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lqr
LQ compensator (full state)
Syntax
[K, X] = lqr(P12) [K, X] = lqr(P, Q, R) [K, X] = lqr(P, Q, R, S)
Arguments
- P12
A state space representation of a linear dynamical system (see syslin)
- P
A state space representation of a linear dynamical system (see syslin)
- Q
Real symmetric matrix, with same dimensions as P.A
- R
full rank real symmetric matrix
- S
real matrix, the default value is
zeros(size(R,1),size(Q,2))
- K
a real matrix, the optimal gain
- X
a real symmetric matrix, the stabilizing solution of the Riccati equation
Description
- Syntax
[K,X]=lqr(P)
Computes the linear optimal LQ full-state gain K for the state space representation P
And instantaneous cost function in l2-norm:- Syntax
[K,X]=lqr(P,Q,R [,S])
Computes the linear optimal LQ full-state gain K for the linear dynamical system P:
And instantaneous cost function in l2-norm:In this case the P.C and P.D components of the system are ignored.
Algorithm
For a continuous plant, if X is the stabilizing solution of the Riccati equation:
(A - B.R-1.S)'.X + X.(A - B.R-1.S) - X.B.R-1.B'.X + Q - S'.R-1.S = 0
the linear optimal LQ full-state gain K is given by
K = -R-1(B'X + S')
For a discrete plant, if X is the stabilizing solution of the Riccati equation:
A'.X.A - X - (A'.X.B + S')(B'.X.B + R)+(B'.X.A + S) + Q = 0
the linear optimal LQ full-state gain K is given by
K = -(B'.X.B + R)+(B'.X.A + S)
An equivalent form for the equation is
with
The gain K
is such that A + B.K
is stable.
The resolution of the Riccati equation is obtained by schur factorization of the 3-blocks matrix pencils associated with these Riccati equations:
For a continuous plant
For a discrete time plant
It is assumed that matrix R or D'D is non singular. |
Examples
Assume the dynamical system formed by two masses connected by a spring and a damper:
A force (where e is a noise) is applied to the big one. Here it is assumed that the deviations from equilibrium positions of the mass dy1 and dy2 positions has well as their derivatives are measured.A state space representation of this system is:
The LQ cost is defined by
The following instructions may be used to compute a LQ compensator of this dynamical system.
// Form the state space model (assume full state output) M = 1; m = 0.2; k = 0.1; b = 0.004; A = [ 0 1 0 0 -k/M -b/M k/M b/M 0 0 0 1 k/m b/m -k/m -b/m]; B = [0; 1/M; 0; 0]; C = eye(4,4); P = syslin("c",A, B, C); //The compensator weights Q_xx=diag([15 0 3 0]); //Weights on states R_uu = 0.5; //Weight on input Kc=lqr(P,Q_xx,R_uu); //form the Plant+compensator system C=[1 0 0 0 //dy1 0 0 1 0];//dy2 S=C*(P/.(-Kc)); //check system stability and(real(spec(S.A))<0) // Check by simulation dt=0.1; t=0:dt:30; u=0.1*rand(t); y=csim(u,t,S,[1;0;0;0]); clf;plot(t',y');xlabel(_("time (s)")) L=legend(["$dy_1$","$dy_2$"]);L.font_size=4;
Reference
Engineering and Scientific Computing with Scilab, Claude Gomez and al.,Springer Science+Business Media, LLC,1999, ISNB:978-1-4612-7204-5
See also
- lqg — LQG compensator
- lqe — linear quadratic estimator (Kalman Filter)
- gcare — Continuous time control Riccati equation
- leqr — H-infinity LQ gain (full state)
- riccati — Solves the matricial Riccati equation (continuous | discrete time domain)
- schur — decomposição (ordenada) de Schur de matrizes e feixes
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