Stability analysis of non-linear dynamical systems

This paper examines the possibility of using non-linear comparison systems in the stability analysis of non-linear dynamical systems. Stability criteria and estimates of the stability region of non-linear comparison systems are established. It is shown that the use of non-linear comparison systems may lead to less conservative results than those obtained by linear ones.     

Analysis of non-linear sampled-data systems with transportation lag†

A detailed investigation is made into the effect of inherent transportation lag on the nature of limit cycles sustained in a second-order sampled-data system containing a relay type non-linearity. The effectiveness of the either signals of sinusoidal, triangular and square wave shape to suppress limit cycles is studied in detail and the advantages and disadvantages of each wave shape are discussed. The method of analysis is via the state transition technique and it is advantageously used to evaluate the response of the system until a final steady-state value is attained by the controlled variable.

The results conclusively prove that the wave shape and the amplitude of the dithering signal have a pronounced influence on the manner of suppression of limit cycles and the final steady-state error. It is clearly shown that systems with larger values of transportation lag call for stronger dithering signals in order to quench the limit cycles that would otherwise persist with a time period which is an integral multiple of the sampling period. It is found, with regard to sinusoidal and triangular dither, that it is possible to exercise smooth control over the transient response to any desired extent by merely varying the amplitude, provided the frequency of dither is at least ten times the frequency of the signal of interest. Finally it is established that it should be possible to choose an optimum amplitude of the dithering signal in respect of settling time or any standard index of performance.     

Stability of non-linear systems via the intrinsic method

A new method, called the intrinsic method, is proposed in this paper to derive suitable Lyapunov functions for a general class of non-linear systems expressed in state variables as n first-order non-linear differential equations. This method, which applies the integration-by-parts procedure, derives a Lyapunov function directly from the differential equations under study.     

Stability criteria for a class of non-linear systems†

A graphical method for investigating the absolute stability of a particular class of systems with many non-linearities is given. The technique is an extension to previous results for the class of systems.     

Self-suppression of limit cycles in non-linear sampled-data systems with transportation lag†

This paper extends the principle of self-suppression of oscillations employed in continuous systems to non-linear sampled-data systems with transportation lag. The principle is based on the proper design of a. minor feedback loop to generate an oscillation with the desired frequency and amplitude. The type of response of the system is shown to depend entirely on the gain of the inner feedback path. Further it is shown that larger values of delay require larger gains to be employed in the inner feedback path to suppress the inherent oscillation. The results clearly indicate that by proper adjustment of the gain a response said to be optimum in regard to Borne performance index can be obtained.     

Stability and performance recovery within discretized non-linear control systems

In this paper, a closed-loop fast sampling analysis of a discretized (emulated) continuous-time controller in a sampled-data environment is presented. The analysis involves a general weighted version of the L_p-norm. This allows a qualitative and quantitative stability and performance analysis for an emulated controller. Several examples are used to show the relevance of these results for the analysis of sampled-data implementations and the computation of quantitative upper limits on the sampling period to achieve recovery of stability and performance.     

Stability of dithered non-linear systems with backlash or hysteresis†

We study the effect of dither on the non-linear element of a single-input single-outout feedback system. We consider non-linearities with memory (backlash, hysteresis), in the feedforward loop; a dither of a given amplitude is injected at the input of the non-linearity. The non-linearity is followed by a linear element with low-pass characteristic. The stability of the dithered system and an approximate equivalent system (in which the non-linearity is a smooth function) are compared. Conditions on the input and on the dither frequency are obtained so that the approximate-system stability guarantees that of the given hysteretic system     

Stability criteria for control systems with one non-linear element†

Liapunov functions are used to develop frequency domain criteria for a class of systems in which the given non–linear characteristic is bounded in both sector and slope. The criteria include the special case when order of numerator and denominator is equal in the linear transfer functions of the class of systems.     

Stability analysis and non-linear behaviour of structural systems using the complex non-linear modal analysis (CNLMA)

Herein, a novel non-linear procedure for producing non-linear behaviour and stable limit cycle amplitudes of non-linear systems subjected to super-critical Hopf bifurcation point is presented. This approach, called Complex Non-Linear Modal Analysis (CNLMA), makes use of the non-linear unstable mode which governs the non-linear dynamic of structural systems in unstable areas. In this study, the computational methodology of CNLMA is presented for the systematic estimation of the non-linear behavior of structural system, and it is applied to a complex system with quadratic and cubic non-linearities. Results from this non-linear method are compared with results obtained by integrating the full original non-linear system. This procedure appears very interesting in regard to computational time, formulation, and also necessitates very few computer resources. Thus the CNLMA is suited to computational implementation for various complex non-linear system with many degrees-of-freedom.     

Stability of non-linear feedback systems with time-varying linear sub-system (plant) †

A new approach is used to examine the stability of feedback systems comprising a time-varying linear subsystem and a time-varying non-linearity. The paper examines three stability aspects of the systems dynamic response: boundedness, unboundedness, and asymptotic decay. In addition to qualitative criteria, the results provide explicit quantitative bounds on the system response. The distinct time-domain method of analysis presented stipulates rather weak a priori restrictions on the nature of the linear and non-linear parts of the system, thus admitting a relatively broad class of systems. In the general time-varying non-linear feedback system considered, the non-linearity may exhibit hysteresis, and the linear subsystem may be non-causal, and may comprise any number of feed-forward differentiators of any order. The criteria obtained are given a simple graphical interpretation. A transformation which trades time-varying gains between plant and non-linearity is introduced. A number of examples demonstrate that the present results can predict stability information not obtainable from other existing criteria. In one example of a system with stationary plant, the present criteria prove superior to the Circle Theorem.     

Cascade connection of non-linear systems

A cascade connection of two non-linear systems described by the Volterra functional series has been investigated. The properties of the Volterra kernels of the equivalent system and a convergence of the resulting Volterra functional series have been examined.     

Controllability of non-linear delay systems

In this paper we examine the approximate controllability in the space W² × L² of non-linear systems with delays in states and controls. If the uncontrolled system is asymptotically stable and if the linear part of the control system is controllable, then the non-linear delay system is approximately controllable. An upper bound on the magnitude of retardation for the system to be controllable is estimated. An example is given to demonstrate the practical verification of the new conditions     

Controllability of non-linear stochastic systems

Complete controllability of a semi-linear stochastic system assuming controllability of the associated linear sytem is studied. It is also shown that a non-linear stochastic system is locally null controllable provided that the corrsponding linearized system is controllable.     

Stability analysis of piecewise non-linear systems and its application to chaotic synchronisation with intermittent control

This paper considers a stability analysis issue of piecewise non-linear systems and applies it to intermittent synchronisation of chaotic systems. First, based on piecewise Lyapunov function methods, more general and less conservative stability criteria of piecewise non-linear systems in periodic and aperiodic cases are presented, respectively. Next, intermittent synchronisation conditions of chaotic systems are derived which extend existing results. Finally, Chua's circuit is taken as an example to verify the validity of our methods.     

Structural identifiability of non-linear systems using linear/non-linear splitting

In this paper, the local state space isomorphism theorem for non-linear systems is used to analyse structural identifiability. In particular it is shown that, under certain restrictions, it is possible to perform a linear/non-linear splitting of the analysis. The relatively straightforward linear analysis then restricts the class of local diffeomorphic transformations as given by the non-linear state space isomorphism theorem. This, in turn, leads to possible simplifications to the subsequent non-linear analysis by providing an efficient means for calculating the local state diffeomorphism.     

Multi-objective non-linear fixed charge transportation problem with multiple modes of transportation in crisp and interval environments

This paper aims to propose an approach based on NSGA-II for solving multi-objective non-linear fixed charge transportation problem with multiple modes of transport in crisp and interval environments. Certain modifications need to be made in the existing NSGA-II configuration to calculate the crowding distance of a solution in the interval environment. Besides, a crossover and a mutation scheme suitable for multiple modes of transportation are developed. In the end, a set of test problems are solved in both environments and some comparative studies are performed restricting the problem to only one mode of transport at a time. Finally, the results of the proposed algorithm are compared with SPEA2.     

Set-point changing for non-linear systems

In many control systems, we sometimes face a situation such that we have to change the set-point of the system in the course of operation. Theoretically ‘set-point changing’ means letting the system be at another equilibrium point. If the system is linear, this operation presents no problem because the properties of linear systems are always global. However, in non-linear systems, this type of global operation has many unsolved problems. We only know experimentally that, if the system is locally stable at every equilibrium point, then we can slowly change the set-point without exciting unstable motion. This paper proposes a theoretically guaranteed method of changing the set-point of non-linear systems and showing a sufficient condition needed for this operation.     

Structural controllability in non-linear systems

Changes with respect to controllability or uncontrollability behaviour of perturbed linear and non-linear systems are examined. The perturbations are considered to be either arbitrary or the result of the action of a transformation group. Certain conclusions concerning the transformation of electrical networks are drawn.     

High-gain observers for non-linear systems

State estimation for non-linear dynamic systems is discussed. A high-gain injection from the output variables is used to attenuate to any desired degree, the effect of the non-linear terms on the estimation errors, which can be made arbitrarily small, from a particular observable form. Some Liapunov arguments are used to study the stability properties of the resulting error dynamics. It is shown that any completely observable multi-input multi-output non-linear system, not necessarily linear in the input variables, can be transformed, by means of a change of coordinates depending on the input variables, in such an observable form. In particular, the solvability of partial differential equations is not needed for the design of this transformation.