Stabilization of systems with asynchronous sensors and controllers

0.88 MB·

by  Masashi Wakaiki

Stabilization of Systems with Asynchronous Sensors
and Controllers
Masashi Wakaiki, Kunihisa Okano, and Joa˜o P. Hespanha
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0 Abstract
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Westudythestabilizationofnetworkedcontrolsystemswithasynchronoussensorsandcontrollers.
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Clockoffsetsbetweenthesensorandthecontrollerareunknownandmodeledasparametricuncertainty.
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2 First we consider MIMO systems and provide a sufficient condition for the existence of controllers that

] are capable of stabilizing the closed-loop system for every clock offset in a given range. For scalar
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systems, we next obtain the maximum length of the offset range for which the system can be stabilized
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. by a linear time-invariant controller. Finally, this bound is compared with the offset bounds that would
s
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[ be allowed if we restricted our attention to static output feedback controllers.

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8 I. INTRODUCTION
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7 In networked control systems, real-time sensor data is often not available at the controller
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. side due to transmission delays. A solution to time-varying delays is that the sensor sends to the
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controller the measurements with time-stamps to provide information on sampling instants [7].
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However, if the local clocks at the sensor and at the controller are not synchronized, then the
:
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i time-stamps and the true sampling instants may not match. Protocols to establish synchronization
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r have been actively studied as surveyed in [14], and synchronization by the global positioning
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system (GPS) or radio clocks has been utilized in practical systems. Nevertheless, synchronizing
Thispaperwaspartiallypresentedatthe2015AmericanControlConference,July1-3,2015,theUSA.Thismaterialisbased
upon work supported by the National Science Foundation under Grant No. CNS-1329650. M. Wakaiki acknowledges Murata
Overseas Scholarship Foundation for the support of this work. K. Okano is supported by JSPS Postdoctoral Fellowships for
Research Abroad. Corresponding author M. Wakaiki. Tel. [phone]. Fax [phone].
The authors are with the Center for Control, Dynamical-systems and Computation (CCDC), University of California, Santa
Barbara, CA 93106-9560 USA (e-mail:{masashiwakaiki, kokano, hespanha}@ece.ucsb.edu).
clocks over networks has fundamental limits [5]. Furthermore, a recent work [9] has shown that
synchronization technologies based on GPS signals may be vulnerable against malicious attacks.
We therefore address control with uncertainty in time.
In this paper, we study the stabilization problem for a system that has asynchronous sensing
and control. The controller side can use the time-stamps but does not know the actual sampling
instants or the clock offset between the sensor and the controller. Our objective is to determine
how large the clock offset can be without compromising the existence of a stabilizing linear
time-invariant (LTI) controller. Moreover, if one exists, we want to design such a controller.
We assume that the clock offset between the sensor and the controller is constant. Clock
properties are affected by environment such as temperature and humidity, but the change of such
propertiesareslowforthetimescalesofinterest.Furthermore,thedifferenceofclockfrequencies
can be ignored. This is justified by noting, e.g., that the time synchronization technique proposed
in [8] achieves the asymptotic error convergence of clock frequencies in the mean square sense
despite random delays.
We formulate the stabilization of systems with clock offsets as the problem of stabilizing
systems with parametric uncertainty, which can be regarded as the simultaneous stabilization
of a family of plants studied in [16, Sec. 5.4] and [17]. However, we had to overcome a few
technical difficulties that distinguish the problem considered here from previously published
results:
Infinitely many plants: We consider a family of plant models that is indexed by a continuous-
valued parameter. Such a family includes infinitely many plants, but the approaches based on
linear matrix inequalities (LMIs), e.g., in [15] exploit the property that the number of plant
models is finite.
Non-linearity of the uncertain parameter: In this work, the uncertain parameter appears in a
non-linear form, and we have a non-convex parameter set. Therefore, we cannot use the LMI-
based techniques in [2] for the robust stabilization of systems with polytopic uncertainties.
Common unstable poles and zeros: Earlier studies on simultaneous stabilization consider a
restricted class of plants. For example, the sufficient condition in [1] is obtained for a family of
plants with no common unstable zeros or poles. The set of plants in [11] has common unstable
zeros (or poles) but all the plants are stable (or minimum-phase). These assumptions are not
satisfied for the systems in the present paper.
Technically,wemakethefollowingcontributionsformulti-inputmulti-output(MIMO)systems
and scalar systems, respectively:
MIMO systems: First we show that an algebraic approach to the simultaneous stabilization
problem provides a necessary and sufficient condition for the existence of a controller that
stabilizes the system for every clock offset in a given range. This necessary and sufficient
condition is not numerically testable, but using a small gain argument, we construct a one-block
H∞ control problem that provides a testable sufficient condition for stabilizability. We also
construct a stabilizing controller from the solution of the H∞ control problem.
Scalar systems: For the special case of scalar systems, our system with clock offsets can
be regarded as an interval system, which has been studied in [6], [13]. In conjunction with
the Nevanlinna-Pick interpolation problem (see, e.g., [3, Chap. 2]), the necessary and sufficient
condition obtained for the MIMO case gives an explicit formula for the exact bound on the clock
offset that can be allowed for stability. We also show that the permissible range of the clock
offset varies depending on the class of the controllers. In particular, the use of LTI controllers
doubles the admissible range of clock offsets with respect to the use of linear static controllers.
Our previous work [12] has considered systems with time-varying clock offsets and has
proposed a stabilization method with causal controllers, based on the analysis of data rate
limitations in quantized control. The stability analysis of systems with variable clock offsets
and saturation has been investigated in [18]. The major difference from these works is that here
we consider only constant offsets and study stabilization limitations that arise from using LTI
controllers.
This paper is based on the conference paper [19]. Here, the preliminary results for single-input
systems is extended to the MIMO case, and we illustrate the relationship between the maximum
offset length and the system’s instability.
The remainder of the paper is organized as follows. Section 2 introduces the closed-loop
system we consider and presents the problem formulation. In Section 3, we obtain a sufficient
condition for the stabilizability of general-order systems. In Section 4, we derive the exact bound
on the permissible clock offset for scalar systems. Section 5 is devoted to stabilizability with
static controllers and the comparison of the offset bounds obtained for LTI controllers and static
controllers.
¯
Notation and definitions: Let Z+ be the set of non-negative integers. Let D, D, and T denote
the open unit disc {z ∈ C : |z| < 1}, the closed unit disc {z ∈ C : |z| ≤ 1}, and the unit circle
{z ∈ C : |z| = 1}, respectively.
We denote by RH∞ the space of all bounded holomorphic real-rational functions in D. Let
us denote the field of fraction of RH∞ by RF∞.
For a commutative ring R, M(R) denotes the set of matrices with entries in R, of whatever
order. For M ∈ M(C), let (cid:107)M(cid:107) be the induced 2-norm. For G ∈ M(RH∞), the RH∞-norm
is defined as (cid:107)G(cid:107) = sup (cid:107)G(z)(cid:107).
∞ z∈D
A pair (N,D) in M(RH∞) is said to be right coprime if the Bezout identity XN +YD = I
holds for some X, Y ∈ M(RH∞). P ∈ M(RF∞) admits a right coprime factorization if
there exist D, N ∈ M(RH∞) such that P = ND−1 and the pair (N,D) is right coprime.
Similarly, a pair (D˜, N˜) in M(RH∞) is left coprime if the Bezout identity N˜X˜ + D˜Y˜ = I
holds for some X˜, Y˜ ∈ M(RH∞). P ∈ M(RF∞) admits a left coprime factorization if there
exist D˜, N˜ ∈ M(RH∞) such that P = D˜−1N˜ and the pair (D˜,N˜) is left coprime. If P is a
scalar-valued function, we use the expressions coprime and coprime factorization.
II. PROBLEM STATEMENT
Consider the following plant:
x˙(t) = Ax(t)+Bu(t), (1)
where x(t) ∈ Rn and u(t) ∈ Rm are the state and the input of the plant, respectively. Using a
sampler and a zero-order hold (ZOH), we connect this continuous-time plant with a discrete-time
controller according to the diagram in Fig. 1.
Let s ,s ,... be sampling instants from the perspective of the controller clock, which, without
1 2
loss of generality, is assumed to be exact. The sensor measures the state x(s ) and sends it to
k
the controller together with a time-stamp. However, since the sensor and the controller share
no global clock, the time-stamp determined by the sensor typically includes an unknown offset
with respect to the controller clock. In this paper, we assume that the offset is constant, that is,
the time-stamp sˆ reported by the sensor is given by
k
sˆk = sk +∆ (k ∈ N) (2)
Discre+zed
 extended
 system
u(t) x(t)
Plant
ZOH
Sampler
u
k+1
Controller
Es+mator
xˆ(t ) (x(s ),sˆ )
k+1 k k
Fig. 1: Closed-loop system with clock offsets.
tk=kh sk� tk+1=(k+1)h sk+�1 tk+2=(k+2)h
sˆk sˆk+1 Time
Fig. 2: Sampling instants s , reported time-stamps sˆ , and updating instants t of the zero-order
k k k
hold.
for some unknown constant ∆ ∈ R. This means that there is a constant offset between the
sensor/controller clocks but no difference in their frequencies, or that this difference is negligible
for the time-scales of interest.
Let h > 0 be the update period of the ZOH. The control signal u(t) is assumed to be
piecewise constant and updated periodically at times tk = kh (k ∈ N) with values uk computed
by a controller: u(t) = u for t ∈ [t ,t ). While the ZOH updates u(t) periodically, the
k k k+1
true sampling times s and the reported sampling times sˆ may not be periodic. However, we
k k
do assume that both s and sˆ do not fall behind t by more than the ZOH update period h.
k k k
Moreover, the controller receives the time-stamped data (x(s ),sˆ ) from the sensor by the next
k k
ZOH-updating instant t . The assumptions are formally stated as follows.
k+1
Assumption 2.1: For every k ∈ Z+, sk,sˆk ∈ [tk,tk+1). Furthermore, x(sk) and sˆk are
available to the controller by time t = t .
k+1
Fig. 2 shows the timing diagram of the sampling instants s , the reported time-stamps sˆ , and
k k
updating instants t of the control inputs.
k
In order to periodically obtain state estimates at time h,2h,3h,..., from the time-stamped
data (x(s ),sˆ ) received non-periodically, we use the following estimator:
k k
˙
xˆ(t) = Axˆ(t)+Bu(t), (t ≤ t < t )
k k+1
(3)
xˆ(sˆk) = x(sk), (k ∈ Z+),
where xˆ ∈ Rn is the estimated state. Note that if sk = sˆk, then x(tk+1) = xˆ(tk+1), and this
estimator would consistently produce xˆ(t) = x(t) for all t. This estimator leads to the discretized
extended system in Fig. 1, with state and input given by
 
x(t )−xˆ(t )
k k
ξk =  , uk = u(tk),
xˆ(t )
k
respectively, which evolve according to
ξ = F ξ +G u , y = H ξ , (4)
k+1 ∆ k ∆ k k ∆ k
where Λ := eAh, Θ := e−A∆ −I, and
 
−ΛΘ −ΛΘ
F∆:= 
Λ(I +Θ) Λ(I +Θ)
 (cid:16) (cid:17) 
Λ (cid:82)he−Aτdτ−(I+Θ)(cid:82)h−∆e−Aτdτ B
G∆:= 0 Λ(I +Θ)(cid:82)h−∆e−0AτdτB  (5)
0
(cid:104) (cid:105)
H := 0 I .
∆
The derivation of (5) is straightforward and can be found in Appendix A.
The objective of the present paper is to determine how large the clock offset ∆ can be allowed
without compromising the existence of a discrete-time LTI controller that stabilizes the closed
loop for every admissible clock offset. Specifically, we want to solve the following problem:
Problem 2.2: Given an interval [∆,∆], determine if there exists a feedback controller that
stabilizes the extended system (4) for every ∆ ∈ [∆,∆]. Furthermore, if one exists, find such a
conroller.
To solve this problem, it is necessary that the extended system (4) has a stabilizing controller
for each ∆, which justifies our second assumption.
Assumption 2.3: For all ∆ ∈ [∆,∆], (F ,G ,H ) is stabilizable and detectable.
∆ ∆ ∆
Remark 2.4: In Problem 2.2, we consider only stability of (4) and thus of the sampled
system with state x(t ),x(t ),.... However, we can easily see from the plant linearity that this
1 2
discrete-time stability leads to x(t), xˆ(t) → 0 (t → ∞); see Appendix B.
Remark 2.5: From the parity interlacing property [21], we can check the simultaneous stabi-
lizability of any two plants in a family of plant models in (4), which gives a necessary condition
for the solvability of Problem 2.2.
Remark 2.6: If A is invertible, then we can rewrite G in (5) as
∆
 
−ΛΘA−1B
G∆ =  . (6)
(Λ(I +Θ)−I)A−1B
In fact, since
(cid:90) h
e−Aτdτ = −(e−Ah −I)A−1,
0
(1,1)-element of G in (5) is given by
∆
(cid:18)(cid:90) h (cid:90) h−∆ (cid:19)
Λ e−AτBdτ −(I +Θ) e−AτBdτ
0 0
(cid:0) (cid:1)
= Λ −(e−Ah −I)A−1B +e−A∆(e−A(h−∆) −I)A−1 B
= Λ(I −e−A∆)A−1B
= −ΛΘA−1B.
After calculating (1,2)-element of G similarly, we obtain (6).
∆
III. ANALYSIS VIA SIMULTANEOUS STABILIZATION
We first consider a general simultaneous stabilization problem not limited to the system
introduced in Section 2.
The transfer function P of the system ξ = Fξ +G u , y = Hξ is usually defined by
k+1 k k k k k
the Z-transform of the system’s impulse response, i.e., H(zI − F)−1G, but in this paper, we
define the transfer function P by P(z) := H(1/z·I−F)−1G for consistency of the Hardy space
theory; see [16, Sec. 2.2] for details. We say that C ∈ M(RF∞) stabilizes P ∈ M(RF∞) if
(I +PC)−1, C(I +PC)−1, and (I +PC)−1P belong to M(RH∞). We recall that when these
three transfer functions belong to M(RH∞), they will have no poles in the closed unit disk.
Consider the family of plants P ∈ M(RF∞) parameterized by θ ∈ S, where S is a nonempty
θ
parameter set, and assume that we have a doubly coprime factorization of P over RH∞ for all
θ
θ ∈ S:
  
˜
Y X D −X
θ θ θ θ
   = I, (7)
˜ ˜ ˜
−N D N Y
θ θ θ θ
where P = N D−1 and P = D˜−1N˜ are a right coprime factorization and a left coprime
θ θ θ θ θ θ
factorization, respectively.
The following theorem gives a necessary and sufficient condition for simultaneous stabiliza-
tion:
Theorem 3.1 ([16], [17]): Given a nonempty set S, consider the plant P having a doubly
θ
coprime factorization (7) for each θ ∈ S. Fix θ ∈ S and define
0
 
˜
(cid:104) (cid:105) (cid:104) (cid:105) Y −N
U V := D˜ N˜  θ0 θ0 (θ ∈ S). (8)
θ θ θ θ ˜
X D
θ0 θ0
Then (V ,U ) is right coprime for every θ ∈ S. Moreover, there exists a controller that stabilizes
θ θ
P for every θ ∈ S if and only if there exists Q ∈ M(RH∞) such that for all θ ∈ S,
θ
(U +V Q)−1 ∈ M(RH∞). (9)
θ θ
Such a stabilizing controller is given by
C := (X˜ +D Q)(Y˜ −N Q)−1. (10)
θ0 θ0 θ0 θ0
Remark 3.2: A left coprime factorization of stabilizing controllers is used in [16, Sec. 5.4]
and [17], whereas we represent controllers by a right coprime factorization in (10). Therefore,
Theorem 3.1 is slightly different from the counterpart in [16, Sec. 5.4] and [17]. The reason
˜
why we use a right coprime factorization is that D of our system (4) and (5) can be explicitly
θ
obtained for all θ, as we shall see Proposition 3.5 below.
It is often not easy to verify that the transfer function Q satisfying (9) exists in a computa-
tionally efficient fashion. To address this challenge, we use a one-block H∞ control problem,
whose solvability is sufficient for the existence of the transfer function Q satisfying (9).
Theorem 3.3: Given a nonempty set S, assume that each plant P (θ ∈ S) has a doubly
θ
coprime factorization (7) such that there exists θ0 ∈ S, L ∈ M(RF∞), and R(θ) ∈ M(R)
˜ ˜
satisfying D = D and
θ θ0
˜ ˜
N (z)−N (z) = R(θ)L(z), (11)
θ θ0
for all θ ∈ S. If there exists Q ∈ M(RH∞) satisfying the following one-block H∞ norm
condition:
1
˜
(cid:107)L(X +D Q)(cid:107) < , (12)
θ0 θ0 ∞ sup (cid:107)R(θ)(cid:107)
θ∈S
then Q satisfies (9), and hence the controller C in (10) stabilizes P for every θ ∈ S.
θ
Remark 3.4: We can solve the problem of finding Q ∈ M(RH∞) in (12) by reducing it
to the Nehari problem. Therefore the sufficient condition in Theorem 3.3 can be checked by
efficient matrix computations; see [4, Chapter 8] for details.
For the system (4) and (5), Proposition 3.5 below will facilitate verifying the assumptions on
˜ ˜
D and N in Theorem 3.3 and give an explicit formula for R and L in (11).
θ θ
˜ ˜
Proof of Theorem 3.3: We define U and V as in (8). Since D = D , it follows from (8)
θ θ θ θ0
˜ ˜ ˜ ˜
and the Bezout identity D Y +N X = I in (7) that
θ0 θ0 θ0 θ0
˜ ˜ ˜ ˜
U = D Y +N X
θ θ θ0 θ θ0
˜ ˜ ˜ ˜ ˜ ˜
= I +((D −D )Y +(N −N )X )
θ θ0 θ0 θ θ0 θ0
˜ ˜ ˜
= I +(N −N )X .
θ θ0 θ0
Moreover, since D˜−1N˜ = N D−1, we obtain
θ0 θ0 θ0 θ0
˜ ˜
V = −D N +N D
θ θ θ0 θ θ0
= −D˜ (N D−1 −D˜−1N˜ )D
θ θ0 θ0 θ θ θ0
= −D˜ (D˜−1N˜ −D˜−1N˜ )D
θ0 θ0 θ0 θ0 θ θ0
˜ ˜
= (N −N )D .
θ θ0 θ0
˜ ˜ ˜
Hence U +V Q = I +(N −N )(X +D Q). In conjunction with the small gain theorem,
θ0 θ θ θ0 θ0 θ0
this equation shows that if
˜ ˜ ˜
(cid:107)(N −N )(X +D Q)(cid:107) < 1 (θ ∈ S), (13)
θ θ0 θ0 θ0 ∞
then (9) holds for all θ ∈ S. From the assumption (11),
˜ ˜ ˜ ˜
(cid:107)(N −N )(X +D Q)(cid:107) ≤ (cid:107)R(θ)(cid:107)·(cid:107)L(X +D Q)(cid:107) .
θ θ0 θ0 θ0 ∞ θ0 θ0 ∞
HenceifQsatisfies(12)forallθ ∈ S,then(13)holds,andhenceP issimultaneouslystabilizable
θ
by C in (10) from Theorem 3.1. (cid:4)
˜
The proposition below shows that the system (4) always satisfies the assumptions on D and
θ
˜
N in Theorem 3.3. We also obtain R and L in (11) without calculating a coprime factorization
θ
of P for all θ ∈ S.
θ
Proposition 3.5: Let Assumption 2.3 hold and suppose that 0 ∈ [∆,∆]. Define the transfer
function P (z) := H (1/z ·I −F )−1G . For all ∆ ∈ [∆,∆], there exists a doubly coprime
∆ ∆ ∆ ∆
factorization (7) such that D˜ = D˜ = I −zeAh, and (11) holds with
∆ 0
R(∆) := (cid:82)∆e−Aτdτ ∈ Rn×n
0 (14)
L(z) := z(z −1)eAhB ∈ (RH∞)n×m.
¯ ¯ ¯
Proof: Let us first obtain another realization (F ,G ,H ) of the system (4). We can
∆ ∆ ∆
transform F into
∆
   
Λ 0 −Θ −I
F¯∆ := T−1F∆T =  , where T :=  .
0 0 I +Θ I
Furthermore, if we define
(cid:90) h (cid:90) h−∆
J := e−Aτdτ, J := e−Aτdτ,
1 2
0 0
then we obtain
 
−ΛJ B
G¯∆ := T−1G∆ =  1 .
−Λ(J −(I +Θ)J )B −ΘΛJ B
1 2 1
¯ ¯ ¯ ¯
Also we have H := H T = [I +Θ I]. We have thus another realization (F ,G ,H ) for
∆ ∆ ∆ ∆ ∆
P .
∆
From detectability in Assumption 2.3, for every ∆ ∈ [∆,∆], there exists a matrix W such
∆
¯ ¯ ˜
that F −W H is Schur stable. Using such a matrix W , we can write D as
∆ ∆ ∆ ∆ ∆
D˜ = I −H¯ (1/z ·I −(F¯ −W H¯ ))−1W ;
∆ ∆ ∆ ∆ ∆ ∆
see, e.g., [16, Theorem 4.19]. Since I+Θ = e−A∆, the inverse of I+Θ exists for all ∆ ∈ [∆,∆].
Set
 
Λ(I +Θ)−1
W∆ :=  .
0
Then we have
 
0 −Λ(I +Θ)−1
¯ ¯
F∆ −W∆H∆ =  ,
0 0