Abstract
In this paper, we provide optimal solutions to two different (but related) input/output design problems involving large-scale linear dynamical systems, where the cost associated to each directly actuated/measured state variable can take different values, but is independent of the input/output performing the task. Under these conditions, we first aim to determine and characterize the input/output placement that incurs in the minimum cost while ensuring that the resulting placement achieves structural controllability/observability. Further, we address a constrained variant of the above problem, in which we seek to determine the minimum cost placement configuration, among all possible input/output placement configurations that ensures structural controllability/observability, with the lowest number of directly actuated/measured state variables. We develop new graph-theoretical characterizations of cost-constrained input selections for structural controllability and properties that enable us to address both problems by reduction to a weighted maximum matching problem — efficiently addressed by algorithms with polynomial time complexity (in the number of state variables). Finally, we illustrate the obtained results with an example.
Highlights
The problem of control systems design, meeting certain desired specifications, is of fundamental importance
We address a constrained variant of the above problem, in which we seek to determine the minimum cost placement configuration, among all possible input/output placement configurations that ensures structural controllability/observability, with the lowest number of directly actuated/measured state variables (Pequito et al, 2013a) as stated in P1
We present the reduction of P1 and P2 to a weighted maximum matching, as provided in Algorithm 1, constrained to the conditions presented in Theorem 2 and Theorem 3, respectively
Summary
The problem of control systems design, meeting certain desired specifications, is of fundamental importance. Possible specifications include (but are not restricted to) controllability and observability. These specifications ensure the capability of a dynamical system (such as chemical process plants, refineries, power plants, and airplanes, to name a few) to drive its state toward a specified goal or infer its present state. To achieve these specifications, the selection of where to place the actuators and sensors assumes a critical importance. We need to consider the cost per actuator/sensor, that depends on its specific functionality and/or its installation and maintenance cost.
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