State Observability and Observers of Linear-Time-Invariant Systems ...

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State Observability and Observers of Linear-Time-Invariant Systems under Irregular Sampling and Sensor Limitations Le Yi Wang, Chanying Li, G. George Yin, Lei Guo, Cheng-Zhong Xu

Abstract—State observability and observer designs are investigated for linear-time-invariant systems in continuous time when the outputs are measured only at a set of irregular sampling time sequences. The problem is primarily motivated by systems with limited sensor information in which sensor switching generates irregular sampling sequences. State observability may be lost and the traditional observers may fail in general, even if the system has a full-rank observability matrix. It demonstrates that if the original system is observable, the irregularly sampled system will be observable if the sampling density is higher than some critical frequency, independent of the actual time sequences. This result extends Shannon’s sampling theorem for signal reconstruction under periodic sampling to system observability under arbitrary sampling sequences. State observers and recursive algorithms are developed whose convergence properties are derived under potentially dependent measurement noises. Persistent excitation conditions are validated by designing sampling time sequences. By generating suitable switching time sequences, the designed state observers are shown to be convergent in mean square, with probability one, and with exponential convergence rates. Schemes for generating desired sampling sequences are summarized. Index Terms—Observability, state observers, irregular sampling, persistent excitation, quantized sensors, strong convergence, mean square convergence.

I. I NTRODUCTION Tate estimation of linear-time-invariant systems is studied when output observations are sampled on a set of nonperiodic and irregular sampling times and the sampled values are corrupted by possibly dependent noises. State observability may be lost and the traditional observers may fail in general, even if the system has a full-rank observability matrix. This paper contributes to this problem in several related aspects: (1) Observability of systems under arbitrary sampling time

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L.Y. Wang is with Department of Electrical and Computer Engineering, Wayne State University, Detroit, Michigan 48202. [email protected]. Research of this author was supported in part by the National Science Foundation under DMS-0624849, and in part by the Air Force Office of Scientific Research under FA9550-10-1-0210. C. Li is with Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong. [email protected]. Research of this author was supported in part by the National Science Foundation under DMS-0624849. G. Yin is with Department of Mathematics, Wayne State University, Detroit, Michigan 48202. [email protected]. Research of this author was supported in part by the National Science Foundation under DMS-0907753, and in part by the Air Force Office of Scientific Research under FA9550-101-0210. L. Guo is with Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China. [email protected] C. Xu is with Department of Electrical and Computer Engineering, Wayne State University, Detroit, Michigan 48202. [email protected].

sequences is established. (2) An observer design procedure is introduced and recursive algorithms are developed. (3) The critical issue of persistent excitation conditions for convergence of state estimation under irregular sampling is resolved. (4) Convergence properties and convergence rates of the designed state observers are derived. Irregular sampling time sequences may be generated passively due to event-triggered sampling or low-resolution signal quantization, or actively by input control or threshold adaptation under binary-valued sensors. While synchronized periodic sampling has been extensively studied, non-uniform sampling has emerged from many applications. Studies on fundamental properties of non-uniform sampling remain an active area of research, see [9] and the references therein for some recent work in this area. Our problem here was initially investigated in [35]. The current paper demonstrates that if the continuoustime system is observable, then within any finite time interval, there exists a critical frequency such that as long as the density of sampling points exceeds the frequency, the initial state can be uniquely reconstructed, regardless the actual sampling time sequences. In this sense, it resolves completely the issue of observability of sampled systems with arbitrary sampling times. The result can be compared to the Nyquist frequency of a signal in Shannon’s sampling theorem under synchronized sampling for signal reconstruction. Binary sensors are used in many practical systems and systems involving communication channels. Due to sensor nonlinearity, the state estimation becomes a nonlinear filtering problem. The added difficulty is that the nonlinearity is nonsmooth and has only two output values. As a result, methodologies that rely on local linearization or small perturbation analysis such as extended Kalman filters are not applicable. Also, under irregular sampling, the sampled systems are time varying, and cannot be lifted to become linear-time invariant systems. Observer design methods for linear-time-invariant systems such as deadbeat design, pole placement, H 2 /H ∞ filtering [6], [8], [16], [28], etc., cannot be directly applied. Early work on optimal nonlinear filtering can be traced back to Stratonovich [30]. The first rigorous derivation was given by Kushner in his seminal paper [19]; see also [20]. Such equations are referred to as Kushner’s equations. Subsequently, the conditional law of the filter was further developed by Zakai [39], later referred to as Zakai’s equations. Nowadays, one of the well-known stochastic PDEs for nonlinear filters is the socalled Duncan-Mortensen-Zakai equation. One main challenge faced by nonlinear filtering problems is that the filters are

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generally infinite dimensional. Much effort has been devoted to finding finite dimensional filters. However, to date, there are only a handful of finite dimensional nonlinear filters known in existence, including the Wonham filter, Benes filter, etc. To deal with nonlinear and time-varying systems and random noises, we use a time domain formulation in a stochastic framework. The main challenges in these problems include: (1) When will the sampled system be observable under irregular sampling sequences? (2) How can the states be estimated on the basis of irregular sampling points? (3) How should the set of sampling times be selected to ensure that persistent excitation conditions for state estimation are validated? (4) What convergence properties can be derived for the state estimates? This paper focuses on resolving these issues. Since this paper deals with convergence analysis in stochastic systems, it is related to many classical treatments of similar topics. We cite [12], [13], [15], [21], [29] for their relevance to this paper, but emphasize that state observability under irregular sampling, state estimation with binary sensors, active observer design, convergence under mixing-type sampling disturbances, and convergence rates of state observers under irregular sampling schemes mark departure of our results from traditional ones. The rest of the paper is organized as follows. Section II formulates the state observation problems pursued in this paper. Section III focuses on observability of systems under arbitrary sampling times. The main result (Theorem 1) resolves this issue completely. The observer design is studied in Section IV. An observer structure is introduced first, which includes a discrete state estimator and a continuous-time state observer with updating schemes. It is shown that if the original system is observable, then the observer can be constructed and implemented. Recursive algorithms are derived. After certain normalization steps in expressing estimation errors, Section V investigates conditions on sampling time sequences that will validate persistent excitation conditions for convergence of state estimation. Section VI is devoted to convergence analysis. Subsection VI-A establishes convergence properties of the state observer in a finite horizon. Mean-squares (MS) convergence and strong (with probability one) convergence are established. Subsection VI-B deals with convergence analysis of the state observer over an infinite horizon. By generating suitable switching time sequences, we show that not only MS and strong convergence properties can be guaranteed, but also the convergence is exponentially fast. Generation of irregular sampling-time sequences is discussed in Section VII, including passive types of signal quantization and event-triggered sampling, and strategies of binary-valued sensors with active input control or threshold adaptation. An illustrative example is provided in Section VIII. Finally, some concluding remarks are given in Section IX. Some related results on identification, state estimation, and fault detection using binary or quantized outputs can be found in [17], [31], [33], [34], [35], [36]. In relation to the existing knowledge on observability of sampled systems, we refer the reader to standard textbooks on digital control systems; see, e.g., [7], [18], [24] for classical synchronized sampling schemes on linear systems, and [1], [11] on nonlinear systems. Shannon’s sampling theorem is a fundamental result on digital

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signal processing, see [25]. References [26], [31] contain more recent studies on observability of sampled systems. Irregular sampling may occur due to event triggered sampling [3], [23] or communication uncertainty and interruptions [12]. Some preliminary results of this paper were reported in [37], [38]. II. P ROBLEM F ORMULATION Consider an MISO (multi-input-single-output) linear timeinvariant system  x(t) ˙ = Ax(t) + Bu(t) t≥0 (1) y(t) = Cx(t),

where A ∈ Rn×n , B ∈ Rn×m , and for t ≥ 0, u(t) ∈ Rm is the control input, x(t) ∈ Rn is the state, and y(t) ∈ R is the system output.1 The initial state is unknown. We are interested in estimation x(t), from some limited observations on y(t). In our setup, the output y(t) is only measured at a sequence of sampling time instants {ti } with measured values γ(ti )2 and noise di γ(ti ) = y(ti ) − di . (2)

We would like to estimate the state x(t) from information on u(t), {ti } and {γ(ti )}. In general, the sampling-time sequence is irregular and non-periodic. In this paper, the switching time sequence will be deterministic. Generation of such irregular sampling-time sequences may be passive as in event-triggered sampling and low-resolution signal quantization. Here, we are mostly motivated by state estimation problems with binary-valued observations. Active control of inputs or thresholds causes system outputs to cross sensor thresholds and the sensor outputs to switch values, generating an irregular sampling-time sequence. As a result, the sequence {ti } will be interchangeably called “sampling-time sequence” or “switching-time sequence.” Generation of such sequences will be discussed in Section VII. It is obvious that state estimation will not be possible if the system is not observable. Also, in this paper, dk is allowed to be dependent. These are stated in the following assumptions. Assumption 1: The following conditions hold. (a) The system is observable, i.e., Wo0 = 0 0 n−1 0 [C (CA) . . . (CA ) ] is full rank. (b) The sensor noise {dk } is a stationary sequence of ρ∗ mixing random variables and Edk = 0. Remark 1: We refer the reader to [32, p. 101] and [2, p. 1466] for a definition of ρ∗ mixing processes. Mixing processes are essentially those whose remote past and distant future are asymptotically independent; different types of mixing processes are characterized by their mixing measures. Loosely speaking, a ρ∗ mixing process is characterized by using covariance as a mixing measure. The class of ρ∗ mixing processes include i.i.d. sequences, stationary martingale difference sequences, and certain φ-mixing processes [10, Section 1 We limit our studies to MISO systems for notational simplicity, although technically inclusion of MIMO cases is quite straightforward. An MIMO system provides more measurement information than MISO systems. Consequently, if one output is sufficient for state estimation, so will be its MIMO counterpart. 2 Using the notation γ(t ) is non standard, but consistent with the interprei tation that it is the threshold value of a binary sensor.

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7.2]. For example, a sequence of i.i.d. random variables {dj } (discrete-time ‘white noise’) is ρ∗ -mixing with covariance Edi dj = 0 for i 6= j. A moving-averaging process of order m driven by a ‘white noise’ is also a ρ∗ -mixing process, where the covariance becomes 0 for any time lag larger than m. Likewise, if dk is a discrete-time Markov chain that is irreducible and aperiodic with a finite state space, then it is also a ρ∗ -mixing process. Here the covariance decays exponentially fast. In addition, ρ∗ -mixing also includes Markov chains with countable spaces satisfying certain conditions, as well as many other infinitely correlated noises as long as their correlations decay sufficiently fast. We should emphasize that our results show that the same convergence properties and convergence rates for i.i.d. noises continue to hold under ρ∗ mixing processes, as such they are not more conservative. For the problems of this paper, practical importance of ρ∗ mixing processes stems from the fact that if an i.i.d. noise passes through a communication channel with finite memory, or infinite but stable decaying memory, the received signal will be corrupted by ρ∗ mixing noises. To simplify notation, in some proofs we adopt the convention of using a generic positive constant κ > 0 to represent unspecified positive constants. Hence, for any a > 0, aκ = κ. This simplifies expressions where actual values of some positive constants are irrelevant to the conclusions. III. O BSERVABILITY OF I RREGULARLY S AMPLED S YSTEMS We start with the system in (1) with noise-free observations (so, y(ti ) = γ(ti ) in this section). The initial state x(0) is unknown. Suppose that the output of the system is sampled at a set of N time instances TN = {ti ≥ 0, i = 1, . . . , N }, generating the set of observations YN = {y(ti ), i = 1, . . . , N }. Observability of the sampled system deals with reconstruction of the state x(t) from output observations YN . Since the system contains no uncertainty and the input is known, this is equivalent to reconstruction of the initial state x(0) from YN . Definition 1: (i) System (1) is said to be TN observable if x(0) can be uniquely determined from any observations YN on TN . (ii) For a given time interval [0, T ] and an integer NT , the system is said to be NT -sample observable if the system is TNT observable for any TNT with 0 ≤ ti ≤ T , i = 1, . . . , NT . We use some examples from [7], [27] to illustrate the key issues involved in observability of the irregularly sampled system. Suppose the system contains complex eigenvalues. Then for some non-zero initial state x(0), the output will be zero infinitely often, say, at ti , i = 1, 2, . . .. If these are the actual sampling times, the output samples will be zeros and provide no information about x(0), rendering the sampled system unobservable with respect to such observation sets. On the other hand, it is well understood that for any given ndimensional observable system, there exists a sufficiently small time interval [0, T ] such that if n sampling times ti ∈ [0, T ], then the system is always Tn observable. In other words, the system is n-sample observable. However, this is not the case when T is not small. The main goal of this section is to establish a general result that resolves the issues illustrated in these examples.

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The main result of this section is the following theorem. Suppose the matrix A in (1) has eigenvalues λj , j = 1, . . . , n. Denote δ := max1≤i,j≤n {=(λi − λj )}, where =(z) is the imaginary part of z ∈ C. For a given time interval [0, T ], define T µT = 2(n − 1) + δ. (3) 2π Theorem 1: Suppose that Assumption 1 (a) is true. Given T > 0, if N ≥ µT , the system in (1) is N -sample observable. Remark 2: This theorem asserts that for a given T > 0, if all sampling points are confined in [0, T ] and the number of samples exceeds µT , the initial state can always be uniquely determined from YN , regardless the actual sampling times. In some sense, the condition on sampling in Theorem 1 is the weakest restriction on a sampling scheme to reconstruct δ := µ, which is x(0). Asymptotically for large T , µTT ≈ 2π independent of the system order n. One may compare this with the Nyquist frequency of Shannon’s sampling theorem for signals. Suppose the largest imaginary part of the eigenvalues of A is ωmax . Then, δ = 2 ωmax and µ = ωmax is precisely π the Nyquist frequency in Hz, if one interprets ωmax as the bandwidth of the system. The result here is more general since it does not require the sampling scheme to be periodic. In the special case of synchronized sampling and pure imaginary eigenvalues, the sampled values may be all zeros if the sampling frequency is below µ (see [27] for details), implying that the observability is lost. In other words, µ is an asymptotically tight bound for large T . The remainder of this section is devoted to proving Theorem 1. Without loss of generality, we may focus only on the zero-input relationship y(t) = CeAt x(0). Under N sampling times TN = {ti , i = 1, . . . , N }, we have YN = {y(ti ), i = 1, . . . , N } with y(ti ) = CeAti x(0), i = 1, . . . , N, which can be written as YN = MN x(0), where     y(t1 ) CeAt1     .. .. YN =  (4)  ; MN =  . . . y(tN )

CeAtN

Consequently, the system is TN observable if and only if MN is full rank. We express eAt in terms of the matrices I, A, A2 , · · · , An−1 , At e = α1 (t)I + α2 (t)A + · · · + αn (t)An−1 , where α(t) = [α1 (t), . . . , αn (t)]0 can be solved by the Lagrange-Hermite interpolation (when the eigenvalues of A are all distinct, the method is called the Sylvester interpolation) [14]. Suppose A has l distinct eigenvalues Pl λi , i = 1, . . . , l of multiplicity mi , respectively. Here, i=1 mi = n. Define the modes3 of the A matrix by  ξ(t) = eλ1 t , · · · ,

tml −1 λl t tm1 −1 λ1 t e , · · · , eλl t , · · · , e (m1 − 1)! (ml − 1)!

0

.

(5)

Then, for any given Ql t > 0, the characteristic polynomial of At is cAt (z) = i=1 (z − λi t)mi . By [14, Section 6.1.14, 3 Although in engineering, it is customary to use real-valued expressions for the modes of A matrix such as {et sin(2t), et cos(2t)} in place of {e(1+2j)t , e(1−2j)t }, development of our results in this section is simplified when the original eigenvalues (possibly complex) are used.

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Pn−1 pp. 390], there is a polynomial r(z) = i=0 ci+1 z i , which satisfies the interpolation conditions: for j = 0, · · · , mi − 1 and i = 1, · · · , l dj ez dj r(z) = = eλi t . (6) dz j z=λi t dz j z=λi t The coefficients ci (t), i = 1 · · · , n, depending on t, are uniquely determined by (6). r(z) is said to interpolate ez and its derivatives at the roots of cAt (z). Let αi (t) = ci (t)ti−1 , i = 1, . . . , n. Then, (6) can be rewritten as Λ0 α(t) = ξ(t), (7) where the n × n matrix Λ0 depends on λi , i = 1, . . . , l and their multiplicities and is invertible due to the uniqueness of solutions of (6) (see [14, pp. 390]). From the proof of [14, Theorem 6.2.9(a)], one has eAt = r(At) = α1 (t)I + α2 (t)A + · · · + αn (t)An−1 .

tj−1 λi t vi,j g(t) = e , (j − 1)! i=1 j=1

(9)

where λi ∈ C, i = 1, · · · , l are the l distinct eigenvalues of A with multiplicity mi and m0 = 0. Recall that a nonlinear function g(t) is said to be non-trivial if g(t) 6≡ 0. In reference to ξ(t), g(t) = ξ 0 (t)v. Since the elements of ξ(t) are linearly independent, for any v 6= 0, g is non-trivial. The following key lemma on the number of zeros of exponential polynomials can be derived from [5, Theorem 3.2.47]. Let µT be defined as in (3). Lemma 1: The number NT of zeros in [0, T ] of a nontrivial exponential polynomial g defined in (9) is bounded by N T ≤ µT . The next lemma is concerned with the rank of MN in (4). Lemma 2: If the system in (1) is observable and N > µT , then MN is full rank. Proof. Suppose A has l distinct eigenvalues λi of multiplicity mi , i = 1, . . . , l. From (8), we have   C  CA    CeAt = [α1 (t), · · · , αn (t)]   = α0 (t)Wo . (10) ..   . CAn−1

0

α (tN )

Then,

[0, T ], i = 1, 2, · · · , N ,   .

MN = ΓN Wo .

(11)

(12)

Since Wo is full rank, we only need to show that ΓN is full rank. However, by (7) Λ0 α(t) = ξ(t), which implies ΓN Λ = ΞN where ΞN

(8)

It should be pointed out that although determination of α(t) from (6) is unique, there may be other α(t) that satisfies (8) but not (6). In fact, in the proof of Theorem 6.2.9(a) of [14], r(z) is selected from interpolation of ez and its derivatives at the roots of the minimal polynomial of At. However, here r(z) interpolates ez and its derivatives at the roots of cAt (z), which still leads to r(At) = eAt by the same argument as in [14, pp. 387]. This indicates that by choosing different polynomials that annihilate At, one may obtain different expressions of eAt in the form of (8). Here we use the particular expression (7), where Λ is invertible. For any given x(0), y(t) = CeAt x(0) is a linear combination of the modes of A. As a result, it belongs to the class of exponential polynomials: for any t ∈ [0, T ] and v = [v1 , · · · , vn ]0 ∈ Cn , let mi l X X

For any N sampling times ti ∈ define  0 α (t1 )  .. ΓN =  .



 ξ 0 (t1 )   .. = . . 0 ξ (tN )

(13)

(14)

Now, by the previous argument, Λ is full rank. Consequently, it remains only to show that ΞN is full rank. For any β ∈ Cn and β 6= 0, define g(t) = ξ 0 (t)β. Since the  elements of  ξ(t) are independent, g(t) is non-trivial. g(t1 )   .. ΞN β =   = 0 means that g(t) has N zeros in .

g(tN ) [0, T ]. However, by Lemma 1, the number of zeros of g(t) is bounded by µT < N . This contradiction implies that β = 0. Since β is arbitrary, this proves that ΞN is full rank. This completes the proof. 

Proof of Theorem 1. The system output can be written as Z t At eA(t−τ ) Bu(τ )dτ = CeAt x(0)+ζ(t), y(t) = Ce x(0)+C t0

where ζ(t) is known. For N sampling times ti ∈ [0, T ] with i = 1, 2, · · · , N , YN = [y(ti ), · · · , y(tN )]0 can be expressed as YN = MN x(0) + ZN , where ZN is known and MN is defined in (4). Since the system is observable and N > µT , by Lemma 2, MN is full rank. Consequently, x(0) can be determined uniquely from YN .  IV. O BSERVER D ESIGN We now study observer design when the output measurements are noise corrupted. A. Observers For both t > t0 and t < t0 , the solution R t to system (1) can be expressed as x(t) = eA(t−t0 ) x(t0 ) + t0 eA(t−τ ) Bu(τ )dτ. Suppose {ti , i = 1, . . . , N } is a sequence of sampling times. The observer is to estimate x(t), tN ≤ t < tN +1 before the next sampling occurs at tN +1 . Since new information about the state is obtained only at the next sampling time tN , a discrete-time state estimator will first generate an estimate zN of the state at tN , which will be used to update the state estimate to x b(tN ) = zN , and then the observer will run open loop in (tN , tN +1 ).

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For ti ≤ tN , we have

ΓN +1 =

γ(ti ) + di = y(ti ) = CeA(ti −tN ) x(tN ) + C

Z

ti

x(tN ) = γ(ti ) − v(ti , tN ) + di ,

ΦN

  = 

VN

CeA(t1 −tN ) .. . CeA(tN −1 −tN ) C 

Z tN eA(ti −τ ) Bu(τ )dτ eA(ti −τ ) Bu(τ )dτ + tN Z tNtN +1 A(ti −tN ) eA(tN −τ ) Bu(τ )dτ = w(ti , tN ) + e =

i = 1, . . . , N. (15)

 = 



 γ(t ) 1  ..   .  , ΓN =    γ(tN−1 ) γ(tN ) 





 , 

Z

ti

tN +1

= w(ti , tN ) + eA(ti −tN ) w(tN , tN +1 ). (16)

As a result, v(ti , tN +1 ) = Cw(ti , tN +1 ) = Cw(ti , tN ) + CeA(ti −tN ) w(tN , tN +1 ) = v(ti , tN ) + CeA(ti −tN ) w(tN , tN +1 )



d1 v(t1 , tN ) .. ..    .  , DN =  . .    dN−1 v(tN−1 , tN ) dN 0

This implies VN +1 =

Then, (15) can be written as ΦN x(tN ) = ΓN − VN + DN .

(17)

Suppose that ΦN is full rank, which will be established in later sections. Then, a least-squares (LS) estimate of x(tN ) is given by zN = (Φ0N ΦN )−1 Φ0N (ΓN − VN ), (18) with estimation error eN = (Φ0N ΦN )−1 Φ0N DN . The observer will have the state estimate updated at tN by x b(tN ) = zN , and will run as an open loop observer for tN ≤ t < tN +1 , x b˙ (t) = Ab x(t) + Bu(t). Due to the state update at tN , x b(t) is discontinuous at tN , but continuously differentiable in (tN , tN +1 ). B. Recursive Algorithms zN can be calculated recursively. Although zN in (18) appears to be in a LS form, the standard recursive LS algorithm is not applicable, due to the fact that ΦN +1 and ΓN +1 − VN +1 are not merely expanded from ΦN and ΓN − VN by adding a new entry. New information at N + 1 consists of the sampling time tN +1 and the measured value γ(tN +1 ), and the following calculated values: MN +1 = eA(tN +1 −tN ) and w(tN , tN +1 ) =

Z

tN

.

tN +1

Define 



w(ti , tN +1 ) Z ti eA(ti −τ ) Bu(τ )dτ =

Since the second R t term is known, it will be denoted by v(ti , tN ) = C tNi eA(ti −τ ) Bu(τ )dτ. This leads to the observations Ce

ΓN γ(tN +1 )

Moreover, eA(ti −τ ) Bu(τ )dτ.

tN

A(ti −tN )



eA(tN −τ ) Bu(τ )dτ.

tN +1

The relationship among ΦN +1 , ΓN +1 , VN +1 and ΦN , ΓN , VN can be derived as follows:     −1 ΦN M N ΦN eA(tN −tN +1 ) +1 , = ΦN +1 = C C



VN + ΦN w(tN , tN +1 ) 0



.

In the following derivation, let PN = (Φ0N ΦN )−1 ,

−1 PN C 0 . KN = MN

(19)

Theorem 2: Suppose that for some N0 , ΦN0 is full rank, and let zN0 , PN0 , and KN0 be specified by (18) and (19), respectively. For N > N0 , zN in (18) can be updated recursively by zN +1 = MN +1 (I − KN +1 CMN +1 )(zN − w(tN , tN +1 )) +PN +1 C 0 γ(tN +1 ) 0 0 0 0 −1 KN +1 = PN MN +1 C (1 + CMN +1 PN MN +1 C ) 0 PN +1 = MN +1 (I − KN +1 CMN +1 )PN MN +1 .   −1 ΦN M N +1 , by the matrix inversion Proof. Since ΦN +1 = C lemma PN+1 = (Φ0N+1 ΦN+1 )−1 −1,0 0 −1 + C 0 C)−1 = (MN+1 ΦN ΦN MN+1 −1 0 0 = MN+1 (PN + MN+1 C 0 CMN+1 )−1 MN+1 0 0 0 = MN+1 (PN − PN MN+1 C (1 + CMN+1 PN MN+1 C 0 )−1 0 ×CMN+1 PN )MN+1 .

Let −1 KN+1 := MN+1 PN+1 C 0 0 0 = (PN − PN MN+1 C 0 (1 + CMN+1 PN MN+1 C 0 )−1 0 0 ×CMN+1 PN )MN+1 C 0 0 = PN MN+1 C 0 (1 + CMN+1 PN MN+1 C 0 )−1 ,

which implies 0 PN +1 = MN +1 (I − KN +1 CMN +1 )PN MN +1 .

Moreover,

Φ0N+1 (ΓN+1 − VN+1  )

   VN + ΦN w(tN , tN+1 ) ΓN − γ(tN+1 ) 0 −1,0 0 −1,0 0 = MN+1 ΦN ΓN + C 0 γ(tN+1 ) − MN+1 ΦN VN −1,0 0 −MN+1 ΦN ΦN w(tN , tN+1 ) −1,0 0 −1,0 −1 = MN+1 ΦN (ΓN − VN ) + C 0 γ(tN+1 ) − MN+1 PN w(tN , tN+1 ). 0

−1, = [MN+1 Φ0N , C 0 ]

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Now, zN+1 = PN+1 Φ0N+1 (ΓN+1 − VN+1 0 −1,0 0 = MN+1 (I − KN+1 CMN+1 )PN MN+1 MN+1 ΦN (ΓN − VN ) 0 +PN+1 C γ(tN+1 ) − MN+1 (I − KN+1 CMN+1 )w(tN , tN+1 ) = MN+1 (I − KN+1 CMN+1 )zN + PN+1 C 0 γ(tN+1 ) −MN+1 (I − KN+1 CMN+1 )w(tN , tN+1 ) = MN+1 (I − KN+1 CMN+1 )(zN − w(tN , tN+1 )) +PN+1 C 0 γ(tN+1 ).



The proof is thus concluded.

V. E STIMATION E RRORS AND P ERSISTENT E XCITATION C ONDITIONS A. Estimation Error Representation and Normalization From (17) and (18), the estimation error for x(tN ) at tN is (Φ0N ΦN )−1 Φ0N DN

e(tN ) = x b(tN ) − x(tN ) =
−1 1 0 1 0 = N r ΦN ΦN N r ΦN D N

(20)

for any 1/2 < r < 1.4 From the expression ΦN given in (16), we define the (negative) time difference τi = ti − tN , 0 = τN > τN −1 = tN −1 − tN > · · · > τ1 = t1 − tN . A typical row of ΦN is CeAτi . By the Cayley-Hamilton Theorem [22], the matrix exponential can be expressed by a polynomial function of A of order at most n − 1. eAt = α1 (t)I + · · · + αn (t)An−1 ,

(21)

where the time functions αi (t) can be derived by the LagrangeHermite interpolation method, see [14], [22] for the algorithms. While the exact calculation of αi (t) is not important here, it needs to be emphasized that αi (t) is a linear combination of the modes of A.5 This implies   C  CA    CeAτi = [α1 (τi ), . . . , αn (τi )]   = ϕ0 (τi )Wo , ..   . n−1 CA (22) where ϕ0 (τi ) = [α1 (τi ), . . . , α n (τ0i )] and Wo is the observϕ (τ1 )   .. ability matrix. Denote ΨN =   . We have . ϕ0 (τN )

ΦN = ΨN Wo .

(23)

As a result,
for any r > 0, eN = −1 1 0 Ψ D . Wo−1 N1r Ψ0N ΨN r N N N Under Assumption 1 (a), Wo−1 exists. Convergence results will be established by the following two sufficient conditions: 1 1 0 0 N r ΨN DN → 0, and N r ΨN ΨN > βI, for some β > 0. 4 This

expression is valid for any r > 0. However, for convergence of Φ0N DN /N r , r > 1/2 is usually required, The boundedness of (Φ0N ΦN /N r )−1 can be established usually for r < 1 only. 5 If A has a real eigenvalue λ of multiplicity m, then the corresponding modes are eλt , teλt ,. . . , tm−1 eλt . If A has a complex pair of eigenvalues σ ± jω of multiplicity m, then the corresponding modes are eσt sin(ωt), eσt cos(ωt), teσt sin(ωt), teσt cos(ωt), . . . , tm−1 eσt sin(ωt), tm−1 eσt cos(ωt).

In principle, for the first condition to be true ΨN must be bounded. Sampling time sequences must be carefully selected to ensure that the second condition is valid. However, ΨN is often unbounded, as shown in the following example. Example 1: Consider a first-order stable system x˙ = −2x+ u. Then, n = 1 and α1 (t) = e−2t , and α1 (τi ) = e−2(ti −tN ) . For any fixed ti , when tN → ∞, τi → −∞ and α1 (τi ) becomes unbounded. One possible remedy is to normalize the error expressions so that the modified sequence becomes bounded. For this exame1 (τi ) with α e1 (τi ) = e−2ti . ple, one may write α1 (τi ) = e2tN α e N where Ψ e N = e−2tN ΨN is bounded, Then, ΨN = e2tN Ψ and 

1 0 ΨN ΨN Nr

−1

1 0 ΨN DN = Nr



e2tN e 0 e ΨN ΨN Nr

−1

1 e0 ΨN DN . Nr

This example is generalized to a normalization procedure below. It is noted that only stable modes of A will cause unboundedness of ΨN since τi < 0. We shall decompose ϕ(τ ) in (22) into two parts ϕ(τ ) = ϕs (τ ) + ϕu (τ ), where ϕs (τ ) contains all modes that are unbounded when τ → −∞ and ϕu (τ ) contains all modes that are uniformly bounded for τ < 0. More concretely, let the distinct eigenvalues of A be Υ = {νi , i = 1, . . . , m} where νi has multiplicity li . Decompose Υ into Υs = {νi ∈ Υ : 0.

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˜ 0 DN /N r takes the form A typical element of Ψ N PN r α ˜ (t , t )d /N . Since α ˜ j (tk , tN ) is doubly indexed j k N k k=1 on tk and tN , this is a triangularly weighted sum of mixing variables dk , and classical results on strong convergence of single-indexed weighted sums of mixing variables cannot be applied. The main convergence results will be given by Lemma 3 and Theorem 4 in Section VI. B. Persistent Excitation Conditions and Sampling Time Sequences e N will be bounded and the remaining With normalization, Ψ key issue is the following persistent excitation condition. PE Condition: For some 1/2 < r < 1,   f (tN ) e 0 e (26) ΨN ΨN > 0, β = inf σmin N Nr

where σmin (H) is the smallest eigenvalue of H for a suitable symmetric matrix H. In a typical parameter estimation problem, the PE condition is imposed on the input, leading to a condition for identificae N is determined tion input design. However, in our problems, Ψ by the matrix A and the sampling sequence only. Hence, the PE condition is a condition on A and the switching time e0 Ψ e sequence {ti }. The PE condition means that (a) Ψ N N is e0 Ψ e N grow at full rank and (b) the smallest eigenvalues of Ψ N least as fast as N r /f (tN ). These are non-trivial conditions. They depend on A, C, and the time sequence. First, (26) is always satisfied under the following conditions. A switching time sequence {ti } in [0, T ] is said to be uniformly spread in [0, T ] if Z T 0 1 0 lim ΦN ΦN = eA t C 0 CeAt dt. (27) N →∞ N 0

The simplest example is when the switching time sequence is equally spaced in [0, T ], for which the limit follows from the Riemann integration. But, there are many other time sequences that also satisfy (27). Proposition 1: If the system is observable and for a fixed T > 0, the time sequence is uniformly spread in [0, T ], then for sufficiently large N , (26) is always satisfied with r = 1. Proof. By hypothesis N1 (Φ0N ΦN ) converges to the integration R T A0 t 0 At e C Ce dt, which is full rank for any T > 0 when 0 (A, C) is observable. Hence, there exists N0 such that for all N > N0 , N1 Φ0N ΦN is uniformly bounded away from 0. Since Wo is full rank and f (tN ) is bounded in a finite interval, (26) is satisfied.  More generally, suppose {τi } is a time sequence in [−T, 0]. Since αj (τ ) are continuous, they are bounded on [−T, 0]. For i, j = 1, . . . , n, suppose λij = PN limN →∞ N1 k=1 αi (τk )αj (τk ) exists. Let M be the n × n symmetric matrix M = [λij ]. Then N1 Ψ0N ΨN → M. As a result, the condition inf N σmin ( N1 Ψ0N ΨN ) > 0 is satisfied if M > 0. On a finite interval, this implies (26). In general, however, (26) is not always satisfied. When the time sequence is causally and sequentially generated, we show by the following examples that (26) may not hold in some choices of time sequences.

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 0 1 , C = [1, 0]. Then −1 0 (A, C) is observable. If the time sequence is ti = i2π, ΦN contains identical rows of C, which will never be full rank for any N . This example motivates some time sequence selections to avoid such situations. Example 4: Consider the same system as in Example 3. Suppose that ti is sequentially generated and approaches 2π. In this case, if we choose the sequence to approach 2π sufficiently fast, the smallest eigenvalues of Ψ0N ΨN become bounded. This implies that N1r Ψ0N ΨN → 0, violating (26). This example shows that even in finite horizon scenarios, one needs to choose the time sequence carefully. We now establish conditions on sampling time sequences that will validate (26). atN for some Case 1: ϕs 6≡ 0. In this case, f (tN ) = tm Ne integer m and a ≥ 0. Let the sampling sequence be selected e 0Ψ e such that for some l > 0, Ψ l l > 0. Then there exists some e e0 Ψ κ > 0, for any N > l, σmin (Ψ N r N ) ≥ κ > 0. Consequently, m atN (26) is satisfied if tN e ≥ βN κ which provides a sufficient condition on the sampling time sequences. Example 5: Returning to Example 1, we have Example 3: Suppose A =

N atN e2tN e 0 e e2tN X −4tk f (tN ) 0 −4t1 e Ψ Ψ = Ψ Ψ = e ≥ e N N N N Nr Nr Nr Nr k=1

As a result, by choosing tN ≥ 2t1 + 21 ln(βN r ) = 2t1 + 1 r 2 ln β + 2 ln N, the PE condition (26) is satisfied. e N = ΨN . All Case 2: ϕs ≡ 0. In this case, f (tN ) = 1 and Ψ 0 e terms in ΨN are either bounded or decaying to 0 exponentially, as tN → ∞. The following example illustrates how the sampling time sequence should be selected. Example 6: Consider a first-order unstable system x˙ = 3x + u. Then, n = 1 and α1 (t) = e3t , and α1 (τi ) = e3(ti −tN ) . PN −6(tN −ti ) 0 . Letting ti = In this case, ΨN ΨN = i=1 e PN −6(t PN i N +1 0 N −ti ) (ln i)/6, ΨN ΨN = e = i=1 i=1 N = 2 , 0 r which implies for any 1/2 < r < 1, ΨN ΨN /N will satisfy (26). We now present a rigorously established sampling time sequence for the PE condition. Suppose that the outputs of the system (1) are sampled at some times {ti } with t1 = 0 and for some constant Te > 0 ti+1 − ti ≥ log(i + 1)− log i and ti ≤ log i + Te,

i ≥ 1. (28) The following lemma shows that the PE condition is indeed satisfied if the outputs are measured at the sampling times defined in (28). Theorem 3: Under Assumption 1 (a) and (28), there is a constant κ1 > 0 such that κ1 N . (29) σmin (Ψ0N ΨN ) ≥ 2[ν+µ(µ−1)] log N This theorem implies that under this sampling time sequence, for any 0 < r < 1, the PE condition (26) is satisfied. The proof of this theorem, which contains some interesting technical results on function zeros and an improved Lojasiewicz inequality, is highly involved and is postponed to Appendix.

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The above examples and theorem indicate a practical guideline in sampling time sequence selections, which will be called Principle of Logarithmic Time Sequences: To satisfy the PE condition (26), the sampling time sequence should take the form of ti = a + b ln i for some positive constants a and b that depend on the system matrix A. VI. C ONVERGENCE A NALYSIS A. Convergence Analysis in Finite Horizon In convergence analysis over a finite horizon, we should fix a time interval [0, T ]. The case of infinite horizon cases will be discussed in the next section. For implementation, time sequences in a finite time interval must be generated causally (no future information is used in selecting the current ti ) and sequentially (ti+1 > ti ). In this case, since ti is monotone and bounded, ti always approaches a limit. As a result, we have ti ∈ [0, T ] and ti % T . However, we also allow the case of pre-determined finite N sampling points. In the later case the statement with a limit “N → ∞” should be understood in the following sense: For any desired N , a sampling scheme can be performed such that at least N sampling times occur in this time interval. This problem will become immaterial for infinite horizon cases in which we always assume that the time sequence is generated causally and sequentially. To proceed, we first establish a rate-of-convergence result regarding the noise. The following result on strong convergence of triangularly weighted sums of ρ∗ -mixing variables is essential for this convergence analysis. Lemma 3: [2, Theorem 2] Let {dk , k ≥ 1} be a ρ∗ -mixing sequence of identically distributed random variables, αp > 1, α > 1/2 , and Ed1 = 0 for α ≤ 1. Suppose that {aN i , 1 ≤ i ≤ N } is an array of real numbers satisfying N X i=1

p

δ

|aN i | = O(N ), 0 < δ < 1.

(30)

PN −1/p

Then N i=1 aN i di → 0 w.p.1 as N → ∞. The following theorem is crucial in obtaining the convergence results as well as large-time behavior of the systems. Theorem 4: Under Assumption 1, for any r > 1/2, 1 ˜0 kΨ DN k → 0 w.p.1 as N → ∞, Nr N

(31)

˜ N is defined by (24). where Ψ Proof. To prove the assertion, it suffices to look at the PN ˜ 0 DN /N r . By the ˜ j (τk )dk /N r of Ψ jth component k=1 α N transformation in (24), α ˜ j is uniformly bounded, |˜ αj | ≤ κ, where κ is a generic positive constant. Select p = 2 and ε = r−1/p = r−1/2. Since 1/2 < r < 1, we have 0 0. It follows PN that for a typical term N1r i=1 α ej (τi )di in (32), N 1 X α ej (τi )di )2 E( r N i=1 N N 1 XX = 2r α ej (τi )e αj (τj )Edi dj N i=1 j=1

≤ This implies

N N κ XX κ |Edi dj | ≤ 2r−1 . N 2r i=1 N

N 2r−1 E(

j≥i

N 1 X α ej (ti )di )2 ≤ κ. N r i=1

(33)

e0 Ψ e −1 exists and Moreover, by (26), ( N1r Ψ N N)   e0 Ψ e −1 ≤ 1 , where σmax (·) is the largest σmax ( N1r Ψ N N) β eigenvalue. This, together with the existence of Wo−1 and (33) implies N 2r−1 Ee0N eN ≤ κ < ∞, (34) for some constant κ  Under some mild conditions, convergence of x b(tN ) to x(tN ) implies that x b(t) − x(t) → 0 for any tN ≤ t < tN +1 . Let Th > 0 be a finite time horizon and e(t) = x b(t) − x(t). Then it is easily seen that e(tN ) = eN . Theorem 6: Under Assumption 1 and (26), for t ∈ [tN , tN + Th ] and tN + Th < tN +1 maxtN ≤t≤tN +Th Ee0 (t)e(t) → 0, N → ∞. Proof. This follows from the fact that for t ∈ [tN , tN + Th ], the estimator runs open loop and e(t) = eA(t−tN ) e(tN ). Since Th is finite and eA(t−tN ) is continuous, there exists a constant κ > 0 such that maxtN ≤t≤tN +Th σmax (eA(t−tN ) ) = κ < ∞. Consequently, for any t ∈ [tN , tN + Th ] Ee0 (t)e(t) ≤ κEe0 (tN )e(tN ) → 0 as N → ∞. 

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B. Convergence Analysis over Infinite Horizons The convergence properties of the previous section are established over a finite time interval. For stability analysis of the closed-loop system, infinite time horizon must be considered. We now investigate observation errors under certain unbounded switching time sequences. In this section, time sequences will always be causally and sequentially generated. 1) Relationship between eN and e(t): error P∞ The observer 0 sequence e(tN ) is said to be in l2 if N =1 Ee (t )e(t )< N N R∞ ∞ and e ∈ L2 (mean square integrable) if 0 Ee0 (t)e(t)dt < ∞. Let |tN +1 − tN | ≤ q(N ). The next lemma claims that for dealing with convergence properties of e(t) we may concentrate on the sequence eN . Lemma 4: If q(N ) is uniformly bounded and tN → ∞, then (a) eN → 0 w.p.1 as N → ∞ implies that e(t) → 0 w.p.1 as t → ∞. (b) eN → 0 in mean square as Np→ ∞ implies that e(t) → 0 in mean square as t → ∞. (c) q(N )eN ∈ l2 implies e ∈ L2 . Proof. (a) Since the observer is running open loop for t ∈ [tN , tN +1 ), we have e(t) = eA(t−tN ) e(tN ). Now if tN +1 −tN is uniformly bounded, there exists a constant κ > 0 such that max

tN ≤t≤tN +1

e0 (t)e(t) ≤ κe0 (tN )e(tN ).

(35)

This and the hypothesis tN → ∞ implies that e(tN ) → 0 as N → ∞ w.p.1 leading to e(t) → 0 w.p.1. as t → ∞. (b) The inequality (35) implies further that ≤ κEe0 (tN )e(tN ). Hence, maxtN ≤t≤tN +1 Ee0 (t)e(t) 0 Ee (tN )e(tN ) → 0, as N → ∞ implies that Ee0 (t)e(t) → 0 as t → ∞. (c) Moreover, Z

∞

Ee0 (t)e(t)dt 0 Z tN = lim Ee0 (t)e(t)dt N→∞ 0 ∞ N Z tj X X ≤ κ lim Ee0 (tj−1 )e(tj−1 )dt ≤ κ q(j)Ee0j ej . N→∞

j=1

tj−1

j=1

p As a result, q(N )e(tN ) ∈ l2 implies that e ∈ L2 . 2) MS and Strong Convergence:



Case 1: ϕs (τ ) 6= 0 In this case, νmin ≤ 0 and hence fs (tN ) → ∞, tN → ∞ or fs (tN ) = 1 (if νmin = 0 and l = 1) as tN → ∞, τi = ti − tN → −∞. Since 1/f (tN ) is bounded, the next theorem follows from Theorem 5. Theorem 7: Under Assumption 1 and (26), if ϕs (τ ) 6= 0, then ζ = supN N 2r−1 Ee0 (tN )e(tN ) < ∞ and e(tN ) → 0 w.p.1 as N → ∞. Case 2: ϕs (τ ) = 0 In this case, ϕ(τ ) = ϕu (τ ) with |ϕu (τ )| → 0, τ → −∞. As a result, when tN → ∞, for any fixed i, ϕj (τi ) = ϕj (ti − tN ) → 0, rendering N1r Ψ0N ΨN → 0 and the PE condition is lost. To resolve this issue, we extract the dominant mode as in Case 1. Since f (tN ) → 0 in this case, the unstable modes may

result in divergence. We now resort to generation of switching time sequences to restore convergence. Assumption 2: For some ν > νmin > 0, the switching-time sequence {tN } satisfies (a) tN → ∞ as N → ∞. (b) For some 0 < r0 < 1/2, e2νtN /N r0 → 0 as N → ∞. (c) |tN +1 − tN | ≤ q(N ) such that for any r1 > 0 ∞ X q(N ) < ∞. (36) N r1 N =1

Remark 3: We show now that a sequence {tN } satisfying Assumption 2 for any chosen r0 can be constructed. This remark also provides a typical choice of such a {tN } sequence. Choose tN from 1 e2νtN = ε , 0 < r0 < 1/2, 0 < ε < min{r0 , 1/2 − r0 }. r 0 N N (37) Then tN = r02ν−ε ln N. Clearly, tN → ∞ and e2νtN /N r0 → 0. Now, r0 − ε ∆tN = tN +1 − tN = (ln(N + 1) − ln N ). 2ν For large N , |∆tN | ≤ q(N ) = O(1/N ) due to lim

N →∞

r0 −ε 2ν (ln(N

+ 1) − ln N ) r0 − ε = . 1/N 2ν

It follows that q(N )/N r1 = O(1/N 1+r1 ), which implies P ∞ r1 < ∞. N =1 q(N )/N Remark 4: Although the generated sampling time sequences are unbounded (i.e., tN → ∞), the sampling intervals (tN − tN −1 ) become progressively smaller. For practical implementation, when the switching sensor has accuracy limit, time delay, sensor noise, there is a limitation on the sampling interval. We can easily accommodate such a limitation by stopping reduction of sampling intervals when they fall below precision levels of the sensors or the control actuators. This stopping rule will result in a small residual and irreducible error. This is not a new phenomenon. For any convergence or stability analysis, measurement precision will impose a limitation. Consider an open-loop system x˙ = 2x + u, the feedback control u = −4x achieves a closed-loop system x˙ = −2x which is exponentially stable. However, if measurement or actuator implementation has limited accuracy, the true feedback will become u = −4x + d with d a bounded error from measurement. Then the closed-loop system becomes x˙ = −2x + d. x will converge exponentially to a bounded set, but not to 0. Our theoretical development will always be subject to such implementation constraints which will not be discussed further. Theorem 8: Under Assumption 1, (26), and 2, with r0 satisfying 0 < r0 < min{2r − 1, 1/2}, (a) e(t) → 0 in the MS sense. (b) e ∈ L2 . Proof. From the definition of ν > νmin , since α ej (τi ) is bounded, we have |˜ αj (τi )/fu (tN )| ≤ κeνtN for some constant κ. (a) Following the derivation steps in (34), we arrive at N 2r−1 Ee0 (tN )e(tN ) ≤ κe2νtN for some constant κ. This leads to e2νtN e2νtN Ee0 (tN )e(tN ) ≤ κ 2r−1 ≤ κ r0 → 0 (38) N N

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by Assumption 2. Now, Lemma 4 implies e(t) → 0 in the MS sense. (b) By (38) and Assumption 2 e2νtN e2νtN 1 = κ ≤ κ r1 N 2r−1 N r0 N r1 N with r1 = 2r − 1 − r0 > 0, which q(N )Ee0N eN ≤ Pimplies ∞ r1 r1 < ∞, 2, κq(N )/N N =1 q(N )/N p . By Assumption 2 2  namely q(N )eN ∈ l . By Lemma 4, e ∈ L . 3) MS Exponential Convergence: We now present conditions under which exponential convergence rates of e(t) can be obtained. We show that e(tN ) converges to 0 exponentially in the MS sense and w.p.1. Case 1: ϕs (τ ) 6= 0 Theorem 9: Under the assumptions of Theorem 7, if the switching time sequence is tN = γ ln N for some γ > 0, then for any 0 < η < (2r − 1)/γ eηtN Ee0 (tN )e(tN ) → 0 as N → ∞. Proof. From tN = γ ln N we obtain N = etN /γ . By Theorem 7, ζ = supN N 2r−1 Ee0 (tN )e(tN ) < ∞, which implies etN (2r−1)/γ Ee0 (tN )e(tN ) ≤ κ < ∞. Since 0 < η < (2r − 1)/γ and tN → ∞, Ee0N eN ≤ κ

e

ηtN

0

Ee (tN )e(tN ) ≤ e

tN (η−(2r−1)/γ)

κ → 0 as N → ∞.

 Case 2: ϕs (τ ) = 0 Theorem 10: Under the assumptions of Theorem 8 and suppose tN is generated from (37), if for some r > 1/2 r e0 Ψ e β = inf N σmin Ψ N N /N > 0, then there exists η > 0 such 0 ηtN that e Ee (tN )e(tN ) → 0 as N → ∞. Proof. By (38) and (37) Ee0 (tN )e(tN ) ≤ κ Let 0


r0 κ N ηγ e 0 ˜ 0N Dn k. kΨN DN k ≤ r1 −ηγ kΨ r 1 N N Choose r1 − ηγ ≥ r0 that is given in Theorem 4, Theorem 4 immediately yields the claim. To prove (ii), since tN → ∞ as N → ∞, for any t > 0, there exists N such that tN ≤ t < tN +1 . Also, keηtN e(tN )k ≤ κ

|t − tN | ≤ |tN +1 − tN | = | ln(N + 1) − ln N | ≤ 1. Since e(t) = eA(t−tN ) e(tN ), tN ≤ t < tN +1 we have eηt ke(t)k ≤ κeηt ke(tN )k ≤ κeη(t−tN ) ≤ κ.  The case ϕs (τ ) = 0 can be directly obtained from Theorem 4 by a similar treatment of Theorem 11 for any switching time sequence with |ti −ti−1 | uniformly bounded, where i ≥ 1. Theorem 12: Under the Assumption 1 and (26), if ϕs (τ ) = 0, there exists some η > 0 for which eηtN ke(tN )k = O(1) w.p.1 as N → ∞ and eηt ke(t)k = O(1) w.p.1 as t → ∞. VII. G ENERATION OF S WITCHING T IME S EQUENCES Estimation of the state utilizes the information from the sampling-time/ switching-time sequence {ti } and the corresponding values γ(ti ). Consequently, it is essential to generate such a sequence. In practical systems, the irregular sampling sequences can be generated via different means, depending on system settings. We list some of these sampling-time generating mechanisms for which the results of this paper may be applied. A. Directly Controlled Sampling Multi-rate sampled systems are a classical topic. Such sampling schemes implicitly assume that sampling times can be designed. We refer the reader to some standard textbooks and monographs [4], [7], [24] for sampled systems with uniform or non-uniform sampling intervals. Classical treatment of this topic is mostly limited to multi-rate sampling and does not seek active utility of such a capability in state estimation. Advanced analog-to-digital conversion units (A-D) allow selection of sampling times beyond periodic sampling schemes. In these cases, the sampling times can be directly controlled. The results of this paper provide some insights and methodologies on how such advanced capabilities might be utilized to achieve fast convergence of state estimation. B. Event-Triggered Sampling and Signal Quantization Departing from traditional sampling mechanisms, sampling times may be event triggered. Typically, if a binary-valued or quantized sensor is used in signal measurement, a sampled value is generated only when the output of such sensors switches values. Each switching of the sensor output represents an event that provides a sampled value of the system output at the switching time. Event-based sampling usually involves fixed thresholds, open-loop stable systems or just a class of bounded signals [3], [23]. For example, photoelectric sensors for positions of robot movements, Hall-effect sensors for speed and acceleration in electrical machines, EGO oxygen sensors in automotive emission control will provide meaningful values of the system outputs only at the time of sensor switching. In a distributed sensor network, a grid of location sensors are allocated in an area, creating a network of checkpoints. When a mobile agent, such as a vehicle, a robot, a soldier, or an airplane, passes through a sensor checkpoint, the event will generate a position signal at that time instant. The set of the time instants forms an irregular sampling-time sequence.

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C. Binary-Valued Sensors with Input Control Suppose that the output of the system (1) is measured by a sensor whose output s(t) ∈ R is binary-valued with a fixed threshold γ. The sensor can be represented by  1, if y(t) ≤ γ; s(t) = S(y(t)) = (39) 0, if y(t) > γ. At a switching time ti , the sensor threshold value γ is an approximation of y(ti ) in that γ = y(ti ) − di , which will be used to estimate the state. Since a binary sensor provides information about y(t) only when it switches, one may actively control the input to cause y(t) to pass the sensor threshold γ. In [35], sampling times are generated by controlling the input so that the sensor output will switch values within a designated time interval, under assumption that the system is state controllable. Assumption 3: 1) The prior information on the initial state x(0) is given by a compact set Ω0 , namely, x(0) ∈ Ω0 . 2) System (1) is completely state controllable and completely state observable. Theorem 13: [35] Under Assumption 3, for any t1 > 0, there exists a causal control input u(t) for t ∈ [0, t1 ] such that s(t) switches at least once in (0, t1 ). Corollary 1: [35] Under Assumption 3, for any t1 > 0 and for any given integer m, there exists a causal control input u(t), t ∈ (0, t1 ) such that s(t) switches at least m times in (0, t1 ). The above results indicate that by actively controlling the input, one can always generate a desired switching-time sequence. D. Binary-Valued Sensors with Threshold Control Another approach of controlled generation of sampling-time sequences is to control the sensor threshold. Suppose that the output of the system (1) is measured by a sensor whose output s(t) ∈ R is binary-valued with a possibly time-varying threshold γ(t). The sensor can be represented by  1, if y(t) ≤ γ(t); s(t) = S(y(t)) = (40) 0, if y(t) > γ(t). One possible scheme of threshold control is illustrated in control problem for a power system shown in Figure 1. The voltage v(t) is the output of a system whose states are to be estimated. The reference voltage vR (t) is generated by a circuit whose value can be adjusted. This can be easily realized by operational amplifiers, DC-DC converters, etc. The binary sensor is a simple sign detector. When the sign detector switches its value at ti , it triggers the reference value vR (ti ) to be coarsely quantized to vbR (ti ) and hence is subject to errors, and sent over the noise-corrupted network to the receiver. The receiver receives w(ti ) = v(ti ) + di as a noise corrupted observation on v(ti ) at the switching time, and has a clock so the time ti is also recorded. The switching time ti and the received value w(ti ) constitute the information for observer design and feedback. Let k · k be a vector norm.

ti

Synchronized Clocks Sign Detector

v(t)

Triggering Event

+

Communication Channel

-

Quantization

v R (t)

w(t i )

vˆ R (t i )

Reference Control

Fig. 1.

Controlled threshold and networking example

Assumption 4: At t = 0, the information on the state x(0) is given by a bounded set Ω. Theorem 14: Under Assumption 4, given a switching time ti , for any T > ti , there exists a continuous threshold function γ(t) for t ∈ [ti , T ] such that s(t) switches at least once in (ti , T ). Proof. Consider first the case y(ti ) > γ(ti ), that is, s(ti ) = 0. We will show that there exists a function γ(t) for t ∈ (ti , T ) such that y(T ) < γ(T ). By continuity, this implies that s(t) switches from 0 to 1 at least once at ti+1 ∈ (ti , T ). For t > 0, y(t) = CeAt x(0) + v(t, 0). Under Assumption 4, there exist known constants α > 0 and β ≥ 0 (β = 0 if the system is stable) such that y(t) ≤ αeβt + v(t, 0),

t > 0.

(41)

Let the threshold function be selected as γ(t) = γ(ti ) + αeηi (t−ti ) + v(t, 0)

(42)

with some ηi > β which is to be specified later. Since v(t, 0) is continuous, γ(t) is continuous. To guarantee y(T ) < γ(T ), we only need to take sufficiently large ηi such that T > which gives

ηi ti |γ(ti )| + , ηi − β α(ηi − β)

(43)

|γ(ti )| . (44) α Then, from (41) and (42), we have y(T ) < γ(T ). Consequently, y(ti+1 ) = γ(ti+1 ) will occur at some ti+1 ∈ (ti , T ). The case of y(ti ) < γ(ti ) can be similarly achieved by (44) if we let γ(t) = γ(ti ) − αeηi (t−ti ) + v(t, 0), where ηi > β satisfies (43), since y(t) ≥ −αeβt + v(t, 0), t > 0.  Theorem 14 can be applied repeatedly to reach the following conclusion. Corollary 2: Under Assumption 4, for any given time interval and for any given integer m, there exists a threshold design γ(t), such that s(t) switches at least m times in the time interval. We should point out that for practical algorithm implementations, we may use different γ(t), as long as the sensor switches frequently to provide information for state estimation. a system A =  Example 7: Consider    with eηi (T −ti ) − eβT ≥ ηi (T − ti ) − βt =

0  0  0 −1

1 0 0 −3

0 1 0 −3

0 0 0   1  , B =  , C 1  2  −2 1

= [1 0 0 0].

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Threshold Design and Sensor Switching

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The input to the system is a step input u(t) = 0, t < 0 and u(t) = 10, t ≥ 0, and initial state is x(0) = [10 0 0 0]0 . This implies that the initial output is y(0) = 10. Initial threshold value is γ(0) = 5. For t > 0, the threshold γ(t) is controlled by increasing its value when s(t) = 0 and decreasing its value when s(t) = 1, leading to two functions of γ(t). In this simulation, both functions are linear, although exponential functions can be used also. Simulation results are shown in Figure 2.

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VIII. A N I LLUSTRATIVE E XAMPLE We provide an example to demonstrate how the methods of this paper may be used. In the example, the time-varying sensor threshold is compared to the noise-corrupted plant output by using a binary comparisor, representing the binary sensor. When its output switches values, the threshold value at that time instant is used as the measured value of the output. Example 8: Consider  now an unstable system x˙ = Ax  +  0 0 1 0 0    0 0 1 0  , B =  1 , Bu, y = Cx with A =     0 2  0 0 1 1 −1 −1 5 2 C = [1, 0, 0, 0] The unknown initial state is x(0) = [10, −8, 0, 0]0. We first run the system open-loop for a short period of time with a constant control input u = 10. The threshold is controlled to increase or decrease on the basis of sensor output switching, using the threshold control algorithms. To take into consideration of the threshold control and time measurement constraints, we impose a bound on the maximum slope of threshold changes at 45 per second. Simulation is performed and the collected data on the binary sequences are used to estimate x(t). Figure 3 depicts the signal trajectories, including the states, the output, the threshold control, and the sensor output. Figure 4 shows state estimation errors as a function of data points.

the sampled system is always observable, independent of the actual times. This finding provides a foundation for state estimation, output feedback, and other control activities that involve irregular sampling due to, for example, communication channel interruptions, event-triggered sampling, or other system constraints. This paper also introduces a method of designing observers for linear time invariant systems with irregular sampling-time sequences, which can be generated either passively in eventtriggered sampling or by actively controlling the system input or sensor threshold when the sensor is binary-valued. By selecting the sampling-time/switching-time appropriately, the persistent excitation condition is validated and convergent state observers can be developed. It is shown that convergence in mean square, with probability one, as well as almost sure exponential rates can be guaranteed when the switching time sequence is suitably generated. The results of this paper can be used in feedback design for stability and performance on the basis of state estimation.

IX. C ONCLUSIONS

A. Improved Lojasiewicz Inequality The smallest eigenvalue of Ψ0N ΨN relies on some lower bounds on modulus of exponential polynomials in the form

This paper has resolved the observability issue: If the density of the sampling times exceeds a critical threshold,

X. A PPENDIX : P ROOF OF T HEOREM 3 The proof of Theorem 3 relies on several technical results, which are of interests by themselves and will be presented first.

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of (9), where f (t) can be viewed as a confinement of a complex exponential polynomial on the real axis. The global Lojasiewicz inequality (see [5, p. 210]) provides a lower bound on the module of an analytic function. It will be extended here to estimate the lower bound in (29). In this subsection, we will derive a key inequality which is essential to estimate the smallest eigenvalue of matrix. First, we need some preliminary knowledge on exponential polynomial. Definition 2: An exponential polynomial of z ∈ C (the set of complex numbers) is an entire function F (z) of the form X F (z) = pj (z)eµj z , (45) 1≤j≤l

where Pnj µj k∈ C, j = 1, . . . , l are distinct, and pj (z) = 6 0 are nonzero polynomials of degree k=0 pj,k z , pj,k = nj . The degree of F (z) is do F :=

l X j=1

nj + l − 1,

(46)

and the norm of F is kF k :=

nj l X X j=1 k=0

|pj,k |.

(47)

g(z) :=

aj (τ−z (g)) = aj (g)eλj z .

βj,k z k−1 eλj z

(48)

j=1 k=1

to z = t, where λj , j = 1, · · · , l are the l distinct eigenvalues of A (also called frequencies of g(z) in complex analysis) P with multiplicity mj , respectively. Note that kj=1 mj = n. The coefficients βj,k ∈ C and here pj (z) =

mj X

βj,k z k−1 = vj,k

k=1

z k−1 , (k − 1)!

(49)

where v = (v1 , · · · , vn ) ∈ Cn with kvk = 1. Obviously, pj (z) 6= 0. According to Definition 2 and (46), the degree of g(z) depends on the coefficients β = {βj,k }. If ∀j, βj,mj 6= 0, then l X (mj − 1) + l − 1 = n − 1. (50) do g = j=1

In general, if the leading coefficient (non-zero βj,k with the Pl highest index k) of pj (z) is βj,kj , then do g = j=1 (kj −1)+ l − 1 ≤ n − 1. We denote the leading coefficient by aj (g) = βj,kj .

(51)

By (47), the norm of g is kgk =

mj l X X

j=1 k=1

where ν = 9do g, µ e = 8 max1≤j≤l (mj − 1), and κ > 0 can be explicitly estimated in terms of λj and mj of pj . However, the leading terms |aj (g)| may be very small even if the norm of g is a given value. So, we make some modifications to the global Lojasiewicz inequality to estimate the modulus of g by kgk. It will be useful in dealing with the eigenvalues of Hermite matrices in the subsequent arguments. The following modification is based on the argument of Grudzinski who obtained an inequality in the form of the global Lojasiewicz inequality without the denominator [5]. The main approach is to introduce the exponential polynomial function τ−z (g)(ω) := g(ω + z), (53) which satisfies

Now, g(t) in (9) is the limitation of mj l X X

0 independent ˜ V )do f . of β such that |g(z)| ≥ κkτ−z (g)kd(z, We now present the main claim of this subsection, which gives an improved Lojasiewicz inequality to suppress the effect of leading term |aj (g)|. Proposition 2: Let V = {z ∈ C : g(z) = 0} = 6 ∅ ˜ z0 ) = inf(1, d(z, V )). Then there exists some j ∈ and d(z, {1, . . . , l} and a positive κ > 0 independent of β such that |g(z)| ≥ κkgk

1 , µ!

(52)

where µ = max1≤j≤l mj − 1. Furthermore, define the supporting function by H(z) := maxj 1, (1 + 4η)i − 1 ≥ 4η for any i ≥ 1. Then, by (58), (62) Parseval’s formula s and Z 2π 1 again, one has kτ−z (pj )k ≥ |ps (eiθ + z)|2 dθ ≥ 2π 0 √ ηγ, which from (56) immediately yields kτ−z (g)k ≥ κkgk kτ−z (pj )ke