Control Of Nonlinear Dynamical Systems Research Articles

A proportional–integral-based robust state-feedback control method for linear parameter-varying systems and its application to aircraft

The integration of advanced linear parameter-varying and classical proportional–integral–derivative control methods has attracted great attention in control of nonlinear dynamic systems. However, linear parameter-varying proportional–integral–derivative control synthesis is a nonconvex bilinear matrix inequality problem. Although the synthesis conditions can be convexified in the case of proportional–integral control, a strong constraint on the structure of matrix variables usually leads to infeasible solutions. In this paper, a linear parameter-varying proportional–integral control design method is proposed to remove that constraint. The approach is based on the assumption of state-feedback proportional–integral control to guarantee the linear matrix inequality variables to be full matrices instead of block diagonal matrices, and extended linear matrix inequalities are proposed to synthesize the controller. This increases the likelihood of finding feasible linear matrix inequality solutions and reduces the conservatism. The proposed method is applied to control longitudinal and lateral dynamics of an F-16 aircraft and promising simulation results are obtained.

Suboptimal Control of Nonlinear Dynamic System with Unlimited Transition Process Time

Abstract In this paper, we consider the issue of synthesizing optimal control of nonlinear dynamic systems in case of a problem with an unspecified time of the transition process. To solve this problem, an algorithmic method for constructing a suboptimal control is suggested. The basis of this method is the behavior of the Hamiltonian on the optimal trajectory, which corresponds to the required condition of the optimal system. For the synthesis of suboptimal control, the original nonlinear model is replaced by an equivalent model with a linear structure with state-dependent parameters. The use of the quadratic quality functional allows to make the transition from solving the Hamilton-Jacobi-Bellman equation to the Riccati-type equation with state-dependent parameters. A computer simulation of a simple nonlinear system, for which the Riccati equation with state-dependent parameters can be solved, has been carried out. Also, for this system a suboptimal control was created with the use of a new algorithmic parametric optimization method. The simulation results confirm the effectiveness of the method. In addition, the constructed suboptimal control is close to the control obtained by solving the Riccati equation.

Impulsive control of a nonlinear dynamical network and its application to biological networks.

The control of nonlinear dynamical systems is always a notable problem in science. According to control theory, suitable inputs for a controllable dynamical system are critical. Previous studies have shown some principles to determine control nodes and design control function. In this work, we propose a new control strategy of nonlinear systems by constructing impulsive control functions, i.e., we can realize the transition from an undesired state to a desired one by controlling appropriately chosen nodes in a discrete manner. In order to demonstrate the effectiveness of the strategy, we apply it to two biological networks: the epithelial-mesenchymal transition (EMT) network and the Notch1-Dll1-Jag1 signaling pathway. The strategy can not only be used to guide pharmacological design in a more feasible form but can also be applied into the fields of biological, medical and other multistable dynamical systems.

Identification and control of dynamical systems using different architectures of recurrent fuzzy system

Fuzzy logic based systems are very widely used for modeling and control of complex non-linear, plants. Fuzzy systems require the knowledge about the structure of the dynamic plant in order to achieve fruitful results. Recurrent Fuzzy systems (RFS) are a variation of fuzzy systems and have the ability to model and control dynamic plants without using the information about the structure of the plant. This paper presents identification and control of non-linear dynamical systems using two different architectures of recurrent fuzzy system (RFS). It highlights the importance of RFS over the conventional type-1 fuzzy based system. The objective of system identification as well as control has been achieved using both the architectures of RFS and the simulation results clearly show their efficiency. This paper also highlights yet another advantage of RFS over the conventional type-1 fuzzy systems which comes into light when dealing with higher order systems. The paper explains how the computational complexity can be greatly reduced by using RFS for higher order dynamical systems. A comparative analysis between the conventional type-1 fuzzy system and the two recurrent fuzzy systems has also been performed.

Exact and approximate controllability of nonlinear dynamic systems in infinite time: The Green's function approach

AbstractA systematic method for establishment of exact and approximate controllability of dynamic systems with nonlinear state constraints within infinite time is developed. Using the recently developed Green's function approach, we derive necessary and sufficient conditions for exact controllability, as well as sufficient conditions for approximate controllability. Conditions for null‐controllability and lack of controllability are established as well. Several possibilities of deriving explicit form for resolving controls are described. Possibilities of heuristic determination of resolving controls is studied in details. The advantages and drawbacks of the method are revealed on specific examples of nonlinear wave and diffusion equations in unbounded domains.

Learning control lyapunov functions from counterexamples and demonstrations

We present a technique for learning control Lyapunov-like functions, which are used in turn to synthesize controllers for nonlinear dynamical systems that can stabilize the system, or satisfy specifications such as remaining inside a safe set, or eventually reaching a target set while remaining inside a safe set. The learning framework uses a demonstrator that implements a black-box, untrusted strategy presumed to solve the problem of interest, a learner that poses finitely many queries to the demonstrator to infer a candidate function, and a verifier that checks whether the current candidate is a valid control Lyapunov function. The overall learning framework is iterative, eliminating a set of candidates on each iteration using the counterexamples discovered by the verifier and the demonstrations over these counterexamples. We prove its convergence using ellipsoidal approximation techniques from convex optimization. We also implement this scheme using nonlinear MPC controllers to serve as demonstrators for a set of state and trajectory stabilization problems for nonlinear dynamical systems. We show how the verifier can be constructed efficiently using convex relaxations of the verification problem for polynomial systems to semi-definite programming (SDP) problem instances. Our approach is able to synthesize relatively simple polynomial control Lyapunov functions, and in that process replace the MPC using a guaranteed and computationally less expensive controller.

Self‐recurrent wavelet neural network–based identification and adaptive predictive control of nonlinear dynamical systems

SummaryIn this paper, the problem of simultaneous identification and predictive control of nonlinear dynamical systems using self‐recurrent wavelet neural network (SRWNN) is addressed. The structure of the SRWNN is a modification of the wavelet neural network (WNN). Unlike WNN, the neurons present in the hidden layer of SRWNN contain the weighted self‐feedback loops. Dynamic back‐propagation algorithm is employed to derive the necessary parameter update equations. To further improve the convergence speed of the parameters, a time‐varying (adaptive) learning rate is used. Four simulation examples are considered for testing the effectiveness of the proposed method. Furthermore, some disturbance rejection tests are also performed on the proposed method. The results obtained through the simulation study confirm the effectiveness of the proposed method.

A new integral inequality and its applications to robust control problems of uncertain nonlinear systems

SummaryIn this paper, a new integral inequality is presented. By combining this integral inequality with adaptive approach, new design methods can be developed to synthesize some adaptive robust control schemes for a large class of uncertain nonlinear systems and to deal with well the unknown nonlinearities appearing in uncertain nonlinear control dynamical systems. As an application of the presented integral inequality to control theory, the robust stabilization problem is considered for a class of uncertain strict‐feedback nonlinear systems with both time‐delay and unknown dead‐zone input nonlinearities. It is shown that there are two main merits in the design method based on the integral inequality presented in this paper. The first one is that one need not estimate and know the unknown nonlinearities to synthesize some stabilizing control schemes. The second one is that the resulting feedback control schemes have rather simple structure.

Measurement and control of nonlinear dynamic systems over the internet (IoT): Applications in remote control of autonomous vehicles

This paper presents a new technique for almost sure asymptotic state tracking, stability and reference tracking of nonlinear dynamic systems by remote controller over the packet erasure channel, which is an abstract model for transmission via WiFi and the Internet. By implementing a suitable linearization method, a proper encoder and decoder are presented for tracking the state trajectory of nonlinear systems at the end of communication link when the measurements are sent through the packet erasure channel. Then, a controller for reference tracking of the system is designed. In the proposed technique linearization is applied when the error between the states and an estimate of these states at the decoder increases. It is shown that the proposed technique results in almost sure asymptotic reference tracking (and hence stability) over the packet erasure channel. The satisfactory performance of the proposed state trajectory and reference tracking technique is illustrated by computer simulations by applying this technique on the unicycle model, which represents the dynamic of autonomous vehicles.

Bio-inspired spiking neural network for nonlinear systems control

Spiking neural networks (SNN) are the third generation of artificial neural networks. SNN are the closest approximation to biological neural networks. SNNs make use of temporal spike trains to command inputs and outputs, allowing a faster and more complex computation. As demonstrated by biological organisms, they are a potentially good approach to designing controllers for highly nonlinear dynamic systems in which the performance of controllers developed by conventional techniques is not satisfactory or difficult to implement. SNN-based controllers exploit their ability for online learning and self-adaptation to evolve when transferred from simulations to the real world. SNN’s inherent binary and temporary way of information codification facilitates their hardware implementation compared to analog neurons. Biological neural networks often require a lower number of neurons compared to other controllers based on artificial neural networks. In this work, these neuronal systems are imitated to perform the control of non-linear dynamic systems. For this purpose, a control structure based on spiking neural networks has been designed. Particular attention has been paid to optimizing the structure and size of the neural network. The proposed structure is able to control dynamic systems with a reduced number of neurons and connections. A supervised learning process using evolutionary algorithms has been carried out to perform controller training. The efficiency of the proposed network has been verified in two examples of dynamic systems control. Simulations show that the proposed control based on SNN exhibits superior performance compared to other approaches based on Neural Networks and SNNs.

Adaptive control over non­linear objects using the robust neural network FCMAC

The paper explores issues related to the application of artificial neural networks (ANN) when solving the problems on identification and control of nonlinear dynamic systems. We have investigated characteristics of the network, which is a result of the application of the apparatus of fuzzy logic in a classical СМАС neural network, which is titled FCMAC ‒ Fuzzy Cerebral Model Arithmetic Computer. We studied influence of the form of receptive fields of associative neurons on the accuracy of identification and control; various information hashing algorithms that make it possible to reduce the amount of memory required for the implementation of a network; robust learning algorithms are proposed allowing the use of a network in systems with strong perturbations. It is shown that the FСМАС network, when selecting appropriate membership functions, can be applied in order to synthesize indirect control systems with and without a reference model; it is more efficient to use it in control systems with the reference model. This sharply reduces the quantity of training pairs and simplifies the coding due to the narrower range of the applied values of input signals. The results obtained are confirmed by simulation modeling of the processes of identification of and control over nonlinear dynamical systems

Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control

This paper presents a class of linear predictors for nonlinear controlled dynamical systems. The basic idea is to lift (or embed) the nonlinear dynamics into a higher dimensional space where its evolution is approximately linear. In an uncontrolled setting, this procedure amounts to numerical approximations of the Koopman operator associated to the nonlinear dynamics. In this work, we extend the Koopman operator to controlled dynamical systems and apply the Extended Dynamic Mode Decomposition (EDMD) to compute a finite-dimensional approximation of the operator in such a way that this approximation has the form of a linear controlled dynamical system. In numerical examples, the linear predictors obtained in this way exhibit a performance superior to existing linear predictors such as those based on local linearization or the so called Carleman linearization. Importantly, the procedure to construct these linear predictors is completely data-driven and extremely simple – it boils down to a nonlinear transformation of the data (the lifting) and a linear least squares problem in the lifted space that can be readily solved for large data sets. These linear predictors can be readily used to design controllers for the nonlinear dynamical system using linear controller design methodologies. We focus in particular on model predictive control (MPC) and show that MPC controllers designed in this way enjoy computational complexity of the underlying optimization problem comparable to that of MPC for a linear dynamical system with the same number of control inputs and the same dimension of the state-space. Importantly, linear inequality constraints on the state and control inputs as well as nonlinear constraints on the state can be imposed in a linear fashion in the proposed MPC scheme. Similarly, cost functions nonlinear in the state variable can be handled in a linear fashion. We treat both the full-state measurement case and the input–output case, as well as systems with disturbances/noise. Numerical examples demonstrate the approach.1

Optimization of Zero-Order TSK-Type Fuzzy System Using Enhanced Particle Swarm Optimizer with Dynamic Mutation and Special Initialization

This paper proposes an enhanced particle swarm optimization with dynamic mutation and special initialization, the EPSODM-I, for improving the accuracy of zero-order TSK-type fuzzy systems design. The EPSODM-I can be regarded as a new population-based evolutionary optimization algorithm. Unlike the generic initialization used in most popular population-based algorithms, the EPSODM-I applies a proposed special initialization method to generate the initial PSO particles for fuzzy system design. The generated initial PSO particles are iteratively improved by a new approach incorporating the dynamic mutation into the existing PSO to provide more diverse search directions. Application examples of the zero-order fuzzy system designs for the tracking control of nonlinear dynamic systems have been simulated to validate the proposed algorithm. In terms of tracking errors, performance comparisons with the typical PSO and different advanced PSO variants verify the superiority of the proposed algorithm. The effects on the convergence rate and optimization accuracy yielded by the proposed special initialization and dynamic mutation have also been discussed and verified crossly in the simulation results.

Comparative Study of Neural Networks for Control of Nonlinear Dynamical Systems with Lyapunov Stability-Based Adaptive Learning Rates

This paper performs the comparative study of two feed-forward neural networks: radial basis function network (RBFN), multilayer feed-forward neural network (MLFFNN) and a recurrent neural network: nonlinear auto-regressive with exogenous inputs (NARX) neural network for their ability to provide an adaptive control of nonlinear systems. Dynamic back-propagation algorithm is used to derive parameter update equations. To ensure stability and faster convergence, an adaptive learning rate is developed in the sense of discrete Lyapunov stability method. Both parameter variation and disturbance signal cases are considered for checking and comparing the robustness of controller. Three simulation examples are considered for carrying out this study. The results so obtained reveal that RBFN-based controller is performing better than that of NARX- and MLFFNN-based controllers.

Optimal control of nonlinear dynamical systems based on a new parallel eigenvalue decomposition approach

SummaryThis manuscript aims to investigate a new approach based on the modal series method and eigenvalue decomposition technique to solve a class of nonlinear optimal control problems. For this purpose, a sequence of decoupled linear two‐point boundary value problems is solved iteratively instead of solving the coupled nonlinear two‐point boundary value problem derived from the maximum principle. The convergence analysis of the suggested technique is also investigated. In addition, the problem that needs to be solved at each iteration is composed of lower‐order decoupled subproblems; hence, they can be solved in parallel. Thus, the new scheme has a parallel computing property improving its computational effectiveness. Numerical simulations and comparative results show that the proposed approach is efficient and provides satisfactory results.

Control Design for Nonlinear Systems via Controller with Quasi-fixed Parameters

Abstract The paper presents an approach to design a control of nonlinear dynamical systems based on the extended linearization technique of a mathematical model of an object such that a given quadratic cost function is minimized. In this case, the coefficients of the nonlinear controller are determined by solving Riccati matrix equation with state dependent parameters. A realizability problem of the controller of this sort is computational complexity of the realtime solution. The proposed approach solves the problem by controller coefficients searching at each control time subinterval of the whole control time interval. The presented methodology is illustrated by designing chemotherapy administration for the cancer treatment. The mathematical model of the cancer growth includes interaction between tumor cells, healthy normal cells and activated immune cells. Numerical results show effectiveness of the solutions obtained.

Heuristic Determination of Resolving Controls for Exact and Approximate Controllability of Nonlinear Dynamic Systems

Dealing with practical control systems, it is equally important to establish the controllability of the system under study and to find corresponding control functions explicitly. The most challenging problem in this path is the rigorous analysis of the state constraints, which can be especially sophisticated in the case of nonlinear systems. However, some heuristic considerations related to physical, mechanical, or other aspects of the problem may allow coming up with specific hierarchic controls containing a set of free parameters. Such an approach allows reducing the computational complexity of the problem by reducing the nonlinear state constraints to nonlinear algebraic equations with respect to the free parameters. This paper is devoted to heuristic determination of control functions providing exact and approximate controllability of dynamic systems with nonlinear state constraints. Using the recently developed approach based on Green’s function method, the controllability analysis of nonlinear dynamic systems, in general, is reduced to nonlinear integral constraints with respect to the control function. We construct parametric families of control functions having certain physical meanings, which reduce the nonlinear integral constraints to a system of nonlinear algebraic equations. Regimes such as time-harmonic, switching, impulsive, and optimal stopping ones are considered. Two concrete examples arising from engineering help to reveal advantages and drawbacks of the technique.

Neural Network Based Semi-Empirical Models for Dynamical Systems Represented by Differential-Algebraic Equations of Index 2

A simulation problem is discussed for nonlinear controlled dynamical systems represented by differential-algebraic equations of index 2. The problem is proposed to be solved in the framework of a neural network based semi-empirical approach combining theoretical knowledge for the object with training tools of artificial neural network field. Special form neural network based semi-empirical models implementing implicit Runge-Kutta method of numerical integration are proposed. The training of the semi-empirical model allows to elaborate the models of aerodynamic coefficients implemented as a part of it. The results of simulation using this elaboration procedure of lift coefficient for reentry hypersonic vehicle are presented.

Semi-empirical Neural Network Based Approach to Modelling and Simulation of Controlled Dynamical Systems

A modelling and simulation approach is discussed for nonlinear controlled dynamical systems under multiple and diverse uncertainties. The main goal is to demonstrate capabilities for semi-empirical neural network based models combining theoretical domain-specific knowledge with training tools of artificial neural network field. Training of the dynamical neural network model for multi-step ahead prediction is performed in a sequential fashion. Computational experiments are carried out to confirm efficiency of the proposed approach.

Differential Equations for Ellipsoidal Estimates of Reachable Sets for a Class of Control Systems with Nonlinearity and Uncertainty

The problem of estimating reachable sets of nonlinear dynamical control systems with combined nonlinearity of quadratic and bilinear types and with uncertainty in initial states and in system parameters is studied. We assume that the uncertainty is of set-membership kind when only the bounding sets for unknown parameters and functions are given. We find the ellipsoidal estimates of reachable sets using the special bilinear-quadratic structure of studied control system. The main result consists in deriving the related differential equations which describe the dynamics of the ellipsoidal estimates of reachable sets of the control system under study, related numerical simulation results are also given.