A Family of Multi-step Derivative-Free Iterations for Solving Nonlinear Systems

International audience

This paper presents how to apply a newly developed general theory for the global transversality and tangency of flows in n-dimensional nonlinear dynamical systems to a 2-D nonlinear dynamical system (i.e. a periodically forced, damped Duffing oscillator). The global tangency and transversality of the periodic and chaotic motions to the separatrix for such a nonlinear system are discussed to help us understand the complexity of chaos in nonlinear dynamical systems. This paper presents the concept that the global transversality and tangency to the separatrix are independent of the Melnikov function (or the energy increment). Chaos in nonlinear dynamical systems makes the exact energy increment quantity to be chaotic no matter if the nonlinear dynamical systems have separatrices or not. The simple zero of the Melnikov function cannot be used to simply determine the existence of chaos in nonlinear dynamical systems. Through this paper, the expectation is that, from now on, one can use the alternative aspect to look into the complexity of chaos in nonlinear dynamical systems. Therefore, in this paper, the analytical conditions for global transversality and tangency of 2-D nonlinear dynamical systems are presented. The first integral quantity increment (i.e. the energy increment) for a certain time interval is achieved for periodic flows and chaos in the 2-D nonlinear dynamical systems. Under the perturbation assumptions and convergent conditions, the Melnikov function is recovered from the first integral quantity increment. A periodically forced, damped Duffing oscillator with a separatrix is investigated as a sampled problem. The corresponding analytical conditions for the global transversality and tangency to the separatrix are obtained and verified by numerical simulations. The switching planes and the corresponding local and global mappings are defined on the separatrix. The mapping structures are developed for local and global periodic flows passing through the separatrix. The mapping structures of global chaos in the damped Duffing oscillator are also discussed. Bifurcation scenarios of the damped Duffing oscillator are presented through the traditional Poincaré mapping section and the switching planes. The first integral quantity increment (i.e. L-function) is presented to observe the periodicity of flows. In addition, the global tangency of periodic flows in such an oscillator is measured by the G-function and G(1)-function, and is verified by numerical simulations. The first integral quantity increment of periodic flows is zero for their complete periodic cycles. Numerical simulations of chaos in such a Duffing oscillator are carried out through the Poincaré mapping sections. The conservative energy distribution, G-function and L-function along the displacement of Poincaré mapping points are presented to observe the complexity of chaos. The first integral quantity increment (i.e. L-function) of chaotic flows at the Poincaré mapping points is nonzero and chaotic. The switching planes of chaos are presented on the separatrix for a better understanding of the global transversality to the separatrix. The switching point distribution on the separatrix is presented and the switching G-function on the separatrix is given to show the global transversality of chaos on the separatrix. The analytical conditions are obtained from the new theory rather than the Melnikov method. The new conditions for the global transversality and tangency are more accurate and independent of the small parameters.

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.

Designing minimum variance controllers (MVC) for nonlinear systems is confronted with many difficulties. The methods which are able to identify MIMO nonlinear systems are scarce, and linear models are not accurate in modeling nonlinear systems. In this paper, Vector ARX (VARX) models are proposed for designing MVC and generalized minimum variance controller (GMVC) for linear and nonlinear systems, and the accuracy of these models in approximating the nonlinear MIMO system is studied. However, the VARX is a linear model. It is shown that this model can identify some kinds of nonlinear systems with any desired accuracy. Therefore, the controller designed by the VARX is accurate, even for these nonlinear systems. The proposed controller is tested on a both linear system and a nonlinear four-tank benchmark process. In spite of the simplicity of designing GMVCs for the VARX models, the results show that the proposed method is accurate and implementable.

<p>Whilst a lot of our strategic focus in the public sector is on linear policy approaches, many systems/ phenomena of importance are defined as non-linear or far from equilibrium. Traditional approaches to linear forecasting have not proved effective for non-linear systems, since non-linear systems follow a different set of rules. Historically, non-linear systems were too hard to forecast, but over recent decades some rules and approaches are starting to emerge. One important and clearly defined category of non-linear systems are those that follow a ‘power-law’ distribution rather than the ‘normal’ distribution, which is often associated with linear systems or systems in equilibrium. My research collects, analyses, and does a comparative analysis of the different power law populations, as well as the main strategic forecasting techniques that can be applied to those populations/ systems. Overall Conclusions and observations. Just as in science and mathematics, there is now a clearly defined separation and understanding of linear and non-linear systems and the rules that apply to each. My thesis has as its central theme, the idea that strategy as a subject also fits this same philosophical separation of approaches, which I have called the strategic planning versus the strategic thinking divide. Strategic planning is essentially the linear approach – being rational and assuming relatively stable conditions. Strategic thinking assumes the world is effectively non-linear and ‘far from equilibrium’. Non-linear approaches mean acknowledging concepts like; punctuated equilibrium, power law ‘log-log’ graphs, ‘scale-free’ characteristics, ‘self organising criticality’, accepting only pattern prediction (including 1/f formulas) and not precise prediction etc. Understanding non-linearity is essential to understand such things as ‘Black Swans’. Luck, serendipity and ‘bounded rationality’ are always involved in non-linear complex adaptive systems, whereas linear systems tend to comply with the so called ‘rational’ traditions in science and economics. Power law statistical distributions can be seen in a wide variety of non-linear natural and man-made phenomena, from earthquakes and solar flares to populations of cities and sales of books. This sheer diversity of effects that have power law distributions is actually an amazing fact that has only become evident over the last decade or so. Since the world contains aspects that are clearly linear and other aspects that are clearly non-linear, it is essential for someone interested in strategy to be able to understand both systems and be able to apply the correct techniques to each approach. The two parts of ‘punctuated equilibrium’ effectively link the two strategic approaches together as there is only one world and not two separate realities. It therefore follows that a strategist needs a good understanding of both strategic planning and strategic thinking, since both are needed for different phases or periods, and perhaps both are needed for any period when you can't tell what phase you are in, which can also happen. I suggest that under a linear phase, the strategic planning approach should be dominant, but supported by strategic thinking (since you never know when events will turn abruptly); whereas in a turbulent non-linear period the strategic thinking approach should be dominant, but supported by strategic planning (since you know that great turbulence will not last). This is a sort of a swapping dominant/ recessive situation, which has a loose parallel in the theory of the left/ right brain split, where it is not wise to use only one style of thinking, since there are two styles which suit different situations. The key is to pick the right thinking style for the right situation. Just as we have one brain, but two thinking styles, so in the strategy toolbox we also have two valid, useful and complimentary general strategic approaches. However for this thesis, I have focused on the non-linear power law aspects of life which have strong implications for strategic thinking, since that is the new area for me as well as one of the new knowledge frontiers for strategy as a subject (and for leadership, politics and many other areas).</p>

The objective of this study is to enhance the responsiveness and precision of control systems through the acceleration of evolutionary programming in the optimization of nonlinear fuzzy control systems. Nonlinear systems are ubiquitous in engineering practice, presenting significant challenges due to their complexity and uncertainty for traditional control methodologies. Fuzzy control, an intelligent control approach rooted in empirical rules, possesses the capability to navigate uncertain and nonlinear issues. However, conventional fuzzy controllers struggle to achieve optimal control performance when confronted with highly complex nonlinear systems. Consequently, this paper introduces the application of accelerated evolutionary programming (RGA) for the optimization of fuzzy controllers, aiming to augment their performance in nonlinear systems. By merging genetic algorithms with a fuzzy adaptive PID controller, the optimized fuzzy controller is better equipped to adapt to the dynamic changes of nonlinear systems, offering robust and efficient control strategies. The results of simulation experiments demonstrate a marked improvement in response time, stability, and error reduction with the fuzzy control system based on RGA optimization. This study offers novel insights and methodologies for the further optimization of control within nonlinear systems, while providing theoretical support for the practical applications of complex systems.

SummaryIn this paper, output‐feedback control strategies are proposed for lower‐triangular nonlinear nonholonomic systems in any prescribed finite time. Specifically, by employing the input‐state–scaling technique, the controlled systems are firstly converted into lower‐triangular nonlinear systems, which makes it possible to study such systems using the high‐gain technique. Then, by introducing a scaling of the state by a function that grows unbounded toward the terminal time and proposing a high‐gain observer–prescribed finite time recovering the system states, the output‐feedback regulation control problem in any prescribed finite time is firstly achieved for nonlinear nonholonomic systems with unknown constant incremental rate. Moreover, by designing another time‐varying high gain, the output‐feedback stabilization control problem in any prescribed finite time is then achieved for nonlinear nonholonomic systems with a time‐varying incremental rate. Finally, a numerical example is introduced to show the effectiveness of proposed control strategies.

Theory of nonlinear control

Block-oriented nonlinear systems have attracted a considerable attention for their flexible structure and practicability. This study proposes a novel multi-innovation stochastic gradient (MISG) algorithm to address the identification problem in input nonlinear systems. This involves applying the inexact line search strategy to determine an appropriate convergence factor at each recursive step. The proposed algorithm tracks the nonlinear system dynamics faster than the conventional MISG algorithm. It is therefore suitable for online identification and can be applied to nonlinear time-varying systems. The concept of auxiliary model identification is also adopted for dealing with unmeasurable variables. The effectiveness of the proposed algorithm is verified through simulated examples.

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

A copula-based Gaussian mixture closure method for stochastic response of nonlinear dynamic systems

This paper introduces an $H_\infty$ fuzzy control synthesis method for a nonlinear large-scale system with a reduced number of linear matrix inequalities (LMIs). It is well known that a nonlinear large-scale system can be transformed to a Takagi–Sugeno (T–S) fuzzy system by using “sector nonlinearity” or “local approximation in fuzzy partition spaces” methods. Next, in order to achieve the fuzzy control design for this T–S fuzzy system, we solve the stabilization conditions that are represented by the LMIs. However, if the number of LMIs is large, the control design process may become very complicated. In this study, based on the Lyapunov method and S -procedure, several theorems are proposed for the synthesis of parallel distributed compensation (PDC)-type fuzzy control such that the nonlinear large-scale system achieves $H_\infty$ control performance, and the number of LMIs to be solved is reduced explicitly. As a result, the control design process will become much easier. Furthermore, if the modeling error between the nonlinear system and T–S fuzzy system exists, the robust $H_\infty$ control performance and the number reduction of LMIs are also achieved by the proposed theorem. Several examples are presented in this paper to show the number reduction effect of LMIs and the effectiveness of the proposed controller synthesis.

In this paper, we address the stabilization problem for nonlinear systems in a critical case. Namely, we study the class of canonical nonlinear systems. Canonical nonlinear systems or chain of power integrators is an important subject of research. Studying such systems is complicated by the fact that they cannot be mapped onto linear systems. Moreover, they have the uncontrollable first approximation. Previous results on smooth stabilization of such systems were obtained under the assumption that the powers in the right-hand side are strictly decreasing. In this work, we consider a case of non-increasing powers in the right-hand side for a three-dimensional system. A popular approach for studying such systems is the backstepping method, which is a method of step-wise stabilization. This method requires a sequential investigation of lower-dimensional subsystems. Backstepping enables the study of a wide range of nonlinear triangular systems but requires technically complex and cumbersome computations. Therefore, a natural question arises about constructing stabilizing controls of a simple form. Polynomial controls can serve as an example of such controls. In the paper, we demonstrate that linear controls can be considered as stabilizing controls. We derive sufficient conditions for the coefficients of the linear control that ensure the asymptotic stability of the zero equilibrium point of the corresponding closed-loop system. The asymptotic stability is proven using the Lyapunov function method, which is found as the sum of squares. The negative definiteness of the Lyapunov function derivative in a neighborhood of the origin guarantees asymptotic stability. In contrast to the case of strictly decreasing powers, additional conditions on the control coefficients, apart from their negativity, emerge. The obtained result extends to a broader class of nonlinear systems through stabilization by nonlinear approximation. This allows the consideration of systems with higher-order terms in the right-hand side. The effectiveness of the applied approach is illustrated by several model examples. The method used in this work to investigate the case of non-increasing powers can be applied to systems of higher dimensions.

Sufficient conditions for the controllability of nonlinear stochastic fractional boundary control systems are established. The equivalent integral equations are derived for both linear and nonlinear systems, and the control function is given in terms of the pseudoinverse operator. The Banach contraction mapping theorem is used to obtain the result. A controllability result for nonlinear stochastic fractional integrodifferential systems is also attained. Examples are included to illustrate the theory.

This paper describes the dynamic behaviour of a coupled system which includes a nonlinear hardening system driven harmonically by a shaker. The shaker is modelled as a linear single degree-of-freedom system and the nonlinear system under test is modelled as a hardening Duffing oscillator. The mass of the nonlinear system is much less than the moving mass of the shaker and thus the nonlinear system has little effect on the shaker dynamics. The nonlinearity is due to the geometric configuration consisting of a mass suspended on four springs, which incline as they are extended. Following experimental validation, the model is used to explore the dynamic behaviour of the system under a range of different conditions. Of particular interest is the situation when the linear natural frequency of the nonlinear system is less than the natural frequency of the shaker such that the frequency response curve of the nonlinear system bends to higher frequencies and thus interacts with the resonance frequency of the shaker. It is found that for some values of the system parameters a complicated frequency response curve for the nonlinear system can occur; closed detached curves can appear as a part of the overall amplitude-frequency response. These detached curves can lie outside or inside the main resonance curve, and a physical explanation for their occurrence is given.