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Linear-Quadratic Regulator (LQR) design


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


[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 x˙=Ax+Bu.

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


The problem data must satisfy:

  • The pair (A,B) is stabilizable.

  • R > 0 and QNR1NT0.

  • (QNR1NT,ABR1NT) has no unobservable mode on the imaginary axis (or unit circle in discrete time).


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:


See Also

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Introduced before R2006a