Convexity of the cost functional in an optimal control problem for a class of positive switched systems

Colaneri, Patrizio; Middleton, Richard H.; Chen, Zhiyong; Caporale, Danilo; Blanchini, Franco

doi:10.1016/j.automatica.2014.02.025

Cited by 81 publications

(49 citation statements)

References 21 publications

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“…is also convex over a finite or infinite time horizon for a piecewise constant u. This follows from [24,Lemma 4] and suggests that an approach inspired by the Model Predictive Control (MPC) framework can be used to compute a timevarying optimal control input for a finite horizon problem. Remark 2: The expression for ∇J 2 in Proposition 4 remains valid for any linear system and any linear operator K: R m → R n×n .…”

Section: Convexity Of Optimal Control Problemsmentioning

confidence: 96%

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

Dhingra

Colombino

Jovanović

2019

IEEE Trans. Control Netw. Syst.

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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,

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Section: Convexity Of Optimal Control Problemsmentioning

confidence: 96%

“…is a convex function of u for every c ∈ R n + [24], [25]. Furthermore, the approach in [17] can be used to cast the problem of unstructured decentralized H ∞ control of positive systems as a semidefinite program (SDP) and [23] can be used to cast it as a Linear Program (LP).…”

Section: (2a)mentioning

confidence: 99%

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

Dhingra

Colombino

Jovanović

2019

IEEE Trans. Control Netw. Syst.

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“…Besides the aforementioned results concerning static-gain state-feedback control of positive linear systems, it has been observed in the literature that synthesis problems for positive linear systems often exhibit interesting convexity properties. For example, it was shown in [13] that positive linear forms on the state variables of a time-varying positive linear system are convex with respect to the diagonals of its state matrix. The authors in [19] established the convexity of a symmetric modification of a class of steady-state disturbance attenuation problems.…”

Section: Introductionmentioning

confidence: 99%

Geometric Programming for Optimal Positive Linear Systems

Ogura

Kishida

Lam

2020

IEEE Trans. Automat. Contr.

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This paper studies the parameter tuning problem of positive linear systems for optimizing their stability properties. We specifically show that, under certain regularity assumptions on the parametrization, the problem of finding the minimum-cost parameters that achieve a given requirement on a system norm reduces to a geometric program, which in turn can be exactly and efficiently solved by convex optimization. The flexibility of geometric programming allows the state, input, and output matrices of the system to simultaneously depend on the parameters to be tuned. The class of system norms under consideration includes the H 2 norm, H ∞ norm, Hankel norm, and Schatten p-norm. Also, the parameter tuning problem for ensuring the robust stability of the system under structural uncertainties is shown to be solved by geometric programming. The proposed optimization framework is further extended to delayed positive linear systems, where it is shown that the parameter tunning problem jointly constrained by the exponential decay rate, the L 1 -gain, and the L ∞ -gain can be solved by convex optimization. The assumption on the system parametrization is stated in terms of posynomial functions, which form a broad class of functions and thus allow us to deal with various interesting positive linear systems arising from, for example, dynamical buffer networks and epidemic spreading processes. We present numerical examples to illustrate the effectiveness of the proposed optimization framework.

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“…In past decades, the stability problems of switched systems have been drawn significant attentions in both theoretical and applied studies, such as biology, 3 chemistry, 4 automotive dynamics control, 5 multiagent systems, 6,7 aerospace engineering, 8,9 traffic control, 10 and so on. Positive switched systems, whose states, inputs, and outputs are all nonnegative values, have been recently received considerable interest in view of their promising applications in different fields (please refer to other works 7,[11][12][13][14][15][16][17][18][19][20] and references cited therein).…”

Section: Introductionmentioning

confidence: 99%

Exponential stability of impulsive positive switched systems with discrete and distributed time‐varying delays

Liu

Zhang

Lü

et al. 2019

Intl J Robust & Nonlinear

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Summary This paper studies the exponential stability problems of discrete‐time and continuous‐time impulsive positive switched systems with mixed (discrete and distributed) time‐varying delays, respectively. By constructing novel copositive Lyapunov‐Krasovskii functionals and using the average dwell time technique, delay‐dependent sufficient conditions for the solvability of considered problems are given in terms of fairly simple linear matrix inequalities. Compared with the most existing results, by introducing an extra real vector, restrictive conditions on derivative of the time‐varying delays (less than 1) are relaxed, thus the obtained improved stability criteria can deal with a wider class of continuous‐time positive switched systems with time‐varying delays. Finally, two simple examples are provided to verify the validity of theoretical results.

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Convexity of the cost functional in an optimal control problem for a class of positive switched systems

Cited by 81 publications

References 21 publications

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

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

Geometric Programming for Optimal Positive Linear Systems

Exponential stability of impulsive positive switched systems with discrete and distributed time‐varying delays

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