1

A Lyapunov framework for nested dynamical systems on multipletime scales with application to converter-based power systems

Irina Subotic, Dominic Groß, Marcello Colombino, and Florian Dorfler

Abstract—In this work, we present a Lyapunov function frame-work for establishing stability with respect to a compact set of anested interconnection of nonlinear dynamical systems orderedfrom slow to fast according to their convergence rates. Theproposed approach explicitly considers more than two time scales,and does not require modeling multiple time scales via scalartime constants. Motivated by the technical results, we develop anovel control strategy for a grid-forming power converter thatconsists of an inner cascaded two-degree of freedom controllerand dispatchable virtual oscillator control as a reference model.The resulting closed-loop converter-based AC power system is inthe form of a nested system with multiple time scales. We applyour technical results to obtain explicit bounds on the controllerset-points, branch powers, and control gains that guaranteealmost global asymptotic stability of the multi-converter ACpower system with respect to a pre-specified solution of the ACpower-flow equations. Finally, we validate the performance of theproposed control structure in a case study using a high-fidelitysimulation with detailed hardware validated converter models.

I. INTRODUCTION

T IME-SCALE separation arguments are ubiquitous incontrol design and analysis of large-scale engineering

systems that contain dynamics on multiple time scales fromdifferent physical domains. Traditionally, singular perturbationtheory has been the standard tool to analyze nonlinear dy-namics that evolve on multiple time scales [1]–[3]. Withinthis framework, stability conditions are typically provided forhyperbolic fixed points of systems with two time scales anda “small” scalar time constant describing the fast time scale.The results can be extended to linear systems with two timescales and a fast time scale modeled by multiple time constants[4], slow-fast control systems with non-hyperbolic fixed points[5], and multiple time scales by successively grouping theminto two time scales (see, e.g., [2]). In contrast, our approachexplicitly considers multiple time scales, stability with respectto a compact set, does not require modeling time scalesvia scalar time constants, and exploits the nested structuretypically exhibited by systems with multiple time scales suchas power systems [6] and biological system [7]. To this end, wedevelop a general Lyapunov function framework for stabilityanalysis of nested nonlinear dynamical systems that can beordered from slow to fast in terms of convergence rates totheir set of steady-states and only depend on the states ofslower systems and the next fastest one.

This work was partially funded by the Swiss Federal Office of Energyunder grant number SI/501707 and the European Union’s Horizon 2020research and innovation programme under grant agreement number 691800.This article reflects only the authors’ views and the European Commission isnot responsible for any use that may be made of the information it contains.

I. Subotic, D. Groß, and F. Dorfler, are with the Automatic ControlLaboratory, ETH Zurich, Switzerland, M. Colombino is with the Electricaland Computer Engineering Department at McGill University, Canada; e-mail:subotici,grodo,dorfler@ethz.ch, marcello.colombino@mcgill.ca

The analysis in this paper is based on a recently developedLyapunov characterization of almost global asymptotic stabil-ity with respect to a compact set presented in [8] that requiresthat the set of states that are unstable but attractive has zeroLebesgue measure. The technical contribution is twofold: first,we develop a Lyapunov characterization of unstable hyperbolicfixed points that have a region of attraction of measure zero;then we provide a Lyapunov framework that results in con-ditions under which the guarantees obtained by applying theaforementioned Lyapunov conditions in [8] to a reduced-ordersystem translate to the full-order nested dynamical system. Ourresults can be interpreted as an extension of the conditions forsystems with two time scales in [9, Ch. 11.5] to multiple nestedsystems and a more general notion of stability. Moreover,we reduce conservatism by allowing for a wider range ofcomparison functions.

Motivated by the transition of power systems towards re-newable energy sources that are connected to the system viapower electronics [10], [11], we apply our technical results tomulti-converter AC power systems. The analysis and control ofpower systems and microgrids is typically based on reduced-order models of various degrees of fidelity that exploit thepronounced time-scale separation between the dynamics ofsynchronous machines and power converters and the transmis-sion network [3], [6], [12]–[14]. While these ad-hoc modelsimplifications have proved themselves useful their validityfor converter-based systems is questionable. For instance,the assumption that the dynamics of transmission lines canbe neglected breaks down for power systems dominated byfast acting power converters [8], [15]–[17]. In this work, wemake the time-scale separation argument rigorous by explicitlyconsidering the interaction of dynamics on different timescales (i.e., converter dynamics, inner controls, line dynamics)and quantifying the parameters (e.g., set-points, control gains,transmission line parameters, etc.) for which stability for theoverall system can be ensured.

The prevalent approach to grid-forming control is so-calleddroop-control [17], [18] and synchronous machine emula-tion [19]–[21]. However, while droop control and machineemulation can provide useful insights, stability guarantees arelocal and typically don’t extend to line dynamics, detailed con-verter models, and operating points with non-zero power flows.In contrast, virtual oscillator control (VOC) ensures almostglobal synchronization [22]–[24] but cannot be dispatched,and almost global asymptotic stability with line dynamics canbe ensured for dispatchable virtual oscillator control (dVOC)[25], [26] for appropriate control gains and power convertersmodeled as controllable voltage sources [8].

Motivated by this result, we consider a converter-basedpower system model that includes dynamic models of voltage

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source converters as well as transmission network dynamics.We develop a control strategy for the converters that usesdispatchable virtual oscillator control (dVOC) as a referencemodel for a cascaded two-degree of freedom voltage andcurrent controller in stationary αβ coordinates instead of localdq coordinates. This allows us to model the whole inverterbased power system as a nested interconnection of subsystems(dVOC, the transmission line dynamics, the inner controlloops) that evolve on different time-scales and to apply ournovel Lyapunov-based stability criterion to obtain conditionson the parameters (i.e., set-points, control gains, and networkparameters) that guarantee almost global asymptotic stabilityof overall system. Finally, we validate the proposed controlarchitecture in a high-fidelity simulation of the hardware set-up described in [26].

The remainder of this section recalls some basic notationand results from graph theory. Section II provides definitionsand a preliminary technical results on almost global asymptoticstability with respect to sets. The main theoretical contri-bution is given in Section III. In Section IV we present adetailed model of a multi-converter AC power system andthe control objectives. Section V presents a cascaded two-degree of freedom control structure that tracks a referenceobtained by dVOC as a reference model and Section VIpresents stability conditions for the multi-converter system.The results are illustrated using a high-fidelity simulation studyin Section VII, and Section VIII provides the conclusions.

NotationWe use R and N to denote the set of real and natural

numbers and define R≥a := x ∈ R|x ≥ a and, e.g.,R[a,b) := x ∈ R|a ≤ x < b. Given θ ∈ [−π, π] the 2Drotation matrix is given by

R(θ) :=

[cos(θ) − sin(θ)sin(θ) cos(θ)

]∈ R2×2.

Moreover, we define the 90 rotation matrix J := R(π/2) thatcan be interpreted as an embedding of the complex imaginaryunit√−1 into R2. Given a matrix A, AT denotes its transpose.

We use ‖A‖ to indicate the induced 2-norm of A. We writeA < 0 (A 0) to denote that A is symmetric and positivesemidefinite (definite). For column vectors x ∈ Rn and y ∈Rm we use (x, y) = [xT, yT]T ∈ Rn+m to denote a stackedvector, and ‖x‖ denotes the Euclidean norm. The absolutevalue of a scalar y ∈ R is denoted by |y|. Furthermore, Indenotes the identity matrix of dimension n, and ⊗ denotes theKronecker product. For any matrix M and any n ∈ N, wedefine Mn = In ⊗M . Matrices of zeros of dimension n×mare denoted by 0n×m and 0n denotes column vector of zerosof length n. We use ‖x‖C := minz∈C‖z − x‖ to denote thedistance of a point x to a set C. We use ϕf (t, x0) to denotethe solution of d

dtx = f(x) at time t ≥ 0 starting from theinitial condition x(0) = x0 at time t0 = 0.

II. ALMOST GLOBAL ASYMPTOTIC STABILITY WITHRESPECT TO A COMPACT SET

Consider the dynamical systemddtx = f(x), (1)

where x ∈ Rn denotes the state vector and f : Rn → Rnis a Lipschitz continuous function. In order to state the mainresults of the paper, we require the following definition ofalmost global asymptotic stability with respect to a set [27].

Definition 1 (Almost global asymptotic stability) The dy-namic system (1) is called almost globally asymptoticallystable with respect to a compact set C ⊂ Rn if

(i) it is almost globally attractive with respect to C, i.e.,

limt→∞‖ϕf (t, x0)‖C = 0 (2)

holds for all x0 /∈ Z , and Z has zero Lebesgue measure,(ii) it is Lyapunov stable with respect to C, i.e., for every

ε ∈ R>0 there exists δ ∈ R>0 such that

‖x0‖C < δ =⇒ ‖ϕf (t, x0)‖C < ε, ∀t ≥ 0. (3)

Next, we recall the definition of comparison functions used toestablish stability properties of dynamical systems [28].

Definition 2 (Comparison functions) A function χc :R≥0 → R≥0 is of class K if it is continuous, strictlyincreasing and χc(0) = 0; it is of class K∞ if it is a K -function and χc(s)→∞ as s→∞.

Next, consider a set U ⊂ Rn that is invariant with respect to (1)(i.e., ϕf (t, x0) ∈ U for all t ∈ R>0 and all x0 ∈ U), satisfiesC ∩ U = ∅, and corresponds to e.g., undesirable equilibria orlimit cycles of (1). In this case global asymptotic stability of(1) with respect to C cannot be established. Instead, the follow-ing Theorem provides a Lyapunov function characterization ofalmost global asymptotic stability with respect to C [8, Th. 1].

Theorem 1 (Lyapunov functions) Consider a compact setC ⊂ Rn and a zero Lebesgue measure set U ⊂ Rn that isinvariant with respect to (1). Moreover, consider a continu-ously differentiable function V : Rn → R>0 and comparisonfunctions χ1, χ2 ∈ K∞ and χ3 ∈ K such that

χ1(‖x‖C) ≤ V(x) ≤ χ2(‖x‖C)d

d tV(x) :=

∂V∂x

f(x) ≤ −χ3(‖x‖C∪U )

holds for all x ∈ Rn. Moreover, let

ZU,f := x0 ∈ Rn| limt→∞‖ϕf (t, x0)‖U = 0.denote the region of attraction of U under (1). If ZU,f haszero Lebesgue measure, the dynamics (1) are almost globallyasymptotically stable with respect to C.

Besides finding a suitable Lyapunov function, the main dif-ficulty in applying Theorem 1 is to verify that the region ofattraction ZU,f of the (unstable) attractive set U , has measurezero. To this end, our first contribution is a Lyapunov-likecondition that characterizes unstable hyperbolic fixed pointswith region of attraction that has zero Lebesgue measure.

Theorem 2 (Unstable hyperbolic fixed point) Consider acontinuously differentiable function f : Rn → Rn, its Jaco-bian Ax? := ∂f(x)

∂x

∣∣x=x?

at an equilibrium x? ∈ Rn, and thelinearized dynamics d

dtxδ = Ax?xδ , where xδ := x − x?.Moreover, consider a quadratic function Vδ := 1

2xTδ Pδxδ . If

3

ddtVδ(xδ) := ∂Vδ

∂xδAx?xδ < 0 holds for all xδ 6= 0n and

there exists x′ ∈ Rn such that Vδ(x′) < Vδ(0n), then x?

is an unstable hyperbolic fixed point of (1), and its region ofattraction Zx?,f has measure zero.

Proof: We first establish that x? is an unstable equilibriumpoint. To this end note that Vδ(x′) < Vδ(0n) implies thatVδ(c′x′) < Vδ(0n) holds for all c′ ∈ R>0 and that Vδ isunbounded from below. Moreover, using d

dtVδ < 0 for allxδ 6= 0n it follows for all c′ ∈ R>0 that Vδ(ϕAx? (t, c′x′))→−∞ and ‖ϕAx? (t, c′x′)‖ → ∞ as t→∞. In other words, 0nis an unstable equilibrium of d

dtxδ = Ax?xδ . Next, assume thatx? is not a hyperbolic fixed point, i.e., that Ax? has eigenvalueswith zero real part. This implies the existence of initial condi-tions xδ,0 6= 0n such that ϕAx? (t, xδ,0) remains bounded forall t ∈ R≥0, but does not converge to the origin 0n. However,because Vδ(ϕAx? (t, xδ,0)) is strictly decreasing in t for allxδ,0 6= 0n it either holds that ‖ϕAx? (t, xδ,0)‖ → 0 as t→∞or Vδ(ϕAx? (t, xδ,0)) → −∞ and ‖ϕAx? (t, xδ,0)‖ → ∞ ast → ∞. Therefore, there exists no initial state xδ,0 6= 0nfor which ϕAx? (t, xδ,0) remains bounded for all t ∈ R≥0,but does not converge to the origin 0n, i.e., Ax? cannot haveeigenvalues with zero real part. Because the equilibrium isunstable, at least one eigenvalue of the linearized system musthave positive real part, and the equilibrium is an unstablehyperbolic fixed point. Finally, because the vector field f iscontinuously differentiable it directly follows from [29, Prop.11] that Zx?,f has Lebesgue measure zero.

III. STABILITY THEORY FOR NESTED SYSTEMS ONMULTIPLE TIME SCALES

In this section we will present a model of nested dynamicalsystems on multiple time-scales and extend the results fromTheorem 1 and Theorem 2 to this class of systems.

A. Nested systems on multiple time scales

Consider the nested dynamical system on N time scalesshown in Figure 1 given by

ddtxi = fi(x1, . . . , xi+1), ∀i ∈ N[1,N−1], (4a)ddtxN = fN (x1, . . . , xN ), (4b)

where xi ∈ Rni denotes the state of the subsystem i ∈N[1,N−1] and fi : Rn1 × . . . × Rni+1 → Rni and fN :Rn1 × . . .×RnN → RnN are Lipschitz continuous functions.Letting x = (x1, . . . , xN ) and f = (f1, . . . , fN ), we obtainthe full system dynamics (1) with n =

∑Ni=1 ni. Broadly

speaking, we assume that the dynamics are ordered from slowto fast convergence to their set of steady-states, i.e., the outerdynamics are the slowest and the inner right dynamics arethe fastest (see Figure 1). To make this argument precise, werecursively define the steady-state maps for each fi, i ∈ N[2,N ].

Assumption 1 (Steady-state maps)Let us define fsN (x1, . . . , xN ) := fN (x1, . . . , xN ). We assumethat there exists a unique steady-state map xsN (x1, . . . , xN−1)

ddtx1=f1

ddtx2=f2

. . . ddtxi=fi

. . . ddtxN=fN

Fig. 1. Block diagram of the nested dynamical system on N time scales

ddtx1=f1

ddtx2=f2

. . . ddtxr=f

sr

. . . ddtxN=fN

Fig. 2. Block diagram of the reduced-order dynamics on r time scales (black)and neglected dynamics (grey).

such that fN (x1, . . . , xN−1, xsN ) = 0. Then for i ∈ N[1,N−1]

we recursively define

fsN−i(x1, . . . , xN−i) := fN−i(x1, . . . , xN−i, xsN−i+1),

and recursively assume that, for all i ∈ N[1,N−2], there existsa unique steady-state map xsN−i(x1, . . . , xN−i−1) such thatfsN−i(x1, . . . , xN−i−1, x

sN−i) = 0.

In other words, fsi denotes the vector field correspondingto the dynamics with index i ∈ N[1,N−1] with the state xi+1

restricted to its steady-state map, i.e., xi+1 = xsi+1. Intuitively,if the systems are ordered from slow to fast convergence totheir set of steady-states, Assumption 1 suggests the naturalmodel reduction procedure that successively replaces fastdynamics by their steady-state maps. Given r ∈ N[1,N ], thisresults in the reduced-order dynamics (see Figure 2) with statevector (x1, . . . , xr) ∈ R

∑ri=1 ni given by

ddt xi = fi(x1, . . . , xi+1), ∀i ∈ N[1,r−1], (5a)ddt xr = fsr (x1, . . . , xr), (5b)xi = xsi (x1, . . . , xi−1), ∀i ∈ N[r+1,N ]. (5c)

We emphasize that (5) defines a reduced-order dynamicalsystem. Next, given a set Ω ⊂ Rn1 , we define the mapping ofΩ under the steady-state maps as

X s(Ω) :=

x1...xN

∈ Rn

∣∣∣∣∣∣∣∣∣x1 ∈ Ω

x2 = xs2(x1)...

xN = xsN (x1, . . . , xN−1)

. (6)

In the remainder of this section we derive conditions that allowus to extend guarantees of the type given in Theorem 1 andTheorem 2 for the reduced-order system d

dt x1 = fs1 (x1) anda set C1 ⊂ Rn1 to the full dynamics (4) and X s(C1) ⊂ Rn.

B. Lyapunov function for nested systems

For all i ∈ N[2,N ] we use yi := xi−xsi ∈ Rni to denote thedifference of xi to its steady-state map xsi . For all ∀i ∈ N[1,N ]

we define the continuously differentiable Lyapunov functionsVi : Rni → R≥0. Given positive constants µi ∈ R≥0 to bedetermined, a Lyapunov function candidate ν : Rn1 × . . . ×RnN → R≥0 for the system (4) is given by

ν := µ1V1(x1) +∑N

i=2µiVi(yi). (7)

4

For clarity of the exposition, we omit the arguments of Vi, xsi ,fi in the remainder. We require the following assumption thatbounds the decrease of the individual Lyapunov functions Viin (7) for their associated reduced-order models and boundstheir increase due to neglecting slower and faster dynamics.

Assumption 2 Given compact sets C1 ⊂ Rn1 and U1 ⊂ Rn1 ,for all i ∈ N[1,N ], there exist positive semidefinite functionsψi : Rni → R≥0 and ψ′i : Rni → R≥0, K functions σisuch that σ1(‖x1‖C1∪U1) ≤ ψ1(x1) +ψ′1(x1) and σi(‖yi‖) ≤ψi(yi)+ψ′i(yi) for all i ∈ N[2,N ], and positive constants α1 ∈R>0, α′1 ∈ R>0 such that

∂V1∂x1

fs1 ≤ −α1ψ1(x1)2 − α′1ψ′1(x1)2,

holds, and for all i ∈ N[2,N ] it holds that

∂Vi∂yi

fsi ≤ −αiψi(yi)2 − α′iψ′i(yi)2.

Moreover, there exists positive constants βi,i+1 ∈ R>0, suchthat

∂V1∂x1

(f1 − fs1

)≤ β1,2ψ1(x1)ψ2(y2)

holds, and for all i ∈ N[2,N−1] it holds that

∂Vi∂yi

(fi − fsi

)≤ βi,i+1ψi(yi)ψi+1(yi+1).

Finally, for all i ∈ N[2,N ], j ∈ N[1,i+1], and k ∈ N[1,i−1],there exists bi,j,k ∈ R>0 such that

−∂Vi∂yi

∂xsi∂xk

fk ≤k+1∑j=2

bi,j,kψi(yi)ψj(yj) + bi,1,kψi(yi)ψ1(x1).

In particular, the first inequality in Assumption 2 bounds thedecrease of the Lyapunov function V1 along the trajectoriesof the reduced-order model d

dtx1 = fs1 (x1). Moreover, fori ∈ N[2,N ] the second inequality bounds the decrease of theLyapunov function candidates Vi in the error coordinates yiunder the assumption that all slower states are constant (i.e.,ddtxj = 0 for all j < i) and all faster states are in theirsteady state (i.e., xj = xsj for all j > i). The remaininginequalities bound the additional terms in the time derivativeof the Lyapunov function ν along the full dynamics (4) thatarise because the slower states are generally not constant andthe faster states are generally not in their steady state.

Note that a Lyapunov-based stability proof requires that theright hand side of the first two inequalities in Assumption 2can be bounded by appropriate comparison functions. If ψ′ =0, then ψi needs to be lower bounded by a K -function andsatisfy the last three inequalities in Assumption 2. In contrast,we only require that ψi + ψ′i is bounded from below by asuitable K -function σi (see, e.g., Theorem 1). This allows foradditional flexibility in choosing ψi to obtain improved bounds(see Section VI-C). For simplicity of notation we define

Definition 3

βi,1 :=∑i−1

k=1bi,1,k ∀i ∈ N[2,N ],

βi,j :=∑i−1

k=j−1bi,j,k ∀i ∈ N[3,N ],∀j ∈ N[2,i−1],

γi := bi,i,i−1 ∀i ∈ N[2,N ].

In other words, βi,i−1, βi−1,i, and γi will be used to bound thedifference between the reduced-order model (5) with r = i−1and the full dynamics d

dtxi = fi. Moreover, αi bounds theconvergence rate of yi under the assumption that all slowerstates are constant all faster states are in their steady state.

Moreover, we define µi used in the Lyapunov function (7)as µi :=

∏i−1j=1

βj,j+1

βj+1,jfor all i ∈ N[2,N ], µ1 = 1, and we define

the symmetric matrix M as follows.

Definition 4 Starting from M1 = α1 the symmetric matrixM ∈ RN×N is recursively defined for all i ∈ N[2,N ] by itsleading principal minors

Mi =

[Mi−1 −βiµi? (αi − γi)µi

],

and βi := (. . . , 12βi,i−3,12βi,i−2, βi,i−1).

Note that the constants βi,j are given by Assumption 2 forj = i+ 1 and Definition 3 for i 6= j.

In the next section we show that the derivative of theLyapunov function ν along the trajectories of (4) is bounded byddtν ≤ −(ψ1, . . . , ψN )TM(ψ1, . . . , ψN ), i.e., ν is decreasingif M is positive definite. The main result of this sectionare two theorems that exploit this fact to establish almostglobal asymptotic stability of the nested system (4) using theLyapunov function ν. Subsequently, we will provide tractableconditions for verifying that M is positive definite (see SectionIII-D).

C. Almost global asymptotic stability of nested systems

We are now ready to state the main result that establishesalmost global asymptotic stability of (4) with respect to a setX s(C1), where X s(·) is defined in (6).

Theorem 3 (Almost global asymptotic stability of nestedsystems) Consider compact sets C1 ⊂ Rn1 and U1 ⊂ Rn1 .Assume that, for all i ∈ N[1,N ], there exists χVi1 ∈ K∞ andχVi2 ∈ K∞ such that χV11 (‖x1‖C1) ≤ V1(x1) ≤ χV12 (‖x1‖C1)

holds and χVi1 (‖yi‖) ≤ Vi(yi) ≤ χ

Vi2 (‖yi‖) holds for all

i ∈ N[2,N ]. Suppose Assumption 1 and 2 hold, M is positivedefinite, and the region of attraction ZX s(U1),f of X s(U1)has measure zero, then the system (4) is almost globallyasymptotically stable with respect to X s(C1).

Proof: Let us consider the Lyapunov function candidateν defined in (7). Using Lemma 1 (given in the appendix) withεi = 1 for all i ∈ N[1,N ], we obtain M = H , M ′ = H ′, and

ddtν ≤ −

ψ1

ψ2

...ψN

T

M

ψ1

ψ2

...ψN

−ψ′1ψ′2...ψ′N

T

M ′

ψ′1ψ′2...ψ′N

.M ′ is a positive definite diagonal matrix, and, if M positivedefinite, there exists a positive constant αM ∈ R>0 such thatM αM and M ′ αM . It follows that

ddtν ≤ −αM

∑Nc

i=1ψ2i + ψ′i

2 ≤ − 12αM

∑Nc

i=1(ψi + ψ′i)

2,

5

where the last inequality follows from the fact that −αM (ψ2i +

ψ′i2) ≤ − 1

2αM (ψi+ψ′i)2 is equivalent to 1

2αM (ψi−ψ′i)2 ≥ 0for all i ∈ N[1,N ]. We conclude that

ddtν ≤ − 1

2αM(σ1(‖x1‖C1∪U1)2 +

∑Nc

i=2σi(‖yi‖)2

). (8)

The right hand side of (8) is positive definite and radiallyunbounded w.r.t X s(C1) ∪ X s(U1), and X s(C1) ∪ X s(U1) iscompact. Using the same steps as in [28, p. 98] there exists afunction χ3 ∈ K such that

ddtν ≤ −χ3(‖x‖X s(C1)∪X s(U1)).

Moreover, under the hypothesis of the theorem, for all i ∈N[1,N ], there exists χ

Vi1 ∈ K∞ and χ

Vi2 ∈ K∞ such that

χV11 (‖x1‖C1) ≤ V1(x1) ≤ χV12 (‖x1‖C1) holds and χVi1 (‖yi‖) ≤Vi(yi) ≤ χ

Vi2 (‖yi‖) holds for all i ∈ N[2,N ]. For j ∈ N[1,2],

we define the functions χj :=∑Nci=1 µiχ

Vij that are positive

definite and radially unbounded w.r.t. the compact set X s(C1).Since it holds that χ1 ≤ ν ≤ χ2, following the same steps asin [28, p. 98] there exist χ1 ∈ K∞ and χ2 ∈ K∞ suchthat χ1(‖x‖X s(C1)) ≤ ν(x) ≤ χ2(‖x‖X s(C1)). Finally, by thehypothesis of the theorem, the region of attraction ZX s(U1) ofX s(U1) has measure zero. With C = X s(C1), U = X s(U1),and V = ν, it follows from Theorem 1 that (4) is almostglobally asymptotically stable with respect to X s(C1).

Theorem 3 requires that the region of attraction X s(U1) hasmeasure zero. This can be verified using the following resultthat relies on the characterization of an unstable hyperbolicfixed point given in Theorem 2 as well as Assumption 2.

Theorem 4 (Region of Attraction) Suppose that fi in (4)is continuously differentiable with linearized dynamics at anequilibrium x? = (x1, . . . , xN ) given by

ddtxδ,N = fδ,i(xδ,1, . . . , xδ,N ) :=

∑N

j=1

∂fN∂xj

∣∣xj=x?j

xδ,j ,

where xδ,j := xj − x?j for all j ∈ N[1,N ], and

ddtxδ,i = fδ,i(xδ,1, . . . , xδ,i+1) :=

∑i+1

j=1

∂fi∂xj

∣∣xj=x?j

xδ,j

for all i ∈ N[1,N−1]. Suppose that Assumption 1 and 2 holdfor the linearized system (i.e., for fδ,i, xδ,i, yδ,i instead of fi,xi, and yi), C1 = ∅ and U1 = 0n1, and Vδ,1 = 1

2xTδ,1P1xδ,1,

Vδ,i = 12y

Tδ,iPiyδ,i for all i ∈ N[2,N ]. Moreover, assume

that M is positive definite. If there exists x′δ,1 such thatVδ,1(x′δ,1) < Vδ,1(0n1

), then the region of attraction Zx?,fof x? has measure zero.

Proof: We define the Lyapunov-like function νδ :=12x

Tδ,1P1xδ,1 +

∑Ni=1

µi2 x

Tδ,iPixδ,i. Following the same steps

as in the proof of Theorem 3, i.e., using Lemma 1 (givenin the appendix) with εi = 1 for all i ∈ N[1,N ], Lemma2, Assumption 2, and Proposition 1, it follows that thereexists a a function χ3 ∈ K such that d

dtνδ ≤ −χ3(‖x‖)holds. Moreover, by the hypothesis of the theorem it holdsthat Vδ,1(x′δ,1) < V1(0n1

) and Vδ,i(0ni) = 0 for all i ∈N[2,N ]. Letting x′δ = (x′δ,1, x

s2, . . . , x

sN ) it directly follows that

νδ(x′1) < νδ(0n) and the theorem follows by noting that νδ

satisfies the conditions Theorem 2.

Note that Theorem 3 and Theorem 4 provide a way toextend results obtained for the reduced-order system d

dtx1 =fs1 (x1) to the full-order system (4). In particular, if there existsa Lyapunov function V1 and Lyapunov-like function Vδ,1 forwhich the conditions of Theorem 1 and Theorem 2 hold, thenTheorem 3 and Theorem 4 require to find Lyapunov functionsVi and Vδ,i for all i ∈ N[2,N ] in the error coordinates yi thatsatisfy Assumption 2.

Moreover, Theorem 3 and Theorem 4 require that M ispositive definite. In the next section we exploit the structureof M to provide a recursive sufficient condition for M to bepositive definite. This simplified recursive condition will beused to provide analytical stability guarantees for the multi-converter power system considered in Section IV.

D. Tractable positivity condition for M

The following condition is necessary and sufficient for Mto be positive definite.

Condition 1 (Positivity condition) For all i ∈ N[2,N ] it holdsthat γi + µiβ

TiM

−1i−1βi < αi, where Mi denotes the i-th

leading minor of the matrix M .

Verifying Condition 1 requires inverting the matrix Mi, i.e.,the complexity of the inequalities that need to be verifiedgrows considerably with N . In contrast, the following re-cursive sufficient condition avoids this issue by introducingadditional variables ci ∈ R>0, that allow us to exploit thestructure of the problem. In particular, the variable ci−1lower bounds the smallest eigenvalue of Mi−1 and can beinterpreted as bound on the convergence rate of the reduced-order dynamics (5) with r = i− 1.

Condition 2 (Sufficient positivity condition) For all i ∈N[2,N ] it holds that

1 <αiβi,i−1ci−1

βi−1,iβTi βi + γiβi,i−1ci−1

, (9)

where c1 := α1 ∈ R>0, αi ∈ R>0, γi ∈ R≥0, βi,j ∈ R>0 forall i ∈ N[2,N ] and j ∈ N[1,i−1], and

ci := 12

(αi − γi +

βi,i−1

βi−1,ici−1 −

√Di

)for all i ∈ N[2,N ] and Di := (αi−γi+βi,i−1

βi−1,ici−1)2+4(βT

i βi−(αi − γi)βi,i−1

βi−1,ici−1).

Condition 2 holds if yi converges fast enough relative to thereduced-order system (5) with r = i− 1. Next, we show that

Condition 2⇒ Condition 1 ⇐⇒ M 0.

Proposition 1 (Positive definite M ) Iff Condition 1 holds,then the matrix M is positive definite. Suppose that Condition2 holds, then ci ∈ R>0 and Mi µici holds for all i ∈ N[2,N ],and the matrix M is positive definite.

Proof: Using Lemma 2 (given in the appendix) it canbe verified that M is positive definite if and only if Mi 0(see Definition 4) holds for all i ∈ N[1,N ], i.e., if and only if

6

Condition 1 holds. Next, using the partitioning of the matrixMi from Definition 4 and applying the Schur complement itcan be verified that Mi µici holds for all i ∈ N[2,N ] if andonly if Mi−1 µici and

αi − γi − ci − βTi (µ−1i Mi−1 − ci)−1βi ≥ 0 (10)

holds for all i ∈ N[2,N ]. The remainder of this proof establishesthat (10) and Mi−1 µici hold for all i ∈ N[2,N ] if Condition2 holds. To this end, we first show that if ci−1 ∈ R>0 andCondition 2 hold, then ci ∈ R>0 and βi,i−1

βi−1,ici−1 − ci ∈ R>0.

In particular, ci ∈ R>0 holds if and only if

(αi − γi +βi,i−1

βi−1,ici−1)2 > Di (11)

holds. Rewriting Di > 0 as Di = (αi−γi−βi,i−1

βi−1,ici−1)2+4βT

i βiit can be verified that (11) is identical to (9). Moreover, bydefinition of ci it follows that βi,i−1

βi−1,ici−1−2ci = γi+

√Di−αi.

Therefore, βi,i−1

βi−1,ici−1 − ci ∈ R>0 holds if ci +

√Di > αi −

γi holds. Using ci ∈ R>0 and ci−1 ∈ R>0 it follows thatβi,i−1

βi−1,ici−1 − ci ∈ R>0 if (11) holds. This proves the claim.

Next, we show that Mi−1 µi−1ci−1, ci−1 ∈ R>0, andCondition 2 guarantee that Mi−1 µici and (10) hold. Inparticular, Mi−1 µi−1ci−1 implies that

µ−1i Mi−1 − ci µi−1µi

ci−1 − ci =βi,i−1

βi−1,ici−1 − ci > 0,

and it follows that Mi−1−µici 0 and (βi,i−1

βi−1,ici−1−ci)−1

(µ−1i Mi−1 − ci)−1. Therefore, (10) holds if

αi − γi − ci − βTi

(βi,i−1

βi−1,ici−1 − ci

)−1βi ≥ 0 (12)

holds. Multiplying (12) by βi,i−1

βi−1,ici−1−ci ∈ R>0 results in the

equivalent condition(βi,i−1

βi−1,ici−1−ci

)(αi−γi−ci

)−βT

i βi ≥ 0and it can be verified that ci defined in Condition 2 is a solutionof this inequality. Thus, Mi−1 µi−1ci−1, ci−1 ∈ R>0, andCondition 2 guarantee that Mi µici holds for all i ∈ N[2,N ].

Next, noting that M1 = µ1c1 ∈ R>0 it follows by inductionthat ci ∈ R>0 and Mi µici holds for all i ∈ N[2,N ]. Finally,applying the Sylvester criterion it can be shown that Mi µicifor all i ∈ N[1,N ] implies that M 0.

E. Connection to singular perturbation theory

To illustrate the main differences to results from singularperturbation theory, the following theorem re-states our resultsin the form typically used in singular perturbation theory.

Theorem 5 (Nested Singular Perturbations) Consider acompact set C1 ⊂ Rn1 . Assume that, for all i ∈ N[1,N ], thereexists χVi1 ∈ K∞ and χ

Vi2 ∈ K∞ such that χV11 (‖x1‖C1) ≤

V1(x1) ≤ χV12 (‖x1‖C1) and χVi1 (‖yi‖) ≤ Vi(yi) ≤ χ

Vi2 (‖yi‖)

for all i ∈ N[2,N ]. If Assumption 1 holds and Assumption2 holds with U1 = ∅, then there exists time constants0 < εN < εN−1 < . . . < ε1 such that the system

εiddtxi = fi(x1, . . . , xi+1), ∀i ∈ N[1,N−1], (13a)

εNddtxN = fN (x1, . . . , xN ), (13b)

is globally asymptotically stable with respect to X s(C1).

The proof is given in the appendix. For N = 2, C1 = 0n1,

ψ′1 = 0 and ψ′2 = 0, the conditions of Theorem 5 areidentical to the conditions given in [9, Theorem 11.3]. Thus,our approach can be interpreted as a generalization of thisresult to multiple time scales, a broader class of comparisonfunctions ψi that reduce conservatism (see Section VI-C), andstability with respect to a compact set. Moreover, it is notalways clear how to pick the parameters εi to model a physicalsystem in the singular perturbation form (13) (see [9, p. 424]).Instead, we bound the convergence rates through Lyapunovfunctions and do not require isolating “small” time constantsεi.

IV. MULTI-CONVERTER POWER SYSTEM AND CONTROLOBJECTIVES

A. Model of an converter based power system

The three-phase power system considered in this articleconsists of N DC/AC converters, interconnected through Mresistive-inductive lines. All electrical quantities in the networkare assumed to be balanced. This allows us to work inαβ coordinates obtained by applying the well-known Clarketransformation to the three-phase variables [30].

In this work we consider an averaged model of a two-levelvoltage source converter consisting of an averaged switchingstage that modulates the DC voltage vDC,k into an AC voltagevm,k = 1

2mkvDC,k, where mk is the modulation signal.The overall setup is depicted in Figure 3. Each converter isconnected to the transmission network via an RLC filter withresistance rf,k ∈ R>0, inductance `f,k ∈ R>0, capacitancecf,k ∈ R>0, and a conductance gf,k ∈ R>0 to ground.The electrical states of the k-th converter (in αβ coordinates)are the output filter current if,k ∈ R2; the terminal voltagevk ∈ R2, and the current injected into the transmissionnetwork io,k ∈ R2. We consider the standard setup in whichthe DC/AC converter is controlled via cascaded current andvoltage control loops (see Section V).

For the purpose of this work we assume that the DCvoltage vDC,k is regulated to be constant via the source feeding

DC/AC

`f,k rf,k

cf,k gf,k

io,kif,k k

+

−

vk

Power inverter Transmission network

t,jk

r t,jk

i t,jk+

−cdc,k vm,k

Reference model(dVOC)

Voltage PI

Controller

+

”Feed-forward”

Current PI

Controller

+

”Feed-forward”

io,k

vkif,k

v

irf,k

vm,k

Reference tracking controllers

Fig. 3. Model of a DC/AC converter with reference model, reference trackingcontroller, and transmission network.

7

the converter. Under this standard assumption, the open-loopdynamics of a single converter are given by

Lfddt if = −Rf if − v + vm, (14a)

Cfddtv = −Gfv − io + if , (14b)

where Rf := diag(rf,kNck=1)⊗ I2, Lf := diag(`f,kNck=1)⊗I2, Cf := diag(cf,kNck=1)⊗I2, and Gf := diag(gf,kNck=1)⊗I2. Moreover, the vector if := (if,1, . . . if,N ) ∈ R2Nc collectsthe output filter currents and vm := (vm,1, . . . vm,N ) ∈ R2Nc

collects the modulated AC voltages.The electrical states of a transmission line are the line

currents it,jk ∈ R2. The electrical parameters of the trans-mission lines are the resistance rt,jk ∈ R>0 and inductance`t,jk ∈ R>0. Moreover, for each transmission line, we definethe impedance matrix Zjk := I2rt,jk + ω0J`t,jk and theadmittance matrix Yjk := Z−1jk .

The network topology is given by the oriented incidencematrix B ∈ −1, 0, 1Nc×Nt of the connected, undirected,weighted graph G = (N , E ,W); where N := N[1,Nc] is theset of nodes, corresponding to the converters; E ⊆ N ×N isthe set of edges (with cardinality |E| = Nt) corresponding tothe transmission lines, and the weights W are given by

‖Yjk‖ := 1√r2t,jk+ω0`2t,jk

. (15)

For the notational simplicity, we associate an index l ∈1, . . . Nt to each of the Nt transmission lines (j, k) ∈ Eand define the extended incidence matrix B := B ⊗ I2.

The transmission network dynamics, in matrix form, aregiven by

LTddt it = −RT it + BTv, (16)

where RT := diag(rlNtl=1) ⊗ I2, LT := diag(`lNtl=1) ⊗I2, and it := (it,1, . . . it,M ) ∈ R2Nt . Moreover, v :=(v1, . . . vN ) ∈ R2Nc denotes the vector of terminal voltagesand io := (io,1, . . . io,N ) ∈ R2Nc collects all currents that flowform the converter into the network and, by Kirchoff’s currentslaw, is given by

io = Bit. (17)

Finally, we define the quasi steady-state of the network. Tothis end, we change variables to a coordinate frame rotatingat the nominal frequency ω0:

v = RN (ω0t)v, i = RM (ω0t)i, io = RN (ω0t)io, (18)

where RN (θ) := IN ⊗ R(θ). In this rotating frame thetransmission network dynamics are given by

LTddt it = −ZT it + BTv, (19)

where ZT = RT + ω0JMLT and JM := IM ⊗ J . The quasi-steady-state model is obtained by considering the steady-statemap of the exponentially stable dynamics (19).

Definition 5 (Quasi-steady-state network model)The quasi-steady-state model of the transmission line cur-rents it and output currents io is given by the steady-statemap ist (v) := Z−1T BTv of (19) and iso(v) := Ynetv withYnet := BZ−1T BT. Moreover, we define the Laplacian L :=

B‖Y ‖lNtl=1BT and the extended Laplacian L := L ⊗ I2 in

αβ coordinates.

In conventional power systems the approximation it = ist (v)is typically justified due to the pronounced time-scale separa-tion between the dynamics of the transmission lines and thedynamics of synchronous machines. However, for converter-based power systems the electromagnetic transients of the lineshave a significant influence on the stability boundaries, and theapproximation is no-longer valid [8], [15]. In the remainderof this work, we make the time-scale separation argumentrigorous and explicitly quantify how large the time-scaleseparation needs to be between the converter, the network,and the converter control.

In the next section we discuss the control objectives in detailbefore presenting the control design in Section V.

B. Control objectives

Before defining the control objectives we define instanta-neous active, reactive, and apparent power as follows.

Definition 6 (Instantaneous Powers) Given the voltage vkand output current io,k at node k ∈ N[1,Nc], we definethe instantaneous active power pk := vTk io,k ∈ R, theinstantaneous reactive power qk := vTkJio,k ∈ R, and theinstantaneous apparent power sk =

√p2k + q2k. Moreover, for

all (j, k) ∈ E , we define the instantaneous active and reactivebranch powers pjk := vTk ij,k and qjk := vTkJij,k.

The control objective is to stabilize the steady-state behaviorspecified by:• Frequency synchronization: Given the nominal syn-

chronous frequency ω0, at steady state it holds that:

ddtvk = ω0Jvk, k ∈ N[1,Nc], (20a)ddt if,k = ω0Jif,k, k ∈ N[1,Nc], (20b)ddt it,k = ω0Jit,k, k ∈ N[1,Nt]. (20c)

• Terminal voltage magnitude: Given the terminal voltagemagnitude set-points v?k > 0, at steady state it holds that:

‖vk‖ = v?k, k ∈ N[1,Nc]. (21)

• Power injection: At steady state, each converter injectsthe prescribed active and reactive power i.e.,

vTk io,k = p?k, vTkJio,k = q?k, k ∈ N[1,Nc]. (22)

Equations (20a), (20b) ensure that, at steady state, the voltagesand currents in the power system evolve as perfect sinusoidalsignals, with frequency ω0. Equations (21) and (22) specify thepower injection and the voltage magnitudes at every node [8].Using the quasi-steady-state network model iso(v) = Ynetv, thelocal non-linear specification on the power injection of everyconverter given by (22) can be expressed as linear, albeit, non-local specification on voltages as follows [8]:• Phase synchronization: Given a relative angle set-pointsθ?k1 ∈ [−π2 , π2 ], at steady state it holds that:

vkv?k−R(θ?k1)

v1v?1

= 0, k ∈ N[1,Nc] \ 1. (23)

8

Where θ?jk := θ?j1 − θ?k1 for all j, k ∈ N[1,Nc] is the relativesteady-state angle between the terminal voltages of the j-thand k-th converter, and we specified phase angles relative tothe first node. Finally, the instantaneous active power, reactivepower, and terminal voltage set-points need to satisfy thepower-flow equations [6].

Condition 3 (Consistent set-points) The set-points p?k ∈ R,q?k ∈ R, v?k ∈ R>0 for active power, reactive power, andvoltage magnitude respectively, are consistent with the powerflow equations, i.e., for all (j, k) ∈ E there exist relative anglesθ?jk ∈ [−π, π] and steady-state branch powers p?jk ∈ R andq?jk ∈ R given by

p?jk:=‖Yjk‖2(v?2k rjk − v?kv?j (rjk cos(θ?jk)+ ω0`jk sin(θ?jk))

),

q?jk:=‖Yjk‖2(v?2k ω0`jk−v?kv?j (ω0`jkcos(θ?jk)− rjk sin(θ?jk))

),

such that p?k =∑

(j,k)∈E p?jk and q?k =

∑(j,k)∈E q

?jk holds for

all k ∈ N[1,N ].

V. DISPATCHABLE VIRTUAL OSCILLATOR CONTROL WITHCASCADED INNER CONTROL LOOPS

In Section IV we introduced a system model for a multi-converter power system and the corresponding control objec-tives. In this section, we will propose a control law that admitsa fully decentralized implementation and consists of twocascaded inner loops that track reference voltage. However,the standard cascaded PI voltage and current control loops areformulated in terms of local rotating reference frames for everyconverter [31]. Typically this approach severely complicatesa nonlinear stability analysis of multi-converter system. Incontrast, we propose a cascaded two-degree of freedom innercontrol structure in stationary (i.e., global non-rotating) coor-dinates that, under suitable assumptions, preserves the stabilityguarantees obtained for dispatchable virtual oscillator controlobtained under the assumption that the terminal voltage v canbe controlled directly [8], [25]. After presenting the controllaw, we re-formulate the closed-loop dynamics and controlobjectives in a global rotating frame, and provide the steady-state maps and reduced-order models required to apply theresults developed in Section III.

A. Dispatchable virtual oscillator control as reference model

We require the following assumption that is commonlymade in the stability analysis of AC power systems.

Assumption 3 (Uniform inductance-resistance ratio) Theratio between the inductance and resistance of every trans-mission line is constant, i.e., for all (j, k) ∈ E it holds that`jkrjk

= ρ ∈ R>0.

This assumption typically holds for transmission lines on thesame voltage level and simplifies the analysis while preservingthe main salient features of the system. We define the angleκ := tan−1(ω0ρ). Note that under Assumption 3 it holds thatL = R(κ)Ynet.

The multi-converter system (without internal converter dy-namics) is almost globally asymptotically stable with respect

to the specifications given in the previous section if theconverter terminal voltages vk evolve according to (see [8])

ddtvk = ω0Jvk + η(Kkvk −R(κ)io,k + ηaΦk(vk)vk), (24)

where η ∈ R>0 and ηa ∈ R>0 are control gains, and Kk andΦk are given by

Kk :=1

v?k2R(κ)

[p?k q?k−q?k p?k

], Φk(vk) := I2 −

‖vk‖2v?2k

I2. (25)

Note that the term ω0Jvk induces a harmonic oscillation ofvk at the nominal frequency ω0. Moreover, Φk(vk)vk can beinterpreted as a voltage regulator, i.e., depending on the signof the normalized quadratic voltage error Φk(vk) the voltagevector vk is scaled up or down. Finally, the term Kkvk −R(κ)io,k can be interpreted either in terms of tracking powerset-points, i.e., (22), or in terms of phase synchronization, i.e.,(23). For details see [8], [25], [26].

To this end, we use (24) as a reference model, i.e., for eachnode k ∈ N[1,Nc] the reference model is defined asddt vk := ω0Jvk + η(Kkvk −R(κ)io,k + ηaΦk(vk)vk), (26)

with Kk and Φk defined as in (25). Next, we design a trackingcontroller that controls the terminal voltage v to track thereference voltage vk.

B. Two-degree-of-freedom reference tracking controllers

We propose two local (i.e., decentralized) cascaded two-degree-of-freedom PI controllers for the filter currents andterminal voltage. An outer loop provides a reference irf,k forfilter currents if,k and ensures that terminal voltage vk istracking the voltage reference signal vk. Furthermore, an innerloop computes the control signal vm,k and guarantees that thefilter current if,k tracks the reference irf,k, provided by theouter loop.

Given the open loop dynamics (14) of the filter; first it isassumed that for the voltage dynamics, given by (14b), thefilter current if,k can be used as a control signal. Then, usingYf,k := ω0Jcf,k + gf,k, the two-degree of freedom voltage PIcontroller irf,k : R2 × R2 × R2 → R2 is defined as

ddtζvk

:= ω0Jζv,k + vk − vk, (27a)

irf,k := Yf,kvk+io,k−Kp,vk(vk−vk)−Ki,vkζvk, (27b)

where the term Yf,kvk compensates the filter admittance lossesat the synchronous steady-state behaviour (20a) with frequencyω0 and ζ

v,kis an integrator state that rotates at the nominal

frequency ω0. We stress that vk, io,k and if,k are obtainedfrom local measurements. The proportional and integral gainsare denoted with Kp,vk ∈ R>0 and Ki,vk ∈ R>0, respectively.

However, because if,k is not an input of the system, thecontroller (27) cannot be applied directly. Therefore, we useanother two-degree of freedom current PI controller

ddtζfk

:= ω0Jζfk+ if,k − irf,k (28a)

vm,k := Zf,kif,k+vk−Kp,if,k(if,k−irf,k)−Ki,if,kζfk, (28b)

that tracks the reference signal irf,k by acting on the controlinput vm,k. The term Zf,kif,k, where Zf,k := ω0J`f,k + rf,k,

9

compensates the filter impedance losses at the synchronoussteady-state behavior (20b) with frequency ω0 and ζ

fkis

an integrator state that rotates at the nominal frequency ω0.Finally, the proportional and integral gains are denoted withKp,fk ∈ R>0 and Ki,fk ∈ R>0, respectively.

C. Closed-loop dynamics in a rotating frame

In order to simplify the analysis, it is convenient to performthe change of variables to a (global) rotating reference framerotating at the nominal frequency ω0. First, we define vectorsv ∈ R2Nc , if ∈ R2Nc , v ∈ R2Nc , as well as, irf ∈ R2Nc , ζ

v∈

R2Nc , and ζf∈ R2Nc that collect the states of the individual

converters and transmission lines (i.e., v := (v1, . . . , vNc)).Next, we define the vector of voltage variables as xv := (v, ζ

v)

and filter current variables as xf := (if , ζf ) and collectall states in the vector x := (v, it, xv, xf ) ∈ Rn, withn := 10Nc + 2Nt. The change of variables to a referenceframe rotating at the nominal frequency ω0 is given by:xirf

io

= R7Nc+Nt(ω0t)

xirfio

. (29)

Next, we define the matrices

K := diag(KkNck=1), Φ(v) := diag(Φ(vk)Nck=1),

Kp,if :=diag(Kp,if,kI2Nck=1), Ki,if :=diag(Ki,if,kI2Nck=1),

Kp,v :=diag(Kp,vkI2Nck=1), Ki,v :=diag(Ki,vkI2Nck=1),

and obtain

ddt v = η(Kv −R(κ)Bit + ηaΦ(v)v)︸ ︷︷ ︸

=:fv(v,it)

, (30a)

ddt it = L−1T (−ZT it + BTv)︸ ︷︷ ︸

=:ft(it,xv)

, (30b)

ddt

[vζv

]︸︷︷︸=xv

=

[C−1f (−Yfv − Bit + if )

v − v

]︸ ︷︷ ︸

=:fv(v,it,xv,xf )

, (30c)

ddt

[ifζf

]︸︷︷︸=xf

=

[−L−1f (−Kp,f (if − irf )−Ki,fζf )

if − irf

]︸ ︷︷ ︸

=:ff (v,it,xv,xf )

, (30d)

where Yf = diag(Yf,kNck=1) and irf is given by

irf = Yfv + Bit −Kp,v(v − v)−Ki,vζf . (31)

The dynamics of the overall system x in the rotating frameare given by d

dtx = fx(x), where fx := (fv, ft, fv, ff ).Additionally, it can be easily noticed that the dynamics fxare given in the normal nested form (4).

D. Steady-state maps and reduced-order dynamics

Letting N = 4, (x1, x2, x3, x4) := (v, it, xv, xf ), and(f1, f2, f3, f4) := (fv, ft, fv, ff ) we can define the steady-state maps xsf , xsv , and ist as in Assumption 1. Moreover, wecan define the corresponding reduced-order dynamics fsv , fst ,

and fsv . The steady-state maps of the closed loop system (30)are given by

xsf (v, it, xv) := (irf (v, it, xv),02Nc), (32a)

xsv(v) := (v,02Nc), (32b)

ist (v) := Z−1t BTv. (32c)

Moreover, we obtain the following reduced-order dynamics

fsv (v, xv) :=

[−C−1f (Kp,v(v − v) +Ki,vζv)

v − v

], (33a)

fst (v, it) := L−1T (−Z−1t it + BTv), (33b)

fsv (v) := η(Kv −R(κ)Ynetv + ηaΦ(v)v

). (33c)

Note that (33a) is obtained by assuming that the currentcontrol (28) perfectly tracks its reference. Moreover, (33b) isobtained by additionally assuming that the voltage control (27)perfectly tracks its reference. Finally, (33c) is the reduced-order model of the closed-loop system obtained by assumingthat the inner controls perfectly track their reference and thatthe transmission network is in its quasi steady state (seeDefinition 5). This is the setup considered in [25].

E. Control objectives in the rotating frame

To formalize the control objectives (20a) to (21), we definethe sets

S =

v ∈ R2Nc

∣∣∣∣ vkv?k = R(θ?k1)v1v?1, ∀k ∈ N[2,Nc]

,

A =v ∈ R2Nc

∣∣ ‖vk‖ = v?k, ∀k ∈ N[1,Nc]

.

The intersection S ∩A of the sets S (synchronization) and A(voltage magnitude) is the desired steady-state behavior of thereference model (26). Moreover, if the voltage v converges toits steady-state, i.e., xv = xs(v) = (v,02Nc), it follows thatv ∈ S ∩A and the voltage v meets the control objectives (23)and (21). Next, note that the control objectives (20b), (20a)and (20c) in the stationary frame correspond to the steady-statemaps (32a), (32b) and (32c) in the rotating frame. Therefore,we can express all specifications stated in Section IV-B in therotating frame as x ∈ X s(S ∩ A).

VI. STABILITY ANALYSIS

In this section we use the results of Section III to analyzestability of the multi-converter AC power system (30) withdispatchable virtual oscillator control as a reference model andcascaded inner tracking controls presented in Section V.

A. Main result

We use the following condition to establish almost globalasymptotic stability of the multi-converter power system.

Condition 4 (Stability Condition) The set-points p?k, q?k, v?kand the steady-state angles θ?jk satisfy Condition 3. Thereexists a maximal steady-state angle θ? ∈ [0, π2 ] such that|θ?jk|≤ θ? holds for all (j, k)∈N[1,Nc]×N[1,Nc]. Furthermore,for all k ∈ N[1,Nc], the line admittances ‖Yjk‖, the stability

10

margin c ∈ R>0, the set-points p?k, q?k, v?k, and the gainsη ∈ R>0 and ηa ∈ R>0 satisfy∑j:(j,k)∈E

‖Yjk‖∣∣∣∣1− v?jv?k cos(θ?jk)

∣∣∣∣+ηa≤ 1+cos(θ?)

2

v?2min

v?2max

λ2(L)−c,

η≤ c

ρ‖Ynet‖(c+ 5‖K − L‖) ,

with v?min := mink∈N[1,Nc]v?k and v?max := maxk∈N[1,Nc]

v?k arethe smallest and largest magnitude set-points, and λ2(L) isthe second smallest eigenvalue of the graph Laplacian L.

Condition 4 is identical to the stability conditions given in[8] were the converters are modeled as controllable voltagesources. Broadly speaking, the first inequality in Condition4 ensures that the network is not to heavily loaded and thesecond inequality ensures that there is a sufficient time-scaleseparation between the dVOC reference model and the linedynamics. The reader is refered to [8, Prop. 2] for an insightfulinterpretation of Condition 4 in terms of power set points,transmission line time constants, and network admittances.

Condition 5 Consider the control gain η and stability marginc that satisfy Condition 4 and the stability margin c2 definedin Condition 1 (with α2 = 1, γ2, β12, and β21 defined inPropositions 7-8). The control gains Kp,vk ∈ R>0, Ki,vk >cf,k of the voltage PI controller satisfy

1 + maxk∈N[1,Nc]

Ki,vkKp,vk

mink∈N[1,Nc]

Ki,vkcf,k

− 1<

4ηc2

‖BR−1T BT‖(1 + 4η2).

Condition 6 Consider the control gains η, Kp,vk , and Ki,vk ,and stability margins c, c1, c2 such that Condition 5 issatisfied. Given the stability margin c3 defined in Condition1 (with α3, γ3 = 0, β32, β31, and β23 as in Propositions9-10), the control gains Kp,fk ∈ R>0, Ki,fk > `f,k of thecurrent PI controller satisfy

1 + maxk∈N[1,Nc]

Ki,fkKp,fk

mink∈N[1,Nc]

Ki,fk`f,k

− 1<

4c3β′43

β34(β′412 + β′42

2 + 4β′432) + c3β′43γ

′4

,

where β34 = maxk∈N[1,Nc]

1Ki,vk

(1 +Ki,vkKp,vk

), β′41 =

maxk∈N[1,Nc]Kp,vk , β′42 = 1

ρ cosκ + ηmaxk∈N[1,Nc]Kp,vk ,

β′43 = ‖Y ′f‖maxk∈N[1,Nc]

Kp,vkcf,k

(1 +Ki,vkKp,vk

) + ‖BL−1T BT‖ +

maxj∈N[1,Nc]Ki,vj , γ

′4 = ‖Y ′fC−1f ‖, and Y ′f := Yf −Kp,v .

Proposition 2 (Stabilizing control gains) Consider set-points p?k, q?k, v?k, and steady-state angles θ?jk that satisfyCondition 3 and suppose that there exists a control gain ηa andstability margin c such that the first inequality of Condition 4holds. Then, there exists control gains η, Kp,vk , Ki,vk , Kp,fk ,and Ki,fk such that Conditions 5-6 hold.

The proof is provided in the appendix. An interpretation ofthe stability conditions is shown in Figure 4. The parameterρ is the time constant of the transmission lines that cannot beinfluenced through control. Moreover, the synchronization gainη is the time constant of the reference model, and the control

Physics

Con

trol

t

Reference tracking

Current

Kp,if,k

Reference tracking

Voltage

Kp,vk

1ms 10ms

Network

ρ

100ms

Reference model

η

Fig. 4. Interpretation of Condition 4 - 6. The reference model needs to besufficiently slow, while the controlled filter current and voltage need to besufficiently fast.

gains Kp,if and Kp,v dominantly influence the convergencerate of the filter current and the termninal voltage closedloop dynamics. Hence, a sufficient time-scale separation mustbe enforced through the control gains. Broadly speaking, thesecond inequality of Condition 4 requires the reference modelto be slow enough compared to the line dynamics, Condition5 implies that the controlled voltages settle sufficiently fastcompared to the transmission line dynamics, and Condition6 requires the controlled filter current to converge faster thanthe terminal voltages. Next, the conditions 4 to 6 reflect thenested structure of the system. In particular, given the networkparameters, control gains and set-points for the referencemodel can be computed that satisfy Condition 4. Furthermore,given the network parameters, the control gains and set-pointsof the reference model, and the filter capacitance, stabilizingvoltage control gains can be found using Condition 5. Finally,for fixed voltage control gains, the current control gains canbe found using Condition 6.

We are now ready to state the main result of the section.

Theorem 6 (Almost global stability of X s(S ∩ A)) Con-sider set-points p?k, q?k, v?k, steady-state angles θ?jk, a stabilitymargin c1, c2 and c3, and control gains α, η, Kp,if,k , Ki,if,k ,Kp,vk and Ki,vk such that Conditions 4 - 6 hold. Then, thedynamics (30) are almost globally asymptotically stable withrespect to X s(S ∩ A), and the origin 0n is an exponentiallyunstable equilibrium.

The proof of Theorem 6 is given in Section VI-G andis obtained by applying Theorem 3 and Theorem 4. Beforepresenting the proof, we present the individual Lyapunov func-tions for the reduced-order system ˙v = fsv (v), the transmissionlines, terminal voltages, and filter currents, and derive boundsas required by Assumption 2.

B. Lyapunov function for the reduced-order system

Given the voltage set-points v?k and relative steady-stateangles θ?k1 for all k ∈ N[1,N ], we define the matrix S :=[v?1R(θ?11)T . . . v?NR(θ?1N )T]T whose null space encodes (23),and the projector PS := I2N− 1∑

v?2iSST onto the nullspace of

S. Then, the Lyapunov function candidate Vv : R2Nc → R≥0for the reduced-order dynamics d

dt v = fs(v) is given by

Vv(v) :=1

2vTPS v +

1

2ηηaα1

Nc∑k=1

(v?k

2 − ‖vk‖2v?k

)2

, (34)

11

where ηa ∈ R>0 is the voltage controller gain and, givenc ∈ R>0, the constant α1 is given by

α1 :=c

5η‖K − L‖2 . (35)

Furthermore, the constants η and c cannot be chosen ar-bitrarily. They must be chosen such that Condition 4 isalways satisfied. Moreover, we define the comparison functionψv : R2Nc → R≥0 as

ψv(v) := η (‖K − L‖‖v‖S + ηa‖Φ(v)v‖) . (36)

Proposition 3 [8, Prop. 3](Lyapunov function for thereference model) Consider Vv(v) defined in (34) and set-points p?k, q?k, v?k, steady-state angles θ?jk, α, c and η suchthat Condition 4 holds. For any η ∈ R>0, there existsχVv1 , χ

Vv2 ∈ K∞ such that

χVv1 (‖v‖S∩A) ≤ Vv(v) ≤ χVv2 (‖v‖S∩A) (37)

holds for all v ∈ R2Nc . Moreover, for all v ∈ R2Nc thederivative of Vv along the trajectories of the reduced-orderdynamics d

dt v = fsv (v) defined in (33c) satisfies

∂Vv∂v

fsv (v) ≤ −α1ψ2v(‖v‖S∩A∪0N ). (38)

C. Lyapunov function for the transmission lines

The Lyapunov function candidate Vt : R2Nt → R≥0 forthe transmission line dynamics is defined using the errorcoordinates yt := it − ist (v) as

Vt(yt) :=ρ

2yTt(BTB + LTBnBTnLT

)yt, (39)

where Bn := Bn ⊗ I2 ∈ RNt×Nt0 , and the columns ofthe matrix Bn span the nullspace of B. Because the graphG is connected it follows from the rank-nullity theoremthat Nt0 := Nt − Nc + 1. Next, consider the functionsψt(yt) = ‖Byt‖, ψ′t(yt) = ‖BnLT yt‖ that exploits theadditional degree of freedom given by ψ′i(yi) in Assumption2 to reduce conservatism in the stability bounds.

Proposition 4 [8, Prop. 4](Lyapunov function for thetransmission lines) Consider Vt(yt) defined in (39), thenthere exists χVt1 , χ

Vt2 ∈ K∞ such that χVt1 (‖yt‖) ≤ Vt(yt) ≤

χVt2 (‖yt‖) holds for all yt ∈ R2Nc . Moreover, for all v ∈ R2Nc

and it ∈ R2Nt it holds for the reduced order dynamics (33b)

∂Vt∂yt

fst (v, yt + ist ) ≤ −ψt(yt)2 − ψ′t(yt)2.

D. Lyapunov function for the terminal voltage

Using the error coordinates yv := xv − xsv , the Lyapunovfunction candidate Vv : R2Nc → R≥0 for the terminal voltageof the converters is defined as

Vv(yv) :=1

2yTv

[K−1p,vCf K−1i,v CfK−1i,v Cf Kp,vK

−1i,v +Ki,vK

−1p,v

]yv. (40)

Moreover, consider the function ψv(yv) := ‖yv‖.

Proposition 5 (Lyapunov function for the terminal volt-age) Consider Lyapunov function the Vv(yv) defined in (40).There exist χVv1 , χ

Vv2 ∈ K∞ such that

χVv1 (‖yv‖) ≤ Vv(yv) ≤ χVv2 (‖yv‖) (41)

holds for all yv ∈ R2Nc . Moreover, for all v ∈ R2Nc , yt ∈R2Nt and yv ∈ R2Nc it holds for the reduced order dynamics(33a)

∂Vv∂yv

fsv (v, yv + xsv) ≤ −α3ψv(yv)2, (42)

where α3 := 1−maxk∈N[1,Nc] cf,kKi,vk

.The proof is provided in the Appendix.

E. Lyapunov function for the converter filter current

Using the error coordinates yf := xf − xsf , the Lyapunovfunction candidate Vf : R4N → R≥0 is defined as

Vf (yf ) :=1

2yTf

[K−1p,fLf K−1i,f LfK−1i,f Lf Kp,fK

−1i,f +Ki,fK

−1p,f

]yf . (43)

Moreover, consider the function ψf (yf ) := ‖yf‖.Proposition 6 (Lyapunov function for the filter current)Consider Lyapunov function Vf (yf ) defined in (43), then thereexist χ

Vf

1 , χVf

2 ∈ K∞ such that

χVf

1 (‖yf‖)≤ Vf (yf ) ≤ χVf2 (‖yf‖) (44)

holds for all yf ∈ R2Nc . Moreover, for all v ∈ R2Nc , it ∈R2Nt and xv ∈ R2Nc and yf ∈ R2Nc it holds for the reduceddynamics (32a)

∂Vf∂yf

fsf (v, yt + ist , yv + xsv, yf + xsf ) ≤ −α4ψf (yf )2, (45)

where α4 := 1−maxk∈N[1,Nc] `f,kKi,fk

.The proof is given in the Appendix.

F. Interactions between the time scales

In addition to establishing that the functions Vv , Vt, Vv ,and Vf are Lyapunov functions when considering the reduced-order system and error dynamics of the faster dynamics inisolation, we require bounds on the interactions between thedifferent time-scales according to Assumption 2. These boundsare given in Propositions 7-12 in the Appendix.

G. Proof of the main result

To prove Theorem 6 we combine the Lyapunov functioncandidates defined in Proposition 3 to 6 as follows

ν := µ1Vv(v) + µ2Vt(yt) + µ3Vv(yv) + µ4Vf (yf ),

where µi is defined as in Theorem 3. Moreover, to show thatthe region of attraction of the origin has measure zero, wedefine the Lyapunov-like function

νδ := Vδ,v(vδ) + µ2Vt(yt) + µ3Vv(yv) + µ4Vf (yf ),

12

where Vδ,v(vδ) := vTδ (PS − 2ηηaα1I2N )vδ . Note that thefunctions fst , fsv , and fsf are linear and the correspondingLyapunov function candidates Vt(yt), Vv(yv), Vf (yf ) arequadratic. We now use Theorem 3 and Theorem 4 to proveTheorem 6.

Proof of Theorem 6: First, observe that the stability condi-tions 4 to 6 ensure that Condition 2 holds and it follows fromProposition 1 that M is positive definite.

Next, we show that the region of attraction of the equilib-rium x? = 0n has measure zero. To this end, note that for allv′δ ∈ S\02Nc it holds that Vδ,v(v′δ) < Vδ,v(02Nc). Moreover,replacing ψv(v) with ψ0

v(vδ) := η(‖K − L‖‖vδ‖S + ηa‖vδ‖)and using the bounds given in Proposition 3, Proposition 7,Proposition 8, Proposition 10, and Proposition 12, it can beverified that Assumption 2 holds for the linearized dynam-ics. The linearized reference voltage dynamics are given byddt vδ := η(Kvδ − R(κ)Bit + ηavδ). Next, we use ψ′v = 0,ψ′v = 0, ψ′f = 0 and note that ‖vδ‖ ≤ η−1a ψ0(vδ),‖yt‖ ≤ ‖yo‖ + ‖yn‖, ψv(yv) = ‖yv‖, and ψf (yf ) = ‖yf‖.Therefore, the conditions of Theorem 4 are satisfied and theregion of attraction of the origin under (30) has measure zero.

Next, we show that the conditions of Theorem 3 are satisfiedfor the dynamics (30), the Lyapunov functions Vv(v), Vt(yt),Vv(yv), Vf (yf ), C1 = S ∩ A, and U1 = 02Nc. To thisend, we note that the Propositions 3 to 12 establish thatAssumption 2 holds and appropriate K∞ functions boundingVv(v), Vt(yt), Vv(yv), and Vf (yf ) from above and belowexist. Next, note that because ψv(v) is positive definiteand radially unbounded with respect to S ∩ A ∪ 02Nc , andS ∩ A ∪ 02Nc is compact, the same steps as in [28, p.98] can be used to show that there exists a K -functionσv(‖v‖S∩A∪02Nc

) ≤ ψv(v). We note that ‖yt‖ ≤ ‖yo‖+‖yn‖,ψv(yv) = ‖yv‖, ψf (yf ) = ‖yf‖, and the region of attractionZ0n,fx of the origin has measure zero. Therefore, theconditions of Theorem 3 are satisfied and the theoremfollows.

VII. ILLUSTRATIVE EXAMPLE

As an illustrative example, we use a high-fidelity PLECSmodel of a single-phase1 converter-based microgrid withω0 = 60 Hz and a resistive load shown in Figure 5 thathas been validated using the hardware testbed described in[26]. In contrast the theoretical analysis, the switching stageof the voltage source converters in the PLECS model are notaveraged and the DC voltage is not constant. Specifically, thesimulation uses the model shown in Figure 6 which consists ofa two-level DC/AC voltage source converter, an RLC outputfilter, a DC-link capacitor, and a DC/DC boost converterused to stabilize the DC voltage. Moreover, the converterswitches are driven via pulse width modulation (with 30 kHzbase frequency). Finally, the controllers are discretized at afrequency of 15 kHz, and the electromagnetic dynamics aresimulated using a variable step ODE solver with a maximumstep size of 1.66 ns.

1The β-components of the three-phase signals are reconstructed using aHilbert transform [26].

2L VSC

`G rG

2L VSC

`G rG

2L VSC

`G rG

rL

Fig. 5. Three two-level voltage source converters (see Figure 6) connectedto a resistive load.

DC/DC

cdc

DC/AC

`f rf

cf

io

−+ vdc

Fig. 6. Two-level DC/AC voltage source converter with RLC output filter,DC-link capacitor, and a DC/DC boost converter used as power supply andto stabilize the DC-link voltage.

In order to illustrate the behavior of a closed-loop systemfor different scenarios, we simulate the sequence of events,shown in Figure 7. Figure 7 (a) shows the black start of themicrogrid using the first and second converter. In this setup,the RLC filter of the converters cannot be disconnected, i.e.,while the third converter is not operating its output filter actsas reactive power load. Next, between t = 1.5s and t = 3.5sthe third converter is pre-synchronized. First, at t = 1.5s thethird converter is activated using only the current PI controller(28) with reference irf,k = Yf,kvk (i.e., as a current sourcesupplying the filter losses). Next, at t = 2s the reference model(26) and voltage PI controller (27) are enabled, and the feed-forward term io,k in the voltage controller is enabled at t =2.5s. Finally, at time t = 3.5s the set points are changed suchthat all converters provide equal power to the load, and thefrequency, terminal voltage, active, and reactive powers allreach their set-points.

A load increase from 125 W to 375 W (without updating theset points) is shown in Figure 7 (b) illustrating power sharingand the droop like behavior of dVOC. Figure 7 (c) shows theresponse of the converters to a the set-point update. Finally,Figure 7 (d) shows the loss of the second converter. Again, thepower sharing and droop like behavior of the frequency andactive power can be observed. Moreover, the first and the thirdconverter again provide reactive power to the output filter ofthe second converter.

VIII. CONCLUSION AND OUTLOOK

In this work, we developed a Lyapunov function frameworkfor stability analysis of nested nonlinear dynamical systemson multiple time scales and obtained conditions for (almost)global asymptotic stability with respect to a set. Our approachexplicitly considers multiple time scales, convergence ratesinstead of scalar time constants, and reduces conservatism.

13

0 2 4

−5

0

5∆ω

[mH

z]

0 2 4

0

50

100

P[W

]

0 2 4

0

0.5

∆‖v

‖[V

]

0 2 4−100

0

100

t [s]

Q[v

ar]

6 6.01 6.02

−5

0

5

6 6.01 6.02

0

50

100

6 6.01 6.02

0

0.5

6 6.01 6.02−100

0

100

t [s]

9 9.1 9.2 9.3

−5

0

5

9 9.1 9.2 9.3

0

50

100

9 9.1 9.2 9.3

0

0.5

9 9.1 9.2 9.3−100

0

100

t [s]

11 11.01 11.02

−5

0

5

11 11.01 11.02

0

50

100

11 11.01 11.02

0

0.5

11 11.01 11.02−100

0

100

t [s]

(a) Black start (b) Load step (c) Set-point change (d) Loss of an inverter

Fig. 7. Simulation results showing the frequency, active power, terminal voltage and reactive power for different events. The colors correspond to the convertersshown in Figure 5.

We applied this technical contribution to a multiple-converterpower system model that includes transmission lines dynam-ics, converter dynamics, cascaded current and voltage controlloops, and dVOC as a reference model. Finally, we obtainexplicit stability conditions on the control gains that enforcethe well-known time scale separation between the differentdynamics, i.e., the dVOC reference model has to be sufficientlyslow relative to the line dynamics and the controlled convertervoltage and current have to be sufficiently fast compared tothe line dynamics. Moreover, the converter current has to befaster than the terminal voltage. Finally, we used a high-fidelitysimulation with detailed converter models (i.e., full switchingand DC side dynamics) to validate the performance of theproposed control strategy.

ACKNOWLEDGMENTS

The authors would like to thank Brian Johnson and Gab-SuSeo for providing the simulation model used in Section VII.

APPENDIX

Lemma 1 (Lyapunov decrease) For all i ∈ N[1,N ] letα(ε)i := ε−1i αi, for all i ∈ N[1,N−1] let β(ε)

i,i+1 := ε−1i βi,i+1,

for all i ∈ N[2,N ] let β(ε)i,1 :=

∑i−1k=1 ε

−1k bi,1,k and γ

(ε)i :=

ε−1i−1bi,i,i−1, and for all i ∈ N[3,N ] and all j ∈ N[2,i−1]

let β(ε)i,j :=

∑i−1k=j−1 ε

−1k bi,j,k. Consider the function νε =

µ(ε)1 V1(x1)+

∑Ni=2 µ

(ε)i Vi(yi) with µ(ε)

i =∏i−1j=1

β(ε)j,j+1

β(ε)j+1,j

for all

i ∈ N[1,N ], and µ(ε)1 = 1. Under Assumption 2, the derivative

of νε along the trajectories of the (13) is bounded by

ddtνε ≤ −

ψ1

ψ2

...ψN

T

H

ψ1

ψ2

...ψN

−ψ′1ψ′2...ψ′N

T

H ′

ψ′1ψ′2...ψ′N

,where H ′ := diag(µ(ε)

i α(ε)′i Ni=1), and H is defined recur-

sively for all i ∈ N[2,N ] starting from H1 = α(ε)1 by

Hi =

[Hi−1 β

(ε)i µ

(ε)i

? (α(ε)i − γ

(ε)i )µ

(ε)i

],

and β(ε)i := (. . . , 12β

(ε)i,i−3,

12β

(ε)i,i−2, β

(ε)i,i−1).

Proof: The time derivative of (7) is given by

ddtνε :=

µ(ε)1

ε1

∂V1∂x1

f1 +∑N−1

i=2

µ(ε)i

εi

∂Vi∂yi

fi +µ(ε)N

εN

∂VN∂yN

fN

14

−∑N

i=2

∑i−1

k=1

µ(ε)i

εk

∂Vi∂yi

∂xsi∂xk

fk

Next, we add and subtract µ(ε)i

εi∂Vi∂yi

fsi for all i ∈ N[1,N−1].Using Assumption 2 and Definition 3 one obtains

µ(ε)1

ε1

∂V1∂x1

fs1 ≤ −µ(ε)1 α

(ε)1 ψ1(x1)2 − µ(ε)

1 α(ε)′1 ψ′1(x1)2,

µ(ε)1

ε1

∂V1∂x1

(f1−fs1

)≤ µ(ε)

1 β(ε)12 ψ1(x1)ψ2(y2),

µ(ε)N

εN

∂VN∂yN

fN ≤−µ(ε)N α

(ε)N ψN (yN )2 − µ(ε)

N α(ε)′N ψ′N (yN )

2,

and for all i ∈ N[2,N−1] one obtains

µ(ε)i

εi

∂Vi∂yi

fsi ≤ −µ(ε)i α

(ε)i ψi(yi)

2 − µ(ε)i α

(ε)′i ψ′i(yi)

2,

µ(ε)i

εi

∂Vi∂yi

(fi − fsi

)≤ µ(ε)

i β(ε)i,i+1ψi(yi)ψi+1(yi+1).

Next, using Assumption 2 and Definition 3, for all i ∈ N[2,N ]

it holds that

−i−1∑k=1

µ(ε)i

εk

∂Vi∂yi

∂xsi∂xk

fk ≤i−1∑k=1

µ(ε)i

εk

k+1∑j=1

bi,j,kψiψj =

= µ(ε)i ψi

( i−1∑k=1

bi,1,kεk︸ ︷︷ ︸

=β(ε)i,1

ψ1 +

i−1∑j=2

i−1∑k=j−1

bi,j,kεk︸ ︷︷ ︸

=β(ε)i,j

ψj +bi,i,i−1εi−1︸ ︷︷ ︸=γ

(ε)i

ψi

),

where ψ1 = ψ1(x1) and ψi = ψi(yi) for all i ∈ N[2,N ]. Usingthese bounds, we can bound d

dtνε by two quadratic forms

ddtνε ≤ −

ψ1

ψ2

...ψN

T

H(µ)

ψ1

ψ2

...ψN

−ψ′1ψ′2...ψ′N

T

H ′

ψ′1ψ′2...ψ′N

,where H ′ := diag(µ(ε)

i α(ε)′i Ni=1) and H is defined recur-

sively for all i ∈ N[2,N ] starting from H(µ)1 = α

(ε)1 by

H(µ)i =

H

(µ)i−1

...

− 12µ

(ε)i β

(ε)i,i−3

− 12µ

(ε)i β

(ε)i,i−2

− 12µ

(ε)i

(µ(ε)i−1

µ(ε)i

β(ε)i−1,i + β

(ε)i,i−1

)? (α

(ε)i − γ

(ε)i )µ

(ε)i

.

The lemma follows by noting thatµ(ε)i−1

µ(ε)i

β(ε)i−1,i = β

(ε)i,i−1.

Lemma 2 (Positive definiteness) Consider a symmetric ma-trix A = ai,jN×N , with ai,i > 0 for all i ∈ N[1,N−1] andai,j = aj,i ≤ 0, for all i ∈ N[1,N ], j ∈ N[1,N ], and i 6= j. Forall k ∈ N[1,N ], let Ak = ai,jk×k denote the k-th leadingprincipal minor of A. A is positive definite if and only if forall k ∈ N[2,N ], Ak is invertible and satisfies

ak,k >

[ ak,1

...ak,k−1

]TA−1k−1

[ a1,k

...ak−1,k

]≥ 0, (47)

where equality holds if

[ ak,1

...ak,k−1

]=

[ a1,k

...ak−1,k

]= 0k−1.

Proof: Using the Schur complement, it can be verifiedthat the k-th minor of A is positive definite if and only if

ak,k −[ ak,1

...ak,k−1

]TA−1k−1

[ a1,k

...ak−1,k

]> 0

and Ak−1 is positive definite. In particular, AN−i is positivedefinite if and only if (47) holds for k = N − i and AN−1−iis positive definite. The Lemma follows by induction over i ∈N[0,N−2].Proof of Theorem 5: Consider the Lyapunov function candidate

νε = µ(ε)1 V1(x1) +

∑Ni=2 µ

(ε)i Vi(yi) with µ(ε)

i =∏i−1j=1

β(ε)j,j+1

β(ε)j+1,j

for all i ∈ N[1,N ], and µ(ε)1 = 1. Using Lemma 1 we obtain

ddtνε ≤ −

ψ1

ψ2

...ψN

T

H

ψ1

ψ2

...ψN

−ψ′1ψ′2...ψ′N

T

H ′

ψ′1ψ′2...ψ′N

.Next, using Lemma 2 it follows that H is positive definite if

γ(ε)i + µ

(ε)i β

(ε)i

TH−1i−1β

(ε)i

T<αiεi, (48)

holds for all i ∈ N[2,N ]. Because the left hand side of (48)only depends on εj for all j < i it directly follows that,for every i ∈ N[2,N ] there exists εi ∈ R>0 with εi < εi−1such that (48) holds. Using the same steps as in the proofof Theorem 3 there exists a function χ3 ∈ K such thatddtν ≤ −χ3(‖x‖X s(C1)) and χ1 ∈ K∞ and χ2 ∈ K∞ suchthat χ1(‖x‖X s(C1)) ≤ ν(x) ≤ χ2(‖x‖X s(C1)). Letting V = νε,C = X s(C1), and U = ∅, it follows from Theorem 1 that thesystem is almost globally asymptotically stable with respectto C = X s(C1). Since U = ∅, (2) holds for all x ∈ Rn, and itfollows that the dynamics are globally asymptotically stable.

Proof of Proposition 2: If the first inequality of Condition 4holds there exists η such that the second inequality of thecondition holds. Next, note that the right-hand side of the in-equality in Condition 5 is positive and independent of the gainsKp,vk and Ki,vk . Moreover, letting ξp = Ki,vk/Kp,vk ∈ R>0

and ξi = Ki,vk/cf,k ∈ R>1 for all k ∈ N[1,N ], the left-handside of the inequality in Condition 5 becomes 1+ξp

ξi−1 ∈ R>0

and can be made arbitrarily small by choosing ξp ∈ R>0 smallenough and ξi ∈ R>1 large enough. Using the same argumentsit can be verified that there always exists Kp,fk ∈ R>0 andKi,fk ∈ R>0 such that Condition 6 holds.

Proof of Proposition 5: Condition 5 implies that 1 −maxk∈N[1,Nc]

cf,kKi,vk

> 0 and it follows that Vv(yv) is positivedefinite. Thus, there exist χVv1 ∈ K∞ and χVv2 ∈ K∞ such that(41) holds. Moreover, we obtain

∂Vv∂yv

fsv (v, yv + xsv) = −yTv[I2N −K−1i,v Cf 02N×2N

02N×2N I2N

]yv,

and the proposition follows immediately.

Proof of Proposition 6: Condition 6 implies that 1 −

15

maxk∈N[1,Nc] `f,kKi,fk

> 0 and it follows that Vf (yf ) is positive

definite. Thus, there exist χVf

1 ∈ K∞ and χVf

2 ∈ K∞ suchthat (44) holds. Moreover, we obtain

∂Vf∂yf

fsf (v, yt + ist , yv + xsv, yf + xsf ) =

= −yTf[(I2N −K−1i,f Lf ) 02N×2N

02N×2N I2N

]yf

and the proposition follows immediately.

Proposition 7 [8, Prop. 5] Let β12 := ‖K − L‖−1. For allv ∈ R2Nc and yt ∈ R2Nt it holds that

∂Vv∂v

(fv − fsv

)≤ β12ψv(v)ψt(yt).

Proposition 8 [8, Prop. 6] Let β21 := ρ‖Ynet‖ and γ2 :=ηρ‖Ynet‖. Then for all v ∈ R2Nc and yt ∈ R2Nt it holds that

−∂Vt∂yt

∂xst∂v

fv ≤ β21ψt(yt)ψv(v) + γ2ψt(yt)2.

Proposition 9 Let β23 := ‖BR−1T BT‖. For all yt ∈ R2Nt andyv ∈ R4N it holds that

∂Vt∂yt

(ft − fst

)≤ β23ψt(yt)ψv(yv).

Proof: Since ft is separable and linear in both arguments,it holds that

∂Vt∂yt

(ft(yt + ist , yv + xsv)− fst (v, yt + ist )

)=∂Vt∂yt

L−1T BTyv=

= ρ(yTt BTB + yTt LTBnBTnLT

)L−1T BTyv.

The proposition immediately follows using BBn = (B ⊗I2)(Bn ⊗ I2) = BBn⊗ I2 = 02Nc×2Nt0 .

Proposition 10 Let β31 := maxk∈N[1,Nc]

cf,kKi,vk

(1 + k ∈N[1,Nc]

Ki,vkKp,vk

) and β32 := ηβ31. Then for all v ∈ R2Nc ,yt ∈ R2Nt and yv ∈ R4N it holds

−∂Vv∂yv

∂xsv∂v

fv ≤ β31ψv(v)ψv(yv) + β32ψit(yt)ψxv (yv).

Proof: Since fv is separable in its arguments and linearin it it holds that fv(v, yt + ist ) = fv(02Nc , yt) + fsv (v) andwe obtain

− ∂Vv∂yv

∂xsv∂v

fv = −yTxv[K−1p,vCf

K−1i,vCf

] (fsv (v)− ηR(κ)Byt

).

Next, note that ‖R(κ)Byt‖ = ‖Byt‖. Moreover, using ‖(K−L)v‖ = ‖(K − L)PS v‖ ≤ ‖K − L‖‖v‖S it holds that‖fsv (v)‖ ≤ ψv(v) and the proposition directly follows.

Proposition 11 Let β34 := maxk∈N[1,Nc]

1Ki,vk

(1 +Ki,vkKp,vk

).For all yv ∈ R4N and yf ∈ R2Nc it holds that

∂Vv∂yv

(fv − fsv

)≤ β34ψv(yv)ψf (yf ).

Proof: Since fv is linear and separable in all its argumentsit holds that fv(v, yt+ ist , yv+xsv, yf +xsf )−fsv (v, yv+xsv) =

fv(02Nc ,02Nt ,02Nc , yf ) = C−1f yf . Consequently,

∂Vv∂yv

fv(02Nc ,02Nt ,02Nc , yf ) = yTxv

[K−1p,vCf 02N×2N

K−1i,vCf 02N×2N

]C−1f yf ,

and the proposition follows immediately

Proposition 12 Let β41 := β4β′41, β42 := β4β

′42, β43 :=

β4β′43, γ4 := β4γ

′4 and β34 = maxk∈N[1,Nc]

1Kp,vk

(1 +Kp,vkKi,vk

), where β4 := maxk∈N[1,Nc]

`f,kKi,fk

(1 +Ki,fkKp,fk

), β′41 =

maxk∈N[1,Nc]Kp,vk , β′42 = 1

ρ cosκ + ηmaxk∈N[1,Nc]Kp,vk ,

β′43 = ‖Y ′f‖maxk∈N[1,Nc]

Kp,vkcf,k

(1 +Ki,vkKp,vk

) + ‖BL−1T BT‖ +

maxj∈N[1,Nc]Ki,vj , γ

′4 = ‖Y ′fC−1f ‖, and Y ′f := Yf − Kp,v .

Then, for all v ∈ R2Nc and yt ∈ R2Nt , yv ∈ R2Nc andyf ∈ R2Nc it holds that

−∂Vf∂yf

(∂xsf∂xv

fv +∂xsf∂xt

ft +∂xsf∂v

fv)≤ β41ψv(v)ψf (yf )+

+ β42ψt(yot )ψf (yf ) + β43ψv(yv)ψf (yf ) + γ4ψf (yf )2,

Proof: Using properties of separability in all argumentsand linearity in some arguments of fv , ft, fv and ff it holdsthat

∂Vf∂yf

(∂xsf∂xv

fv +∂xsf∂xt

ft +∂xsf∂v

fv)

=

− yTf[K−1p,fLfK−1i,f Lf

]( [(Yf −Kp,v)C

−1f 02Nc×2Nc

]yf+

− (Yf −Kp,v)C−1f

[Kp,v Ki,v

]yv+

−[Ki,v − BL−1T BT 02Nc×2Nc

]yv+

−(B(

1

ρI2M + JMω0) + ηR(κ)B

)yt +Kp,vf

sv (v)

).

The proposition follows directly from the properties of Kro-necker product, i.e., B( 1

ρI2M + ω0JM ) = ( 1ρI2N + ω0JN )B

and ‖fsv (v)‖ ≤ ψv(v) (see the proof of Proposition 11).

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