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lqr

LQ compensator (full state)

Calling Sequence

`[K,X]=lqr(P12)`

Arguments

P12

`syslin` list (state-space linear system)

K,X

two real matrices

Description

`lqr` computes the linear optimal LQ full-state gain for the plant `P12=[A,B2,C1,D12]` in continuous or discrete time.

`P12` is a `syslin` list (e.g. `P12=syslin('c',A,B2,C1,D12)`).

The cost function is l2-norm of `z'*z` with `z=C1 x + D12 u` i.e. `[x,u]' * BigQ * [x;u]` where

```[C1' ]               [Q  S]
BigQ= [    ]  * [C1 D12] = [    ]
[D12']               [S' R]```

The gain `K` is such that `A + B2*K` is stable.

`X` is the stabilizing solution of the Riccati equation.

For a continuous plant:

`(A-B2*inv(R)*S')'*X+X*(A-B2*inv(R)*S')-X*B2*inv(R)*B2'*X+Q-S*inv(R)*S'=0`
`K=-inv(R)*(B2'*X+S)`

For a discrete plant:

`X=A'*X*A-(A'*X*B2+C1'*D12)*pinv(B2'*X*B2+D12'*D12)*(B2'*X*A+D12'*C1)+C1'*C1;`
`K=-pinv(B2'*X*B2+D12'*D12)*(B2'*X*A+D12'*C1)`

An equivalent form for `X` is

`X=Abar'*inv(inv(X)+B2*inv(r)*B2')*Abar+Qbar`

with `Abar=A-B2*inv(R)*S'` and `Qbar=Q-S*inv(R)*S'`

The 3-blocks matrix pencils associated with these Riccati equations are:

```discrete                           continuous
|I   0    0|   | A    0    B2|         |I   0   0|   | A    0    B2|
z|0   A'   0| - |-Q    I    -S|        s|0   I   0| - |-Q   -A'   -S|
|0   B2'  0|   | S'   0     R|         |0   0   0|   | S'  -B2'   R|```

Caution: It is assumed that matrix R is non singular. In particular, the plant must be tall (number of outputs >= number of inputs).

Examples

```A=rand(2,2);B=rand(2,1);   //two states, one input
Q=diag([2,5]);R=2;     //Usual notations x'Qx + u'Ru
Big=sysdiag(Q,R);    //Now we calculate C1 and D12
[w,wp]=fullrf(Big);C1=wp(:,1:2);D12=wp(:,3:\$);   //[C1,D12]'*[C1,D12]=Big
P=syslin('c',A,B,C1,D12);    //The plant (continuous-time)
[K,X]=lqr(P)
spec(A+B*K)    //check stability
norm(A'*X+X*A-X*B*inv(R)*B'*X+Q,1)  //Riccati check
P=syslin('d',A,B,C1,D12);    // Discrete time plant
[K,X]=lqr(P)
spec(A+B*K)   //check stability
norm(A'*X*A-(A'*X*B)*pinv(B'*X*B+R)*(B'*X*A)+Q-X,1) //Riccati check```