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# lqr

LQ compensator (full state)

### Syntax

[K,X]=lqr(P12)

[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:### 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 byFor a discrete plant, if is the stabilizing solution of the Riccati equation:

the linear optimal LQ full-state gain K is given byAn 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 force (where is a noise) is applied to the big one. Here it is assumed that the deviations from equilibrium positions of the mass and 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

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