Stabilization of linear time-varying systems

For linear time-varying systems with bounded system matrices we discuss the problem of stabilizability by linear state feedback. It is shown that an optimal control approach yields a criterion in terms of the cost for stabilizability. The constants appearing in the criterion of optimality allow for the distinction of exponential and uniform exponential stabilizability. We show that the system is completely controllable if, and only if, the Lyapunov exponent is arbitrarily assignable by a suitable feedback. For uniform exponential stabilizability and the assignability of the Bohl exponent this property is known. Also, dynamic feedback does not provide more freedom to address the stabilization problem

Stabilizability of systems with exponential dichotomy

In this paper we introduce the concept of controllability into a closed subspace for time-varying linear systems. Various characterizations are given and the dual relation is discussed. This concept is used to present a necessary and sufficient condition for the stabilizability of systems with exponential dichotomy

On the stabilization of persistently excited linear systems

We consider control systems of the type $\dot x = A x +\alpha(t)bu$, where $u\in\R$, $(A,b)$ is a controllable pair and $\alpha$ is an unknown time-varying signal with values in $[0,1]$ satisfying a persistent excitation condition i.e., \int_t^{t+T}\al(s)ds\geq \mu for every $t\geq 0$, with $0<\mu\leq T$ independent on $t$. We prove that such a system is stabilizable with a linear feedback depending only on the pair $(T,\mu)$ if the eigenvalues of $A$ have non-positive real part. We also show that stabilizability does not hold for arbitrary matrices $A$. Moreover, the question of whether the system can be stabilized or not with an arbitrarily large rate of convergence gives rise to a bifurcation phenomenon in dependence of the parameter $\mu/T$

Discrete-time systems with time-varying time delay: Stability and stabilizability

This paper deals with the class of linear discrete-time systems with varying time delay. The problems of stability and stabilizability for this class of systems are considered. Given an upper bound and a lower bound on the time-varying delay, sufficient conditions for checking the stability of this class of systems are developed. A control design algorithm is also provided. All the results developed in this paper are in the LMI formalism which makes their solvability easier using existing tools. A numerical example is provided to show the effectiveness of the established results

Mean Square Capacity of Power Constrained Fading Channels with Causal Encoders and Decoders

This paper is concerned with the mean square stabilization problem of discrete-time LTI systems over a power constrained fading channel. Different from existing research works, the channel considered in this paper suffers from both fading and additive noises. We allow any form of causal channel encoders/decoders, unlike linear encoders/decoders commonly studied in the literature. Sufficient conditions and necessary conditions for the mean square stabilizability are given in terms of channel parameters such as transmission power and fading and additive noise statistics in relation to the unstable eigenvalues of the open-loop system matrix. The corresponding mean square capacity of the power constrained fading channel under causal encoders/decoders is given. It is proved that this mean square capacity is smaller than the corresponding Shannon channel capacity. In the end, numerical examples are presented, which demonstrate that the causal encoders/decoders render less restrictive stabilizability conditions than those under linear encoders/decoders studied in the existing works.Comment: Accepted by the 54th IEEE Conference on Decision and Contro

Stabilization over power-constrained parallel Gaussian channels

This technical note is concerned with state-feedback stabilization of multi-input systems over parallel Gaussian channels subject to a total power constraint. Both continuous-time and discrete-time systems are treated under the framework of H2 control, and necessary/sufficient conditions for stabilizability are established in terms of inequalities involving unstable plant poles, transmitted power, and noise variances. These results are further used to clarify the relationship between channel capacity and stabilizability. Compared to single-input systems, a range of technical issues arise. In particular, in the multi-input case, the optimal controller has a separation structure, and the lower bound on channel capacity for some discrete-time systems is unachievable by linear time-invariant (LTI) encoders/decoder

Stabilization of systems with asynchronous sensors and controllers

We study the stabilization of networked control systems with asynchronous sensors and controllers. Offsets between the sensor and controller clocks are unknown and modeled as parametric uncertainty. First we consider multi-input linear systems and provide a sufficient condition for the existence of linear time-invariant controllers that are capable of stabilizing the closed-loop system for every clock offset in a given range of admissible values. For first-order systems, we next obtain the maximum length of the offset range for which the system can be stabilized by a single controller. Finally, this bound is compared with the offset bounds that would be allowed if we restricted our attention to static output feedback controllers.Comment: 32 pages, 6 figures. This paper was partially presented at the 2015 American Control Conference, July 1-3, 2015, the US