Abstract
In this paper we investigate realizability of discrete time linear dynamical systems (LDSs) in fixed state space dimension. We examine whether there exist different Θ = (A,B,C,D) state space realizations of a given Markov parameter sequence Y with fixed B, C and D state space realization matrices. Full observation is assumed in terms of the invertibility of output mapping matrix C. We prove that the set of feasible state transition matrices associated to a Markov parameter sequence Y is convex, provided that the state space realization matrices B, C and D are known and fixed. Under the same conditions we also show that the set of feasible Metzler-type state transition matrices forms a convex subset. Regarding the set of Metzler-type state transition matrices we prove the existence of a structurally unique realization having maximal number of non-zero off-diagonal entries. Using an eigenvalue assignment procedure we propose linear programming based algorithms capable of computing different state space realizations. By using the convexity of the feasible set of Metzler-type state transition matrices and results from the theory of non-negative polynomial systems, we provide algorithms to determine structurally different realization. Computational examples are provided to illustrate structural non-uniqueness of network-based LDSs.
Highlights
Many problems in computer science and engineering involve sequences of real-valued multivariate observations
A discrete time linear dynamical system (LDS) in state space representation is given by a tuple Θ = (A, B, C, D) and the associated system of difference equations (DEs) is as follows: x(k + 1) = Ax(k) + Bu(k), x(0) = x0, (1)
By some simple linear dynamical system models we show that the set of feasible state transition matrices A(Y, B, C, D) is not necessary unique and structurally different dynamically equivalent realizations can be computed
Summary
Many problems in computer science and engineering involve sequences of real-valued multivariate observations. The distributed, locally connected structure of dynamical systems poses important requirements towards efficient computational approaches, e.g. distributed controller synthesis methods over traditional centralized control algorithms [33, 9] It can be observed in many dynamical systems that the underlying network structure is topologically non-unique i.e., different interconnection (graph) patterns can be encoded by the same dynamical equations [1, 2]. Using the eigenvalue assignment procedure we formulate a convex optimization based procedure that can be efficiently employed to find different realizations of LDSs. Assuming the Metzler property and making use of the convexity of the feasible set of system matrices we provide algorithms capable of determining structurally different dynamically equivalent state space realizations
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