# Documentation

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## Typical Loop Shapes, S and T Design

Consider the multivariable feedback control system shown in the following figure. In order to quantify the multivariable stability margins and performance of such systems, you can use the singular values of the closed-loop transfer function matrices from r to each of the three outputs e, u, and y, viz.

`$\begin{array}{l}S\left(s\right)\stackrel{def}{=}{\left(I+L\left(s\right)\right)}^{-1}\\ R\left(s\right)\stackrel{def}{=}K\left(s\right){\left(I+L\left(s\right)\right)}^{-1}\\ T\left(s\right)\stackrel{def}{=}L\left(s\right){\left(I+L\left(s\right)\right)}^{-1}=I-S\left(s\right)\end{array}$`

where the L(s) is the loop transfer function matrix

 $L\left(s\right)=G\left(s\right)K\left(s\right).$ (2-1)

Block Diagram of the Multivariable Feedback Control System

The two matrices S(s) and T(s) are known as the sensitivity function and complementary sensitivity function, respectively. The matrix R(s) has no common name. The singular value Bode plots of each of the three transfer function matrices S(s), R(s), and T(s) play an important role in robust multivariable control system design. The singular values of the loop transfer function matrix L(s) are important because L(s) determines the matrices S(s) and T(s).

### Robustness in Terms of Singular Values

The singular values of S(jω) determine the disturbance attenuation, because S(s) is in fact the closed-loop transfer function from disturbance d to plant output y — see Block Diagram of the Multivariable Feedback Control System. Thus a disturbance attenuation performance specification can be written as

 $\overline{\sigma }\left(S\left(j\omega \right)\right)\le |{W}_{1}^{-1}\left(j\omega \right)|$ (2-2)

where $|{W}_{1}^{-1}\left(j\omega \right)|$ is the desired disturbance attenuation factor. Allowing $|{W}_{1}\left(j\omega \right)|$ to depend on frequency ω enables you to specify a different attenuation factor for each frequency ω.

The singular value Bode plots of R(s) and of T(s) are used to measure the stability margins of multivariable feedback designs in the face of additive plant perturbations ΔA and multiplicative plant perturbations ΔM, respectively. See the following figure.

Consider how the singular value Bode plot of complementary sensitivity T(s) determines the stability margin for multiplicative perturbations ΔM. The multiplicative stability margin is, by definition, the "size" of the smallest stable ΔM(s) that destabilizes the system in the figure below when ΔA = 0.

Taking $\overline{\sigma }\left({\Delta }_{M}\left(j\omega \right)\right)$ to be the definition of the "size" of ΔM(), you have the following useful characterization of "multiplicative" stability robustness:

### Multiplicative Robustness:

The size of the smallest destabilizing multiplicative uncertainty ΔM(s) is:

`$\overline{\sigma }\left({\Delta }_{M}\left(j\omega \right)\right)=\frac{1}{\overline{\sigma }\left(T\left(j\omega \right)\right)}.$`

The smaller is $\overline{\sigma }\left(T\left(j\omega \right)\right)$, the greater will be the size of the smallest destabilizing multiplicative perturbation, and hence the greater will be the stability margin.

A similar result is available for relating the stability margin in the face of additive plant perturbations ΔA(s) to R(s) if you take $\overline{\sigma }\left({\Delta }_{A}\left(j\omega \right)\right)$ to be the definition of the "size" of ΔA() at frequency ω.

The size of the smallest destabilizing additive uncertainty ΔA is:

`$\overline{\sigma }\left({\Delta }_{A}\left(j\omega \right)\right)=\frac{1}{\overline{\sigma }\left(R\left(j\omega \right)\right)}.$`

As a consequence of robustness theorems 1 and 2, it is common to specify the stability margins of control systems via singular value inequalities such as

 $\overline{\sigma }\left(R\left\{j\omega \right\}\right)\le |{W}_{2}^{-1}\left(j\omega \right)|$ (2-3)
 $\overline{\sigma }\left(T\left\{j\omega \right\}\right)\le |{W}_{3}^{-1}\left(j\omega \right)|$ (2-4)

where |W2()| and |W3()| are the respective sizes of the largest anticipated additive and multiplicative plant perturbations.

It is common practice to lump the effects of all plant uncertainty into a single fictitious multiplicative perturbation ΔM, so that the control design requirements can be written

`$\frac{1}{{\sigma }_{i}\left(S\left(j\omega \right)\right)}\ge |{W}_{1}\left(j\omega \right)|;\text{ }{\overline{\sigma }}_{i}\left(T\left[j\omega \right]\right)\le |{W}_{3}^{-1}\left(j\omega \right)|$`

as shown in Singular Value Specifications on L, S, and T.

It is interesting to note that in the upper half of the figure (above the 0 dB line),

`$\underset{¯}{\sigma }\left(L\left(j\omega \right)\right)\approx \frac{1}{\overline{\sigma }\left(S\left(j\omega \right)\right)}$`

while in the lower half of Singular Value Specifications on L, S, and T (below the 0 dB line),

`$\underset{¯}{\sigma }\left(L\left(j\omega \right)\right)\approx \overline{\sigma }\left(T\left(j\omega \right)\right).$`

This results from the fact that

`$S\left(s\right)\stackrel{def}{=}{\left(I+L\left(s\right)\right)}^{-1}\approx L{\left(s\right)}^{-1}$`

if $\underset{¯}{\sigma }\left(L\left(s\right)\right)\gg 1$, and

`$T\left(s\right)\stackrel{def}{=}L\left(s\right){\left(I+L\left(s\right)\right)}^{-1}\approx L\left(s\right)$`

if $\overline{\sigma }\left(L\left(s\right)\right)\ll 1$.

Singular Value Specifications on L, S, and T

Thus, it is not uncommon to see specifications on disturbance attenuation and multiplicative stability margin expressed directly in terms of forbidden regions for the Bode plots of σi(L()) as "singular value loop shaping" requirements, either as specified upper/lower bounds or as a target desired loop shape — see the preceding figure.

### Guaranteed Gain/Phase Margins in MIMO Systems

For those who are more comfortable with classical single-loop concepts, there are the important connections between the multiplicative stability margins predicted by $\overline{\sigma }\left(T\right)$ and those predicted by classical M-circles, as found on the Nichols chart. Indeed in the single-input/single-output case,

`$\overline{\sigma }\left(T\left(j\omega \right)\right)=|\frac{L\left(j\omega \right)}{1+L\left(j\omega \right)}|$`

which is precisely the quantity you obtain from Nichols chart M-circles. Thus, ${‖T‖}_{\infty }$ is a multiloop generalization of the closed-loop resonant peak magnitude which, as classical control experts will recognize, is closely related to the damping ratio of the dominant closed-loop poles. Also, it turns out that you can relate ${‖T‖}_{\infty }$, ${‖S‖}_{\infty }$ to the classical gain margin GM and phase margin θM in each feedback loop of the multivariable feedback system of Block Diagram of the Multivariable Feedback Control System via the formulas:

`$\begin{array}{l}{G}_{M}\ge 1+\frac{1}{{‖T‖}_{\infty }}\\ {G}_{M}\ge 1+\frac{1}{1-\frac{1}{{‖S‖}_{\infty }}}\\ {\theta }_{M}\ge 2{\mathrm{sin}}^{-1}\left(\frac{1}{2{‖T‖}_{\infty }}\right)\\ {\theta }_{M}\ge 2{\mathrm{sin}}^{-1}\left(\frac{1}{2{‖T‖}_{\infty }}\right).\end{array}$`

(See [6].) These formulas are valid provided ${‖S‖}_{\infty }$ and ${‖T‖}_{\infty }$ are larger than 1, as is normally the case. The margins apply even when the gain perturbations or phase perturbations occur simultaneously in several feedback channels.

The infinity norms of S and T also yield gain reduction tolerances. The gain reduction tolerance gm is defined to be the minimal amount by which the gains in each loop would have to be decreased in order to destabilize the system. Upper bounds on gm are as follows:

`$\begin{array}{l}{g}_{M}\le 1-\frac{1}{{‖T‖}_{\infty }}\\ {g}_{M}\le \frac{1}{1+\frac{1}{{‖S‖}_{\infty }}}.\end{array}$`

### Using LOOPSYN to Do H-Infinity Loop Shaping

The command `loopsyn` lets you design a stabilizing feedback controller to optimally shape the open loop frequency response of a MIMO feedback control system to match as closely as possible a desired loop shape Gd. The basic syntax of the `loopsyn` loop-shaping controller synthesis command is:

```K = loopsyn(G,Gd) ```

Here `G` is the LTI transfer function matrix of a MIMO plant model, `Gd` is the target desired loop shape for the loop transfer function `L=G*K`, and `K` is the optimal loop-shaping controller. The LTI controller `K` has the property that it shapes the loop `L=G*K` so that it matches the frequency response of `Gd` as closely as possible, subject to the constraint that the compensator must stabilize the plant model `G`.