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

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

subject to the system dynamics

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

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

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

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 .

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

Remarks

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:

Limitations

The problem data must satisfy:

• The pair (A,B) is stabilizable.

• R > 0 and .

• has no unobservable mode on the imaginary axis (or unit circle in discrete time).

See Also

care, dlqr, lqgreg, lqrd, lqry, lqi

lqi

lqrd

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