Critical stability constraints for discrete-time linear systems

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.

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 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.

In this paper we present a technique to stabilize discrete-time linear systems with bounded inputs. Based on optimal control techniques, we construct a continuous bounded state feedback which leads to global asymptotic stabilization for the case where the open-loop system has all its eigenvalues with modulus less than or equal to one. If the open-loop system has eigenvalues with modulus greater than one, a region of attraction of the origin is obtained. The resulting state feedback can be seen as a pointwise linear feedback with state-dependent gains, which are defined in terms of a non-linear algebraic equation.

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.

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.

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

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 geometric properties of the canonical parameter space are used to derive necessary conditions for constraining zeros to lie within or on a central circular region of radius α in the z-plane. These necessary conditions are then compared with the sufficient conditions for the stability of linear discrete time systems.

This paper studies the problem of stabilization of discrete-time linear systems with input delay and saturation nonlinearity. By exploring some further intricate properties of the recently developed parametric Lyapunov equation-based low-gain feedback design approach, solutions are proposed to solve the problems by both state feedback and output feedback. This new approach is not only simpler than the existing method that is based on the eigenstructure assignment technique, but also provides explicit conditions on the low-gain parameter to guarantee the stability of the closed-loop system. Moreover, it is possible by adjusting the low-gain parameter online to achieve global results when the system is subject to both input saturation and time-delay. Also, the delay in the input is allowed to be time-varying in some cases.

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

This paper addresses the problem of observer-based stabilization of discrete-time linear systems in presence of parameter uncertainties and $\ell_{2}$ -bounded disturbances. We propose a new variant of the classical two-steps LMI approach. In the first step, we use a slack variable technique to solve the optimization problem resulting from the stabilization problem by a static state feedback. In the second step, a part of the slack variable obtained is incorporated in the $\mathcal{H}_{\infty}$ observer-based stabilization problem, to calculate simultaneously the Lyapunov matrix and the observer-based controller gains. Numerical evaluation by Monte Carlo is presented to show the superiority of the proposed Modified Two-Steps Method (MTSM) from LMI feasibility point of view.

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