This paper addresses difference flatness for structured LTI discrete-time systems. Two forms of necessary and sufficient conditions for an output to be a structural flat output are given. First, a preliminary result algebraically defines a flat output in terms of invariant zeros regardless whether an LTI system is structured or not. Next, the conditions are expressed in terms of graphical conditions to define a structural flat output. Checking for the graphical conditions calls for algorithms that have polynomial-time complexity and that are commonly used for digraphs. The tractability of the conditions is illustrated on several examples.

In this article, we present a reinforcement learning-based scheme for secondary frequency control of lossy inverter-based microgrids. Compared with the existing methods in the literature, we relax the common restrictions on the system, i.e., being lossless, and the transmission lines and loads to have known constant impedances. The proposed secondary frequency control scheme does not require a priori information about system parameters and can achieve frequency synchronization within an ultimate bound in the presence of dominantly resistive and/or inductive line and load impedances, model parameter uncertainties, and time varying loads and disturbances. First, using Lyapunov theory, a feedback control is formulated based on the unknown dynamics of the microgrid. Next, a performance function is defined based on cumulative costs toward achieving convergence to the nominal frequency. The performance function is approximated by a critic neural network in real-time. An actor network is then simultaneously learning a parameterized approximation of the nonlinear dynamics and optimizing the approximated performance function obtained from the critic network. Furthermore, using the Lyapunov approach, the uniformly ultimate boundedness of the closed-loop frequency error dynamics and the networks' weight estimation errors are shown.

We present a privacy-preserving distributed reinforcement learning-based control scheme to address the problem of frequency control and economic dispatch in power generation systems. The proposed control approach requires neither a priori system model knowledge nor the mathematical formulation of the generation cost functions. Due to not requiring the generation cost models, the control scheme is capable of dealing with scenarios in which the cost functions are hard to formulate and/or non-convex. Furthermore, it is privacy-preserving, i.e. none of the units in the network needs to communicate its cost function and/or control policy to its neighbors. To realize this, we propose an actor-critic algorithm with function approximation in which the actor step is performed individually by each unit with no need to infer the policies of others. Moreover, in the critic step each generation unit shares its estimate of the local measurements and the estimate of its cost function with the neighbors, and via performing a consensus algorithm, a consensual estimate is achieved. The performance of our proposed control scheme, in terms of minimizing the overall cost while persistently fulfilling the demand and fast reaction and convergence of our distributed algorithm, is demonstrated on a benchmark case study.

In this paper we consider the problem of controlling a limited number of target nodes of a network. Equivalently, we can see this problem as controlling the target variables of a structured system, where the state variables of the system are associated to the nodes of the network. We deal with this problem from a different point of view as compared to most recent literature. Indeed, instead of considering controllability in the Kalman sense, that is, as the ability to drive the target states to a desired value, we consider the stronger requirement of driving the target variables as time functions. The latter notion is called functional target controllability. We think that restricting the controllability requirement to a limited set of important variables justifies using a more accurate notion of controllability for these variables. Remarkably, the notion of functional controllability allows formulating very simple graphical conditions for target controllability in the spirit of the structural approach to controllability. The functional approach enables us, moreover, to determine the smallest set of steering nodes that need to be actuated to ensure target controllability, where these steering nodes are constrained to belong to a given set. We show that such a smallest set can be found in polynomial time. We are also able to classify the possible actuated variables in terms of their importance with respect to the functional target controllability problem.

In this paper we study linear structured systems described by means of system matrices of which only the zero/non-zero structure is known and where the non-zeros are supposed to have independent values. The structure of linear structured systems can be represented by means of various types of graphs, like directed graphs or dynamic graphs. Here we use both type of graphs because they enable us to formulate and study certain controllability properties in a uniform and straightforward way. In this paper we extend the results of a previous paper containing a partial characterisation of the fixed part of the controllable subspace of linear structured systems. This fixed part is defined as the part of the controllable subspace that is independent of the values to the non-zeros, and therefore can be seen as the robust part of the controllable subspace. It turns out that, by considering the generic dimension of the controllable subspace, a characterisation of the fixed part can be obtained. The latter dimension equals the size of the minimal set of nodes in the dynamic graph that separates between the set of input nodes and the set of final state nodes. Computing the supremal of such minimal separating sets, we are capable of characterising the fixed part. In the paper we indicate how this supremal minimal separating set can be obtained insightfully and efficiently using the recursive nature of the dynamic graph. Our results are illustrated by some meaningful examples.

In this paper, we present a reinforcement learning control scheme for optimal frequency synchronization in a lossy inverter-based microgrid. Compared to the existing methods in the literature, we relax the restrictions on the system, i.e. being a lossless microgrid, and the transmission lines and loads to have constant impedances. The proposed control scheme does not require a priori information about system parameters and can achieve frequency synchronization in the presence of dominantly resistive and/or inductive line and load impedances, model parameter uncertainties, time varying loads and disturbances. First, using Lyapunov theory a feedback control is formulated based on the unknown dynamics of the microgrid. Next, a performance function is defined based on cumulative rewards towards achieving convergence to the nominal frequency. The performance function is approximated by a critic neural network in real-time. An actor network is then simultaneously learning a parameterized approximation of the nonlinear dynamics and optimizing the approximated performance function obtained from the critic network. The performance of our control scheme is validated via simulation on a lossy microgrid case study in the presence of disturbances.

In this paper we consider (large and complex) interconnected networks. We assume that each state/node, not belonging to a set of forbidden nodes of the network, can be selected to act as a steering node, meaning that such a node then is influenced by its own individual control. We aim to achieve structural controllability and we present a classification of the associated steering nodes as being essential (always required to be present), useful (present in certain configurations) and useless (never necessary in whatever configuration). The classification is based on two types of decomposition that naturally show up in the context of the two conditions (connection condition and rank condition) for structural controllability. The underlying methods are related to well-known and efficient network algorithms.

In this paper we study the controllability of interconnected networks that are described by means of structured linear systems with state-like and control variables. We assume that the systems operate in discrete time with the set of integers as the time axis. Further, we assume that the state-like variables for their evolution only depend on recent values of their neighbours with, however, unknown weight factors. These recent values may be one step back in time, but also more steps. This yields a description of the systems by means of matrices containing fixed zeros and free parameters, together with a time lag structure. Knowing the dependency and lag structure, we represent (the structure of the) systems by means of weighted directed graphs and study questions concerning their structural controllability, where the latter has to be defined in an appropriate way, i.e., in behavioural sense. We provide a necessary and sufficient characterization of structural controllability of our systems using a graph representation. The obtained characterization makes use of well-known and efficient algorithms from graph theory. We prove that in this context finding the minimal number of driver (controller) nodes is an NP-hard problem. The concepts and results of the paper are illustrated on academic examples and on a gene regulatory network.

In this paper we discuss a generalization of power algorithms over max-plus algebra. We are interested in finding such a generalization starting from various existing power algorithms. The resulting algorithm can be used to determine the so-called generalized eigenmode of any square regular matrix over max-plus algebra. In particular, the algorithm can be applied in the case of regular reducible matrices in which the existing power algorithms can not be used to compute eigenvalues and corresponding eigenvectors.

In this paper we consider interconnected networks that are described by means of structured linear systems with state and control variables. We represent these systems, whose matrices contain fixed zeros and free parameters, by means of directed graphs and study questions concerning controllability and the controllable subspace. We show in this paper that the controllable subspace can have a part that will be present for almost all values of the free parameters. It actually is a subspace of the controllable subspace and will be referred to as the fixed controllable subspace. The subspace can then be seen as a kind of robustly controllable part of the system. Indeed, it is a subspace in the state space with the generic property that states in it can be steered in an arbitrary way. We derive a characterization of the fixed controllable subspace using the graph representation. The obtained characterization makes use of well-known algorithms from optimization and networks theory. To get some more insight in the components in the fixed part, we also give a representation of the structured linear systems by means of bipartite graphs. Using the Dulmage–Mendelsohn decomposition, we are able to decompose our structured systems in such a way that in some special cases, the fixed controllable subspace can be obtained directly from the decomposition.