Function Reference: ncfsyn

Function File: [K, N, gamma, info] = ncfsyn (G, W1, W2, factor)

Loop shaping H-infinity synthesis. Compute positive feedback controller using the McFarlane/Glover loop shaping design procedure [1]. Using a precompensator W1 and/or a postcompensator W2, the singular values of the nominal plant G are shaped to give a desired open-loop shape. The nominal plant G and shaping functions W1, W2 are combined to form the shaped plant, Gs where Gs = W2 G W1. We assume that W1 and W2 are such that Gs contains no hidden modes. It is relatively easy to approximate the closed-loop requirements by the following open-loop objectives [2]:

For disturbance rejection make
large; valid for frequencies at which
For noise attenuation make
small; valid for frequencies at which
For reference tracking make
large; valid for frequencies at which
For robust stability to a multiplicative output perturbation
small; valid for frequencies at which
```
.
```

Then a stabilizing controller Ks is synthesized for shaped plant Gs. The final positive feedback controller K is then constructed by combining the

H-infinity

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-infinity

problem formulation of minimizing the

H-infinity

norm of the frequency weighted gain from disturbances on the plant input and output to the controller input and output as follows:

                              -1            -1            -1
min ||N(K) ||  ,    N = |W1   |(I - K G)   |W1   G W2   |
 K            oo          |W2 G |

[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.
W1: 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.
W2: 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: 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.
N: State-space model of the closed loop depicted below.
info: Structure containing additional information.
info.gamma: L-infinity norm of N. gamma = norm (N, inf).
info.emax: Nugap robustness. emax = inv (gamma).
info.Gs: Shaped plant. Gs = W2 * G * W1.
info.Ks: Controller for shaped plant. Ks = ncfsyn (Gs).
info.rcond: 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 SLICOT SB10ID, SB10KD and SB10ZD by courtesy of NICONET e.V.

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.

Package: control