2.2. MODELING UNCERTAINSYSTEMS 15
as specifying a maximum percentage error between Pnom and everyother element of P.
The system Pnom(s) is the element of Pthat comes from ∆ = 0 and is called the
nominal system. In otherwords, for ∆ = 0, the input-output relationship is
y(s)=P
nom(s)u(s). As ∆ deviates from zero (but remains bounded insize), the
nominal system is multiplied by (I+∆W
m
(s)). Wm(s)is a frequency weighting function
which allows us the specify the maximum effect of the perturbation for each frequency.
Including Wm(s) allows us to model Pwith ∆ being bounded by one. Any
normalization of ∆ is simply included in Wm(s).
We often assume that ∆ is also linear and time-invariant. This means that ∆(ω)is
simply an unknown, complex valued matrix at each frequency,ω.Ifk∆k
∞≤1, then, at
each frequency,σmax(∆(ω )) ≤1. Section 2.2.3 gives a further discussion on the pros
and cons of considering ∆ to be linear, time-invariant.
Now consider an example of this approachfrom a Nyquist point of view. Asimple first
order SISO system with multiplicative output uncertaintyis mo deled as
y(s)=(I+W
m
(s)∆)Pnom(s)u(s),
where
Pnom(s)=1+0.05s
1+sand Wm(s)=0.1+0.2s
1+0.05s.
Figure 2.3 illustrates the set of systems generated by a linear time-invariant∆,
k∆k∞≤1.
At each frequency,ω, the transfer function of every element of P, lies within a circle,
centered at Pnom(ω), of radius |Pnom(ω)Wm(ω)|. Notethat for cer tain frequencies the
disks enclose the origin. This allows us to consider perturbed systems that are
non-minimum phase even though the nominal system is not.
It is worth pointingout that Pis still a mo del; in this case a set of regions in the
Nyquist plane. This is model set is now able to describe a larger set of system behaviors
than a single nominal model. There is still an inevita ble mismatch betweenany model
(robust control model set or otherwise)and the b ehaviorsof a physical system.