ltru, ltry (Robust Control Toolbox)

LQG loop transfer recovery.

[af,bf,cf,df,svl] = ltru(A,B,C,D,Kc,Xi,Th,r,w) 
[af,bf,cf,df,svl] = ltry(A,B,C,D,Kf,Q,R,q,w) 
[ssf,svl] = ltru(ss,Kc,Xi,Th,r,w,svk) 
[ssf,svl] = ltry(ss,Kf,Q,R,q,w,svk)

Description
Given a plant with transfer function , ltru implements the Doyle-Stein procedure for recovering full-state-feedback loop transfer function

via a Kalman-filter-reconstructed state feedback with fictitious plant noise. Singular-value Bode plots of the reconstructed-state loop transfer function matrix

are computed and displayed for each value of the fictitious noise intensity parameter r, so that you can observe the loop-transfer recovery as r increases,

Input variables are:

A,B,C,D :=

, (the plant)

ss = optional system matrix form

Kc -- full-state feedback gain matrix

Xi -- nominal plant noise intensity

Th -- measurement noise intensity

r -- row vector of intensities of fictitious plant noise

-- frequencies for Bode plot

The row vector r contains a set of "recovery" gains (e.g.,

). ltru will iteratively compute the singular-value Bode plot of the loop gain F(s)G(s) and plot the associated curves in front of the user, until the gain vector r runs out of its entries.

The controller F(s) corresponding to the last value of r is returned in state-space (af,bf,cf,df). All the singular values (MIMO case) or the Nyquist loci (SISO case) will be returned in variable svl. The frequency response of L(j

) is stored in svk which can be passed into ltru as an optional input and displayed superimposed on the frequency plots of L_r(j

ltry does the "dual" problem, i.e., recovery of the observer loop transfer function

Examples
Consider the fighter design example in [2]. Apply the LTR procedure ltry to the plant model, and let the observer be a Kalman-Bucy filter.

The initial cost and recovery gains used in this example are

The singular value Bode plot is shown in Figure 1-13.

The LQG/LTR loop transfer function converges in the limit (as q increases to

) to C (Is - A)^-¹ K_f, which is the KBF loop transfer function.

Algorithm
The controller F(s) is computed as

where, in ltru, the Kalman filter gain is

and

satisfies the Kalman filter Riccati equation

In ltry gain Kc is computed as

where P satisfies the full-state Riccati equation

The theory is documented in [1].

Figure 1-13: Example of LQG/LTR at Plant Output.

Limitations
The ltru procedure may fail for nonminimum phase plants or for plants with number of control inputs exceeds the number of measurement outputs. The dual procedure ltry may fail for nonminimum phase plants or for plants with fewer inputs than outputs.

See Also
h2lqg, hinf, hinfdemo, lqg, ltrdemo

References
[1] J. Doyle and G. Stein "Multivariable Feedback Design: Concepts for a Classical/Modern Synthesis," IEEE Trans. on Automat. Contr., AC-26, pp. 4-16, 1981.

[2] M. G. Safonov, A. J. Laub, and G. Hartmann, "Feedback Properties of Multivariable Systems: The Role and Use of Return Difference Matrix," IEEE Trans. of Automat. Contr., AC-26, pp. 47-65, 1981.

lqg mksys, vrsys, issystem