We consider the effect of parametric uncertainty on properties of Linear Time Invariant systems. Traditional approaches to this problem determine the worst-case gains of the system over the uncertainty set. Whilst such approaches are computationally
tractable, the upper bound obtained is not necessarily informative in terms of assessing the influence of the parameters on the system performance. We present theoretical results that lead to simple, convex algorithms producing parametric bounds on the $mathcal{L}_2$-induced input-to-output and state-to-output gains as a function of the uncertain parameters. These bounds provide quantitative information about how the uncertainty affects the system.
In this paper, we consider the problem of model order reduction of stochastic biochemical networks. In particular, we reduce the order of (the number of equations in) the Linear Noise Approximation of the Chemical Master Equation, which is often used
to describe biochemical networks. In contrast to other biochemical network reduction methods, the presented one is projection-based. Projection-based methods are powerful tools, but the cost of their use is the loss of physical interpretation of the nodes in the network. In order alleviate this drawback, we employ structured projectors, which means that some nodes in the network will keep their physical interpretation. For many models in engineering, finding structured projectors is not always feasible; however, in the context of biochemical networks it is much more likely as the networks are often (almost) monotonic. To summarise, the method can serve as a trade-off between approximation quality and physical interpretation, which is illustrated on numerical examples.
This paper addresses the problem of model reduction for dynamical system models that describe biochemical reaction networks. Inherent in such models are properties such as stability, positivity and network structure. Ideally these properties should b
e preserved by model reduction procedures, although traditional projection based approaches struggle to do this. We propose a projection based model reduction algorithm which uses generalised block diagonal Gramians to preserve structure and positivity. Two algorithms are presented, one provides more accurate reduced order models, the second provides easier to simulate reduced order models. The results are illustrated through numerical examples.
SOSTOOLS v3.00 is the latest release of the freely available MATLAB toolbox for formulating and solving sum of squares (SOS) optimization problems. Such problems arise naturally in the analysis and control of nonlinear dynamical systems, but also in
other areas such as combinatorial optimization. Highlights of the new release include the ability to create polynomial matrices and formulate polynomial matrix inequalities, compatibility with MuPAD, the new MATLAB symbolic engine, as well as the multipoly toolbox v2.01. SOSTOOLS v3.00 can interface with five semidefinite programming solvers, and includes ten demonstration examples.
