Nonlinear Systems Evolving with State Suprema as Multi-Mode Multi-Dimensional (M3D) Systems: Analysis & Observation

Preprints, 5th IFAC Conference on Analysis and Design of Hybrid Preprints, 5th IFAC Conference on Analysis and Design of Hybrid Preprints, 5th IFAC Conference on Analysis and Design of Hybrid Systems Preprints, Systems 5th IFAC Conference on Analysis and Design of Hybrid October 14-16, 2015. Georgia Tech, Atlanta, USAonline at www.sciencedirect.com Available Systems Systems October 14-16, 2015. Georgia Tech, Atlanta, USA October October 14-16, 14-16, 2015. 2015. Georgia Georgia Tech, Tech, Atlanta, Atlanta, USA USA

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Nonlinear Systems Evolving with State Nonlinear Systems Evolving with State Nonlinear Systems Evolving with State Nonlinear Systems Evolving with State Suprema as Multi-Mode Multi-Dimensional Suprema as Multi-Mode Multi-Dimensional Suprema as Multi-Mode Multi-Dimensional 3 3 D) Systems: Analysis & Observation (M (M Analysis & Observation 3 D) Systems: (M (M 3 D) D) Systems: Systems: Analysis Analysis & & Observation Observation

∗ ∗∗ Aftab Ahmed ∗ Erik I. Verriest ∗∗ Aftab Ahmed ∗ Erik I. Verriest ∗∗ ∗∗ ∗ Aftab Ahmed Erik I. Verriest Aftab Ahmed ∗ Erik I. Verriest ∗∗ ∗ ∗ Georgia Institute of Technology, GA-30332 USA Georgia Institute of Technology, GA-30332 USA ∗ ∗ of ∗ Georgia Institute ([email protected]). Georgia Institute of Technology, Technology, GA-30332 GA-30332 USA USA ([email protected]). ∗∗ ([email protected]). of Technology, GA-30332 USA ∗∗ Georgia Institute ([email protected]). ∗∗ Georgia Institute of Technology, GA-30332 USA ∗∗ ∗∗ Georgia ([email protected]) Institute of Technology, GA-30332 Georgia ([email protected]) Institute of Technology, GA-30332 USA USA ([email protected]) ([email protected]) Abstract: This paper considers the class of systems described by functional differential Abstract: This This paper paper considers considers the the class class of of systems systems described described by by functional functional differential differential Abstract: equations involving the sup-operator. This feature makes these systems infinite dimensional Abstract: This paper considers the class of systems described by functional differential equations involving the sup-operator. This feature makes these systems infinite dimensional equations involving the This feature makes these systems infinite dimensional and nonlinear. We the higher order case partial state, x(t), equations involving the sup-operator. sup-operator. feature these the systems infinite and nonlinear. We consider consider the general generalThis higher order makes case where where the partial state, dimensional x(t), lies lies in in n nonlinear. and We consider the general higher order case where the partial state, x(t), in R . The state space for such systems is an infinite-dimensional Banach space equipped with and nonlinear. We consider thesystems generalishigher order case where the partial state, x(t), lies lies in Rnnnn . The state space for such an infinite-dimensional Banach space equipped with The space systems is Banach space equipped with R the convergence topology. We elucidate that such systems may be modeled as MultiThe state state space for for such such systems is an an infinite-dimensional infinite-dimensional Banach space equipped with R .. uniform the uniform convergence topology. We elucidate that such systems may be modeled as Multi3 the uniform convergence topology. We elucidate that such systems may be modeled as MultiMode Multi-Dimensional (M The stability and observer design for such systems 3 D) systems. the uniform convergence topology. We elucidate that such systems may be modeled as MultiMode Multi-Dimensional (M 333 D) systems. The stability and observer design for such systems is is Mode Multi-Dimensional (M The and observer design for systems is studied, the framework, the the new of sup Mode Multi-Dimensional (M D) D) systems. systems. The stability stability and using observer for such such is studied, the first first using using Razumikhin’s Razumikhin’s framework, the second second using thedesign new concept concept ofsystems sup based based 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, studied, the first using Razumikhin’s framework, usingdemonstrate the new concept of sup based output injection. Simulation results are shown atthe thesecond end which the effectiveness, output injection. Simulation results are shown the validity and usefulness of the proposed output injection. Simulation results are observer shown at atscheme. the end end which which demonstrate demonstrate the the effectiveness, effectiveness, validity and usefulness of the proposed observer scheme. validity and usefulness of the proposed observer scheme. validity and usefulness of the proposed observer scheme. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Systems Systems with with state state suprema, suprema, nonlinear nonlinear systems, systems, Multi-Mode Multi-Mode Multi-Dimensional Multi-Dimensional Keywords: 3 Keywords: Systems with state suprema, nonlinear systems, Multi-Mode Multi-Dimensional (M D) systems, observer design, Razumikhin framework. 3 Keywords: Systems with state suprema, nonlinear systems, Multi-Mode Multi-Dimensional observer design, Razumikhin framework. (M 3 3 D) systems, (M D) systems, systems, observer observer design, design, Razumikhin Razumikhin framework. framework. (M 3 D) constant current generator. 1. constant current generator. 1. INTRODUCTION INTRODUCTION AND AND MOTIVATION MOTIVATION constant current generator. 1. INTRODUCTION AND MOTIVATION 11 qq 11 constant current generator. 1. INTRODUCTION AND MOTIVATION (3) x(θ) + − x(t) ˙ = − 11 w(t) qq sup 11 x(t) We investigate the behavior of the following system evolvw(t) (3) sup x(θ) + x(t) − x(t) ˙ = − T T T t−τ ≤θ≤t x(θ) + T We investigate the behavior of the following system evolvw(t) (3) sup x(t) − x(t) ˙˙ = − T T t−τ ≤θ≤t w(t) (3) sup x(θ) + x(t) − x(t) = − We investigate the behavior of the following system evolving with state suprema, T T T We investigate the behavior of the following system evolvt−τ ≤θ≤t t−τ ≤θ≤t ing with state suprema, T T T In the above equation T ∈ R, q ∈ R and τ ∈ R are t−τ ≤θ≤t + T ing with state suprema, In the above equation T ∈ R, q ∈ R and τ ∈ R are + are x(t) ˙˙ suprema, = C T x(θ) (1) ing with state the equation T R, qq ∈ and ττ ∈ R + are constants which characterize x(t) = Ax(t) Ax(t) + +B B t−τsup sup (1) In T x(θ) In the above above equation T ∈ ∈ the R, object. ∈ R R The and state ∈ variable R+ T + constants which characterize the object. The state variable ≤θ≤t C T x(θ) x(t) ˙˙ = Ax(t) + B sup (1) t−τ ≤θ≤t C x(t) = Ax(t) + B sup C x(θ) (1) constants which characterize the object. The state variable x(t) and the driving term (forcing function) w(t) physically constants which characterize the object. The state variable t−τ ≤θ≤t n t−τ ≤θ≤t x(t) and the driving term (forcing function) w(t) physically where x(t) ∈ R represents the state vector of the system, t−τ ≤θ≤t x(t) and driving term (forcing function) w(t) represent the regulated voltage and the perturbation effect where n×n x(t) ∈ Rnnnn represents the state vector of the system, x(t) and the the driving term (forcing function) w(t) physically physically n represent the regulated voltage and the perturbation effect where x(t) ∈ R represents the state vector of the system, A ∈ R is a constant time-invariant matrix and B ∈ R n×n n, represent the regulated voltage and the perturbation effect where x(t) ∈ R represents the state vector of the system, respectively at any arbitrary instant of time t. Notice that A ∈ R is a constant time-invariant matrix and B ∈ R , n×n n represent the regulated voltage and the perturbation effect n×n n is a constant time-invariant matrix and B ∈ Rn respectively at any arbitrary instant of time t. Notice that A ∈ R , n×n n C ∈ R constant vectors. τ ∈ R is the length of n are + respectively at any arbitrary instant of time t. Notice that A ∈ R is a constant time-invariant matrix and B ∈ R , (3) not only involves the unknown function x but also its C ∈ Rnnn are constant vectors. τ ∈ R+ is the length of respectively at any arbitrary instant of time t. Notice that (3) not only involves the unknown function x but also its C ∈ R are constant vectors. τ ∈ R is the length of + the memory of the sup functional. Notice that (1) is not + C ∈ R are constant vectors. τ ∈ R is the length of (3) not only involves the unknown function x but also its maximum value over an interval of past history of length + the memory of the sup functional. Notice that (1) is not maximum (3) not only involves the unknown function x but also its value over an interval of past history of length the memory of the sup functional. Notice that (1) is not an Ordinary Differential Equation (ODE) but is indeed a the memory of the sup functional. Notice but thatis(1) is nota maximum value over an interval of past history of length τmaximum and is therefore indeed an infinite-dimensional system. an Ordinary Differential Equation (ODE) indeed value over an interval of past history of length τ and is therefore indeed an infinite-dimensional system. an Ordinary Differential Equation (ODE) but is indeed a Functional Equation (FDE). Examples of such indeed an system. an OrdinaryDifferential Differential Equation (ODE) but is indeed a ττ and Functional Differential Equation (FDE). Examples of such and is is therefore therefore indeed an infinite-dimensional infinite-dimensional system. Functional Differential Equation (FDE). Examples of such Another example of historical significance, for systems are stuck float and rachet where the decision is Functional Differential Equation (FDE). Examples of such example of historical significance, for systems systems systems are stuck float and rachet where the decision is Another example of historical significance, for systems evolving state the Hausrath systems are stuck float and where the decision is to the maximum value of state variable. Another with example of suprema, historical is significance, for equation systems systems are by stuck and rachet rachet is Another evolving with state suprema, is the Hausrath equation to be be made made by thefloat maximum value where of the the the statedecision variable. evolving with state suprema, is the Hausrath equation quoted in Hale (1977) as, to be made by the maximum value of the state variable. System (1) can also be equivalently written in the following evolving with state suprema, is the Hausrath equation to be made by also the be maximum valuewritten of theinstate variable. quoted in Hale (1977) as, System (1) can equivalently the following System (1) be (1977) as, fashion. ˙˙ in = −ζx(t) ζζ sup |x(θ)|, ζζ > 0, tt ≥ 0. (4) quotedx(t) in Hale Hale (1977)+ as, System (1) can can also also be equivalently equivalently written written in in the the following following quoted fashion.  x(t) = −ζx(t) + sup > 0, ≥ 0. (4) fashion. t−τ ≤θ≤t |x(θ)|, x(t) ˙ = Ax(t) + Bu(t)   x(t) ˙ = −ζx(t) + ζ sup |x(θ)|, ζ > 0, t ≥ 0. (4) fashion. t−τ ≤θ≤t  x(t) ˙ = −ζx(t) + ζ sup |x(θ)|, ζ > 0, t ≥ 0. (4) ˙ = Ax(t) + Bu(t)  x(t) t−τ ≤θ≤t ≤θ≤t t−τ x(t) ˙ = Ax(t) + Bu(t)  T Σ : (2) This equation possesses a richer structure than (3). Here ≤θ≤t x(t) ˙ Ax(t) +C Bu(t) u(t) = sup T x(θ). Σuu :  (2) This equation possesses t−τ a richer structure than (3). Here  u(t) = sup C x(θ). T Σ (2) equation aa richer structure than (3). Here the over history can go negative t−τ ≤θ≤t C T u(t) sup T x(θ). Σuuu ::  (2) This Thissupremum equation possesses possesses richer structure than Here  the supremum over the the past past history can never never go(3). negative ≤θ≤t C x(θ). u(t) = = t−τ sup can t−τ ≤θ≤t ≤θ≤t the supremum over the past history can never go negative t−τ because of the modulus operator. However, the regularity In this format, it be thought of as a closed-loop the supremum over the past history can never go negative t−τ ≤θ≤t because of the modulus operator. However, the regularity In this format, it can be thought of as a closed-loop because of operator. However, the regularity and of are inferior that of In it can be thought of as closed-loop feedback system in which control ∈ R at because of the the modulus modulus operator. regularity In this this format, format, it can bethe thought ofpolicy as a a u(t) closed-loop and smoothness smoothness of its its solution solution areHowever, inferior to tothe that of (3). (3). feedback system in which the control policy u(t) ∈ R at and smoothness of its solution are inferior to that (3). feedback system in which the control policy u(t) ∈ R at any instant of time t depends on the supremum of the and smoothness of its solution are inferior to that of com(3). feedback system in which the control policy u(t) ∈ofRthe at Hadeler (1979) modeled the vision process in theof any instant of time t depends on the supremum Hadeler (1979) modeled the vision process in the comany instant of time t depends on the supremum of the weighted linear combination of the states over the past Hadeler (1979) modeled the vision process in the comany instant of time t depends on the supremum of the pound eye of horseshoe by the FDE weighted linear combination of the states over the past Hadeler (1979) thecrab vision in the comeye of aamodeled horseshoe crab by process the following following FDE weighted linear combination of states history length of weighted linear of combination of the the states over over the the past past pound pound eye of aa horseshoe crab by the following FDE evolving with state suprema: history interval interval of length ττ units units of time. time. pound eye of horseshoe crab by the following FDE evolving with state suprema: history interval of length τ units of time. history interval of length τ units of time. evolving with state suprema: The theoretical results and investigations of FDEs with x(t) ˙ = −δx(t) + p sup (x(τ (t)), c), δ, p ∈ R, c < evolving with state suprema: The theoretical results and investigations of FDEs with x(t) ˙ = −δx(t) + p t−τ ≤τ sup(t)≤t(x(τ (t)), c), δ, p ∈ R, c < 0 0 The theoretical results and investigations of FDEs with “suprema” opens the door to enormous possibilities for x(t) ˙ = −δx(t) + p sup (x(τ The theoretical results and investigations of FDEs with x(t) ˙ = −δx(t) + p sup(t)≤t (x(τ (t)), (t)), c), c), δ, δ, p p∈ ∈ R, R, cc < <0 0 t−τ ≤τ “suprema” opens the door to enormous possibilities for t−τ ≤τ (t)≤t t−τ (5) “suprema” opens the door to possibilities for their applications world processes and phenomena t−τ ≤τ ≤τ (t)≤t (t)≤t “suprema” opens to thereal door to enormous enormous possibilities for (5) their applications to real world processes and phenomena (5) their applications to real world processes and phenomena where the state x is related to the activation potential (Bainov and Hristova (2011)). (5) their applications to real world processes and phenomena where the state x is related to the activation potential (Bainov and Hristova (2011)). where the state x is related to the activation potential (Bainov and Hristova (2011)). above a certain threshold produced in sensory cell where state xthreshold is relatedproduced to the activation (Bainov and Hristova (2011)). above the a certain in sensorypotential cell by by above aa certain threshold produced in sensory cell by light. The reciprocal of δδ accounts for response 2. LITERATURE SURVEY above certain threshold produced inthe sensory celltime by light. The reciprocal of accounts for the response time 2. LITERATURE SURVEY light. The reciprocal of δδ accounts for the response time 2. constant. τ stands for the lateral inhibition delay with light. The reciprocal of accounts for the response time 2. LITERATURE LITERATURE SURVEY SURVEY constant. τ stands for the lateral inhibition delay with a a constant. ττ stands the lateral inhibition delay with value about 100 Popov constant. stands for for the of delayfunction with a a typical experimental experimental value oflateral aboutinhibition 100 msec. msec. The The function Popov (1966) (1966) encountered encountered the the following following historical historical equaequa- typical experimental value of about 100 The Popov (1966) encountered the historical equasup(x(τ (t)), c) is termed rectifier function. tion while studying the voltage regulation problem of aa typical typical experimental valueas ofthe about 100 msec. msec. The function function Popov (1966) encountered the following following historical equasup(x(τ (t)), c) is termed as the rectifier function. tion while studying the voltage regulation problem of tion sup(x(τ (t)), (t)), c) c) is is termed termed as as the the rectifier rectifier function. function. tion while while studying studying the the voltage voltage regulation regulation problem problem of of aa sup(x(τ

Copyright © 2015, IFAC 2015 242 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © IFAC 2015 242 Copyright © IFAC 2015 242 Peer review under responsibility of International Federation of Automatic Copyright © IFAC 2015 242Control. 10.1016/j.ifacol.2015.11.182

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Σc :

Fig. 1. Compound Eye of an Arthropod The schematic structure of the compound eye of an arthropod is shown in Fig. 1. Light stimulation creates depolarizing graded potentials in insect photoreceptors (as opposed to hyperpolarizing in vertebrate rods and cones). Action potentials do not exist, generally, although they may have a role in photoreceptors of some species (e.g. in the cockroach, Heimonen et al. (2006)). The signals are processed in the first synaptic layer, the lamina, and in the further neural centers (e.g. the medulla) in a retinotopic fashion. Ivanov et al. (2002) established some very useful and interesting links between the York 3/2 conditions, the Halanay inequality and scalar differential equations with maxima. Gonzalez and Pinto (2002) deduced asymptotic equilibrium for a certain class of scalar FDEs with maxima. They use the basic inequality of Gronwall-Bellman type, fixed point methods and contractive operators. In Bohner et al. (2013), the authors investigate parametric stability for nonlinear differential equations with “maxima” in terms of two measures. The authors obtain several sufficient conditions for parametric stability as well as uniform parametric stability. A comparison with scalar ODEs is also offered in the paper. 3. NOVELTY AND CONTRIBUTION In this paper we investigate and solve the problem of stability analysis, controller synthesis and observer design for general higher order systems with dynamics evolving with state suprema. We use Razumikhin function based platform for the analysis and synthesis. We emphasize that the infinite-dimensional systems with suprema possess a rich structure and share some of the features of Multi-Mode Multi-Dimensional (M 3 D) systems. The state space for the characterization of trajectories and Cauchy problem is the infinite dimensional Banach space endowed with the topology of uniform convergence. Very little is known about such class of systems from the literature of systems and controls perspective. These systems, though complex, nonlinear and non-smooth in nature, form an important class of infinite-dimensional systems and are therefore worth researching. 4. STATE SPACE & THE CAUCHY PROBLEM The Cauchy problem for the evolution of the infinite dimensional state can be characterized as follows. 243

243

  x(t) ˙ = Ax(t) + B  x(t) = ψ(t);

sup

C T x(θ);

t−τ ≤θ≤t

∀t ≥ 0

(6)

∀t ∈ [−τ, 0].

Where A ∈ Rn×n is a constant matrix and B ∈ Rn , C ∈ Rn are constant vectors. x(t) ∈ Rn is the state vector and ψ(t) ∈ C([−τ, 0]; Rn ) is the initial infinite dimensional history function living in the Banach function space. Here C([−τ, 0]; Rn ) denotes the Banach space of continuous functions mapping the interval [−τ, 0] to Rn with the topology of uniform convergence. This means that the norm of an element φ in this function space is defined by the following norm. (7) φC = sup φ(θ) θ∈[−τ,0]

Nonlinear Nature: The sup operator present in (1) fails to satisfy both of the defining characteristics of linearity viz., additivity and homogeneity. Therefore, we establish that systems evolving with state suprema in general and of the form (1) in particular are nonlinear infinite dimensional dynamical systems. More precisely and rigorously, the sup functional is a sublinear functional or a Banach functional. 5. MULTI-MODE MULTI-DIMENSIONAL (M 3 D) SYSTEMS CHARACTER In this section, we elucidate that systems evolving with state suprema exhibit a Multi-Mode Multi-Dimensional character. This concept can be easily grasped once we unravel the behavior or action of the sup operator. There is an intrinsic mode switching in this system, such a switched system is referred to as Auto-Hybrid M 3 D system. The mode switching is not triggered by an external agency, rather it is an inherent feature of the system itself. Verriest (2012) gives a detailed analysis of the dynamics of M 3 D hybrid systems. Such systems have modes of different dimensions, while the overall state space of the hybrid system is defined as a sheaf. The implications of the behavior at the mode transitions is investigated in detail for pseudo- continuous systems. An M 3 D system is called pseudo- continuous (p-M 3 D) if instantaneous switching via higher dimensional modes does not have any effect upon return to the lower dimensional mode. Canonical forms and parameterizations are derived for pM 3 D systems. Such systems may be actively controlled (exo-M 3 D), or passively switched via a fixed switching surface (auto-M 3 D). 5.1 The Action of the sup Operator First we want to investigate and unravel the behavior of the sup operator. We discuss in detail two separate cases. First the infinite memory case is analyzed and then the one with finite memory. Case I: Infinite Length Memory Fig. 2 elucidates the behaviour of the sup operator by taking into account infinite length memory i.e. τ → ∞. The initial history is represented by ψ. x(t) represents the original function (solid line) and z(t) = supt−τ ≤θ≤t x(θ) represents the action of the sup which is portrayed by

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a dotted line. Notice that when x(t) is monotonically increasing i.e. (x˙ ≥ 0), z(t) follows x(t). However, when x(t) starts decreasing z(t) remains held at previous value until x(t) ≥ z(t). This action can be compactified by observing that z(t) = x(t − τ (t)) where τ (t) is a special type of time-varying delay with rate unity as depicted in the bottom sub figure in Fig. 2. This special unit rate delay is either zero or of sawtooth type with unity slope. Notice that the variable delay function has a jump discontinuity and as a result is a piecewise continuous function of time.

x(t) is monotonically increasing (from B to C) i.e. (x˙ ≥ 0), z(t) follows x(t). However, when x(t) starts decreasing z(t) remains held at previous value until the memory is exhausted (point D). From point D onwards, z is a shifted version (τ delayed) of x till point E where x again starts rising and z follows (overlaps) x. This behavior can be systematically captured by observing that z(t) = x(t − τ (t)) where τ (t) is a special type of time-varying delay with rate unity, as depicted in the bottom sub figure in Fig. 3. This special unit rate delay is either zero or of sawtooth type with unity slope or constant. In this case, we have an additional feature of constant delay. Notice again that the delay has a jump. In general the piecewise continuous time-varying delay function τ (t) looks of trapezoidal or sawtooth type or zero. Notice that the initial history ψ ∈ C does play a significant role in the output waveform of the sup functional.

Fig. 2. The action of the sup operator with infinite memory A more thorough and meticulous observation of Fig. 2 reveals that system(1) with infinite memory can be expressed in as Multi Mode system as follows. In fact there are two modes, respectively of dimension n and 2n.  Mode I:      x(t) ˙ = (A + BC T )x(t); x(t) ∈ Ω1 ⊂ Rn    Mode II: (8) Σ∞ :  x(t) ˙ = Ax(t) + BC T x∗ ;          x˙ ∗ = 0; x = x∗ ∈ Ω2 ⊂ R2n x

Here t∗ represents the instant of time at which the maximum of the state trajectory x∗ = x(t∗ ) is attained and afterwards the trajectory goes down. There is not an infinite dimensional mode in this case. However if C is fixed, then by changing variables and making C T x a state variable, its dimension is only n + 1. Mode II exhibits a switching to Mode I if (C T x) > C T x∗ . On the other d (C T x) < 0, with hand, Mode I switches to Mode II if dt ∗ transition x → (x, X = x). The transition from Mode II back to Mode I is trivial i.e., x∗ → x. Notice that the mode transitions form an integral part of the hybrid system description. Therefore, it is concluded that the infinite memory based sup system can be modeled and visualized as an M 3 D hybrid system with two distinct finite dimensional modes. This is indeed an interesting and appealing feature! There is no sup-operator in the M 3 D description, but its state space description ( a sheaf) gets more involved! Case II: Finite Length Memory Fig. 3 shows the behavior of the sup operator for a finite length memory of length τ units of time. This case is a bit more complex. The initial history is represented by ψ from A to B. x(t) represents the original function (solid line) and z(t) = supt−τ ≤θ≤t x(θ) represents the action of the sup which is shown by a dotted line. Notice that when 244

Fig. 3. The action of the sup operator with finite memory of length τ time units In the light of the above mentioned discussion, system (1) with finite length memory can be expressed as the following M 3 D system.  Mode I:      x(t) ˙ = (A + BC T )x(t); x(t) ∈ Ω1 ⊂ Rn     Mode II:     x(t) ˙ = Ax(t) + BC T x∗ ; ΣM 3 D :    x  ∗  ∈ Ω2 ⊂ R2n x ˙ = 0; x =   x∗      Mode III:    x(t) ˙ = Ax(t) + BC T x(t − τ ); x(t) ∈ Ω3 ⊂ C (9) Notice that Mode I is a finite dimensional mode. The trajectory in this mode lives in a state space which is a subset of the well known Euclidean space Rn . The individual characteristic behavior or the eigen structure of this mode is described by the modal equation, det(λI − A − BC T ) = 0. Mode II is also a finite dimensional mode. Here t∗ represents the time at which the trajectory achieves its supremum. Notice that the second term in this mode i.e. BC T x(t∗ ) acts as a driving agency or forcing function. The trajectories in this mode lie in a state space which is a sub partition of Rn . The independent characteristic behavior of the trajectories in this mode is governed by the characteristic equation det(λI − Ac ) = 0

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 A BC T ∈ R2n×2n . The third mode is 0 0 an infinite-dimensional mode. Here, the state space for the trajectories is a subset of the Banach function space C([−τ, 0]; Rn ). This mode in itself exhibits the modal characteristic equation of the form det(λI −A−BC T e−λτ ) = 0. This equation is transcendental in nature. where Ac =



An alternative but more complex M 3 D structure will be as follows.  Mode I:      x(t) ˙ = (A + BC T )x(t); x(t) ∈ Ω1 ⊂ Rn       Mode II: τ ΣM 3 D : x(t) ˙ = Ax(t) + BC T x(t − τ (t)); x(t) ∈ Ω2 ⊂ C;    τ (t) ≤ τ ; τ˙ = 1∀t ≥ 0     Mode III:     x(t) ˙ = Ax(t) + BC T x(t − τ ); x(t) ∈ Ω3 ⊂ C (10) Notice that in this structure, Mode II is also an infinitedimensional mode with a special time-varying delay with delay rate τ˙ being exactly unity. Transition structure and mode switching conditions are needed but not mentioned here. Remark 1: Notice that the above discussed M 3 D system in not a pseudo-continuous Multi-Mode Multi Dimensional (p − M 3 D) system because Mode III requires infinite memory due to its infinite dimensional nature whereas Mode II is finite dimensional. Mode II is finite dimensional, but needs the history it leaves behind (to properly initiate Mode III). Due to this fact, the entire system is not a p − M 3 D system unlike the one discussed in Verriest (2012). 5.2 Asymptotic Stability of the Equilibrium Point of (6) We now use the Lyapunov-Razumikin (LR) theorem which is a very powerful theorem in the context of stability analysis of systems characterized by FDEs (Hale and Lunel (1993)) and derive the sufficient conditions for the asymptotic stability of the trivial steady state (equilibrium point: x = 0) of the sup based system (6). Unlike the Lyapunov-Krasovskii (LK) functional based approach in an infinite dimensional setting, this theorem uses functions which are relatively easier to handle with. This theorem also gives sufficient conditions for the stability. Consider the Lyapunov function V : Rn → R+ described by V (x) = xT P x where P = P T > 0. We have for x ∈ Rn , α(x) ≤ V (x) ≤ β(x), with α(x) = λmin (P )x2 and β(x) = λmax (P )x2 . Define η : R+ → R+ by η(r) = ρ2 r, r ∈ R+ and ρ > 1. This ensures one of the requirements of the LR theorem that η(r) > r, ∀r > 0. Let x(t) be the solution trajectory of (6), such that for t ≥ 0, θ ∈ [−τmax , 0], V (x(t + θ)) < η(V (x(t)) ⇒ V (x(t + θ)) < ρ2 (V (x(t)) T

⇒ x(t + θ)2P < ρ2 x(t)2P sup

t−τ ≤θ≤t

Now, we have for t ≥ 0, T  ∂V dx V˙ (x(t)) = ∂x dt dx = ∇V (x). dt = D(xT (t)P x(t)) = x˙ T (t)P x(t) + xT (t)P x(t) ˙ = (Ax(t) + B

sup

C T x(θ))T P x(t)

t−τ ≤θ≤t

+ xT (t)P (Ax(t) + B

C T x(θ))

sup t−τ ≤θ≤t

= xT (t)(AT P + P A)x(t) + (B T P x + xT P B)

C T x(θ)

sup t−τ ≤θ≤t

≤ xT (t)(AT P + P A)x(t) + (B T P x + xT P B)

C T x(θ)

sup

t−τ ≤θ≤t

≤ xT (t)(AT P + P A)x(t)

+ (B T P x + xT P B)

≤ xT (t)(AT P + P A)x(t)

C T x(θ)

sup t−τ ≤θ≤t

+ (B T P x + xT P B)C

≤ xT (t)(AT P + P A)x(t)

sup

x(θ)

t−τ ≤θ≤t

+ (B T P x + xT P B)Cρx(t)

≤ xT (t)(AT P + P A)x(t) + 2ρBCP x(t)2

= −xT (t)Qx(t) + 2ρBCP x(t)2 ≤ −(λmin (Q) − 2ρBCP )x(t)2 < 0 if λmin (Q) > 2ρBCP .

(11) Since V (x(t)) > 0 and V˙ (x(t)) < 0 whenever V (x(t + δ)) ≤ η(V (x(t))), ∀δ ∈ [−τ, 0], all the conditions of LR theorem are satisfied. We, therefore, conclude that the equilibrium point (origin) of (6) is asymptotically stable. Furthermore, since V (x) = xT P x = x2P i.e. V → ∞ as xP → ∞. Therefore, V is radially unbounded, and by definition this implies that the origin is a global attractor. The above result is very useful and can be stated as the following theorem. Theorem 1. The infinite-dimensional system evolving with state suprema characterized by (6) is uniformly globally asymptotically stable if given any symmetric and positive definite matrix Q (Q = QT > 0) there exists a scalar ρ > 1 such that λmin (Q) > 2ρBCP  (12) T where the matrix P = P > 0 precisely satisfies the following algebraic Lyapunov equation. AT P + P A + Q = O (13) Remark 2: Notice that the above theorem requires that it is mandatory that the sup free system is asymptotically stable. In other  words, the matrix A must be Hurwitz. Also since P  = λmax (P T P ) is the induced or spectral norm of P , the upper bound on ρ in (13) can also be stated as follows.

2 T

⇒ x (t + θ)P x(t + θ) < ρ x (t)P x(t)

⇒

245

x(θ) < ρx(t) 245

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λmin (Q)  (14) 2BC λmax (P T P ) Remark 2: We can express V˙ (x) in the above analysis as follows. V˙ (x(t)) ≤ −ζV (x(t)) (15) where ζ = λmin (Q) − 2ρBCP . ρ<

The corollary given below is an immediate consequence of the above result. Corollary 1. The dynamical system with suprema given by (6) is exponentially stable if A is Hurwitz and there exists a scalar ρ > 1 such that (13) and (14) hold. 5.3 Controller Synthesis and the Stabilization Problem Consider the following problem.   x(t) ˙ = Ax(t) + B sup C T x(θ) + Γu(t);   t−τ ≤θ≤t Σf b : u(t) = Kx(t)    x(t) = ψ(t); ∀t ∈ [−τ, 0].

We will use the concept of sup Based Output Injection to construct the observer for the recovery of the state x(t) in the same spirit as in Ahmed and Verriest (2015). Let x ˆ(t) represent the estimated state, the infinite-dimensional estimator (being a replica of the state dynamics) will be governed by the dynamics as follows. x ˆ˙ (t) = Aˆ x(t) + Γw(t) + L (y(t) − yˆ(t)) (20) where the last term is the correction term with L ∈ Rn being the gain matrix of the estimator and the estimated output yˆ(t) is precisely given by, yˆ(t) = sup C T x ˆ(θ) (21) t−τ ≤θ≤t

Fig. 4 gives a basic conceptual block diagram to elucidate ∀t ≥ 0 the observation scheme using the concept of sup based output injection.

(16) Suppose that the unforced system (6) is not stable and we want to solve the stabilization problem i.e., we want to design a control policy u = Kx such that the resulting closed-loop feedback system (16) is asymptotically stable. Here K represents the static state feedback gain matrix. Notice that a necessary condition for this problem is that the pair (A, Γ) must be controllable. We express our result as the following theorem. Theorem 2. The infinite-dimensional system evolving with state suprema characterized by (16) is uniformly globally asymptotically stabilizable if (A, Γ) is controllable and given any symmetric and positive definite matrix Q ∈ Rn×n (Q = QT > 0) and a row vector K ∈ R1×n there exists a scalar ρ > 1 such that λmin (Q) > 2ρBCP  (17) where the matrix P = P T > 0 precisely satisfies the following algebraic Lyapunov equation. (A + ΓK)T P + P (A + ΓK) + Q = O (18) Proof: The proof an immediate consequence of the application of Theorem 1 to the closed loop system (16).  6. THE OBSERVATION PROBLEM We express the dynamics of the system and measurements as a setup or formulation for our observer problem as follows.  Dynamics of the Plant Model:     ˙ = Ax(t) + Γw(t)  x(t) Observation or Measurement Equation:   T    y(t) = sup C x(θ)

y(t) represents the nonlinear measurement at any arbitrary instant of time t and τ > 0 denotes the memory associated with the sup functional.

(19)

t−τ ≤θ≤t

Here x(t) ∈ Rn is the state vector, w(t) ∈ Rp is the driving force (control input or some known perturbation or disturbance), the state coupling matrix A ∈ Rn×n , the input coupling matrix (read in matrix) Γ ∈ Rn×p and C ∈ Rn the readout or measurement matrix. The scalar 246

Fig. 4. Illustrative Block Diagram of sup Based Observer Defining e(t) = x(t) − x ˆ(t) as the estimation error or observation error at any arbitrary time instant t, we get the observer error dynamics as follows.   T T ˆ(θ) e(t) ˙ = Ae(t) − L sup C x(θ) − sup C x ⇒ e(t) ˙ ≤ Ae(t) + L

+

⇒ e(t) ˙ ≤ Ae(t) + L+

t−τ ≤θ≤t

sup t−τ ≤θ≤t

sup t−τ ≤θ≤t



t−τ ≤θ≤t

C x(θ) − C T x ˆ(θ)



T

C T e(θ)





(22) (23)

Notice that the first inequality (22) is obtained from the preceding equation by using the subadditivity/difference property of the sup functional i.e., sup(x1 (θ) − x2 (θ)) ≥ sup x1 (θ) − sup x2 (θ). θ∈Θ

θ∈Θ

θ∈Θ

In all the above inequalities, L+ stands for the vector L but with a restriction that all the entries strictly positive. So our observer gain matrix L cannot have zero or negative elements. This positivity preserves the error inequality and is needed to make the observer design proof tractable using Theorem 1. Theorem 3. The infinite-dimensional system of error dynamics evolving with error suprema characterized by (23) is asymptotically stable if the pair (C T , A) is observable and given any symmetric and positive definite matrix S (S = S T > 0) there exists a scalar ρ > 1 such that λmin (S)  (24) L+  < 2ρC λmax (RT R)

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Aftab Ahmed et al. / IFAC-PapersOnLine 48-27 (2015) 242–247

247

where the matrix R = RT > 0 is precisely a solution the following algebraic Lyapunov equation. AT R + RA + S = O (25) Proof: The proof relies on the fact that the observer design is basically a dual problem of the control synthesis. The result is based on Razumikhin framework and application of Theorem 1 to the error dynamics (23).  7. SIMULATION RESULTS We illustrate the applicability and effectiveness of our sup based output injection technique for the observer design by the following academic example. Example: Consider the following second order nonautonomous coupled system with sup based measurement.  x˙ (t) = x1 (t) − x2 (t)   1 x˙ 2 (t) = 5x1 (t) − 3x2 (t) + 0.5w(t)   y(t) = 1.75 sup x1 (θ)

Fig. 5. Actual, Observed State Trajectories and Estimation Error Profiles some simulation results were shown to express the benefits of our analysis and observer design for the higher order systems evolving with state suprema. ACKNOWLEDGEMENTS

t−1≤θ≤t



 t+1 We take φ(t) = ∈ C, ∀t ∈ [−1, 0] and 1 + 0.25 cos(4.5t)   ˆ = 0.85 − 0.1 sin(t) ∈ C, ∀t ∈ [−1, 0] as the initial φ(t) t + 1.75 infinite-dimensional history functions for the actual states and the observed states respectively. Clearly, our system and measurement matrices    are as follows:   1 −1 0 1.75 A= , Γ= , C= . 5 −3 0.5 0 It is easy to check that A is Hurwitz and the pair (C T , A) is observable. For simplicity, the matrix S was taken as the Identity matrix and ρ > 1 was picked  as ρ =2. Solving 1.5 2 the Lyapunov equation yields R = with the 2 3.5 spectral norm, R = 4.7361. The unit Heaviside function was taken as the known perturbation/disturbance w(t). Using Theorem 3, we designed the observer as L = L+ =   0.0125 . This observer gain matrix L+ ensures that 0.0185 L+  = 0.023 < 0.03016 and thus all the requirements of Theorem 3 are totally satisfied. Fig. 5 portrays the actual state trajectories, the observed state trajectories and the estimation errors of the states. We can clearly see that the error profiles are asymptotically stable. In other words, the estimated states are faithfully tracking the actual states as time progresses. 8. CONCLUDING REMARKS In this paper we investigated and solved the problem of stability analysis, controller synthesis and observer design for general higher order systems with dynamics evolving with state suprema. We used Razumikhin function based platform for the analysis and synthesis. The state space for the characterization of trajectories and Cauchy problem is the infinite dimensional Banach space endowed with the topology of uniform convergence. It was also elucidated that such infinite dimensional systems possess a rich structure and exhibit the features of M 3 D systems. Finally, 247

Aftab Ahmed is a Ph.D. student and a Fulbright scholar at Georgia Institute of Technology. He gratefully acknowledges the Institute of International Education (IIE). The authors also deeply acknowledge the thorough review and highly constructive comments of the anonymous reviewers. REFERENCES Ahmed, A. and Verriest, E.I. (2015). Observer based temperature control of a tank: From constant to nonlinear state-dependent delay. Proceedings of the 12th IFAC Workshop on Time Delay Systems, Ann Arbor, MI, USA, 197–202. Bainov, D.D. and Hristova, S.G. (2011). Differential Equations with Maxima. CRC Press, New York. Bohner, M.J., Georgieva, A.T., and Hristova, S.G. (2013). Nonlinear differential equations with “maxima”: Parametric stability in terms of two measures. Applied Mathematics & Information Sciences, 7, 41–48. Gonzalez, P. and Pinto, M. (2002). Asymptotic equilibrium for certain type of differential equations with maximum. Proyecciones, 21, 9–19. Hadeler, K.P. (1979). Delay Equations in Biology. Springer, New York. Hale, J.K. (1977). Theory of Functional Differential Equations. Springer, New York. Hale, J.K. and Lunel, S.M.V. (1993). Introduction to Functional Differential Equations. Springer-Verlag, New York. Heimonen, K., Salmela, I., Kontiokari, P., and Weckstrom, M. (2006). Large functional variability in cockroach photoreceptors: Optimization to low light levels. The Journal of Neuroscience, 26, 13454–13462. Ivanov, A., Liz, E., and Trofimchuk, S. (2002). Halanay inequality, york 3/2 stability criterion, and differential equations with maxima. Tohoku Math. Journal, 54, 277– 295. Popov, E.P. (1966). Automatic Regulation and Control. 2nd Edition: (In Russian), Moscow. Verriest, E.I. (2012). Pseudo-continuous multidimensional multi-mode systems. Discrete Event Dynamic Systems, 22, 27–59.