Optimal Stabilization Problem Research Articles

Linear quadratic optimal control turnpike in finite and infinite dimension: Two-term expansion of the value function

In this paper, we consider a linear quadratic (LQ) optimal control problem in both finite and infinite dimensions. We derive an asymptotic expansion of the value function as the fixed time horizon T tends to infinity. The leading term in this expansion, proportional to T, corresponds to the optimal value attained through the classical turnpike theory in the associated static problem. The remaining terms are associated with optimal stabilization problems towards the turnpike.

Optimal control for networked control system with Markovian packet loss and delay

AbstractThis paper focuses on the optimal output‐feedback control and stabilization of a networked control system, where both time delay and packet loss are involved. Different from most previous work, the packet loss is Markovian, and the time delay occurs between the actuator and controller. The difficulty is rooted in the failure of the separation principle. The main tools are solving the delayed forward‐backward stochastic difference equations (D‐FBSDEs) and convergent analysis. Resorting to solving D‐FBSDEs and completing the square, we acquire the analytical solution, including the necessary and sufficient solvability condition and the explicit controller, of the finite‐horizon optimal control problem. Based on the finite‐horizon result and convergent analysis, we then obtain the analytical solution of the infinite‐horizon optimal control and stabilization problems. In addition, by analyzing the convergence of the optimal estimator, we establish a constraint relationship between the packet loss probability and the system matrix . The results are evaluated by numerical examples.

Adaptive optimal ℓ∞‐induced robust stabilization of minimum phase SISO plant under bounded disturbance and coprime factor perturbations

AbstractThis paper addresses the problem of optimal robust stabilization of a discrete‐time minimum‐phase plant in the framework of robust control theory in the setup and under poor a priori information. Coefficients of the transfer function of the plant nominal model with stable zeros are unknown and belong to a known bounded polyhedron in the space of coefficients. The gains of coprime factor perturbations of the plant and the upper bound of external disturbance are also unknown. The problem under consideration is to design adaptive controller that minimizes, with the prescribed accuracy, the worst‐case asymptotic upper bound of the output. Solution of the problem is based on set‐membership estimation of unknown parameters and treating the control criterion as the identification criterion. A hard nonconvex problem of on‐line computation of optimal estimates is reduced, under additional nonrestrictive assumption, to a linear‐fractional programming via a nonlinear transformation of estimated parameters. Despite the non‐identifiability of the unknown parameters, the proposed adaptive controller guarantees, with the prescribed accuracy, the same optimal asymptotic upper bound of the output of adaptive system as the optimal controller for the plant with known parameters. In addition to the optimality of adaptive control, the proposed solution provides on‐line verification/validation of current estimates and a priori assumptions.

Correction of the optimal regulator based on solving the inverse problem of optimal stabilization with vector control

Objective. The problem of designing controllers that implement a given programmed movement of a controlled object and the problem of determining the movement of a dynamic system are two main problems in classical control theory. This article discusses the solution of direct and inverse optimal stabilization problems. The state vector is assumed to be completely available for measurement.Method. Based on the optimality ratio linking the weight coefficients of the quadratic quality functional and the optimal gain matrix, which closes the control object, it is proposed to use a numerical method for determining the functional matrices. Mathematical models of autonomous fully controlled objects were used for the study, the formation of which was carried out randomly, in particular, according to the normal distribution law.Result. The initial stage of the solution is associated with modal synthesis, the result of which is a proportional regulator that provides stabilization of the control object by the location of the poles of the synthesized system. The next step is to determine the weighting coefficients of the functional by numerically solving the optimality ratio. The final stage is the solution of the direct optimal stabilization problem, which is based on the Lagrange variational problem. As a result, the optimal regulator is calculated, which, when switched on in a closed system instead of a modal one, reduces the duration of the transient process.Conclusion. The proposed approach of the authors allows minimizing to a certain extent the transients of the adjusted control system.

ON OPTIMAL STABILIZATION OF PART OF VARIABLES OF ROTARY MOVEMENT OF A RIGID BODY WITH ONE FIXED POINT IN THE CASE OF SOPHIA KOVALEVSKAYA

An optimal stabilization problem for part of variables of rotary movement of a rigid body with one fixed point in the Sophia Kovalevskaya's case is discussed in this work. The differential equations of motion of the system are given and it is shown that the system may rotate around Ox with a constant angular velocity. Taking this motion as unexcited, the differential equations for the corresponding excited motion were drawn up. Then the system was linearized and a control action was introduced along one of the generalized coordinates. The optimal stabilization problem for part of the variables was posed and solved. The graphs of optimal trajectories and optimal control were constructed.

Optimal control and stabilization for linear mean‐field system with indefinite quadratic cost functional

AbstractIn this paper, we consider an optimal control and stabilization problem of linear mean‐field (MF) system, where the quadratic cost functional is allowed to be indefinite. Inspired by the equivalent cost functional method, we introduce a subset, which helps us to investigate the convergence property of generalized differential Riccati equations arising in indefinite mean‐field linear quadratic (MF‐LQ) problems with finite horizon. A coupled generalized algebraic Riccati equation (GARE) is thus obtained. More importantly, the solution pair of corresponding GARE can be decomposed into a solution pair of a coupled standard algebraic Riccati equation (SARE) and a matrix pair in the subset we introduce. Thus, an equivalent relationship is established between GARE and SARE. With this equivalence, we derive necessary and sufficient conditions to stabilize linear MF‐system with indefinite weighting matrices in mean‐square sense.

Stabilization of Stochastic Dynamical Systems of a Random Structure with Markov Switches and Poisson Perturbations

An optimal control for a dynamical system optimizes a certain objective function. Here, we consider the construction of an optimal control for a stochastic dynamical system with a random structure, Poisson perturbations and random jumps, which makes the system stable in probability. Sufficient conditions of the stability in probability are obtained, using the second Lyapunov method, in which the construction of the corresponding functions plays an important role. Here, we provide a solution to the problem of optimal stabilization in a general case. For a linear system with a quadratic quality function, we give a method of synthesis of optimal control based on the solution of Riccati equations. Finally, in an autonomous case, a system of differential equations was constructed to obtain unknown matrices that are used for the construction of an optimal control. The method using a small parameter is justified for the algorithmic search of an optimal control. This approach brings a novel solution to the problem of optimal stabilization for a stochastic dynamical system with a random structure, Markov switches and Poisson perturbations.

FINITE APPROXIMATIONS OF THE PROBLEM OF OPTIMAL IMPULSE STABILIZATION FOR A SYSTEM WITH AFTEREFFECT

A method for constructing finite-dimensional approximations for the optimal impulse stabilizing control of a linear autonomous system of differential equations with aftereffect is proposed. Auxiliary finite-dimensional problems of optimal stabilization of linear autonomous systems of ordinary differential equations with non-impulse controls are used. A procedure for reducing the dimension of a system of algebraic matrix Riccati equations is described.

Optimal Stabilization of Periodic Orbits

In this contribution the optimal stabilization problem of periodic orbits is studied in a symplectic geometry setting. For this, the stable manifold theory for the point stabilization case is generalized to the case of periodic orbit stabilization. Sufficient conditions for the existence of a normally hyperbolic invariant manifold (NHIM) of the Hamiltonian system are derived. It is shown that the optimal control problem has a solution if the related periodic Riccati equation has a unique periodic solution. For the analysis of the stable and unstable manifold a coordinate transformation is used which is moving along the orbit. As an example, an optimal control problem is considered for a spring-mass oscillator system, which should be stabilized at a certain energy level.

Observer-Based Adaptive Inverse Optimal Output Regulation for a Class of Uncertain Nonlinear Systems

This paper addresses the adaptive inverse optimal output regulation problem for a class of uncertain nonlinear systems driven by an exosystem. The unknown parameters, internal disturbances, and unmeasured states are contained in the nonlinear system. Firstly, the output regulation problem is decomposed into a feedforward control design problem which can be solved by the internal model based on the output regulation theory, and an adaptive inverse optimal stabilization problem. Then an auxiliary system is designed, and a new state observer related to the auxiliary system is given. By combining adaptive control technology and inverse optimal control method, a novel adaptive output feedback inverse optimal controller is developed to make the output of the system track the reference signal fast. With this control strategy, all the signals of the closed-loop system are uniformly ultimately bounded (UUB), and the newly well-defined cost functional which is connected with the auxiliary system and the controller can be minimized. Finally, a simulation case is put forward to verify the feasibility of the newly raised controller and the state observer.

Comparative Study of Markov Chain Filtering Schemas for Stabilization of Stochastic Systems under Incomplete Information

The object under investigation is a controllable linear stochastic differential system affected by some external statistically uncertain piecewise continuous disturbances. They are directly unobservable but assumed to be a continuous-time Markov chain. The problem is to stabilize the system output concerning a quadratic optimality criterion. As is known, the separation theorem holds for the system. The goal of the paper is performance analysis of various numerical schemes applied to the filtering of the external Markov input for system stabilization purposes. The paper briefly presents the theoretical solution to the considered problem of optimal stabilization for systems with the Markov jump external disturbances: the conditions providing the separation theorem, the equations of optimal control, and the ones defining the Wonham filter. It also contains a complex of the stable numerical approximations of the filter, designed for the time-discretized observations, along with their accuracy characteristics. The approximations of orders 12, 1, and 2 along with the classical Euler–Maruyama scheme are chosen for the comparison of the Wonham filter numerical realization. The filtering estimates are used in the practical stabilization of the various linear systems of the second order. The numerical experiments confirm the significant influence of the filtering precision on the stabilization performance and superiority of the proposed stable schemes of numerical filtering.

Linear–Quadratic Optimal Control for Discrete-Time Mean-Field Systems With Input Delay

The linear–quadratic (LQ) optimal control and stabilization problems for mean-field systems with input delay (MFSID) are investigated in this article. The necessary and sufficient solvability conditions for LQ control of MFSID are first given in terms of a convexity condition and the solvability of equilibrium conditions. Consequently, by solving the associated mean-field forward and backward stochastic difference equations, the optimal control is derived in terms of the solution of a modified Riccati equation. Furthermore, for the infinite-horizon case, the stabilization problem for MFSID is studied, and the necessary and sufficient stabilizability conditions are obtained. We show that MFSID can be mean square stabilizable if and only if a modified algebraic Riccati equation admits a unique positive-definite solution.

Decentralized control for multi-sensors networked systems with different transmission delays and packet dropouts

In this paper, the linear quadratic (LQ) optimal decentralized control and stabilization problems are investigated for multi-sensors networked control systems (MSNCSs) with multiple controllers of different information structure. Specifically, for a MSNCS, in view of the packet dropouts and the transmission delays, each controller may access different information sets. To begin with, the sufficient and necessary solvability conditions for the LQ decentralized control problems are developed. Consequently, for the purpose of deriving the optimal decentralized control strategy, an innovative orthogonal decomposition method is proposed to decouple the forward and backward stochastic difference equations (FBSDEs) from the maximum principle. In the following, we show that the optimal decentralized controller can be calculated according to a set of Riccati-type equations. Finally, a stabilizing controller is derived for the stabilization problem.

L2-Loss nonparallel bounded support vector machine for robust classification and its DCD-type solver

In this paper, we study a new classification methodology to enhance the robustness and convergence of the nonparallel support vector machine (NPSVM), namely L2-Loss nonparallel bounded SVM (L2-NPBSVM). We first define a L2-Loss and an adjustable hinge loss for NPSVM. Then, using the two loss functions, we propose the L2-NPBSVM algorithm. Both the L2-loss and the adjustable hinge loss are insensitive to the feature noise around the decision boundary. In addition, the L2-Loss can make full use of the margin distribution information in the training samples. The margin distribution is more important than margin maximization for generalization performance. Furthermore, an additional regularization term is added into the objective functions of L2-NPBSVM. It can ensure the global solution and stability of optimization problems. Thus, the classification performance of L2-NPBSVM can be further improved. In addition, in order to shorten the training time, the dual coordinate descent (DCD) algorithm is described and analyzed to optimize L2-NPBSVM. Numerical experiments on different datasets demonstrate that our L2-NPBSVM has the advantages of strong robustness, strong generalization ability and fast convergence.

Continuous Differentiability of the Value Function of Semilinear Parabolic Infinite Time Horizon Optimal Control Problems on L2(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(\\Omega )$$\\end{document} Under Control Constraints

An abstract framework guaranteeing the local continuous differentiability of the value function associated with optimal stabilization problems subject to abstract semilinear parabolic equations subject to a norm constraint on the controls is established. It guarantees that the value function satisfies the associated Hamilton–Jacobi–Bellman equation in the classical sense. The applicability of the developed framework is demonstrated for specific semilinear parabolic equations.

Minimax stabilization of the line-of-sight of an inertial object on a moving base in the presence of friction force

The solution of the problem of optimal stabilization of the line-of-sight of an inertial object in the vicinity of the program trajectory is given. The motion of this line is described by a system of nonlinear differential equations of the fourth order. The system of equations is linearized in the vicinity of the desired motion mode. In the problem solved, the perturbations are represented as deviations of the initial position from zero, as well as in the form of constant perturbations. The stabilization is carried out by means of linear feedback. In the problem, the feedback coefficients are calculated as optimal for the worst possible disturbances. Calculations are performed in two ways: a search of all possible combinations of parameters with a given sampling step and a parallel genetic algorithm.

Event-triggered intelligent critic control with input constraints applied to a nonlinear aeroelastic system

In this paper, we establish an event-triggered intelligent control scheme with a single critic network, to cope with the optimal stabilization problem of nonlinear aeroelastic systems. The main contribution lies in the design of a novel triggering condition with input constraints, avoiding the Lipschitz assumption on the inverse hyperbolic tangent function. Based on an improved weight updating criterion that eliminates the requirement of initial admissible control, the control law is obtained approximately by online training of a single critic network. The Lyapunov stability and the Zeno phenomenon of the closed-loop system are analysed. The feasibility of the established algorithm is verified by applying it to an optimal stabilization task of a nonlinear aeroelastic system. The results reveal that the developed approach can handle input-constrained optimal control problems, with performance comparable to the time-based method that updates control inputs at each instant, while reducing the computational and communication's load.

Decentralized stabilization of large-scale stochastic nonlinear systems with time-varying powers

In this paper, we study the decentralized stabilization problem for large-scale high-order stochastic nonlinear systems with time-varying powers. By using the backstepping design technique, a new decentralized state-feedback controller is constructed to ensure the closed-loop system is globally asymptotically stable (GAS) in probability. Then we further redesign a new optimal controller to solve the decentralized inverse optimal stabilization (IOS) problem. Specifically, our redesigned stabilizing backstepping controller is not only globally asymptotically stable for the closed-loop system but also optimal for the meaningful cost function. Finally, a simulation example is given to illustrate the effectiveness of the designed controllers.

Optimal control and stabilization for linear continuous‐time mean‐field systems with delay

This paper studies optimal control and stabilization problems for continuous-time mean-field systems with input delay, which are the fundamental development of control and stabilization problems for mean-field systems. There are two main contributions: (1) To the best of the authors' knowledge, the present paper is the first to establish the necessary and sufficient solvability condition for this kind of optimal control problem with input delay, and to derive the analytical form of an optimal controller through overcoming the obstacle that separation principle no longer holds for multiplicative-noise systems. (2) For the stabilization problem, under the assumption of exact observability, it is strictly proven that the system is stabilizable if and only if the algebraic Riccati equation has a unique positive definite solution.

Optimal damping concept: features and applications

This work presents new ideas related to the synthesis of nonlinear stabilizing control laws in the framework of optimization approach. The focus is done on the optimal damping concept, proposed by V.I. Zubov in the early 60s of the last century. This theory is used to reduce significant computational costs in solving optimal stabilization problems. The essential features of the aforementioned concept are taken into account, allowing the construction of new methods for practical synthesis of control systems with desired dynamic properties. Various modern aspects of the optimal damping theory’s practical implementation are discussed. Special attention is paid to the specific choice of the functional to be damped to provide the desirable stability and performance features of the closed-loop connection.