Multiple model approach to linear parameter varying time-delay system identification with EM algorithm

This paper is concerned with the identification problem of linear parameter varying (LPV) time-delay systems. Due to inherent nonlinearity, the industrial processes are often approximately described by an LPV model constructed by synthesizing multiple local models. Time-delay is commonly experienced in industrial processes and it can be parameter varying or constant in the process model. The multiple model identification of LPV systems with parameter varying or constant time-delay is formulated in the scheme of the expectation-maximization (EM) algorithm and the parameter varying property and the time-delay property of the process are handled simultaneously. The irrigation channel example and high purity distillation column example are used to present the effectiveness of the proposed method.     

Performance analysis of the recursive parameter estimation algorithms for multivariable Box–Jenkins systems

Two auxiliary model based recursive identification algorithms, a generalized extended stochastic gradient algorithm and a recursive generalized extended least squares algorithm, are developed for multivariable Box–Jenkins systems. The basic idea is to use the auxiliary models to estimate the unknown noise-free outputs of the system and to replace the unmeasurable terms in the information vectors with their estimates. We prove that the estimation errors given by the proposed algorithms converge to zero under the persistent excitation condition. Finally, an example is provided to show the effectiveness of the proposed algorithms.     

State space model identification of multirate processes with time-delay using the expectation maximization

This paper presents the problems of state space model identification of multirate processes with unknown time delay. The aim is to identify a multirate state space model to approximate the parameter-varying time-delay system. The identification problems are formulated under the framework of the expectation maximization algorithm. Through introducing two hidden variables, a new expectation maximization algorithm is derived to estimate the unknown model parameters and the time-delays simultaneously. The effectiveness of the proposed algorithm is validated by a simulation example.     

Robust identification of nonlinear time-delay system in state-space form

This paper puts forward a robust identification solution for nonlinear time-delay state-space model (NDSSM) with contaminated measurements. To enhance the robustness of the developed method for outliers, the heavy-tailed Laplace distribution is employed to describe and protect the output measurement process. The undetermined time-delay is considered to be uniformly distributed and the boundary of it is known as a priori. In the developed solution, the uncertain time-delay is treated as a latent process variable and it is iteratively calculated with the expectation–maximization (EM) algorithm. The EM algorithm is actually an iterative optimization algorithm and it is effective for the hidden variable problems. The particle filter is introduced to numerically approximate the cost function (Q-function) in the EM algorithm since it is difficult to calculate directly. The efficacy of the developed solution is evaluated via a numerical test and a two-link robotic manipulator.     

Balanced truncation for discrete time-delay systems via the interpretation of system energy

In this paper, the balanced truncation method is investigated for discrete time-delay systems. We show that the energy associated with the system controllability and observability can be characterized via the delay Lyapunov matrices, similar to the case of continuous time-delay systems. Then, we balance the system via a coordinate transformation in order to retain the delay structure of systems naturally. In this way, the balanced truncation method is conducted to obtain structure-preserving reduced models. Further, we provide an efficient process to compute a low-rank approximation to delay Lyapunov matrices based on the equivalent expression of discrete time-delay systems, which enables an approximate but fast execution of the proposed method. The stability of reduced models is also discussed in the paper. Finally, numerical examples are simulated to verify the feasibility and efficiency of the proposed method.     

Dissipative state observer design for nonlinear time-delay systems

This paper is concerned with the design of dissipative state observers for a family of time-delay nonlinear systems. The Dissipativity method, proposed by one of the authors for delay-free nonlinear systems, is extended here to a class of time-delay nonlinear systems. The design method consists in decomposing the time-delay estimation error dynamics into a time-delay linear subsystem and a time-varying memoryless nonlinearity, connected in a negative feedback loop. By using some storage functionals, both delay-independent and delay-dependent dissipativity criteria are derived in order to guarantee the exponential convergence property of the observer. The exponential stability of the estimation error is then ensured, assuming that the nonlinearity is dissipative with respect to a quadratic supply rate and the linear part is designed, through the observer gains, to be dissipative with respect to a complementary supply rate. The design conditions are formulated in terms of tractable bilinear (BMI’s) or linear matrix inequalities (LMI’s). An interesting advantage is that the proposed dissipative design extends and generalizes under a unified framework several methods available in the literature, since a wide diversity of nonlinearities can be considered. Numerical examples are provided to demonstrate the effectiveness of the theoretical results.     

Robust identification of Wiener time-delay system with expectation-maximization algorithm

This paper considers the parameter estimation for Wiener time-delay systems with the output data contaminated with outliers. The time-delay and corrupted output data bring great challenges to the parameter estimation problem. The statistical model of the estimation problem is constructed based on the Laplace distribution and the identification problem is formulated in the scheme of the expectation-maximization (EM) algorithm. The negative effect of outliers imposed on the parameter estimation problem is sufficiently suppressed and the unknown time-delay and model parameters can be estimated simultaneously. The simulation example is given to demonstrate the effectiveness of the proposed algorithm.     

Exact analysis for the identification of non-minimum phase processes

Analytical expressions are derived from a single symmetrical relay feedback test for the identification of open loop stable and unstable non-minimum phase processes. The second derivative of the limit cycle output helps in finding the time delay of the process. The derived expressions can be used to identify the process models with up to five unknowns. The effectiveness of the proposed identification method is verified through different simulation results.     

A generalized multiple-integral inequality and its application on stability analysis for time-varying delay systems

This paper is concerned with the stability analysis of time-delay systems. Lyapunov–Krasovskii functional method is utilized to obtain stability criteria in the form of linear matrix inequalities. The main purpose is to obtain less conservative stability criteria by reducing the estimation gap of the time derivative of the constructed Lyapunov–Krasovskii functional. First, a generalized multiple-integral inequality is put forward based on Schur complement lemma. Then, some special cases of the proposed generalized multiple-integral inequality are given to estimate single and double integral terms in the derivative of Lyapunov–Krasovskii functional. Furthermore, less conservative stability criteria are derived. Finally, three examples are given to illustrate the improvement of the proposed criteria.     

Tracking control algorithms for plants with input time-delays based on state and disturbance predictors and sub-predictors

The novel control algorithm for linear time invariant plants with input time-delays and presence of external disturbances is proposed. The algorithm based on the state and disturbance predictors ensures the tracking control with unknown reference model parameters. The accuracy in steady state depends on the highest derivative of disturbance and reference model signals, therefore, the magnitude of these signals can be sufficiently large. Further, the proposed algorithm are extended on the state and disturbance sub-predictors which implement multi-step prediction. Compared with the predictor based algorithm the sub-predictor based algorithm allows to control plants with a larger input time-delay and allows to predict the disturbance in less time. The sufficient conditions in terms of linear matrix inequalities (LMIs) provide the estimate of the maximum time-delay that preserves the closed-loop system stability. Numerical examples illustrate the efficiency of the designed method compared with some existing ones.     

Consensus control via iterative learning for distributed parameter models multi-agent systems with time-delay

Most of the available results of iterative learning control (ILC) are that solve the consensus problem of lumped parameter models multi-agent systems. This paper considers the consensus control problem of distributed parameter models multi-agent systems with time-delay. By using the knowledge between neighboring agents, considering time-delay problem in the multi-agent systems, a distributed P-type iterative learning control protocol is proposed. The consensus error between any two agents in the sense of L² norm can converge to zero after enough iterations based on proposed ILC law. And then we extend these conclusions to Lipschitz nonlinear case. Finally, the simulation result shows the effectiveness of the control method.     

Sampled-data control for time-delay systems

The sampled-data systems are hybrid ones involving continuous time and discrete time signals, which makes the traditional analysis and synthesis methodologies of time-delay systems unable to be directly used in the cases of hybrid systems with time-delay. The primary disadvantages of current design techniques of sampled-data control are their inabilities to deal effectively with time-delay and the model uncertainty. In this paper, we generalized the analysis methodology of time-delay systems to that of the hybrid systems with time-delay and uncertainty, which developed a design procedure of sampled-data control for time-delay systems. Asymptotic stability of the time-delay hybrid systems was developed. The time-delay dependent robust sampled-data control for the time-varying delay of an uncertain linear system was then discussed. The results were described as linear matrix inequalities, which can be solved using newly released LMITool.     

Adaptive pseudospectral methods for solving constrained linear and nonlinear time-delay optimal control problems

In this paper, we first develop an adaptive shifted Legendre–Gauss (ShLG) pseudospectral method for solving constrained linear time-delay optimal control problems. The delays in the problems are on the state and/or on the control input. By dividing the domain of the problem into a uniform mesh based on the delay terms, the constrained linear time-delay optimal control problem is reduced to a quadratic programming problem. Next, we extend the application of the adaptive ShLG pseudospectral method to nonlinear problems through quasilinearization. Using this scheme, the constrained nonlinear time-delay optimal control problem is replaced with a sequence of constrained linear-quadratic sub-problems whose solutions converge to the solution of the original nonlinear problem. The method is called the iterative-adaptive ShLG pseudospectral method. One of the most important advantages of the proposed method lies in the case with which nonsmooth optimal controls can be computed when inequality constraints and terminal constraints on the state vector are imposed. Moreover, a comparison is made with optimal solutions obtained analytically and/or other numerical methods in the literature to demonstrate the applicability and accuracy of the proposed methods.     

Hierarchical recursive generalized extended least squares estimation algorithms for a class of nonlinear stochastic systems with colored noise

This paper considers the parameter identification problem of a bilinear state space system with colored noise based on its input-output representation. An input-output representation of a bilinear state-space system is derived for the parameter identification by eliminating the state variables in the model, and a recursive generalized extended least squares algorithm is presented for estimating the parameters of the obtained model. Furthermore, a three-stage recursive generalized extended least squares algorithm is proposed for reducing the computational cost. The validity of the proposed method is evaluated through a numerical example.     

Identification of Wiener–Hammerstein models based on variational bayesian approach in the presence of process noise

This paper addresses the identification of Wiener–Hammerstein (WH) models in the presence of process and measurement noises, which has not been well studied yet in the existing works. To achieve an unbiased estimation, the model parameters are obtained by maximizing the likelihood function, which is solved in the expectation-maximization framework. Due to the difficulty of computing the posterior distributions of the latent variables of WH models, variational Bayes (VB) is used here, and a method for approximating the posterior distributions based on Monte Carlo integral is proposed in VB framework. To the best of our knowledge, it is the first time to use VB approach for WH model identification. Two simulation examples demonstrate the effectiveness of the proposed method. Moreover, the proposed method is used for a WH benchmark problem, and the results show that it improves the identification performance.     

Event-triggered consensus of nonlinear multi-agent systems with stochastic switching topology

This paper is concerned with a leader-follower consensus problem for networked Lipschitz nonlinear multi-agent systems. An event-triggered consensus controller is developed with the consideration of discontinuous state feedback. To further enhance the robustness of the proposed controller, modeling uncertainty and switching topology are also considered in the stability analysis. Meanwhile, a time-delay equivalent approach is adopted to deal with the discrete-time control problem. Particularly, a sufficient condition for the stochastic stabilization of the networked multi-agent systems is proposed based on the Lyapunov functional method. Furthermore, an optimization algorithm is developed to derive the parameters of the controller. Finally, numerical simulation is conducted to demonstrate the effectiveness of the proposed control algorithm.     

Identification and control of delayed SISO systems through pattern search methods

In this study, an approach to identify and control stable, unstable and integrating systems with unknown delays, framed on the generalized Pattern Search Method is presented. The proposed method inherits the global convergence properties of the generalized Pattern Search Method, allowing us to make a stability analysis of the proposed approach and delay identification capabilities. The proposed approach identifies the delay and guarantees closed-loop stability, which could be a difficult task since in unstable and integrating cases, open-loop experiments are not feasible. Simulation examples show the usefulness of the proposed strategy proving that the scheme is capable of identifying the delay and stabilizing the system even with long delay.     

Disturbance rejection for time-delay systems based on the equivalent-input-disturbance approach

This paper presents a disturbance rejection method for time-delay systems. The configuration of the control system is constructed based on the equivalent-input-disturbance (EID) approach. A modified state observer is applied to reconstruct the state of the time-delay plant. A disturbance estimator is designed to actively compensate for the disturbances. Under such a construction of the system, both matched and unmatched disturbances are rejected effectively without requiring any prior knowledge of the disturbance or inverse dynamics of the plant. The presentation of the closed-loop system is derived for the stability analysis and controller design. Simulation results demonstrate the validity and superiority of the proposed method.     

T–S fuzzy control for dithered nonlinear singularly perturbed systems with multiple time delays

This paper is concerned with the stability problem of nonlinear multiple time-delay singularly perturbed (NDSP) systems. To overcome the effect of modeling error between the reduced-order model of the NDSP plant and Takagi–Sugeno (T–S) fuzzy models, a robustness design of model-based fuzzy control is proposed in this study. A stability criterion in terms of Lyapunov’s direct method is derived to guarantee the asymptotic stability of NDSP systems. According to this criterion, a model-based fuzzy controller is then synthesized via the technique of parallel distributed compensation (PDC) to stabilize the NDSP system. If the designed fuzzy controller cannot stabilize the NDSP system, a high-frequency signal, commonly referred to as dither, is simultaneously introduced to stabilize it. Based on the relaxed method, the NDSP system can be stabilized by regulating appropriately the parameters of dither. If the dither’s frequency is high enough, the output of the dithered reduced system and that of its corresponding mathematical model – the relaxed reduced system – can be made as close as desired. This makes it possible to obtain a rigorous prediction of the stability of the dithered reduced system based on the one of the relaxed reduced system.