MATLAB_LMI工具箱使用教程算例及论文原文

【实例简介】
MATLAB_LMI工具箱使用教程算例及论文原文
运行上述程序后,在命令窗口 ()输入: 得出: ()输入: ()输入 得出 输入: 得出 (4)输入: 得出: 输入: 得出: ()输入 得出 ()输入 得出 与论文中结果保持一致! PERGAMON Automatica35(1999)1443-1451 www.elsevier.com/locate/automatica Technical Communique H state feedback control for generalized continuous/discretc timc-dclay system Jong hae Kim", *, Hon g Bae park Sensor Technology Research Center, Kyungpook National University, Taegu, 702-701, South Korea School of Electronic and Electrical Engineering, Kyungpook National University, Taegu, 702-701, South Korea Received 5 February 1998; revised 20 July 1998; received in final form 3 February 1999 Abstract In this paper, we consider the problem of designing h state feed back 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 solvcd on the basis of lincar matrix inequality(LMi) tcchniquc considcring time delays. The sufficient condition for the cxistcncc of controller and H state feedback controller design methods are presented. Also, using some changes of variables and Schur complements, the obtained sufficient 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. C 1999 Elsevier Science Ltd. All rights reserved Keywords: H control; State feedback; Delayed system; Linear matrix inequality 1. Introduction the state delayed system. And the work proposed by Choi and Chung(1995) was extended to the problem of Since the time delay is frequently a source of instability memoryless H controller design for linear systems with and encountered in various engineering systems such as delayed state and control using the Riccati equation chemical proccsscs, long transmission lincs in pncumatic approach. But not only thcir works (Lcc ct al, 1994; Choi systems, etc, the study of time-delay systems has received and Chung, 1995, 1996) but also other results were con considerable attention over the past years. Because some servative in pre-determination of some starting values works of analytic H controller design method(see e.g. determined whether there exists a positive-definite solu Doyle et al., 1989; Gahinet, 1996) and software toolbox tion, and were not considered delayed state and control Gahinet et al., 1995) have been developed, many state input in the controlled signal output. Also Niculescu feed back controller design methods of time-delay sys-(1995) presented H memoryless control with an a-stab tems were presented(Shen et al., 1991; Lee et al., 1994; ility constraint for time-delays systems using the linear Mahmoud and Al-Muthairi, 1994; Choi and Chung, matrix inequality (Lmi approach However, the work 1995, 1996; Kim et al., 1996; Ge et al, 1996; Yu et al., did not consider time-varying delay in states and control 1996). Lee et al. (1994) presented a memoryless H con- inputs For a linear system with time-varying delay in all troller design which is a delay-independent stabilizer for 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 Corresponding author. Tel. +82-53-940-8848: fax: +82-53 (1996) proposed robust controller design method for 950-6827; e-mail: kimjh(@strc kyungpook ackI uncertain systems with time delays using the LMi was recommended for publication in revised form by Editor Peter approach. But the work did not trcat disturbance at- Dorato tenuation H problem. Also some starting variables 0005-1098799 S-see front matter C) 1999 Elsevier Science Ltd. All rights reserved PII:S0005-1098(99)00038-2 1444 J.H. Kim, H.B. Park/ Automatica 35(1999)1443-1457 are pre-selected in order to obtain positive-definite 2. Continuous time controller design solutions. this is restrictive in terms of the existence of positive-definite solution. Therefore, this paper presents Consider a continuous time linear system with time controller design methods without choice of some vari varying delays a bles and considcrs the robust H control problem of parameter uncertain time-delay systems in Corollaries i(t)=Ax(t)+ Aax(t-di(t)+ Biw(t)+ B2u(t) 1 and 2 The first aim of this paper is to find solutions at a time Bau(t-d2(t) without the pre-selection of some variables using LMI technique. Recently, many works(Xie and souza, 1992: z(t)=Cx(t)+ Cax(t-d,(t))+ d1w(t)+ D2u(t) Garcia et al, 1994; Yuan et al., 1996) related robust problem or robust H problem against parameter uncer- Dau(t -d2(t) (1) tainties were presented. Also, robust control problem x(t)=0,t<0,x(0)=x with time-delay( Choi and Chung, 1996; Kim et al., 1996; Mahmoud and Al-Muthairi. 1994: Shen et al.. 1991 where x(t er is the state, u(t)Er is the control input, Jeung et al. 1996), H control problem with time-delay w tEr is the disturbance input. which belongs to (Lcc ct al., 1994; Choi and Chung, 1995; Niculescu, L [o, oo), and z(ERP is the controlled signal output 1995), and robust H control problem with time delay And we assume that all states are measurable. In here (Yuet al., 1996; Ge et al., 1996)were proposed. However, time-varying delays are satisfied with many related works treated controller design method in continuous time case only. Therefore, it is important 0≤d1(t)<∞,d1(1)≤B;1<1.i=1,2. to deal with a controller design method in discrete time case because most of engineering systems are As an H controller of the time-delay system(1),we controlled by digital computcr. Garcia ct al.(1994) propose a continuous timc statc fccdback law presented a robust stabilization of discrete time linear systems with norm-bounded time-varying uncertainty, u(t)=Kx(t and Yuan et al.(1996) proposed a robust H control for linear discrete time systems with norm-bounded When wc apply control (3)to thc timc-dclay systcm(1), time-varying uncertainty but they did not consider time the closed-loop system from w(t) to z(t) is given by delay. Therefore, the second objective of this paper is to present an H state feedback controller design xi(t)=Akx(t)+dx(t-d(t)+bw(t) method of discrete time-delay systems. Also, the prop eller design method can be extended into +BKx(t-d2(1), the problem of robust H state feedback controller z()=crx(t)+ Cdx(t-d,(t))+ duw(t) design method for parameter uncertain time-delay sys tcms through somc manipulations using thc cxisting dakx(t-d2(t), results(Xie and Souza, 1992, Gu, 1994; Garcia et aL, 1994; Kokame et al., 1995; Yuan et al., 1996). To find where, Ak=A+B2K and CK=C+D,2K an h state feedback controller we consider the bounded real lemma for the closed-loop system as LMI Lemma 1. For a given constant >0, system(1) is quad problems rulically stable with un H norm bound y by che controller In this paper, we propose H state feedback controller (3) if there exist positive-definite matrices P,RI, and design methods of the generalized time-delay systems R, such that in continuous time and discrete time cases, respectivel The existence conditions and the design methods of state feedback H controllers are given. Through AkP+PAk+R+KR2k Pad pBd PB ck some changes of variables and Schur complements, AlP R 00 CI the obtaincd sufficient condition is changed into an LMi form in terms of each finding variable. The BaP 0 R 2 state feedback controller can be easily obtained BiP I D using LMI Toolbox (Gahinet et al, 1995) because the transformed suficient condition is an lmi form in terms of variables. The state feed back I o control- ler guarantees not only the quadratic stability of the closcd-loop systcm but also the H norm bound holds for time delays(2 ). In here, r =(1-B)R;i=1,2, within a 2. are positive-definite matrices H. Kim. H B. Park/Au Proof. Firstly, we define a Lyapunov functional as which ensures the quadratic stability of the closed-loop system (4). In the next pl ume the zero initial V(x():=x(n)Px()+ condition and let us introd x()rx()da [z()2z()-y2w(1)w(O x(a'kr2kx(t)d t-d2(t hen for any nonzero n()∈L2[0.∞ And it is noticed that condition (5) implies AkP+ PAk+rI+kRak PAd PBa Is [z(t)z(t-w(t) w()+v(x(t))]dt AdP R10 BiP 0R2 ≤|[z(()-y2w()yw()+(x()dr (12) and further substituting Eq. 9) into Eq(12)and let Taking the derivative of the Lyapunov functional (6) 9(t=[x(t) x(t-d,(t))x(t-d2(t))'k w(t)], then along the solution of Eg. (4) yields J s 5(Zs(t)dt, (13) V(x(t))=x(t)TPx(t)+x(t)Px(t) +x(o'r1x(0+x(0KR2Kx(O where is defined (1-d1(t)x(t-d1(t)R1x(t-a1(1) PAd+ckcd AdP+Cdc (1-B1)R1 (1-a2(1)x(t-d2()KR2kx(t-d2() BiP+DaCk BIP+ DI, CK Di, Ca which is negative-definite when the matrix(Kreinder and PBa+crda PBI+CKDI1 Jameson, 1972) CID CdDl (14) (x(r)=刘(n)Px()+x(1)PX() DD4-(1-B2)R2 DIDI +x(t)Rx(t)+x(t)'KR2kx(t D 221+DM,D11 where H=AkP+ PAk+CkCk+R,+KR2K.This (1-阝1)x(t-d1(t)R1x(t-d1(t) Z<Oin Eg (14) implies 2(t)2<7w(t)l2 for any non (1-B2)x(n-d2()KR2Kx(-d2() w(t)∈L2[0,∞). Thcrcforc, when Z<0,t≥0,thc system (1)is quadratically stable with an Hnorm (9) bound y by controller (3). Using Schur complements Boyd et al, 1994), Z <O in Eg(14)is transformed into When assuming the zero input, we have Eq.(5).口 Theorem 1. Consider the continuous time-delay system(1) For a given positive constant if there exist positive x(t=d,(t) definite matrices @, S1, S2, and a matrix M such that Kx(t-d(t) AkP+ PAK+Ri+KR2k PAa PBa B1-?2ID10 Adp R U/2D11U30 (15) BiP R M O x(t) x(t-d()<0 (10) holds for the time delays(2), then eq (1)is quadratical Kx(t-d2(t)) stable with an H norm bound y by controller (3). In here, 144 J.H. Kim, H.B. Park/ Automatica 35(1999)1443-1457 some terms are defined as follows AlP +Par tPArIATP PB, PB U1=2A+AQ+MB2+ B2M+(1-B)AdS,ad R 1-P2)B4S U2=M Di2+oc +(1-bi Ads, cd +(1-B2)-B4S2D +(1-f1)-1CaS1Ca+(1-B2)1D4S Ck+PAGRiICd K M=KP-l 11 Q=P 0(19) I+ CaRi R 1.2. (16) R Proof. Using Schur complements and some changes of variables, the proof is completed. The inequality of Eq ( 5) Is equivalent to AkP+PAk PAGRIIAdP+ PBaR2 BIP PB AkP+ PAK+ri PAd pB R R CK+PARi Cd+ PBGR2 Da PB CK K I+ Cari Cd+ dr2 da 0 R R (20) * R K+AkP-1+ ARIAd +BR2lBa B AkP+ PAk PAl PB PB R10 半 R 半 x+AR1CⅡ+BR21D}P-1k D 11 Ⅰ+CR DaRdA KT R 0 0 R Da 0 0(18) where,* mean symmetric terms. Using some changes of R 0 KP,0=P, and Si=R -R Fg(21)is changed to Fq(15). J.H. Kim, H.B. Park/ Automatica 35(1999)1443-1457 1447 Ey . (15)is an LMI form in terms of 0, M, S, and s In here, w(t), z(t), and n are the additional disturbance Therefore. the continuous time h state feedback con input, the additional controlled signal output, and posit- troller K can be calculated from the M= KP-1 after ive real number Unknown matrix is defined as finding the LMI solutions, 2, M,S1, and S2, from Eq (15). USing LMI Toolbox(Gahinct ct al., 1995), thc F(t)∈9:={F(1:F(t)F()≤l, solutions can be easily obtained at a time because the elements of F(t) Eq(15)is an LMI form in terms of variables arc Lcbcsguc measurable Therefore, the robust h state feed back controller design Corollary 1. For the same controller, the generalized con- problem for parameter uncertain delay system can be tinuous parameter uncertain system with time-varying solved using the proposed method delay in states and control inputs Example 1. Consider a generalized continuous time x(t)=「A+△A(t)x(t)+「A+△A(t)x(t-d1(t) delay system +[B4+△B(m)u(t+[Ba+△Bat)]u(t-d2(t) 0.20.1 x(t)+ x(t-di(t) 00.1 +[B+△B,(O)]w(), 0.1 0.1 )+||u(l)+ (-l2() z(t)=[cz+aci(t)x(t)+ [Cid+ aczdtjx(t-di(t) 0.1 0.1 +[Deu+ denju(t) 1x()+[0.10.1]x(t-d1(t) lDed+ adzdtju(t-d,(t) +(1)+0.1u(t-d2(1) γ=1,d1(t)=2+0.2cost,d2(t)=5+0.2sin(3t) +[D2m+△D2(t)]w(t) From the solutions satisfying Eq. (15) and changes of △A(t)△B(t)△B,(t)△Aat)△Bat) variables(16), all solutions are obtained at a time as follows. ACi(t) AD2 t) ADaw(t) ACedt) ADed(t) 25.8838 8.7399 F(t[er Eu ew edx eau] 8.73994.2486 0.2676 0.0082 can be transformed into the system without the parameter R 000820.2200 uncertainties M=[-1.1255-1.4670] i(t)= Ax(t)+Agx(t-d,(t)) R 2 0.2177 Buu(t)+ bu(t-d(t))+[Bw HI Therefore, the continuous time state feedback gain is wIt K=[-1631143.60441 z() The obtained controller guarantees the stability for time x(t)+ x(t-d,(D) 2t) varying delays and satisfies II norm bound of the E E closed-loop system D p()+ u(t-d2() 3. Discrete time controller design Consider a discrete time linear system with time delays D x(k+ 1)=Ax(k)+ Adx(k-d1) E (t) +B,w(k)+b2u(k)+ Bau( -d z(k)=Cx(k)+ Cax(k-d1)+Diw(k) under preserving quadratic stability and H norm bound through some manipulations using the existing results(see e.g +D12(k)+Dal(k-d2), (26) Xie and Souza, 1992; Gu, 1994, Kokame et aL., 1995). x(k)=0,k<o, x(O) 【实例截图】
【核心代码】