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Linear-quadratic (LQ) state-feedback regulator for state-space system
[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 such that:
For a continuous 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.
The problem data must satisfy:
The pair
is stabilizable.
and
.
has no unobservable mode on the
imaginary axis.
care, dlqr, lqgreg, lqrd, lqry
![]() | lqgreg | lqrd | ![]() |
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