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Mixed-sensitivity loop shaping lets you design an
*H*_{∞} controller by simultaneously shaping the frequency
responses for tracking and disturbance rejection, noise reduction and robustness, and controller
effort. This technique is a useful way to balance the necessary tradeoff
between performance and robustness. To use this technique, you convert your desired
responses into up to three weighting functions that the `mixsyn`

command uses to synthesize the controller.

`mixsyn`

designs a controller *K* for your plant
*G*, assuming the standard control configuration of the following
diagram.

To do so, the function appends the weighting functions you provide,
*W*_{1}(*s*),
*W*_{2}(*s*), and
*W*_{3}(*s*), to the control system, as
shown in the following diagram.

`mixsyn`

treats the problem as an
*H*_{∞} synthesis problem (see `hinfsyn`

). It analyzes the weighted control system as
*LFT*(*P*,*K*), where *P*
is an augmented plant *P* such that {*z*;*e*} =
*P*{*w*;*u*}, as shown in the following diagram.

The transfer function from *w* to *z* can be expressed
as

$$M\left(s\right)=\left[\begin{array}{c}{W}_{1}S\\ {W}_{2}KS\\ {W}_{3}T\end{array}\right],$$

where

*S*= (*I*+*GK*)^{–1}is the sensitivity function.*KS*is the transfer function from*w*to*u*(the control effort).*T*= (*I*–*S*) =*GK*(*I*+*GK*)^{–1}is the complementary sensitivity function.

`mixsyn`

seeks a controller *K* that minimizes
||*M*(*s*)||_{∞}, the
*H*_{∞} norm (peak gain) of *M*. To do
so, it invokes `hinfsyn`

on the augmented plant ```
P =
augw(G,W1,W2,W3)
```

.

For loop gain *L* = *GK*, to achieve good reference
tracking and disturbance rejection, you typically want high loop gain at low frequency. To
achieve robustness and attenuation of measurement noise, you typically want *L*
to roll off at high frequency. This loop shape is equivalent to small *S* at
low frequency and small *T* at high frequency.

For mixed-sensitivity loop shaping, you choose weighting functions to specify those target
shapes for *S* and *T* as well as the control effort
*KS*. The *H*_{∞} design
constraint,

$${\Vert M\left(s\right)\Vert}_{\infty}={\Vert \left[\begin{array}{c}{W}_{1}S\\ {W}_{2}KS\\ {W}_{3}T\end{array}\right]\Vert}_{\infty}\le \text{\hspace{0.17em}}\text{\hspace{0.17em}}1,$$

means that

$$\begin{array}{c}{\Vert S\Vert}_{\infty}\le \left|{W}_{1}^{-1}\right|\\ {\Vert KS\Vert}_{\infty}\le \left|{W}_{2}^{-1}\right|\\ {\Vert T\Vert}_{\infty}\le \left|{W}_{3}^{-1}\right|.\end{array}$$

Therefore, you set the weights equal to the reciprocals of the desired shapes for
*S*, *KS*, and *T*. In particular,

For good reference-tracking and disturbance-rejection performance, choose

*W*_{1}large inside the control bandwidth to obtain small*S*.For robustness and noise attenuation, choose

*W*_{3}large outside the control bandwidth to obtain small*T*.To limit control effort in a particular frequency band, increase the magnitude of

*W*_{2}in this frequency band to obtain small*KS*.

`mixsyn`

returns the minimum
||*M*(*s*)||_{∞} in the output argument
`gamma`

. For the returned controller *K*, then,

$$\begin{array}{c}{\Vert S\Vert}_{\infty}\le \gamma \left|{W}_{1}^{-1}\right|\\ {\Vert KS\Vert}_{\infty}\le \gamma \left|{W}_{2}^{-1}\right|\\ {\Vert T\Vert}_{\infty}\le \gamma \left|{W}_{3}^{-1}\right|.\end{array}$$

If you do not want to restrict control effort, you can omit
*W*_{2}. In that case, `mixsyn`

minimizes the *H*_{∞} norm of

$$M\left(s\right)=\left[\begin{array}{c}{W}_{1}S\\ {W}_{3}T\end{array}\right].$$

You can use `makeweight`

to create weighting functions with the
desired gain profiles. The following example illustrates how to choose and create weighting
functions for controller design with `mixsyn`

.

Do not choose weighting functions with poles very close to *s* = 0
(*z* = 1 for discrete-time systems). For instance, although it might seem
sensible to choose *W*_{1} = 1/*s* to
enforce zero steady-state error, doing so introduces an unstable pole that cannot be
stabilized, causing synthesis to fail. Instead, choose *W*_{1} = 1/(*s* +
*δ*). The value *δ* must be small but not very small compared to
system dynamics. For instance, for best numeric results, if your target crossover frequency is
around 1 rad/s, choose *δ* = 0.0001 or 0.001. Similarly, in discrete time,
choose sample times such that system and weighting dynamics are not more than a decade or two
below the Nyquist frequency.

Load a plant model for a mixed-sensitivity${\mathit{H}}_{\infty}$ controller design. This two-input, two-output, six-state model is described in the example Loop Shaping of HIMAT Pitch Axis Controller.

load mixsynExampleData G size(G)

State-space model with 2 outputs, 2 inputs, and 6 states.

To design a controller for performance and robustness, shape the sensitivity and complementary sensitivity functions. Choose weights that are the inverse of the desired shapes.

To achieve good reference-tracking and disturbance-rejection performance, shape *S* to be small inside the control bandwidth, which means choosing `W1`

large at low frequency, rolling off at high frequency. For this example, specify `W1`

with:

Low-frequency gain of about 30 dB (33 in absolute units)

High-frequency gain of about –6 dB (0.5 in absolute units)

0 dB crossover at about 5 rad/s.

W1 = makeweight(33,5,0.5);

For robustness and noise attenuation, shape `T`

to be small outside the control bandwidth, which means choosing `W3`

large at high frequency.

W3 = makeweight(0.5,20,20);

Examine both weighting functions. Their inverses are the target shapes for `S`

and `T`

.

bodemag(W1,W3) yline(0,'--'); legend('W1','W3','0 dB') grid on

Because `S + T = I`

, `mixsyn`

cannot make both `S`

and `T`

small (less than 0 dB) in the same frequency range. Therefore, when you specify weights, there must be a frequency band in which both `W1`

and `W3`

are below 0 dB.

Use `mixsyn`

to compute the optimal mixed-sensitivity controller with these weights. For this example, impose no penalty on controller effort by setting `W2`

to `[]`

.

[K,CL,gamma] = mixsyn(G,W1,[],W3); gamma

gamma = 0.7331

The resulting `gamma`

, which is the peak singular value across all frequencies, is well below 1, indicating that the closed-loop system meets the design requirements. Examine the resulting system responses. First, compare the resulting sensitivity `S`

and complementary sensitivity `T`

to the corresponding weighting functions `W1`

and `W3`

.

L = G*K; I = eye(size(L)); S = feedback(I,L); T= I-S; sigma(S,'b',W1,'b--',T,'r',W3,'r--',{0.1,1000}) legend('S','W1','T','W3')

The plot shows that `S`

and `T`

achieve the desired loop shape, where `S`

is large inside the control bandwidth and a is small outside the control bandwidth.

To see how mixed-sensitivity loop-shaping achieves the goals of classic loop shaping, compare the open-loop response `L`

to the weighting functions. `L ~ W1`

where `W1`

is large, and `L ~ 1/W3`

where `W3`

is large.

sigma(L,'b',W1,'r--',1/W3,'g--',{0.1,1000}) legend('L','W1','1/W3')