Showing papers in "Siam Journal on Control and Optimization in 1992"
TL;DR: Convergence with probability one is proved for a variety of classical optimization and identification problems and it is demonstrated for these problems that the proposed algorithm achieves the highest possible rate of convergence.
Abstract: A new recursive algorithm of stochastic approximation type with the averaging of trajectories is investigated. Convergence with probability one is proved for a variety of classical optimization and identification problems. It is also demonstrated for these problems that the proposed algorithm achieves the highest possible rate of convergence.
1,970 citations
TL;DR: For the observation or control of solutions of second-order hyperbolic equation in this paper, Ralston's construction of localized states [Comm. Pure Appl. Math, 22 (1969), pp.
Abstract: For the observation or control of solutions of second-order hyperbolic equation in $\mathbb{R}_t \times \Omega $, Ralston’s construction of localized states [Comm. Pure Appl. Math., 22 (1969), pp. ...
1,510 citations
TL;DR: In this paper, the authors considered the problem of finding an adapted pair of pairs (Phi, Psi )(x,t) uniquely solving the nonlinear stochastic partial differential equation.
Abstract: This paper studies the following form of nonlinear stochastic partial differential equation: \[ \begin{gathered} - d\Phi _t = \mathop {\inf }_{v \in U} \left\{ {\frac{1}{2}\sum_{i,j} {\left[ {\sigma \sigma ^ * } \right]_{ij} (x,v,t)} \partial _{x_i x_j } \Phi _t (x) + \sum_i {b_i (x,v,t)} \partial _{x_i } \Phi _t (x) + L(x,v,t)} \right. \hfill \\ \qquad \qquad \left. { + \sum_{i,j} {\sigma_{ij}(x,v,t)\partial _{x_i } \Psi _{j,t} (x)} } \right\}dt - \Psi _t (x)dW_t ,\quad \Phi _T (x) = h(x), \hfill \\ \end{gathered}\] where the coefficients $\sigma _{ij} $, $b_i $, L, and the final datum h may be random. The problem is to find an adapted pair $(\Phi ,\Psi )(x,t)$ uniquely solving the equation. The classical Hamilton–Jacobi–Bellman (HJB) equation can be regarded as a special case of the above problem. An existence and uniqueness theorem is obtained for the case where $\sigma $ does not contain the control variable v. An optimal control interpretation is given. The linear quadratic case is discussed as well.
325 citations
TL;DR: The linear convergence of both the gradient projection algorithm of Goldstein and Levitin and Polyak, and a matrix splitting algorithm using regular splitting, is established, which does not require that the cost function be strongly convex or that the optimal solution set be bounded.
Abstract: Consider the problem of minimizing, over a polyhedral set, the composition of an affine mapping with a strictly convex essentially smooth function. A general result on the linear convergence of descent methods for solving this problem is presented. By applying this result, the linear convergence of both the gradient projection algorithm of Goldstein and Levitin and Polyak, and a matrix splitting algorithm using regular splitting, is established. The results do not require that the cost function be strongly convex or that the optimal solution set be bounded. The key to the analysis lies in a new error bound for estimating the distance from a feasible point to the optimal solution set.
220 citations
TL;DR: In this paper, a representation for all controllers that satisfy an ∞-type constraint is derived for time-varying systems, and a solution to the design problem exists if these equations have a solution on the optimization interval.
Abstract: A representation formula for all controllers that satisfy an $\mathcal{L}^\infty $-type constraint is derived for time-varying systems. It is now known that a formula based on two indefinite algebraic Riccati equations may be found for time-invariant systems over an infinite time support (see [J. C. Doyle et al., IEEE Trans. Automat. Control, AC-34 (1989), pp. 831–847]; [K. Glover and J. C. Doyle, Systems Control Lett., 11 (1988), pp. 167–172]; [K. Glover et al., SIAM J. Control Optim., 29 (1991), pp. 283–324]; [M. Green et al., SIAM J. Control Optim., 28 (1990), pp. 1350–1371]; [D. J. N. Limebeer et al., in Proc. IEEE conf. on Decision and Control, Austin, TX, 1988]; [G. Tadmor, Math. Control Systems Signal Processing, 3 (1990), pp. 301–324]). In the time-varying case, two indefinite Riccati differential equations are required. A solution to the design problem exists if these equations have a solution on the optimization interval. The derivation of the representation formula illustrated in this paper mak...
208 citations
TL;DR: In this paper, a stochastic search method is proposed for finding a global solution to the discrete optimization problem in which the objective function must be estimated by Monte Carlo simulation, and it is shown under mild conditions that the Markov chain is strongly ergodic.
Abstract: In this paper a stochastic search method is proposed for finding a global solution to the stochastic discrete optimization problem in which the objective function must be estimated by Monte Carlo simulation. Although there are many practical problems of this type in the fields of manufacturing engineering, operations research, and management science, there have not been any nonheuristic methods proposed for such discrete problems with stochastic infrastructure. The proposed method is very simple, yet it finds a global optimum solution. The method exploits the randomness of Monte Carlo simulation and generates a sequence of solution estimates. This generated sequence turns out to be a nonstationary Markov chain, and it is shown under mild conditions that the Markov chain is strongly ergodic and that the probability that the current solution estimate is global optimum converges to one. Furthermore, the speed of convergence is also analyzed.
204 citations
TL;DR: In this paper, a set of extended quadratic controller normal forms of linearly controllable nonlinear systems is given, which is the generalization of the Brunovsky form of linear systems.
Abstract: In this paper, a set of extended quadratic controller normal forms of linearly controllable nonlinear systems is given, which is the generalization of the Brunovsky form of linear systems. A set of invariants under the quadratic changes of coordinates and feedbacks is found. It is then proved that any linearly controllable nonlinear system is linearizable to second degree by a dynamic state feedback.
183 citations
TL;DR: In this paper, an optimal boundary control problem for the Navier-Stokes equations is presented, where the control is the velocity on the boundary, which is constrained to lie in a closed, convex subset of the boundary.
Abstract: An optimal boundary control problem for the Navier–Stokes equations is presented The control is the velocity on the boundary, which is constrained to lie in a closed, convex subset of $H^{{1 / 2}} $ of the boundary A necessary condition for optimality is derived Computations are done when the control set is actually finite-dimensional, resulting in an application to viscous drag reduction
149 citations
TL;DR: In this article, a technique different from the boundary layer method is developed to deal with singularly perturbed optimal control problems, which is applicable in particular in the case when the optima...
Abstract: A technique different from the boundary layer method is developed to deal with singularly perturbed optimal control problems. The technique is applicable, in particular, in the case when the optima...
113 citations
TL;DR: Explicit algebraic conditions for the suboptimality of some parameter in the output measurement control problem are presented in this paper, which is used to compute the optimal value by quadratically convergent algorithms and to solve the almost disturbance decoupling problem with internal stability.
Abstract: Explicit algebraic conditions are presented for the suboptimality of some parameter in the $H_\infty $-optimization problem by output measurement control. Apart from two strict properness conditions, no artificial assumptions restrict the underlying system. In particular, the plant may have zeros on the imaginary axis or at infinity. These suboptimality characterizations are applied to show how to compute the optimal value by quadratically convergent algorithms and to solve the almost disturbance decoupling problem with internal stability.
108 citations
TL;DR: In this article, the authors considered an infinite horizon investment-consumption model in which a single agent consumes and distributes his wealth in two assets, a bond and a stock, and the problem of maximization of the total utility from consumption was treated.
Abstract: This paper considers an infinite horizon investment-consumption model in which a single agent consumes and distributes his wealth in two assets, a bond and a stock. The problem of maximization of the total utility from consumption is treated. State (amount allocated in assets) and control (consumption, rates of trading) constraints are present. It is shown that the value function is the unique viscosity solution of a system of variational inequalities with gradient constraints.
TL;DR: In this paper, a coprime factorization approach to the synthesis of internally stabilizing controllers for a given system such that the $\mathcal{H}_\infty $ norm of the closed loop is strictly less than a given bound is presented.
Abstract: This paper develops a coprime factorization approach to the synthesis of internally stabilizing controllers for a given system such that the $\mathcal{H}_\infty $ norm of the closed loop is strictly less than a given bound.By the use of coprime factorizations, it is shown that the $\mathcal{H}_\infty $ control problem is fundamentally related to the so-called analytic systems considered by Helton et al. [Regional Conference Series in Mathematics 68, 1987]. Such problems admit a solution if and only if a certain J-lossless factorization exists. Interpreted in the $\mathcal{H}_\infty $ control context, this means that for a controller of the requisite type to exist the plant must admit a certain J-lossless coprime factorization. The full $\mathcal{H}_\infty $ synthesis problem requires that two nested J-lossless factorizations exist. The results are independent of whether discrete time or continuous time systems are being considered.It is then shown that J-lossless factorization is equivalent to the existen...
TL;DR: In this paper, algebraic tests are derived for the suboptimality of some parameter in the $H ∞ $-optimization problem by state-feedback, where the finite zero structure of the plant is not restricted.
Abstract: Algebraic tests are derived for the suboptimality of some parameter in the $H_\infty $-optimization problem by state-feedback, where the finite zero structure of the plant is not restricted. As an application of these characterizations, a quadratically convergent algorithm for the computation of the optimal value is presented. The suboptimality tests are based on new solvability criteria for general algebraic Riccati inequalities that are of independent interest.
TL;DR: In this paper, the authors developed a theory of feedback stabilization for SISO transfer functions over a general integral domain which extends the well-known coprime factorization approach to stabilization.
Abstract: This paper develops a theory of feedback stabilization for SISO transfer functions over a general integral domain which extends the well-known coprime factorization approach to stabilization. Necessary and sufficient conditions for stabilizability of a transfer function in this general setting are obtained. These conditions are then refined in the special cases of unique factorization domains (UFDs), Noetherian rings, and rings of fractions. It is shown that these conditions can be naturally interpreted geometrically in terms of the prime spectrum of the ring. This interpretation provides a natural generalization to the classical notions of the poles and zeros of a plant.The set of transfer functions is topologized so as to restrict to the graph topology of Vidyasagar [IEEE Trans. Automatic Control, AC-29 (1984), pp. 403–418], when the ring is a Bezout domain. It is shown that stability of a feedback system is robust in this topology when the ring is a UFD.This theory is then applied to the problem of sta...
TL;DR: In this article, it was proved that the sequence of recursive estimators generated by Ljung's scheme combined with a suitable restarting mechanism converges under certain conditions with rate O(M (n^{{{ - 1} / 2}} ), where the rate is measured by the $L_q $-norm of the estimation error for any $1 \leq q < \infty $.
Abstract: It is proved that the sequence of recursive estimators generated by Ljung’s scheme combined with a suitable restarting mechanism converges under certain conditions with rate $O_M (n^{{{ - 1} / 2}} )$, where the rate is measured by the $L_q $-norm of the estimation error for any $1 \leq q < \infty $.
TL;DR: In this paper, it was shown that the existence of an internally stabilizing controller that makes the $H_ ∞ $ norm strictly less than 1 is related to stabilizing solutions to two algebraic Riccati equations.
Abstract: This paper is concerned with the discrete time $H_\infty $ control problem with measurement feedback. It follows that, as in the continuous time case, the existence of an internally stabilizing controller that makes the $H_\infty $ norm strictly less than 1 is related to the existence of stabilizing solutions to two algebraic Riccati equations. However, in the discrete time case, the solutions of these algebraic Riccati equations must satisfy extra conditions.
TL;DR: In this article, the equivalence of a nonlinear system with a bilinear system, or, more generally, a linear time-dependent system, plus an output injection, was investigated.
Abstract: This paper, following the purpose of synthesis of observers for nonlinear systems, investigates the question of equivalence of a nonlinear system with a bilinear system, or, more generally, a linear time-dependent system, plus an output injection. In a previous work by the same authors, such questions have already been dealt with from the local point of view. The goal herein is to examine the global situation. Using basic facts from algebraic topology, it is shown that in the single output case, whenever the possibility to bilinearize up to output injection holds locally everywhere, it also holds globally.
TL;DR: In this paper, where and how solutions associated to a differential inclusion can or cannot enter a given target is studied, where partitions of the target boundary are associated with the dynamic of the system and the behaviour of these solutions is qualitatively described in terms of viability and invariance kernels of sets.
Abstract: Where and how solutions associated to a differential inclusion can or cannot enter a given target is studied For this purpose, partitions of the target boundary are associated with the dynamic of the system The behaviour of these solutions is qualitatively described in terms of viability and invariance kernels of sets These kernels determine points such that there exist (respectively, all) solutions starting at these points remain in a given set of constraints The sets that are reached in finite time by viable solutions to the system are also studied Finally, some applications to control systems with one target are provided, and the concept of semipermeable barriers will be generalized
TL;DR: In this paper, an equivalence between realizability of input/output (i/o) operators by rational control systems and high-order algebraic differential equations for pairs was shown.
Abstract: An equivalence is shown between realizability of input/output (i/o) operators by rational control systems and high-order algebraic differential equations for (i/o) pairs. This generalizes, to nonli...
TL;DR: Using Hamilton's principle, a nonlinear system of partial differential equations describing the dynamics of a network of vibrating strings is derived in this paper, where an equilibrium is linearized to obtain a linear...
Abstract: Using Hamilton’s principle a nonlinear system of partial differential equations describing the dynamics of a network of vibrating strings is derived. An equilibrium is linearized to obtain a linear...
TL;DR: In this article, the Riccati difference equation (RDE) for the filtering problem is studied and the existence, stabilizability, and attractiveness properties of the real symmetric solutions that remain bounded on $( - ∞, + ∞ )$ (infinite-time solutions) are investigated.
Abstract: This paper studies the time-varying Riccati difference equation (RDE) for the filtering problem. In particular, existence, stabilizability, and attractiveness properties of the real symmetric solutions that remain bounded on $( - \infty , + \infty )$ (infinite-time solutions) are investigated. Under the assumption of uniform detectability, conditions for the existence of the maximal and stabilizing solutions are given. Analogous results are worked out for the minimal and antistabilizing solutions by making reference to the uniform antidetectability notion. Moreover, it is shown that, under uniform observability, the set of all symmetric infinite-time solutions constitute an infinite number of lattices with common minimal and maximal elements.
TL;DR: In this article, it was shown that the full-order system is exponentially stable for sufficiently small values of the perturbation parameter, and its rate of convergence approaches that of the reduced-order systems as the parameter approaches zero.
Abstract: This paper establishes some results and properties related to the exponential stability of general dynamical systems and, in particular, singularly perturbed systems. For singularly perturbed systems it is shown that if both the reduced-order system and the boundary-layer system are exponentially stable, then, provided that some further regularity conditions are satisfied, the full-order system is exponentially stable for sufficiently small values of the perturbation parameter $\mu $, and its rate of convergence approaches that of the reduced-order system $(\mu = 0)$ as $\mu $ approaches zero. Exponentially decaying norm bounds are given for the “slow” and “fast” components of the full-order system trajectories. To achieve this result, a new converse Lyapunov result for exponentially stable systems is presented.
TL;DR: The present paper provides an intuitive design-dependent characterization of the cause/effect notion of causality and suggests a framework for the optimization of constrained nonsequential stochastic control problems.
Abstract: In control theory, the usual notion of causality---that, at all times, a system's output (action) only depends on its past and present inputs (observations)---presupposes that all inputs and outputs can be ordered, a priori, in time. In reality, many distributed systems (those subject to deadlock, for instance), are not sequential in this sense.
In a previous paper (part I) [SIAM J. Control Optim., 30 (1992), pp. 1447--1475], the relationship between a less restrictive notion of causality, deadlock-freeness, and the design-independent properties of a potentially nonsequential generic stochastic control problem formulated within the framework of Witsenhausen's intrinsic model was explored. In the present paper (part II) the properties of individual designs are examined. In particular, a property of a design's information partition that is necessary and sufficient to ensure its deadlock-freeness is identified and shown to be sufficient to ensure its possession of an expected reward. It is also shown, by example, that there exist nontrivial deadlock-free designs that cannot be associated with any deadlock-free information structure.
The first result provides an intuitive design-dependent characterization of the cause/effect notion of causality and suggests a framework for the optimization of constrained nonsequential stochastic control problems. The second implies that this characterization is finer than existing design-independent characterizations, including properties C (Witsenhausen) and CI (part I).
TL;DR: In this paper, a quadratic optimal control for a linear stochastic evolution equation with unbounded coefficients was solved using the dynamic programming approach and attention was focused on the Riccati equation.
Abstract: This paper solves a quadratic optimal control for a linear stochastic evolution equation with unbounded coefficients. It is assumed that the stochastic noise depends both on the state and on the control. The dynamic programming approach is used and attention is focused on the Riccati equation. In §§5 and 6 some attractivity and maximality properties of the solutions of the algebraic Riccati equation are proved and it is shown that, in some special cases, there exists a maximal solution.
TL;DR: In this paper, the dynamic disturbance decoupling problem for nonlinear systems is introduced and a local solution of this problem is obtained in the case that the system under consideration is invertible.
Abstract: In analogy with the dynamic input-output decoupling problem the dynamic disturbance decoupling problem for nonlinear systems is introduced. A local solution of this problem is obtained in the case that the system under consideration is invertible. The solution is given in algebraic as well as in geometric terms. The theory is illustrated by means of two examples: a mathematical one and an example of a voltage frequency controlled induction motor.
TL;DR: In this paper, the authors considered the smooth fit for a class of one-dimensional singular stochastic control problems allowing the system to be of nonlinear diffusion type and proved the existence and the uniqueness of a convex $C^2 $-solution to the corresponding variational inequality.
Abstract: This paper considers the principle of smooth fit for a class of one-dimensional singular stochastic control problems allowing the system to be of nonlinear diffusion type The existence and the uniqueness of a convex $C^2 $-solution to the corresponding variational inequality are obtained It is proved that this solution gives the value function of the control problem, and the optimal control process is constructed As an example of the degenerate case, it is proved that the conclusion is also true for linear systems, and the explicit formula for the smooth fit points is derived
TL;DR: In this paper, the non-regular dynamic disturbance decoupling problem for nonlinear control systems is introduced and a local solution is given by means of a constructive algorithm that is based on Singh's algorithm and the clamped dynamics algorithm.
Abstract: The nonregular dynamic disturbance decoupling problem for nonlinear control systems is introduced. A local solution is given by means of a constructive algorithm that is based on Singh’s algorithm and the clamped dynamics algorithm. Further studied is the nonlinear model matching problem that is defined as follows: given a nonlinear control system, to be referred to as the plant, and another nonlinear control system, to be referred to as the model, can a compensator for the plant be found in such a way that the input-output behavior of the compensated plant matches that of the model? By proving that the solvability of the nonlinear model matching problem is equivalent to the solvability of an associated nonregular dynamic disturbance decoupling problem, a complete local solution of this problem can be established.
TL;DR: In this paper, a connection between nearness to unstabilizability of a stabilizable pair of matrices and the singularity of the symmetric positive definite solution to an associated algebraic Riccati equation is established.
Abstract: A connection is established between nearness to unstabilizability of a stabilizable pair $(A,B)$ of matrices and nearness to singularity of the symmetric positive definite solution to an associated algebraic Riccati equation. From this result, computable upper and lower bounds are derived for the distance of $(A,B)$ to the nearest uncontrollable pair. Numerical tests confirm the validity of the method and potential applications are discussed.
TL;DR: In this paper, the problem of tuning natural frequencies of a linear system by a memoryless controller is considered, and the main result is an exact combinatorial characterization of the nilpotency index of the cohomology ring of the real Grassmannian.
Abstract: This paper considers the problem of tuning natural frequencies of a linear system by a memoryless controller. Using algebro-geometric methods it is shown how it is possible to improve current sufficiency conditions.The main result is an exact combinatorial characterization of the nilpotency index of the $\bmod 2$ cohomology ring of the real Grassmannian. Using this characterization, new sufficiency results for generic pole assignment for the linear system with m-inputs, p-outputs, and McMillan degree n are given. Among other results it is shown that \[2.25 \cdot \max (m,p) + \min (m,p) - 3 \geq n\] is a sufficient condition for generic real pole placement, provided $\min (m,p) \geq 4$.
TL;DR: In this paper, boundary or distributed stationary control problems are studied in relation to an elliptic operator and state and control constraints, and different kinds of conditions are formulated to prove the existence of a decoupled optimality system and Lagrange multipliers.
Abstract: In this paper boundary or distributed stationary control problems are studied in relation to an elliptic operator and state and control constraints. Different kinds of conditions are formulated to prove the existence of a decoupled optimality system and Lagrange multipliers.