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Optimal performance of networked control systems under the packet dropouts and channel noise
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Optimal performance of networked control systems under the packet dropouts and channel noise$ Xi-Sheng Zhan b,a, Jie Wu a,n, Tao Jiang b, Xiao-Wei Jiang b a b
Department of Control Science and Engineering, Hubei Normal University, Huangshi, 435002, P.R. China Department of Electronics and Information Engineering, Huazhong University of Science and Technology, Wuhan, 430074, P.R. China
art ic l e i nf o
a b s t r a c t
Article history: Received 3 December 2014 Received in revised form 10 March 2015 Accepted 26 May 2015 This paper was recommended for publication by Prof. Y. Chen
The optimal tracking performance of single-input single-output (SISO) discrete-time networked control systems (NCSs) with the packet dropouts and channel noise is studied in this paper. The communication channel is characterized by three parameters: the packet dropouts, channel noise and the encoding and decoding. The explicit expression of the optimal tracking performance is obtained by using the spectral factorization. It is shown that the optimal tracking performance dependents on the nonminimum phase zeros, unstable poles of the given plant, as well as the packet dropout probability, channel noise and the encoding and decoding. The optimal tracking performance is improved by two-parameter compensator. Finally, a typical example is given to illustrate the theoretical results. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Networked control systems Optimal tracking performance Packet dropout probability Nonminimum phase zeros Encoding and decoding
1. Introduction In recent years, more and more researchers are interested in NCSs, for example [1–6] and references therein. Because of the limitation of the network resource, time-delay caused by data transmission and/or packet dropout will inevitably degrade control performance of the NCSs, or even cause the control systems instable. The technologies about modeling of the NCSs and stabilization analysis are now fairly mature [7-10]. The problem of stabilization about uncertain NCSs with random but bounded delays is discussed in paper [11]. The problem of modeling and stabilization of a wireless NCSs with packet loss and time delay was considered in paper [12]. The tracking controller design problem for discrete-time networked predictive control systems with uncertain Markov delays was proposed in paper [13]. In spite of the significant progress in those studies, the more inspiring and challenging issues of control optimal performance under such network environment remain largely open; for example, optimal
☆ This work was partially supported by the National Natural Science Foundation of China under Grants 61471163, 61472122 and 61370093, the Postdoctoral Science Foundation of China under Grant 2013M540581, the Emphasis Foundation of Department of Education of Hubei province under Grant D20152502, the National Natural Science Foundation of Hubei Province under Grant 2014CFC1105. n Corresponding author. E-mail address: [email protected] (J. Wu).
tracking design and attainable tracking performance of NCSs constrained by the communication links in the feedback loop. Performance limitations of control systems resulting from nonminimum phase zeros and unstable poles of given systems have been known for a long time [14–17]. The tracking performance achievable via feedback was studied in [18] with respect to SISO stable systems. The result was extended to multi-input multioutput (MIMO) unstable systems in [15], and it was found that the minimal tracking error not only dependents on the nonminimum phase zeros location of a given system, but also dependents on the input signal may interact with those zeros, i.e., the angles between the input and zero directions. The performance limitations for linear feedback control systems in the presence of plant uncertainty was investigated in paper [17]. In recent years, the topic of optimal tracking performance has been extended to NCSs [19,20]. This paper [20] studied optimal tracking performance issues for MIMO linear time-invariant systems under networked control with limited bandwidth and additive colored white Gaussian noise channel. The optimal tracking problem for SISO networked systems over a communication channel with packet dropouts was studied in paper [21]. The optimal tracking and regulation performance of discrete-time MIMO linear time-invariant systems was investigated in paper [22]. The best tracking problem for SISO NCSs with network-induced delay and bandwidth constraints was studied in paper [23]. It is noted that those results provide useful guidelines in design of networked control systems including design of communication channels. However, it was showed in
http://dx.doi.org/10.1016/j.isatra.2015.05.012 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Zhan X-S, et al. Optimal performance of networked control systems under the packet dropouts and channel noise. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.012i
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2
[19,21,23] that in order to attain the minimal tracking error, the channel input of NCSs is often required to have an infinite power in the optimal tracking problem. This requirement cannot be met in general in practice. Thus the channel input power of NCSs should be considered in the performance index to address this issue. Meanwhile, the proposed results do not consider the technology of encoding and decoding. The technology of channel encoding and decoding is a very important part of network communication, which has the ability to check for errors or correct for errors. In this paper, we consider the optimal tracking problem in terms of both the tracking error energy and the channel input energy, and meanwhile we consider communication link over the packet dropouts, channel noise and the encoding and decoding of a communication, which are more realistic models of communication link than those in [19,21,23]. In this paper, we study the optimal tracking performance issues pertaining to SISO discrete-time NCSs with considering the packet dropouts, channel noise and the encoding and decoding of communication channel. The objective is to minimize the tracking error between the output and the reference signals of a feedback system under the constraint of channel input power. The optimal tracking performance is attained by stabilizing one or twoparameter compensators. The tracking error is defined in an square error sense. The tracking performance index is given by the weighted sum between the power of the tracking error and the channel input. The obtained results show the optimal tracking performance of networked control systems which determined by plant internal structure and networked parameters, no matter what compensator is adopted, which will provide good guidance for the design of networked control systems. This paper is organized as follows. The problem formulations are introduced in Section 2. The optimal tracking performance by one-parameter and two-parameter compensators with packet dropouts and channel noise under channel power constraint in feedback loop is proposed in Section 3. An example is given to illustrate the theoretical results in Section 4. The conclusions are presented in Section 5.
( dr ¼
0 1
if the system output is not successfully transmitted to the controller if the system output is successfully transmitted to the controller
where the stochastic variable dr A R is a Bernoulli distributed white sequence with Probfdr ¼ 0g ¼ q;
Probfdr ¼ 1g ¼ 1 q;
where q is the packet dropout probability. The signal r,y and u represent, respectively, the reference input, the system output and the system input, whose Z transforms ~ The reference signal r is a random signal, and are r~ ; y~ and u. Efj rðkÞj g ¼ 0; Efj rðkÞj 2 g ¼ σ 21 . Channel noise signal n is a zero mean Gaussian white noise, Efj nðkÞj 2 g ¼ σ 22 . The signals of r, n, dr are uncorrelated with each other in this paper. According to Fig. 1, the signal y represents the channel input. For the given reference signal r, the tracking error of NCSs is ~ According to Fig. 1, we can obtain defined as e~ ¼ r~ y. ~ u~ ¼ KðzÞ r~ dr y~ A 1 ðzÞn ; y~ ¼ Gu; ~ e~ ¼ r~ y~ ¼ r~ Gu:
ð2:1Þ
According to the calculation methods of [22], a straightforward is given as
2. Problem formulation H
For any vector u, we denote its conjugate transpose by u . The expectation operator is defined as Efg. The open unit disc, the closed unit disc, the exterior of the closed unit disc and the unit c circle are denoted by D : ¼ fz : j zj o 1g, D : ¼ fz : j zj r 1g, D : ¼ fz : j zj 4 1g and ∂D : ¼ fz : j zj ¼ 1g, respectively. We define the Hilbert space L2 L2 9 f : f ðzÞ measurable in D; Z π 1 J f ðejθ Þ J 2 dθ o1 : J f J 22 : ¼ 2π π The space L2 is the Hilbert space with inner product Z π H f ðejθ Þgðejθ Þ dθ:
1 f;g : ¼ 2π
In other words, for any f A H2 and g A H2? , we have f ; g ¼ 0. Finally, RH1 denotes the set of all stable, proper, rational transfer function. We consider SISO discrete-time NCSs with packet dropouts, encoding and decoding and channel noise of the communication channel as depicted in Fig. 1, where the problem is to obtain the optimal tracking performance of NCSs. In this setup, G denotes the plant model and K denotes the single-degree-of-freedom compensator, whose Z-transforms are GðzÞ and KðzÞ, respectively. The communication channel is characterized by three parameters: the transfer functions A and A 1 , the channel noise signal n and the parameter dr . The transfer functions A and A 1 are used to model the encoding and decoding, and are assumed to be nonminimum phase. The parameter dr denotes whether or not a packet is dropped
S ejω ¼ ð1−
G ejω K ejω A−1 ejω G ejω K ejω ÞSre þ Sne ; 1 þ ð1−qÞG ejω K ejω 1 þ ð1−qÞG ejω K ejω
Sre is the frequency characteristics from r toe and Sne is the frequency characteristics from n to e. According to [22] and r,n, dr are uncorrelated with each other in this paper, we can obtain
E J e~ J 22 ¼ J 1
GK A 1 GK J 22 σ 21 þ J J 2σ2 : 1 þ ð1 qÞGK 1 þ ð1 qÞGK 2 2
ð2:2Þ
Similarly, we can obtain
E J y~ J 22 ¼ J
GK A 1 GK J 22 σ 21 þ J J 2 σ2 : 1 þð1 qÞGK 1 þ ð1 qÞGK 2 2
ð2:3Þ
π
which further induces the L2 norm J f J 22 : ¼ f ; f . For any f, g A L2 , they are orthogonal if f ; g ¼ 0. It is well known [24] that L2 can be decomposed into two orthogonal subspaces H2 and H2? given n c H2 : ¼ f : f ðzÞ analytic in D ; Z π 1 J f J 22 : ¼ sup J f ðrejθ Þ J 2 dθ o 1 : r 4 1 2π π H2? : ¼ f : f ðzÞ analytic in D; Z π 1 2 J f ðrejθ Þ J dθ o 1 : J f J 22 : ¼ sup r o 1 2π π
Fig. 1. Networked control systems with channel noise and packet dropout.
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3. Optimal tracking performance with channel input power constraint
Thus Jn ¼
Generally speaking, the channel input is required to satisfy the power constraint E J y~ J 2 o Γ , for some predetermined input power level Γ 4 0. A power constraint such as the above inequality may arise either from electronic hardware limitations or regulatory constraints introduced to minimize interference to other communication system users. In this paper, the tracking performance index of the discrete-time networked control systems considering channel input energy constraint is defined as
J : ¼ ð1 εÞE J e~ J 22 þ ε Ef J y~ J 22 g Γ ; ð3:4Þ where 0 r ε o 1, which represented the trade-off between the optimal tracking error and the channel input power. We define optimal tracking performance of discrete-time NCSs with packet dropouts, encoding and decoding and channel noisy of the communication channel under considering channel input power constraint as J n , and the optimal tracking performance J n is attained when the controller is chosen among all possible stabilizing controllers (denoted by K).
Q A RH1
where m
iaj i ¼ 1
Therefore (
! GK A 1 GK J 22 σ 21 þ J J 22 σ 22 K AK 1 þ ð1 qÞGK 1 þ ð1 qÞGK !) 1 GK A GK 2 2 2 2 J σ þJ J σ ð3:5Þ εΓ : þε J 1 þ ð1 qÞGK 2 1 1 þ ð1 qÞGK 2 2 ð1 εÞ J 1
J n ¼ inf
For the rational transfer function ð1 qÞG, Let its coprime factorization be given by ð1 qÞG ¼ NM
1
;
ð3:6Þ
where N; M A RH1 , and satisfy the Bezout identity MX NY ¼ 1;
ð3:7Þ
2
Theorem 1. For given discrete-time NCSs as depicted Fig. 1, if ð1 qÞG can be decomposed as (3.6), the optimal performance under channel input power constraint can be expressed as n X 1 j si j 2 1 j sj j 2 2 J n Zð1 εÞ σ1 s j si 1 i¼1 2 m 1 j pi j 2 1 j pj j 2 Lz 1 ðpj Þ X 1 þ σ 21 b j bi ð1 qÞ2 j ¼ 1 p j pi 1 m 1 j pi j 2 1 j pj j 2 γ j γ H X 1 j þ σ 22 εΓ ; b j bi ð1 qÞ2 j ¼ 1 p j pi 1
bj ¼ ∏
K AK
) pffiffiffiffiffiffiffiffiffiffi 1 εð q þ MX MQNÞ 2 2 N ðY MQ Þ 2 2 J 2 σ 2 εΓ : J pffiffiffi J 2σ1 þ J ð1 qÞA εðY MQNÞ ð1 qÞ 1
It is clear that in order to obtain J n ; Q must be appropriately selected.
n
J ¼ inf J:
( inf
3
pj pi ; 1 pj p i
γ j ¼ Om 1 ðpj ÞLz 1 ðpj Þ:
Proof. In order to calculate the J n , we denote pffiffiffiffiffiffiffiffiffiffi ( 1 1 εð q þ MX MQN Þ 2 2
p ffiffiffi J n1 ¼ inf 2σ1 ; Q A RH1 ð1 qÞ2 εðY MQ ÞN J n2 ¼
inf
Q A RH1
J ð1 qÞ 1 A 1 N ðY MQ Þ J 22 σ 22 :
According to (3.7), we can rewrite J n1 8 pffiffiffiffiffiffiffiffiffiffi 1 εð q þ 1 þ NY MQN Þ > < 1 pffiffiffi 2 2
J n1 ¼ inf 2σ1 : εðYN MQN Þ 2 Q A RH1 > :ð1 qÞ
where X; Y A RH1 . It is well known that every stabilizing compensator K can be characterized by Youla parameterization [25] Y MQ ; Q A RH1 : K: ¼ K:K¼ ð3:8Þ X NQ
Because Lz is the all pass factors, J n1 can be expressed 8 pffiffiffiffiffiffiffiffiffiffi < 1 1 ε ð1 qÞL 1 þN m Y MQN m
z n 2 σ 2 : J 1 ¼ inf pffiffiffi 2 1 Q A RH1 :ð1 qÞ2 εðYNm MQNm Þ
It is also well known that a nonminimum phase transfer function could factorize a minimum phase part and an all pass factor [25]
According to (3.10), we can obtain
N ¼ Lz N m ;
M ¼ Bp M m ;
A ¼ A z Om ;
ð3:9Þ
where Lz ; Bp and Az are the all pass factors, N m ; Om and M m are the minimum phase parts. Lz includes all the zeros of the plant outside c the unit circle si A D ; i ¼ 1; …; n ., Bp includes all the c poles of the plant outside the unit circle pj A D ; i ¼ 1; …; m, and Az includes all zeros of the code outside the unit circle c oi A D ; i ¼ 1; …; no . We consider coprime factorization of Lz ðsÞ; Az and BpðsÞ respectively as z si ; i ¼ 1 1 siz n
Lz ¼ ∏
z pi ; j ¼ 1 1 piz m
Bp ¼ ∏
n0
Az ¼ ∏
z oi
i ¼ 1 1 oiz
:
ð3:10Þ
n
Lz 1 ð1Þ ¼ ∏ s i : i¼1
Because Lz 1 ðzÞ Lz 1 ð1Þ is in H2? , and ð1 qÞLz 1 ð1Þ þ Nm Y MQN m is in H2 , conversely. Hence pffiffiffiffiffiffiffiffiffiffi 1 1 εð1 qÞ Lz 1 ðzÞ Lz 1 ð1Þ 2 2 n J1 ¼ 2 σ1 ð1 qÞ2 0 pffiffiffiffiffiffiffiffiffiffi 1 ε ð1 qÞLz 1 ð1Þ þ N m Y MQN m 2 2 1 σ : inf þ 2 1 2 Q A RH pffiffiffi 1 ð1 qÞ εðYNm MQNm Þ ð3:11Þ n
In order to calculate the J 1 , we denote 3.1. Optimal performance of one-parameter compensator with channel input power constraint According to (3.5)–(3.8), we can get
N ðY MQ Þ 2 2 q 1 1 ð1 εÞ þ MX NQM J 22 σ 21 þ J σ 1 q 1 q 1q ð1 qÞA 2 2
Y MQ N ðY MQ Þ 2 2 þε J N J 22 σ 21 þ J J σ g εΓ : 1q ð1 qÞA 2 2
n
J ¼
inf
Q A RH1
pffiffiffiffiffiffiffiffiffiffi 1 ε ð1 qÞLz 1 ð1Þ þ N m Y MQN m 2 2 σ J 11 ¼ inf 2 1 pffiffiffi ð1 qÞ2 Q A RH1 εðYN m MQN m Þ ! pffiffiffiffiffiffiffiffiffiffi n 1 ε ðq 1Þ ∏ s þ N YB 1 M QN m m i m p 2 2 i¼1 1 σ : inf ¼ 2 1 2 Q A RH pffiffiffi 1 1 ð1 qÞ ε YN m Bp M m QN m n
1
ð3:12Þ
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pffiffiffiffiffiffiffiffiffiffi 1 εð1 qÞ L 1 ðzÞ L 1 ð1Þ 2 2 z z J 12 ¼ 2σ1 ð1 qÞ2 0
2 m 1 j pi j 2 1 j pj j 2 Lz 1 ðpj Þ X þ σ 21 ; ð1 qÞ2 i;j ¼ 1 b j bi p j pi 1
1
n
1
where
Based on a partial fraction procedure, we can obtain YN m Bp 1 ¼
m X 1 p j z Yðpj ÞNm ðpj Þ j¼1
z pj
bj m
þ R1 ;
ð3:13Þ
pj pi : 1 pj p i Because Az and Lz are the all pass factors, we can rewrite J n2 m
bj ¼ ∏
iaj i;j ¼ 1
p p
where R1 A RH1 ; bj ¼ ∏ 1 j p pi . According to (3.7) and Mðpj Þ ¼ 0, j i iaj i ¼ 1 we can obtain 1 Yðpj Þ ¼ N m ðpj ÞLz 1 ðpj Þ:
ð3:14Þ
J n2 ¼
1 1 inf J Om N m YBp 1 N m QM m J 22 σ 22 : ð1 qÞ2 Q A RH1
According to (3.12)–(3.14), we can get 0 1 m Yðp ÞN m ðp Þ n X pffiffiffiffiffiffiffiffiffiffi j j 1 1 1 ε@ðq 1Þ ∏ s i þ Bp ðzÞ Bp ð1Þ þ R2 M m QNm A bj i¼1 j¼1 1 0 1 J 22 σ 21 ; J n11 ¼ inf J m Yðp ÞN m ðp Þ ð1 qÞ2 Q A RH1 pffiffiffi X j j 1 1 @ A ε Bpk ðzÞ Bp ð1Þ þR2 M m QN m bj j¼1
where Bp 1 ðzÞ ¼
1 pjz ; z pj
Bp 1 ð1Þ ¼ p j ;
R2 ¼
m X j¼1
pj
Yðpj ÞN m ðpj Þ þ R1 : bj
It is note that the Bp 1 ðzÞ Bp 1 ð1Þ A H 2? , it follows that 0 pffiffiffiffiffiffiffiffiffiffi 1 n 1 εððq 1Þ ∏ s i þ R2 Þ C B 1 i¼1 B C J n11 ¼ inf J B pffiffiffi C A Q A RH1 ð1 qÞ2 @ εR2
þ
pffiffiffiffiffiffiffiffiffiffi ! 1ε pffiffiffi M m QN m J 22 σ 21
ε
m Yðp ÞN m ðp Þ pffiffiffiffiffiffiffiffiffiffi X j j Bp 1 ðzÞ Bp 1 ð1Þ 1ε bj j¼1
1 J m Yðp ÞN m ðp Þ ð1 qÞ2 pffiffiffi X 1 j j ε Bp ðzÞ Bp 1 ð1Þ bj j¼1
J 22 σ 21
An appropriate Q can be selected such that 0 pffiffiffiffiffiffiffiffiffiffi 1 n pffiffiffiffiffiffiffiffiffiffi ! ε s þ R Þ ð ð q 1 Þ ∏ 1 2 i B C 1ε i¼1 B pffiffiffi C M m QN m ¼ 0: B C pffiffiffi @ εR2 A ε
According to the same method as J n1 , we can obtain m 1 j pi j 2 1 j pj j 2 γ j γ H X 1 j J n2 ¼ σ 22 ; ð1 qÞ2 i;j ¼ 1 b j bi p j pi 1 1 ðpj ÞLz 1 ðpj Þ According to the J n1 and J n2 , we can obtain where γ j ¼ Om n J n X 1 j si j 2 1 j sj j 2 2 J n Z ð1 εÞ σ1 s j si 1 i;j ¼ 1 2 m 1 j pi j 2 1 j pj j 2 Lz 1 ðpj Þ X 1 þ σ 21 ð1 qÞ2 i;j ¼ 1 b j bi p j pi 1 m 1 j pi j 2 1 j pj j 2 γ j γ H X 1 j þ σ 22 εΓ ð1 qÞ2 i;j ¼ 1 b j bi p j pi 1
where m
bj ¼ ∏
iaj i;j ¼ 1
pj pi ; 1 pj p i
γ j ¼ Om 1 ðpj ÞLz 1 ðpj Þ:
The proof is now completed.□ n
Therefore, J 11 is obtained
2 m 1 j pi j 2 1 j pj j 2 Lz 1 ðpj Þ X σ 21 ; J 11 ¼ b j bi ð1 qÞ2 j ¼ 1 p j pi 1 1
n
ð3:15Þ
where m
bj ¼ ∏
iaj i ¼ 1
pj pi : 1 pj p i
It follows by the similar arguments as in [26], that we can obtain J n12 n X 1 j si j 2 1 j sj j 2 2 J n12 ¼ ð1 εÞ σ1 s j si 1 i;j ¼ 1 According to (3.11) and (3.15), J n1 is obtained J n1 ¼ ð1 εÞ
n X 1 j si j 2 1 j sj j 2 2 σ1 s j si 1 i;j ¼ 1
Fig. 2. Networked control systems with two-parameter compensator.
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3.2. Optimal performance of two-parameter compensator with channel input power constraint Consider SISO discrete-time NCSs with packet dropouts, encoding and decoding and channel noise of the communication channel as depicted in Fig. 2, where the problem is to obtain the optimal tracking performance of NCSs by two-parameter or twodegree-of-freedom compensator. In Fig. 2, ½K 1 K 2 denotes the two-degree-of-freedom compensator, whose Z-transforms is ½K 1 ðzÞfK 2 ðzÞ. The other variables are the same as Fig. 1. According to Fig. 2, we can obtain that ~ r þ nÞA 1 K 2 : y~ ¼ Gr~ K 1 þGðyAd According to the Lemma 1 and the calculation methods of [22], a straightforward is given as Sðejω Þ ¼ 1
Gðejω ÞK 1 ðejω Þ A 1 ðejω ÞGðejω ÞK 2 ðejω Þ Sre Sne : 1 ð1 qÞGðejω ÞK 2 ðejω Þ 1 ð1 qÞGðejω ÞK 2 ðejω Þ
Sre is the frequency characteristics from r to e and Sne is the frequency characteristics from n to e. According to [22] and the signals of r,n, dr are uncorrelated with each other in this paper, we can obtain
E J e~ J 22 ¼ J 1
GK 1 A 1 GK 2 J 22 σ 21 þ J J 2σ2: 1 ð1 qÞGK 2 1 ð1 qÞGK 2 2 2
Similarly, we can obtain
E J y~ J 22 ¼ J
GK 1 A 1 GK 2 J 2σ2 þ J J 2σ2 : 1 ð1 qÞGK 2 2 1 1 ð1 qÞGK 2 2 2
We define the optimal tracking performance of discrete-time NCSs by two-parameter compensators with packet dropouts, encoding and decoding and channel noise of the communication channel under considering channel input power constraint J n , and the optimal tracking performance J n is attained when the controller is chosen among all possible stabilizing controllers (denoted by K): J n ¼ inf J: K AK
Therefore (
GK 1 A 1 GK 2 J 22 σ 21 þ J J 2σ2 J ¼ inf ð1 εÞ J 1 K AK 1 ð1 qÞGK 2 1 ð1 qÞGK 2 2 2 !) GK 1 A 1 GK 2 J 22 σ 21 þ J J 22 σ 22 þε J εΓ : 1 ð1 qÞGK 2 1 ð1 qÞGK 2
!
5
Theorem 2. For given discrete-time NCSs by Fig. 2, assume that the plant has many poles of the plant outside the unit circle c pj A D ; i ¼ 1; …; m and zeros of the plant outside the unit circle c si A D ; i ¼ 1; …; n, then the optimal performance by two-parameter compensators under channel input power constraint can be expressed as n X 1 j si j 2 1 j sj j 2 2 J n ¼ ð1 εÞ σ 1 þ εð1 εÞσ 21 λi λHi s j si 1 i;j ¼ 1 m 1 j pi j 2 1 j pj j 2 γ j γ H X 1 j þ σ 22 εΓ ð1 qÞ2 i;j ¼ 1 b j bi p j pi 1
where m
bj ¼ ∏
iaj i;j ¼ 1
pj pi ; 1 pj p i
n
λi ¼ ∏ s i ; i¼1
γ j ¼ Om 1 ðpj ÞLz 1 ðpj Þ:
From the expression in Theorem 2, the optimal tracking performance of discrete-time NCSs consists of two parts, one depends on the nonminimum phase zeros of the given plant, the reference input signal, and the other depends on the unstable poles of the given plant, as well as the packet dropouts probability, encoding and decoding and channel noise of the communication channel. The tracking performance is improved by two-parameter compensator scheme. The results show how the optimal tracking performance is limited by the packet dropouts probability and channel noise of the communication channel. Proof. Because Lz and Az are the all pass factors, it follows that " pffiffiffiffiffiffiffiffiffiffi # 1 εðð1 qÞLz 1 N m Q Þ 2 2 1 n J ¼ inf J pffiffiffi J 2σ1 εN m Q ð1 qÞ2 Q A RH1 þ
1 ð1 qÞ2
inf
D A RH1
1 J Om N m ðY MDÞ J 22 σ 22 εΓ :
In order to obtain J n , we define pffiffiffiffiffiffiffiffiffiffi 1 εðð1 qÞLz 1 N m Q Þ 2 2 1 p ffiffiffi J 2 σ1 ; inf J J n3 ¼ εN m Q ð1 qÞ2 Q A RH1
ð3:17Þ
n
It is well known that all stabilizing two-parameter compensators K can be characterized by the Youla parameterization [25] K : ¼ K : K ¼ ½K 1 K 2 ¼ ðX DN Þ 1
½QY DM; Q A RH1 ; D A RH1 : ð3:16Þ According to (3.6), (3.7) and (3.16), we can rewrite J n
1 N ðY MDÞ 2 2 NQ ‖22 σ 21 þ ‖ ‖2 σ 2 J n ¼ inf ð1 εÞ ‖1 Q ;D A RH1 1q ð1 qÞA
NQ 2 2 NðY MDÞ 2 2 ‖ σ þ‖ ‖ σ εΓ : þε ‖ 1q 2 1 ð1 qÞA 2 2 By simple modification, J n can be expressed as " pffiffiffiffiffiffiffiffiffiffi # 1 εð1 q NQ Þ 2 2 1 p ffiffiffi inf ‖ Jn ¼ ‖2 σ 1 εNQ ð1 qÞ2 Q A RH1 þ
1 ð1 qÞ2
inf
D A RH1
‖A 1 N ðY MDÞ‖22 σ 22 εΓ :
It is clear that in order to obtain J n , Q ; D must be appropriately selected.
J n4 ¼
1 ð1 qÞ2
inf
D A RH1
1 J Om N m ðY MDÞ J 22 σ 22 :
ð3:18Þ
According to (3.17), we can rewrite J n3 as J n3 ¼
pffiffiffiffiffiffiffiffiffiffi 1 ε ð1 qÞ Lz 1 ðzÞ Lz 1 ð1Þ þ Lz 1 ð1Þ qLz 1 ð1Þ N m Q J σ 21 : J ffiffiffi p ð1 qÞ2 Q A RH1 εN m Q 1
inf
Because ð1 qÞ Lz 1 ðzÞ Lz 1 ð1Þ is in H2? , and conversely, other part is in H2 . Hence pffiffiffiffiffiffiffiffiffiffi 1 εð1 qÞ Lz 1 ðzÞ Lz 1 ð1Þ 2 2 1 n J 2 σ1 J3 ¼ J ð1 qÞ2 0 pffiffiffiffiffiffiffiffiffiffi 1 1 ε Lz ð1Þ qLz 1 ð1Þ N m Q 1 J 22 σ 21 : þ inf J pffiffiffi εN m Q ð1 qÞ2 Q A RH1 In order to calculate J n3 , define pffiffiffiffiffiffiffiffiffiffi 1 εð1 qÞ Lz 1 ðzÞ Lz 1 ð1Þ 2 2 1 J n31 ¼ J 2σ1: J ð1 qÞ2 0 pffiffiffiffiffiffiffiffiffiffi 1 1 ε Lz ð1Þ qLz 1 ð1Þ N m Q 1 n J 22 σ 21 : inf J pffiffiffi J 32 ¼ ð1 qÞ2 Q A RH1 εN m Q
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It follows by the same method as J n12 , that n pffiffiffiffiffiffiffiffiffiffi X 1 j si j 2 1 j sj j 2 2 n σ1: J 31 ¼ 1 ε s j si 1 i¼1
Remark 3.1. In Theorem 2, if we do not consider the communication constraints, then Jn ¼
By simple modification, J n32 can be expressed as 0 1 pffiffiffiffiffiffiffiffiffiffi n pffiffiffiffiffiffiffiffiffiffi ! 1ε 1 B 1 ε ∏ s i ðq 1Þ C n p ffiffiffi J 32 ¼ inf J N m Q J 22 σ 21 : þ @ A i ¼ 1 ε ð1 qÞ2 Q A RH1 0 According to [25], we introduce an inner-outer factorization " pffiffiffiffiffiffiffiffiffiffi # 1ε pffiffiffi N m ¼ Δi Δ0 :
ε
To find the optimal Q, introduce 2 3 ΔTi ð zÞ 5: ψ 94 T I Δi ð zÞΔi ð zÞ Then, we have ψ Ti ψ i ¼ I, and it follows that 20
n
J 32 ¼
1
inf ð1 qÞ2 Q A RH1
6B J ψ 4@
3 1 0 pffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffi n 1ε 1 ε ∏ s i ðq 1Þ C B pffiffiffi C 7 AN m Q 5J 22 σ 21 : Aþ@ ε i¼1 0
By a simple calculation, we can get 0 1 pffiffiffiffiffiffiffiffiffiffi n ε s ð q 1 Þ 1 ∏ 1 i C TB inf J Δi @ J n32 ¼ A þ Δ0 Q J 22 σ 21 i¼1 ð1 qÞ2 Q A RH1 0 0 1 pffiffiffiffiffiffiffiffiffiffi n 1 ε ∏ s i ðq 1Þ C 2 2 1 T B þ J I Δi Δi @ A J 2σ1 i¼1 ð1 qÞ2 0
ð3:19Þ
1 pffiffiffiffiffiffiffiffiffiffi n ε s ðq 1Þ 1 ∏ i C TB According to Q A RH1 , and inf Q A RH1 J Δi @ A i¼1 0 þ Δ0 Q J 22 can be made arbitrarily small by properly choosing Q A RH1 . A simple calculation of the second term of (3.19) yields J n32 ¼ εð1 εÞσ 21 λi λi ; H
where λi ¼ ∏ni¼ 1 s i Similarly to J n2 , we can obtain J n4 m 1 j pi j 2 1 j pj j 2 γ j γ H X 1 j n J4 ¼ σ 22 ; ð1 qÞ2 i;j ¼ 1 b j bi p j pi 1 1 ðpj ÞLz 1 ðpj Þ. Therefore, we can write J n where γ j ¼ Om n X 1 j si j 2 1 j sj j 2 2 J n ¼ ð1 εÞ σ 1 þ εð1 εÞσ 21 λi λHi s s 1 j i i;j ¼ 1
m 1 j pi j 2 1 j pj j 2 γ j γ H X j þ σ 22 εΓ ð1 qÞ2 i;j ¼ 1 b j bi p j pi 1 1
where m
iaj i;j ¼ 1
pj pi ; 1 pj p i
n
λi ¼ ∏ s i ; i¼1
j i
Remark 3.1 shows that the optimal tracking performance of NCSs are constrained by the nonminimum phase zeros of a given plant without communication channel. The optimal tracking performance of NCSs are the same to [27] without considering communication constraints. 4. Illustrative example In this section, an example is given to illustrate the effectiveness of the obtained theoretic results. [28] considered a leaderfollower multi-agent system, the position and velocity information of leader are considered as the reference signal, and the goal of design the optimal controller is to get the minimal tracking error between the leader and the follower. But, owing to the structural characteristics of follower and the communication constraint between the leader and the follower, the minimal tracking error becomes zero is impossible. Thus, we focus on the study of the relationship among the tracking performance, structural characteristics of followers and communication parameters (packet dropouts, channel noise and encoding and decoding in this paper). Consider the model of follower with transfer function by GðzÞ ¼
0
bj ¼ ∏
n m 1 j pi j 2 1 j pj j 2 γ j γ H X X 1 j si j 2 1 j sj j 2 2 1 j σ1 þ σ 22 : s j si 1 ð1 qÞ2 i;j ¼ 1 b j bi p p 1 i;j ¼ 1
γ j ¼ Om 1 ðpj ÞLz 1 ðpj Þ:
The proof is now completed.□ In the case that there is no communication channel in feedback path, one has the following remark.
zk ; ðz þ 0:5Þðz 3Þ
where k A ð110Þ. This plant is nonminimum phase. The unstable pole of a given plant is located at p1 ¼ 3. For any k 41, it has nonminimum phase zero of a given plant, which is located at z1 ¼ k. The spectral density of reference models is σ 21 ¼ 0:5. The communication channel parameters is given: the transfer function 3 of encoding is denote A ¼ zzþ0:2 , the spectral density of channel 2 noise is σ 2 ¼ 0:3, the packet dropout probability is q ¼ 0:5. The trade-off is ε ¼ 0:2. The channel input power is Γ ¼ 10. From Theorem 1, the following optimal tracking performance is 2 obtained J Z 0:4ðk 1Þ þ 17:84ð1k3k Þ2 2. 3 The optimal tracking performance about SISO NCSs with the different nonminimum phase zeros and considering or without considering encoding and decoding is shown in Fig. 3. It can be observed from Fig. 3 that the optimal tracking performance approaches infinity when the nonminimum phase(NMP) zero moves closer to the unstable pole. It can also be seen from Fig. 3 that the optimal tracking performance can be improved by considering encoding and decoding of the communication channel in the feedback control system. The encoding, channel noise and packet dropouts appear often in NCSs, which will inevitably degrade control performance of the networked control systems, or even cause the control systems unstable. The optimal tracking problem for NCSs over a communication channel with packet dropouts was studied in [21], which only considered the packet dropouts and used two-parameter compensator scheme. In Theorem 2, we study the optimal tracking performance of NCSs with considering the encoding, channel noise and packet dropouts of a communication channel by applying twoparameter compensator scheme, and consider the channel input energy constraint. From Theorem 2, the following optimal tracking 2 performance is obtained J ¼ 0:4ðk 1Þ þ 1:84ð1k3k Þ2 þ 0:08k 2. 3 The optimal tracking performance about SISO NCSs with the different NMP zeros and considering one or two parameter compensator is shown in Fig. 4. It can be observed from Fig. 4 that the optimal tracking performance approaches infinity when
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7
10000
J* considering encoding and decoding 9000
J* without considering encoding and decoding
Tracking error J
*
8000 7000 6000 5000 4000 3000 2000 1000 0
1
2
3
4
5
6
7
8
9
10
The nonminimum phase zeros location k Fig. 3. Optimal tracking performance of networked control system with one parameter compensator.
7000 *
J Theorem 1 *
J Reference[21]
6000
*
J Theorem 2
* Tracking error J
5000 4000 3000 2000 1000 0
−1000
1
2
3
4
5
6
7
8
9
10
The nonminimum phase zeros location k Fig. 4. Optimal tracking performance of networked control system with two parameter compensator.
7000
J* considering coded−decoded *
J without considering coded−decoded
6000
* Tracking error J
5000
4000
3000
2000
1000
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
packet dropout probability q Fig. 5. Optimal tracking performance of networked control system.
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the NMP zero moves closer to the unstable pole. It can also be seen from Fig. 4 that the optimal tracking performance can be improved by two parameter compensator. The same situation result can be observed from Fig. 4 that the obtained optimal tracking performance is better than [21]. In order to illustrate the packet dropouts probability affecting the optimal tracking performance of NCSs. We assume that the given plant has the NMP zeros at z1 ¼ 2:5. From Theorem 2, the following optimal tracking performance is obtained J n ¼ 0:6 1 2 þ90:04ð1 qÞ . The optimal tracking performance about SISO NCSs with the different the packet dropouts probability is shown in Fig. 5. It can be seen from Fig. 5 that the optimal tracking performance has been degraded because of the packet dropouts probability of the communication channel in the feedback control system. 5. Conclusions In this paper, the problem of optimal tracking performance with packet dropouts and channel noise under the channel input power constraint has been studied. The network constraints under consideration are packet dropouts, channel noise and encoding and decoding. Explicit expressions of the optimal tracking error have been obtained by using one or two parameter compensator with or without communication constraints in the feedback path. It is shown that the optimal tracking performance can be described by a sum of two parts, one dependent on the nonminimum phase zeros of the given plant, as well as the reference input signal, and the other dependent on the nonminimum phase zeros and unstable poles the given plant, as well as the packet dropouts probability, channel noise and the encoding and decoding. The result shows how the packet dropouts probability and channel noise of communication channel may fundamentally constrain a control system's tracking capability. An example has been given to illustrate the obtained results. References [1] Zhang H, Shi Y, Mehr MS, Robust. H1 PID control for multivariable networked control systems with disturbance/noise attenuation. Int J Robust Nonlinear Control 2012;22(2):183–204. [2] Shi Y, Yu B. Output feedback stabilization of networked control systems with random delays modeled by Markov chains. IEEE Trans Autom Control 2009;54 (7):1668–74. [3] Qiu JB, Feng G, Gao HJ. Non-synchronized state estimation of multi-channel networked nonlinear systems with multiple packet dropouts via T–S fuzzy affine dynamic models. IEEE Trans Fuzzy Syst 2011;19(1):75–90. [4] Ding SX, Zhang P, Yin S, Ding EL. An integrated design framework of faulttolerant wireless networked control systems for industrial automatic control applications. IEEE Trans Ind Inf 2013;9(1):462–71.
[5] Qiu JB, Feng G, Gao HJ. Fuzzy-model-based piecewise H1 static output feedback controller design for networked nonlinear systems. IEEE Trans Fuzzy Syst 2010;18(5):919–34. [6] Guan ZH, Yang CX, Huang J. Stabilization of networked control systems with short or long random delays: a new multirate method. Int J Robust Nonlinear Control 2010;20(16):1802–16. [7] Wang ZD, Yang FW, Daniel WC, Robust XHLiu. H1 control for networked systems with random packet losses. IEEE Trans Syst, Man, Cybern, Part B: Cybern 2007;37(4):916–24. [8] Jiang S, Fang HJ. H1 static output feedback control for nonlinear networked control systems with time delays and packet dropouts. ISA Trans 2013;46 (2):233–7. [9] Dong HL, Wang ZD, James L, Gao HJ. Fuzzy-model-based robust fault detection with stochastic mixed time-delays and successive packet dropouts. IEEE Trans Syst, Man, Cybern—Part B 2012;42(2):365–76. [10] Hu SL, Yue D. Event-triggered control design of linear networked systems with quantizations. ISA Trans 2012;51(1):153–62. [11] Yang CX, Guan ZH, Huang J. Stochastic fault tolerant control of networked control systems. J Frankl Inst 2009;346(10):1006–20. [12] Bai JJ, Su HY, Gao JF, Sun T. Modeling and stabilization of a wireless network control system with packet loss and time delay. J Frankl Inst 2012;349 (7):2420–30. [13] Zhang H, Shi Y, Wang JM. Observer-based tracking controller design for networked predictive control systems with uncertain Markov delays. Int J Control 2013;86(10):1824–36. [14] Wang BX, Guan ZH, Yuan FS. Optimal tracking and two-channel disturbance rejection under control energy constraint. Automatica 2011;47(4):733–8. [15] Chen J, Hara S, Chen G. Best tracking and regulation performance under control energy constraint. IEEE Trans Autom Control 2003;48(8):1320–36. [16] Guan ZH, Wang BX, Ding L. Modified tracking performance limitations of unstable linear SIMO feedback control systems. Automatica 2014;50(1):262–7. [17] Goodwin G, Salgado M, Yuz J. Performance limitations for linear feedback systems in the presence of plant uncertainty. IEEE Trans Autom Control 2003;48(8):1312–9. [18] Morari M, Zafiriou E. Robust process control. Englewood Cliffs, NJ: PrenticeHall; 1989. [19] Guan ZH, Zhan XS, Feng G. Optimal tracking performance of mimo discretetime systems with communication constraints. Int J Robust Nonlinear Control 2012;22(13):1429–39. [20] Guan ZH, Chen CY, Feng G. Optimal tracking performance limitation of networked control systems with limited bandwidth and additive colored white gaussian noise. IEEE Trans Circuits Syst I: Regul Pap 2013;60(1):189–98. [21] Zhan XS, Guan ZH, Zhang XH, Yuan FS. Optimal tracking performance and design of networked control systems with packet dropout. J Frankl Inst 2013;350(10):3205–16. [22] Jiang XW, Guan ZH, Yuan FS, Zhang XH. Performance limitations in the tracking and regulation problem for discrete-time systems. ISA Trans 2014;53 (2):251C257. [23] Zhan XS, Guan ZH, Zhang XH, Yuan FS. Best tracking performance of networked control systems based on communication constraints. Asian J Control 2014;16(4):1155–63. [24] Francis BA. A Course in H1 Control theory. In: Lecture notes in control and information sciences. Berlin: Springer; 1987. [25] Zhou KM, Doyle JC.Robust and optimal control. New Jersey: Perntice-Hall, Inc.; 1995. [26] Toker O, Chen J, Qiu L. Tracking performance limitations in LTI multivariable discrete-time systems. IEEE Trans Circuits Syst-I 2002;49(5):657–70. [27] Wang HN, Guan ZH, Ding L. Limitations on minimum tracking energy for SISO plants. In: Proceedings of Chinese control and decision conference; 2009. p. 1432–7. [28] Hu JP, Feng G. Distributed tracking control of leader–follower multi-agent systems under noisy measurement. Automatica 2010;46(8):1382–7.
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Optimal performance of networked control systems under the packet dropouts and channel noise$ Xi-Sheng Zhan b,a, Jie Wu a,n, Tao Jiang b, Xiao-Wei Jiang b a b
Department of Control Science and Engineering, Hubei Normal University, Huangshi, 435002, P.R. China Department of Electronics and Information Engineering, Huazhong University of Science and Technology, Wuhan, 430074, P.R. China
art ic l e i nf o
a b s t r a c t
Article history: Received 3 December 2014 Received in revised form 10 March 2015 Accepted 26 May 2015 This paper was recommended for publication by Prof. Y. Chen
The optimal tracking performance of single-input single-output (SISO) discrete-time networked control systems (NCSs) with the packet dropouts and channel noise is studied in this paper. The communication channel is characterized by three parameters: the packet dropouts, channel noise and the encoding and decoding. The explicit expression of the optimal tracking performance is obtained by using the spectral factorization. It is shown that the optimal tracking performance dependents on the nonminimum phase zeros, unstable poles of the given plant, as well as the packet dropout probability, channel noise and the encoding and decoding. The optimal tracking performance is improved by two-parameter compensator. Finally, a typical example is given to illustrate the theoretical results. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Networked control systems Optimal tracking performance Packet dropout probability Nonminimum phase zeros Encoding and decoding
1. Introduction In recent years, more and more researchers are interested in NCSs, for example [1–6] and references therein. Because of the limitation of the network resource, time-delay caused by data transmission and/or packet dropout will inevitably degrade control performance of the NCSs, or even cause the control systems instable. The technologies about modeling of the NCSs and stabilization analysis are now fairly mature [7-10]. The problem of stabilization about uncertain NCSs with random but bounded delays is discussed in paper [11]. The problem of modeling and stabilization of a wireless NCSs with packet loss and time delay was considered in paper [12]. The tracking controller design problem for discrete-time networked predictive control systems with uncertain Markov delays was proposed in paper [13]. In spite of the significant progress in those studies, the more inspiring and challenging issues of control optimal performance under such network environment remain largely open; for example, optimal
☆ This work was partially supported by the National Natural Science Foundation of China under Grants 61471163, 61472122 and 61370093, the Postdoctoral Science Foundation of China under Grant 2013M540581, the Emphasis Foundation of Department of Education of Hubei province under Grant D20152502, the National Natural Science Foundation of Hubei Province under Grant 2014CFC1105. n Corresponding author. E-mail address: [email protected] (J. Wu).
tracking design and attainable tracking performance of NCSs constrained by the communication links in the feedback loop. Performance limitations of control systems resulting from nonminimum phase zeros and unstable poles of given systems have been known for a long time [14–17]. The tracking performance achievable via feedback was studied in [18] with respect to SISO stable systems. The result was extended to multi-input multioutput (MIMO) unstable systems in [15], and it was found that the minimal tracking error not only dependents on the nonminimum phase zeros location of a given system, but also dependents on the input signal may interact with those zeros, i.e., the angles between the input and zero directions. The performance limitations for linear feedback control systems in the presence of plant uncertainty was investigated in paper [17]. In recent years, the topic of optimal tracking performance has been extended to NCSs [19,20]. This paper [20] studied optimal tracking performance issues for MIMO linear time-invariant systems under networked control with limited bandwidth and additive colored white Gaussian noise channel. The optimal tracking problem for SISO networked systems over a communication channel with packet dropouts was studied in paper [21]. The optimal tracking and regulation performance of discrete-time MIMO linear time-invariant systems was investigated in paper [22]. The best tracking problem for SISO NCSs with network-induced delay and bandwidth constraints was studied in paper [23]. It is noted that those results provide useful guidelines in design of networked control systems including design of communication channels. However, it was showed in
http://dx.doi.org/10.1016/j.isatra.2015.05.012 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
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[19,21,23] that in order to attain the minimal tracking error, the channel input of NCSs is often required to have an infinite power in the optimal tracking problem. This requirement cannot be met in general in practice. Thus the channel input power of NCSs should be considered in the performance index to address this issue. Meanwhile, the proposed results do not consider the technology of encoding and decoding. The technology of channel encoding and decoding is a very important part of network communication, which has the ability to check for errors or correct for errors. In this paper, we consider the optimal tracking problem in terms of both the tracking error energy and the channel input energy, and meanwhile we consider communication link over the packet dropouts, channel noise and the encoding and decoding of a communication, which are more realistic models of communication link than those in [19,21,23]. In this paper, we study the optimal tracking performance issues pertaining to SISO discrete-time NCSs with considering the packet dropouts, channel noise and the encoding and decoding of communication channel. The objective is to minimize the tracking error between the output and the reference signals of a feedback system under the constraint of channel input power. The optimal tracking performance is attained by stabilizing one or twoparameter compensators. The tracking error is defined in an square error sense. The tracking performance index is given by the weighted sum between the power of the tracking error and the channel input. The obtained results show the optimal tracking performance of networked control systems which determined by plant internal structure and networked parameters, no matter what compensator is adopted, which will provide good guidance for the design of networked control systems. This paper is organized as follows. The problem formulations are introduced in Section 2. The optimal tracking performance by one-parameter and two-parameter compensators with packet dropouts and channel noise under channel power constraint in feedback loop is proposed in Section 3. An example is given to illustrate the theoretical results in Section 4. The conclusions are presented in Section 5.
( dr ¼
0 1
if the system output is not successfully transmitted to the controller if the system output is successfully transmitted to the controller
where the stochastic variable dr A R is a Bernoulli distributed white sequence with Probfdr ¼ 0g ¼ q;
Probfdr ¼ 1g ¼ 1 q;
where q is the packet dropout probability. The signal r,y and u represent, respectively, the reference input, the system output and the system input, whose Z transforms ~ The reference signal r is a random signal, and are r~ ; y~ and u. Efj rðkÞj g ¼ 0; Efj rðkÞj 2 g ¼ σ 21 . Channel noise signal n is a zero mean Gaussian white noise, Efj nðkÞj 2 g ¼ σ 22 . The signals of r, n, dr are uncorrelated with each other in this paper. According to Fig. 1, the signal y represents the channel input. For the given reference signal r, the tracking error of NCSs is ~ According to Fig. 1, we can obtain defined as e~ ¼ r~ y. ~ u~ ¼ KðzÞ r~ dr y~ A 1 ðzÞn ; y~ ¼ Gu; ~ e~ ¼ r~ y~ ¼ r~ Gu:
ð2:1Þ
According to the calculation methods of [22], a straightforward is given as
2. Problem formulation H
For any vector u, we denote its conjugate transpose by u . The expectation operator is defined as Efg. The open unit disc, the closed unit disc, the exterior of the closed unit disc and the unit c circle are denoted by D : ¼ fz : j zj o 1g, D : ¼ fz : j zj r 1g, D : ¼ fz : j zj 4 1g and ∂D : ¼ fz : j zj ¼ 1g, respectively. We define the Hilbert space L2 L2 9 f : f ðzÞ measurable in D; Z π 1 J f ðejθ Þ J 2 dθ o1 : J f J 22 : ¼ 2π π The space L2 is the Hilbert space with inner product Z π H f ðejθ Þgðejθ Þ dθ:
1 f;g : ¼ 2π
In other words, for any f A H2 and g A H2? , we have f ; g ¼ 0. Finally, RH1 denotes the set of all stable, proper, rational transfer function. We consider SISO discrete-time NCSs with packet dropouts, encoding and decoding and channel noise of the communication channel as depicted in Fig. 1, where the problem is to obtain the optimal tracking performance of NCSs. In this setup, G denotes the plant model and K denotes the single-degree-of-freedom compensator, whose Z-transforms are GðzÞ and KðzÞ, respectively. The communication channel is characterized by three parameters: the transfer functions A and A 1 , the channel noise signal n and the parameter dr . The transfer functions A and A 1 are used to model the encoding and decoding, and are assumed to be nonminimum phase. The parameter dr denotes whether or not a packet is dropped
S ejω ¼ ð1−
G ejω K ejω A−1 ejω G ejω K ejω ÞSre þ Sne ; 1 þ ð1−qÞG ejω K ejω 1 þ ð1−qÞG ejω K ejω
Sre is the frequency characteristics from r toe and Sne is the frequency characteristics from n to e. According to [22] and r,n, dr are uncorrelated with each other in this paper, we can obtain
E J e~ J 22 ¼ J 1
GK A 1 GK J 22 σ 21 þ J J 2σ2 : 1 þ ð1 qÞGK 1 þ ð1 qÞGK 2 2
ð2:2Þ
Similarly, we can obtain
E J y~ J 22 ¼ J
GK A 1 GK J 22 σ 21 þ J J 2 σ2 : 1 þð1 qÞGK 1 þ ð1 qÞGK 2 2
ð2:3Þ
π
which further induces the L2 norm J f J 22 : ¼ f ; f . For any f, g A L2 , they are orthogonal if f ; g ¼ 0. It is well known [24] that L2 can be decomposed into two orthogonal subspaces H2 and H2? given n c H2 : ¼ f : f ðzÞ analytic in D ; Z π 1 J f J 22 : ¼ sup J f ðrejθ Þ J 2 dθ o 1 : r 4 1 2π π H2? : ¼ f : f ðzÞ analytic in D; Z π 1 2 J f ðrejθ Þ J dθ o 1 : J f J 22 : ¼ sup r o 1 2π π
Fig. 1. Networked control systems with channel noise and packet dropout.
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3. Optimal tracking performance with channel input power constraint
Thus Jn ¼
Generally speaking, the channel input is required to satisfy the power constraint E J y~ J 2 o Γ , for some predetermined input power level Γ 4 0. A power constraint such as the above inequality may arise either from electronic hardware limitations or regulatory constraints introduced to minimize interference to other communication system users. In this paper, the tracking performance index of the discrete-time networked control systems considering channel input energy constraint is defined as
J : ¼ ð1 εÞE J e~ J 22 þ ε Ef J y~ J 22 g Γ ; ð3:4Þ where 0 r ε o 1, which represented the trade-off between the optimal tracking error and the channel input power. We define optimal tracking performance of discrete-time NCSs with packet dropouts, encoding and decoding and channel noisy of the communication channel under considering channel input power constraint as J n , and the optimal tracking performance J n is attained when the controller is chosen among all possible stabilizing controllers (denoted by K).
Q A RH1
where m
iaj i ¼ 1
Therefore (
! GK A 1 GK J 22 σ 21 þ J J 22 σ 22 K AK 1 þ ð1 qÞGK 1 þ ð1 qÞGK !) 1 GK A GK 2 2 2 2 J σ þJ J σ ð3:5Þ εΓ : þε J 1 þ ð1 qÞGK 2 1 1 þ ð1 qÞGK 2 2 ð1 εÞ J 1
J n ¼ inf
For the rational transfer function ð1 qÞG, Let its coprime factorization be given by ð1 qÞG ¼ NM
1
;
ð3:6Þ
where N; M A RH1 , and satisfy the Bezout identity MX NY ¼ 1;
ð3:7Þ
2
Theorem 1. For given discrete-time NCSs as depicted Fig. 1, if ð1 qÞG can be decomposed as (3.6), the optimal performance under channel input power constraint can be expressed as n X 1 j si j 2 1 j sj j 2 2 J n Zð1 εÞ σ1 s j si 1 i¼1 2 m 1 j pi j 2 1 j pj j 2 Lz 1 ðpj Þ X 1 þ σ 21 b j bi ð1 qÞ2 j ¼ 1 p j pi 1 m 1 j pi j 2 1 j pj j 2 γ j γ H X 1 j þ σ 22 εΓ ; b j bi ð1 qÞ2 j ¼ 1 p j pi 1
bj ¼ ∏
K AK
) pffiffiffiffiffiffiffiffiffiffi 1 εð q þ MX MQNÞ 2 2 N ðY MQ Þ 2 2 J 2 σ 2 εΓ : J pffiffiffi J 2σ1 þ J ð1 qÞA εðY MQNÞ ð1 qÞ 1
It is clear that in order to obtain J n ; Q must be appropriately selected.
n
J ¼ inf J:
( inf
3
pj pi ; 1 pj p i
γ j ¼ Om 1 ðpj ÞLz 1 ðpj Þ:
Proof. In order to calculate the J n , we denote pffiffiffiffiffiffiffiffiffiffi ( 1 1 εð q þ MX MQN Þ 2 2
p ffiffiffi J n1 ¼ inf 2σ1 ; Q A RH1 ð1 qÞ2 εðY MQ ÞN J n2 ¼
inf
Q A RH1
J ð1 qÞ 1 A 1 N ðY MQ Þ J 22 σ 22 :
According to (3.7), we can rewrite J n1 8 pffiffiffiffiffiffiffiffiffiffi 1 εð q þ 1 þ NY MQN Þ > < 1 pffiffiffi 2 2
J n1 ¼ inf 2σ1 : εðYN MQN Þ 2 Q A RH1 > :ð1 qÞ
where X; Y A RH1 . It is well known that every stabilizing compensator K can be characterized by Youla parameterization [25] Y MQ ; Q A RH1 : K: ¼ K:K¼ ð3:8Þ X NQ
Because Lz is the all pass factors, J n1 can be expressed 8 pffiffiffiffiffiffiffiffiffiffi < 1 1 ε ð1 qÞL 1 þN m Y MQN m
z n 2 σ 2 : J 1 ¼ inf pffiffiffi 2 1 Q A RH1 :ð1 qÞ2 εðYNm MQNm Þ
It is also well known that a nonminimum phase transfer function could factorize a minimum phase part and an all pass factor [25]
According to (3.10), we can obtain
N ¼ Lz N m ;
M ¼ Bp M m ;
A ¼ A z Om ;
ð3:9Þ
where Lz ; Bp and Az are the all pass factors, N m ; Om and M m are the minimum phase parts. Lz includes all the zeros of the plant outside c the unit circle si A D ; i ¼ 1; …; n ., Bp includes all the c poles of the plant outside the unit circle pj A D ; i ¼ 1; …; m, and Az includes all zeros of the code outside the unit circle c oi A D ; i ¼ 1; …; no . We consider coprime factorization of Lz ðsÞ; Az and BpðsÞ respectively as z si ; i ¼ 1 1 siz n
Lz ¼ ∏
z pi ; j ¼ 1 1 piz m
Bp ¼ ∏
n0
Az ¼ ∏
z oi
i ¼ 1 1 oiz
:
ð3:10Þ
n
Lz 1 ð1Þ ¼ ∏ s i : i¼1
Because Lz 1 ðzÞ Lz 1 ð1Þ is in H2? , and ð1 qÞLz 1 ð1Þ þ Nm Y MQN m is in H2 , conversely. Hence pffiffiffiffiffiffiffiffiffiffi 1 1 εð1 qÞ Lz 1 ðzÞ Lz 1 ð1Þ 2 2 n J1 ¼ 2 σ1 ð1 qÞ2 0 pffiffiffiffiffiffiffiffiffiffi 1 ε ð1 qÞLz 1 ð1Þ þ N m Y MQN m 2 2 1 σ : inf þ 2 1 2 Q A RH pffiffiffi 1 ð1 qÞ εðYNm MQNm Þ ð3:11Þ n
In order to calculate the J 1 , we denote 3.1. Optimal performance of one-parameter compensator with channel input power constraint According to (3.5)–(3.8), we can get
N ðY MQ Þ 2 2 q 1 1 ð1 εÞ þ MX NQM J 22 σ 21 þ J σ 1 q 1 q 1q ð1 qÞA 2 2
Y MQ N ðY MQ Þ 2 2 þε J N J 22 σ 21 þ J J σ g εΓ : 1q ð1 qÞA 2 2
n
J ¼
inf
Q A RH1
pffiffiffiffiffiffiffiffiffiffi 1 ε ð1 qÞLz 1 ð1Þ þ N m Y MQN m 2 2 σ J 11 ¼ inf 2 1 pffiffiffi ð1 qÞ2 Q A RH1 εðYN m MQN m Þ ! pffiffiffiffiffiffiffiffiffiffi n 1 ε ðq 1Þ ∏ s þ N YB 1 M QN m m i m p 2 2 i¼1 1 σ : inf ¼ 2 1 2 Q A RH pffiffiffi 1 1 ð1 qÞ ε YN m Bp M m QN m n
1
ð3:12Þ
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pffiffiffiffiffiffiffiffiffiffi 1 εð1 qÞ L 1 ðzÞ L 1 ð1Þ 2 2 z z J 12 ¼ 2σ1 ð1 qÞ2 0
2 m 1 j pi j 2 1 j pj j 2 Lz 1 ðpj Þ X þ σ 21 ; ð1 qÞ2 i;j ¼ 1 b j bi p j pi 1
1
n
1
where
Based on a partial fraction procedure, we can obtain YN m Bp 1 ¼
m X 1 p j z Yðpj ÞNm ðpj Þ j¼1
z pj
bj m
þ R1 ;
ð3:13Þ
pj pi : 1 pj p i Because Az and Lz are the all pass factors, we can rewrite J n2 m
bj ¼ ∏
iaj i;j ¼ 1
p p
where R1 A RH1 ; bj ¼ ∏ 1 j p pi . According to (3.7) and Mðpj Þ ¼ 0, j i iaj i ¼ 1 we can obtain 1 Yðpj Þ ¼ N m ðpj ÞLz 1 ðpj Þ:
ð3:14Þ
J n2 ¼
1 1 inf J Om N m YBp 1 N m QM m J 22 σ 22 : ð1 qÞ2 Q A RH1
According to (3.12)–(3.14), we can get 0 1 m Yðp ÞN m ðp Þ n X pffiffiffiffiffiffiffiffiffiffi j j 1 1 1 ε@ðq 1Þ ∏ s i þ Bp ðzÞ Bp ð1Þ þ R2 M m QNm A bj i¼1 j¼1 1 0 1 J 22 σ 21 ; J n11 ¼ inf J m Yðp ÞN m ðp Þ ð1 qÞ2 Q A RH1 pffiffiffi X j j 1 1 @ A ε Bpk ðzÞ Bp ð1Þ þR2 M m QN m bj j¼1
where Bp 1 ðzÞ ¼
1 pjz ; z pj
Bp 1 ð1Þ ¼ p j ;
R2 ¼
m X j¼1
pj
Yðpj ÞN m ðpj Þ þ R1 : bj
It is note that the Bp 1 ðzÞ Bp 1 ð1Þ A H 2? , it follows that 0 pffiffiffiffiffiffiffiffiffiffi 1 n 1 εððq 1Þ ∏ s i þ R2 Þ C B 1 i¼1 B C J n11 ¼ inf J B pffiffiffi C A Q A RH1 ð1 qÞ2 @ εR2
þ
pffiffiffiffiffiffiffiffiffiffi ! 1ε pffiffiffi M m QN m J 22 σ 21
ε
m Yðp ÞN m ðp Þ pffiffiffiffiffiffiffiffiffiffi X j j Bp 1 ðzÞ Bp 1 ð1Þ 1ε bj j¼1
1 J m Yðp ÞN m ðp Þ ð1 qÞ2 pffiffiffi X 1 j j ε Bp ðzÞ Bp 1 ð1Þ bj j¼1
J 22 σ 21
An appropriate Q can be selected such that 0 pffiffiffiffiffiffiffiffiffiffi 1 n pffiffiffiffiffiffiffiffiffiffi ! ε s þ R Þ ð ð q 1 Þ ∏ 1 2 i B C 1ε i¼1 B pffiffiffi C M m QN m ¼ 0: B C pffiffiffi @ εR2 A ε
According to the same method as J n1 , we can obtain m 1 j pi j 2 1 j pj j 2 γ j γ H X 1 j J n2 ¼ σ 22 ; ð1 qÞ2 i;j ¼ 1 b j bi p j pi 1 1 ðpj ÞLz 1 ðpj Þ According to the J n1 and J n2 , we can obtain where γ j ¼ Om n J n X 1 j si j 2 1 j sj j 2 2 J n Z ð1 εÞ σ1 s j si 1 i;j ¼ 1 2 m 1 j pi j 2 1 j pj j 2 Lz 1 ðpj Þ X 1 þ σ 21 ð1 qÞ2 i;j ¼ 1 b j bi p j pi 1 m 1 j pi j 2 1 j pj j 2 γ j γ H X 1 j þ σ 22 εΓ ð1 qÞ2 i;j ¼ 1 b j bi p j pi 1
where m
bj ¼ ∏
iaj i;j ¼ 1
pj pi ; 1 pj p i
γ j ¼ Om 1 ðpj ÞLz 1 ðpj Þ:
The proof is now completed.□ n
Therefore, J 11 is obtained
2 m 1 j pi j 2 1 j pj j 2 Lz 1 ðpj Þ X σ 21 ; J 11 ¼ b j bi ð1 qÞ2 j ¼ 1 p j pi 1 1
n
ð3:15Þ
where m
bj ¼ ∏
iaj i ¼ 1
pj pi : 1 pj p i
It follows by the similar arguments as in [26], that we can obtain J n12 n X 1 j si j 2 1 j sj j 2 2 J n12 ¼ ð1 εÞ σ1 s j si 1 i;j ¼ 1 According to (3.11) and (3.15), J n1 is obtained J n1 ¼ ð1 εÞ
n X 1 j si j 2 1 j sj j 2 2 σ1 s j si 1 i;j ¼ 1
Fig. 2. Networked control systems with two-parameter compensator.
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3.2. Optimal performance of two-parameter compensator with channel input power constraint Consider SISO discrete-time NCSs with packet dropouts, encoding and decoding and channel noise of the communication channel as depicted in Fig. 2, where the problem is to obtain the optimal tracking performance of NCSs by two-parameter or twodegree-of-freedom compensator. In Fig. 2, ½K 1 K 2 denotes the two-degree-of-freedom compensator, whose Z-transforms is ½K 1 ðzÞfK 2 ðzÞ. The other variables are the same as Fig. 1. According to Fig. 2, we can obtain that ~ r þ nÞA 1 K 2 : y~ ¼ Gr~ K 1 þGðyAd According to the Lemma 1 and the calculation methods of [22], a straightforward is given as Sðejω Þ ¼ 1
Gðejω ÞK 1 ðejω Þ A 1 ðejω ÞGðejω ÞK 2 ðejω Þ Sre Sne : 1 ð1 qÞGðejω ÞK 2 ðejω Þ 1 ð1 qÞGðejω ÞK 2 ðejω Þ
Sre is the frequency characteristics from r to e and Sne is the frequency characteristics from n to e. According to [22] and the signals of r,n, dr are uncorrelated with each other in this paper, we can obtain
E J e~ J 22 ¼ J 1
GK 1 A 1 GK 2 J 22 σ 21 þ J J 2σ2: 1 ð1 qÞGK 2 1 ð1 qÞGK 2 2 2
Similarly, we can obtain
E J y~ J 22 ¼ J
GK 1 A 1 GK 2 J 2σ2 þ J J 2σ2 : 1 ð1 qÞGK 2 2 1 1 ð1 qÞGK 2 2 2
We define the optimal tracking performance of discrete-time NCSs by two-parameter compensators with packet dropouts, encoding and decoding and channel noise of the communication channel under considering channel input power constraint J n , and the optimal tracking performance J n is attained when the controller is chosen among all possible stabilizing controllers (denoted by K): J n ¼ inf J: K AK
Therefore (
GK 1 A 1 GK 2 J 22 σ 21 þ J J 2σ2 J ¼ inf ð1 εÞ J 1 K AK 1 ð1 qÞGK 2 1 ð1 qÞGK 2 2 2 !) GK 1 A 1 GK 2 J 22 σ 21 þ J J 22 σ 22 þε J εΓ : 1 ð1 qÞGK 2 1 ð1 qÞGK 2
!
5
Theorem 2. For given discrete-time NCSs by Fig. 2, assume that the plant has many poles of the plant outside the unit circle c pj A D ; i ¼ 1; …; m and zeros of the plant outside the unit circle c si A D ; i ¼ 1; …; n, then the optimal performance by two-parameter compensators under channel input power constraint can be expressed as n X 1 j si j 2 1 j sj j 2 2 J n ¼ ð1 εÞ σ 1 þ εð1 εÞσ 21 λi λHi s j si 1 i;j ¼ 1 m 1 j pi j 2 1 j pj j 2 γ j γ H X 1 j þ σ 22 εΓ ð1 qÞ2 i;j ¼ 1 b j bi p j pi 1
where m
bj ¼ ∏
iaj i;j ¼ 1
pj pi ; 1 pj p i
n
λi ¼ ∏ s i ; i¼1
γ j ¼ Om 1 ðpj ÞLz 1 ðpj Þ:
From the expression in Theorem 2, the optimal tracking performance of discrete-time NCSs consists of two parts, one depends on the nonminimum phase zeros of the given plant, the reference input signal, and the other depends on the unstable poles of the given plant, as well as the packet dropouts probability, encoding and decoding and channel noise of the communication channel. The tracking performance is improved by two-parameter compensator scheme. The results show how the optimal tracking performance is limited by the packet dropouts probability and channel noise of the communication channel. Proof. Because Lz and Az are the all pass factors, it follows that " pffiffiffiffiffiffiffiffiffiffi # 1 εðð1 qÞLz 1 N m Q Þ 2 2 1 n J ¼ inf J pffiffiffi J 2σ1 εN m Q ð1 qÞ2 Q A RH1 þ
1 ð1 qÞ2
inf
D A RH1
1 J Om N m ðY MDÞ J 22 σ 22 εΓ :
In order to obtain J n , we define pffiffiffiffiffiffiffiffiffiffi 1 εðð1 qÞLz 1 N m Q Þ 2 2 1 p ffiffiffi J 2 σ1 ; inf J J n3 ¼ εN m Q ð1 qÞ2 Q A RH1
ð3:17Þ
n
It is well known that all stabilizing two-parameter compensators K can be characterized by the Youla parameterization [25] K : ¼ K : K ¼ ½K 1 K 2 ¼ ðX DN Þ 1
½QY DM; Q A RH1 ; D A RH1 : ð3:16Þ According to (3.6), (3.7) and (3.16), we can rewrite J n
1 N ðY MDÞ 2 2 NQ ‖22 σ 21 þ ‖ ‖2 σ 2 J n ¼ inf ð1 εÞ ‖1 Q ;D A RH1 1q ð1 qÞA
NQ 2 2 NðY MDÞ 2 2 ‖ σ þ‖ ‖ σ εΓ : þε ‖ 1q 2 1 ð1 qÞA 2 2 By simple modification, J n can be expressed as " pffiffiffiffiffiffiffiffiffiffi # 1 εð1 q NQ Þ 2 2 1 p ffiffiffi inf ‖ Jn ¼ ‖2 σ 1 εNQ ð1 qÞ2 Q A RH1 þ
1 ð1 qÞ2
inf
D A RH1
‖A 1 N ðY MDÞ‖22 σ 22 εΓ :
It is clear that in order to obtain J n , Q ; D must be appropriately selected.
J n4 ¼
1 ð1 qÞ2
inf
D A RH1
1 J Om N m ðY MDÞ J 22 σ 22 :
ð3:18Þ
According to (3.17), we can rewrite J n3 as J n3 ¼
pffiffiffiffiffiffiffiffiffiffi 1 ε ð1 qÞ Lz 1 ðzÞ Lz 1 ð1Þ þ Lz 1 ð1Þ qLz 1 ð1Þ N m Q J σ 21 : J ffiffiffi p ð1 qÞ2 Q A RH1 εN m Q 1
inf
Because ð1 qÞ Lz 1 ðzÞ Lz 1 ð1Þ is in H2? , and conversely, other part is in H2 . Hence pffiffiffiffiffiffiffiffiffiffi 1 εð1 qÞ Lz 1 ðzÞ Lz 1 ð1Þ 2 2 1 n J 2 σ1 J3 ¼ J ð1 qÞ2 0 pffiffiffiffiffiffiffiffiffiffi 1 1 ε Lz ð1Þ qLz 1 ð1Þ N m Q 1 J 22 σ 21 : þ inf J pffiffiffi εN m Q ð1 qÞ2 Q A RH1 In order to calculate J n3 , define pffiffiffiffiffiffiffiffiffiffi 1 εð1 qÞ Lz 1 ðzÞ Lz 1 ð1Þ 2 2 1 J n31 ¼ J 2σ1: J ð1 qÞ2 0 pffiffiffiffiffiffiffiffiffiffi 1 1 ε Lz ð1Þ qLz 1 ð1Þ N m Q 1 n J 22 σ 21 : inf J pffiffiffi J 32 ¼ ð1 qÞ2 Q A RH1 εN m Q
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It follows by the same method as J n12 , that n pffiffiffiffiffiffiffiffiffiffi X 1 j si j 2 1 j sj j 2 2 n σ1: J 31 ¼ 1 ε s j si 1 i¼1
Remark 3.1. In Theorem 2, if we do not consider the communication constraints, then Jn ¼
By simple modification, J n32 can be expressed as 0 1 pffiffiffiffiffiffiffiffiffiffi n pffiffiffiffiffiffiffiffiffiffi ! 1ε 1 B 1 ε ∏ s i ðq 1Þ C n p ffiffiffi J 32 ¼ inf J N m Q J 22 σ 21 : þ @ A i ¼ 1 ε ð1 qÞ2 Q A RH1 0 According to [25], we introduce an inner-outer factorization " pffiffiffiffiffiffiffiffiffiffi # 1ε pffiffiffi N m ¼ Δi Δ0 :
ε
To find the optimal Q, introduce 2 3 ΔTi ð zÞ 5: ψ 94 T I Δi ð zÞΔi ð zÞ Then, we have ψ Ti ψ i ¼ I, and it follows that 20
n
J 32 ¼
1
inf ð1 qÞ2 Q A RH1
6B J ψ 4@
3 1 0 pffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffi n 1ε 1 ε ∏ s i ðq 1Þ C B pffiffiffi C 7 AN m Q 5J 22 σ 21 : Aþ@ ε i¼1 0
By a simple calculation, we can get 0 1 pffiffiffiffiffiffiffiffiffiffi n ε s ð q 1 Þ 1 ∏ 1 i C TB inf J Δi @ J n32 ¼ A þ Δ0 Q J 22 σ 21 i¼1 ð1 qÞ2 Q A RH1 0 0 1 pffiffiffiffiffiffiffiffiffiffi n 1 ε ∏ s i ðq 1Þ C 2 2 1 T B þ J I Δi Δi @ A J 2σ1 i¼1 ð1 qÞ2 0
ð3:19Þ
1 pffiffiffiffiffiffiffiffiffiffi n ε s ðq 1Þ 1 ∏ i C TB According to Q A RH1 , and inf Q A RH1 J Δi @ A i¼1 0 þ Δ0 Q J 22 can be made arbitrarily small by properly choosing Q A RH1 . A simple calculation of the second term of (3.19) yields J n32 ¼ εð1 εÞσ 21 λi λi ; H
where λi ¼ ∏ni¼ 1 s i Similarly to J n2 , we can obtain J n4 m 1 j pi j 2 1 j pj j 2 γ j γ H X 1 j n J4 ¼ σ 22 ; ð1 qÞ2 i;j ¼ 1 b j bi p j pi 1 1 ðpj ÞLz 1 ðpj Þ. Therefore, we can write J n where γ j ¼ Om n X 1 j si j 2 1 j sj j 2 2 J n ¼ ð1 εÞ σ 1 þ εð1 εÞσ 21 λi λHi s s 1 j i i;j ¼ 1
m 1 j pi j 2 1 j pj j 2 γ j γ H X j þ σ 22 εΓ ð1 qÞ2 i;j ¼ 1 b j bi p j pi 1 1
where m
iaj i;j ¼ 1
pj pi ; 1 pj p i
n
λi ¼ ∏ s i ; i¼1
j i
Remark 3.1 shows that the optimal tracking performance of NCSs are constrained by the nonminimum phase zeros of a given plant without communication channel. The optimal tracking performance of NCSs are the same to [27] without considering communication constraints. 4. Illustrative example In this section, an example is given to illustrate the effectiveness of the obtained theoretic results. [28] considered a leaderfollower multi-agent system, the position and velocity information of leader are considered as the reference signal, and the goal of design the optimal controller is to get the minimal tracking error between the leader and the follower. But, owing to the structural characteristics of follower and the communication constraint between the leader and the follower, the minimal tracking error becomes zero is impossible. Thus, we focus on the study of the relationship among the tracking performance, structural characteristics of followers and communication parameters (packet dropouts, channel noise and encoding and decoding in this paper). Consider the model of follower with transfer function by GðzÞ ¼
0
bj ¼ ∏
n m 1 j pi j 2 1 j pj j 2 γ j γ H X X 1 j si j 2 1 j sj j 2 2 1 j σ1 þ σ 22 : s j si 1 ð1 qÞ2 i;j ¼ 1 b j bi p p 1 i;j ¼ 1
γ j ¼ Om 1 ðpj ÞLz 1 ðpj Þ:
The proof is now completed.□ In the case that there is no communication channel in feedback path, one has the following remark.
zk ; ðz þ 0:5Þðz 3Þ
where k A ð110Þ. This plant is nonminimum phase. The unstable pole of a given plant is located at p1 ¼ 3. For any k 41, it has nonminimum phase zero of a given plant, which is located at z1 ¼ k. The spectral density of reference models is σ 21 ¼ 0:5. The communication channel parameters is given: the transfer function 3 of encoding is denote A ¼ zzþ0:2 , the spectral density of channel 2 noise is σ 2 ¼ 0:3, the packet dropout probability is q ¼ 0:5. The trade-off is ε ¼ 0:2. The channel input power is Γ ¼ 10. From Theorem 1, the following optimal tracking performance is 2 obtained J Z 0:4ðk 1Þ þ 17:84ð1k3k Þ2 2. 3 The optimal tracking performance about SISO NCSs with the different nonminimum phase zeros and considering or without considering encoding and decoding is shown in Fig. 3. It can be observed from Fig. 3 that the optimal tracking performance approaches infinity when the nonminimum phase(NMP) zero moves closer to the unstable pole. It can also be seen from Fig. 3 that the optimal tracking performance can be improved by considering encoding and decoding of the communication channel in the feedback control system. The encoding, channel noise and packet dropouts appear often in NCSs, which will inevitably degrade control performance of the networked control systems, or even cause the control systems unstable. The optimal tracking problem for NCSs over a communication channel with packet dropouts was studied in [21], which only considered the packet dropouts and used two-parameter compensator scheme. In Theorem 2, we study the optimal tracking performance of NCSs with considering the encoding, channel noise and packet dropouts of a communication channel by applying twoparameter compensator scheme, and consider the channel input energy constraint. From Theorem 2, the following optimal tracking 2 performance is obtained J ¼ 0:4ðk 1Þ þ 1:84ð1k3k Þ2 þ 0:08k 2. 3 The optimal tracking performance about SISO NCSs with the different NMP zeros and considering one or two parameter compensator is shown in Fig. 4. It can be observed from Fig. 4 that the optimal tracking performance approaches infinity when
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7
10000
J* considering encoding and decoding 9000
J* without considering encoding and decoding
Tracking error J
*
8000 7000 6000 5000 4000 3000 2000 1000 0
1
2
3
4
5
6
7
8
9
10
The nonminimum phase zeros location k Fig. 3. Optimal tracking performance of networked control system with one parameter compensator.
7000 *
J Theorem 1 *
J Reference[21]
6000
*
J Theorem 2
* Tracking error J
5000 4000 3000 2000 1000 0
−1000
1
2
3
4
5
6
7
8
9
10
The nonminimum phase zeros location k Fig. 4. Optimal tracking performance of networked control system with two parameter compensator.
7000
J* considering coded−decoded *
J without considering coded−decoded
6000
* Tracking error J
5000
4000
3000
2000
1000
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
packet dropout probability q Fig. 5. Optimal tracking performance of networked control system.
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the NMP zero moves closer to the unstable pole. It can also be seen from Fig. 4 that the optimal tracking performance can be improved by two parameter compensator. The same situation result can be observed from Fig. 4 that the obtained optimal tracking performance is better than [21]. In order to illustrate the packet dropouts probability affecting the optimal tracking performance of NCSs. We assume that the given plant has the NMP zeros at z1 ¼ 2:5. From Theorem 2, the following optimal tracking performance is obtained J n ¼ 0:6 1 2 þ90:04ð1 qÞ . The optimal tracking performance about SISO NCSs with the different the packet dropouts probability is shown in Fig. 5. It can be seen from Fig. 5 that the optimal tracking performance has been degraded because of the packet dropouts probability of the communication channel in the feedback control system. 5. Conclusions In this paper, the problem of optimal tracking performance with packet dropouts and channel noise under the channel input power constraint has been studied. The network constraints under consideration are packet dropouts, channel noise and encoding and decoding. Explicit expressions of the optimal tracking error have been obtained by using one or two parameter compensator with or without communication constraints in the feedback path. It is shown that the optimal tracking performance can be described by a sum of two parts, one dependent on the nonminimum phase zeros of the given plant, as well as the reference input signal, and the other dependent on the nonminimum phase zeros and unstable poles the given plant, as well as the packet dropouts probability, channel noise and the encoding and decoding. The result shows how the packet dropouts probability and channel noise of communication channel may fundamentally constrain a control system's tracking capability. An example has been given to illustrate the obtained results. References [1] Zhang H, Shi Y, Mehr MS, Robust. H1 PID control for multivariable networked control systems with disturbance/noise attenuation. Int J Robust Nonlinear Control 2012;22(2):183–204. [2] Shi Y, Yu B. Output feedback stabilization of networked control systems with random delays modeled by Markov chains. IEEE Trans Autom Control 2009;54 (7):1668–74. [3] Qiu JB, Feng G, Gao HJ. Non-synchronized state estimation of multi-channel networked nonlinear systems with multiple packet dropouts via T–S fuzzy affine dynamic models. IEEE Trans Fuzzy Syst 2011;19(1):75–90. [4] Ding SX, Zhang P, Yin S, Ding EL. An integrated design framework of faulttolerant wireless networked control systems for industrial automatic control applications. IEEE Trans Ind Inf 2013;9(1):462–71.
[5] Qiu JB, Feng G, Gao HJ. Fuzzy-model-based piecewise H1 static output feedback controller design for networked nonlinear systems. IEEE Trans Fuzzy Syst 2010;18(5):919–34. [6] Guan ZH, Yang CX, Huang J. Stabilization of networked control systems with short or long random delays: a new multirate method. Int J Robust Nonlinear Control 2010;20(16):1802–16. [7] Wang ZD, Yang FW, Daniel WC, Robust XHLiu. H1 control for networked systems with random packet losses. IEEE Trans Syst, Man, Cybern, Part B: Cybern 2007;37(4):916–24. [8] Jiang S, Fang HJ. H1 static output feedback control for nonlinear networked control systems with time delays and packet dropouts. ISA Trans 2013;46 (2):233–7. [9] Dong HL, Wang ZD, James L, Gao HJ. Fuzzy-model-based robust fault detection with stochastic mixed time-delays and successive packet dropouts. IEEE Trans Syst, Man, Cybern—Part B 2012;42(2):365–76. [10] Hu SL, Yue D. Event-triggered control design of linear networked systems with quantizations. ISA Trans 2012;51(1):153–62. [11] Yang CX, Guan ZH, Huang J. Stochastic fault tolerant control of networked control systems. J Frankl Inst 2009;346(10):1006–20. [12] Bai JJ, Su HY, Gao JF, Sun T. Modeling and stabilization of a wireless network control system with packet loss and time delay. J Frankl Inst 2012;349 (7):2420–30. [13] Zhang H, Shi Y, Wang JM. Observer-based tracking controller design for networked predictive control systems with uncertain Markov delays. Int J Control 2013;86(10):1824–36. [14] Wang BX, Guan ZH, Yuan FS. Optimal tracking and two-channel disturbance rejection under control energy constraint. Automatica 2011;47(4):733–8. [15] Chen J, Hara S, Chen G. Best tracking and regulation performance under control energy constraint. IEEE Trans Autom Control 2003;48(8):1320–36. [16] Guan ZH, Wang BX, Ding L. Modified tracking performance limitations of unstable linear SIMO feedback control systems. Automatica 2014;50(1):262–7. [17] Goodwin G, Salgado M, Yuz J. Performance limitations for linear feedback systems in the presence of plant uncertainty. IEEE Trans Autom Control 2003;48(8):1312–9. [18] Morari M, Zafiriou E. Robust process control. Englewood Cliffs, NJ: PrenticeHall; 1989. [19] Guan ZH, Zhan XS, Feng G. Optimal tracking performance of mimo discretetime systems with communication constraints. Int J Robust Nonlinear Control 2012;22(13):1429–39. [20] Guan ZH, Chen CY, Feng G. Optimal tracking performance limitation of networked control systems with limited bandwidth and additive colored white gaussian noise. IEEE Trans Circuits Syst I: Regul Pap 2013;60(1):189–98. [21] Zhan XS, Guan ZH, Zhang XH, Yuan FS. Optimal tracking performance and design of networked control systems with packet dropout. J Frankl Inst 2013;350(10):3205–16. [22] Jiang XW, Guan ZH, Yuan FS, Zhang XH. Performance limitations in the tracking and regulation problem for discrete-time systems. ISA Trans 2014;53 (2):251C257. [23] Zhan XS, Guan ZH, Zhang XH, Yuan FS. Best tracking performance of networked control systems based on communication constraints. Asian J Control 2014;16(4):1155–63. [24] Francis BA. A Course in H1 Control theory. In: Lecture notes in control and information sciences. Berlin: Springer; 1987. [25] Zhou KM, Doyle JC.Robust and optimal control. New Jersey: Perntice-Hall, Inc.; 1995. [26] Toker O, Chen J, Qiu L. Tracking performance limitations in LTI multivariable discrete-time systems. IEEE Trans Circuits Syst-I 2002;49(5):657–70. [27] Wang HN, Guan ZH, Ding L. Limitations on minimum tracking energy for SISO plants. In: Proceedings of Chinese control and decision conference; 2009. p. 1432–7. [28] Hu JP, Feng G. Distributed tracking control of leader–follower multi-agent systems under noisy measurement. Automatica 2010;46(8):1382–7.
Please cite this article as: Zhan X-S, et al. Optimal performance of networked control systems under the packet dropouts and channel noise. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.012i