計算Real-μ及實數不確定性問題強健穩定控制器設計之區間法

Abstract

雖然奇異值強健測試可用來分析系統實數參數的變化，但是計算一個含有實數不確定性之系統轉移矩陣的結構化奇異值(mu-範數)是非常困難。目前為止，MATLAB mu-tool的軟體工具程式是最廣泛被使用做為結構化奇異值計算的數值方法，然而它只能求得mu-範數的上限和下限，故本文中，吾人提出一個由區間分析所發展出來可計算mu-範數的數值法。 在實務控制系統的設計上，經常會有一些設計參數是不確定，這些不確定參數或許會隨著溫度、溼度或其他環境變數改變，而這些改變將對系統的穩定性產生衝擊。吾人也將使用這個全域最佳法 - 區間法，來計算這些不確定參數的穩定裕度。 mu-合成法可用來設計實數不確定性問題之強健穩定控制器，在本文中，吾人使用區間演算法計算Real-mu，同時並以一個簡單的範例來說明，以mu-合成法處理實數不確定性問題時如何使用區間演算法並結合陡降法設計強健穩定控制器。

Singular value robustness tests can be used to analyze real parameter variations, but it is difficult to calculate the structured singular value (μ-norm) of a system transfer matrix with real uncertainty. Thus far, the MATLAB (Matrix Laboratory) with μ-tools toolbox software package is the most widely used method for numerical computation of structured singular values. However, it can only obtain the lower and upper bounds of the μ-norm. In this paper, we present a method developed from interval analysis which can compute the true value of the μ-norm in numerical computation. Frequently in practical control system design, some designing parameters are uncertain. These uncertain parameters may vary with temperature, humidity or other environmental variable, and these variations will have an impact on the stability of the system. We also use the global optimal method － interval method, by which stability margins of these uncertain parameters can be computed. m-synthesis can be used to design a robust stabilizing controller for real uncertainty problems. In this paper, we use the interval algorithm to calculate the structured singular value with respect to real uncertainty blocks. By a simple design example, we illustrate how to use the interval algorithm incorporate with steepest-descent method in μ-synthesis to design a robust stabilizing controller for real uncertainty problems.