ReviewFrom static output feedback to structured robust static output feedback: A survey
Introduction
Static output feedback design is a theoretically challenging issue in control theory and it has attracted considerable attention due to its great importance in practice. However, so far, there has been no exact solution to this prominent problem which can guarantee the design of static output feedback or determine that such a feedback does not exist. The fact is that the problem is intrinsically a Bilinear Matrix Inequality (BMI) problem which is generally NP-hard (Toker & Ozbay, 1995); furthermore, it becomes non-smooth in the case of problem formulation in the space of the controller parameters (Toscano, 2013).
To solve the static output feedback design problem, well-known bilinear matrix inequality (BMI) solvers such as the commercial software package PENBMI (Henrion, Lofberg, Kocvara, Stingl, 2005, Kocvara, & Stingl) and the free open-source MATLAB toolbox PENLAB (Fiala, Kocvara, & Stingl, 2013) can be applied. The algorithms behind these solvers combines the ideas of the (exterior) penalty and (interior) barrier methods with the augmented Lagrangian approach (Kocvara & Stingl, 2003). These solvers can locally solve all kinds of BMI problems, including static output feedback. Since our aim is to survey dedicated static output feedback design methods, we no longer discuss these general BMI approaches. Note however that BMI solvers most often fail to provide a solution for the static output feedback BMI problems, and the choice of an initial guess is very crucial for these solvers.
The only survey dedicated to static output feedback has been conducted in Syrmos, Abdallah, Dorato, and Grigoriadis (1997). Since then, the past two decades have witnessed much theoretical progress on static output feedback design which has not been covered in that survey. A large amount of research has been carried out on the development of the static output feedback controllers according to Lyapunov theory via linear matrix inequality based (LMI-based) approaches (e.g. Agulhari, Oliveira, Peres, 2012, Apkarian, Noll, Tuan, 2003, Arzelier, Gryazina, Peaucelle, Polyak, 2010, Arzelier, Peaucelle, 2002, Benton, Smith, 1998, Cao, Sun, 1998, Cao, Sun, Mao, 1998, Dabboussi, Zrida, 2012, Dong, Yang, 2013, Du, Yang, 2008, Ebihara, Hagiwara, 2003, Ebihara, Tokuyama, Hagiwara, 2004, Geromel, de Souza, Skelton, 1998b, Ghaoui, Oustry, Ait-Rami, 1997, Ghaoui, Balakrishnan, 1994, Grigoriadis, Beran, 2000, Grigoriadis, Skelton, 1996, Hassibi, How, Boyd, 1999, Iwasaki, 1999, Iwasaki, Skelton, 1995b, Karimi, Sadabadi, 2013, Kim, Moon, Kwon, 2007, Koroglu, Falcone, 2014, Lee, Lee, Kwon, 2006, Leibfritz, 2001, Mehdi, Boukas, Bachelier, 2004, Moreira, Oliveira, Peres, 2011, Noll, Torki, Apkarian, 2004, Peaucelle, Arzelier, 2001a, Sadabadi, Karimi, 2013a, Sadabadi, Karimi, 2013b, Sadabadi, Karimi, 2015, Sadeghzadeh, 2014, Tran Dinh, Gumussoy, Michiels, Diehl, 2012). Most of these methods present an iterative algorithm in which a set of LMIs is iteratively repeated until some certain termination criteria are met. In addition to the Lyapunov-based approaches, there exist non-Lyapunov-based static output feedback control strategies (see, e.g. Apkarian, 2013, Apkarian, Bompart, Noll, 2007, Apkarian, Noll, 2006, Arzelier, Deaconu, Gumussoy, Henrion, 2011, Burke, Henrion, Overton, 2006b, Chesi, 2014, Gumussoy, Henrion, Millstone, Overton, 2009, Gumussoy, Overton, 2008, Peretz, 2016).
The objective of this paper is to provide a comprehensive review on the existing static output feedback design methods. The main focus is on pure stabilizing static output feedback design with no other specification. But the paper also addresses the problem of structured feedback, simultaneous stabilization, multi-performance, and robust control design. All methods and approaches described in the survey are gathered in order to provide a comprehensive classification. All results have been reinterpreted and rewritten so as to fit a common notation/framework. The notation uniformization allows a simplified overview on the differences and resemblances of the results. It allows as well to provide direct extensions of the existing results for example using system duality. Due to the fact that fixed-order dynamic output-feedback can equivalently be transformed into static output feedback by introducing an augmented plant (Ghaoui et al., 1997), this survey paper can also be used for fixed/low-order control design problem.
The paper is organized as follows. Section 2 presents problem statement and main difficulties associated with stabilizing static output feedback design and its extensions to structured feedback, simultaneous stabilization, multi-performance, and robust control. The five sections that follow provide our classification of SOF design methods. Section 3 focuses on special cases where under specific structures of the open-loop system, the SOF problem becomes convex. Section 4 reviews the available literature on iterative LMI heuristics for the intrinsically BMI nature of SOF design. Section 5 covers the heuristics related to a reformulation of the SOF design as LMIs with rank constraints. While all the previous sections describe results build out of classical Lyapunov conditions, Section 6 is devoted to methods with decoupled Lyapunov matrices that have better characteristics with respect to robustness. Section 7 exposes alternative approaches which are non-Lyapunov-based. All the classes of results are analyzed in terms of their known or claimed numerical characteristics, well as in terms of their ability to address the structured feedback, simultaneous multi-performance, and robustness issues. The paper ends with global concluding remarks in Section 8.
The notation used in this paper is standard. In particular, matrices I and 0 are the identity matrix and the zero matrix of appropriate dimensions, respectively. The symbol ⋆ denotes symmetric blocks in block matrices. The symbols AT, A⊥, ‖A‖F, and are respectively notations for the transpose of A, the maximal rank perpendicularity such that Frobenius norm of A, and the unique nonnegative-definite square root of positive-definite matrix A. For symmetric matrices, P > 0 (P < 0) indicates the positive-definiteness (the negative-definiteness).
Section snippets
Main SOF stabilization problem
Consider a linear time-invariant (LTI) continuous-time system
and a static output feedback controller
where is the state, in the control input, and is the output of the system. The state-space matrices A, B, C, and the control gain K are of appropriate dimensions. The closed-loop system is described as follows:
and its stability is equivalent to that of the dual system
Theorem 1 The following statements are
Convex cases
From our study of the literature we have been able to extract five types of results in which a particular structure of the data enables to convert the non-convex static output feedback design to a convex optimization problem. The first of these results is the well known state-feedback case (and its dual case related to observer design).
Proposition 1 In the two following cases a change of variables involving the Lyapunov matrix makes the problem convex: In the full-actuation case, if there exist and L
Iterative LMI heuristics
As seen in the previous section, only special cases provide convex LMI conditions for the design of SOF gains. Meaning that only subsets of all possible SOF gains can be designed with these methods. Some of these subsets can be empty, even when there exists some stabilizing gain. To go further, one can address the original problem that happens to be bilinear in the variables. A natural approach, largely explored in the literature, is to proceed as in Proposition 5 by freezing some of the
Heuristics for solving LMIs with rank constraints
As seen in the last proposition, there is a high potential of algorithms that would be based on the two dual constraints (5) and (6). It is the case of all the algorithms that follow which are based on the following reformulation of the problem, where the first two conditions are noting but (6) and (5) after applying the elimination lemma (Iwasaki, Skelton, 1994, Iwasaki, Skelton, 1995a, Skelton, Iwasaki, Grigoriadis, 1998) and the last two guarantee that and hence the SOF problem has a
Methods with decoupled Lyapunov matrices
In the previous sections we have recalled many heuristic techniques based on the formulations (5) and (6) that in general do not allow during iterations to optimize simultaneously over the control gain and the Lyapunov matrices. The reason for this difficulty is in the products between these decision matrices in the matrix inequalities. In the following, we expose two alternative formulations that allow decoupling of the control gains (or some matrices which in turn allow to build the SOF
Non-Lyapunov-based approaches
In addition to Lyapunov-based approaches, there exist non-Lyapunov-based methods. These approaches focus on solving the following optimization problem: where α is the spectral abscissa of the closed loop system, i.e. the maximum real part of its eigenvalues. The above optimization problem is non-convex and non-smooth. In fact, the lack of convexity and smoothness of the spectral abscissa and other similar performance criteria make the above optimization problem difficult to solve (
Conclusions
This paper reviews the existing methods for the design of a static output feedback. It does not claim to provide an exhaustive list of the many contributions on the topic, but we believe it covers all main methods. The survey proposes a comprehensive classification of these methods: convex cases; iterative LMIs for the BMI problem; iterative LMIs for the rank constraint formulation; achievable improvements with decoupled Lyapunov matrices; non-Lyapunov approaches. The global conclusion is that
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