Modern Control Engineering

In the robust control theory, we define unstructured uncertainty as. Since the

exact description of is unknown, we use an estimate of (as to the magnitude

and phase characteristics) and use this estimate in the design of the controller that sta-

bilizes the control system. Stability of a system with unstructured uncertainty can then

be examined by use of the small gain theorem to be given following the definition of the

norm.

Norm. The norm of a stable single-input–single-output system is the largest

possible amplification factor of the steady-state response to sinusoidal excitation.

For a scalar (s), gives the maximum value of. It is called the norm.

See Figure 10–41.

In robust control theory we measure the magnitude of the transfer function by the

norm. Assume that the transfer function is proper and stable. [Note that a

transfer function is called proper if is limited and definite. If = 0, it

is called strictly proper.] The norm of is defined by

means the maximum singular value of. ( means .) Note that

the singular value of a transfer function is defined by

where is the ith largest eigenvalue of and it is always a non-negative real

value. By making smaller, we make the effect of input won the output zsmaller.

It is frequently the case that instead of using the maximum singular value , we use

the inequality

and limit the magnitude of (s)by. To make the magnitude of small, we choose

a small and require that g 7 £ (^7) q 6 g.
£ g 7 £ (^7) q
7 £ (^7) q 6 g
7 £ (^7) q
7 £ (^7) q

li(££) ££

si(£)= 2 li(£*£)

£

s[£(jv)] [£(jv)] s smax

7 £ (^7) q=s[£(jv)]

Hq £(s)

£(s) £(q) £(q)

Hq £(s)

£ 7 £ (^7) q £(jv) Hq

H Hq

Hq

¢(s) ¢(s)

¢(s)

808 Chapter 10 / Control Systems Design in State Space

F(s)

|F ||F||

(jv

)| in db

v z

v

Figure 10–41

Bode diagram and

the norm .Hq 7 £ (^7) q
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