Linear quadratic optimal control theory is the most mature, systematic and widely used branch of optimal control theory.
State equations and output equations for linear time-variant systems:$$\begin{cases} \dot X(t)=A(t)X(t)+B(t)U(t) & ,X(t_0)=X_0\\ Y(t)=C(t)X(t)\end{cases}$$
Set the expected output to $z(t)$,the output error vector:$e(t)=z(t)-y(t)$
Introducing quadratic performance function:$J=\frac{1}{2}e^T(t_f)Fe(t_f)+\frac{1}{2}\int_{t_0}^{t_f}[X^T(t)Q(t)X(t)+U^T(t)R(t)U(t)]dt$
Objective: Look for an optimal control input $U^*(t)$ to minimize the indicator function.
Therefore, there are:
This is called the ** state adjustment problem**
This is called output adjustment problem
Linear time-invariant system equation of state:$\dot X(t)=A(t)X(t)+B(t)U(t)$,initial conditions:$X(t_0)=X_0,t\in[t_0,t_f]$
Quadratic performance index function:$$J=\frac{1}{2}X^T(t_f)FX(t_f)+\frac{1}{2}\int_{t_0}^{t_f}[X^T(t)Q(t)X(t)+U^T(t)R(t)U(t)]dt$$
Necessary and Sufficient Conditions for Optimal Control Input $U^*(t)$:$$U^*(t)=-R^{-1}(t)B^T(t)P(t)X^*(t)=-K(t)X^*(t)$$
The optimal trajectory is:$$\dot X^*(t)=A(t)X^*(t)+B(t)U^*(t)=[A(t)-B(t)K(t)]X^*(t)$$
The state transition equation is:$$X^*(t)=X_0e^{\int_0^t(A(x)+B(x)K(x))dx}$$
The optimal performance function is:$$J^*=\frac{1}{2}X_0^TP(t_0)X_0$$
where, P(t) satisfies:$$\begin{cases}-\dot P(t)+P(t)A(t)+A^T(t)P(t)-P(t)B(t)K(t)=-Q(t)\\ P(t_f)=F\end{cases}$$
Linear time-invariant system equation of state:$\dot X(t)=AX(t)+BU(t)$,initial conditions:$X(t_0)=X_0,t\in[t_0,\inf]$
Quadratic performance function:$$J=\frac{1}{2}\int_{t_0}^{\inf}[X^T(t)Q(t)X(t)+U^T(t)R(t)U(t)]dt$$
Necessary and Sufficient Conditions for Optimal Control Input $U^*(t)$:$$U^*(t)=-R^{-1}B^TPX^*(t)=-KX^*(t)$$
The optimal trajectory is:$$\dot X^*(t)=AX^*(t)+BU^*(t)=[A-BK]X^*(t)$$
The state transition equation is:$$X^*(t)=X_0e^{(A+BK)t}$$
The optimal performance function is:$$J^*=\frac{1}{2}X_0^TPX_0$$
where, P(t) satisfies:$$PA+A^TP-PBK=-Q$$
Matlab code
continious-time algebraic Riccati equations:
[P,L,K]=care(A,B,Q,R);
R(defaut I);K=$R^{-1}B^TP$;L(pole points)
linear-quadratic regulator
[K,P,L]=lqr(A,B,Q,R)
A=[1 0;1 1];B=[1;0];
EigenValue=eig(A)
Q=[10 0;0 10];
K=lqr(A,B,Q,1);
sys=ss(A-B*K,B,[1 0],[]);
EigenValueMod=eig(sys.a)
step(sys)
EigenValue = 1 1 EigenValueMod = -3.1421 -1.4584
Tracked reference variable $\widetilde{y}$ is the output of the linear time-invariant system:$\begin{cases} \dot Z=FZ & ,Z(t_0)=Z_0\\ \widetilde{y}=HZ\end{cases}$
satisfy:$\sum(A,B)$ fully controllable,$\sum(A,C)$ fully observable,$\sum(F,H)$ fully observable。
Quadratic performance function:$J(u)=\int_0^{\inf} [(Y-\widetilde{y})^TQ(Y-\widetilde y)+U^TRU]dt$
where $\bar{X}=\begin{bmatrix}X\\Z\end{bmatrix},\bar{A}=\begin{bmatrix}A&O\\O&F\end{bmatrix},\bar{B}=\begin{bmatrix}B\\O\end{bmatrix},\bar{Q}=\begin{bmatrix}C^TQC&-C^TQH\\-H^TQC&H^TQH\end{bmatrix},\bar{R}=R$
Therefore, there is an optimal tracking control system input $U^*$:$$U^*(t)=-K^*_1X-K^*_2Z$$ where $\begin{cases}K^*_1=R^{-1}B^TP\\K^*_2=R^{-1}B^TP_{12}\end{cases}$
The robustness indicators for measuring a system are: phase angle margin, gain margin,non-linear feedback tolerance
! To be continued