First-order Nonlinear Systems Research Articles

Lyapunov, Adaptive, and Optimal Design Techniques for Cooperative Systems on Directed Communication Graphs

This paper presents three design techniques for cooperative control of multiagent systems on directed graphs, namely, Lyapunov design, neural adaptive design, and linear quadratic regulator (LQR)-based optimal design. Using a carefully constructed Lyapunov equation for digraphs, it is shown that many results of cooperative control on undirected graphs or balanced digraphs can be extended to strongly connected digraphs. Neural adaptive control technique is adopted to solve the cooperative tracking problems of networked nonlinear systems with unknown dynamics and disturbances. Results for both first-order and high-order nonlinear systems are given. Two examples, i.e., cooperative tracking control of coupled Lagrangian systems and modified FitzHugh–Nagumo models, justify the feasibility of the proposed neural adaptive control technique. For cooperative tracking control of the general linear systems, which include integrator dynamics as special cases, it is shown that the control gain design can be decoupled from the topology of the graphs, by using the LQR-based optimal control technique. Moreover, the synchronization region is unbounded, which is a desired property of the controller. The proposed optimal control method is applied to cooperative tracking control of two-mass–spring systems, which are well-known models for vibration in many mechanical systems.

Degrees of freedom analysis and parameter optimization of state feedback of first-order affine singular control for bioprocess systems

The state feedback optimization of first-order affine nonlinear singular control systems is formulated as a time-invariant parameter optimization problem. Degrees of freedom (DF) is applied to the differential balance equations with initial control or its time derivatives and switching times as parameters, and uncovered that the state feedback optimization problem has −2 DF, clarifying admissible singular control structures. Using these two novel concepts, we considered end-point optimization problems in fed-batch fermentation problem to illustrate the state feedback optimization of the feed flow rate.

Lyapunov-type inequalities for the first-order nonlinear Hamiltonian systems

In this paper, we establish several new Lyapunov-type inequalities for the first-order nonlinear Hamiltonian system { x ′ ( t ) = α ( t ) x ( t ) + β ( t ) | y ( t ) | μ − 2 y ( t ) , y ′ ( t ) = − γ ( t ) | x ( t ) | ν − 2 x ( t ) − α ( t ) y ( t ) , which generalize or improve all related existing ones.

Nonlinear adaptive learning control for unknown time‐varying parameters and unknown time‐varying delays

AbstractA new adaptive learning control approach is proposed for a class of first‐order nonlinear systems with two unknown time‐varying parameters and an unknown time‐varying delay. By reconstructing the system equation, all unknown time‐varying terms, including the time‐varying delay, are combined into an unknown periodic time‐varying vector, which is estimated by a periodic adaptive mechanism. By constructing a Lyapunov–Krasovskii‐like composite energy function (CEF), we prove the boundedness of all signals and the convergence of the tracking error. The results are extended to two classes of high‐order nonlinear systems with mixed parameters. Three simulation examples are provided to illustrate the effectiveness of the control algorithms proposed in this paper.Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Probability density function for stochastic response of non-linear oscillation system under random excitation

Probability density function (PDF) of stochastic responses is a critical topic in uncertainty analysis. In this paper, orthogonal decomposition technique was extended to discuss non-stationary response of non-linear oscillator under random excitation. The PDF of stochastic reponses is represented by a set of standardized multivariable orthogonal polynomials. According to the Galerkin scheme, the original problem, which has to solve the Fokker–Planck–Kolmogorov (FPK) equation, was converted to a first-order linear ordinary differential equation, in terms of unknown time-dependent coefficients. Then, stationary and non-stationary PDFs of uncertainty responses were obtained. In numerical examples, first-order and second-order non-linear systems exposed to the Gaussian white noise were considered. Finally, the accuracy of the proposed method was demonstrated through appropriate comparisons to Monte-Carlo simulation and analytical results.

Unbounded solutions of a nonlinear parabolic equation

The article investigates the equation $$ {u_t}{\text{ = }}{\left( {u{u_x}} \right)_x}{\text{ + }}\left( {u - {u_0}} \right)\left( {u - {u_1}} \right){\text{,}}\quad \quad {u_1} > {u_0} > 0. $$ We derive a sufficient condition on the initial function of the corresponding Cauchy problem ensuring that the solution is unbounded (i.e., does not exist globally). We also construct a family of exact solutions of the form $$ p(t){\text{ + }}q(t)\cos \frac{x}{{\sqrt 2 }} $$ , where the coefficients p(t ) and q(t ) satisfy a first-order nonlinear dynamical system. A number of unboundedness conditions are obtained for the solutions of this family.

Neural network approximation for periodically disturbed functions and applications to control design

This paper addresses the approximation problem of functions affected by unknown periodically time-varying disturbances. By combining Fourier series expansion into multilayer neural network or radial basis function neural network, we successfully construct two kinds of novel approximators, and prove that over a compact set, the new approximators can approximate a continuously and periodically disturbed function to arbitrary accuracy. Then, we apply the proposed approximators to disturbance rejection in the first-order nonlinear control systems with periodically time-varying disturbances, but it is straightforward to extend the proposed design methods to higher-order systems by using adaptive backstepping technique. A simulation example is provided to illustrate the effectiveness of control schemes designed in this paper.

Cone invariance and rendezvous of multiple agents

In this article is presented a dynamical systems framework for analysing multi-agent rendezvous problems and characterize the dynamical behaviour of the collective system. Recently, the problem of rendezvous has been addressed considerably in the graph theoretic framework, which is strongly based on the communication aspects of the problem. The proposed approach is based on the set invariance theory and focusses on how to generate feedback between the vehicles, a key part of the rendezvous problem. The rendezvous problem is defined on the positions of the agents and the dynamics is modelled as linear first-order systems. These algorithms have also been applied to non-linear first-order systems. The rendezvous problem in the framework of cooperative and competitive dynamical systems is analysed that has had some remarkable applications to biological sciences. Cooperative and competitive dynamical systems are shown to generate monotone flows by the classical Muller—Kamke theorem, which is analysed using the set invariance theory. In this article, equivalence between the rendezvous problem and invariance of an appropriately defined cone is established. The problem of rendezvous is cast as a stabilization problem, with a the set of constraints on the trajectories of the agents defined on the phase plane. The n-agent rendezvous problem is formulated as an ellipsoidal cone invariance problem in the n-dimensional phase space. Theoretical results based on set invariance theory and monotone dynamical systems are developed. The necessary and sufficient conditions for rendezvous of linear systems are presented in the form of linear matrix inequalities. These conditions are also interpreted in the Lyapunov framework using multiple Lyapunov functions. Numerical examples that demonstrate application are also presented.

Exact Analytic Solutions of the Plastic Spin Equations in Simple Shear

We prove that the first-order nonlinear differential system governing the plastic spin response in simple shear (Dafalias’s equations) is reduced to an equivalent equation of the Abel normal form by means of admissible functional transformations. In similar Abel equations result also the original and the generalized Volterra differential systems, describing the problem of two populations conflicting with one another. The above reduced Abel equations do not admit exact analytic solutions in terms of known (tabulated) functions, since only very special cases of these types of equations can be analytically solved in parametric form. We provide a mathematical solution methodology leading to the construction of exact implicit analytic solutions of the above-mentioned type of equations. Since the plastic spin nonlinear differential system results in a special unsolvable form of Abel’s equation of the normal form, we perform the exact implicit analytic solution of this system too.

Optimal tracking control of nonlinear dynamical systems

This paper presents a simple methodology for obtaining the entire set of continuous controllers that cause a nonlinear dynamical system to exactly track a given trajectory. The trajectory is provided as a set of algebraic and/or differential equations that may or may not be explicitly dependent on time. Closed-form results are also provided for the real-time optimal control of such systems when the control cost to be minimized is any given weighted norm of the control, and the minimization is done not just of the integral of this norm over a span of time but also at each instant of time. The method provided is inspired by results from analytical dynamics and the close connection between nonlinear control and analytical dynamics is explored. The paper progressively moves from mechanical systems that are described by the second-order differential equations of Newton and/or Lagrange to the first-order equations of Poincaré, and then on to general first-order nonlinear dynamical systems. A numerical example illustrates the methodology.

Symmetries, integrals, and three-dimensional reductions of Plebañski’s second heavenly equation

We study symmetries and conservation laws for Plebanski’s second heavenly equation written as a first-order nonlinear evolutionary system which admits a multi-Hamiltonian structure. We construct an optimal system of one-dimensional subalgebras and all inequivalent three-dimensional symmetry reductions of the original four-dimensional system. We consider these two-component evolutionary systems in three dimensions as natural candidates for integrable systems.

On the computation of steady hopper flows III: Model comparisons

Gravity flows of granular materials through hoppers are considered. For hoppers shaped as general nonaxisymmetric cones, i.e., “pyramids”, the flow inherits some simplified features from the geometry: similarity solutions can be constructed. Using two different plasticity laws, namely Matsuoka–Nakai and von Mises, those solutions are obtained by solving first-order nonlinear partial differential algebraic systems for stresses, velocities, and a plasticity function.A pseudospectral discretization is applied to both models and the resulting flow fields are examined. Some similarities are found, but important differences appear, especially with regard to velocities near the wall and normal wall stresses. Preliminary comparisons with recent experiments [J.F. Wambaugh, R.P. Behringer, Asymmetry-induced circulation in granular hopper flows, in: R. Garcia-Rojo, H.J. Herrmann, S. McNamara (Eds.), Powders and Grains, 2005, pp. 915–918] based on the present results indicate that for slow granular flows the lesser known Matsuoka–Nakai plasticity law yields better results than more common models based on a von Mises criterion.

Dynamical Modeling of Turbine Flow Meters

Turbine flow meters can be represented by a simple first-order nonlinear system adequately with two instrument constants: the meter scale-factor and the dynamic factor. Conventionally only the scale-factor is determined during calibration. This paper presents a method to ascertain the dynamic factor, so that the dynamic response of the flow meter is also taken into account.

Multi-Hamiltonian structure of Plebanski's second heavenly equation

We show that Plebanski's second heavenly equation, when written as a first-order nonlinear evolutionary system, admits multi-Hamiltonian structure. Therefore by Magri's theorem it is a completely integrable system. Thus it is an example of a completely integrable system in four dimensions.

Design of controller using variable transformations for a nonlinear process with dead time

In this work, a globally linearized controller (GLC) for a first-order nonlinear system with dead time is proposed. This is similar to the GLC proposed by Ogunnalke [Ind. Eng. Chem. Process. Des. Dev. 25, 241–248 (1986)] for nonlinear systems without dead time. Two methods are proposed. One is based on the Smith prediction from the model in the transformed domain and the other is based on Newton's extrapolation method. The simulation study is made on the conical tank level process and the results are compared with those obtained using a conventional PI controller and the Smith PI controller based on the transfer function model about the operating point 39%. Finally, experimental results on the laboratory conical tank level process are also given.

Implementation of Predictive Model Based Control on Programmable Logic Controller

Model-based predictive control (MBPC) is one of the first and most popular non-standard control methods. Although MBPC has around 30 year history, it was usually implemented in industrial control systems using PC computers. Technological progress made control equipment more powerful last years. Modem programmable logic controllers (PLCs) are now capable to run more computation consuming algorithms. The paper presents an application of one of the commercially used MBPC control methods - Predictive Functional Control (PFC). The standard first-order nonlinear system with polynomial approximation is proposed as internal model. The application was made for flow control of the fluid in the installation of chemical reactor. Presented application shows that nonlinear control can be necessary even at a very low level in the control hierarchy.

Nonlinear Control Systems: Analysis and Design

Introduction. 1.1 Linear Time-Invariant Systems. 1.2 Nonlinear Systems. 1.3 Equilibrium Points. 1.4 First-Order Autonomous Nonlinear Systems. 1.5 Second-Order Systems: Phase-Plane Analysis. 1.6 Phase-Plane Analysis of Linear Time-Invariant Systems. 1.7 Phase-Plane Analysis of Nonlinear Systems. 1.8 Higher-Order Systems. 1.9 Examples of Nonlinear Systems. 1.10 Exercises. Mathematical Preliminaries. 2.1 Sets. 2.2 Metric Spaces. 2.3 Vector Spaces. 2.4 Matrices. 2.5 Basic Topology. 2.6 Sequences. 2.7 Functions. 2.8 Differentiability. 2.9 Lipschitz Continuity. 2.10 Contraction Mapping. 2.11 Solution of Differential Equations. 2.12 Exercises. Lyapunov Stability I: Autonomous Systems. 3.1 Definitions. 3.2 Positive Definite Functions. 3.3 Stability Theorems. 3.4 Examples. 3.5 Asymptotic Stability in the Large. 3.6 Positive Definite Functions Revisited. 3.7 Construction of Lyapunov Functions. 3.8 The Invariance Principle. 3.9 Region of Attraction. 3.10 Analysis of Linear Time-Invariant Systems. 3.11 Instability. 3.12 Exercises. Lyapunov Stability II: Nonautonomous Systems. 4.1 Definitions. 4.2 Positive Definite Functions. 4.3 Stability Theorems. 4.4 Proof of the Stability Theorems. 4.5 Analysis of Linear Time-Varying Systems. 4.6 Perturbation Analysis. 4.7 Converse Theorems. 4.8 Discrete-Time Systems. 4.9 Discretization. 4.10 Stability of Discrete-Time Systems. 4.11 Exercises. Feedback Systems. 5.1 Basic Feedback Stabilization. 5.2 Integrator Backstepping. 5.3 Backstepping: More General Cases. 5.4 Examples. 5.5 Exercises. Input-Output Stability. 6.1 Function Spaces. 6.2 Input-Output Stability. 6.3 Linear Time-Invariant Systems. 6.4 Lp Gains for LTI Systems. 6.5 Closed Loop Input-Output Stability. 6.6 The Small Gain Theorem. 6.7 Loop Transformations. 6.8 The Circle Criterion. 6.9 Exercises. Input-to-State Stability. 7.1 Motivation. 7.2 Definitions. 7.3 Input-to-State Stability (ISS) Theorems. 7.4 Input-to-State Stability Revisited. 7.5 Cascade Connected Systems. 7.6 Exercises. Passivity. 8.1 Power and Energy: Passive Systems. 8.2 Definitions. 8.3 Interconnections of Passivity Systems. 8.4 Stability of Feedback Interconnections. 8.5 Passivity of Linear Time-Invariant Systems. 8.6 Strictly Positive Real Rational Functions. Exercises. Dissipativity. 9.1 Dissipative Systems. 9.2 Differentiable Storage Functions. 9.3 QSR Dissipativity. 9.4 Examples. 9.5 Available Storage. 9.6 Algebraic Condition for Dissipativity. 9.7 Stability of Dissipative Systems. 9.8 Feedback Interconnections. 9.9 Nonlinear L2 Gain. 9.10 Some Remarks about Control Design. 9.11 Nonlinear L2-Gain Control. 9.12 Exercises. Feedback Linearization. 10.1 Mathematical Tools. 10.2 Input-State Linearization. 10.3 Examples. 10.4 Conditions for Input-State Linearization. 10.5 Input-Output Linearization. 10.6 The Zero Dynamics. 10.7 Conditions for Input-Output Linearization. 10.8 Exercises. Nonlinear Observers. 11.1 Observers for Linear Time-Invariant Systems. 11.2 Nonlinear Observability. 11.3 Observers with Linear Error Dynamics. 11.4 Lipschitz Systems. 11.5 Nonlinear Separation Principle. Proofs. Bibliography. List of Figures. Index.

Variable universe stable adaptive fuzzy control of a nonlinear system

A kind of stable adaptive fuzzy control of a nonlinear system is implemented based on the variable universe method proposed first in [1]. First of all, the basic structure of variable universe adaptive fuzzy controllers is briefly introduced. Then the contraction-expansion factor which is a key tool of the variable universe method is defined by means of the integral regulation idea, and then a kind of adaptive fuzzy controller is designed by using such a contraction-expansion factor. The simulation on the first-order nonlinear system is done, and as a result, its simulation effect is quite good in comparison with the corresponding result in [2,3]. Second, it is proved that the variable universe adaptive fuzzy control is asymptotically stable by use of Liapunov theory. The simulation on a second-order nonlinear system shows that its simulation effect is also quite good in comparison with the corresponding result in [2]. Besides, a useful tool, called symbolic factor, is proposed, which may be of universal significance. It can greatly reduce the setting time and enhance the robustness of the system.

A regulator for a class of unknown continuous-time nonlinear systems

We introduce a control law for a class of unknown nonlinear continuous-time systems in which full state measurements are available. We show that as long as a certain feedback gain parameter is sufficiently large, the closed-loop system is stable. Furthermore, the magnitude of the control is bounded in the limit. As a corollary to the main result, we show that first-order nonlinear systems in the class we consider can be stabilized using linear proportional integral controllers, and we give explicit tuning rules for these controllers.

Neural network-based model reference adaptive control system

In this paper, an approach to model reference adaptive control based on neural networks is proposed and analyzed for a class of first-order continuous-time nonlinear dynamical systems. The controller structure can employ either a radial basis function network or a feedforward neural network to compensate adaptively the nonlinearities in the plant. A stable controller-parameter adjustment mechanism, which is determined using the Lyapunov theory, is constructed using a sigma-modification-type updating law. The evaluation of control error in terms of the neural network learning error is performed. That is, the control error converges asymptotically to a neighborhood of zero, whose size is evaluated and depends on the approximation error of the neural network. In the design and analysis of neural network-based control systems, it is important to take into account the neural network learning error and its influence on the control error of the plant. Simulation results showing the feasibility and performance of the proposed approach are given.