Stability Of Discrete Systems Research Articles

The Frequency-Domain Criterion for Quadratic Stability of Discrete Systems with Switching between Three Linear Subsystems

The Frequency-Domain Criterion for Quadratic Stability of Discrete Systems with Switching between Three Linear Subsystems

Convergence analysis of ABC algorithm based on difference model

In the cross-mutation phase of the Artificial Bee Colony optimization algorithm (ABC algorithm), the value of parameter φ is very important to the stability of ABC algorithm, however, most literatures did not specify the origin of the value range of parameter φ. Through the theoretical analysis of the stability of the ABC algorithm, appropriate parameters can be selected for the algorithm under different convergence conditions and different application conditions, and then the rapid convergence of the algorithm can be achieved, so the stability analysis of the algorithm is very worth exploring. This paper firstly introduces the ABC algorithm, and then differentiates the motion trajectory equations of employed bees in the ABC algorithm to obtain a differential model; then uses the discrete system stability theory to analyze the stability of the system, and obtains the relationship between the value range of the parameters in the cross-mutation stage and the stability of the algorithm. Finally, the correctness of the conclusion will be verified by experiments using the test function. The results obtained show that ABC algorithm with φ∈(−2,0) converges faster, and has better global optimization performance than ABC algorithm with other values of φ, which are investigated in this paper.

A linear matrix inequality-based criterion for realizing externally interfered discrete system using saturation arithmetic

The article deals with stability analysis of externally interfered discrete systems with overflow nonlinearities. The proposed criterion uses passivity property to ensure the exponential stability of discrete systems in presence of saturation overflow nonlinearity. The established condition not only provides the convergence rate of the discrete system but also mitigates the influence of external disturbance on the system. It is shown that the system with saturation nonlinearities exhibits passive behaviour under external disturbance and is exponentially stable when the disturbance vanishes. The established criterion guarantees the discrete system employing saturation arithmetic to be free of limit-cycle overflow oscillations. The conditions reported are implemented in linear matrix inequality terms and are quite novel in comparison with previous works. The usefulness of the proposed approach is validated via several numerical examples.

Digital Predictive Twins for Virtual Stability Analyzers

Identification methods are presented for real-time development of discrete intelligent predictive models of dynamic processes for electric power systems. It is shown that digital models created at each time instant based on machine learning can effectively predict the possibility of stability loss for a wide class of nonlinear dynamic processes. The stability of discrete systems is studied on the basis of the Gramian method. In this paper, the stability indices of systems are determined using energy functional. Spectral expansions of functional are obtained, which makes it possible to reveal dominant modes that affect the energy of oscillations in the modes of operation of systems near the stability boundary.

Research of the transient processes for discrete control systems

The dynamic characteristics of discrete systems, as in the theory of continuous systems, include transfer functions, time (impulse, transient) and frequency characteristics. When determining time characteristics, a single discrete impulse and a single discrete function are used as typical signals. The transient characteristic of a linear discrete system is the response to the single discrete function under zero initial conditions. The study of discrete systems begins with an assessment of their fundamental operability, that is, stability. The stability of continuous linear systems is determined by the behavior of the transient component of the general solution of a homogeneous differential equation of the system under study. Similarly, the problem is solved for discrete systems described by linear difference equations. A discrete system is called stable if the transient process tends to zero over time. The analysis of the discrete control system on the stability and accuracy of the transient process is conducted. Conclusion: in a discrete system stability, accuracy and control quality depend on the parameters of the gain system and the time.

Investigation of asphalt mixture internal structure consistency in accelerated discrete element models

The discrete element method (DEM) requires an enormous amount of computational resources when applied to asphalt mixture simulation. Reducing material modulus is recognized as an efficient method to cut down the computational cost. However, over-reduced material modulus would case unacceptable calculation errors for DEM models, and the allowable modulus reduction range is not clear. Based on the small displacement assumption in the DEM modeling, the overlap ratio between particles should be small enough to ensure efficient force transition and discrete system stability. Combining this assumption with time step determination and force-displacement law, a theoretical inference that specified a material modulus reduction range without causing an exceeded overlap ratio can be derived. To verify this theoretical inference, a three-step DEM compaction model, including gravity fall, static compaction, and gyratory compaction processes was established for asphalt mixtures. Three groups of asphalt mixtures with different mixture designs were tested both in the laboratory and in simulation. To evaluate the effects of the reduced material modulus on the internal-structure of asphalt mixture specimen under three compaction states, this study proposed 4 categories of internal-structure indexes in respect of specimen’s height, average coordination number, rotation angles of coarse aggregates, and spatial distribution of mastic particles. In the presented case, when the modulus reduction was 1/100 times, the maximum simulation error of internal-structure indexes (except the average coordinate number) was around 4% compared to the control group. Meanwhile, the model’s calculation efficiency was increased by 8–10 times depending on the mixture design.

Dominant Pole Based Approximation for Discrete Time System

This paper presents an effective procedure for model order reduction of discrete time control system. The exact model derived from complex dynamic systems proves to be very complicated for analysis, control and design. This necessity brings about using a tool known as model order reduction technique or model simplification. A novel mixed method has been implemented in this paper for reducing the order of the large scale dynamic discrete system. Dominant pole based pole clustering method has been used to derive the coefficients of denominator polynomial while Padé approximation has been applied to obtain the coefficients of numerator polynomial of the reduced order model. The proposed method is quite simple and able to generate a stable reduced order model from high order stable discrete systems. The dominancy of poles has been decided by values of the ratio of residue to its pole. The pole is considered dominant which have larger ratio value. An illustrative example has been considered to show the various reduction steps. The result obtained confirms the effectiveness of the approach.

Stabilization of Discrete System with Bounded Parameters

In this paper stabilization problem of linear discrete single input, single output plant by affine stabilizator is considered. It is assumed that stabilizing vector is bounded and its values are changed in a given box. We use the Schur-Szegö parameters (reflection coefficients) and obtain conditions for nonexistence and existence of a stabilizing vector.

Asymptotic behaviours and stability of zero dynamics in nonlinear discrete-time systems using Taylor approach and multirate input

ABSTRACTIt is well known that the existence of unstable sampled zero dynamics is recognised as a major barrier in many control problems. When the usual digital control with zero-order hold (ZOH) or fractional-order hold (FROH) input is used, unstable sampled zero dynamics inevitably appear in the discrete-time model even though the continuous-time system with relative degree more than or equal to three is of minimum phase. In this paper, we show how an approximate sampled-data model can be obtained for nonlinear systems by the use of multirate input and hold such as a generalised sample hold function (GSHF) in order that discrete zero dynamics of the resulting model can be arbitrarily placed. Furthermore, the properties of sampled zero dynamics are studied and the conditions for ensuring the stability of sampling zero dynamics of the desired model are derived. The results presented here generalise well-known notion of sampling zero dynamics from the linear case to nonlinear systems, and GSHF can provide some advantages over ZOH or FROH in terms of stability of discrete system zero dynamics.

Augmented Lyapunov-Krasovskii Functional Approach to Stability of Discrete Systems With Time-Varying Delays

This paper investigates the problem of delay-dependent stability for discrete-time systems with time-varying delays. A novel augmented Lyapunov-Krasovskii functional is proposed in deriving stability criteria in which the feasible region is enhanced. Also, an improved summation inequality is developed and applied to find the lower bound of summation inequalities. Via two numerical examples, improved results will be shown by comparing with maximum delay bounds.

On Properties of Maximal Set of External Practical Stability of Discrete Systems

On Properties of Maximal Set of External Practical Stability of Discrete Systems

Pyragas stabilization of discrete systems via delayed feedback with periodic control gain

In this paper a method for stabilization of unstable periodic solutions of dynamic systems is given. It is based on the delayed feedback with pulse periodic gain. The obtained algorithm is applicable if the linearized system around the cycle has any number of eigenvalues larger than unity. The method is illustrated with the numerical experiment for Henon map.

Investigation of Deviation of Trajectories of Linear Stable Discrete Systems from Monotonously Decreasing Curve on Free Motion in the Case of Multiple Real Positive Eigenvalues of their Matrices

Investigation of Deviation of Trajectories of Linear Stable Discrete Systems from Monotonously Decreasing Curve on Free Motion in the Case of Multiple Real Positive Eigenvalues of their Matrices

Pyragas stabilization of discrete systems with delayed feedback and pulse periodic gain

A method for stabilizing periodic solutions of dynamical systems is proposed. It is based on the introduction into a system of delayed feedback with pulse periodic gain with period equal to twice the period of the orbit to be stabilized. This approach allows one to overcome the restrictions that arise in the case of constant-gain stabilization. The algorithm obtained can be applied for any number of multipliers greater than one of a system linearized in the neighborhood of a cycle. The method is illustrated by numerical experiments for various discrete systems. Unstable cycles of a cubic map and of the Lozi and the Ricker maps are stabilized

Modified Schur-Cohn Criterion for Stability of Delayed Systems

A modified Schur-Cohn criterion for time-delay linear time-invariant systems is derived. The classical Schur-Cohn criterion has two main drawbacks; namely, (i) the dimension of the Schur-Cohn matrix generates some round-off errors eventually resulting in a polynomial ofswith erroneous coefficients and (ii) imaginary roots are very hard to detect when numerical errors creep in. In contrast to the classical Schur-Cohn criterion an alternative approach is proposed in this paper which is based on the application of triangular matrices over a polynomial ring in a similar way as in the Jury test of stability for discrete systems. The advantages of the proposed approach are that it halves the dimension of the polynomial and it only requires seeking real roots, making this modified criterion comparable to the Rekasius substitution criterion.

Stability analysis of uncertain discrete systems with two additive time delays

This paper presents new results of stability for discrete systems with two additive time-delays. By using new Lyapunov function; new conditions are established for the asymptotic and or robust stability. The stability of systems with multiple additive delays is also studied. The results are expressed in terms of linear matrix inequalities. The conditions are shown, via two examples, to be much less conservative than some existing results.

Stability and Stabilization for Discrete System With Probabilistic Nonlinearities

This paper investigates stability and stabilization of discrete systems with probabilistic nonlinearities and time-varying delay. New characters of the nonlinearities, the probability of the nonlinearities happening between different bounds, are used to build new type of system model, which can help us make a full use of the inner variation information of the nonlinearities. With the help of the new characters, new system model is proposed. Then, sufficient conditions for the mean square stability of the system can be obtained by using the Lyapunov functional approach and linear matrix inequalities technique. An example is proposed to illustrate the efficiency of the proposed method.

A new result on the delay-dependent stability of discrete systems with time-varying delays

SUMMARY This paper proposes an improvement to the delay-dependent stability of discrete systems with time-varying delays. The approach is based on the observation that the positive definiteness of a chosen Lyapunov–Krasovskii functional does not necessarily require all the involved symmetric matrices to be positive definite, which has been overlooked in the literature. The derived delay-dependent stability conditions are in terms of linear matrix inequalities. It is theoretically proved that our results are less conservative than the corresponding ones obtained by requiring the positive definiteness of all the symmetric matrices in a chosen Lyapunov–Krasovskii functional. The importance of the present approach is that a great number of delay-dependent analysis and synthesis results obtained by the aforementioned requirement in the literature can be improved by the present approach without introducing any new decision variables. Copyright © 2013 John Wiley & Sons, Ltd.

Robust Stability of Discrete System with Non-Linear Impulsive Controller

Robust Stability of Discrete System with Non-Linear Impulsive Controller

Stabilization of discrete systems by dynamic regulator

We consider the system $$\begin{gathered} x_{k + 1} = A_k x_k + b_k u_k , \hfill \\ u_{k + 1} = m_k^* x_k ,k = 1,2,..., \hfill \\ \end{gathered} $$ where Ak ∈ ℝn×n, bk ∈ ℝn, and mk ∈ ℝn. We assume that Ak is a Frobenius matrix, the last component of vector bk is zero, and all entries of Ak and bk are bounded for all k. Lyapunov quadratic function with diagonal matrix of coefficients is used to find coefficients mk and restrictions on coefficients bk which make the system globally asymptotically stable.