1. For $disturbance\; rejection$ make \(\underline{\sigma}(W_2 G W_1)\) large; valid for frequencies at which \(\underline{\sigma}(G_S) \gg 1\)
2. For $noise\; attenuation$ make \(\overline{\sigma}(W_2 G W_1)\) small; valid for frequencies at which \(\overline{\sigma}(G_S) \ll 1\)
3. For $reference\; tracking$ make \(\underline{\sigma}(W_2 G W_1)\) large; valid for frequencies at which \(\underline{\sigma}(G_S) \gg 1\)
4. For $robust\; stability$ to a multiplicative output perturbation \(G_p = (I + \Delta) G\) make \(\overline{\sigma}(W_2 G W_1)\) small; valid for frequencies at which \(\overline{\sigma}(G_S) \ll 1\)

Then a stabilizing controller Ks is synthesized for shaped plant Gs. The final positive feedback controller K is then constructed by combining the \(H_{\infty}\) controller Ks with the shaping functions W1 and W2 such that K = W1 Ks W2. In [1] is stated further that the given robust stabilization objective can be interpreted as a \(H_{\infty}\) problem formulation of minimizing the \(H_{\infty}\) norm of the frequency weighted gain from disturbances on the plant input and output to the controller input and output as follows: $$ \underset{K}{\min} \, || N(K) ||_{\infty}, $$$$ N = | W_{1}^{-1}; W_2 G | \ (I - K G)^{-1} \ | W_1, \ G W_{2}^{-1} | $$

[K, N] = ncfsyn (G, W1, W2, f) The function ncfsyn - the somewhat cryptic name stands for $normalized\; coprime\; factorization\; synthesis$ - allows the specification of an additional argument, factor f. Default value f = 1 implies that an optimal controller is required, whereas f > 1 implies that a suboptimal controller is required, achieving a performance that is f times less than optimal.

Inputs

G

LTI model of plant.

LTI model of precompensator. Model must be SISO or of appropriate size. An identity matrix is taken if W1 is not specified or if an empty model [] is passed.

LTI model of postcompensator. Model must be SISO or of appropriate size. An identity matrix is taken if W2 is not specified or if an empty model [] is passed.

factor = 1 implies that an optimal controller is required. factor > 1 implies that a suboptimal controller is required, achieving a performance that is factor times less than optimal. Default value is 1.

Outputs

K

State-space model of the H-infinity loop-shaping controller. Note that K is a $positive$ feedback controller.

State-space model of the closed loop depicted below.

Structure containing additional information.

L-infinity norm of N. gamma = norm (N, inf).

Nugap robustness. emax = inv (gamma).

Shaped plant. Gs = W2 * G * W1.

Controller for shaped plant. Ks = ncfsyn (Gs).

Estimates of the reciprocal condition numbers of the Riccati equations and a few other things. For details, see the description of the corresponding SLICOT routine.

Block Diagram of N

             ^ z1              ^ z2
             |                 |
  w1  +      |   +--------+    |            +--------+
 ----->(+)---+-->|   Ks   |----+--->(+)---->|   Gs   |----+
        ^ +      +--------+          ^      +--------+    |
        |                        w2  |                    |
        |                                                 |
        +-------------------------------------------------+
 

Algorithm
Uses SB10KD and SB10ZD, Copyright (c) 2020, SLICOT, available under the BSD 3-Clause (License and Disclaimer).

References
[1] D. McFarlane and K. Glover, A Loop Shaping Design Procedure Using H-infinity Synthesis, IEEE Transactions on Automatic Control, Vol. 37, No. 6, June 1992.
[2] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control: Analysis and Design: Second Edition. Wiley, Chichester, England, 2005.

Source Code: ncfsyn