## Simultaneous stabilization of linear systems

### Vincent D. Blondel

Springer Verlag, Heidelberg, 1994
Lecture Notes in Control and Information Sciences, ISBN: 3-540-19862-8

Review in Zentralblatt MATH (795.93083):

"The central question investigated in this beautiful research monograph is the problem of simultaneous stabilization: Let p_1, p_2, ..., p_k be k linear systems. Under what conditions is it possible to find a controller C which is stabilizing for each p_i (i= 1,... , k).
The traditional examples of the k systems include nominal systems, with its failed modes, an airplane which has natural operating points (it takes off, flies and lands) etc. Unexplored but potentially fruitful are systems of dynamics of different national incomes, p_1, p_2, ..., p_k. Can the U.N. -- World Bank provide a stabilizing controller C for each p_i.

The author provides the current state of attempts to solve this problem. Simultaneous stabilizability question for k=2 is reported to be fully solved. There exists no tractable condition for simultaneous stabilization of three or more linear systems. Thus if we allow only finite combination of rational, logical and sign test operations on the coefficient of the k (k > 2)$systems, it is not possible to find necessary and sufficient conditions for simultaneous stabilization. The text is very lucid and carefully written with an excellent summary and bibliography at the end of each chapters. It contains a mathematical appendix. The English'' prologue and Epilogue is beautiful. It succeeded in its attempt to explain mathematical theory to the first man in the street. This research monograph is recommended to control theorists, applied mathematicians and systems engineers." From the introduction: "This monograph focuses on the question of simultaneous stabilization of scalar linear systems: Let$p\sb 1,p\sb 2,\cdots, p\sb k$be$k$scalar linear systems. Under what condition does there exist a controller$c$that is stabilizing for each$p\sb i (i=1,\cdots,k)$? We are most interested in the existence question and we make only a few remarks on constructive procedures. The volume consists of 6 chapters and 3 appendices. The chapters are as independent of each other as possible. They all start with a short introduction and end with a concluding section that summarizes the content and gives bibliographical references. Chapter 2: Stabilization. This chapter contains all the basic definitions: stabilization, strong, bistable and simultaneous stabilization, etc. There exist several conceptual ways to look at stabilization and we present two of these: the algebraic factorization approach and the geometrical avoidance interpretation. Chapter 3: Youla-Kucera parametrization. The Youla-Kucera parametrization of the set of all stabilizing controllers of a given system is the stepping stone to many control design strategies. It is presented here and used to derive equivalences between strong, bistable and simultaneous stabilization. Chapter 4: Necessary conditions. A rational function that has no poles in the right half plane has no poles on the positive real axis. This elementary observation is the guideline used in the chapter to obtain necessary conditions for simultaneous stabilization. If all the closed-loop transfer functions associated to the systems$p\sb i (i=1,\cdots,k)$in feedback with a controller$c$have no poles in the right half plane, then this controller is such that all the closed loop transfer functions have no poles on the positive real axis. Such a controller is said to be$\bold R\sb {+\infty}$-stabilizing for$p\sb i (i=1,\cdots,k)$. The existence of an$\bold R\sb {+\infty}$-stabilizing controller is thus a necessary condition for the existence of a stabilizing controller. In this chapter we derive tractable necessary and sufficient conditions for simultaneous$\bold R\sb {+\infty}$-stabilizability of$k$systems and for strong and bistable$\bold R\sb {+\infty}$-stabilization. These conditions encompass all known necessary conditions and are expressed in the form of interlacement properties on poles and zeros of the systems and of related rational functions. Chapter 5: Sufficient conditions. In this chapter we show that the necessary conditions derived in Chapter 4 are also sufficient for simultaneous stabilization of two systems and for strong stabilization. Then we show that this property does not flow on to bistable stabilization or to simultaneous stabilization of more than two systems: three systems that are simultaneously$\bold R\sb {+\infty}$-stabilizable are not guaranteed simultaneous stabilizable. The conditions given in Chapter 4 for bistable or simultaneous stabilization of$3$or more systems are not sufficient. The proof of this result makes crucial use of results from analytic function theory. In the last two sections we provide examples of sufficient conditions that are related to the avoidance concept given in Chapter 2 and to$H\sb \infty\$ control design.

Chapter 6: Necessary and sufficient conditions. The result contained in this closing chapter is that the simultaneous stabilization question is rationally undecidable: it is not possible to find necessary and sufficient conditions for simultaneous stabilization of three or more systems that involve only a finite combination of rational operations (additions, subtractions, multiplications and divisions), logical operations (and' and or') and sign test operations (equal to, greater than, greater than or equal to, etc.) on the coefficients of the three systems."