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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 [1]-[2]. 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
*W*_{1} and
*W*_{2}. The function assumes the positive feedback
configuration of the following illustration.

To specify negative feedback, replace *G* by –*G*. The
controller *K _{s}* stabilizes a family of systems given
by a ball of uncertainty in the normalized coprime factors of the shaped plant

`ncfsyn`

is obtained as `[`

computes the Glover-McFarlane `K`

,`CL`

,`gamma`

,`info`

] = ncfsyn(`G`

)*H*_{∞} normalized
coprime factor loop-shaping controller `K`

for the plant
`G`

, with *W*_{1} =
*W*_{2} = *I*. `CL`

is the closed-loop system from the disturbances
*w*_{1} and
*w*_{2} to the outputs
*z*_{1} and
*z*_{2}. The function also returns the
*H*_{∞} performance `gamma`

, and a
structure containing additional information about the result.

While

`ncfmargin`

assumes a negative-feedback loop, the`ncfsyn`

command designs a controller for a positive-feedback loop. Therefore, to compute the margin using a controller designed with`ncfsyn`

, use`[marg,freq] = ncfmargin(G,K,+1)`

.

The returned controller *K* =
*W*_{1}*K _{s}*

$$\gamma \left({K}_{s}\right)={\Vert \left[\begin{array}{c}I\\ {K}_{s}\end{array}\right]{(I-{G}_{s}{K}_{s})}^{-1}[I,{G}_{s}]\Vert}_{\infty}={\Vert \left[\begin{array}{c}I\\ {G}_{s}\end{array}\right]{(I-{K}_{s}{G}_{s})}^{-1}[I,{K}_{s}]\Vert}_{\infty}.$$

The optimal performance is the minimal cost

$$\gamma :=\underset{{K}_{s}}{\mathrm{min}}\gamma \left({K}_{s}\right).$$

Suppose that
*G _{s}*=

$${\tilde{G}}_{s}=(N+{\Delta}_{1}){(M+{\Delta}_{2})}^{-1}$$

where Δ_{1}, Δ_{2} are a stable pair
satisfying

$${\Vert \left[\begin{array}{c}{\Delta}_{1}\\ {\Delta}_{2}\end{array}\right]\Vert}_{\infty}<MARG:=\frac{1}{\gamma}.$$

The closed-loop *H*_{∞}-norm objective has the
standard signal gain interpretation. Finally it can be shown that the controller,
*K _{s}*, does not substantially affect the loop shape
in frequencies where the gain of

[1] McFarlane, Duncan C., and Keith Glover, eds. *Robust
Controller Design using Normalized Coprime Factor Plant Descriptions*. Vol. 138.
Lecture Notes in Control and Information Sciences. Berlin/Heidelberg: Springer-Verlag, 1990.
https://doi.org/10.1007/BFb0043199.

[2] McFarlane, D., and K. Glover, “A Loop Shaping Design Procedure
using H_{∞} Synthesis,” *IEEE Transactions on Automatic
Control,* no. 6 (June 1992): pp. 759–69. https://doi.org/10.1109/9.256330.

[3] Vinnicombe, Glenn. “Measuring Robustness of Feedback Systems.” PhD Dissertation, University of Cambridge, 1992.

[4] Zhou, Kemin, and John Comstock Doyle. *Essentials of
Robust Control*. Upper Saddle River, NJ: Prentice-Hall, 1998.