Adaptive Output Tracking Control Research Articles

Adaptive Output Feedback Tracking Control for Nonlinear Systems with Unknown Growth Rate

Article Adaptive Output Feedback Tracking Control for Nonlinear Systems with Unknown Growth Rate Manman Yuan 1,2,*, and Wei Qian 1,2,* 1 School of Electrical Engineering and Automation, Henan Polytechnic University, Jiaozuo 454003, China 2 Henan Key Laboratory of Intelligent Detection and Control of Coal Mine Equipment, Jiaozuo 454003, China * Correspondence: ymmamhappy@163.com; qwei@hpu.edu.cn Received: 4 September 2023 Accepted: 7 November 2023 Published: 26 March 2024 Abstract: In this paper, the problem of adaptive output feedback tracking is considered for a class of nonlinear systems with lower-triangular structures. A novel dynamic gain is introduced to deal with the unknown growth rate. By coupling the dynamic gain with the observer and the controller, an adaptive output tracking controller is developed, which can guarantee that all signals of the closed-loop system are globally bounded. Finally, the effectiveness of the presented control scheme is illustrated by a numerical example.

Adaptive Output Feedback Tracking Control for Nonlinear Systems with Unknown Growth Rate

Article Adaptive Output Feedback Tracking Control for Nonlinear Systems with Unknown Growth Rate Manman Yuan 1,2,*, and Wei Qian 1,2,* 1 School of Electrical Engineering and Automation, Henan Polytechnic University, Jiaozuo 454003, China 2 Henan Key Laboratory of Intelligent Detection and Control of Coal Mine Equipment, Jiaozuo 454003, China * Correspondence: ymmamhappy@163.com; qwei@hpu.edu.cn Received: 4 September 2023 Accepted: 7 November 2023 Abstract: In this paper, the problem of adaptive output feedback tracking is considered for a class of nonlinear systems with lower-triangular structures. A novel dynamic gain is introduced to deal with the unknown growth rate. By coupling the dynamic gain with the observer and the controller, an adaptive output tracking controller is developed, which can guarantee that all signals of the closed-loop system are globally bounded. Finally, the effectiveness of the presented control scheme is illustrated by a numerical example.

Adaptive Robust Output Tracking of Uncertain Strict-Feedback Nonlinear Systems with Full State Constraints via Control Schemes with Simple Structure

In this paper, the problem of output tracking control is considered for a class of uncertain strict-feedback nonlinear systems with virtual control coefficients and full state constraints. It is assumed only that the nonlinear uncertainties of the systems are some smooth functions enough to guarantee the existence of solutions to the differential equations which describe uncertain systems, and that the virtual control coefficients are unknown. Based on backstepping algorithm, by combining the barrier Lyapunov function with an integral inequality reported in the current control literature, a design method is presented whereby a simple adaptive robust output tracking control scheme can be synthesized. The presented design method need not know all the nonlinear upper bounds of uncertainties, which are repeatedly employed at each step of backstepping procedure. It is also shown that the output tracking error of the uncertain nonlinear systems with both unknown virtual control coefficients and full state constraints can converge uniformly exponentially towards a ball which can be as small as desired. Finally, a numerical example is given, and the corresponding simulations are also implemented to demonstrate the simplicity of the proposed method and the validity of the theoretical results.

Governor-parameterized barrier function for safe output tracking with locally sensed constraints

This paper considers output tracking with time-varying constraints, obtained from online sensor measurements, relevant to autonomous system navigation in unknown environments. Inspired by reference governor techniques, we introduce a virtual governor system, whose state specifies an output regulation point for the actual system and is controlled to adaptively track an output reference without violating the constraints. Our main contribution is a governor-parameterized barrier function (PBF) that quantifies the trade-off between safety (distance from constraint violation) and system energy (output-regulation Lyapunov function). The PBF defines a local safe set that varies as the system-governor state or the sensor measurements change. This safe set induces a governor control law, which guarantees safe and stable output tracking for the real system. We demonstrate our adaptive output-tracking controller on a feedback-linearizable system navigating in an unknown environment, where obstacle distances are measured online.

Adaptive Output Tracking Control of Piecewise Affine Systems With Prescribed Performance

In this article, we investigate the adaptive output tracking control for multiinput–multioutput piecewise affine systems with prescribed performance. Both direct and indirect adaptation approaches are studied. Given a desired trajectory, both control approaches ensure the output tracking error to be confined within a performance bound, which prescribes the steady-state tracking error as well as the transient behavior, such as decaying rate and overshoot. We establish novel common Lyapunov functions without solving the conventional Lyapunov equations. Based on these common Lyapunov functions, the stability of the closed-loop system under arbitrary switching is established. Furthermore, the parameter convergence for both direct and indirect approaches is proved under the persistently exciting condition of the input signals. The dynamic gain adjustment technique is incorporated to counter the singularity problem in the indirect adaptation case. Finally, the numerical simulation validates the effectiveness and correctness of the proposed approaches in both direct and indirect adaptation cases.

Adaptive finite-time direct fuzzy control for a nonlinear system with an unknown control gain based on an observer

An adaptive fuzzy output tracking control scheme is presented for a class of strict-feedback systems with an unknown control gain in this paper. An ideal controller that can achieve finite-time stabilization of the filtered tracking error system is approximated by an ideal fuzzy logic system. The actual fuzzy function is used as the controller to estimate the ideal fuzzy system via on-line adjustment of system parameters. A τ-observer ( i.e., the control error observer) is designed to manage the unknown fuzzy control error. Unlike previous studies, new parameter adaptive laws are constructed in this study, and derivative calculations in the filtered tracking error system are not required. An adaptive direct fuzzy controller based on a τ-observer is thus proposed. The practical finite-time stabilization of the output tracking error and the boundedness of all states in the closed-loop system are also ensured. Finally, the effectiveness of the proposed control criterion is verified via two simulation examples.

Adaptive neural network output tracking control of uncertain switched nonlinear systems: An improved multiple Lyapunov function method

For a class of uncertain switched nonlinear systems, the adaptive neural network(NN) output tracking control problem is investigated by using the neural network technique in this paper. The considered switched nonlinear systems involve unknown control coefficients, external disturbances, and unmodeled dynamics merged in the full-states. An improved multiple Lyapunov function method is developed through relaxing the traditional multiple Lyapunov function conditions. A feasible state-dependent switching signal and an adaptive NN output tracking switching controller are designed such that the output tracking error converges to an arbitrarily small neighborhood of the origin, and all the signals in the closed-loop system remain within a bounded region. It is proved that the positive definiteness of Lyapunov functions and the solvability assumption of the adaptive NN output tracking control problem for all the subsystems are unnecessary. An application example of the mass-spring-damper system and a numerical example are given to illustrate the effectiveness of the proposed algorithm.

Adaptive Fuzzy Fast Finite-Time Dynamic Surface Tracking Control for Nonlinear Systems

In this paper, we investigate the adaptive fast finite-time tracking control problem for a class of uncertain nonlinear strict-feedback systems by using backstepping technique and fast finite-time stable theory. Dynamic surface control approach is introduced to reduce the computational complexity because of the repeated differentiation of virtual signals in the traditional backstepping algorithm. By employing the approximation of fuzzy logic systems, a fuzzy-based adaptive fast finite-time output tracking control approach is presented, which can guarantee the convergence of tracking error and the boundedness of all closed-loop signals in the fast finite-time. In the final, the validity of the developed control method is proved by the simulation results.

Adaptive Intelligent Control for Input and Output Constrained High-Order Uncertain Nonlinear Systems

This article reports this article on the problem of adaptive fuzzy output tracking control for a category of high-order nonlinear systems with input saturation, output constraint, and serious uncertainties. A high-order barrier Lyapunov function and an auxiliary Hyperbolic Tangent function is employed to deal with output constraint and input saturation, respectively. By incorporating a backstepping design technique, adaptive fuzzy control, and adding a power integrator, a novel control scheme is designed to ensure that all states in the resulting closed-loop system are bounded and the tracking error converges to a bounded compact set. Moreover, two simulations are conducted to verify the effectiveness of the design method.

Implicit function based adaptive control of non-canonical form discrete-time nonlinear systems

This paper presents a new study on adaptive state feedback output tracking control problem for uncertain discrete-time nonlinear systems in a general non-canonical form. Time-advance operations on the output of such systems result in the output dynamics being nonlinearly dependent on the control input and unknown parameters, which leads to three technical issues: implicit relative degree; nonlinearly parameterized uncertainties; and non-affine control input. To address these issues, this paper first employs feedback linearization and implicit function theory to construct a relative degree dependent normal form; then proposes an adaptive parametric reconstruction based method to simultaneously deal with linearly and nonlinearly parameterized uncertainties in the output dynamics; and finally constructs a key implicit function equation to derive a unique adaptive control law which ensures closed-loop stability and asymptotic output tracking. An explicitly iterative solution based adaptive control law is also proposed to ensure closed-loop stability and bounded output tracking within any degree of accuracy. The simulation verifies the effectiveness of the proposed adaptive control method.

Robust adaptive radial‐basis function neural network‐based backstepping control of a class of perturbed nonlinear systems with unknown system parameters

SummaryIn this article, the robust adaptive output tracking control problem is addressed for a class of nonlinear systems with nonlinear dynamics and unknown system parameters. The nonlinear dynamics including internal parameter uncertainties and external disturbances are formulated as time‐varying state/input‐dependent perturbations. Radial‐basis function neural networks (RBFNNs) are developed to approximate the perturbations. A robust adaptive RBFNN‐based output feedback control strategy against the perturbations is developed by using backstepping technique with immeasurable states and without knowing any system parameter. Based on Lyapunov stability theorem, the asymptotic output tracking results of the closed‐loop nonlinear system are obtained in the presence of perturbations, immeasurable states, and unknown system parameters. The efficacy of the proposed adaptive RBFNN‐based output feedback control strategy is validated by simulation in a DC–DC buck converter system.

Output tracking control for TRMS based on time receding optimal observation of disturbances

The paper presents an approach to design the adaptive output tracking controller for disturbed Twin Rotor Multi-Input Multi-Output System (TRMS) by using a time receding observer of functional disturbances for compensative control purpose, without using conventional methods as a neural network. To do this, first the disturbed Euler-Lagrange model of TRMS is converted to an equivalent bilinear form. And then, secondly an optimal disturbances estimator for this disturbed bilinear system is constructed based on time receding minimizing their effect. The complete output tracking controller for TRMS is created then by combining an exact linearization controller with the proposed disturbances estimation mechanism. Simulation results show that the here suggested controller meet completely the expected output tracking performances.

A New Nussbaum-Type Function and its Application in the Control of Uncertain Strict-Feedback Systems

In this study, an adaptive fuzzy output tracking control scheme is proposed for uncertain strict-feedback systems with unknown control directions. Nussbaum-type functions are designed, with which the effects of multiple unknown control directions can be handled. Compared with the existing Nussbaum-type functions dealing with the same problem, the proposed Nussbaum-type functions have smaller amplitudes that are more conducive to the design of the controller. Fuzzy functions are used to estimate the unknown terms by combining adaptive laws with backstepping procedure. Adaptive fuzzy controller is constructed to guarantee that the output tracking error system is asymptotically stable and all states in the closed-loop system are bounded. Finally, numerical simulation studies are presented to illustrate the effectiveness of the proposed criteria.

Robust tracking control of robot manipulators with friction and variable loads without velocity measurement: A switched control strategy

This article addresses the adaptive-based robust output feedback tracking control for robot manipulators with friction and alternating unknown loads. A switched nonlinear system is first established to model the friction and parameter variations, caused by the load change. Under arbitrary load changings, an adaptive [Formula: see text] tracking control strategy is proposed to ensure link position tracking, in the presence of uncertainties and external disturbances. Then, for bounded external disturbances, a novel robust adaptive output tracking control strategy is developed, which guarantees all the closed-loop signals are bounded and tracking error is driven to zero. Unlike some previous studies, the proposed algorithms do not require velocity measurements, and the unknown switched parameters and disturbances are neither required to be periodic nor to have known bounds. A simulation study is also given to demonstrate the analytically proved properties of the proposed schemes.

Comparative study on resistance and displacement based adaptive output tracking control strategies for resistance spot welding

Abstract To achieve high quality resistance spot welds (RSW) through intelligent manufacturing of automotive body structures, adaptive control for RSW processes becomes imperative and has gained global attention. Tracking ideal process signal profiles predetermined offline is a commonly-used strategy to realize the adaptive control of RSW and is found effective in compensating for the adverse impacts caused by shunting and electrode wear. However, the validity of this control strategy under other types of disturbance is still unproven. In this article, typical abnormal welding conditions, i.e. initial sheet gaps and edge proximity, were designed to test the performance of output tracking control strategies for steel RSW. Two types of process signals including dynamic resistance and electrode displacement were tracked to approach the referenced signal profiles through the adjustment of weld parameters. Comparative studies were conducted based on metallographic analysis and mechanical tests to validate the two tracking control strategies. Results demonstrated that gap and edge proximity conditions decreased the nugget diameter and the tensile-shear strength of the welds. When satisfactory tracking of the reference resistance signal was realized under these two types of abnormal conditions, weld properties are further degraded, owing to a reduction in welding current. On the contrary, tracking the electrode displacement signal mitigated the negative influences from the disturbance through increasing welding current properly. When the disturbance was too severe, the displacement-based strategy raised the risk of expulsion. The findings suggest that tracking the dynamic resistance signal is not optimal for adaptive control of steel RSW, while tracking the electrode displacement signal is more effective. However, to achieve a more satisfactory adaptive control result, the output tracking control should be used in combination with expulsion suppression techniques.

Adaptive Nonlinear Output Tracking Control With Rejection of Unmatched Biased Sinusoidal Disturbances for Nonlinear Systems

An adaptive nonlinear output tracking control method is proposed to reject unmatched biased sinusoidal disturbances for nonlinear systems that are in the strict feedback form. The proposed method consists of a disturbance observer (DOB) and an adaptive nonlinear controller. The DOB is in the form of a high-pass filter, and it estimates biased sinusoidal disturbances with unknown frequencies. The performance of the DOB is investigated in the time and frequency domains. As the DOB cannot accurately estimate biased sinusoidal disturbances with unknown frequencies, an adaptive nonlinear controller is designed to compensate for the disturbance estimation errors and ensure output tracking. The adaptive nonlinear controller is developed via the backstepping procedure. Thus, matching conditions are not significant in this design. The discontinuous “sgn” function is applied to the controller to compensate for the disturbance estimation errors. An update law of the control gain of the discontinuous “sgn” function is proposed to suppress the disturbance estimation errors without the knowledge of their upper bounds. Low-pass filters are embedded in the controller to smooth the derivative of the discontinuous “sgn” function. Consequently, multiple unmatched biased sinusoidal disturbances are compensated by the proposed method without any knowledge of the disturbances. The proposed method can reject multiple biased sinusoidal disturbances while it guarantees the constraint of the tracking error in steady-state response. Furthermore, the proposed method can reject multiple biased sinusoidal disturbances.

Neural Networks-Based Adaptive Exponential Quasi-Passification and Output Tracking Control for Uncertain Switched Nonlinear Systems

In this paper, we mainly study the adaptive exponential quasi-passivity and adaptive tracking control of lower triangular uncertain switched nonlinear systems, even though the adaptive output tracking control problem of individual subsystem is unsolvable. First, the exponential quasipassivity concept is proposed to describe the energy changing of the overall switched nonlinear systems without the exponential quasi-passivity property of all the subsystems. Then, for switched nonlinear systems, the semiglobally uniformly ultimate boundedness is achieved by using exponential quasipassivity. Second, this result is applied to solve adaptive tracking control problem uncertain switched nonlinear systems in lower-triangular form. A new adaptive tracking control technique is developed by combining quasi-passification methods with adaptive backstepping techniques. The unknown nonlinear functions are approximated by the radial basis function neural networks. In contrast to the existing results, the multiple storage functions method reduces the conservativeness caused by a common Lyapunov function for all subsystems. Finally, the effectiveness of the proposed method is verified by an example.

Neural Adaptive Fault Tolerant Control of Nonlinear Fractional Order Systems Via Terminal Sliding Mode Approach

This article proposes an adaptive neural output tracking control scheme for a class of nonlinear fractional order (FO) systems in the presence of unknown actuator faults. By means of backstepping terminal sliding mode (SM) control technique, an adaptive fractional state-feedback control law is extracted to achieve finite time stability along with output tracking for an uncertain faulty FO system. The unknown nonlinear terms are approximated by radial-basis function neural network (RBFNN) with unknown approximation error upper bound. Using convergence in finite time and fractional Lyapunov stability theorems, the finite time stability and tracking achievement are proved. Finally, the proposed fault tolerant control (FTC) approach is validated with numerical simulations on two fractional models including fractional Genesio–Tesi and fractional Duffing's oscillator systems.

Adaptive neural output-feedback control for nonstrict-feedback time-delay fractional-order systems with output constraints and actuator nonlinearities

This study addresses the issue of the adaptive output tracking control for a category of uncertain nonstrict-feedback delayed incommensurate fractional-order systems in the presence of nonaffine structures, unmeasured pseudo-states, unknown control directions, unknown actuator nonlinearities and output constraints. Firstly, the mean value theorem and the Gaussian error function are introduced to eliminate the difficulties that arise from the nonaffine structures and the unknown actuator nonlinearities, respectively. Secondly, the immeasurable tracking error variables are suitably estimated by constructing a fractional-order linear observer. Thirdly, the neural network, the Razumikhin Lemma, the variable separation approach, and the smooth Nussbaum-type function are used to deal with the uncertain nonlinear dynamics, the unknown time-varying delays, the nonstrict feedback and the unknown control directions, respectively. Fourthly, asymmetric barrier Lyapunov functions are employed to overcome the violation of the output constraints and to tune online the parameters of the adaptive neural controller. Through rigorous analysis, it is proved that the boundedness of all variables in the closed-loop system and the semi global asymptotic tracking are ensured without transgression of the constraints. The principal contributions of this study can be summarized as follows: (1) based on Caputo’s definitions and new lemmas, methods concerning the controllability, observability and stability analysis of integer-order systems are extended to fractional-order ones, (2) the output tracking objective for a relatively large class of uncertain systems is achieved with a simple controller and less tuning parameters. Finally, computer-simulation studies from the robotic field are given to demonstrate the effectiveness of the proposed controller.

Adaptive Robust Backstepping Output Tracking Control for a Class of Uncertain Nonlinear Systems Using Neural Network

The problem of robust output tracking is studied for a class of uncertain nonlinear systems in the presence of structure uncertainties, external disturbances, and unknown time-varying virtual control coefficients. In this study, it is supposed that the upper bounds of external disturbances and that the upper and lower bounds of unknown time-varying virtual control coefficients are unknown. By employing a simple structure neural network (NN), the unknown structure uncertainties are approximated. A class of backstepping approach-based adaptive robust controllers is synthesized for such uncertain nonlinear systems. By making use of Lyapunov functional approach, it is also shown that the proposed adaptive robust backstepping output tracking controller can guarantee the tracking error between the system output and the desired reference signal to converge asymptotically to zero. Finally, two numerical examples are given to demonstrate the effectiveness of the proposed controller.