Robust Control Toolbox™

Loop shaping design using Glover-McFarlane method

Syntax

• [K,CL,GAM,INFO]=ncfsyn(G) 
  [K,CL,GAM,INFO]=ncfsyn(G,W1) 
  [K,CL,GAM,INFO]=ncfsyn(G,W1,W2)
  [K,CL,GAM,INFO]=ncfsyn(G,W1,W2,'ref')
  

Description

ncfsyn is a method for designing controllers that uses a combination of loop shaping and robust stabilization as proposed in McFarlane and Glover [1]-[2]. The first step is for you to select a pre- and post-compensator W1 and W2, so that the gain of the `shaped plant' G_s:= W2GW1 is sufficiently high at frequencies where good disturbance attenuation is required and is sufficiently low at frequencies where good robust stability is required. The second step is to use ncfsyn to compute an optimal positive feedback controllers K.

The optimal Ks has the property that the sigma plot of the shaped loop

Ls=W2*G*W1*Ks

matches the target loopshape G_s optimally, roughly to within plus or minus 20*log10(GAM) decibels. The number margin GAM=1/ncfmargin(Gs,K) and is always greater than 1. GAM gives a good indication of robustness of stability to a wide class of unstructured plant variations, with values in the range 1<GAM<3 corresponding to satisfactory stability margins for most typical control system designs.

[K,CL,GAM,INFO]=ncfsyn(G,W1,W2,'ref')computes the Glover-McFarlane H normalized co-prime factor loop-shaping controller K, with a reference command, for lti plant G, weights W1 and W2 if the 'ref'option is included. The closed-loop system returned, CL, represents the transfer matrix from the reference and disturbance to the feedback error and output of W1.

Algorithm

K=W2*Ks*W1, where Ks =K is an optimal H controller that simultaneously minimizes the two H cost functions

Roughly speaking, this means for most plants that

(W2GW1 K), db = (W2GW1), db ± , db

(KW2GW1), db = (W2GW1), db ± , db,

so you can use the weights W1 and W2 for loopshaping. For a more precise bounds on loopshaping accuracy, see Theorem 16.12 of Zhou and Glover [1].

Theory ensures that if G_s=NM-1is a normalized coprime factorization (NCF) of the weighted plant model G_s satisfying

G_s=N(jw)*N(jw) + M(jw)*M(jw) = I,

then the control system will remain robustly stable for any perturbation to the weighted plant model G_s that can be written

for some stable pair _{1, 2} satisfying

• .

The closed-loop H-norm objective has the standard signal gain interpretation. Finally it can be shown that the controller, K, does not substantially affect the loop shape in frequencies where the gain of W2GW1 is either high or low, and will guarantee satisfactory stability margins in the frequency region of gain cross-over. In the regulator set-up, the final controller to be implemented is K=W1KW2.

Input arguments

G	LTI plant to be controlled
W1,W2	Stable minimum-phase LTI weights, either SISO or MIMO. Default is W1=I, W2=I
`ref'	Reference input to controller. Default is no reference input is included.

Output arguments

K	LTI controller K= W1*Ks*W2
CL	, LTI H optimal closed loop
GAM	H optimal cost = = hinfnorm(CL) 1
INFO	structure array containing additional information

Additional output INFO fields

INFO.emax	nugap robustness emax=1/GAM=ncfmargin(Gs,-Ks)=
INFO.Gs	'shaped plant' Gs=W2*G*W1
INFO.Ks	Ks = K = NCFSYN(Gs) = NCFSYN(W2*G*W1)

:

[MARG,FREQ] = ncfmargin(G,K,TOL) calculates the normalized coprime factor/gap metric robust stability margin assuming negative feedback.

MARG.

where G and K are LTI plant and controller, and TOL (default=.001) is the tolerance used to compute the H norm. FREQ is the peak frequency -- i.e., the frequency at which the infinity norm is reached to within TOL.

Algorithm

See the McFarlane and Glover [1]-[2] for details.

Example

The following code shows how ncfsyn can be used for loop-shaping. The achieved loop G*K has a sigma plot is equal to that of the target loop G*W1 to within plus or minus 20*log10(GAM) decibels.

• s=zpk('s');
  G=(s-1)/(s+1)^2;
  W1=0.5/s; 
  [K,CL,GAM]=ncfsyn(G,W1);
  sigma(G*K,'r',G*W1,'r-.',G*W1*GAM,'k-.',G*W1/GAM,'k-.')
  
  
  

Figure 5-15: Achieved loop G*K and shaped loop Gs, ±20log(GAM) db

Reference

[1]  McFarlane, D.C., and K. Glover, Robust Controller Design using Normalised Coprime Factor Plant Descriptions, Springer Verlag, Lecture Notes in Control and Information Sciences, vol. 138, 1989.

[2]  McFarlane, D.C., and K. Glover, "A Loop Shaping Design Procedure using Synthesis," IEEE Transactions on Automatic Control, vol. 37, no. 6, pp. 759- 769, June 1992.

[3]  Vinnicombe, G., "Measuring Robustness of Feedback Systems," PhD dissertation, Department of Engineering, University of Cambridge, 1993.

[4]  Zhou, K., and J.C. Doyle, Essentials of Robust Control. NY: Prentice-Hall, 1998.

See Also
gapmetric   Computes the gap and the Vinnicombe gap metric

hinfsyn     H controller synthesis

loopsyn     H - loop shaping controller synthesis

ncfmargin   Normalized coprime stability margin of the plant-controller feedback loop

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