Mu Analysis and Synthesis Toolbox

ncfsyn synthesizes an H controller to robustly stabilize a family of systems given by a ball of uncertainty in the normalized coprime factors of the system description

cf2sys calculates a SYSTEM/CONSTANT/VARYING matrix from a coprime factorization.

emargin calculates the normalized coprime factor/gap metric robust stability margin b(P,K) as defined in the "Loop Shaping Using H· Synthesis" section in Chapter 3.

Syntax

• [sysk,emax,sysobs] = ncfsyn(sysgw,factor,opt) 
  sysout = cf2sys(sysrcf) 
  emarg = emargin(sys_g,sysw,tol)
  

Description
A method of designing controllers is to use a combination of loop shaping and robust stabilization as proposed in McFarlane and Glover. The first step is to design a pre- and post-compensator W₁(s) and W₂, so that the gain of W₂(s)P(s)W₁(s) 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 design a feedback controller, K, so that

which will also give robust stability of the perturbed weighted plant

where NM^-1 = W₂PW₁is a normalized coprime factorization satisfying N(jw)*N(jw) + M(jw)*M(jw) = I. This stability margin is always less than 1 and gives a good indication of robust stability to a wide class of unstructured perturbations, with values of > 0.2 - 0.3 generally satisfactory.

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 W₂PW₁ 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 W₁KW₂.

When the option 'ref' is specified, the controller includes an extra set of reference inputs as proposed in Vinnicombe, 1993, and should be implemented as u = sysk *


where y = sysgw * u and r is a reference input. The closed-loop response will then be y = Nr, where N is the numerator of a normalized right coprime factor of sysgw.

Input arguments

sysgw	the weighted system to be controlled
factor	=1 implies that an optimal controller is required. >1 implies that a suboptimal controller is required achieving a performance FACTOR less than optimal.
opt	'ref' the controller includes an extra set of reference input and should be implemented as in Figure 3-12 on page 3-? (optional).

Output arguments

sysk	H loopshaping controller
emax	Stability margin as an indication robustness to unstructured perturbations. emax is always less than 1 and values of emax greater than 0.3 generally indicate good robustness margins.
sysobs	H loopshaping observer controller. This variable is created only if factor>1 and opt = 'ref'

    

cf2sys creates a SYSTEM/CONSTANT/VARYING matrix from a coprime factorization.

emargin calculates the normalized coprime factor/gap metric robust stability margin b(P,K) as defined in the "Loop Shaping Using H· Synthesis" section in Chapter 3. The default value for the tolerance tol supplied to the hinfnorm computation is 0.001.

Algorithm
See the McFarlane and Glover reference for details.

Reference
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.

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.

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

See Also
gap, hinfsyn, hinfnorm, nugap

musynfit, musynflp, muftbtch

nd2sys, zp2sys