A stability property and its application to discrete-time nonlinear system control

Abstract

Aktmct-An inequality on the solution of discrete-time nonlinear systems is established. A special case of this inequality leads to a stability result that is the counterpart of the Kelemen theorem on continuous-time systems. Application of this inequality gives some results on stability, regulation, and tracking of the discrete-time nonlinear systems.

Optimal control of a particular class of singularly perturbed nonlinear discrete-time systems

“…Therefore, as shown in [11], an application of the Gronwall-Bellman inequality in the Appendix shows that z(k) k b 2 c 1 1 + c1b3 k k j=1 1 + c1b3 0j 0j ky(j 0 1)k thus kz(k)k b2c1 k k j=1 0j ky(j 0 1)k where = + cb 3 . Note that can be made less that one by suitably choosing b 3 .…”

“…Substituting (10) into (1) and (2) and expanding the Taylor series expansion about (0, 0), we obtain k+2 x f (k + 1) = k+1 @f @x x f + @f @y y f + 2k+2 R 1 (x f ; y f ) (11) k+1 y f (k + 1) = k+1 @g @x x f + @g @y y f + 2k+2 R 2 (x f ; y f ) (12) where lim (x; y)! (0; 0) R1(x; y); R2(x; y) = 0.…”

A stability property of nonlinear sampled-data systems with slowly varying inputs

“…Several researchers have investigated the behavior of nonlinear systems under the influence of slowly varying input signals for both continuous-time [4], [5], [7] and discrete-time [3] cases. Essentially, the notion is that if the system possesses a manifold of exponentially or asymptotically stable constant operating points (equilibria) corresponding to constant values of the input signal, then an initial state close to this manifold and a slowly varying input signal yield a trajectory that remains close to the manifold.…”

On the discrete-time robust nonlinear servomechanism problem

“…= n + p (9) for all λ given by ([3], [4], [11] and [14] From the results of the discrete-time servomechanism problem in [11], [14] and [21], it is not difficult to deduce that, under assumption A1, and suppose that the close-loop system (5) (12) (ii) the closed-loop system (5) satisfies R2 if there exists a sufficiently smooth…”