Asymptotic Tracking Research Articles

Stability Analysis of Pressure and Penetration Rate in Rotary Drilling System

The purpose of this paper is to stabilize the annular pressure profile throughout the wellbore continuously while drilling. A new nonlinear dynamical system is developed and a controller is designed to stabilize the annular pressure and achieve asymptotic tracking by applying feedback control of the main pumps. Hence, the paper studies the control design for the well known Managed Pressure Drilling system (MPD). MPD provides a closedloop drilling process in which pore pressure, formation fracture pressure, and bottomhole pressure are balanced and managed at the surface. Although, responses must provide a solution for critical downhole pressures to preserve drilling efficiency and safety. Our MPD scheme is elaborated in reference to a nontrivial backstepping control procedure and the effectiveness of the proposed control laws are shown by simulations.

Asymptotic Trajectory Tracking of Autonomous Bicycles via Backstepping and Optimal Control

This letter studies the trajectory tracking problem for an autonomous bicycle that is a non-minimum phase, strongly nonlinear system. As compared with most existing methods dealing only with approximate trajectory tracking, this letter proposes a constructive design to achieve asymptotic trajectory tracking. More specifically, under the assumption that the desired trajectory is generated by a virtual bicycle, a novel asymptotic trajectory tracking controller design scheme is presented. Firstly, the nonlinear dynamics of the autonomous bicycle is established. Secondly, it is decomposed into two interconnected subsystems: a tracking subsystem and a balancing subsystem. For the tracking subsystem, the popular backstepping approach is applied to determine the propulsive force of the bicycle. For the balancing subsystem, optimal control is applied to determine the steering angular velocity of the handlebar in order to balance the bicycle and align the bicycle with the desired yaw angle. Thirdly, to tackle the strong coupling between the tracking and the balancing systems, the small-gain technique is applied for the first time to prove the asymptotic stability of the closed-loop bicycle system. Finally, the efficacy of the proposed exact trajectory tracking control methodology is validated by numerical simulations.

Asymptotic Trajectory Tracking of Underactuated Non-minimum Phase Marine Vessels

This paper presents a trajectory tracking scheme for a class of underactuated non-minimum phase marine vessels. Through proper coordinates transformations, the vessels tracking model can be transformed into two-input-two-output strict-feedback form for which general backstepping method can be applicable to solve the control problem. To avoid possible singularity problem in the recursive control design, an AMO (asymptotic modification of orientation) concept is applied. In this case, the vessels are forced to track this modified trajectory instead of directly tracking the predefined reference trajectory. Since the vessels’ tracking model is transformed into two-input-two-output strict-feedback form with invertible control gain matrix, the non-minimum phase problem can be easily handled by the proposed method. Proposed tracking method can guarantee the asymptotic stability of closed-loop system in terms of polar coordinates. Simulation studies are also carried out to demonstrate the effectiveness of presented tracking scheme.

Position Control of Flexible Joint Carts Using Adaptive Generalized Dynamics Inversion

This paper presents the design and implementation of Adaptive Generalized Dynamic Inversion (AGDI) to track the position of a Linear Flexible Joint Cart (LFJC) system along with vibration suppression of the flexible joint. The proposed AGDI control law will be comprised of two control elements. The baseline (continuous) control law is based on principle of conventional GDI approach and is established by prescribing the constraint dynamics of controlled state variables that reflect the control objectives. The control law is realized by inverting the prescribed dynamics using dynamically scaled Moore-Penrose generalized inversion. To boost the robust attributes against system nonlinearities, parametric uncertainties and external perturbations, a discontinuous control law will be augmented which is based on the concept of sliding mode principle. In discontinuous control law, the sliding mode gain is made adaptive in order to achieve improved tracking performance and chattering reduction. The closed-loop stability of resultant control law is established by introducing a positive define Lyapunov candidate function such that semi-global asymptotic attitude tracking of LFJC system is guaranteed. Rigorous computer simulations followed by experimental investigation will be performed on Quanser's LFJC system to authenticate the feasibility of proposed control approach for its application to real world problems.

Fuzzy adaptive asymptotic tracking of fractional order nonlinear systems with uncertain disturbances

<p style='text-indent:20px;'>In this article, an adaptive asymptotic tracking control scheme is proposed for fractional order nonlinear systems (FONSs) with time-varying disturbance. By introducing some well defined smooth functions and the bounded estimation approach, the effects caused by the unknown virtual control coefficients (UVCC) and unknown nonlinear functions are counteracted. For the UVCC, we only need to assume that their lower bounds are positive constants. Fuzzy logic systems (FLSs) are applied to approximate unknown nonlinear functions. Moreover, the fractional directed Lyapunov method is used to prove that the tracking error asymptotically converges to zero. Finally, an illustrative simulation example is applied to verify the superior performance of the presented control algorithms.</p>

A New Adaptive Controller for Nonlinear Systems with Uncertain Virtual Control Gains

This paper addresses the adaptive asymptotic tracking control problem for nonlinear systems whose virtual control gains are unknown nonlinear functions of system states. Only in the first step, the Nussbaum gain technique is utilized to handle the uncertain virtual control gain. In the remaining steps, virtual control gains are dealt with by constructing novel control laws without the approximation of the uncertain nonlinear functions and external disturbances by neural networks or fuzzy logic. New adaptive laws are defined to compensate for unknown virtual control gains, uncertain parameters, and external disturbances. Finally, an adaptive tracking controller is designed and applied to the control of a 3‐order robot system, which guarantees the boundedness of all the signals in the closed‐loop system and asymptotic stability of the tracking error.

Exponential Stability With RISE Controllers

A class of continuous robust controllers termed Robust Integral of the Sign of the Error (RISE) have been published over the past two decades as a means to yield asymptotic tracking error convergence and asymptotic identification of time-varying uncertainties, for classes of nonlinear systems that are subject to sufficiently smooth bounded exogenous disturbances and/or modeling uncertainties. Despite the wide application of RISE-based techniques, an open question that has eluded researchers during this time-span is whether the asymptotic tracking error convergence is also uniform or exponential. This question has remained open due to certain limitations in the traditional construction of a Lyapunov function for RISE-based error systems. In this letter, new insights on the construction of a Lyapunov function are used that result in an exponential stability result for RISE-based controllers. As an outcome of this breakthrough, the inherent learning capability of RISE-based controllers is shown to yield exponential identification of state-dependent disturbances/uncertainty.

Lyapunov-Derived Control and Adaptive Update Laws for Inner and Outer Layer Weights of a Deep Neural Network

Lyapunov-based real-time update laws are well-known for neural network (NN)-based adaptive controllers that control nonlinear dynamical systems using single-hidden-layer NNs. However, developing real-time weight update laws for deep NNs (DNNs) remains an open question. This letter presents the first result with Lyapunov-based real-time weight adaptation laws for each layer of a feedforward DNN-based control architecture, with stability guarantees. Additionally, the developed method allows nonsmooth activation functions to be used in the DNN to facilitate improved transient performance. A nonsmooth Lyapunov-based stability analysis proves global asymptotic tracking error convergence. Simulation results are provided for a nonlinear system using DNNs with leaky rectified linear unit (LReLU) and hyperbolic tangent activation functions to demonstrate the efficacy of the developed method.

Asymptotic Tracking Control on Profiling Posture of Sprayer Boom Systems With Position Uncertainties

An adaptive and robust backstepping method based on wavelet network approximation was proposed to solve the problems of load variation, unmodeled uncertainties, physical parameter perturbation and external disturbance in the sprayer boom profiling system. Firstly, a complete mathematical model for the boom system with uncertainties, unknowns and nonlinear terms was established and transformed into a state space form with strict feedback. Secondly, the designed wavelet primitive was used to construct the neural network, to approximate the virtual equivalent control part of the backstepping method under the condition that the optimal error is bounded. The adaptive update law was selected to estimate the unknown parameters. The robust compensation term was introduced to reduce the adverse effect of the composite interference on the system. The input command signal order requirement was reduced. Finally, suitable functions were constructed by means of the Lyapunov stability theory, to prove that the position tracking error of the closed-loop system asymptotically converges to the origin. The simulation results show that, the proposed control method can realize the rapid maneuver adjustment of the sprayer boom position and posture, and effectively enhance the robust stability and control accuracy of the boom system.

Adaptive Actuator Failure Compensation on the Basis of Contraction Metrics

This letter develops an adaptive actuator failure compensation method for nonlinear systems with unmatched parametric uncertainty based on contraction metrics. The proposed method, which is constructed by benefiting from the recent achievements on contraction metrics based adaptive control techniques, ensures the closed-loop stability and asymptotic tracking of the desired trajectory in the presence of actuator failures. In particular, a sufficient convex condition is derived for constructing a valid metric, by which a quadratic program-based controller is obtained to determine the inputs of the actuators. The introduced method is more general than the common adaptive actuator failure compensation methods, as it does not require the system to have an identical relative degree for all inputs and be transformable into the parametric strick-feedback or feedback linearization form. Besides, it can be enriched with learning-based algorithms and common robust modifications. Simulation results are presented to verify the effectiveness of the proposed controller.

Stability analysis of pressure and penetration rate in rotary drilling system

<p style='text-indent:20px;'>The purpose of this paper is to stabilize the annular pressure profile throughout the well bore continuously while drilling. A new nonlinear dynamical system is developed and a controller is designed to stabilize the annular pressure and achieve asymptotic tracking by applying feedback control of the main pumps. Hence, the paper studies the control design for the well known Managed Pressure Drilling system (MPD). MPD provides a closed loop drilling process in which pore pressure, formation fracture pressure, and bottomhole pressure are balanced and managed at surface. Although, responses must provide a solution for critical downhole pressures to preserve drilling efficiency and safety. Our MPD scheme is elaborated in reference to a nontrivial backstepping control procedure, and the effectiveness of the proposed control laws is shown by simulations.</p>

Adaptive Decentralized Asymptotic Tracking Control for Large-Scale Nonlinear Systems With Unknown Strong Interconnections

An adaptive decentralized asymptotic tracking control scheme is developed in this paper for a class of large-scale nonlinear systems with unknown strong interconnections, unknown time-varying parameters, and disturbances. First, by employing the intrinsic properties of Gaussian functions for the interconnection terms for the first time, all extra signals in the framework of decentralized control are filtered out, thereby removing all additional assumptions imposed on the interconnections, such as upper bounding functions and matching conditions. Second, by introducing two integral bounded functions, asymptotic tracking control is realized. Moreover, the nonlinear filters with the compensation terms are introduced to circumvent the issue of “explosion of complexity”. It is shown that all the closed-loop signals are bounded and the tracking errors converge to zero asymptotically. In the end, a simulation example is carried out to demonstrate the effectiveness of the proposed approach.

Integral Reinforcement Learning for Tracking in a Class of Partially Unknown Linear Systems With Output Constraints and External Disturbances

In this paper, the H∞ tracking control problem of partially unknown linear systems with output constraints and disturbance is studied by the reinforcement learning (RL) method. Firstly, an augmented system is established based on the reference trajectory dynamics and target system dynamics, and a special cost function is established to realize asymptotic tracking. In addition, the barrier function (BF) is used to transform the augmented system, and the output constraints is realized simultaneously by minimizing the quadratic cost function of the transformed system. Using only the obtained data and part of the system dynamics, the optimal control strategy and the worst disturbance strategy are obtained by using the integral reinforcement learning (IRL). Rigorous stability analysis shows that the proposed method can make the trajectory of system states converge, and the output of the control strategy can make the tracking error asymptotically stable. Finally, a simulation example is conducted to verify the effectiveness of the proposed algorithm.

Ultra-Local Model-Based Intelligent Robust Control of PKMs: Theory and Simulations

In this paper, we propose a novel Intelligent Robust Control (IRC) suitable for controlling highly nonlinear Multiple-Input-Multiple-Output (MIMO) systems. The proposed IRC scheme takes advantage of the Robust Integral of the Sign of the Error (RISE) control law and Model-Free Control (MFC). The MFC scheme is mainly composed of: (i) a nonlinear function estimated from an ultra-local model representing the input-output behavior of the system, (ii) the νth derivative of the reference trajectory as a feedforward term, and (iii) a feedback control term. MFC is characterized by its simple concept and its ability to compensate for the modeled and unmodeled system dynamics through its nonlinear compensation term. The proposed IRC approach consists of redesigning the feedback term of MFC scheme based on RISE feedback law to further improve its robustness against external disturbances and to guarantee a semi-global asymptotic tracking despite the presence of disturbances and uncertainties. Numerical simulations under different operating conditions have been conducted on T3KR parallel manipulator, in a pick-and-throw task, to validate the relevance of the proposed IRC strategy. The comparison with a model-based feedforward RISE and a feedforward super-twisting sliding mode control, by exploiting different performance indices, confirms the superiority of the proposed IRC approach.

Distributed Consensus Control of Networked Robotic Systems With Dynamic Leader Under Time-Varying Delays and Denial-of-Service Attacks

Consensus of multi-agent systems has a wide range of applications, such as coordination, cooperative transportation, and synchronization. This study investigates consensus control in the task space of multi-robotic systems modeled using Euler-Lagrange equations with a moving leading under time-varying delays and denial-of-service (DoS) attacks, respectively. Two control algorithms are studied when networked robotic systems are interconnected over directed spanning trees with a constant-velocity leader as the root. First, we use the Lyapunov-Razumikhin theorem to address a condition of the maximum time delay for nonuniform time-delay communication, thereby ensuring asymptotic tracking of the network. Second, we propose a novel resilient control framework to deal with DoS attacks. Note that designing the control algorithm for DoS attacks is difficult because the attacks cause signal discontinuity. The proposed resilient controller can release the DoS duration condition, which is a common requirement in many related works. It is verified that the consensus errors converge to the origin under the proposed control frameworks. We present experimental results to demonstrate the effectiveness of the proposed controllers.

Real-Time Modular Deep Neural Network-Based Adaptive Control of Nonlinear Systems

A real-time deep neural network (DNN) adaptive control architecture is developed for uncertain control-affine nonlinear systems to track a time-varying desired trajectory. A Lyapunov-based analysis is used to develop adaptation laws for the output-layer weights and develop constraints for inner-layer weight adaptation laws. Unlike existing works in neural network and DNN-based control, the developed method establishes a framework to simultaneously update the weights of multiple layers for a DNN of arbitrary depth in real-time. The real-time controller and weight update laws enable the system to track a time-varying trajectory while compensating for unknown drift dynamics and parametric DNN uncertainties. A nonsmooth Lyapunov-based analysis is used to guarantee semi-global asymptotic tracking. Comparative numerical simulation results are included to demonstrate the efficacy of the developed method.

Distributed Continuous-Time Algorithms for Time-Varying Constrained Convex Optimization

This paper is devoted to the distributed continuous-time optimization problems with time-varying objective functions and time-varying constraints. Different from most studied distributed optimization problems with time-invariant objective functions and constraints, the optimal solutions in this paper are time varying and form a trajectory. First, for the case where there exist only time-varying nonlinear inequality constraints, we present a distributed control algorithm that consists of a sliding-mode consensus part and a Hessian-based optimization part coupled with the log-barrier penalty functions. The algorithm can guarantee the asymptotical tracking of the optimal solution with a zero tracking error. Second, we extend the previous result to the case where there exist not only time-varying nonlinear inequality constraints but also linear equality constraints. An extended algorithm is presented, where quadratic penalty functions are introduced to account for the equality constraints and an adaptive control gain is designed to remove the restriction on knowing the upper bounds on certain information. The asymptotical convergence of the extended algorithm to the vicinity of the optimal solution is studied under suitable assumptions. The effectiveness of the proposed algorithms is illustrated in simulation. In addition, one proposed algorithm is applied to a multi-robot multi-target navigation problem with experimental demonstration on a multi-crazyflie platform to validate the theoretical results.

Neural network-based robust integral error sign control for servo motor systems with enhanced disturbance rejection performance

Uncertain dynamics and unknown time-varying disturbances always exist in servo systems and deteriorate tracking accuracy significantly. To tackle the problem, this paper presents a novel adaptive robust control scheme based on neural networks and the robust integral of the sign of the error (RISE) method. In the proposed scheme, a new neural network compensator is developed, where a reference-driven neural network and an error-driven neural network are employed to compensate for uncertain system dynamics and unknown time-varying disturbances, respectively. And an RISE-based robust feedback controller is designed to suppress uncompensated dynamics. Asymptotic tracking control of the servo system with uncertain dynamics and unknown time-varying disturbances is guaranteed by using the Lyapunov theory. Comparative experiments and simulations with different reference signals and various types of external disturbances were conducted based on a linear motor-driven stage. Experimental and simulational results verify the superior tracking performance and powerful disturbance rejection ability of the proposed method.

Robust indirect adaptive control of electromechanical servo systems with uncertain time-varying parameters

This paper focuses on high-accuracy motion control for electromechanical servo systems with time-varying parameter estimation. A robust indirect adaptive controller based on a nonlinear X-swapping technique is proposed to achieve the online estimation of the time-varying parameters. The time-varying parametric uncertainties can be well compensated via the adaptive feedforward cancellation technique, and the residual model uncertainties are suppressed by a nonlinear robust control law. The Lyapunov-based stability analysis illustrates that the proposed method obtains a prescribed estimating result and an excellent asymptotic tracking performance. Extensive experimental results are acquired to affirm the applicability of the proposed control strategy.

High Precision Motion Control of Electro-Mechanical Launching Platform with Modeling Uncertainties: A New Integrated Error Constraint Asymptotic Design

For the demands of a high precision motion control of an uncertain electro-mechanical launching platform, a novel integrated error constraint asymptotic control in the presence of parametric uncertainties and uncertain disturbance is proposed, of which the barrier function method and a continuous asymptotic control design are integrated for the first time. The former technique can effectively avoid excessive tracking errors at the transient phase, which is caused by the disturbance and the large uncertain system parameters’ deviation between the initial estimated value and the actual value, by selecting a proper barrier threshold, while the latter technique can handle the uncertain disturbance to achieve asymptotic tracking. A rigorous stability analysis is given to illustrate the theoretical performance. In addition, as a supplementary measure, repetitive control is employed to estimate and compensate the possible periodic-like disturbance under certain conditions. Two experimental cases on a prototype of a launching platform demonstrate the effectiveness of the proposed controller.