Guaranteed Cost Control for a Class of Uncertain Stochastic Impulsive Systems with Markovian Switching

Abstract

This article is concerned with the problem of guaranteed cost control for a class of uncertain stochastic impulsive systems with Markovian switching. To the best of our knowledge, it is the first time that such a problem is investigated for stochastic impulsive systems with Markovian switching. For an uncontrolled system, the conditions in terms of certain linear matrix inequalities (LMIs) are obtained for robust stochastical stability and an upper bound is given for the cost function. For the controlled systems, a set of LMIs is developed to design a linear state feedback controller which can stochastically stabilize the class of systems under study and guarantee the given cost function to have an upper bound. Further, an optimization problem with LMI constraints is formulated to minimize the guaranteed cost of the closed-loop system. Finally, a numerical example is provided to show the effectiveness of the proposed method.     

The maximum principles for partially observed risk-sensitive optimal controls of Markov regime-switching jump-diffusion system

ABSTRACT

This paper studies partially observed risk-sensitive optimal control problems with correlated noises between the system and the observation. It is assumed that the state process is governed by a continuous-time Markov regime-switching jump-diffusion process and the cost functional is of an exponential-of-integral type. By virtue of a classical spike variational approach, we obtain two general maximum principles for the aforementioned problems. Moreover, under certain convexity assumptions on both the control domain and the Hamiltonian, we give a sufficient condition for the optimality. For illustration, a linear-quadratic risk-sensitive control problem is proposed and solved using the main results. As a natural deduction, a fully observed risk-sensitive maximum principle is also obtained and applied to study a risk-sensitive portfolio optimization problem. Closed-form expressions for both the optimal portfolio and the corresponding optimal cost functional are obtained.     

Deterministic optimal control,given only a partial observation of the initial state

The deterministic linear-system, quadratic-cost optimal control problem is considered when the only state information available is a partial linear observation of the initial statex ₀. Thus, it is only known that the initial condition belongs to a particular linear variety. A control function is found which is optimal, in the sense (roughly) that (i) it can be computed using available information aboutx ₀ and (ii) no other control function which can be found using that information gives lower cost than it does for every initial condition that could have given rise to the information. The optimal control can be found easily from the conventional Riccati equation of optimal control. Applications are considered in the presence of unknown exponential disturbances and to the case with a sequence of partial state observations.     

Multi-model predictive control of Hammerstein-Wiener systems based on balanced multi-model partition

ABSTRACT

Model analysis of Hammerstein-Wiener systems has been made, and it is found that the included angle is applicable to such systems to measure the non-linearity. Then, a dichotomy gridding algorithm is proposed based on the included angle. Supporting by the gridding algorithm, a balanced multi-model partition method is put forward to partition a Hammerstein-Wiener system into a set of local linear models. For each linear model, a linear model predictive controller (MPC) is designed. After that, a multi-MPC is composed of the linear MPCs via soft switching. Thus, a complex non-linear control problem is transformed into a set of linear control problems, which simplifies the original control problem and improves the control performance. Two non-linear systems are built into Hammerstein-Wiener models and investigated using the proposed methods. Simulations demonstrate that the proposed gridding and partition methods are effective, and the resulted multi-MPC controller has satisfactory performance in both set-point tracking and disturbance rejection control.     

A NON-LINEAR ℓ 1 PROBLEM

Abstract

In this note we consider a non-linear problem posed to the author in a private communication with Per Enflo: If a normed space X contains a non-linear ? ₁ ⁽ⁿ⁾-cube (where n ≥ 2 is a natural number) does it necessarily contain a linear isometric copy of ? ₁ ⁽ⁿ⁾?

We exhibit a strong regularity property of non-linear ? ₁ ⁽ⁿ⁾ cubes and apply it to obtain an affirmative answer to Enflo's problem in the setting X = l _∞ ^(m) that, moreover, coincides precisely with well known linear theory.     

A Mixed Linear Quadratic Optimal Control Problem with a Controlled Time Horizon

A mixed linear quadratic (MLQ) optimal control problem is considered. The controlled stochastic system consists of two diffusion processes which are in different time horizons. There are two control actions: a standard control action \(u(\cdot )\) enters the drift and diffusion coefficients of both state equations, and a stopping time \(\tau \) , a possible later time after the first part of the state starts, at which the second part of the state is initialized with initial condition depending on the first state. A motivation of MLQ problem from a two-stage project management is presented. It turns out that solving an MLQ problem is equivalent to sequentially solve a random-duration linear quadratic (RLQ) problem and an optimal time (OT) problem associated with Riccati equations. In particular, the optimal cost functional can be represented via two coupled stochastic Riccati equations. Some optimality conditions for MLQ problem is therefore obtained using the equivalence among MLQ, RLQ and OT problems. In case of seeking the optimal time in the family of deterministic times (even through somewhat restrictive, such seeking is still reasonable from practical standpoint), we give a more explicit characterization of optimal actions.     

Filtering,Smoothing and M-ary Detection with Discrete Time Poisson Observations

Abstract

In this article, we solve a class of estimation problems, namely, filtering smoothing and detection for a discrete time dynamical system with integer-valued observations. The observation processes we consider are Poisson random variables observed at discrete times. Here, the distribution parameter for each Poisson observation is determined by the state of a Markov chain. By appealing to a duality between forward (in time) filter and its corresponding backward processes, we compute dynamics satisfied by the unnormalized form of the smoother probability. These dynamics can be applied to construct algorithms typically referred to as fixed point smoothers, fixed lag smoothers, and fixed interval smoothers. M-ary detection filters are computed for two scenarios: one for the standard model parameter detection problem and the other for a jump Markov system.     

Optimization of a time-variable,constrained control matrix

The problem posed is to choose, in a optimal manner, a time-variable, bounded, linear transformation defining the velocity of a state point inn-dimensional space in terms of the state. The two-point boundary-value problem which arises from an application of the Pontryagin maximum principle is explicitly solvable; hence, a formula is derived showing that the optimal trajectories in state space are equiangular spirals in two-dimensional subspaces ofR ⁿ and also describing the boundary of the set of attainability. This formula is used to solve the problem of minimal-time transfer between any two given points, and the optimal control is specified both as an open-loop and a closed-loop controller. The solutions to the problem of maximizing a linear payoff function of the final state and of maximizing the angle of rotation of the state vector about the origin are also given.     

Exact Finite Difference Scheme for an Advection-Reaction Equation

We construct an exact finite difference scheme for a non-linear PDE having linear advection and an odd-cubic reaction term. This construction is based on the fact that the general initial-value problem for the equation can be completely solved. We also give a detailed discussion of the mathematical structure of the exact finite difference equation.     

Stochastic adaptive control for continuous-time linear systems with quadratic cost

An adaptive control problem is formulated and solved for a completely observed, continuous-time, linear stochastic system with an ergodic quadratic cost criterion. The linear transformationsA of the state,B of the control, andC of the noise are assumed to be unknown. Assuming only thatA is stable and that the pair (A, C) is controllable and using a diminishing excitation control that is asymptotically negligible for an ergodic, quadratic cost criterion it is shown that a family of least-squares estimates is strongly consistent. Furthermore, an adaptive control is given using switchings that is self-optimizing for an ergodic, quadratic cost criterion.This research was partially supported b y NSF Grants ECS-9102714, ECS-9113029, and DMS-9305936.     

Direct Approach to the Minimization of the Maximal Stress over an Arch Structure

The problem of minimizing the maximal stress over an arch structure is studied within linear elastic thin shell theory, the design variable being the shape of the arch. As the maximal stress cost is nondifferentiable, a nonsmooth analysis approach is developed.The pointwise stress minimization problem is studied. Then, a proof of the subdifferentiability of the maximal stress cost is given, the rapid computation of a subgradient being done by means of an adjoint state method in a very weak sense.Numerical results are presented. They show that the optimal shape obtained for the L ²-norm of displacements is slightly different from the one obtained for the maximal stress cost.     

Existence for an abstract control problem in banach spaces

This paper is concerned with existence in L¹ for a control problem with a lower semicontinuous cost functional associated with a general linear process of causal type in Banach space.

The state system considered includes several standard infinite dimensional systems such as: linear evolution equation, wave equation, linear functional differentional equation. In particular, linear distributed parameter systems are included. The existence results are consequences of an intricate analysis regarding the lower semicontinuity and compactness of level sets of Bolza functional in L¹ spaces. All these results are given under minimal requirements and using sharp functional methods. Specialized to finite dimensional problems the results given here extend some recent results of Ioffe and Rockafellar, but the proofs are quite different.     

The symmetry principle in continuum mechanics

For problems of the mechanics of an anisotropic inhomogeneous continuum, theorems are given concerning the uninterrupted symmetrical and antisymmetrical analytical continuation of the solution through the plane part of the boundary surface of the medium. Theorems are given for two types of mechanics problem; in the first of these both symmetrical and antisymmetrical continuations of the solution are allowed, while in the second only symmetrical continuation of the solution is allowed. Problems of the first type include problems which are reduced to linear thermoelastic dynamic differential equations of motion of an inhomogeneous anisotropic medium possessing a plane of elastic symmetry, to linear thermoelastic dynamic differential equations of motion of an inhomogeneous Cosserat medium, to non-linear differential equations describing the static elastoplastic stress state of a plate, etc. The second type includes problems which are reduced to non-linear differential equations describing geometrically non-linear strains of shells, to Navier–Stokes equations, etc. These theorems extend the principle of mirror reflection (the Riemann–Schwartz principle of symmetry) to linear and non-linear equations of continuum mechanics. The uninterrupted continuation of the solutions is used to solve problems of the equilibrium state of bodies of complex shape.     

A Local Relaxation Approach for the Siting of Electrical Substations

The siting and sizing of electrical substations on a rectangular electrical grid can be formulated as an integer programming problem with a quadratic objective and linear constraints. We propose a novel approach that is based on solving a sequence of local relaxations of the problem for a given number of substations. Two methods are discussed for determining a new location from the solution of the relaxed problem. Each leads to a sequence of strictly improving feasible integer solutions. The number of substations is then modified to seek a further reduction in cost. Lower bounds for the solution are also provided by solving a sequence of mixed-integer linear programs. Results are provided for a variety of uniform and Gaussian load distributions as well as some real examples from an electric utility. The results of gams/dicopt, gams/sbb, gams/baron and cplex applied to these problems are also reported. Our algorithm shows slow growth in computational effort with the number of integer variables.     

Risk-sensitive average continuous-time Markov decision processes with unbounded rates

Abstract

In this paper we study the risk-sensitive average cost criterion for continuous-time Markov decision processes in the class of all randomized Markov policies. The state space is a denumerable set, and the cost and transition rates are allowed to be unbounded. Under the suitable conditions, we establish the optimality equation of the auxiliary risk-sensitive first passage optimization problem and obtain the properties of the corresponding optimal value function. Then by a technique of constructing the appropriate approximating sequences of the cost and transition rates and employing the results on the auxiliary optimization problem, we show the existence of a solution to the risk-sensitive average optimality inequality and develop a new approach called the risk-sensitive average optimality inequality approach to prove the existence of an optimal deterministic stationary policy. Furthermore, we give some sufficient conditions for the verification of the simultaneous Doeblin condition, use a controlled birth and death system to illustrate our conditions and provide an example for which the risk-sensitive average optimality strict inequality occurs.     

Numerical treatment of singularly perturbed two point boundary value problems using initial-value method

In this paper, we describe an initial-value method for linear and nonlinear singularly perturbed boundary value problems in the interval [p,q]. For linear problems, the required approximate solution is obtained by solving the reduced problem and one initial-value problems directly deduced from the given problem. For nonlinear problems the original second-order nonlinear problem is linearized by using quasilinearization method. Then this linear problem is solved as previous method. The present method has been implemented on several linear and non-linear examples which approximate the exact solution. We also present the approximate and exact solutions graphically.     

Local search algorithm for universal facility location problem with linear penalties

The universal facility location problem generalizes several classical facility location problems, such as the uncapacitated facility location problem and the capacitated location problem (both hard and soft capacities). In the universal facility location problem, we are given a set of demand points and a set of facilities. We wish to assign the demands to facilities such that the total service as well as facility cost is minimized. The service cost is proportional to the distance that each unit of the demand has to travel to its assigned facility. The open cost of facility i depends on the amount z of demand assigned to i and is given by a cost function \(f_i(z)\). In this work, we extend the universal facility location problem to include linear penalties, where we pay certain penalty cost whenever we refuse serving some demand points. As our main contribution, we present a (\(7.88+\epsilon \))-approximation local search algorithm for this problem.     

On the controllability of a robot arm

Considered is the rotation of a robot arm or rod in a horizontal plane about an axis through the arm's fixed end and driven by a motor whose torque is controlled. The model was derived and investigated computationally by Sakawa and co-authors in [7] for the case that the arm is described as a homogeneous Euler beam. The resulting equation of motion is a partial differential equation of the type of a wave equation which is linear with respect to the state, if the control is fixed, and non-linear with respect to the control. Considered is the problem of steering the beam, within a given time interval, from the position of rest for the angle zero into the position of rest under a certain given angle. At first we show that, for every L²-control, there is exactly one (weak) solution of the initial boundary value problem which describes the vibrating system without the end condition. Then we show that the problem of controllability is equivalent to a non-linear moment problem. This, however, is not exactly solvable. Therefore, an iteration method is developed which leads to an approximate solution of sufficient accuracy in two steps. This method is numerically implemented and demonstrated by an example. © 1998 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Exponential Stability in Mean Square for a General Class of Discrete-Time Linear Stochastic Systems

Abstract

The problem of the mean square exponential stability for a class of discrete-time linear stochastic systems subject to independent random perturbations and Markovian switching is investigated. The case of the linear systems whose coefficients depend both to present state and the previous state of the Markov chain is considered. Three different definitions of the concept of exponential stability in mean square are introduced and it is shown that they are not always equivalent. One definition of the concept of mean square exponential stability is done in terms of the exponential stability of the evolution defined by a sequence of linear positive operators on an ordered Hilbert space. The other two definitions are given in terms of different types of exponential behavior of the trajectories of the considered system. In our approach the Markov chain is not prefixed. The only available information about the Markov chain is the sequence of probability transition matrices and the set of its states. In this way one obtains that if the system is affected by Markovian jumping the property of exponential stability is independent of the initial distribution of the Markov chain.

The definition expressed in terms of exponential stability of the evolution generated by a sequence of linear positive operators, allows us to characterize the mean square exponential stability based on the existence of some quadratic Lyapunov functions.

The results developed in this article may be used to derive some procedures for designing stabilizing controllers for the considered class of discrete-time linear stochastic systems in the presence of a delay in the transmission of the data.