPAK!P#R1#KTR2K AT KPAd AT KPBd AT dPAK !R1#AT d PAdAT d PBd BT d PAKBT d PAd!R2#BT d PBd (0.(32) where x(k)3Rn is the state, u(k)3Rm is the control input, w(k)3Rl is the disturbance input, that is an l2sequence each of whose component has norm less than one, and z(k)3Rp is the controlled signal output. And we assume that all states are measurable. In here, positive integer time delays are satised with 04di(R,i1, 2.(27) As an H= controller of time-delay system (26), we pro- pose a state feedback law u(k)Kx(k).(28) When we apply the control (28) to the time-delay system (26), the closed-loop system from w(k) to z(k) is given by x(k#1)AKx(k)#Adx(k!d1)#B1w(k) #BdKx(k!d2), z(k)CKx(k)#Cdx(k!d1)#D11w(k) #DdKx(k!d2)(29) where, AKA#B2K and CKC#D12K. Lemma 2. For a given c0, the system (26) is quadrati- cally stable with an H= norm bound c by the controller (28) if there exist positive-de,nite matrices P, R1, and R2such that ! P1AKAd BdB10 AT K !P#R1#KTR2K000CT K AT d 0!R100CT d BT d 00!R20DT d BT 1 000!c2I DT 11 0CKCdDdD11!I (0(30) holds for the time delays (27). In here, R1and R2are positive-de,nite matrices. Proof. Firstly, we dene a Lyapunov functional as (x(k):x(k)TPx(k)# k1 + i/kd1 x(i)TR1x(i) # k1 + i/kd2 x(i)TKTR2Kx(i).(31) And it is noticed that condition (30) implies Taking the di!erence of the Lyapunov functional (31) yields *k(x(k#1)!(x(k) x(k#1)TPx(k#1) !x(k)T(P!R1!KTR2K)x(k) !x(k!d1)TR1x(k!d1) !x(k!d2)TKTR2Kx(k!d2).(33) When