loopsyn

Purpose

H optimal controller synthesis for LTI plant

Syntax

• [K,CL,GAM,INFO]=loopsyn(G,Gd)
  [K,CL,GAM,INFO]=loopsyn(G,Gd,RANGE) 
  

Description

loopsyn is an H optimal method for loopshaping control synthesis. It computes a stabilizing Hcontroller K for plant G to shape the sigma plot of the loop transfer function GK to have desired loop shape G_d with accuracy =GAM in the sense that if ₀ is the 0db crossover frequency of the sigma plot of G_d(j), then, roughly,

(6-16)

(6-17)

The STRUCT array INFO returns additional design information, including a MIMO stable min-phase shaping pre-filter W, the shaped plant G_s = GW, the controller for the shaped plant K_s=WK, as well as the frequency range {_min,_max} over which the loop shaping is achieved

Input Argument	Description
G	LTI plant
Gd	Desired loop-shape (LTI model)
RANGE	(optional, default {0,Inf}) Desired frequency range for loop-shaping, a 1-by-2 cell array {min,max}; max should be at least ten times min

.

Output Argument	Description
K	LTI controller
CL= G*K/(I+GK)	LTI closed-loop system
GAM	Loop-shaping accuracy (GAM 1, with GAM=1 being perfect fit
INFO	Additional output information

INFO.W	LTI pre-filter W satisfying (Gd)=(GW) for all ; W is always minimum-phase.
INFO.Gs	LTI shaped plant: Gs = GW.
INFO.Ks	LTI controller for the shaped plant: Ks=WK.
INFO.range	{min,max} cell-array containing the approximate frequency range over which loop-shaping could be accurately achieved to with accuracy G. The output INFO.range is either the same as or a subset of the input range.

Algorithm

Using the GCD formula of Le and Safonov [1], loopsyn first computes a stable-minimum-phase loop-shaping, squaring-down prefilter W such that the shaped plant G_s = GW is square, and the desired shape G_d is achieved with good accuracy in the frequency range {_min,_max} by the shaped plant; i.e.,

• (G_d) (G_s) for all {_min,_max}.

Then, loopsyn uses the Glover-McFarlane [2] normalized-coprime-factor control synthesis theory to compute an optimal "loop-shaping" controller for the shaped plant via Ks=ncfsyn(Gs), and returns K=W*Ks.

If the plant G is a continuous time LTI and

1. G has a full-rank D-matrix, and
2. no finite zeros on the j-axis, and
3. {_min,_max}=[0,],

then GW theoretically achieves a perfect accuracy fit (G_d) = (GW) for all frequency . Otherwise, loopsyn uses a bilinear pole-shifting bilinear transform [3] of the form

• Gshifted=bilin(G,-1,'S_Tust',[min,max]),
  

which results in a perfect fit for transformed Gshifted and an approximate fit over the smaller frequency range [_min,_max] for the original unshifted G provided that _max >> _min. For best results, you should choose _max to be at least 100 times greater than _min. In some cases, the computation of the optimal W for Gshifted may be singular or ill-conditioned for the range [_min,_max], as when Gshifted has undamped zeros or, in the continuous-time case only, Gshifted has a D-matrix that is rank-deficient); in such cases, loopsyn automatically reduces the frequency range further, and returns the reduced range [_min,_max] as a cell array in the output INFO.range={_min,_max}

Examples

The following code generates the optimal loopsyn loopshaping control for the case of a 5-state, 4-output, 5-input plant with a full-rank non-minimum phase zero at s=+10. The result in shown in Figure 6-10.

• rand('seed',0);randn('seed',0); 
  s=tf('s'); w0=5; Gd=5/s;           % desired bandwith w0=5
  G=((s-10)/(s+100))*rss(3,4,5);     % 4-by-5 non-min-phase plant
  [K,CL,GAM,INFO]=loopsyn(G,Gd);  
  sigma(G*K,'r',Gd*GAM,'k-.',Gd/GAM,'k-.',{.1,100})  % plot result
  

This figure shows that the LOOPSYN controller K optimally fits

• sigma(G*K) = sigma(Gd)±GAM  %  dB
  

In the above example, GAM = 2.0423 = 6.2026 dB.

Figure 6-10: LOOPSYN controller

The loopsyn controller K optimally fits sigma(G*K). As shown in Figure 6-10, it is sandwiched between sigma(Gd/GAM) and sigma(Gd*GAM) in accordance with the inequalities in Equation 6-16 and Equation 6-17. In the this example, GAM = 2.0423 = 6.2026 db.

Limitations

The plant G must be stabilizable and detectable, must have at least as many inputs as outputs, and must be full rank; i.e,

• size(G,2) size(G,1)
• rank(freqresp(G,w)) = size(G,1) for some frequency w.

The order of the controller K can be large. Generically, when G_d is given as a SISO LTI, then the order N_K of the controller K satisfies

N_K = N_Gs + N_W
= N_yN_Gd + N_RHP + N_W
= N_yN_Gd + N_RHP + N_G

where

• N_y denotes the number of outputs of the plant G.
• N_RHP denotes the total number of nonstable poles and nonminimum-phase zeros of the plant G, including those on the stability boundary and at infinity.
• N_G, N_Gs, N_Gd and N_W denote the respective orders of G, G_s, G_d and W.

Model reduction can help reduce the order of K -- see reduce and ncfmr.

References

[1]  Le, V.X., and M.G. Safonov. Rational matrix GCD's and the design of squaring-down compensators--a state space theory. IEEE Trans. Autom.Control, AC-36(3):384-392, March 1992.

[2]  Glover, K., and D. McFarlane. Robust stabilization of normalized coprime factor plant descriptions with H-bounded uncertainty. IEEE Trans. Autom. Control, AC-34(8):821-830, August 1992.

[3]  Chiang, R.Y., and M.G. Safonov. H synthesis using a bilinear pole-shifting transform. AIAA J. Guidance, Control and Dynamics, 15(5):1111-1115, September-October 1992.

See Also

loopsyn_demo  A demo of this function

mixsyn      H mixed-sensitivity controller synthesis

ncfsyn      H normalized coprime factor controller synthesis

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