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Notions and a Passivity Tool for Switched DAE Systems

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Notions and a Passivity Tool for Switched DAE Systems Notions and a Passivity Tool for Switched DAE Systems Pablo Na˜ nez˜ 1, Ricardo G. Sanfelice2, and Nicanor Quijano1 Abstract— This paper proposes notions and a tool for pas- DAE systems and switched DAE systems with inputs and sivity properties of non-homogeneous switched Differential outputs1. More precisely, we characterize non homogeneous Algebraic Equation (DAE) systems and their relationships with linear switched DAE systems as a class of hybrid DAE stability and control design. Motivated by the lack of results on input-output analysis (such as passivity) for switched DAE systems. systems and their interconnections, we propose to model non- As a motivation for the study of switched DAE systems homogeneous switched DAE systems as a class of hybrid with inputs and outputs, we employ the DC-DC boost con- systems, modeled here as hybrid DAE systems with linear verter [5]. First, we model each mode of operation using the flows. Passivity and its variations are defined for switched DAE switched DAE representation in (1). We study its passivity systems and methods relying on storage functions are proposed. The main contributions of this paper are: 1) passivity and properties and solve the problem of set-point tracking of the detectability concepts for switched DAE systems, 2) links of the output voltage, in this case the voltage at the capacitor, using aforementioned passivity and detectability properties to stabi- a passivity-based controller. The main contribution of this lization via static output-feedback. Our results are illustrated paper is a tool that allows one to link the passivity properties in a power system, namely, the DC-DC boost converter, whose of a switched DAE system to the asymptotic stability of the model involves DAEs and requires feedback control. system using a static state-feedback controller. Building from I. INTRODUCTION the invariance principles for hybrid DAE systems in [2] and The characterization of a system behavior based on the the passivity and detectability notions for hybrid systems relationship between the energy injected and dissipated by in [3] and given a static output-feedback control law that a system is known as passivity. A system that stores and satisfies some mild conditions and a (flow- or jump-) passive dissipates energy without generating energy on its own is said switched DAE with respect to a set of interest, we show that to be passive. The physical interpretation of energy makes the control law renders such set asymptotically stable. Due to passivity an intuitive tool to assert the stability properties space constraints, strict and output versions of the passivity of any system with inputs and outputs. There is plenty of properties described in this document and its relationship to literature that documents dissipativity and passivity, from zero-input stability of switched and hybrid DAE systems, as definitions, sufficient conditions for stability, to passivity well as proofs are not included in this document. based control [1]. Passivity using storage functions, which The notation used throughout the paper is as follows. We for the case of zero inputs guarantee asymptotic stability. define R≥0 := [0; ) and N := 0; 1;::: . Given vectors n m 1> > > f g Passivity properties are also very useful when analyzing ν R ;! R , [ν ! ] is equivalent to (ν; !), where ( )2> denotes2 the transpose operation. Given a function f : interconnection of systems. · Rm Rn, its domain of definition is denoted by dom f, This paper pertains to the study and design of hybrid ! i.e., dom f := x Rm f(x) is defined . The range of and switched Differential Algebraic Equations (DAEs) using f 2 j g passivity tools. In particular, a switched DAE is given as f is denoted by rge f, i.e., rge f := f(x) x dom f . The right limit of the function f isf definedj as 2f +(x) :=g _ −1 Eσξ = Aσξ + Bσu (1a) limν!0+ f(x + ν) if it exists. The notation f (r) stands y = h (ξ; u); (1b) for the r level set of f on dom f, i.e., f −1(r) := z σ − f 2 dom f f(z) = r . Given two functions f : Rm Rn where σ : [0; ) Σ is the switching signal and Σ is j m ng ! 1 ! and h : R R , f(x); h(x) denotes the inner product a finite discrete set. The results in [2] allow us to model between f and!h at xh. We denotei the distance from a vector homogeneous and autonomous switched DAE systems as n n y R to a closed set R by y A, which is given hybrid DAE systems. We extend the results in [2] and [3] 2 A ⊂ j j n×n by y A := infx2A x y . Given a matrix P R , the to allow the analysis of the passivity properties of hybrid j j j − j 2 determinant of P is denoted by det P . Given n N, the matrix 0 n×n denotes the zero matrix,2 while 1P. Na˜ nez˜ and N. Quijano are with Universidad de los Andes, Bogota,´ n R n n n×n2 × Colombia. pa.nanez49, [email protected]. Research In R denotes the n n identity matrix. 2 × by P. Nanez has been partially supported by COLCIENCIAS under contract The remainder of this paper is organized as follows. In 567. This work has been supported in part by project ALTERNAR, Acuerdo 005, 07/19/13, CTeI-SGR-Narino,˜ Colombia. Section III, the required modeling background is presented. 2R. G. Sanfelice is with the Department of Computer Engi- neering, University of California, Santa Cruz, CA 95064, USA. 1It is important to clarify that, for a solution to (1a) to exist, at each [email protected]. This research has been partially supported by change in σ it is required to map the state previous to the switching instant the National Science Foundation under CAREER Grant no. ECS-1450484 to a point in the space defined by the algebraic conditions of the subsequent and Grant no. CNS-1544396, and by the Air Force Office of Scientific mode. These resets of the state can be computed by the so-called consistency Research under Grant no. FA9550-16-1-0015. projectors in Definition 3.3 [4, Definition 3.7]. In Section IV, a description of hybrid DAE systems with vL vD +/ +/ inputs/outputs is presented, which is followed in Section V L − d − vC by the introduction of the passivity and stability definitions iD iL iS i R for such systems. Also Section V-B revisits the motiva- D + tional Example II, where the definitions and the results in vcc S c − Sections V through V-B are exercised. vD λ (a) Boost converter circuit. (b) v − i curve. II. MOTIVATIONAL EXAMPLE D D Fig. 1. DC-DC boost converter circuit and current-voltage characteristic curve of the diode. We consider the DC-DC boost converter shown in Fig- ure 1(a) and model it as a switched DAE as in (1). More differential-algebraic equations as follows2: interestingly, using concepts of passivity, a given set-point for the voltage, and an appropriate selection of inputs and Mode 1 (σ = 1) Mode 2 (σ = 2) outputs, we will show that a passivity-based control law d d dt vcc = 0 dt vcc = 0 renders a set of interest asymptotically stable. d 1 d 1 c dt vC = iD R vC c dt vC = iD R vC d − d − There is a fair amount of literature related to the control L dt iL = vL L dt iL = vL of DC-DC boost converters from many perspectives. The au- 0 = iL iS iD 0 = i i i − − L − S − D thors in [6] follow an energy-based hybrid control approach 0 = iS 0 = iD to design controllers for impulsive dynamical systems. In [5], 0 = vcc vL vD vC 0 = vcc vL d − − − − the authors propose a Control Lyapunov Function (CLF) dt vD = 0 0 = vD + vC approach for the control of the DC-DC boost converter. Mode 3 (σ = 3) Mode 4 (σ = 4) Following the models therein, the converter is composed by d d dt vcc = 0 dt vcc = 0 an inductor L, a capacitor c, a resistor R, a voltage source d 1 d 1 c dt vC = iD R vC c dt vC = iD R vC vcc, a switch S, and a diode d. The voltage across the d − d − L dt iL = vL L dt iL = vL inductor, diode, and capacitor are denoted as vL, vD, and 0 = iL iS iD 0 = iL iS iD vC , respectively. The current through the inductor, switch, − − − − 0 = iS 0 = vD + vC and diode are denoted as iL, iS, and iD, respectively. In 0 = vcc vL vD vC 0 = vcc vL this example, we consider the model of the diode depicted − − − d − 0 = iL v = 0 in Figure 1(b), where λ is the forward bias voltage of the dt D diode. Now, consider the switching signal σ : [0; ) Σ, 1 ! The DC-DC boost converter is designed (and controlled) to where each element in Σ represents a mode of operation of deliver a desired DC voltage at its output, which is typically the boost converter. the voltage of the capacitor (vC ). The set-point for the The state conditions where each one of the modes is valid voltage is denoted as u R and is treated as a new input. are as follows: To proceed with a CLF2 approach, we add an extra state e that is reset to the value of u at switching instants. The • Mode 1: (Switch is open and diode is conducting) In dynamics of the state e are given by e_ = 0 during flows and this mode the current through the diode is positive.
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