Robustness of Distributed Averaging Control in Power Systems: Time Delays & Dynamic Communication Topology

Johannes Schiffer a Florian Dörfler b Emilia Fridman c,d

aSchool of Electronic & Electrical Engineering, University of Leeds, United Kingdom, LS2 9JT bAutomatic Control Laboratory, Swiss Federal Institute of Technology (ETH) Z¨urich, Switzerland, 8092

cTel Aviv University, Tel Aviv 69978, Israel

dITMO University, 49 Avenue Kronverkskiy, 197101 Saint Petersburg, Russia

Abstract

Distributed averaging-based integral (DAI) controllers are becoming increasingly popular in power system applications. The literature has thus far primarily focused on disturbance rejection, steady-state optimality and adaption to complex physical system models without considering uncertainties on the cyber and communication layer nor their effect on robustness and performance. In this paper, we derive sufficient delay-dependent conditions for robust stability of a secondary-frequency-DAI- controlled power system with respect to heterogeneous communication delays, link failures and packet losses. Our analysis takes into account both constant as well as fast-varying delays, and it is based on a common strictly decreasing Lyapunov- Krasovskii functional. The conditions illustrate an inherent trade-off between robustness and performance of DAI controllers. The effectiveness and tightness of our stability certificates are illustrated via a numerical example based on Kundur’s four- machine-two-area test system.

Key words: Power system stability, cyber-physical systems, time delays, distributed control

1 Introduction successfully cope with these changes, the control and operation paradigms of today’s power systems have 1.1 Motivation to be adjusted to the new conditions. The increasing complexity in terms of network dynamics and number Power systems worldwide are currently experiencing of active network elements renders centralized and in- drastic changes and challenges. One of the main driving flexible approaches inappropriate creating a clear need factors for this development is the increasing penetra- for robust and distributed solutions with plug-and-play tion of distributed and volatile renewable generation capabilities [2]. The latter approaches require a com- interfaced to the network with power electronics ac- bination of advanced control techniques with adequate arXiv:1607.07743v2 [math.OC] 6 Feb 2017 companied by a reduction in synchronous generation. communication technologies. This results in power systems being operated under more and more stressed conditions [1]. In order to Multi-agent systems (MAS) represent a promising framework to address these challenges [3]. A popular ? The project leading to this manuscript has received fund- distributed control strategy for MAS are distributed ing from the European Union’s Horizon 2020 research and averaging-based integral (DAI) algorithms, also known innovation programme under the Marie Sk lodowska-Curie as consensus filters [4,5] that rely on averaging of in- grant agreement No. 734832, the Ministry of Education tegral actions through a communication network. The and Science of Russian Federation (project 14.Z50.31.0031), distributed character of this type of protocol has the ETH Zürich funds, the SNF Assistant Professor Energy advantage that no central computation unit is needed Grant #160573 and the Israel Science Foundation (Grant and the individual agents, i.e., generation units, only No. 1128/14). have to exchange information with their neighbors [6]. Email addresses: [email protected] (Johannes Schiffer), [email protected] (Florian Dörfler), [email protected] (Emilia Fridman). One of the most relevant control applications in power

Preprint submitted to Automatica 7 February 2017 systems is frequency control which is typically divided 1.3 Contributions into three hierarchical layers: primary, secondary and tertiary control [7]. In the present paper, we focus on The present paper addresses both the cyber and the secondary control which is tasked with the regulation of physical aspects of DAI frequency control by deriv- the frequency to a nominal value in an economically effi- ing conditions for robust stability of nonlinear DAI- cient way and subject to maintaining the net area power controlled power systems under communication uncer- balance. The literature on secondary DAI frequency con- tainties. With regards to delays, we consider constant trollers addressing these tasks is reviewed in the follow- as well as fast-varying delays. The latter are a com- ing. mon phenomenon in sampled data networked control systems, due to digital control [26,27,28] and as the 1.2 Literature Review on DAI Frequency Control network access and transmission delays depend on the actual network conditions, e.g., in terms of congestion DAI algorithms have been proposed previously to ad- and channel quality [29]. In addition to delays, in prac- dress the objectives of secondary frequency control tical applications the topology of the communication in bulk power systems [8,9,10,11] and also in micro- network can be time-varying due to message losses and grids (i.e., small-footprint power systems on the low link failures [30,5,31]. This can be modeled by a switch- and medium voltage level) [12,6,13]. They have been ing communication network [30,5]. Thus, the explicit extended to achieve asymptotically optimal injections consideration of communication uncertainties leads to [14,15], and have also been adapted to increasingly a switched nonlinear power system model with (time- complex physical system models [16,17]. The closed- varying) delays the stability of which is investigated in loop DAI-controlled power system is a cyber-physical this paper. system whose stability and performance crucially relies on nearest-neighbor communication. Despite all recent More precisely, our main contributions are as follows. advances, communication-based controllers (in power First, we derive a strict Lyapunov function for a nom- systems) are subject to considerable uncertainties such inal DAI-controlled power system model without com- as message delays, message losses, and link failures munication uncertainties, which may also be of indepen- [18,2] that can severely reduce the performance – or dent interest. Second, we extend this strict Lyapunov even affect the stability – of the overall cyber-physical function to a common Lyapunov-Krasovskii functional system. Such cyber-physical phenomena and uncertain- (LKF) to provide sufficient delay-dependent conditions ties have not been considered thus far in DAI-controlled for robust stability of a DAI-controlled power system power systems. with dynamic communication topology as well as heterogeneous constant and fast-varying delays. Our stabil- For microgrids, the effect of communication delays on ity conditions can be verified without exact knowledge secondary controllers has been considered in [19] for the of the operating state and reflect a fundamental trade- case of a centralized PI controller, in [20] for a centralized off between robustness and performance of DAI control. PI controller with a Smith predictor as well as a model Third and finally, we illustrate the effectiveness of the predictive controller and in [21] for a DAI-controlled mi- derived approach on a numerical benchmark example, crogrid with fixed communication topology. In all three namely Kundur’s four-machine-two-area test system [7, papers, a small-signal (i.e., linearization-based) analysis Example 12.6]. of a model with constant delays is performed.

In [22,23] distributed control schemes for microgrids The remainder of the paper is structured as follows. In are proposed, and conditions for stability under time- Section 2 we recall some preliminaries on algebraic graph varying delays as well as a dynamic communication theory and introduce the power system model employed topology are derived. However, both approaches are for the analysis. The DAI control is motivated and in- based on the pinning-based controllers requiring a troduced in Section 3, where we also derive a suitable master-slave architecture. Compared to the DAI con- error system. A strict Lyapunov function for the closed- troller in the present paper, this introduces an additional loop DAI-controlled power system is derived in Section 4. uncertainty as the leader may fail (see also Remark 1 Based on this Lyapunov function, we then construct a in [23]). In addition, the analysis in [22,23] is restricted common LKF for DAI-controlled power systems with to the distributed control scheme on the cyber layer constant and fast-varying delays in Section 5. A numer- and neglects the physical dynamics. Moreover, the con- ical example is provided in Section 6. The paper is control in [22] is limited to power sharing strategies, and cluded with a brief summary and outlook on future work secondary frequency regulation is not considered. in Section 7.

The delay robustness of alternative distributed sec- Notation. We deﬁne the sets R≥0 := x R x 0 , { ∈ | ≥ } ondary control strategies (based on primal-dual decom- R>0 := x R x > 0 and R<0 := x R x < 0 . position approaches) has been investigated for constant For a set{ , ∈ denotes| } its cardinality and{ ∈ [ ]k| denotes} delays and a linearized power system model in [24,25]. the set ofV all|V| subsets of that contain k elements.V Let V

2 n 2 x = col(xi) R denote a vector with entries xi for i i, k [ ] there exists an ordered sequence of nodes ∈ ∈ { } ∈ N 1, . . . , n , 0n the zero vector, 1n the vector with all en- from i to k such that any pair of consecutive nodes in the { } tries equal to one, In the n n identity matrix, 0n×n the sequence is connected by a power line represented by an × n n matrix with all entries equal to zero and diag(ai) an admittance, i.e., the electrical network is connected. × n n diagonal matrix with diagonal entries ai R. Like- × ∈ wise, A = blkdiag(Ai) denotes a block-diagonal matrix n×n We consider a heterogeneous generation pool consisting with block-diagonal matrix entries Ai. For A R , ∈ of rotational synchronous generators (SGs) and inverter- A > 0 means that A is symmetric positive definite. interfaced units. The former are the standard equipment The elements below the diagonal of a symmetric main conventional power networks and mostly used to con- trix are denoted by . We denote by W [ h, 0], h R>0, ∗ − ∈ nect fossil-fueled generation to the network. Compared the Banach space of absolutely continuous functions φ : to this, most renewable and storage units are connected n ˙ n [ h, 0] R , h R>0, with φ L2( h, 0) and with via inverters, i.e., power electronics equipment, to the − → ∈ ∈ − 0.5 R 0 2 grid. We assume that all inverter-interfaced units are the norm φ W = maxθ∈ a,b φ(θ) + φ˙ dθ . k k [ ] | | −h fitted with droop control and power measurement fil- Also, f denotes the gradient of a function f : Rn R. ∇ → ters, see [35,36]. This implies that their dynamics admit a mathematically equivalent representation to SGs 2 Preliminaries [36,37]. Hence, the dynamics of the generation unit at the i-th node, i , considered in this paper is given by ∈ N 2.1 Algebraic Graph Theory θ˙i = ωi, d d 2 (1) An undirected graph of order n is a tuple = ( , ), Miω˙ i = Di(ωi ω ) + Pi GiiVi + ui Pi, where = 1, . . . , n is the set of nodes andG N[ E]2, − − − − = eN, . . . ,{ e , is the} set of undirected edges,E⊆ i.e.,N the 1 m where D is the damping or (inverse) droop coef- elementsE { of are} subsets of that contain two elements. i R>0 ficient, ωd∈ is the nominal frequency, P d is the In the contextE of the presentN work, each node in the graph R>0 i R active power∈ setpoint and G V 2,G , represents∈ represents a generation unit. The adjacency matrix ii i ii R≥0 the (constant active power) load at the∈i-th node 1 . Fur- |N |×|N | has entries a = a = 1 if an edge betweenA ∈i R ik ki thermore, u : is a control input and M and k exists and a = 0 otherwise. The degree of a node i R≥0 R i R>0 ik is the (virtual) inertia→ coefficient, which in case∈ of an i is defined as d = P|N | a . The Laplacian matrix i k=1 ik inverter-interfaced unit is given by Mi = τP Di, where of an undirected graph is given by = , where i |N |×|N | L D − A τPi R>0 is the low-pass filter time constant of the = diag(di) R . An ordered sequence of nodes ∈ D ∈ power measurement filter, see [39,36]. Following stan- such that any pair of consecutive nodes in the sequence dard practice in power systems, all parameters are as- is connected by an edge is called a path. A graph sumed to be given in per unit [7]. The active power flow is called connected if for all pairs i, k [ ]2 thereG n Pi : R R is given by exists a path from i to k. The Laplacian{ } matrix ∈ N of an → L undirected graph is positive semidefinite with a simple X zero eigenvalue if and only if the graph is connected. Pi = Bik ViVk sin(θik), k∈Ni | | The corresponding right eigenvector to this simple zero eigenvalue is 1n, i.e., 1n = 0n [32]. We refer the reader to [33,32] for furtherL information on graph theory. where we have introduced the short-hand θik = θi θk. For a detailed modeling of the system components,− the reader is referred to [40,36]. 2.2 Power Network Model To derive a compact model representation of the power We consider a Kron-reduced [34,7] power system model system, it is convenient to introduce the matrices composed of n 1 nodes. The set of network nodes is ≥ denoted by = 1, . . . , n . Following standard prac- n×n n×n N { } D = diag(Di) R ,M = diag(Mi) R , tice, we make the following assumptions: the line admit- ∈ ∈ tances are purely inductive and the voltage amplitudes Vi R>0 at all nodes i are constant [7]. To each ∈ ∈ N 1 node i , we associate a phase angle θi : R≥0 R For constant voltage amplitudes, any constant power load ∈ N → can equivalently be represented by a constant impedance and denote its time derivative by ωi = θ˙i, which rep- load, i.e., to any constant P R>0 and constant V R>0, resents the electrical frequency at the i-th node. Un- ∈ ∈2 there exists a constant G R>0, such that P = GV . On der the made assumptions, two nodes i and k are con- larger time scales, the loads∈ may not be constant but fol- nected via a nonzero susceptance Bik R<0. If there is ∈ low regular (e.g., daily) fluctuations that can be accurately no line between i and k, then Bik = 0. We denote by represented by internal models in the controllers [8,11,38]. i = k Bik = 0 the set of neighboring nodes For the time-scales of interest to us, a constant load model ofN the {i-th∈ node. N | Furthermore,6 } we assume that for all suffices and, accordingly, our controllers contain integrators.

3 and the vectors Building upon [12,15,10], we consider the following secondary frequency control scheme for the system (2) n n θ = col(θi) R , ω = col(ωi) R , net ∈d 2 n ∈ n P = col(Pi GiiVi ) R , u = col(ui) R . u = p, − ∈ ∈ − d (5) p˙ = K(ω 1nω ) KA Ap. Also, we introduce the potential function U : Rn R, − − L → X We refer to the control (5) as distributed averaging in- U(θ) = Bik ViVk cos(θik). n×n − {i,k}∈[N ]2 | | tegral (DAI) control law in the sequel. Here, K R is a diagonal gain matrix with positive diagonal∈ entries, Then, the dynamics (1), i , can be compactly writ- = > Rn×n is the Laplacian matrix of the undi- ten as ∀ ∈ N rectedL L and∈ connected communication graph over which the individual generation units can communicate with θ˙ = ω, each other, and A is a positive definite diagonal matrix (2) d net with the element Aii > 0 being a coefficient accounting Mω˙ = D(ω 1nω ) + P + u θU(θ). − − − ∇ for the cost of secondary control at node i. It has been shown in [12,15,10] that the control law (5) is a suit- Observe that due to symmetry of the power flows P , i able secondary frequency control scheme for the system s d > (2), i.e., it can achieve ω = ω despite unknown (con- 1 θU(θ) = 0. (3) n ∇ stant) loads; see [8,11] for variations of the control (5) for dynamic load models. In addition to secondary frequency control, the control law (5) can also ensure that 3 Nominal DAI-Controlled Power System the power injections of all generation units satisfy the Model with Fixed Communication Topology identical marginal cost requirement in steady-state, i.e., and no Delays ∗ ∗ Aiiu = Akku for all i , k , (6) In this section, the employed secondary control scheme i k ∈ N ∈ N is motivated and introduced. Subsequently, we derive the considered resulting nominal closed-loop system, i.e., where Aii and Akk are the respective diagonal entries of without delays and switched topology. For this model, we the matrix A. construct a suitable error system and a strict Lyapunov function in Section 4, both of which are instrumental to The nominal closed-loop system resulting from combin- establish the robust stability results under communica- ing (2) with (5) is given by tion uncertainties in Section 5. θ˙ = ω, 3.1 Secondary Frequency Control: Objectives and Dis- Mω˙ = D(ω ωd1 ) + P net U(θ) p, (7) tributed Averaging Integral (DAI) Control n θ − − d − ∇ − p˙ = K(ω 1nω ) KA Ap. − − L The whole power system is designed to work at, or at least very close to, the nominal network frequency ωd [7]. However, by inspection of a synchronized solution (i.e., a To formalize our main objective, it is convenient to in- ∗ s solution with constant uniform frequencies ω = ω 1n, troduce the notion below. constant control input u∗ and constant phase angle dif- ∗ ferences θik) of the system (2), we have that Definition 1 (Synchronized motion) The system (7) admits a synchronized motion if it has a solution for > > s d net ∗ ∗ 0 = 1n Mω˙ = 1n ( D1n(ω ω )+P +u θU(θ )), all t 0 of the form − − −∇ ≥ which with (3) implies that ∗ ∗ ∗ ∗ s ∗ n θ (t) = θ + ω t, ω = ω 1n, p R , 0 ∈ 1> net ∗ s d n (P + u ) where ωs and θ∗ n such that ω = ω + > . (4) R 0 R 1n D1n ∈ ∈ ∗ ∗ π 2 net θ0,i θ0,k < i , k i. Note that the loads GiiVi contained in P are usually | − | 2 ∀ ∈ N ∀ ∈ N > net s unknown. Hence, in general 1n P = 0 and, thus, ω = ωd, unless the additional control signal6 u∗ accounts for6 the power imbalance. Control schemes which yield such u and, consequently, ensure ωs = ωd are termed secondary Note that [10, Lemma 4.2] implies that the system (7) frequency controllers. has at most one synchronized motion col(θ∗, ω∗, p∗)

4 (modulo 2π). That motion also satisﬁes the identical As a consequence, the coordinate ζ can be expressed by marginal cost requirement (6) and is characterized by means of θ and the parameter ζ¯0. Hence,

> net 1 1 ∗ − 1n P 1 − −1 11 p = K 2 W p¯ + K 2 A 1nζ p = αA n, α = > −1 (8) 1n A 1n √µ 1 1 − 1 −1 > −1 d s d ∗ ∗ = K 2 W p¯ + K 2 A 1n(1 A (θ 1nω t) + ζ¯ ) . and, hence, ω = ω , see (4) with u = p . µ n − 0 − (14) 3.2 A Useful Coordinate Transformation Accordingly, we define the matrix We introduce a coordinate transformation that is fun- > 1 1 (n−1)×(n−1) damental to establish our robust stability results. Recall ¯ = W K 2 A AK 2 W R , > L L ∈ that 1n = 0. This results in an invariant subspace of the p-variables,L which makes the construction of a strict that corresponds to the communication Laplacian ma- Lyapunov function for the system (7) difficult. There- trix after scaling and projection. Note that is con- L L fore, we seek to eliminate this invariant subspace through nected by assumption, and thus ¯ is positive definite. L an appropriate coordinate transformation. To this end and inspired by [30,31,41], we introduce the variables In the reduced coordinates, the dynamics (7) become p¯ R(n−1) and ζ R via the transformation ∈ ∈ θ˙ = ω, " # d net p¯ > − 1 h 1 i Mω˙ = D(ω ω 1n)+P θU(θ) 2 √1 − 2 −1 = K p, = W µ K A 1n , (9) − − −∇ ζ W W 1 1 − 1 −1 > −1 d K 2 W p¯ + K 2 A 1n(1 A (θ 1nω t) + ζ¯ ) , − µ n − 0 1 > − 1 − 2 −1 2 n×(n−1) ˙ 2 where µ = K A 1n 2 and W R p¯ = W K p˙ has orthonormalk columns thatk are all orthogonal∈ to > 1 d = W K 2 (ω 1 ω ) ¯p,¯ − 1 −1 > − 1 −1 n K 2 A 1n, i.e., W K 2 A 1n = 0(n−1). Hence, the − − L (15) transformation matrix Rn×n is orthogonal, i.e., W ∈ where we have used (14) and the fact that 1n = 0n. 1 1 1 L > > − 2 −1 > − 2 −1 Note that the transformation (9) removes the invariant =WW + K A 1n1n K A =In. (10) WW µ subspace span(1n) of a synchronized motion of (7) in the θ-variables and shifts it to the invariant subspace − 1 −1 Accordingly,p ¯ is a projection of p on the subspace or- span(K 2 A ), where it is factored out in the orthogo- − 1 −1 − 1 thogonal to K 2 A 1n scaled by K 2 . Furthermore, nal reduced-orderp ¯-variables.

1 ζ = 1>K−1A−1p (11) 3.3 Error States √µ n For the subsequent analysis, we make the following stan- can be interpreted as the scaled average secondary con- dard assumption [37,10]. trol injections of the network. Indeed, from (7) together with the fact that 1> = 0, we have that n L Assumption 2 (Existence of synchronized motion) The closed-loop system (15) possesses a synchronized 1 > −1 −1 1 > −1 d motion. ζ˙ = 1 K A p˙ = 1 A (ω 1nω ), √µ n √µ n − With initial time t0 = 0, as well as which by integrating with respect to time and recalling ∗ > − 1 ∗ (11) yields p¯ = W K 2 p ,

1 > −1 d −1 see (9), we introduce the error coordinates ζ = 1 A (θ θ0 1nω t + K p0) √µ n − − (12) d ∗ 1 > −1 d ω˜ = ω 1nω , p˜ =p ¯ p¯ , = 1 A (θ 1nω t) + ζ¯0, − − √µ n − Z t ˜ ∗ ∗ θ = θ θ = θ0 θ0 + ω˜(s)ds, − − 0 where −1 > − − ˜ ζ¯ = √µ 1 A 1(K 1p θ ). (13) x = col θ, ω,˜ p˜ . 0 n 0 − 0

5 Then the dynamics (15) become where > 0 is a positive real and sufficiently small parameter. The Lyapunov function (17) is based on the ˙ > θ˜ =˜ω, classic kinetic and potential energy terms ω Mω and ∗ ∗ U(θ) [42] written in error coordinates, a Bregman con- Mω˜˙ = Dω˜ ˜U(θ˜ + θ ) + ˜U(θ ) − − ∇θ ∇θ struction to center the Lyapunov function as in [11,8], a 1 1 −1 > −1 (16) Chetaev-type cross term between the (incremental) po- K 2 W p˜ A 1n1 A θ,˜ − − µ n tential and kinetic energies [43], and a quadratic term for > 1 the secondary control inputs (also in error coordinates). p˜˙ =W K 2 ω˜ ¯p˜ − L We have the following result. ∗ and x = 0(3n−1) is an equilibrium point of (16). Fur- thermore, in error coordinates the potential function Proposition 3 (Stability of the nominal system) n Consider the system (16) with Assumption 2. The U : R R reads → function V in (17) is a strict Lyapunov function for X ∗ ˜ ∗ ˜ ∗ the system (16). Furthermore, x = 0(3n−1) is locally U(θ + θ ) = Bik ViVk cos(θik + θik) − {i,k}∈[N ]2 | | asymptotically stable. with ˜ ∗ ˜ ∗ ˜ ∗ ∂U(θ + θ ) ∗ ∂U(θ + θ ) PROOF. We first show that the function V in (17) is θ˜U(θ+θ ) = , θ˜U(θ ) = ˜ . ∇ ∂θ˜ ∇ ∂θ˜ θ=0n locally positive definite. It is easily verified that

V ∗ = 0(3n−1). Recall from Section 3.1 that any synchronized motion of ∇ x s d ∗ the system (7) satisfies ω = ω and that p is uniquely Moreover, the Hessian of V evaluated at x∗ is given by given by (8). Thus, for a fixed value of ζ¯0 it follows from (13) together with (14) that asymptotic stability of x∗  2 1 −1 > −1  implies convergence of the solutions col(θ, ω, p) of the U x∗ + A 1n1 A E12 0 ∇θ˜ | µ n 2 original system (7) with initial conditions that satisfy 2   V ∗ =  M 0  , ∇ x  ∗  −1 > − − ζ¯ = √µ 1 A 1(K 1p θ ) I(n−1) 0 n 0 − 0 ∗ ∗ ∗ ∗ ∗ to a synchronized motion col(θ , ω , p ), the initial an- where gles θ∗ of which satisfy 2 2 0 E = AM U x∗ + U x∗ MA 12 ∇θ˜ | ∇θ˜ | ¯ −1 > −1 −1 ∗ ∗ and 0 denotes a zero matrix of appropriate dimension. ζ0 = √µ 1n A (K p θ0). 2 ∗ − Under the standing assumptions, θ˜U x is a Laplacian ∇ | 2 ¯ ∗ As this holds true for any value of ζ0 and the dynamics matrix of an undirected connected graph. Hence, θ˜U x ∗ 2 ∇ | (16) are independent of ζ¯ , asymptotic stability of x ∗ 1 0 is positive semidefinite with ker( θ˜U x ) = span( n). implies convergence of all solutions of the original system −1 > −1 ∇ | Furthermore, A 1n1n A is positive semidefinite and (7) to a synchronized motion. −1 > −1 2 ker(A 1n1 A ) ker( U x∗ ) = 0n. In addition, M n ∩ ∇θ˜ | is a diagonal matrix with positive diagonal entries. Thus, 2 4 Stability Analysis of the Nominal Closed- all block-diagonal entries of V x∗ are positive definite. Loop System with a Strict Lyapunov Function This implies that there is a sufficiently∇ small ∗ > 0 such ∗ 2 that for all ]0, ] we have that V ∗ > 0. We ∈ ∗ ∇ x To pave the path for the analysis in Section 5, we start by choose such . Therefore, x is a strict minimum of V. investigating stability of an equilibrium x∗ of the nominal (without delays and with constant communication The time derivative of V along solutions of (16) is given topology) closed-loop system (16). by

The proposition below provides a stability proof for an  1  equilibrium of the system (16) by employing the follow- A 0.5AD 0.5AK 2 W >  ing strict Lyapunov function candidate V˙ = ξ  D 0.5E 0 ξ, (18) −  ∗ − 22 n×(n−1)  ¯ ∗ > ∗ 1 > V =U(θ˜ + θ ) θ˜ ˜U(θ ) + ω˜ Mω˜ ∗ ∗ L − ∇θ 2 1 > − 1 > where we have used the property that + (1 A 1θ˜)2 + p˜ p˜ (17) 2µ n 2 > ∗ ∗ ˜ ∗ ∗ > + ω˜ AM( ˜U(θ˜ + θ ) ˜U(θ )), ( ˜U(θ + θ ) ˜U(θ )) 1n = 0 ∇θ − ∇θ ∇θ − ∇θ

6 and defined the shorthand on the existence, continuity, or boundedness ofτ ˙ik(t) [27,28]. The resulting control error eik is then computed E = (AM 2U(θ˜ + θ∗) + 2U(θ˜ + θ∗)AM), (19) as 22 ∇θ˜ ∇θ˜ as well as eik(t) = Aiipi(t τik(t)) Akkpk(t τik(t)), − − − ˜ ∗ ∗ ξ = col θ˜U(θ + θ ) θ˜U(θ ), ω,˜ p˜ . i.e., the protocol is only executed after the message from ∇ − ∇ node k arrives at node i. Note that we allow for asym- Note that the matrix E only depends on the cosines metric delays, i.e., τik(t) = τki(t). Furthermore, as stan- 22 dard in sampled-data networked6 control systems [27,28], of the angles θ.˜ Hence, there is a positive real constant ˜ n the delay τik(t) may be piecewise-continuous in t. Also, γ, such that E22 γIn for all θ R . Furthermore, A, ¯ ≤ ∈ we assume that the switches in topology don’t modify D and are positive definite matrices. Thus, Lemma the delays between two connected nodes. 11 in AppendixL A implies that there is a sufficiently small ∗∗ (0, ∗] such that for any (0, ∗∗] the ∈ ∈ In order to write the resulting closed-loop system com- matrix on the right-hand side of (18) is positive definite n×n pactly, we introduce the matrices T`,m R , m = ˙ 0 ∈ and, thus, V < 0 for all ξ = (3n−1). Consequently, 1,..., 2 ` , where ` is the number of edges of the undi- by choosing such , V is a strict6 Lyapunov function for |E | |E | ∗ rected graph with index ` = σ(t) of the dynamic the system (16) and, by Lyapunov’s theorem [44], x is communication network of the DAI∈ M control (5), m de- asymptotically stable, completing the proof. notes the information flow from node i to k over the edge i, k with delay τm = τik and all elements of T`,m are zero{ } besides the entries 5 Stability of the Closed-Loop System with Dy- namic Communication Topology and Delays t`,m,ii = 1, t`,m,ik = 1. (20) −

This section is dedicated to the analysis of cyber-physical As we allow for τik = τki, we require 2 ` matrices T`,m aspects in the form of communication uncertainties on in order to distinguish6 between the delayed|E | information the performance of the DAI-controlled power system ﬂow from k to i (with delay τik) and that from k to i (with model (16). delay τki). Note that by summing over all T`,m we recover the full Laplacian matrix of the communication network 5.1 Modeling and Problem Formulation corresponding to the topology index ` = σ(t) , i.e., ∈ M

2|E`| Recall from Section 1 that the most relevant practical X communication uncertainties in the context of DAI con- ` = T`,m. L trol are message delays, message losses and link failures. m=1 We follow the approaches in [30,5,31,27,28] to model these phenomena. With the above considerations, the closed-loop system Thus, link failures and packet losses are modeled by a dy- (7) becomes the switched nonlinear delay-differential namic communication network with switched communi- system cation topology σ(t), where σ : R≥0 is a switching G → M signal and = 1, 2, . . . , ν , ν R>0, is an index set. θ˙ = ω, M { } ∈ The finite set of all possible network topologies amongst d net Mω˙ = D(ω ω 1n) + P θU(θ) p, = n nodes is denoted by Γ = 1, 2,... ν . The − −  − ∇ −  Laplacian|N | matrix corresponding to{G the indexG `G=}σ(t) 2|E`| > n×n ∈ d X is denoted by ` = = ( `) . We employ p˙ = K(ω 1nω ) KA  T`,mAp(t τm) , ` R − − − theM following standardL L assumptionL G on∈ [30,5]. m=1 Gσ(t) (21) Assumption 4 (Uniformly connected communication topologies) The communication topology σ(t) where τm(t) [0, hm], m = 1,..., 2 ` are fast-varying G ∈ |E | is undirected and connected for all t R≥0. delays and T corresponds to the m-th delayed (di- ∈ `,m rected) channel of the `-th communication topology cor- With regards to communication delays, we suppose responding to the topology index ` = σ(t) of the that a message sent by generation unit k to the dynamic communication network of the DAI∈ control M (5). generation unit i over the communication∈ N chan- ∈ N nel (i.e., edge) i, k is affected by a fast-varying delay We are interested in the following problem. { } τik : R≥0 [0, h], h R≥0, where the qualifier ”fast- varying” means→ that there∈ are no restrictions imposed

7 Problem 5 (Conditions for robust stability) Con- 5.2 Stability of the Closed-Loop System with Dynamic ¯ sider the system (21). Given hm R≥0, m = 1,..., 2 , Communication Topology and Delays ∈ E ¯ = max`∈M ` , derive conditions under which the so- E |E | lutions of the system (21) converge asymptotically to a We analyze stability of equilibria of the system (22) synchronized motion. with dynamic communication topology and time delays. The result is formulated for fast-varying piecewise- As in the nominal scenario, we make the assumption continuous bounded delays τm(t) [0, hm] with hm ¯ ∈ ∈ below. R≥0, m = 1,..., 2 . For the case of constant delays, stability can be verifiedE via the same conditions. Assumption 6 (Existence of synchronized motion) The closed-loop system (21) possesses a synchronized To streamline our main result, we recall from Section 3 motion. that the matrix A can be used to achieve the objective of identical marginal costs. As a consequence, the choice of From (14) it follows that for any m = 1,..., 2 ¯, A influences the corresponding equilibria of the system E (22), see (8). But the equilibria are independent of the integral control gain matrix K. Hence, we may chose 1 1 − 1 −1 p(t τm) =K 2 W p¯(t τm) + K 2 A 1nζ(t τm) , K as a free tuning parameter to ensure stability of the − − µ − √ closed-loop system (22). The proof of the proposition below is given in Appendix B. which together with the fact that by construction, see 1 0 (20), T`,m n = n implies that Proposition 7 (Robust stability) Consider the system with Assumptions 4 and 6. Fix and as well 1 (22) A D 2 ¯ T`,mAp(t τm) = T`,mAK W p¯(t τm). as some hm R≥0, m = 1,..., 2 . Select K such that − − ∈ E for all `,m and ¯` defined in (23), respectively (24), ` = T L (n−1)×(n−1) Hence, with Assumption 6 and by following the steps in 1,..., , there exist matrices Sm > 0 R , Section 3, we represent the system (21) in reduced-order |M| (n−1)×(n−1) ∈(n−1)×(n−1) Rm > 0 R and S12,m R sat- error coordinates as, cf., (16), isfying ∈ ∈

˜˙   θ =˜ω, Ψ11 Ψ12 0 Ψ14 ˙ ˜ ∗ ∗   Mω˜ = Dω˜ θ˜U(θ + θ ) + θ˜U(θ )  Ψ Ψ Ψ  − − ∇ ∇  22 23 24  1 1 Ψ = ∗ > 0, (25) 2 −1 > −1 ˜   K W p˜ A 1n1 A θ, (22)  R + SS12 + S  − − µ n  ∗ ∗  2|E| R + S + Ψ¯ 44 > 1 X ∗ ∗ ∗ p˜˙ =W K 2 ω˜ `,mp˜(t τm), − T − m=1 where where we defined, analogous to ¯, R = blockdiag(Rm),S = blockdiag(Sm), L 2E¯ X > 1 1 (n−1)×(n−1) S = blockdiag(S ), R¯ = h2R , `,m = W K 2 AT`,mAK 2 W R , (23) 12 12,m j j T ∈ j=1 1 > 1 which satisfies Ψ = D K 2 W RW¯ K 2 , Ψ = ¯` ¯`R¯ ¯`, 11 − 22 L − L L ¯ > ¯ Ψ44 = blockdiag `,mR `,m , 2|E`| −T T X 1 1 1 h i > 2 2 ¯ `,m = W K A ÀK W =: `. (24) Ψ12 = K 2 W R¯ ¯`, Ψ23 = S ...S ¯ , T L L L − 1 2E m=1 h i h i ¯ ¯ ¯ ¯ Ψ14 = Ψ14,1 ... Ψ14,2E¯ , Ψ24 = Ψ24,1 ... Ψ24,2E¯ , Note that, by assumption, the graph associated to ` is 1 L ¯ 2 ¯ connected and thus ¯` is positive definite for any ` . Ψ14,m = K W R `,m, L ∈ M − T Ψ¯ 24,m = ¯`R¯ `,m Sm 0.5 `,m, By construction, the system (22) has an equilibrium L T − − T (26) ∗ ˜ point z = col(θ, ω,˜ p˜) = 0(3n−1). Furthermore, by the same arguments as in Section 3.1, asymptotic stability 0 denotes a zero matrix of appropriate dimensions and of z∗ implies convergence of the solutions of the system (21) to a synchronized motion. As a consequence of this " # RS12 fact, we provide a solution to Problem 5 by studying 0. (27) stability of z∗. R ≥ ∗

8 ∗ Then the equilibrium z = 0 is locally uniformly Hence, by continuity, for any given ˆ` > 0 and R > 0 (3n−1) L asymptotically stable for all fast-varying delays τm(t) we can find a small enough κ, such that Φ22 > 0. In ∈ [0, hm]. addition, D > 0 and (27) is satisfied with strict inequality by assumption. Therefore, the matrix Ψ in (28) is The stability certificate (25), (27) is based on a LKF de- a parameter-dependent composite matrix of the form rived from the Lyapunov function (17), and it is fairly stated in Lemma 11. Consequently, Lemma 11 implies that for given h, A, D and `, ` = 1, . . . , ν, there is tight (see Section 6). Note that the evaluation of the cer- L tificate (25), (27) and the corresponding controller tun- always a sufficiently small gain κ such that there exist matrices ,R and S satisfying conditions (25), (27). ing inherently is centralized and requires complete sys- S 12 tem information. However, the stability certificate (25), (27) can be also made wieldy for a practical plug-and- play control implementation by trading off controller The claim in Corollary 8 is in a very similar spirit to the performance for robust stability. The following corollary result obtained in [30,5] for the standard linear consen- makes this idea precise for the case of uniform delays, i.e., sus protocol with delays. In essence, Corollary 8 shows ¯ τm(t) = τ(t) [0, h], hm = h, h R≥0, m = 1,..., 2 . ∈ ∈ E that there is a trade-off between delay-robustness, i.e., feasibility of conditions (25), (27) and a high gain matrix Corollary 8 (Performance-robustness-trade-off) K for the DAI controller (5). Aside from displaying an Consider the system (22) with Assumptions 4 and 6. inherent performance-robustness trade-off, Corollary 8 Fix A and D as well as some h R≥0. Suppose that allows us to certify robust stability (for any switched ∈ τm(t) = τ(t) [0, h] and that (27) is satisfied with strict communication topology) based merely on sufficiently ∈ n×n inequality. Set K = κ , where κ R≥0 and R>0 small control gains and without evaluating a linear ma- is a diagonal matrix withK positive diagonal∈ entries.K ∈ Then trix inequality in a centralized fashion. there is κ > 0 sufficiently small, such that the equilib- ∗ rium z = 0(3n−1) is locally uniformly asymptotically We remark that it is also possible to obtain a completely stable for all fast-varying delays τ(t) [0, h]. decentralized (though in general more conservative) tun- ∈ ing criterion by further bounding the matrices in (28), respectively (25). PROOF. With K = κ , we have that K Remark 9 Conditions (25), (27) are linear matrix in- 2E¯ equalities that can be efficiently solved via standard nu- X `,m = κ ˆ`,m, ¯` = κ ˆ` = κ ˆ`,m, merical tools, e.g., [45]. Furthermore, instead of checking T T L L T m=1 (25), (27) for all ¯` it also suffices to do so for their convex hull [46]. In addition,L checking the convex hull gives a with `,m and ¯` defined in (23), respectively (24). Fur- robustness criterion for all unknown topology configura- thermore,T we writeL the free parameter matrix S as S = tions within the chosen convex hull. Compared to the re- κ , > 0. Consequently, the matrices in (26) also be- lated results on stability of delayed port-Hamiltonian sys- S S come κ-dependent. Then, for τm(t) = τ(t) h the ma- tems with application to microgrids with delays [47,48], trix Ψ in (25) can be written as ≤ the conditions (25), (27) are independent of the specific ∗ equilibrium point z .     D 0 0 0 Φ Φ 0 Φ 11 12 14 ˆ−1 ˆ     Remark 10 Note that ` 1/λmax( `)I(n−1), where  0 0 0   Φ22 Φ24 ˆ L ≥ L Ψ =  ∗  + κ  ∗ −S  , (28) λmax( `) denotes the maximum eigenvalue of the matrix     L  RS12   ˆ Hence, shows that the admissible  ∗ ∗   ∗ ∗ S S  `, ` = 1, . . . , ν. (30) largestL gain κ in order for Φ (defined in (29)) to be pos- R Φ44 22 ∗ ∗ ∗ ∗ ∗ ∗ itive definite is constrained by the maximum eigenvalue of all matrices ˆ This fact is confirmed by the numer- where `. ical experimentsL in Section 6. Similar observations have 2 1 > 1 2 1 1 also been made numerically for consensus systems [31]. Φ = h 2 WRW 2 , Φ = h κ 2 2 WR ˆ`, 11 − K K 12 K L 2 2 Φ = ˆ` h κ ˆ`R ˆ`, Φ = h κ ¯`R ¯`, 22 L − L L 44 S − L L 2 1 1 2 6 Numerical Example Φ14 = h κ 2 2 WR ˆ`, Φ24 = h κ ˆ`R ˆ` 0.5 ˆ`. − K L L L − S − (29)L We illustrate the effectiveness of the proposed approach Note that Φ can be written as on Kundur’s four-machine-two-area test system [7, Ex- 22 ample 12.6]. The considered test system is shown in Fig. 1. The employed parameters are as given in [7, Ex- −1 2 ˆ` ˆ h κR ˆ`. (30) L L` − L ample 12.6] with the only difference that we set the

9 Table 1 damping constants to Di = 1/(0.05 2π 60) pu (with re- · · Values of phase angles at considered equilibria z∗,1 to z∗,3. spect to the machine loading SSG = [700, 700, 719, 700] ∗,1 MVA). The system base power is SB = 900 MVA, the θ [rad] [0.224, 0.117, 0.076, 0.189] π/2 − − · base voltage is V = 230 kV and the base frequency is ∗,2 B θ [rad] [0.3, 0.4, 0.5, 0.4] 3π/8 ωB = 1 rad/s. − − · θ∗,3 [rad] [0.3, 0.4, 0.5, 0.4] π/2 − − · For the tests, we consider the four different communi- Electrical network cation network topologies shown in Fig. 1. Topology 1 G 7 8 9 is the nominal topology, representing a ring graph. In G1 1 5 6 110 km 110 km 10 11 3 G3 25 km 10 km 10 km 25 km topology 2, respectively 3 and 4, one of the links ∼ ∼ is broken.G Note that eachG of the configurationsG is connected and, hence, the dynamic communication network L7 L9 given by the topologies Γ = 1, 2, 3, 4 satisfies As- 2 4 sumption 4. Furthermore, we{G set AG =G diag(G }S /S ) and SG B G2 G4 K = κ , where = 0.05A−1 and κ is a free tuning ∼ ∼ parameter.K K Topologies of dynamic communication network

We consider an exemplary scenario with fast-varying de- G1 G3 G1 G3 G1 G3 G1 G3 lays τm [0, hm]s, hm = 2s, m = 1,..., 2 ¯. With the ∈ E G1 G2 G3 G4 chosen parameters, we check conditions (25), (27). The G2 G4 G2 G4 G2 G4 G2 G4 numerical implementation is done in Yalmip [45]. In order to identify the maximum admissible gain κ, we select an initial value for κ of κinit = 2.0 and iteratively Fig. 1. Kundur’s two-area-four-machine test system taken decrease the value of κ until conditions (25), (27) are from [7, Example 12.6] and below the four different topologies of the switched communication network. satisfied. The obtained feasible gain is κfeas = 1.544. For the nominal operating point z∗,1 the maximum fea- The simulation results shown in Fig. 2 confirm the fact sible gain obtained for fast-varying delays with hm = that the systems’ trajectories converge to a synchro- 2s via simulation experiments is κfeas,sim = 6.330. For nized motion if κfeas = 1.544. Following standard prac- higher values of κ, the system exhibits limit cycling tice in sampled-data networked control systems [27,28], behavior. The value of κfeas,sim = 6.330 is about 4.1 the time-varying delays are implemented as piecewise- times larger than the value of κfeas = 1.544 obtained continuous signals. For the present simulations, we have via Proposition 7. In the case of z∗,2 and with the same used the rate transition and variable time delay blocks delays the maximum feasible gain identified via simu- in Matlab/Simulink with a sampling time Ts = 2ms to lation is κfeas,sim = 2.856, which is about 1.85 times generate the fast-varying delays. During the simulation, larger as the value of κfeas = 1.544 obtained from con- the communication topology switches randomly every ditions (25), (27). The maximum feasible gain is further 0.5s. Variations in the switching interval did not lead to reduced in the last operating scenario with equilibrium ∗,3 meaningful changes in the system’s behavior. This shows z , where we obtain κfeas,sim = 1.698. This value is that the DAI control is very robust to dynamic changes only 1.1 times larger than the value of κfeas = 1.544 ob- in the communication topology. Furthermore, our exper- tained from conditions (25), (27). We observe a very sim- iments confirm the observation in Remark 10 and [31] ilar behavior in the case of constant delays. This shows that for a given upper bound on the delay the feasibil- that - for the investigated scenarios - our (equilibrium- ity of conditions (25), (27) is highly dependent on the independent) conditions are fairly conservative at equi- largest eigenvalue of the Laplacian matrices. libria corresponding to less stressed operating conditions, while they are almost exact under highly stressed We verify the conservativeness of the sufficient condi- operating conditions. tions (25), (27) in simulation. Recall that the conditions are equilibrium-independent. Hence, if feasible they guarantee (local) stability of any equilibrium point satisfying the standard requirement of the equilibrium angle differences being contained in an arc of length π/2, 7 Conclusions cf. Assumption 6. Therefore, we consider three different operating conditions: at first, the nominal equilibrium We have considered the problem of robust stability of point z∗,1 reported in [7, Example 12.6] and subse- a DAI-controlled power system with respect to cyber- quently two operating points z∗,2 and z∗,3 under more physical uncertainties in the form of constant and fast- stressed conditions, i.e., with some angle differences be- varying communication delays as well as link failures and ing closer to the ends of the π/2-arc. The angles for all packet losses. The phenomena of link failures and packet three scenarios are given in Table 1. losses lead to a time-varying communication topology

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12 As B11 is positive deﬁnite by assumption, B11 > 0 for we have that any > 0. Furthermore, by applying the Schur comple- 2|E | ment to C we obtain ` > > X V˙ = ω˜(t) Dω˜(t) p˜(t) `,mp˜(t τm) − − T − m=1 > −1 A22 + B22 B12B11 B12 , 2|E`| − > > X = ω˜(t) Dω˜(t) p˜(t) `,mp˜(t) − − T m=1 which, as A22 > 0 by assumption, is positive deﬁnite 2|E`| Z t (B.3) for small enough . The latter together with B11 > 0 > X +p ˜(t) `,m p˜˙(s)ds implies that C > 0 for small enough , completing the T t−τ proof. m=1 m > > = ω˜(t) Dω˜(t) p˜(t) ¯`p˜(t) − − L 2|E`| Z t > X +p ˜(t) `,m p˜˙(s)ds, T m=1 t−τm

B Proof of Proposition 7 where we have used (24) to obtain the last equality and recall that, for any ` , ¯` > 0. Furthermore, ∈ M L We give the proof of Proposition 7. The proof is inspired > > ˙ ,m =p ˜(t) Smp˜(t) p˜(t hm) Smp˜(t hm). (B.4) by [31,27,47,48] and established by constructing a com- V1 − − − mon LKF for the system (22). Consider the LKF with m = 1,..., 2 ¯, By using E Z t−τm Z t p˜(t hm) =p ˜(t) p˙(s)ds p˙(s)ds, 2E¯ 2E¯ − − − X X t−hm t−τm =V + ,m + ,m, V V1 V2 m=1 m=1 and deﬁning Z t > (B.1) ,m = p˜(s) Smp˜(s)ds, t−τ t V1 Z m Z t−hm η¯ = col p˜(t), p˜˙(s)ds, p˜˙(s)ds , Z t m > t−hm t−τm 2,m = hm (hm + s t)p˜˙(s) Rmp˜˙(s)ds. V t−h − m (B.4) is equivalent to

(n−1)×(n−1)   where V is defined in (17) and Sm R as well 0 Sm Sm (n−1)×(n−1) ∈ − − as Rm R are positive definite matrices. ˙ >   ∈ 1,m = η¯m  Sm Sm  η¯m. (B.5) V − ∗  Sm Recall that the proof of Proposition 3 implies that with ∗ ∗ Assumption 2 there is a sufficiently small > 0, such Also, by differentiating ,m we obtain that V is locally positive definite with respect to z∗. Ac- V2 cordingly, for this value of > 0 also is positive definite t ∗ V Z with respect to z , since 1,m is positive definite and so ˙ ˙ > ˙ 2 ˙ > ˙ V 2,m = hm p˜(s) Rmp˜(s)ds + hmp˜(t) Rmp˜(t). is 2,m, which can be seen in the following reformulation V − t−hm V (B.6) By following [27,28], we reformulate the first term on the Z 0 Z t > right-hand side of (B.6) as 2,m = hm p˜˙(s) Rmp˜˙(s) dsdφ. V −h t φ m + Z t > hm p˜˙(s) Rmp˜˙(s)ds = − t−h Next, we inspect the time derivative of along solutions m Z t−τm Z t V > > of the system (22). To this end, we at first assume = 0. hm p˜˙(s) Rmp˜˙(s)ds hm p˜˙(s) Rmp˜˙(s)ds For that case by recalling (18) together with the fact that − t−hm − t−τm (B.7)

Z t Condition (27) is feasible by assumption. Thus, applying p˜(t τm) =p ˜(t) p˜˙(s)ds, (B.2) − − t−τm Jensen’s inequality together with Lemma 1 in [27], see

13 ¯ also [50], to both right-hand side terms in (B.7) yields where η R(n+(1+4E)(n−1)), the following estimate for the first term on the right- ∈ hand side of (B.6) η = col (˜ω(t), p˜(t), η1, η2) , ! Z t−τ1 Z t−τ2E¯ Z t η = col p˜˙(s)ds, . . . , p˜˙(s)ds , > 1 hm p˜˙(s) Rmp˜˙(s)ds t−h1 t−h2E¯ (B.12) − t−hm Z t Z t ! " t−τ #> " #" t−τ # R m ˙ R m ˙ η2 = col p˜˙(s)ds, . . . , p˜˙(s)ds t−h p˜(s)ds Rm S12,m t−h p˜(s)ds m m t−τ1 t−τ ¯ R t R t . 2E ≤ − p˜˙(s)ds Rm p˜˙(s)ds t−τm ∗ t−τm (B.8) and Ψ is defined in (25). Therefore, if conditions (25), (27) are satisfied, then ˙ 0. V ≤ In order to rewrite the second term on the right-hand 2 ˙ > ˙ ˙ ˙ side of (B.6), i.e., hmp˜(t) Rmp˜(t), at first we replace p˜ As of now, = 0 and is not strict, i.e., not negative by its explicit vector field in (22). This yields definite in all state variables.V Yet, as the system (22) is non-autonomous, we need a strict common LKF in order > to establish the asymptotic stability claim. By making " #" 1 1 1 #" # 2 > 2 2 > ω˜ K WRmW K K WRm ω˜ use of Proposition 3, this can be achieved without major p˜˙(t) Rmp˜˙(t) = − , > 1 difficulties as follows. From (18) in the proof of Propo- ψ RmW K 2 Rm ψ − sition 3 together with (B.11), we have that for = 0, (B.9) 6 P2|E| where ψ = j=1 `,jp˜(t τj). Next, we apply the iden- " # ! T − 0n×n 0n×(n+(1+4E¯)(n−1)) tity (B.2) to obtain ˙ ξ¯> + Ξ ξ,¯ V ≤ − Ψ ∗ 2|E| 2|E`| ! (B.13) X X Z t ψ = `,jp˜(t τj) = `,j p˜(t) p˜˙(s)ds . T − T − j=1 j=1 t−τj where

∗ ∗ ξ¯=col ˜U(θ˜(t) + θ ) ˜U(θ ), ω˜(t), p˜(t), η , η , Furthermore, we recall the fact (24) which implies that ∇θ −∇θ 1 2

 1  2|E`| A 0.5AD 0.5AK 2 W 0 0 X `,jp˜(t) = ¯`p˜(t).   T L  0.5E22 0 0 0 j=1 ∗ −    Ξ =  0 0 0 , ∗ ∗  Then, by introducing the short-hand vector  0 0 ∗ ∗ ∗  0   ∗ ∗ ∗ ∗ 2|E`| Z t X E22 is defined in (19) and 0 denotes a zero matrix of ap- ηˆ = col ω,˜ p˜(t), `,j p˜˙(s)ds , T propriate dimensions. Recall that if conditions (25), (27) j=1 t−τj are satisfied, then Ψ > 0 for all ¯`, ` . Thus, following the proof of Proposition 3, LemmaL ∈ 11 M in Appendix A (B.9) is equivalent to implies that there is a sufficiently small > 0 such that the matrix sum on the right-hand side of (B.13) is positive definite and, at the same time, is locally positive p˜˙(t)>R p˜˙(t) m definite. Consequently, there existsV a real γ > 0 such  1 > 1 1 1  K 2 WRmW K 2 K 2 WRm ¯` K 2 WRm that ˙ (t) γ z(t) , z(t) = col(θ˜(t), ω˜(t), p˜(t)). Local − L V ≤ − k k ∗ >   uniform asymptotic stability of z follows by invoking =η ˆ  ¯`Rm ¯` ¯`Rm  η.ˆ  ∗ L L −L  the standard Lyapunov-Krasovskii theorem [27,28] and Rm arguments from [51] for systems with piecewise-conti- ∗ ∗ (B.10) nuous delays. By direct inspection, if (25), (27) are satisfied for some hm R≥0, then they are also satisfied ∈ for any τm(t) [0, hm), which in particular includes the Finally, by collecting the terms (B.3), (B.5), (B.8) and ∈ case of constant delays, i.e.,τ ˙m = 0. This completes the (B.10), ˙ can be upper-bounded by proof. V

˙ η>Ψη, (B.11) V ≤ −