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.
- nonlinear control systems
- simultaneous H∞ control
- state feedback
- storage functions
- strict-feedback form
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 is the state,
where , and , are smooth functions with θ
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 such that the state feedback controller
internally stabilizes the systems in Eq. (3) simultaneously; and, for each
2.2. Control storage functions
For ensuring the continuity of the obtained simultaneous
3. Main results
For a single system, it has been shown in reference  that the existence of CSFs is a necessary and sufficient condition for the existence of
It is easy to show that we can find a function and a positive definite function such that
Similarly, we can find functions , and positive definite function , such that
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 is dominated by a continuous function if there exists a constant
By the backstepping method, we can show that
After some manipulations, we have
Therefore, is a CSF of
As is dominated by , we can choose a
Now we prove that
where the continuous function with is such that if
Note that such always exists since is dominated by . Clearly,
This implies that
To derive simultaneous
is a simultaneous
Since , if we can show that
In this case,
In this case, since
Similarly, in this case we can show that
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:
It can be shown that
Let γ = 3. It can be verified that
is positive, definite, and radially unbounded. By choosing
is a common CSF for the three systems in Eq. (10). For
and (with β1 = β2 = β3 = 0.1)
is a simultaneous
In this chapter, a systematic way for constructing simultaneous
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