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SIAM J. on Control and Optimization

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2006

Volume 44, Issue 6, pp. 1923-2299


Controller Design via Nonsmooth Multidirectional Search

Pierre Apkarian and Dominikus Noll

SIAM J. Control Optim. 44, pp. 1923-1949 (27 pages) | Cited 16 times

Online Publication Date: July 26, 2006

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We propose an algorithm which combines multidirectional search (MDS) with nonsmooth optimization techniques to solve difficult problems in automatic control. Applications include static and fixed-order output feedback controller design, simultaneous stabilization, $H_2/H_\infty$-synthesis, and much else. We show how to combine direct search techniques with nonsmooth descent steps in order to obtain convergence certificates in the presence of nonsmoothness. Our technique is efficient when small and medium size controllers for plants with large state dimension are sought. Our numerical testing includes several benchmark examples. For instance, our algorithm needs 0.41\,s to compute a static output feedback stabilizing controller for the Boeing 767 flutter benchmark problem [E. E. J. Davison, IFAC Technical Committee Reports, Pergamon Press, Oxford, 1990], a system with 55 states. The first static controller without performance specifications for this system was obtained in [J. Burke, A. Lewis, and M. Overton, SIAM J. Optim., 15 (2003), pp. 751-779].

On the Controllability of a Fractional Order Parabolic Equation

Sorin Micu and Enrique Zuazua

SIAM J. Control Optim. 44, pp. 1950-1972 (23 pages) | Cited 6 times

Online Publication Date: July 26, 2006

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The null-controllability property of a $1-d$ parabolic equation involving a fractional power of the Laplace operator, $(-\Delta)^\alpha$, is studied. The control is a scalar time-dependent function $g=g(t)$ acting on the system through a given space-profile $f=f(x)$ on the interior of the domain. Thus, the control $g$ determines the intensity of the space control $f$ applied to the system, the latter being given a priori. We show that, if $\alpha\leq 1/2$ and the shape function $f$ is, say, in $L^2$, no initial datum belonging to any Sobolev space of negative order may be driven to zero in any time. This is in contrast with the existing positive results for the case $\alpha >1/2$ and, in particular, for the heat equation that corresponds to $\alpha=1$. This negative result exhibits a new phenomenon that does not arise either for finite-dimensional systems or in the context of the heat equation.
On the contrary, if more regularity of the shape function $f$ is assumed, then we show that there are initial data in any Sobolev space $H^m$ that may be controlled. Once again this is precisely the opposite behavior with respect to the control properties of the heat equation in which, when increasing the regularity of the control profile, the space of controllable data decreases.
These results show that, in order for the control properties of the heat equation to be true, the dynamical system under consideration has to have a sufficiently strong smoothing effect that is critical when $\alpha=1/2$ for the fractional powers of the Dirichlet Laplacian in $1-d$. The results we present here are, in nature and with respect to techniques of proof, similar to those on the control of the heat equation in unbounded domains in [S. Micu and E. Zuazua, Trans. Amer. Math. Soc., 353 (2000), pp. 1635-1659] and [S. Micu and E. Zuazua, Portugal. Math., 58 (2001), pp. 1-24].
We also discuss the hyperbolic counterpart of this problem considering a fractional order wave equation and some other models.

State Feedback $H_\infty$ Control for a Class of Nonlinear Stochastic Systems

Weihai Zhang and Bor-Sen Chen

SIAM J. Control Optim. 44, pp. 1973-1991 (19 pages) | Cited 42 times

Online Publication Date: July 26, 2006

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This paper discusses the $H_{\infty}$ control problem for a class of nonlinear stochastic systems with both state- and disturbance-dependent noise. By means of Hamilton--Jacobi equations, both infinite and finite horizon nonlinear stochastic $H_\infty$ control designs are developed.
Some results on nonlinear $H_\infty$ control of deterministic systems are generalized to a stochastic setting. We introduce some useful concepts such as "zero-state observability" and "zero-state detectability" which, together with the stochastic LaSalle invariance principle, yield some valuable consequences in infinite horizon nonlinear stochastic $H_\infty$ control.

A Parametrization of Solutions of the Discrete-time Algebraic Riccati Equation Based on Pairs of Opposite Unmixed Solutions

Harald K. Wimmer

SIAM J. Control Optim. 44, pp. 1992-2005 (14 pages) | Cited 2 times

Online Publication Date: July 26, 2006

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The paper describes the set of solutions of the discrete-time algebraic Riccati equation. It is shown that each solution is a combination of a pair of opposite unmixed solutions. There is a one-to-one correspondence between solutions and invariant subspaces of the closed loop matrix of an unmixed solution. The results of the paper provide an extended counterpart of the parametrization theory of continuous-time algebraic Riccati equations by Willems, Coppel, and Shayman.

Linear Programming Approach to Deterministic Long Run Average Problems of Optimal Control

Vladimir Gaitsgory and Sergey Rossomakhine

SIAM J. Control Optim. 44, pp. 2006-2037 (32 pages) | Cited 10 times

Online Publication Date: July 26, 2006

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We establish that deterministic long run average problems of optimal control are "asymptotically equivalent" to infinite-dimensional linear programming problems (LPPs) and we establish that these LPPs can be approximated by finite-dimensional LPPs, the solutions of which can be used for construction of the optimal controls. General results are illustrated with numerical examples.

Bifurcations of 1-parameter Families of Control-affine Systems in the Plane

Bronislaw Jakubczyk and Witold Respondek

SIAM J. Control Optim. 44, pp. 2038-2062 (25 pages) | Cited 3 times

Online Publication Date: July 26, 2006

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We define bifurcations of control-affine systems in the plane and classify all generic 1-parameter bifurcations at control-regular points. More precisely, we classify topological bifurcations of invariants of usual feedback equivalence. Such bifurcations form six different classes: two bifurcations of equilibrium sets, two bifurcations of critical sets, and two bifurcations of pairs of invariants. We also classify all generic 1-parameter families of control-affine systems with respect to orbital feedback equivalence.

Option Pricing With Markov-Modulated Dynamics

A. Jobert and L. C. G. Rogers

SIAM J. Control Optim. 44, pp. 2063-2078 (16 pages) | Cited 16 times

Online Publication Date: July 26, 2006

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Markov-modulated models for equity prices have recently been extensively studied in the literature. In this paper, we apply some old results on the Wiener--Hopf factorization of Markov processes to a range of option-pricing problems for such models. The first example is the perpetual American put, where the exact (numerical) solution is obtained without discretizing any PDE. We then show how the methodology of Rogers and Stapleton [Finance Stoch., 2 (1997), pp. 3-17] can be used to tackle finite-horizon problems and illustrate the methodology by pricing European, American, single barrier, and double barrier options under Markov-modulated dynamics.

Supervisory Control of Discrete Event Systems with CTL* Temporal Logic Specifications

Shengbing Jiang and Ratnesh Kumar

SIAM J. Control Optim. 44, pp. 2079-2103 (25 pages) | Cited 4 times

Online Publication Date: July 26, 2006

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The supervisory control problem of discrete event systems with temporal logic specifications is studied. The full branching time logic of CTL* is used for expressing specifications of discrete event systems. The control problem of CTL* is reduced to the decision problem of CTL*. A small model theorem for the control of CTL* is obtained. It is shown that the control problem of CTL* (resp., CTL) is complete for deterministic double (resp., single) exponential time. A sound and complete supervisor synthesis algorithm for the control of CTL* is provided. Special cases of the control of computation tree logic (CTL) and linear-time temporal logic are also studied.

Existence of Optimal Policies for Semi-Markov Decision Processes Using Duality for Infinite Linear Programming

Diego Klabjan and Daniel Adelman

SIAM J. Control Optim. 44, pp. 2104-2122 (19 pages) | Cited 2 times

Online Publication Date: July 26, 2006

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Semi-Markov decision processes on Borel spaces with deterministic kernels have many practical applications, particularly in inventory theory. Most of the results from general semi-Markov decision processes do not carry over to a deterministic kernel since such a kernel does not provide "smoothness." We develop infinite dimensional linear programming theory for a general stochastic semi-Markov decision process. We give conditions, general enough to allow deterministic kernels, for solvability and strong duality of the resulting linear programs. By using the developed linear programming theory we give conditions for the existence of a stationary deterministic policy for deterministic kernels, which is optimal among all possible policies.

A Representation Theorem for the Error of Recursive Estimators

László Gerencsér

SIAM J. Control Optim. 44, pp. 2123-2188 (66 pages) | Cited 3 times

Online Publication Date: July 26, 2006

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The ultimate objective of this paper is to develop new techniques that can be used for the analysis of performance degradation due to statistical uncertainty for a wide class of linear stochastic systems. For this we need new technical tools similar to those used in [L. Gerencsér, Statist. Plann. Inference, 41 (1994), pp. 303-325]. The immediate technical objective is to extend the previous technical results to the Djereveckii--Fradkov--Ljung scheme with enforced boundedness. Our starting point is a standard approximation of the estimation error used in the asymptotic theory of recursive estimation. Tight control of the difference between the estimation error and its standard approximation, referred to as residuals, is a crucial point in our applications. The main technical advance of the present paper is a set of strong approximation theorems for three closely related recursive estimation algorithms in which, for any $q \ge 1$, the $L_q$-norms of the residual terms are shown to tend to zero with rate $N^{-1/2-\varepsilon}$ with some $\varepsilon > 0$. This is a significant extension of previous results for the recursive prediction error or RPE estimator of ARMA processes given in [L. Gerencsér, Systems Control Lett., 21 (1993), pp. 347-351]. Two useful corollaries will be derived. In the first a standard transform of the estimation-error process for the basic recursive estimation method, Algorithm CR\@, will be shown to be $L$-mixing, while in the second the asymptotic covariance matrix of the estimator for the same method will be given. Applications to multivariable adaptive prediction and the minimum-variance self-tuning regulator for ARMAX systems will be described.

State Feedback Impulse Elimination for Singular Systems over a Hermite Domain

Daniel Cobb

SIAM J. Control Optim. 44, pp. 2189-2209 (21 pages)

Online Publication Date: July 26, 2006

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We reduce the problem of impulse elimination via state feedback in singular differential equations to algebra. Our results are developed for systems over an arbitrary Hermite domain. We show that the established theories for the time-invariant and the real analytic time-varying settings can be unified in this way. Besides the constant and real analytic functions, several other function rings are considered. Our algebraic theory is applied to these cases, providing solutions to the impulse elimination problem for classes of systems not previously studied. In particular, our work allows the restriction of the feedback matrix to certain function rings.

High-Gain State Feedback Analysis Based on Singular System Theory

Daniel Cobb and Jacob Eapen

SIAM J. Control Optim. 44, pp. 2210-2232 (23 pages)

Online Publication Date: July 26, 2006

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We consider linear, time-invariant state-space systems under high-gain state feedback. The analysis is couched in terms of singular system theory and Grassman manifolds. Our work is distinguished from that of other authors by the fact that we do not allow a gain-dependent state coordinate change. Simple necessary and sufficient conditions are proven under which a singular system is a high-gain limit of a given state-space system. It is shown that the feedback matrix achieves a limit on an appropriate Grassmanian, so infinite gains constitute well-defined mathematical objects. The special cases of minimum-order stable and zeroth-order limits are studied in depth, including an analysis of solution behavior. Finally, the classical "cheap control" problem is interpreted within the context of our results.

Rendezvous in Higher Dimensions

Steve Alpern and Vic Baston

SIAM J. Control Optim. 44, pp. 2233-2252 (20 pages) | Cited 1 time

Online Publication Date: July 26, 2006

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Two players are placed on the integer lattice $Z^{n}$ (consisting of points in $n$-dimensional space with all coordinates integers) so that their vector difference is of length 2 and parallel to one of the axes. Their aim is to move to an adjacent node in each period, so that they meet (occupy same node) in least expected time $R\left( n\right),$ called the rendezvous value. We assume they have no common notion of directions or orientations (i.e., no common notion of "clockwise"). We extend the known result $R\left( 1\right) =3.25$ of Alpern and Gal to obtain $R\left( 2\right) =197/32=\allowbreak 6.\,\allowbreak16,$ and the bounds $2n\leq R\left( n\right) \leq\left( 32n^{3}+12n^{2}-2n-3\right) /12n^{2}.$ For $n=2$ we characterize the set of all optimal strategies and show that none of them simultaneously maximizes the probability of meeting by time $t$ for all $t.$ This behavior differs from that found by Anderson and Fekete, and the authors, for the related problem where the players are initially placed at diagonals of one of the squares of the lattice $Z^{2}.$

What Periodic Signals Can an Exponentially Stabilizable Linear Feedforward Control System Asymptotically Track?

Eero Immonen and Seppo Pohjolainen

SIAM J. Control Optim. 44, pp. 2253-2268 (16 pages) | Cited 5 times

Online Publication Date: July 26, 2006

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We study asymptotic tracking and rejection of continuous periodic signals in the context of exponentially stabilizable linear infinite-dimensional systems. Our reference signals are in Sobolev-type spaces $H(\omega_n,f_n)$ and they (as well as the disturbance signals) are generated by an infinite-dimensional exogenous system. We show that there exists a feedforward controller which achieves output regulation if and only if the so-called regulator equations are satisfied and a decomposability condition holds. For SISO systems this result allows us to completely answer the question posed in the title: We show that if the stabilized plant does not have transmission zeros at the frequencies $i\omega_n$ of the reference signals, then all reference signals in $H(\omega_n,f_n)$ can be asymptotically tracked in the presence of disturbances if and only if \[ \bigl(H_K(i\omega_n)^{-1}[1-H_d(n)]f_n^{-1}\bigr)_{n \in I} \in \ell^2. \] Here $H_K(i\omega_n)$, $n \in I$, is the transfer function of the stabilized plant evaluated at $i\omega_n$, and $(H_d(n))_{n \in I}$ is a sequence of disturbancecoefficients for the stabilized plant. Moreover, the sequence $(f_n)_{n\in I}$ consists of weights for the Fourier coefficients of the reference signals. We give four examples to illustrate the theory.

On Iterative Solutions of General Coupled Matrix Equations

Feng Ding and Tongwen Chen

SIAM J. Control Optim. 44, pp. 2269-2284 (16 pages) | Cited 44 times

Online Publication Date: July 26, 2006

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In this paper we study coupled matrix equations, which are encountered in many systems and control applications. First, we extend the well-known Jacobi and Gauss--Seidel iterations and present a large family of iterative methods, which are then applied to develop iterative solutions to coupled Sylvester matrix equations. The basic idea is to regard the unknown matrices to be solved as parameters of a system to be identified and to obtain the iterative solutions by applying a hierarchical identification principle. Next, we generalize the Sylvester equations to general coupled matrix equations, and propose a gradient-based iterative algorithm for the solutions, using a block-matrix inner product---the star $(\star)$ product; we prove that the iterative algorithm always converges to the (unique) solutions for any initial values. One advantage of the algorithms proposed is that they require less storage space in implementation than existing numerical methods. Finally, we test the algorithms and show their effectiveness using numerical examples.

Some New Regularity Properties for the Minimal Time Function

Giovanni Colombo, Antonio Marigonda, and Peter R. Wolenski

SIAM J. Control Optim. 44, pp. 2285-2299 (15 pages) | Cited 5 times

Online Publication Date: July 26, 2006

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A minimal time problem with linear dynamics and convex target is considered. It is shown, essentially, that the epigraph of the minimal time function $T(\cdot)$ is $\varphi$-convex (i.e., it satisfies a kind of exterior sphere condition with locally uniform radius), provided $T(\cdot)$ is continuous. Several regularity properties are derived from results in [G. Colombo and A. Marigonda, Calc. Var. Partial Differential Equations, 25 (2005), pp. 1-31], including twice a.e. differentiability of $T(\cdot)$ and local estimates on the total variation of $DT$.
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