The shifted Legend re approach to non-linear system analysis and identification

Applications of the shifted Legendre polynomial expansion to the analysis and identification of the non-linear system described by a Hammerstein model which consists of a single-valued non-linearity followed by a linear plant are studied. For the analysis, by using the shifted Legendre polynomial expansion, the solution of a non-linear state equation is reduced to the solution of a linear algebraic matrix equation. For the identification, through the shifted Legendre expansions of the measured input-output data, the unknown parameters of both the linear plant and the characterization of the non-linear element are estimated using the least-squares method. Algorithms are presented. Numerical examples are given to illustrate the use of this approach.     

Shifted Jacobi series solution of simultaneous linear distributed systems

A double Jacobi series is introduced to approximate functions of two independent variables. It is then applied to analysing simultaneous linear distributed parameter systems. The solution for the coefficient matrices can be obtained directly from a Kronecker product formula. The method is algebraic and computer oriented. One illustrative example is given for demonstration. Very satisfactory results are obtained, owing to the rapid convergence of the shifted Jacobi series.     

Optimal control using an algebraic method for control-affine non-linear systems

A deterministic optimal control problem is solved for a control-affine non-linear system with a non-quadratic cost function. We algebraically solve the Hamilton–Jacobi equation for the gradient of the value function. This eliminates the need to explicitly solve the solution of a Hamilton–Jacobi partial differential equation. We interpret the value function in terms of the control Lyapunov function. Then we provide the stabilizing controller and the stability margins. Furthermore, we derive an optimal controller for a control-affine non-linear system using the state dependent Riccati equation (SDRE) method; this method gives a similar optimal controller as the controller from the algebraic method. We also find the optimal controller when the cost function is the exponential-of-integral case, which is known as risk-sensitive (RS) control. Finally, we show that SDRE and RS methods give equivalent optimal controllers for non-linear deterministic systems. Examples demonstrate the proposed methods.     

Jacobi–Gauss–Lobatto collocation approach for non-singular variable-order time fractional generalized Kuramoto–Sivashinsky equation

This paper introduces the non-singular variable-order (VO) time fractional version of the generalized Kuramoto–Sivashinsky (GKS) equation with the aid of fractional differentiation in the Caputo–Fabrizio sense. The Jacobi–Gauss–Lobatto collocation technique is developed for solving this equation. More precisely, the derivative matrix of the classical Jacobi polynomials and the VO fractional derivative matrix of the shifted Jacobi polynomials (which is obtained in this study) together with the collocation technique are used to transform the solution of problem into the solution of an algebraic system of equations. Numerical simulations for several test problems have been shown to accredit the established algorithm.

     

Application and comparison of different identification schemes under industrial conditions

Two new identification procedures based on different strategies are presented to determine a mathematical model for the crude iron quality of a blast furnace process. The first algorithm estimates the parameters of a common linear difference equation model based on identification results obtained from separate series of data with given model structure. Prior to complete estimation of the parameters, the second method automatically detects the significant terms of a general model to optimally characterize a linear or non-linear system. The process is described by a general discrete polynomial expansion of past and present input signals and past output signals. This algorithm is based on orthogonalization and application of several information criteria. A comparison of the prediction accuracy of the linear model obtained from the first method with the linear and non-linear model resulting from the second algorithm and with models developed by other identification procedures is presented and some experiences from this application are discussed.     

Solution of linear two-point boundary-value problems via shifted Chebyshev series

In this paper, the shifted Chebyshev polynomial functions approximation is extended to solve the linear ordinary differential equation of the two-point boundary-value problem. The linear ordinary differential equation of boundary-value problems are reduced to the linear functional differential equation of the initial-value problem. A new time-domain approach to the derivation of a Chebyshev transformation matrix is presented. Using the derived Chebyshev transformation matrix together with the Chebyshev integration matrix, the solution of the linear functional ordinary differential equation of initial-value problem can be obtained via shifted Chebyshev series. Two examples are given and the satisfactory computational results are compared with those of the exact solution.     

H ∞ control for non-linear stochastic systems: the output-feedback case

The H _∞ output-feedback control problem for non-linear stochastic systems is considered. A solution for a large class of non-linear stochastic systems is introduced (including non-linear diffusion systems as a subclass). This solution is based on a bounded real lemma for non-linear stochastic systems that was previously established via a stochastic dissipativity concept. The theory yields sufficient conditions for the closed-loop system to possess a prescribed L ₂-gain bound in terms of two Hamilton Jacobi inequalities: one that is associated with the state feedback part of the problem is n-dimensional (where n is the underlying system's state dimension) and the other inequality that stems from the estimation part is 2n-dimensional. Both stationary and non–stationary systems are considered. Stability of the closed-loop system is established, both in the mean-square and the in-probability senses. As the solution to the Hamilton Jacobi inequalities may, in general, lead to a non–realisable state estimator, a modification of the associated 2n-dimensional Hamilton Jacobi inequality is made in order to circumvent this realisation problem, while preserving the system's L ₂-gain bound. For time-invariant systems, the problem of robust output-feedback is considered in the case of norm-bounded uncertainties. A solution is then derived in terms of linear state-dependent matrix inequalities.     

Contraction analysis of non-linear distributed systems

Contraction theory is a comparatively recent dynamic analysis and non-linear control system design tool based on an exact differential analysis of convergence. This paper extends contraction theory to local and global stability analysis of important classes of non-linear distributed dynamics, such as convection-diffusion-reaction processes, Lagrangian and Hamilton–Jacobi dynamics, and optimal controllers and observers. By contrast with stability proofs based on energy dissipation, stability and convergence can be determined for energy-based systems excited by time-varying inputs.

The Hamilton–Jacobi–Bellman controller and a similar optimal non-linear observer design are studied based on explicitly computable conditions on the convexity of the cost function. These stability conditions extend the well-known conditions on controllability and observability Grammians for linear time-varying systems, without requiring the unknown transition matrix of the underlying differential dynamics.     

A tau method approach for the diffusion equation with nonlocal boundary conditions

An approximate method for solving the diffusion equation with nonlocal boundary conditions is proposed. The method is based upon constructing the double shifted Legendre series to approximate the required solution using Legendre tau method. The differential and integral expressions which arise in the diffusion equation with nonlocal boundary conditions are converted into a system of linear algebraic equations which can be solved for the unknown coefficients. Numerical examples are included to demonstrate the validity and applicability of the method and a comparison is made with existing results.     

Finite-horizon differential games for missile–target interception system using adaptive dynamic programming with input constraints

In this paper, the problem of intercepting a manoeuvring target within a fixed final time is posed in a non-linear constrained zero-sum differential game framework. The Nash equilibrium solution is found by solving the finite-horizon constrained differential game problem via adaptive dynamic programming technique. Besides, a suitable non-quadratic functional is utilised to encode the control constraints into a differential game problem. The single critic network with constant weights and time-varying activation functions is constructed to approximate the solution of associated time-varying Hamilton–Jacobi–Isaacs equation online. To properly satisfy the terminal constraint, an additional error term is incorporated in a novel weight-updating law such that the terminal constraint error is also minimised over time. By utilising Lyapunov's direct method, the closed-loop differential game system and the estimation weight error of the critic network are proved to be uniformly ultimately bounded. Finally, the effectiveness of the proposed method is demonstrated by using a simple non-linear system and a non-linear missile–target interception system, assuming first-order dynamics for the interceptor and target.     

Identification of discrete Hammerstein systems by the Fourier series regression estimate

The identification of a single-input, single-output (SISO) discrete Hammerstein system is studied. Such a system consists of a non-linear memoryless subsystem followed by a dynamic, linear subsystem. The parameters of the dynamic, linear subsystem are identified by a correlation method and the Newton-Gauss method. The main results concern the identification of the non-linear, memoryless subsystem. No conditions are imposed on the functional form of the non-linear subsystem, recovering the non-linear using the Fourier series regression estimate. The density-free pointwise convergence Of the estimate is proved, that is.algorithm converges for all input densities The rate of pointwise convergence is obtained for smooth input densities and for non-linearities of Lipschitz type.Globle convergence and its rate are also studied for a large class of non-linearities and input densities     

On Popov criteria for SPFM systems

The problem of the identification of a non-linear continuous system is to a considerable extent a problem of representation of the structure of the system. A simple decomposition of a stationary single input — single output non-linear Volterra system is given, separating this system into a linear state equation and a non-linear (polynomial) output equation.     

Recursive identification of linear and non-linear systems

The paper develops recursive techniques for off-line identification of linear and nonlinear systems. It is shown that if the system is linear and time invariant, impulse response characterization of the system coupled with an orthogonal series approximation can be utilized for the purpose stated above. The techniques of adaptive Kalman filtering are shown to be applicable, which besides permitting recursive evaluation of the coefficients, lead to a number of important advantages. In the second part of the study, the proposed method is extended for recursive identification of a class of non-linear systems which can be represented as a cascade combination of a linear dynamical system and a non-linear zero memory system. The method of Volterra series representation of such systems is utilized. Results are illustrated through numerical examples in each case.     

Solution of control problems by multi-dimensional non-stationary systems by the diagonalization of the matrix of the linear part of the motion equation system

An approach to the solution of control problems by multi-dimensional non-stationary systems (linear or non-linear) is given. The linear part of the motion equation system (the equation system of the first approximation motion) is transformed canonically to a form where the solution of the transformed equation system can be represented in quadratures.     

Analysis of time-delay systems using the Galerkin method

This paper is concerned with the analysis of a time-varying, variable time-delay system using the Galerkin method. The approximate solution is expanded in a set of shifted Legendre polynomials with unknown expansion coefficients. A delay matrix is derived to manipulate the time delay. The original delay-differential equation is converted to a set of algebraic equations through setting the weighted integrals of the residual to zero. An approximate shifted Legendre series solution to the time delay system is thus obtained by solving the resultant algebraic equations. Numerical results are provided to illustrate the applicability of the method.     

Accuracy and convergence of a finite element algorithm for laminar boundary layer flow

The Galerkin-weighted residuals formulation is employed to derive an implicit finite element solution algorithm for a generally non-linear initial-boundary value problem. Solution accuracy and convergence with discretization refinement are quantized in several error norms, for the non-linear parabolic partial differential equation system governing laminar boundary layer flow, using linear, quadratic and cubic functions. Richardson extrapolation is used to isolate integration truncation error in all norms, and Newton iteration is employed for all equation solutions performed in double-precision. The mathematical theory supporting accuracy and convergence concepts for linear elliptic equation appears extensible to the non-linear equations characteristic of laminar boundary layer flow.     

Numerical solution of fractional delay differential equation by shifted Jacobi polynomials

In this paper, the fractional delay differential equation (FDDE) is considered for the purpose to develop an approximate scheme for its numerical solutions. The shifted Jacobi polynomial scheme is used to solve the results by deriving operational matrix for the fractional differentiation and integration in the Caputo and Riemann–Liouville sense, respectively. In addition to it, the Jacobi delay coefficient matrix is developed to solve the linear and nonlinear FDDE numerically. The error of the approximate solution of proposed method is discussed by applying the piecewise orthogonal technique. The applicability of this technique is shown by several examples like a mathematical model of houseflies and a model based on the effect of noise on light that reflected from laser to mirror. The obtained numerical results are tabulated and displayed graphically.     

Graphical analysis of a class of non-linear feedback control systems with step input†

The class of systems considered in this investigation is a cascade combination of a linear memory system and a non-linear no-memory system in the forward path of a unity feedback control system. The output of the non-linear no-memory system is assumed to be a polynomial function of the input. Regardless of the exact nature of the non-linearity, the objective of this method of analysis is to predict the behaviour of higher-order non-linear systems with different initial conditions for step inputs.

Two different cascade combinations of linear and non-linear blocks in the forward path are considered. For both configurations a similar non-linear differential equation is obtained for some variable in the system. The non-linear differential equation is further reduced to a first-order equation, explicitly independent of the independent variable, time t. Treating all other coefficients as parameters and eliminating each in turn, finally the required phase-plane trajectory is obtained.     

Non-linear impulsive dynamical systems. Part II: Stability of feedback interconnections and optimality

In a companion paper (Nonlinear Impulsive Dynamical Systems. Part I: Stability and Dissipativity) Lyapunov and invariant set stability theorems and dissipativity theory were developed for non-linear impulsive dynamical systems. In this paper we build on these results to develop general stability criteria for feedback interconnections of non-linear impulsive systems. In addition, a unified framework for hybrid feedback optimal and inverse optimal control involving a hybrid non-linear-non-quadratic performance functional is developed. It is shown that the hybrid cost functional can be evaluated in closed-form as long as the cost functional considered is related in a specific way to an underlying Lyapunov function that guarantees asymptotic stability of the non-linear closed-loop impulsive system. Furthermore, the Lyapunov function is shown to be a solution of a steady-state, hybrid Hamilton‐Jacobi‐Bellman equation.     

Obtaining real solutions for non-linear systems using certain transformation techniques†

Certain non-linear systems can be transformed into equivalent linear systems using power law transformations of the dependent variable. The conditions for obtaining a real solution using this technique are examined for an example non-linear oscillator. Although the requirement of a real solution can place restrictions on the parameters of the equation, a technique is suggested whereby a real solution can be obtained to an approximating equation which differs from the actual equation by an arbitrarily small amount.