Mikhailov Stability Criterion Research Articles

Modified Mikhailov stability criterion for non-commensurate fractional-order neutral differential systems with delays

This paper studies the asymptotic stability of non-commensurate fractional-order neutral differential systems with constant delays. To do this, we propose a modified Mikhailov stability criterion. Our work not only generalizes the existing results in the literature but also provides a rigorous mathematical basis for the frequency domain analysis method concerning fractional-order systems with delays. Specific examples and numerical illustrations are also provided to demonstrate the validity of the obtained result.

Mikhailov stability criterion for fractional commensurate order systems with delays

In this paper, we present the extension of the Mikhailov stability criterion to linear fractional commensurate order systems with delays of the retarded type. The extension is obtained by generalizing the Mikhailov stability criterion of fractional commensurate order and integer order delay systems. The validity of the results is illustrated by means of several examples.

On the robust stability of commensurate fractional-order systems

Based on a recent generalised version of the Mikhailov stability criterion, this paper presents a Kharitonov–like test for a class of linear fractional–order systems described by transfer functions whose coefficients are subject to interval uncertainties. To this purpose, first the transfer function is associated with an integer-order complex polynomial function of the generalised frequency (i.e. the current coordinate along the boundary radii of the instability sector) whose coefficients are uncertain. Then the geometrical form of the value set of this characteristic polynomial is determined from the direct examination of its monomial terms. To show how the test operates, it is finally applied to two fractional–order transfer functions whose coefficients belong to given intervals.

Proportional–integral–derivative stabilization of complex conjugate-order systems

Proportional–integral–derivative (PID) stabilization is an important control design strategy that provides the designer with all PID controller set which results control system stability absolutely. In this way, the designer has a wide range of freedom to obtain the controller that meets the desired criteria. This process is particularly advantageous in situations where it is difficult to clearly define the design criteria at the beginning of the design or to make a balanced decision among the design criteria. The main objective of this paper is to present for the first time a PID stabilization method for complex conjugate-order systems, which is a new type of system for the control community and has been little studied on. The method is based on obtaining of stability/instability regions using the D-decomposition method in the controller parameter space graphically. These regions are formed by stability boundaries that are defined as real root, infinite root and complex root boundaries. The stability of the regions is determined using generalized modified Mikhailov stability criterion that is a powerful stability tool of the system theory. The simulation results indicate that the presented stabilization method is effective and practically useful in the analysis and control of the complex conjugate-order systems.

Modified Mikhailov stability criterion for continuous-time noncommensurate fractional-order systems

This paper presents the modified Mikhailov stability criterion, which can be effectively used in stability analysis for continuous-time noncommensurate fractional-order systems. The main advantage of the proposed methodology is that the stability analysis of noncommensurate fractional-order systems leads to the same computational complexity as for the commensurate order ones. Simulation examples confirm the usefulness of the introduced methodology.

A modified Mikhailov stability criterion for a class of discrete-time noncommensurate fractional-order systems

This paper introduces an extension of the Mikhailov stability criterion to a class of discrete-time noncommensurate fractional-order systems using the nabla fractional-order Grünwald-Letnikov difference. The new stability analysis methods proposed in the paper are computationally simple and can be effectively used both for commensurate and noncommensurate fractional-order systems. The main advantage of the proposed methodology is the fact that the stability analysis of noncommensurate fractional-order systems leads to exactly the same computational complexity as for the commensurate-order ones. Simulation examples confirm usefulness of the proposed methodology.

Some Insights on the Asymptotic Stabilization of a Class of SISO Marginally Stable Systems Using One Delay Block

In this paper, the stability of a class of marginally stable SISO systems is studied by applying a single one delay block as a feedback controller. More precisely, we consider an open-loop system with no zeros and whose poles are located exactly on the imaginary axis. Furthermore, a control law formed uniquely by a proportional gain and a delayed behavior is proposed for its closed-loop stabilization. The main ideas are based on a detailed analysis of the characteristic quasi-polynomial of the closed-loop system as the controller parameters (gain, delay) are varied. More precisely, by using the Mikhailov stability criterion, for a fixed delay value, we compute some gain margin guaranteeing the closed-loop stability. The particular case when the characteristic roots of the open-loop system are equidistantly distributed on the imaginary axis is also addressed. Finally, an illustrative example shows the effectiveness of the approach.

Model order reduction of discrete-time interval system based on Mikhailov stability criterion

Analysis of higher-order interval system is an appealing topic of research demanding an appropriate methodology to reduce them to their lower approximate. This paper attempts to fulfill the requirement by introducing two different algorithms interlaced with the property of Mikhailov stability criterion. Precisely, the stated approach derives the reduced order denominator polynomial, and two separate techniques compute the numerator polynomials. Overall the two proposed algorithms are justified for their novelty through their implication over the numerical examples followed by their rational acceptability to a real-time test system.

Modification of Mikhailov stability criterion for fractional commensurate order systems

In this paper we present the modification of the Mikhailov stability criterion for linear fractional commensurate order systems. The modification consists in determining the appropriate measure for the total argument change depending on the highest fractional order αn=nα of the system and not only on the integer n as stated in the literature. The validity of the result is illustrated by means of several examples.

Bifurcation Analysis of Fractional Order Single Cell with Delay

This paper presents the bifurcation analysis of fractional order model of delayed single cell which is proposed for delayed cellular neural networks with respect to the time delay τ. The bifurcation points, time delay τc, are determined by modified Mikhailov stability criterion for a range of fractional delayed cell order 0.3 ≤ q < 1. Numerical results obtained from Adams–Bashforth–Moulton method demonstrate that the supercritical Hopf bifurcation occurs in the system.

Exact stability test of neutral delay differential equations via a rough estimation of the testing integral

In this paper, we study the problem of stability test of neutral delay differential equations. Firstly we prove that the Mikhailov stability criterion and its equivalent integral form for ordinary differential equations hold for neutral delay differential equations in general form. The criteria are simple in form that is easy for numerical implementation. However, the criteria are characterized by an auxiliary function associated with the characteristic function, not by the characteristic function itself. To reduce the computational complexity, we further prove that a Mikhailov-type criterion in terms of the characteristic function holds. With this new criterion, the stability of a given neutral delay differential equation can be tested with a rough estimation of the testing integral. Thus, the computational complexity and computational cost can be greatly reduced. As two applications of these criteria, we firstly propose a numerical scheme for calculating the rightmost characteristic root(s) as well as the characteristic roots other than the rightmost roots of a given neutral delay differential equation, demonstrated with two examples. Then we derive a graphical stability criterion. With this graphical stability criterion, it is not required to know the exact curve of the Nyquist plot, but just to know whether the Nyquist plot encircles the origin of the complex plane or not.

Stability analysis of continuous-time linear systems consisting of n subsystems with different fractional orders

Abstract. The stability problem of continuous-time linear systems described by the state equation consisting of n subsystems with different fractional orders of derivatives of the state variables has been considered. The methods for asymptotic stability checking have been given. The method proposed in the general case is based on the Argument Principle and it is similar to the modified Mikhailov stability criterion known from the stability theory of natural order systems. The considerations are illustrated by numerical examples.

Stability and dynamics of isothermal CMSMPR crystallizers

Detailed analysis of stability and bifurcation properties and of the dynamic behaviour of isothermal continuous mixed suspension, mixed product removal (CMSMPR) crystallizers is presented using the moment equation model. It is shown that isothermal CMSMPR crystallizers may exhibit not only limit cycle oscillations but, in the case of magma-dependent nucleation, also saddle point instabilities. Applying the Mikhailov stability criterion, two equivalent sets of necessary and sufficient conditions are derived for checking the stability of steady states. Stability maps are presented in the planes of different pairs of system parameters. It is shown that saddle node and boundary bifurcation, as well as bifurcation at infinity may arise in the case of magma-dependent secondary nucleation. Oscillations of the population density and distribution functions are also presented and analysed, approximating the size distribution by gamma distribution. In dynamic states, the crystal size distribution may differ significantly from the exponential one, while the mean values of oscillations of supersaturation are lower than the corresponding steady state values. A causal loop diagram between the variables reveals that instabilities are created by the interactions of the autoinhibition generated negative feedback and a positive feedback between the four leading moments of crystal size.

Discrete-Time Models and Stability of Distributed Parameter Systems

The partial differential equations describing distributed parameter systems may often be reduced to transcendental transfer functions with the aid of appropriate boundary conditions. In the analysis and synthesis of closed loop systems, the transcendental transfer functions have to be approximated in a suitable manner. In this paper, discrete-time model of distributed parameter systems is obtained. The model employs a sample and hold circuit in the loop. The response of the model system is compared with the response obtained by approximating the transcendental transfer function by root factor and other approximations. The stability of linear and nonlinear systems with distributed parameters is investigated by employing the Mikhailov stability criterion.