Adaptive control of non‐linear systems with unknown virtual control coefficients

This paper deals with state feedback adaptive control of parametric‐strict‐feedback (triangular) non‐linear systems with unknown virtual control coefficients. A priori knowledge of the signs of the virtual coefficients is not required, and control signals and adaptive laws are smooth. Asymptotic tracking of smooth reference signals is achieved while all the variables remain bounded. The proposed algorithms make use of backstepping and tuning functions, and enlarge the class of non‐linear systems with unknown parameters for which asymptotic output tracking can be achieved. Copyright © 2000 John Wiley & Sons, Ltd.     

Adaptive least mean squares block Volterra filters

Adaptive filtering has found many applications in situations where the underlying signals are changing or unknown. While linear filters are simple from implementation and conceptual points of view, many signals are non‐linear in nature. Non‐linear filters based on truncated Volterra expansions can effectively model a large number of systems. Unfortunately, the resulting input auto‐moment matrix is ill conditioned, which results in a slow convergence rate. This paper proposes a class of block adaptive Volterra filters in which the input sequences are Hadamard transformed to improve the condition number of the input auto‐moment matrix and consequently improve the convergence rate. This is achieved by the decorrelation effect produced by the orthogonality of the transform. Since Hadamard transformation employs only ±1's, the additional required computational and implementation burdens are few. The effect of additive white Gaussian noise is introduced. Simulation experiments are given to illustrate the improved performance of the proposed method over the conventional Volterra LMS method. Copyright © 2001 John Wiley & Sons, Ltd.     

Direct adaptive control for non‐linear uncertain systems with exogenous disturbances

A direct adaptive non‐linear control framework for multivariable non‐linear uncertain systems with exogenous bounded disturbances is developed. The adaptive non‐linear controller addresses adaptive stabilization, disturbance rejection and adaptive tracking. The proposed framework is Lyapunov‐based and guarantees partial asymptotic stability of the closed‐loop system; that is, asymptotic stability with respect to part of the closed‐loop system states associated with the plant. In the case of bounded energy L₂ disturbances the proposed approach guarantees a non‐expansivity constraint on the closed‐loop input–output map. Finally, several illustrative numerical examples are provided to demonstrate the efficacy of the proposed approach. Copyright © 2002 John Wiley & Sons, Ltd.     

Stochastic H∞ control problem with state‐dependent noise for weakly coupled large‐scale systems

In this paper, stochastic H_∞ state feedback control with state‐dependent noise for weakly coupled large‐scale systems is discussed. After establishing the asymptotic structure of the stochastic algebraic Riccati equation (SARE), a new iterative algorithm that combines the Newton's method with the fixed‐point algorithm is derived for the first time. As a result, both the quadratic convergence and the reduced‐order computation in the same dimension of the subsystems are attained. Copyright © 2007 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.     

A combined approach for the identification of continuous non‐linear systems

The identification of a non‐linear continuous output‐only system from a time series is considered for the case that the functional form of the model is not known beforehand. To estimate both functions and parameters, a combination of non‐parametric modelling based on non‐linear regression and parametric modelling based on a multiple shooting algorithm is proposed. This strategy to determine non‐linear differential equations is exemplified on experimental data from a chaotic circuit where an accurate reconstruction of the observed attractor is obtained. Copyright © 2003 John Wiley & Sons, Ltd.     

Estimation algorithm for system with non‐Gaussian multiplicative/additive noises based on variational Bayesian inference

This paper considers estimation algorithms for linear and nonlinear systems contaminated by non‐Gaussian multiplicative and additive noises. Based on the variational idea, in order to derive optimal estimation algorithms, we combine the multiplicative noise with states as the joint parameters to estimate. The application of variational Bayesian inference to joint estimation of the state and the multiplicative noise is established. By treating the states as unknown quantities as well as the multiplicative noise, there are now correlations between the states and multiplicative noise in the posterior distribution. There are two main goals in Bayesian learning. The first is approximating the marginal likelihood (PDF of multiplicative noise) to perform model comparison. The second is approximating the posterior distribution over the states (also called a system model), which can then be used for prediction. The two goals constitute the iterative algorithm. The rules for determining the loop is the Kullback‐Leibler divergence between the true distribution of state and a chosen fixed tractable distribution, which is used to approximate the true one. The iterative algorithm is deduced, which is initialized based on the idea of sampling. Meanwhile, the convergence analysis of the proposed iterative algorithm is presented. The numerical simulation results in a comparison between the proposed method and these existing classic algorithms in the context of nonlinear hidden Markov models, state‐space models, and target‐tracking models with non‐Gaussian multiplicative noise demonstrate the superiorities, not only in speed, precision, and computation load but also in the ability to process non‐Gaussian complex noise.     

A switching periodic adaptive control approach for time‐varying parameters with unknown periodicity

Periodic variations are encountered in many real systems, which can exist in the system parameters, as a disturbance or as the tracking objective. However, there exist a great number of situations where the periodicity is not known in advance. Hence, how to compensate for the effects of time‐varying parameters with unknown periodicity remains a challenge for the controller design. In this paper, we proposed a switching periodic adaptive control approach for continuous‐time nonlinear parametric systems with periodic uncertainties in which the period and bound are not known in advance. We utilized a fully saturated periodic adaptation law to identify the unknown periodic parameters in a pointwise manner. In addition, we provided a logic‐based switching scheme to estimate the unknown period and bound online simultaneously. By virtue of Lyapunov stability analysis, we show that the asymptotic convergence can be guaranteed irrespective of the initial conditions. Finally, we carried out numerical simulations to demonstrate the efficacy of the switching periodic adaptive control algorithm. The proposed approach can be applied to parametric nonlinear systems with time‐varying parameters of unknown periodicity irrespective of the types of periodic uncertainties. Copyright © 2015 John Wiley & Sons, Ltd.     

An indirect adaptive pole‐placement control for MIMO discrete‐time stochastic systems

In this paper, an indirect adaptive pole‐placement control scheme for multi‐input multi‐output (MIMO) discrete‐time stochastic systems is developed. This control scheme combines a recursive least squares (RLS) estimation algorithm with pole‐placement control design to produce a control law with self‐tuning capability. A parametric model with a priori prediction outputs is adopted for modelling the controlled system. Then, a RLS estimation algorithm which applies the a posteriori prediction errors is employed to identify the parameters of the model. It is shown that the implementation of the estimation algorithm including a time‐varying inverse logarithm step size mechanism has an almost sure convergence. Further, an equivalent stochastic closed‐loop system is used here for constructing near supermartingales, allowing that the proposed control scheme facilitates the establishment of the adaptive pole‐placement control and prevents the closed‐loop control system from occurring unstable pole‐zero cancellation. An analysis is provided that this control scheme guarantees parameter estimation convergence and system stability in the mean squares sense almost surely. Simulation studies are also presented to validate the theoretical findings. Copyright © 2005 John Wiley & Sons, Ltd.     

Discrete‐time periodic adaptive control for parametric systems with non‐sector nonlinearities

In this paper, a periodic adaptive control approach is proposed for a class of discrete‐time parametric systems with non‐sector nonlinearities. The proposed periodic adaptive control law is characterized by either one‐period delayed parametric updating or two‐period delayed parametric updating when input gain contains periodic unknowns. Logarithmic‐type discrete Lyapunov function is employed to handle the difficulties caused by the uncertainties that do not satisfy the linear growth condition. Some extensions to nonlinear systems with multiple unknown parameters and time‐varying input gain, tracking tasks, as well as higher‐order systems in canonical form, are also discussed. Copyright © 2013 John Wiley & Sons, Ltd.     

Finite‐model adaptive control using an LS‐like algorithm

Adaptive control problem of a class of discrete‐time nonlinear uncertain systems, of which the internal uncertainty can be characterized by a finite set of functions, is formulated and studied by using an least squares (LS)‐like algorithm to design the feedback control law. For the finite‐model adaptive control problem, this algorithm is proposed as an extension of counterpart of traditional LS algorithm. Stability in sense of pth mean for the closed‐loop system is proved under a so‐called linear growth assumption, which is shown to be necessary in general by a counter‐example constructed in this paper. The main results have been also applied to parametric cases, which demonstrate how to bridge the non‐parametric case and parametric case. Copyright © 2006 John Wiley & Sons, Ltd.     

Indirect adaptive control of a class of marine vehicles

A nonlinear adaptive framework for bounded‐error tracking control of a class of non‐minimum phase marine vehicles is presented. The control algorithm relies on a special set of tracking errors to achieve satisfactory tracking performance while guaranteeing stable internal dynamics. First, the design of a model‐based nonlinear control law, guaranteeing asymptotic stability of the error dynamics, is presented. This control algorithm solves the tracking problem for the considered class of marine vehicles, assuming full knowledge of the system model. Then, the analysis of the zero‐dynamics is carried out, which illustrates the efficacy of the chosen set of tracking errors in stabilizing the internal dynamics. Finally, an indirect adaptive technique, relying on a partial state predictor, is used to address parametric uncertainties in the model. The resulting adaptive control algorithm guarantees Lyapunov stability of the errors and parameter estimates, as well as asymptotic convergence of the errors to zero. Numerical simulations illustrate the performance of the adaptive algorithm. Copyright © 2009 John Wiley & Sons, Ltd.     

Cumulant‐based approach to FIR system identification

In this paper, a non‐recursive approach is developed for estimating the coefficients of a moving average (MA) model from only third‐order cumulant statistics of a finite realization of the observations of the output data. The signal observations may be noisy. The excitation signal is assumed to be zero mean, non‐Gaussian stationary sequence that is not observed. The noise is additive and may be coloured Gaussian or non‐Gaussian. This novel technique is based on forming a third‐order cumulant composite data matrix. The method presented here requires the solution of a system of linear equations, which can be achieved using least‐squares methods. The proposed approach is illustrated via computer simulations and is shown to be consistent. Copyright © 2003 John Wiley & Sons, Ltd.     

Adaptive control for non‐negative and compartmental dynamical systems with applications to general anesthesia

Non‐negative and compartmental dynamical system models are composed of homogeneous interconnected subsystems or compartments which exchange variable non‐negative quantities of material with conservation laws describing transfer, accumulation, and elimination between the compartments and the environment. These models are widespread in biological and physiological sciences and play a key role in understanding these processes. In this paper, we develop a direct adaptive control framework for linear uncertain non‐negative and compartmental systems. The proposed framework is Lyapunov‐based and guarantees partial asymptotic set‐point regulation; that is, asymptotic set‐point stability with respect to part of the closed‐loop system states associated with the plant. In addition, the adaptive controller guarantees that the physical system states remain in the non‐negative orthant of the state space. Finally, a numerical example involving the infusion of the anesthetic drug propofol for maintaining a desired constant level of depth of anesthesia for non‐cardiac surgery is provided to demonstrate the efficacy of the proposed approach. Copyright © 2003 John Wiley & Sons, Ltd.     

Convergence analysis of the RLS identification algorithm with exponential forgetting in stationary ARX‐structures

The recursive least‐squares (RLS) identification algorithm is often extended with exponential forgetting as a tool for parameter estimation in time‐varying stochastic systems. The statistical properties of the parameter estimates obtained from such an extended RLS‐algorithm depend in a non‐linear way on the time‐varying characteristics and on the forgetting factor. In this paper, the RLS‐estimator with exponential forgetting is applied to time‐invariant Gaussian autoregressions with second‐order stationary external inputs, i.e.to Gaussian ARX‐processes. Approximate expressions for the asymptotic bias and covariance of the parameter estimates when the forgetting factor tends to one and time to infinity are given, showing that the bias is non‐zero and that the covariance function decays exponentially with a rate that is given by the forgetting factor. The orders of magnitude of the errors in the asymptotic expressions are also derived. Copyright © 1999 John Wiley & Sons, Ltd.     

Modeling of non‐linear systems by FWNNs and their intelligent control

By utilizing some of the important properties of wavelets like denoising, compression, multiresolution along with the concepts of fuzzy logic and neural network, two fuzzy wavelet neural networks (FWNNs) are proposed for approximating any arbitrary non‐linear function, hence, identifying a non‐linear system. We have fuzzified the output of DWT block, which receives the given inputs, in the proposed two methods: one using compression property and other using multiresolution property. We present a new type of fuzzy neuron model, each non‐linear synapse of which is characterized by a set of fuzzy implication rules with singleton weights in their consequents. It is shown that noise and disturbance in the reference signal are reduced with wavelets and also the variation of somatic gain, the parameter that controls the slope of the activation function in the neural network, leads to more accurate output. Identification results are found to be accurate and speed of their convergence is fast. Next, we simulate a control system for keeping output at a desired level by using the identified models. Two self‐learning controllers are designed in this simulation. One is a self‐learning fuzzy PI controller and other is a NN controller. Simulation results show that the NN controller is more adaptive and fast. Copyright © 2005 John Wiley & Sons, Ltd.     

Convergence analysis of blind image deconvolution via dispersion minimization

A new non‐linear adaptive filter called blind image deconvolution via dispersion minimization has recently been proposed for restoring noisy blurred images blindly. This is essentially a two‐dimensional version of the constant modulus algorithm that is well known in the field of blind equalization. The two‐dimensional extension has been shown capable of reconstructing noisy blurred images using partial a priori information about the true image and the point spread function in a variety of situations by means of simulations. This paper analyses the behaviour of the algorithm by investigating the static properties of the cost function and the dynamic convergence of the parameter estimates. The theoretical results are supported with computer simulations. Copyright © 2006 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.     

On simplex‐based piecewise‐linear approximations of non‐linear mappings

Simplex‐based piecewise‐linear (PWL) approximations of non‐linear mappings are needed when the robust PWL analysis is used to directly solve non‐linear equations. This paper proposes a straightforward technique for transforming the well‐known approximations into another form. This new form is computationally more efficient, since it preserves the sparse structure of the original Jacobian matrix. Furthermore, this new form of PWL approximation explicitly relates the simplex‐based PWL analysis to the conventional formulation of the Katzenelson algorithm. The proposed transform technique is also extended to treat groupwise‐separable mappings and, finally, non‐separable but sparse mappings that arise in real‐life simulation of large electronic circuits. In this paper, all these (transformed) simplex‐based PWL approximations are compared in terms of their generality and efficiency. The computational efficiency of the PWL approximation that utilizes sparsity is validated with realistic simulations. Copyright © 2005 John Wiley & Sons, Ltd.     

Filtering,smoothing and prediction using a control‐loop spectral factorization method for coloured noise

A method for the linear least‐squares estimation of random signals contaminated with random noise that uses a new method of spectral factorization is shown. It is shown that the optimal filter can be written entirely in terms of the two spectral factors of signal plus noise and noise‐alone, and can be applied to the general case of coloured and white additive noise. The method of spectral factorization used is novel and uses control‐system methodology. Copyright © 2012 John Wiley & Sons, Ltd.     

A note on global convergence of adaptive control in a discrete‐time non‐linear case

Adaptive control is applied to a particular class of SISO discrete‐time non‐linear systems. Global boundedness and convergence are obtained by introducing a modification to a classical adaptive scheme. Copyright © 1999 John Wiley & Sons, Ltd.