The linear periodic output regulation problem

The problem of asymptotic output regulation for linear systems driven by time-varying, T-periodic exosystems is considered in this paper. Necessary and sufficient condition for its solvability based on the existence of periodic solutions of differential Sylvester equations are derived. These conditions constitute a generalization to the periodic case of the celebrated algebraic regulator equations of Francis. A general algorithm for the synthesis of an error-feedback regulator is given. For the case of minimum-phase systems, it is shown that the regulator design can be carried out without the knowledge of the Floquet decomposition of the exosystem, thus extending significantly the applicability of the general result. The more challenging issue of robust regulation by error feedback is also addressed, and solved under a stronger observability condition.     

The convergence of a differential-difference scheme of gas dynamic equations in Lagrangian mass variables

The convergence to a smooth solution of a completely conservative differential-difference scheme of gas dynamic equations in Lagrangian mass variables with sources (sinks) is investigated for the case of the ideal gas. It is proved that for the class of sufficiently smooth solutions of the differential problem the solution of the difference problem converges in the mesh norm L ₂ and that the rate of convergence is O(h ²).     

Convergence and computation of describing functions using a recursive Volterra series

Presented in this paper is a comparison of algorithms for computing an approximation to the sinusoidal input describing function (SIDF) for the nonlinear differential equation ?(t)+b₁y(t)+b₂u²(t)y(t) = K(u?(t)+b₃u(t)) The importance of this nonlinear differential equation comes from the context of nonlinear feedback controller design. Specifically, this equation is either a linear lead or lag controller (depending on the coefficient values) augmented with a nonlinear, polynomial type term. Consequently, obtaining a SIDF representation of this nonlinear differential equation or creating a process to obtain SIDFs for other similar differential equations, will facilitate nonlinear controller design using classical loop shaping tools. The two SIDF approximations studied include the well‐established harmonic balance method and a Volterra series based algorithm. In applying the Volterra series, several theoretical issues were addressed including the development of a recursive solution that calculates high order Volterra transfer functions, and the guarantee of convergence to an arbitrary accuracy. Throughout the paper, case studies are presented. Copyright © 2004 John Wiley & Sons, Ltd.     

Suboptimal stochastic linear feedback control of linear systems with state- and control-dependent noise: The incomplete information case

The stochastic regulation problem for linear systems with state- and control-dependent noise and a noisy linear output equation is considered. The optimal quadratic cost output-feedback control law in a class of linear controllers is found. This problem was first addressed in the early 1970s and solved, in the complete information case, by Wonham. In this paper we give the solution of the problem in the incomplete information case, that is, for a linear output equation corrupted by Gaussian noise. Moreover, a different method is used here, giving the solution in a more direct way even in the complete information case.     

Global stability for separable nonlinear delay differential equations

Consider the following separable nonlinear delay differential equation

, where we assume that, there is a strictly monotone increasing function f(x) on (−∞, +∞) such that

In this paper, to the above separable nonlinear delay differential equation, we establish conditions of global asymptotic stability for the zero solution. In particular, for a special wide class of f(x) which contains a case of f(x) = e^x−1, we give more explicit conditions. Applying these, we offer conditions of global asymptotic stability for solutions of nonautonomous logistic equations with delays.     

Patchy solution of a Francis–Byrnes–Isidori partial differential equation

The solution to the nonlinear output regulation problem requires one to solve a first‐order partial differential equation, known as the Francis–Byrnes–Isidori equations. In this paper, we propose a method to compute approximate solutions to the Francis–Byrnes–Isidori equations when the zero dynamics of the plant are hyperbolic and the exosystem is two dimensional. With our method, we are able to produce approximations that converge uniformly to the true solution. Our method relies on the periodic nature of two‐dimensional analytic center manifolds. Copyright © 2012 John Wiley & Sons, Ltd.     

An inequality governing nonlinear H∞ control

This note gives necessary and sufficient conditions for solving a reasonable version of the nonlinear H^∞ control problem. The most objectionable hypothesis is elegant and holds in the linear case, but every possibly may not be forced for nonlinear systems. What we discover in distinction to Isidori and Astolfi (1992) and Ball et al. (1993) is that the key formula is not a (nonlinear) Riccati partial differential inequality, but a much more complicated inequality mixing partial derivatives and an approximation theoretic construction called the best approximation operator. This Chebeshev-Riccati inequality when specialized to the linear case gives the famous solution to the H^∞ control problem found in Doyle et al. (1989). While complicated the Chebeshev-Riccati inequality is (modulo a considerable number of hypotheses behind it) a solution to the nonlinear H^∞ control problem. It should serve as a rational basis for discovering new formulas and compromises. We follow the conventions of Ball et al. (1993) and this note adds directly to that paper.     

Risk-sensitive control of Markov jump linear systems: Caveats and difficulties

In this technical note, we revisit the risk-sensitive optimal control problem for Markov jump linear systems (MJLSs). We first demonstrate the inherent difficulty in solving the risk-sensitive optimal control problem even if the system is linear and the cost function is quadratic. This is due to the nonlinear nature of the coupled set of Hamilton-Jacobi-Bellman (HJB) equations, stemming from the presence of the jump process. It thus follows that the standard quadratic form of the value function with a set of coupled Riccati differential equations cannot be a candidate solution to the coupled HJB equations. We subsequently show that there is no equivalence relationship between the problems of risk-sensitive control and H ^∞ control of MJLSs, which are shown to be equivalent in the absence of any jumps. Finally, we show that there does not exist a large deviation limit as well as a risk-neutral limit of the risk-sensitive optimal control problem due to the presence of a nonlinear coupling term in the HJB equations.

     

Efficient MFS Algorithms for Inhomogeneous Polyharmonic Problems

In this work we develop an efficient algorithm for the application of the method of fundamental solutions to inhomogeneous polyharmonic problems, that is problems governed by equations of the form Δ ^ℓ u=f, ℓ∈ℕ, in circular geometries. Following the ideas of Alves and Chen (Adv. Comput. Math. 23:125–142, 2005), the right hand side of the equation in question is approximated by a linear combination of fundamental solutions of the Helmholtz equation. A particular solution of the inhomogeneous equation is then easily obtained from this approximation and the resulting homogeneous problem in the method of particular solutions is subsequently solved using the method of fundamental solutions. The fact that both the problem of approximating the right hand side and the homogeneous boundary value problem are performed in a circular geometry, makes it possible to develop efficient matrix decomposition algorithms with fast Fourier transforms for their solution. The efficacy of the method is demonstrated on several test problems.     

Robust performance control of uncertain nonlinear systems: A self-tuning approach

The problem of output regulation with a guaranteed H^∞ performance, besides robust stability, for a class of feedback linearizable nonlinear systems via self-tuning controllers is investigated. The H^∞ performance consists of the desired disturbance attenuation and internal finite L²-gain stability. We show that, under the perturbations of matched parametric uncertainties, the sufficient condition for the existence of a self-tuning controller reduces to the solvability of a single Hamilton-Jacobi-Isaacs inequality (or algebraic Riccati equation) which indicates that the design can be performed as if the system uncertainties were absent. Under certain situations, the sufficient condition is also necessary. Once a solution of this nonlinear differential inequality (or algebraic equation) is available, a desired self-tuning controller with gradient-type parameter estimator can easily be constructed. The present work falls into the category of singular nonlinear H^∞ control since the desired H^∞ performance does not require any penalty on control input variables. The results also provide an immediate application to H^∞ self-tuning model reference control in linear systems under full state measurement.     

Decomposition and stability of linear systems with multiplicative noise

This paper concerns a representation of solutions and the stability of linear systems with multiplicative white noise, which is described by a vector Ito stochastic differential equation. The solution can be represented as a finite product of exponential matrices if Lie algebra generated by system matrices is solvable. If Lie algebra is not solvable, it is shown by the decomposition principle of Lie algebra that the problem of solving an equation can be reduced to the problem of solving a set of equations, whose corresponding Lie algebra is simple. Noting the structure of the sample solution, we present a technique of obtaining asymptotic stability conditions of sample solutions w.p.1, in the pth-order moment and in the pth-mean moment. The necessary and/or sufficient conditions of stability in some stochastic sense are obtained under certain conditions.     

Coupling of optimal variation of parameters method with Adomian’s polynomials for nonlinear equations representing fluid flow in different geometries

In this study, we describe a modified analytical algorithm for the resolution of nonlinear differential equations by the variation of parameters method (VPM). Our approach, including auxiliary parameter and auxiliary linear differential operator, provides a computational advantage for the convergence of approximate solutions for nonlinear boundary value problems. We consume all of the boundary conditions to establish an integral equation before constructing an iterative algorithm to compute the solution components for an approximate solution. Thus, we establish a modified iterative algorithm for computing successive solution components that does not contain undetermined coefficients, whereas most previous iterative algorithm does incorporate undetermined coefficients. The present algorithm also avoid to compute the multiple roots of nonlinear algebraic equations for undetermined coefficients, whereas VPM required to complete calculation of solution by computing roots of undetermined coefficients. Furthermore, a simple way is considered for obtaining an optimal value of an auxiliary parameter via minimizing the residual error over the domain of problem. Graphical and numerical results reconfirm the accuracy and efficiency of developed algorithm.

     

Volterra series for solving weakly non-linear partial differential equations: application to a dissipative Burgers’ equation

A method to solve weakly non-linear partial differential equations with Volterra series is presented in the context of single-input systems. The solution x(z,t) is represented as the output of a z-parameterized Volterra system, where z denotes the space variable, but z could also have a different meaning or be a vector. In place of deriving the kernels from purely algebraic equations as for the standard case of ordinary differential systems, the problem turns into solving linear differential equations. This paper introduces the method on an example: a dissipative Burgers'equation which models the acoustic propagation and accounts for the dominant effects involved in brass musical instruments. The kernels are computed analytically in the Laplace domain. As a new result, writing the Volterra expansion for periodic inputs leads to the analytic resolution of the harmonic balance method which is frequently used in acoustics. Furthermore, the ability of the Volterra system to treat other signals constitutes an improvement for the sound synthesis. It allows the simulation for any regime, including attacks and transients. Numerical simulations are presented and their validity are discussed.     

Global nonlinear output regulation: Convergence-based controller design

In this paper we present output-feedback controllers solving the global output regulation problem for a class of nonlinear systems. The proposed controllers are based on the notion of convergent systems. The presented solution extends well-established results on the linear output regulation problem and the local nonlinear output regulation problem to the global case. For Lur’e systems, which are not necessarily in the output-feedback form, the proposed controllers can be found by solving the regulator equations and certain linear matrix inequalities. For systems in the output-feedback form with uncertain parameters and uncertain nonlinearities we provide a robust regulator that does not rely on the minimum phaseness assumption on the system, which is crucial in the previous regulator designs for output-feedback systems. The results are illustrated by examples.     

Global -gain design for a class of nonlinear systems

It is known that the so-called control problem of a nonlinear system is locally solvable if the corresponding problem for the linearized system can be solved by linear feedback. In this paper we prove that this condition suffices to solve also a global control problem, for a fairly large class of nonlinear systems, if one is free to choose a state-dependent weight of the control input. Using a two-way (backward and forward) recursive induction argument, we simultaneously construct, starting from a solution of the Riccati algebraic equation, a global solution of the Hamilton–Jacobi–Isaacs partial differential equation arising in the nonlinear control, as well as a state feedback control law that achieves global disturbance attenuation with internal stability for the nonlinear systems.     

Global L2-gain design for a class of nonlinear systems

It is known that the so-called H_∞ control problem of a nonlinear system is locally solvable if the corresponding problem for the linearized system can be solved by linear feedback. In this paper we prove that this condition suffices to solve also a globalH_∞ control problem, for a fairly large class of nonlinear systems, if one is free to choose a state-dependent weight of the control input. Using a two-way (backward and forward) recursive induction argument, we simultaneously construct, starting from a solution of the Riccati algebraic equation, a global solution of the Hamilton–Jacobi–Isaacs partial differential equation arising in the nonlinear H_∞ control, as well as a state feedback control law that achieves global disturbance attenuation with internal stability for the nonlinear systems.     

Anisotropic <Emphasis Type="Italic">ε</Emphasis>-optimal model reduction for linear discrete time-invariant system

We consider an ε-optimal model reduction problem for a linear discrete time-invariant system, where the anisotropic norm of reduction error transfer function is used as a performance criterion. For solving the main problem, we state and solve an auxiliary problem of H ₂ ε-optimal reduction of a weighted linear discrete time system. A sufficient optimality condition defining a solution to the anisotropic ε-optimal model reduction problem has the form of a system of cross-coupled nonlinear matrix algebraic equations including a Riccati equation, four Lyapunov equations, and five special-type nonlinear equations. The proposed approach to solving the problem ensures stability of the reduced model without any additional technical assumptions. The reduced-order model approximates the steady-state behavior of the full-order system.     

Decentralized multivariable voltage and speed regulator for large-scale power systems with guarantee of stability and transient performance

This paper presents a novel procedure for the design of decentralized regulators for large power systems with a formal proof of ‘global’ stability. The distinctive feature of the solution is that both voltage and rotor speed dynamics are regulated simultaneously contrary to most of the solutions proposed so far in the literature. First, the traditional multimachine power system algebraic-differential equations are reformatted into suitable state equations, more appropriate for modern control tools. Secondly, a voltage and speed controller based on this model is proposed. The design consists of first cancelling some of the dynamical model non-linearities using non-linear excitation and valve input. The resulting subsystems are stabilized by auxiliary controls with linear and non-linear components. The non-linear component, which uses local signals to dominate those with interconnections, is derived from a stability criterion involving the Lyapunov function of the entire power system. The gains of the linear component are computed from the solution of an algebraic Riccati equation similar to the one involved in the full information H _∞ problem. These gains guarantee that effects of interconnection signals on voltage and speed dynamics are considerably reduced. The benefit of the proposed scheme is that the voltage regulation characteristic ensures a good post-fault voltage profile which helps improve rotor oscillations damping. Simulation results on a realistic power system confirm that the system stability is considerably improved in presence of severe contingencies.     

The solution of a nonclassic problem for one-dimensional hyperbolic equation using the decomposition procedure

In this article, we propose a new approach for solving an initial–boundary value problem with a non-classic condition for the one-dimensional wave equation. Our approach depends mainly on Adomian's technique. We will deal here with new type of nonlocal boundary value problems that are the solution of hyperbolic partial differential equations with a non-standard boundary specification. The decomposition method of G. Adomian can be an effective scheme to obtain the analytical and approximate solutions. This new approach provides immediate and visible symbolic terms of analytic solution as well as numerical approximate solution to both linear and nonlinear problems without linearization. The Adomian's method establishes symbolic and approximate solutions by using the decomposition procedure. This technique is useful for obtaining both analytical and numerical approximations of linear and nonlinear differential equations and it is also quite straightforward to write computer code. In comparison to traditional procedures, the series-based technique of the Adomian decomposition technique is shown to evaluate solutions accurately and efficiently. The method is very reliable and effective that provides the solution in terms of rapid convergent series. Several examples are tested to support our study.     

Linear Operator Inequalities for Strongly Stable Weakly Regular Linear Systems

We consider the question of the existence of solutions to certain linear operator inequalities (Lur'e equations) for strongly stable, weakly regular linear systems with generating operators A, B, C, 0. These operator inequalities are related to the spectral factorization of an associated Popov function and to singular optimal control problems with a nonnegative definite quadratic cost functional. We split our problem into two subproblems: the existence of spectral factors of the nonnegative Popov function and the existence of a certain extended output map. Sufficient conditions for the solvability of the first problem are known and for the case that A has compact resolvent and its eigenvectors form a Riesz basis for the state space, we give an explicit solution to the second problem in terms of A, B, C and the spectral factor. The applicability of these results is demonstrated by various heat equation examples satisfying a positive-real condition. If (A, B) is approximately controllable, we obtain an alternative criterion for the existence of an extended output operator which is applicable to retarded systems. The above results have been used to design adaptive observers for positive-real infinite-dimensional systems. Date received: July 25, 1997. Date revised: February 10, 2001.