Loop shaping design using Glover-McFarlane method
ncfsyn implements a method for designing controllers that
uses a combination of loop shaping and robust stabilization as proposed in -. The function computes the Glover-McFarlane H∞
normalized coprime factor loop-shaping controller K for a plant
G with pre-compensator and post-compensator weights
W2. The function assumes the positive feedback
configuration of the following illustration.
To specify negative feedback, replace G by –G. The
controller Ks stabilizes a family of systems given
by a ball of uncertainty in the normalized coprime factors of the shaped plant Gs =
W2GW1. The final controller K returned by
ncfsyn is obtained as K =
computes the Glover-McFarlane H∞ normalized
coprime factor loop-shaping controller
info] = ncfsyn(
K for the plant
G, with W1 =
W2 = I.
CL is the closed-loop system from the disturbances
w2 to the outputs
z2. The function also returns the
H∞ optimal cost
a structure containing additional information about the result.
The following code shows how
ncfsyn can be used for loop-shaping.
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-.')
The singular value plot of the achieved loop
G*K is equal to that of the target loop
G*W1 to within plus or minus
GAM (in dB).
Plant, specified as a dynamic system model such as a state-space
ss) model. If
G is a generalized
state-space model with uncertain or tunable control design blocks, then
ncfsyn uses the nominal or current value of those elements.
G must have the same number of inputs and outputs.
W1— Pre-compensator weight
eye(N)(default) | LTI model
Pre-compensator weight, specified as:
N is the
number of inputs or outputs in
SISO minimum-phase LTI model. In this case,
ncfsyn uses the
same weight for every loop channel.
MIMO minimum-phase LTI model of the same I/O dimensions as
Select a pre-compensator and post-compensator weights W1 and W2 such that the gain of the shaped plant Gs = W2GW1 is sufficiently high at frequencies where good disturbance attenuation is required, and sufficiently low at frequencies where good robust stability is required.
W2— Post-compensator weight
Post-compensator weight, specified as the identity matrix
or a SISO or MIMO LTI model. The considerations for specifying
are the same as those for
CL— Optimal closed-loop system
Optimal closed-loop system from the disturbances w1 and w2 to the outputs z1 and z2, returned as a state-space model. The closed-loop system is given by:
gamma— H∞ optimal cost
H∞ optimal cost, returned as a positive scalar
value greater than 1. The optimal cost is
hinfnorm(CL). The optimal
controller Ks is such that the singular-value
plot of the shaped loop Ls =
W2GW1Ks optimally matches the target loop shape
Gs to within a factor of
gamma is related to the normalized coprime stability margin of
the system by
gamma = 1/ncfmargin(Gs,-K). Thus,
gamma gives a good indication of robustness of stability to a
wide class of unstructured plant variations, with values in the range 1 <
gamma < 3 corresponding to satisfactory stability margins for
most typical control system designs.
ncfmargin assumes a negative-feedback
ncfsyn command designs a controller for a positive-feedback
loop. Therefore, to compute the margin using a controller designed with
The returned controller K = W1KsW2, where Ks is an optimal H∞ controller that minimizes the H∞ cost
The optimal performance is the minimal cost
Suppose that Gs=NM–1, where N(jω)*N(jω) + M(jω)*M(jω) = I, is a normalized coprime factorization (NCF) of the weighted plant model Gs. Then, theory ensures that the control system remains robustly stable for any perturbation to Gs of the form
where Δ1, Δ2 are a stable pair satisfying
The closed-loop H∞-norm objective has the standard signal gain interpretation. Finally it can be shown that the controller, Ks, 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=W1KsW2.
 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.
 Zhou, K., and J.C. Doyle, Essentials of Robust Control. NY: Prentice-Hall, 1998.