Structured Decentralized Control of Positive Systems With Applications to Combination Drug Therapy and Leader Selection in Directed Networks

Abstract

We study a class of structured optimal control problems in which the main diagonal of the dynamic matrix is a linear function of the design variable. While such problems are in general challenging and nonconvex, for positive systems we prove convexity of the H2 and H∞ optimal control formulations which allow for arbitrary convex constraints and regularization of the control input. Moreover, we establish differentiability of the H∞ norm when the graph associated with the dynamical generator is weakly connected and develop a customized algorithm for computing the optimal solution even in the absence of differentiability. We apply our results to the problems of leader selection in directed consensus networks and combination drug therapy for HIV treatment. In the context of leader selection, we address the combinatorial challenge by deriving upper and lower bounds on optimal performance. For combination drug therapy, we develop a customized subgradient method for efficient treatment of diseases whose mutation patterns are not connected.(A ≥ 0) if A has positive (nonnegative) entries and A 0 (A 0) if A is symmetric and positive (semi)-definite. We define the sparsity pattern of a vector u, sp (u), as the set of indices for which u i is nonzero, u 1 := i |u i | is the 1 norm, and K † : R n×n → R m is the adjoint of a linear operator K: R m → R n×n if it satisfies, X, K(u) = K † (X), u for all u ∈ R m and X ∈ R n×n .Definition 1 (Graph associated with a matrix): G(A) = (V, E) is the graph associated with a matrix A ∈ R n×n , with the set of nodes (vertices) V := {1, . . . , n} and the set of edges E := {(i, j)| A ij = 0}, where (i, j) denotes an edge pointing from node j to node i.Definition 2 (Strongly connected graph): A graph (V, E) is strongly connected if there is a directed path between any two distinct nodes in V.Definition 3 (Weakly connected graph): A graph (V, E) is weakly connected if replacing its edges with undirected edges results in a strongly connected graph.Definition 4 (Balanced graph): A graph (V, E) is balanced if, for every node i ∈ V, the sum of edge weights on the edges pointing to node i is equal to the sum of edge weights on the edges pointing from node i.Definition 5: A dynamical system is positive if, for any nonnegative initial condition and any nonnegative input, the output is nonnegative for all time. A linear time-invariant system,

Geometric Programming for Optimal Positive Linear Systems

“…Under these assumptions, for all nonnegative initial condition x(0) and nonnegative input signal u(t) (t ≥ 0), the values of the state x(t) and output y(t) remain nonnegative at every time instant t. Also, we say that the system Σ θ is internally stable if the matrix A(θ ) is Hurwitz stable. The parametrized positive model Σ θ arises in various contexts including drug therapy and leader selection [18], as well as dynamical buffer networks [43] and networked epidemics [34], [41] (see Sections IX and VIII for these examples, respectively). In this paper, we consider the following general parameter optimization problem:…”

“…The authors in [14] showed the convexity of the power norm of output signals with respect to the diagonals of the state matrix. The authors in [18] presented an intrinsic convexity property of H 2 and H ∞ state-feedback control problems for positive linear systems. A similar result is obtained in [15] for robust state-feedback stabilization under structured uncertainties.…”

Regularized Identification with Internal Positivity Side-Information

“…Luenberger pioneered the system theoretic approach to positive systems with his seminal work [8] in the 1980s. Since then various subjects of system theory aspects have been tackled, e.g., realization theory [7], controllability and reachability [9], observability and observer design [10], robust stability [11], [12], positive stabilization [13], [14], fault detection and estimation [15], decentralized and distributed control [16], [17], and, large scale positive systems and scalable control [18], [19].…”

Optimal and automated decentralised converter control design in more electrical aircraft power electronics embedded grids

“…Compared to previous works, the proposed approach is not limited to investigating the stability of the network, but aims to automatically design controllers for each of the individual inverters that compose the power system. Starting from the global state space model of the system, local converter controllers are synthesised by solving an optimal H 2 structured control problem [21]. The power of this approach lies in the ability of handling complex control problems, whilst always returning a stable controller with great robustness to system disturbances and parametric uncertainty [22].…”