Minimizing a convex combination of the overshoot and undershoot
Introduction
As before, it is still a hard task to check the feasibility of given time-domain design specifications and to merge them into controller synthesis, even though we are now armed with more advanced results in the area of time-domain optimal control techniques such as L1 [1] and L∞ [6], [11], [12]. In a tracking system design with a fixed reference, for example, we have nontrivial questions:
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Is there any controller such that the resulting tracking error has no overshoot (positive error)? If many, which one gives the smallest value? If none, what is the best result we can achieve?
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Often controllers designed for a small overshoot give a large undershoot and vice versa. What are the quantitative aspects of this phenomenon?
The motivation and objective of our study come from the two open questions above. We choose a new cost function that discriminates positive and negative errors under the L∞ framework developed in [6]. Having an asymmetric cost function, we can obtain useful answers to the above questions against increased mathematical difficulties.
This paper is divided into five sections. Section 2 gives a mathematical preliminary. We define our problem in Section 3.1 and develop its dual formulation in Section 3.2. In Section 3.3, we study on the existence of the optimal solution and specify its structure. In Section 3.4, we examine an interrelation of the overshoot and undershoot. We give a numerical example in Section 4 and the conclusion in Section 5.
Section snippets
Mathematical preliminary
Let Lp, 1⩽p⩽∞, denote the space of Lebesque measurable functions from to R, the set of real number, with bounded ∥·∥p norms. For a x∈Lp, we define its positive and negative projections;Clearly, it holds that 0⩽∥x±∥⩽∥x∥1, ∥x+∥1+∥x−∥1=∥x∥1 and ∥x∥1=∥x+∥1⇔x⩾0.
Let RH∞ denote the set of proper and stable transfer functions. Throughout this paper, with a function , its Laplace transform is denoted by . Similarly, given a , g denotes its
Problem definition
Overall our problem formulation comes from that of L∞ optimal control problem developed in [6]. Remarkable differences include a new cost function and corresponding modification in rational approximation results (Lemma 1 below). First, we shortly summarize L∞ problem formulation.
Consider a generalized system configuration in Fig. 1, where G is a finite dimensional linear time-invariant plant and all signals , , , are assumed to be scalar valued.
From the YJBK parameterization, every
An illustrative example
Our problem is to minimize fλ(φ) where , G=NM−1=(s−2/s+1)/(s−1/s+1) and , the unit step, and K is the controller to be designed.
Following standard procedures we have the next SPM problem:Since r.d., Lemma 1 implies that there exist a rational controller whose performance is arbitrary close to the optimal value.
With two real zeros s1=1 and s2=2, we haveand the dual SPM problem, with
Conclusion
We have dealt with the problem of minimizing a convex combination of the overshoot and undershoot of SISO continuous time system due to a fixed input. It was shown that every optimal solution, if existing, is a sum of delayed step functions whose positive and negative magnitude are generally different. In addition, we analytically explained the interplay of the overshoot and undershoot in controller synthesis. We believe our results are significant generalizations of the L∞ optimal problem.
We
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