Search for: All records

Modern control theory provides us with a spectrum of methods for studying the interconnection of dynamic systems using input-output properties of the interconnected subsystems. Perhaps the most advanced framework for such inputoutput analysis is the use of Integral Quadratic Constraints (IQCs), which considers the interconnection of a nominal linear system with an unmodelled nonlinear or uncertain subsystem with known input-output properties. Although these methods are widely used for Ordinary Differential Equations (ODEs), there have been fewer attempts to extend IQCs to infinitedimensional systems. In this paper, we present an IQC-based framework for Partial Differential Equations (PDEs) and Delay Differential Equations (DDEs). First, we introduce infinitedimensional signal spaces, operators, and feedback interconnections. Next, in the main result, we propose a formulation of hard IQC-based input-output stability conditions, allowing for infinite-dimensional multipliers. We then show how to test hard IQC conditions with infinite-dimensional multipliers on a nominal linear PDE or DDE system via the Partial Integral Equation (PIE) state-space representation using a sufficient version of the Kalman-Yakubovich-Popov lemma (KYP). The results are then illustrated using four example problems with uncertainty and nonlinearity. 

— This paper proposes a method for certifying the local asymptotic stability of a given nonlinear Ordinary Differential Equation (ODE) by using Sum-of-Squares (SOS) programming to search for a partially quadratic Lyapunov Function (LF). The proposed method is particularly well suited to the stability analysis of ODEs with high dimensional state spaces. This is due to the fact that partially quadratic LFs are parametrized by fewer decision variables when compared with general SOS LFs. The main contribution of this paper is using the Center Manifold Theorem to show that partially quadratic LFs that certify the local asymptotic stability of a given ODE exist under certain conditions. 

The immune response is a dynamic process by which the body determines whether an antigen is self or nonself. The state of this dynamic process is defined by the relative balance and population of inflammatory and regulatory actors which comprise this decision making process. The goal of immunotherapy as applied to, e.g. Rheumatoid Arthritis (RA), then, is to bias the immune state in favor of the regulatory actors - thereby shutting down autoimmune pathways in the response. While there are several known approaches to immunotherapy, the effectiveness of the therapy will depend on how this intervention alters the evolution of this state. Unfortunately, this process is determined not only by the dynamics of the process, but the state of the system at the time of intervention - a state which is difficult if not impossible to determine prior to application of the therapy. To identify such states we consider a mouse model of RA (Collagen-Induced Arthritis (CIA)) immunotherapy; collect high dimensional data on T cell markers and populations of mice after treatment with a recently developed immunotherapy for CIA; and use feature selection algorithms in order to select a lower dimensional subset of this data which can be used to predict both the full set of T cell markers and populations, along with the efficacy of immunotherapy treatment. 

The Partial Integral Equation (PIE) framework provides a unified algebraic representation for use in analysis, control, and estimation of infinite-dimensional systems. However, the presence of input delays results in a PIE representation with dependence on the derivative of the control input, u˙. This dependence complicates the problem of optimal state-feedback control for systems with input delay – resulting in a bilinear optimization problem. In this paper, we present two strategies for convexification of the H∞-optimal state-feedback control problem for systems with input delay. In the first strategy, we use a generalization of Young's inequality to formulate a convex optimization problem, albeit with some conservatism. In the second strategy, we filter the actuator signal – introducing additional dynamics, but resulting in a convex optimization problem without conservatism. We compare these two optimal control strategies on four example problems, solving the optimization problem using the latest release of the PIETOOLS software package for analysis, control and simulation of PIEs. 

Inspired by the recently proposed Partial Integral Equality(PIE) representation for linear delay systems, this paper proposes a fuzzy-PIE representation for T-S fuzzy systems with delays for the first time. Inspired by the free-weighting matrix technique, this paper introduces the free-weighting Partial Integral (PI) operators. Based on the novel representation and free-weighting PI operators, the stability issue is investigated for the T-S fuzzy systems with delays. The corresponding conditions are given as Linear Partial Inequality (LPI) and can be solved by the MATLAB toolbox PIETOOLS. Compared with the existing results, our method has no need of the bounding technique and a large amount of matrix operation. The numerical examples show the superiority of our method. This paper adds to the expanding field of LPI approach to fuzzy systems with delays.