Modeling and control of complex dynamic systems: Applied mathematical aspects

The concept of complex dynamic systems arises in many varieties, including the areas of energy generation, storage and distribution, ecosystems, gene regulation and health delivery, safety and security systems, telecommunications, transportation networks, and the rapidly emerging research topics seeking to understand and analyse. Such systems are often concurrent and distributed, because they have to react to various kinds of events, signals, and conditions. They may be characterized by a system with uncertainties, time delays, stochastic perturbations, hybrid dynamics, distributed dynamics, chaotic dynamics, and a large number of algebraic loops. This special issue provides a platform for researchers to report their recent results on various mathematical methods and techniques for modelling and control of complex dynamic systems and identifying critical issues and challenges for future investigation in this field. This special issue amazingly attracted one-hundred-and eighteen submissions, and twenty-eight of them are selected through a rigorous review procedure

Stability analysis of a general class of singularly perturbed linear hybrid systems

Motivated by a real problem in steel production, we introduce and analyze a general class of singularly perturbed linear hybrid systems with both switches and impulses, in which the slow or fast nature of the variables can be mode-dependent. This means that, at switching instants, some of the slow variables can become fast and vice-versa. Firstly, we show that using a mode-dependent variable reordering we can rewrite this class of systems in a form in which the variables preserve their nature over time. Secondly, we establish, through singular perturbation techniques, an upper bound on the minimum dwell-time ensuring the overall system's stability. Remarkably, this bound is the sum of two terms. The first term corresponds to an upper bound on the minimum dwell-time ensuring the stability of the reduced order linear hybrid system describing the slow dynamics. The order of magnitude of the second term is determined by that of the parameter defining the ratio between the two time-scales of the singularly perturbed system. We show that the proposed framework can also take into account the change of dimension of the state vector at switching instants. Numerical illustrations complete our study

Feedback Control of an Exoskeleton for Paraplegics: Toward Robustly Stable Hands-free Dynamic Walking

This manuscript presents control of a high-DOF fully actuated lower-limb exoskeleton for paraplegic individuals. The key novelty is the ability for the user to walk without the use of crutches or other external means of stabilization. We harness the power of modern optimization techniques and supervised machine learning to develop a smooth feedback control policy that provides robust velocity regulation and perturbation rejection. Preliminary evaluation of the stability and robustness of the proposed approach is demonstrated through the Gazebo simulation environment. In addition, preliminary experimental results with (complete) paraplegic individuals are included for the previous version of the controller.Comment: Submitted to IEEE Control System Magazine. This version addresses reviewers' concerns about the robustness of the algorithm and the motivation for using such exoskeleton

Notions, Stability, Existence, and Robustness of Limit Cycles in Hybrid Systems

This paper deals with existence and robust stability of hybrid limit cycles for a class of hybrid systems given by the combination of continuous dynamics on a flow set and discrete dynamics on a jump set. For this purpose, the notion of Zhukovskii stability, typically stated for continuous-time systems, is extended to the hybrid systems. Necessary conditions, particularly, a condition using a forward invariance notion, for existence of hybrid limit cycles are first presented. In addition, a sufficient condition, related to Zhukovskii stability, for the existence of (or lack of) hybrid limit cycles is established. Furthermore, under mild assumptions, we show that asymptotic stability of such hybrid limit cycles is not only equivalent to asymptotic stability of a fixed point of the associated Poincar\'{e} map but also robust to perturbations. Specifically, robustness to generic perturbations, which capture state noise and unmodeled dynamics, and to inflations of the flow and jump sets are established in terms of $\mathcal{KL}$ bounds. Furthermore, results establishing relationships between the properties of a computed Poincar\'{e} map, which is necessarily affected by computational error, and the actual asymptotic stability properties of a hybrid limit cycle are proposed. In particular, it is shown that asymptotic stability of the exact Poincar\'{e} map is preserved when computed with enough precision. Several examples, including a congestion control system and spiking neurons, are presented to illustrate the notions and results throughout the paper.Comment: 26 pages. Version submitted for revie

A framework for modeling and analysis of dynamical properties of spiking neurons

A hybrid systems framework for modeling and analysis of robust stability of spiking neurons is proposed. The framework is developed for a population of n interconnected neurons. Several well-known neuron models are studied within the framework, including both excitatory and inhibitory simplified Hodgkin-Huxley, Hopf, and SNIPER models. For each model, we characterize the sets that the solutions to each system converge to. Using Lyapunov stability tools for hybrid systems, the stability properties for each case are established. An external stimuli is introduced to the simplified Hodgkin-Huxley model to achieve a global asymptotic stability property. Due to the regularity properties of the data of the hybrid models considered, the asserted stability properties are robust to small perturbations. Simulations provide insight on the results and the capabilities of the proposed framework. © 2014 American Automatic Control Council

Optimum timing for integrated pest management: Modelling rates of pesticide application and natural enemy releases

Many factors including pest natural enemy ratios, starting densities, timings of natural enemy releases, dosages and timings of insecticide applications and instantaneous killing rates of pesticides on both pests and natural enemies can affect the success of IPM control programmes. To address how such factors influence successful pest control, hybrid impulsive pest–natural enemy models with different frequencies of pesticide sprays and natural enemy releases were proposed and analyzed. With releasing both more or less frequent than the sprays, a stability threshold condition for a pest eradication periodic solution is provided. Moreover, the effects of times of spraying pesticides (or releasing natural enemies) and control tactics on the threshold condition were investigated with regard to the extent of depression or resurgence resulting from pulses of pesticide applications. Multiple attractors from which the pest population oscillates with different amplitudes can coexist for a wide range of parameters and the switch-like transitions among these attractors showed that varying dosages and frequencies of insecticide applications and the numbers of natural enemies released are crucial. To see how the pesticide applications could be reduced, we developed a model involving periodic releases of natural enemies with chemical control applied only when the densities of the pest reached the given Economic Threshold. The results indicate that the pest outbreak period or frequency largely depends on the initial densities and the control tactics