A unified framework for design and analysis of networked and quantized control systems

Transcription

1 1 A unified framework for design and anaysis of networked and quantized contro systems Dragan ešić and Danie Liberzon Abstract We generaize and unify a range of recent resuts in quantized contro systems QCS) and networked contro systems CS) iterature and provide a unified framework for controer design for contro systems with quantization and time scheduing via an emuation-ike approach. A crucia step in our proofs is finding an appropriate Lyapunov function for the quantization/time-scheduing protoco which verifies its uniform goba exponentia stabiity UGES). We construct Lyapunov functions for severa representative protocos that are commony found in the iterature, as we as some new protocos not considered previousy. Our approach is fexibe and amenabe to further extensions which are briefy discussed. I. ITRODUCTIO Contro of systems over band-imited channes is currenty attracting a ot of attention in the contro community due to a range of emerging contro appications. In such systems, certain contro oops contain channes in which ony a finite amount of data can be transferred at any transmission instant, and this eads to communication constraints that cannot be ignored in the controer design. Currenty, there are two main approaches to modeing band-imited communication channes in contro oops: i) the channe is digita and due to the finite word ength effects ony a finite number of bits can be transmitted over the channe at any transmission instant. The main issue in contro stabiization) of systems with such channes is that of quantization and we use the term quantized contro systems QCS) to denote systems exhibiting this feature see, for instance, [1], [4], [17], [6] and references cited therein); ii) the channe is a seria bus and ony a subset of sensors and/or actuators can transmit their data over the channe at each transmission instant. In this case, the sensor/actuator data is transmitted in packets and typicay the quantization effects are ignored in the anaysis see [8], [9], [11], [19], [4], [7] and references therein). The main issue in this cass of systems is time scheduing of transmissions of various signas in the system and these systems are often referred to in the iterature as networked contro systems CS). These two modeing approaches evoved separatey in the iterature with itte cross-referencing or cross-fertiization. However, a very simiar controer design approach has been D. ešić is with Department of Eectrica and Eectronic Engineering, University of Mebourne, Parkvie, 305, Victoria, Austraia. Emai: d.nesic@ee.unimeb.edu.au. Supported by the Austraian Research Counci under the Discovery Grants and Austraian Professoria Feow schemes. D. Liberzon is with Coordinated Science Laboratory, University of Iinois at Urbana-Champaign, Urbana, IL 61801, USA. Emai: iberzon@uiuc.edu. Supported by the SF ECCS grant. proposed for both QCS and CS and this approach consists of the foowing steps: i) ignore the communication constraints of the channes quantization or time scheduing) and design a controer using the cassica techniques; ii) design/choose a particuar quantizer or partition of the sensor/actuator vector, as we as an agorithm protoco) that governs the quantization or time-scheduing during the operation of the system; iii) determine a sufficienty sma upper bound on the intertransmission intervas, the so-caed maxima aowabe transmission interva MATI), that guarantees the stabiity of the system. For instance, this controer design approach was proposed in [7] for CS and it was used in [13] for QCS. We note that this approach is a natura generaization of the emuation approach to controer design for samped-data systems, see e.g. []. We mention that, whie there have been no systematic attempts to unify formuations and techniques from the CS and QCS iterature, some specific designs combining quantization and time scheduing have been proposed, for instance, in [6], [17], [5], [16]. We beieve that this fact confirms the need for deveoping a genera framework encompassing a arge cass of such protocos, which is the goa of this work. In the seque, we refer to systems that combine time scheduing with quantization as networked and quantized contro systems QCS). The main purpose of this paper is to unify the controer design approaches mentioned above for QCS and CS which naturay eads to the foowing contributions: i) We provide a unified framework for the emuation design approach which covers genera QCS and which is fexibe, genera and amenabe to further extensions and modifications; ii) Our unified controer emuation framework brings two seemingy unreated areas QCS and CS) under one umbrea and faciitates a cross-fertiization between them. For instance, we show that the notion of uniformy gobay exponentiay stabe UGES) protocos that was introduced in the CS iterature see [19]) has a natura interpretation in QCS. We beieve that this is the first time in the iterature that this connection is made. Moreover, the sma-gain theorems used in stabiity proofs of CS in [19] provide new insights for QCS considered, for instance, in [1] and [13]; iii) We show that many existing protocos for CS or QCS) as we as new protocos for QCS) are UGES in the sense of our definition, and we provide constructions of Lyapunov functions for these protocos in Section IV. On one hand, this anaysis provides a competey new view of box and zoom protocos found in the quantization iterature. On the other hand, this eads to an anaysis of genuiney new protocos for QCS that combine

2 quantization and time scheduing, such as the TOD with box protocos considered in Section IV-C. iv) An outcome of our unified approach is a range of emuation-type resuts stated in Section V which provide anaytic bounds on the MATI that guarantee stabiity of various casses of CS, QCS and QCS with various protocos. We note that these bounds are essentia in appying the emuation procedure and they provide an insight into how the controer and the protoco affect the stabiity of the overa cosed-oop system. The paper is organized as foows. Section II contains the mathematica preiminaries and background resuts that are adapted from [19]. In Section III we iustrate that modes of a range of CS and QCS that arise in the iterature, as we as QCS not considered previousy, are specia cases of a particuar cass of systems with umps to which our main resuts appy. We do not attempt to be exhaustive in this section and there may be other casses of systems that fit our framework. We show that the ump equation in this mode depends soey on the network protoco. In Section IV, which contains our main technica contributions, we show that numerous protocos found in CS, QCS and some new protocos for QCS that we introduce are Lyapunov UGES in the sense defined in Section II. In Section V we combine the resuts from preceding sections to state expicit bounds on MATI that guarantee stabiity of the cosed oop with emuated controers in various situations. Some possibe generaizations and concusions are presented in the ast section. Severa auxiiary resuts are stated and proved in the Appendix. II. PRELIMIARIES AD BACKGROUD RESULTS R and denote, respectivey, the sets of rea and natura numbers. R 0 denotes the set of non-negative rea numbers. Given t R 0 and a piecewise continuous function f : R R n, we use the notation ft + ) := im s t fs). The Eucidean norm on R n is denoted by ; sometimes we wi aso use the infinity norm x := max{ x i : 1 i n} on R n. The corresponding induced matrix norms are denoted respectivey as and. Given a piecewise continuous signa ϕ : J R n, where J is a subinterva of [t 0, ), we define its L norm as foows: ϕ := sup s J ϕs). If ϕ ) is defined on [t 0, ) and there exists K 0 such that ϕ K, then we write ϕ L. A function γ : R 0 R 0 is said to be of cass K if it is continuous, zero at zero, stricty increasing and unbounded. A function β : R 0 R 0 R 0 is said to be of cass KL if for each s 0 the function βs, ) is decreasing to zero and for each fixed t 0 the function β, t) is of cass K. A function β is said to be of cass exp- KL if there exist K, c > 0 such that βs, t) = K exp ct)s. To shorten notation, we often use x, y) := x T y T ) T. We write diag{a 1,..., A } for the bock-)diagona matrix with the indicated eements on the diagona and zeros esewhere. For the systems we consider in this paper, a monotonicay increasing sequence of times t i R 0 is given where i and t 0 = 0. Moreover, we assume that there exist ε > 0 and τ > ε such that 1 ε t i t i 1 τ i. 1) We set the stage by outining the emuation approach and recaing a resut that foows from [19]. The emuation approach was pioneered in [7] for CS without disturbances and further deveoped in [19] for systems with disturbances. The same approach was pursued in [13] and esewhere for QCS. Here we propose to appy this approach to QCS. The first step in the emuation approach is to design a controer for a given pant ignoring the network i.e. quantization and/or time scheduing). amey, given a pant one first designs a nomina controer ẋ = ft, x, u) ) u = kt, x). 3) However, in the presence of time scheduing and/or quantization, the state x is not directy avaiabe for contro. So, one instead coses the oop with the controer u = kt, ˆx) 4) where ˆx is an estimate of x generated using quantized state vaues transmitted over the network. In Section III we wi expain how ˆx is to be generated in various cases. To mode the systems arising in this way, we consider the foowing cass of systems with umps which is centra to our approach: ẋ = ft, x, z) t [t i 1, t i ] 5) ż = gt, x, z) t [t i 1, t i ] 6) zt + i ) = hi, xt i), zt i )), 7) where x R n x, z R n z, t i satisfy 1) and 0 < ε < τ. Here 5) gives the cosed-oop pant dynamics, whie z incudes the network-induced error variabes as we as some other auxiiary variabes needed to impement the quantization procedure. We wi show in Section III that severa known casses of CS and QCS, as we as their QCS generaizations, can be written in the above form. Our approach wi appy to a of these different systems in a unified manner. We adopt terminoogy from [7] and refer to τ as the maximum aowabe transmission interva MATI). We note that even for the CS case, the above mode is more genera than the one considered in [19] because the ump equation is aowed to depend expicity on x. For a precise definition of soutions to such systems, pease refer to [19]. We can assume that the system ẋ = ft, x, 0) 8) is stabe in an appropriate sense. ote that 8) is the nomina cosed-oop system without the quantization and/or time scheduing i.e. no bandwidth imitations), obtained from 5) by setting z 0. Indeed, we wi see in the next section 1 The bound ε t i+1 t i guarantees that the system does not exhibit Zeno soutions. This assumption can be reaxed by requiring that the sequence t i be stricty increasing and im i t i =.

3 3 that 8) is the same as ), 3). However, we need to expicity characterize robustness of the pant with respect to z viewed as an exogenous input. We do this by empoying the foowing standard) notions; see, e.g., [] for time-invariant systems and [5] for time-varying systems. Definition 1 The system ẋ = ft, x, z) is input-to-state stabe ISS) from z to x if there exist functions κ K and β KL such that for a t 0 0, x 0 R n x, z L and each corresponding soution x ), we have that xt) β x 0, t t 0 ) + κ z ) t t 0 0. If the system is ISS with κs) = γ s, γ 0 a inear function and β, ) an exp-kl function, then we say that the system is ISS with a inear gain and an exp-kl function. A system with state x and no input z 0) is uniformy gobay asymptoticay stabe UGAS) if for a x 0 R n x and a corresponding soutions x ), we have that xt) β x 0, t t 0 ) t t 0 0. The system is uniformy gobay exponentiay stabe UGES) if the above hods with a cass exp-kl function β. Motivated by the resuts in [19], we refer to the ump equation 7) as the network protoco. Our main resuts in Section V are presented for a arge cass of UGES protocos that are characterized in Definition stated beow. To define this cass of protocos, we introduce an auxiiary discrete-time system: z + = hi, x, z) i, 9) where h comes from 7); here z + is a shorthand for zi + 1) whie the variabes on the right-hand side are evauated at time step i. We refer to 9) as a discrete-time system induced by the protoco 7), or simpy as a protoco. ote that we treat x in the above equation as an input. Centra to this paper is the foowing cass of protocos : Definition We say that the protoco 7) is uniformy gobay exponentiay stabe UGES) with Lyapunov function W if there exist W : R nz R 0, ρ [0, 1) and a 1, a > 0 such that we have a 1 z W i, z) a z 10) W i + 1, hi, x, z)) ρw i, z), 11) for a i, x R nx and a z R nz. In other words, the protoco 7) is UGES with Lyapunov function W if W is a UGES Lyapunov function for the system 9). Sometimes we wi simpy refer to such protocos as Lyapunov UGES or ust UGES protocos when the Lyapunov function W is not being specified. Section IV contains many exampes of various time-scheduing and quantization protocos, as we as proofs of their UGES properties. Linear bounds are used This definition was first used in [19] for a smaer cass of protocos of the form z + = hi, z) whose right hand side is independent of x. However, with the definition we use here, a main resuts in [19] sti hod for the system 5) 7). in 10) instead of more standard quadratic ones in order to streamine the cacuations. This does not introduce a oss of generaity since, given a function with quadratic bounds, we can ust take its square root in order to obtain another function satisfying 10) we wi see such constructions in Section IV). The foowing resut is a coroary of main resuts in [19] and in Section V it wi be used to state a range of new emuation resuts for casses of CS, QCS and QCS that are introduced in Section III. Theorem 1 Consider the system 5) 7) with 1). Suppose that the foowing conditions hod: i) System 5) is ISS from z to x with inear gain κs) = γ s, γ 0; ii) Inequaities 10), 11) hod, i.e. the protoco 7) is Lyapunov UGES with a Lyapunov function W i, z); iii) For some L, c 0, we have that W from item ii) and g in 6) satisfy W i, z) z, gt, x, z) LW i, z) + c x 1) for amost a z R nz, x R nx and t 0. iv) MATI in 1) satisfies τ ε, τ ) where τ := 1 ) L + L n cγ/a1 ρl + cγ/a 1, 13) ε 0, τ ) is arbitrary, L and c come from 1), γ from item i), a 1 from 10), and ρ from 11). Then, the system 5) 7) is UGAS. Moreover, if the system 5) is ISS from z to x with inear gain and exp-kl function β, then the system 5) 7) is UGES. The main idea behind the proof of this resut is to use items ii), iii) and iv) to show that the z-subsystem is ISS with respect to x, and then use item i) and invoke an ISS smagain theorem see, e.g., [10]) to prove stabiity of the overa cosed-oop system; the MATI bound 13) ensures that the sma-gain condition hods. Theorem 1 is genera enough to cover many cases of interest and iustrate the unifying nature of our approach. We emphasize, however, that using resuts of [19] one can state a range of more genera or aternative stabiity resuts, such as resuts based on L p stabiity and an appropriate sma-gain theorem, which are omitted for space reasons. III. SYSTEM MODELS In this section we demonstrate that modes of various CS and QCS that were previousy considered in the iterature, as we as QCS that were not considered in the iterature, can be written in the form 5), 6), 7). This modeing framework is centra to our approach as it aows us to unify, generaize, and compare many resuts in the iterature. Indeed, we first show that modes of CS considered in [19], [7] and references cited therein are a specia case of 5), 6), 7). This fact was aready observed in [19] so we ust quicky repeat the necessary steps for competeness and to iustrate the unifying nature of our resuts. Second, we show that modes of QCS with box quantization protocos considered in [13],

4 4 [14], [6] and the references therein are a specia case of 5), 6), 7). As far as we are aware, this is the first time that QCS are modeed in this particuar form. Third, we show that one can combine genera time-scheduing and box quantization protocos that are respectivey used in CS and QCS mentioned above in order to obtain a more genera QCS mode that is sti a specia case of 5), 6), 7). Exampes of such schemes can be found in [6], [17], and [16]. Finay, we show that one can combine zoom protocos considered in [15] with the time-scheduing protocos in [19] to obtain a mode that is again a specia case of 5), 6), 7). These exampes iustrate the unifying nature of our approach and we emphasize that the same framework coud be usefu in other situations not considered here. In order to simpify the presentation, we consider static state feedback controers and pants without disturbances, and we assume that ony the state is sent over the network. Each of these simpifying assumptions can be reaxed. For instance, the CS resuts in [7] were presented for dynamic output feedback controers where both the contro input and pant output are sent over the network. These resuts were further generaized in [19] to cover genera systems with exogenous disturbances. Output quantization was addressed in [13], and the QCS framework deveoped in [1] covers both input and output quantization. Exogenous disturbances in QCS are the subect of [15]. However, since no unified treatment of CS and QCS has been attempted in the iterature, we choose to avoid these more chaenging scenarios in order to make our modes more transparent. A. CS For the purpose of comparison and to iustrate the unifying nature of our resuts, we reproduce the cass of modes for CS considered in [19], [7] and show that it is a specia case of 5), 6), 7). In this and the foowing subsections, we aways assume that 1) hods. Consider a genera noninear pant ) where x R n. It was proposed in [7] to first design a nomina controer 3) ignoring the effects of the network, so that the cosed-oop system ), 3) is stabe in an appropriate sense. The designed controer is then impemented over the network in the foowing manner. We assume that the vector x is partitioned into, 1 n different subvectors enumerated from 1 to, i.e., x = x 1, x,..., x ). We refer to the i th subvector as the i th node. At each transmission time t i, the protoco gives access to the network to one of the nodes i {1,,..., }. We consider genera noninear CS of the foowing form: ẋ = ft, x, kt, ˆx)) t [t i 1, t i ] ˆx = 0 t [t i 1, t i ] ˆxt + i ) = xt i) + hi, et i )) 14) where ˆx is the vector of most recenty transmitted pant state vaues via the network, e := ˆx x is the network-induced error, and we cosed the oop with the controer 4). We assume that ˆx is hed constant between the transmission instants i.e. we use a zero-order hod in each node). The function h is typicay such that, if the th node gets access to the network at some transmission time t i, the corresponding part of the error vector is reset to zero. However, neither of these assumptions zeroorder hod and resetting the error to zero) is necessary in the present framework, and both wi be reaxed ater in order to hande quantization effects. Rewriting the system in x, e) coordinates, we obtain the foowing CS mode: where ẋ = ft, x, e) t [t i 1, t i ] 15) ė = gt, x, e) t [t i 1, t i ] 16) et + i ) = hi, et i)), 17) ft, x, e) := ft, x, kt, x+e)); gt, x, e) := ft, x, e). 18) The system 15), 16), 17) has exacty the same form as 5), 6), 7) if we think of e in 16), 17) as z in 6), 7). The error vector e modes the effects of the network and, since we assumed that CS has nodes, it can be partitioned as e = e 1, e,..., e ). We refer to the ump equation 17) as a time-scheduing protoco. In [19], time-scheduing protocos with the maps h of the foowing form were considered: hi, e) = I Ψs))e, 19) where s = si, e) : R n {1,..., } is some scheduing function, Ψs) = diag{δ 1s I n1,..., δ s I n }, 0) is the number of nodes, δ i is the standard Kronecker deta, and I n are identity matrices of dimension n, with =1 n = n. The above mode of the protoco assumes that if the node is transmitted at time t i over the network, then e t + i ) = 0. Exampes of this cass of protocos are given in Section IV-A. B. QCS We now consider the mode for QCS described in [13] for inear pants and in [14] for noninear pants. Modes considered esewhere e.g., [6], [17], [6]) are very simiar. However, in the QCS iterature these modes have previousy not been written in the way we do it here. Let the pant be given by ), where x R n. We assume that a nomina feedback aw 3) is given. When the state is quantized, this feedback aw is not impementabe. Instead, the contro aw wi depend not on the state x but on its estimate, ˆx, which we generate as foows. In between the times t i, we et ˆx = ft, ˆx, u) 1) which is a copy of the pant dynamics. The initia condition can be arbitrary, e.g., ˆxt 0 ) = 0. At each t i, we reset ˆx to a new vaue obtained from the quantized measurement. The quantizer, q, is defined by three parameters: an integer > 1 a given constant which defines the number of quantization eves), ˆx R n the current state estimate), and ξ R 0 an auxiiary variabe which defines the size of the quantization regions). Consider a hypercubic box centered at ˆx with edges ξ and divide it into n equa smaer sub-boxes in each

5 5 dimension), numbered from 1 to n in some specific way. 3 We et qx) be the number of the sub-box that contains x provided that x is indeed contained in one of them). The new vaue of ˆx is then defined to be the center of this box. This can be described by a ump equation of the form ˆxt + i ) = gi, qxt i)), ˆxt i ), ξt i )). We aso update ξ to refect the fact that the size of the box known to contain x was divided by as the resut of the above procedure: ξt + i ) = ξt i). Unti the time t i+1, we propagate ξ according to some differentia equation ξ = g ξ t, ξ) and then the procedure is repeated. We can think of ˆx and ξ as being impemented synchronousy on both ends of the communication channe, i.e., in the encoder and the decoder, starting from some known initia vaues. Having defined ˆx, we can now aso cose the oop using the contro aw 4). It is convenient to rewrite the system dynamics in terms of the quantization-induced error e := ˆx x. Suppressing the equations for ˆx and instead writing the equations for e as before, we arrive at the cosed-oop dynamics in the form ẋ = ft, x, e) t [t i 1, t i ] ) ė = g e t, x, e) t [t i 1, t i ] 3) ξ = g ξ t, ξ) t [t i 1, t i ] 4) et + i ) = h ei, xt i ), et i ), ξt i )), 5) ξt + i ) = h ξξt i )) 6) where f is given by 18), g e t, x, e) := ft, x+e, kt, x+e)) ft, x, kt, x+e)) 7) and h e i, x, e, ξ) := gi, qx), x + e, ξ) x. This is simiar to 15), 16), 17) and becomes a specia case of 5), 6) and 7) if we define z := e, ξ). ote that the equations for e are not actuay impemented, but ony used for anaysis. The above definition of z aows us to fit QCS modes as we as QCS modes considered beow) to the form 5), 6) and 7), and consequenty enabes us to appy Definition and Theorem 1 off-the-shef. However, we wi see in Remark 6 that sometimes one might want to treat e and ξ separatey instead of combining them into a singe vector. To make sure that the quantizer is we defined, we need to assume that the initiaization of ξ and its evoution during continuous fows and umps fufis the foowing. 3 In [13] and [14], the tota number of sub-boxes is, i.e., 1/n in each dimension. Whie our present choice of notation somewhat simpifies the formuas that foow, it is a trivia matter to adapt the resuts to the notation used in [13], [14]. Simiary, we coud et ξ be an n-vector instead of a scaar and thus aow more genera rectiinear boxes, as done, e.g., in [6], [6]. Assumption 1 A bound on the initia state x0) is known and ξ and ˆx are such that et) ξt) t 0. 8) This assumption basicay means that the quantizer wi never saturate. It is easy to enforce this assumption when the pant is inear; cf. [13]. In fact, since the quantized measurements are ony taken at the times t i, it is enough to require that 8) be satisfied at these times, and then we can set g ξ = 0. Remark 1 QCS protocos considered in [1], [13] and esewhere do not assume a known initia bound on the state, and instead impement an initia zooming-out stage whose goa is to obtain such a bound in some finite time. Since the size of the quantization box must grow during this zooming-out stage, the corresponding protocos are not UGES and so our anaysis does not directy appy to them. However, the MATI bounds we provide in this paper remain accurate for these systems since, as shown in [1], [13], zooming out can be impemented with a very coarse quantizer or a very arge MATI. C. QCS with box protocos In this subsection, we combine quantization with time scheduing and show that the cosed-oop system can be written in the form 5), 6), 7). As a specia case, when there is ony one node = 1), we obtain the previous QCS mode. We note aso that a sighty different cass of quantization protocos with zoom studied in [1], [1], [15] can aso be incorporated into our framework see Section III-D beow). We again start with the pant ) and the nomina contro aw 3). The next step is to impement this controer over a band-imited channe that wi invove quantization and time scheduing. amey, we appy the contro aw 4), where ˆx is an estimate of x generated as expained next. In between the transmission times, ˆx is obtained by running a copy of the pant, given by 1), in the encoder and decoder. As in Section III-A, we partition the state vector as x = x 1, x,..., x ), where is the number of nodes and each x has dimension n. The vector ˆx is partitioned accordingy as ˆx = ˆx 1, ˆx,..., ˆx ), and the error vector e := ˆx x 9) is partitioned as e = e 1, e,..., e ). We assume that for each {1,,..., }, we are given a quantizer q with parameters, ˆx R n, ξ R 0 defined as expained in Section III-B. Here we take to be the same for a, but this is not necessary see aso footnote 3). We assume that at any transmission time t i, a quantized vaue of ony one x, {1,,..., } wi be transmitted over the network. To decide which component x we are going to transmit, we use a scheduing function s : R n R n R {1,,..., }: s = si, x, e, ξ). 30) Then, we quantize the component x sti)t i ) using the corresponding quantizer q sti), and send the quantized vaue over the network to the decoder. To simpify the notation, we write

6 6 the quantized vaue simpy as qx sti )t i )) since it is aways cear which quantizer is being used. The decoder takes the transmitted quantized vaue and resets the vaue of ˆx sti) to be the center of the box corresponding to qx sti )t i )), whie keeping a other components of ˆx unchanged. In other words, at transmission times we have ˆx t + i ) = { g i, qx sti)t i )), ˆx sti)t i ), ξ sti)t i )) if = st i ) ˆx t i ) if st i ) which eads to e t + i ) = { h i) if = st i ) e t i ) if st i ) 31) where h i) = h i, x sti )t i ), e sti )t i ), ξ sti )t i )) and h i, x s, e s, ξ s ) := g i, qx s ), x s + e s, ξ s ) x s. We find it convenient to rewrite 31) in the foowing form: et + i ) = I Ψst i)))et i ) + Ψst i ))Hi) =: h e i, xt i ), et i ), ξt i )), 3) where Ψs) was defined in 0), Hi) = Hi, x sti)t i ), e sti)t i ), ξ sti)t i )), and h 1 i, x s, e s, ξ s ) Hi, x s, e s, ξ s ) :=. h i, x s, e s, ξ s ) 33) At each transmission time we need to adust the size of the box ξ ) that corresponds to the component x that is being transmitted. In the seque, we consider update aws of the foowing form: ξt + i ) = h ξi, st i ), ξt i )). 34) In particuar, ξ sti ) is typicay divided by as before. In between the transmission times, the vector ξ = ξ 1, ξ,..., ξ ) is propagated according to some differentia equation of the form 4). To summarize, the cosed-oop dynamics can be written as ẋ = ft, x, e) t [t i 1, t i ] ė = g e t, x, e) t [t i 1, t i ] ξ = g ξ t, ξ) t [t i 1, t i ] 35) et + i ) = h ei, xt i ), et i ), ξt i )) ξt + i ) = h ξi, st i ), ξt i )), where f and g e are given by 18) and 7), and s is given in 30). ote that we are again suppressing the dynamics of ˆx, which are superfuous in view of 9). Finay, it is cear that this system can be written in the form 5), 6) and 7) by defining z := e, ξ). To make sure that the quantizer does not saturate and a the previous constructions are vaid, we impose the foowing assumption see aso Remark 1 earier). Assumption A bound on the initia state x0) is known and ξ and ˆx are such that e i t) ξ i t) i {1,..., }, t 0. 36) More detais on how to satisfy this assumption wi be given ater when we speciaize to inear pants in Section V). D. QCS with zoom protocos In this section we consider another cass of quantization protocos and combine them with time-scheduing protocos. These protocos were considered for the case of quantization ony) in [15]. That work in turn buids on the framework deveoped in [1], [1]; the modes considered in these earier papers differ ony sighty from the ones we use in [15] and here. These quantization protocos are actuay quite simiar to the box protocos that we discussed in the previous subsections. The main difference is that typicay, these zoom protocos do not strive to hande quantizers with as few quantization regions as possibe, and consequenty their impementation is somewhat more static; in particuar, the center of the quantizer does not evove according to a pant mode between the transmission times as it does for box protocos. Some other differences between the two quantizer casses are purey notationa, and these are refected here to make a better ink with the existing iterature. Assume that we are given quantizers q : R n Q R n, {1,..., }, where each Q is a finite or countabe) set of quantization points. Unike in Subsection III-C, these quantizers do not have a moving center. Instead of describing them using the integer parameter, we assume the foowing. Assumption 3 There exist stricty positive numbers M and, {1,..., } such that the foowing hods for a {1,..., } and z R n : z M = q z ) z z > M = q z ) > M. This formaism, proposed in [1], aows for quantization regions of genera shapes, and incorporates as a specia case rectiinear regions considered earier in [4], [1]. Each quantizer q aso has a zoom parameter µ > 0, whose roe is simiar to that of ξ from Subsection III-C: q x, µ ) := µ q x µ Writing µ = µ 1, µ,..., µ ), we define the overa quantizer as qx, µ) = q 1 x, µ),..., q x, µ) ). Quantization is then combined with time scheduing in the same way as in Subsection III-C. At each t i, the vaue of ˆx sti ) is reset to q sti )x sti ), µ sti )). Here we et ˆx = 0 between transmission times, which is actuay more consistent with Section III-A. The vector of zoom parameters µ is aso hed constant between transmission times, whie at transmission times we update it as foows: where µt + i ) = Ω inµt i ) ) Ω in = diag{ω 1,..., Ω }.

7 7 and Ω 0, 1) for each. Rewriting everything in x, e)- coordinates as before, we obtain cosed-oop dynamics of the form ẋ = ft, x, e) t [t i 1, t i ] ė = g e t, x, e) t [t i 1, t i ] µ = 0 t [t i 1, t i ] et + i ) = I Ψst i)))et i ) + Ψst i )) qxt i ), µt i )) xt i ) ) µt + i ) = Ω inµt i ), 37) where e is given by 9), Ψ is given by 0), and the scheduing function s has the same roe as in the previous subsection and in genera we have s = si, x, e, µ). 38) To avoid quantizer saturation, we need the foowing. Assumption 4 A bound on the initia state x0) is known and µ is such that for a {1,..., } we have x t) µ t) M t 0. If M = for a {1,..., } i.e., a quantizers have infinite range), then Assumption 4 aways hods. For finiterange quantizers, this assumption pays a roe simiar to that of Assumption, and it is not difficut to enforce it at east for inear pants see [1]). Remark 1 sti appies here. We aso note that in the presence of unknown externa disturbances, we coud not hope to satisfy Assumption 4 or Assumption ) from some time onward, and so more genera quantization protocos woud need to be empoyed; see [15] for more detais. IV. UGES PROTOCOLS In this section we consider various exampes of protocos that may arise in Subsections III-A, III-B, III-C and III-D. More importanty, we show for each of these protocos that they are UGES in the sense of Definition. It wi become apparent from our proofs that many other protocos, not considered here, can be treated in a simiar manner. In Section V, the technica proofs presented in this section are combined with Theorem 1 to obtain a range of new resuts for emuation of casses of systems considered in Section III. This genera approach is nove for any system that invoves quantization with or without time scheduing. A. CS protocos In this subsection, we consider protocos of the form e + = I Ψs))e = hi, e) s = si, e), 39) which arise in CS considered in Subsection III-A; the function Ψ is defined by 0). We typicay assume that if a node transmits at time t i, then e is reset to zero at time t + i, i.e., e t + i ) = 0. In other words, we ignore possibe quantization effects in this subsection. However, we emphasize that this assumption is not needed in genera and this wi become cear in the seque. We present two exampes of protocos from [19], [7] and quote resuts from [19] that show that these protocos are UGES. Besides serving to iustrate the unifying nature of our resuts, the RR and TOD protocos that we present next are used in the seque to generate severa genuiney new protocos that combine quantization and time scheduing. Hence, resuts presented here are usefu for our subsequent anaysis. 1) Round robin RR) protoco: The simpest timescheduing protoco is round robin in which the node is transmitted periodicay with period, where is the tota number of nodes. Such protocos are most widey studied in the iterature see, for instance, [8], [9], [11]) and the anaysis we present was given in [19]. We specify now the protoco for this situation. First, we note that the scheduing function becomes in this case: s = si) = if i = + k for some k = 0, 1,, ) The foowing resut was proved in [19]. We present the proof for competeness and because severa proofs in the seque rey on this computation. Proposition The RR protoco 39), 40) is UGES with the Lyapunov function W i, e) := k=i φk, i, e), where φ denotes the soution of the discrete-time system 39), 40) at time k starting at time i and initia condition e. In particuar, we can take a 1 = 1, a = 1, and ρ =. Proof of Proposition : Consider the Lyapunov function V i, e) = φk, i, e), 41) k=i where it is obvious that e = φ i, i, e) V i, e) for a e and a i. Moreover, note that φk, i, e) = 0 for a i and a k i +. Using this and aso the fact that hi, e) e, we can write that V i, e) e 4) for a e and a i. Furthermore, we have that V i + 1, hi, e)) = V i, e) e 1 V i, e), for a e and a i, where 4) was used in the ast step. Finay, with the definition W i, e) := V i, e) we obtain that 1, 10) and 11) hod with a 1 = 1, a =, and ρ = which competes the proof. ) Try-once-discard TOD) protoco: In this section we consider the Try-Once-Discard TOD) time-scheduing protoco proposed by Wash et a in [7], whose stabiity was anaysed in [19]. In TOD protoco, the scheduing function takes the foowing form: s = se) = min{arg max e i }. 43) i In other words, we compare the norms of errors e i and then transmit the node i that corresponds to the argest error. If severa errors are the same, we transmit the node i with a minimum index priority to transmit in this case can be chosen differenty). It was expained in [7] how TOD protoco can be

8 8 impemented in Contro Area etwork CA); TOD cannot be directy impemented in many wired and any wireess networks. The foowing resut was proved in [19]. Proposition 3 The TOD protoco 39), 43) is UGES with the Lyapunov function W e) := e. In particuar, we can take 1 a 1 = a = 1 and ρ =. Proof of Proposition 3: It is obvious that 10) hods with a 1 = a = 1 since W e) = e. Consider arbitrary e and suppose without oss of generaity that e 1 max [,] e, hence se) = 1. Then, we can write: e 1 = max [1,] e 1 e = 1 e. 44) =1 Using the properties of TOD protoco, we can write: I Ψe))e = 1 δ 1se) )e = e =1 = e e 1. = 45) Finay, using 44) and 45) we have W e + ) e e = e = W e), which competes the proof. B. QCS protocos We now prove that the quantization box protoco from Section III-B is UGES. As far as we are aware, this is the first anaysis of stabiity properties for this protoco, taken separatey from the continuous-time dynamics. However, protocos of this kind have been widey used in the iterature as encoder/decoder modes of communication channes between the pant and the controer; see, e.g., [13], [14], [6], [17], [1], [6] athough the focus in most of these works is somewhat different from the Lyapunov-based emuation design pursued here). The protoco is given by e + = h e i, x, e, ξ) ξ + = ξ. 46) In view of Assumption 1 and other constructions in Section III-B, it is easy to show that there exists a d 1 0 such that h e in 46) satisfies h e i, x, e, ξ) d 1 ξ i, x, e, ξ, 47) where we can take d 1. The proof that we present beow, athough straightforward, is important since severa other proofs in the seque use simiar constructions. = n Proposition 4 Suppose that 47) hods. Then, the box quantization protoco 46) is UGES with the Lyapunov function W e, ξ) := ε e + ξ, where ε 0, ρ) and ρ = 1 d 1. In particuar, we can take a 1 = min{1, ε}, a = 1 + ε, and ρ = εd Proof of Proposition 4: To show UGES of the protoco 46) we first note that its mode is a cascade of two systems. It is easy to see that the ξ-subsystem is UGES since with Uξ) := ξ we can write 4 : Uξ + ) = 1 ξ = 1 Uξ). Moreover, the e-subsystem is ISS, uniformy in x, when ξ is regarded as the input and, in this case, one-step dead-beat stabe when ξ = 0). Take V e) := e and use 47) to write V e + ) = h e i, x, e, ξ) d 1 ξ = d 1 Uξ). By defining W e, ξ) := εv e) + Uξ) we have min{ε, 1} e, ξ) W e, ξ) 1 + ε) e, ξ). Finay, we can write: W e +, ξ + ) = εv e + ) + Uξ + ) εd ) Uξ) εd ) W e, ξ), which competes the proof since our choice of ε guarantees that ρ = εd < 1. C. Box protocos for QCS The focus of this subsection are protocos of the form e + = I Ψs))e + Ψs)Hi, x s, e s, ξ s ) = h e i, x, e, ξ) ξ + = h ξ i, s, ξ) 48) s = si, x, e, ξ) that arise in Subsection III-C. The functions H and h ξ depend on the quantization procedure, whereas the scheduing function s depends on the time-scheduing procedure. Hence, the protoco 48) combines time scheduing and quantization. In view of 33) and Assumption, there exists a d 0 such that Hi, qx), e, ξ) d ξ i, x, e, ξ. 49) In what foows, this is a we need to know about the function H. Various combinations of scheduing with box protocos are possibe and we consider two such protocos where we combine the RR and TOD time-scheduing protocos with the box quantization protoco. We are not aware of this cass of protocos having been systematicay studied in the iterature. 1) RR protoco with quantization: In this section we combine the RR protoco considered in Subsection IV-A with the box quantization protoco considered in [13]. A somewhat simiar but more compicated) protoco was considered in [17]. The function s in 48) is given by 40) and the function Ψ is defined in 0). We know that H satisfies 49). We et h ξ in 48) be given by h ξ i, s, ξ) = I Ψsi)))ξ + 1 Ψsi))ξ, 50) 4 ote that ξ ) in 4) is non-negative by definition and in principe we coud use Uξ) = ξ. However, in order to be consistent with our definition of UGES protocos, we need to show UGES of the auxiiary discrete-time system 46) which is defined on R n R and that is why we use Uξ) := ξ.

9 9 where n s is the number of quantization eves for the corresponding quantizer q s ), Ψs) := diag{δ 1s,..., δ s } 51) note that the matrices Ψ and Ψ have different dimensions); δ i is the Kronecker symbo; denotes the number of nodes used in the time scheduing. This means that ξ si) is divided by whie other components of ξ remain unchanged. To summarize, the discrete-time system induced by the protoco is e + = I Ψsi)))e + Ψsi))Hi, x si), e si), ξ si) ) ξ + = I Ψsi)))ξ + 1 Ψsi))ξ = h ξ i, ξ). 5) ext, we show that the protoco 5) is Lyapunov UGES and in the proof we construct an appropriate Lyapunov function W for this protoco. Proposition 5 Suppose that 49) hods. Then, the protoco 5) is Lyapunov UGES. In particuar, the inequaities 10) and 11) hod with a 1 = min{1, ε}, a = ε + 1 { } ), 1 and ρ = max, εd + ρ, where ε and ρ = +1 0, 1 ρ d. Proof of Proposition 5: We construct the Lyapunov function for the protoco 5) in severa steps. First, we construct a Lyapunov function for the ξ-subsystem. Then, we construct an ISS Lyapunov function for the e-subsystem when ξ is regarded as an input. Finay, we combine the two functions since the system is a cascade of two subsystems) into a Lyapunov function of the overa system. Thus the proof foows the pattern of the proof of Proposition 4, athough the constructions and cacuations are significanty more difficut. Consider the ξ-subsystem in 5), and denote the soution of this system at time k starting from ξ at time i as φ ξ k, i, ξ). Consider the function Ũi, ξ) = φ ξ k, i, ξ), 53) k=i where it is obvious that ξ = φ ξ i, i, ξ) Ũi, ξ) for a ξ R, i. Moreover, note that for arbitrary i and a k [i +, i + + 1) 1], = 0, 1,... we have that ) 1 φ ξ k, i, ξ) ξ and hence we can write for a i and ξ that ) 1 Ũi, ξ) ξ = =0 Furthermore, we have that 1 1 ξ = 1 ξ. Ũi + 1, h ξ i, ξ)) = Ũi, ξ) ξ 1 ) 1 Ũi, ξ) = + 1 Ũi, ξ) for a ξ R and a i. Defining Ui, ξ) := we have for a i, ξ) that Ũi, ξ), ξ Ui, ξ) ã ξ 54) Ui + 1, h ξ i, ξ)) ρui, ξ), 55) where ã = 1, ρ := +1. ote that 0 < ρ < 1 because > 1 and 1. ext, we find an ISS Lyapunov function for the e-subsystem in 5) when ξ is regarded as an input. To do this we first consider the system e + = I Ψsi)))e =: hi, e). A Lyapunov function W for this system is given in Proposition. We reabe this function as V i, e). Then, for a i and e we have: e V i, e) e V i + 1, hi, 1 e)) V i, e). Moreover, V is gobay Lipschitz with the Lipschitz constant equa to see Lemma in the Appendix). We show that the same function is an ISS Lyapunov function for the e-subsystem in 5): V i + 1, h e i, x, e, ξ)) = V i + 1, hi, e))+ V i + 1, h e i, x, e, ξ)) V i + 1, hi, 1 e)) V i, e) + V i + 1, h e i, x, e, ξ)) V i + 1, hi, e)) 1 V i, e) + h e i, x, e, ξ) hi, e) 56) 1 = V i, e) + Ψsi))Hi, x si), e si), ξ si) ) 1 V i, e) + d Ui, ξ), where we used the goba Lipschitz property of V, 49), and 54). Finay, we et W i, e, ξ) := εv i, e) + Ui, ξ) where ε is stricty positive. We can write for a i, e, ξ that min{1, ε} e, ξ) W i, e, ξ) ε ) + e, ξ). 1 Thus 10) hods since ε > 0, 1 and > 1. Moreover, denoting W + := W i + 1, h e i, x, e, ξ), h ξ i, ξ)), we show that 11) hods by using 55) and 56) as foows: W + = εv i + 1, h e i, x, e, ξ)) + Ui + 1, h ξ i, ξ)) 1 ε V i, e) + εd + ρ ) Ui, ξ) { 1 max, εd } εv ) + ρ i, e) + Ui, ξ) = ρw i, e, ξ), { where ρ := max 1, εd + ρ ote that ρ < 1 since ε < 1 ρ d. } and ρ was defined earier.

10 10 ) TOD protoco with quantization: In this subsection we consider a combination of TOD protoco and quantization. We beieve that this protoco has not been considered previousy in the iterature. The protoco is given by: e + = I Ψs))e + Ψs)Hi, x s, e s, ξ s ) = h e i, x, e, ξ) ξ + = I Ψs))Mse), ξ) + 1 Ψs)ξ = h ξ e, ξ) 57) where Ψ is defined in 0), Ψs) is defined in 51), s is defined in 43), H satisfies 49), and Ms, ξ) = m 1 s, ξ),..., m s, ξ)), where m s, ξ) := min{ξ s, ξ }. In other words, the updated vaue of ξ for any {1,..., } satisfies the foowing: ξ ξ + if = s = ξ s if ξ ξ s and s 58) ξ if ξ ξ s and s. The above protoco first compares the errors e i in individua nodes and then transmits a quantized version of the measurement in the node with the argest error. ote that the ξ corresponding to this node is divided by after the transmission. Moreover, since we aready know which node has the argest error, any of the ξ i s corresponding to other nodes that are arger than ξ are reset to be equa to ξ reca Assumption ). In this manner the quantization protoco is working more efficienty. Remark Whie not immediatey obvious, the protoco 57), 58) is quite a natura way to combine TOD with a box quantization protoco. We next comment on a few aternative reated protocos. First, note that a naive update for ξ as foows: ξ + = I Ψs))ξ + 1 Ψs)ξ where s is defined in 43) may not work and, in particuar, we were unabe to prove that this modified protoco is Lyapunov UGES. On the other hand, consider the foowing modification of the box protoco for QCS given by 57): e + = I Ψs))e + Ψs)Hi, e s, ξ s ) 59) ξ + = I Ψs))ξ + 1 Ψs)ξ 60) s = min{arg max ξ i } 61) i which was considered in [6] see aso [16]). The functions Ψ, Ψ and H are the same as in 57). ote that this protoco appies the TOD time scheduing based on the vaues of ξ i instead of the vaues of e i that were used in 57). We note that this protoco is not UGES in the sense of our Definition. Indeed, consider the foowing initia condition: e = 0, e,..., e ); ξ = ξ 1, 0, 0,..., 0), 6) where ξ 1 > 0 and there exists {, 3,..., } such that e 0. Denote the e and ξ parts of the soution of the system 59) 61) initiaized at e, ξ ) as φ e k, e, ξ ) and φ ξ k, e, ξ ) respectivey. Since ξ 1 > 0 and ξ = 0, =, 3,..., we have that for a time s 1 node 1 wi aways transmit) and as a consequence we have for a k : ) φ e k, e, ξ ) = e 0; φ ξ k, e, ξ ξ1 ) =, 0, 0,..., 0. k Since φ e does not converge to zero, the system is not UGES in the sense of Definition. evertheess, stabiity of inear QCS with this protoco was proved in [6] under appropriate conditions. Hence, our definition of UGES protocos is too strong in this case and Theorem 1 cannot be appied directy. We wi discuss in the next section how to modify our resuts to dea with this kind of situation see Remark 6). We prove next that the protoco 57) is UGES. Proposition 6 Suppose that 49) hods. Then, the protoco 57) is Lyapunov UGES. In particuar, the inequaities 10) and 11) hod with a 1 = min{1, ε}, a = 1 + ε, and ρ = max max }, and α 0, 1) is arbitrary. { 1 { +α 1, },, εd + ρ where ε α +α 0, 1 ρ d ), ρ = Proof of Proposition 6: We construct a Lyapunov function for the system 57) in severa steps. First, we show that the ξ-subsystem is UGES, uniformy in e or in other words, exponentiay ISS with zero gain when e is considered as an input to the system). Let the Lyapunov function candidate be Uξ) := ξ. In the seque we often use the foowing fact: 1 ξ max ξi. 63) i Let α 0, 1) be arbitrary. Let {1,..., } be such that ξ = max i ξ i and consider two cases. CASE 1: ξ s α max i ξ i = αξ. Using 58), 63) we have: U h ξ e, ξ)) = ξ + ) + k ξ + k ) ξ s + k ξ k α ξ + ξk = ξ 1 α ) max ξi i k ξ 1 α ξ + α 1 U ξ) =: ρ 1U ξ). ote that ρ 1 < 1 for a 1. CASE : ξ s > α max i ξ i = αξ. Using 58), 63) we have: U h ξ e, ξ)) = k=1ξ + k ) = ξ + k ) + 1 k s ξk + 1 ξ s = ξk 1 1 ) ξs k s k=1 ξ 1 1 ) α max ξi i ξ 1 1 ) α 1 ξ = α + α U ξ) =: ρ U ξ). ote that ρ < 1 whenever > 1. ξ s

11 11 By defining ρ := max{ρ 1, ρ }, we can write that for a ξ, e we have: Uh ξ e, ξ)) ρuξ). ext, we ook at e by defining V e) = e. We showed in Proposition 3 that the foowing hods: 1 V I Ψe))e) V e). Hence, using 49), 57) and the triange inequaity we can write: V h e i, x, e, ξ)) = I Ψe))e + Ψe)Hi, x s, e s, ξ s ) V I Ψe))e) + Ψe)Hi, x s, e s, ξ s ) 1 V e) + duξ) We have proved that the e-subsystem is ISS with the ISS Lyapunov function V e). We combine the two Lyapunov functions U and V to get an overa Lyapunov function in the form W ξ, e) = εv e) + Uξ). The remaining steps of the proof are the same as in the proof of Proposition 5. D. Zoom protocos for QCS The focus of this subsection are protocos of the form e + = I Ψs))e + Ψs)qx, µ) x) =: h e i, x, e, µ) µ + = Ω in µ =: h µ µ) 64) s = si, x, e, µ) that arise in Subsection III-D. The protoco 64) combines time scheduing and quantization in a different manner from the protocos considered in the previous subsection. We are not aware of this cass of protocos having been previousy considered in the iterature. In the specia case when = 1 no time scheduing) these protocos were considered in [15]. We wi combine RR and TOD protocos with the zooming quantization protoco and show that both of these combined protocos are UGES. A direct consequence of these resuts is that the zooming-in protocos considered in [15] are UGES. 1) RR with zoom protocos: In this section, we consider the protoco 64) with Ψ defined by 0) and s defined by 40). In this case, we can prove: Proposition 7 Suppose that Assumptions 3 and 4 hod and max Ω < 1. Then, the protoco 64) with 40) is Lyapunov UGES. In particuar, the inequaities 10) and 11) hod with: ) ε 0, 1 max Ω, a 1 = min{1, ε}, a = 1 + ε, max ρ = max { 1, ε max } + max Ω. Proof of Proposition 7: The system is a cascade of an ISS system and a UGES system, and the proof proceeds simiary to that of Proposition 5. Let the function 41) be denoted as V. We show that Ṽ i, e) := V i, e) is the ISS Lyapunov function for the e-subsystem. ote first that we have e Ṽ i, e) e. Moreover, since Ṽ is gobay Lipschitz with Lipschitz constant equa to see Lemma in the Appendix), we obtain 1 Ṽ i + 1, e + ) = Ṽ + max µ It is aso easy to see that the function Uµ) := µ satisfies Uµ + ) max Ω µ = max Ω Uµ). We define W i, x, e) = εv i, e) + Uµ). Then we have min{1, ε} e, µ) W i, e, µ) 1 + ε ) e, µ). Finay, we have W i + 1, e +, µ + ) εṽ i + 1, e+ ) + Uµ + ) 1 ε Ṽ i, e) + ε max Uµ) + max Ω Uµ) { 1 max, ε } ε max + max Ω Ṽ i, e) + Uµ) ) = ρw i, e, µ) and ρ < 1 because ε < 1 max Ω max. ) TOD with zoom protocos: In this case, we consider 64) where Ψ is defined in 0) and s is defined in 43). We can prove the foowing: Proposition 8 Suppose that Assumptions 3 and 4 hod and max Ω < 1. Then, the protoco 64) with 43) is UGES with the Lyapunov function W e, µ) := ε e + µ. In particuar, the inequaities 10) and 11) hod with: ε 0, 1 max Ω max ρ = max { 1 ), a 1 = min{1, ε}, a = 1 + ε,, ε max } + max Ω. Proof of Proposition 8: The system is again a cascade connection of an ISS and a UGES system. Indeed, it is easy to see that V e) = e is an ISS Lyapunov function for the e-subsystem: V h e x, e, µ)) = I Ψe))e + Ψe)qx, µ) x) I Ψe))e + Ψe)qx, µ) x) 1 V e) + max ) µ. Moreover, using Uµ) := µ we can show UGES of the µ- subsystem: Uh µ µ)) = Ω in µ max Ω )Uµ). Combining these two Lyapunov functions as before gives us the Lyapunov function for the overa protoco in the form W e, µ) = εv e) + Uµ). Remark 3 ote that when = 1 one node ony), we have in particuar that Ψe) = I and therefore: e + = qx, µ) x = h e x, e, µ) µ + = Ω in µ = h µ µ),