Affine Nonlinear Systems Research Articles

Input-to-state Stability of Nonlinear Positive Systems

In this paper, input-to-state stability (ISS), as a useful tool for robust analysis, is first applied to continuous-time and discrete-time nonlinear positive systems. For continuous-time and discrete-time positive systems, some new definitions of ISS are introduced. Different from the usual ISS definitions for nonlinear systems, our ISS definitions can fully reflect the positiveness requirements of states and inputs of the positive systems. By introducing the max-separable ISS Lyapunov functions, some ISS criterions are given for general nonlinear positive systems. Based on that, the ISS criterions for linear positive systems and affine nonlinear homogeneous systems are given. Through them, the ISS properties can be judged directly from the differential and algebraic characteristics of the systems. Simulation examples verify the validity of our results.

A New Hamilton–Jacobi Differential Game Framework for Nonlinear Estimation and Output Feedback Control

In this paper, we develop a new framework for designing state estimators/filters and output measurement feedback controllers for affine nonlinear systems in state space. The problems are formulated as zero-sum differential games, and sufficient conditions for their solvability are given in terms of Hamilton–Jacobi–Isaacs equations (HJIEs). These HJIEs are new, in the sense that they are both state-dependent and measurement output dependent. This allows for the filter and observer gains to be optimized over all possible nonlinear gains. Examples and simulation results are also presented to support the theory.

Nonlinear Model Predictive Control for Mobile Robot Using Varying-Parameter Convergent Differential Neural Network

The mobile robot kinematic model is a nonlinear affine system, which is constrained by velocity and acceleration limits. Therefore, the traditional control methods may not solve the tracking problem because of the physical constraint. In this paper, we present the nonlinear model predictive control (NMPC) algorithm to track the desired trajectory based on neural-dynamic optimization. In the proposed algorithm, the NMPC scheme utilizes a new neural network named the varying-parameter convergent differential neural network (VPCDNN) which is a Hopfifield-neural network structure with respect to the differential equation theory to solve the quadratic programming (QP) problem. The new network structure converges to the global optimal solution and it is more efficient than traditional numerical methods. In the simulation, we verify that the proposed method is able to successfully track reference trajectories with a two-wheel mobile robot. The experimental validation has been conducted in simulation and the results show that the proposed method is able to precisely track the trajectory maintaining a high robustness based on the VPCDNN solver.

Conditions for fixed-time stability and stabilization of continuous autonomous systems

This work presents Lyapunov analysis conditions for fixed-time stability, a property where all the system’s trajectories converge exactly to zero in a finite amount of time that is independent of the system’s initial condition. Necessary and sufficient conditions for fixed-time stability without taking into account the regularity of the settling-time function are presented first. Next, a characterization for fixed-time stability with continuous settling-time functionis introduced. A particular form of the characterizing functions follows, it allows to establish more constructive conditions and in order to obtain a converse result, the concept of complete fixed-time stability is introduced. A set of academic examples and an example of allocation of mobile agents illustrate the given concepts. Finally, a sufficient condition for fixed-time stabilization of nonlinear affine systems is obtained.

Robust $${{\cal H}_\infty}$$ Control for Switched Nonlinear Port-controlled Hamiltonian Systems

In the paper, the robust $${{\cal H}_\infty}$$ control for switched nonlinear port-controlled Hamiltonian (SNPCH) systems with external disturbances is investigated via the multiple Lyapunov functions method. Under two cases that the SNPCH system is subject to actuator saturation (AS) and the SNPCH system is without AS, a design approach for the switching robust $${{\cal H}_\infty}$$ feedback controller is developed to attenuate the external disturbances. The sufficient conditions are established for $${{\cal H}_\infty}$$ control of the corresponding closed-loop systems based on Hamilton-Jacobi inequality techniques. Furthermore, via the Hamiltonian realization method, the obtained results are applied to solve the $${{\cal H}_\infty}$$ control for switched nonlinear affine systems with external disturbances. Finally, two numerical examples are presented to illustrate the effectiveness of the proposed $${{\cal H}_\infty}$$ control methods.

An Analysis of IRL-Based Optimal Tracking Control of Unknown Nonlinear Systems with Constrained Input

In this paper, a comparison is addressed between two methods, that is, the optimal tracking control methods of unknown nonlinear systems with and without constrained input. Firstly, the optimal tracking problem for a class of affine nonlinear system is formulated. The tracking cost functions are also defined, both for the two methods. The two methods are proved to be equivalent as the actuator bound is large enough. Integral reinforcement learning (IRL) algorithm is employed to solve the optimal tracking problem by using only system data. To facilitate the implementation of the IRL algorithm, the actor-critic neural network technique and the least squares method are employed in approximating the unknown weights iteratively. In the simulation, a detailed comparison is given to demonstrate the relationship between the two methods in the aspects of control input and tracking cost value.

Design of Modified Repetitive Controller for T–S Fuzzy Systems

A repetitive controller contains a pure-delay positive-feedback loop that makes it difficult to stabilize a strictly proper system. A low-pass filter is inserted in a repetitive controller to relax the stability condition of the modified repetitive-control system at the cost of degrading the tracking performance. In this study, a modified repetitive-control approach is developed, which reaches a balance between the stability and tracking performance for a class of affine nonlinear systems based on the Takagi–Sugeno fuzzy model. First, a 2D model is established to adjust continuous control and discrete learning actions preferentially induced by exploiting the 2D property in a repetitive-control process. Then, the Lyapunov stability theory and 2D system theory are used to derive a sufficient stability condition in the form of linear matrix inequalities to design parallel-distributed-compensation-based state-feedback controllers. Finally, an application-oriented example is used, and a comparison is performed to show that an extra variable is introduced such that the developed method has a better tracking performance.

Data-driven finite-horizon optimal tracking control scheme for completely unknown discrete-time nonlinear systems

This paper proposes finite-horizon optimal tracking control approach based on data for completely unknown discrete-time nonlinear affine systems. First, the identifier is designed by input and output data, which is used to identify system function and system model. And based on tracking error, the system function is transformed to the augmentation system with finite-time optimal performance. In finite time, by minimizing the performance index function, the iterative approximate dynamic programming (ADP) is utilized to solve Hamilton–Jacobi–Bellman (HJB) equation. The idea is carried by the policy iterative (PI) based on the model neural network, which makes the iterative control of the augmentation system available at the each step. At the same time, the action neural network is utilized to acquire the approximate optimal tracking control law and the critic neural network is used for approximating the optimal performance index function for the augmentation system. Afterwards, the paper show the analysis process that the convergence and stability for the iterative ADP algorithm and the weight estimation errors based on the PI, respectively. The end of the paper, a simulation example is applied to show the theoretical results and proposed approach.

Pole Path Assignment of Constrained SISO Affine Nonlinear Systems

This paper proposes a feedback linearization control scheme for affine nonlinear systems with a constrained control signal where a fast response is required. Instead of placing the linearized system poles at exact locations, ray-like paths in the left hand plane are selected where the poles can freely vary to efficiently exploit the admissible input signal range to increase the speed of response. A stability test for the proposed method is derived via the multivariable circle criterion and the Kalman–Yakubovich–Popov lemma. Simulation results demonstrate how the method significantly increases the speed of response in spite of the input constraints compared to fixed pole placement.

Estimation of the Attraction Domain for an Affine System with Constrained Vector Control Closed by the Linearizing Feedback

Nonlinear affine systems with constrained vector control that are represented in a canonical (normal) form and are closed by feedbacks linearizing the system in a neighborhood of the origin, are considered. For the nonlinear closed-loop system, the problem is set to construct an estimate of the attraction domain of an equilibrium position. A method for constructing an estimate of the attraction domain, which is based on results of absolute stability theory, is suggested. The estimate is sought as a Cartesian product of positive invariant sets of the subsystems composing the system. In the case of ellipsoidal invariant sets, construction of the estimate reduces to solving a system of linear matrix inequalities. The discussion is illustrated by numerical examples.

A new control method for global stabilisation of translational oscillator with rotational actuator

ABSTRACTA translational oscillator with a rotational actuator (TORA) is an underactuated mechanical system with two degrees of freedom. This paper presents a new method to solve the global stabilisation control problem for this underactuated system. First, a set of suitable state variables is chosen for the TORA. It transforms the system into a cascade affine nonlinear system. Then, a positive-definite Lyapunov function is constructed based on the structural characteristics of the system. A control law is designed to guarantee that the derivative of the Lyapunov function is non-positive. Finally, the global stability of the closed-loop control system is analysed using the Lasalle invariance principle. The condition on the control parameters is presented to globally stabilise the TORA at the origin. The effectiveness and robustness of the method are verified by a numerical example.

ADP-Based Online Tracking Control of Partially Uncertain Time-Delayed Nonlinear System and Application to Wheeled Mobile Robots.

In this paper, an adaptive dynamic programming-based online adaptive tracking control algorithm is proposed to solve the tracking problem of the partial uncertain time-delayed nonlinear affine system with uncertain resistance. Using the discrete-time Hamilton-Jacobi-Bellman function, the input time-delay separation lemma, and the Lyapunov-Krasovskii functionals, the partial state and input time delay can be determined. With the approximation of the action and critic, and resistance neural networks, a near-optimal controller and appropriate adaptive laws are defined to guarantee the uniform ultimate boundedness of all signals in the target system, and the tracking error convergence to a small compact set to zero. A numerical simulation of the wheeled mobile robotic system is presented to verify the validity of the proposed method.

Optimization of sampling intervals for tracking control of nonlinear systems: A game theoretic approach

This paper presents a near optimal adaptive event-based sampling scheme for tracking control of an affine nonlinear continuous-time system. A zero-sum game approach is proposed by introducing a novel performance index. The optimal value function, i.e., the solution to the associatedHamilton–Jacobi–Issac (HJI) equation is approximated using a functional link neural network (FLNN) with event-based aperiodic state feedback information as inputs. The saddle point approximated optimal solution is employed to design the near optimal event-based control policy and the sampling condition. An impulsive weight update scheme is designed to guarantee local ultimate boundedness of the closed-loop parameters, which is analyzed via extension of Lyapunov stability theory for the impulsive hybrid dynamical systems. Zeno-freeness of the event-sampling scheme is enforced and its effect on stability is analyzed. Finally, numerical simulation results are included to corroborate the analytical design, which shows a 48.82% reduction of feedback communication and computational load.

Adaptive Fault-Tolerant Control of Uncertain Switched Nonaffine Nonlinear Systems With Actuator Faults and Time Delays

This paper addresses the adaptive fault-tolerant control (FTC) problem of uncertain switched nonaffine nonlinear systems with actuator failures and time delays. Unlike the literature employing the FTC methods, the novel nonlinear fault compensate function with adjustable parameter factor, whose norm is used for approximation with fuzzy logic systems, is constructed based on the mean value theorem. Accordingly, a standard transformation scheme from the multi-input and single-output nonaffine nonlinear system to the corresponding affine nonlinear system is first to be given; then, the designed adaptation mechanism with state-dependent switching strategy can effectively eliminate the effects of multiple time delays and actuator faults, including outage and stuck modes. Furthermore, by using the Lyapunov–Krasovskii stability analysis method, it is proved that the proposed adaptive switched controller can guarantee the states of the overall closed-loop systems to asymptotically converge to zero. Finally, the simulation results are displayed to verify the feasibility of the presented approach.

Inverse optimal control of a class of affine nonlinear systems

This paper proposes a systematic formulation of inverse optimal control (IOC) law based on a rather straightforward reduction of control Lyapunov function (CLF), applicable to a class of second-order nonlinear systems affine in the input. This method exploits the additional design degrees of freedom resulting from the non-uniqueness of the state dependent coefficient (SDC) formulation, which is widely used in pseudo-linear control techniques. The applicability of the proposed approach necessitates an apparently effortless SDC formulation satisfying an SDC matrix criterion in terms of the structure and characteristics of the state matrix, [Formula: see text]. Subsequently, a sufficient condition for the global asymptotic stability (g.a.s) of the closed-loop system is established. The SDC formulations conforming to the sufficient condition ensure the existence and determination of a smooth radially unbounded polynomial CLF of the form [Formula: see text], while offering a benevolent choice for the gain matrix [Formula: see text], in the CLF. The direct relationship between the gain matrix [Formula: see text] and state weighing matrix [Formula: see text] ensures optimization of an equivalent [Formula: see text]. This feature enables one to rightfully choose the gain matrix [Formula: see text] as per the performance requisites of the system. Finally, the application of the proposed methodology for the speed control of a permanent magnet synchronous motor validates the efficacy and design flexibility of the methodology.

Design of robust <i>H</i><SUB align="right">∞ fuzzy output feedback controller for affine nonlinear systems: fuzzy Lyapunov function approach

In this paper, we propose a new systematic approach based on non-quadratic Lyapunov function and technique of introducing slack matrices, for a class of affine nonlinear systems with disturbance. To achieve the goal, first, the affine nonlinear system is represented via Takagi-Sugeno (T-S) fuzzy bilinear model. Subsequently, the robust H∞ controller is designed based on parallel distributed compensation (PDC) scheme. Then, the stability conditions are derived in terms of linear matrix inequalities (LMIs) by utilising Lyapunov function. Moreover, some slack matrices are proposed to reduce the conservativeness of the LMI stability conditions. Finally, for illustrating the merits and verifying the effectiveness of the proposed approach, the application of an isothermal continuous stirred tank reactor (CSTR) for Van de Vusse reactor is discussed in details.

Design of robust H_ fuzzy output feedback controller for affine nonlinear systems: Fuzzy Lyapunov function approach

In this paper, we propose a new systematic approach based on non-quadratic Lyapunov function and technique of introducing slack matrices, for a class of affine nonlinear systems with disturbance. To achieve the goal, first, the affine nonlinear system is represented via Takagi-Sugeno (T-S) fuzzy bilinear model. Subsequently, the robust H∞ controller is designed based on parallel distributed compensation (PDC) scheme. Then, the stability conditions are derived in terms of linear matrix inequalities (LMIs) by utilising Lyapunov function. Moreover, some slack matrices are proposed to reduce the conservativeness of the LMI stability conditions. Finally, for illustrating the merits and verifying the effectiveness of the proposed approach, the application of an isothermal continuous stirred tank reactor (CSTR) for Van de Vusse reactor is discussed in details.

Optimal Control Method for Robot-Tracking Based on Control-Lyapunov-Function

A control Lyapunov function (CLF)-based optimal control method is proposed to solve an affine nonlinear system with input limitation for the wheeled robot. Motivated by Lyapunov stability theory, viability theory, and exact linearization and sliding mode control, the construction of the CLF uses the information of the multi-order derivatives of the system outputs. The result shows the stability depends on the controller coefficients and the input limitation. Thus, the sufficient conditions are given and proved for stable controller design. The proposed control method provides a new robust regulation map to an affine nonlinear system for high-dimensional trajectory tracking. A simulation of multi-agent wheeled robot formation is designed to illustrate the application of the proposed controller. Meanwhile, the validation and feasibility of the controller are also verified by the simulation.

Convergence Time Calculation for Supertwisting Algorithm and Application for Nonaffine Nonlinear Systems

In this study, an accurate convergence time of the supertwisting algorithm (STA) is proposed to build up a framework for nonaffine nonlinear systems’ finite‐time control. The convergence time of the STA is provided by calculating the solution of a differential equation instead of constructing Lyapunov function. Therefore, precise convergence time is presented instead of estimation of the upper bound of the algorithm’s reaching time. Regardless of affine or nonaffine nonlinear systems, supertwisting control (STC) provides a general solution based on virtual control law skill ensuring the output of the systems converges to the origin point at exact time. Benchmark tests are simulated to demonstrate the effectiveness and efficiency of the algorithm.

On two isomorphic Lie algebroids associated with feedback linearization

We present two Lie algebroids linked to the construction of the linearizing output of an input affine nonlinear system. The algorithmic development of the linearizing output proceeds inductively, and each stage has two structures, namely a codimension one foliation defined through an integrable 1-form ω, and a transversal vectorfield g to the foliation. Each integral manifold of the vectorfield g defines an equivalence class of points. Due to transversality, a leaf of the foliation is chosen to represent these equivalence classes. A Lie groupoid is defined with its base given as the particular chosen leaf and with the product induced by the pseudogroup of diffeomorphisms that preserve equivalence classes generated by the integral manifolds of g. Two Lie algebroids associated with this groupoid are then defined. The theory is illustrated with an example using polynomial automorphisms as particular cases of diffeomorphisms and shows the relation with the Jacobian conjecture.