M. S. Mousa

Quantization and overflow effects in digital implementations of linear dynamic controllers

The stability of a class of single-input/single-output (SISO) digital-feedback control systems is investigated. The systems considered contain a linear dynamic plant, a digital controller, and suitable D/A (digital-to-analog) and A/D converters. The design of such systems is usually accomplished by ignoring the nonlinear effects caused by quantization and overflow truncation. Simulations of specific examples establish the quantization in such systems can lead to loss of asymptotic stability of the origin. Obtained results prove that when quantization is taken into account (but overflow is neglected), one only has convergence to some (small) neighborhood about the origin. The results also prove that for initial conditions sufficiently far from the origin, overflow effects can lead to unbounded solutions. In particular, this is the case for observable systems with at least one eigenvalue in the open right-half plane (RHP). The case of systems with no eigenvalues in the open RHP, but with eigenvalues on the j omega -axis, is still unresolved. >

On asymptotic behaviour of the difference equation\(X_{N + 1} = \alpha + \frac{{X_{N - 1} ^P }}{{X_N ^P }}\)

AbstractIn this paper, we investigate local stability, oscillation and boundeness character of positive solutions of the difference equation

Stability analysis of interconnected dynamical systems: Hybrid systems involving operators and difference equations

x_{n + 1} = \alpha + \frac{{x_{n - 1} ^p }}{{x_n ^p }},n = 0,1,...

On asymptotic behaviour of the difference equation xn+1=α+xn-kxn

under specified conditions.

On the recursive sequences xn+1=-αxn-1β±xn

We address the stability analysis of composite hybrid dynamical feedback systems of the type depicted in Fig. 1, consisting of a block (usually the plant) which is described by an operator L and of a finite-dimensional block described by a system of ordinary differential equations (usually the controller). We establish results for the well-posedness, attractivity, asymptotic stability, uniform boundedness, asymptotic stability in the large, and exponential stability in the large for such systems. The hypotheses of these results are phrased in terms of the I/O properties of L and in terms of the Lyapunov stability properties of the subsystem described by the indicated ordinary differential equations. The applicability of our results is demonstrated by means of general specific examples (involving C 0 -semigroups, partial differential equations, or integral equations which determine L ).

On asymptotic behaviour of the difference equation

We address the stability analysis of interconnected feedback systems of the type depicted in Fig. 1, which consists of a linear interconnection of l subsystems. Each subsystem is a feedback system in its own right, consisting of a local plant (which is described by an operator L_i ) and of a digital controller (which is described by a system of difference equations and which includes A/D and D/A converters). We establish conditions for the attractivity, asymptotic stability, asymptotic stability in the large, and boundedness of solutions for such systems. The hypotheses of our results are phrased in terms of the I/O properties of the operators L_i and of the entire interconnected system, and in terms of the Lyapunov stability properties of digital controllers described by the indicated difference equations. In all cases, our results allow a stability analysis of complex interconnected systems in terms of the qualitative properties of the simpler free subsystems and in terms of the properties of the system interconnecting structure. The applicability of our results is demonstrated by means of a specific example (Fig. 2).

On asymptotic behaviour of the difference equation X N+1 = α + X N-1 P /X N P

The stability of SISO digital feedback systems is considered. These systems contain a linear dynamic plant, a digital controller and suitable D/A and A/D connections. The design of such a system is normally accomplished by ignoring the nonlinear effects caused by quantization and overflow truncation. It is shown (by simulations of specific examples) that quantization can lead to loss of asymptotic stability. Instead, our results prove that one has convergence to a small neighborhood of the origin. For large initial conditions it is shown that overflow effects can lead to unbounded solutions.