Explicit construction of a Barabanov norm for a class of positive planar discrete-time linear switched systems
Explicit construction of a Barabanov norm for a class of positive planar discrete-time linear switched systems
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DOI：
10.1016/j.automatica.2011.09.028
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发表时间：
2012
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影响因子：
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通讯作者：
Ron Teichner;M. Margaliot
中科院分区：
文献类型：
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作者：
Ron Teichner;M. Margaliot
我们考虑离散时间线性切换系统在任意切换下的稳定性。解决这个问题的一个有效方法是基于研究“最不稳定”开关定律（MUSL）。如果对应于 MUSL 的切换系统的解收敛到原点，则切换系统对于任何切换律都是稳定的。 MUSL 可以使用最优控制技术来表征。这种变分方法产生了 Hamilton-Jacobi-Bellman 方程，描述了 MUSL 下切换系统的行为。该方程的解有时被称为切换系统的巴拉巴诺夫范数。尽管巴拉巴诺夫范数被广泛研究，但似乎很少有实际以封闭形式计算它的例子。在本文中，我们考虑一类特殊的正平面离散时间线性切换系统，并为相应的 Barabanov 范数和 MUSL 提供闭合形式表达式。该范数中的单位圆是平行四边形。
We consider the stability under arbitrary switching of a discrete-time linear switched system. A powerful approach for addressing this problem is based on studying the “most unstable” switching law (MUSL). If the solution of the switched system corresponding to the MUSL converges to the origin, then the switched system is stable for any switching law. The MUSL can be characterized using optimal control techniques. This variational approach leads to a Hamilton–Jacobi–Bellman equation describing the behavior of the switched system under the MUSL. The solution of this equation is sometimes referred to as a Barabanov norm of the switched system. Although the Barabanov norm was studied extensively, it seems that there are few examples where it was actually computed in closed-form. In this paper, we consider a special class of positive planar discrete-time linear switched systems and provide a closed-form expression for a corresponding Barabanov norm and a MUSL. The unit circle in this norm is a parallelogram.