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Accepted Manuscript
Event-based reference tracking control of discrete-time nonlinear
systems via delta operator method
Rongjun Liu, Junfeng Wu, Dan Wang
PII:
DOI:
Reference:
S0016-0032(18)30500-3
https://doi.org/10.1016/j.jfranklin.2018.07.022
FI 3569
To appear in:
Journal of the Franklin Institute
Received date:
Revised date:
Accepted date:
16 November 2017
1 May 2018
31 July 2018
Please cite this article as: Rongjun Liu, Junfeng Wu, Dan Wang, Event-based reference tracking control
of discrete-time nonlinear systems via delta operator method, Journal of the Franklin Institute (2018),
doi: https://doi.org/10.1016/j.jfranklin.2018.07.022
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ACCEPTED MANUSCRIPT
Event-based reference tracking control of discrete-time nonlinear systems
via delta operator method
Rongjun Liua , Junfeng Wua,∗, Dan Wangb
of Measure-Control Technology and Communication Engineering, Harbin University of Science and Technology,
Harbin 150022 , P.R. China
b College of Automation, Harbin Engineering University, Harbin 150001, P.R. China
CR
IP
T
a College
Abstract
AN
US
This paper investigates the event-based tracking control for delta-sampling systems with a reference model.
Takagi-Sugeno (T-S) fuzzy model is used to approximate the nonlinearity. The delta operator is used to
implement the discrete-time system. The event trigger is adopted for saving the network resources and the
controller forces, and its detection period is designed with the same period of the delta-sampling period. Since
the measurement is delayed from the sensor to the event-trigger, the methodology of time-delay systems,
called the scaled small gain theorem, is applied for the system stability analysis. The reference output
tracking controller is designed to ensure the stability of the resulting system in H∞ sense. The optimization
conditions of the desired H∞ event-based tracking controller are synthesized, and the simulation example
validates its effectiveness finally.
M
Keywords: Event-triggered control, output tracking control, fuzzy systems, delta operator.
1. Introduction
AC
CE
PT
ED
Tacking control has been one of the hot spots in control engineering field. In the past decades, the
tracking control problems of linear/nonilear control systems, uncertain systems, switching systems, and
networked control systems, have been well studied [1, 2, 3, 4, 5]. The aim of output tracking control is to
design a controller to force the plant to track a output signal of a given reference model. Generally, the
real plants are usually complex with nonlinearity and uncertainties including the parameter perturbation
and disturbance noise. Therefore, the modeling of the plants are primarily important for accurate sensor
sampling and controller design. As is known, Takagi-Sugeno (T-S) fuzzy models [6] have been fully applied
for the representation of nonlinear systems since the convenience of model approximation and controller
realization [7]. In the frame of the T-S fuzzy model, several fuzzy rules based on the fuzzy sets are built
with corresponding linear subsystems. In this way, nonlinearity and uncertainties can be approximated in
pieces, and then a defuzzified plant can be yielded. The application of the T-S fuzzy model can be found for
different kind of the control systems such as the static output feedback control [8], state feedback control
[9, 10], observer-based control[11, 12, 13].
In digital control systems and the communications fields, high-speed signal processing can store informative data and deal with the fast signal transmission [14, 15, 16]. The analysis methods based on traditional
shift operator can handle the intractable digital control problems when fast sampling frequency are executed
[17, 18]. To formulate the systems of the continuous-time form and the discrete-time form in a unit frame,
delta operator was detailed in [17, 19]. With a fast sampling period based on delta operator, the finite word
∗ Corresponding
author
Email addresses: 641445222@qq.com (Rongjun Liu), wu_jf@hrbust.edu.cn (Junfeng Wu), bigbigfaceshop@163.com
(Dan Wang)
Preprint submitted to Journal of The Franklin Institute
August 13, 2018
ACCEPTED MANUSCRIPT
length performance can operate well. The following definition gives the formulation of delta operator [17]:
(
d x(t ), h = 0,
dtk k
δx(tk ) =
(1)
x(tk +h)−x(tk )
, h 6= 0,
h
AC
CE
PT
ED
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AN
US
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where tk = kh with h the sampling period, and δx (tk ) is the delta operator of x(tk ). x(tk ) is the state variable
of a real system. Obviously, with a small sampling period, the control system closes to a continuous-time
system. Thus, the characteristics of the stability and performance can be well responded like the continuoustime systems. By utility of the delta operator, typical control methods, such as feedback control [20, 21],
fault-tolerant control [22], disturbance rejection control [23] and tracking control [24], have designed for the
linear and nonlinear systems. For instance, in terms of the discrete-time fuzzy systems, the authors in [25]
coped with the H∞ control problem based on delta operator. The work in [26] presented a saturated control
scheme by using the fuzzy models and delta operator. For the discrete-time nonlinear systems based on
delta operator, an output tracking control scheme was detailed in [24] using T-S fuzzy models.
Recently, in contrast of sending information with a fixed period, the event-triggering communication
strategy has been built under the consideration of saving energy economy and reducing transmissions task
in the network resource [27, 28]. Frequent signal transmissions can deplete sensor or controller batteries
more easily, especially in the case when networks are utilizing the periodic communication paradigm. The
general setting of an event-triggering mechanism consists of an event detector unit and a sender unit. The
detector unit monitors a user-defined event-triggering condition, and instructs the sender unit to transmit
new measurements on violation of the condition. It deserves to be mentioned that the authors in [29] discussed the transmission of a a discrete event-triggered communication of fuzzy-model-based systems. In [30],
an event-driven control strategy was presented for the consensus of multi-agent systems with fixed/switching
topologies. The work in [31] introduced a methodology in terms of a class of time-delay system to handle the
event-triggered control of networked control systems. For the multi-area power systems with communication
delays, load frequency control was detailed in [32] based on an event-triggering condition. In terms of the
sampling systems based on delta operator, however few techniques are developed to the event-triggering
mechanism. Meanwhile, event-based tracking control of the nonlinear system still reminds an open problem
to explore. Motivated from the above discussion, we attempt to explore an event-based tracking controller
for the nonlinear systems of delta-sampling form.
In this work, we focus on dealing with the problems of the event-triggering mechanism design and
the full state-feedback tracking controller synthesis. The contribution follows that: i) the event-triggering
mechanism based on delta-sampling form is presented with the event-detecting period being concordant
with the delta-operator sampling period, ii) the stability criterion based on scaled small gain theorem is
developed for the resulting event-driven control system, and iii) an optimal event-based output tracking
controller is designed for the desired H∞ performance with a predefined reference system. The rest of
the paper follows that Section 2 explain the system descriptions, Section 3 presents the mentioned control
methods, and Section 4 gives a illustrative example for validation. The conclusion is detailed in Section 5.
Notation: `2 [0, ∞) is the space of square-integrable vector functions over [0, ∞). ∆1 ◦ ∆2 denotes the
series connection of the mappings ∆1 and ∆2 . For matrix X ∈ Rn×n , [X]s is used to denote X + X T
for simplicity. The superscripts “X T ” and “X −1 ” are respectively the matrix transpose and inverse of X.
The transposed elements in the matrix symmetric positions are denoted by “?”. “I” is an appropriately
dimensioned identity matrix if not specified.
2. Problem Formulation
Based on the formulation of delta operator (1), we establish the following T-S fuzzy systems
Plant Rule i : IF g1 (tk ) is Si1 , and · · · , and gp (tk ) is Sip , THEN
δx(tk ) = Ai x(tk ) + Bi u(tk ) + Bwi w(tk ),
y(tk ) = Ci x(tk ) + Di u(tk ) + Dwi w(tk ),
2
(2)
ACCEPTED MANUSCRIPT
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where x(tk ) ∈ Rn and u(tk ) ∈ Rm are the system state and system control input, y(tk ) ∈ Rm and w(tk ) ∈ Rl
are the system controllable output and the bounded external disturbance input. Matrices Ai , Bi , Bwi , Ci ,
Di and Dwi are known. Sij and gj (tk ) are respectively the fuzzy set and the premise variable (i = 1, 2, · · · , q,
j = 1, 2, · · · , p). The defuzzified expression of model (2) gives

q
P

 δx(tk ) =
vi (g(tk )) [Ai x(tk ) + Bi u(tk ) + Bwi w(tk )] ,



i=1


q
P
y(tk ) =
vi (g(tk )) [Ci x(tk ) + Di u(tk ) + Dwi w(tk )] ,
(3)

i=1


q

P


µi (g(tk )),
 vi (g(tk )) = µi (g(tk ))/
i=1
where µi (g(tk )) =
p
Q
Sij (gj (tk )),
j=1
q
P
i=1
vi (g(tk )) = 1, and 0 ≤ µi (g(tk )) ≤ 1 is the membership function
T
AN
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corresponding to i-th rule for fuzzy set Sij .
In the event trigger unit, the event detector monitors the state x(tk ) with the sampling period h. Let
the current detected state as x(ti ) (i = 0, 1, 2, . . . , ∞), and the latest detected and triggered one as x(tik )
(ik ∈ N, i0 = 0, k ∈ N is the number of triggering). The following event trigger condition (4) is adopted to
trigger new measurement over the network once x(tik +j ) satisfies
(x(tik +j ) − x(tik )) (x(tik +j ) − x(tik )) > ρxT (tik )x(tik ),
(4)
M
where ρ is a user-defined error tolerance. Assume that the initial state x(t0 ) is transmitted (known), and
then the next triggered time instant tik+1 is determined by
tik+1 = tik + max dk , h ,
(5)
where
n
o
T
dk = min tj | (x(tik +j ) − x(tik )) (x(tik +j ) − x(tik )) > ρxT (tik )x(tik ) .
ED
j≥1
PT
Remark 1. Actually, Zeno phenomena [33, 34, 35] are a common problem in event-triggering mechanism
for the control loop. Motivated by the seting in [33], we modify the positive real number as the sampling
period h in (5) to refrain from the Zeno behavior in the presented event-triggered control loop, since the
minimum step of the triggering interval is the sampling period via the delta operator in the control loop.
CE
Let sk is the time instant when the measurement x(sk ) arrives at the ZOH. As a result, on time interval
[sk , sk+1 ) for k ∈ N+ , we obtain that the following inequality always holds for j = 1, 2, · · ·
T
(x(tik +j ) − x(tik )) (x(tik +j ) − x(tik )) ≤ ρxT (tik )x(tik ).
(6)
AC
Considering the delayed state induced by the event-triggering mechanism, let dk = tik −tk be the delayed
time of k-th triggered measurement between the event trigger and the ZOH. Assume dm = min{dk |k =
0, 1, 2, . . . , ∞} and dM = max{dk |k = 0, 1, 2, . . . , ∞} be the minimum and the maximum delay, respectively.
On interval [sk , sk+1 ) for k ∈ N+ , define the following sub-intervals:

I1 = [sk , sk + h),



 I2 = [sk + h, sk + 2h),
..

.


 ?
Ijk = [sk + (jk? − 1)h, sk+1 ),
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with jk? = min+ {j | sk + (j − 1)h ≥ sk+1 }, and functions
j∈N
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
tk − tik , tk ∈ I1 ,



 tk − tik +1 , tk ∈ I2 ,
τk =
..

.



tk − tik +(jk? −1) , tk ∈ Ijk? ,

x(tik ) − x(tik ), tk ∈ I1 ,



 x(tik ) − x(tik +1 ), tk ∈ I2 ,
e(tk ) =
..

.



x(tik ) − x(tik +(jk? −1) ), tk ∈ Ijk? .
(7)
(8)
Then from (7), we know τk is a piecewise linear function, and the interval [sk , sk+1 ) can be represented as
?
[sk , sk+1 ) :=
jk
[
Ij .
j=0
AN
US
Considering the measurement transmitted through the network each time, one can obtain that it always
holds from (7) that
dm ≤ τk < dk + h ≤ dM + h.
Moreover, from (7) and (8) we know that the input signal x̄(tk ) of the controller is
x̄(tk ) = x(tik ) = e(tk ) + x(tk − τk ), tk ∈ [sk , sk+1 ).
And then the condition (6) can be updated as
T
(9)
M
eT (tk )e(tk ) ≤ ρ (e(tk ) + x(tk − τk )) (e(tk ) + x(tk − τk ))
ED
for any tk ∈ [sk , sk+1 ). Consequently, the received signal of the ZOH equals to the triggered the signal at
event-triggering instants t0 < ti1 < ti2 < · · · < tik < · · · .
In this paper, we consider the reference model and the reference output yr (tk ) ∈ Rm as follows
δxr (tk ) = Exr (tk ) + r(tk ),
(10)
yr (tk ) = F xr (tk ),
CE
PT
where xr (tk ) and r(tk ) are respectively the reference state and the reference input which is bounded. E and
F are constant matrices with E Hurwitz.
Based on the triggered measurement x(tik ), using the parallel distributed compensation, we design the
following event-based tracking controller to force the system (2) to track the reference output (see the
reference in (10)).
AC
u(tk ) =
=
q
X
i=1
q
X
i=1
vi (g(tk )) [Ki x(tik ) + Kri xr (tk )]
(11)
vi (g(tk )) [Ki (e(tk ) + x(tk − τk )) + Kri xr (tk )] ,
(12)
where tk ∈ [sk , sk+1 ), matrices Ki and Kri are the tracking controller parameters.
T
Let the tracking error ỹ(tk ) = y(tk )−yr (tk ) and state z(tk ) = xT (tk ) xT
. Considering (3)-(11),
r (tk )
for any tk ∈ [sk , sk+1 ) the closed-loop system gives

q P
q
P

 δz(tk ) =
vi vj Āij z(tk ) + B̄dij z (tk − τk ) + B̄ij $(tk ) ,

i=1 j=1
(13)
q P
q
P


vi vj C̄ij z(tk ) + D̄dij z (tk − τk ) + D̄ij $(tk ) ,
 ỹ(tk ) =
i=1 j=1
4
ACCEPTED MANUSCRIPT
T
and
where vi = vi (g(tk )) for simplicity, $(tk ) = eT (tk ) wT (tk ) rT (tk )
Ai Bi Krj
Bi Kj 0
Bi Kj Bwi 0
Āij =
, B̄dij =
, B̄ij =
,
0
E
0
0
0
0
I
C̄ij = Ci + Di Krj − F , D̄dij = DKj 0 , D̄ij = DKj Dwi 0 .
Before further we introduce the following definition and lemmas.
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Definition 1. [36] The stability of a delta operator system can be ensured when the following condition
holds:
1. V (x(tk )) ≥ 0, with equality if and only if x(tk ) = 0;
2. δV (x(tk )) = [V (x(tk + h)) − V (x(tk ))]/h < 0, where V (x(tk )) is a Lyapunov function in δ-domain.
The first condition gives the stability criterion in terms of the Lyapunov functions both in s-domain and
z-domain. With respect to the condition 2, there exists
dV (x(tk ))
V (x (tk + h)) − V (x(tk ))
=
< 0,
h→0
h
dtk
lim δV (x(tk )) = lim
h→0
AN
US
when h → 0. It corresponds to the Lyapunov function in z-domain. When h = 1, there exists δV (x(tk )) =
[V (x(tk + 1)) − V (x(tk ))]/1 = V (x(tk + 1)) − V (x(tk )) < 0. Hence, the Lyapunov function in δ-domain
implies traditional z-domain when h = 1.
Lemma 1. [36] For any of the vector functions a (tk ) and b(tk ) it holds that
δ a(tk )b(tk ) = δa(tk ) · b(tk ) + a(tk ) · δb(tk ) + h · δa(tk ) · δb(tk ).
Lemma 2. For any matrix Z > 0 and any positive integers q and q0 with 1 ≤ q0 ≤ q, it holds that
xT (l) X
l=q0
q
X
M
q
X
l=q0
x (l) ≤ (q − q0 + 1)
q
X
xT (l) Xx (l) .
l=q0
ED
Lemma 3. (Scaled small gain theorem) [37, 38, 39] Consider the following interconnection system
β(tk ) = Γ · α(tk ),
(14)
α(tk ) = ∆ · β(tk ),
(15)
CE
PT
where operator Γ mapping α(tk ) ∈ Rα to β(tk ) ∈ Rβ . Subsystem (14) is a known system and the subsystem
(15) is an unknown one but the operator ∆ ∈ = , k∆k∞ ≤ 1. Given that subsystem
(14) is internally
stable, the interconnection system (14)–(15) is asymptotically stable for all ∆ ∈ = if Nβ ◦ Γ ◦ Nα−1 ∞ < 1,
where nonsingular matrices {Nα , Nβ } ∈ X, and X , {Nα , Nβ } ∈ Rα×α · Rβ×β : kNα ◦ Γ ◦ Nβ−1 k∞ < 1 .
To represent the time-delay system to an interconnection system [40], we define
AC
α(tk ) =
dM
1
[2z (tk − τk ) − (z(tk − dm ) + z(tk − dM − h))]
+ h − dm
to approximate z (tk − τk ). Further, letting d = dM + h − dm , it results that
1
2
α(tk ) =
z (tk − τk ) − [z(tk − dm ) + z (tk − dM − h)]
d
2


−τk /h−1
−dm /h−1
X
X
2 h
h
= 
δz (tk + lh) −
δz (tk + lh)
d 2
2
l=−dM /h−1
=
1
dM /h − dm /h
i=−τk /h
−dm /h−1
X
φ (l) δz (tk + lh) ,
l=−dM /h
5
(16)
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where α(tk ) is the estimating error, and
φ (l) =
1, (l ≤ τk /h) ,
−1, (l > τk /h) .
According to Lemma 3, we write the system (13) as




δz(tk )
z̃(tk )
 β(tk )  = Γ(tk ) ·  α(tk )  ,
ỹ(tk )
γ$(tk )
z T (tk ) z T (tk − dm ) z T (tk − dM − h)

Ā(tk ) =
C̄(tk ) =
B̄(tk ) =
q X
q
X
i=1 j=1
q X
q
X
i=1 j=1
q
q X
X
, tk = −dM /h, −dM /h + 1, · · · , 0, and
1
d
2 B̄d (tk )
2 B̄d (tk )
1
d
B̄
(t
)
d
k
2
2 B̄d (tk )
1
d
D̄
(t
)
d
k
2
2 D̄d (tk )
q
q
XX
1
2 B̄d (tk )
1
2 B̄d (tk )
1
2 D̄d (tk )
Ā(tk )
Γ(tk ) =  Ā(tk )
C̄(tk )
T
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(18)

γ −1 B̄(tk )
γ −1 B̄(tk )  ,
γ −1 D̄(tk )
AN
US
where z̃(tk ) =
α(tk ) = ∆(tk ) · β(tk ),
(17)
vi vj B̄dij ,
vi vj Āij , B̄d (tk ) =
vi vj C̄ij , D̄d (tk ) =
vi vj B̄ij , D̄(tk ) =
vi vj D̄di ,
i=1 j=1
q X
q
X
vi vj D̄ij .
i=1 j=1
M
i=1 j=1
i=1 j=1
q X
q
X
ED
P−dm /h−1
1
and it can be specified the mapping ∆(tk ) : β(tk ) → α(tk ) = dM /h+1−d
i=−dM /h+1 φ (i) β (tk + ih)
m /h
according to the mapping in (15). Referring to [40, 41], we know operator k∆(tk )k∞ ≤ 1. Meanwhile, we
present the following lemma for stability analysis basis.
PT
Lemma 4. Suppose system (17) is internally stable. The closed-loop system (14) is asymptotically stable
in
for the mapping ∆(tk ), if there exists strictly positive-definite {Nα , Nβ } ∈ X such that
an H∞ sense
Nβ ◦ Γ ◦ Nα−1 < 1.
∞
CE
Proof. In terms of the interconnection system (17)–(18), it results form Lemma 3 that
2
2
2
2
kβ(tk )k2 + kỹ(tk )k2 < kα(tk )k2 + kγ$(tk )k2 .
(19)
AC
P−dm /h−1
1
Since α(tk ) = dM /h+1−d
i=−dM /h+1 φ (i) δz (tk + ih), it derives kỹk2 ≤ γ k$(tk )k2 ∀ 0 6= $ ∈ `2 [0, ∞),
m /h
which means interconnection system (17) with an H∞ performance γ for ∆(tk ) ∈ X. This ends the proof.
Therefore, in the next step we will analyze the stability of the system (17), i.e. the stability of the
following system
δz(tk ) = Ā(tk )z(tk ) + 21 B̄d (tk ) [z(tk − dm ) + z(tk − dM − h) + dα(tk )] + B̄(tk )γ$(tk ),
(20)
ỹ(tk ) = C̄(tk )z(tk ) + 21 D̄d (v) [z(tk − dm ) + z(tk − dM − h) + dα(tk )] + D̄(tk )γ$(tk ).
Hence, the aim of this work is to analyze the stability of the resulting augmented system (20), and then
design the parameters of the event trigger and the tracking controller. Additionally, the term z (tk − τk ) in
(13) is actually a time-varying delay term. In this paper we apply the so called analysis methodology in
terms of the “ time-delay system” to analyze and synthesize the resulting event-based control system (20).
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3. Main Results
Firstly, based on the resulting time-delay control system (20), we present the following theorem for a
stability criterion using Lyapunov functional approach.
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Theorem 1. System (17)–(18) is asymptotically stable in an H∞ sense, if there exist symmetric matrices
P > 0, Y > 0, U1 > 0, U2 > 0, W1 > 0, W2 > 0 and Z > 0 such that
Φ1 Φ2
Υ=
< 0,
(21)
? Φ3
where
Φ1 =
−Z
?
0
−I
, Φ2 =
0
0
X Ā(tk )
C̄(tk )
1
X B̄d (tk )
2
1
D̄ (t )
2 d k
1
X B̄d (tk )
2
1
D̄ (t )
2 d k
Ω1 = (h − 2)P + dm W1 + (dM + h)W2 ,
d
X B̄d (tk )
2
d
D̄ (t )
2 d k
X B̄(tk )
D̄(tk )
AN
US
1
1
W1 −
W2 ,
Ω2 = [P Ā(tk )]s + (dM /h + 1 − dm /h + 1)Y + U1 + U2 −
dm
dM + h

1
1
d
Ω1 P Ā(tk )
P B̄d (tk )
P B̄d (tk )
P B̄d (tk )
2
2
2
1
1
1
d
1
 ?
P B̄d (tk ) + dm W1
P B̄d (tk ) + dM +h W2
P B̄d (tk )
Ω2
2
2
2

 ?
?
− 41 Y − U1 − d1m W1
− 14 Y
− d4 Y

Φ3 = 
1
1
− d4 Y
?
?
− 4 Y − U2 − dM +h W2
 ?

2
d
 ?
?
?
?
− 4 Y −Z
?
?
?
?
?
Proof. Select Lyapunov-Krasovskii functional candidate V (tk ) =
4
P
P B̄(tk )
P B̄(tk )
0
0
0
−γ 2 I
,





.



Vl (tk ) for system (20), where
V1 (tk ) = z T (tk )P z(tk ),
dM /h+1
i=dm /h j=1
dm /h
V3 (tk ) = h
X
dM /h+1
z T (tk − ih)U1 z(tk − ih) + h
PT
i=1
z T (tk − jh)Y z(tk − jh),
ED
i
X X
V2 (tk ) = h
M
l=1
dm /h
V4 (tk ) = h2
X
i
X
i=1
δz(tk − jh)W1 δz(tk − jh) + h2
CE
i=1 j=1
X
z T (tk − ih)U2 z (tk − ih) ,
dM /h+1
i
X X
i=1
j=1
β(tk − jh)W2 β(tk − jh).
Using Lemma 1 with system (20), δV1 (tk ) results that
AC
δV1 (tk ) = 2δ T (z(tk ))P z(tk ) + hδ T (z(tk )) P δ(z(tk ))
n
= hδ T (z(tk ))P δ (z(tk )) + 2z T (tk )P Ā(tk )z(tk )
o
1
+ B̄d (tk ) [z(tk − dm ) + z(tk − dM − h) + dα(tk )] + B̄(tk )$(tk ) .
2
7
(22)
ACCEPTED MANUSCRIPT
By the delta operator manipulation, it follows that
δV2 (tk ) ≤ (dM /h + 1 − dm /h + 1) z T (tk )Y z(tk ) − z T (tk − αT ) Y z (tk − αT )
= (dM /h + 1 − dm /h + 1) z T (tk )Y z(tk )
T
1
d
1
− z(tk − dm ) + z (tk − dM − h) + α(tk )
2
2
2
1
1
d
×Y
z(tk − dm ) + z(tk − dM − h) + α(tk ) ,
2
2
2

dm /h
X
1
z T (tk − (i − 1) h) U1 z (tk − (i − 1) h)
δV3 (tk ) = · h 
h
i=1
X
i=1
z T (tk − (i − 1) h) U2 z (tk − (i − 1) h)
dm /h
i=1
dM /h+1
z T (tk − ih)U1 z (tk − ih) +
X
T
T
i=1

z T (tk − ih) U2 z(tk − ih)
AN
US
−
X
CR
IP
T
dM /h+1
+
(23)
= z (tk ) (U1 + U2 ) z(tk ) − z (tk − dm )U1 z(tk − dm )
− z T (tk − dM − h)U2 z (tk − dM − h) .
Using Lemma 2, δV4 (tk ) results that
(24)
δV4 (tk ) ≤ β T (tk ) ((dm /h)hW1 + (dM /h + 1) hW2 ) β(tk )
dm /h
dm /h
X
h X T
β (tk − ih) W1
β(tk − ih)
dm /h i=1
i=1
−
h
dM /h + 1
M
−
dM /h+1
β T (tk − ih) W2
ED
X
i=1
dM /h+1
X
i=1
β (tk − ih)
PT
= δ T (z(tk )) (dm W1 + dM W2 ) δ(z(tk ))
1
T
−
[z(tk ) − z(tk − dm )] W1 [z(tk ) − z(tk − dm )]
dm
1
T
[z(tk ) − z(tk − dM − h)] W2 [z(tk ) − z (tk − dM − h)] .
−
dM + h
CE
(25)
AC
For any P > 0, it always holds that
n
0 = −2δ T (z(tk ))P δz(tk ) − Ā(tk )z(tk )
o
1
− B̄d (tk ) [z(tk − dm ) + z(tk − dM − h) + dα(tk )] − B̄(tk )$(tk ) .
2
(26)
Further, recalling Lemma 4, we consider the following inequality from (19) for H∞ performance analysis.
z=
∞
X
T
β (tk ) Xβ(tk ) − αT (tk )Xα(tk ) + ỹ T (tk )ỹ(tk ) − γ 2 $T (tk )$(tk ) .
(27)
k=0
With ϑ =
δ T (z(tk ))
z T (tk ) z T (tk − dm ) z T (tk − dM − h) αT (tk ) γ$T (tk )
8
T
, under the zero
ACCEPTED MANUSCRIPT
initial condition, it results that
z ≤ z + V (∞) − V (0)
∞
X
T
=
β (tk )Xβ(tk ) − αT (tk )Xα(tk ) + ỹ T (tk )ỹ(tk ) − γ 2 $T (tk )$(tk ) + δV (tk )
=
k=0
∞
X
k=0
−1
ϑT (tk )(Φ3 − ΦT
2 Φ1 Φ2 )ϑ(tk ).
CR
IP
T
−1
According to Schur complement, Υ < 0 is equivalent to Φ3 − ΦT
2 Φ1 Φ2 < 0. Hence z < 0 under the
zero initial condition. With Lemma 4 we know that kỹ(tk )k2 < kγ$(tk )k2 holds and the H∞ performance
index γ guaranteed. Additionally, considering α(tk ) γ$(tk ) = 0, it follows δV (tk ) < 0 for any nonzero
states. That is to say the asymptotic stability of system (20) can be ensured. It ends the proof. In terms of the conditions (21), for a more relaxed stability condition of system (20), the following
corollary is developed.
AN
US
Corollary 1. System (17)–(18) is asymptotically stable in an H∞ sense, if there exist proper symmetric
matrices P > 0, Y > 0, U1 > 0, U2 > 0, W1 > 0, W2 > 0 and Z > 0, for i, j = 1, 2, · · · , q such that:
Υii < 0,
(28)
1
1
Υii + (Υij + Υji ) < 0, (i 6= j) ,
q−1
2

Υij
?
?
?
?
?
1
P B̄di
2
1
1
P
B̄
+
W2
di
2
dM +h
1
−4Y
− 41 Y − U2 − dM1+h W2
?
?
d
P B̄di
2
d
P B̄di
2
− d4 Y
− d4 Y
2
− d4 Y − Z
?
P B̄i
P B̄i
0
0
0
−γ 2 I
X B̄i
,
D̄i
1
1
W1 −
W2 .
= P Āij + ĀT
ij P + (dM /h + 1 − dm /h + 1) Y + U1 + U2 −
dm
dM + h
Φ2ij
Φ3ij
, Φ2ij =
0
0
X Āij
C̄ij
1
X B̄di
2
1
D̄
2 di
1
X B̄di
2
1
D̄
2 di
d
X B̄di
2
d
D̄
2 di
PT
Ω2ij




=


 ?
?
Φ1
=
?
1
P B̄di
2
1
P
B̄
+ d1m W1
di
2
1
− 4 Y − U1 − d1m W1
P Āij
Ω2ij
?
?
ED
Φ3ij
Ω1
?
?
?
M
where
(29)





,



Proof. Firstly, based on (28)–(29) we know Υij < 0 holds for i, j = 1, 2, · · · , q, using the results in [42].
q P
q
P
Moreover, substituting the defined matrices of system (20) into Υij yields
vi vj Υij = Υ. Hence, Υ < 0
i=1 j=1
CE
holds, and the same conclusion in Theorem 1 can be obtained. This ends the proof. AC
Remark 2. The conditions synthesized in Theorem 1 and Corollary 1 are actually with some couplings of
the tracking controller gains Ki and Kri . To find some solutions to the controller gains, we develop the
conditions in Corollary 1 by matrix transformation technique.
Theorem 2. System (17)–(18) is asymptotically stable with an H∞ performance level γ, if there exist
appropriately dimensioned symmetric matrices Q > 0, R > 0, Ȳ > 0, Ū1 > 0, Ū2 > 0, W̄1 > 0, W̄2 > 0,
Z̄ > 0, Mj and Lj for i, j = 1, 2, · · · , q the conditions (31) and (32) hold. The H∞ performance level γ can
be optimized by
min ρ = γ 2 subject to
(30)
Ῡii < 0,
1
1
Ῡii +
Ῡij + Ῡji < 0, (i 6= j)
q−1
2
9
(31)
(32)
ACCEPTED MANUSCRIPT
where
Φ̄2ij =

Φ̄3ij
with




=



Φ̄1
?
Φ̄2ij
Φ̄3ij
0 ΘAij
0 ΘCij
Ω̄1
?
?
?
?
?
, Φ̄1 =
Z̄ − 2Θ0
?
0
−I
1
d
2 ΘBdi
2 ΘBdi
1
d
Θ
Ddi
2
2 ΘDdi
1
2 ΘBdi
1
1
Θ
2 Bdi + dm W1
− 41 Ȳ − Ū1 − d1m W̄1
1
2 ΘBdi
1
2 ΘDdi
ΘAij
Ω̄2ij
?
?
?
?
,
ΘBi
ΘDi
,
1
2 ΘBdi
1
1
2 ΘBdi + dM +h W2
− 14 Ȳ
1
− 4 Ȳ − Ū2 − dM1+h W̄2
?
?
?
d
2 ΘBdi
d
2 ΘBdi
− d4 Ȳ
− d4 Ȳ
d2
− 4 Ȳ − Z̄

ΘBi
ΘBi
0
0
0
−γ 2 I




,



CR
IP
T
Ῡij =
?
?
?
U11 U12
U21 U22
Ȳ =
, Ū1 =
, Ū2 =
, ΘDdi = Ddi Q 0 ,
?
U13
?
U23
W11 W12
W21 W22
X1 X2
W̄1 =
, W̄2 =
, Z̄ =
,
?
W13
?
W23
? X3
Ai Q + Bi Mj Bi Nj
Bdi Q 0
Q 0
ΘAij =
, ΘBdi =
, Θ0 =
,
0
ER
0
0
0 R
Bdi Mj Bwi 0
ΘBij =
, ΘCij = Ci Q + Di Mj Di Nj − F R ,
0
0
I
ΘDij = Bdi Mj Dwi 0 , Ω̄1 = (h − 2)Θ0 + dm W̄1 + dM + hW̄2 ,
1
1
W̄1 −
W̄2 .
Ω̄2ij = ΘAij + ΘTAij + (dM /h + 1 − dm /h + 1) Ȳ + Ū1 + Ū2 −
dm
dM + h
Y2
Y3
M
AN
US
Y1
?
ED
If the conditions (31) and (32) hold, then the controller gains in (11) are determined by
Kj = Mj Q−1 , Krj = Lj R−1 .
(33)
CE
PT
Proof. Following the proof of Corollary 1, for i, j = 1, 2, · · · , q we have Ῡij < 0 from (31) and (32). Let
−1
P̄ −T Z̄ P̄ −1 ,
P̄ = P −1 = diag {Q, R}, and define Y = P̄ −T Ȳ P̄ −1 , U1 = P̄ −T Ū1 P̄ −1 , U2 = P̄ −T Ū
2 P̄−1 , Z =
−1
−T
−1
−1
−T
W1 = P̄ W̄
, W2 = P̄ W̄2 P̄ . Then by the congruence transformation diag P̄ , I, P̄ , P̄ −1 , P̄ −1 ,
1 P̄
P̄ −1 , P̄ −1 , I to Ῡij < 0, we have
Φ̃1 Φ̃2ij
< 0,
(34)
? Φ3ij
AC
where Mj = Kj Q and Lj = Krj R from (33), and
X − 2P
0
0 P Āij
Φ̃1 =
, Φ̃2ij =
?
−I
0 C̄ij
1
2 P B̄dij
1
2 D̄dij
1
2 P B̄dij
1
2 D̄dij
d
2 P B̄dij
d
2 D̄dij
P B̄ij
D̄ij
.
T
Besides, for any X > 0, since (X − P ) X −1 (X − P ) ≥ 0 it results −P X −1 P ≤ X − 2P . Using this
property, it yieldsΥij < 0 in Corollary 1, by performing the congruence transformation via
diag XP −1 , I, I, I, I, I, I, I . The proof is completed. Remark 3. Compared with the existing work [24, 43, 2], the proposed controller is an event-triggered one
for the output tracking control. It can save the control cost and communication resource since its the communication between the controller the plant is aperiodic. The simulations in Section 4 will give the illustration
of this virtue. Besides, the controller gains designed in this paper can be changed (subject to the conditions
synthesized in Theorem 2) when differen sampling periods of the plant are used in practice.
10
ACCEPTED MANUSCRIPT
Remark 4. Since solution of the synthesised conditions in Theorem 2 is subject to the sampling period
h, different sampling periods may yield different optimal γ. It depends on the specific circumstances for a
real control system. In simulations, we will use a fixed sampling period for a clear comparison between the
presented event-based controller and the conventional time-driven controller.
4. Simulation Example
CR
IP
T
We provide the mass-spring mechanical system [43] as the plant for the demonstration of the proposed
event-based tracking controller. Let s denote the displacement from a reference point. According to the
dynamics equation
ms̈ + Ff + Fs = u, where Ff = cṡ with c > 0 stands for the viscous damping force,
Fs = k 1 + a2 s2 s with constants k and a stands for the restoring force of the spring, m stands for the
mass, and u stands for the external control input. Thus, the dynamics equation is
AN
US
ms̈ + cẋ + k ṡ + ka2 s3 = u,
T
T . The state-space equation yields
= s ṡ
Let the state variable vector x = x1 x2
ẋ1
0
1
x1
0
=
+
u.
ẋ2
−k − ka2 x21 −c
x2
1
The system output is give by y = x1 . Considering the noise in the state process, we apply the following
fuzzy systems to approximate the plant with h = 0.1 s.
Fuzzy Rule 1 : IF x21 (tk ) is D̄, THEN
δx(tk ) = A1 x(tk ) + B1 u(tk ) + Bw1 w(tk ),
y(tk ) = C1 x(tk ) + D1 u(tk ) + Dw1 w(tk ).
M
¯ THEN
Fuzzy Rule 2 : IF x21 (tk ) is d,
δx(tk ) = A2 x(tk ) + B2 u(tk ) + Bw2 w(tk ),
y(tk ) = C2 x(tk ) + D2 u(tk ) + Dw2 w(tk ).
ED
The membership functions are chosen as
v1 x21 (tk ) = x21 − d¯ / D̄ − d¯ ,
v2 x21 (tk ) = 1 − v1 x21 (tk ) ,
AC
CE
PT
where D̄ = 0.5 and d¯ = 0 are the upper bound and lower bound of x21 , respectively. With c = 0.4 kg/s,
a2 = 0.9, m = 1 kg, k = 1.1 kg/s2 , we have
−0.0631 0.9825
0.0395
0.4
A1 =
, B1 =
, Bw1 =
,
−1.5671 −0.4561
0.9825
0.2
−0.0435 0.9830
0.0396
0.6
A2 =
, B2 =
, Bw2 =
,
−1.0813 −0.4367
0.9830
0.3
T
C1 = 1 0
, D1 = 0.1, Dw1 = 0.01, C2 = C1 , D2 = D1 , Dw2 = Dw1 .
The reference model is chosen as
δxr (tk ) = −0.9516xr (tk ) + r(tk ),
yr (tk ) = 0.15xr (tk ).
Suppose dm = 0.2 s and dM = 2.0 s and set the event trigger parameter ρ = 1.5. Solving the optimization
problem in Theorem 2, we have γmin = 0.7381, and the presented event-triggered controller:
2
X
u(tk ) =
vi Ki x(tik ) + Kri xr (tk ) ,
i=1
11
ACCEPTED MANUSCRIPT
with the gains
K1 =
K2 =
−1.8641
−2.6672
−1.2257
−1.1334
, Kr1 = 0.3421,
, Kr3 = 4.2457.
AN
US
CR
IP
T
To clear illustrate the efficacy of the designed event-triggered controller, we compare the presented
controller with the conventional controller in [24]. Meanwhile, we use the same setting and system parameters
of the simulated systems for the both controllers. Assumed that w(tk ) = 0.1rand · sin (20tk ), r(tk ) =
0.2 sin (20tk ). Then the simulation results are illustrated in Figs. 1–4.
The control inputs used in the two cases are given in Fig. 1. Fig. 2 shows the event-triggered intervals
at each triggered instant of the event-triggered controller. From Fig. 2 we know there are 26 times (including the one at the initial instant) of triggering in the event trigger unit. That is to say it saves almost
(20/h − 26)/(20/h) ≈ 98% cost for controller updating. Fig. 3 shows that y(tk ) tracks yq (tk ) well under
the conventional controller in [24] (shown in Fig. 3(a)) and the designed event-triggered controller (shown
in Fig. 3(b)). Accordingly, Fig. 4 depicts both tracking errors ỹ. The tracking errors are almost the same.
Obviously, the tracking performance under the presented controller is almost as well as the conventional one
while it saves much control cost.
1
0.5
M
−0.5
−1
−1.5
ED
C ontroller signa l u
0
PT
−2
T he conventio nal contro ller in [2 4 ]
−2.5
0
5
10
tk
15
20
Figure 1: Control input u(tk ).
AC
CE
−3
T he designed event-triggered co ntro ller
5. Conclusion
The problem of the event-based tracking control of discrete-time nonlinear systems has been handled in
this article. The delta-sampling plant based on T-S fuzzy model has been formulated. The event-triggering
mechanism has been designed with the same period of the delta-sampling period. For the resulting timedelay system, Lyapunov stability theory and the scaled small gain theorem have been based for the stability
analysis. The reference output tracking controller has been designed to ensure the stability of the resulting
system in H∞ sense. Finally, the designed event-based control has been successfully validated by the massspring mechanical tracking control system. Based on the developed results, future work will be concerned
12
ACCEPTED MANUSCRIPT
2
1.8
1.6
CR
IP
T
1.4
τk
1.2
1
0.8
0.6
AN
US
0.4
0.2
0
0
5
10
tk
15
20
(a)
0.8
M
Figure 2: Triggered intervals of the event trigger at each triggered instant.
ED
y(tk ) under the
co ntroller in [24 ]
yr (tk ) under the
co ntroller in [24 ]
0.7
y(tk ) under the
desig ned co ntro ller
yr (tk ) under the
desig ned co ntro ller
0.7
0.6
PT
0.6
(b)
0.8
0.5
0.5
0.4
0.4
CE
0.3
0.3
0.2
AC
0.2
0.1
0.1
0
0
−0.1
−0.1
0
5
10
tk
15
20
−0.2
0
5
Figure 3: Outputs y(tk ) and yr (tk ).
13
10
tk
15
20
ACCEPTED MANUSCRIPT
1
Under the controller in [2 4]
Under the presented contro ller
0.6
CR
IP
T
Tra cking error ỹ
0.8
0.4
0.2
−0.2
0
AN
US
0
5
10
tk
15
20
Figure 4: Tracking error e(tk ).
M
with the control case when stochastic process and parametric perturbation existing in the tracking control
systems.
ED
Acknowledgment
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PT
This work was supported by the Harbin Applied Technology Research and Development Project (excellent
academic leader) (No. 2014RFXYJ051), China.
AC
CE
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