SIAM Digital Library
 
 
 

SIAM J. on Control and Optimization

All Journal content published prior to 1997 is part of LOCUS.

Search Issue | RSS Feeds RSS
Previous Issue

2007

Volume 45, Issue 6, pp. 1931-2304


Global Smooth Solutions and Exponential Stability for a Nonlinear Beam

Peng‐Fei Yao and George Weiss

SIAM J. Control Optim. 45, pp. 1931-1964 (34 pages) | Cited 3 times

Online Publication Date: January 08, 2007

Full Text: | Download PDF

Show Abstract
In this paper we consider a dynamical system with boundary input and output describing the bending vibrations of a quasi‐linear beam, where the nonlinearity comes from Hooke’s law. First we derive an existence result for short‐time solutions of the system of equations. Then we show that the structure of the boundary input and output forces the system to admit global solutions at least when the initial data and the boundary input are small in a certain sense. In particular, we prove that the norm of the state of the system decays exponentially if the input becomes zero after a finite time (the input being zero can be understood as a boundary feedback).

From Differential Calculus to 0‐1 Topological Optimization

Maatoug Hassine, Sophie Jan, and Mohamed Masmoudi

SIAM J. Control Optim. 45, pp. 1965-1987 (23 pages)

Online Publication Date: January 08, 2007

Full Text: | Download PDF

Show Abstract
The topological asymptotic expansion gives the variation of a cost function when a small hole is created in a domain. This approach leads to very powerful algorithms in topological optimization. Unfortunately, these asymptotic expansions are obtained with the use of complicated mathematical tools. The goal of this paper is to provide a straightforward way to derive a topological asymptotic expansion using a classical gradient. We will illustrate this general approach by some numerical experiments for the elasticity and the Stokes problems.

Coprime Factorization and Dynamic Stabilization of Transfer Functions

Kalle M. Mikkola

SIAM J. Control Optim. 45, pp. 1988-2010 (23 pages) | Cited 3 times

Online Publication Date: January 08, 2007

Full Text: | Download PDF

Show Abstract
It is known that a matrix‐valued transfer function $P$ has a stabilizing dynamic controller $Q$ (i.e., $[\begin{smallmatrix} I & -Q \\ -P & I \end{smallmatrix}]^{-1} \in {\rm H}^{\infty}$) iff $P$ has a right (or left) coprime factorization. We show that the same result is true in the operator‐valued case. Thus, the standard Youla–Bongiorno parameterization applies to every dynamically stabilizable function. We then derive further equivalent conditions, one of them being that $P$ has a stabilizing controller with internal loop; this and some others are new even in the scalar‐valued case. We also establish certain related results. For example, we extend the classical results on coprime factorization and partial feedback (measurement‐feedback) stabilization to nonrational transfer functions. All our results apply in both discrete‐ and continuous‐time settings, except that in the latter it is not clear whether the controller $Q$ can always be chosen so that it is “continuous‐time proper” (holomorphic and bounded on a right half‐plane) unless, e.g., $P(z)\rightarrow0$ as $\operatorname{Re} z\rightarrow+\infty$.

Simplified Formula for a Controller in Optimal Control Problems

Jovan Stefanovski

SIAM J. Control Optim. 45, pp. 2011-2034 (24 pages)

Online Publication Date: January 12, 2007

Full Text: | Download PDF

Show Abstract
We derive a simplified frequency‐domain formula for a controller transfer matrix for LQ optimal output feedback control of stochastic systems. For this purpose we apply a generalization of the Wiener–Hopf method. The generalization is characterized by the following three properties: (1) We generalize the usual operator $\{\cdot\}_+$ on rational matrices in the traditional Wiener–Hopf approach. The same algorithms as for computation of $\{\cdot\}_+$ are applicable to compute the new operator. (2) The matrices of the Youla–Kučera parametrization do not appear in the optimal controller transfer matrix ${\bf C}$ (they appear only in its derivation), even if the plant is unstable. (3) Unlike the traditional Wiener–Hopf method [D. C. Youla, H. A. Jabr, and J. J. Bongiorno, Jr., IEEE Trans. Automat. Control, AC‐21 (1976), pp. 319–338], where the spectral factors in the two spectral factorizations are both stable and minimum phase, our spectral factors need to be minimum phase only. Finally, three state‐space applications of the formula are presented.

On the Parametrization of All Regularly Implementing and Stabilizing Controllers

C. Praagman, H. L. Trentelman, and R. Zavala Yoe

SIAM J. Control Optim. 45, pp. 2035-2053 (19 pages) | Cited 5 times

Online Publication Date: January 12, 2007

Full Text: | Download PDF

Show Abstract
In this paper we deal with problems of controller parametrization in the context of behavioral systems. Given a full plant behavior, a subbehavior of the manifest plant behavior is called regularly implementable if it can be achieved as the controlled behavior resulting from the interconnection of the full plant behavior with a suitable controller behavior, in such a way that the controller does not impose restrictions that are already present in the plant. We establish a parametrization of all controllers that regularly implement a given behavior. We also obtain a parametrization of all stabilizing controllers.

On the Intersection of a Clarke Cone with a Boltyanskii Cone

Alberto Bressan

SIAM J. Control Optim. 45, pp. 2054-2064 (11 pages)

Online Publication Date: January 12, 2007

Full Text: | Download PDF

Show Abstract
We provide an example of two closed sets $S_1,S_2\subset\R^4$ such that $S_1\cap S_2=\{0\}$. Yet, at the origin, a Boltyanskii tangent cone $C_1$ to $S_1$ and the Clarke tangent cone $C_2$ to $S_2$ are strongly transversal. This settles a question originally proposed by H. Sussmann.

On the Internal Model Structure for Infinite‐Dimensional Systems: Two Common Controller Types and Repetitive Control

Eero Immonen

SIAM J. Control Optim. 45, pp. 2065-2093 (29 pages) | Cited 2 times

Online Publication Date: January 12, 2007

Full Text: | Download PDF

Show Abstract
In this paper we shall extend and amplify the recent theory of controllers employing the so‐called internal model structure. For exponentially stable closed loop control systems this structure has been shown in another paper to be necessary and sufficient for robust output regulation, also in infinite‐dimensional spaces. Here we shall derive conditions under which two controller types occurring frequently in applications have the internal model structure. Under these conditions robust output regulation is achieved—also in infinite‐dimensional spaces—if the closed loop system is sufficiently stable. In the case that the closed loop system is only strongly (but not exponentially) stable, it is sometimes possible to obtain conditional robustness. This means that asymptotic tracking/disturbance rejection is not destroyed by small perturbations so long as closed loop stability also persists. Our results allow for infinite‐dimensional plants, controllers, and exogenous systems, and as an application of such a general setting we shall consider generalized repetitive control for exponentially stable infinite‐dimensional SISO systems.

Weighted Admissibility and Wellposedness of Linear Systems in Banach Spaces

Bernhard H. Haak and Peer Chr. Kunstmann

SIAM J. Control Optim. 45, pp. 2094-2118 (25 pages) | Cited 6 times

Online Publication Date: January 12, 2007

Full Text: | Download PDF

Show Abstract
We study linear control systems in infinite‐dimensional Banach spaces governed by analytic semigroups. For $p\in[1,\infty]$ and $\alpha\in\mathbb{R}$ we introduce the notion of $L^p$‐admissibility of type $\alpha$ for unbounded observation and control operators. Generalizing earlier work by Le Merdy [J. London Math. Soc. (2), 67 (2003), pp. 715–738] and Haak and Le Merdy [Houston J. Math., 31 (2005), pp. 1153–1167], we give conditions under which $L^p$‐admissibility of type $\alpha$ is characterized by boundedness conditions which are similar to those in the well‐known Weiss conjecture. We also study $L^p$‐wellposedness of type $\alpha$ for the full system. Here we use recent ideas due to Pruss and Simonett [Arch. Math. (Basel), 82 (2004), pp. 415–431]. Our results are illustrated by a controlled heat equation with boundary control and boundary observation where we take Lebesgue and Besov spaces as state space. This extends the considerations in [C. I. Byrnes et al., J. Dynam. Control Systems, 8 (2002), pp. 341–370] to non‐Hilbertian settings and to $p\neq2$.

Conjugate Points in Formation Constrained Optimal Multi‐Agent Coordination: A Case Study

Jianghai Hu, Maria Prandini, and Claire Tomlin

SIAM J. Control Optim. 45, pp. 2119-2137 (19 pages) | Cited 2 times

Online Publication Date: January 12, 2007

Full Text: | Download PDF

Show Abstract
In this paper, an optimal coordinated motion planning problem for multiple agents subject to constraints on the admissible formation patterns is formulated. Solutions to the problem are reinterpreted as distance minimizing geodesics on a certain manifold with boundary. A geodesic on this manifold may fail to be a solution for different reasons. In particular, if a geodesic possesses conjugate points, then it will no longer be distance minimizing beyond its first conjugate point. We study a particular instance of the formation constrained optimal coordinated motion problem, where a number of initially aligned agents tries to switch positions by rotating around their common centroid. The complete set of conjugate points of a geodesic naturally associated with this problem is characterized analytically. This allows us to prove that the geodesic will not correspond to an optimal coordinated motion when the angle of rotation exceeds a threshold that decreases to zero as the number of agents increases. Moreover, infinitesimal perturbations that improve the performance of the geodesic after it passes the conjugate points are also determined, which, interestingly, are characterized by a certain family of orthogonal polynomials.

Characterization of Delay‐Independent Stability and Delay Interference Phenomena

Wim Michiels and Silviu‐Iulian Niculescu

SIAM J. Control Optim. 45, pp. 2138-2155 (18 pages) | Cited 1 time

Online Publication Date: January 12, 2007

Full Text: | Download PDF

Show Abstract
The problem of the asymptotic stability independent of delays for a class of linear systems including multiple delays is addressed. Both cases where the delays are allowed to vary independently of each other and where they are restricted to a one‐dimensional subspace of the delay‐parameter space are considered. It the latter case it turns out that the resulting dependency between the delays (rationally independent, rationally dependent, commensurate) plays an important role. The stability conditions are expressed in terms of the spectral properties of some appropriate complex matrices. As a consequence of the stability study, a complete characterization of the delay interference phenomenon is given. Furthermore, a connection is established with the stability theory for continuous‐time delay‐difference equations, subjected to delay perturbations. Various illustrative examples complete the paper.

On Markov Games with Average Reward Criterion and Weakly Continuous Transition Probabilities

Heinz‐Uwe Küenle

SIAM J. Control Optim. 45, pp. 2156-2168 (13 pages) | Cited 5 times

Online Publication Date: January 22, 2007

Full Text: | Download PDF

Show Abstract
In this paper we consider two‐person zero‐sum stochastic games with the average reward criterion and Borel state and action spaces. A geometric drift condition is assumed. We show that the optimality (Shapley) equation has a unique solution if the transition probability function is weakly continuous, the stage reward is lower semicontinuous, and the set‐valued mappings of admissible actions satisfy some semicontinuity assumptions. Furthermore, the minimizing player has an optimal stationary strategy and the maximizing player has an ϵ‐optimal stationary strategy for every $\varepsilon > 0 $.

Convergent Numerical Scheme for Singular Stochastic Control with State Constraints in a Portfolio Selection Problem

Amarjit Budhiraja and Kevin Ross

SIAM J. Control Optim. 45, pp. 2169-2206 (38 pages) | Cited 2 times

Online Publication Date: January 22, 2007

Full Text: | Download PDF

Show Abstract
We consider a singular stochastic control problem with state constraints that arises in problems of optimal consumption and investment under transaction costs. Numerical approximations for the value function using the Markov chain approximation method of Kushner and Dupuis are studied. The main result of the paper shows that the value function of the Markov decision problem (MDP) corresponding to the approximating controlled Markov chain converges to that of the original stochastic control problem as various parameters in the approximation approach suitable limits. All our convergence arguments are probabilistic; the main assumption that we make is that the value function be finite and continuous. In particular, uniqueness of the solutions of the associated HJB equations is neither needed nor available (in the generality under which the problem is considered). Specific features of the problem that make the convergence analysis nontrivial include unboundedness of the state and control space and the cost function; degeneracies in the dynamics; mixed boundary (Dirichlet–Neumann) conditions; and presence of both singular and absolutely continuous controls in the dynamics. Finally, schemes for computing the value function and optimal control policies for the MDP are presented and illustrated with a numerical study.

Performance Recovery in Digital Implementation of Analogue Systems

Guofeng Zhang, Tongwen Chen, and Xiang Chen

SIAM J. Control Optim. 45, pp. 2207-2223 (17 pages) | Cited 1 time

Online Publication Date: February 02, 2007

Full Text: | Download PDF

Show Abstract
In this paper, the generalized bilinear transformation (GBT) is proposed. Compared with the traditional bilinear, zero‐order hold (ZOH) and first‐order hold transformations, one advantage of GBT is that it may convert unstable poles (zeros) to stable poles (zeros). It is proved that controllability and observability are invariant under GBT. After that, it is shown that the performance of a sampled‐data system obtained via GBT approaches that of the analogue system as the underlying sampling period goes to zero. Performance studied here is characterized in terms of internal stability and $\ell_{p}$ induced norms for all $1 \leq p \leq \infty$. This results extends the main results in [G. Zhang and T. Chen, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, Suppl., 2003, pp. 28–33] and [G. Zhang and T. Chen, Automatica J. IFAC, 40 (2004), pp. 327–330] from SISO to MIMO and also removes the limitation on the “A” matrix of the system. Finally, an example is employed to compare digital implementations via GBT and the ZOH transformation.

Pathwise Stochastic Control Problems and Stochastic HJB Equations

Rainer Buckdahn and Jin Ma

SIAM J. Control Optim. 45, pp. 2224-2256 (33 pages) | Cited 3 times

Online Publication Date: February 02, 2007

Full Text: | Download PDF

Show Abstract
In this paper we study a class of pathwise stochastic control problems in which the optimality is allowed to depend on the paths of exogenous noise (or information). Such a phenomenon can be illustrated by considering a particular investor who wants to take advantage of certain extra information but in a completely legal manner. We show that such a control problem may not even have a “minimizing sequence,” but nevertheless the (Bellman) dynamical programming principle still holds. We then show that the corresponding Hamilton–Jacobi–Bellman equation is a stochastic partial differential equation, as was predicted by Lion and Souganidis [C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), pp. 735–741]. Our main device is a Doss–Sussmann‐type transformation introduced in our previous work [Stochastic Process. Appl., 93 (2001), pp. 181–204] and [Stochastic Process. Appl., 93 (2001), pp. 205–228]. With the help of such a transformation we reduce the pathwise control problem to a more standard relaxed control problem, from which we are able to verify that the value function of the pathwise stochastic control problem is the unique stochastic viscosity solution to this stochastic partial differential equation, in the sense of [Stochastic Process. Appl., 93 (2001), pp. 181–204] and [Stochastic Process. Appl., 93 (2001), pp. 205–228].

Adaptive Digital Control of Hammerstein Nonlinear Systems with Limited Output Sampling

Feng Ding, Tongwen Chen, and Zenta Iwai

SIAM J. Control Optim. 45, pp. 2257-2276 (20 pages) | Cited 22 times

Online Publication Date: February 13, 2007

Full Text: | Download PDF

Show Abstract
This paper is motivated by the practical control considerations that nonlinearity is abundant in industrial processes and output sampling rates are often limited due to hardware constraints. In particular, for a Hammerstein nonlinear sampled‐data system in which the output sampling period is an integer multiple of the input updating period, we derive, by using a polynomial transformation technique, a mathematical model which is suitable for parameter estimation with dual‐rate measurement data. Further, we present an adaptive control scheme for such a dual‐rate nonlinear system; the parameter estimation–based adaptive algorithm can achieve virtually asymptotically optimal control and ensure that the closed‐loop system is stable and globally convergent. The simulation results are included.

Output Stabilization via Nonlinear Luenberger Observers

L. Marconi, L. Praly, and A. Isidori

SIAM J. Control Optim. 45, pp. 2277-2298 (22 pages) | Cited 9 times

Online Publication Date: February 13, 2007

Full Text: | Download PDF

Show Abstract
The present paper addresses the problem of the existence of an (output) feedback law that asymptotically steers to zero prescribed outputs, while keeping all state variables bounded, for any initial conditions in a given compact set. The problem can be viewed as an extension of the classical problem of semiglobally stabilizing the trajectories of a controlled system to a compact set. The problem also encompasses a version of the classical problem of output regulation. Under only a weak minimum phase assumption, it is shown that there exists a controller solving the problem at hand. The paper is deliberately focused on theoretical results regarding the existence of such a controller. Practical aspects involving the design and the implementation of the controller are left to a forthcoming work.

Dynamic Programming for Ergodic Control of Markov Chains under Partial Observations: A Correction

Vivek S. Borkar

SIAM J. Control Optim. 45, pp. 2299-2304 (6 pages)

Online Publication Date: February 20, 2007

Full Text: | Download PDF

Show Abstract
A gap in the author’s work on dynamic programming for ergodic control of partially observed Markov chains [V. S. Borkar, SIAM J. Control Optim., 39 (2000), pp. 673–681] is pointed out and a correction is provided.
Close

Your Recent History Recent History Help

Recently Viewed

  1. On the Lavrentiev Phenomenon for Optimal Control Problems with Second-Order Dynamics
  2. Sufficient Conditions for the Optimality of a Stochastic Control
  3. On Controllability by Means of Two Vector Fields
  4. Simulating Size-constrained Galton–Watson Trees
  5. Locally Testable Codes Require Redundant Testers
© SIAM By using SIAM Publications Online you agree to abide by the Terms and Conditions of Use. Banner art adapted from a figure by Hinke M. Osinga and Bernd Krauskopf (University of Bristol, UK).
close