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There are several ways of defining norms of a scalar signal *e*(*t*)
in the time domain. We will often use the 2-norm, (*L*_{2}-norm), for
mathematical convenience, which is defined as

$${\Vert e\Vert}_{2}:={\left({\displaystyle {\int}_{-\infty}^{\infty}e{\left(t\right)}^{2}dt}\right)}^{\frac{1}{2}}.$$

If this integral is finite, then the signal *e* is *square
integrable*, denoted as *e* ∊
L_{2}. For vector-valued signals

$$e\left(t\right)=\left[\begin{array}{c}{e}_{1}\left(t\right)\\ {e}_{2}\left(t\right)\\ \vdots \\ {e}_{n}\left(t\right)\end{array}\right],$$

the 2-norm is defined as

$$\begin{array}{c}{\Vert e\Vert}_{2}:={\left({\displaystyle {\int}_{-\infty}^{\infty}{\Vert e\left(t\right)\Vert}_{2}^{2}dt}\right)}^{\frac{1}{2}}\\ ={\left({\displaystyle {\int}_{-\infty}^{\infty}{e}^{T}\left(t\right)e\left(t\right)dt}\right)}^{\frac{1}{2}}.\end{array}$$

In µ-tools the dynamic systems we deal with are exclusively linear, with state-space model

$$\left[\begin{array}{c}\dot{x}\\ e\end{array}\right]=\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]\left[\begin{array}{c}x\\ d\end{array}\right],$$

or, in the transfer function form,

*e*(*s*)* = T*(*s*)*d*(*s*)*, T*(*s*)*:=
C(sI – A)*^{–1}*B
+ D *

Two mathematically convenient measures of the transfer matrix *T*(*s*)
in the frequency domain are the matrix H_{2} and *H*_{∞} norms,

$$\begin{array}{l}{\Vert T\Vert}_{2}:={\left[\frac{1}{2\pi}{\displaystyle {\int}_{-\infty}^{\infty}{\Vert T\left(j\omega \right)\Vert}_{F}^{2}d\omega}\right]}^{\frac{1}{2}}\\ {\Vert T\Vert}_{\infty}:=\underset{\omega \in R}{\mathrm{max}\overline{\sigma}}\left[T\left(j\omega \right)\right],\end{array}$$

where the Frobenius norm (see the MATLAB^{®} `norm`

command) of a complex matrix *M* is

$${\Vert M\Vert}_{F}:=\sqrt{\text{Trace}\left({M}^{*}M\right)}.$$

Both of these transfer function norms have input/output time-domain
interpretations. If, starting from initial condition *x*(0)
= 0, two signals *d* and *e* are
related by

$$\left[\begin{array}{c}\dot{x}\\ e\end{array}\right]=\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]\left[\begin{array}{c}x\\ d\end{array}\right],$$

then

For

*d*, a unit intensity, white noise process, the steady-state variance of*e*is ∥*T*∥_{2}.The

*L*_{2}(or RMS) gain from,*d*→*e*$$\underset{d\ne 0}{\mathrm{max}}\frac{{\Vert e\Vert}_{2}}{{\Vert d\Vert}_{2}}$$

is equal to ∥

*T*∥_{∞}. This is discussed in greater detail in the next section.

Any performance criterion must also account for

Relative magnitude of outside influences

Frequency dependence of signals

Relative importance of the magnitudes of regulated variables

So, if the performance objective is in the form of a matrix
norm, it should actually be a *weighted norm*

∥*W _{L}*

where
the weighting function matrices *W _{L}* and

**Unweighted MIMO System**

Suppose *T* is a MIMO stable linear system,
with transfer function matrix *T*(*s*).
For a given driving signal $$\tilde{d}\left(t\right)$$,
define $$\tilde{e}$$ as
the output, as shown below.

Note that it is more traditional to write the diagram in Unweighted MIMO System: Vectors from Left to Right with the arrows going from left to right as in Weighted MIMO System.

**Unweighted MIMO System: Vectors from Left
to Right**

The two diagrams shown above represent the exact same system. We prefer to write these block diagrams with the arrows going right to left to be consistent with matrix and operator composition.

Assume that the dimensions of *T* are *n _{e}* ×

$$\beta :={\Vert T\Vert}_{\infty}:=\underset{\omega \in R}{\mathrm{max}\overline{\sigma}\left[T\left(j\omega \right)\right]}.$$

Now consider a response, starting from initial condition equal to 0. In that case, Parseval's theorem gives that

$$\frac{{\Vert \tilde{e}\Vert}_{2}}{{\Vert \tilde{d}\Vert}_{2}}=\frac{{\left[{\displaystyle {\int}_{0}^{\infty}{\tilde{e}}^{T}\left(t\right)}\text{\hspace{0.17em}}\tilde{e}\left(t\right)dt\right]}^{\frac{1}{2}}}{{\left[{\displaystyle {\int}_{0}^{\infty}{\tilde{d}}^{T}\left(t\right)}\text{\hspace{0.17em}}\tilde{d}\left(t\right)dt\right]}^{\frac{1}{2}}}\le \beta .$$

Moreover, there are specific disturbances *d* that
result in the ratio $${\Vert \tilde{e}\Vert}_{2}/{\Vert \tilde{d}\Vert}_{2}$$ arbitrarily
close to β. Because of this, ∥*T*∥_{∞} is
referred to as the *L*_{2} (or
RMS) gain of the system.

As you would expect, a sinusoidal, steady-state interpretation
of ∥*T*∥_{∞} is
also possible: For any frequency $$\overline{\omega}\in R$$,
any vector of amplitudes $$a\in {R}_{{n}_{d}}$$,
and any vector of phases $$\varphi \in {R}^{{n}_{d}}$$,
with ∥* a*∥

$$\tilde{d}\left(t\right)=\left[\begin{array}{c}{a}_{1}\mathrm{sin}\left(\overline{\omega}t+{\varphi}_{1}\right)\\ \vdots \\ {a}_{{n}_{d}}\mathrm{sin}\left(\overline{\omega}t+{\varphi}_{{n}_{d}}\right)\end{array}\right].$$

Applying this input to the system *T* results
in a steady-state response $${\tilde{e}}_{ss}$$ of
the form

$${\tilde{e}}_{ss}\left(t\right)=\left[\begin{array}{c}{b}_{1}\mathrm{sin}\left(\overline{\omega}t+{\varphi}_{1}\right)\\ \vdots \\ {b}_{{n}_{e}}\mathrm{sin}\left(\overline{\omega}t+{\varphi}_{{n}_{e}}\right)\end{array}\right].$$

The vector $$b\in {R}^{{n}_{e}}$$ will
satisfy ∥*b*∥_{2} ≤
β. Moreover, β, as defined in Weighted MIMO System, is the smallest
number such that this is true for every ∥*a*∥_{2} ≤
1, $$\overline{\omega}$$,
and * ϕ*.

Note that in this interpretation, the vectors of the sinusoidal
magnitude responses are unweighted, and measured in Euclidean norm.
If realistic multivariable performance objectives are to be represented
by a single MIMO ∥·∥_{∞} objective
on a closed-loop transfer function, additional scalings are necessary.
Because many different objectives are being lumped into one matrix
and the associated cost is the norm of the matrix, it is important
to use frequency-dependent weighting functions, so that different
requirements can be meaningfully combined into a single cost function.
Diagonal weights are most easily interpreted.

Consider the diagram of Weighted MIMO System, along with Unweighted MIMO System: Vectors from Left to Right.

Assume that *W _{L}* and

$${W}_{L}=\left[\begin{array}{cccc}{L}_{1}& 0& \dots & 0\\ 0& {L}_{2}& \dots & 0\\ \vdots & \vdots & \ddots & \vdots \\ 0& 0& \dots & {L}_{{n}_{e}}\end{array}\right],\text{\hspace{1em}}{W}_{R}=\left[\begin{array}{cccc}{R}_{1}& 0& \dots & 0\\ 0& {R}_{2}& \dots & 0\\ \vdots & \vdots & \ddots & \vdots \\ 0& 0& \dots & {R}_{{n}_{d}}\end{array}\right].$$

**Weighted MIMO System**

Bounds on the quantity ∥*W _{L}*

$${\tilde{e}}_{ss}\left(t\right)=\left[\begin{array}{c}{\tilde{e}}_{1}\mathrm{sin}\left(\overline{\omega}t+{\varphi}_{1}\right)\\ \vdots \\ {\tilde{e}}_{{n}_{e}}\mathrm{sin}\left(\overline{\omega}t+{\varphi}_{{n}_{d}}\right)\end{array}\right]$$ | (5-1) |

satisfies

$$\sum _{i=1}^{{n}_{e}}{\left|{W}_{{L}_{i}}\left(j\overline{\omega}\right){\tilde{e}}_{i}\right|}^{2}}\le 1$$

for all sinusoidal input signals $$\tilde{d}$$ of the form

$$\tilde{d}\left(t\right)=\left[\begin{array}{c}{\tilde{d}}_{1}\mathrm{sin}\left(\overline{\omega}t+{\varphi}_{1}\right)\\ \vdots \\ {\tilde{d}}_{{n}_{e}}\mathrm{sin}\left(\overline{\omega}t+{\varphi}_{{n}_{d}}\right)\end{array}\right]$$ | (5-2) |

satisfying

$$\sum _{i=1}^{{n}_{d}}\frac{{\left|{\tilde{d}}_{i}\right|}^{2}}{{\left|{W}_{{R}_{i}}\left(j\overline{\omega}\right)\right|}^{2}}}\le 1$$

if and only if ∥*W _{L}*

This approximately (*very* approximately
— the next statement is not actually correct) implies that
∥*W _{L}*

$$\left|{\tilde{d}}_{i}\right|\le \left|{W}_{{R}_{i}}\left(j\overline{\omega}\right)\right|$$

the steady-state error components will satisfy

$$\left|{\tilde{e}}_{i}\right|\le \frac{1}{\left|{W}_{{L}_{i}}\left(j\overline{\omega}\right)\right|}.$$

This shows how one could pick performance weights to reflect the desired
frequency-dependent performance objective. Use *W _{R}* to
represent the relative magnitude of sinusoids disturbances that might
be present, and use

Remember, however, that the weighted *H*_{∞} norm
does *not* actually give element-by-element bounds
on the components of $$\tilde{e}$$ based
on element-by-element bounds on the components of $$\tilde{d}$$.
The precise bound it gives is in terms of Euclidean norms of the components of $$\tilde{e}$$ and $$\tilde{d}$$ (weighted
appropriately by *W _{L}*(

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