# 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 = 0the 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 = 0the 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

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 |

### 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

*dy*and

_{1}*dy*positions has well as their derivatives are measured.

_{2}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 — [ordered] Schur decomposition of matrix and pencils

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