Optimal Stabilization Problem Research Articles

Asymptotically stable critic designs for approximate optimal stabilization of nonlinear systems subject to mismatched external disturbances

This paper addresses approximate optimal stabilization problems for nonlinear systems in the presence of mismatched external disturbances via asymptotically stable critic designs. By establishing the nonlinear disturbance observer, the corresponding information is utilized to construct the online updated cost function, which reflects the real-time disturbances, regulation and control simultaneously. With the help of the proper cost function, the Hamilton–Jacobi–Bellman equation is solved by employing a critic neural network, whose weight vector is guaranteed to be asymptotically stable with nested tuning laws. The approximate optimal control is derived to guarantee the closed-loop system to be ultimately uniformly bounded based on the Lyapunov stability theorem. The effectiveness of the developed stabilization scheme is verified via simulations of two numerical examples.

Time-optimal control problem with two final points for a kinematic model of an UAV

We study the time-optimal control problem for an unmanned aerial vehicle (drone) moving in the plane of a constant altitude. A kinematic model is considered where the angular velocity is a control. Such a system is described by Markov-Dubins equations; a large number of works are devoted to solving different optimal and admissible control and stabilization problems for such models. In the papers [T. Maillot, U. Boscain, J.-P. Gauthier, U. Serres, Lyapunov and minimum-time path planning for drones, J. Dyn. Control Syst., V. 21 (2015)] and [M.A.~Lagache, U. Serres, V. Andrieu, Minimal time synthesis for a kinematic drone model, Mathematical Control and Related Fields, V. 7 (2017)] the time optimal control problem is solved where the drone must reach a given unit circle in the minimal possible time and stay on this circle rotating counterclockwise. In particular, in the mentioned works it is shown that is this case the problem is simplified; namely, the problem becomes two-dimensional. In the present paper we consider a natural generalization of the formulation mentioned above: in our problem, the drone must reach a given unit circle in the minimal possible time and stay on this circle, however, both rotating directions are admissible. That is, the drone can rotate clockwise or counterclockwise; the direction is chosen for reasons of minimizing the time of movement. Such a reformulation leads to the time-optimal control problem with two final points. In the paper, we obtain a complete solution of this time-optimal control problem. In particular, we show that the optimal control takes the values $\pm1$ or $0$ and has no more than two switchings. If the optimal control is singular, i.e., contains a piece $u=0$, then this piece is unique and the duration of the last piece equals $\pi/3$; moreover, in this case the optimal control ins non-unique and the final point can be $(0,1)$ as well as $(0,-1)$. If the optimal control is non-singular, i.e., takes the values $\pm1$, then it is unique (except the case when the duration of the last piece equals $\pi/3$) and the optimal trajectory entirely lies in the upper or lower semi-plane. Also, we give a solution of the optimal synthesis problem.

Об одной задаче оптимальной стабилизации полёта квадрокоптера

Об одной задаче оптимальной стабилизации полёта квадрокоптера

О задаче оптимальной стабилизации одной системы линейных нагруженных дифференциальных уравнений

О задаче оптимальной стабилизации одной системы линейных нагруженных дифференциальных уравнений

Pareto Optimal Decentralized Control of a Rotor in Electromagnetic Bearings

The problem of the optimal stabilization of a vertical rigid rotor rotating in two electromagnetic bearings is considered. A quantitative estimate of transient processes is based on the maximal deviation of the rotor from the equilibrium state and the maximal value of the current intensity in the windings of electromagnets. The use of linear matrix inequalities allow synthesizing the Pareto optimal control laws in the form of linear state-feedback, as well as decentralized control laws, which are defined by the displacements of the rotor in each electromagnetic bearing. The Pareto optimal criteria for centralized and decentralized control laws are compared. It is shown that utilization of simple easily realizable decentralized control laws only slightly worsens transient processes.

Control for networked control systems with remote and local controllers over unreliable communication channel

This paper is concerned with the problems of optimal control and stabilization for networked control systems (NCSs), where the remote controller and the local controller operate the linear plant simultaneously. The main contributions are two-fold. Firstly, a necessary and sufficient condition for the finite horizon optimal control problem is given in terms of the two Riccati equations. Secondly, it is shown that the system without the additive noise is stabilizable in the mean square sense if and only if the two algebraic Riccati equations admit the unique solutions, and a sufficient condition is given for the boundedness in the mean square sense of the system with the additive noise. Numerical examples about unmanned aerial vehicles model are shown to illustrate the effectiveness of the proposed algorithm.

Learning-Based Adaptive Optimal Tracking Control of Strict-Feedback Nonlinear Systems.

This paper proposes a novel data-driven control approach to address the problem of adaptive optimal tracking for a class of nonlinear systems taking the strict-feedback form. Adaptive dynamic programming (ADP) and nonlinear output regulation theories are integrated for the first time to compute an adaptive near-optimal tracker without any a priori knowledge of the system dynamics. Fundamentally different from adaptive optimal stabilization problems, the solution to a Hamilton-Jacobi-Bellman (HJB) equation, not necessarily a positive definite function, cannot be approximated through the existing iterative methods. This paper proposes a novel policy iteration technique for solving positive semidefinite HJB equations with rigorous convergence analysis. A two-phase data-driven learning method is developed and implemented online by ADP. The efficacy of the proposed adaptive optimal tracking control methodology is demonstrated via a Van der Pol oscillator with time-varying exogenous signals.

Output stabilization of distributed bilinear systems

This paper studies regional stabilization of a distributed bilinear system evolving on a spatial domain Ω. Sufficient conditions for regional weak, strong and exponential stabilization are given. Also we discuss a regional optimal stabilization problem. The obtained results are illustrated by examples and simulations.

Neural network based optimal adaptive attitude control of near-space vehicle with system uncertainties and disturbances

In this paper, a neural network based optimal adaptive attitude control scheme is derived for the near-space vehicle with uncertainties and external time-varying disturbances. Firstly, radial basis function neural network (RBFNN) approximation method and nonlinear disturbance observer (NDO) are used to tackle the system uncertainties and external disturbances, respectively. Subsequently, a feedforward control input under backstepping control frame with RBFNN and NDO is designed to transform the optimal tracking control problem into an optimal stabilization problem. Then, a single online approximation based adaptive method is used to learn the Hamilton–Jacobi–Bellman equation to obtain the corresponding optimal controller. As a result, the compound controller consists of feedforward control input and optimal controller which can ensure that the near-space vehicle attitude angles are able to track reference signals in an optimal way. Lyapunov stability analysis method is used to show that all the closed-loop system signals are uniformly ultimately bounded. Finally, simulation results show the effectiveness of the proposed optimal attitude control scheme.

Regional Feedback Stabilization for Infinite Semilinear Systems

In this paper, we treat the problem of regional optimal stabilization of a class of nonlinear systems by using a switching feedback. Firstly, we proof that the switching control exponentially stabilizes the system on a subregion including the whole domain. Secondly, under a perturbation of the control operator, we show the robustness of our result. In the last part, the stabilizing feedback is characterized by the minimization of a regional cost even under a small perturbation. We conclude by giving different applications.

On the Optimal Stabilization of an Integral Manifold

By the method of Lyapunov functions, we study the problem of optimal stabilization of an analytically defined integral manifold in the class of stochastic differential equations in the case where random perturbations belong to the class of processes with independent increments.

ДИСКРЕТНАЯ ПРОЦЕДУРА ОПТИМАЛЬНОЙ СТАБИЛИЗАЦИИ ПЕРИОДИЧЕСКИХ ЛИНЕЙНЫХ СИСТЕМ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ

We propose procedure to solve the optimal stabilization problem for linear periodic systems of differential equations. Stabilizing controls, formed as a feedback, are defined by the system states at the fixed instants of time. Equivalent discrete-time linear periodic problem of optimal stabilization is considered. We propose a special procedure for the solution of discrete periodic Riccati equation. We investigate the relation between continuous-time and discrete-time periodic optimal stabilization problems. The proposed method is used for stabilization of mechanical systems.

ALGORITHM FOR SOLVING THE PROBLEM OF OPTIMAL STABILIZATION BY OUTPUT AND THEIR APPLICATION

Abstract The paper is devoted to the solutions of high-accuracy of the synthesis problem of optimal regulators with the frequency-time method. The problem for finding the optimal regulators by output variable in the stationary and periodic cases is considered and an iteration scheme for solution such problems supported symbolic computation is proposed. The solutions of high-accuracy of the matrix algebraic equations of Riccati, Lyapunov and Sylvester are given. At first the inverse optimal stabilization problem by output variable is investigated. Using the method of linear matrix inequalities the computational algorithms are proposed. The obtained results have been applied to practice, including to the optimal control problem used in oil extraction of gas lift method.

Inverse Problems of Optimal Stabilization with Scalar Control

Inverse Problems of Optimal Stabilization with Scalar Control

OPTIMAL STABILIZATION OF ROTATIONAL MOTION OF A RIGID BODY AROUND ITS CENTER OF GRAVITY

The present work considers the optimal stabilization problem in rotational motion of a rigid body around its center of gravity. The case of the Euler rotational motion of a rigid body around a fixed point is considered. The optimal stabilization problem of the considered motion is assumed and solved. Input controls are introduced in the direction of the generalized coordinates, full controllability of linear approximation of the system is checked. Besides the optimal stabilization problem of the system on classical sense is solved, optimal Lyapunov function, optimal controls and value of functional are obtained.

On Epsilon-Stability in Optimization

We study stability of optimization problems in the separated locally convex topological vector space setting. We use the concepts of 𝜖-stability, dual 𝜖-stability, and 𝜖-duality gap, and establish geometric characterizations of these notions by an epigraphical analysis approach. Under a constraint qualification involving quasi relative interiors, we obtain some criteria for 𝜖-stability and 𝜖-duality gap, which are shown to be useful for obtaining relevant stability and duality theorems in infinite dimensional optimization. We apply our approach to study cone constrained problems, Fenchel duality, conjugate duality, and the subdifferentiability of functions associated with epigraphical type sets. Several duality and stability results of recent publications can be deduced from and sometimes improved by a geometric characterization of 𝜖-stability in a unifying way.

Output Feedback Control and Stabilization for Multiplicative Noise Systems With Intermittent Observations

This paper mainly focuses on the optimal output feedback control and stabilization problems for discrete-time multiplicative noise system with intermittent observations. The main contributions of this paper can be concluded as follows. First, different from the previous literatures, this paper overcomes the barrier of the celebrated separation principle for stochastic control problems of multiplicative noise systems. Based on the measurement process, the optimal estimation is presented, and by using dynamic programming principle, the optimal output feedback controller is designed with feedback gain based on the given coupled Riccati equations. Second, the necessary and sufficient stabilization conditions for multiplicative noise system with intermittent observation in the mean square sense are developed for the first time. Finally, the novel results developed in this paper can be applied to solve the output feedback control and stabilization problems for general networked control system of user datagram protocol network case. The range of packet losses rate and the allowable maximum packet losses rate are presented explicitly.

Optimal stabilization of bodies in electromagnetic suspensions without measurements of their location

The optimal stabilization problem is considered for bodies in electromagnetic suspensions. To solve this problem, we form a linear stationary control law for the linearized system. This law is based on the feedback principle and uses the measuring of the current intensity in the circuit of the electromagnet, while the location of the body and its velocity are not measured. The optimality criterion is the generalized H ∞-norm of the linearized system: it characterizes the extinguishing level for perturbations generated by external actions and unknown initial conditions. To compute the feedback parameters, the technique of linear matrix inequalities is applied. We provide mathematical simulation examples for the dynamics of a body in an electromagnetic suspension.

Kinematic problem of optimal nonlinear stabilization of angular motion of a rigid body

The problem of optimal transfer of a rigid body to a prescribed trajectory of preset angular motion is considered in the nonlinear statement. (The control is the vector of absolute angular velocity of the rigid body.) The functional to be minimized is a mixed integral quadratic performance criterion characterizing the general energy expenditure on the control and deviations in the state coordinates.

Data-driven approximate value iteration with optimality error bound analysis

Features of the data-driven approximate value iteration (AVI) algorithm, proposed in Li et al. (2014) for dealing with the optimal stabilization problem, include that only process data is required and that the estimate of the domain of attraction for the closed-loop is enlarged. However, the controller generated by the data-driven AVI algorithm is an approximate solution for the optimal control problem. In this work, a quantitative analysis result on the error bound between the optimal cost and the cost under the designed controller is given. This error bound is determined by the approximation error of the estimation for the optimal cost and the approximation error of the controller function estimator. The first one is concretely determined by the approximation error of the data-driven dynamic programming (DP) operator to the DP operator and the approximation error of the value function estimator. These three approximation errors are zeros when the data set of the plant is sufficient and infinitely complete, and the number of samples in the interested state space is infinite. This means that the cost under the designed controller equals to the optimal cost when the number of iterations is infinite.