Mathematical control of complex systems

Copyright © 2013 ZidongWang et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

STABILITY AND PERFORMANCE OF NETWORKED CONTROL SYSTEMS

Network control systems (NCSs), as one of the most active research areas, are arousing comprehensive concerns along with the rapid development of network. This dissertation mainly discusses the stability and performance of NCSs into the following two parts. In the first part, a new approach is proposed to reduce the data transmitted in networked control systems (NCSs) via model reduction method. Up to our best knowledge, we are the first to propose this new approach in the scientific and engineering society. The "unimportant" information of system states vector is truncated by balanced truncation method (BTM) before sending to the networked controller via network based on the balance property of the remote controlled plant controllability and observability. Then, the exponential stability condition of the truncated NCSs is derived via linear matrix inequality (LMI) forms. This method of data truncation can usually reduce the time delay and further improve the performance of the NCSs. In addition, all the above results are extended to the switched NCSs. The second part presents a new robust sliding mode control (SMC) method for general uncertain time-varying delay stochastic systems with structural uncertainties and the Brownian noise (Wiener process). The key features of the proposed method are to apply singular value decomposition (SVD) to all structural uncertainties, to introduce adjustable parameters for control design along with the SMC method, and new Lyapunov-type functional. Then, a less-conservative condition for robust stability and a new robust controller for the general uncertain stochastic systems are derived via linear matrix inequality (LMI) forms. The system states are able to reach the SMC switching surface as guaranteed in probability 1 by the proposed control rule. Furthermore, the novel Lyapunov-type functional for the uncertain stochastic systems is used to design a new robust control for the general case where the derivative of time-varying delay can be any bounded value (e.g., greater than one). It is theoretically proved that the conservatism of the proposed method is less than the previous methods. All theoretical proofs are presented in the dissertation. The simulations validate the correctness of the theoretical results and have better performance than the existing results

Stability Analysis and Control of Several Classes of Logical Dynamic Systems and the Applications in Game Theory

With the rapid development of complex networks, logical dynamic systems have been commonly used mathematical models for simulating Genetic Regulatory Networks (GRNs) and Networked Evolutionary Games (NEGs), which have attracted considerable attention from biology, economy and many other fields. By resorting to the Semi-Tensor Product (STP) of matrices, logical dynamic systems can be equivalently converted into discrete time linear systems with algebraic forms. Based on that, this thesis analyzes the stability and studies the control design problems of several classes of logical dynamic systems. Moreover, the obtained results are applied to investigate the control and optimization problems of NEGs. The main results of this thesis are the following. • The stability and event-triggered control for a class of k-Valued Logical Networks (KVLNs) with time delays are studied. First, some necessary and sufficient con- ditions are obtained to detect the stability of Delayed k-Valued Logical Networks (DKVLNs). Second, the global stabilization problem under event-triggered control is considered, and some necessary and sufficient conditions are presented for the sta- bilization of Delayed k-Valued Logical Control Networks (DKVLCNs). Moreover, an algorithm is proposed to construct all the event-triggered state feedback controllers via antecedence solution technique. • The robust control invariance and robust set stabilization problems for a class of Mix- Valued Logical Control Networks (MVLCNs) with disturbances are studied. First, a calculation method for the Largest Robust Control Invariant Set (LRCIS) contained in a given set is introduced. Second, based on the Robust Control Invariant Subset (RCIS) obtained, the robust set stabilization of MVLCNs is discussed, and some new results are presented. Furthermore, the design algorithm of time-optimal state feedback stabilizers via antecedence solution technique is derived. • The robust set stability and robust set stabilization problems for a class of Probabilis- tic Boolean Control Networks (PBCNs) with disturbances are studied. An algorithm to determine the Largest Robust Invariant Set (LRIS) with probability 1 of a given set for a Probabilistic Boolean Network (PBN) is proposed, and the necessary and sufficient conditions to detect whether the PBN is globally finite-time stable to this invariant set with probability 1 are established. Then, the PBNs with control inputs are considered, and an algorithm for LRCIS with probability 1 is provided, based on which, some necessary and sufficient conditions for finite-time robust set stabiliza- tion with probability 1 of PBCNs are presented. Furthermore, the design scheme of time-optimal state feedback stabilizers via antecedence solution technique is derived. • The stabilization and set stabilization problems for a class of Switched Boolean Con- trol Networks (SBCNs) with periodic switching signal are studied. First, algebraic forms are constructed for SBCNs with periodic switching signal. Second, based on the algebraic formulations, the stabilization and set stabilization of SBCNs with peri- odic switching signal are discussed, and some new results are presented. Furthermore, constructive procedure of open loop controllers is given, and the design algorithms of switching-signal-dependent state feedback controllers via antecedence solution tech- nique are derived. • The dynamics and control problems for a class of NEGs with time-invariant delay in strategies are studied. First, algebraic forms are constructed for Delayed Networked Evolutionary Games (DNEGs). Second, based on the algebraic formulations, some necessary and sufficient conditions for the global convergence of desired strategy pro- file under a state feedback event-triggered controller are presented. Furthermore, the constructive procedure and the number of all valid event-triggered state feedback controllers are derived, which can make the game converge globally. • The evolutionary dynamics and optimization problems of the networked evolutionary boxed pig games with the mechanism of passive reward and punishment are studied. First, an algorithm is provided to construct the algebraic formulation for the dynamics of this kind of games. Then, the impact of reward and punishment parameters on the final cooperation level of the whole network is discussed

Stochastic Systems: Modeling, Optimization, and Applications

The special issue of Mathematical Problems in Engineering deals with the issues of modeling, optimization, and applications associated with stochastic systems. This special issue provides a forum for researchers and practitioners to publish quality research work on modeling, optimization approaches, and their applications in the context of theory analysis and engineering developments. The accepted papers in this special issue include stochastic stability, stabilization and control optimization, stochastic optimization, particle swarm optimization, modeling and identification methods, signal processing, and robust filtering. The issue includes thirty-nine papers out of which six consider the stability and stabilization problems of stochastic systems. Twelve papers cover the problems of the controller design and relevant optimization algorithms

Output control with Internet-in-the-loop : An event-driven realization

International audienceThis work is devoted to the remote feedback control of a linear process with "Internet in the loop". In such a networked control situation, variable and unpredictable delays arise, which may decrease the global performance or destabilize the system. Our aim is to obtain the best performance despite the variation of the network QoS (quality of service). The considered application is based on a Master-Slave structure. The Slave is a light mobile robot, that receives the control data and sends its sampled position via a UDP protocol. A Master computer realizes the remote control, based on a remote observer and a state feedback. The global strategy is without buffers. The packets are time-stamped so the Master detects the variable time delays (the network QoS). This information is used to adapt its observer/controller gains and guarantee the best possible performances. The design of this gain scheduling strategy relies on Lyapunov-Krasovskii functionals with an LMI optimization which guarantees the stability even with packet losses. Experimental results are provided

H ∞ sliding mode observer design for a class of nonlinear discrete time-delay systems: A delay-fractioning approach

Copyright @ 2012 John Wiley & SonsIn this paper, the H ∞  sliding mode observer (SMO) design problem is investigated for a class of nonlinear discrete time-delay systems. The nonlinear descriptions quantify the maximum possible derivations from a linear model, and the system states are allowed to be immeasurable. Attention is focused on the design of a discrete-time SMO such that the asymptotic stability as well as the H ∞  performance requirement of the error dynamics can be guaranteed in the presence of nonlinearities, time delay and external disturbances. Firstly, a discrete-time discontinuous switched term is proposed to make sure that the reaching condition holds. Then, by constructing a new Lyapunov–Krasovskii functional based on the idea of ‘delay fractioning’ and by introducing some appropriate free-weighting matrices, a sufficient condition is established to guarantee the desired performance of the error dynamics in the specified sliding mode surface by solving a minimization problem. Finally, an illustrative example is given to show the effectiveness of the designed SMO design scheme