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State Observers and Stabilization of Systems with Binary-Valued Observations
Proceedings of the 15th IFAC Symposium on System Identification Saint-Malo, France, July 6-8, 2009
State Observers and Stabilization of Systems with Binary-Valued Observations Le Yi Wang ∗ Chanying Li ∗∗ George G. Yin ∗∗∗ Lei Guo ∗∗∗∗ Cheng-Zhong Xu † ∗
ECE Dept., Wayne State University, Detroit, Michigan 48202. ECE Dept. Wayne State University, Detroit, Michigan 48202. ∗∗∗ Math. Dept., Wayne State University, Detroit, Michigan 48202. ∗∗∗∗ AMSS, Chinese Academy of Sciences, Beijing 100190, China. † ECE Dept. Wayne State University, Detroit, Michigan 48202. ∗∗
Abstract: This paper introduces an observer for linear time invariant systems in continuous time whose outputs are measured only by binary-valued sensors. The traditional observers will fail in general, even if the system has a full-rank observability matrix. It is shown that by controlling the sensor threshold, it is always possible to cause the output to cross the sensor threshold within a designated time interval. Information on the time instants of threshold crossing is then used to develop state estimates. Convergence properties of the state observer are established. It is shown that by generating appropriate switching time sequences, state observers are convergent in the mean squares sense, with probability one, and exponentially. Combined with a state feedback using the state estimates, this state estimator leads to a stabilizing output feedback. 1. INTRODUCTION
can then be used to develop state estimates. Convergence properties of the state observer are established. Combined with a state feedback using the state estimates, this state estimator leads to a stabilizing output feedback.
Binary-valued sensors are used in many practical systems. More importantly, when a network is involved in a control loop, signals must be transmitted with quantized values, which can be modeled as a quantized sensor. In this paper, we use a networked feedback to explain physical interpretations and motivations for our problem formulations and results. Utility of binary sensors poses substantial difficulties since only very limited information is available for system modeling, identification, estimation, control, and fault detection. Some related results on identification, state estimation, and fault detection using binary or quantized outputs can be found in Kim et al (2005); Koutsoukos (2003); Sur and Paden (1998); Wang et al (2003); Wang and Yin (2007); Wang et al (2008). The traditional observers will fail in general, even if the system has a full-rank observability matrix. In Wang et al (2008), the control input is used to generate switching time sequences so that additional information can be obtained to estimate the internal states. However, when the observer is used in a feedback control, the control input must be used both for generating switching time sequences for state observation and for feedback control. This paper introduces a method of controlling the threshold to obtain information on system outputs. This method leaves the control input for feedback design, hence it separates observer design from feedback and greatly reduces complications in control design. It is shown that by controlling the sensor threshold, it is always possible to cause the output to cross the sensor threshold within a designated time interval. Information on the time sequence of threshold crossing and the corresponding threshold values
978-3-902661-47-0/09/$20.00 © 2009 IFAC
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A binary-valued sensor with controlled thresholds may be viewed as a controlled sampler. We refer the reader to some standard textbooks and monographs, e.g., Chen and Francis (1996); Phillips and Nagle (1995) for sampled systems with uniform or non-uniform sampling intervals. The method of this paper may be viewed as an active controlled and event triggered sampling. The main goal of such a threshold control is to ensure convergence of estimates and stability of the closed-loop system. Since the method can deal with large sampling errors effectively, it allows the sampler to use coarse quantization of the reference values over a network, saving network resources. As such it can be used in sensor networks and smart sensors in which network resources are scarce. The main challenges in these problems include: (1) How should threshold control be designed? (2) How can the states be estimated? (3) What convergence properties can be derived for the state estimates? (4) How can the system be controlled to achieve stability and performance for the closed-loop system? This paper will focus 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 Chung (1974); Hall and Heyde (1980); Kushner and Yin (2003); Guo (1995); Huang and Guo (1990); Stout (1974) for their relevance to this paper. For stabilization of sampled systems, traditional feedback design with either full information state observation or regular sensor output observations is a mature area. This paper deals with binary sensors, threshold design, convergence under mixing-type sampling
10.3182/20090706-3-FR-2004.0044
15th IFAC SYSID (SYSID 2009) Saint-Malo, France, July 6-8, 2009 disturbances, and convergence rates in MS and almost sure senses, which are new topics in this area. The paper is organized as follows. Section 2 formulates the state observation problems pursued in this paper. The observer design is studied in Section 3. An observer structure is introduced first, which includes a threshold control module, a discrete state estimator, and a continuous-time state observer with updating schemes. It is shown that if the system is observable, then the observer can be constructed and implemented. Threshold design is presented such that the sensor can provide persistent information for state estimation. Section 4 establishes convergence properties of the state observer in a finite horizon. Mean-squares (MS) and strong (with probability one) convergence is established. Section 5 deals with convergence analysis of the state observer over an infinite horizon, which is an essential requirement for stabilization of an unstable plant. By generating suitable switching time sequences, we show that not only MS and strong convergence properties can be guaranteed, but also exponential convergence rates. Section 6 combines the observer and state feedback to arrive at an output feedback system. Stability of the overall system is shown to hold in the MS sense, with probability one, and in exponential speed. Finally, some concluding remarks are given in Section 7. All proofs and most examples are omitted due to space limitations. The full version of the paper may be obtained from the authors. 2. PROBLEM FORMULATION Consider an MISO (multi-input-single-output) linear timeinvariant system ( x(t)= ˙ Ax(t) + Bu(t) y(t)= Cx(t) (1) s(t)= S(y(t)),
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. Let (Ω, ϕ.P ) be the probability space, on which the disturbance {dk } is defined. Denote ϕn = σ{dk : k ≤ n} and ϕn = σ{dk : k ≥ n} as the σ-algebras generated by {dk : k ≤ n} and {dk : k ≥ n}, respectively. Assumption 1. The following conditions hold. (1) Wo′ = [C ′ (CA)′ . . . (CAn−1 )′ ] is full rank, i.e., the system is observable. (2) The sensor noise {dk } is a stationary sequence such that Edk = 0 and for some µ > 0, E|dk |2+µ < ∞. For l > 0, let φ(l) = sup E 1/q |P (B|ϕn ) − P (B)|q , B∈ϕn+l 2+µ 1+µ .
where q = {dk } is a φ-mixing process with mixing measure defined by φ(·) above satisfying P∞ µ/(1+µ) φ (j) < ∞. j=1
Remark 2. Note that the sensor noises di are not required to be independent or identically distributed. This sequence is only required to be zero mean, have finite (2 + µ)th moment, and dependence between di and dj decays relatively fast when i and j are farther apart. The mixing condition is modeled after (Ethier and Kurtz, 1986, Section 7.2). For many observer design problems, a slightly stronger condition with the process {di } being bounded seems to be a reasonable assumption. The rationale of it is: Truncated normal noise is frequently used in practice. In such a case, for j ≥ i, by using the well-known mixing inequalities (Billingsley, 1968, p. 170), we have X X |Edi dj | ≤ κ φ(j − i) < ∞. j≥i
where A ∈ Rn×n , B ∈ Rn×m , u(t) ∈ Rm is the control input, x(t) ∈ Rn is the state, and y(t) ∈ R is the system output. y(t) is measured by a sensor whose output s(t) ∈ R is binary-valued with threshold γ(t), which is a design variable in this paper. The sensor can be represented by 1, if y(t) ≤ γ(t); s(t) = S(y(t)) = (2) 0, if y(t) > γ(t).
j≥i
In this paper, we adopt the convention of using a generic positive constant κ > 0 to represent unspecified positive constants. Hence, for any a > 0, aκ = κ. We would like to estimate the state x(t) from information on u(t), {ti } and {γ(ti )}, and to derive an output feedback scheme to stabilize the system. It is easy to understand that traditional passive-type observers will fail even if the system is observable. For instance, suppose u is zero and the threshold is fixed at a value γ0 . There exists a set of initial states from which the sensor output will never switch, rendering a set of unobservable states under binary-valued sensors. This implies that active or controlled observer design is needed. In Wang et al (2008), we introduced the method of controlled switching by using control u to generate information for state reconstruction. This approach, however, has a drawback in applications to feedback systems since u will be used in dual purposes of feedback control and state observation.
At a switching time ti , the sensor threshold value γ(ti ) is an approximation of y(ti ) γ(ti ) = y(ti ) − di (3) which will be used to estimate the state. The measurement disturbance {di } at switching times represents combined effects of typical output measurement noises, threshold implementation inaccuracy (switching time and value), quantization errors in representing the thresholds and channel noises in communicating the threshold values. By allowing potentially large di , data transmission complexity can be reduced.
This paper introduces the threshold control method for state observation, leaving u to be designed solely for feedback control. On the other hand, this method requires communication of γ(ti ) at the event of sensor output switching, and as a result carries increased complexity. By allowing switching errors di in our algorithm development, we permit γ(ti ) be quantized with relatively large errors so that communication complexity can be reduced.
The sequence of switching times ti depends on the unknown system state x(t) and the threshold control γ(t). Section 3 contains discussions on how such a sequence might be generated via threshold control. In this paper, the switching time sequence will be deterministic and can be generated with a certain level of accuracy through threshold control. It is obvious that state estimation will
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15th IFAC SYSID (SYSID 2009) Saint-Malo, France, July 6-8, 2009 u
x
B
x
y
C
S J (t)
A
Then, (4) can be written as ΦN x(tN ) = ΓN − VN + DN .
s
Suppose that ΦN is full rank, which will be established in late sections. Then, a least-squares estimate of x(tN ) is given by zN = (Φ′N ΦN )−1 Φ′N (ΓN − VN ), (5) with estimation error eb(tN ) = (Φ′N ΦN )−1 Φ′N DN . The observer will have the state estimate updated at the switching time tN by x b(tN ) = zN , and will run as an open loop observer for tN ≤ t < tN +1 , x b˙ = Ab x + Bu. Due to the state update at the switching time tN , x b(t) is discontinuous at tN , but continuously differentiable in (tN , tN +1 ).
Threshold Control
Threshold
xˆ
B
Switching time
xˆ
Switching Uncertainty
Output Estimate
A
yˆ (t i ) State Update
xˆ (t i )
Discrete State Estimator
State Observer
Fig. 1. Observer Structures 3. OBSERVER DESIGN The main idea here is to develop a causal control strategy for the sensor threshold γ(t) based on observations on s(t) such that switching of the sensor output can be actively generated. Since each output switching provides additional information about y(ti ) = Cx(ti ) from (3), state estimation can be performed by using the sequence of observations. Control strategy for γ(t) and algorithms for state estimation will be the focus of this section. Convergence properties of the algorithms will be established in next two sections. 3.1 Observers The state solution to the system (1) can be expressed in the form, valid for both t > t0 and t < t0 , Z t t0
Suppose {ti , i = 1, . . . , N } is a sequence of switching times. The observer is to estimate x(t), tN ≤ t < tN +1 before the next switching occurs at tN +1 .
The observer has a structure shown in Figure 1. Since new information about the state is obtained only at the time of switching 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 ). For ti ≤ tN , if the sensor switches value at ti , we have Z ti γ(ti ) + d(ti ) = y(ti ) = CeA(ti −tN ) x(tN ) + C
eA(ti −τ ) Bu(τ )dτ.
Corollary 5 indicates that generation of a switching time can be controlled within a designated small time interval. In the subsequent development, a statement like “tN = ln N ” should be understood as “tN can be controlled to lie in an arbitrarily small interval around ln N .” Since the subsequent results remain valid for all tN within a sufficiently small interval around ln N , for notational simplicity and without loss of generality, we will use tN = ln N in statements and derivations.
tN
Since the second term is known, it will be denoted by Z ti eA(ti −τ ) Bu(τ )dτ. v(ti , tN ) = C tN
This leads to the switched observations
CeA(ti −tN ) x(tN ) = γ(ti ) − v(ti , tN ) + d(ti ),
i = 1, . . . , N.
(4)
Since this paper does not consider input noises, v(ti , tN ) is known. Define A(t1 −tN )
4. CONVERGENCE ANALYSIS IN FINITE TIME
Ce
ΦN
ΓN
=
; A(tN −1 −tN ) .. .
Ce
Since a binary sensor provides information about y(t) only when it switches, it is essential that the threshold is designed appropriately to cause y(t) to pass the sensor threshold γ(t). Let k · k be a vector norm. Assumption 3. At t = 0, the information on the state x(0) is given by a bounded set Ω. Theorem 4. Under Assumption 3, 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 ). We briefly summarizes the basic idea of threshold control. Suppose y(ti ) > γ(ti ). For t > 0, y(t) = CeAt x(0)+ v(t, 0). Under Assumption 3, there exist known constants α > 0 and β ≥ 0 such that y(t) ≤ αeβt + v(t, 0), t > 0. (6) Let the threshold function be selected as γ(t) = γ(ti ) + αeηi (t−ti ) + v(t, 0) (7) with some ηi > β. Take sufficiently large ηi such that T > ηi ti /(ηi − β) + |γ(ti )|/(α(ηi − β)) which gives eηi (T −ti ) − eβT ≥ ηi (T − ti ) − βt = |γ(ti )|/α. Then, from (6) and (7), we have y(T ) < γ(T ). Corollary 5. Under Assumption 3, 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.
eA(t−τ ) Bu(τ )dτ.
x(t) = eA(t−t0 ) x(t0 ) +
3.2 Threshold Design
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.
C γ(t1 ) − v(t1 , tN ) d(t1 ) . . .. .. ; DN = . = γ(t d(tN−1 ) N−1 ) − v(tN−1 , tN ) γ(tN ) d(tN )
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 >
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15th IFAC SYSID (SYSID 2009) Saint-Malo, France, July 6-8, 2009 Without loss of generality, suppose the eigenvalue ν1 has the real part νmin with the highest multiplicity l. Define νmin tN the dominant mode f (tN ) = tl−1 if φs 6≡ 0, and N e f (tN ) = 1 if φs ≡ 0. Let α ej (τi ) = αj (τi )/f (tN ), which leads to ϕ(τ e i ) = ϕ(τi )/f (tN ). ϕ(τ e i ) is uniformly bounded in τ < 0. Now, define e N = ΨN /f (tN ). Ψ (11) This leads to, for any r > 0,
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 ր T . However, we also allow the case of predetermined finite N sampling points. Consequently, in the later case the statement with a limit “N → ∞” should be understood in the following sense: For any desired N , a threshold control can be performed such that at least N switching times occur in this time interval. This is to ensure that causality is maintained and also all switching times are in a uniformly bounded interval. This problem will become immaterial for infinite horizon cases in which we always assume that the time sequence is generated causally and sequentially.
1 1 e(tN )= Wo−1 ( r Ψ′N ΨN )−1 r Ψ′N DN N N 1 1 ′ e Ψ e N )−1 1r Ψ e ′N DN . = W −1 ( Ψ f (tN ) o N r N N
Theorem 6. Under Assumption 1, for any r > 1/2, 1 ˜′ kΨ DN k = O(1) w.p.1 as N → ∞, (12) Nr N ˜ N is defined by (11). where Ψ Remark 7. In proving the above theorem, the mixing condition in Assumption 1 is used in an essential way. In fact, the assumption and the monotonicity of φ(i) imply that N φµ/(1+µ) (N ) → 0 as N → ∞. We only outlined the main ideas and omitted the details. Assumption 8. (1) For some T > 0, |τi | ≤ T , i = 1, 2, . . .. (2) For some r > 1/2, 1 e′ e β = inf σmin ( r Ψ ΨN ) > 0, N N N where σmin (H) is the smallest eigenvalue of H for a suitable matrix H. Theorem 9.. Under Assumptions 1 and 8, the following assertions hold: ζ = sup N 2r−1 Ee′ (tN )e(tN ) < ∞; (13)
4.1 Convergence From (5), the estimation error for x(tN ) at the switching time tN is e(tN )= x b(tN ) − x(tN ) = (Φ′N ΦN )−1 Φ′N DN 1 1 = ( Φ′N ΦN )−1 Φ′N DN . N N From the expression A(t1 −tN ) Ce
ΦN
=
. ..
Ce
A(tN −1 −tN )
C
,
we define the (negative) time interval τi = ti − tN , 0 = τN > τN −1 = tN −1 − tN > · · · > τ1 = t1 − tN . Then, a typical row of ΦN is CeAτi . By the Cayley-Hamilton Theorem Ogata (2002), 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 , (8) where the time functions αi (t) can be derived by the Sylvester interpolation method, see Horn and Johnson (1991); Ogata (2002) for the algorithms. This implies CeAτi
= [α1 (τi ), . . . , αn (τi )]
C CA .. . CAn−1
= ϕ′ (τi )Wo ,
N
e(tN ) → 0 w.p.1 as N → ∞. Under some mild conditions, convergence of x b(tN ) to x(tN ) implies that x b(t) converges to x(t). Let Th > 0 be a finite time horizon and e(t) = x b(t) − x(t). Theorem 10.. Under Assumptions 1 and 8, for t ∈ [tN , tN + Th ], tN + Th < tN +1 max Ee′ (t)e(t) → 0, N → ∞.
(9)
tN ≤t≤tN +Th
4.2 PE-Type Conditions
where ϕ′ (τi ) = [α1 (τi ), . . . , αn (τi )] and Wo is the observability matrix. Denote ΨN = [ϕ(τ1 ), . . . , ϕ(τN )]′ . We have ΦN = ΨN Wo . (10)
The key condition for Theorem 9 is Assumption 8: inf N σmin ( N1r Ψ′N ΨN ) > 0 which may be viewed as a type of persistent excitation conditions (PE). In a typical parameter estimation problem, the PE condition is imposed on the input, leading to a condition for identification input design. However, in our problems here, ΨN cannot be designed. Consequently, this becomes a system condition. It is observed that ϕ(t) is determined by the matrix A only. Hence, the PE is a condition on A and the switching time sequence {ti }.
Let the eigenvalues of A be {νi , i = 1, . . . , n} and define νmin = min Re νi , where Re denotes the real part. If A is unstable, it has at least one unstable eigenvalue, but may contain some stable modes (if νmin < 0)) or marginally stable modes (νmin = 0 but imaginary eigenvalues are simple). Otherwise, if νmin > 0, then A has no stable modes. It turns out that this distinction is important for threshold control design and convergence analysis.
Assumption 8 means that (1) Ψ′N ΨN is full rank and (2) the smallest eigenvalues of Ψ′N ΨN grow at least as fast as N r . These are non-trivial conditions. They depend on A, C, and the time sequence. But Assumption 8 is always satisfies if a switching time sequence {ti } in [0, T ] is uniformly spread in [0, T ] Z T ′ 1 ′ lim κ ΦN ΦN = eA t C ′ CeAt dt. (14) N →∞ N 0
To facilitate this distinction, we shall decompose ϕ(τ ) in (9) into two parts ϕ(τ ) = ϕs (τ ) + ϕu (τ ), where ϕs (τ ) contains all modes of A corresponding to eigenvalues of A with non-positive real parts, and ϕu (τ ) contains all modes of A corresponding to eigenvalues in the right half plane. 1 1 It should be emphasized that τ takes negative values. As a result, for a stable mode, such as te−t , the corresponding φ(τ ) = τ e−τ → ∞ as τ → −∞.
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15th IFAC SYSID (SYSID 2009) Saint-Malo, France, July 6-8, 2009 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 (14). Proposition 11. 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 , Assumption 8 is always satisfied with r = 1.
for any fixed i, ϕj (τi ) = ϕj (ti − tN ) → 0, rendering 1 ′ N r ΨN 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 14. For some ν > νmin > 0, the switching time sequence tN has the following properties: (1) tN → ∞, N → ∞. (2) For some 0 < r0 < 1/2,
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 the following limit exists N 1 X αi (τk )αj (τk ). λij = lim N →∞ N
e2νtN /N r0 → 0 as N → ∞. (3) |tN +1 − tN | ≤ q(N ) such that for any r1 > 0 ∞ X q(N )/N r1 < ∞.
k=1
(15)
N =1
Let M be the n × n symmetric matrix M = [λij ]. Then N1 Ψ′N ΨN → M. As a result, the condition inf N σmin ( N1 Ψ′N ΨN ) > 0 is satisfied if M > 0.
Remark 15.. We show now that a sequence {tN } satisfying Assumption 14 for any chosen r0 can be constructed. This remark also provides a typical choice of such a {tN } sequence. Choose tN from e2νtN 1 = ǫ , 0 < r0 < 1/2, 0 < ǫ < min{r0 , 1/2 − r0 }. N r0 N (16) Then tN = r02ν−ǫ ln N.
In general, Assumption 8 is not always satisfied. We can show that for a finite T > 0, if the switching time sequence ti → T and is selected appropriately, Assumption 8 holds true for any observable systems Li et al (2009). 5. CONVERGENCE OVER INFINITE HORIZONS
Clearly, tN → ∞ and
e2νtN N r0
→ 0. Now, r0 − ǫ ∆tN = tN +1 − tN = (ln(N + 1) − ln N ). 2ν For large N , |∆tN | ≤ q(N ) = O(1/N ) due to r0 −ǫ (ln(N + 1) − ln N ) r0 − ǫ lim 2ν = . N →∞ 1/N 2ν It follows that q(N )/N r1 = O(1/N 1+r1 ), which implies P∞ q(N ) N =1 N r1 < ∞. Theorem 16. Under Assumptions 1, 8 and 14, with r0 satisfying 0 < r0 < min{2r − 1, 1/2}, (1) e(t) → 0 in the MS sense. (2) e ∈ L2 .
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, which will be essential for stabilization of the closed-loop system. In this section, time sequences will always be causally and sequentially generated. The observer error sequence e(tN ) is said to be in l2 P∞ ′ 2 if N =1 Ee (t RN∞)e(tN′ ) < ∞ and e ∈ L (mean square integrable) if 0 Ee (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 12. If q(N ) is uniformly bounded and tN → ∞, then (1) e(tN ) → 0, N → ∞ w.p.1 implies e(t) → 0, t → ∞ w.p.1. (2) e(tN ) → 0, N → ∞ in the MS sense implies e(t) → 0, t → ∞ in the MS sense. (3) p q(N )e(tN ) ∈ l2 implies e ∈ L2 .
5.2 MS Exponential Convergence Case 1: ϕs (τ ) 6= 0 Theorem 17.. Under the assumptions of Theorem 13, if the switching time sequence is tN = γ ln N for some γ > 0, then for any 0 < η < (2r − 1)/γ eηtN Ee′ (tN )e(tN ) → 0 as N → ∞. Case 2: ϕs (τ ) = 0 Theorem 18. Under the assumptions of Theorem 16 and suppose tN is generated from (16), if for some r > 1/2 1 e′ e β = inf σmin ( r Ψ ΨN ) > 0, N N N then there exists η > 0 such that eηtN Ee′ (tN )e(tN ) → 0 as N → ∞.
5.1 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 9. Theorem 13.. Under Assumption 1 and 8, if ϕs (τ ) 6= 0, then ζ = sup N 2r−1 Ee′ (tN )e(tN ) < ∞
5.3 Almost Sure Exponential Convergence
N
and
By a certain choice of switching time sequences, Theorem 6 implies exponential convergence in the almost sure sense. Suppose that the switching time sequence is tN = γ ln N which implies etN = N γ .
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 → ∞,
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15th IFAC SYSID (SYSID 2009) Saint-Malo, France, July 6-8, 2009 7. CONCLUSIONS
Theorem 19. Under the Assumptions 1 and 8, if ϕs (τ ) 6= 0, then there exists some η > 0 such that
This paper introduces a method of designing observers and output feedback for linear time invariant systems with binary-valued observations. By controlling sensor thresholds, this method provides a controlled and eventtriggered sampling process for the system output such that convergent state observers and stabilizing output feedback can be developed. It is shown that stability in mean square, with probability one, as well as almost sure exponential stability can be guaranteed when the switching time sequence is suitably generated. Although this paper only presents stabilization results, feedback design for performance can be similarly implemented on the basis of state estimation.
(i) eηtN ke(tN )k = O(1) w.p.1 as N → ∞. (ii) eηt ke(t)k = O(1) w.p.1 as t → ∞. The case ϕs (τ ) = 0 can be directly obtained from Theorem 6 by a similar treatment of Theorem 19 for any switching time sequence with |ti − ti−1 | uniformly bounded, where i ≥ 1. Theorem 20. Under the Assumptions 1 and 8, 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 → ∞.
REFERENCES
6. BINARY OUTPUT FEEDBACK
K.J. Astrom and B. Wittenmark, Computer-Controlled Systems: Theory and Design, 3rd Ed., Prentice Hall Information and System Sciences Series, Prentice Hall, 1996. P. Billingsley, Convergence of Probability Measures, J. Wiley, New York, NY, 1968. T.W. Chen and B.A. Francis, Optimal Sampled-Data Control Systems, Springer, 1996. Y.S. Chow, Local convergence of martingales and law of large numbers, Ann. Math. Statist., 36 (1965), 552-558. K.L. Chung, A Course in Probability Theory, 2nd Edition, Academic Press, 1974. C.S. G¨ unt¨ urk, One-bit sigma-delta quantization with exponential accuracy, Communications on Pure and Applied Mathematics, Volume 56, Issue 11, pp. 1608 - 1630, 2003. S.N. Ethier and T.G. Kurtz, Markov Processes: Characterization and Convergence, J. Wiley, New York, NY, 1986. L. Guo, Convergence and logarithm laws of self-tuning regulators, Automatica, Vol.31, No.3, pp.435-450, 1995. P. Hall and C.C. Heyde, Martingale Limit Theory and Its Application, Academic Press, New York, NY, 1980. R.A. Horn and C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991. D.W. Huang and L. Guo, Estimation of nonstationary ARMAX models based on Hannan-Rissanen method, The Annals of Statistics, Vol.18, No.4, pp.1729-1756, 1990. R.Z. Khasminskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Alphen aan den Rijn, Netherlands, 1980. W. Kim, K. Mechitov, J.Y. Choi, S.K. Ham, On tracking objects with binary proximity sensors, Information Processing in Sensor Networks, Fourth International Symposium pp. 301 - 308, 2005. X. Koutsoukos, Estimation of hybrid systems using discrete sensors, Proc. 42nd IEEE Conf. Decision and Control, pp. 155-160, Dec. 2003. H.J. Kushner and G. Yin, Stochastic Approximation and Recursive Algorithms and Applications, 2nd Ed., Springer-Verlag, New York, 2003. J. Lamperti, Stochastic Processes, Springer-Verlag, New York, 1977. C.Y. Li, L.Y. Wang, G. Yin, L. Guo, C.Z. Xu, Convergence rates of state estimation for LTI systems with binary-valued observations, Submitted 2009. K. Ogata, Morden Control Engineering, 4th Edition, Prentice Hall, 2002. C. L. Phillips and H.T. Nagle, Digital Control System Analysis and Design, 3rd Ed., Prentice Hall, 1995. J. Raisch, Controllability and observability of simple hybrid control systems-FDLTI plants with symbolic measurements and quantized control inputs, CONTROL’94, PP. 595-600, March 21-24, 1994. W.F. Stout, Almost Sure Convergence, Academic Press, New York, 1974. J. Sur and B.E. Paden, State observer for linear time-invariant systems with quantized output, ASME J. Dynamic Systems, Measurement, and Control, Vol. 120, 1998. L.Y. Wang, J.F. Zhang, and G. Yin, System identification using binary sensors, IEEE Trans. Automat. Control, 48, pp. 1892-1907, 2003. L.Y. Wang and G. Yin, Asymptotically efficient parameter estimation using quantized output observations, Automatica, 43, 11781191, 2007. L.Y. Wang, G.H. Xu, and G. Yin, State Reconstruction for Linear Time-Invariant Systems with Binary-Valued Output Observations, Systems & Control Letters, 57, pp. 958-963, 2008. G. Yin, On extensions of Polyak’s averaging approach to stochastic approximation, Stochastics Stochastics Rep., 36 (1991), 245-264.
The observer provides a state estimate x b(t) which can be used to construct a feedback. Together, they form a binary output feedback. The feedback control is u(t) = −K x b(t) which leads to the closed-loop system x˙ = Ax − BK x b = (A − BK)x − BKe = Ac x + Bc e (17) where e = x b −x is the observation error, and Ac = A−BK is stable. Assumption 21. (1) The system is stablizable. (2) K is designed such that A − BK is stable. Under Assumption 21, the expression (17) shows that the closed-loop system is a stable system driven by the observation error process e(t). We first present the MS stability of the closed-loop system, namely Ex′ (t)x(t) → 0, t → ∞. Theorem 22. Under Assumptions of Theorems 16 and Assumption 21, the state of the closed-loop system satisfies: (1) it is square integrable, x ∈ L2 . (2) Ex′ (t)x(t) → 0, t → ∞. For analysis of strong stability x(t) → 0 w.p.1. we shall first simplify some expressions. The state of the closedloop system (17) can be solved as Z t Ac t x(t) = e x(0) + eAc (t−τ ) Bc e(τ )dτ. 0
Since the first term is deterministic and decays to 0 exponentially, it will not affect analysis of strong stability. Hence, assume x(0) = 0 and x(t) is simplified to Z t x(t) = eAc (t−τ ) Bc e(τ )dτ. (18) 0
For any vector norm k · k Z t kx(t)k ≤ κ e−a(t−τ ) ke(τ )kdτ 0
for some κ > 0 and a > 0 since the closed-loop system is stable. Theorem 23. Under Assumptions of either Theorem 13 or Theorem 16, there exists an η0 > 0 such that with probability one, kx(t)k ≤ κe−η0 t as t → ∞.
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State Observers and Stabilization of Systems with Binary-Valued Observations Le Yi Wang ∗ Chanying Li ∗∗ George G. Yin ∗∗∗ Lei Guo ∗∗∗∗ Cheng-Zhong Xu † ∗
ECE Dept., Wayne State University, Detroit, Michigan 48202. ECE Dept. Wayne State University, Detroit, Michigan 48202. ∗∗∗ Math. Dept., Wayne State University, Detroit, Michigan 48202. ∗∗∗∗ AMSS, Chinese Academy of Sciences, Beijing 100190, China. † ECE Dept. Wayne State University, Detroit, Michigan 48202. ∗∗
Abstract: This paper introduces an observer for linear time invariant systems in continuous time whose outputs are measured only by binary-valued sensors. The traditional observers will fail in general, even if the system has a full-rank observability matrix. It is shown that by controlling the sensor threshold, it is always possible to cause the output to cross the sensor threshold within a designated time interval. Information on the time instants of threshold crossing is then used to develop state estimates. Convergence properties of the state observer are established. It is shown that by generating appropriate switching time sequences, state observers are convergent in the mean squares sense, with probability one, and exponentially. Combined with a state feedback using the state estimates, this state estimator leads to a stabilizing output feedback. 1. INTRODUCTION
can then be used to develop state estimates. Convergence properties of the state observer are established. Combined with a state feedback using the state estimates, this state estimator leads to a stabilizing output feedback.
Binary-valued sensors are used in many practical systems. More importantly, when a network is involved in a control loop, signals must be transmitted with quantized values, which can be modeled as a quantized sensor. In this paper, we use a networked feedback to explain physical interpretations and motivations for our problem formulations and results. Utility of binary sensors poses substantial difficulties since only very limited information is available for system modeling, identification, estimation, control, and fault detection. Some related results on identification, state estimation, and fault detection using binary or quantized outputs can be found in Kim et al (2005); Koutsoukos (2003); Sur and Paden (1998); Wang et al (2003); Wang and Yin (2007); Wang et al (2008). The traditional observers will fail in general, even if the system has a full-rank observability matrix. In Wang et al (2008), the control input is used to generate switching time sequences so that additional information can be obtained to estimate the internal states. However, when the observer is used in a feedback control, the control input must be used both for generating switching time sequences for state observation and for feedback control. This paper introduces a method of controlling the threshold to obtain information on system outputs. This method leaves the control input for feedback design, hence it separates observer design from feedback and greatly reduces complications in control design. It is shown that by controlling the sensor threshold, it is always possible to cause the output to cross the sensor threshold within a designated time interval. Information on the time sequence of threshold crossing and the corresponding threshold values
978-3-902661-47-0/09/$20.00 © 2009 IFAC
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A binary-valued sensor with controlled thresholds may be viewed as a controlled sampler. We refer the reader to some standard textbooks and monographs, e.g., Chen and Francis (1996); Phillips and Nagle (1995) for sampled systems with uniform or non-uniform sampling intervals. The method of this paper may be viewed as an active controlled and event triggered sampling. The main goal of such a threshold control is to ensure convergence of estimates and stability of the closed-loop system. Since the method can deal with large sampling errors effectively, it allows the sampler to use coarse quantization of the reference values over a network, saving network resources. As such it can be used in sensor networks and smart sensors in which network resources are scarce. The main challenges in these problems include: (1) How should threshold control be designed? (2) How can the states be estimated? (3) What convergence properties can be derived for the state estimates? (4) How can the system be controlled to achieve stability and performance for the closed-loop system? This paper will focus 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 Chung (1974); Hall and Heyde (1980); Kushner and Yin (2003); Guo (1995); Huang and Guo (1990); Stout (1974) for their relevance to this paper. For stabilization of sampled systems, traditional feedback design with either full information state observation or regular sensor output observations is a mature area. This paper deals with binary sensors, threshold design, convergence under mixing-type sampling
10.3182/20090706-3-FR-2004.0044
15th IFAC SYSID (SYSID 2009) Saint-Malo, France, July 6-8, 2009 disturbances, and convergence rates in MS and almost sure senses, which are new topics in this area. The paper is organized as follows. Section 2 formulates the state observation problems pursued in this paper. The observer design is studied in Section 3. An observer structure is introduced first, which includes a threshold control module, a discrete state estimator, and a continuous-time state observer with updating schemes. It is shown that if the system is observable, then the observer can be constructed and implemented. Threshold design is presented such that the sensor can provide persistent information for state estimation. Section 4 establishes convergence properties of the state observer in a finite horizon. Mean-squares (MS) and strong (with probability one) convergence is established. Section 5 deals with convergence analysis of the state observer over an infinite horizon, which is an essential requirement for stabilization of an unstable plant. By generating suitable switching time sequences, we show that not only MS and strong convergence properties can be guaranteed, but also exponential convergence rates. Section 6 combines the observer and state feedback to arrive at an output feedback system. Stability of the overall system is shown to hold in the MS sense, with probability one, and in exponential speed. Finally, some concluding remarks are given in Section 7. All proofs and most examples are omitted due to space limitations. The full version of the paper may be obtained from the authors. 2. PROBLEM FORMULATION Consider an MISO (multi-input-single-output) linear timeinvariant system ( x(t)= ˙ Ax(t) + Bu(t) y(t)= Cx(t) (1) s(t)= S(y(t)),
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. Let (Ω, ϕ.P ) be the probability space, on which the disturbance {dk } is defined. Denote ϕn = σ{dk : k ≤ n} and ϕn = σ{dk : k ≥ n} as the σ-algebras generated by {dk : k ≤ n} and {dk : k ≥ n}, respectively. Assumption 1. The following conditions hold. (1) Wo′ = [C ′ (CA)′ . . . (CAn−1 )′ ] is full rank, i.e., the system is observable. (2) The sensor noise {dk } is a stationary sequence such that Edk = 0 and for some µ > 0, E|dk |2+µ < ∞. For l > 0, let φ(l) = sup E 1/q |P (B|ϕn ) − P (B)|q , B∈ϕn+l 2+µ 1+µ .
where q = {dk } is a φ-mixing process with mixing measure defined by φ(·) above satisfying P∞ µ/(1+µ) φ (j) < ∞. j=1
Remark 2. Note that the sensor noises di are not required to be independent or identically distributed. This sequence is only required to be zero mean, have finite (2 + µ)th moment, and dependence between di and dj decays relatively fast when i and j are farther apart. The mixing condition is modeled after (Ethier and Kurtz, 1986, Section 7.2). For many observer design problems, a slightly stronger condition with the process {di } being bounded seems to be a reasonable assumption. The rationale of it is: Truncated normal noise is frequently used in practice. In such a case, for j ≥ i, by using the well-known mixing inequalities (Billingsley, 1968, p. 170), we have X X |Edi dj | ≤ κ φ(j − i) < ∞. j≥i
where A ∈ Rn×n , B ∈ Rn×m , u(t) ∈ Rm is the control input, x(t) ∈ Rn is the state, and y(t) ∈ R is the system output. y(t) is measured by a sensor whose output s(t) ∈ R is binary-valued with threshold γ(t), which is a design variable in this paper. The sensor can be represented by 1, if y(t) ≤ γ(t); s(t) = S(y(t)) = (2) 0, if y(t) > γ(t).
j≥i
In this paper, we adopt the convention of using a generic positive constant κ > 0 to represent unspecified positive constants. Hence, for any a > 0, aκ = κ. We would like to estimate the state x(t) from information on u(t), {ti } and {γ(ti )}, and to derive an output feedback scheme to stabilize the system. It is easy to understand that traditional passive-type observers will fail even if the system is observable. For instance, suppose u is zero and the threshold is fixed at a value γ0 . There exists a set of initial states from which the sensor output will never switch, rendering a set of unobservable states under binary-valued sensors. This implies that active or controlled observer design is needed. In Wang et al (2008), we introduced the method of controlled switching by using control u to generate information for state reconstruction. This approach, however, has a drawback in applications to feedback systems since u will be used in dual purposes of feedback control and state observation.
At a switching time ti , the sensor threshold value γ(ti ) is an approximation of y(ti ) γ(ti ) = y(ti ) − di (3) which will be used to estimate the state. The measurement disturbance {di } at switching times represents combined effects of typical output measurement noises, threshold implementation inaccuracy (switching time and value), quantization errors in representing the thresholds and channel noises in communicating the threshold values. By allowing potentially large di , data transmission complexity can be reduced.
This paper introduces the threshold control method for state observation, leaving u to be designed solely for feedback control. On the other hand, this method requires communication of γ(ti ) at the event of sensor output switching, and as a result carries increased complexity. By allowing switching errors di in our algorithm development, we permit γ(ti ) be quantized with relatively large errors so that communication complexity can be reduced.
The sequence of switching times ti depends on the unknown system state x(t) and the threshold control γ(t). Section 3 contains discussions on how such a sequence might be generated via threshold control. In this paper, the switching time sequence will be deterministic and can be generated with a certain level of accuracy through threshold control. It is obvious that state estimation will
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15th IFAC SYSID (SYSID 2009) Saint-Malo, France, July 6-8, 2009 u
x
B
x
y
C
S J (t)
A
Then, (4) can be written as ΦN x(tN ) = ΓN − VN + DN .
s
Suppose that ΦN is full rank, which will be established in late sections. Then, a least-squares estimate of x(tN ) is given by zN = (Φ′N ΦN )−1 Φ′N (ΓN − VN ), (5) with estimation error eb(tN ) = (Φ′N ΦN )−1 Φ′N DN . The observer will have the state estimate updated at the switching time tN by x b(tN ) = zN , and will run as an open loop observer for tN ≤ t < tN +1 , x b˙ = Ab x + Bu. Due to the state update at the switching time tN , x b(t) is discontinuous at tN , but continuously differentiable in (tN , tN +1 ).
Threshold Control
Threshold
xˆ
B
Switching time
xˆ
Switching Uncertainty
Output Estimate
A
yˆ (t i ) State Update
xˆ (t i )
Discrete State Estimator
State Observer
Fig. 1. Observer Structures 3. OBSERVER DESIGN The main idea here is to develop a causal control strategy for the sensor threshold γ(t) based on observations on s(t) such that switching of the sensor output can be actively generated. Since each output switching provides additional information about y(ti ) = Cx(ti ) from (3), state estimation can be performed by using the sequence of observations. Control strategy for γ(t) and algorithms for state estimation will be the focus of this section. Convergence properties of the algorithms will be established in next two sections. 3.1 Observers The state solution to the system (1) can be expressed in the form, valid for both t > t0 and t < t0 , Z t t0
Suppose {ti , i = 1, . . . , N } is a sequence of switching times. The observer is to estimate x(t), tN ≤ t < tN +1 before the next switching occurs at tN +1 .
The observer has a structure shown in Figure 1. Since new information about the state is obtained only at the time of switching 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 ). For ti ≤ tN , if the sensor switches value at ti , we have Z ti γ(ti ) + d(ti ) = y(ti ) = CeA(ti −tN ) x(tN ) + C
eA(ti −τ ) Bu(τ )dτ.
Corollary 5 indicates that generation of a switching time can be controlled within a designated small time interval. In the subsequent development, a statement like “tN = ln N ” should be understood as “tN can be controlled to lie in an arbitrarily small interval around ln N .” Since the subsequent results remain valid for all tN within a sufficiently small interval around ln N , for notational simplicity and without loss of generality, we will use tN = ln N in statements and derivations.
tN
Since the second term is known, it will be denoted by Z ti eA(ti −τ ) Bu(τ )dτ. v(ti , tN ) = C tN
This leads to the switched observations
CeA(ti −tN ) x(tN ) = γ(ti ) − v(ti , tN ) + d(ti ),
i = 1, . . . , N.
(4)
Since this paper does not consider input noises, v(ti , tN ) is known. Define A(t1 −tN )
4. CONVERGENCE ANALYSIS IN FINITE TIME
Ce
ΦN
ΓN
=
; A(tN −1 −tN ) .. .
Ce
Since a binary sensor provides information about y(t) only when it switches, it is essential that the threshold is designed appropriately to cause y(t) to pass the sensor threshold γ(t). Let k · k be a vector norm. Assumption 3. At t = 0, the information on the state x(0) is given by a bounded set Ω. Theorem 4. Under Assumption 3, 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 ). We briefly summarizes the basic idea of threshold control. Suppose y(ti ) > γ(ti ). For t > 0, y(t) = CeAt x(0)+ v(t, 0). Under Assumption 3, there exist known constants α > 0 and β ≥ 0 such that y(t) ≤ αeβt + v(t, 0), t > 0. (6) Let the threshold function be selected as γ(t) = γ(ti ) + αeηi (t−ti ) + v(t, 0) (7) with some ηi > β. Take sufficiently large ηi such that T > ηi ti /(ηi − β) + |γ(ti )|/(α(ηi − β)) which gives eηi (T −ti ) − eβT ≥ ηi (T − ti ) − βt = |γ(ti )|/α. Then, from (6) and (7), we have y(T ) < γ(T ). Corollary 5. Under Assumption 3, 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.
eA(t−τ ) Bu(τ )dτ.
x(t) = eA(t−t0 ) x(t0 ) +
3.2 Threshold Design
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.
C γ(t1 ) − v(t1 , tN ) d(t1 ) . . .. .. ; DN = . = γ(t d(tN−1 ) N−1 ) − v(tN−1 , tN ) γ(tN ) d(tN )
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 >
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15th IFAC SYSID (SYSID 2009) Saint-Malo, France, July 6-8, 2009 Without loss of generality, suppose the eigenvalue ν1 has the real part νmin with the highest multiplicity l. Define νmin tN the dominant mode f (tN ) = tl−1 if φs 6≡ 0, and N e f (tN ) = 1 if φs ≡ 0. Let α ej (τi ) = αj (τi )/f (tN ), which leads to ϕ(τ e i ) = ϕ(τi )/f (tN ). ϕ(τ e i ) is uniformly bounded in τ < 0. Now, define e N = ΨN /f (tN ). Ψ (11) This leads to, for any r > 0,
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 ր T . However, we also allow the case of predetermined finite N sampling points. Consequently, in the later case the statement with a limit “N → ∞” should be understood in the following sense: For any desired N , a threshold control can be performed such that at least N switching times occur in this time interval. This is to ensure that causality is maintained and also all switching times are in a uniformly bounded interval. This problem will become immaterial for infinite horizon cases in which we always assume that the time sequence is generated causally and sequentially.
1 1 e(tN )= Wo−1 ( r Ψ′N ΨN )−1 r Ψ′N DN N N 1 1 ′ e Ψ e N )−1 1r Ψ e ′N DN . = W −1 ( Ψ f (tN ) o N r N N
Theorem 6. Under Assumption 1, for any r > 1/2, 1 ˜′ kΨ DN k = O(1) w.p.1 as N → ∞, (12) Nr N ˜ N is defined by (11). where Ψ Remark 7. In proving the above theorem, the mixing condition in Assumption 1 is used in an essential way. In fact, the assumption and the monotonicity of φ(i) imply that N φµ/(1+µ) (N ) → 0 as N → ∞. We only outlined the main ideas and omitted the details. Assumption 8. (1) For some T > 0, |τi | ≤ T , i = 1, 2, . . .. (2) For some r > 1/2, 1 e′ e β = inf σmin ( r Ψ ΨN ) > 0, N N N where σmin (H) is the smallest eigenvalue of H for a suitable matrix H. Theorem 9.. Under Assumptions 1 and 8, the following assertions hold: ζ = sup N 2r−1 Ee′ (tN )e(tN ) < ∞; (13)
4.1 Convergence From (5), the estimation error for x(tN ) at the switching time tN is e(tN )= x b(tN ) − x(tN ) = (Φ′N ΦN )−1 Φ′N DN 1 1 = ( Φ′N ΦN )−1 Φ′N DN . N N From the expression A(t1 −tN ) Ce
ΦN
=
. ..
Ce
A(tN −1 −tN )
C
,
we define the (negative) time interval τi = ti − tN , 0 = τN > τN −1 = tN −1 − tN > · · · > τ1 = t1 − tN . Then, a typical row of ΦN is CeAτi . By the Cayley-Hamilton Theorem Ogata (2002), 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 , (8) where the time functions αi (t) can be derived by the Sylvester interpolation method, see Horn and Johnson (1991); Ogata (2002) for the algorithms. This implies CeAτi
= [α1 (τi ), . . . , αn (τi )]
C CA .. . CAn−1
= ϕ′ (τi )Wo ,
N
e(tN ) → 0 w.p.1 as N → ∞. Under some mild conditions, convergence of x b(tN ) to x(tN ) implies that x b(t) converges to x(t). Let Th > 0 be a finite time horizon and e(t) = x b(t) − x(t). Theorem 10.. Under Assumptions 1 and 8, for t ∈ [tN , tN + Th ], tN + Th < tN +1 max Ee′ (t)e(t) → 0, N → ∞.
(9)
tN ≤t≤tN +Th
4.2 PE-Type Conditions
where ϕ′ (τi ) = [α1 (τi ), . . . , αn (τi )] and Wo is the observability matrix. Denote ΨN = [ϕ(τ1 ), . . . , ϕ(τN )]′ . We have ΦN = ΨN Wo . (10)
The key condition for Theorem 9 is Assumption 8: inf N σmin ( N1r Ψ′N ΨN ) > 0 which may be viewed as a type of persistent excitation conditions (PE). In a typical parameter estimation problem, the PE condition is imposed on the input, leading to a condition for identification input design. However, in our problems here, ΨN cannot be designed. Consequently, this becomes a system condition. It is observed that ϕ(t) is determined by the matrix A only. Hence, the PE is a condition on A and the switching time sequence {ti }.
Let the eigenvalues of A be {νi , i = 1, . . . , n} and define νmin = min Re νi , where Re denotes the real part. If A is unstable, it has at least one unstable eigenvalue, but may contain some stable modes (if νmin < 0)) or marginally stable modes (νmin = 0 but imaginary eigenvalues are simple). Otherwise, if νmin > 0, then A has no stable modes. It turns out that this distinction is important for threshold control design and convergence analysis.
Assumption 8 means that (1) Ψ′N ΨN is full rank and (2) the smallest eigenvalues of Ψ′N ΨN grow at least as fast as N r . These are non-trivial conditions. They depend on A, C, and the time sequence. But Assumption 8 is always satisfies if a switching time sequence {ti } in [0, T ] is uniformly spread in [0, T ] Z T ′ 1 ′ lim κ ΦN ΦN = eA t C ′ CeAt dt. (14) N →∞ N 0
To facilitate this distinction, we shall decompose ϕ(τ ) in (9) into two parts ϕ(τ ) = ϕs (τ ) + ϕu (τ ), where ϕs (τ ) contains all modes of A corresponding to eigenvalues of A with non-positive real parts, and ϕu (τ ) contains all modes of A corresponding to eigenvalues in the right half plane. 1 1 It should be emphasized that τ takes negative values. As a result, for a stable mode, such as te−t , the corresponding φ(τ ) = τ e−τ → ∞ as τ → −∞.
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15th IFAC SYSID (SYSID 2009) Saint-Malo, France, July 6-8, 2009 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 (14). Proposition 11. 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 , Assumption 8 is always satisfied with r = 1.
for any fixed i, ϕj (τi ) = ϕj (ti − tN ) → 0, rendering 1 ′ N r ΨN 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 14. For some ν > νmin > 0, the switching time sequence tN has the following properties: (1) tN → ∞, N → ∞. (2) For some 0 < r0 < 1/2,
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 the following limit exists N 1 X αi (τk )αj (τk ). λij = lim N →∞ N
e2νtN /N r0 → 0 as N → ∞. (3) |tN +1 − tN | ≤ q(N ) such that for any r1 > 0 ∞ X q(N )/N r1 < ∞.
k=1
(15)
N =1
Let M be the n × n symmetric matrix M = [λij ]. Then N1 Ψ′N ΨN → M. As a result, the condition inf N σmin ( N1 Ψ′N ΨN ) > 0 is satisfied if M > 0.
Remark 15.. We show now that a sequence {tN } satisfying Assumption 14 for any chosen r0 can be constructed. This remark also provides a typical choice of such a {tN } sequence. Choose tN from e2νtN 1 = ǫ , 0 < r0 < 1/2, 0 < ǫ < min{r0 , 1/2 − r0 }. N r0 N (16) Then tN = r02ν−ǫ ln N.
In general, Assumption 8 is not always satisfied. We can show that for a finite T > 0, if the switching time sequence ti → T and is selected appropriately, Assumption 8 holds true for any observable systems Li et al (2009). 5. CONVERGENCE OVER INFINITE HORIZONS
Clearly, tN → ∞ and
e2νtN N r0
→ 0. Now, r0 − ǫ ∆tN = tN +1 − tN = (ln(N + 1) − ln N ). 2ν For large N , |∆tN | ≤ q(N ) = O(1/N ) due to r0 −ǫ (ln(N + 1) − ln N ) r0 − ǫ lim 2ν = . N →∞ 1/N 2ν It follows that q(N )/N r1 = O(1/N 1+r1 ), which implies P∞ q(N ) N =1 N r1 < ∞. Theorem 16. Under Assumptions 1, 8 and 14, with r0 satisfying 0 < r0 < min{2r − 1, 1/2}, (1) e(t) → 0 in the MS sense. (2) e ∈ L2 .
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, which will be essential for stabilization of the closed-loop system. In this section, time sequences will always be causally and sequentially generated. The observer error sequence e(tN ) is said to be in l2 P∞ ′ 2 if N =1 Ee (t RN∞)e(tN′ ) < ∞ and e ∈ L (mean square integrable) if 0 Ee (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 12. If q(N ) is uniformly bounded and tN → ∞, then (1) e(tN ) → 0, N → ∞ w.p.1 implies e(t) → 0, t → ∞ w.p.1. (2) e(tN ) → 0, N → ∞ in the MS sense implies e(t) → 0, t → ∞ in the MS sense. (3) p q(N )e(tN ) ∈ l2 implies e ∈ L2 .
5.2 MS Exponential Convergence Case 1: ϕs (τ ) 6= 0 Theorem 17.. Under the assumptions of Theorem 13, if the switching time sequence is tN = γ ln N for some γ > 0, then for any 0 < η < (2r − 1)/γ eηtN Ee′ (tN )e(tN ) → 0 as N → ∞. Case 2: ϕs (τ ) = 0 Theorem 18. Under the assumptions of Theorem 16 and suppose tN is generated from (16), if for some r > 1/2 1 e′ e β = inf σmin ( r Ψ ΨN ) > 0, N N N then there exists η > 0 such that eηtN Ee′ (tN )e(tN ) → 0 as N → ∞.
5.1 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 9. Theorem 13.. Under Assumption 1 and 8, if ϕs (τ ) 6= 0, then ζ = sup N 2r−1 Ee′ (tN )e(tN ) < ∞
5.3 Almost Sure Exponential Convergence
N
and
By a certain choice of switching time sequences, Theorem 6 implies exponential convergence in the almost sure sense. Suppose that the switching time sequence is tN = γ ln N which implies etN = N γ .
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 → ∞,
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15th IFAC SYSID (SYSID 2009) Saint-Malo, France, July 6-8, 2009 7. CONCLUSIONS
Theorem 19. Under the Assumptions 1 and 8, if ϕs (τ ) 6= 0, then there exists some η > 0 such that
This paper introduces a method of designing observers and output feedback for linear time invariant systems with binary-valued observations. By controlling sensor thresholds, this method provides a controlled and eventtriggered sampling process for the system output such that convergent state observers and stabilizing output feedback can be developed. It is shown that stability in mean square, with probability one, as well as almost sure exponential stability can be guaranteed when the switching time sequence is suitably generated. Although this paper only presents stabilization results, feedback design for performance can be similarly implemented on the basis of state estimation.
(i) eηtN ke(tN )k = O(1) w.p.1 as N → ∞. (ii) eηt ke(t)k = O(1) w.p.1 as t → ∞. The case ϕs (τ ) = 0 can be directly obtained from Theorem 6 by a similar treatment of Theorem 19 for any switching time sequence with |ti − ti−1 | uniformly bounded, where i ≥ 1. Theorem 20. Under the Assumptions 1 and 8, 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 → ∞.
REFERENCES
6. BINARY OUTPUT FEEDBACK
K.J. Astrom and B. Wittenmark, Computer-Controlled Systems: Theory and Design, 3rd Ed., Prentice Hall Information and System Sciences Series, Prentice Hall, 1996. P. Billingsley, Convergence of Probability Measures, J. Wiley, New York, NY, 1968. T.W. Chen and B.A. Francis, Optimal Sampled-Data Control Systems, Springer, 1996. Y.S. Chow, Local convergence of martingales and law of large numbers, Ann. Math. Statist., 36 (1965), 552-558. K.L. Chung, A Course in Probability Theory, 2nd Edition, Academic Press, 1974. C.S. G¨ unt¨ urk, One-bit sigma-delta quantization with exponential accuracy, Communications on Pure and Applied Mathematics, Volume 56, Issue 11, pp. 1608 - 1630, 2003. S.N. Ethier and T.G. Kurtz, Markov Processes: Characterization and Convergence, J. Wiley, New York, NY, 1986. L. Guo, Convergence and logarithm laws of self-tuning regulators, Automatica, Vol.31, No.3, pp.435-450, 1995. P. Hall and C.C. Heyde, Martingale Limit Theory and Its Application, Academic Press, New York, NY, 1980. R.A. Horn and C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991. D.W. Huang and L. Guo, Estimation of nonstationary ARMAX models based on Hannan-Rissanen method, The Annals of Statistics, Vol.18, No.4, pp.1729-1756, 1990. R.Z. Khasminskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Alphen aan den Rijn, Netherlands, 1980. W. Kim, K. Mechitov, J.Y. Choi, S.K. Ham, On tracking objects with binary proximity sensors, Information Processing in Sensor Networks, Fourth International Symposium pp. 301 - 308, 2005. X. Koutsoukos, Estimation of hybrid systems using discrete sensors, Proc. 42nd IEEE Conf. Decision and Control, pp. 155-160, Dec. 2003. H.J. Kushner and G. Yin, Stochastic Approximation and Recursive Algorithms and Applications, 2nd Ed., Springer-Verlag, New York, 2003. J. Lamperti, Stochastic Processes, Springer-Verlag, New York, 1977. C.Y. Li, L.Y. Wang, G. Yin, L. Guo, C.Z. Xu, Convergence rates of state estimation for LTI systems with binary-valued observations, Submitted 2009. K. Ogata, Morden Control Engineering, 4th Edition, Prentice Hall, 2002. C. L. Phillips and H.T. Nagle, Digital Control System Analysis and Design, 3rd Ed., Prentice Hall, 1995. J. Raisch, Controllability and observability of simple hybrid control systems-FDLTI plants with symbolic measurements and quantized control inputs, CONTROL’94, PP. 595-600, March 21-24, 1994. W.F. Stout, Almost Sure Convergence, Academic Press, New York, 1974. J. Sur and B.E. Paden, State observer for linear time-invariant systems with quantized output, ASME J. Dynamic Systems, Measurement, and Control, Vol. 120, 1998. L.Y. Wang, J.F. Zhang, and G. Yin, System identification using binary sensors, IEEE Trans. Automat. Control, 48, pp. 1892-1907, 2003. L.Y. Wang and G. Yin, Asymptotically efficient parameter estimation using quantized output observations, Automatica, 43, 11781191, 2007. L.Y. Wang, G.H. Xu, and G. Yin, State Reconstruction for Linear Time-Invariant Systems with Binary-Valued Output Observations, Systems & Control Letters, 57, pp. 958-963, 2008. G. Yin, On extensions of Polyak’s averaging approach to stochastic approximation, Stochastics Stochastics Rep., 36 (1991), 245-264.
The observer provides a state estimate x b(t) which can be used to construct a feedback. Together, they form a binary output feedback. The feedback control is u(t) = −K x b(t) which leads to the closed-loop system x˙ = Ax − BK x b = (A − BK)x − BKe = Ac x + Bc e (17) where e = x b −x is the observation error, and Ac = A−BK is stable. Assumption 21. (1) The system is stablizable. (2) K is designed such that A − BK is stable. Under Assumption 21, the expression (17) shows that the closed-loop system is a stable system driven by the observation error process e(t). We first present the MS stability of the closed-loop system, namely Ex′ (t)x(t) → 0, t → ∞. Theorem 22. Under Assumptions of Theorems 16 and Assumption 21, the state of the closed-loop system satisfies: (1) it is square integrable, x ∈ L2 . (2) Ex′ (t)x(t) → 0, t → ∞. For analysis of strong stability x(t) → 0 w.p.1. we shall first simplify some expressions. The state of the closedloop system (17) can be solved as Z t Ac t x(t) = e x(0) + eAc (t−τ ) Bc e(τ )dτ. 0
Since the first term is deterministic and decays to 0 exponentially, it will not affect analysis of strong stability. Hence, assume x(0) = 0 and x(t) is simplified to Z t x(t) = eAc (t−τ ) Bc e(τ )dτ. (18) 0
For any vector norm k · k Z t kx(t)k ≤ κ e−a(t−τ ) ke(τ )kdτ 0
for some κ > 0 and a > 0 since the closed-loop system is stable. Theorem 23. Under Assumptions of either Theorem 13 or Theorem 16, there exists an η0 > 0 such that with probability one, kx(t)k ≤ κe−η0 t as t → ∞.
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