Abstract
Based on the control storage function approach, a constructive method for designing simultaneous H∞ controllers for a collection of nonlinear control systems in strict-feedback form is developed. It is shown that under mild assumptions, common control storage functions (CSFs) for nonlinear systems in strict-feedback form can be constructed systematically. Based on the obtained common CSFs, an explicit formula for constructing simultaneous H∞ controllers is presented. Finally, an illustrative example is provided to verify the obtained theoretical results.
Keywords
- nonlinear control systems
- simultaneous H∞ control
- state feedback
- storage functions
- strict-feedback form
1. Introduction
The simultaneous
All the results mentioned earlier are derived for linear systems case. Till now, only few results have been reported about simultaneous
2. Problem formulation and preliminaries
In this section, the simultaneous
2.1. Problem formulation
Consider a collection of nonlinear control systems:
where
where
Suppose that the following assumption holds.
It is clear that we can always find a positive (semi)definite function
The objective of this chapter is to find a continuous function
internally stabilizes the systems in Eq. (3) simultaneously; and, for each
2.2. Control storage functions
Here we review some important concepts about the CSF method introduced in references [7, 9].
For ensuring the continuity of the obtained simultaneous
3. Main results
For a single system, it has been shown in reference [7] that the existence of CSFs is a necessary and sufficient condition for the existence of
Let
It is easy to show that we can find a function
For
Similarly, we can find functions
Then, it is clear that the function
is positive definite, and radially unbounded.
Now, we discuss the existence of common CSFs for the systems in Eq. (3). For convenience, we say that a continuous function
By the backstepping method, we can show that
After some manipulations, we have
where
Therefore,
As
Notice that
This shows that
Now we prove that
where the continuous function
Note that such
This implies that
To derive simultaneous
where β
is a simultaneous
A.
Since
Then, with the controller defined in Eq. (8), all the closed-loop systems satisfy the
In this case,
In this case, since
Similarly, in this case we can show that
These discussions imply that Eq. (9) holds. That is, all the possible closed-loop systems satisfy the
B. Internal stability
To prove internal stability, notice that Eq. (6) implies that, along the trajectories of system
That is, for each
This shows that all the closed-loop systems are internally stable.
4. An illustrative example
Consider the following nonlinear systems:
where
It can be shown that
with
Let γ = 3. It can be verified that
Then,
is positive, definite, and radially unbounded. By choosing
Therefore,
is a common CSF for the three systems in Eq. (10). For
and (with β1 = β2 = β3 = 0.1)
From
is a simultaneous
5. Conclusions
In this chapter, a systematic way for constructing simultaneous
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