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Control over a communication channel with random noise and delays夡 Stefano Battilotti ∗ Dipartimento di Informatica e Sistemistica “Antonio Ruberti”, Italy Received 31 May 2006; received in revised form 18 May 2007; accepted 29 May 2007 Available online 27 September 2007

Abstract We study the problem of controlling a general class of nonlinear systems through a memoryless channel with constant delay. The remote controller receives the delayed measurements from the controlled plant and transmits back to the plant a control law, designed according to a certainty equivalence strategy. The closed-loop system trajectories are convergent to zero in probability and square integrable, despite the presence of uncertainties and square integrable noise. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Remote control; Measurement noise; Time delays; Stabilization in probability

1. Introduction and discussion of the main result The problem of studying the properties of systems connected by communication channels has received increasing attention in the very recent years. In this paper we study the problem of controlling a plant when there is a communication channel connecting the sensor to the controller and the controller to the actuators. This kind of problems arise when plant and controller are far apart or located in different places. A study from a general point of view has been done in Tatikonda, Sahai, and Mitter (2004) and Hristu-Varsakelis and Zhang (2005) for linear systems and no communication channel between the controller and the actuators (see also the literature therein included). A LQG problem has been examined in a sequential rate distortion (SRD) framework, including the design of encoder/decoder. The work of Tatikonda et al. (2004) is focused on stochastic linear systems and no delay is considered. In this paper we do not consider the presence of an encoder/decoder and assume that the communication channel is a memoryless vector channel and the plant is nonlinear. In Tatikonda et al. (2004) the encoder/decoder and controller design is 夡 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Dragan Nesic under the direction of Editor Hassan Khalil. ∗ Tel.: +39 06 44585356; fax: +39 06 44585367. E-mail address: [email protected]

0005-1098/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2007.05.025

accomplished through stochastic kernels by minimizing the SRD of the decoding error. Our objective is to achieve global stabilization in probability by estimating the state of the plant and feeding it back to the controlled plant. As far as we know, ours is the ﬁrst global result on the stabilization of nonlinear stochastic systems through a memoryless channel (with no data rate limitations). Sufﬁcient conditions for mean square stability of impulsive nonlinear stochastic system, driven by a renewal process, have been given in Hespanha and Teel (2006). A data rate-limited local stability result has been presented in Nair and Evans (2000) and Nair, Evans, Mareels, and Moran (2004). While Tatikonda et al. (2004), Nair and Evans (2000) and Nair et al. (2004) are focused on minimal data rate issues, we seek for stabilizing controllers and optimal or constrained performance is not considered. Moreover, we consider also the presence of a transmission delay through the channel. Stabilization of a chain of integrators with input delay has been studied in Mazenc, Mondié, and Niculescu (2003); in this paper we pursue a generalization of this result to nonlinear stochastic systems with both input and measurement delays. We present the different parts of our problem. (a) Plant: We consider a probability space (, A, P) and the following class of stochastic systems: dxj = [xj +1 + j s (x, xn+1 )]dt + j (x, xn+1 )d, j = 1, . . . , n

(1)

S. Battilotti / Automatica 44 (2008) 348 – 360

with states x = (x1 , . . . , xn )T , xi ∈ R, control u := xn+1 ∈ R and an l-dimensional standard Wiener process adapted to a given ﬁltration {Ft } ⊂ A of -algebras. We assume that (H1) the functions j s , j : Rn ×R → R are locally Lipschitz continuous and such that for j = 1, . . . , n and h = j + 1, . . . , n + 1

(2)

(3)

for all x ∈ R , xn+1 , xh ∈ R, for some smooth functions aj sh : Rn−j +2 → R (here j s |xh = xh denotes the function j s (x, xn+1 ) evaluated for xh = xh ). n

The functions j s , j model the “uncertainty” of the plant. In particular, if j s (x, xn+1 ) and j (x, xn+1 ) are smooth, vanish at the origin and dxdj j s (x, xn+1 ) = dxdj j (x, xn+1 ) = 0 for all x ∈ Rn , then (2) and the last two equalities of (3) are satisﬁed (upper triangular systems or feedforward systems). On the other hand, aj s,j +1 (0, 0, . . . , 0) = 0 in (3) requires that for xl = 0, l = j + 2, . . . , n + 1, the incremental ratio of j s and xj +1 can be made as small as j calculated between xj +1 and desired in norm by making small xj +1 and xj +1 . For smooth j s and j this means that j s and j do not contain for xl =0, l = j + 2, . . . , n + 1, linear terms in xj +1 . Systems (1) satisfying (H1) include a wide class of nonlinear systems, among which systems with control saturations. The following example: dx1 = (x2 + x22 sin(x1 x2 ))dt + x2 x3 d, dx2 = x3 dt + x32 cos(x1 x3 )d, dx3 = x4 dt + x43 cos(x1 x3 )d

(H2) (t) is a zero-mean, sample continuous, strongly Markov stochastic real-valued process adapted to {Ft } and E ∞ { − 2 ( )d } < ∞ (here E denotes expectation); (H3) the functions j m : Rn → R are locally Lipschitz continuous and such that for j =1, . . . , n and h=j +1, . . . , n+1

j m |xh =0,h=j +1,...,n = 0

(4)

satisﬁes (H1) with a1s2 = 2(x2 + x2 )2 + 2x24 [(sin(x1 x2 ) − 2 x2 ))/(x2 − x2 )] , a2s3 = (x3 + x3 )2 and a3s4 = 18[1 + sin(x1 2 2 2 x4 ) + x4 ] . (x4 − (b) Memoryless vector forward channel: Each state xj , j = 1, . . . , n, of the system is transmitted from the plant to the controller through a communication channel, affected by noise and subject to a constant delay with values in a compact set. The terminology “vector channel” refers to the fact that the alphabet of transmitted symbols is Rn (Tatikonda et al., 2004) and “forward” refers to the fact that the symbol is transmitted from the plant to the controller. The controller receives for all t 0

The functions j m and the process model the uncertainty added by the channel. The function j m models how a transmission error, depending on the transmitted data themselves, is propagated through the channel: exactly according to the paradigm pointed out in (6). For simplicity of calculations, in this paper we consider additive stochastic noise and deterministic uncertainty as in (5). However, all the results of this paper remain unchanged by modeling also multiplicative stochastic noise or time-varying (but bounded) uncertainty as j (t) = xj (t − ) + j m (x(t − ), t − )(t − ), which is a more realistic model over communication channels. Relative to example (4), the functions 1m = x1 ,

(5)

2m = x2 + x34 sin(x2 ),

3m = x3

(7)

satisfy (H3) with a2m3 = (x3 + x3 )2 (x32 + x32 )2 . (c) Remote controller: The noisy measurements (5) are received by a remote controller unit at the other end of the channel. By using the noisy measurements (5), an estimate of the state x is calculated on-line and a certainty equivalence strategy is applied to design a control law xn+1 for (1). (d) Memoryless backward channel: The control law xn+1 is transmitted through a communication channel from the controller back to the plant. The terminology “backward” refers to the fact that the symbol is transmitted from the controller back to the plant. For the sake of simplicity and since the presence of noise and delay in the backward channel can be treated in the same way as in the forward channel, in this paper we will assume that the remote controller is directly connected to the plant. Our task is to design a state estimator plus controller to drive the trajectories of (1) asymptotically to zero in probability despite the uncertainty and noise added by the communication channels. The √ main result of this paper is the following. Let G(r) = r/ 1 + r 2 . Theorem 1. Under assumptions (H1)–(H3) there exist hj ∈ (0, 1), Rj 1, j = 1, . . . , n, such that the trajectories x(t) of (1) with

j (t) = xj (t − ) + j m (x(t − )) + (t − ), j = 1, . . . , n,

(6)

xh ∈ R and for some smooth functions for all x ∈ Rn , aj mh : Rn−j +1 → R .

aj s,j +1 (0, 0, . . . , 0) = 0, j s |xh =0,h=j +1,...,n+1 = j |xh =0,h=j +1,...,n+1 = 0

where

|j m − j m |xh =xh |2 |xh − xh |2 aj mh ( xh , xj +1 , . . . , xn ),

2 2 |j s − j s |xh = xh | + j − j |xh = xh

|xh − xh |2 aj sh ( xh , xj +1 , . . . , xn+1 ),

349

u(t) = −

G(n (t)) , 2Rn

(8)

350

S. Battilotti / Automatica 44 (2008) 348 – 360

where n (t), t 0, satisﬁes the following differential equations: 1 1 ˙ 1 (t) = (h1 − 1)G(1 (t)) + G( 1 (t) − 1 (t)) , 2R1 h1 1 1 ˙ j (t) = (hj − 1)G(j (t)) + G( j (t) − j (t) 2Rj hj G(j −1 (t)) + ) , j = 2, . . . , n, (9) 2Rj −1 and j (t), j =1, . . . , n, are as in (5), are square integrable and P{lim supt→∞ x(t) = 0} = 1 for all F0 -measurable initial conditions. Lipschitz continuity of the functions j s , j m , j , j = 1, . . . , n, and Lipschitz continuity and boundedness of G(r) for all r guarantee the existence of an a.s. unique, continuous, Ft -adapted (strongly) Markov solution of (1)–(9) (see further remarks in Section 1.4). Lipschitz continuity is a standard assumption for the existence of trajectories and the boundedness of G(r) is here crucial for the functions on the right of ˙ j (t), j =1, . . . , n, being locally Lipschitz continuous uniformly with respect to . Since G(r) is bounded, the a.s. boundedness of is not required: bounded innovation assures bounded effect of unbounded noise. From now on for simplicity we will omit F0 -measurability of the initial conditions. Theorem 1 can be interpreted as follows: the plant with dynamics (1) transmits through the forward channel the state x, a remote controller calculates an estimate of x as in (9) and transmits the control law xn+1 (t) = −G(n (t))/(2Rn ) through the backward channel back to the plant. Theorem 1 can be also generalized to the case of multi-input systems, but for simplicity here we limit ourselves to one input. 1.1. Control of networks The model adopted in this paper is too far simple for capturing the complexity of a communication channel. Two directions can be pursued for proving results like Theorem 1 for more realistic networks. The ﬁrst one consists of modeling the network as an impulsive stochastic system driven by a renewal process (Hespanha, 2006), which is nothing but a jump-diffusion equation with state-dependent intensity. This allows us to model state reset and packet dropouts to use the Îto formula and Lyapunov-based arguments to derive sufﬁcient conditions for mean square stability. The second one consists of modeling the network as a stochastic system switching with a Markov chain with given transition probabilities (Battilotti & De Santis, 2005). No state reset is considered, while packet dropout can be modeled as “absence” of control. The stability analysis of the stochastic switching system can be performed by using the inﬁnitesimal generator of the system “frozen” at the i state of the Markov chain. Thus, if the values of the time delay and the system parameters vary according to a Markov chain as it can be the case of a network model, the stability analysis can be performed as if the delay and the system parameters would be constant. In this sense, Theorem 1 can be

used or modiﬁed accordingly for the stability analysis of a network with delays and random noise as long as the expectation of the dwell time (the time from one switching to the next one) is sufﬁciently large (Battilotti & De Santis, 2005). In the present setup, since we can use the same Lyapunov function W (i) for each system “frozen” at the i state of the Markov chain (i.e. it is a “common” Lyapunov function), no condition on the dwell time is needed and Theorem 1 can be readily extended to systems (1) switching according to a Markov chain. 1.2. Stability and convergence in probability: related notions Various notions of asymptotic stability in probability can be found in Khas’minskii (1980). These notions assume that only a Wiener process affects the system: as clear from (1) and our assumptions the system is also affected by an additive/multiplicative square integrable noise (not necessarily a Wiener process). A more general notion has been introduced in Battilotti (2005): whatever the initial state is and as long as the noise “energy” is sufﬁciently “small”, there is a “small risk” for which the trajectory zz0 (t) of the closed-loop system will not enter a given neighborhood of the origin. In this paper we want to allow “large” values of the noise “energy” and, in addition, large time delays. To this aim, we introduce the following deﬁnition. Deﬁnition 2. We say that a system : dz(t) = f (z(t), z(t − )), Υ (t))dt + g(z(t), z(t − ), Υ (t))d(t), with square integrable Υ (t) over [−, ∞), is globally asymptotically convergent in probability (GACP) if zz0 (t) is square integrable and P{lim supt→∞ zz0 (t) = 0} = 1 for all z0 . With Deﬁnition 2 in mind we see that one of the main result of Theorem 1 is to prove that the closed-loop system (1)–(8) is GACP. 1.3. Iterative design and innovations nesting For achieving stability of the closed-loop system, we implement a “dynamic” backstepping design in which xj +1 is used as control for each dynamics dzj , j = 1, . . . , n, where zj := xj − xj is the backstepping coordinate, and for each state zj we implement a robust (against uncertainties, noise and measurement delays) estimator to obtain an estimate of zj . This estimate is ﬁnally used to design the control xj +1 according to a certainty equivalence strategy. Having stipulated that xj +1 = −G(j )/2Rj , j = 1, . . . , n − 1, it is important to note the “nested” structure of the controller (9), in the sense that the innovation j − xj − j and, thus, the dynamics of ˙ j depends on j −1 through the term xj . This technique, introduced in Battilotti (2005) (innovations nesting), comes naturally from the “split and control” technique performed on the overall system by splitting it into n one-dimensional dynamics dzj , j = 1, . . . , n, controlled through xj with measurement j := j − xj .

S. Battilotti / Automatica 44 (2008) 348 – 360

1.4. Why are bounded control and bounded innovations important? Bounded backstepping with full state information has been studied for deterministic systems in Mazenc and Iggidr (2001). A natural consequence of the backstepping procedure is the nested structure of the controller. Since in our approach xj +1 is xj +1 =−G(j )/2Rj used as control for dzj , the boundedness of is crucial due to the presence of terms O(|xj +1 |) in j s and j (by (3)), as x22 and x44 in dx1 and, respectively, dx3 of (4) (O(|xj +1 |) means inﬁnitesimals with order greater than |xj +1 |). The technique of bounded innovations G( j − xj − j ) in (9) has been introduced in the recent paper Battilotti (2005) and limits the effect of measurement noise. An important fact about the opportunity of using bounded innovations as in (9) is also that the conditions of existence of trajectories in some probabilistic sense are satisﬁed for the closed-loop (1)–(9), uniformly w.r.t. . These conditions guarantee the existence of a unique, continuous, Ft -adapted (strongly) Markov solution of (1)–(9) (Mao, Matasov, & Piunovskiy, 2000). 1.5. Delay compensation Backstepping design with full state information and input delay has been studied for deterministic systems in Mazenc et al. (2003), using Lyapunov–Krasovskii functionals whose derivative along the system trajectories is negative deﬁnite. In this paper, we follow this approach although we have to cope with many additional difﬁculties such as noise compensation and model uncertainties. Moreover, we introduce a new technique which consists of resynchronizing states and delayed measurements considering the mismatch between them as an “uncertainty” (Section 3.2). As a consequence we obtain that the amplitude of the control has to be taken as small as possible on account of the delays to cope with. 1.6. Organization of the paper The GACP property of the closed-loop system plant + channel + controller is implied by the GACP property of the systems P1 and P2 (see Section 5 for deﬁnitions of P1 and P2 ). We will prove Theorem 1 by proving the GACP property of P1 and P2 . The system P1 is given by plant (1) with (9) and Υ (t) = (t − ), while the system P2 is given by plant (1) with (8) and Υ (t) = ((t − )zT (t − ) − T (t))T (here z and are the column vectors with jth component zj and j , respectively). The stability analysis of P1 and P2 is performed in a general framework, which consists of the following key points: (I) Deﬁne an n-dimensional system as having states, inputs and measurements (Section 2); both the inputs and measurements can be distinguished in endogenous (i.e. controls or measurements made on the system) and exogenous (i.e. from outside the system). (II) Split into one-dimensional systems j , j = 1, . . . , n, (Section 3) each one with a state, some inputs (of which only one control) and measurements (of which only

351

one endogenous measurement); since the measurement are delayed, we consider the delayed system j obtained from j with t replaced by t − . This allows us to resynchronize j with respect to its delayed measurement and to consider the mismatch between the delayed measurement delay and states as an “uncertainty”. (III) Find a one-dimensional measurement feedback controller Cj and a Lyapunov function Wj for each one-dimensional system j according to a certainty equivalence design (Section 4). (IV) Take the interconnection Cj , j = 1, . . . , n, as candidate controller C for : Section 5.1. The GACP properties of the closed-loop system ◦ C are proved by suitably selecting the parameters of the controllers Cj , j = 1, . . . , n, in such a way to compensate for the cross j , j = 1, . . . , n, where terms in each inﬁnitesimal generator LW Wj is a Lyapunov–Krasovskii functional deﬁned from Wj : Section 5.2. 1.7. Notations • Rs be the vector space of s-dimensional real column vectors; R+ (resp. R ) denotes the set of positive (resp. nonnegative) real numbers; In is the n × n identity matrix; Rn×n denotes the set√of n × n matrices. • v= v T v denotes the Euclidean norm of any given vector v √ and vA := v T Av for any positive semideﬁnite matrix A; Fr denotes the Frobenius norm Tr{ T }= Tr{ T }. • For any continuous function f : Rq × Rl → Rr , (s, r) → f (s, r), we denote by f (z, r) or f |s=z the function f (s, r) with s = z. For any given functions hj : Rq → R, j = 1, 2, we say that h1 (s) is of the same order as h2 (s) if there exists

1 , 2 > 0 such that 1 h2 (s) h1 (s) 2 h2 (s) for all s and we write h1 ∼ h2 . Moreover, we say that h1 (s) is less or equal to h2 (s) if there exists > 0 such that h1 (s) h2 (s) for all s and we write h1 h2 . 2. Complex systems as interconnection of simpler systems Following Battilotti (2005), we consider any n-dimensional system such as (4)–(7) as the interconnection of n onedimensional systems, each one with (I) state z; (II) control input v and some exogenous inputs (inputs affecting the system such as states of other interconnected systems or exogenous processes like ; we denote by the vector of the inputs); (III) Wiener process ; (IV) endogenous measurement (measurement available from the system) and some exogenous measurements (measurements available from other interconnected system); we denote by the vector of the measurements; (IV) uncertainty (, ) ( is the vector of state and measurement uncertainty as j s and j m , while is the vector of stochastic uncertainty as j ); (V) a set of constraints M among state x, inputs and measurements ; we write (x, , ) ∈ M. For technical reasons, we keep the Wiener process apart from exogenous inputs. It is important for the controller design to evaluate the effect of the uncertainty (, ) on the system under the constraints M. This motivates the following deﬁnition.

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S. Battilotti / Automatica 44 (2008) 348 – 360

Deﬁnition 3 (Incremental rate). We will say that a system with states x ∈ Rn , inputs ∈ I ⊆ Rm × Rr and measurements ∈ Z ⊆ Rp ×Rs , uncertainty (, ) ∈ Rq ×Rn and constraints M has (smooth) incremental rate z if there exist a nonempty subvector z of (x T T )T and a (smooth) function z : Rn × I × Z → R such that (|z=0 − )2 + |z=0 − 2Fr z (x, , )z2

(10)

for all (x, , ) ∈ M. If |x,v,=0 = 0 and since − |x=0 + |x=0 − |x,v=0 + |x,v=0 − |x,v,=0 for a system with states x, inputs and measurements , uncertainty (, ) and constraints M, and the same decomposition applies to Fr , we expect to have the following general relation among and , on one side, and x, , on the other, under the constraints M () 2

2

+ 4 2Fr

x (x, )x2 + v ()v2 + j (x, , )2j , ∀(x, , ) ∈ M,

j = 1, . . . , n, to have zj as “nominal part” of the measurement xj +1 |), thus j . Since by (2) each j i and j contains terms O(| to control dzj through xj +1 it is important to keep xj +1 as small as possible. This accounts for introducing the control constraints {| xi+1 | i ∈ (0, 1], i = 1, . . . , n}. Moreover, it is important to note that, since the Wiener process does not affect the measurements j , j = 1, . . . , n, and as long as we assume that xj , j = 1, . . . , n, is a smooth function of j −1 (see (A) of Section 5), by the Itô rule we have dzj =dxj − x˙ j dt where x˙ j is the time derivative of xj . From now on we will assume this. It is easy to see that, in backstepping coordinates zj , j =1, . . . , n, and with measurement change j , j = 1, . . . , n, (1)–(5) can be split into n one-dimensional systems of the form j :

dzj (t) = ( xj +1 (t) + j s (t)) dt + j (t) d(t), j (t) = zj (t − ) + j m (t − ) + (t − )

(12)

xj +1 with xn+1 := xn+1 , exogenous with state zj , control x2 , . . . , xj , x˙ j , xj +2 , . . . , inputs z1 , . . . , zj −1 , zj +1 , . . . , zn , xn+1 and , uncertainties j , j m and (11)

j ∈J

where j is the jth element of , j ∈ J := {1, . . . , r}, and x : Rn × Z → R , v : Z → R+ and j : Rn × I × Z → R are (smooth) incremental rates (“rescaled” by the square of the smooth function : Z → R+ ). The inequality (10) means that the uncertainty (, ) (“rescaled” by 2 ) is known up to the square of the states and the inputs, weighted by the corresponding incremental rates evaluated under the constraints M. Note that the incremental rate v is assumed (without loss of generality) to be a positive real function. Moreover, the factor 4 in (10) is only for convenience and can be incorporated in the incremental rates and scaling. Deﬁnition 4 (Incremental rates and scaling of ). We will say that a system with states x ∈ Rn , inputs ∈ I ⊆ Rm × Rr and measurements ∈ Z ⊆ Rp × Rs , uncertainty (, ) ∈ Rq × Rn×n and constraints M has (smooth) incremental rates x , v and j , j ∈ J , with scaling if (10) holds for some (smooth) nonnegative functions x : Rn × Z → R , j : Rn × I × Z → R and positive v , : Z → R+ . Throughout the paper, when an incremental rate is not explicitly cited in the context, we consider it equal to zero. 3. Splitting and resynchronization 3.1. Splitting In this section, using the general framework introduced in Section 2, we split the dynamics (1)–(5) into n one-dimensional systems having a “canonical” form. To do this we change coordinate xj into backstepping coordinate zj := xj − xj , j = 1, . . . , n, x1 := 0, xn+1 := xn+1 , to use xj +1 as control for dzj instead of using xj +1 (which is a state for (1)) for controlling j := j − xj , dxj . Also we change each measurement j into

x˙ j j s := j s + zj +1 − ns := ns − x˙ n ,

if j = 1, . . . , n − 1,

if j = n

(13)

with x˙ 1 := 0 and constraints {| xi+1 | i ∈ (0, 1], i=1, . . . , n}. 3.2. Resynchronizing states and delayed measurements The measurement delay can be interpreted as a control delay by considering zj (t − ) instead of zj (t) in j , thus obtaining the system j : xj +1 (t − ) + j s (t − )) dt + j (t − ) d (t), dzj (t) = ( j (t) = zj (t) + j m (t − ) + j (t),

(14)

xj +1 (t − ), with state zj (t) := zj (t − ), delayed control measurement j (t), (t) := (t − ) and Wiener process (t) := (t − ). Moreover, the delayed control xj +1 (t − ) can be seen as an “uncertainty” for j as follows: j :

dzj (t) = ( xj +1 (t) + j s, (t)) dt + j (t) d (t), j (t) = zj (t) + j m, (t) + (t),

(15)

j s (t − ) + xj +1 (t − ) − xj +1 (t), where j s, (t) = j m, (t) = j m (t − ) and j (t) = j (t − ). Note that j has state zj , control xj +1 , exogenous inputs z1, , . . . , zj −1, , zj +1, , . . . , zn , x2 , . . . , xj , . . ., xj , xj +2 , . . . , xn+1 , xn+1 , and, in addition, x2 − x2 , . . . , xn+1, − xi (t) := xi (t − ), uncertainties j s, , j s, , j m, , and constraints {| xi+1 |, | xi+1, | i ∈ (0, 1], i = 1, . . . , n}. We will distinguish the incremental rates of j from those of j by adding a subscript . By using (H1), (H3) and Lemma x − xh 6 we ﬁnd jxhi (j , . . . , h−1 ), j h (j , . . . , h−1 ) > 0, i z

h = j + 1, . . . , n + 1, i = s, m, and smooth jji : Rn−h+1 → R , h = j, . . . , n, i = s, m, j = 1, . . . , n, such that (53–55)

S. Battilotti / Automatica 44 (2008) 348 – 360

hold true for all xj , xj , j = 1, . . . , n, xj , xj , x˙ j , ∈ R, xj |j −1 . As a consequence j =1, . . . , n+1, for which | xj |, |

x − xh

of (54) and (55) xj hi , j h i

, h = j + 1, . . . , n + 1, i = s, m, x˙

zjhi , h = j, . . . , n, i = s, m, j = 1, . . . , n, and j sj = 4 and j m = 2 are the incremental rates of j (the other ones are set to zero as stipulated at the end of Section 2). Note also the inputs x˙ j and xj − xj are left by now not speciﬁed. When in the proof of Theorem 1 we will deﬁne xj as a function of j −1 or, respectively, zj −1 (see (9)) we will consider zj −1, , zj −1, − j −1 and j −1 − j −1, as exogenous inputs for j and rewrite x˙ j and xj − xj in terms of the incremental rates of zj −1, , zj −1, − j −1 and j −1 − j −1, .

353

Rj 1 1 such that for all Rj 1 Rj 1 the controller Cj :

xj +1 = −

1 G(j ), 2Rj 1

1 1 ˙ j = (hj s − 1)G(j ) + G( j − j ) 2Rj 1 hj s

has levels j , hjjs and Wj (zj , j ) =

1 + zj2 +

1 + (zj − j )2 − 2

(20)

j i j 2j i

(21)

satisﬁes LWj − j s zj2 − j m (zj − j )2 +

4. Controlling one-dimensional dynamics is simpler than controlling the whole system The next step is to derive one-dimensional controllers for systems like j in (15). In this section we obtain feedback controllers Cj and a Lyapunov function Wj for any onedimensional system j like (15). Throughout, we will denote by j i , i ∈ Jj , any one of the exogenous inputs of j . A measurement feedback controller for j is deﬁned as xj +1 = Fj (j ),

dj = [Hj (j ) + xj +1 + Gj ( j − j )] dt (16)

with j ∈ R, smooth Hj : R → R and Gj : R → R, vanishing xj +1 = at zero. A state feedback controller for j is deﬁned as Fj (zj ). Note the structure of the controller (16) is based on a certainty equivalence principle, consisting of replacing in the state feedback controller xj +1 = Fj (j ) an estimate j of zj . We say that the controller (16) has control input level jf and innovations level j m (or simply levels (jf , j m )) if there exist jf , j m ∈ (0, ∞] such that

Gj ( j − j )j m ∈ (0, ∞],

i∈Jj

with j s ∼

1 − hj s , Rj 1 (1 + zj2 )

j m ∼

1 − 16hj s Rj 1 hj s [1 + (zj − j )2 ] (22)

and incremental rates

j ji :=

Rj 1 j i [ + hj m j jmi ]. hj s j s

∀ j , j .

(17)

The numbers jf and j m characterize the maximum level allowed for the control input xj +1 and, respectively, for the innovations j − j feedback in the control loop by (16). The following result can be proved as a combination of Theorems 3 and 4 of Battilotti (2005) and tells us how to select a measurement feedback controller for j . Given d = f (, ) dt + g(, ) d and smooth real-valued W () we recall that jW j2 W LW := f + (1/2)Tr g T 2 g . j j

(23)

5. Proof of Theorem 1 The proof of Theorem 1 is quite technical and it is structured into steps. It is readily seen that the system given by the plant, channels and controller is GACP if the following is shown: (A) the system P1 : P1 xj +1 + j s, ) dt + j d , dzj = (

j = zj + j m, + ,

Fj (j )jf ∈ (0, ∞],

(19)

(24)

1 G(j ), j = 1, . . . , n, 2Rj 1 1 ˙ j = (hj − 1)G(j ) + G( j − j ) , 2Rj hj

xj +1 = −

where j s, P1 = j s, for j = 1, . . . , n, is GACP with Υ = ; note that this by deﬁnition implies square integrability of zj and zj − j , j = 1, . . . , n, over [0, ∞); (B) the system P2 : xj +1 + j s, ) dt + j d , dzj = ( P2

(18)

Theorem 5. Let j = 1, . . . , n. For any system j like (15) satisfying (53)–(55) for each 1 , . . . , n ∈ (0, 1] and for all xh , xh , h = 1, . . . , n, xh , x˙ h , ∈ R, h = 1, . . . , n + 1, such xh | h−1 , there exist j , hj s , hj m ∈ (0, 1) and that | xh |, |

xj +1 = −

1 G(zj ), 2Rj

j = 1, . . . , n,

(25)

P2 P2 j s, if j n − 1, ns, = ns, + xj +1 |n − where j s, = T xj +1 , is GACP with Υ = ( zn − n ) .

354

S. Battilotti / Automatica 44 (2008) 348 – 360

While system (24) is nothing but j , j = 1, . . . , n, with measurement feedback controller 1 j − j ) 2Rj , ˙ j = (hj − 1)G(j ) + G( hj xj +1 = −G(j )/2Rj ,

j = 1, . . . , n,

(26)

system (25) is nothing but j , j =1, . . . , n, with state feedback controller xj +1 = −G(zj )/2Rj , xn+1 = −G(n )/2Rj .

j = 1, . . . , n − 1, (27)

We prove only that (24) is GACP (the proof that (25) is GACP goes in the same way). We list our steps with remarks: A.1. For each system j , j = 1, . . . , n, in (12) we apply Theorem 5 to obtain a measurement feedback controller Cj and a Lyapunov function Wj (Section 5.1). We take the interconnection C of Cj , j = 1, . . . , n, as measurement feedback controller for the interconnection of j , j =1, . . . , n. This is nothing but (26). We also compute the inﬁnitesimal generator LWj in (28) and the incremental rates in (31)–(32). A.2. To prove that (24) is GACP, we ﬁnd a sequence of compact sets, collapsing to the origin, such that each compact set is hit with probability one by the trajectories of (24), the probability that the limsup of the trajectories for t → ∞ being outside this compact set is small and the probability that the “tail” of the integrals of the squared trajectories being large is small. A.3. To this aim, we start our stability analysis with the closedloop n ◦Cn . The effect of the measurement delay is seen as an “uncertainty” and related to the increments n (t) − n (t). The increments (n (t) − n (t))2 are bounded by t an integral t−2(n+1) n (t) dr a.e. (Lemma 7), which together with some other terms in LWn can be rewritten n n for some Lyapunov–Krasovskii functional W as LW (see (36)). In this way the effect of the measurement delay can be coped with through a Lyapunov-based analysis. A.4. We deﬁne a closed interval [0, cn ] in (34). The typically upper triangular structure assumed in (H1) and (H3), the square integrability of assumed in (H2) and the boundedness of the innovations in (26) are such that Wn (t) is regular (i.e. no ﬁnite escape time) and hits with probability one (i.e. it is recurrent relatively to) [0, cn ]. This is proved in Claims B.1.0 (thanks to Lemma 9) and B.1.1. A key tool analysis is the Dynkin’s formula (Oksendal, 1985, Theorem 7.10). This formula gives in this case a relation bei (t)} and LW i (t). However, since the validtween E{W ity of the Dynkin’s formula is conditioned to bounded trajectories, this formula is mainly used with “stopped” trai ( [cn ,k] (t)), k > cn , where [cn ,k] (t) is a stopjectories W ping time. The upper stopping value k ensures bounded trajectories, while the lower stopping value cn is such that i (t) is negative so that E{W i (t)} in nonincreasing LW (see (40)).

A.5. Another consequence of (H1)–(H3) is that while Wn (t) approaches [0, cn ], the other processes Wj (t), j = 1, . . . , n − 1, are regular. Roughly speaking, although the measurement feedback delay may be large, we have all the time we need to drive Wn (t) inside [0, cn ] while Wj (t), j = 1, . . . , n − 1, being a.s. bounded in the meanwhile. Moreover, larger the delay is, smaller the control amplitude should be (see the condition Rj21 in Lemma 7 and note that the gains of (26) go as 1/Rj 1 ). A.6. From the preceding points, by Dynkin’s formula and ˘ Lemma 10 (which is a consequence of Ceby˘ sev inequality) we conclude that the probability of large values ∞ of Wnwn ,0 (t) and lims→∞ s (Wnwn ,0 (r)) dr is small (Claims B.1.2 and B.1.3). Next, we consider i, ◦ Ci , i =n−1, n, and deﬁne a composite Lyapunov–Krasovskii (n−1) for i, ◦ Ci , i = n − 1, n (see (42)), functional W n and Wn−1, . From here we go to reason as in using W points A.3–A.5 and consider next an increasing number of systems i, ◦ Ci , i = j, . . . , n − 1, n (the proof is detailed only for i, ◦ Ci , i = n − 1, n: Claims B.2.0–B.2.2). A.7. After n − 1 iterations we obtain a composite Lyapunov– (1) for i, ◦ Ci , i = 1, . . . , n and Krasovskii functional W from this we conclude that the probability of large values ∞ of lim sups→∞ Wlw i ,0 (t) and lims→∞ s (Wlw l ,0 (r))dr, l ∈ {1, . . . , n}, is small (see (51)–(52)). All the above items together lead to the proof that (24) is GACP (Section 5.2). In the same way (without proof), we obtain the controller (27) and prove that also (25) is GACP by using the fact zj − j , j = 1, . . . , n, and are square integrable. This concludes the proof of Theorem 1. We assume that the reader is familiar with the notion of regularity and inaccessibility (Khas’minskii, 1980). 5.1. Design of controller (26) We begin with point A.1. Apply Theorem 5 to each j , j = 1, . . . , n, in (15) and obtain j , hj s , hj m ∈ (0, 1] and Rj 1 > 0 such that for all Rj 1 Rj 1 the controller (19) denoted by Cj , has levels (j , j / hj s ) and z

e

LWj − [j s − jj ]zj2 − [j m − jj ]ej2 z

e

+ jj−1, zj2−1, + jj−1, ej2−1, n

+

z

l=j +1

+

x

2 [jl zl2 +j l+1 xl+1 ]+

n+1 l=j +2

x − xl

j l

j l=j −1

−l

j l

( xl − xl )2 + j 2

with z11 = e11 = 0 = 0 = 0, ej = zj − j , Wj (zj , j ) = 1 + zj2 + 1 + ej2 − 2

(l −l )2

(28)

(29)

S. Battilotti / Automatica 44 (2008) 348 – 360

and stability margins j s ∼ 1/[Rj 1 (1 + zj2 )],

j m ∼ 1/[Rj 1 (1 + ej2 )].

(30)

Moreover, by (23) and Lemma 7, since |G( 2l=1 sl )| 2l=1 |G(sl )| and |G(s1 ) − G(s2 )||G(s1 − s2 )| for all s1 , s2 , and rewriting xl2 and ( xl − xl )2 , l = 3, . . . , n + 1, in terms of the exogenous inputs zl−1, , el−1, and l−1, − l−1 , z

e

z

x

z

x

l+1 l+1 l 1l , 1l ∼ R11 [1sl + 1m + 1s + 1m ],

l = 2, . . . , n,

1 ∼ R11 [1s + 1m ], R11 x x − xl+1 − xl+1 [j l+1, + j l+1, ], s m 2 Rl1

−l

∼

−1

∼ 1/[R11 (1 + (1 − 1 )2 )],

1l

11

l = 2, . . . , n, (31)

e

l = j − 1, j ,

z

j ∼ Rj 1 [j s + j m + 1],

jl ∼ Rj 1 /[Rj4−1,1 (1 + zl2 )],

−j −1

z

e

j j −1,

∼ Rj 1 /[Rj4−1,1 (1 + (j −1, − j −1 )2 )], Rj 1 1 1 j −j j ∼ + 4 , Rj 1 Rj −1,1 1 + (j − j )2 z

z

x

x

l+1 jl , jl ∼ Rj 1 [jls + jlm + j l+1 s + j m + 1],

xl+1, − xl+1

Rj 1 [j s ∼ 2 Rl1

x

+ j l+1, m

− xl+1

+ 1]

1 + (l − l )2

(32) (1)

e

j m − jj j m /2

(35)

t−2(n+1)

where j (t) denotes j (t − ). By (30) we can select εj 1 such that z

Let t s 2(n + 1). Note that G2 (s) = (s 2 ) for all s and since (s) is nondecreasing for all s 0 2j =1 (sj ) 2j =1 ( 1 + sj − 1) ( 2j =1 ( 1 + sj − 1)) for all sj 0, j = 1, 2. From (29), (32), (33), with j = n, and Lemma 8, we infer the (2) (1) , ∞), existence of εi εi , continuous Rl1 : [1, ∞) → [Rl1 l = 1, . . . , n, R11 (ε) · · · Rn1 (ε) for all ε 1, bn 1 and (2) ln > 0 such that for all ε εi , i=1, . . . , n, (33) for j =1, . . . , n holds with 2 8bn Rn1 4(n + 1) ln 1 cn := 1+ < 1, , 4 2 Rn1 (n + 1) Rn−1,1 ln Rn−1,1 2 1 + Rn1 ln | 1 1 , (34) 4(n + 1)bn Rn1 + 4 6 4 2Rn1 Rn−1,1 Rn1 Rn−1,1 ln (Wn ) bn Rn1 + 4 (Wn−1, ) + bn Rn1 2 Rn1 Rn−1,1 t + n (r) dr a.e.

,

l = j + 1 . . . , n,

j s − jj j s /2,

B.1. We begin our stability analysis with n ◦ Cn , by using LWn , our assumptions (H1)–(H3), the boundedness of the innovations in (26) and modeling the effect of the measurement delay as an “uncertainty”, related to the increments n (t) − n (t) (A.3–A.5). We need the following claim, proved in the Appendix.

LWn −

l = j + 1, . . . , n, − j l l

• For simplicity and whenever possible, we denote equivalently W (z(t)) by W (t). Also, → means “tends to” and ↑ (↓) means “monotonically increasingly (decreasingly) tends to” and by (s) we denote the function s/[1 + s].

Claim B.1.0. Wnwn ,s (t) is regular for all s 0 and wn > 0.

and for j = 2, . . . , n jl ∼ Rj 1 /[Rj4−1,1 (1 + el2 )],

355

(33)

(1)

for all ε εj . At the end of these calculations, we obtain the controller (9) as the interconnection of Cj , j = 1, . . . , n. 5.2. (24) is GACP Few more notations before proceeding: ∞ • By Ms () for s − we denote E{ s (s)2 ds} (which exists since is square integrable over [−, ∞) by (H2)) and by W w,s (t) we denote W (t) with W (s) = wi . • For any closed interval ⊂ R we denote by := / }, with = ∞ a.e. if P{W w,s (t) ∈ inf t s {t : W w,s (t) ∈ , ∀t s} = 1 (the ﬁrst exit time from ). • For any pair of closed intervals 1 , 2 ⊂ R by 1 (t, 1 ) the “stopping” time min{ 1 , 2 , t}, t s.

n

along the trajectories of (1)–(24), with n (t) = Rbn6Rn1 [ n−1,1

l=n−1

R2 (Wl (t)) + 2 (t)] + Rb3n [1 + R 4 n1 ][(Wn (t)) + 2 (t)]. Thus, n1 n−1,1

since Rn−1,1 Rn1 1, from (34)–(36) we obtain a.e. n ) 4(n + 1) l b ( W R n n n1 n − LW 1+ (Wn−1, ) + 4 2 Rn1 Rn−1,1 Rn−1,1 + bn Rn1 [1 + 12(n + 1)]2

(36)

t n = Wn + 2 t with W t−2(n+1) ( r n () d) dr. On application of the Dynkin’s formula to (36) and taking into account the square integrability of over [−, ∞) we obtain for all w n , k cn n ,s w E{W ( [cn ,k] (t))} n

− cn E{ [cn ,k] (t) − s} + n ( wn , Ms− ())

(37)

for n ( wn , Ms− ()) := w n +bn Rn1 [1+12(n+1)Ms− ())]. n ,s w Thus, on account of the nonnegativity of W (·), we conclude n

356

S. Battilotti / Automatica 44 (2008) 348 – 360

from (37) that for all w n , k cn

hold true with

E{ [cn ,k] (t) − s} 8Rn1 n ( wn , Ms− ())/cn .

(38)

cn−1 :=

The following claim is proven in the Appendix.

2 8bn−1 Rn−1,1 4 Rn−2,1

From (37) it also follows that for all w n , k cn (39)

LWn−1, −

(40)

From this we obtain the following claim, which is proved in the Appendix. Claim B.1.2. For each n > 0 there exists dn > 0 such that Pr{lim supt→∞ Wnwn ,0 (t) > dn } < n for all wn 0. In the same way as in (36), from (37) and the nonnegativity

(t) n ,s n ,s w w (·) we conclude that E{ s [cn ,k] (W (r)) dr} of W n n s− (8Rn1 / ln ) n ( wn , M ()) for all w n−1 , k cn−1 . This, by following the proof of Claim B.1.2, leads to:

along the trajectories of (1)–(24), with n−1 (t) = n

[

l=n−2 (Wl (t))

n−1 (s) =

max

Wn (zn ,n ) r

s x

b

×

w

,s

n−1 Claim B.2.0. Wn−1, (t) is regular for all wn−1 > 0 and s 0.

Let t s 2(n+1). From (29), (32) and (33), with j =n−1, (3) (2) and Lemma 8, we infer as in (36) the existence of εi εi , i = 1, . . . , n, continuous Rl1 : [1, ∞) → [Rl1 , ∞), l = 1, . . . , n, R11 (ε) · · · Rn1 (ε) for all ε 1, bn−1 1 and ln−1 > 0 such (3) that for all εi εi , i = 1, . . . , n, (33) for j = 1, . . . , n and (36)

1

1

+ 16(n + 1)]2

→ is continuous and nondecreasThe function n−1 : ing. From the properties of n−1 we get the following claim, which is proven in the Appendix.

Rn−1,1 n ][ l=n−1 4 Rn−2,1 +2 (t)]. Thus, 2

2 Rn−1,1

1+ 4 + 4 6 Rn−2,1 Rn−1,1 Rn−2,1 1 n ) + bn−1 Rn−1,1 [n−1 (W n ) + 2 (W Rn1

x

R

R

+

n−1, ) ln−1 (W 2Rn−1,1 4(n + 1) bn−1 Rn−1,1 1+ (Wn−2, ) + 2 4 Rn−2,1 Rn−2,1 n ) + 4(n + 1)bn−1 + bn−1 Rn−1,1 n−1 (W

n+1 n+1 + n−1,s + n−1,m + 1)

R

bn−1 [1 3 Rn−1,1

n−1, − LW

zn n [(zn−1,s + n−1,m

× (Wn (zn , n ) + 2)2 + n−1, ] dr.

+ 2 (t)] +

bn−1 Rn−1,1 6 Rn−2,1

(Wl (t)) + 2 (r)] + n−1R 2n−1,1 [(Wn (t)) n1 n (·) and n−1 (s) and (s) are nondecreasing since Wn (·) W for all s 0, as in (36) we get

B.2. Our next step is to consider i, ◦ Ci , i = n − 1, n (see A.6). Let s+1

ln−1 (Wn−1 ) bn−1 Rn−1,1 + (Wn−2, ) 4 Rn−1,1 Rn−2,1

t−2(n+1)

Claim B.1.3. For each n > 0 there exists dn > 0 such that ∞ n ,s w (r)) dr > dn } < n for all wn 0. Pr{lims→∞ s (W n

(41)

+ bn−1 Rn−1,1 n−1 (Wn )[(Wn ) + 2 ] t + n−1 (r) dr a.e.

Since P{ n (cn ) < ∞} = 1 by Claim B.1.1, we have [cn ,k] (t) ↑

[cn ,∞) (t) a.e. then, on account of Fatou’s lemma and by letting k → ∞ in (39), for all w n cn n ,s w E{W ( [cn ,∞) (t))} n ( wn , Ms− ()). n

4(n + 1) 1+ < 1, 2 Rn−2,1

ln−1 2(n + 1)bn−1 2Rn−1,1 2 Rn−1,1 1 1 × 1+ 4 , + 4 6 Rn−2,1 Rn−1,1 Rn−2,1

n cn , in other Claim B.1.1. P{ [cn ,∞) < ∞} = 1 for all w n ,s w (t) is recurrent relatively to the compact set words, W n [0, cn ].

n ,s w E{W ( [cn ,k] (t))} n ( wn , Ms− ()). n

a.e.

t n−1, = Wn−1, + 2 t with W t−2(n+1) r n−1 (r) dr. We have all we need to deﬁne a composite Lyapunov– (n−1) for i, ◦ Ci , i = n − 1, n (see Krasovskii functional W n and Wn−1, . Let n−1 (s) = 2[n−1 (s) + (42)), using W 6bn−1 [ R 6 1

n−2,1

+

1 [1 4 Rn−1,1

+

2 Rn−1,1

4 Rn−2,1

]+

bn−1 Rn−1,1 Rn1 1 2 ]] ln Rn1

(note

that n−1 (s) 1 for all s 0 since n−1 (s) has the same property) and (n−1) = n−1 W (n−1) , W n−1, + 2n−1 (dn )W n W (n−1) = W

(42)

S. Battilotti / Automatica 44 (2008) 348 – 360

all w n−1 , k cn−1

with

n , ) an−1 (W ˙ n−1 = − n−1 , 2n−1 (dn )dn

n−1 (0) > 0,

s

n−1, )+2n−1 (dn )(W n ) (W (n−1) ) and n−1 (s) 1 Since (W for all s 0, there exists ε(4) ε(3) , i = 1, . . . , n, continuous , ∞), l = 1, . . . , n, R (ε) · · · R (ε) Rl1 : [1, ∞) → [Rl1 11 n1 (4) for all ε 1, bn−1 1 and ln−1 > 0 such that for all εi εi , i = 1, . . . , n, (33) for j = 1, . . . , n, (36) and (42) hold true with ln−1 bn Rn1 n−1 (dn ) 4(n + 1) 1+ , 3 2 8 Rn−1,1 Rn−1,1

(n−1) ) (n−1) n−1 − ln−1 (W LW Rn−1,1 4(n + 1) bn−1 Rn−1,1 1+ (Wn−2, ) + 4 2 Rn−2,1 Rn−2,1

(46)

Following the proof of Claim B.1.2, we conclude from (45) by Lemma 10 that for each n−1 > 0 there exists dn−1 > 0 such that n−1 ,0 w < n−1 (t) > d , A (47) Pr lim sup W n−1 n n−1, t→∞

for all w n−1 0 and from (46) that

lim

s→∞ s

∞

(n−1) w n−1 ,0 n−1 ((W ) (r)) dr > dn−1 , An <

Pr (43)

0

lim sup Wiwi ,0 (t) > dn−1 , An ; i t→∞

∈ {n − 1, n}

n−1 ,0 (n−1) )w Pr lim sup(W (t) > dn−1 , An

(49)

t→∞

for all wn−1 , wn , w(n−1) , w n−1 0. We conclude from (47) that Pr{lim supt→∞ Wiwi ,0 (t) > dn−1 , An ; i ∈ {n − 1, n}} < n−1 for all wn−1 , wn 0. Thus, by Claim B.1.2 and since Pr{A} Pr{A, B}Pr{B} − Pr{A, B} for all events A, B, Pr{lim sup Wiwi ,0 (t) > dn−1 ; i ∈ {n − 1, n}} t→∞

Pr n−1 (0)n−1 (t)1∀t 0,

< n−1 + n

(44)

Since r 1 ⇒ r(s)(rs) for all s 0, by application of Dynkin’s formula to (43) and taking into account the square integrability of over [−, ∞), as a consequence of (44) we obtain for all w n−1 , k cn−1 n−1 ,s (n−1) )w ( [cn−1 ,k] (t))} E{IAn (W

cn−1 E{IAn ( [cn−1 ,k] (t) − s)} 8Rn−1,1 ∞ + n−1 w n−1 , E IAn 2 (r)n−1 (r) dr

(r)) dr

for all w n−1 0. By (44) ∞ and since n−1 (s) 1 for all s 0, then if n−1 (0) = exp( 0 a(s) ds a.e.

a(s) ds , An

lim n−1 (t) = 1, An = 1.

)

(48)

a.e. along the trajectories of j ◦ Cj , j = n − 1, n and for some ln−1 > 0. Moreover, denoting by An the event {s > Tn−1 := n ,0 w sup{t 0 : W (t) > dn }}, it is easy to see that n

t→∞

IAn ((W

s

Pr

+ [bn−1 Rn−1,1 (n−1 (dn ) + 16(n + 1))

∞

n−1 ,s (n−1) w

(8Rn−1,1 / ln−1 )n−1 ∞ 2 × w n−1 , E IAn (r)n−1 (r) dr .

n ) + 2 ]bn−1 Rn−1,1 . − n−1 (dn ), 0}[(W

+2n−1 (dn )bn Rn1 (1 + 12(n + 1))]2

[cn−1 ,k] (t)

E

n , ) = max{n−1 (W n ) an−1 (W

Pr n−1 (0) = exp

357

(50)

for all wn−1 , wn 0. We conclude that: Claim B.2.1. For each n−1 > 0 there exists dn−1 > 0 such that Pr{lim supt→∞ Wiwi ,0 (t) > dn−1 ; i ∈ {n − 1, n}} < n−1 for all wn−1 , wn 0. In the same way, as in Claim B.2.1 and from (48) we get:

−

(45)

s

for n−1 ( wn−1 , r) := w n−1 + [bn−1 Rn−1,1 (n−1 (dn ) + 16(n + 1)) + 2n−1 (dn )bn Rn1 (1 + 12(n + 1))]r. In the same way as (n−1) (·) we conclude for in (45), from the nonnegativity of W

Claim B.2.2. For each n−1 > 0 there exists dn−1 > 0 ∞ such that Pr{lim sups→∞ s (Wiwi ,0 (r)) dr > dn−1 ; i ∈ {n − 1, n}} < n−1 for all wn−1 , wn 0. i ,0 w Let Ai the event {s > Ti−1 := sup{t 0 : W (t) > di }} for i di > 0. Using induction on the number i = 2, . . . , n of systems i ◦ Ci considered and following the proof of Claims B.i.j , i = 1, 2, j = 0, 1, 2, 3, we obtain that W1w1 ,s (t) is regular for

358

S. Battilotti / Automatica 44 (2008) 348 – 360

60

5

50

4

40

3

30

2

20

1

10

0

0

-1 0

500

1000

1500

2000

2500

0

500

1000

1500

2000

2500

Fig. 1. Closed-loop system states x1 (t), x2 (t).

all w1 > 0 and s 0 and for each 1 , d1 > 0 Pr lim sup Wlw i ,0 (t) > d1 ; l ∈ {1, . . . , n} < 1 , Pr

t→∞

∞

lim

s→∞ s

x x −

(51)

(Wlw l ,0 (r)) dr > d1 ; l ∈ {1, . . . , n} < 1 (52)

for all w1 , . . . , wn 0. Note that (51) and (52) hold for all 1 , d1 > 0 since as a consequence of the expression of 1 we can take c1 (having the corresponding role of cn LW in (36) and cn−1 in (42)) any positive number. Since Wl , l = 1, . . . , n, is a proper function of its arguments, by letting 1 , d1 → 0 in (51) we get Pr{lim supt→∞ Wlw i ,0 (t) = 0; l = 1, . . . , n} = 1. Thus, by letting 1 , d1 → 0 in (52) we also ∞ get Pr{lims→∞ s (Wlw l ,0 (r)) dr = 0; l = 1, . . . , n} = 1, which, since (s)s/2 for all 0 s 1, implies square integrability of the trajectories of (24). This with Claims B.i.0, i =1, . . . , n−1, and sinceWl , l =1, . . . , n, is a proper function of its arguments, prove that (24) is GACP. 6. Example Consider dx1 (t)=x2 (t) dt +x22 (t) d(t), dx2 (t)=u(t) dt, with delayed measurements 1 (t) = x1 (t − 5) + x24 (t − 5) sin(x1 (t − 5)) + (t − 5) and 2 (t) = x2 (t − 5) + (t − 5), standard Wiener process (t) and channel noise (t) = e−t w(t), w(t) a gaussian process with zero mean and variance 104 . As shown by the simulations in Fig. 1 with x1 (−5) = 4, x2 (−5) = 5, 1 (0)=0, 2 (0)=0, the convergence of the closed-loop system trajectories to zero is guaranteed despite the channel noise (t) and delay = 5.

h (j , . . . , h−1 ) > 0, there exist jxhi (j , . . . , h−1 ), j h i zh h=j +1, . . . , n+1, i=s, m, and smooth j i : Rn−h+1 → R , h = j, . . . , n, i = s, m, j = 1, . . . , n, such that for j = 1, . . . , n

x lim j +1 (j ) = 0, j →0+ j s

(53)

[j s |X=X + zj +1, − x˙ j ]2 + 4j |X 2 n

4| x˙ j |2 +

h=j +1

+

n+1

z

zh2 jh s (zh , . . . , zn )

xh2 xj hs (j , . . . , h−1 )

h=j +1

+

n+1

xh x −

( xh − xh )2 j hs

(j , . . . , h−1 ),

(54)

h=j +1

[j m |X=X + ]2 22 +

+

n h=j +1

n

z

zh2 jh m (zh , . . . , zn )

xh2 xj hm (j , . . . , h−1 )

h=j +1

+

n

x − xh

( xh − xh )2 j hm

(j , . . . , h−1 )

(55)

h=j +1

Appendix For brevity, denote xi = xi , i = 1, . . . , n, by X = X and h . xi , i = h, . . . , n, by Xh = Zh + X xi = zi + Lemma 6. Let xj , xj , j = 1, . . . , n, xj , xj , x˙ j , ∈ R, j = 1, . . . , n + 1, x1 = x˙ 1 = x1 := 0, xn+1 := xn+1 , xn+1, := xn+1, and zj := xj − xj , j = 1, . . . , n. Under assumptions (H1) and (H3) for each 1 , . . . , n ∈ (0, 1]

xj , xj , x˙ j , ∈ R, j =1, . . . , n+ for all xj , xj , j =1, . . . , n, xj | j −1 . 1, such that | xj |, | Proof. Follow the same steps of Lemma A.1 of Battilotti (2007). Lemma 7. Let xj , xj , j = 1, . . . , n, xj , xj , ∈ R, j = 1, . . . , n + 1, x1 = x1 := 0, xn+1 := xn+1 , xn+1, := xn+1,

S. Battilotti / Automatica 44 (2008) 348 – 360

and zj := xj − xj , j = 1, . . . , n. Under assumption (H3) G2 (ej + j m |X=X + ) G2 (ej ) +

n l=j +1

n

+

xl2 +

l=j +1

( xl − xl )2 + G2 ( )

(56)

l=j +1

for all xj , xj , ej , j = 1, . . . , n, xj , xj , x˙ j , x˙ j , ∈ R, xj |j −1 . j = 1, . . . , n, such that | xj |, | Proof. Follow the same steps of Lemma A.2 of Battilotti (2007). Lemma 8. Let j = 1, . . . , n, s 2(n + 1). Under assumptions (H1)–(H3) and if Rj21 , j = 1, . . . , n, for all t s along the trajectories of j -(19) and i = j, . . . , n (i (t) − i (t))2

1 Rj21

t

⎧ n ⎨

t−2(n+1) ⎩ i=j

⎫ ⎬ (Wi (r)) + 2 (r) dr ⎭

a.e. (57)

Proof. Fix any trajectory of j -(19). By integrating (19) between t − and t and using the Schwarz inequality we get a.e. t t 1 (j (t) − j (t))2 2 G2 (j (r)) dr + G2 (ej (r) Rj 1 t− t− +j m |X=X (r) + (r)) dr with ej = zj − j . Taking into account that xl = −(1/(2Rl1 )) G(l−1 ) and xl = −(1/(2Rl1 ))G(l−1, ), using (56) and since G2 (s1 ) + G2 (s2 )(W (s1 , s2 )) for all s1 , s2 , we get a.e. t 1 (j (t) − j (t))2 j (t) + 2 G2 (j (r) − j (r)) dr Rj 1 t− t n with j (t) = (1/Rj21 ) t− [ ni=j (Wi (r)) + i=j +1 G(i (r)) − i (r))2 + 2 (r)] dr. On account of Gronwall’s lemma and since G2 (s)s 2 for all s, it follows that ⎡ t n 1 2 ⎣ (j (t) − j (t)) 2 (Wi (r)) Rj 1 t−2 i=j +

n

⎤ (i (r))−i (r))2 +2 (r)⎦ dr. (58)

i=j +1

Using (58) iteratively we obtain (57).

smooth proper function such that LV (z)aV (z) + b() for all z, and for some a 0 and continuous and nondecreasing b : R → R . Then, zz0 ,s (t) is regular. Proof. Combine the arguments of Theorem III.4.1 and Lemma III.8.1 of Khas’minskii (1980).

G2 (zl2 )

n

359

Lemma 9. Let : dz(t) = f (z(t), (t)) dt + g(z(t), (t)) d, with z ∈ Rv , some exogenous Rq -valued process such that P{supt s (t) < ∞} = 1 for all s 0. Let V : Rv → Rv be a

Lemma 10. Let W (t) be a separable, nonnegative and sample continuous real-process such that E{IB W (t)} 0 for all t s 0 and for some event B. We have P{supt s W (t) > , B} 0 for all t s 0.

˘ Proof. Use the Ceby˘ sev inequality, separability and sample continuity of W (t). Proof of Claim B.1.0. Since Wn (zn , n ) is proper it is sufﬁcient to prove that zn (t) and n (t) are regular. Let Sn (zn , n ) = zn2 + 2n . Then, taking into account the form of the controller Cn and the incremental rates of n (Section 3) we have LSn an1 Sn + an2 [1 + 2 ] for some an1 , an2 0. Since ∞ (t) is square integrable over [−, ∞) by (H2), then P{ s 2 ( ) d < ∞} = 1 for all s 0 and, consequently, P{supt s | (t)| < ∞} = 1. On application of Lemma 9, with V = Sn , a = an1 and b() = an2 [ + 1] and = , we conclude that zn (t)and n (t) are regular. Proof of Claim B.1.1. Since Wnwn ,s (t) is regular by Claim B.1.0 and t t n () d dr t−2(n+1) r

2(n + 1)bn Rn1

1 6 Rn−1,1

R2 1 + 4 1 + 4 n1 [M− () + 4(n + 1)22 ] Rn1 Rn−1,1 n ,s w (t) is regular. It follows that for all t s, also W n 0 [cn ,k] ↑ [cn ,∞) a.e. as k → ∞. Letting k → ∞ and on account of Fatou’s lemma, from (38) we get for all t s and w n cn E{ [cn ,k] (t) − s} ↑ E{ [cn ,∞) (t) − ˘ wn , Ms− ()/cn . Using this and Ceby˘ sev ins}8Rn1 n ( equality, for all t > s and wn cn

P{ [cn ,∞) t}P{ [cn ,∞) (t) t} wn , Ms− ())/[cn (t − s)]. 8Rn−1,1 n (

(59)

The events At = { [cn ,∞) t} are monotonically decreasing as t increases. Thus, on account of the continuity from above of P and letting t → ∞ in (59) we ﬁnally obtain P{ [cn ,∞) t} ↓ P{ [cn ,∞) = ∞} = 0 which proves our claim. (1)

(1)

Proof of Claim B.1.2. Let cn be such that 0 < cn < cn and t s and set n =2n (cn , Ms− ()). Moreover, to ﬁx the ideas assume that wn = n . Using the arguments of Lemma IV.2.1 of

360

S. Battilotti / Automatica 44 (2008) 348 – 360

Khas’minskii (1980), we prove that wn ,s ( (1) (t))} E{W n [c ,∞) n

s− n (n , Ms− ()) := (1) ()). n (cn , M

(60) (1)

By inaccessibility of the origin [c(1) ,∞) → ∞ as cn (1) cn

n

→

0. Since (60) holds for any ∈ (0, cn ), we get on (1) account of Fatou’s lemma by letting cn → 0 in (60) w ,s (1) n s− E{W (t)}n (cn , M ()). Thus, by Lemma 10 and the n square integrability of (t) over [−, ∞), for each n > 0 there t exist t ∗ s and dn > 0 such that (1) n (cn , M ( ))/dn < n wn ,s (t) > dn } < n . By letting for all t t ∗ and Pr{supt t ∗ W n t ∗ → ∞ in (60),we prove our claim on account of the continuity of Pr. This proves Claim B.1.2 for wn = n . The case 0 < wn < n and [0,cn ,∞) = ∞ a.e. is trivial and the case 0 < wn < n and [0,cn ,∞) < ∞ or wn > n follows from the strong Markov property. Proof of Claim B.2.0. Since Wn−1 (zn−1, , n−1 ) is proper it is sufﬁcient to prove that zn−1, (t) and n−1 (t) are reg2 2 ular. Let Sn−1 (zn−1, , n−1 ) = zn−1, + n−1 . Then, taking into account the form of the controller Cn−1 in (19) and the incremental rates of n−1, (Section 3) we have LSn−1 an−1,1 Sn + an−1,2 (zn , n )[1 + 2 ] for some an−1,1 0 and continuous and nondecreasing an−1,2 : R → R . Since (by (H2)) is square integrable over [−, ∞) then P{supt s 2 (t) < ∞} = 1 for all s 0. Moreover, by Claim B.1.0 and letting n → 0 and dn → ∞ in Claim B.1.2 it w ,s follows that P{supt s Wnn (t) < ∞}E = 1 for all s 0 and, since Wn (zn , n ) is proper, P{supt s |zn (t)| < ∞} = 1 and P{supt s |n (t)| < ∞} = 1 for all s 0. On application of Lemma 9, with V =Sn−1 , a =an−1,1 , b()=an−1,2 ()[1+ 2 ] and =( zn n )T , we obtain that zn−1, (t) and n−1 (t) are regular. References Battilotti, S. (2005). Stochastic stabilization of nonlinear systems in feedforward form with noisy outputs. IEEE Transactions on Automatic Control, 50, 100–105.

Battilotti, S. (2007). Lyapunov-based design of iISS feedforward systems with uncertainty and noisy measurements. SIAM Journal on Control and Optimization, 46, 84–115. Battilotti, S., & De Santis, (2005). Dwell time controllers for stochastic systems with switching Markov chain. Automatica, 41, 923–934. Hespanha, J. P. (2006). A model for stochastic hybrid system with application to communication networks. Nonlinear Analysis, 62, 1353–262. Hespanha, J. P., & Teel, A. R. (2006). Stochastic impulsive systems driven by renewal processes. In MNTS Proceedings. Hristu-Varsakelis, D., & Zhang, L. (2005). LQG Control under limited communication. In Proceedings of the IEEE conference on decision control (pp. 57–62). Khas’minskii, R. Z. (1980). Stochastic stability of differential equations. Sjithoff & Noordhoff: Rockville. Mazenc, F., & Iggidr, A. (2001). Backstepping with bounded feedbacks for systems in feedback form. In Fifth IFAC symposium on nonlinear control systems, St. Petersburg. Mao, X., Matasov, A., & Piunovskiy, A. B. (2000). Stochastic differential delay equations with Markovian switching. Bernoulli, 6, 73–90. Mazenc, F., Mondié, S., & Niculescu, S. I. (2003). Global asymptotic stabilization for chains of integrator with delay in the input. IEEE Transactions on Automatic Control, 48, 57–62. Nair, G. N., & Evans, R. J. (2000). Stabilization with data-rate-limited feedback: Tightest attainable bounds. Systems & Control Letters, 41, 49–76. Nair, G. N., Evans, R. J., Mareels, I. M. Y., & Moran, W. (2004). Topological feedback entropy and nonlinear stabilization. IEEE Transactions on Automatic Control, 49, 1585–1597. Oksendal, B. (1985). Stochastic differential equations. Berlin, Heidelberg: Springer. Tatikonda, S., Sahai, A., & Mitter, S. K. (2004). Stochastic linear control over a communication channel. IEEE Transactions on Automatic Control, 49, 1549–1561.

Stefano Battilotti was born in 1962 in Rome. He received the laurea degree in 1987 and the Ph.D. degree in 1992 from the University of Rome “La Sapienza”. He was appointed Assistant Professor in 1994 and Associate Professor in 1998 by the Dipartimento di Informatica e Sitemistica “Antonio Ruberti”. Since 2005 he is a Professor of Automatic Control in the same department. He is a Senior member of IEEE. His research interests are focused on the stabilization of nonlinear deterministic and stochastic systems. He has authored more than 30 papers in archival journals and the book “Noninteracting control with stability for nonlinear systems” (Springer-Verlag, 1994).

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