Class Of Nonlinear Control Systems Research Articles

A Simple Linear Controller Design to Ensure Global Exponential Stability for a Class of Uncertain Fifth-Order Nonlinear Systems

This paper intends to explore the controller design problem for a class of uncertain fifth-order nonlinear control systems. Combining the theory of differential and integral inequalities, a linear controller will be proposed to promote a class of nonlinear control systems with multiple uncertainties to achieve the goal of global exponential stability. In addition, the guaranteed exponential convergence rate of such uncertain nonlinear systems will also be precisely calculated. Finally, some numerical simulation results will also be presented to verify and illustrate the correctness of this main theorem and the design process of the controller

Motion planning and stabilization of nonholonomic systems using gradient flow approximations

Nonlinear control-affine systems described by ordinary differential equations with time-varying vector fields are considered in the paper. We propose a unified control design scheme with oscillating inputs for solving the trajectory tracking and stabilization problems under the bracket-generating condition. This methodology is based on the approximation of a gradient-like dynamics by trajectories of the designed closed-loop system. As an intermediate outcome, we characterize the asymptotic behavior of solutions of the considered class of nonlinear control systems with oscillating inputs under rather general assumptions on the generating potential function. These results are applied to examples of nonholonomic trajectory tracking and obstacle avoidance.

Output tracking control design with anti-disturbance rejection for modified repetitive nonlinear control systems

ABSTRACTThis paper intends to analyse the disturbance rejection and output tracking control for a class of nonlinear control systems with disturbances. In this connection, we present a method that combines a proportional-integral observer and nonlinear-equivalent-input-disturbance estimator for superior disturbance rejection performance. Specifically, the nonlinear-equivalent-input-disturbance estimator comprises of equivalent-input-disturbance estimator and nonlinear feedback term, which is employed to estimate and reject the disturbances from the nonlinear system. Notably, the proportional-integral loop in the proportional-integral observer reduces the estimation inaccuracy of the nonlinear-equivalent-input-disturbance. Then the estimated disturbance is intertwined into the repetitive control input to compensate it efficiently. In order to obtain the required results, the proposed control system is converted into a two-dimensional modified repetitive control system to describe the learning and control actions. In particular, the proposed controller enables to adjust the gains directly to improve the learning and control performance and as a result, the tracking accuracy increases. Using a general Lyapunov–Krasovskii functional, singular value decomposition technique and linear matrix inequalities approach, a design algorithm for establishing proportional-integral observer and feedback gains is developed for the system under consideration. Finally, simulation results are given to illustrate the developed method's validity and superiority.

Aizerman Conjectures for a Class of Multivariate Positive Systems

The Aizerman Conjecture predicts stability for a class of nonlinear control systems on the basis of linear system stability analysis. The conjecture is known to be false in general. Here, a number of Aizerman conjectures are shown to be <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">true</i> for a class of internally positive multivariate systems, under a natural generalisation of the classical sector condition and, moreover, guarantee positivity in closed loop. These results are stronger and/or more general than existing results. The paper relates the obtained results to other, diverse, results in the literature.

New Results on Input Delay Tolerance for a Class of Nonlinear Control Systems

This paper studies the problem of when a globally asymptotically stabilizable (GAS) non-homogeneous system by state or output feedback is globally tolerant to input delay. With the aid of the new lemma on the combination of “linear growth and global Lipschitz” property in homogeneous norm reported by Yu and Lin [2023], we prove that for a class of non-homogeneous systems that are not locally exponentially stabilizable (LES) but dominated by a homogeneous system of degree zero, GAS by nonsmooth homogeneous state or output feedback implies GAS of the closed-loop system with input delay, i.e., the property of global input delay tolerance (GIDT). We then illustrate applications of the obtained GIDT results to a class of time-delay non-homogeneous systems with uncontrollable/unobservable linearization.

An Algorithm for Finding Feedback in a Problem with Constraints for One Class of Nonlinear Control Systems

We consider the construction of a feedback according to the Kalman algorithm for a continuous nonlinear control system on a finite time interval with control constraints where the right-hand side of the dynamics equations is linear in control and linearizable in the vicinity of the zero equilibrium position. The solution of an auxiliary optimal control problem with a quadratic functional is used for this task by analogy with the SDRE approach. Because this approach is used in the literature to find suboptimal synthesis in optimal control problems with a quadratic functional with formally linear systems where all coefficient matrices in differential equations and criteria can contain state variables, on a finite time interval it becomes necessary to solve a complicated matrix differential Riccati equations with state-dependent coefficient matrices. Due to the nonlinearity of the system this issue significantly increases the number of calculations for obtaining the coefficients of the gain matrix in the feedback and for obtaining synthesis with a given accuracy in comparison with the Kalman algorithm for linear-quadratic problems. The proposed synthesis construction algorithm is constructed using the extension principle proposed by V.F. Krotov and developed by V.I. Gurman and allows one not only to expand the scope of the SDRE approach to nonlinear control problems with control constraints in the form of closed inequalities, but also to propose a more efficient computational algorithm for finding the matrix of feedback gains in control problems on a finite interval. This article establishes the correctness of the application of the extension principle by introducing analogs of the Lagrange multipliers, which depend on the state and time, and also derives a formula for the suboptimal value of the quality criterion. The presented theoretical results are illustrated by calculating suboptimal feedbacks in the problems of managing three-sector economic systems.

When does stabilizability imply the existence of infinite horizon optimal control in nonlinear systems?

The paper addresses an existence problem for infinite horizon optimal control when the system under control is exponentially stabilizable or stable. Classes of nonlinear control systems for which infinite horizon optimal controls exist are identified in terms of stability, stabilizability, detectability and growth conditions. The result then applies to estimate the existence region of stable manifolds in the associated Hamiltonian systems. Applications of the results also include the analysis for turnpike property in nonlinear finite horizon optimal control problems by a geometric approach.

Algorithm for Finding Feedback in a Problem with Constraints for One Class of Nonlinear Control Systems

For a continuous nonlinear control system on a finite time interval with control constraints, where the right-hand side of the dynamics equations is linear in control and linearizable in the vicinity of the zero equilibrium position, we consider the construction of a feedback according to the Kalman algorithm. For this, the solution of an auxiliary optimal control problem with a quadratic functional is used by analogy with the SDRE approach.Since this approach is used in the literature to find suboptimal synthesis in optimal control problems with a quadratic functional with formally linear systems, where all coefficient matrices in differential equations and criteria can contain state variables, then on a finite time interval it becomes necessary to solve a complicated matrix differential Riccati equations, with state-dependent coefficient matrices. This circumstance, due to the nonlinearity of the system, in comparison with the Kalman algorithm for linear-quadratic problems, significantly increases the number of calculations for obtaining the coefficients of the gain matrix in the feedback and for obtaining synthesis with a given accuracy. The proposed synthesis construction algorithm is constructed using the extension principle proposed by V. F. Krotov and developed by V. I. Gurman and allows not only to expand the scope of the SDRE approach to nonlinear control problems with control constraints in the form of closed inequalities, but also to propose a more efficient computational algorithm for finding the matrix of feedback gains in control problems on a finite interval. The article establishes the correctness of the application of the extension principle by introducing analogs of the Lagrange multipliers, depending on the state and time, and also derives a formula for the suboptimal value of the quality criterion. The presented theoretical results are illustrated by calculating suboptimal feedbacks in the problems of managing three-sector economic systems.

Sequential predictors for delay-compensating feedback stabilization of bilinear systems with uncertainties

We construct delay-compensating input-to-state stabilizing feedback controllers for a class of nonlinear control systems that include bilinear systems that have pointwise delays in their inputs. Our new approach for delay compensation does not require constructing or estimating distributed terms in the formulas for the stabilizing control laws. We allow arbitrarily long constant input delays. We illustrate our findings in a power system example.

Data‐driven (Bilinear) identification and reduction

AbstractIdentifying dynamical systems from measured data is an important step towards accurate modeling and control. Model order reduction (MOR) constitutes a class of methods that can be used to replace large, complex models of dynamical processes with simpler, smaller models. The reduced‐order models (ROMs) can be then used for further tasks such as control, design, and simulation. One typical approach for projection‐based model reduction for both linear and nonlinear dynamical systems is by employing interpolation. Projection‐based methods require access to the internal dynamics of the system which is not always available. The aim here is to compute ROMs without having access to the internal dynamics, by means of a realization independent method. The proposed methodology will fall into the broad category of data‐driven approaches. The method under consideration, which will be referred to as the Loewner framework (LF), was originally introduced by the third author. Based on data, LF identifies state‐space models in a direct way. In the original setup, the framework relies on compressing the full data set to extract dominant features and, at the same time, to eliminate the inherent redundancies. In the broader class of nonlinear control systems, the LF has been already extended to certain classes with a special structure such as quadratic or bilinear systems. The LF in combination with Volterra series‐(VS) offers a non‐intrusive approximation method that is capable of identifying bilinear models from input‐output time‐domain measurements. This method uses harmonic inputs which establish a natural way for data acquisition. The Fourier transform associates these measurements with the derived generalized frequency response functions GFRFs and the LF makes the connection with system theory. In the linear case, the LF associates the so‐called S‐parameters with the linear transfer function by interpolating in the frequency domain. The goal of the proposed method is to extend identification to the case of bilinear systems from time‐domain measurements and to approximate other general nonlinear systems by means of the Carleman bilinearizarion scheme. By identifying the linear contribution with the LF, a considerable reduction is achieved by means of the SVD. The fitted linear system has the same McMillan degree as the original linear system. Then, the performance of the linear model is improved by augmenting a special nonlinear structure. In a nutshell, we learn reduced‐dimension bilinear models directly from a potentially large‐scale system that is simulated/sampled in the time domain. This is done by fitting first a linear model, and afterwards, by fitting the corresponding bilinear operator which will improve the fitting performance.

Minimal controllability time for systems with nonlinear drift under a compact convex state constraint

In this paper we estimate the minimal controllability time for a class of non-linear control systems with a bounded convex state constraint. An explicit expression is given for the controllability time if the image of the control matrix is of co-dimension one. A lower bound for the controllability time is given in the general case. The technique is based on finding a lower dimension system with the similar controllability properties as the original system. The controls corresponding to the minimal time, or time close to the minimal one, are discussed and computed analytically. The effectiveness of the proposed approach is illustrated by a few examples.

Symbolic Padé representation of stabilizing regulators for a class of nonlinear control systems with a parameter

An approach for the construction of a parametric set of stabilizing controls for a class of pseudo-linear control systems with a parameter that varies on a certain interval of the positive semi-axis is described. The algorithm is based on a combination of the State-dependent Riccati equation approach (SDRE) for feedback control design, matrix Padé approximations and intellectual choice theory. Firstly, asymptotic expansions for the solutions of certain matrix algebraic state-dependent Riccati equations in the vicinity of the boundary points of the interval of the parameter variation are constructed. The obtained asymptotic expansions are matched into one symbolic construction using two-point diagonal Padé approximations and as a result a parametric set of solutions for the entire domain of parameter variation is obtained. The resulting controllers have the form of the Kalman regulators and can be used for systems with different control gains coefficients. The symbolic and parametric representation of the regulators allows to increase the efficiency of calculations and to propose algorithms for the intellectual selection of the parameters taking into account the changes in the characteristics of motion in real time.

Stability analysis of the high-order nonlinear extended state observers for a class of nonlinear control systems

The nonlinear extended state observer (ESO) is a novel observer for a class of nonlinear control system. However, the non-smooth structure of the nonlinear ESO makes it difficult to measure the stability. In this paper, the stability problem of the nonlinear ESO is considered. The describing function (DF) method is adopted to analyze the stability of high-order nonlinear ESOs. The main result of the paper shows the existence of the self-oscillation and a sufficient stability condition for high-order nonlinear ESOs. Based on the analysis results, we give a simple and fast parameter tuning method for the nonlinear ESO and the active disturbance rejection control (ADRC). Realistic application simulations show the effectiveness of the proposed parameter tuning method.

Stabilization of a class of nonlinear control systems via a neural network scheme with convergence analysis

In this paper, the stability of a class of nonlinear control systems is analyzed. We first construct an optimal control problem by inserting a suitable performance index; this problem is referred to as an infinite horizon problem. By a suitable change of variable, the infinite horizon problem is reduced to a finite horizon problem. We then present a feedback controller designing approach for the obtained finite horizon control problem. This approach involves a neural network scheme for solving the nonlinear Hamilton Jacobi Bellman equation. By using the neural network method, an analytic approximate solution for value function and a suboptimal feedback control law are achieved. A learning algorithm based on a dynamic optimization scheme with stability and convergence properties is also provided. Some illustrative examples are employed to demonstrate the accuracy and efficiency of the proposed plan. As a real-life application in engineering, the stabilization of a micro-electromechanical system is studied.

НЕЛИНЕЙНАЯ МОДЕЛЬ СИСТЕМЫ ЭЛЕКТРОПИТАНИЯ ПОГРУЖНЫХ ОБЪЕКТОВ С УЧЕТОМ ИЗМЕНЕНИЯ ДЛИНЫ КАБЕЛЬ-ТРОСА

Актуальность работы определяется необходимостью создания адекватных математических моделей систем электропитания сложного погружного технологического оборудования, например электроцентробежных насосов и автономных подводных аппаратов, получающих энергию переменного тока по достаточно протяженной кабельной линии. Приемлемая адекватность характеристик модели и реальной системы электропитания данного класса объектов может быть достигнута в результате учета изменяемой длины кабельной линии и различных режимов работы погружного оборудования. Использование передаточных функций в качестве промежуточных моделей основных элементов системы электропитания и операторного метода является достаточно приемлемым решением для адекватного отражения процессов изменения напряжения на нагрузке. Далее путем преобразования передаточной функции записывается нелинейная математическая модель системы электропитания в пространстве переменных состояния, которая позволяет аналитическим образом решить задачу робастной стабилизации неизмеряемого напряжения на полезной электромеханической нагрузке. Цель работы: построение нелинейной математической модели системы электропитания с передачей энергии переменного тока по кабель-тросу с учетом изменения его длины в пространстве переменных состояния. Методы: операторный метод, методы пространства состояния и математического моделирования. Результаты. Разработана методика определения параметров устройства компенсации реактивной мощности в кабель-тросе для увеличения его пропускной способности при передаче мощности на дальние расстояния при повышенной частоте. Определена аналитическая зависимость, позволяющая рассчитать индуктивность компенсирующего устройства во всем диапазоне изменения длины кабель-троса. Для построения оптимального регулятора получены полиномы, позволяющие определять коэффициенты передаточных функций при изменении длины кабеля и сопротивления нагрузки системы электропитания. На основании передаточных функций, полученных для различных режимов работы системы электропитания, определены параметры схемы замещения, позволяющей записать нелинейную математическую модель системы электропитания пониженного порядка. Проведена сравнительная оценка результатов моделирования системы электропитания с передачей энергии переменного тока по кабель-тросу, подтверждающая возможность ее описания моделью пониженного порядка в пространстве состояний. Выводы. В работе предложена методика определения параметров устройства компенсации реактивной мощности, позволяющего увеличить пропускную способность кабель-троса. На основании передаточных функций, коэффициенты которых зависят от длины кабель-троса, записана упрощенная нелинейная математическая модель системы электропитания погружного технологического оборудования. В результате полученная модель системы электропитания может быть использована для последующего синтеза оптимального регулятора напряжения в классе нелинейных систем управления.

Interval Sliding Mode Observer Based Incipient Sensor Fault Detection With Application to a Traction Device in China Railway High-Speed

This paper proposes an interval sliding mode observer (ISMO) and an incipient sensor faults detection method for a class of nonlinear control systems with observer unmatched uncertainties. The interval bounds for continuous nonlinear functions and new injection functions are constructed to design ISMOs. An incipient fault detection framework with newly designed residual and threshold generators is proposed. The detectability is then studied, and a set of sufficient detectable conditions are presented. Applications to an electrical traction device used in China Railway High-speed (CRH) are presented to verify the effectiveness of the proposed incipient sensor fault detection methodology.

Carleman State Feedback Control Design of a Class of Nonlinear Control Systems

We consider optimal feedback control design for nonlinear control systems with polynomial right-hand sides. The control objective is to minimize a quadratic cost functional over all state feedback control laws with polynomial structure subject to dynamics of the system. First, we utilize ideas from Carleman linearization to lift a given finite-dimensional nonlinear system into an infinite-dimensional linear system. Finite-order truncations of the resulting infinite-dimensional linear system are investigated and connections between (local) stability properties of the original nonlinear system and its finite-order truncations are established. We show that the optimal feedback control design can be approximated and cast as an optimization problem with bilinear matrix equation constraints. Through several simulations, we show that this approximate method can be efficiently implemented using the Alternating Direction Method of Multipliers (ADMM) methods.

Solving multiobjective random interval programming problems by a capable neural network framework

In this paper, the stability of a class of nonlinear control systems is analyzed. We first construct an optimal control problem by inserting a suitable performance index, which this problem is referred to as an infinite horizon problem. By a suitable change of variable, the infinite horizon problem is reduced to a finite horizon problem. We then present a feedback controller designing approach for the obtained finite horizon control problem. This approach involves a neural network scheme for solving the nonlinear Hamilton Jacobi Bellman (HJB) equation. By using the neural network method, an analytic approximate solution for value function and suboptimal feedback control law is achieved. A learning algorithm based on a dynamic optimization scheme with stability and convergence properties is also provided. Some illustrative examples are employed to demonstrate the accuracy and efficiency of the proposed plan. As a real life application in engineering, the stabilization of a micro electro mechanical system is studied.

Compositional Abstraction for Networks of Control Systems: A Dissipativity Approach

In this paper, we propose a compositional scheme for the construction of abstractions for networks of control systems by using the interconnection matrix and joint dissipativity-type properties of subsystems and their abstractions. In the proposed framework, the abstraction, itself a control system (possibly with a lower dimension), can be used as a substitution of the original system in the controller design process. Moreover, we provide a procedure for constructing abstractions of a class of nonlinear control systems by using the bounds on the slope of system nonlinearities. We illustrate the proposed results on a network of linear control systems by constructing its abstraction in a compositional way without requiring any condition on the number or gains of the subsystems. We use the abstraction as a substitute to synthesize a controller enforcing a certain linear temporal logic specification. This example particularly elucidates the effectiveness of dissipativity-type compositional reasoning for large-scale systems.

Control landscapes for a class of non-linear dynamical systems: sufficient conditions for the absence of traps

We establish three tractable, jointly sufficient conditions for the control landscapes of non-linear dynamical systems to be trap free. Trap free landscapes ensure that local optimization methods (such as gradient ascent) can achieve monotonic convergence to the objective in both simulations and in practical circumstances. These results extend prior research primarily regarding the Schrödinger equation to a broader class of non-linear control problems encompassing both quantum and other dynamical systems. This outcome elucidates that these previous conclusions on quantum control landscapes were not specifically contingent upon any features unique to quantum dynamics. As an illustration of the new general results we demonstrate that they encompass end-point objectives for a general class of non-linear control systems having the form of a linear time invariant term with an additional state dependent non-linear term. Within this large class of non-linear control problems, each of the three sufficient conditions is shown to hold for all but a null set of cases. We establish a Lipschitz condition for two of these sufficient conditions and under specific circumstances we explicitly find the associated Lipschitz constants. A detailed numerical investigation using the D-MOPRH gradient control optimization algorithm is presented for a particular example amongst this family of systems. The numerical results confirm the trap free nature of the landscapes of such systems.