Bifurcations in the frequency response of nonlinear feedback control systems

In the previous paper, the authors described the statistical studies on the response of nonlinear control systems subjected to an ergodic stationary Gaussian random input. This paper is devoted to the fundamental approaches to the non-stationary random time series which are considered to arise quite often for the automatic control systems in practice. Firstly, the response of nonlinear control systems subjected to a non-stationary random signal is evaluated. Secondly, for an example, a non-stationary Orenstein-Uhlenbeck process which is obtained by using the method of Fokker-Planck is treated. Detail illustrations are shown by several examples.

In recent years, the non-stationary random processes have attracted considerable attentions in the field of adaptive or self-optimizing control, but little seems to have been done for those cases. In this paper, a very simple but fundamental case of non-stationary random process of response of nonlinear control systems is treated. That is, the analytical study on the evaluation of the response of nonlinear control systems subjected to a suddenly applied stationary random signal is described The two methods are established here by introducing some extensive method of the equivalent linearization technique. A graphical method for evaluating the non-stationary random response is proposed and such procedures are explained by typical examples in detail.

Theory of nonlinear control

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.

An adaptive system for linear systems with unknown parameters is a nonlinear system. The analysis of such adaptive systems requires similar techniques to analyse nonlinear systems. Therefore it is natural to treat adaptive control as a part of nonlinear control systems. Nonlinear and Adaptive Control Systems treats nonlinear control and adaptive control in a unified framework, presenting the major results at a moderate mathematical level, suitable for MSc students and engineers with undergraduate degrees. Topics covered include introduction to nonlinear systems; state space models; describing functions for common nonlinear components; stability theory; feedback linearization; adaptive control; nonlinear observer design; backstepping design; disturbance rejection and output regulation; and control applications, including harmonic estimation and rejection in power distribution systems, observer and control design for circadian rhythms, and discrete-time implementation of continuous-time nonlinear control laws.

One of the primary concerns in industrial motor drive systems is the suppression of the vibration caused by the elasticity of mechanical load. This paper is intended to clarify the conditions required to obtain the desirable damping by a simple control system through the analysis of the frequency response of a two-mass control system, considering the plant of left-coprime factorized form. The paper also presents some useful findings for practical design of the vibration suppression control, showing that if the above-mentioned conditions are satisfied, sufficient control performance can be obtained by a two degrees of freedom controller of the same structure as that for rigid load, and that we are compelled to reduce the stability margin in the vibration suppression control where the anti-resonant frequency and resonant frequency are near to each other.

A dynamic physical system often undergoes phase transitions in response to fluctuations induced on system parameters. For example, hurricane activity is the climate system's response initiated by a liquid-vapor phase transition associated with non-linearly coupled fluctuations in the ocean and the atmosphere. Because our quantitative knowledge about highly non-linear dynamic systems is very meager, scientists often resort to linear regression techniques such as Least Absolute Deviation (LAD) to learn the non-linear system's response (e.g., hurricane activity) from observed or simulated system's parameters (e.g., temperature, precipitable water, pressure). While insightful, such models still offer limited predictability, and alternatives intended to capture non-linear behaviors such as Stepwise Regression are often controversial in nature. In this paper, we hypothesize that one of the primary reasons for lack of predictability is the treatment of an inherently multi-phase system as being phase less. To bridge this gap, we propose a hybrid approach that first predicts the phase the system is in, and then estimates the magnitude of the system's response using the regression model optimized for this phase. Our approach is designed for systems that could be characterized by multi-variate spatio-temporal data from observations, simulations, or both.

In this paper we study a broad class of nonlinear control systems containing a single memoryless nonlinear element. We present conditions under which there exists a unique periodic response, with a given period, to an arbitrary periodic input with the same period, and we derive an upper bound on the mean-square error incurred by applying the describing-function technique. The expression for the error reflects the intuitive engineering arguments that are often employed to justify the use of the describing-function method. Conditions are also presented under which subharmonic response components and self-sustained oscillations cannot occur.

Efficiency requirements often cause physical models of musical instruments to be vastly simplified when implemented, sometimes to the point where the model loses certain ‘‘correct’’ behaviors, such as realistic attacks or even stability. In the case of instability, many implementations which use fixed-point arithmetic implicitly rely on the limiting behavior of the number system to contain the amplitude of the signal. Certain models, such as clarinet models, tend to continue to work well in the presence of this instability, due to the fact that the resulting clipping does not significantly alter the spectral content of the signal. Other models, however, are significantly altered by the limiting. In certain cases, a simple nonlinear control system can be placed into the model (without significantly increasing the computation requirements) to serve the oscillation amplitude to a desired amplitude envelope. This control system improves the stability of the system, which in saturating number systems also reduces distortion. The control system can also improve other performance behaviors, such as attacks.

Output feedback nonlinear control of an aeroelastic system with unsteady aerodynamics

Fundamentals of the theory of non-linear pluse control systems

: This report is concerned primarily with assessing the state of the art of nonlinear control as a prelude to the solution of the actual specification problem. As an introduction, the classical methods of nonlinear analysis are discussed, and the reasons for the inadequacy of these techniques for automatic control systems are explained. The two generally known methods of analyzing the stability of autonomous nonlinear control systems, namely phase plane analysis and the describing function, are discussed. The concept of the state variable and the state space is introduced in some detail. The stability of the nonautonomous system and the response of autonomous and nonautonomous systems are covered. As a criterion for specification, the time optimum system is stressed, and distinction is made between the solution of the time optimum problem as a performance index and the synthesis of the optimum switching boundaries. The phase plane is discussed for forced systems and the work of Wiener is mentioned in connection with the response of nonlinear systems to random inputs.

A EROSPACE engineering applications greatly stimulated the development of optimal control theory during the 1950s and 1960s, where the objective was to drive the system states in such a way that some defined cost was minimized. This turned out to have very useful applications in the design of regulators (where some steady state is to be maintained) and in tracking control strategies (where some predetermined state trajectory is to be followed). Among such applications was the problem of optimal flight trajectories for aircraft and space vehicles. Linear optimal control theory in particular has been very well documented and widely applied, where the plant that is controlled is assumed linear and the feedback controller is constrained to be linear with respect to its input. However, the availability of powerful low-cost microprocessors has spurred great advantages in the theory and applications of nonlinear control. The competitive era of rapid technological change, particularly in aerospace exploration, now demands stringent accuracy and cost requirements in nonlinear control systems. This has motivated the rapid development of nonlinear control theory for application to challenging, complex, dynamical real-world problems, particularly those that bear major practical significance in aerospace, marine, and defense industries. Infinite-time horizon nonlinear optimal control (ITHNOC) presents a viable option for synthesizing stabilizing controllers for nonlinear systems by making a state-input tradeoff, where the objective is to minimize the cost given by a performance index. The original theory of nonlinear optimal control dates from the 1960s. Various theoretical and practical aspects of the problem have been addressed in the literature over the decades since. In particular, the continuous-time nonlinear deterministic optimal control problem associated with autonomous (time-invariant) nonlinear regulator systems that are affine (linear) in the controls has been studied by many authors. The long-established theory of optimal control offers quite mature and well-documented techniques for solving this control-affine nonlinear optimization problem, based on dynamic programming or calculus of variations, but their application is generally a very tedious task. Bellman’s dynamic programming approach reduces to solving a nonlinear first-order partial differential equation (PDE), expressed by the Hamilton–Jacobi–Bellman (HJB) equation. The solution to the HJB equation gives the optimal performance/cost value (or storage) function and determines an optimal control in feedback form under some smoothness assumptions. Alternatively, in the classical calculus of variations, optimal control problems can be characterized locally in terms of the Hamiltonian dynamics arising from Pontryagin’s minimum principle. These are the characteristic equations of the HJB PDE, which result in a nonlinear, constrained two-point boundary value problem (TPBVP) that, in general, can only be solved by successive approximation of the optimal control input using iterative numerical techniques for each set of initial conditions. Numerically, even though the nonlinear TPBVP is somewhat easier to solve than the HJBPDE, control signals can only be determined offline and are thus best suited for feedforward control of plants for which the state trajectories are known a priori. Therefore, contrary to the dynamic programming approach, the resultant control law is not generally in feedback form. Open-loop control, however, is sensitive to random disturbances and requires that the initial state be on the optimal trajectory. In contrast, nonlinear optimal feedback has inherent robustness properties (inherent in the sense that it is obtained by ignoring uncertainty and disturbances). The potential difficulty with the HJB approach is that no efficient algorithm is available to solve the PDE when it is nonlinear and the problem dimension is high, making it impossible to derive exact expressions for optimal controls for most nontrivial problems of interest. The optimal can only be computed in special cases, such as linear dynamics and quadratic cost, or very low-dimensional systems. In particular, if the plant is linear time invariant (LTI) and the (infinite-time) performance index is quadratic, then the corresponding HJB equation for this infamous linear-quadratic regulator (LQR) problem reduces to an algebraic Riccati equation (ARE). Contrary to the well-developed and widely applied theory and computational tools for theRiccati equation (for example, see [1]), theHJB equation is difficult, if not impossible, to solve for most practical applications. The exact solution for the optimal control policies is very complex

A nonlinear adaptive output feedback control approach based on LOLIMOT is proposed for a class of nonaffine system in the non-strict feedback form with unknown nonlinearities in this paper. In order to overcome the disadvantages of output feedback control, LOLIMOT which can identify the nonlinear system accurately is applied. During the control process, backstepping approach is utilized to design the adaptive output feedback controller and the internal pseudo-controller is used as one-step predictor. The effectiveness of proposed scheme is shown by simulation on a proper nonlinear system.