Control Of Nonlinear Dynamical Systems Research Articles

Artificial neural network‐based adaptive control for nonlinear dynamical systems

SummaryThis research article presents an artificial neural network (ANN)‐based indirect adaptive control method for nonlinear dynamical systems. In this article, a modified Elman recurrent neural network (MERNN) is proposed as an identifier and controller for controlling nonlinear systems. The architecture of the proposed controller is a modified form of the existing Elman recurrent neural network. The parameter training of ANN‐based controllers is obtained by using the most popular optimization algorithm which is known as the back‐propagation algorithm. A comparative study includes Elman, Diagonal, Jordan, feed‐forward neural network (FFNN), and radial basis function network (RBFN)‐based controllers to compare with the proposed MERNN controller. To determine the controller's robustness, parameter variations, and disturbance signals have been considered. The performance analysis of the proposed controller is illustrated by two simulation examples. The simulation results reveal that MERNN can not only identify the unknown dynamics of the plant but also adaptively control it compared to the others.

Non-Quadratic Proxy Functions in Mirror Descent Method Applied to Designing of Robust Controllers for Nonlinear Dynamic Systems with Uncertainty

Non-Quadratic Proxy Functions in Mirror Descent Method Applied to Designing of Robust Controllers for Nonlinear Dynamic Systems with Uncertainty

Exponential super-twisting control for nonlinear systems with unknown polynomial perturbations

The study focuses on the control of nonlinear dynamic systems in the presence of parameter uncertainties, unmodeled dynamics, and external disturbances. The lumped perturbation is assumed to be bounded within a polynomial in the system state with the polynomial parameters and degrees unknown a priori such that it accommodates a quite wider range dynamic systems. Based on the studies in recent super-twisting algorithm designs and the idea from adaptive sliding mode control for nonlinear systems with uncertainties, we propose a novel adaptive super-twisting algorithm with exponential reaching law, or exponential super-twisting algorithm (ESTA), for the high-stability and acceptable accuracy control of the aimed nonlinear dynamics. The stability analysis and practical finite-time (PFT) convergence are proven using Lyapunov theory and an intuitive analysis of the control behaviour. Simulations are performed to compare the proposed ESTA with the existing super-twisting method and the traditional proportional integral differential control. The simulation results demonstrate the effectiveness of the proposed ESTA in terms of the fastest settling time and the smallest overshoot.

Prescribed-Time Optimal Control of Nonlinear Dynamical Systems With Application to a Coupled Tank System

Prescribed-Time Optimal Control of Nonlinear Dynamical Systems With Application to a Coupled Tank System

Optimal Control Approach To Non-Linear Robust Control System With Two Uncertainties And Matching Condition

In this paper, we introduce the robust control problem (RCP) of non-linear uncertain systems with the matching condition. A relationship has been developed between the robustness with perturbations (uncertainties) and the optimality condition of the current control problem. A computational algorithm for resilient control of nonlinear dynamic systems is proposed, as well as its equivalence to a specific optimal control problem (OCP.

ATTAINABLE SETS OF INTEGRAL CONSTRAINED SEIR CONTROL SYSTEM WITH NONLINEAR INCIDENCE

In this survey, we consider the dynamics of a contagious disease spread by employing a nonlinear dynamical control system of differential equations. It considers treatment and vaccination as key control parameters to discern their influence on disease control. The study, approximate the attainable sets of a given control system and presents visual results, while also discussing potential biological applications of their findings.

Синтез квазиоптимальных законов управления на основе принципа декомпозиции и редукции задачи Лагранжа к изопериметрической в условиях неопределенности

The decomposition principle serves as the basis for methods for constructing universal laws of control of mechanical systems that ensure the stability of any target movement from a set of permissible ones and do not contain information about the dynamic parameters of the control object. It is known that the use of reduction of the Lagrange problem to an isoperimetric one makes it possible to obtain an effective solution to the problem of synthesis of nonlinear control laws, the structure of which determines the variant of nonlinear correction of the control laws of the decomposition principle, which provides an increase in performance. This happens due to the fact that the solution of the game problem is obtained under the assumption of the worst variant of disturbing forces. With a known structure of the controlled system or the availability of information about external forces based on their measurement, the correction of the control law makes it possible to improve the quality of control in comparison with the traditional solution. The purpose of this work is to increase the efficiency of control of nonlinear dynamic systems under uncertainty conditions based on the use of a quasi-optimal control structure obtained on the basis of the maximum condition of the generalized power function and the principle of release, and to analyze the effectiveness of the resulting solution in terms of speed using numerical simulation.

Adaptive recurrent neural network for uncertainties estimation in feedback control system

In this paper, a recurrent neural network (RNN) is used to estimate uncertainties and implement feedback control for nonlinear dynamic systems. The neural network approximates the uncertainties related to unmodeled dynamics, parametric variations, and external disturbances. The RNN has a single hidden layer and uses the tracking error and the output as feedback to estimate the disturbance. The RNN weights are online adapted, and the adaptation laws are developed from the stability analysis of the controlled system with the RNN estimation. The used activation function, at the hidden layer, has an expression that simplifies the adaptation laws from the stability analysis. It is found that the adaptive RNN enhances the tracking performance of the feedback controller at the transient and steady state responses. The proposed RNN based feedback control is applied to a DC–DC converter for current regulation. Simulation and experimental results are provided to show its effectiveness. Compared to the feedforward neural network and the conventional feedback control, the RNN based feedback control provides good tracking performance.

MULTI-MOTOR DRIVE NON-PARAMETRIC BLACK-BOX FUZZY MODEL

The multi-motor drives, the typical examples of which are continuous lines (CL) for tension processing of various materials are complex and coupled MIMO higher-order nonlinear systems, which parameters are difficult to identify. The article focuses on a method for the design of a non-parametric black-box fuzzy model of a continuous production line with emphasis on minimal knowledge on the modelled system. The proposed fuzzy model structure is based on the state space representation of the dynamic system in discrete form, which only requires a suitable set of information on its input/output relations. The properties of the proposed CL fuzzy model were verified by experimental measurements on a CL laboratory model. The obtained results have confirmed the rightness and effectiveness of the fuzzy model design method that can be applied not only in the field of industry technologies with CL but also in modeling and control of nonlinear dynamic systems.

Robust adaptive control of nonlinear dynamic systems using hybrid sliding mode regressive neural learning technique

Purpose The proposed Sliding Mode Control-Global Regressive Neural Network (SMC-GRNN) algorithm is an integration of Global Regressive Neural Network (GRNN) and Sliding Mode Control (SMC). Through this integration, a novel structure of GRNN is designed to enable online and. This structure is then combined with SMC to develop a stable adaptive controller for a class of nonlinear multivariable uncertain dynamic systems.Design/methodology/approach In this study, a new hybrid (SMC-GRNN) control method is innovatively developed.Findings A novel structure of GRNN is designed that can be learned online and then be integrated with the SMC to develop a stable adaptive controller for a class of nonlinear uncertain systems. Furthermore, Lyapunov stability theory is utilized to ensure the hidden-output weighting values of SMC-GRNN adaptively updated in order to guarantee the stability of the closed-loop dynamic system. Eventually, two different numerical benchmark tests are employed to demonstrate the performance of the proposed controller.Originality/value A novel structure of GRNN is originally designed that can be learned online and then be integrated with the sliding mode SMC control to develop a stable adaptive controller for a class of nonlinear uncertain systems. Moreover, Lyapunov stability theory is innovatively utilized to ensure the hidden-output weighting values of SMC-GRNN adaptively updated in order to guarantee the stability of the closed-loop dynamic system.

Estimation and stochastic control of nonlinear dynamic systems over the AWGN channel: Application in tele‐presence and tele‐operation of autonomous vehicles

AbstractThis paper is concerned with tele‐presence and tele‐operation of autonomous vehicles over the Additive White Gaussian Noise (AWGN) channel. This wireless communication channel is subject to transmission noise and transmission power constraint. An encoder, decoder and controller are proposed by implementing a novel linearization method for linearizing the nonlinear dynamic systems at operating points. For the linearized systems, the encoder, decoder and controller are implemented that were previously proposed for output tracking as well as stability of linear dynamic systems. Two novel linearization methods are proposed and their satisfactory performances are proved for output tracking as well as stability of nonlinear dynamic system. The first method is based on the fixed linearization rate and the second one is based on the variable (optimal) linearization rate. The decoder that is based on the second method is in fact the extended Kalman filter with the optimal linearization rate. The performances of these two methods are compared with each other and the other proposed methods by implementing them to the problems of the tele‐presence and tele‐operation of autonomous vehicles over the AWGN channel. Their satisfactory performances are illustrated. It is illustrated that the second method has higher computational complexity with slightly better performance and is applicable to a larger class of reference trajectories.

Online Reinforcement Learning Control of Nonlinear Dynamic Systems: A State-action Value Function Based Solution

In this paper, we present an online reinforcement learning-based solution to the optimal control problem of continuous-time nonlinear input-affine systems. The proposed approach contains a concurrent identifier that estimates time derivatives of states of the system in some arbitrary points. The identifier is utilized to simulate a so-called Bellman error in some unvisited points. The simulated errors together with errors obtained along the trajectory of the system are used to estimate the state-action value function, which is then employed to derive the estimated optimal controller. The designed approach does not explicitly require the input dynamics, which is hard to segregate it from the drift dynamics in optimal regulation problems. In addition, the simulated Bellman errors relax the confining persistence of excitation condition, which is needed for convergence in deterministic systems. A Lyapunov-based analysis was conducted to derive convergence conditions. Simulation studies demonstrated the effectiveness of the developed control scheme.

Nonlinear dynamical systems to study epileptic seizures and extract average amount of mutual information from encephalographs – Part I

In this study, we apply the non-linear dynamical systems theory for the assessment built on recurrence-quantification analysis technique for characterizing—differentiating non-linear electro encephalograph (EEG) signals dynamics. The technique offers convenient quantifiable data plus information over normal, tumultuous, or probability and statistical stochastic properties of inherent systems dynamics theory. The R.Q.A-established processes as the quantifiable mathematical features of non-linear electroencephalograph signals dynamics. Average amount of mutual information (AAMI) applied to compute highly applicable feature-manifestation sub-sets out of R.Q.A-built centered-features. The chosen features were then fed into the computer using artificial intelligence based neural net-works for clustering the data of encephalograph-signals to identify ictic(i.e.,ictal), inter ictal, followed by state of controls. The study is implemented by validating R.Q.A with a data base for various issues of categorization. Results showed that the combination of five selected features created on AAMI attained the precision of100% and proves dominance of R.Q.A. Nonlinear dynamical control systems theory and analysis techniques centered on R.Q.A can be used as an appropriate methodology for distinguishing the non-linear systems dynamics of encephalograph signals data also epileptic seizures tracing.

New Approach to the Synthesis of Optimal Terminal Control of Nonlinear Dynamic Systems

The problem of constructing common solutions to terminal control problems of nonlinear systems is considered here. Previously proven positions are used that the optimal trajectory is an envelope of a parametric family of surfaces (a parametric family of singular curves), and that optimal control can be found on this family. The fact that at each point of the optimal trajectory the vector-function of Lagrange factors is tangent to it, but also tangent to the singular curve, is played out here. A constructive method of constructing singular curves based on conditional separation of variables in the Hamilton-Jacobi equation is given. The " free" parameters of singular curves are based on the condition of minimizing the terminal functionality, which avoids an explicit solution to the boundary problem for a class of nonlinear dynamic systems, and simplifies computational algorithms. Singular curves are described by a reduced (abbreviated) mathematical model. Thus, to synthesize the law of optimal control, we must use the complete (original) mathematical model of the dynamic system, but to calculate it at one time or another, it is enough reduced model. This consideration defines the principle of informational dualism. An illustrative example is given. It has been shown that this approach can be used to solve some classes of differential games.

Convex Optimization-Based Techniques for Trajectory Design and Control of Nonlinear Systems with Polytopic Range

This paper presents new techniques for the trajectory design and control of nonlinear dynamical systems. The technique uses a convex polytope to bound the range of the nonlinear function and associates with each vertex an auxiliary linear system. Provided controls associated with the linear systems can be generated to satisfy an ordering constraint, the nonlinear control is computable by the interpolation of controls obtained by convex optimization. This theoretical result leads to two numerical approaches for solving the nonlinear constrained problem: one requires solving a single convex optimization problem and the other requires solving a sequence of convex optimization problems. The approaches are applied to two practical problems in aerospace engineering: a constrained relative orbital motion problem and an attitude control problem. The solve times for both problems and approaches are on the order of seconds. It is concluded that these techniques are rigorous and of practical use in solving nonlinear trajectory design and control problems.

Non-positive barrier function: a new notion of barrier function for state-safety control of nonlinear dynamical systems

The analysis and control of system state safety are a research hotspot in the field of control. This article has thoroughly investigated some issues related to safety criteria with application to the safety analysis, diagnosis, and safety control of a class of dynamical systems with their unique set of available states via a new notion of barrier functions. Inspired by reciprocal barrier functions and zeroing barrier functions, a type of non-positive barrier function (NPBF) is designed and the safety criteria are proposed, guaranteeing that the unique set of available states is forward-invariant to ensure system safety at all times. Then, the control barrier function is established based on NPBFs for a class of dynamical control systems and used to design the safety controller. Conclusively, this is a first-of-its-kind study to apply safety criteria and control barrier functions to safeguard a class of continuously stirred tank reactors.

The use of digital twins to remotely update feedback controllers for the motion control of nonlinear dynamic systems

The use of a digital twin to update a feedback controller is considered, and this is illustrated using simulations of a position-controlled dynamical system with a time-varying nonlinear element. The feedback control system consists of a dc motor driving the displacement of a three degree of freedom structure through a lead screw that is subject to backlash, whose gap angle changes over time due to wear for example. The backlash is shown to destabilise the feedback loop when using a PID controller designed for the linear system. However, stability can then be re-established by including a dead zone within the controller. The design of the dead zone depends on the extent of the gap angle in the backlash and is a trade-off between stabilising the system and avoiding excessive steady-state errors and undesired transient behaviour. When the backlash gap angle changes significantly from the one used to design the dead zone, both performance and stability are affected. Therefore, a digital twin of the system is used to estimate the backlash gap angle as it changes with time and this estimate is used to re-design the dead zone, which is then communicated back to update the controller. The digital twin and controller-design process can be implemented offline and remotely from the physical twin and the real-time controller via an asynchronous link with variable time delays. As a result of combining the feedback loop and digital twin, good performance and stability are maintained at all times.

A new result on PD controller design for second order nonlinear uncertain systems

Under what condition a proportional-derivative (PD) controller can stabilize globally second order nonlinear uncertain systems? It is an important and attractive question. This paper will broaden the range of a function space F which is used to describe the uncertainty of nonlinear systems in literature, such that the corresponding result still holds for nonlinear dynamical control systems with more uncertainty. The method is based on some related results in the research about Markus-Yamabe Conjecture.

Nonlinear Dynamics

The modeling and control of nonlinear dynamic systems is challenging in mathematics and engineering [...]

Orbital decompositions and integrable pseudosymmetries of control systems

Local decompositions of nonlinear dynamical control systems are investigated. We use orbital equivalences to transform systems into decomposed forms. In the language of infinite jets, every control system is interpreted as an infinite dimensional manifold. Each orbital decomposition of the system defines an invariant distribution on this manifold. A local basis of this distribution is an integrable pseudosymmetry of the system. An algorithm for constructing orbital decompositions is given and demonstrated by the example of the Kapitsa pendulum system. It is also shown how to use orbital decompositions to plan trajectories.