Positivity-preserving H∞ model reduction for positive systems

This is the post-print version of the Article - Copyright @ 2011 ElevierThis paper is concerned with the model reduction of positive systems. For a given stable positive system, our attention is focused on the construction of a reduced-order model in such a way that the positivity of the original system is preserved and the error system is stable with a prescribed H∞ performance. Based upon a system augmentation approach, a novel characterization on the stability with H∞ performance of the error system is first obtained in terms of linear matrix inequality (LMI). Then, a necessary and sufficient condition for the existence of a desired reduced-order model is derived accordingly. Furthermore, iterative LMI approaches with primal and dual forms are developed to solve the positivity-preserving H∞ model reduction problem. Finally, a compartmental network is provided to show the effectiveness of the proposed techniques.The work was partially supported by GRF HKU 7137/09E

Order Reduction of the Chemical Master Equation via Balanced Realisation

We consider a Markov process in continuous time with a finite number of discrete states. The time-dependent probabilities of being in any state of the Markov chain are governed by a set of ordinary differential equations, whose dimension might be large even for trivial systems. Here, we derive a reduced ODE set that accurately approximates the probabilities of subspaces of interest with a known error bound. Our methodology is based on model reduction by balanced truncation and can be considerably more computationally efficient than the Finite State Projection Algorithm (FSP) when used for obtaining transient responses. We show the applicability of our method by analysing stochastic chemical reactions. First, we obtain a reduced order model for the infinitesimal generator of a Markov chain that models a reversible, monomolecular reaction. In such an example, we obtain an approximation of the output of a model with 301 states by a reduced model with 10 states. Later, we obtain a reduced order model for a catalytic conversion of substrate to a product; and compare its dynamics with a stochastic Michaelis-Menten representation. For this example, we highlight the savings on the computational load obtained by means of the reduced-order model. Finally, we revisit the substrate catalytic conversion by obtaining a lower-order model that approximates the probability of having predefined ranges of product molecules.Comment: 12 pages, 6 figure

Positive state-bounding observer for interval positive systems under L1 performance

In this paper, the positive observer problem is investigated for interval positive systems under the L1-induced performance. To estimate the state of positive systems, a pair of state-bounding positive observers is designed. A novel characterization is first proposed under which the augmented system is stable and satisfies the L1-induced performance. Necessary and sufficient conditions are then presented to design the observers. The results obtained in this paper are expressed in terms of linear programming problems, and can be easily solved by standard software. In the end, we present a numerical example to show the effectiveness of the derived design procedures. © 2014 TCCT, CAA.postprin

H∞ model reduction for positive systems

This paper is concerned with the model reduction of positive systems. For a given stable positive system, our attention is focused on the construction of a reduced-order model in such a way that the positivity of the original system is preserved and the error system is stable with a prescribed H∞ performance. Based upon a system augmentation approach, a novel characterization on the stability with H∞ performance of the error system is first obtained in terms of linear matrix inequality (LMI). Then, a necessary and sufficient condition for the existence of a desired reduced-order model is derived accordingly. A significance of the proposed approach is that the reduced-order system matrices can be parametrized by a positive definite matrix with flexible structure, which is fully independent of the Lyapunov matrix; thus, the positivity constraint on the reduced-order system can be readily embedded in the model reduction problem. Finally, a numerical example is provided to show the effectiveness of the proposed techniques.published_or_final_versionThe 2010 American Control Conference (ACC2010), Baltimore, MD., 30 June-2 July 2010. In American Control Conference Proceedings, 2010, p. 6244-624

An Augmented Lagrangian Method for the Optimal H

This paper treats the computational method of the optimal H∞ model order reduction (MOR) problem of linear time-invariant (LTI) systems. Optimal solution of MOR problem of LTI systems can be obtained by solving the LMIs feasibility coupling with a rank inequality constraint, which makes the solutions much harder to be obtained. In this paper, we show that the rank inequality constraint can be formulated as a linear rank function equality constraint. Properties of the linear rank function are discussed. We present an iterative algorithm based on augmented Lagrangian method by replacing the rank inequality with the linear rank function. Convergence analysis of the algorithm is given, which is distinct to the now available heuristic methods. Numerical experiments for the MOR problems of continuous LTI system illustrate the practicality of our method

Frequency-limited H∞ model reduction for positive systems

In this paper, the problem of frequency-limited H∞ model reduction for positive linear time-invariant systems is investigated. Specifically, our goal is to find a stable positive reduced-order model for a given positive system such that the H∞ norm of the error system is bounded over a frequency interval of interest. A new condition in terms of matrix inequality is developed for characterizing the frequency-limited H∞ performance. Then an equivalent parametrization of a positive reduced-order model is derived, based on which, an iterative algorithm is constructed for optimizing the reduced-order model. The algorithm utilizes coarse reduced-order models resulting from (generalized) balanced truncation as the initial value. Both continuous- and discrete-time systems are considered in the same framework. Numerical examples clearly show the effectiveness and advantages of the proposed model reduction method

Decentralized control of compartmental networks with H∞ tracking performance

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A symmetry approach for balanced truncation of positive linear systems

We consider model order reduction of positive linear systems and show how a symmetry characterization can be used in order to preserve positivity in balanced truncation. The reduced model has the additional feature of being symmetric