dlqr

Linear-quadratic (LQ) state-feedback regulator for discrete-time state-space system

Syntax

[K,S,e] = dlqr(A,B,Q,R,N)


Description

[K,S,e] = dlqr(A,B,Q,R,N) calculates the optimal gain matrix K such that the state-feedback law

$$u\left[n\right]=-Kx\left[n\right]$$

minimizes the quadratic cost function

$$J(u)={\displaystyle \sum _{n=1}^{\infty }(x{\left[n\right]}^{T}Qx\left[n\right]+u{\left[n\right]}^{T}Ru\left[n\right]+2x{\left[n\right]}^{T}Nu}\left[n\right])$$

for the discrete-time state-space model

$$x\left[n+1\right]=Ax\left[n\right]+Bu\left[n\right]$$

The default value N=0 is assumed when N is omitted.

In addition to the state-feedback gain K, dlqr returns the infinite horizon solution S of the associated discrete-time Riccati equation

$$A^{T}SA-S-(A^{T}SB+N){(B^{T}SB+R)}^{-1}(B^{T}SA+N^T)+Q=0$$

and the closed-loop eigenvalues e = eig(A-B*K). Note that K is derived from S by

$$K={(B^{T}SB+R)}^{-1}(B^{T}SA+N^T)$$

Limitations

The problem data must satisfy:

• The pair (A,B) must be stabilizable.

• R must be positive definite.

• $$\left[\begin{array}{cc}Q& N\\ N^T& R\end{array}\right]$$ must be positive semidefinite (equivalently, $$Q-N{R}^{-1}{N}^T\ge 0$$).

• $$(Q-N{R}^{-1}{N}^T,A-B{R}^{-1}{N}^T)$$ must have no unobservable mode on the unit circle.

Version History

Introduced before R2006a