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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*).

The singular values of *S*(*j*ω) 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

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

where $$\left|{W}_{1}^{-1}\left(j\omega \right)\right|$$ is the desired disturbance attenuation factor. Allowing $$\left|{W}_{1}\left(j\omega \right)\right|$$ 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 *Δ _{A}* and multiplicative plant perturbations

Consider how the singular value Bode plot of complementary sensitivity
*T*(*s*) determines the stability margin for multiplicative
perturbations *Δ _{M}*. The multiplicative stability margin is, by definition, the "size" of the
smallest stable

**Additive/Multiplicative Uncertainty**

Taking $$\overline{\sigma}\left({\Delta}_{M}\left(j\omega \right)\right)$$ to be the definition of the "size" of *Δ _{M}*(

The size of the smallest destabilizing multiplicative uncertainty *Δ _{M}*(

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

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 *Δ _{A}*(

The size of the smallest destabilizing additive uncertainty *Δ _{A}* 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

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

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

where |*W*_{2}(*jω*)| and |*W*_{3}(*jω*)| 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 *Δ _{M}*, so that the control
design requirements can be written

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

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),

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

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

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

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 $$\underset{\xaf}{\sigma}\left(L\left(s\right)\right)\gg 1$$, 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
*σ _{i}*(

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,

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

which is precisely the quantity you obtain from Nichols chart
*M*-circles. Thus, $${\Vert T\Vert}_{\infty}$$ 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 $${\Vert T\Vert}_{\infty}$$, $${\Vert S\Vert}_{\infty}$$ to the classical gain margin *G _{M}* and
phase margin

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

(See [6].) These formulas are valid provided $${\Vert S\Vert}_{\infty}$$ and $${\Vert T\Vert}_{\infty}$$ 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
g _{m}* is defined to be the minimal amount by which the gains in
each loop would have to be

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

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`

.

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