1. Introduction
Recently, as real systems become more complex, many people have concerned about interconnected
systems (1), which effectively represent the complexity of a system through the composition with
subsystems and interconnection. Due to the structure of interconnected systems, some
additional problems should be considered, such as high dimensionality, structural
constraints of the controller, and uncertain or unknown information about the interconnections.
To solve these problems, a decentralized control technique, which is separately developed
for each subsystem while considering interconnections, is feasible. For this reason,
decentralized control techniques for interconnected systems have been actively studied
and presented so far (2-7).
Compared to a lot of research of the decentralized control, the decentralized observer
design is still lacking. Most of the previous observer design techniques were also
proposed by merging with the control technique (8-10), for exactly known systems (11,12), or for asymptotically stable systems (13,14). However, in the case of interconnected systems, system model is not able to be exactly
known due to the uncertain or unknown interconnection. In (15), the decentralized fuzzy observer is designed for interconnected systems with unknown
interconnection. However, this paper have only presented for sampled-data observing
technique, but have not studied for exact decentralized observer for discrete-time
interconnected systems. In (16), the decentralized fuzzy observer is designed for interconnected systems with unknown
interconnection, but this paper have not considered parametric uncertainties of subsystems.
Also, research is still lacking for observer design that considers observation performance
for interconnected systems. Especially, the detailed study is needed for the decentralized
observer design technique of discrete-time nonlinear interconnected systems with unknown
interconnection and parametric uncertainties.
Motivated by the above problems of the previous techniques, in this paper, the decentralized
fuzzy observer design technique is proposed for discrete-time nonlinear interconnected
systems with parametric uncertainties. Based on the Takagi–Sugeno (T–S) fuzzy model,
we consider that the interconnected system is composed of fuzzy subsystems and unknown
interconnections. Also, for the decentralized fuzzy observer design, the interconnection
bound to attenuation degree ratio (IAR) is defined and the observer design problem
is stated. By using the error dynamics and the observation performance function, the
sufficient condition with minimization problem is formulated for the decentralized
observer design, and its condition is derived into linear matrix inequalities (LMIs).
Finally, the validity and effectiveness of the proposed technique is demonstrated
by the simulation example.
This paper is organized as follows: Section 2 describes the decentralized fuzzy observer
and the observer design problem. The LMI conditions with minimization problem based
on the error dynamics are proposed in Section 3. In Section 4, it shows an numerical
example. Finally, the conclusions are given in Section 5.
Notation: The subscript $i$ denotes fuzzy rule indices, and the subscript $k$ denotes
the subsystem index. The notation $(\bullet)^{T}$ denotes the transpose of the argument
and the notations $\ast$ is used for the transposed element in symmetric position.
2. Preliminaries
We consider a discrete-time nonlinear interconnected system composed of $n$ Takagi-Sugeno
(T-S) fuzzy subsystems, which is described for $k$th subsystem as follows:
where $x_{k}(t)\in bold{R}^{m_{k}}$ and $y_{k}(t)\in bold{R}^{l_{k}}$ are the state
and output vectors of $k$th subsystem, respectively, $\Gamma_{kip}$ is a fuzzy set
for $p\in I_{w}=\{1,\:2,\:\cdots ,\:w\}$, $i\in I_{r}=\{1,\:2,\:\cdots ,\:r\}$ and
$k\in I_{n}=\{1,\:2,\:\cdots ,\:n\}$. and $x_{k}(t)={c o l}\left\{x_{1}(t),\:x_{2}(t),\:\cdots
,\:x_{n}(t)\right\}$ is the whole state variable of the interconnected system, Also,
$A_{ki}\in bold{R}^{m_{k}\times m_{k}}$ and $C_{k}\in bold{R}^{l_{k}\times m_{k}}$
are the linear matrices of the $k$th subsystem, and the uncertain matrix $\Delta A_{ki}$
is structured as following assumption:
Assumption 1: The matrix $\Delta A_{ki}$ for parametric uncertainties is represented
as follows:
$\Delta A_{ki}=D_{ki}F_{ki}(t)E_{1ki}$
where $D_{ki}$ and $E_{ki}$ are known real constant matrices of compatible dimensions,
and $F_{ki}(t)$ is an unknown matrix function with Lebesgue-measurable elements and
with $(F_{ki}(t))^{T}F_{ki}(t)\le I$.
Also, the interconnection function $h_{k}(x(t))$ is an unknown vector function with
the following assumption:
Assumption 2: The vector function $h_{k}(x(t))$ is unknown, but satisfies the following
quadratic inequality:
$\left .\left(h_{k}(x(t))\right)^{T}h_{k}(x(t))\le\alpha^{2}x(t)^{T}H_{k}^{T}H_{k}x(t)\right
.$
where $\alpha >0$ is a bound scalar of the interconnection term, and $H_{k}$ is a
given constant matrix with appropriate dimension.
Using center-average defuzzification, a product inference, and a singleton fuzzifier
to the fuzzy IF-THEN rule eq(1), the $k$th subsystem can be inferred as follows:
where
$\mu_{ki}(z_{k}(t))=\dfrac{\omega_{ki}(z_{k}(t))}{\sum_{i=1}^{r}\omega_{ki}(z_{k}(t))}$,
$\omega_{ki}(z_{k}(t))=\prod_{p=1}^{q}\Gamma_{kip}(z_{kp}(t))$
in which $\Gamma_{kip}(z_{kp}(t))$ is the fuzzy membership grade of $z_{kp}(t)$ in
$\Gamma_{kip}$.
Based on the $k$th subsystem eq(2), the decentralized fuzzy observer can be supposed in the following forms:
where $\hat x_{k}(t)$ and $\hat y_{k}(t)$ are the estimated state and output variables,
respectively, and $L_{ki}$ is the observer gain matrices with appropriate dimension.
Remark 1: In this paper, the decentralized fuzzy observer is considered for the measurable
premise variable case. It means that premise variable $z_{k}(t)$ can be obtained by
the output variable $y_{k}(t)$, directly. If not, the decentralized fuzzy observer
is designed based on the non-measurable premise variable case. However, it is out
of the scope.
Now, to design the observer gain, the estimation error is considered as $e_{k}(t)=x_{k}(t)-\hat
x_{k}(t)$, and its dynamics is developed as follows:
where
$\Phi_{k}(t)=\sum_{i=1}^{r}\mu_{ki}(z_{k}(t))\left(A_{ki}-L_{ki}C_{k}\right)$,
$\Delta A_{k}(t)=\sum_{i=1}^{r}\mu_{ki}(z_{k}(t))\Delta A_{ki}$.
From the fuzzy interconnected systems composed of the closed-loop subsystems eq(4), the decentralized fuzzy observer design problem can be stated as follows:
Problem 1: Find the observer gain matrix $L_{ki}$ such that
1) The estimation error dynamics eq(4) with $h_{k}(x(t))=0$ and $\Delta A_{ki}=0$ is asymptotically stable.
2) There exists the attenuation degree $\gamma\ge 0$ satisfying norm inequality $∥
e(t)∥\le\gamma ∥ x(t)∥$, where $e_{k}(t)={c o l}\left\{e_{1}(t),\:e_{2}(t),\:\cdots
,\:e_{n}(t)\right\}$ is the whole estimation error.
3) The IAR $\gamma /\alpha$ is minimized.
3. Main Results
This section shows the decentralized fuzzy observer design method based on the error
dynamics eq(4) considering Problem 1. To achieve the design purpose, we define the following performance
function:
Definition 1: The performance function
$J=\sum_{t=0}^{\infty}\left(e(t)^{T}e(t)-\gamma^{2}x(t)^{T}x(t)\right)$
is said to be an IAR guaranteed function. If there exists the decentralized fuzzy
observer gain $L_{ki}$ such that the IAR guaranteed function satisfies $J\le 0$, we
guarantee the existence of the attenuation degree $\gamma$ and the decentralized fuzzy
observer satisfies norm inequality $∥ e(t)∥\le\gamma ∥ x(t)∥$.
Also, before proceeding to our main results, following lemmas have to be considered
for the proof.
Lemma 1 (17): For any real matrices $X$ and $Y$ with appropriate dimensions, the following inequality
is always satisfied:
$X^{T}Y+Y^{T}X\le\sigma X^{T}X+\sigma^{-1}Y^{T}Y$
where $\sigma$ is a positive constant.
Lemma 2 (18): Given some constant matrices $D$ and $E$, and some symmetric constant matrices $P_{1}$
and $P_{2}$ of appropriate dimensions, the following inequality holds:
$\left[\begin{matrix}-P_{1} & * \\ D F E & -P_{2}\end{matrix}\right]<0$
where the matrix $F$ satisfies $F^{T}F\le I$, if and only if
$\left[\begin{matrix}-P_{1} & * & * & * \\ 0 & -P_{2} & * & * \\ E & 0 & -\epsilon
I & * \\ 0 & D^{T} & 0 & -\epsilon^{-1}\end{matrix}\right]<0$
for some $\epsilon >0$,
On the basis of the above lemmas, the sufficient condition for the decentralized fuzzy
observer design based of the discrete-time nonlinear interconnected system composed
of fuzzy subsystems eq(2) is summarized as follows.
Theorem 1: Consider the nonlinear interconnected system composed of fuzzy subsystems
eq(2) and the decentralized fuzzy observer eq(3). If there exist scalar $\delta$ and $\beta$, and matrices $N_{ki}$ and $P_{k}=P_{k}^{T}>0$
such that following optimal problem with the LMI is satisfied:
where
$\Phi_{ki}=P_{k}A_{ki}-N_{ki}C_{k}$, $\rho =\dfrac{\delta}{n\lambda_{H}}-\beta$,
and $\lambda_{H}$ is a maximum eigenvalue of matrices $H_{k}^{T}H_{k}$ for $k\in I_{n}$,
and $\epsilon_{i}$ is a given constant scalar for $(k,\:i)\in I_{n}\times I_{n}$,
then decentralized fuzzy observer eq(3) well estimates the nonlinear interconnected system eq(2), the decentralized fuzzy observer gain $L_{ki}=P_{k}^{-1}N_{ki}$, and $\delta =\gamma^{2}/(\alpha^{2}+1)$.
Proof: Consider the Lyapunov functional condidate as $V(t)=\sum_{k=1}^{n}e_{k}(t)^{T}P_{k}e_{k}(t)$
with $P_{k}=P_{k}^{T}>0$. Then, we have
Based on Lemma 1, we know that
$\left(\Phi_{k}(t)e_{k}(t)+\Delta A_{k}(t)x_{k}(t)\right)^{T}P_{k}h_{k}(x(t))$
$+h_{k}(x(t))^{T}P_{k}\left(\Phi_{k}(t)e_{k}(t)+\Delta A_{k}(t)x_{k}(t)\right)$
$\le\sigma^{-1}\left(\Phi_{k}(t)e_{k}(t)+\Delta A_{k}(t)x_{k}(t)\right)^{T}P_{k}^{2}$
$\times\left(\Phi_{k}(t)e_{k}(t)+\Delta A_{k}(t)x_{k}(t)\right)+\sigma h_{k}(x(t))^{T}h_{k}(x(t))$
with $\sigma\ge 0$. Thus, from inequality eq(7), we can obtain the followings:
If there exists some scalar $\beta$ such that $P_{k}-\beta I<0$, then inequality eq(8) is represented as
From Assumption 2, we can further rewrite inequality eq(9) as
$(9)\le\sum_{k=1}^{n}\left(\Phi_{k}(t)e_{k}(t)+\Delta A_{k}(t)x_{k}(t)\right)^{T}\left(P_{k}+\sigma^{-1}P_{k}^{2}\right)$
$\times\left(\Phi_{k}(t)e_{k}(t)+\Delta A_{k}(t)x_{k}(t)\right)$
$-\sum_{k=1}^{n}e_{k}(t)^{T}(P_{k}-I)e_{k}(t)$
$-\sum_{k=1}^{n}n\lambda_{H}(\sigma +\beta)x_{k}(t)^{T}x_{k}(t)$
$+\sum_{k=1}^{n}\left((\sigma +\beta)\lambda_{H}(\alpha^{2}+1)x(t)^{T}x(t)-\dfrac{\gamma^{2}}{n}x(t)^{T}x(t)\right)$.
Thus, the sufficient condition for $V(t+1)-V(t)$ $+\sum_{k=1}^{n}\left(e_{k}(t)^{T}e_{k}(t)-\gamma^{2}x_{k}(t)^{T}x_{k}(t)\right)<0$
can be rewritten as
Applying the Schur complement to eq(10) yields
By substituting eq(12) into eq(13), using Lemma 2, applying Congruence transformation with ${diag}\left\{I,\: I,\: P_{k},\:
P_{k},\: I,\: I\right\}$ and denoting $P_{k}L_{ki}=N_{ki}$, $\delta =\gamma^{2}/(\alpha^{2}+1)$,
and $\rho =\left(\delta /(n\lambda_{H})\right)-\beta$, we can obtain the LMI eq(5). Thus, if LMIs eq(5) and eq(6) are satisfied, it follows:
Then, we get $J<0$ by summing inequality eq(14) from $t=0$ to $\infty$. Thus, LMIs eq(5) and eq(6) with optimal problem guarantees the existence $\gamma$ and minimization of IAR. Also,
if $h_{k}(x(t))=0$ and $\Delta A_{ki}=0$, LMIs eq(5) is simplified as
By applying the Schur complement and Congruence transformation with ${diag}\left\{I,\:
P_{k}^{-1},\: P_{k}^{-1}\right\}$, the sufficient condition eq(15) is represented as
$-P_{k}+I+\left(A_{ki}-L_{ki}C_{k}\right)^{T}(P_{k}+\beta^{-1}I)\left(A_{ki}-L_{ki}C_{k}\right)PREC
0$.
Then, the above inequality guarantees the asymptotic stability of the estimation error
dynamics eq(4) with $h_{k}(x(t))=0$ and $\Delta A_{ki}=0$.
■
Remark 2: To obtain LMI eq(5), it is necessary for process to dispose of uncertain matrix $\Delta A_{k}(t)$ from
inequality eq(13) of Proof, Also, this process is possible by applying Lemma 2, and so is omitted here.
Remark 3: Theorem 1 presents the decentralized fuzzy observer design technique with
the minimization of the IAR. It means that the minimized error can be obtained in
the fixed interconnection bound, or the maximum interconnection bound can be found
in the fixed attenuation degree of the estimation error.
4. Simulation
In this section, we provide simulation results to demonstrate the effectiveness of
the proposed decentralized fuzzy observer design method. To simulate, we consider
the nonlinear double mass-spring system (19) connected by a spring as follows:
$\ddot\zeta_{k}(t)=-f_{k}(\zeta_{k}(t))-h_{k}(\zeta_{k}(t))$
$y_{k}(t)=\zeta_{k}(t)$
where
$f_{1}(\zeta_{1}(t))=0.01\zeta_{1}(t)+0.67\zeta_{1}(t)^{3}$
$f_{2}(\zeta_{2}(t))=0.02\zeta_{2}(t)+0.63\zeta_{2}(t)^{3}$
and $h_{k}(\zeta(t))=\alpha(0.1\zeta_{k}(t)-0.1\zeta_{l}(t))$, which is the interconnection
function by the constant $\alpha$, with $\zeta(t)=\left[\zeta_{1}(t)^{T} \quad \zeta_{2}(t)^{T}\right]^{T}$
for $\left\{(k,\:l)\in I_{2}| k\ne l\right\}$. Then, by choosing $x_{k}(t)=\left[\begin{matrix}{lll}x_{k
1}(t) & x_{k 2}(t)\end{matrix}\right]^{T}=\left[\begin{matrix}{ll}\zeta_{k}(t) & \dot{\zeta}_{k}(t)\end{matrix}\right]^{T}$,
using the membership function $\mu_{k1}(\zeta_{k}(t))=1-\left(\zeta_{k}(t)^{2}/M_{k}^{2}\right)$
and $\mu_{k2}(\zeta_{k}(t))=\zeta_{k}(t)^{2}/M_{k}^{2}$ where $M_{k}=10$ is the maximum
value of $\left |\zeta_{k}(t)\right |$, and approximate discretizaition, the discrete-time
double mass-spring system can be represented by the following fuzzy subsystem:
$x_{k}(t+1)=\sum_{i=1}^{2}\mu_{ki}(z_{k}(t))\left(\left(I+A_{ki}T\right)x_{k}(t)+T
h_{k}(t)\right)$
$y_{k}(t)=C_{k}x_{k}(t)$
where
$A_{11}=\left[\begin{matrix}0 & 1 \\ -0.01 & 0\end{matrix}\right]$, $A_{12}=\left[\begin{matrix}
0& 1\\-0.68& 0\end{matrix}\right]$,
$A_{21}=\left[\begin{matrix} 0& 1\\-0.02& 0\end{matrix}\right]$, $A_{22}=\left[\begin{matrix}
0& 1\\-0.65& 0\end{matrix}\right]$,
$C_{1}=C_{2}=\left[\begin{matrix} 1& 0\end{matrix}\right]$,
$H_{1}=\left[\begin{matrix} 0& 0& 0& 0\\-0.1& 0& 0.1& 0\end{matrix}\right]$, $H_{2}=\left[\begin{matrix}
0& 0& 0& 0\\0.1& 0& -0.1& 0\end{matrix}\right]$.
To use Theorem 1, we assume that a sampling period is $T=0.1$, the system matrix $A_{ki}$
has parametric uncertainties of $0.1$ and the value of the constant $\epsilon_{i}$
is $100$. by choosing the initial condition $x_{1}(0)=\left[\begin{matrix} 1& 0.1\end{matrix}\right]^{T}$,
$x_{2}(0)=\left[\begin{matrix} 1& 0.1\end{matrix}\right]^{T}$ and $\hat x_{k}(0)=\left[\begin{matrix}
0& 0\end{matrix}\right]^{T}$ , and solving LMIs eq(5) and eq(6), we get the decentralized fuzzy observer gains as follows:
$L_{11}=[\begin{aligned}1.0758\\0.7572\end{aligned}]$, $L_{12}=[\begin{aligned}1.0758\\0.6903\end{aligned}]$,
$L_{21}=[\begin{aligned}1.0758\\0.7564\end{aligned}]$, $L_{22}=[\begin{aligned}1.0758\\0.6934\end{aligned}]$.
Time responses of each subsystem are shown in Fig 1-4 with $\alpha =10$. As shown in figures, we can guarantee that the proposed decentralized
fuzzy observer has good estimation performance for the nonlinear interconnected system.
The results of estimation error are also shown in Fig 5. To show the effectiveness of the proposed technique, we compare the performance
of Theorem 1 and previous fuzzy observers without considering parametric uncertainties
or interconnection by using the performance measure function
$P =\sum_{t=100}^{1000}P(t)=\sum_{t=100}^{1000}\sum_{k=1}^{n}\sum_{v=1}^{2}(x_{kv}(t)-\hat
x_{kv}(t))^{2}$
in Table 1 and Fig 6. From the results of Table 1, Thus, we know that the proposed technique has better observing performance than
others.
그림. 1. $x_{11}(t)$의 시간응답: 상태변수 $x_{11}(t)$ (실선), 추정상태변수 $\hat x_{11}(t)$ (파선).
Fig. 1. Time Responses of $x_{11}(t)$: original $x_{11}(t)$ (solid) and estimated
$\hat x_{11}(t)$ (dashed).
그림. 2. $x_{12}(t)$의 시간응답: 상태변수 $x_{12}(t)$ (실선), 추정상태변수 $\hat x_{12}(t)$ (파선).
Fig. 2. Time Responses of $x_{12}(t)$: original $x_{12}(t)$ (solid) and estimated
$\hat x_{12}(t)$ (dashed).
그림. 3. $x_{21}(t)$의 시간응답: 상태변수 $x_{21}(t)$ (실선), 추정상태변수 $\hat x_{21}(t)$ (파선).
Fig. 3. Time Responses of $x_{21}(t)$: original $x_{21}(t)$ (solid) and estimated
$\hat x_{21}(t)$ (dashed).
그림. 4. $x_{22}(t)$의 시간응답: 상태변수 $x_{22}(t)$ (실선), 추정상태변수 $\hat x_{22}(t)$ (파선).
Fig. 4. Time Responses of $x_{22}(t)$: original $x_{22}(t)$ (solid) and estimated
$\hat x_{22}(t)$ (dashed).
그림. 5. 추정오차 $e(t)$의 시간응답: $e_{11}(t)$ (실선), $e_{12}(t)$ (파선), $e_{21}(t)$ (쇄선), and
$e_{22}(t)$ (점선).
Fig. 5. Time responses of estimation error $e(t)$: $e_{11}(t)$ (solid), $e_{12}(t)$
(dashed), $e_{21}(t)$ (dash-dotted), and $e_{22}(t)$ (dotted).
그림. 6. 성능평가함수 $P(t)$의 시간응답: 정리 1 (실선), 퍼지 관측기 (18) (파선), 분산 퍼지 관측기 (19) (점선).
Fig. 6. Time responses of performance measure function $P(t)$: Theorem 1 (solid),
fuzzy observer (20) (dashed), and decentralized fuzzy observer (16) (dotted).
표 1. 성능 비교
Table 1. Performance Comparison
|
Performance measure function
|
Results
|
|
Theorem 1
|
0.4529
|
|
fuzzy observer (20)
|
0.8357
|
|
decentralized fuzzy observer (16)
|
0.5028
|
5. Conclusion
In this paper, the decentralized fuzzy observer design technique has been proposed
for discrete-time nonlinear interconnected systems. For considering interconnected
systems, T-S fuzzy subsystems with paramteric uncertainties and unknown interconnection
have been adopted. The observer is designed to minimize IAR by using the error dynamics
and the defined performance function. The sufficient condition for oberver design
is provided in the LMI format. The numerical example has been shown to prove the advantage
of the developed method.