lqry

Form linear-quadratic (LQ) state-feedback regulator with output weighting

Syntax

[K,S,e] = lqry(sys,Q,R,N)


Description

Given the plant

$$\begin{array}{l}\dot{x}=Ax+Bu\\ y=Cx+Du\end{array}$$

or its discrete-time counterpart, lqry designs a state-feedback control

$$u=-Kx$$

that minimizes the quadratic cost function with output weighting

$$J(u)={\displaystyle {\int }_0^{\infty }(y^{T}Qy+u^{T}Ru+2y^{T}Nu)dt}$$

(or its discrete-time counterpart). The function lqry is equivalent to lqr or dlqr with weighting matrices:

$$\left[\begin{array}{cc}\overline{Q}& \overline{N}\\ {\overline{N}}^T& \overline{R}\end{array}\right]=\left[\begin{array}{cc}{C}^T& 0\\ D^T& I\end{array}\right]\left[\begin{array}{cc}Q& N\\ N^T& R\end{array}\right]\left[\begin{array}{cc}C& D\\ 0& I\end{array}\right]$$

[K,S,e] = lqry(sys,Q,R,N) returns the optimal gain matrix K, the Riccati solution S, and the closed-loop eigenvalues e = eig(A-B*K). The state-space model sys specifies the continuous- or discrete-time plant data (A, B, C, D). The default value N=0 is assumed when N is omitted.

Examples

See LQG Design for the x-Axis for an example.

Limitations

The data $$A,B,\overline{Q},\overline{R},\overline{N}$$ must satisfy the requirements for lqr or dlqr.