Interval-excitation Condition Research Articles

Practical fixed‐time composite‐learning control for full‐state constraint strict‐feedback non‐linear systems: A dynamic regressor extension and mixing based approach

AbstractA practical fixed‐time composite learning control scheme, by combining dynamic regressor extension and mixing (DREM) parameter identification algorithm and adaptive dynamic surface control (DSC) technique, is proposed for a class of strict‐feedback non‐linear systems subjected to linear‐in‐parameters uncertainties and full‐state constraint. To address the problem of state constraint, a non‐linear transformation function is introduced to convert the originally constrained non‐linear system into an unconstrained one. Meanwhile, the hyperbolic tangent function is employed to avoid singularity issues that often appeared in the traditional fixed‐time (FXT) control designs. In order to relax the requirement of persistency of excitation condition, a modified FXT‐DREM parameter identification approach with an interval excitation condition is constructed by introducing a three‐layer transformation technique derived from the classical DREM algorithm. Then, the modified FXT‐DREM parameter identification algorithm is seamlessly integrated into the adaptive DSC framework, resulting in a composite‐learning control scheme. By employing Lyapunov stability analysis, the fixed‐time convergence of both the parameter estimation error and the trajectory tracking error is proved. Finally, the effectiveness of the proposed design is demonstrated through simulation test.

An I & I Adaptive Redesign Approach for Asymptotic Stability without PE

This paper proposes an adaptation redesign approach of the immersion and invariance (I&I) adaptive control to achieve asymptotic stability of the controlled plant and the parameter estimator at the desired equilibrium. The key idea is to employ the technique of generalized parameter estimation-based observer on the parameter estimation error dynamics by applying the indirect I & I adaptive control scheme, yielding a linear regression equation, from which the adaptive law can be redesigned. As a result, it is shown that globally exponential parameter convergence can be guaranteed under an interval excitation (IE) condition, which is much weaker than the conventionally required persistent excitation. Under a stabilizability assumption, an adaptation-redesigned feedback control law can be designed to achieve global asymptotic stability at the desired equilibrium point under the IE condition. The proposed adaptive control approach is applied to a class of parameteric strict-feedback systems without overparameterization, which is needed by the standard I & I adaptive control.

Locally Weighted Learning Robot Control With Improved Parameter Convergence

An adaptive robot control approach that can achieve an accurate approximation of the complex robot dynamics without knowledge unlearning in the case of multiple sequential tasks is proposed in this article. A locally weighted learning (LWL) model with automatic structure growth according to the size of the learning domain is introduced to approximate unknown robot dynamics, and a composite learning technique with regressor extension is applied to improve parameter convergence. The LWL ensures that learning in one area of the learning domain does not cause unlearning in another area, and the composite learning theoretically guarantees the identification performance of the LWL model. By the exploitation of both stored data in memory and instantaneous data, parameter convergence in the LWL model is achieved under a more achievable interval-excitation condition than the stringent persistent-excitation condition. This further enhances the trajectory tracking performance for multiple sequential tasks, which is generally not achievable by existing approximation-based adaptive robot control approaches. Experimental studies have verified the superiority of the proposed approach over prevalent approaches.

Adaptive Tracking Control of Hydraulic Systems With Improved Parameter Convergence

Most recent studies on adaptive hydraulic tracking control focus on the trajectory tracking performance while the parameter convergence property is often unsatisfying. This article proposes a composite learning adaptive position tracking controller with improved parameter convergence for electro-hydraulic servo systems. In the composite learning, a prediction error is formulated to exploit input–output memory data, and parameter estimates are driven simultaneously by tracking and prediction errors. Practical exponential stability of the closed-loop system, which implies the convergence of both the tracking and parameter estimation error, is established by a more realizable interval-excitation condition than the stringent persistent–excitation condition. Therefore, superior trajectory tracking is obtained compared with the classical adaptive hydraulic control. Besides, the initial fluid control volumes of hydraulic systems are assumed to be unknown <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> , which enhances the generality of the proposed control approach. The abovementioned two properties are generally not achievable in prevalent approaches to adaptive hydraulic control. Moreover, noisy acceleration signals and the time derivatives of pressure signals are not needed in the proposed approach, which improves its robustness against measurement noise. Extensive experimental results verify its superiority over currently available ones.

Composite learning sliding mode control of uncertain nonlinear systems with prescribed performance

This paper explores the prescribed performance tracking control problem of nonlinear systems with triangular structure. To obtain the desired transient performance and precise estimations of uncertain terms, the techniques of neural network control, sliding mode control and composite learning control are incorporated into the proposed control method. The presented control strategy can ensure the tracking error converges to a prescribed small residual set. Compared with the persistent excitation condition required in the conventional adaptive control, the interval excitation condition needed in the proposed control approach is weak, which guarantees that the radial basis function neural networks approximate the unknown nonlinear terms more accurately. Finally, two simulation examples are exploited to manifest the effectiveness of the proposed approach.

Overparameterized model parameter recovering with finite‐time convergence

SummaryIn this article, we consider the online parameter estimation problem for overparameterized linear regression models. Many estimation methods require at least the interval excitation condition, which is not always fulfilled for such models, especially in normal operation. To relax the required excitation level, we detect linearly dependent columns in the regressor, obtain the reduced model and estimate parameters in a finite time. On the last step, the parameters of the initial model are recovered. The proposed method efficiency is demonstrated on a simple overparameterized model and a magnetic levitation system.

Disturbance Frequency Estimation for an LTV System

We consider the frequency estimation problem for a sinusoidal disturbance acting on a linear time-varying system, where only the input and output signals are available. We propose a novel parametrization that translates this problem into a linear regression model with unknown parameters. The frequency estimation is performed using the Dynamic Regressor Extension and Mixing procedure and an algebraic finite-time estimator. The resulting scheme provides finite-time frequency estimation under the interval excitation condition. The role of the tuning coefficients in satisfying the convergence condition is also discussed. Simulations illustrate the applicability of the proposed solution.

Composite learning control of robotic systems: A least squares modulated approach

Most current studies of adaptive robot control concentrate on parameter convergence in the steady state, while parameter convergence rates are rarely investigated. This paper proposes a least-squares modulated composite learning robot control based on Moore–Penrose pseudoinverse to improve the performance of parameter convergence. In the composite learning, a prediction error is constructed based on online historical data and regressor extension, and both the prediction and tracking errors are applied to update parameter estimates such that accurate and smooth parameter estimation is obtained under a weak excitation condition termed interval excitation (IE). The distinctive features of the proposed method include: (1) Asymptotic stability of the closed-loop system is proven without the IE condition; (2) exponential stability is proven and balanced and easily tunable rates of parameter convergence are achieved under the IE condition, where the rates are independent of unpredictable excitation levels in different regressor channels. These two features are generally not achievable with the existing adaptive robot control methods. Experimental results on an industrial manipulator have demonstrated the effectiveness and superiority of the proposed approach.

Composite Learning Robot Control With Friction Compensation: A Neural Network-Based Approach

Friction is one of the significant obstacles that hinders high-performance robot tracking control because accurate friction modeling and effective compensation are challenging issues. To address this problem, in this paper, we propose a modified neural network (NN) structure with additional jump approximation activation functions to model the inherent discontinuous friction in robotic systems, this structure allows us to improve the NN approximation accuracy without using too many NN nodes. The modeling accuracy is theoretically guaranteed by a composite learning technique, it explores both online historical data and instantaneous data to achieve NN weight convergence under a much weaker interval-excitation condition than the stringent persistent-excitation condition. Furthermore, a partitioned NN technique is used to handle a problem caused by variable substitution when formulating the prediction error for composite learning. This technique also helps us to alleviate the requirements regarding the inertial matrix inversion and joint acceleration signals. The practical exponential stability of the closed-loop system is proved under the more realizable interval-excitation condition. Experimental results demonstrate the effectiveness and superiority of the proposed approach.

Composite Learning Adaptive Dynamic Surface Control of Fractional-Order Nonlinear Systems.

Adaptive dynamic surface control (ADSC) is effective for solving the complexity problem in adaptive backstepping control of integer-order nonlinear systems. This article focuses on the ADSC design for parametric uncertain fractional-order nonlinear systems (FONSs). In each backstepping step, the virtual controller is driven to pass through a fractional dynamic surface whose fractional-order derivative can be calculated easily. An ADSC law that ensure tracking error convergence is designed. The proposed ADSC requires a stringent condition called persistent excitation (PE) to achieve parameter convergence. To relax this limitation, a prediction error is defined by using online recorded data and instantaneous data, and a composite learning law is proposed to utilize both the prediction error and the tracking error. Then, a composite learning ADSC (CLADSC) method is developed to guarantee tracking error convergence and accurate parameter estimation under an interval excitation condition that is weaker than the PE one. Finally, an illustrative example is presented to show the performance of our methods.

Composite learning adaptive backstepping control using neural networks with compact supports

SummaryThe ability to learn is crucial for neural network (NN) control as it is able to enhance the overall stability and robustness of control systems. In this study, a composite learning control strategy is proposed for a class of strict‐feedback nonlinear systems with mismatched uncertainties, where raised‐cosine radial basis function NNs with compact supports are applied to approximate system uncertainties. Both online historical data and instantaneous data are utilized to update NN weights. Practical exponential stability of the closed‐loop system is established under a weak excitation condition termed interval excitation. The proposed approach ensures fast parameter convergence, implying an exact estimation of plant uncertainties, without the trajectory of NN inputs being recurrent and the time derivation of plant states. The raised‐cosine radial basis function NNs applied not only reduces computational cost but also facilitates the exact determination of a subregressor activated along any trajectory of NN inputs so that the interval excitation condition is verifiable. Numerical results have verified validity and superiority of the proposed approach.

On parameter convergence in least squares identification and adaptive control

SummaryLeast squares estimation is appealing in performance and robustness improvements of adaptive control. A strict condition termed persistent excitation (PE) needs to be satisfied to achieve parameter convergence in least squares estimation. This paper proposes a least squares identification and adaptive control strategy to achieve parameter convergence without the PE condition. A modified modeling error that utilizes online historical data together with instant data is constructed as additional feedback to update parameter estimates, and an integral transformation is introduced to avoid the time derivation of plant states in the modified modeling error. On the basis of these results, a regressor filtering–free least squares estimation law is proposed to guarantee exponential parameter convergence by an interval excitation condition, which is much weaker than the PE condition. And then, an identification‐based indirect adaptive control law is proposed to establish exponential stability of the closed‐loop system under the interval excitation condition. Illustrative results considering both identification and control problems have verified the effectiveness and superiority of the proposed approach.

Composite learning fuzzy synchronization for incommensurate fractional‐order chaotic systems with time‐varying delays

SummaryThis paper presents a composite learning fuzzy control to synchronize two different uncertain incommensurate fractional‐order time‐varying delayed chaotic systems with unknown external disturbances and mismatched parametric uncertainties via the Takagi‐Sugeno fuzzy method. An adaptive controller together with fractional‐order composite learning laws is designed based on both a parallel distributed compensation technology and a fractional Lyapunov criterion. The boundedness of all variables in the closed‐loop system and the Mittag‐Leffler stability of tracking error can be guaranteed. T‐S fuzzy systems are provided to tackle unknown nonlinear functions. The distinctive features of the proposed approach consist in the following: (1) a supervisory control law is designed to compensate the lumped disturbances; (2) both the prediction error and the tracking error are used to estimate the unknown fuzzy system parameters; (3) parameter convergence can be ensured by an interval excitation condition. Finally, the feasibility of the proposed control strategy is demonstrated throughout an illustrative example.

Adaptive Parameter Estimation and Control Design for Robot Manipulators With Finite-Time Convergence

For parameter identifications of robot systems, most existing works have focused on the estimation veracity, but few works of literature are concerned with the convergence speed. In this paper, we developed a robot control/identification scheme to identify the unknown robot kinematic and dynamic parameters with enhanced convergence rate. Superior to the traditional methods, the information of parameter estimation error was properly integrated into the proposed identification algorithm, such that enhanced estimation performance was achieved. Besides, the Newton–Euler (NE) method was used to build the robot dynamic model, where a singular value decomposition-based model reduction method was designed to remedy the potential singularity problems of the NE regressor. Moreover, an interval excitation condition was employed to relax the requirement of persistent excitation condition for the kinematic estimation. By using the Lyapunov synthesis, explicit analysis of the convergence rate of the tracking errors and the estimated parameters were performed. Simulation studies were conducted to show the accurate and fast convergence of the proposed finite-time (FT) identification algorithm based on a 7-DOF arm of Baxter robot.

Composite learning robot control with guaranteed parameter convergence

Parameter convergence is desirable in adaptive control as it enhances the overall stability and robustness properties of the closed-loop system. However, a stringent condition termed persistent excitation (PE) must be satisfied to guarantee parameter convergence in the conventional adaptive control. This paper provides the first result of parameter convergence without the PE condition for adaptive control of a general class of robotic systems. More specifically, we develop a composite learning robot control (CLRC) strategy to achieve fast and accurate parameter estimation under a condition termed interval excitation (IE) which is much weaker than the PE condition. In the composite learning, a time-interval integral of a filtered regressor is utilized to construct a prediction error such that the time derivation of plant states is not necessary, and both the prediction error and a filtered tracking error are employed to update the parameter estimate. The closed-loop system is proven to be globally exponentially stable under the IE condition. Robustness against external disturbances of the CLRC is analyzed in the Lyapunov sense. An illustrative example shows the effectiveness and superiority of the proposed approach.

Model reference composite learning control without persistency of excitation

Parameter convergence is desirable in adaptive control as it brings several attractive features, including accurate online modelling, exponential tracking, and robust adaptation without parameter drift. However, a strong persistent-excitation (PE) condition must be satisfied to guarantee parameter convergence in the conventional adaptive control. This study proposes a model reference composite learning control strategy to guarantee parameter convergence without the PE condition. In the composite learning, an integral at a moving-time window is applied to construct a prediction error, an integral transformation is derived for avoiding the time derivation of plant states in the calculation of the prediction error, and both the tracking error and the prediction error are applied to update parametric estimates. Global exponential stability of the closed-loop system is established under an interval-excitation condition which is much weaker than the PE condition. Compared with a concurrent learning technique that has the same aim as this study, the proposed composite learning technique avoids the usage of singular value maximisation and fixed-point smoothing resulting in a considerable reduction of computational cost. Numerical results have verified effectiveness and superiority of the proposed control strategy.

Composite Learning Fuzzy Control of Uncertain Nonlinear Systems

Function approximation accuracy and computational cost are two major concerns in approximation-based adaptive fuzzy control. In this paper, a model reference composite learning fuzzy control strategy is proposed for a class of affine nonlinear systems with functional uncertainties. In the proposed approach, a modified modeling error that utilizes data recorded online is defined as a prediction error, a linear filter is applied to estimate time derivatives of plant states, and both the tracking error and the prediction error are exploited to update parametric estimates. It is proven that the closed-loop system achieves semiglobal practical exponential stability by an interval-excitation condition which is much weaker than a persistent-excitation condition. Compared with a concurrent learning approach that has the same aim as this study, the computational cost of the proposed approach is significantly reduced for the guarantee of accurate function approximation. An illustrative example of aircraft wing rock control has been provided to verify effectiveness of the proposed control strategy.

Composite Learning From Adaptive Dynamic Surface Control

In the conventional adaptive control, a stringent condition named persistent excitation (PE) must be satisfied to guarantee parameter convergence. This technical note focuses on adaptive dynamic surface control for a class of strict-feedback nonlinear systems with parametric uncertainties, where a novel technique coined composite learning is developed to guarantee parameter convergence without the PE condition. In the composite learning, online recorded data together with instantaneous data are applied to generate prediction errors, and both tracking errors and prediction errors are utilized to update parametric estimates. The proposed approach is also extended to an output-feedback case by using a nonlinear separation principle. The distinctive feature of the composite learning is that parameter convergence can be guaranteed by an interval-excitation condition which is much weaker than the PE condition such that the control performance can be improved from practical asymptotic stability to practical exponential stability. An illustrative example is used for verifying effectiveness of the proposed approach.