A New Sum-of-Squares Design Framework for Robust Control of Polynomial Fuzzy Systems With Uncertainties
A New Sum-of-Squares Design Framework for Robust Control of Polynomial Fuzzy Systems With Uncertainties
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DOI：
10.1109/tfuzz.2015.2426719
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发表时间：
2016-02
影响因子：
11.9
通讯作者：
Kazuo Tanaka;Motoyasu Tanaka;Ying-Jen Chen;Hua O. Wang
中科院分区：
文献类型：
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作者：
Kazuo Tanaka;Motoyasu Tanaka;Ying-Jen Chen;Hua O. Wang
针对不确定多项式模糊系统的鲁棒控制问题，提出了一种新的平方和（SOS）设计框架。根据SOS的概念，推导出了两类鲁棒镇定条件。一类是保证多项式模糊控制系统全局渐近稳定的全局SOS鲁棒镇定条件。二是半奇异SOS鲁棒镇定条件。后者适用于多项式模糊控制难以保证全局渐近稳定的复杂系统。本文所导出的SOS鲁棒镇定条件的主要特点是表示为关于多项式李雅普诺夫函数参数和多项式反馈增益的非凸形式。由于从非凸SOS设计条件到凸SOS设计条件的典型转换通常会导致一些保守问题，因此本文提出的新设计框架给出了避免保守问题的关键思想。第一个关键的想法是，我们直接解决非凸SOS设计条件，而不适用于典型的转换。第二个关键思想是我们引入了一个所谓的共正性概念。这些想法提供了一些优势，除了放松。为了有效地解决我们的SOS鲁棒稳定条件，我们引入了一个梯度算法制定为一个最小化的SOS多项式的时间导数的上界的优化问题，可以被视为多项式李雅普诺夫函数的候选人。三个设计实例说明了所提出的设计框架的有效性和适用性。实例表明，我们的新SOS设计框架的优点，现有的线性矩阵不等式方法和现有的凸SOS方法。
This paper presents a new sum-of-squares (SOS, for brevity) design framework for robust control of polynomial fuzzy systems with uncertainties. Two kinds of robust stabilization conditions are derived in terms of SOS. One is global SOS robust stabilization conditions that guarantee the global and asymptotical stability of polynomial fuzzy control systems. The other is semiglobal SOS robust stabilization conditions. The latter is available for very complicated systems that are difficult to guarantee the global and asymptotical stability of polynomial fuzzy control systems. The main feature of all the SOS robust stabilization conditions derived in this paper are to be expressed as nonconvex formulations with respect to polynomial Lyapunov function parameters and polynomial feedback gains. Since a typical transformation from nonconvex SOS design conditions to convex SOS design conditions often results in some conservative issues, the new design framework presented in this paper gives key ideas to avoid the conservative issues. The first key idea is that we directly solve nonconvex SOS design conditions without applying the typical transformation. The second key idea is that we bring a so-called copositivity concept. These ideas provide some advantages in addition to relaxations. To solve our SOS robust stabilization conditions efficiently, we introduce a gradient algorithm formulated as a minimizing optimization problem of the upper bound of the time derivative of an SOS polynomial that can be regarded as a candidate of polynomial Lyapunov functions. Three design examples are provided to illustrate the validity and applicability of the proposed design framework. The examples demonstrate advantages of our new SOS design framework for the existing linear matrix inequality approaches and the existing convex SOS approach.