Inverse Optimal Stabilization Research Articles

Inverse Optimal Adaptive Neural Control for State-Constrained Nonlinear Systems.

Optimizing a performance objective during control operation while also ensuring constraint satisfactions at all times is important in practical applications. Existing works on solving this problem usually require a complicated and time-consuming learning procedure by employing neural networks, and the results are only applicable for simple or time-invariant constraints. In this work, these restrictions are removed by a newly proposed adaptive neural inverse approach. In our approach, a new universal barrier function, which is able to handle various dynamic constraints in a unified manner, is proposed to transform the constrained system into an equivalent one with no constraint. Based on this transformation, a switched-type auxiliary controller and a modified criterion for inverse optimal stabilization are proposed to design an adaptive neural inverse optimal controller. It is proven that optimal performance is achieved with a computationally attractive learning mechanism, and all the constraints are never violated. Besides, improved transient performance is obtained in the sense that the bound of the tracking error could be explicitly designed by users. An illustrative example verifies the proposed methods.

Decentralized Adaptive Neural Inverse Optimal Control of Nonlinear Interconnected Systems.

Existing methods on decentralized optimal control of continuous-time nonlinear interconnected systems require a complicated and time-consuming iteration on finding the solution of Hamilton-Jacobi-Bellman (HJB) equations. In order to overcome this limitation, in this article, a decentralized adaptive neural inverse approach is proposed, which ensures the optimized performance but avoids solving HJB equations. Specifically, a new criterion of inverse optimal practical stabilization is proposed, based on which a new direct adaptive neural strategy and a modified tuning functions method are proposed to design a decentralized inverse optimal controller. It is proven that all the closed-loop signals are bounded and the goal of inverse optimality with respect to the cost functional is achieved. Illustrative examples validate the performance of the methods presented.

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.

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.

Adaptive Fuzzy Inverse Optimal Fixed-Time Control of Uncertain Nonlinear Systems

Most existing methods on optimal finite-time control are restricted to a complex design and learning procedure, and only practical finite-time stable is ensured, which greatly limits the desirable performance of optimal and finite-time control. To solve the problem, an adaptive fuzzy fixed-time inverse approach is first proposed in this article, which achieves the optimized performance without recourse to Hamilton–Jacobi–Bellman equations and improves practical finite/fixed-time stable to fixed-time stable. Technically, to overcome the inverse optimal design difficulty of a nonlinear fixed-time controller, a series of singularity-avoidance functions and a Sontag-type function are incorporated to design a specified form of auxiliary controller, based on which an inverse optimal fixed-time controller is designed. Then, by introducing a two-Lyapunov functions method, it is proved that inverse optimal stabilization is ensured and the tracking error goes to a prescribed interval asymptotically within a fixed-time. Effectiveness of the proposed methods are illustrated by two examples.

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.

Stochastic Nonlinear Prescribed-Time Stabilization and Inverse Optimality

We solve the prescribed-time mean-square stabilization and inverse optimality control problems for stochastic strict-feedback nonlinear systems by developing a new nonscaling backstepping design scheme. A key novel design ingredient is that the time-varying function is not used to scale the coordinate transformations and is only suitably introduced into the virtual controllers. The advantage of this approach is that a simpler controller results and the control effort is reduced. By using this method, we design a new controller to guarantee that the equilibrium at the origin of the closed-loop system is prescribed-time mean-square stable. Then, we redesign the controller and solve the prescribed-time inverse optimal mean-square stabilization problem, with an infinite gain margin. Specifically, the designed controller is not only optimal with respect to a meaningful cost functional but also globally stabilizes the closed-loop system in the prescribed-time. Finally, two simulation examples are given to illustrate the stochastic nonlinear prescribed-time control design.

Bounded and inverse optimal formation stabilization of second-order agents

This paper formulates and solves a new problem of both bounded and inverse optimal formation stabilization control for a group of second-order dynamic mobile agents with collision avoidance and limited sensing range. The control design is based on new Lyapunov functions, new non-zero convergence and dominating lemmas, new pairwise collision avoidance functions, and forwarding and inverse optimal control design methods. The proposed formation stabilization control design guarantees no collision between any agents, “almost global” asymptotic stability of desired equilibrium points and instability of undesired equilibrium points, an infinite gain margin, bounded controls by a pre-specified constant, and minimization of a cost function that penalizes both stabilization errors and the control inputs without having to solve a Hamilton–Jacobi–Bellman or Hamilton–Jacobi–Isaacs equation.

An inverse optimal approach to ship course-keeping control

Abstract This paper deals with the ship course tracking control problem in a novel inverse optimal control approach. The inverse optimal stabilization problem and inverse optimal gain assignment problem are firstly extended to general systems affine in the control with unknown control gain. It is shown that a sufficient condition to solve the inverse optimal control problem is the existence of a stabilization control law in a special form for a corresponding auxiliary system. Then, by employing backstepping technique, control laws are designed which solve the inverse optimal stabilization, inverse optimal adaptive stabilization and inverse optimal adaptive gain assignment problem of ship course control system, respectively. Simulations are included to illustrate the effectiveness of the proposed control algorithms.

Stability in probability and inverse optimal control of evolution systems driven by Levy processes

This paper first develops a Lyapunov-type theorem to study global well-posedness &#x0028 existence and uniqueness of the strong variational solution &#x0029 and asymptotic stability in probability of nonlinear stochastic evolution systems &#x0028 SESs &#x0029 driven by a special class of Levy processes, which consist of Wiener and compensated Poisson processes. This theorem is then utilized to develop an approach to solve an inverse optimal stabilization problem for SESs driven by Levy processes. The inverse optimal control design achieves global well-posedness and global asymptotic stability of the closed-loop system, and minimizes a meaningful cost functional that penalizes both states and control. The approach does not require to solve a Hamilton-Jacobi-Bellman equation &#x0028 HJBE &#x0029. An optimal stabilization of the evolution of the frequency of a certain genetic character from the population is included to illustrate the theoretical developments.

Inverse Optimal Stabilization of Dynamical Systems by Wiener Processes

Abstract This paper formulates and solves new problems of inverse optimal stabilization and inverse optimal stabilization with gain assignment for nonlinear systems by Wiener processes. First, a theorem is developed to design inverse optimal stabilizers (i.e., covariance matrix multiplied by variance of Wiener processes), where it does not require to solve a Hamilton–Jacobi–Belman equation. Second, another theorem is developed to design inverse optimal stabilizers with gain assignment for nonlinear systems perturbed by both nonvanishing deterministic and stochastic (Wiener processes) disturbances without having to solve a Hamilton–Jacobi–Isaacs equation.

State‐feedback stabilization and inverse optimal control for stochastic high‐order nonlinear systems with time‐varying powers

AbstractThis paper considers the stability and optimality problems associated with stochastic high‐order nonlinear systems with time‐varying powers. This research extends previous studies on the plant with constant powers. In this paper, we firstly construct a state‐feedback controller to ensure that the equilibrium at the origin of the closed‐loop system is globally asymptotically stable (GAS) in probability. Then we redesign a new optimal controller with infinite gain margin to solve the inverse optimal stabilization (IOS) problem. Specifically, the optimal controller is not only optimal with respect to a meaningful cost functional but also globally asymptotically stabilizes the closed‐loop system. The efficiency of the designed controllers is demonstrated by a simulation example.

Inverse optimal control of stochastic systems driven by Lévy processes

This paper formulates and solves new problems of inverse optimal stabilization and inverse optimal gain assignment for stochastic nonlinear systems driven by Lévy processes. First, a theorem is developed to design inverse optimal stabilizers based on inverse pre-optimal stabilization controllers for stochastic systems with known noise characteristics, where it does not require to solve a Hamilton–Jacobi–Bellman equation. Second, another theorem is developed to design inverse optimal gain assignment controllers for stochastic systems with unknown noise characteristics, where there is no need to solve a Hamilton–Jacobi–Isaacs equation. Third, the results are applied to design inverse optimal stabilizers for Euler–Lagrange systems.

Inverse optimal control of evolution systems and its application to extensible and shearable slender beams

An optimal &#x0028 practical &#x0029 stabilization problem is for mulated in an inverse approach and solved for nonlinear evolution systems in Hilbert spaces. The optimal control design ensures global well-posedness and global practical K&#x221E -exponential stability of the closed-loop system, minimizes a cost functional, which appropriately penalizes both state and control in the sense that it is positive definite &#x0028 and radially unbounded &#x0029 in the state and control, without having to solve a Hamilton-Jacobi-Belman equation &#x0028 HJBE &#x0029. The Lyapunov functional used in the control design explicitly solves a family of HJBEs. The results are applied to design inverse optimal boundary stabilization control laws for extensible and shearable slender beams governed by fully nonlinear partial differential equations.

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 Optimal Attitude Stabilization of Flexible Spacecraft with Actuator Saturation

This paper presents a new robust inverse optimal control strategy for flexible spacecraft attitude maneuvers in the presence of external disturbances and actuator constraint. A new constrained attitude controller for flexible spacecraft is designed based on the Sontag-type formula and a control Lyapunov function. This control law optimizes a meaningful cost functional and the stability of the resulting closed-loop system is ensured by the Lyapunov framework. A sliding mode disturbance observer is used to compensate unknown bounded external disturbances. The ultimate boundedness of estimation error dynamics is guaranteed via a rigorous Lyapunov analysis. Simulation results are provided to demonstrate the performance of the proposed control law.

Output-Feedback and Inverse Optimal Control of a Class of Stochastic Nonlinear Systems with More General Growth Conditions

This paper investigates the problem of output-feedback stabilization for a class of stochastic nonlinear systems in which the nonlinear terms depend on unmeasurable states besides measurable output. We extend linear growth conditions to power growth conditions and reduce the control effort. By using backstepping technique, choosing a high-gain parameter, an output-feedback controller is designed to ensure the closed-loop system to be globally asymptotically stable in probability, and the inverse optimal stabilization in probability is achieved. The efficiency of the output-feedback controller is demonstrated by a simulation example.

Finite Time Inverse Optimal Stabilization for Stochastic Nonlinear Systems

This paper deals with finite time inverse optimal stabilization for stochastic nonlinear systems. A concept of the stochastic finite time control Lyapunov function (SFT-CLF) is presented, and a control law for finite time stabilization for the closed-loop system is obtained. Furthermore, a sufficient condition is developed for finite time inverse optimal stabilization in probability, and a control law is designed to ensure that the equilibrium of the closed-loop system is finite time inverse optimal stable. Finally, an example is given to illustrate the applications of theorems established in this paper.

State feedback control of a class of high‐order stochastic nonlinear systems

SUMMARYThis paper poses and solves a new problem of state feedback stabilization for a class of high‐order stochastic nonlinear systems in which the power order restriction and growth condition are relaxed to a more general form. Based on the ideas of the homogeneous systems theory and the adding of a power integrator technique, a state feedback controller is constructed to ensure that the equilibrium at the origin of the closed‐loop system is globally asymptotically stable in probability and that the problem of inverse optimal stabilization in probability is solved. The efficiency of the state feedback controller is demonstrated by a simulation example. Copyright © 2011 John Wiley & Sons, Ltd.

Further Results on Output-Feedback Stabilization for a Class of Stochastic Nonlinear Systems

For a class of stochastic nonlinear systems in which the drift and diffusion vector fields depend on the unmeasurable states besides the measurable output, the research purpose of this technical note is to find the maximum value interval of observer gain for the desired control performance. For this interval, the designed output-feedback controller ensures that the equilibrium at the origin is globally asymptotically stable in probability, and the inverse optimal stabilization in probability is achieved.