lqr

Linear-Quadratic Regulator (LQR) design

Syntax

[K,S,e] = lqr(SYS,Q,R,N) [K,S,e] = LQR(A,B,Q,R,N)

Description

[K,S,e] = lqr(SYS,Q,R,N) calculates the optimal gain matrix K.

For a continuous time system, the state-feedback law u = –Kx minimizes the quadratic cost function

$J (u) = \int_{0}^{\infty} (x^{T} Q x + u^{T} R u + 2 x^{T} N u) d t$

subject to the system dynamics

$\dot{x} = A x + B u .$

In addition to the state-feedback gain K, lqr returns the solution S of the associated Riccati equation

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

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

$K = R^{- 1} (B^{T} S + N^{T}) .$

For a discrete-time state-space model, u[n] = –Kx[n] minimizes

$J = \sum_{n = 0}^{\infty} {x^{T} Q x + u^{T} R u + 2 x^{T} N u}$

subject to x[n + 1] = Ax[n] + Bu[n].

[K,S,e] = LQR(A,B,Q,R,N) is an equivalent syntax for continuous-time models with dynamics $\dot{x} = A x + B u .$

In all cases, when you omit the matrix N, N is set to 0.

Limitations

The problem data must satisfy:

The pair (A,B) is stabilizable.
R > 0 and $Q - N R^{- 1} N^{T} \geq 0$ .
$(Q - N R^{- 1} N^{T}, A - B R^{- 1} N^{T})$ has no unobservable mode on the imaginary axis (or unit circle in discrete time).

Tips

lqr supports descriptor models with nonsingular E. The output S of lqr is the solution of the Riccati equation for the equivalent explicit state-space model:

$\frac{d x}{d t} = E^{- 1} A x + E^{- 1} B u$

Documentation

lqr

Syntax

Description

Limitations

Tips

See Also

Control System Toolbox Documentation

Support