On the Complexity of the Constrained Input Selection Problem for Structural Linear Systems

This paper studies the problem of, given the structure of a linear-time invariant system and a set of possible inputs, finding the smallest subset of input vectors that ensures system's structural controllability. We refer to this problem as the minimum constrained input selection (minCIS) problem, since the selection has to be performed on an initial given set of possible inputs. We prove that the minCIS problem is NP-hard, which addresses a recent open question of whether there exist polynomial algorithms (in the size of the system plant matrices) that solve the minCIS problem. To this end, we show that the associated decision problem, to be referred to as the CIS, of determining whether a subset (of a given collection of inputs) with a prescribed cardinality exists that ensures structural controllability, is NP-complete. Further, we explore in detail practically important subclasses of the minCIS obtained by introducing more specific assumptions either on the system dynamics or the input set instances for which systematic solution methods are provided by constructing explicit reductions to well known computational problems. The analytical findings are illustrated through examples in multi-agent leader-follower type control problems

Consistent Approximations for the Optimal Control of Constrained Switched Systems

Though switched dynamical systems have shown great utility in modeling a variety of physical phenomena, the construction of an optimal control of such systems has proven difficult since it demands some type of optimal mode scheduling. In this paper, we devise an algorithm for the computation of an optimal control of constrained nonlinear switched dynamical systems. The control parameter for such systems include a continuous-valued input and discrete-valued input, where the latter corresponds to the mode of the switched system that is active at a particular instance in time. Our approach, which we prove converges to local minimizers of the constrained optimal control problem, first relaxes the discrete-valued input, then performs traditional optimal control, and then projects the constructed relaxed discrete-valued input back to a pure discrete-valued input by employing an extension to the classical Chattering Lemma that we prove. We extend this algorithm by formulating a computationally implementable algorithm which works by discretizing the time interval over which the switched dynamical system is defined. Importantly, we prove that this implementable algorithm constructs a sequence of points by recursive application that converge to the local minimizers of the original constrained optimal control problem. Four simulation experiments are included to validate the theoretical developments

Multicell Coordinated Beamforming with Rate Outage Constraint--Part I: Complexity Analysis

This paper studies the coordinated beamforming (CoBF) design in the multiple-input single-output interference channel, assuming only channel distribution information given a priori at the transmitters. The CoBF design is formulated as an optimization problem that maximizes a predefined system utility, e.g., the weighted sum rate or the weighted max-min-fairness (MMF) rate, subject to constraints on the individual probability of rate outage and power budget. While the problem is non-convex and appears difficult to handle due to the intricate outage probability constraints, so far it is still unknown if this outage constrained problem is computationally tractable. To answer this, we conduct computational complexity analysis of the outage constrained CoBF problem. Specifically, we show that the outage constrained CoBF problem with the weighted sum rate utility is intrinsically difficult, i.e., NP-hard. Moreover, the outage constrained CoBF problem with the weighted MMF rate utility is also NP-hard except the case when all the transmitters are equipped with single antenna. The presented analysis results confirm that efficient approximation methods are indispensable to the outage constrained CoBF problem.Comment: submitted to IEEE Transactions on Signal Processin