Strong Structural Controllability and Observability of Linear Time-Varying Systems

In this note we consider continuous-time systems x'(t) = A(t) x(t) + B(t) u(t), y(t) = C(t) x(t) + D(t) u(t), as well as discrete-time systems x(t+1) = A(t) x(t) + B(t) u(t), y(t) = C(t) x(t) + D(t) u(t) whose coefficient matrices A, B, C and D are not exactly known. More precisely, all that is known about the systems is their nonzero pattern, i.e., the locations of the nonzero entries in the coefficient matrices. We characterize the patterns that guarantee controllability and observability, respectively, for all choices of nonzero time functions at the matrix positions defined by the pattern, which extends a result by Mayeda and Yamada for time-invariant systems. As it turns out, the conditions on the patterns for time-invariant and for time-varying discrete-time systems coincide, provided that the underlying time interval is sufficiently long. In contrast, the conditions for time-varying continuous-time systems are more restrictive than in the time-invariant case.Comment: This work has been accepted for publication in the IEEE Trans. Automatic Control. v2: Section IV (observability) added; plus minor modifications; accepted versio

Minimizing Inputs for Strong Structural Controllability

The notion of strong structural controllability (s-controllability) allows for determining controllability properties of large linear time-invariant systems even when numerical values of the system parameters are not known a priori. The s-controllability guarantees controllability for all numerical realizations of the system parameters. We address the optimization problem of minimal cardinality input selection for s-controllability. Previous work shows that not only the optimization problem is NP-hard, but finding an approximate solution is also hard. We propose a randomized algorithm using the notion of zero forcing sets to obtain an optimal solution with high probability. We compare the performance of the proposed algorithm with a known heuristic [1] for synthetic random systems and five real-world networks, viz. IEEE 39-bus system, re-tweet network, protein-protein interaction network, US airport network, and a network of physicians. It is found that our algorithm performs much better than the heuristic in each of these cases

A Graphical Characterization of Structurally Controllable Linear Systems with Dependent Parameters

One version of the concept of structural controllability defined for single-input systems by Lin and subsequently generalized to multi-input systems by others, states that a parameterized matrix pair $(A, B)$ whose nonzero entries are distinct parameters, is structurally controllable if values can be assigned to the parameters which cause the resulting matrix pair to be controllable. In this paper the concept of structural controllability is broadened to allow for the possibility that a parameter may appear in more than one location in the pair $(A, B)$. Subject to a certain condition on the parameterization called the "binary assumption", an explicit graph-theoretic characterization of such matrix pairs is derived

A linear time algorithm to verify strong structural controllability

We prove that strong structural controllability of a pair of structural matrices $(\mathcal{A},\mathcal{B})$ can be verified in time linear in $n + r + \nu$, where $\mathcal{A}$ is square, $n$ and $r$ denote the number of columns of $\mathcal{A}$ and $\mathcal{B}$, respectively, and $\nu$ is the number of non-zero entries in $(\mathcal{A},\mathcal{B})$. We also present an algorithm realizing this bound, which depends on a recent, high-level method to verify strong structural controllability and uses sparse matrix data structures. Linear time complexity is actually achieved by separately storing both the structural matrix $(\mathcal{A},\mathcal{B})$ and its transpose, linking the two data structures through a third one, and a novel, efficient scheme to update all the data during the computations. We illustrate the performance of our algorithm using systems of various sizes and sparsity

Temperature states in Powder Bed Fusion additive manufacturing are structurally controllable and observable

Powder Bed Fusion (PBF) is a type of Additive Manufacturing (AM) technology that builds parts in a layer-by-layer fashion out of a bed of metal powder via the selective melting action of a laser or electron beam heat source. The technology has become widespread, however the demand is growing for closed loop process monitoring and control in PBF systems to replace the open loop architectures that exist today. Controls-based models have potential to satisfy this demand by utilizing computationally tractable, simplified models while also decreasing the error associated with these models. This paper introduces a controls theoretic analysis of the PBF process, demonstrating models of PBF that are asymptotically stable, stabilizable, and detectable. We show that linear models of PBF are structurally controllable and structurally observable, provided that any portion of the build is exposed to the energy source and measurement, we provide conditions for which time-invariant PBF models are classically controllable/observable, and we demonstrate energy requirements for performing state estimation and control for time-invariant systems. This paper therefore presents the foundation for an effective means of realizing closed loop PBF quality control