lqr

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:

Remark In this case the P.C and P.D componants of the system are ignored

Algorithm

• For a continuous plant, if is the stabilizing solution of the Riccati equation:

  
  the linear optimal LQ full-state gain K is given by
  

• For a discrete plant, if is the stabilizing solution of the Riccati equation:

  
  the linear optimal LQ full-state gain K is given by
  

  An equivalent form for the equation is

  
  with
  

The gain K is such that 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

  

Caution

It is assumed that matrix or is non singular.

Remark

If the full state of the system is not available, An estimator can be built using the lqe or the lqg function.

Examples

Assume the dynamical system formed by two masses connected by a spring and a damper:

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 — Riccati equation
• schur — [ordered] Schur decomposition of matrix and pencils