An optimal control approach to spacecraft rendezvous on elliptical orbit

The linear quadratic optimal control problem under quadratic constraints is an optimization problem over a generally non-convex set. Yakubovich [8] and Megretski [4] have studied this problem, and they show how it may be translated into a two-stage, convex optimization problem. In this paper we study the linear quadratic control problem under quadratic constraints for generalized first-order systems. As in the state-space case the linear quadratic control problem without quadratic constraints may be solved in terms of a linear matrix inequality. Subsequently, we use the results from [8] to derive a linear matrix inequality characterizing the linear quadratic optimal behaviour under quadratic constraints.

Linear Quadratic Optimal Control of a Wave Equation with Boundary Damping and Pointwise Control Input

A comprehensive treatment of the discrete-time, steady-state linear quadratic Gaussian stochastic control problem which is based entirely on the z-transfer-function point of view is presented. The optimal solution is found for the general multi-variable case by exploiting the properties of the newly defined discrete generalized spectral factor. The proposed method of solution is developed independently of the results found by the time-domain approach. This method can easily cope with cases of discrete stationary coloured signals and dynamical operators in the performance index and its computational superiority over existing time-domain techniques is discussed.

A comprehensive treatment of the steady-state linear quadratic Gaussian stochastic control problem which is based on the transfer-function point of view is presented. The optimal solution is found, for the general multivariable case, by exploiting the properties of the newly defined generalized spectral factors. The exact equivalence between the results found by the time domain and the transfer-function approaches is established, even for the case of unstable systems. The proposed method of solution can easily cope with cases of stationary coloured signals, dynamical operators in the performance index and singular noise autocorrelation and control weighting matrices ; its computational superiority over existing time-domain techniques is demonstrated.

One Kind of Fully Coupled Linear Quadratic Stochastic Control Problem with Random Jumps

On the stochastic linear quadratic control problem with piecewise constant admissible controls

The (multi-player) linear quadratic state feedback control problem for index one descriptor systems

The linear quadratic tracking control problem is studied for a class of discrete-time uncertain Markov jump linear systems with time-varying conditional distributions. The controller is designed under the assumption that it has no access to the true states of the Markov chain, but rather it depends on the Markov chain state estimates. To deal with uncertainty, the transition probabilities of Markov state estimates between the different operating modes of the system are considered to belong in an ambiguity set of some nominal transition probabilities. The estimation problem is solved via the one-step forward Viterbi algorithm, while the stochastic control problem is solved via minimax optimization theory. An optimal control policy with some desired robustness properties is designed, and a maximizing time-varying transition probability distribution is obtained. A numerical example is given to illustrate the applicability and effectiveness of the proposed approach.

Suboptimal linear quadratic tracking control for multi-agent systems

In this paper we consider infinite-time linear quadratic Gaussian control problems for non-autonomous systems with possibly unbounded coefficients of their inhomogeneous components. It is shown that under standard stabilizability and detectability conditions, along with assumptions on inhomogeneous term growth, there exists an optimal control law. This control law of well-known feedback form appears to be optimal with respect to proposed extended criteria. The applied criteria expand the notions of long-run average and pathwise long-run average, commonly used for ergodic control. We also investigate the discounted tracking problems, where the cost incorporates a discount function penalizing deviations from reference trajectories.

This paper analyzes one kind of linear quadratic (LQ) stochastic control problem of forward backward stochastic control system associated with Lévy process. We obtain the explicit form of the optimal control, then prove it to be unique, and get the linear feedback regulator by introducing one kind of generalized Riccati equation. Finally, we discuss the solvability of the generalized Riccati equation, and its existence and uniqueness of the solutions are proved in a special case.

The paper considers the linear quadratic(LQ) control problem for the Itô-type stochastic system with input delays. Due to simultaneous appearances of diffusion terms (dependent on the system state and control input) as well as delays in the dynamic system, the problem is very involved and outstanding at present. Our main idea is to pursue the explicit optimal cost of the problem and to exploit the interplay between the original problem and its equivalent abstract description. Therefore, we not only provide a causal and adapted controller based on generalized Riccati equations with the same dimension as the system (without delays) but also present the sufficient and necessary condition under which the optimal control problem for the abstract stochastic system is solvable.

A new indirect adaptive control scheme based on recurrent neural networks is proposed. The certainty equivalence principle is used to combine an adaptive law with a control structure derived from the linear quadratic (LQ) control problem. The proposed approach includes two sets of neural networks each with two feedback connected layers to solve for two types of algebraic matrix Riccati equations. One is for the Kalman filter and the other one is for the LQ controller design. The gradient algorithm is used as an adaptive law for identifying plant parameters and is used as the update rule for neural networks.

A stochastic linear quadratic (LQ) control problem is indefinite when the cost weighting matrices for the state and the control are allowed to be indefinite. Indefinite stochastic LQ theory has been extensively developed and has found interesting applications in finance. However, there remains an outstanding open problem, which is to identify an appropriate Riccati-type equation whose solvability is {\it equivalent} to the solvability of the indefinite stochastic LQ problem. This paper solves this open problem for LQ control in a finite time horizon. A new type of differential Riccati equation, called the generalized (differential) Riccati equation, is introduced, which involves algebraic equality/inequality constraints and a matrix pseudoinverse. It is then shown that the solvability of the generalized Riccati equation is not only sufficient, but also necessary, for the well-posedness of the indefinite LQ problem and the existence of optimal feedback/open-loop controls. Moreover, all of the optimal controls can be identified via the solution to the Riccati equation. An example is presented to illustrate the theory developed.

This paper is concerned with a general linear quadratic (LQ) control problem of mean-field backward stochastic differential equation (BSDE). Here, the weighting matrices in cost functional are allowed to be indefinite. Necessary and sufficient conditions for optimality are obtained via a mean-field backward-forward stochastic differential equation (BFSDE). By investigating the connections with LQ problems of mean-field forward systems and taking some limiting procedures, we establish the solvabilities of corresponding Riccati equations when the cost functional is uniformly convex. Subsequently, we develop general procedures of constructing explicit representations of optimal control and optimal cost via two Riccati equations, a mean-field BSDE and an adjoint equation. Moreover, sufficient conditions for the uniform convexity of cost functional are also proposed in terms of Riccati equations, which have not been considered in existing literatures for backward systems. Two examples are provided to illustrate our theoretical results.