The modern approach to characterizing closed-loop performance objectives is to
measure the size of certain closed-loop transfer function matrices using various
matrix norms. Matrix norms provide a measure of how large output signals can get for
certain classes of input signals. Optimizing these types of performance objectives
over the set of stabilizing controllers is the main thrust of recent optimal control
theory, such as *L*_{1},
*H*_{2},
*H _{∞}*, and optimal control.
Hence, it is important to understand how many types of control objectives can be
posed as a minimization of closed-loop transfer functions.

Consider a tracking problem, with disturbance rejection, measurement noise, and
control input signal limitations, as shown in Generalized and Weighted Performance Block Diagram.
*K* is some controller to be designed and
*G* is the system you want to control.

**Typical Closed-Loop Performance Objective**

A reasonable, though not precise, design objective would be to design
*K* to keep tracking errors and control input signal small
for all reasonable reference commands, sensor noises, and external force
disturbances.

Hence, a natural performance objective is the closed-loop gain from exogenous
influences (reference commands, sensor noise, and external force disturbances) to
regulated variables (tracking errors and control input signal). Specifically, let
*T* denote the closed-loop mapping from the outside
influences to the regulated variables:

You can assess performance by measuring the gain from *outside influences
to regulated variables*. In other words, good performance is
associated with *T* being small. Because the closed-loop system
is a multiinput, multioutput (MIMO) dynamic system, there are two different aspects
to the gain of *T*:

Spatial (

*vector*disturbances and*vector*errors)Temporal (dynamic relationship between input/output signals)

Hence the performance criterion must 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}*

The closed-loop performance objectives are formulated as weighted closed-loop
transfer functions that are to be made small through feedback. A generic
example, which includes many relevant terms, is shown in block diagram form in
Generalized and Weighted Performance Block Diagram. In the diagram,
*G* denotes the plant model and *K* is
the feedback controller.

**Generalized and Weighted Performance Block
Diagram**

The blocks in this figure might be scalar (SISO) and/or multivariable (MIMO),
depending on the specific example. The mathematical objective of
*H*_{∞} control is to make the
closed-loop MIMO transfer function
*T _{ed}* satisfy
∥

Performance requirements on the closed-loop system are transformed into the
*H*_{∞} framework with the
help of *weighting* or *scaling*
functions. Weights are selected to account for the relative magnitude of
signals, their frequency dependence, and their relative importance. This is
captured in the figure above, where the weights or scalings
[*W _{cmd}*,

Signal |
Meaning |
---|---|

$${\tilde{d}}_{1}$$ |
Normalized reference command Typical reference command in physical units |

$${\tilde{d}}_{2}$$ |
Normalized exogenous disturbances Typical exogenous disturbances in physical units |

$${\tilde{d}}_{3}$$ |
Normalized sensor noise Typical sensor noise in physical units |

$${\tilde{e}}_{1}$$ |
Weighted control signals Actual control signals in physical units |

$${\tilde{e}}_{2}$$ |
Weighted tracking errors Actual tracking errors in physical units |

$${\tilde{e}}_{3}$$ |
Weighted plant errors Actual plant errors in physical units |

**W _{cmd}**

*W _{cmd}* is included in

$${W}_{cmd}=\frac{3}{\frac{1}{2\cdot 2\pi}s+1}.$$

**W _{model}**

*W _{model}* represents a desired ideal
model for the closed-looped system and is often included in problem formulations
with tracking requirements. Inclusion of an ideal model for tracking is often
called a

$${W}_{model}=\frac{{\omega}^{2}}{{s}^{2}+2\zeta \omega +{\omega}^{2}}$$

for specific desired natural frequency ω and desired damping ratio ζ. Unit conversions might be necessary to ensure exact correlation between the ideal model and the closed-loop system. In the fighter pilot example, suppose that roll-rate is being commanded and 10º/second response is desired for each inch of stick motion. Then, in these units, the appropriate model is:

$${W}_{model}=10\frac{{\omega}^{2}}{{s}^{2}+2\zeta \omega +{\omega}^{2}}.$$

**W _{dist}**

*W _{dist}* shapes the frequency content
and magnitude of the exogenous disturbances affecting the plant. For example,
consider an electron microscope as the plant. The dominant performance objective
is to mechanically isolate the microscope from outside mechanical disturbances,
such as ground excitations, sound (pressure) waves, and air currents. You can
capture the spectrum and relative magnitudes of these disturbances with the
transfer function weighting matrix

**W _{perf1}**

*W _{perf}_{1}*
weights the difference between the response of the closed-loop system and the
ideal model

**W _{perf2}**

*W _{perf2}* penalizes variables
internal to the process

**W _{act}**

*W _{act}* is used to shape the penalty
on control signal use.

**W _{snois}**

*W _{snois}* represents frequency domain
models of sensor noise. Each sensor measurement feedback to the controller has
some noise, which is often higher in one frequency range than another. The

**H _{sens}**

*H _{sens}* represents a model of the
sensor dynamics or an external antialiasing filter. The transfer functions used
to describe

This generic block diagram has tremendous flexibility and many control
performance objectives can be formulated in the
*H*_{∞} framework using this
block diagram description.

Performance and robustness tradeoffs in control design were discussed in the
context of multivariable loop shaping in Tradeoff Between Performance and Robustness. In the *H _{∞}* control design
framework, you can include robustness objectives as additional disturbance to error
transfer functions — disturbances to be kept small. Consider the following
figure of a closed-loop feedback system with additive and multiplicative uncertainty
models.

The transfer function matrices are defined as:

$$\begin{array}{l}TF{\left(s\right)}_{{z}_{1}\to {w}_{1}}={T}_{I}\left(s\right)=KG{\left(I+GK\right)}^{-1}\\ TF{\left(s\right)}_{{z}_{2}\to {w}_{2}}=K{S}_{O}\left(s\right)=K{\left(I+GK\right)}^{-1}\end{array}$$

where *T _{I}*(

The *H*_{∞} control robustness
objective is now in the same format as the performance objectives, that is, to
minimize the *H*_{∞} norm of the
transfer matrix from *z*,
[*z*_{1},*z*_{2}],
to *w*,
[*w*_{1},*w*_{2}].

Weighting or scaling matrices are often introduced to shape the frequency and
magnitude content of the sensitivity and complementary sensitivity transfer function
matrices. Let *W _{M}* correspond to the
multiplicative uncertainty and

The multiplicative weighting or scaling
*W _{M}* represents a percentage error in
the model and is often small in magnitude at low frequency, between 0.05 and 0.20
(5% to 20% modeling error), and growing larger in magnitude at high frequency, 2 to
5 ((200% to 500% modeling error). The weight will transition by crossing a magnitude
value of 1, which corresponds to 100% uncertainty in the model, at a frequency at
least twice the bandwidth of the closed-loop system. A typical multiplicative weight
is

$${W}_{M}=0.10\frac{\frac{1}{5}s+1}{\frac{1}{200}s+1}.$$

By contrast, the additive weight or scaling
*W _{A}* represents an absolute error that
is often small at low frequency and large in magnitude at high frequency. The
magnitude of this weight depends directly on the magnitude of the plant model,

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 _{cmd}* =
1/

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