Performance limitations of networked control systems with quantization and packet dropouts

Performance limitations of networked control systems with quantization and packet dropouts

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ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Contents lists available at ScienceDirect

ISA Transactions journal homepage: www.elsevier.com/locate/isatrans

Research article

Performance limitations of networked control systems with quantization and packet dropouts$ Xiao-Wei Jiang a,b, Xian-He Zhang a,n, Zhi-Hong Guan b, Li Yu c, Huai-Cheng Yan d a

College of Mechatronics and Control Engineering, Hubei Normal University, Huangshi 435002, PR China College of Automation, Huazhong University of Science and Technology, Wuhan 430074, PR China c School of Electronic Information and Communications, Huazhong University of Science and Technology, Wuhan 430074, PR China d School of Information Science and Engineering, East China University of Science and Technology, Shanghai 200237, PR China b

art ic l e i nf o

a b s t r a c t

Article history: Received 31 May 2016 Received in revised form 26 November 2016 Accepted 19 December 2016

This paper investigates the problem of optimal tracking performance of networked control systems (NCSs) with quantization and packet-dropouts. The system under consideration is linear time-invariant (LTI), multi-input multi-output (MIMO), where an H2 norm of error signal between the reference input and the system output is used as the tracking performance index. The impacts of packet-dropouts in the communication channel and the quantized input and output are studied. The goal is to obtain the minimal error in tracking a random signal, by searching through all possible stabilizing two-parameter controllers. It is shown that, the minimum value of tracking error is closely related to the reference input signal direction, the non-minimum phase zeros and unstable poles of the given plant, including the locations and directions. We also demonstrated the quantization error and the packet-dropouts may degrade the tracking performance. A typical example is given to evaluate the theoretical results. & 2017 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Networked control systems Packet-dropouts Optimal tracking Quantization

1. Introduction In the past decade, the performance limitations of networked control systems (NCSs) has gained considerable attention [1,2], which can help researchers to understand how the performance of the systems may be intrinsically constrained by the characteristics of the plant. For tracking performance of the classical systems, it is generally known that the minimal tracking error depends upon the non-minimum phase zero, the unstable poles and the time delays in the plant [3]. NCSs with distributed sensors, controllers and actuators have been produced as the rapid development of network technology [4–6]. These systems have significant advantages in engineering applications such as networked direct current motor and telemedicine [7,8]. However, the controllers and the plants to be controlled in the NCSs often communicate with each other in a non-ideal manner due to long distance communication channels. Thus many factors such as time-delay [9,10], packet-dropouts [11–13], quantization [14,15] will inevitably bring some adverse effects on the performance of system, and even worse they may ☆ This work was partially supported by the National Natural Science Foundation of China under Grants 61472122, 61603128 and 61602163, and the Postdoctoral Science Foundation of China under Grant 2015T80800. n Corresponding author. E-mail address: [email protected] (X.-H. Zhang).

cause the systems instability. In [16], assuming only one node can access the network and send its information the authors investigate the stability of NCSs with time-varying transmission intervals and time-varying transmission delays. A new linear delayed delta operator switched system model has been proposed in [17] to describe the networked control systems with packetdropouts and network-induced delays, and a verification theorem has been given to show the solvability of the stabilization conditions by solving a class of finite linear matrix inequalities (LMIs). Other related works can also be found in [18–20]. The results obtained in previous works have provided valuable insight into about the relationship between stability, performance and communication constraints. However, it should be noted that signal quantization is an essential part of the communication process, and packet-dropouts are typical features associated with the networked control systems. Thus, it is meaningful and significant to reveal the quantitative relationship between the minimal tracking error and communication constraint. The goal of this work is to adopt the two-parameter controllers to investigate the optimal tracking performance of the networked control systems with quantization and packet-dropouts. The tracking performance can be measured by the energy of the error signal between the output of the plant and the reference signal. Two cases are considered in this paper. In the first case, the system sensor is far away from the plant while the controller is near to the plant. In the second case, the setting is the opposite. The adopted model can be

http://dx.doi.org/10.1016/j.isatra.2016.12.006 0019-0578/& 2017 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Jiang X-W, et al. Performance limitations of networked control systems with quantization and packet dropouts. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2016.12.006i

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2

found in many real systems. For example, in a robot surgery system with remote monitoring, the patient is the plant and the robot is the controller. The remote expert obtains information via the network transmission, and the instruction of the expert is then returned to the robot via the network transmission. Main contributions of this paper can be summarized as follows. Firstly, explicit expressions are given to show the relationship between tracking performance and intrinsic characteristics of the plant. Secondly, by using additive white noise to model the quantization error, and a binary stochastic process to model the packet the quantization and packet dropouts' effects on tracking performance are quantitatively revealed. Thirdly, the results obtained in this paper may give some guidance for the design of optimal controllers. The remainder of this paper is organized as follows. In Section 2, the notations are defined, the Youla parameterization of stabilizing controllers is introduced, and a brief narrative of all-pass factors of the non-minimum phase transfer function matrices is provided. Section 3 provides the formulation and solution of the problem of optimal tracking performance with quantization output and packetdropouts. In Section 4, the optimal tracking performance with quantization input and packet-dropouts is further investigated. An illustrative example is given in Section 5. Finally, the paper is concluded in Section 6.

the communication channel is denoted by α and thus 0 ≤ α < 1. For the rational transfer function matrix (1 − α ) G (z ), the right and the left coprime factorizations are given by

˜ −1N˜ , (1 − α ) G (z ) = NM−1 = M

(2.1)

˜ ∈  ∞ and satisfy the double Bezout identity where N , M , N˜ , M

⎡ X˜ − Y˜ ⎤ ⎡ M Y ⎤ = I, ⎢ ⎥ ˜ ⎦ ⎢⎣ N X ⎥⎦ ⎣ − N˜ M

(2.2)

for some X , Y , X˜ , Y˜ ∈  ∞. Then, all the stabilizing two-parameter compensators K can be characterized by the following set [21]:

{

(

 := K : K = [ K1 K2 ] = X˜ − RN˜

−1 ⎡

)

}

˜ ⎤⎦, Θ, R ∈  ∞ . ⎣ Θ Y˜ − RM

(2.3)

where K1 and K2 are two independent controllers that will be designed. Assume that G is right invertible, which implies that G(z) has a right inverse for some z. For a right-invertible G(z), each of its nonminimum phase zeros is also one for N(z). Denote si ∈ + , i = 1, … , Ns as the non-minimum phase zeros of G(z), where Ns is the number of non-minimum phase zeros, and ηi are the corresponding unitary zero direction vectors. Then it is possible to factorize N(z) as

N (z ) = L (z ) Nm (z ),

(2.4)

where L(z) is an all-pass factor and Nm(z) is the minimum phase part of N(z). A useful all-pass factor is given by

2. Preliminaries

⎤⎡ H ⎤ ⎡ 1 − s¯i z − si 0⎥ ⎢ ηi ⎥ ⎢ ⎡ ⎤ , L (z ) = ∏ L i (z ), L i (z ) = ⎣ ηi Ui ⎦ ⎢ 1 − si 1 − s¯i z ⎥ ⎢ UH⎥ i=1 ⎢⎣ 0 I ⎥⎦ ⎣ i ⎦ Ns

The notations used throughout in this paper are described as follows. z¯ denotes the conjugate of a complex number z. The transpose and conjugate transpose of a vector u and a matrix A are denoted by uT , uH and AT , AH , respectively. The open unit disc is denoted by D :={ z ∈ C: z < 1}, the closed unit disc is denoted by

{

}

D¯ :={ z ∈ C: z ≤ 1}, the unit circle is denoted by ∂D := z ∈ C : z = 1 ,

and the complement of D¯ by D¯ c :={ z ∈ C: z > 1}. Moreover, let · 2 denote the Euclidean vector norm and · F the Frobenius norm, ‖G‖2F :=tr (GH G ). The Hilbert space  2 is defined as ⎧ ⎛ 1  2 :=⎨ G: G (z ) measurable in ∂D , G 2 :=⎜ ⎝ 2π ⎩ ⎪



1 G〉:= 2π

π

π

∫−π

⎞1/2 ‖G (e jθ )‖2F dθ ⎟ ⎠

⎫ < ∞⎬ , ⎭ ⎪

with the inner product 〈F , where F ∫−π and G are the transfer function matrices in the Hilbert space. It is well known that  2 admits an orthogonal decomposition into the subspaces  2 and  ⊥2 , where ⎧ ¯ c , G :=sup ⎛⎜ 1  2 :=⎨ G : G (z ) analytic in D 2 ⎪ r > 1 ⎝ 2π ⎩ ⎪

⎞1/2

π

∫−π ‖G (re jθ )‖2F dθ⎟⎠

where Ui are matrices that together with ηi form a unitary matrix. If the plant G(z) has unstable poles pk ∈ + , k = 1, … , Np , where Np is the number of the unstable poles and Λ is a real diagonal matrix, ˜ (z ) Λ as then it is possible to factorize M

˜ (z ) Λ = M ˜ m (z ) B˜(z ), M

⎫ ⎬, < ∞⎪ ⎭ ⎪

(2.6)

˜ m (z ) is the minimum phase and B˜ (z ) is an all-pass factor. where M Specifically, B˜ (z ) can be constructed as follows:



tr (F H (e jθ ) G (e jθ )) dθ ,

(2.5)

Np

B˜(z ) =

∏ B˜k (z), k=1

⎡ z −pk 0⎤ ⎡ w H ⎤ k ⎥ B˜k (z ) = ⎡⎣ wk Wk ⎤⎦ ⎢ 1 −p¯k z ⎥ ⎢ , H ⎢⎣ 0 I ⎥⎦ ⎢⎣ Wk ⎥⎦

where wi are unitary vectors obtained by factorizing the zeros one at a time, and Wi are the matrices that together with wi form a unitary matrix. The tracking performance index of the system is defined as

{

}

J :=E e (k )T e (k ) ,

(2.7)

and ⎧ ⎛ 1  2⊥ :=⎨ G : G (z ) analytic in D, G 2 :=sup ⎜ ⎪ r < 1 ⎝ 2π ⎩ ⎪

π

∫−π

G (re jθ )

2 F

⎫ ⎞1/2 ⎬. < ∞⎪ dθ ⎟ ⎠ ⎭ ⎪

It follows that for any F ∈  2 and G ∈  ⊥2 , we have F , G = 0. It is worth pointing out that the same notation · 2 will be used to denote these norms, and the meaning of each of these norms will be cleared from the context. Let  ∞ denote the set of all stable, proper, and rational transfer function matrices. The expectation operator is denoted by E { ·}. Finally,

cos ∠(u, v):=

uH v u

v

where e (k ) = y (k ) − r (k ). The minimum tracking error achievable by all possible stabilizing controllers is then determined as

J * := inf J , K1, K2∈ 

where  denotes the set of all stabilizing two-parameter controllers.

3. Tracking performance with quantization output and packet dropouts

.

where ∠(u, v ) is the principal angle between the two subspaces spanned by u and v. In this section, some important factorizations are described that will be frequently used. The packet-dropouts' probability in

In the section, the problem under consideration is depicted in Fig. 1, in which G denotes the plant and ⎡⎣ K1 K2 ⎤⎦ are the twoparameter controllers. Q is used to model the uniform quantizer, which takes uniform quantization interval as shown in Fig. 2. The quantizer is a crucial part in the process of signal transmission. dr

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in Assumption 3.1 is used to model the packet-dropouts in the communication channel. The reference signal under consideration in this paper is a discrete-time step signal [22],

3

yq yq

y

2

y

yq

⎧ r0 v, k ≥ 0 r (k ) = ⎨ k<0 ⎩ 0,

y

yq

2

where the magnitude r0 of the reference signal r(k) is a random variable with zero mean and unit variance and v is a constant unitary vector representing the reference signal input direction. The Z-transform of r(k) is ∞

r^ (z ) =

∑ r0 (k) z−kv. k=0

The uniform quantizer will be adopted as follows. y is a sequence of inputs to the uniform quantizer. The quantized outputs can be seen as an input y suffering from an additive noise n. That is, the quantization output yq can be described as yq = y + n, where n is a random variable over ( − Δ/2, Δ/2], and Δ is the so-called quantization interval. Bennett [23] first showed that the uniform quantization noise can be reasonably modeled as additive white noise under certain conditions. These conditions are (1) the uniform quantizer does not overload, which means that the quantizer magnitude error is guaranteed to be no larger than Δ/2, (2) the quantizer has a large number of levels, (3) Δ between the levels is small, and (4) the input probability density function is smooth. These are the preconditions of the discussion in this paper. The quantization noise is given by

n (k ) = (n1, n2 , …, nm

)T .

The noise signals in different channels are statistically independent with identical uniform distribution over [ − Δi /2, Δi /2], where Δi is the quantization interval in channel i. We define 3

Λ = 6 diag (Δ1 , Δ2 , … , Δm ). In order to facilitate the analysis, the following assumptions are made. Assumption 3.1. (Description of dr): The signal dr is a binary stochastic process that models the packet-dropouts in the communication channel

⎧ 0 if packet drop occurs at time k, dr (k ) = ⎨ ⎩ 1 if no packet drop occurs at time k, P { dr (k ) = 0} = α . Assumption 3.2. The reference signal r is uncorrelated with quantization noise n and packet dropouts dr in the communication channel. For the two provided vector zero mean wide sense stationary (w.s.s.) processes, x and y, their covariance and the associated cross-spectrum are defined as Rxy (τ ) = E x (k + τ ) yT (k ) , and

{

K1 K 2

}

y

u

Fig. 2. The uniform quantization. ∞

Sxy (e jω ) = ∑k =−∞ Rxy (k ) e−jωk , respectively, provided the sum converges. Given x as above, its variance is defined as σx2 = tr Rxx (0), and its power spectral density (PSD) is defined as Sx (e jω ) = Sxx (e jω ). We can observe that if x always has a positive rational spectrum, then it is possible to find a stable, minimum phase and biproper transfer function matrix Ωx (z ) such that σx2 =

1 2π

π

2

∫−π Ωx (e jω) dω=

‖Ωx (z )‖22 . Consequently, the following lemma is given before main results presenting.

^ are the zero mean w.s.s. Lemma 3.1. [24] Assume that y^ , x^ , z^ and w stochastic processes that dr is a sequence of independent and identically distributed (i.i.d.) Bernoulli random variables with parameter 1 − α , and that G(z) is the transfer function matrix of a proper LTI system. Then jω jω jω (1) If y^ = x^ + z^ , then S yw ^ ^ (e ) = S xw ^ ^ (e ) + S zw ^ ^ (e ) ; (2) If y^ = G (z ) x^ , then S ^ ^ (e jω ) = G (e jω ) S ^ ^ (e jω ); yw

xw

jω jω jω (3) If y^ = G (z ) dr x^ , then S yw ^ ^ (e ) = (1 − α ) G (e ) S xw ^ ^ (e ) .

To obtain the minimum tracking error variance, an expression for the tracking error PSD, Se (e jω ) will be derived first. From Fig. 1, it follows that

e=r−y

with a probability distribution given by P { dr (k ) = 1} = 1 − α and

r

y

G

= r − G (z ) u = r − G (z )(K1 (z ) r − K2 (z ) dr yq ) = r − G (z )(K1 (z ) r − K2 (z ) dr (y + n)) = r − G (z ) K1 (z ) r + G (z ) K2 (z ) dr n + G (z ) K2 (z ) dr (r − e). Therefore Lemma 3.1 and straightforward algebra gives

{

Se (e jω ) = Sα (e jω ) [I − G (e jω ) K1 (e jω ) + (1 − α ) G (e jω ) K2 (e jω )] Sre (e jω )

}

+ (1 − α ) G (e jω ) K2 (e jω ) Sne (e jω ) ,

(3.1)

(z ))−1.

where Sα (z ) = (I + (1 − α ) G (z ) K2 In order to evaluate Sre (e jω ), the authors proceed as follows: from 3.1 and Lemma 3.1 it is clear that

{

Ser (e jω ) = Sα (e jω ) [I − G (e jω ) K1 (e jω ) + (1 − α ) G (e jω ) K2 (e jω )] Sr (e jω )

communication link

yq

Q quantizer

dr Fig. 1. Tracking with quantization output and packet dropouts.

}

+ (1 − α ) G (e jω ) K2 (e jω ) Snr (e jω ) .

(3.2)

According to Assumption 3.2 and 3.2, we have S re (e jω) = SαH (e jω)[I − G (e jω) K1 (e jω) + (1 − α ) G (e jω) K2 (e jω)]H S rH (e jω). (3.3)

Expression for Sne (e jω ) can be derived using a similar reasoning, yielding

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4 H Sne (e jω ) = (1 − α ) ⎡⎣ Sα (e jω ) G (e jω ) K2 (e jω ) ⎤⎦ SnH (e jω ).

(3.4)

J1* =

Substituting (3.4) and (3.3) into (3.1), we obtain [18]

σe2

2 = ‖Sα (z ) ⎡⎣ I − G (z ) K1 (z ) + (1 − α ) G (z ) K2 (z ) ⎤⎦ r^ (z )‖22 + ( 1 − α )

‖Sα (z ) G (z ) K2 (z ) Λ‖22 ,

(3.5)

where V = diag (γ1, γ2, … , γn ). Define J :=σe2, then the tracking performance is equivalent to minimize

⎧ v J * = inf ⎨ Sα (z ) ⎡⎣ I − G (z ) K1 (z ) + (1 − α ) G (z ) K2 (z ) ⎤⎦ K1, K2∈  ⎩ z−1 ⎫ + (1 − α )2 Sα (z ) G (z ) K2 (z ) Λ 22 ⎬ , ⎭

i=1

J1* =

=

inf

Θ ∈  ∞

⎛ ⎞ ⎛ Ns ⎞ ⎜ L −1 (z ) − 1 ⎜ ∏ L i (z ) ⎟ Nm (z ) Θ (z ) ⎟ v ⎜ ⎟ ⎜ 1 ⎟z−1 1 − α ⎝ i=1 ⎠ ⎝ ⎠

inf

Θ ∈  ∞

(L1−1 (z ) − I )



v

[ I − Sα (z ) G (z ) K1 (z ) ] z − 1

2 2

⎫ 2 + ( 1 − α ) ‖Sα (z ) G (z ) K2 (z ) Λ‖22 ⎬ . ⎭

(L1−1 (1) − I )



⎛ ⎞ ⎛ Ns ⎞ ⎜ I − 1 ⎜ ∏ L i (z ) ⎟ Nm (z ) Θ (z ) ⎟ v ∈  2. ⎜ ⎟ ⎜ ⎟z−1 1 − α ⎝ i=1 ⎠ ⎝ ⎠

2

2

+ (3.7)



(L i−1 (z ) − I )

i =1

Θ∈  ∞

Ns

J1* =



˜ (z )) Λ‖2 . inf ‖N (z )(Y˜ (z ) − R (z ) M 2

R ∈  ∞

(L i−1 (z ) − I )

i=1 Ns

=

2



(I − L i (Z ))

i=1 2

+ 2

inf

Θ ∈  ∞

⎛ ⎞ v 1 Nm (z ) Θ (z ) ⎟ ⎜I − ⎝ ⎠z−1 1−α

2

. 2

(3.9)

2

= 0. 2

2

v z−1

v z−1

2

2

. 2

Using the expression in (2.5), J1* can be explicitly evaluated as follows. First, note that

(I − L i (z )) inf

, 2

Consequently

R ∈  ∞

Θ∈  ∞

v z−1

⎛ ⎞ v 1 ⎜I − Nm (z ) Θ (z ) ⎟ ⎝ ⎠z−1 1−α

inf

˜ (z )) Λ‖2 . inf ‖N (z )(Y˜ (z ) − R (z ) M 2

⎡ ⎤ v 1 N (z ) Θ (z ) ⎥ ⎢⎣ I − ⎦z−1 1−α

2

Note that Nm(z) is right invertible and a proper Θ (z ) can be chosen such that

Define

J2* =

2

Ns

Thus the problem under consideration reduces to

J1* =

inf

Θ ∈  ∞

⎛ ⎞ ⎛ Ns ⎞ ⎜ I − 1 ⎜ ∏ L i (z ) ⎟ Nm (z ) Θ (z ) ⎟ v 1 ⎟ ⎜ ⎟ z−1 1 − α ⎜⎝ i = 1 ⎠ ⎝ ⎠

and accordingly J1* =

Sα (z ) G (z ) = G (z )(I − K2 (z )(1 − α ) G (z ))−1 1 ˜ )M ˜ −N˜ ⎤⎦−1 = NM−1 ⎡⎣ I − (X˜ − RN˜ )−1(Y˜ − RM 1−α 1 ˜ ˜ −N˜ − RN˜ ) ⎤⎦−1 = NM−1 ⎡⎣ I − (X˜ − RN˜ )−1(YM 1−α 1 ˜ ˜ −N ⎤⎦−1 = NM−1 ⎡⎣ (X˜ − RN˜ )−1(X˜ − YM 1−α 1 −1 = NM−1 ⎡⎣ (X˜ − RN˜ )−1M−1⎤⎦ 1−α 1 = N (X˜ − RN˜ ). 1−α

+

1 z−1

J1* = (L1−1 (1) − I ) v

It then follows from (2.1) and (2.3) that

Θ∈  ∞

. 2

v ∈  2⊥. z−1



= G (z )(I − K2 (z )(1 − α ) G (z ))−1.

inf

2

Furthermore, Θ (z ) can be selected such that

Sα (z ) G (z ) = (I − (1 − α ) G (z ) K2 (z ))−1G (z )

J* =

2

Since L i (1) = I , it follows that

From push-through rule of transfer function

⎡ ⎤ v 1 N (z ) Θ (z ) ⎥ ⎢⎣ I − ⎦z−1 1−α

2

v z−1

Proof. From (3.6) and the definition of Sα (z ), Jn can be rewritten as ⎪

. 2

By the fact that  2 and  ⊥2 are orthogonal complements in  2, it follows that

* are given in the proof. where J21 and J22

⎧ J * = inf ⎨ K1, K2∈  ⎩

2

⎛ ⎞ ⎛ Ns ⎞ 1 ⎜ ∏ Li (z) ⎟⎟ Nm (z ) Θ (z ) ⎟⎟ v + ⎜⎜ I − ⎜ 1 − α ⎝ i=1 ⎠ ⎝ ⎠z−1

(3.6)

si 2 − 1 * cos2 ∠(ηi , v) + J21 + J22 si − 1 2



Θ∈  ∞

⎛ ⎞ v 1 ⎜I − L (z ) Nm (z ) Θ (z ) ⎟ ⎝ ⎠z−1 1−α

Noting that Li(z) is all-pass, J1* can be further written as

2

Theorem 3.1. For a MIMO system, assume that the plant has nonminimum phase zeros si ∈ + , i = 1, … , Ns and unstable poles pi ∈ + , i = 1, … , Np , then the best achievable tracking performance for the system depicted in Fig. 1 is given by

J* =

inf

2

where  denotes the set of all stabilizing two-parameter controllers.

Ns

the two minimization problems in (3.8). Then from (2.4) we have

2

,

v s 2−1 1 . = ηi H v i η z−1 1 − si i 1 − s¯i z

(

)

Since 1/1 − s¯i z ∈  2 and based on the partial fraction procedure, we have

2

(3.8)

Thus J * = J1* + J2* and in what follows, the authors explicitly solve

1 1 = − 1 − s¯i z s¯i z



∑ n= 0

⎛ 1 ⎞n ⎟ . ⎜ ⎝ s¯i z ⎠

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Therefore

J21 = 2

Ns



(I − L i (z ))

i=1

v z−1

Ns

=

∑ |ηiH v|22 i=1

2

Ns

=

∑ i=1 Ns

=

∑ i=1

∞ ⎛ 1 ⎞−2n ( |si |2 − 1)2 ∑ ⎟ ⎜ 2 2 ⎝ s¯ z ⎠

1 − si si

n= 0

2 ⎛ ⎞ ⎜ p − 1⎟ ‖Λω Np ‖22 ⎝ Np ⎠ 2 ⎛ ⎞ ⎜ p − 1⎟ Λ I − (1 + p¯ Np ) ω Np ω NHp ω Np− 1 ⎝ Np − 1 ⎠

(

+

i

1

2

−1

Np

(3.10)



=

k=1

2

2

(3.12)

* = J22

˜ m (z ) L−1 (z ) ( ΛΦ − R1 (z ) ) + Nm (z ) R (z ) M

inf

R ∈  ∞

2

Ns

=

∑ ⎡⎣ Hi Li−1(z) Gi ( ΛΦ − R1(si ) ) ⎤⎦ + R2 (z)

inf

R ∈  ∞

i=1 2

R ∈  ∞

˜ m (z ) + Nm (z ) R (z ) M

˜ −1

˜ m (z )∥2 . ∥ N (z ) Y˜ (z ) ΛB (z ) − N (z ) R (z ) M 2

2 Ns

=

From double Bezout identity, the following is obtained: −1

)

⎞ ⎟ ω1 ⎟ ⎠

2 ⎛ ⎞ ⎜ p − 1⎟ ‖Λξ Np+ 1 − k ‖22 . ⎝ Np + 1 − k ⎠

˜ m (z ) B˜(z )‖2 inf ‖N (z ) Y˜ (z ) Λ − N (z ) R (z ) M 2

inf

2

* . By selecting R(z) properly, Next, we will focus on the calculation of J22 the authors obtain

R ∈  ∞

R ∈  ∞

2

)

(

si 2 − 1 cos2 ∠(ηi , v). si − 1 2

˜ (z ) Λ‖2 . inf ‖N (z ) Y˜ (z ) Λ − N (z ) R (z ) M 2

J2* =

)

⎛ Np − 1 Λ ⎜⎜ ∏ I − (1 + p¯ Np+ 1 − l ) ω Np+ 1 − l ω NHp+ 1 − l ⎝ l= 1

˜ Λ admits an all-pass It can be easily known from (2.6) that M factorization, thus we have

=

(p

+⋯+

si 2 − 1 H 2 η v 1 − si 2 i 2

Next, the calculation of J2* is presented. Firstly, J2* can be rewritten as

J2* =

5



inf

R ∈  ∞

i=1

⎡ H L −1 (z ) − L −1 (∞) G ΛΦ − R (s ) ⎤ + R (z ) i( 1 i )⎦ 3 i ⎣ i i

(

)

2

−1

N (z ) Y˜ (z ) ΛB˜ (z ) = − ΛB˜ (z ) + R1 (z ),

˜ m (z ) + Nm (z ) R (z ) M

˜ m (z ) ∈  ∞. Note that B˜ −1 (z ) − B˜ −1 (∞) ∈  ⊥ where R1 (z ) = X (z ) M i i 2 ˜ m (z ) ∈  2 , and R(z) can be selected such that ΛΦ − R1 (z ) + N (z ) R (z ) M the authors further obtain

2 Ns

2

i=1

2

∑ ⎡⎣ Hi ( Li−1(z) − Li−1(∞) ) Gi ( ΛΦ − R1(si ) ) ⎤⎦

=

, −1

J2* = =

where R2 (z ) ∈  ∞, R3 (z ) ∈  ∞, and R1 (si ) = ΛB˜ (si ). Thus

−1

inf ‖ − ΛB˜ (z ) + R1 (z ) − N (z ) R (z ) Mm (z )‖22

Ns

R ∈  ∞



* = J22

inf ‖Λ − ( R1 (z ) − N (z ) R (z ) Mm (z ) ) B˜(z )‖22

R ∈  ∞

⎛ −1 ⎞ −1 = Λ ⎜ B˜ Np (z ) − B˜ Np (z ) ⎟ ⎝ ⎠

2 2

i, j = 1, i ≠ j

⎛ −1 ⎞ −1 −1 + ΛB˜ Np (∞) ⎜ B˜ Np− 1 (z ) − B˜ Np− 1 (∞) ⎟ ⎝ ⎠

⎛ −1 ⎞ −1 −1 −1 −1 + ⋯ + ΛB˜ Np (∞) B˜ Np− 1 (∞)⋯B˜2 (∞) ⎜ B˜1 (z ) − B˜1 (∞) ⎟ ⎝ ⎠ +

2

−1 −1 × ηj ηjH Gj Λ (Φ − B˜ (sj ))(Φ − B˜ (si )) H ΛGiH ηi ,

2

(3.13)

and J2* can be obtained from (3.11), (3.12) and (3.13). Hence the proof is completed. □

2 2

Next, in order to gain further conceptual insights into this result, a number of special cases of interest are discussed.

inf ‖ΛΦ − R1 (z ) + N (z ) R (z ) Mm (z )‖22 .

R ∈  ∞

Corollary 3.1. For a SISO system, the result of Theorem 3.1 reduces to

Define

Ns

*, J2* = J21 + J22

(3.11)

J* =

∑ i=1

where

⎛ Np ⎞ 2 ⎛ ⎞ si 2 − 1 2 ∠(η , v) + β 2 ⎜ ⎜ p ⎟ + J ⎟, cos 1 − ∑ + − i N 1 k 0 p ⎜ ⎟ ⎝ ⎠ si − 1 2 ⎝ k=1 ⎠

where

(

−1 −1 J21 = Λ B˜ Np (s ) − B˜ Np (∞)

+⋯+

)

2 2

( (∞) ( B˜

−1 −1 −1 + ΛB˜ Np (∞) B˜ Np− 1 (s ) − B˜ Np− 1 (∞)

−1 −1 −1 ΛB˜ Np (∞) B˜ Np− 1 (∞)⋯B˜2

and

* = J22

( si 2 − 1)(|sj |2 − 1)(si − 1)(sj − 1) H H ηi Hi Hj (s¯i − 1)(s¯j − 1)(s¯i sj − 1)

−1 1 (s )



−1 B˜1 (∞)

)

)

2

, 2

Ns

2 2

J0 =

∑ i, j = 1, i ≠ j

( si 2 − 1)(|sj |2 − 1)(si − 1)(sj − 1) ¯ Hi Hj (s¯i − 1)(s¯j − 1)(s¯i sj − 1)

⎛ × Gj ⎜⎜ ⎝

Np

⎞⎛

Np



k=1

⎠⎝

k=1



∏ ( − p¯k ) − B˜−1(sj ) ⎟⎟⎜⎜ ∏ ( − pk ) − B˜−1(si )⎟⎟HGiH,

and the remaining notations have been defined above.

˜ m (z )‖2 . inf ‖ΛΦ − R1 (z ) + N (z ) R (z ) M 2

R ∈  ∞

The calculation of J21 yields

Corollary 3.2. If G(z) in Fig. 1 is a MIMO plant with a single nonminimum phase zero s with direction η and a single unstable pole p with direction ω, then the result of Theorem 3.1 degenerates to

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6

J* =

s2 − 1 (sp − 1) 2 (p2 − 1) H 2 cos2 ∠(η, v) + ‖ω Λ‖2 . 2 s−1 (s − p)2

dr quantizer

r

This expression shows that the tracking performance is very sensitive to quantization if the nonminimum phase zero is close to the unstable pole. Remark 3.1. The result of Theorem 3.1 that is the optimal value of the tracking performance mainly results from the optimal design of the two-parameter controllers. The ‘optimal’ here means the controller corresponding to the performance limitation values, which are firstly proposed and investigated in [30]. Actually, the key procedure is to compute the control parameter Θ (z ) and R(z). It should be noted that Nm(z) is right invertible and Θ (z ) can be obtained from the term

inf

Θ∈  ∞

⎛ ⎞ v 1 ⎜I − Nm (z ) Θ (z ) ⎟ ⎝ ⎠z−1 1−α

K1 K 2

y

uq

G communication link

Fig. 3. Tracking with quantization input and packet dropouts.

Ns

si 2 − 1 cos2 ∠(ηi , v) si − 1 2



J* =

i=1



+

i, j ∈ I

2

= 0,

u

Q

(1 − si 2 )(1 − |sj |2 ) tr (ΛGmH (si ) G m (si ) Λ) s¯i + sj

where I is an index set defined by I :={i: N˜ (si ) Λ = 0}.

2

* , that is and R(z) can be selected from the calculation process of J22 a proper R(z) can be chosen such that

Proof. According to (4.1), Jn can be rewritten as J* =

˜ m (s )‖2 = 0, inf ‖R3 (z ) + Nm (z ) R (z ) M 2

R ∈  ∞

where R3 (z ) have been defined above. Then, the two-parameter controllers K1 (z ) and K2 (z ) can be designed based on the Youla parameterization (2.3). It can also be known that the packet dropouts' probability α is considered in the design of the optimal controllers in order to guarantee the optimal tracking performance.

⎧ ⎨ ⎪ K1, K2∈  ⎩ inf



v

[ I − (1 − α) Sα (z ) G (z ) K1 (z ) ] z − 1

2 2

⎫ + (1 − α )2‖Sα (z ) G (z ) Λ‖22 ⎬ . ⎪ ⎭ ⎪

It then follows from (2.1), (2.3) and (3.7) that J* =

2

v

[ I − N (z ) Θ (z ) ] z − 1

inf

Θ∈  ∞

+ 2

inf ‖N (z )(X˜ (z ) − R (z ) N˜ (z )) Λ‖22 .

R ∈  ∞

Define

(

)

2

J3 = N (z ) X˜ (z ) − R (z ) N˜ (z ) Λ . 2

n

4. Tracking performance with quantization input and packet dropouts In this section, the control structure depicted in Fig. 3 is discussed, in which is quantizer located in the feed-forward channel. By using the similar analysis in Section 3, it follows from Fig. 3 that

e = r − G (z ) dr K1 (z ) r − G (z ) dr n + G (z ) dr K2 (z )(r − e) Thus we have

⎧ v J * = inf ⎨ Sα (z ) ⎡⎣ I − (1 − α ) G (z )(K1 (z ) + K2 (z )) ⎤⎦ K1, K2∈  ⎩ z−1 ⎫ + (1 − α )2‖Sα (z ) G (z ) Λ‖22 ⎬ . ⎭



inf J3 ,

R ∈  ∞

If define f (z )≔N˜ (z ) Λ, then we have f (si ) = 0 for any i ∈ I. Because f (z) is right invertible, it admits a factorization f (z ) = g (z ) b (z ), where g(z) is minimum phase and b(z) is a scalar all-pass factor that can be expressed as

∏ i∈I

z − si . 1 − s¯i z

2

(4.1)

Ns

=T+

J3* =

b (z ) =

2

Lemma 4.1. [25] Let L be defined by (2.5). Then, for any X ∈  ∞, the equality

L−1X

Note that the first term of J have been obtained in the previous section, thus the authors only need to consider the following minimization problem:

L N−z1 (si )⋯L i−+11 (si ) L i−1L i−−11 (si )⋯L1−1 (zi ) Z (si )

i=1

holds for some T ∈  ∞. The lemma can easily be proved by using mathematics inductive method. Then we are ready to present the main results in this section.

On the other hand, it should be noted from double Bezout identity ˜ (z ) − N (z ) Y˜ (z ) = I . Then in that N (z ) X˜ (z ) = X (z ) N˜ (z ), and X (z ) M light of (2.4), one has

J3* = =

inf ‖ ( X (z ) − N (z ) R (z ) ) N˜ (z ) Λ‖22

R ∈  ∞

inf ‖L−1 (z ) X (z ) g (z ) − Nm (z ) R (z ) g (z )‖22 .

R ∈  ∞

(4.2)

It then follows that

L−1 (z ) X (z ) N˜ (z ) Λ − Nm (z ) Y˜ (z ) P (z ) Λ = G m (z ) Λ, where Gm (z ) = L−1 (z ) G (z ) = Nm (z ) M (z ), and thus

L−1 (z ) X (z ) g (z ) b (z ) z = si = G m (si ) Λ.

(4.3)

Based on Lemma 4.1, the following can be obtained: Theorem 4.1. For a MIMO system, assume that the plant has unstable poles pi ∈ + , i = 1, … , Np , and nonminimum phase zeros zi ∈ + , i = 1, … , Nz , then the best achievable tracking performance for the system depicted in Fig. 3 is given by

L−1 (z ) X (z ) g (z ) = T (z ) +

∑ L N−s1 (si )⋯L i−+11 (z i ) L i−1L i−−11 (si )⋯L1−1 (si ) X (si ) g (si ), i∈I

where T (z ) ∈  ∞. Let z → ∞ that will give

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L−1 (∞) X (∞) g (∞) = T (∞) +

Then from (4.1) and (4.5), and considering the result from the previous section, Theorem 4.1 can be obtained and thus the proof is completed. □

∑ L N−s1(si )⋯Li−+11(si ) Li−1(∞) Li−−11(si ) i∈I

⋯L1−1 (si ) X (si ) g (si ). Then from (4.2) and with the all-pass factor b(z), we have

Remark 4.1. If the interconnection of the plant and the controller is transparent, that is there are no quantization and packet dropouts in the communication channel, then the results of Theorems (3.1) and (4.1) reduce to

J3* = ‖L−1 (z ) X (z ) g (z ) − L−1 (∞)(∞) X (∞) g (∞) + L−1 (∞)(∞) X (∞) g (∞) − Nm (z ) R (z ) g (z )‖22

Ns

∑ L N−s1(si )⋯Li−+11(si ) ( Li−1(z) − Li−1(∞) ) Li−−11(si )

=

J* =

i∈I

. 2

Note that

∑ L N−s1(si )⋯Li−+11(si ) ( Li−1 − Li−1(∞) ) Li−−11(si )⋯L1−1(si ) X (si ) g (si ) ∈  2⊥, i∈I

and R (z ) ∈  ∞ can be chosen such that

L−1 (∞) X (∞) g (∞) + T (z ) − T (∞) − Nm (z ) R (z ) g (z ) ∈  2, and

inf ‖L−1 (∞) X (∞) g (∞) + T (z ) − T (∞) − Nm (z ) R (z ) g (z )‖22 = 0.

R ∈  ∞

Thus 2

∑ L N−s1 (si )⋯L i−+11 (si ) ( L i−1 (z ) − L i−1 (∞) ) L i−−11 (si )⋯L1−1 (si ) X (si ) g (si ) i∈I

2

∑ L N−s1 (si )⋯L i−+11 (si ) ηi ηiH (1 − si )(1 −

=

2)

si L i−−11 (si )⋯L1−1 (si ) X (si ) g (si ) (1 − s¯i )(z − si )

i∈I

On the other hand, a direct calculation of L−1X (z ) g (z )

L−1 (z ) X (z ) g (z ) z = si = −

∏ i, j ∈ I, i ≠ j

ηi ηi H

z = si

2

. 2

yields

1 − si −1 L i − 1 (si )⋯L1−1 (si ) X (si ) g (si ). 1 − s¯i

5. Illustrative example

(4.4) In this section, an illustrative example is given to show the effectiveness of the obtained theoretic results. Consider the plant with transfer function matrix given by

(1 − si )(1 − si 2 ) −1 L i − 1 (si )⋯L1−1 (si ) X (si ) g (si ) (1 − s¯i )(z − si ) (1 − si 2 ) 1 − si s¯j −1 L (z ) X (z ) g (z ) z = si (z − si ) si − sj

L N−s1 (si )⋯L i−+11 (si ) ηi ηi H

∏ i, j ∈ I, i ≠ j



= −

i, j ∈ I , i ≠ j

⎛ 1 ⎜ z +0.1 ⎜ z −0.5 G (z ) = ⎜ z +0.1 ⎜ ⎜ 0 ⎝

(1 − si 2 ) G m (si ) Λ. (z − si )

By invoking Cauchy's theorem, the following can be obtained:

1 z − si

2

= 2

1 . s¯i + sj

Thus

∑ ∏ i ∈ I j ∈ I, i ≠ j

(1 − si 2 ) G m (si ) Λ (z − si ) 2

=

∑ i, j ∈ I

(1 − si

( 1 − s ) tr ( ΛG s¯ + s 2) i

j

j

2

),

H m (si ) G m (si ) Λ

(4.5)



1

0

z 2 +0.1z +0.2 ⎟ 1

0

z 2 +0.1z +0.2

z −1.5 (z −k)(z +0.5)

0

⎟ ⎟. ⎟ ⎟ ⎠

The plant is right-invertible and has a non-minimum phase zero at z¼ 1.5 with output zero direction vector η = (0, 0, 1)T . Also, it has an unstable pole at z¼k when k > 1 with the right pole direction vector ω = (0, 1, 0)T . Based on the all-pass factorization, we can arrive at

⎛ 1 ⎜ z +0.1 ⎜ z −0.5 G m (z ) = ⎜ z +0.1 ⎜ ⎜ 0 ⎝

2

J3* =

Remark 4.2. It should be noted that the communication factor has some influence on stability and performance of NCSs, but the quantitative relationship is an interesting and meaningful problem needed to be solved. Stability analysis of networked control systems with quantization has been studied in [14,15,27,28], and some linear matrix inequalities are given to guarantee the stability of NCSs. Compared with these results, some explicit expressions have been given in this paper to clarify how quantization error and packet-dropouts affect the tracking performance of NCSs. These results are obtained by frequency domain analysis method including factorizations of transfer function matrices, Youla parametrization of all stabilizing controllers. Packet dropouts are another typical feature of NCSs, how to quantitatively reveal the relationship between tracking performance and packet-dropouts from frequency domain method may be a challenging and meaningful problem. In [19–22,29], authors have not considered packet dropouts.

si − sj −1 L N (si )⋯L i−+11 (si ) 1 − si s¯j s

Then it follows from (4.3) and (4.4) that

= −

si 2 − 1 cos2 ∠(ηi , v). si − 1 2

It can be observed that the result is in accordance with [3,26,30], in which traditional control systems have been investigated, without considering communication factors such as packet-dropouts. The quantization and packet-dropouts' effects on tracking * in Theorem 3.1. performance can be known from J21 and J22

2

J3* =

∑ i=1

⋯L1−1 (si ) X (si ) g (si ) + L−1 (∞) X (∞) g (∞) + T (z ) − T (∞) − Nm (z ) R (z ) g (z )

7

0 0 1 −1.5z (z −k)(z +0.5)

1



z 2 +0.1z +0.2 ⎟ 1 z 2 +0.1z +0.2

0

⎟ ⎟. ⎟ ⎟ ⎠

Firstly, we set Λ = diag (0.01, 0.02, 0.03) and select the different input vectors v as

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8 9 8

v=v 2 v=v

7

Tracking error [J*]

6. Conclusions

v=v 1 3

6 4.4 5

4.3 4.2

4

4.1

3

4 1.48

2

1.485

1.49

1 0 1.3

1.35

1.4

1.45

1.5

Location of p

Fig. 4. J * with respect reference input directions. 500 Λ=Λ 1

450

Λ=Λ

2

Λ=Λ 3

400 350 Tracking error [J*]

In this paper, the optimal tracking performance of the networked control systems (NCSs) has been studied with quantization and packet dropouts. Based on the Youla parametrization of two-parameter controllers, explicit expressions of the minimal tracking error have been obtained for the NCSs with quantization and packet dropouts in the feedback or feed-forward channel. It has been demonstrated that the tracking performance critically depends on the characteristics of the reference signal and communication channel, the nonminimum phase zeros and the unstable poles of the plant, including the locations and directions. Furthermore, the tracking performance may be severely degraded by the quantization errors and the packet dropouts. Theoretical analysis has been finally verified by a numerical example. It should be noted that the explicit expressions provided in this paper demonstrate how the tracking performance of the NCSs may be severely degraded by the quantization errors, but how to design the optimal quantizer is an interesting and important problem. Moreover, the system under consideration is linear time-invariant (LTI), but nonlinear systems or coupled control system are more general. It is another interesting and challenge problem to study optimal performance of nonlinear systems by using frequency domain methods.

References

300 250 200 150 100 50 0 1.3

1.4

1.5

1.6

1.7

1.8

Location of p

Fig. 5. J * with respect to quantization errors.

⎛ 0⎞ v1 = ⎜⎜ 1⎟⎟, ⎝ 0⎠

v2 =

1 2

⎛ 1⎞ ⎜ ⎟, ⎜ 0⎟ ⎝ 1⎠

v3 =

⎛ 1⎞ 1 ⎜ ⎟ 2 . 6 ⎜⎝ ⎟⎠ 1

The simulation results are illustrated as follows. Secondly, set the input vector v = v1 and choose different values of quantization noise as

⎛ 0.1 0 0 ⎞ ⎜ ⎟ Λ1 = ⎜ 0 0.2 0 ⎟, ⎝ 0 0 0.3⎠

⎛ 0.2 0 0 ⎞ ⎜ ⎟ Λ2 = ⎜ 0 0.3 0 ⎟ ⎝ 0 0 0.5⎠

and

Λ3 = 0.

The simulation results are illustrated as follows. The relationship between the tracking performance and the reference input can be observed from Fig. 4. The non-minimum phase zero and the reference directions are parallel for v = v1, but for v = v2 and v = v3, they are not parallel which implies that if k > 1, then the tracking error will be bigger due to the effect from the non-minimum phase zero and the unstable located closely. The similar simulation can be seen in Fig. 5, it can also be seen that the quantization errors worsen the tracking performance like the way they are demonstrated in Theorems 3.1 and 4.1. A special case is Λ = Λ3, which means the quantization error will be zero. Thus the tracking performance will become better. However, it is known that this is an ideal situation because the quantization is an imperative part in the communication.

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Please cite this article as: Jiang X-W, et al. Performance limitations of networked control systems with quantization and packet dropouts. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2016.12.006i