A Linear Programming Approach to the Synthesis of Fixed-Structure Controllers
A Linear Programming Approach to the Synthesis of Fixed-Structure Controllers
复制标题
固定结构控制器综合的线性规划方法
DOI：
10.1109/tac.2008.927790
复制
发表时间：
2008
影响因子：
6.8
通讯作者：
S. Bhattacharyya
中科院分区：
文献类型：
--
作者：
Waqar A. Malik;S. Darbha;S. Bhattacharyya
本文介绍了一种新的固定结构和固定阶控制器的综合方法。在许多实际应用中需要这样的控制器。一类固定结构控制器的综合问题可以归结为确定一个真实的控制器参数向量（或简单地说，一个控制器）<i>K</i>=（<sub>k1</sub>，<sub>k2</sub>，.<i></i><i></i>，<i>k</i><sub>t</sub>），使得给定的一组真实的或复多项式的形式<i>P</i>（<i>s</i>，<i>K</i>）：=<i>Po</i>（<i>s</i>）+<i>k</i><sub>1</sub><sub>P1</sub>（<i>s</i>）+...<i></i>+<i>k</i><sub>t</sub><i>P</i><sub>t</sub>（<i>s</i>）是Hurwitz。闭环系统的稳定性要求一个真实的特征多项式是Hurwitz，而几个性能标准可以通过确保一族复多项式是Hurwitz来满足。本文的一个新特点是利用Hurwitz多项式的交错性（IP）来构造稳定控制器集合的任意紧近似。这是通过系统地构造<i>K</i>中的线性不等式组来完成的。线性不等式的可行集的并集提供了所有控制器<i>K</i>的集合的近似，其使<i>P</i>（<i>s</i>，<i>K</i>）Hurwitz。当线性不等式的集合数增加并趋于无穷大时，我们证明了可行集的并集<i>趋于</i>所需结构的<i>所有</i>稳定控制器的集合。在构造线性不等式集合时使用的主要工具是厄米-比勒定理、笛卡尔符号规则及其推广。我们提供的例子所提出的方法的适用性的合成固定阶稳定控制器。
This paper describes a new approach to the synthesis of fixed-structure and fixed-order controllers. Such controllers are required in many practical applications. A broad class of fixed-structure controller synthesis problems can be reduced to the determination of a real controller parameter vector (or simply, a controller) <i>K</i>=(<i>k</i> <sub>1</sub>, <i>k</i> <sub>2</sub>, ... , <i>k</i> <sub>t</sub>), so that a given set of real or complex polynomials of the form <i>P</i>(<i>s</i>,<i>K</i>):=<i>Po</i>(<i>s</i>)+<i>k</i> <sub>1</sub> <i>P</i> <sub>1</sub>(<i>s</i>)+... +<i>k</i> <sub>t</sub> <i>P</i> <sub>t</sub>(<i>s</i>) is Hurwitz. The stability of the closed-loop system requires a real characteristic polynomial to be Hurwitz, while several performance criteria can be satisfied by ensuring that a family of complex polynomials is Hurwitz. A novel feature of this paper is the exploitation of the interlacing property (IP) of Hurwitz polynomials to construct arbitrarily tight approximations of the set of stabilizing controllers. This is done by systematically constructing sets of linear inequalities in <i>K</i>. The union of the feasible sets of linear inequalities provides an approximation of the set of all controllers <i>K</i>, which render <i>P</i>(<i>s</i>, <i>K</i>) Hurwitz. As the number of sets of linear inequalities increases and approaches infinity, we show that the union of the feasible sets <i>approaches</i> the set of <i>all</i> stabilizing controllers of the desired structure. The main tools that are used in the construction of the sets of linear inequalities are the Hermite-Biehler theorem, Descartes' rule of signs, and its generalization. We provide examples of the applicability of the proposed methodology to the synthesis of fixed-order stabilizing controllers.