Reduction of SDPs in H∞ control of SISO systems and performance limitations analysis
Reduction of SDPs in H∞ control of SISO systems and performance limitations analysis
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SISO 系统 H∞ 控制中 SDP 的减少及性能限制分析
DOI：
10.1109/cdc.2016.7798342
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
N. Sebe
中科院分区：
文献类型：
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作者：
Hayato Waki;Y. Ebihara;N. Sebe
在基于SDP的H∞控制中，我们在通过各种软件求解SDP时经常遇到数值困难。根据经验可知，如果手头的 SDP 或其对偶没有内点可行解，就会出现这种数值困难，并且 H ∞ 控制中的某些 SDP 确实存在这种情况。为了构思一种在具体问题设置中解决此类数值困难的方法，在本文中，我们重点关注传递函数 (1 + PK)-1P 的 H∞ 控制问题的对偶 SDP 并将其简化。更准确地说，通过主动利用对象 P 的不稳定零点（非最小相位零点）信息，我们将原始对偶 SDP 简化为一组简化的 SDP，每个简化 SDP 及其对偶都有内点可行解。这样，我们通过数值实验证明了SDP软件可以完成可靠的数值计算。另一方面，一旦我们获得了简化的SDP，就可以将它们进一步简化为由不稳定零点确定的矩阵最大奇异值的计算。这样，如果不稳定零点的数量适中，我们就可以获得最佳可实现的 H∞ 性能或其不稳定零点下界的解析表达式。关键词：H∞控制、SDP、数值可靠性、最佳可实现性能。
In SDP-based H∞ control, we often encounter numerical difficulties when solving SDPs by various pieces of software. It is empirically known that such numerical difficulty occurs if an SDP at hand or its dual has no interior point feasible solutions, and this is indeed the case of some SDPs in H∞ control. To conceive a way for getting around such numerical difficulties in a concrete problem setting, in this paper, we focus on the dual SDP for the H∞ control problem of the transfer function (1 + PK)-1P and simplify it. More precisely, by actively using the information of unstable zeros (non-minimum phase zeros) of the plant P, we reduce the original dual SDP into a set of simplified SDPs each of which and its dual have interior point feasible solutions. In this way, we show by numerical experiments that reliable numerical computation can be done by SDP software. On the other hand, once we have obtained simplified SDPs, it becomes possible to further reduce them into the computation of maximum singular values of matrices determined by unstable zeros. In this way, if the number of unstable zeros is moderate, we can obtain analytical expressions of the best achievable H∞ performance or its lower bounds in terms of the unstable zeros. Keywords: H∞ control, SDP, numerical reliability, best achievable performance.