Adaptive robust boundary control of shaft vibrations under perturbations with unknown upper bounds

This paper presents robust and adaptive boundary control designs to stabilize the two‐dimensional vibration of hybrid shaft model. The hybrid shaft is mathematically represented by a set of partial differential equations, governing the shaft vibrations, coupled to ordinary differential equations, describing rigid body spinning and dynamic boundary conditions. The control objective is to stabilize the transverse vibrations of the perturbed shaft while regulating the spinning rate. To achieve this, the paper first establishes robust boundary control laws that fulfil the control objective in the presence of modeling uncertainties and external disturbances operating over the shaft domain and boundary. Lyapunov‐based analyses show that the proposed robust control exponentially stabilizes the shaft with vanishing distributive perturbations, while assuring ultimately bounded vibrations in the case of nonvanishing perturbations. Then, adaptive control philosophy is utilized to achieve redesigned robust controllers that only use online adaptation of control gains without acquiring the knowledge of bounds on perturbations, as well as dynamic parameters. An advantage of this design is avoiding an overconservative robust control law, which may induce poor stability and chattering in tackling system perturbations with unknown upper bounds. Simulations through finite element method illustrate the results. Copyright © 2014 John Wiley & Sons, Ltd.     

State estimation of system with bounded uncertain parameters: Interval multimodel approach

The objective of this study is the analysis of dynamic systems represented by a multimodel expression with variable parameters. Changes in these parameters are unknown but bounded. Since it is not possible to estimate these parameters over time, the simulation of such systems requires the consideration of all possible values taken by these parameters. More precisely, the goal is to determine, at any moment, the smallest set containing all the possible values of the state vector simultaneously compatible with the state equations and with a priori known bounds of the uncertain parameters. This set will be characterized by two trajectories corresponding to the lower and upper limits of the state at every moment. This characterization can be realized by a direct simulation of the system, given the bounds of its parameters. It can also be implemented with a Luenberger‐type observer, fed with the system measurements.     

Use of prior bounds on time constants and steady‐state gain in recursive parameter bounding

Computation of parameter bounds of a linear dynamical system, given input–output observations and bounds on model‐output error, has been developed as an alternative to classical parameter estimation using least squares, maximum likelihood or the prediction error method. When bounds on time‐domain plant behaviour are known in advance, they can be used to develop prior parameter bounds for discrete‐time rational transfer‐function parameters. These bounds can be used to initialize standard parameter‐bounding algorithms which process input–output observations to update the exact polytope feasible set or one of its outer bounding approximations such as an ellipsoid, orthotope or parallelotope. This paper presents a method to compute such prior bounds from bounds on time constants and steady‐state (dc) gain, often available from the physics of the system or from previous experience. The method finds subsets making up the prior feasible parameter set, recursively in model order, for any configuration of the pole ranges. An analysis leading to measures of the value of prior bounds, in terms of their chances of remaining active when new bounds derived from observations are imposed, is presented. A simulation study compares polytope updating with and without such initial bounds. The simulations investigate the influence of the tightness of time‐constant and steady‐state‐gain bounds in reducing the volume of the feasible sets obtained as observations are processed. The effects of initial bound tightness and signal‐to‐noise ratio on survival time of the prior bounds are also examined. Copyright © 2002 John Wiley & Sons, Ltd.     

Dual high‐gain‐based adaptive output‐feedback control for a class of nonlinear systems

We propose an adaptive output‐feedback controller for a general class of nonlinear triangular (strict‐feedback‐like) systems. The design is based on our recent results on a new high‐gain control design approach utilizing a dual high‐gain observer and controller architecture with a dynamic scaling. The technique provides strong robustness properties and allows the system class to contain unknown functions dependent on all states and involving unknown parameters (with no magnitude bounds required). Unlike our earlier result on this problem where a time‐varying design of the high‐gain scaling parameter was utilized, the technique proposed here achieves an autonomous dynamic controller by introducing a novel design of the observer, the scaling parameter, and the adaptation parameter. This provides a time‐invariant dynamic output‐feedback globally asymptotically stabilizing solution for the benchmark open problem proposed in our earlier work with no magnitude bounds or sign information on the unknown parameter being necessary. Copyright © 2007 John Wiley & Sons, Ltd.     

Adaptive control of stochastic nonlinear systems with Markovian switching

This paper is concerned with the problem of adaptive control for a class of stochastic nonlinear systems with Markovian switching, where the upper bounds of nonlinearities of stochastic Markovian jump systems are assumed to be unknown. Firstly, an adaptation law is developed to estimate these unknown parameters. Then, a class of adaptive state feedback controller is proposed such that not only the estimated errors are bounded almost surely but also, the states of the resulting closed‐loop system are asymptotically stable almost surely. Finally, a numerical example is given to show the validity of the results.Copyright © 2012 John Wiley & Sons, Ltd.     

Sliding mode disturbance observer control based on adaptive synchronization in a class of fractional‐order chaotic systems

In this paper, a fractional‐order Dadras‐Momeni chaotic system in a class of three‐dimensional autonomous differential equations has been considered. Later, a design technique of adaptive sliding mode disturbance‐observer for synchronization of a fractional‐order Dadras‐Momeni chaotic system with time‐varying disturbances is presented. Applying the Lyapunov stability theory, the suggested control technique fulfils that the states of the fractional‐order master and slave chaotic systems are synchronized hastily. While the upper bounds of disturbances are unknown, an adaptive regulation scheme is advised to estimate them. The recommended disturbance‐observer realizes the convergence of the disturbance approximation error to the origin. Finally, simulation results are presented in one example to demonstrate the efficiency of the offered scheme on the fractional‐order Dadras‐Momeni chaotic system in the existence of external disturbances.     

State controllability computation technique for linear time‐varying systems by using Taylor series approximation

This article presents a technique to determine the controllability Grammian matrix (CGM) for linear time‐varying systems by using truncated Taylor polynomial vector and the operational matrix of integration. An important property of this algorithm is that it starts by integrating the Lyapunov differential matrix equation in terms of the CGM. However, the algorithm does not use the mathematical integration processes actually, but uses the truncated Taylor polynomial vector and the operational matrix of integration. Thus, the problem is reduced to solving a linear set of algebraic equations with constant coefficients consisting of the Taylor polynomial constant coefficients of each of the CGM elements. Numerical results and error curves are given to illustrate the improvements achieved by the proposed algorithm. Copyright © 2008 John Wiley & Sons, Ltd.     

Adaptive switching control of uncertain fractional systems: Application to Chua's circuit

Since the introduction of fractional‐order differential equations, there has been much research interest in synthesis and control of oscillatory, periodic, and chaotic fractional‐order dynamical systems. Therefore, in this article, the problem of stabilization and control of nonlinear three‐dimensional perturbed fractional nonlinear systems is considered. The major novelty of this article is handling partially unknown dynamics of nonlinear fractional‐order systems, as well as coping with input saturation along the existence of model variations and high‐frequency sensor noises via just one control input. The method supposes no known knowledge on the upper bounds of the uncertainties and perturbations. It is assumed that the working region of the input saturation function is also unknown. After the introduction of a simple finite‐time stable nonlinear sliding manifold, an adaptive control technique is used to reach the system variables to the sliding surface. Rigorous stability discussions are adopted to prove the convergence of the developed sliding mode controller. The findings of this research are illustrated using providing computer simulations for the control problem of the chaotic unified system and the fractional Chua's circuit model.     

Simultaneous input and parameter estimation with input observers and set‐membership parameter bounding: theory and an automotive application

The paper addresses an on‐line, simultaneous input and parameter estimation problem for a first‐order system affected by measurement noise. This problem is motivated by practical applications in the area of engine control. Our approach combines an input observer for the unknown input with a set‐membership algorithm to estimate the parameter. The set‐membership algorithm takes advantage of a priori available information such as (i) known bounds on the unknown input, measurement noise and time rate of change of the unknown input; (ii) the form of the input observer in which the unknown parameter affects only the observer output; and (iii) the input observer error bounds for the case when the parameter is known exactly. The asymptotic properties of the algorithm as the observer gain increases are delineated. It is shown that for accurate estimation the unknown input needs to approach the known bounds a sufficient number of times (these time instants need not be known). Powertrain control applications are discussed and a simulation example based on application to engine control is reported. A generalization of the basic ideas to higher order systems is also elaborated. Copyright © 2006 John Wiley & Sons, Ltd.     

An adaptive regulation method for a class of uncertain time‐delay systems

In this paper, a new adaptive robust stabilization scheme is proposed for uncertain neutral time‐delay systems. No upper bounds on the uncertainties are assumed to be available. An update law is first used to find estimates of these upper bounds. A state‐feedback controller is then designed, which is shown to stabilize the underlying system under some mild conditions. The asymptotic stability of the state trajectories is proved using the Lyapunov–Krasovskii approach. An example is provided, which demonstrates the efficacy of the proposed adaptive control scheme. Copyright © 2012 John Wiley & Sons, Ltd.     

Blind identification of sparse Volterra systems

This paper is concerned with blind identification for single‐input single‐output Volterra systems with finite order and memory with the second‐order and the third‐order statistics. For the full‐sized Volterra system (i.e. all its kernels are nonzero) excited by unknown independently and identically distributed stationary random sequences, it is shown that blind identifiability does not hold in the second‐order moment (SOM) and the third‐order moment (TOM) domain. However, under some sufficient conditions, a class of truncated sparse Volterra systems, where some kernels are restricted to being zero, can be identified blindly and more Volterra parameters can be estimated in TOM than in SOM. Numerical examples illustrate the effectiveness of the proposed methods. Copyright © 2007 John Wiley & Sons, Ltd.     

Wave digital simulation of passive systems in linear state‐space form

This paper deals with wave digital modeling of passive state‐space models. The set of differential equations must be of linear state‐space form, but all parameters can be time‐variant and/or nonlinear. For such state‐space models, a canonical internally passive reference circuit is presented and used for deriving wave digital structures. In order to show the usability, special solutions for important basic linear time‐variant models are compared with wave digital simulation results. Moreover, the wave digital modeling of a nonlinear and time‐variant oscillator is discussed. Especially for a lossless oscillator an implementation is proposed, which preserves energy under finite‐arithmetic conditions. This is verified by comparing simulation results with the analytical solution of a gravity pendulum. Copyright © 2009 John Wiley & Sons, Ltd.     

Adaptive estimation and rejection of unknown sinusoidal disturbances through measurement feedback for a class of non‐minimum phase non‐linear MIMO systems

This paper develops an adaptive estimation method to estimate unknown disturbances in a class of non‐minimum phase non‐linear MIMO systems. The unknown disturbances are generated by an unknown linear exosystem. The frequencies, phases and amplitudes of the disturbances are unknown, the only available information of the disturbances is the number of distinctive frequencies. The system considered in this paper is a class of MIMO non‐linear systems in the output feedback form which can be non‐minimum phase. The proposed estimation algorithm provides exponentially convergent estimates of system states, unknown disturbances in the system and frequencies of the disturbances characterized by the eigenvalues of the exosystem. Moreover, based on the stabilization controller for the disturbance free system, the estimates of the disturbances are used to solve the disturbance rejection problem. The unknown disturbances are compensated completely with the stability of the whole closed‐loop system. Copyright © 2006 John Wiley & Sons, Ltd.     

Adaptive output tracking for a class of non‐linear time‐delay systems

In this paper the output tracking control problem for a class of non‐linear time delay systems with some unknown constant parameters is addressed. Such a problem is solved in the case that the non‐linear time‐delay system has full delay relative degree and stable internal dynamics. It is supposed moreover that the output and its time derivatives until n?1, where n is the length of the state vector (euclidean part), do not depend explicitly on the unknown parameters. This work is the first step towards the application of the methodologies of adaptive control for non‐linear delayless systems, based on tools of differential geometry, to non‐linear time‐delay systems too. Copyright © 2004 John Wiley & Sons, Ltd.     

Parameter bounds for a class of discrete bilinear systems from records with bounded output errors

Parameter bounding offers a useful alternative to point parameter estimation methods when either statistical hypotheses on the errors are not met or uncertainties are better characterized in a deterministic way (e.g. systematic, round-off, truncation errors). So far, many efforts have been devoted to the problem of parameter bounding in linear systems, where exact parameter uncertainty intervals can be computed. In contrast, only partial results have been found for general non-linear parametrization, namely either upper or lower bounds on parameter uncertainties can be evaluated. In this paper we derive approximate parameter uncertainty intervals for a class of discrete bilinear systems with bounded output errors. This work is based on a linear input-output parametrization and previous results on bounded errors-in-variables models. For an extensively simulated example, central estimates by means of the bounded errors-in-variables approach and least squares estimates are computed and compared.     

Reliable H∞ filtering for discrete time‐delay Markovian jump systems with partly unknown transition probabilities

In this paper, the reliable H_∞ filtering problem is studied for a class of discrete nonlinear Markovian jump systems with sensor failures and time delays. The transition probabilities of the jumping process are assumed to be partly unknown. The failures of sensors are quantified by a variable taking values in a given interval. The time‐varying delay is unknown with given lower and upper bounds. The purpose of the addressed reliable H_∞ filtering problem is to design a mode‐dependent filter such that the filtering error dynamics is asymptotically mean‐square stable and also achieves a prescribed H_∞ performance level. By using a new Lyapunov–Krasovskii functional and delay‐partitioning technique, sufficient delay‐dependent conditions for the existence of such a filter are obtained. The filter gains are characterized in terms of the solution to a convex optimization problem that can be easily solved by using the semi‐definite programme method. A numerical example is provided to demonstrate the effectiveness of the proposed design approach. Copyright © 2011 John Wiley & Sons, Ltd.     

Adaptive robust control of a class of dynamic delay systems with unknown uncertainty bounds

In this paper, a robust adaptive controller is developed for a class of uncertain dynamic systems with time‐varying delays and subject to uncertainties whose bounds are unknown but their functional properties are known. It is shown that if a constraint on the norm of the matrix associated with the delayed state is met, then the adaptive controller designed guarantees that all solutions of the class of systems considered converge to a ball with any prespecified exponential convergence rate towards it. Finally, an example is included to illustrate the results developed in this paper. Copyright © 2001 John Wiley & Sons, Ltd.     

An adaptive switching scheme for uncertain discrete time‐delay systems

In this paper, an adaptive switching control algorithm is proposed for the stabilization of uncertain discrete‐time systems with time‐varying delay. It is assumed that the time delay is unknown and time varying, nonetheless bounded with a known bound. It is supposed that the system is highly uncertain, and that a set of controllers are designed (off‐line) to stabilize the system in the whole uncertain parameter space; subsequently, a switching scheme is developed to stabilize the uncertain time‐delay system. A thorough stability analysis for the uncertain time‐delay system under the mentioned control scheme is provided. Furthermore, an upper bound on the allowable rate of change of the system parameters and delay is obtained. Simulation results are presented to show the efficacy of the proposed switching scheme. Copyright © 2008 John Wiley & Sons, Ltd.     

Global adaptive finite‐time stabilization of a class of switched nonlinear systems with parametric uncertainty

This paper investigates the global adaptive finite‐time stabilization of a class of switched nonlinear systems, whose subsystems are all in p (p≤1) normal form with unknown control coefficients and parametric uncertainties. The restrictions on the power orders and the nonlinear perturbations are relaxed. By using the parameter separation technique, the uncertain parameters are separated from nonlinear functions. A systematic design procedure for a common state feedback controller and a switching adaptive law is presented by employing the backstepping methodology. It is proved that the closed‐loop system is finite‐time stable under arbitrary switching by utilizing the common Lyapunov function. Finally, with the application to finite‐time control of chemical reactor systems, the effectiveness of the proposed method is demonstrated. Copyright © 2015 John Wiley & Sons, Ltd.