Control System Toolbox

lqr

Design linear-quadratic (LQ) state-feedback regulator for state-space systems

Syntax

• [K,S,e] = lqr(SYS,Q,R)
  [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 such that:

For a continuouse time system, the state-feedback law 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). Note that is derived from by

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 dx/dt=Ax+Bu.

In all cases, the default value N=0 is assumed when N is omitted.

Limitations

The problem data must satisfy:

• The pair is stabilizable.
• and .
• has no unobservable mode on the imaginary axis.

See Also
care        Solve continuous Riccati equations

dlqr        State-feedback LQ regulator for discrete plant

lqgreg      Form LQG regulator

lqrd        Discrete LQ regulator for continuous plant

lqry        State-feedback LQ regulator with output weighting

lqgreg

lqrd

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