Optimal Control Of Nonlinear Systems Research Articles

Fuzzy optimal control for nonlinear systems with time-varying delay via sampled-data controller

This work is devoted to a fuzzy sampled-data optimal control problem for nonlinear systems with time-varying delay in Takagi–Sugeno fuzzy form. Based on the Lyapunov–Krasovskii functional theory, some novel stability criteria are established. A fuzzy sampled-data controller is developed to ensure that the fuzzy closed-loop sampled-data control system is asymptotically stable and the guaranteed cost performance is also minimized. It is shown that the obtained results are less conservative in the stability analysis. Two examples of the computer-simulated truck-trailer system and the continuous stirred tank reactor system are provided to show the effectiveness and the merits of the proposed design.

An analytical approach to optimal control of nonlinear systems with input constraints

In this paper, a novel optimal control method with the analytical solution strategy is developed for a wide class of nonlinear systems with input constraints. First, an equivalent constrained optimisation problem is formulated by performing a quadratic performance index including the control inputs and the predicted tracking errors. Then, the problem is analytically solved by using the Kerush-Kuhn-Tucker (KKT) theorem to find the optimal control law. For comparison, a computational technique based on calling the genetic algorithm (GA) is also provided to online solving the developed optimisation problem. The proposed control method with two solutions is applied on a mathematical example and a chemical reactor which is a multi-input multi-output (MIMO) system. The results show that the proposed KKT-based predictive controller is effective from different aspects. Importantly, it is very fast, easy to solve and suitable for online implementation compared with the conventional nonlinear model predictive control (NMPC) method.

An analytical approach to optimal control of nonlinear systems with input constraints

In this paper, a novel optimal control method with the analytical solution strategy is developed for a wide class of nonlinear systems with input constraints. First, an equivalent constrained optimisation problem is formulated by performing a quadratic performance index including the control inputs and the predicted tracking errors. Then, the problem is analytically solved by using the Kerush-Kuhn-Tucker (KKT) theorem to find the optimal control law. For comparison, a computational technique based on calling the genetic algorithm (GA) is also provided to online solving the developed optimisation problem. The proposed control method with two solutions is applied on a mathematical example and a chemical reactor which is a multi-input multi-output (MIMO) system. The results show that the proposed KKT-based predictive controller is effective from different aspects. Importantly, it is very fast, easy to solve and suitable for online implementation compared with the conventional nonlinear model predictive control (NMPC) method.

A Fast Iterative Approach for Optimal Control of Nonlinear Systems

To improve computational efficiency, a fast iterative solution method is proposed to solve optimization problems of nonlinear systems in this paper. The main idea is to extract the main linear part from the nonlinear system, and then regard the remaining nonlinear part as disturbance, so that the converted system can be solved by combining the optimal solution of the linear system with the fast iterative solution proposed. After the fast iterative solution method is introduced, its effectiveness is validated through a numerical example and an application which is engine energy-saving control while taking vehicle speed tracking into consideration. All of the simulation results show that the proposed iterative algorithm has the advantages of fast convergence speed and high accuracy.

Trajectory tracking error using fractional order time-delay recurrent neural networks using Krasovskii-Lur’e functional for Chua’s circuit via inverse optimal control

This paper presents an application of a Fractional Order Time Delay Neural Networks to chaos synchronization. The two main methodologies, on which the approach is based, are fractional order time-delay recurrent neural networks and the fractional order inverse optimal control for nonlinear systems. The problem of trajectory tracking is studied, based on the fractional order Lyapunov-Krasovskii and Lur’e theory, that achieves the global asymptotic stability of the tracking error between a delayed recurrent neural network and a reference function is obtained. The method is illustrated for the synchronization, the analytic results we present a trajectory tracking simulation of a fractional order time-delay dynamical network and the Fractional Order Chua’s circuits

Optimal Nonlinear Control Using Hamilton–Jacobi–Bellman Viscosity Solutions on Unstructured Grids

The optimal control of nonlinear systems is traditionally obtained by the application of the Pontryagin minimum principle. Despite the success of this methodology in finding the optimal control for complex systems, the resulting open-loop trajectory is guaranteed to be only locally optimal. Furthermore, the computation of open-loop solutions is computationally intensive, which rules out its application in feedback controllers, reducing its robustness against disturbances. In principle, these issues can be addressed by solving the Hamilton–Jacobi–Bellman (HJB) partial differential equation (PDE). However, the space complexity of the problem is exponential with respect to the number of dimensions of the system. Moreover, the value function of the HJB equation may be nondifferentiable, which renders traditional PDE solution methods impractical. Therefore, extant methods are suitable only for special problem classes such as those involving affine systems or where the value function is differentiable. To deal with these issues, this work introduces a methodology for the solution of the HJB equation for general nonlinear systems that combines PDE viscosity solutions, quasi–Monte Carlo grids, and kriging regression to implement globally optimal nonlinear feedback controllers for practical applications. The effectiveness of the method is illustrated with smooth and nondifferentiable problems with finite and infinite horizons.

An iterative method for optimal feedback control and generalized HJB equation

In this paper, a new iterative method is proposed to solve the generalized Hamilton-Jacobi-Bellman &#x0028 GHJB &#x0029 equation through successively approximate it. Firstly, the GHJB equation is converted to an algebraic equation with the vector norm, which is essentially a set of simultaneous nonlinear equations in the case of dynamic systems. Then, the proposed algorithm solves GHJB equation numerically for points near the origin by considering the linearization of the non-linear equations under a good initial control guess. Finally, the procedure is proved to converge to the optimal stabilizing solution with respect to the iteration variable. In addition, it is shown that the result is a closed-loop control based on this iterative approach. Illustrative examples show that the update control laws will converge to optimal control for nonlinear systems.

Backstepping-based adaptive predictive optimal control of nonlinear systems with application to missile–target engagement

In this paper, an adaptive predictive optimal control scheme for a class of block strict-feedback nonlinear systems is proposed by integrating the adaptive dynamic programming (ADP) technique, predictive control and backstepping method. The basic idea is that designing the virtual and actual controls of backstepping is the optimized solutions of corresponding subsystems. Firstly, the virtual control input is derived for the subsystem by utilizing ADP technique, in which a critic neural network (NN) is constructed to approximate the solution of the associated Hamilton–Jacobi–Bellman (HJB) equation. Then, to further reduce the computational complexity, the actual controller is given in an analytical form by using continuous-time predictive approach. Theoretical analysis guarantees the stability of the closed-loop system by Lyapunov method. Finally, the effectiveness of the proposed adaptive predictive optimal control scheme is validated through an application to missile–target engagement.

Optimal control of nonlinear systems with separated linear part via quadratic criteria

We present a method for synthesis of optimal control with feedback of nonlinear systems with separated linear part via quadratic criteria. This method is based on a special method of successive approximations, which allows, under standard assumptions, to find optimal control within any finite time interval and to get the procedure of its construction. An example is provided for applying this method for synthesis of control of a system which is similar to Watt’s centrifugal governor.

Optimal control for sampled-data nonlinear systems with multiple time-delays in delta domain with application to a quarter-car suspension

ABSTRACTThe paper deals with an optimal control for nonlinear systems with multiple time delays in delta domain. By constructing sequences for optimal control problem, we demonstrate the optimal control and the optimal state trajectories by proving the convergence of these sequences. The optimal control is achieved from deriving a Riccati equation and adjoint differential equations. The compensation term in the designed controller compensates the effect which is produced by nonlinearity, time delay, and disturbance. A quarter-car suspension model with actuator and measurement delays is adopted to illustrate the effectiveness of the proposed control. The advantage of controller in delta domain is obvious than that of the discrete-time domain.

Neural Network-Based Model-Free Adaptive Near-Optimal Tracking Control for a Class of Nonlinear Systems.

In this paper, the receding horizon near-optimal tracking control problem about a class of continuous-time nonlinear systems with fully unknown dynamics is considered. The main challenges of this problem lie in two aspects: 1) most existing systems only restrict their considerations to the state feedback part while the input channel parameters are assumed to be known. This paper considers fully unknown system dynamics in both the state feedback channel and the input channel and 2) the optimal control of nonlinear systems requires the solution of nonlinear Hamilton-Jacobi-Bellman equations. Up to date, there are no systematic approaches in the existing literature to solve it accurately. A novel model-free adaptive near-optimal control method is proposed to solve this problem via utilizing the Taylor expansion-based problem relaxation, the universal approximation property of sigmoid neural networks, and the concept of sliding mode control. By making approximation for the performance index, it is first relaxed to a quadratic program, and then, a linear algebraic equation with unknown terms. An auxiliary system is designed to reconstruct the input-to-output property of the control systems with unknown dynamics, so as to tackle the difficulty caused by the unknown terms. Then, by considering the property of the sliding-mode surface, an explicit adaptive near-optimal control law is derived from the linear algebraic equation. Theoretical analysis shows that the auxiliary system is convergent, the resultant closed-loop system is asymptotically stable, and the performance index asymptomatically converges to optimal. An illustrative example and experimental results are presented, which substantiate the efficacy of the proposed method and verify the theoretical results.

Influence of flame dynamics on the optimal control of combustion with uncertainties

ABSTRACTA non-linear optimal control system is developed for attenuating thermos-acoustic pressure oscillations in a combustion chamber where the unsteady flow field associated with the internal instability is modelled in terms of a finite set of assumed modes of the non-linear in-homogeneously driven pressure wave field. A distributed set of auxiliary fuel injectors are used to control the thermo-acoustic pressure field with pressure measurements at independent locations. The mathematical model of the thermo-acoustic pressure field with multiple fuel injectors is developed. The flame dynamics is also modelled and its states are included in the design of the controller. Extensive simulations are performed to reveal the features of the thermo-acoustic pressure field within the combustion chamber and qualitatively compared with published data. The controller is synthesized on the basis of a non-linear quadratic Gaussian approach as well as a non-linear H∞ controller approach. The performances of the two controllers are compared. The influence of the flame dynamics is assessed by comparing the closed-loop performance by including the lag effects of a simplified flame transfer function. In particular it is shown that the non-linear H∞ controller approach is well-suited for the control of the combustion chamber pressure field as it has the structure of comb or multiple-notch filter and can attenuate acoustic tones associated with the combustion instabilities.

Robust Optimal Control of Nonlinear Systems With System Disturbance During Feedback Disruption

AbstractIn this paper, a robust optimal control problem of nonlinear systems with system disturbance during feedback disruption is considered. This is an extended work of previous time‐delay optimal control results, by adding external disturbance in the considered system. It is shown that there exists an optimal input signal which keeps the performance error within the specified bound for the longest time. Then, it is shown that such an optimal input signal can be approximated by an implementable bang‐bang input signal in terms of control performance. Two examples are given for illustration.

Optimistic planning with an adaptive number of action switches for near-optimal nonlinear control

We consider infinite-horizon optimal control of nonlinear systems where the control actions are discrete, and focus on optimistic planning algorithms from artificial intelligence, which can handle general nonlinear systems with nonquadratic costs. With the main goal of reducing computations, we introduce two such algorithms that only search for constrained action sequences. The constraint prevents the sequences from switching between different actions more than a limited number of times. We call the first method optimistic switch-limited planning (OSP), and develop analysis showing that its fixed number of switches S leads to polynomial complexity in the search horizon, in contrast to the exponential complexity of the existing OP algorithm for deterministic systems; and to a correspondingly faster convergence towards optimality. Since tuning S is difficult, we introduce an adaptive variant called OASP that automatically adjusts S so as to limit computations while ensuring that near-optimal solutions keep being explored. OSP and OASP are analytically evaluated in representative special cases, and numerically illustrated in simulations of a rotational pendulum. To show that the algorithms also work in challenging applications, OSP is used to control the pendulum in real time, while OASP is applied for trajectory control of a simulated quadrotor.

A NN-Based Hybrid Intelligent Algorithm for a Discrete Nonlinear Uncertain Optimal Control Problem

A discrete-time nonlinear optimal control system with uncertain perturbation is studied in this paper. A value-iteration-based hybrid intelligent algorithm (HIA), incorporating heuristic dynamic programming algorithm and uncertain simulation, is presented. And some characters of the HIA are provided. Two neural networks are used in the algorithm: a critic network is aimed at approximating the objective function, as well as an active network is used to obtain an uncertain optimal control law. As an application, a schistosomiasis compartment SI model is solved to illustrate the feasibility and effectiveness of the HIA.

Data-based robust optimal control of continuous-time affine nonlinear systems with matched uncertainties

In this paper, the robust optimal control of continuous-time affine nonlinear systems with matched uncertainties is investigated by using a data-based integral policy iteration approach. It is a natural extension of the traditional optimal control design, under the framework of adaptive dynamic programming (ADP) method, to robust optimal control of nonlinear systems with matched uncertainties. In theoretical aspect, by increasing a feedback gain to the optimal controller of the nominal system, the robust controller of the matched uncertain system is obtained, which also achieves optimality with a newly well-defined cost function. When regarding the implementation, the data-based integral policy iteration algorithm is used to solve the Hamilton–Jacobi–Bellman equation corresponding to the nominal system with completely unknown dynamics information. Then, the actor-critic technique based on neural networks and least squares implementation method are employed to facilitate deriving the optimal control law iteratively, so that the closed-form expression of the robust optimal controller is available. Additionally, two simulation examples with application backgrounds are presented to illustrate the effectiveness of the established robust optimal control scheme. In summary, it is important to note that the result developed in this paper broadens the application scope of ADP-based optimal control approach to more general nonlinear systems possessing dynamical uncertainties.

Optimal control of nonlinear systems with input constraints using linear time varying approximations

We propose a new method to solve input constrained optimal control problems for autonomous nonlinear systems affine in control. We then extend the method to compute the bang-bang control solutions under the symmetric control constraints. The most attractive aspect of the proposed technique is that it enables the use of linear quadratic control theory on the input constrained linear and nonlinear systems. We illustrate the effectiveness of our technique both on linear and nonlinear examples and compare our results with those of the literature.

Online identifier–actor–critic algorithm for optimal control of nonlinear systems

SummaryIn this paper, a novel identifier–actor–critic optimal control scheme is developed for discrete‐time affine nonlinear systems with uncertainties. In contrast to traditional adaptive dynamic programming methodology, which requires at least partial knowledge of the system dynamics, a neural‐network identifier is employed to learn the unknown control coefficient matrix working together with actor–critic‐based scheme to solve the optimal control online. The critic network learns the approximate value function at each step. The actor network attempts to improve the current policy based on the approximate value function. The weights of all neural networks are updated at each sampling instant. Lyapunov theory is utilized to prove the stability of closed‐loop system. It shows that system states and neural network weights are uniformly ultimately bounded. Finally, simulations are provided to illustrate the effectiveness of the developed method. Copyright © 2016 John Wiley & Sons, Ltd.

Inverse optimal controller for nonlinear systems with convex input constraints

Abstract: In this paper, we propose an inverse optimal controller design method for nonlinear systems with convex input constraints. The proposed design method constructs a state feedback controller as an optimal solution of a parametric convex optimization problem. The necessary and sufficient condition for inverse optimality is given by the Karush-Kuhn-Tucker (KKT) condition. We also show that the Lagrange multiplier for the optimization problem characterizes the asymptotically stabilizable domain and the robustness of the proposed controller. The effectiveness of the proposed method is confirmed by computer simulation.

Multiple Model Adaptive Tracking Control Based on Adaptive Dynamic Programming

Adaptive dynamic programming (ADP) has been tested as an effective method for optimal control of nonlinear system. However, as the structure of ADP requires control input to satisfy the initial admissible control condition, the control performance may be deteriorated due to abrupt parameter change or system failure. In this paper, we introduce the multiple models idea into ADP, multiple subcontrollers run in parallel to supply multiple initial conditions for different environments, and a switching index is set up to decide the appropriate initial conditions for current system. By taking this strategy, the proposed multiple model ADP achieves optimal control for system with jumping parameters. The convergence of multiple model adaptive control based on ADP is proved and the simulation shows that the proposed method can improve the transient response of system effectively.