Nonsmooth variational analysis and related computational methods are powerful tools that can be effectively applied to identify local minimizers of nonconvex optimization problems arising in fixed-order controller design. We support this claim by applying nonsmooth analysis and methods to a challenging "Belgian chocolate" stabilization problem posed in 1994: find a stable, minimum phase, rational controller that stabilizes a specified second-order plant. Although easily stated, this particular problem remained unsolved until 2002, when a solution was found using an eleventh-order controller. Our computational methods find a stabilizing third-order controller without difficulty, suggesting explicit formulas for the controller and for the closed loop system, which has only one pole with multiplicity 5. Furthermore, our analytical techniques prove that this controller is locally optimal in the sense that there is no nearby controller with the same order for which the closed loop system has all its poles further left in the complex plane. Although the focus of the paper is stabilization, once a stabilizing controller is obtained, the same computational techniques can be used to optimize various measures of the closed loop system, including its complex stability radius or Hinfin performance

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Stabilization via Nonsmooth, Nonconvex Optimization

We show that the simultaneous stabilizability of three linear systems, that is the question of knowing whether three linear systems are simultaneously stabilizable, is rationally undecidable. By this we mean that it is not possible to find necessary and sufficient conditions for simultaneous stabilization of the three systems in terms of expressions involving the coefficients of the three systems and combinations of arithmetical operations (additions, subtractions, multiplications, and divisions), logical operations (''and'' and ''or''), and sign test operations (equal to, greater than, greater than or equal to,...).

Simultaneous Stabilizability of Three Linear Systems Is Rationally Undecidable

Brief paper: PID controller synthesis for a class of unstable MIMO plants with I/O delays

The simultaneous stabilization (resp., asymptotic stabilization) of a countable family of control systems consists of finding a control which stabilizes (resp., asymptotically stabilizes) all the systems in the family. In this paper, we introduce a new method which enables us to show that, given any countable family of stabilizable nonlinear systems, there exists a continuous state feedback law which simultaneously stabilizes (not asymptotically) the family. Then, by enriching this method, we prove that any finite family of stabilizable linear time invariant (LTI) systems can be simultaneously exponentially stabilized by means of nonlinear time-varying state feedback. We also derive sufficient conditions for the simultaneous asymptotic stabilizability of countably infinite families of LTI systems. Finally, sufficient conditions for the simultaneous asymptotic stabilizability of finite families of nonlinear systems are provided and used for the simultaneous asymptotic stabilization of certain pairs of nonlinear homogeneous systems.

Simultaneous Stabilization of Linear and Nonlinear Systems by Means of Nonlinear State Feedback

Simultaneous stabilization deals with the following question: given a finite number of LTI plants P/sub 1/,P/sub 2/,...P/sub k/ does there exist a single LTI controller C such that each of the feedback interconnections (P/sub i/,C) (i=1,2,...,k) is internally stable? This paper presents a new methodology for the design of simultaneously stabilizing controllers for two or more plants that satisfy a sufficient condition. A classic result from simultaneous-stability theory is invoked to cast the sufficient condition as a linear matrix inequality (LMI). It is shown that in this setting, the problem of design of simultaneously stabilizing controllers can be reduced to that of a standard H/sub /spl infin// control problem. The technique developed is applied to the design of a fault-tolerant controller for lane-keeping control of automated vehicles. The controller makes the system insensitive to a failure in either one of two lateral error measuring sensors used for lane-keeping control. Experimental results confirm the efficacy of the design and reinforce analytical predictions of performance.

Design of simultaneously stabilizing controllers and its application to fault-tolerant lane-keeping controller design for automated vehicles

The problem of the simultaneous stabilizability of a finite family of single-input, single-output time-invariant systems by a time- invariant controller is studied. The link between stabilization and avoidance is shown and is used to derive necessary conditions for the simultaneous stabilization of k plants. These necessary conditions are proved to be, in general, not sufficient. This result also disproves a long- standing conjecture on the stabilizability condition of a single plant with a stable minimum phase controller. The main result is to show that, unlike the case of two plants, the existence of a simultaneous stabilizing controller for more than two plants is not guaranteed by the existence of a controller such that the closed loops have no real unstable poles.

Simultaneous Stabilization of Three or More Plants: Conditions on the Positive Real Axis Do Not Suffice

Let Aℝ([MATHEMATICAL DOUBLE-STRUCK CAPITAL D]) denote the set of functions belonging to the disc algebra having real Fourier coefficients. We show that Aℝ([MATHEMATICAL DOUBLE-STRUCK CAPITAL D]) has Bass and topological stable ranks equal to 2, which settles the conjecture made by Brett Wick in [18]. We also give a necessary and sufficient condition for reducibility in some real algebras of functions on symmetric domains with holes, which is a generalization of the main theorem in [18]. A sufficient topological condition on the symmetric open set [MATHEMATICAL DOUBLE-STRUCK CAPITAL D] is given for the corresponding real algebra Aℝ([MATHEMATICAL DOUBLE-STRUCK CAPITAL D]) to have Bass stable rank equal to 1.

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On the stable rank and reducibility in algebras of real symmetric functions

Let denote a compact real symmetric subset of and let denote the real Banach algebra of all real symmetric continuous functions on that are analytic in the interior of , endowed with the supremum norm. We characterize all unimodular pairs in which are reducible. In addition, for an arbitrary compact in , we give a new proof (not relying on Banach algebra theory or elementary stable rank techniques) of the fact that the Bass stable rank of is 1. Finally, we also characterize all compact real symmetric sets such that , respectively , has Bass stable rank 1.

Reducibility in AR(K), CR(K), and A(K)

Let A R denote the set of functions from the disk algebra having real Fourier coefficients. Generalizing a result of Quadrat we show that every unstabilizable multi-input multi-output plant is as close as we want to a stabilizable multi-input multi-output plant in the product topology.

R. Rupp

Papers

Simultaneous Stabilization of Three or More Plants: Conditions on the Positive Real Axis Do Not Suffice

On the stable rank and reducibility in algebras of real symmetric functions

Reducibility in AR(K), CR(K), and A(K)

On the density of stabilizable plants in the class of unstabilizable plants: The real symmetric disk algebra case

Geometrische Optik