ScienceDirect IFAC-PapersOnLine 48-27 (2015) 242–247 Available online at www.sciencedirect.com 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2015.11.182 © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Aftab Ahmed * Erik I. Verriest ** * Georgia Institute of Technology, GA-30332 USA (aftab.ahmed@gatech.edu). ** Georgia Institute of Technology, GA-30332 USA (erik.verriest@ece.gatech.edu) Nonlinear Systems Evolving with State Suprema as Multi-Mode Multi-Dimensional (M 3 D) Systems: Analysis & Observation Abstract: This paper considers the class of systems described by functional differential equations involving the sup-operator. This feature makes these systems infinite dimensional and nonlinear. We consider the general higher order case where the partial state, x(t), lies in R n . The state space for such systems is an infinite-dimensional Banach space equipped with the uniform convergence topology. We elucidate that such systems may be modeled as Multi- Mode Multi-Dimensional (M 3 D) systems. The stability and observer design for such systems is studied, the first using Razumikhin’s framework, the second using the new concept of sup based output injection. Simulation results are shown at the end which demonstrate the effectiveness, validity and usefulness of the proposed observer scheme. Keywords: Systems with state suprema, nonlinear systems, Multi-Mode Multi-Dimensional (M 3 D) systems, observer design, Razumikhin framework. 1. INTRODUCTION AND MOTIVATION We investigate the behavior of the following system evolv- ing with state suprema, ˙ x(t)= Ax(t)+ B sup t-τ ≤θ≤t C T x(θ) (1) where x(t) ∈ R n represents the state vector of the system, A ∈ R n×n is a constant time-invariant matrix and B ∈ R n , C ∈ R n are constant vectors. τ ∈ R + is the length of the memory of the sup functional. Notice that (1) is not an Ordinary Differential Equation (ODE) but is indeed a Functional Differential Equation (FDE). Examples of such systems are stuck float and rachet where the decision is to be made by the maximum value of the state variable. System (1) can also be equivalently written in the following fashion. Σ u :    ˙ x(t)= Ax(t)+ Bu(t) u(t)= sup t-τ ≤θ≤t C T x(θ). (2) In this format, it can be thought of as a closed-loop feedback system in which the control policy u(t) ∈ R at any instant of time t depends on the supremum of the weighted linear combination of the states over the past history interval of length τ units of time. The theoretical results and investigations of FDEs with “suprema” opens the door to enormous possibilities for their applications to real world processes and phenomena (Bainov and Hristova (2011)). 2. LITERATURE SURVEY Popov (1966) encountered the following historical equa- tion while studying the voltage regulation problem of a constant current generator. ˙ x(t)= - 1 T x(t) - q T sup t-τ ≤θ≤t x(θ)+ 1 T w(t) (3) In the above equation T ∈ R, q ∈ R and τ ∈ R + are constants which characterize the object. The state variable x(t) and the driving term (forcing function) w(t) physically represent the regulated voltage and the perturbation effect respectively at any arbitrary instant of time t. Notice that (3) not only involves the unknown function x but also its maximum value over an interval of past history of length τ and is therefore indeed an infinite-dimensional system. Another example of historical significance, for systems evolving with state suprema, is the Hausrath equation quoted in Hale (1977) as, ˙ x(t)= -ζx(t)+ ζ sup t-τ ≤θ≤t |x(θ)|,ζ> 0,t ≥ 0. (4) This equation possesses a richer structure than (3). Here the supremum over the past history can never go negative because of the modulus operator. However, the regularity and smoothness of its solution are inferior to that of (3). Hadeler (1979) modeled the vision process in the com- pound eye of a horseshoe crab by the following FDE evolving with state suprema: ˙ x(t)= -δx(t)+ p sup t-τ ≤τ (t)≤t (x(τ (t)),c), δ, p ∈ R,c< 0 (5) where the state x is related to the activation potential above a certain threshold produced in sensory cell by light. The reciprocal of δ accounts for the response time constant. τ stands for the lateral inhibition delay with a typical experimental value of about 100 msec. The function sup(x(τ (t)),c) is termed as the rectifier function.