A randomized approximation algorithm for the minimal-norm static-output-feedback problem

Abstract

A new randomized algorithm is suggested, for extracting static-output-stabilizing-feedbacks, with approximately minimal-norm, for LTI systems. The algorithm has two similar stages, where in the first one the feasibility problem is solved, and in the second one the optimization problem is solved. The formulation is unified for the feasibility and for the optimization problems, as well as for continuous-time or discrete-time systems. The method is demonstrated by applying it to the hard (conjectured to be NP-hard) problem of the minimal-gain static-output-stabilizing-feedback, and to the hard (conjectured to be NP-hard) problem of regional pole-placement via static-output-feedback in non-convex or unconnected regions. A proof of convergence (in probability) that captures the two rounds of the algorithm is given, and complexity analysis is provided, under some mild assumptions.

Introduction

The problem of finding necessary and sufficient conditions, for the existence of static-output-stabilizing-feedbacks (SOF), which can be computed in reasonable (i.e. polynomial) time, has a long history (see  Syrmos, Abdallah, Dorato, and Grigoradis (1997) for a survey). The problem is known to be NP-hard if structural-constraints or bounds are imposed on the entries of the controllers (see  Blondel and Tsitsiklis (1997) and Nemirovskii (1993)), but unknown and suspected to be NP-hard, otherwise. Obviously, in real-life, when one searches a minimal-norm SOF, it is always done in a bounded region. Thus, minimal-gain SOF with bounded entries is obviously NP-hard problem. Pole-placement and simultaneous stabilization via SOF are also NP-hard (see  Fu (2004) and  Blondel and Tsitsiklis (1997), resp.). Thus, practically, one should expect that only approximation or randomized algorithms will be able to cope with these problems.

Many problems can be reduced to the constrained SOF problem. These include the reduction of the minimal-degree dynamic-feedback problem, robust or decentralized stability via static-feedback and PID controllers to this problem (see  Blondel and Tsitsiklis (1997), Mesbahi (1999) and  Zheng, Wang, and Lee (2002), resp.). A formulation of the reduced-order $H_{\infty }$ filter problem as constrained SOF problem, is considered in  Borges, Calliero, Oliveira, and Peres (2011).

The solution of the SOF problem is important for systems which models structural dynamics, and naturally needs a static feedback that can be built into the structure (e.g. buildings and bridges subject to earthquakes, strong winds and unpredicted dynamic loadings—see  Spencer and Sain (1997), Xu and Teng (2002), Yang, Lin, and Jabbari (2003) and  Polyak, Khlebnikov, and Shcherbakov (2013), where it is shown that optimal SOF’s, although costless, may achieve similar performance as optimal dynamic feedbacks). Note that the long-term memory of dynamic feedbacks is useless in the case of unpredicted dynamic loadings.

The suggested algorithm that will be called the Ray-Shooting (RS) algorithm, does not make use of any heavy tools like Semi-Definite Programming (SDP) or Linear, or Bilinear Matrix Inequalities (LMI’s and BMI’s resp.). Note that the existence of BMI’s solutions is NP-hard problem (see  Toker and Özbay (1995)). The least-rank dynamic-output-feedback problem is considered in  Mesbahi (1999) using SDP. This can be solved with the RS method, by applying a Binary-Search on the rank and using its formulation as a SOF problem, with much less computational efforts and without the need of finding any feasible initial point. Also, the problem of optimal abscissa via SOF can be solved with the RS method, by applying Binary-Search on the abscissa.

In  Vidyasagar and Blondel (2001) the problem of a common Lyapunov matrix, and the problems of static-stability and simultaneous static stability, are treated, using the probabilistic method (i.e. the “generate and check” strategy). The article deals with the problem of counting the minimal number of samples that guarantee a given probability threshold of success, through the use of the Chernoff bound and bounds on the Vapnik–Chervonenkis dimension (VC-dimension) of the problem, without knowing the specific distribution of successful examples and without using the structure of the given system (i.e. the specific $A,B,C$ matrices). In this article, it is shown that at least $max(\frac{16}{{ϵ}^2}ln\left(\frac{4}{\delta }\right),\frac{32d}{{ϵ}^2}ln\left(\frac{32e}{{ϵ}^2}\right))$ samples are needed to guarantee with $1-\delta$ confidence that the empirical probability is $ϵ$ uniformly-close to the exact probability (with $ϵ,\delta \in (0,1)$), where $d$ is the VC-dimension of the problem, where for the SOF problem it is shown there that $d\le 4p^{2}ln\left(4e(2p^2+p)\right)$ ($p$ being the state-space dimension). Thus, for $p=10,ϵ=\delta =0.01$ at least $1.3538\cdot {10}^{10}$ samples are needed ($d\le 3093$).

One way to overcome this “curse of dimensionality” is presented in  Tempo, Calafiore, and Dabbene (2005) and  Tempo and Ishii (2007). Concerning the problem of Robust Stability via SOF, it is proved in  Tempo et al. (2005) that the empirical performance measure is ${ϵ}_1$ probability-close to the exact performance with probability at least $1-{ϵ}_2$ and confidence at least $1-\delta$, if one takes $M\ge \frac{ln(2/\delta )}{ln(1/(1-{ϵ}_2))}$ samples of the controller parameters and $N\ge \frac{ln\left(\frac{4M}{\delta }\right)}{2{ϵ}_1^2}$ samples of the system uncertainty (${ϵ}_1,{ϵ}_2,\delta \in (0,1)$). Thus, for ${ϵ}_1={ϵ}_2=\delta =0.01$ one needs $M\ge 528$ samples of the controller parameters and $N\ge \text{61,303}$ samples of the system uncertainty, which results in 32,367,984 evaluations of the performance measure. The RS algorithm seems to practically overcome this obstacle and at least suggests another way to cope with this problem.

The structure of the article is as follows: in Section  2, we set notions, definitions, and give some lemmas. In Section  3, we introduce a lemma which provides the basis for the RS algorithm. We next introduce an approximation algorithm that applies the RS method again, in order to find a minimal-gain SOF. In Section  4, we consider the Alternating-Projections (AP) algorithm introduced in  Yang and Orsi (2006), which solves the problem of pole-placement via static-feedback, and we revise this algorithm with some improvement. We also consider the Hide-And-Seek algorithm, introduced in  Bélisle (1992), which solves the global optimization problem of continuous functions on compact domains. In Section  5, we compare the results of the RS algorithm with the algorithms: AP, Hide-And-Seek, Mixed LMI/Randomized Method (see  Arzelier, Gryazina, Peaucelle, and Polyak (2010)), HIFOO (see  Gumussoy, Henrion, Millstone, and Overton (2009)) and HINFSTRUCT (see  Apkarian and Noll (2006)). In Section  6, we revise a proof for the convergence in probability of the RS algorithm and discuss its complexity under some reasonable assumptions. In Section  7, we conclude with some remarks concerning the comparison between the algorithms and discuss the limitations of the RS method.