Robust Delay-Independent/Dependent Stabilization of Uncertain Time-Delay Systems by Variable Structure Control 187 ii = 1; setlmis([]) P =lmivar(1,[2 1]); R1=lmivar(1,[2 1]); R2=lmivar(1,[2 1]); lmiterm([-1 1 1 P],ii,ii) lmiterm([-2 1 1 R1],ii,ii) lmiterm([4 1 1 P],1,A0til','s') lmiterm([4 1 1 R1],ii,ii) lmiterm([4 2 2 R1],-ii,ii) lmiterm([4 1 2 P],1,A1hat) LMISYS=getlmis; [copt,xopt]=feasp(LMISYS); P=dec2mat(LMISYS,xopt,P); R1=dec2mat(LMISYS,xopt,R1); evlmi=evallmi(LMISYS,xopt); [lhs,rhs]=showlmi(evlmi,4); lhs P eigP=eig(P) R1 eigR1=eig(R1) eigsLHS=eig(lhs) BTP=B'*P BTPB=B'*P*B invBTPB=inv(B'*P*B) normG1 = norm(G1) A2 clear; clc; A0=[-1 0.7; 0.3 1]; A1=[-0.1 0.1; 0 0.2]; A2=[0.2 0; 0 0.1]; B=[1; 1] setlmis([]) P =lmivar(1,[2 1]); R1=lmivar(1,[2 1]); R2=lmivar(1,[2 1]); Geq=inv(B'*P*B)*B'*P A0hat=A0-B*G*A0 A1hat=A1-B*G*A1 A2hat=A2-B*G*A2 G= place(A0hat,B,[-4.2 6i -4.2+.6i]) A0til=A0hat-B*G1 Robust Control, Theory and Applications 188 eigA0til=eig(A0til) eigA0hat=eig(A0hat) eigA1hat=eig(A1hat) eigA2hat=eig(A2hat) ii = 1; lmiterm([-1 1 1 P],ii,ii) lmiterm([-2 1 1 R1],ii,ii) lmiterm([-3 1 1 R2],ii,ii) lmiterm([4 1 1 P],1,A0til','s') lmiterm([4 1 1 R1],ii,ii) lmiterm([4 1 1 R2],ii,ii) lmiterm([4 2 2 R1],-ii,ii) lmiterm([4 1 2 P],1,A1hat) lmiterm([4 1 3 P],1,A2hat) lmiterm([4 3 3 R2],-ii,ii) LMISYS=getlmis; [copt,xopt]=feasp(LMISYS); P=dec2mat(LMISYS,xopt,P); R1=dec2mat(LMISYS,xopt,R1); R2=dec2mat(LMISYS,xopt,R2); evlmi=evallmi(LMISYS,xopt); [lhs,rhs]=showlmi(evlmi,4); lhs eigsLHS=eig(lhs) P eigP=eig(P) R1 R2 eigR1=eig(R1) eigR2=eig(R2) BTP=B'*P BTPB=B'*P*B invBTPB=inv(B'*P*B) % recalculate Geq=inv(B'*P*B)*B'*P A0hat=A0-B*G*A0 A1hat=A1-B*G*A1 A2hat=A2-B*G*A2 G= place(A0hat,B,[-4.2 6i -4.2+.6i]) A0til=A0hat-B*G1 eigA0til=eig(A0til) eigA0hat=eig(A0hat) eigA1hat=eig(A1hat) eigA2hat=eig(A2hat) ii = 1; Robust Delay-Independent/Dependent Stabilization of Uncertain Time-Delay Systems by Variable Structure Control 189 setlmis([]) P =lmivar(1,[2 1]); R1=lmivar(1,[2 1]); R2=lmivar(1,[2 1]); lmiterm([-1 1 1 P],ii,ii) lmiterm([-2 1 1 R1],ii,ii) lmiterm([-3 1 1 R2],ii,ii) lmiterm([4 1 1 P],1,A0til','s') lmiterm([4 1 1 R1],ii,ii) lmiterm([4 1 1 R2],ii,ii) lmiterm([4 2 2 R1],-ii,ii) lmiterm([4 1 2 P],1,A1hat) lmiterm([4 1 3 P],1,A2hat) lmiterm([4 3 3 R2],-ii,ii) LMISYS=getlmis; [copt,xopt]=feasp(LMISYS); P=dec2mat(LMISYS,xopt,P); R1=dec2mat(LMISYS,xopt,R1); R2=dec2mat(LMISYS,xopt,R2); evlmi=evallmi(LMISYS,xopt); [lhs,rhs]=showlmi(evlmi,4); lhs eigsLHS=eig(lhs) P eigP=eig(P) R1 R2 eigR1=eig(R1) eigR2=eig(R2) BTP=B'*P BTPB=B'*P*B invBTPB=inv(B'*P*B) normG1 = norm(G1) A3 clear; clc; A0=[-0.228 2.148 -0.021 0; -1 -0.0869 0 0.039; 0.335 -4.424 -1.184 0; 0 0 1 0]; A1=[ 0 0 -0.002 0; 0 0 0 0.004; 0.034 -0.442 0 0; 0 0 0 0]; B =[-1.169 0.065; 0.0223 0; 0.0547 2.120; 0 0]; setlmis([]) P =lmivar(1,[4 1]); R1=lmivar(1,[4 1]); G=inv(B'*P*B)*B'*P A0hat=A0-B*G*A0 Robust Control, Theory and Applications 190 A1hat=A1-B*G*A1 G1= place(A0hat,B,[ 5+.082i 5 082i 2 3]) A0til=A0hat-B*G1 eigA0til=eig(A0til) eigA0hat=eig(A0hat) eigA1hat=eig(A1hat) %break ii = 1; lmiterm([-1 1 1 P],ii,ii) lmiterm([-2 1 1 R1],ii,ii) lmiterm([4 1 1 P],1,A0til','s') lmiterm([4 1 1 R1],ii,ii) lmiterm([4 2 2 R1],-ii,ii) lmiterm([4 1 2 P],1,A1hat) LMISYS=getlmis; [copt,xopt]=feasp(LMISYS); P=dec2mat(LMISYS,xopt,P); R1=dec2mat(LMISYS,xopt,R1); evlmi=evallmi(LMISYS,xopt); [lhs,rhs]=showlmi(evlmi,4); lhs P eigP=eig(P) R1 eigR1=eig(R1) eigsLHS=eig(lhs) BTP=B'*P BTPB=B'*P*B invBTPB=inv(B'*P*B) gnorm=norm(G) A4 clear; clc; A0=[2 0 1; 1.75 0.25 0.8; -1 0 1] A1=[-1 0 0; -0.1 0.25 0.2; -0.2 4 5] B =[0;0;1] %break h1=1.0; setlmis([]); P=lmivar(1,[3 1]); Geq=inv(B'*P*B)*B'*P A0hat=A0-B*Geq*A0 A1hat=A1-B*Geq*A1 eigA0hat=eig(A0hat) eigA1hat=eig(A1hat) Robust Delay-Independent/Dependent Stabilization of Uncertain Time-Delay Systems by Variable Structure Control 191 DesPol = [-2.7 8+.5i 8 5i]; G= place(A0hat,B,DesPol) A0til=A0hat-B*G eigA0til=eig(A0til) R1=lmivar(1,[3 1]); S1=lmivar(1,[3 1]); T1=lmivar(1,[3 1]); lmiterm([-1 1 1 P],1,1); lmiterm([-1 2 2 R1],1,1); lmiterm([-2 1 1 S1],1,1); lmiterm([-3 1 1 T1],1,1); lmiterm([4 1 1 P],(A0til+A1hat)',1,'s'); lmiterm([4 1 1 S1],h1,1); lmiterm([4 1 1 R1],h1,1); lmiterm([4 1 1 T1],1,1); lmiterm([4 1 2 P],-1,A1hat*A0hat); lmiterm([4 1 3 P],-1,A1hat*A1hat); lmiterm([4 2 2 R1],-1/h1,1); lmiterm([4 3 3 S1],-1/h1,1); lmiterm([4 4 4 T1],-1,1); LMISYS=getlmis; [copt,xopt]=feasp(LMISYS); P=dec2mat(LMISYS,xopt,P); R1=dec2mat(LMISYS,xopt,R1); S1=dec2mat(LMISYS,xopt,S1); T1=dec2mat(LMISYS,xopt,T1); evlmi=evallmi(LMISYS,xopt); [lhs,rhs]=showlmi(evlmi,4); lhs,h1,P,R1,S1,T1 eigsLHS=eig(lhs) % repeat clc; Geq=inv(B'*P*B)*B'*P A0hat=A0-B*Geq*A0 A1hat=A1-B*Geq*A1 eigA0hat=eig(A0hat) eigA1hat=eig(A1hat) G= place(A0hat,B,DesPol) A0til=A0hat-B*G eigA0til=eig(A0til) setlmis([]); P=lmivar(1,[3 1]); R1=lmivar(1,[3 1]); S1=lmivar(1,[3 1]); T1=lmivar(1,[3 1]); Robust Control, Theory and Applications 192 lmiterm([-1 1 1 P],1,1); lmiterm([-1 2 2 R1],1,1); lmiterm([-2 1 1 S1],1,1); lmiterm([-3 1 1 T1],1,1); lmiterm([4 1 1 P],(A0til+A1hat)',1,'s'); lmiterm([4 1 1 S1],h1,1); lmiterm([4 1 1 R1],h1,1); lmiterm([4 1 1 T1],1,1); lmiterm([4 1 2 P],-1,A1hat*A0hat); lmiterm([4 1 3 P],-1,A1hat*A1hat); lmiterm([4 2 2 R1],-1/h1,1); lmiterm([4 3 3 S1],-1/h1,1); lmiterm([4 4 4 T1],-1,1); LMISYS=getlmis; [copt,xopt]=feasp(LMISYS); P=dec2mat(LMISYS,xopt,P); R1=dec2mat(LMISYS,xopt,R1); S1=dec2mat(LMISYS,xopt,S1); T1=dec2mat(LMISYS,xopt,T1); evlmi=evallmi(LMISYS,xopt); [lhs,rhs]=showlmi(evlmi,4); lhs,h1,P,R1,S1,T1 eigLHS=eig(lhs) NormP=norm(P) G NormG = norm(G) invBtPB=inv(B'*P*B) BtP=B'*P eigP=eig(P) eigR1=eig(R1) eigS1=eig(S1) eigT1=eig(T1) A5 clear; clc; A0=[-4 0; -1 -3]; A1=[-1.5 0; -1 -0.5]; B =[ 2; 2]; h1=2.0000; setlmis([]); P=lmivar(1,[2 1]); Geq=inv(B'*P*B)*B'*P A0hat=A0-B*Geq*A0 A1hat=A1-B*Geq*A1 eigA0hat=eig(A0hat) Robust Delay-Independent/Dependent Stabilization of Uncertain Time-Delay Systems by Variable Structure Control 193 eigA1hat=eig(A1hat) % DesPol = [ 8+.5i 8 5i]; G= place(A0hat,B,DesPol); avec = [2 0.1]; G = avec; A0til=A0hat-B*G1 eigA0til=eig(A0til) R1=lmivar(1,[2 1]); S1=lmivar(1,[2 1]); T1=lmivar(1,[2 1]); lmiterm([-1 1 1 P],1,1); lmiterm([-1 2 2 R1],1,1); lmiterm([-2 1 1 S1],1,1); lmiterm([-3 1 1 T1],1,1); lmiterm([4 1 1 P],(A0til+A1hat)',1,'s'); lmiterm([4 1 1 S1],h1,1); lmiterm([4 1 1 R1],h1,1); lmiterm([4 1 1 T1],1,1); lmiterm([4 1 2 P],-1,A1hat*A0hat); lmiterm([4 1 3 P],-1,A1hat*A1hat); lmiterm([4 2 2 R1],-1/h1,1); lmiterm([4 3 3 S1],-1/h1,1); lmiterm([4 4 4 T1],-1,1); LMISYS=getlmis; [copt,xopt]=feasp(LMISYS); P=dec2mat(LMISYS,xopt,P); R1=dec2mat(LMISYS,xopt,R1); S1=dec2mat(LMISYS,xopt,S1); T1=dec2mat(LMISYS,xopt,T1); evlmi=evallmi(LMISYS,xopt); [lhs,rhs]=showlmi(evlmi,4); lhs,h1,P,R1,S1,T1 eigsLHS=eig(lhs) % repeat Geq=inv(B'*P*B)*B'*P A0hat=A0-B*Geq*A0 A1hat=A1-B*Geq*A1 eigA0hat=eig(A0hat) eigA1hat=eig(A1hat) G = avec; A0til=A0hat-B*G eigA0til=eig(A0til) setlmis([]); P=lmivar(1,[2 1]); R1=lmivar(1,[2 1]); S1=lmivar(1,[2 1]); Robust Control, Theory and Applications 194 T1=lmivar(1,[2 1]); lmiterm([-1 1 1 P],1,1); lmiterm([-1 2 2 R1],1,1); lmiterm([-2 1 1 S1],1,1); lmiterm([-3 1 1 T1],1,1); lmiterm([4 1 1 P],(A0til+A1hat)',1,'s'); lmiterm([4 1 1 S1],h1,1); lmiterm([4 1 1 R1],h1,1); lmiterm([4 1 1 T1],1,1); lmiterm([4 1 2 P],-1,A1hat*A0hat); lmiterm([4 1 3 P],-1,A1hat*A1hat); lmiterm([4 2 2 R1],-1/h1,1); lmiterm([4 3 3 S1],-1/h1,1); lmiterm([4 4 4 T1],-1,1); LMISYS=getlmis; [copt,xopt]=feasp(LMISYS); P=dec2mat(LMISYS,xopt,P); R1=dec2mat(LMISYS,xopt,R1); S1=dec2mat(LMISYS,xopt,S1); T1=dec2mat(LMISYS,xopt,T1); evlmi=evallmi(LMISYS,xopt); [lhs,rhs]=showlmi(evlmi,4); lhs,h1,P,R1,S1,T1 eigsLHS=eig(lhs) NormP=norm(P) G NormG = norm(G) invBtPB=inv(B'*P*B) BtP=B'*P eigsP=eig(P) eigsR1=eig(R1) eigsS1=eig(S1) eigsT1=eig(T1) 8. 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[...]... increased (see Eqs (8)-(10)), and vice versa 3 Controlled system, variable filter and sliding mode control 3.1 Controlled system This paper deals with next nth order nonlinear differential equation n x( ) = f ( x ) + b ( x ) u, (13) 202 Robust Control, Theory and Applications y=x, (14) where x = [ x , x , , x( n − 1) ]T is state vector of the system In this paper, it is assumed that a part of states, y( = x... Prentice Hall Part 4 Selected Trends in Robust Control Theory 10 Robust Controller Design: New Approaches in the Time and the Frequency Domains Vojtech Veselý, Danica Rosinová and Alena Kozáková Slovak University of Technology Slovak Republic 1 Introduction Robust stability and robust control belong to fundamental problems in control theory and practice; various approaches have been proposed to cope with... SMC method 212 Robust Control, Theory and Applications The way to improve the control performance and to clarify the stability of the proposed method theoretically has been remained Proposed method Actor-Critic + SMC Conventional method SMC PID Complete observation Actor-Critic Complete observation Incomplete observ + s.v.f.* Complete observation m-max [kg] 2.081 2.377 11.788 4.8 06 1 .66 8 m-min [kg]... 198 Robust Control, Theory and Applications control problems comes from the fact that a number of RL algorithms, e.g Q-learning (Watkins et al (1992)) and actor-critic learning (Wang et al (2002)) and Obayashi et al (2008)), do not require knowledge or identification/learning of the system dynamics On the other hand, remarkable characteristics of SMC method are simplicity of its design method, good robustness... due to linearization and approximation, etc A control system is robust if it is insensitive to differences between the actual plant and its model used to design the controller To deal with an uncertain plant a suitable uncertainty model is to be selected and instead of a single model, behaviour of a whole class of models is to be considered Robust control theory provides analysis and design approaches... conventional SMC method, next one is the PID control method 6 Conclusion A robust reinforcement learning method using the concept of the sliding mode control was mainly explained Through the inverted pendulum control simulation, it was verified that the robust reinforcement learning method using the concept of the sliding mode control has good performance and robustness comparing with the conventional... simplicity of its design method, good robustness and stability for deviation of control conditions Recently, a few researches as to robust reinforcement learning have been found, e.g., Morimoto et al (2005) and Wang et al (2002) which are designed to be robust for external disturbances by introducing the idea of H∞ control theory (Zhau et al (19 96) ), and our previous work (Obayashi et al (2009)) is... Man, and Cybernetics, Vol.32, No.4, August, pp.483-492 X S Wang & Y H Cheng & J Q Yi (2007) “A fuzzy Actor–Critic reinforcement learning network”, Information Sciences, 177, pp.3 764 -3781 C Watkins & P Dayan (1992).”Q-learning,” Machine learning, Vol.8, pp.279-292 K Zhau & J.C.Doyle & K.Glover (19 96) Robust optimal control , Englewood Cliffs NJ, Prentice Hall Part 4 Selected Trends in Robust Control Theory. .. with the model-based control theory, such as optimal control, sliding mode control (SMC), H ∞ control and so on The control systems designed through such above control theories have some advantages, that is, the good nature which its adopted theory has originally, robustness, less required iterative learning number which is useful for fragile system controller design not allowed a lot of iterative procedure... 0 2 4 6 8 10 Time[sec] Fig 15 Comparison of the porposed method with incomplete observation, the conventional actor-critic method and PID method for the angle, θ 210 Robust Control, Theory and Applications Figure 15 shows the comparison of the porposed method with incomplete observation, the conventional actor-critic method and PID method for the angle, θ In this figure, the proposed method and PID . sliding-mode controller for boost-type converters with a wide range of operating conditions, IEEE Transactions on Industrial Electronics, Vol. 54, No. 6, pp. 32 76- 32 86. Robust Control, Theory and Applications. method with the model-based control theory, such as optimal control, sliding mode control (SMC), H ∞ control and so on. The control systems designed through such above control theories have some. filter and sliding mode control 3.1 Controlled system This paper deals with next nth order nonlinear differential equation. ( ) () () , n x f bu=+xx (13) Robust Control, Theory and Applications