On dynamics, complementarity and passivity: electrical networks with ideal diodes
W.P.M.H. Heemels, M.K. C¸ amlıbel∗ J.M. Schumacher Dept. of Electrical Engineering Dept. of Econometrics Eindhoven University of Technology Tilburg University P.O. Box 513, 5600 MB Eindhoven P.O. Box 90153, 5000 LE Tilburg The Netherlands The Netherlands e-mail: w.p.m.h.heemels,k.camlibel @tue.nl e-mail: [email protected] { }
Abstract In this paper we study linear passive electrical circuits mixed with ideal diodes and voltage/current sources within the framework of linear complementarity systems. Linear complementarity systems form a subclass of hybrid dynamical systems and as such questions about existence and uniqueness of solution trajectories are non-trivial and will be investigated here. The nature of the behaviour is analyzed and characterizations of the inconsistent states of the network are presented. Also explicit jump rules from these inconsistent states are given in various forms. Finally, these results lead to a generalization of the notion of passivity to linear complementarity systems.
Keywords: Hybrid systems, complementarity, well-posedness, ideal diodes, passivity
1 Introduction erns the actual evolution of the network’s vari- ables. At a mode transition (a diode going from The systems studied in this paper fall within the conduction to blocking or vice versa) the set of class of linear complementarity systems (LCS) with equations changes and a reset (jump) of system’s external inputs. Linear complementarity systems variables may occur (think of the instantaneous consist of combinations of linear time-invariant dy- discharge of a capacitor directly connected to a namical systems and complementarity conditions diode). The model leads to a description with vary- as appearing in the linear complementarity prob- ing continuous (mode) dynamics and discrete ac- lem of mathematical programming [4]. These sys- tions like mode transitions and re-initializations. tems were introduced in [15] and further studied The mode transitions are triggered by certain in- in [3, 9, 10, 12, 16]. However, in all these papers equalities expressing for instance that the current the situation with nonzero (discontinuous) exter- through an ideal diode is always nonnegative. This nal inputs is not considered and as such will be indicates that the issue of existence and uniqueness studied in this paper for the first time. In par- of solution trajectories is non-trivial. Besides well- ticular, we will focus on LCS that satisfy a pas- posedness, we characterize the inconsistent states sivity condition on the underlying state space de- (i.e., states from which discontinuities and Dirac scription. In this way, the particular applications impulses occur) in several equivalent ways. Next at hand are linear electrical networks with ideal to these extensions of previous results in [3] to the diodes and current/voltage sources. In this con- nonzero input case, the main results present ex- text complementarity modelling has been used be- plicit expressions for the jumps (re-initializations) fore in e.g. [2,11] for the simulation and verification of the state vector, which have interesting physical of large-scale networks. interpretations. Finally, we introduce a concept of LCS are nonlinear discontinuous hybrid dynam- passivity for an LCS and state a sufficient condition ical systems. This can be illustrated by the be- for this. haviour of networks with ideal diodes. The “mode” (also called “configuration” or “topology”) of the The proofs of the results stated in this paper can circuit is determined by the “discrete state” of the be found in [8]. diodes (blocking or conducting), which changes in Throughout the paper, R denotes the real num- time. To each mode a different set of differential bers, R := [0, ) the nonnegative real numbers, + ∞ and algebraic equations is associated which gov- 2(t0,t1) the square integrable functions on (t0,t1), andL the Bohl functions (i.e. functions having ra- B ∗Also at the Dept. of Econometrics, Tilburg University, tional Laplace transforms) defined on (0, ). The P.O. Box 90153, 5000 LE Tilburg, The Netherlands. On (i) ∞ leave from Dept. of Electrical Eng., Istanbul Technical Uni- distribution δt stands for the i-th distributional versity, 80626 Maslak Istanbul, Turkey. derivative of the Dirac impulse supported at t. The dual cone of a set Rn is defined by 3 Linear passive networks with ideal diodes Rn Q⊆ ∗ = x x>y > 0 for all y . For a pos- Linear electrical networks consisting of (linear) re- Qitive integer{ ∈ k,| the set k¯ is defined∈Q} as 1, 2,...,k . { } sistors, inductors, capacitors, gyrators, transform- For a matrix A the notation posA is used to indi- ers (RLCGT), ideal diodes and current and/or volt- cate all positive combinations of the columns of A, age sources can be formulated by the complemen- i.e., posA := v v = αiA i for some αi > 0 { | i • } tarity formalism. Indeed, the RLCGT-network is with A i denoting the Pi-th column of A. The or- • k given by the state space description thogonality u>y = 0 between two vectors u R and y Rk is denoted by u y. As usual, we∈ say x˙(t)=Ax(t)+Bu(t)+Ew(t) (3a) ∈ ⊥ Rn n that a triple (A, B, C) with A × is mini- y(t)=Cx(t)+Du(t)+Fw(t) (3b) ∈ n 1 mal, when the matrices [B AB ... A − B] and n 1 z(t)=Gx(t)+Hu(t)+Jw(t) (3c) [C> A>C> ...(A>) − C>] have full rank. Finally, we define the linear complementarity under suitable conditions (the network does not problem LCP(q, M) (see [4] for a survey) with data contain loops with only capacitors and voltage gen- k k k q R and M R × by the problem of finding erators or nodes with the only elements incident ∈ ∈ z Rk such that 0 z q+Mz 0. The solution being inductors and current generators). See chap- ∈ 6 ⊥ > set of LCP(q, M) will be denoted by SOL(q, M). ter 4 in [1] for more details. In (3) A, B, C, Many numerical algorithms (appropriate in differ- D, E, F , G, H and J are real matrices of ap- ent situations) are available for solving LCPs. For propriate dimensions. The variables x(t) Rn, an overview the reader is referred to [4, 11]. (u(t),y(t)) Rk+k and (w(t),z(t)) Rp+∈p are the state variable,∈ the connection variables∈ to the diodes and the variables corresponding to the ex- 2 Passivity for linear systems ternal ports (connected to the sources) on time t, We start by recalling the notion of passivity for a respectively. To be more specific, the pair (ui,yi) linear time-invariant system. denotes the voltage-current variables at the con- nections to the diodes, i.e., for i =1,... ,k Definition 2.1 [17] Consider a system ui = Vi,yi = Ii or ui = Ii,yi = Vi, (4) (A, B, C, D) described by the equations − − where Vi and Ii are the voltage across and current x˙(t)=Ax(t)+Bu(t) (1a) through the i-th diode, respectively (adopting the y(t)=Cx(t)+Du(t), (1b) usual sign convention for ideal diodes). The ideal diode characteristic is described by the relations where x(t) Rn, u(t) Rk, y(t) Rk and A, B, C, V 0,I 0, V =0orI =0 ,i=1,... ,k ∈ ∈ ∈ i 6 i > i i and D are matrices of appropriate dimensions. The { } (5) quadruple (A, B, C, D) is called passive,ordissipa- tive with respect to the supply rate u>y, if there and is shown in Figure 1. n exists a nonnegative function V : R R+, called 7→ Ii a storage function, such that for all t0 6 t1 and all k+n+k V time functions (u, x, y) 2 (t0,t1) satisfying + i (1) the following inequality∈L holds: Ii V t1 i V (x(t0)) + u>(t)y(t)dt > V (x(t1)). Zt0
This inequality is called the dissipation inequality. Figure 1: The ideal diode characteristic.
By combining (3) and (5) by eliminating Vi and Theorem 2.2 [17] Assume that (A, B, C) is min- Ii by using (4) the following system description is imal. Then (A, B, C, D) is passive if and only if the obtained: matrix inequalities x˙(t)=Ax(t)+Bu(t)+Ew(t) (6a) A>K + KA KB C> y(t)=Cx(t)+Du(t)+Fw(t) (6b) K = K> > 0, − 6 0 B>K C (D + D>) − − z(t)=Gx(t)+Hu(t)+Jw(t) (6c) (2) 0 y(t) u(t) 0. (6d) 6 ⊥ > 1 have a solution. Moreover, V (x)= 2 x>Kx de- Since (6a)-(6c) is a model for a RLCGT- fines a quadratic storage function if and only if K multiport network consisting of resistors, capac- satisfies (2). itors, inductors, gyrators and transformers, the quadruple speaking, an initial solution is only valid temporar- ily as it will satisfy the system’s equations only un- C DF (A, [BE], , ) (7) til the next switch of one of the diodes. At this G HJ point we only allow Bohl functions (combinations of sines, cosines, exponentials and polynomials) as is passive (or in the terms of [17], dissipative with inputs. This is not a severe restriction as we con- respect to the supply rate u>y + w>z). sider initial solutions in this section. In the global The following technical assumption will be used solution concept we will allow the inputs to be con- often in this paper. Its latter part is standard in catenations of Bohl functions, which may conse- the literature on dissipative dynamical systems, see quently even be discontinuous. e.g. [17]. Definition 4.4 The distribution (u, x, y) Assumption 3.1 B has full column rank and k+n+k is said to be an initial solution to (6a),∈ (A, B, C) is a minimal representation. Bimp (6b) and (6d) with initial state x0 and input w if ∈B 4 Solution concept To define a solution concept, it is natural to employ 1.x ˙ = Ax+Bu+Ew+x0δ0 and y = Cx+Du+Fw the distributional theory, since the abrupt changes as equalities of distributions. in the trajectories can be adequately modelled by 2. u and y are initially nonnegative. impulses. To do so, we need to recall the definition of a Bohl distribution and an initial solution [10]. 3. for all i k¯, either ui =0oryi = 0 as equali- ties of distributions.∈ Definition 4.1 We call u a Bohl distribution,if l i (i) i A justification for restricting the set of initial so- u = uimp +ureg with uimp = i=0 u− δ0 for u− R ∈ and ureg . We call uimpPthe impulsive part of lutions to the space of Bohl distributions is given ∈B u and ureg the regular part of u. The space of all in [3, Lemma 3.9] and [7, Lemma 3.3]. It is shown Bohl distributions is denoted by imp. there that the mode dynamics given by a set of lin- B ear DAEs (1 and 3 in the definition above) has a It seems natural to call a (smooth) Bohl func- unique solution, which is necessarily a Bohl distri- tion u initially nonnegative if there exists an ε>0 bution. such that u(t) 0 for all t [0,ε). Note that > ∈ a Bohl function u is initially nonnegative if and 5 Well-posedness only if there exists a σ0 R such that its Laplace ∈ The statements in the sections 5 and 6 are exten- transformu ˆ(σ) > 0 for all σ > σ0. Hence, there is a connection between small time values for time sions of the corresponding results in [3, 7], which functions and large values for the indeterminate s deal with the input free case only. Consider (6) in in the Laplace transform. This fact is closely re- which the additional variable z is omitted for the lated to the well-known initial value theorem (see moment (as it does not play a role in existence and e.g. [5]). The definition of initial nonnegativity for uniqueness of solution trajectories), i.e. look at Bohl distributions will be based on this observation (see also [9, 10]). x˙(t)=Ax(t)+Bu(t)+Ew(t) (8a) y(t)=Cx(t)+Du(t)+Fw(t) (8b) Definition 4.2 We call a Bohl distribution u ini- 0 6 y(t) u(t) > 0. (8c) tially nonnegative, if its Laplace transformu ˆ(s) sat- ⊥ isfiesu ˆ(σ) > 0 for all sufficiently large real σ. Proposition 5.1 Consider an LCS with external inputs given by (8) such that (A, B, C, D) is pas- Remark 4.3 To relate the definition to the time sive and Assumption 3.1 is satisfied. Define := domain, note that a scalar-valued Bohl distribu- Rk Q SOL(0,D)= v 0 6 v Dv > 0 and let ∗ tion u without derivatives of the Dirac impulse (i.e. be the dual cone{ ∈ of .| ⊥ } Q 0 0 uimp = u δ for some u R) is initially nonnega- Q ∈ n tive if and only if 1. For arbitrary initial state x0 R and any input w , there exists∈ exactly one 0 1. u > 0, or initial solution,∈B which will be denoted by x0,w x0,w x0,w 2. u0 = 0 and there exists an ε>0 such that (u , x , y ). u (t) 0 for all t [0,ε). reg > ∈ 2. No initial solution contains derivatives of the x0,w 0 With these notions we can recall the concept of Dirac distribution. Moreover, uimp = u δ0, x0,w x0,w 0 0 an initial solution [10], which is used as a “build- x =0and y = Du δ0 for some u imp imp ∈ ing block” for the global solution concept. Loosely Rk. n 3. For all x0 R and w it holds that at which w is not continuous is called the collec- ∈ 0 ∈B 0 Cx0 + Fw(0) + CBu ∗, where u δ0 is tion of discontinuity points of w and is denoted by x0∈Q,w d the impulsive part of u . Γw = θi . Note that the right-continuity is just a normalization,{ } which will simplify the notation in 4. The initial solution (ux0,w, xx0,w, yx0,w) is the sequel. The separation of the transition points smooth (i.e., has a zero impulsive part) if and of a piecewise Bohl function by a positive constant only if Cx0 + Fw(0) ∗. α is required to prevent the system from showing ∈Q Zeno behaviour due to Zeno input trajectories. To The proposition gives explicit conditions for ex- present the global existence result, we define the istence and uniqueness of solutions to a class of following distribution space. hybrid dynamical systems of the complementarity type. Similar statements for general hybrid sys- Definition 5.3 The distribution space [0,T) tems are hard to come by (cf. [13] for partial re- L2,δ sults). Note that the first statement of the propo- is defined as the set of all u = uimp + ureg, where θ θ R sition by itself does not immediately guarantee the uimp = θ Γ u δθ for u with Γ a finite subset ∈ ∈ of [0,T),P and ureg 2[0,T) existence of a solution on a time interval with pos- ∈L itive length. The reason is that an initial solution with a non-zero impulsive part may only be valid Theorem 5.4 Consider the LCS given by (8) such at the time instant on which the Dirac distribu- that (A, B, C, D) is passive and Assumption 3.1 is tion is active. If the impulsive part of the (unique) satisfied. Moreover, let the initial state x0, T>0 initial solution is equal to u0δ , the state after re- d 0 and w be specified and let Γ := θi 0 ∈PB Fw { } initialization is equal to x0 + Bu [6, 10]. The oc- be the set of discontinuity points associated with currence of infinitely many jumps at t = 0 with- Fw. Then (8) has a unique solution (u, x, y) out any smooth continuation on a positive length k+n+k ∈ 2,δ [0,T) on [0,T) with initial state x0 and in- time interval is in principle not excluded. However, Lput w in the following sense. the third and fourth claim in the proposition ex- 1 clude this particular instance of Zeno behaviour : 1. x˙ = Ax+Bu+Ew+x0δ0 and y = Cx+Du+Fw if smooth continuation is not directly possible from hold as equalities of distributions x0, it is possible after one re-initialization (jump). 0 d Indeed, since Cx0 + Fw(0) + CBu ∗, it fol- 2. For any interval (a, b) such that (a, b) ΓFw = ∈Q ∅ ∩ lows from the fourth claim that the initial solution the restriction x (a,b) is continuous. 0 | corresponding to x0 + Bu and input w is smooth. This initial solution satisfies the equations (8) on an d 3. For each θ 0 ΓFw the corresponding θ ∈{θ }∪θ interval of the form (0,ε) with ε>0 by definition impulse (u δθ,x δθ,y δθ) is equal to the im- and hence, we proved a local existence and unique- pulsive part of the unique initial solution3 to ness result. This result will be extended to obtain (8) with initial state xreg(θ ):=limt θ xreg(t) − ↑ global existence of solutions. Before we can formu- (taken equal to x0 for θ =0) and input t late such a theorem, we need to define the allowable w(t θ). 7→ input functions and a global solution concept. − 4. 0 u (t) y (t) 0 for almost all t 6 reg ⊥ reg > ∈ Definition 5.2 A function w :[0, ) R is (0,T). called piecewise Bohl,ifw is right-continuous∞ 7→ 2 and there exists a countable collection Γw = τi An important observation of the theorem above (0, ) and an α>0 such that { }⊂ is that jumps can only occur at the initial time ∞ instant (t = 0) and on discontinuity points of Fw. τi+1 > τi + α, and Hence, if Fw is continuous, jumps of the state can • only occur at the initial time instant. for every i there exists a v with • w = v . ∈B |(τi,τi+1) |(τi,τi+1) 6 (In)consistent initial states The set of piecewise Bohl functions is denoted by Definition 6.1 We call an initial state x0 con- . PB sistent with respect to the input w for the sys- tem (8), if the corresponding initial solution We call the collection Γ = τ the set of tran- w i (ux0,w, xx0,w, yx0,w) is smooth. A state x is called sition points associated with w{. The} subset of τ 0 { i} inconsistent with respect to w, if it is not consistent 1Zeno behaviour in a hybrid system means that there for w. is an infinite number of discrete events (mode transitions and/or re-initializations) in a finite time interval. 3Note that we shift time over θ to be able to use the 2This means that for all τ [0, ) the limit definition of an initial solution, which is only given for an ∈ ∞ limt τ w(t)=w(τ). initial condition at t =0. ↓ Corollary 6.2 Consider an LCS given by (8) such The re-initialized state xx0,w(0+) is the unique that (A, B, C, D) is passive and Assumption 3.1 is solution of the following ordinary LCP. satisfied. Define := SOL(0,D) and let ∗ be Q Q 1 the dual cone of . The following statements are v = N >Cx0 + N >Fw(0) + N >CK− C>Nλ equivalent. Q (11a) 0 v λ 0 (11b) 1. x is consistent with respect to w for (8). 6 ⊥ > 0 ∈B and u0 follows similarly as in (ii). 2. Cx + Fw(0) ∗. 0 ∈Q (iv) The jump multiplier u0 is the unique mini- 3. LCP(Cx0 + Fw(0),D) has a solution. mizer of
4. Cx0 + Fw(0) pos(I, D), where I is the 1 ∈ − Minimize 2 (x0 + Bv)>K(x0 + Bv)+v>Fw(0) identity matrix. (12a) Subject to v (12b) The above corollary gives several tests for de- ∈Q termining whether an initial state is consistent or inconsistent. In particular, the network is impulse- Observe that (i) is a generalized LCP, which uses Rk free, if SOL(0,D)= 0 (or, in terms of [4], if D a cone instead of the usual positive cone + [4, p. { } Rk Q k k is an Ro-matrix). Note that in this case ∗ = . 31]. Indeed, in case = R and thus = R Q + ∗ + In case the matrix [CF] has full row rank, this (9) reduces to an ordinaryQ LCP. StatementQ (ii) condition is also necessary. expresses the fact that among the admissible re- initialized states p (admissible in the sense that 7 Characterizations of jumps smooth continuation is possible after the reset, i.e. Cp+ Fw(0) ∗) the nearest one is chosen in the In this section we will study the jump phenomena sense of the∈Q metric defined by any arbitrary stor- more extensively as we are interested in a general- age function corresponding to (A, B, C, D). A sim- ized notion for passivity for the network including ilar situation is encountered in mechanical systems the diodes. with inelastic impacts [14, p. 75], where it has been called “a principle of economy.” Finally, (iv) states Theorem 7.1 Let an LCS be given by (8) such that in case Fw(0) = 0, the jump multiplier sat- that (A, B, C, D) is passive and Assumption 3.1 isfies the complementarity conditions (i.e., v ) ∈Q is satisfied. Define := SOL(0,D) and let ∗ and minimizes the internal energy (expressed by Q Q 1 be the dual cone of . Consider the initial so- the storage function 2 x>Kx) after the jump. Note x ,w x ,w xQ,w lution (u 0 , x 0 , y 0 ) corresponding to initial that x0 +Bv is the re-initialized state when the im- n state x0 R and input w . Moreover, denote pulsive part is equal to vδ0. Under the assumption ∈ x0,w ∈B x ,w the impulsive part of u by u0δ . The following of x 0 (0+) x imB, it can be shown that the imp 0 − 0 ∈ equivalent characterizations can be given for u0. two optimization problems are actually each other’s dual (see e.g. page 117 in [4]). (i) The jump multiplier u0 is uniquely determined by the generalized LCP (see [4] page 31 on 8 Passivity of a complementarity system complementarity problems over cones) In this section, we consider (6) with F = 0. The 0 0 assumption F = 0 is made to prevent the situation u Cx0 + Fw(0) + CBu ∗ (9) Q3 ⊥ ∈Q that the input w may cause impulsive motions and state jumps for times t>0. (ii) The re-initialized state xx0,w(0+) := x0,w limt 0 xreg (t) is the unique minimizer of ↓ Definition 8.1 The LCS given by
1 x˙(t)=Ax(t)+Bu(t)+Ew(t) (13a) Minimize [p x0]>K[p x0] (10a) 2 − − y(t)=Cx(t)+Du(t) (13b) subject to Cp + Fw(0) ∗, (10b) ∈Q z(t)=Gx(t)+Hu(t)+Jw(t) (13c) where K is any solution to (2) and thus 0 6 u(t) y(t) > 0 (13d) 1 ⊥ V (x)= 2 x>Kx is a storage function for (A, B, C, D). The multiplier u0 is uniquely de- is called dissipative with respect to the supply x0,w 0 termined from x (0+) = x0 + Bu . rate w>z, if there exists a nonnegative function V : Rn R , called a storage function, such that 7→ + (iii) The cone is equal to Nv v > 0 and for all 0