《现代控制理论基础》第五章(讲义) 性能指标J为 J=I(x'Ox+u'Ru)dr 式中 Q=010R=[] 001 求黎卡提方程的正定矩阵R、最优反馈增益矩阵κ和矩阵所的特征值。 利用 MATLAB Program5.3,可求解该问题。 MATLAB Programs.3 %--------Design of quadratic optimal regulatorsystem %*****Determination of feedback gain matrix K for quadratic *** Enter state matrix A and control matrix b***** 00;1 %******Enter matrices Q and r of the quadratic performance %******The optimal feedback gain matrix K, solution P of Riccati %equation, and closed-loop poles( that is, the eigenvalues %of A-BK )can be obtained by entering the following K, P,Egr(A, B, Q, R) 0.0143 0.1107 0.0676 4.2625 2.4957 0.0143 2.4957 2.8150 0.1107 0.0143 0.1107 0.0676 -0.9859+1.7110 -1.9859-1.7110i [例5.13]考虑与例12.7中讨论的相同的系统。该系统的状态空间表达式为 Ax+ Bu 式中 010 A=001B=|0C=poqD=01《现代控制理论基础》第五章(讲义) 10 性能指标 J 为 式中 求黎卡提方程的正定矩阵 R、最优反馈增益矩阵 K 和矩阵 A-BK 的特征值。 利用 MATLAB Program 5.3，可求解该问题。 MATLAB Program5.3 %--------Design of quadratic optimal regulatorsystem------- %*****Determination of feedback gain matrix K for quadratic *****Enter state matrix A and control matrix B***** A=[0 0;0 0 1;-35 -27 -9]; B-[0;0;1] %******Enter matrices Q and R of the quadratic performance %index ***** Q=[1 0 0;0 1 0;0 0 1]; R=[1]; %******The optimal feedback gain matrix K, solution P of Riccati %equation,and closed-loop poles(that is,the eigenvalues %of A-BK)can be obtained by entering the following %command***** [K,P,E]=[qr(A,B,Q,R)] K= 0.0143 0.1107 0.0676 P= 4.2625 2.4957 0.0143 2.4957 2.8150 0.1107 0.0143 0.1107 0.0676 E= -5.0958 -0.9859+1.7110I -1.9859-1.7110i ------------------------------------------------------------------ [例 5.13] 考虑与例 12.7 中讨论的相同的系统。该系统的状态空间表达式为 x  = Ax + Bu y = Cx + Du 式中 , 1 0 0, [0] 1 0 0 , 0 2 3 0 0 1 0 1 0 = =           =           − − A = B C D J （x Qx + u Ru）dt  =  0 , 1 0 0 1 0 1 0 1 0 0 =           Q = R