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# lqi

Linear quadratic integral compensator (full state)

### Syntax

[K,X]=lqi(P,Q,R [,S])

### Arguments

- P
The plant state space representation (see syslin) with nx states, nu inputs and ny outputs.

- Q
Real nx+ny by nx+ny symmetric matrix,

- R
full rank nu by nu real symmetric matrix

- S
real nx+ny by nu matrix, the default value is zeros(nx+ny,nu)

- K
a real matrix, the optimal gain

- X
a real symmetric matrix, the stabilizing solution of the Riccati equation

### Description

This function computes the linear quadratic integral full-state gain K for the plant P. The associated system block diagram is:

The plant P is given by its state space representation

The cost function in l2-norm is: where and is the integrator(s) state(s);### Algorithm

The lqi function solves the lqr problem for the augmented plant

### Caution

It is assumed that matrix is non singular.

### Remark

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

### Examples

Linear quadratic integral controller of a simplified disk drive using state observer.

//Disk drive model G=syslin("c",[0,32;-31.25,-0.4],[0;2.236068],[0.0698771,0]); t=linspace(0,20,2000); y=csim("step",t,G); //State estimator Wy=1; Wu=1; S=0; Q=G.B*Wu*G.B'; R=Wy+G.D*S + S'*G.D+G.D*Wu*G.D'; S=G.B*Wu*G.D'+S; //State estimator [Kf,X]=lqe(G,Q,R,S); Gx=observer(G,Kf); //LQI compensator wy=100; Q= wy*sysdiag(G.C'*G.C,1); R=1/wy; Kc=lqi(G,Q,R); //full controller K=lft([1;1]*(-Kc(1:2)*Gx(:,[2 1])+Kc(3)*[1/%s 0]),1);//e-->u //Full system H=(-K*G)/.(1);// full system transfer function y=csim("step",t,H); clf;plot(t,y)

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