Nonaffine Systems Research Articles

Approximate Optimal Tracking Control of Nondifferentiable Signals for a Class of Continuous-Time Nonlinear Systems.

In this article, for a class of continuous-time nonlinear nonaffine systems with unknown dynamics, a robust approximate optimal tracking controller (RAOTC) is proposed in the framework of adaptive dynamic programming (ADP). The distinguishing contribution of this article is that a new Lyapunov function is constructed, by using which the derivative information of tracking errors is not required in computing its time derivative along with the solution of the closed-loop system. Thus, the proposed method can make the system states follow nondifferentiable reference signals, which removes the common assumption that the reference signals have to be continuous for tracking control of continuous-time nonlinear systems in the literature. The theoretical analysis, simulation, and application results well illustrate the effectiveness and superiority of the proposed method.

Quantitative Data-Driven Adaptive Iterative Learning Control: From Trajectory Tracking to Point-to-Point Tracking.

This article reconsiders the data quantization problem in iterative learning control (ILC) for nonlinear nonaffine systems from four aspects: 1) use of available additional control knowledge; 2) different tracking tasks; 3) adaptation to uncertainties; and 4) data-driven design and analysis framework. An iterative linear data model (iLDM) is established first to represent the nonlinear nonaffine system for subsequent control algorithm design and analysis under a data-driven framework. A quantitative data-driven adaptive ILC (QDDAILC) is then developed using quantized tracking errors based on the nonlifted iLDM and, thus, additional available input information from previous time instants can be utilized to improve control performance. The parameter estimation derived from an adaptive updating law makes the learning gain of the QDDAILC adjustable, therefore improving the robustness to uncertainties. Due to the coupled dynamics among inputs and tracking errors, a new double-dynamics analysis method is introduced besides the contraction mapping principle to show error convergence. A quantized data-driven adaptive point-to-point ILC (QDDAPTPILC) is further presented using partial quantized measurements at the specified instants for multi-intermediate-point tracking. Simulation examples verify theoretical results and illustrate that the QDDAPTPILC outperforms the QDDAILC for multi-intermediate-point tracking tasks because it removes the unnecessary constraints.

Nonlinear Observer Design for Load Torque Estimation of Induction Motors

The paper introduces a design of a nonlinear observer for the estimation of load torque of induction motors based on the output of the stator currents. The observer is built in the synchronous reference frame d-q with three main steps. First, the nonlinear observer for the class of non-affine control systems is reviewed. Second, the induction motor model is written in an appropriate form as a non-affine control system with canonical form for the design of the nonlinear observer. The system state is composed of the stator currents, speed and load torque. Based on the stator currents, the rotor flux leakages are estimated via an open loop and used for the load torque observer. The observer is designed with a constant gain which is calculated by considering the trajectory of the rotor flux leakages. Both the load torque and rotor speed can be estimated by using the proposed observer. Finally, simulation and experiment are carried to validate the effectiveness and robustness of the proposed observer.

Convergence Analysis of Sampled-Data ILC for Locally Lipschitz Continuous Nonlinear Nonaffine Systems With Nonrepetitive Uncertainties

This article investigates a challenging open problem of convergence analysis of sampled-data iterative learning control for locally Lipschitz continuous (LLC) nonlinear nonaffine continuous-time systems. The restrictive repetitive conditions are relaxed by taking the LLC nonlinear nonaffine systems with iteratively variant initial states, reference trajectories, and exogenous disturbances into consideration. A new convergence analysis method is proposed by incorporating the mathematical induction with the contraction mapping principle (MI-CMP) to prove the boundedness of system variables, and bounded convergence of tracking error for a finite iteration as the first step. Then, the mathematical proof by contradiction is utilized along with the MI-CMP to further prove the robust convergence of the tracking error in any iteration to an error bound related to the bounds of the iteration-varying uncertainties and the sampling period. It is also shown that the tracking error converges to a smaller bound if the iteration-varying uncertainties disappear, even to zero if the sampling period approaches zero. In addition, an algorithm is provided to optimize the learning operator under the convergence condition. Theoretical results are further verified by simulations.

Observer-based data-driven iterative learning control

By considering non-repetitive uncertainties, an observer-based data-driven iterative learning control (ObDDILC) is proposed in this article for non-linear non-affine systems in the existence of non-repeatable disturbances, random initial values, and input constraints. Aiming to addressing the non-affine and non-linear characteristics of the systems, a linear data model (LDM) is constructed in iteration domain without introducing any physical interpretation but only for the purpose of the subsequent algorithm design and analysis. The iteration-varying initial values and disturbances are incorporated as a total non-repetitive uncertainty of the LDM. Both an iterative learning observer and a parameter estimator are proposed to address the total non-repetitive uncertainties and unknown parameters of the established LDM, respectively. Then, an observer-based learning control law is developed using the estimated output to compensate the impact of the non-repetitive uncertainties on the control performance, where a saturated function is employed to deal with the input constraints. The convergence of proposed ObDDILC is proved by using contraction mapping as the basic tool. All of the algorithms are designed and analysed without dependence of any model information except I/O data. The theoretical results are tested by simulations.

Adaptive Regulation of Discrete-Time Nonaffine Systems With Parametric Uncertainty

This article investigates the problem of global adaptive stabilization for discrete-time nonlinearly parameterized systems with nonaffine inputs. Using the notion of passivity and the idea of small bounded feedback, we present a discrete-time adaptive controller that globally adaptively stabilizes a class of nonaffine systems with unknown parameters, under the assumptions that the unforced dynamics are stable and the controllabilitylike rank conditions hold. The latter is shown to be characterizable by the vector fields $f(x,0,\theta)$ and $\frac{\partial f}{\partial u}(x, 0, \theta)$ , together with their “Lie derivatives” and “Lie brackets” in discrete time.

Design of a nonsingular adaptive fuzzy backstepping controller for electrostatically actuated microplates

This paper adopts some alternative strategies to design a nonlinear controller for double electrostatically actuated microplates. The novel design is carried out to solve the singularity problem reported in many articles due to the use of the Taylor expansion to simplify the electrostatic force. The nonlinear governing partial differential equation is converted to the modal equation using the Galerkin method. Then, based on the Lyapunov stability criterion, a fuzzy backstepping controller facilitated by prescribed performance functions is applied to the non-affine system to extend the travel range beyond the pull-in region and capture the structural and nonstructural uncertainties that exist in the practical systems. The present work also aims to bring satisfactory transient and steady-state performance indices to the system. Moreover, unknown time-varying delays as the indispensable part of practical systems are considered in the proposed control scheme to suppress the delays occurring in the measurement of the states by constructing Lyapunov–Krasovskii function. The accuracy of the modal equation in both the static and dynamic analysis is verified through a meshless method as a direct solution of the partial differential equation. The proposed controller guarantees that all the closed-loop signals are semi-globally, uniformly ultimately bounded, and the error evolves within the decaying prescribed bounds. Finally, the proposed controller demonstrates its feasibility to extend the travel range within and beyond the pull-in range despite the unknown uncertainties and time-varying delays which exist in the system.

Voltage Regulation of DC-Microgrid With PV and Battery

This paper studies voltage regulation and maximum power point tracking (MPPT) control for a DC-microgrid that includes a photovoltaic (PV) panel, battery, constant resistance and constant power loads. A dynamic model of the DC-microgrid system described by a multi-input and multi-output nonlinear system with non-affine inputs is derived. Based on this nonlinear dynamic model, we use output regulation theory to design a local state feedback controller that regulates voltages to prescribe set points and maximizes PV power output. Global set point regulation is also studied using passive system theory for non-affine systems and Lyapunov stability theory. The effectiveness of the proposed control schemes is investigated using simulations when changes in both illumination and load occur.

A new adaptive neural control scheme for hypersonic vehicle with actuators multiple constraints

A new adaptive neural control method, with the actuators multiple constraints of amplitude and rate into consideration, is proposed in this paper for the flexible air-breathing hypersonic vehicle (AHV). In order to better reflect the characteristics of the actual AHV model, we regard the AHV as a completely unknown non-affine system in the control law design process, which is different from the existing AHV control methods, thus ensuring the reliability of the designed control law. On the basis of the implicit function theorem, the radial basis function neural network (RBFNN) is introduced to approximate the model. Meanwhile, the minimum learning parameter algorithm is adopted to adaptively adjust the weight vector of RBFNN, then the design of the ideal control law is completed. When the amplitude and rate of the actuator are saturated, the designed novel auxiliary error compensation system is used to effectively compensate for the ideal control law, and the stability of the closed-loop control system is proved via the Lyapunov stability theory. In addition, to avoid the “explosion of terms” problem in the control law design process, the finite-time-convergence-differentiator is introduced to accurately estimate the differential signal. Finally, the effectiveness of the control method designed in this paper is verified by simulation.

Observer-Based Sampled-Data Model-Free Adaptive Control for Continuous-Time Nonlinear Nonaffine Systems With Input Rate Constraints

A sampled-data model-free adaptive control (SMFAC) strategy is proposed for continuous-time nonlinear nonaffine systems with input rate constraints. By using differential and integral mean value theorems as two basic mathematic tools, a sampled-data local dynamic linearization method is proposed at first to transform the continuous-time nonlinear nonaffine model into a sampled-data nonlinear affine I/O model, including a linear parametric term affined to the control input and a nonlinear uncertainty term. On this basis, we consequently propose an observer-based SMFAC (ObSMFAC) scheme, including a sampled-data parameter estimator to estimate the unknown partial derivatives and a sampled-data observer to estimate the residual nonlinear uncertainty, respectively. Note that the sampling period is incorporated explicitly in the proposed ObSMFAC which enhances the control performance by reducing its negative influence on the system stability. The constraint on the input rate is also considered in the control law as the transition condition of the input updating algorithms. The convergence of the proposed ObSMFAC is proved by using the contraction mapping principle. The simulation study demonstrates the theoretical results.

Adaptive recursive sliding-mode dynamic surface and its event-triggered control of uncertain non-affine systems

An adaptive neural network control strategy and its event-triggered controller are designed for non-affine pure-feedback uncertain systems based on nonlinear gains and recursive sliding-mode dynamic surfaces. A kind of nonlinear gain functions is introduced into the traditional framework of dynamic surface control (DSC) with recursive sliding-mode to make a compromise between control accuracy and transient performance. Radial basis function (RBF) neural networks (NNs) are adopted to approximate unknown functions at each step, and novel adaptive update laws with leakage terms of σ-modification is constructed. To reduce the action number of the actuator, an extended control strategy in an event-triggered manner is proposed. By the Lyapunov function, it is proven that both of the two control strategies can force the tracking error arbitrarily small and guarantee all the signals in the closed-loop system uniformly ultimately bounded. Finally, simulation results are provided to verify the effectiveness of the proposed control strategy.

Characteristic Model-Based Adaptive Control for a Class of MIMO Uncertain Nonaffine Nonlinear Systems Governed by Differential Equations

This paper addresses the difficulty of designing a controller for a class of multi-input multi-output uncertain nonaffine nonlinear systems governed by differential equations. We first derive the first-order characteristic model composed of a linear time-varying uncertain system for such nonaffine systems and then design an adaptive controller based on this first-order characteristic model for position tracking control. The designed controller exhibits a simple structure that can effectively avoid the controller singularity problem. The stability of the closed-loop system is analyzed using the Lyapunov method. The effectiveness of our proposed method is validated with a numerical example.

Distributed output‐feedback finite‐time tracking control of nonaffine nonlinear leader‐follower multiagent systems

SummaryThe distributed output‐feedback tracking control for a class of networked multiagents in nonaffine pure‐feedback form is investigated in this article. By introducing a low‐pass filter and some auxiliary variables, we first transform the nonaffine system into the affine form. Then, the finite‐time observer is designed to estimate the states of the newly derived affine system. By applying the fraction dynamic surface control approach and the neural network‐based approximation technique, the distributed output‐feedback control laws are proposed and it is proved that the tracking errors converge to an arbitrarily small bound around zero in finite time. Finally, some simulation examples are provided to confirm the effectiveness of the developed method.

Design of Adaptive Robust Controller for Second-Order Non-affine Systems with Input Saturation

In this paper, two techniques are introduced to transform an uncertain second-order non-affine system under input saturation to an equivalent affine model. First, this affine model is obtained using the linearization technique with respect to the system inputs. In the case of the system being non-differentiable or its derivative with respect to the input being zero, this technique cannot be used. Then, to cope with this problem, an average dynamical model is suggested using the Filippov’s model and the pulse width modulation. Afterward, an adaptive sliding mode controller is designed to overcome the approximation errors appeared in this transformation and inherent system uncertainties. Stability analysis is also derived using the Lyapunov theory. Ultimately, the capability of the proposed method is shown through MATLAB simulations.

Deterministic learning from neural control for uncertain nonlinear pure‐feedback systems by output feedback

SummaryThe essence of intelligence lies in the acquisition/learning and utilization of knowledge. However, how to implement learning in dynamical environments for nonlinear systems is a challenging issue. This article investigates the deterministic learning (DL) control problem for uncertain pure‐feedback systems by output feedback, which achieves the human‐like learning and control in a simple way. To reduce the complexity of control design and analysis, first, by combining an appropriate system transformation, the original pure‐feedback system is transformed into a simple normal nonaffine system. An observer is then introduced to estimate the transformed system states. Based on the backstepping and dynamic surface control techniques, a simple adaptive neural control scheme is first developed to guarantee the finite time convergence of the tracking error using only one neural network (NN) approximator. Second, through DL, the exponential convergence of the NN weights is obtained with the satisfaction of partial persistent excitation condition. Thus, locally accurate approximation/learning of the transformed unknown system dynamics is achieved and stored as constant NNs. Finally, by utilizing the stored knowledge, an experience‐based controller is constructed and a novel learning control scheme is further proposed to improve the control performance without any further adaptation online for the estimate neural weights. Simulation results have been given to illustrate that the proposed scheme not only can learn and memorize knowledge like humans but also can utilize experience to achieve superior control performance.

Enhanced data-driven optimal iterative learning control for nonlinear non-affine discrete-time systems with iterative sliding-mode surface

In this work, an enhanced data-driven optimal iterative learning control (eDDOILC) is proposed for nonlinear nonaffine systems where a new iterative sliding mode surface (ISMS) is designed to replace the traditional tracking error in the controller design and analysis. It is the first time to extend the sliding mode surface to the iteration domain for systems operate repetitively over a finite time interval. By virtual of the new designed ISMS, the control design becomes more flexible where both the time and the iteration dynamics can be taken into account. Before proceeding to the controller design, an iterative dynamic linear data model is built between two consecutive iterations to formulate the linear input-output data relationship of the repetitive nonlinear nonaffine discrete-time system. The linear data model is virtual and does not have any physical meanings, which is very different to the traditional mechanism mathematical model. In the sequel, the eDDOILC is proposed by designing an objective function with respect to the proposed two-dimensional ISMS. Rigorous proof is provided to show the convergence of the proposed eDDOILC method. Furthermore, the results have been extended to a multiple-input multiple-output (MIMO) nonaffine nonlinear discrete-time repetitive system. In general, the proposed eDDOILC is data-driven where no explicit model information is included. It is illustrated that the presented eDDOILC is effective when applied to the nonlinear nonaffine uncertain systems.

Event-triggered constrained control with DHP implementation for nonaffine discrete-time systems

This paper proposes an event-based near-optimal control algorithm for nonaffine discrete-time systems with constrained inputs. The method is derived from the dual heuristic dynamic programming (DHP) technique. The challenge caused by saturating actuators is overcome by using a nonquadratic performance index. Then, the event-based control technique is used to decrease the amount of computation. Meanwhile, the stability analysis is provided. It illustrates that the proposed event-based method can asymptotically stabilize the nonaffine systems by using the Lyapunov method. Furthermore, the stability conditions and the design process of the event-based controller are established. The event-based DHP algorithm is implemented by constructing three neural networks, namely, the model network, the critic network, and the action network. Finally, simulation studies are conducted to demonstrate the applicability and the performance of the proposed method.

Adaptive neural network control for a class of MIMO non-affine uncertain systems with input dead-zone nonlinearity and external disturbance

This paper studies an adaptive tracking control for a class of multi-input multi-output (MIMO) non-affine nonlinear systems, with input dead-zone nonlinearity and external disturbances. By using the mean-value theorem, the system model is transformed into an affine form so as the difficulty in controlling non-affine systems is overcome. In the proposed control design, neural networks (NNs) are used to approximate the unknown nonlinearities based on their universal approximation properties. To compensate for approximation errors and external disturbances, an adaptive robust control term is introduced. In comparison with existing approaches, the structure of the designed controller is considerably simpler, and can handle a wider range of nonlinear systems. The stability of the closed-loop system is investigated by using Lyapunov theory. The simulation results illustrate the proposed method.

Adaptive Neural Event-Triggered Control of MIMO Pure-Feedback Systems With Asymmetric Output Constraints and Unmodeled Dynamics

In this paper, the issue of adaptive neural event-triggered control (ETC) is studied for uncertain block-structure multi-input multi-output (MIMO) constrained non-affine nonlinear systems with unmodeled dynamics. A dynamic signal produced by the auxiliary system based on the property of unmodeled dynamics is employed to solve the dynamical disturbances. The unknown continuous function obtained at each step of recursion is estimated by using radial basis function neural networks (RBFNNs). Utilizing logarithmic function as an invertible mapping, the uncertain constrained MIMO non-affine system is changed into a novel unconstrained block-structure MIMO nonaffine system. Using improved dynamic surface control (DSC) strategy, adaptive event-triggered control scheme is developed for the transformed non-affine system based on relative threshold mechanism. According to the Lyapunov method, all the signals in the closed-loop system are shown to be semi-globally uniformly ultimately bounded (SGUUB). Output constraint requirements are not triggered, and Zeno behavior is avoided. A constrained pure-feedback system and a kind of 2-DOF flexible manipulator system are used to illustrate the theoretical findings.

Data-Based Nonaffine Optimal Tracking Control Using Iterative DHP Approach

Abstract In this paper, a data-based optimal tracking control approach is developed by involving the iterative dual heuristic dynamic programming algorithm for nonaffine systems. In order to gain the steady control corresponding to the desired trajectory, a novel strategy is established with regard to the unknown system function. Then, according to the iterative adaptive dynamic programming algorithm, the updating formula of the costate function and the new optimal control policy for unknown nonaffine systems are provided to solve the optimal tracking control problem. Moreover, three neural networks are used to facilitate the implementation of the proposed algorithm. In order to improve the accuracy of the steady control corresponding to the desired trajectory, we employ a model network to directly approximate the unknown system function instead of the error dynamics. Finally, the effectiveness of the proposed method is demonstrated through a simulation example.