Stability Analysis Of Nonlinear Systems Research Articles

Stability analysis of nonlinear systems with delayed impulses: a generalised average dwell-time scheme

This paper studies the input-to-state stability (ISS) and uniform stability for impulsive nonlinear systems with delayed impulses, where delays are flexible between two consecutive impulsive instants. Illustrating effects caused by delays, concepts of generalised average impulsive delay and generalised reverse average impulsive delay are introduced, and more applicable in practice. Moreover, sufficient conditions for stability properties are formulated with destabilising and stabilising impulses through a general candidate ISS-Lyapunov function and the generalised conditions. It is shown that the double effects of delays in impulses (i.e. stabilising an initial unstable system, or contrarily, destabilising an initial stable system) coincide with the existing results using the exponential candidate ISS-Lyapunov function. Finally, two examples are provided to demonstrate our developed techniques.

Stability analysis of nonlinear systems in the presence of event‐triggered impulsive control

AbstractThis note puts forward continuous event‐triggered impulsive control (CETIC) and dynamic event‐triggered impulsive control (DETIC) to discuss a class of (integral) input‐to‐state stability (iISS, ISS) for nonlinear systems (NSs), where the impulse sequences are produced by certain predesigned event‐triggering conditions. Different from traditionary event‐triggered control (ETC), CETIC indicates that a controller will be stimulated only if the given state‐dependent event condition is invoked. There exist no transfer of control between two continuous impulse triggered instants. Compared with the traditional static ETC, the DETIC can efficaciously lessen controller update and dramatically save energy at the same decay rate. In addition, all sample path solutions (SPS) for the system have the lowest time between events guaranteed to be positive. Utilizing CETIC strategy, we get some Lyapunov conditions to effectually avoid infinite triggering behavior and obtain the ISS‐type stability of the investigative systems. Then, one applies the theoretical consequences to NSs and derive a class of ETC mechanism with impulsive control gains (ICG) by linear matrix inequalities (LMIs). Because of the existence of timer, the DETIC strategy naturally excludes Zeno phenomena. Furthermore, conclusions in this thesis permit the upper bound estimation for the differential Lyapunov function coefficient to be time‐varying function instead of a constant in certain extant results, which means the criteria of the Lyapunov technique in this paper is less conservative and looser. Eventually, three examples with related simulations are demonstrated to indicate the rationalization and usefulness of our conclusion.

Stability Analysis of Nonlinear Systems with Impulsive Perturbations: Application to Hopfield Neural Networks

Stability Analysis of Nonlinear Systems with Impulsive Perturbations: Application to Hopfield Neural Networks

Homogeneous systems stabilization based on convex embedding

The paper develops control algorithms for a class of affine nonlinear systems using the so-called canonical homogeneous representation. It is demonstrated that such a representation exists for any homogeneous vector field bounded on the unit sphere. It is shown that canonical homogeneous representation is useful for LMI-based control design and stability analysis of nonlinear systems. Theoretical results are supported by numerical examples.

Metaheuristic Solution for Stability Analysis of Nonlinear Systems Using an Intelligent Algorithm with Potential Applications

In this article, we provide a metaheuristic-based solution for stability analysis of nonlinear systems. We identify the optimal level set in the state space of these systems by combining two optimization phases. This set is in a definite negative region of the time derivative for a polynomial Lyapunov function (LF). Then, we consider a global optimization problem stated in two phases. The first phase is an external optimization to search for a definite positive LF, whose derivative is definite negative under linear matrix inequalities. The candidate LF coefficients are adjusted using a Jaya metaheuristic optimization algorithm. The second phase is an internal optimization to ensure an accurate estimate of the attraction region for each candidate LF that is optimized externally. The key idea of the algorithm is based mainly on a Jaya optimization, which provides an efficient way to characterize accurately the volume and shape of the maximal attraction domains. We conduct numerical experiments to validate the proposed approach. Two potential real-world applications are proposed.

Simplified Method of Stability Analysis of Nonlinear Systems without Using of Lyapunov Concept

Simplified Method of Stability Analysis of Nonlinear Systems without Using of Lyapunov Concept

Performance improvement and Lyapunov stability analysis of nonlinear systems using hybrid optimization techniques

AbstractUsing Hybrid optimization algorithms for nonlinear systems analysis is a novel approach. It is a powerful technique that uses the exploitation ability of one algorithm and the exploration ability of another algorithm, to find the best solution. Literature survey reveals that hybrid algorithms not only show quality response but also give faster convergence of error for nonlinear systems. In this paper, hybrid optimization techniques based proportional integral derivative (PID) controller is used for benchmark problems: Continuous stirred tank reactor (CSTR), Inverted pendulum and blood glucose system. Two recent hybrid algorithms: Particle swarm optimization‐Gravitational search algorithm (PSO‐GSA) and Particle swarm optimization‐Grey wolf algorithm (PSO‐GWO) are implemented to control the temperature and concentration of CSTR, pendulum angle of inverted pendulum, glucose concentration and insulin level of blood glucose system. In PID and PSOGWO algorithms, the exploration abilities of GSA and GWO combined with the exploitation ability of PSO have been used. The performance of these algorithms is then compared with individual PSO, GSA, and GWO algorithms proving their superiority. Stability is ensured using the Lyapunov approach while the robustness of the systems is checked using the parameter perturbation technique. Simulation results show substantial improvement in the performance of these systems by using these meta‐heuristic hybrid optimization techniques. A comparative analysis of these algorithms has also been done.

Nonlinear forced vibration and stability analysis of nonlinear systems combining the IHB method and the AFT method

The incremental harmonic balance (IHB) method is a very powerful tool for analyzing nonlinear forced vibrations in structures subjected to periodic forces. In this paper, a methodology is proposed to interpret the stability of the calculated solutions and the nonlinear vibrations of nonlinear systems by combining the IHB method and the Alternating Frequency/Time (AFT) method. The main purpose of combining the IHB method and the AFT method is to analyze the nonlinear vibration of systems with complex nonlinearities. For the mass matrix, damping matrix, and linear stiffness matrix, the Galerkin procedure is performed according to the classical IHB method. The nonlinear force matrix and its derivative matrix are calculated in the time domain and transformed into the frequency domain by Fourier transform. By multiplying these matrices with a transformation matrix, they are transformed into matrices that can be used in the IHB method. Three example problems are proposed to prove the effectiveness of the suggested method and the results compared with the results calculated using numerical integration (NI) and the results presented in previous literature. The results are very close. The proposed method can analyze nonlinear vibrations of systems with complex nonlinearities as well as geometric nonlinearities.

T–S Fuzzy Sampled-Data Control for Nonlinear Systems With Actuator Faults and Its Application to Wind Energy System

This article concerns the problem of stability analysis of nonlinear systems under the sampled-data control (SDC) with actuator faults. The addressed nonlinear systems can be expressed by a number of linear arrangements, based on the Takagi–Sugeno (T–S) fuzzy technique. The time-dependent SDC with actuator faults of the addressed nonlinear systems delineates based on the input model of Markovian variability. The time-scheduled Lyapunov functional and the looped-functional are taken together in the production of a Lyapunov functional candidate. Based on a free matrix-based integral inequality and the Lyapunov functional, some sufficient linear matrix inequality conditions are delivered to guarantee the mean-square asymptotic stability under <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> performance of the considered sampled-data T–S fuzzy systems with actuator faults. Finally, we demonstrate the feasibility and strength of our proposed method through some examples.

Nonlinear Complementary Filter for Attitude Estimation by Fusing Inertial Sensors and a Camera.

Using a standalone camera for pose estimation has been quite a standard task. However, the point correspondence-based algorithms require at least four feature points in the field of view. This paper considers the situation that there are only two feature points. Focusing on the attitude estimation, we propose to fuse a camera with low-cost inertial sensors based on a nonlinear complementary filter design. An implicit geometry measurement model is derived using two feature points in an image. This geometry measurement is fused with the angle rate measurement and vector measurement from inertial sensors using the proposed nonlinear complementary filter with only two parameters to be adjusted. The proposed nonlinear complementary filter is posed directly on the special orthogonal group SO(3). Based on the theory of nonlinear system stability analysis, the proposed filter ensures locally asymptotic stability. A quaternion-based discrete implementation of the filter is also given in this paper for computational efficiency. The proposed algorithm is validated using a smartphone with built-in inertial sensors and a rear camera. The experimental results indicate that the proposed algorithm outperforms all the compared counterparts in estimated accuracy and provides competitive computational complexity.

New results on delay-dependent stability for nonlinear systems with two additive time-varying delays

The problem of stability analysis for nonlinear systems with two additive time-varying delays is revisited. New stability criteria are acquired by utilizing Lyapunov-Krasovskii methodology with abundant time-varying delay information. Firstly, four augmented single integral terms are introduced to obtain less conservative stability conditions. Secondly, single- and double- integral terms are developed through employing integral intervals decomposition approach and the bounds of them are estimated combined with more accurate integral inequalities. Thirdly, a new nonlinear optimization technology is established to deal with the problem of nonlinear time-delay inequalities in stability criterion. Finally, an application example for the load frequency control (LFC) scheme in power systems and a typical numerical example are provided to show the advantages and the validity of the proposed approach.

An Improved Stability Theorem for Nonlinear Systems on Time Scales With Application to Multi-Agent Systems

In this brief, stability analysis of nonlinear systems on time scales is further investigated. The traditional stability theorems require the time-scale derivative of related Lyapunov function to be negative, which are conservative. Based on the induction principle on time scales, an improved stability theorem is proposed in this brief by introducing the time-scale type uniformly asymptotically stable function. It is shown that this improved stability theorem weakens the negativity restriction on the time-scale derivative of related Lyapunov function. Then, this improved stability theorem is applied to solve consensus problem of multi-agent systems on time scales. A numerical example is given to illustrate the effectiveness of the theoretic results.

Exponential stability analysis of nonlinear systems with bounded gain error

Considering technology limitation or device restriction in practical application, we formulate new nonlinear systems with bounded gain error, which contain switched control and impulsive control. We then investigate the exponential stability of the considered systems. Finally, the effectiveness of the proposed criteria is confirmed via an example based on Chua’s oscillator.

Lyapunov stability analysis for nonlinear systems with state-dependent state delay

This paper addresses the stability problem for systems with state-dependent state delay (delay which involves the state of the system). Different from the time-dependent delay, the state dependence of the delay makes the value of delay dependent on the state change, which indicates that it is impossible to exactly know a priori how far in the history the state-information is needed. We apply the Lyapunov stability theory to obtain sufficient conditions for exponential stability of the zero equilibrium. Then we apply those results to some specific examples to illustrate the effectiveness of the approach and our general results. A class of stabilizing memoryless controllers for a second-order system with state-dependent state delay is also proposed.

Sampled-data controller design and stability analysis for nonlinear systems with input saturation and disturbances

In modern control theory, it is more applicable for digital computers using a sampled-data controller than a continuous-time controller. Meanwhile, input saturation and disturbance inevitably exist in control systems. Thus, how to propose an applicable sampled-data controller for nonlinear systems considering input saturation together with external disturbances is essential for applications. To deal with this control problem, a novel auxiliary control signal is designed in this paper. Then, by using Lyapunov method, the control design process is simplified compared with the traditional backstepping technique, and only one Lyapunov function is needed. In addition, the state and disturbance observers are proposed with the new auxiliary control signal to track the unavailable states and external disturbances. Finally, we propose a novel controller reflecting the sampled-data output-feedback for the nonlinear systems to ensure the system stability. The effectiveness of proposed control method in this paper is confirmed through two sets of numerical simulation results.

New Criteria for Guaranteed Cost Control of Nonlinear Fractional-Order Delay Systems: a Razumikhin Approach

The Krasovskii–Lyapunov second method provides a powerful approach to stability analysis of nonlinear systems; however, it is not always effectively applied for fractional-order systems (FOSs) with delay. In this paper, we investigate the problem of guaranteed cost control of fractional-order delay systems subject to nonlinear perturbations and parametric time-varying uncertainties. By using fractional Razumikhin theorem, new sufficient conditions are derived for designing a guaranteed cost controller, which not only makes the closed-loop system asymptotically stable but also guarantees an adequate cost level of performance. Compared with the existing results on the integer-order control systems, our results are more effective and convenient for testing and application. The proposed approach allows us to derive stability criteria of linear uncertain FOSs with delay. Finally, numerical examples are given to show the effectiveness of the obtained results.

A multiple-comparison-systems method for distributed stability analysis of large-scale nonlinear systems

Lyapunov functions provide a tool to analyze the stability of nonlinear systems without extensively solving the dynamics. Recent advances in sum-of-squares methods have enabled the algorithmic computation of Lyapunov functions for polynomial systems. However, for general large-scale nonlinear networks it is yet very difficult, and often impossible, both computationally and analytically, to find Lyapunov functions. In such cases, a system decomposition coupled to a vector Lyapunov functions approach provides a feasible alternative by analyzing the stability of the nonlinear network through a reduced-order comparison system. However, finding such a comparison system is not trivial and often, for a nonlinear network, there does not exist a single comparison system. In this work, we propose a multiple comparison systems approach for the algorithmic stability analysis of nonlinear systems. Using sum-of-squares methods we design a scalable and distributed algorithm which enables the computation of comparison systems using only communications between the neighboring subsystems. We demonstrate the algorithm by applying it to an arbitrarily generated network of interacting Van der Pol oscillators.

Lyapunov Stability Analysis of Switching Controllers in Presence of Sliding Modes and Parametric Uncertainties With Application to Pneumatic Systems

This paper concerns the control and the stability analysis of pneumatic actuators, which are nowadays of widespread use in the industry. A problem related to the use of such actuators is the so-called stick-slip, due to the presence of dry friction in the system. In this paper, we provide an empirical switching control law that avoids this phenomenon as well as a general approach to the stability analysis of nonlinear systems that will let us prove the stability of the closed-loop system. The approach is based on casting the closed-loop system into a piecewise-affine form and finding a Lyapunov function for it. Such an approach will be able to cope with the special features of the controlled pneumatic system model, namely, the presence of sliding modes, a whole equilibrium set, and uncertainties on the values of a few parameters. At the end of this paper, we will show how such a method can be successfully applied to our experimental setup under several hypotheses.

Theoretical analysis and modeling of flow instability in a mini-channel evaporator

Pressure drop oscillations in micro/mini-channel evaporators and corresponding flow instabilities, temperature fluctuations have received copious of investigations during the last decade. This paper presents a transient lumped model and theoretical analysis for the pressure drop oscillation in a mini-channel evaporator. Based on the model, the effects of saturation temperature, heat and mass flux on the oscillation are investigated. Experimental studies of ammonia and water flow boiling instabilities are conducted. The mini-channel evaporator consists of 4 parallel 1×1.1mm channels with a uniformly heated length of 250mm. A nonlinear system stability analysis is presented. Apart from upstream compressibility, the inlet sub-cooling degree has a significant effect on the pressure drop oscillation. A maximum allowable inlet sub-cooling degree causing no pressure drop oscillation is proposed. The oscillation period is comprehensively studied, and it is found that the upstream compressible volume required sustaining the oscillation decreases with channel length/diameter ratio dramatically. Despite this, the internal compressibility of the long channel is insufficient to sustain the pressure drop oscillation. In addition, the mass flow rate of the upstream pump can greatly affect the oscillation and the flow boiling system may show different behaviors due to the variation of upstream mass flow rate.

Adaptive Control? But is so Simple!

In a recent paper, a few pioneers of adaptive control review the classical model reference adaptive control (MRAC) concept, where the designer is basically supposed to conceive a model of the same order as the (possibly very large) plant, and then build an adaptive controller such that the plant is stable and ultimately follows the behavior of the model. Basically, adaptive control methods based on model following assume full-state feedback or full-order observers or identifiers. These assumptions, along with supplementary prior knowledge, allowed the first rigorous proofs of stability with adaptive controllers, which at the time was a very important first result. However, in order to obtain this important mathematical result, the developers of classical MRAC took the useful scalar Optimal Control feedback signal and made it into an adaptive gain-vector of basically of the same order as the plant, which again had to multiply the plant state-vector in order to finally end with another scalar adaptive control feedback signal. It is quite known today, however, what happens when this requirement is not satisfied, and when "unmodeled dynamics" distorts the controller based on these ideal assumptions. Even though much effort has been invested to maintain stability in spite of so-called "unmodeled dynamics," in some applications, such as large flexible structures and other real-world applications, even if one can assume that the order of the plant is known, one just cannot implement a controller of the same order as the plant (or even a "nominal" or a "dominant" part of the plant), before even mentioning the complexity of such an adaptive controller. Without entering the argument around their special reserve in relation to claimed efficiency of the particular L1-Adaptive Control methodology, this paper first shows that, after the first successful proof of stability and even under the same basic full-state availability assumption, the adaptive control itself can be reduced to just one adaptive gain (which multiplies one error signal) in single-input-single-output (SISO) systems and, as a straightforward extension, an m*m gain matrix in an m-input-m-output (MIMO) plant. Not only is stability not affected, but actually the simplified scheme also gets rid of most seemingly "inherent" problems of the adaptive control represented by classical MRAC. Moreover, proofs of stability have all been based on the so-called Barbalat's lemma which seems to require very strict uniform continuity of signals. The apparent implications are that any discontinuity, such as square-wave input commands or just some occasionally discontinuous disturbance, may put stability of adaptive control in danger, without even mentioning such things as impulse response. Instead, based on old yet amazingly unknown extensions of LaSalle's Invariance Principle to nonautonomous nonlinear systems, recent developments in stability analysis of nonlinear systems have mitigated or even eliminated most apparently necessary prior conditions, thus adding confidence in the robustness of adaptive scheme in real world situations.