Nonlinear Controller Design via Inferenced-Augmentation of Equivalent Linearized System

A survey of the authors' recent contributions are given to a new multi-constrained and multi-criteria optimization approach to the design of optimal compensators for general MIMO nonlinear feedback control systems in several practical considerations including such as robust stabilization with the presence of uncertainty, tracking and model matching, and disturbance rejection problems. First, the general framework for nonlinear closed-loop feedback systems is described in a Banach space setting in the time domain. Then, several typical optimal feedback design problems are formulated. Moreover, existence, uniqueness and characteristics theorems are established. Finally, a convergent recursive algorithm for solving the general constrained optimization is included.

It is generally impossible to obtain the exact response of nonlinear control systems by the integral method. It is necessary, therefore, to use the approximate methods such as perturbation, iteration or others, when we intend to study the systems by the analytical method. The solution will be represented by infinite series, and the first several terms will afford an approximate solution. Then, the convergency of the solution and the accuracy of the approximate solution must be estimated, being compared with the numerical solution, which has a significant meaning in the theoretical study of nonlinear control systems. The author devised a modified Runge-Kutta method. He proved that the mcdified method is less troublesome in calculation with better accuracy than the conventional method in the illustrative analysis of the indicial response of gain-saturated second order servomechanism. He believes that the modified method will be superior to the conventional one in the general nonlinear gain systems or, moreover, in the general nonlinear control systems.

Non-intrusive frequency response analysis of nonlinear systems with interval uncertainty: A comparative study

Specific features of nonlinear systems determined by the energy interaction of their coordinates are considered. To assess these features, the potential energy surfaces are considered, on which the system dynamics is presented using the trajectories of motion of an image point. The specific features of these trajectories are determined by the topological characteristics of the potential energy surface in the configuration space (local surface curvatures). Unlike linear systems, nonlinear systems are characterized by the presence on the energy surface of additional extremal curvatures and characteristic points: in addition to elliptic points, parabolic and hyperbolic points appear, which determine the character of geodetic lines near these extrema and points. The specific features of the geodetic lines determine the character of free and forced vibrations in the system, its dynamic features, and the presence of nonlinear effects in the system.

This paper proposes a novel cerebellar model articulation controller (CMAC) based controller design method for nonlinear mechanical systems. CMAC is employed to approximate and compensate the uncertainties induced by inaccurate modelling of the system to be controlled. A mathematical model of CMAC in which the adjustable parameters appear linearly is derived and the stability of the overall system is proved using the Lyapunov theory. The authors also extend the proposed scheme to obtain robustness under the bounded disturbances and the approximation error of the CMAC. Although the control problem of a simple nonlinear system is considered in this work, the proposed scheme can easily be extended to more general nonlinear systems.

Dantzig (1963), Orchard-Hays (1968) and other pioneers have helped to develop linear programming theory and applications. Additionally an elegant solution technique is available for linear systems of equations. However there is a need to be able to solve the general non-linear model and non-linear systems of equations. Multi-stage Monte Carlo optimization along with the absolute value transformations presented here show some promise in solving difficult non-linear problems. They also perform well on many linear problems and tend to streamline the solution process and give the practitioner more freedom to develop accurate non-linear models and systems. The fundamental theorem of linear programming tells us that the optimal solution is at a ‘corner point’ of the n-dimensional constrained solution space. The simplex technique goes along the edges of the constrained region from corner point to corner point until it finds the optimal solution. However, with multistage Monte Carlo for non-linear problems, the s...

The method using interval observers to estimate parameters have been verified in linear system and simple nonlinear cooperative system (elements of Jacobian matrix on the non-diagonal are non-negative). We can see the concept about cooperative system in references, but it has some limitations for reality system. This paper tries to use interval observer method for more general nonlinear system which is not a cooperative system. The Jacobian matrix of the observer can meet the condition of cooperative system by properly choosing the gain of the observer, hence the observer becomes a monotonic observer. By using this method, one can determine whether the parameter interval contains the value of the system parameter. The feasibility of this idea is illustrated by the simulation example using a nonlinear dynamic system model.

The authors prove that generically a nonlinear autonomous system on a compact d-dimensional manifold has an observer of dimension 2d+1. This is a consequence of the result that, for a generic d-dimensional nonlinear system, the state can be reconstructed on the basis of the 2d first derivatives of the output. >

This is the first part of a paper on an innovative approach of studying general nonlinear structural dynamic systems. In this part, the theoretical background and some numerical simulations are presented. Firstly, a frequency domain nonlinear compensation method is proposed. It can be used to recover the linear path of a general non-linear system as long as the Volterra expansion is held between the input and the output. Secondly, a new nonlinear element location scheme is developed based on the system linear path identification. Lastly, as a natural extension in the development of the contents of this paper, an in-situ nonlinear restoring force state mapping technique is proposed. The use of these new methods is demonstrated through several numerical examples of either single or multiple DOF systems incorporating some typical types of nonlinearities.

The receptance method has been proved to be effective for active vibration control of linear systems, theoretically and experimentally. A remarkable advantage of this method is that it does not require any prior knowledge of the system matrices, which are typically obtained from the finite element method. In this paper, a modified receptance method is proposed for obtaining the time-varying receptances of a general nonlinear system; the method is then applied to a structurally nonlinear aeroelastic system for active flutter suppression. A two-dimensional aeroelastic model with a polynomial nonlinearity in the pitching stiffness is considered. The Runge-Kutta method is used for numerical simulations. The poles of the closed-loop system are assigned to improve the system stability; both fixed and varied poles cases are studied. The effects of poles values, weight coefficients in indicator function, and control limitations are discussed. Results show that the aforementioned parameters affect both the system response and the input of the control surface, and the modified receptance method is effective for active flutter suppression of the nonlinear aeroelastic system by rational pole assignments.

ABSTRACTThis paper addresses the identification of Nonlinear (NL) systems using a linearization approach, introducing a combined estimator to tackle this challenge. We assume that the unknown NL system operates around a stable, slowly varying (nominal) operating point. The system trajectory is then perturbed slightly via small input perturbations. While the system's operating point evolves slowly over time, the small (broadband) input signal excites the system dynamics. We demonstrate that the NL system's response to these small perturbations can be approximated by a Linear Parameter‐Varying (LPV) system model. Furthermore, we show that this LPV model represents the linearized version of the unknown NL system around the operating point. A new parametrization for the LPV model coefficients, referred to as “gradient‐parameterized” LPV coefficients, is introduced, establishing a structural relationship between the coefficients. This structural relationship reduces the number of parameters to be estimated and ensures that the LPV model always corresponds to the linearized form of the NL system. Additionally, we demonstrate that this LPV model structure allows for the unique reconstruction of the NL system model through symbolic integration, resulting in a closed‐form nonlinear Ordinary Differential Equation (ODE). This integration introduces a second structural relationship, linking the LPV model to the NL model. By leveraging these two structural relationships, we reformulate the problem of NL system identification via linearization as a combined estimation problem, leading to a unified LPV‐NL estimation framework. This approach utilizes all available data, including perturbation data (linear response) and the varying operating point (NL response). The proposed approach employs a combined estimator to locally identify the joint LPV‐NL model within the neighborhood of the system's operating point. Although the estimated NL model is localized around this nominal trajectory, an experimental approach is proposed to reduce the linear approximation error, thereby enlarging the validity region of the final model. Finally, we present a numerical example to illustrate the performance of the proposed method.

For general uncertain nonlinear time-delay systems, a domination design approach to robust adaptive controller based on Lyapunov-Razumikhin function is provided. To deal with the obstacle brought by the Lyapunov-Razumikhin function with the Razumikhin condition, a new parameter variable is introduced into the adaptive controller design instead of the conventional parameter variable. Based on Lyapunov-Razumikhin stability theory, it is shown that the proposed robust adaptive controller can guarantee the global uniform ultimate boundedness of all the signals in the resulting closed-loop dynamical system. In addition, it is also shown that the proposed design scheme can be applied to recursive design for the robust adaptive stabilizing controller of general uncertain nonlinear time-delay systems with triangular structure.

In this paper, the problem about optimal iterative learning control of general nonlinear discrete-time systems has been studied. Based on contraction mapping principle, if some mild assumptions are given, we can obtain some sufficient conditions for the existence of optimal iterative learning control of general nonlinear discrete-time systems. These conditions are generalizations of the existence conditions of optimal iterative learning control of linear discrete-time systems.

Geometrically nonlinear transverse steady-state periodic forced vibration of multi-degree-of-freedom discrete systems with a distributed nonlinearity

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