Linear-in-the-parameters Research Articles

Accelerated Gradient Approach For Deep Neural Network-Based Adaptive Control of Unknown Nonlinear Systems.

Recent connections in the adaptive control literature to continuous-time analogs of Nesterov's accelerated gradient method have led to the development of new real-time adaptation laws based on accelerated gradient methods. However, previous results assume that the system's uncertainties are linear-in-the-parameters (LIP). To compensate for non-LIP uncertainties, our preliminary results developed a neural network (NN)-based accelerated gradient adaptive controller to achieve trajectory tracking for nonlinear systems; however, the development and analysis only considered single-hidden-layer NNs. In this article, a generalized deep NN (DNN) architecture with an arbitrary number of hidden layers is considered, and a new DNN-based accelerated gradient adaptation scheme is developed to generate estimates of all the DNN weights in real-time. A nonsmooth Lyapunov-based analysis is used to guarantee the developed accelerated gradient-based DNN adaptation design achieves global asymptotic tracking error convergence for general nonlinear control affine systems subject to unknown (non-LIP) drift dynamics and exogenous disturbances. A comprehensive set of simulation studies are conducted on a two-state nonlinear system, a robotic manipulator, and a complex 20-D nonlinear system to demonstrate the improved performance of the developed method. Our simulation studies demonstrate enhanced tracking and function approximation performance from both DNN architectures and accelerated gradient adaptation.

Distributed Adaptive Synchronization in Euler–Lagrange Networks With Uncertain Interconnections

In this work we propose a new practical synchronization protocol for multiple Euler Lagrange (EL) systems without structural linear-in-the-parameters (LIP) knowledge of the uncertainty and where the agents can be interconnected before control design by unknown state-dependent interconnection terms. This setting is meant to overcome two standard a priori assumptions in the literature concerning uncertainty with LIP structure and absence of interaction among agents before designing the synchronization protocol. To overcome these assumptions, we propose an adaptive distributed control mechanism having the purpose of estimating the coefficients of the resulting state-dependent uncertainty structure.

Adaptive prescribed‐time tracking control for uncertain nonlinear systems with full state constraints

SummaryThis paper investigates the adaptive tracking control in a prescribed‐time for a class of uncertain nonlinear systems with a full state constraint. Firstly, in order to restrict the state constraints, the barrier Lyapunov function is employed and parameter uncertainty is handled based on linear‐in‐the‐parameters (LIP). A prescribed‐time adaptive controller is then designed by backstepping to ensure that the closed‐loop system can reach zero in a prescribed finite time under any given initial conditions. The prescribed‐time is independent of the design of the parameters. Finally, two simulation examples verify the effectiveness and usability of the proposed method.

A New Class of Efficient Adaptive Filters for Online Nonlinear Modeling

Nonlinear models are known to provide excellent performance in real-world applications that often operate in non-ideal conditions. However, such applications often require online processing to be performed with limited computational resources. To address this problem, we propose a new class of efficient nonlinear models for online applications. The proposed algorithms are based on linear-in-the-parameters (LIP) nonlinear filters using functional link expansions. In order to make this class of functional link adaptive filters (FLAFs) efficient, we propose low-complexity expansions and frequency-domain adaptation of the parameters. Among this family of algorithms, we also define the partitioned-block frequency-domain FLAF, whose implementation is particularly suitable for online nonlinear modeling problems. We assess and compare frequency-domain FLAFs with different expansions providing the best possible tradeoff between performance and computational complexity. Experimental results prove that the proposed algorithms can be considered as an efficient and effective solution for online applications, such as the acoustic echo cancellation, even in the presence of adverse nonlinear conditions and with limited availability of computational resources.

Interpolated Individual Weighting Subband Volterra Filter for Nonlinear Active Noise Control

Nonlinear active noise control (NLANC) is always a challenge in algorithm design. Most of the NLANC algorithms are based on the linear-in-the-parameters (LIP) filters, and no work is conducted on subband adaptive filter (SAF). In this brief, a novel nonlinear SAF, termed the Volterra filtered-x normalized SAF (VFxNSAF), is proposed for NLANC, whose input vector is obtained according to the size of the analysis filter bank. Moreover, the virtual microphone strategy is applied to VFxNSAF to suppress the noise at a remote location. To mitigate the computational burden, the interpolated IWVFxNSAF with individual weighting (IW) (I-IWVFxNSAF) is proposed, in which the quadratic kernel is obtained by the interpolated filter. Simulation results corroborate the superiority of the VFxNSAF and I-IWVFxNSAF algorithms for NLANC.

Model reference safety‐critical adaptive control for a class of nonlinearly parameterized systems

AbstractSafety is the most fundamental problem of safety‐critical systems. Safety control addresses the problem whether a given unsafe region of the state space can be avoided by a specific control‐input. Moreover, linearly parameterized dynamical system is a general assumption in most safety‐critical adaptive control literature; however, unknown parameters in real systems are usually nonlinear. The control problem of nonlinearly parameterized systems is really difficult without linear‐in‐the‐parameters (LIP) assumptions, which tends to be complicated and computationally intensive. In this paper, a novel model reference safety‐critical adaptive control (MRAC) approach is proposed for a class of nonlinearly parameterized systems. The proposed approach involves a novel controller architecture with a modified update law, which specifically filters out the unsafe behavior (the system is in a state which the system cannot operate normally, i.e., the given unsafe region), while preserving favorable tracking capability and robustness. The novelty of this paper is that the nonlinearly parameterized systems can be enforced safety, without LIP assumptions and complex calculations. Most importantly, this approach is effective for nonlinearly parameterized systems as well as linearly parameterized systems. Finally, three illustrative numerical examples are presented to demonstrate the effectiveness of the proposed design approach.

Robust Gradient-Based Adaptive Control of Nonlinearly Parametrized Plants

Most of the contributions in adaptive control literature assume that the system dynamics is linearly parametrizable, and a certainty equivalence principle is exploited to guarantee global stability and asymptotic convergence of tracking error to zero. Although linear-in-the-parameters (LIP) assumption is reasonable for a large class of dynamics, there exists a considerable number of real world systems, involving complex dynamics, where nonlinear parametrizations are inevitable. Previous research has shown that classical gradient-based adaptive designs with the certainty equivalence principle do not perform satisfactorily for nonlinear-in-the-parameter (NLIP) systems, and may, in fact, cause instability in many situations. This letter is an attempt toward addressing this issue by a novel non-certainty equivalence adaptive control design, where the classical gradient-based adaptive algorithm is used to tackle the LIP component of the NLIP dynamics, while a robust compensator, appended to the controller, accounts for the linearization error. The designed controller ensures global uniformly ultimately bounded stability of the error dynamics. Simulation results on a model of biochemical process, involving NLIP dynamics, are provided to validate the theoretical development.

Adaptive neural network-based satellite attitude control in the presence of CMG uncertainty

An attitude tracking controller is developed for control moment gyroscope (CMG)-actuated satellites, which is shown to achieve accurate attitude tracking in the presence of unmodeled external disturbance torques, parametric uncertainty, and nonlinear CMG disturbances. Since the disturbances/uncertainties do not all satisfy the typical linear-in-the-parameters (LP) assumption, a neural network (NN) is included in the control development. The innovation of the result is the development of a Lyapunov-based design/analysis that indicates exponential convergence to an arbitrarily small domain. The result is obtained despite the characteristics of the uncertainty; the nonvanishing disturbance terms; and the fact that the control input is premultiplied by a non-square, time-varying, nonlinear, uncertain matrix. In addition to the Lyapunov-based analysis, experimental results demonstrate the performance of the developed controller.

A multi-output two-stage locally regularized model construction method using the extreme learning machine

This paper investigates the construction of linear-in-the-parameters (LITP) models for multi-output regression problems. Most existing stepwise forward algorithms choose the regressor terms one by one, each time maximizing the model error reduction ratio. The drawback is that such procedures cannot guarantee a sparse model, especially under highly noisy learning conditions. The main objective of this paper is to improve the sparsity and generalization capability of a model for multi-output regression problems, while reducing the computational complexity. This is achieved by proposing a novel multi-output two-stage locally regularized model construction (MTLRMC) method using the extreme learning machine (ELM). In this new algorithm, the nonlinear parameters in each term, such as the width of the Gaussian function and the power of a polynomial term, are firstly determined by the ELM. An initial multi-output LITP model is then generated according to the termination criteria in the first stage. The significance of each selected regressor is checked and the insignificant ones are replaced at the second stage. The proposed method can produce an optimized compact model by using the regularized parameters. Further, to reduce the computational complexity, a proper regression context is used to allow fast implementation of the proposed method. Simulation results confirm the effectiveness of the proposed technique.

Efficient adaptive identification of linear-in-the-parameters nonlinear filters using periodic input sequences

This paper introduces computationally efficient NLMS and RLS adaptive algorithms for identifying non-recursive, linear-in-the-parameters (LIP) nonlinear systems using periodic input sequences. The algorithms presented in the paper are exact and require a real-time computational effort of a single multiplication, an addition and a subtraction per input sample. The transient, steady state, and tracking behavior of the algorithms as well as the effect of model mismatch is studied in the paper. The low computational complexity, good performance and broad applicability make the approach of this paper a valuable alternative to the current techniques for nonlinear system identification.

A novel automatic two-stage locally regularized classifier construction method using the extreme learning machine

This paper investigates the design of a linear-in-the-parameters (LITP) regression classifier for two-class problems. Most existing algorithms generally learn a classifier (model) from the available training data based on some stopping criterions, such as the Akaike's final prediction error (FPE). The drawback here is that the classifier obtained is then not directly obtained based on its generalization capability. The main objective of this paper is to improve the sparsity and generalization capability of a classifier, while reducing the computational expense in producing it. This is achieved by proposing an automatic two-stage locally regularized classifier construction (TSLRCC) method using the extreme learning machine (ELM). In this new algorithm, the nonlinear parameters in each term, such as the width of the Gaussian function and the power of a polynomial term, are firstly determined by the ELM. An initial classifier is then generated by the direct evaluation of these candidates models according to the leave-one-out (LOO) misclassification rate in the first stage. The significance of each selected regressor term is also checked and insignificant ones are replaced in the second stage. To reduce the computational complexity, a proper regression context is defined which allows fast implementation of the proposed method. Simulation results confirm the effectiveness of the proposed technique.

Fast automatic two-stage nonlinear model identification based on the extreme learning machine

It is convenient and effective to solve nonlinear problems with a model that has a linear-in-the-parameters (LITP) structure. However, the nonlinear parameters (e.g. the width of Gaussian function) of each model term needs to be pre-determined either from expert experience or through exhaustive search. An alternative approach is to optimize them by a gradient-based technique (e.g. Newton's method). Unfortunately, all of these methods still need a lot of computations. Recently, the extreme learning machine (ELM) has shown its advantages in terms of fast learning from data, but the sparsity of the constructed model cannot be guaranteed. This paper proposes a novel algorithm for automatic construction of a nonlinear system model based on the extreme learning machine. This is achieved by effectively integrating the ELM and leave-one-out (LOO) cross validation with our two-stage stepwise construction procedure [1]. The main objective is to improve the compactness and generalization capability of the model constructed by the ELM method. Numerical analysis shows that the proposed algorithm only involves about half of the computation of orthogonal least squares (OLS) based method. Simulation examples are included to confirm the efficacy and superiority of the proposed technique.

Adaptive discrete neural observer design for nonlinear systems with unknown time‐delay

AbstractThis paper focuses on the adaptive observer design for nonlinear discrete‐time MIMO systems with unknown time‐delay and nonlinear dynamics. The delayed states involved in the system are arguments of a nonlinear function and only the estimated delay is utilized. By constructing an appropriate Lyapunov–Krasovskii function, the delay estimation error is considered in the observer parameter design. The proposed method is then extended to the system with a nonlinear output measurement equation and the delayed dynamics. With the help of a high‐order neural network (HONN), the requirement for a precise system model, the linear‐in‐the‐parameters (LIP) assumption of the delayed states, the Lipschitz or norm‐boundedness assumption of unknown nonlinearities are removed. A novel converse Lyapunov technical lemma is also developed and used to prove the uniform ultimate boundedness of the proposed observer. The effectiveness of the proposed results is verified by simulations. Copyright © 2010 John Wiley & Sons, Ltd.

Modular Adaptive Control of Uncertain Euler–Lagrange Systems With Additive Disturbances

A novel adaptive nonlinear control design is developed which achieves modularity between the controller and the adaptive update law. Modularity between the controller/update law design provides flexibility in the selection of different update laws that could potentially be easier to implement or used to obtain faster parameter convergence and/or better tracking performance. For a general class of linear-in-the-parameters (LP) uncertain Euler-Lagrange systems subject to additive bounded non-LP disturbances, the developed controller uses a model-based feedforward adaptive term in conjunction with the recently developed robust integral of the sign of the error (RISE) feedback term. Modularity in the adaptive feedforward term is made possible by considering a generic form of the adaptive update law and its corresponding parameter estimate. This generic form of the update law is used to develop a new closed-loop error system and stability analysis that does not depend on nonlinear damping to yield the modular adaptive control result.

BACKLASH COMPENSATION WITH FILTERED PREDICTION IN DISCRETE TIME NONLINEAR SYSTEMS BY DYNAMIC INVERSION USING NEURAL NETWORKS

ABSTRACTA dynamics inversion compensation scheme is designed for control of nonlinear discrete‐time systems with input backlash. This paper extends the dynamic inversion technique to discrete‐time systems by using a filtered prediction, and shows how to use a neural network (NN) for inverting the backlash nonlinearity in the feedforward path. The technique provides a general procedure for using NN to determine the dynamics preinverse of an invertible discrete time dynamical system. A discrete‐time tuning algorithm is given for the NN weights so that the backlash compensation scheme guarantees bounded tracking and backlash errors, and also bounded parameter estimates. A rigorous proof of stability and performance is given and a simulation example verifies performance. Unlike standard discrete‐time adaptive control techniques, no certainty equivalence (CE) or linear‐in‐the‐parameters (LIP) assumptions are needed.

Fuzzy system as parameter estimator of nonlinear dynamic functions

In this paper we use adaptive fuzzy systems as intelligent identification systems for nonlinear time-varying plants. A new technique to design the fuzzy system that relies on the minimization of a loss function is presented. The design technique uses the centers of the fuzzy sets (labels) at the antecedent part of the rule base as the estimated parameters. This parametrization has the Linear In The Parameters (LITPs) characteristic that allows standard parameter estimation technique to be used to estimate the parameters of the fuzzy system. The combination of the fuzzy system and the estimation method then performs as a nonlinear estimator. If several fuzzy sets are defined for the input variables at the antecedent part, the fuzzy system ("fuzzy estimator") then behaves as a collection of nonlinear estimators where different rules' regions have different parameters. The proposed scheme is potentially capable of estimating the parameters of highly nonlinear plants. Simulation examples, which use plants with highly nonlinear gain, show the power of the proposed estimation scheme in comparison to estimation using the linear model.

Fuzzy System as a Parameter Estimatior of Nonliner Dynamic Functions

In this paper Adaptive Fuzzy Systems are used as intelligent identification systems for nonlinear plants. A new technique to design the fuzzy system that relies on the minimisation of a loss function is presented. The design technique uses the centres of the fuzzy sets (labels) at the antecedent part of the Rule Base as the estimated parameters. This parametrisation has the Linear in The Parameters (LITPs) characteristic that allows standard parameter estimation technique to be used to estimate the parameters of the fuzzy system. The combination of the fuzzy system and the estimation method then performs as a nonlinear estimator. If several fuzzy sets are defined for the input variables at the antecedent part, the fuzzy system (“fuzzy estimator”) then behaves as a collection of nonlinear estimators where different rule's regions have different parameters. The proposed scheme is potentially capable of estimating the parameters of highly nonlinear plants. Simulation examples, which use plants with highly nonlinear gain, show the power of the proposed estimation scheme in comparison to estimation using linear model.