On reducing the topological entropy of linearized nonlinear systems

Abstract

The topological entropy is a measure that quantifies the unstable of a dynamical systems. Specifically, in the case of linear systems, the topological entropy is defined as the sum of the real parts of the unstable eigenvalues (continuous-time systems) or the logarithm of the product of the magnitudes of the unstable eigenvalues (discrete-time systems), see e.g. [10], [1], [12], [3]. It is well-known that the topological entropy plays a key role in control engineering. Indeed, as it has been shown in the literature, the knowledge of the topological entropy is required to establish the existence of stabilizing feedback controllers for linear systems in the presence of communications constraints. For instance, [7] shows that a single-input discrete-time system is stabilizable in the presence of a quantizer if and only if the topological entropy is smaller than a function of the quanti- zation density. Also, [2] shows that a single-input continuous- time system is stabilizable in the presence of noise if and only if the topological entropy is smaller than a function of the signal-to-noise ratio (SNR). Other results similarly related to the topological entropy are proposed in [13], [8], [11], [9], [14]. Recent works have also addressed the determination of the worst-case topological entropy in uncertain systems, see e.g. [5]. Hence, reducing the topological entropy is a problem of fundamental importance in order to achieve stabilizability. This problem has been addressed for linear systems in [6]. In the case of nonlinear systems, the topological entropy is associated with the linearization around an equilibrium point of interest. This means that, contrary to the case of linear systems, the topological entropy in nonlinear systems is also a function of the system input, see e.g. [4]. This paper addresses a class of synthesis problems for nonlinear systems where the target is to determine an operating scenario for reducing the topological entropy associated with the linearization around some equilibrium points of interest over a set of admissible system inputs. Specifically, the operating scenario may be represented by a set of system parameters to be selected or by a controller to be implemented. Each of these, together with the admissible system inputs, affect the possible equilibrium points and the corresponding possible linearizations of the nonlinear system around them. The synthesis problem consists of determining an operating scenario such that the maximum value of the topological entropy associated with all the possible linearizations is not larger than a prescribed threshold.