Stability of linear discrete-time systems: geometric considerations

In this note, we investigate the stabilization of discrete-time linear systems with an unknown time-varying delay. Existing literature on the stabilization of time-delay systems require some knowledge, such as an upper bound, of the delay. In this paper, by proposing a switched low-gain feedback control law, we are able to achieve asymptotic stabilization of the system without any knowledge of the delay. In addition, the proposed switched low-gain feedback law also significantly improves the closed-loop system performance in terms of overshoot and convergence speed in comparison with the traditional fixed gain design. Simulation study illustrates the effectiveness of our theoretical results.

This paper considers stabilization of discrete-time linear systems, where network exists for transmitting the sensor and controller information, and arbitrary bounded packet loss occurs between the sensor and controller, and between the controller and actuator. The stabilization of this system is transformed into the robust stabilization of a set of systems. A novel stability result is given for this system, which is specially applied on model predictive control (MPC) that explicitly considers the satisfaction of input and state constraints. A synthesis approach of MPC, parameterizing the infinite horizon control moves into a single state feedback law, is proposed. A simulation example is given to illustrate the effectiveness of the proposed MPC.

Finite-time stability and stabilization of linear discrete time-varying stochastic systems

We study the self-triggered stabilization of discretetime linear systems with quantized state measurements.In the networked control system we consider, sensors may be spatially distributed and be connected to a self-triggering mechanism through finite data-rate channels.Each sensor independently encodes its measurements and sends them to the self-triggering mechanism.The self-triggering mechanism integrates quantized measurement data and then computes sampling times.Assuming that the closedloop system is stable in the absence of quantization and selftriggered sampling, we propose a joint design method of an encoding scheme and a self-triggering mechanism for stabilization.To deal with data inaccuracy due to quantization, the proposed self-triggering mechanism uses not only quantized data but also an upper bound of quantization errors, which is shared with a decoder.

Quantized stabilization of wireless networked control systems with packet losses

This paper deals with the problem of robust stabilization for linear discrete-time systems under non-linear perturbations. Without limitation to single-input systems and restriction on the structures of both the positive definite matrix and the corresponding variable, a new sufficient condition is presented which guarantees the stability of the closed-loop system and at the same time maximizes the bound on the non-linearity. In addition, our main result is extended to discrete time-delay systems. All the conditions are given within the framework of linear matrix inequalities. The effectiveness of the methods is illustrated with three examples.

The paper addresses feedback stabilization of discrete-time linear systems with multiple sensors, controllers, and actuators. The information is transmitted between them via a limited capacity deterministic communication network with arbitrary topology, which may be dynamically changed by nodes authorized to switch channels. The transferred messages may be delayed, corrupted or even lost; they may interfere and collide with each other. It is shown that the criterion for stabilizability is given by the network rate region, which answers to the question: how much information can be reliably transmitted from one set of points to another set of points. The system is stabiliazible if and only if a certain vector characterizing its rate of instability in the open-loop lies in the interior of the rate domain. In general, the rate domain of the primal network is not relevant here and a certain extension of the network should be employed. Furthermore, the relevant rate domain corresponds to a special subclass of decoders. In some cases, all decoders can be considered and this domain equals the standard one.

: This report considers a feedback control law for linear time-varying and time invariant discrete systems based on a receding horizon concept applied to a minimum energy problem with fixed terminal constraints. The control law is shown to be asymptotically stable and to result in a new method for stabilizing linear time-varying systems as well as extending some well known methods for stabilizing time invariant systems. In particular, the stabilizing gains of the feedback control law are obtained from the solution to a discrete Riccati equation over an arbitrary finite time interval, which is relatively easy to compute. The gain matrix reduces to a constant matrix for linear time invariant systems. Some stability results will turn out to be special cases of these results. The results parallel those for linear continuous time systems, although the technical details are tedious and more involved.

This paper investigates the stabilization problem for discrete-time linear controllable systems subject to actuator saturation. In the case of completely known plants, a control design technique based on a time varying sliding surface is proposed ensuring stabilization under the assumption of availability of all state measurements. Further, in the presence of matched disturbances with known constant bound, a discrete time sliding mode controller is proposed ensuring plant practical stabilization, and a conservative estimate of the attraction domain is given. Theoretical results have been validated by experimental data using a twin-rotor system.

Critical stability constraints are a small set of conditions that are enough to maintain the stability of a system when some parameters are perturbed from a nominal stable setting. The paper uses a recently introduced efficient integer-preserving (IP) form of the Bistritz test to derive critical constraints for stability of discrete-time linear systems. The new procedure produces polynomial (rather than rational) constraints of particularly low degree whose variates are the free parameters (or the literal coefficients) of the system's characteristic polynomial. Comparison with the modified Jury test, also an efficient IP stability test, shows that the constraints are obtained with less computation and, more contributing to simplicity, the constraints appear as polynomials with degrees lower by a factor of two.

A computationally efficient and simple method is suggested for determining the stability of digital systems. The new method is then compared with other existing methods.

Stabilization of linear discrete-time systems with actuator saturation

Stabilization of linear discrete-time systems over resource-constrained networks under dynamical multiple description coding scheme

2021 Benjamin Franklin Medal in Mechanical Engineering presented to C. Daniel Mote, Jr., Ph.D.

Establishing robust stability of discrete-time systems with time-varying uncertainty: The Gram-SOS approach