An iterative method for suboptimal control of a class of nonlinear time-delayed systems

This work presents an approximate solution method for the infinite-horizon nonlinear time-delay optimal control problem. A variational iteration method (VIM) is applied to design feedforward and feedback optimal controllers. By using the VIM, the original optimal control is transformed into a sequence of nonhomogeneous linear two-point boundary value problems (TPBVPs). The existence and uniqueness of the optimal control law are proved. The optimal control law obtained consists of an accurate linear feedback term and a nonlinear compensation term which is the limit of an adjoint vector sequence. The feedback term is determined by solving Riccati matrix differential equation. By using the finite-step iteration of a nonlinear compensation sequence, we can obtain a suboptimal control law. Simulation results demonstrate the validity and applicability of the VIM.     

Finding the optimal control of linear systems via He's variational iteration method

In this article, He's variational iteration method has been applied to find the optimal control of linear systems, approximately. Numerical results are given for several test examples involving scalar and second-order systems to demonstrate the applicability and efficiency of the method.     

An efficient two-step iterative method for solving a class of complex symmetric linear systems

In this paper, a new two-step iterative method called the two-step parameterized (TSP) iteration method for a class of complex symmetric linear systems is developed. We investigate its convergence conditions and derive the quasi-optimal parameters which minimize the upper bound of the spectral radius of the iteration matrix of the TSP iteration method. Meanwhile, some more practical ways to choose iteration parameters for the TSP iteration method are proposed. Furthermore, comparisons of the TSP iteration method with some existing ones are given, which show that the upper bound of the spectral radius of the TSP iteration method is smaller than those of the modified Hermitian and skew-Hermitian splitting (MHSS), the preconditioned MHSS (PMHSS), the combination method of real part and imaginary part (CRI) and the parameterized variant of the fixed-point iteration adding the asymmetric error (PFPAE) iteration methods proposed recently. Inexact version of the TSP iteration (ITSP) method and its convergence properties are also presented. Numerical experiments demonstrate that both TSP and ITSP are effective and robust when they are used either as linear solvers or as matrix splitting preconditioners for the Krylov subspace iteration methods and they have comparable advantages over some known ones for the complex symmetric linear systems.     

Initial condition of costate in linear optimal control using convex analysis

The Pontryagin Maximum Principle is one of the most important results in optimal control, and provides necessary conditions for optimality in the form of a mixed initial/terminal boundary condition on a pair of differential equations for the system state and its conjugate costate. Unfortunately, this mixed boundary value problem is usually difficult to solve, since the Pontryagin Maximum Principle does not give any information on the initial value of the costate. In this paper, we explore an optimal control problem with linear and convex structure and derive the associated dual optimization problem using convex duality, which is often much easier to solve than the original optimal control problem. We present that the solution to the dual optimization problem supplements the necessary conditions of the Pontryagin Maximum Principle, and elaborate the procedure of constructing the optimal control and its corresponding state trajectory in terms of the solution to the dual problem.     

On a class of suboptimal controls for infinite-dimensional bilinear systems

The suboptimal control of a bilinear system is considered with respect to a quadratic cost criterion. The feedback control is in the space of formal power series on a Hilbert space.     

A second-order maximum principle for discrete-time bilinear control systems with applications to discrete-time linear switched systems

A powerful approach for analyzing the stability of continuous-time switched systems is based on using optimal control theory to characterize the “most unstable” switching law. This reduces the problem of determining stability under arbitrary switching to analyzing stability for the specific “most unstable” switching law. For discrete-time switched systems, the variational approach received considerably less attention. This approach is based on using a first-order necessary optimality condition in the form of a maximum principle (MP), and typically this is not enough to completely characterize the “most unstable” switching law. In this paper, we provide a simple and self-contained derivation of a second-order necessary optimality condition for discrete-time bilinear control systems. This provides new information that cannot be derived using the first-order MP. We demonstrate several applications of this second-order MP to the stability analysis of discrete-time linear switched systems.     

Stabilization of discrete-time Markovian jump linear systems via time-delayed and impulsive controllers

This paper discusses the stabilization problem for a class of discrete-time Markovian jump linear systems with time-delayed and impulsive controllers. The delay in the mode signal is assumed to be constant, while that in the system state may be time-varying. First, a hybrid controller with delay and impulses is introduced for the discrete-time stochastic systems. Then, some sufficient conditions are proposed for the design of a hybrid controller such that the closed-loop system is stochastically stable. Finally, a numerical example is provided to illustrate the effectiveness of the proposed result.     

Optimal control for multi-stage and continuous-time linear singular systems

In this paper, optimal control problems for multi-stage and continuous-time linear singular systems are both considered. The singular systems are assumed to be regular and impulse-free. First, a recurrence equation is derived according to Bellman's principle of optimality in dynamic programming. Then, by applying the recurrence equation, bang-bang optimal controls for the control problems with linear objective functions subject to two types of multi-stage singular systems are obtained. Second, employing the principle of optimality, a equation of optimality for settling the optimal control problem subject to a class of continuous-time singular systems is proposed. The optimal control problem may become simpler through solving this equation of optimality. Two numerical examples and a dynamic input–output model are presented to show the effectiveness of the results obtained.     

An eigenvalue based approach for the stabilization of linear time-delay systems of neutral type

An eigenvalue based approach for the stabilization of linear neutral functional differential equations is presented, which extends the recently developed continuous pole placement method for delay equations of retarded type. The approach consists of two steps. First the stability of the associated difference equation is determined and a procedure is applied to compute the supremum of the real parts of its characteristic roots, which corresponds to computing the radius of the essential spectrum of the solution operator of the neutral equation. No restrictions are made on the dimension of the system and the number of delays. Also the effect of small delay perturbations is explicitly taken into account. As a result of this first step the stabilization problem of the neutral equation is reduced to a problem involving only a finite number of characteristic roots. As a second step, stabilization is achieved by shifting the rightmost or unstable characteristic roots to the left half plane in a quasi-continuous way, by applying small changes to the controller parameters, and meanwhile monitoring other characteristic roots with a large real part. A numerical example is presented.     

Improved ellipsoidal bound of reachable sets for time-delayed linear systems with disturbances

We consider the problem of finding an ellipsoidal reachable set that bounds the states of time-delayed linear systems with bounded peak disturbances. The considered time-delay is time-varying but it has an upper bound in magnitude and rate. Based on the modified Lyapunov-Krasovskii(L-K) type functional, we derive a delay-dependent result expressed in the form of matrix inequalities containing only one non-convex scalar. The result is extended to uncertain time-delayed linear systems. Finally, we show the usefulness of our result by a comparative numerical example.     

Observer-based adaptive control for uncertain time-delay systems

In this paper, we will focus on investigating the observer-based controlling problem of time-delay systems. First, we investigate a class of simple time-delay systems. The corresponding adaptive observer and controller are designed, which are both independent of the time-delays. Based on Lyapunov stability theory, we prove that the closed-loop system is asymptotically stable. Next we further consider the interconnected time-delay system case. The corresponding adaptive observer and controller are designed, we prove that the resulting closed-loop system is also asymptotically stable. Simulations on controlling time-delay systems and interconnected systems are investigated, and the results show that the designed controllers are feasible and efficient.     

Suboptimal stochastic linear feedback control of linear systems with state- and control-dependent noise: The incomplete information case

The stochastic regulation problem for linear systems with state- and control-dependent noise and a noisy linear output equation is considered. The optimal quadratic cost output-feedback control law in a class of linear controllers is found. This problem was first addressed in the early 1970s and solved, in the complete information case, by Wonham. In this paper we give the solution of the problem in the incomplete information case, that is, for a linear output equation corrupted by Gaussian noise. Moreover, a different method is used here, giving the solution in a more direct way even in the complete information case.     

Suboptimal control for nonlinear systems: a successive approximation approach

This paper presents a successive approximation approach (SAA) designing optimal controllers for a class of nonlinear systems with a quadratic performance index. By using the SAA, the nonlinear optimal control problem is transformed into a sequence of nonhomogeneous linear two-point boundary value (TPBV) problems. The optimal control law obtained consists of an accurate linear feedback term and a nonlinear compensation term which is the limit of an adjoint vector sequence. By using the finite-step iteration of the nonlinear compensation sequence, we can obtain a suboptimal control law. Simulation examples are employed to test the validity of the SAA.     

An explicit formula for the suboptimal control of linear systems

The suboptimal control of a linear system can be expressed in terms of the system parameters by a simple, explicit formula. At the same time, the suboptimal state (trajectory) is obtained. An example is worked out to illustrate the accuracy as compared to the Riccati solution.     

Reliable passive control for singular systems with time-varying delays

In this paper, the problem of reliable passive control is investigated for singular systems with time-varying delays. The aim of the addressed reliable passive control problem is to design a state feedback controller such that, for all possible actuator failures, the resultant closed-loop system is regular, impulse-free, exponentially stable, and passive. A delay-dependent condition is established to guarantee the considered system to be regular, impulse-free, exponentially stable, and passive. Based on the derived condition, the reliable passive control problem is solved, and an explicit expression for the desired controller is given. Numerical examples are provided to demonstrate the effectiveness and feasibility of our results.     

Approximate analytical solutions of systems of PDEs by homotopy analysis method

In this paper, the homotopy analysis method (HAM) is applied to obtain series solutions to linear and nonlinear systems of first- and second-order partial differential equations (PDEs). The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of series solutions. It is shown in particular that the solutions obtained by the variational iteration method (VIM) are only special cases of the HAM solutions.     

Robust adaptive control for interval time-delay systems

1 Introduction The problem of time-delay is commonly encountered in various engineering systems, such as electric, pneumatic and hydraulic networks, long transmission lines, etc., and it is also a great source of systems instability and poor perfor- mance. During the last decades, we have seen an increasing interest for the control of this class of systems and many results have reported in the literature [1～5]. On the other hand, it has been recognized that nonlinear uncertainties are unavoid…     

An iterative approach to the optimal co-design of linear control systems

This paper investigates the optimal co-design of both physical plants and control policies for a class of continuous-time linear control systems. The optimal co-design of a specific linear control system is commonly formulated as a nonlinear non-convex optimisation problem (NNOP), and solved by using iterative techniques, where the plant parameters and the control policy are updated iteratively and alternately. This paper proposes a novel iterative approach to solve the NNOP, where the plant parameters are updated by solving a standard semi-definite programming problem, with non-convexity no longer involved. The proposed system design is generally less conservative in terms of the system performance compared to the conventional system-equivalence-based design, albeit the range of applicability is slightly reduced. A practical optimisation algorithm is proposed to compute a sub-optimal solution ensuring the system stability, and the convergence of the algorithm is established. The effectiveness of the proposed algorithm is illustrated by its application to the optimal co-design of a physical load positioning system.