Positive Descriptor Time-varying Discrete-time Linear Systems and Their Asymptotic Stability

The positivity and asymptotic stability of the descriptor time-varying discrete-time linear systems are addressed. The Weierstrass-Kronecker theorem on the decomposition of the regular pencil is extended to the time-varying discrete-time descriptor linear systems. Using the extension necessary and sufficient conditions for the positivity of the systems are established. Sufficient conditions for asymptotic stability of the positive systems are presented. The effectiveness of the tests is demonstrated on the example

Robust moving horizon H∞ control of discrete time-delayed systems with interval time-varying delays

In this study, design of a delay-dependent type moving horizon state-feedback control (MHHC) is considered for a class of linear discrete-time system subject to time-varying state delays, norm-bounded uncertainties, and disturbances with bounded energies. The closed-loop robust stability and robust performance problems are considered to overcome the instability and poor disturbance rejection performance due to the existence of parametric uncertainties and time-delay appeared in the system dynamics. Utilizing a discrete-time Lyapunov-Krasovskii functional, some delay-dependent linear matrix inequality (LMI) based conditions are provided. It is shown that if one can find a feasible solution set for these LMI conditions iteratively at each step of run-time, then we can construct a control law which guarantees the closed-loop asymptotic stability, maximum disturbance rejection performance, and closed-loop dissipativity in view of the actuator limitations. Two numerical examples with simulations on a nominal and uncertain discrete-time, time-delayed systems, are presented at the end, in order to demonstrate the efficiency of the proposed method

Total Stability Properties Based on Fixed Point Theory for a Class of Hybrid Dynamic Systems

Es reproducción del documento publicado en http://dx.doi.org/10.1155/2009/826438Robust stability results for nominally linear hybrid systems are obtained from total stability theorems for purely continuous-time and discrete-time systems by using the powerful tool of fixed point theory. The class of hybrid systems dealt consists, in general, of coupled continuous-time and digital systems subject to state perturbations whose nominal (i.e., unperturbed) parts are linear and, in general, time-varying. The obtained sufficient conditions on robust stability under a wide class of harmless perturbations are dependent on the values of the parameters defining the over-bounding functions of those perturbations. The weakness of the coupling dynamics in terms of norm among the analog and digital substates of the whole dynamic system guarantees the total stability provided that the corresponding uncoupled nominal subsystems are both exponentially stable. Fixed point stability theory is used for the proofs of stability. Ageneralization of that result is given for the case that sampling is not uniform. The boundedness of the state-trajectory solution at sampling instants guarantees the global boundedness of the solutions for all time. The existence of a fixed point for the sampled state-trajectory solution at sampling instants guarantees the existence of a fixed point of an extended auxiliary discrete system and the existence of a global asymptotic attractor of the solutions which is either a fixed point or a limit n globally stable asymptotic oscillation.Ministerio de Educación (Projecto DPI2006-00714); Gobierno Vasco (GIC07143-IT-269-07 y SAIOTEK S-PE08UN15

Control over communication networks : modeling, analysis, and synthesis

The focus of this work is on dynamical systems that are controlled over a communication network, also denoted as Networked Control Systems (NCSs). Such systems consist of a continuous-time plant and a discrete-time controller that are connected via a communication network, such as e.g. controller area network (CAN), wireless networks, or internet. Advantages of the use of such a network are a reduction of installation and maintenance costs and a flexible architecture. The reduction of the costs is achieved by using one (shared) processor to control multiple plants, instead of using dedicated processors for each plant. Adding or removing plants or controllers to the network is easy, which explains the benefit in terms of a flexible architecture of the control system. Moreover, the use of wireless networks obviously allows to separate the controller and plant physically. Typical applications of NCSs are mobile sensor networks, remote surgery, automated highway systems, and the cooperative control of unmanned aerial vehicles. Disadvantages of the use of such networks are the occurrence of time-varying delays, time-varying sampling intervals, and packet dropouts, i.e. loss of data. Moreover, time-varying sampling intervals and delays may also result from other sources than the communication network. Namely, in many high-tech embedded systems, the processor is used for both the control computation and other software tasks, such as interrupt and error handling. This leads to variation in the computation time or variation in the moment of asking for new sensor data, resulting in variable sampling intervals. The amount of variation depends on the chosen software implementation, the chosen architecture, and the processor load. A control design that can deal with the variation in the time-delays, sampling intervals, and the occurrence of packet dropout is important for the multidisciplinary design of high-tech systems. Namely, such robustness properties of the control design represent a relaxation on the demands from control engineering on the software and communication network design. In this thesis, a discrete-time model for linear NCSs is derived that considers time-varying delays, time-varying sampling intervals, and packet dropouts. Based on this model, examples of the destabilizing effect of variations in the delay and variations in the sampling intervals are given to show the necessity of stability conditions that consider the effects of time-varying delays, time-varying sampling intervals, and packet dropouts. To derive such stability conditions, upper and lower bounds of time-varying delays and sampling intervals are assumed, as well as a maximum number for the subsequent packet dropouts. Based on these assumptions, sufficient conditions in terms of linear matrix inequalities (LMIs) are derived that guarantee global asymptotic stability of the NCS. Two different control strategies, i.e. state feedback control and state-feedback control including past control input information are considered. For both control approaches, conditions in terms of LMIs are given for the controller synthesis problem and a comparison of the applicability of both control approaches is made. Besides the stability analysis and controller synthesis conditions, the intersample behavior is investigated to ensure stability of the continuous-time system between the sampling instants. An extension to the stability analysis conditions is given that can be used to solve the approximate tracking problem for NCSs with time-varying delays and sampling intervals and packet dropouts. Only approximate tracking can be achieved because the time-varying delays, sampling intervals, packet dropouts, and the use of a zero-order hold between the controller and actuator cause an inexact feedforward, which induces a perturbation on the tracking error dynamics. Sufficient conditions for the input-tostate stability of the tracking error dynamics are provided and an upper bound for the tracking error is given as a function of the plant properties, the control design, and the bounds on the delays, the sampling interval and the number of subsequent packet dropouts. To validate the obtained stability and controller synthesis conditions experiments are performed on a typical motion control example. First, measurements are performed to validate the stability region, i.e. all stabilizing controllers, for constant time-delays. Second, the destabilizing effect of time-variation of the delays is shown in experiments. Third, the obtained stabilizing controllers for time-varying delays, with constant sampling intervals are validated. A comparison between the stability regions for constant delays and time-varying delays shows that the stability conditions developed in this thesis are not overly conservative. The delay combinations that result in instability in the measurements confirm this observation

Incremental Dissipativity based Control of Discrete-Time Nonlinear Systems via the LPV Framework

Unlike for Linear Time-Invariant (LTI) systems, for nonlinear systems, there exists no general framework for systematic convex controller design which incorporates performance shaping. The Linear Parameter-Varying (LPV) framework sought to bridge this gap by extending convex LTI synthesis results such that they could be applied to nonlinear systems. However, recent literature has shown that naive application of the LPV framework can fail to guarantee the desired asymptotic stability guarantees for nonlinear systems. Incremental dissipativity theory has been successfully used in the literature to overcome these issues for Continuous-Time (CT) systems. However, so far no solution has been proposed for output-feedback based incremental control for the Discrete-Time (DT) case. Using recent results on convex analysis of incremental dissipativity for DT nonlinear systems, in this paper, we propose a convex output-feedback controller synthesis method to ensure closed-loop incremental dissipativity of DT nonlinear systems via the LPV framework. The proposed method is applied on a simulation example, demonstrating improved stability and performance properties compared to a standard LPV controller design.Comment: Accepted to 60th Conference on Decision and Control, Austin, 202

H∞ Preview Control of a Class of Uncertain Discrete-Time Systems

This paper investigates the problem of H∞ preview tracking control with robust performance for uncertain discrete-time systems. In order to avoid applying the difference operator to the time-varying matrix, by taking advantage of the difference between the system state variables, input variables, and the corresponding auxiliary variables, instead of the usual difference between system states, an augmented error system including previewed information is constructed, which converts the tracking problem into a regulator problem. A sufficient condition based on the free-weighting matrices technique and the Lyapunov stability theory is derived for the robust asymptotic stability of uncertain systems. Moreover, a state feedback control law with preview action design method is obtained via linear matrix inequality (LMI) approach. Based on these, a state observer for preview control systems is formulated. Previewable reference signals are fully utilized through reformulation of the output equation while designing the state observer. The proposed construction method of augmented error system is applicable to uncertain discrete-time system in which the uncertainties are general. Also an integrator is introduced to ensure the closed-loop system tracking performance with no static error. The numerical results also show the effectiveness of the preview control law for uncertain systems in the paper