ltrsyn (Robust Control Toolbox™)

Robust Control Toolbox™

ltrsyn

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LQG loop transfer-function recovery (LTR) control synthesis

Syntax

[K,SVL,W1] = ltrsyn(G,F,XI,THETA,RHO)
[K,SVL,W1] = ltrsyn(G,F,XI,THETA,RHO,W)
[K,SVL,W1] = ltrsyn(G,F,XI,THETA,RHO,OPT)
[K,SVL,W1] = ltrsyn(G,F,XI,THETA,RHO,W,OPT)

Description

[K,SVL,W1] = ltrsyn(G,F,XI,TH,RHO) computes a reconstructed-state output-feedback controller K for LTI plant G so that K*G asymptotically `recovers' plant-input full-state feedback loop transfer function L(s) = F(Is-A)^-1B+D; that is, at any frequency w>0, max(sigma(K*G-L, w)) 0 as rho , where L= ss(A,B,F,D) is the LTI full-state feedback loop transfer function.

[K,SVL,W1] = ltrsyn(G,F1,Q,R,RHO,'OUTPUT') computes the solution to the `dual' problem of filter loop recovery for LTI plant G where F is a Kalman filter gain matrix. In this case, the recovery is at the plant output, and max(sigma(G*K-L, w)) 0 as rho , where L1 denotes the LTI filter loop feedback loop transfer function L1= ss(A,F,C,D).

Only the LTI controller K for the final value RHO(end)is returned.

Inputs

G
LTI plant

F
LQ full-state-feedback gain matrix

XI
plant noise intensity,
or, if OPT='OUTPUT' state-cost matrix XI=Q,

THETA
sensor noise intensity
or, if OPT='OUTPUT' control-cost matrix THETA=R,

RHO
vector containing a set of recovery gains

W
(optional) vector of frequencies (to be used for plots); if input W is not supplied, then a reasonable default is used

Outputs

K
K(s) -- LTI LTR (loop-transfer-recovery) output-feedback, for the last element of RHO (i.e., RHO(end))

SVL
sigma plot data for the `recovered' loop transfer function if G is MIMO or, for SISO G only, Nyquist loci SVL = [re(1:nr) im(1:nr)]

W1
frequencies for SVL plots, same as W when present

Inputs
`G`	LTI plant
`F`	LQ full-state-feedback gain matrix
`XI`	plant noise intensity, or, if `OPT='OUTPUT'` state-cost matrix `XI=Q`,
`THETA`	sensor noise intensity or, if `OPT='OUTPUT'` control-cost matrix `THETA=R`,
`RHO`	vector containing a set of recovery gains
`W`	(optional) vector of frequencies (to be used for plots); if input `W` is not supplied, then a reasonable default is used

Outputs
`K`	K(s) -- LTI LTR (loop-transfer-recovery) output-feedback, for the last element of `RHO` (i.e., `RHO(end))`
`SVL`	sigma plot data for the `recovered' loop transfer function if `G` is MIMO or, for SISO `G` only, Nyquist loci `SVL = [re(1:nr) im(1:nr)]`
`W1`	frequencies for SVL plots, same as `W` when present

Algorithm

For each value in the vector RHO, [K,SVL,W1] = ltrsyn(G,F,XI,THETA,RHO) computes the full-state-feedback (default OPT='INPUT') LTR controller

where K_c=F and K_f=lqr(A',C',XI+RHO(i)*B*B',THETA). The `fictitious noise' term RHO(i)*B*B' results in loop-transfer recovery as RHO(i) . The Kalman filter gain is where capital sigma satisfies the Kalman filter Riccati equation . See [1] for further details.

Similarly for the `dual' problem of filter loop recovery case, [K,SVL,W1] = ltrsyn(G,F,Q,R,RHO,'OUTPUT') computes a filter loop recovery controller of the same form, but with K_f=F is being the input filter gain matrix and the control gain matrix K_c computed as K_c=lqr(A,B,Q+RHO(i)*C'*C,R).

Figure 5-12: Example of LQG/LTR at Plant Output.

Example

s=tf('s');G=ss(1e4/((s+1)*(s+10)*(s+100)));[A,B,C,D]=ssdata(G);
F=lqr(A,B,C'*C,eye(size(B,2)));
L=ss(A,B,F,0*F*B);
XI=100*C'*C; THETA=eye(size(C,1)); 
RHO=[1e3,1e6,1e9,1e12];W=logspace(-2,2);
nyquist(L,'k-.');hold;
[K,SVL,W1]=ltrsyn(G,F,XI,THETA,RHO,W);