Chebyshev series approach to system identification,analysis and optimal control

The problems of identification, analysis and optimal control have been recently studied via orthogonal functions. The particular orthogonal functions used up to now are the Walsh, the block-pulse and the Laguerre functions. In this paper, the Chebyshev functions are introduced and solutions for the aforementioned problems are established. The algorithms proposed are analogous to those already derived for the Walsh, block-pulse and Laguerre functions. The Chebyshev series approach presented here appears to have certain advantages over other orthogonal series, and they may therefore be more suitable for the study of the problems of identification, analysis and optimal control.     

Orthogonal functions approach to optimal control of delay systems with reverse time terms

Using block-pulse functions (BPFs)/shifted Legendre polynomials (SLPs) a unified approach for computing optimal control law of linear time-varying time-delay systems with reverse time terms and quadratic performance index is discussed in this paper. The governing delay-differential equations of dynamical systems are converted into linear algebraic equations by using operational matrices of orthogonal functions (BPFs and SLPs). The problem of finding optimal control law is thus reduced to the problem of solving algebraic equations. One example is included to demonstrate the applicability of the proposed approach.     

A delay system approach to networked control systems with limited communication capacity

This paper addresses the problem of quantized feedback control for networked control systems (NCSs). Firstly, with consideration of the effect of network conditions, such as network-induced delays, data packet dropouts and signal quantization, the sampled-data model of closed-loop feedback system based on the updating sequence of the event-driven holder is formulated, from which a continuous system with two additive delay components in the state is developed. Subsequently, by making use of a novel interval delay system approach, the stability analysis and control synthesis for NCSs with two/one static quantizer are solved accordingly. Finally, two illustrative examples are given to show the effectiveness and advantage of the method.     

Finite-time stabilization of nonlinear systems via impulsive control with state-dependent delay

This paper focuses on the issue of finite-time stability for a general form of nonlinear systems subject to state-dependent delayed impulsive controller. Based on the Lyapunov theory and the impulsive control theory, sufficient conditions for finite-time stability (FTS) and finite-time contractive stability (FTCS) are obtained. Additionally, we apply theoretical results to finite-time synchronization of chaotic systems and design the effective state-dependent delayed impulsive controllers in terms of techniques of linear matrix inequality (LMI). Finally, we present two numerical examples of finite-time synchronization of cellular neural networks and Chua’s circuit to verify the effectiveness of our results.     

A novel structure for the optimal control of bilateral teleoperation systems with variable time delay

The purpose of designing a controller for a teleoperation system is achieving stability and optimal operation in the presence of factors such as time delay, system disturbance and modeling errors. In this article three new schemes for teleoperation systems are suggested using an optimal control to reduce the error of tracking between the master and slave systems. In the first scheme optimal controller has been designed in both the master and slave subsystems and by a suitable combination of the output signals of both controllers and exerting it to the slave, it has tried to create the best performance with regard to tracking. In the second scheme, as in the first one, optimal controller is applied to both the master and slave systems and the output of each controller is then applied to its own system, and by changing the system parameters and weighting factors, it has tried to reduce the tracking error between the master and the slave subsystems. In the third structure optimal control is applied to the master. In all three structures the positions of master-slave are compared together and controlling signals are applied to the master or slave so that they can track each other in the least possible time. In all schemes the effectiveness of the system is shown through the simulations and they are compared with each other.     

Fuzzy control of dithered chaotic systems via neural-network-based approach

This paper presents an effective approach for controlling chaos. First, a neural-network (NN) model is employed to approximate the chaotic system. Then, a linear differential inclusion (LDI) state-space representation is established for the dynamics of an NN model. Based on the LDI state-space representation, a fuzzy controller is proposed to tame the chaotic system. If the designed fuzzy controller cannot suppress the chaos, a high frequency signal, commonly called dithers, is simultaneously injected into the chaotic system. According to the relaxed method, an appropriate dither is introduced to steer the chaotic motion into a periodic orbit or a steady state. If the frequency of dither is high enough, the trajectory described by the dithered chaotic system and that of its corresponding mathematical model—the relaxed system can be made as close as desired. This phenomenon enables us to get a rigorous prediction of the dithered chaotic system’s behavior by obtaining the behavior of the relaxed system. Finally, a numerical example with simulations is given to illustrate the concepts discussed throughout this paper.     

Fault tolerant cooperative control for affine multi-agent systems: An optimal control approach

The goal of this paper is to propose an optimal fault tolerant control (FTC) approach for multi-agent systems (MASs). It is assumed that the agents have identical affine dynamics. The underlying communication topology is assumed to be a directed graph. The concepts of both inverse optimality and partial stability are further employed for designing the control law fully developed in the paper. Firstly, the optimal FTC problem for linear MASs is formulated and then it is extended to MASs with affine nonlinear dynamics. To solve the Hamilton-Jacobi-Bellman (HJB) equation, an Off-policy Reinforcement Learning is used to learn the optimal control law for each agent. Finally, a couple of numerical examples are provided to demonstrate the effectiveness of the proposed scheme.     

Design of optimal PID controller for multivariable time-varying delay discrete-time systems using non-monotonic Lyapunov-Krasovskii approach

This paper is concerned with stability analysis and stabilization of time-varying delay discrete-time systems in Lyapunov-Krasovskii stability analysis framework. In this regard, a less conservative approach is introduced based on non-monotonic Lyapunov-Krasovskii (NMLK) technique. The proposed method derives time-varying delay dependent stability conditions based on Lyapunov-Krasovskii functional (LKF), which are in the form of linear matrix inequalities (LMI). Also, a PID controller designing algorithm is extracted based on obtained NMLK stability condition. The stability of the closed loop system is guaranteed using the designed controller. Another property that is important along with the stability, is the optimality of the controller. Thus, an optimal PID designing technique is introduced in this article. The proposed method can be used to design optimal PID controller for unstable multi-input multi-output time-varying delay discrete-time systems. The proposed stability and stabilization conditions are less conservative due to the use of non-monotonic decreasing technique. The novelty of the paper comes from the consideration of non-monotonic approach for stability analysis of time-varying delay discrete-time systems and using obtained stability conditions for designing PID controller. Numerical examples and simulations are given to evaluate the theoretical results and illustrate its effectiveness compared to the existing methods.     

Optimal control of linear delay systems via hybrid of block-pulse and Legendre polynomials

A method for finding the optimal control of a linear time varying delay system with quadratic performance index is discussed. The properties of the hybrid functions which consists of block-pulse functions plus Legendre polynomials are presented. The operational matrices of integration, delay and product are utilized to reduce the solution of optimal control to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.     

Containment control of multi-agent systems via a disturbance observer-based approach

This paper deals with the containment control problem for multi-agent systems with exogenous disturbances. A disturbance observer-based control approach is employed to estimate the disturbances generated by an exogenous system. Consequently, distributed disturbance observer-based containment control protocols are proposed by using the state feedback control and the output feedback control, respectively. Furthermore, with the help of algebraic graph theory and Lyapunov stability theory, sufficient conditions are established to ensure that multi-agent systems with exogenous disturbances can achieve containment control via the disturbance observer-based approach. Finally, the effectiveness of our theoretical results is verified by providing numerical simulation examples.     

Stabilization of time delay systems with saturations via PDE predictor boundary control design

This paper addresses the stabilization issue of linear time delay system with input saturation and distinct input delays via predictor feedback boundary control algorithm by employing transport partial differential equations (PDEs). First, the addressed ordinary differential equation (ODE) system with input delay is equivalently represented as a cascade of an ODE and transport PDEs. Second, by employing the backstepping Volterra integral transformation technique, the equivalent cascade system is transformed into a stable target system, whose kernels are solved by the constraints satisfying transport PDEs. Third, based on the boundary conditions of the obtained invertible transformation, the proposed feedback control law can be formulated. Fourth, by applying semigroup operator theory, the well-posedness of the resulting system is proved and consequently, novel exponential stability conditions of the addressed system are established. Then, the domain of attraction region under the given actuator saturation constraints is estimated with the help of the solution of obtained stability conditions. Finally, a demonstrative simulation example is offered to verify the feasibility and usefulness of the results.     

Analysis and parameter estimation of bilinear systems via Chebyshev polynomials

The operational properties of the integration and product of Chebyshev polynomials are used in the analysis of bilinear systems by the approximation of time functions by truncated Chebyshev series. The operational properties are also applied to determine the unknown parameters of a general bilinear system from the input-output data. Examples with excellent results are given.     

A maximum principle approach to optimal control for one-dimensional hyperbolic systems with several state variables

The maximum principle developed by Sloss et al. [Optimal control of structural dynamic systems in one space dimension using a maximum principle, J. Vibr. Control 11 (2005) 245–261] is used to determine the optimal control functions for a class of one-dimensional distributed parameter structures. The distributed parameter structures are governed by systems of fourth order hyperbolic equations with constant coefficients. A quadratic performance index is formulated as the cost functional of the problem and can be used to represent the energy of the structure and the force spent in the control process. The developed maximum principle establishes a theoretical foundation for the solution of the optimal control problem and relates the optimal control vector to an adjoint variable vector. The method of solution is outlined which involves reducing the original problem to a system of ordinary differential equations. The solution of the general problem is given and a structural control problem is solved to illustrate the solution procedure. The effectiveness of the proposed control solution is shown by comparing the behavior of controlled and uncontrolled systems.     

Numerical solutions of optimal control for linear Volterra integrodifferential systems via hybrid functions

By applying hybrid functions of general block-pulse functions and Legendre polynomials, linear Volterra integrodifferential systems are converted into a system of algebraic equations. The approximate solutions of linear Volterra integrodifferential systems are derived. Using the results we obtain the optimal control and state as well as the optimal value of the objective functional. The numerical examples illustrate that the algorithms are valid.     

Automatic generation control of two-area electrical power systems via optimal fuzzy classical controller

The interconnected large-scale power systems are liable to performance degradation under the presence of sudden small load demands, parameter ambiguity and structural changes. Due to this, to supply reliable electric power with good quality, robust and intelligent control strategies are extremely requisite in automatic generation control (AGC) of power systems. Hence, this paper presents an output scaling factor (SF) based fuzzy classical controller to enrich AGC conduct of two-area electrical power systems. An implementation of imperialist competitive algorithm (ICA) is made to optimize the output SF of fuzzy proportional integral (FPI) controller employing integral of squared error criterion. Initially the study is conducted on a well accepted two-area non-reheat thermal system with and without considering the appropriate generation rate constraint (GRC). The advantage of the proposed controller is illustrated by comparing the results with fuzzy controller and bacterial foraging optimization algorithm (BFOA)/genetic algorithm (GA)/particle swarm optimization (PSO)/hybrid BFOA-PSO algorithm/firefly algorithm (FA)/hybrid FA-pattern search (hFA-PS) optimized PI/PID controller prevalent in the literature. The proposed approach is further extended to a newly emerged two-area reheat thermal-PV system. The superiority of the method is depicted by contrasting the results of GA/FA tuned PI controller. The proposed control approach is also implemented on a multi-unit multi-source hydrothermal power system and its advantage is established by Correlating its results with GA/hFA-PS tuned PI, hFA-PS/grey wolf optimization (GWO) tuned PID and BFOA tuned FPI controllers. Finally, a sensitivity analysis is performed to demonstrate the robustness of the proposed method to broad changes in the system parameters and size and/or location of step load perturbation.     

New distributed adaptive protocols for uncertain nonlinear leader-follower multi-agent systems via a repetitive learning control approach

In this paper, the distributed consensus problem of leader-follower multi-agent systems with unknown time-varying coupling gains and parameter uncertainties are investigated, and the fully distributed protocols with the adaptive updating laws of periodic time-varying parameters are designed by using a repetitive learning control approach. By virtue of algebraic graph theory, Barbalat’s lemma and an appropriate Lyapunov-Krasovskii functional, it is shown that each follower agent can asymptotically track the leader even though the dynamic of the leader is unknown to any of them, i.e., the global asymptotic consensus can be achieved. At last, a simulation example is given to illustrate the feasibility and efficiency of the proposed protocols.     

On the set-valued approach to optimal control of sliding mode processes

This paper is devoted to a theoretic framework for a general optimal control problem (OCP) associated with the classic sliding mode process. The sliding dynamic behavior is interpreted here as a special kind of additional constraints related to the main optimization problem. We are specially interested in the development of some adequate constructive approximations of the original OCPs. The mathematical approach based on the set-valued analysis allows to study the discontinuity of sliding mode dynamics in the abstract setting. Moreover, we also establish some sensitivity properties of the optimal solutions. The obtained results provide an universal analytical tool for the corresponding conceptual approximation schemes related to the original OCPs. The constructive approximations proposed in this paper are numerically stable and can be applied to various classes of optimal control processes governed by the affine control systems.     

A delay distribution based stability analysis and synthesis approach for networked control systems

Communication delays in networked control systems (NCSs) has been shown to have non-uniform distribution and multifractal nature. This paper proposes a delay distribution based stability analysis and synthesis approach for NCSs with non-uniform distribution characteristics of network communication delays. A stochastic control model related with the characteristics of communication networks is established to describe the NCSs. Then, delay distribution-dependent NCS stability criteria are derived in the form of linear matrix inequalities (LMIs). Also, the maximum allowable upper delay bound and controller feedback gain can be obtained simultaneously from the developed approach by solving a constrained convex optimization problem. Numerical examples showed that the results derived from the proposed method are less conservativeness than those derived from the existing methods.     

Direct solution of nonlinear optimal control problems using quasilinearization and Chebyshev polynomials

In this paper, a numerical method to solve nonlinear optimal control problems with terminal state constraints, control inequality constraints and simple bounds on the state variables, is presented. The method converts the optimal control problem into a sequence of quadratic programming problems. To this end, the quasilinearization method is used to replace the nonlinear optimal control problem with a sequence of constrained linear-quadratic optimal control problems, then each of the state variables is approximated by a finite length Chebyshev series with unknown parameters. The method gives the information of the quadratic programming problem explicitly (The Hessian, the gradient of the cost function and the Jacobian of the constraints). To show the effectiveness of the proposed method, the simulation results of two constrained nonlinear optimal control problems are presented.     

New approaches to stability analysis for time-varying delay systems

In this paper, two new estimation approaches namely delay-dependent-matrix-based (DDMB) reciprocally convex inequality approach and DDMB estimation approach, are introduced for stability analysis of time-varying delay systems. Different from existing estimation techniques with constant matrices, the estimation approaches are with delay-dependent matrices, which can employ more free matrices and utilize more information of both time delay and its derivative. Based on the estimation approaches, less conservative stability criteria with lower computational complexity are derived in the form of linear matrix inequalities (LMIs). Finally, two numerical examples are given to illustrate the advantages of the proposed methods.