Linear quadratic regulator problem with positive controls

Heemels, W.P.M.H.; Eijndhoven, S.J.L. van; Stoorvogel, Anton A.

doi:10.23919/ecc.1997.7082364

Cited by 16 publications

(23 citation statements)

References 13 publications

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“…2 and 3) that guaranteed this condition, a more elegant solution would be to incorporate this particular constraint when solving the LQR problem. Nevertheless, this is a topic of current investigation (Heemels et al 1998;Chen and Zhou 2004;Hu and Zhou, 2005) that requires additional work before it may be applied.…”

Section: Extensions and Future Workmentioning

confidence: 98%

Energy-based stochastic control of neural mass models suggests time-varying effective connectivity in the resting state

Sotero

Shmuel

2011

J Comput Neurosci

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Several studies posit energy as a constraint on the coding and processing of information in the brain due to the high cost of resting and evoked cortical activity. This suggestion has been addressed theoretically with models of a single neuron and two coupled neurons. Neural mass models (NMMs) address mean-field based modeling of the activity and interactions between populations of neurons rather than a few neurons. NMMs have been widely employed for studying the generation of EEG rhythms, and more recently as frameworks for integrated models of neurophysiology and functional MRI (fMRI) responses. To date, the consequences of energy constraints on the activity and interactions of ensembles of neurons have not been addressed. Here we aim to study the impact of constraining energy consumption during the resting-state on NMM parameters. To this end, we first linearized the model, then used stochastic control theory by introducing a quadratic cost function, which transforms the NMM into a stochastic linear quadratic regulator (LQR). Solving the LQR problem introduces a regime in which the NMM parameters, specifically the effective connectivities between neuronal populations, must vary with time. This is in contrast to current NMMs, which assume a constant parameter set for a given condition or task. We further simulated energy-constrained stochastic control of a specific NMM, the Wilson and Cowan model of two coupled neuronal populations, one of which is excitatory and the other inhibitory. These simulations demonstrate that with varying weights of the energy-cost function, the NMM parameters show different time-varying behavior. We conclude that constraining NMMs according to energy consumption may create more realistic models. We further propose to employ linear NMMs with time-varying parameters as an alternative to traditional nonlinear NMMs with constant parameters.

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Section: Extensions and Future Workmentioning

confidence: 98%

Energy-based stochastic control of neural mass models suggests time-varying effective connectivity in the resting state

Sotero

Shmuel

2011

J Comput Neurosci

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“…for the system (1), (2) such that the closed-loop system (15) becomes strictly Metzlerian stable if the following LP has a feasible solution with respect to the variables w = [ w 1 w 2 . .…”

Section: A Metzlerian Stabilizationmentioning

confidence: 99%

“…Theorem 2: [21] There exist a state feedback control law (14) for the system (1), (2) such that the closed-loop system (15) becomes strictly Metzlerian stable if the following LMI has a feasible solution with respect to the variables Y and Z…”

Section: A Metzlerian Stabilizationmentioning

confidence: 99%

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Constrained stabilization with maximum stability radius for linear continuous-time systems

Shafai

Ghadami

Oghbaee

2013

52nd IEEE Conference on Decision and Control

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This Paper considers the problem of constrained stabilization of linear continuous-time systems by state feedback control law. The goal is to solve this problem under positivity constraint which means that the resulting closed-loop systems are not only stable, but also positive. We focus on the class of linear continuous-time positive systems (Metzlerian systems) and use the interesting properties of Metzler matrix to provide the necessary ingredients for the main results of the paper. First, some necessary and sufficient conditions are presented for the existence of controllers satisfying the Metzlerian constraint, and the constrained stabilization is solved using linear programming (LP) or linear matrix inequality (LMI). A major objective is to formulate the constrained stabilization problem with the aim of maximizing the stability radius. We show how to solve this problem with an additional LMI formulation.

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“…For finite-horizon problems, convex duality was utilized by Rockafellar and Wolenski [28] and [14]. To a certain extent, convex analysis has been used directly in the study of CLQR by Di Blasio [7], and Heemels, Eijndhoven, and Stoorvogel [19], but duality has not been fully taken advantage of. We attempt to do it here, working directly with CLQR.…”

Section: Introductionmentioning

confidence: 99%

Continuous time constrained linear quadratic regulator - convex duality approach

Goebel

Subbotin

Proceedings of the 2005, American Control Conference, 2005.

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A continuous time infinite horizon linear quadratic regulator with input constraints is studied. On the theoretical side, optimality conditions, both in the open loop and feedback form, are shown together with smoothness of the value function and local Lipschitz continuity of the optimal feedback. Arguments are self-contained, use basic ideas of convex conjugacy, and in particular, use a dual optimal control problem. A method of calculating the optimal and stabilizing feedback without relying on discrete optimization is outlined.

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Linear quadratic regulator problem with positive controls

Cited by 16 publications

References 13 publications

Energy-based stochastic control of neural mass models suggests time-varying effective connectivity in the resting state

Energy-based stochastic control of neural mass models suggests time-varying effective connectivity in the resting state

Constrained stabilization with maximum stability radius for linear continuous-time systems

Continuous time constrained linear quadratic regulator - convex duality approach

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