Constrained off-line synthesis approach of model predictive control for networked control systems with network-induced delays.

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Constrained off-line synthesis approach of model predictive control for networked control systems with network-induced delays Xiaoming Tang a,n, Hongchun Qu a, Ping Wang a, Meng Zhao b a Key Laboratory of Industrial Internet of Things & Networked Control, Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing 400065, PR China b College of Automation, Chongqing University, Chongqing 400044, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 12 December 2013 Received in revised form 31 August 2014 Accepted 15 November 2014 This paper was recommended for publication by Qing-Guo Wang

This paper investigates the off-line synthesis approach of model predictive control (MPC) for a class of networked control systems (NCSs) with network-induced delays. A new augmented model which can be readily applied to time-varying control law, is proposed to describe the NCS where bounded deterministic network-induced delays may occur in both sensor to controller (S–A) and controller to actuator (C–A) links. Based on this augmented model, a sufficient condition of the closed-loop stability is derived by applying the Lyapunov method. The off-line synthesis approach of model predictive control is addressed using the stability results of the system, which explicitly considers the satisfaction of input and state constraints. Numerical example is given to illustrate the effectiveness of the proposed method. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Model predictive control Networked control systems Robust control Lyapunov method Network-induced delays

1. Introduction Networked control systems (NCSs) are control systems in which the control loop is closed over a real-time network [1–4]. The insertion of networks brings not only the advantages, but also some challenging problems, and conventional control theories for point-to-point control systems must be reevaluated before applying them to the NCSs. In the past few years, various methodologies have been proposed for modeling, stability analysis, and controller design for NCSs [5–11,33,26,27,15,14]. Network-induced delay is known as the major source of degrading the performance and even causing instability of NCSs. In order to overcome the adverse effect of network-induced delay in NCSs, the uncertain system approach [12,13], the impulsive system approach [16], the stochastic system approach [17–21,24,25], and time delay system approach [22] have been addressed in the literature. For data transmission existing in both sensor to controller (S–C) and controller to actuator (C–A) links, it is more difficult to study NCS with timevarying control law than that with time-invariant controller. One of the difficulties lies in the modeling of such NCS. For the timeinvariant controller, the control law is static and one can establish the model of such NCS by combining the S–C and C–A delays, see

n

Corresponding author. E-mail address: [email protected] (X. Tang).

[22,23]. However, for the time-varying control law, such as MPC, the controller works as an intelligent unit and calculates the control strategy based on the estimations of delayed steps of the current received data and the time of the control command arriving at the actuator, and hence, the approach of combining the delays in both links is not available. In the present paper, by respectively considering the bounded deterministic network-induced delays in S–C and C–A links, a novel augmented closed-loop model is provided which can be readily applied to MPC and other time-varying control laws. MPC is one of the most popular control methods in industrial process control field, and the defining feature of MPC is handling the physical constraints in a systematic manner. A synthesis approach of MPC is the one which guarantees the closed-loop stability, i.e., the closed-loop system is stable whenever the optimization problem is feasible [36]. The synthesis approaches of MPC (see [28,29]) can significantly improve the control performance, especially when system uncertainties exist. The network-induced delays in NCS naturally introduce the uncertainties and hence, it is interesting and necessary to extend these approaches to the networked environment. Unfortunately, although there are many nice papers consider the stabilization of NCS with MPC (see, [30–32,34,37–39]), limited works have been found on the synthesis approaches of MPC. This situation motivates our present investigation. This paper aims to give an off-line synthesis approach of MPC for NCS with network-induced delays in both S–C and C–A links. The most pertinent works to this paper are our previous works

http://dx.doi.org/10.1016/j.isatra.2014.11.007 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Tang X, et al. Constrained off-line synthesis approach of model predictive control for networked control systems with network-induced delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.11.007i

X. Tang et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

2

[35,36]. Ding [35] has provided the on-line synthesis approach of MPC for NCS with packet loss by extending the results in [28]. Tang and Ding [36] have investigated the on-line synthesis approaches of MPC for NCS where the packet loss and data quantization are coexisting, by generalizing the results in [35] and combing the sector bound approach in [33]. However, both works in [36,35] do not consider the network-induced delay which is considered as the major source of causing instability of NCSs. The reasons of preventing us to tackle the adverse effect of network-induced delays in the previous two papers lie in the following two aspects. First, since the property of network-induced delay process is completely distinct from that of packet loss, how to establish the model of such NCS needs to be further investigated. Actually, according to our study, the model of NCS with packet loss is only a special case of NCS with network-induced delays (case x^ ¼ x, where x^ is the received sensed state by the controller). The NCS model established in this paper incorporates the adverse effects of packet loss and network-induced delays simultaneously. Second, the different properties between the packet loss and networkinduced delays make the main difficulty to obtain the synthesis approach of MPC. For example, in [35], the real-time condition on the augmented state of MPC, which is one of the crucial conditions in guaranteeing the recursive feasibility of synthesis approach of MPC, can be easily obtained by introducing “Assumption 9” that enables the controller to know whether or not the previous sent values have been applied. However, due to the properties of network-induced delays, the controller is difficult to know when the control law has been implemented and the initial condition cannot be easily established as in [35]. In the present paper, we obtain the real-time conditions on the augmented state by construction all the possible values of the augmented state, which is also suitable for [35] when “Assumption 9” is not applied and we think is one of the major difficulties overcome for extending the synthesis approach of MPC to network-induced delay environment in this paper. Further, there is an unavoidable fact in [36,35] that the online synthesis approach of MPC involves heavy computational burden which may prevent its practical application. As comparison, this paper provides an off-line synthesis approach of MPC by generalizing the procedure in [29]. The off-line synthesis approach of MPC makes the computational burden significantly reduced, which is preferable for such a practical NCS and we think could be useful for readers. Notation: I is the identity matrix with appropriate dimension. For any vector x and matrix W, ‖x‖2W ≔xT Wx. The superscript T denotes the transpose for vectors or matrices. xðk þ ijkÞ is the value of vector x at a future time k þ i predicted at time k. The symbol ðnÞ is used to induce symmetric matrices.

where A and B are constant matrices of appropriate dimensions; uðkÞ A Rm and xðkÞ A Rn are input and state of the plant, respec^ A Rm and xðkÞ ^ A Rn are output and input of the contively; uðkÞ troller, respectively. The assumptions for NCS are

2. Modeling of NCS with network-induced delays

^ l Þ is xðkjl Þ. For convenience, define kjl as the index such that xðj

The structure of NCS is depicted in Fig. 1, the plant is xðk þ 1Þ ¼ AxðkÞ þ BuðkÞ;

kZ0

ð1Þ

A1 The sensor is clock driven, i.e., it sends the sensed state x(k) at each k; A2 The controller is event driven, i.e., at time k, the controller reads ^ ^ xðkÞ and calculates uðkÞ if and only if buffer 1 receives data; A3 The actuator is event driven, i.e., at time k, the actuator reads u (k) and updates the input if and only if buffer 2 receives data; A4 In S–C link, the sensed state x(k) is transmitted at each k, and the bounded network-induced delays may occur; ^ A5 In C–A link, the control move uðkÞ is transmitted at each k, and the bounded network-induced delays may occur; A6 If the actuator does not receive any data, the input uðk 1Þ is utilized; A7 The sensed state and control signal are marked with timestamp.

2.1. Data transmissions in S–C link In S–C link, since the network-induced delay is bounded and may be larger than one sampling period, three usual cases may occur at time k: (a) Only one data packet arrives at buffer 1; (b) More than one data packet arrives at buffer 1; (c) There is no data packet arrives at buffer 1. In case (a), the controller reads the arrived data and calculates the control move. In case (b), the controller reads the newest data for calculation, while the others are discarded. In case (c), the controller does not calculate. It is shown that the discarded sensed states do not affect the controller. The following time sequences and steps are introduced to describe the data transmission process in S–C link and identify the sensed states which affect the controller. (1) Assume that x(k) arrives at buffer 1 at time j k , where j k is a non-negative integer. Then, the time points fj 1 ; j 2 ; j 3 ; …g are assembled to form time sequence J 1 . (2) For any three adjacent time points fj a ; j a þ 1 ; j a þ 2 g A J 1 , if j a þ 1 j a Z 0 and j a þ 2 j a þ 1 Z1, then j a þ 1 ’s form the sequence J 2 ≔fj1 ; j2 ; j3 ; …g which affects the closed-loop system. Specially, if j 2 j 1 Z 1, then j 1 A J 2 .

Example 1. Fig. 2 shows an example of data transmissions from the sensor to the actuator. Different time sequences are illustrated as follows. First, it is easy to show that J 1 ¼ f1; 2; 2; 5; 6; 6; 7; 8; 9; 9; 11; 11; …g. Second, by deleting xð1Þ; xð4Þ; xð8Þ; xð10Þ, which are not read by the controller, J 2 ¼ f1; 2; 5; 6; 7; 8; 9; 11; …g is obtained. 2.2. Data transmissions in C–A link

Fig. 1. Networked control systems with network-induced delays.

The data transmission process in C–A link is similar to that in S–C link. When only one data packet arrives at buffer 2 during one sampling interval, the arrived data packet is utilized to update the control input; When more than one data packet arrives at buffer 2 during one sampling interval, the actuator only reads the most recent data for utilization; When no new data packet arrives buffer 2 during one sampling interval, the previous control input acts on the plant. Hence, only a part of control moves affect the actuator.



3

It is shown that fjr ; ır g is a pair which satisfies jr ¼ ζ ır and jr r ır ; r Z 1. Example 3. Continue with Examples 1 and 2. J ¼ f1; 2; 5; 8; 9; 11; …g; I ¼ f1; 4; 7; 8; 10; 11; …g. Any il in I 2 , but not in I , corre^ ζ i Þ which does not affect the closed-loop system. For sponds to uð l example, at k ¼5, the actuator reads the same u^ as at k¼4. Hence, ^ uð4Þ does not affect the closed-loop system. Moreover, any js in J 2 , ^ s Þ which does not affect the closedbut not in J , corresponds to xðj ^ ^ loop system. For example, uð6Þ and uð7Þ are discarded in buffer 2. ^ ^ Hence, xð6Þ, xð7Þ do not affect the closed-loop system. Proposition 1. For the bounded delay process in Definitions 1 and 2,

Fig. 2. An example of packets transmitted and utilized.

The following two time sequences and steps are given to describe the data transmission process in C–A link and identify the control moves which affect the actuator. ^ (1) Assume that uðkÞ arrives at buffer 2 at time ı k , where ı k is a non-negative integer. Then, the time points fı 1 ; ı 2 ; ı 3 ; …g are assembled to form time sequence I 1 . (2) For any three adjacent time points fı b ; ı b þ 1 ; ı b þ 2 g A I 1 , if ı b þ 1 ı b Z 0 and ı b þ 2 ı b þ 1 Z 1, then ı b þ 1 ’s form the sequence I 2 ≔fi1 ; i2 ; i3 ; …g which affect the actuator. Specially, if ı 2 ı 1 Z 1, then ı 1 A I 2 . For convenience, we define ζ il as the index such that uðil Þ is ^ ζ i Þ. It is shown that ζ i A J 2 . uð l l Example 2. Continue with Example 1. It is shown that I 1 ¼ f1; 4; 5; 5; 7; 8; 8; 8; 10; 11; 11; …g; I 2 ¼ f1; 4; 5; 7; 8; 10; 11; …g. Appar^ ^ ^ ^ ently, uð3Þ, uð6Þ, uð7Þ and uð10Þ are discarded. Let d1 ≔maxjl A J 2 ðjl þ 1 jl Þ, d2 ≔maxil A I 2 ðil þ 1 il Þ be the maximum delay upper bounds in S–C and C–A links, respectively. If d1 ¼ d2 ¼ 1, then there is no delay. Definition 1. Bounded delay process is defined as fηðjl Þ≔ jl þ 1 jl jjl A J 2 g, fρðil Þ≔il þ 1 il jil A I 2 g where ηðjl Þ and ρðil Þ take values in the finite set D1 ≔f1; 2; …; d1 g and D2 ≔f1; 2; …; d2 g, respectively. Definition 2. The bounded delay process in Definition 1 is said to be arbitrary if ηðjl Þ and ρðil Þ take values in D1 and D2 arbitrarily, respectively. 2.3. Time sequences which affect the closed-loop system Based on the analyzes in Sections 2.1 and 2.2, we find the data sequences which affect the controller and the actuator, respectively. However, some data in the two sequences may not affect the closed-loop system. There are two reasons that contribute to this. First, if fuðil Þ; uðil þ 1 Þ; …; uðil þ c Þg are the same u^ calculated at ζ il , then fuðil þ 1 Þ; …; uðil þ c Þg do not affect the closed-loop system. ^ s Þ is discarded in buffer 2, then xðj ^ s Þ does not affect Second, if uðj the closed-loop system. We define I ≔fı1 ; ı2 ; ı3 ; …g and J ≔ fj1 ; j2 ; j3 ; ⋯g the time sequences which affect the closed-loop system, and I ; J are obtained by (1) I D fı_1 ; ı_2 ; ı_3 ; …g, J Dfj_1 ; j_2 ; _j 3 ; …g; (2) for any integer c 4 0, if ζ ic þ 1 a ζ ic , then ic þ 1 A I , otherwise ic þ 1 2 = I . Specially, i1 A I ; (3) for js A J 2 , if js does not correspond to any ζ il , then js 2 =J, otherwise js A J .

jr kjr þ1 A D1

ð2Þ

ır jr þ 1 A D2

ð3Þ

ır kjr þ1 A D3 ≔f1; 2; …; d1 þd2 1g

ð4Þ

ır þ 1 kjr A D≔f1; 2; …; 2d1 þd2 2g:

ð5Þ

2.4. The new closed-loop model of NCS In this part, based on the above analyzes, we establish the model of NCS with network-induced delays. The resulting NCS model lays a foundation for the synthesis approach of networked MPC. ^ where K A Rmn is Suppose the networked controller is u^ ¼ K x, ^ l Þ. For to be designed. At each jl, the controller calculates uðj ^ ^ l Þ is sent into the network. At ır , uðır Þ acts jl r k o jl þ 1 , uðkÞ ¼ uðj on the plant, and the control input will not change until uðır þ 1 Þ updates the actuator. Consider the control input uðır Þ ¼ ^ r Þ ¼ Kxðkjr Þ, and the definitions of J and I , the closed-loop K xðj system for all k Z ı1 becomes xðk þ 1Þ ¼ Ak ır þ 1 xðır Þ þ Bk ır þ 1 Kxðkjr Þ

ð6Þ

∑js¼10 As B

for any integer j 4 0. The where ır rk oır þ 1 , ır A I , Bj ¼ purpose of this paper is to construct controller u^ ¼ K x^ so that (6) is stable. By considering all the possible values in Proposition 1, the system in (6) can be expressed as Case ð1Þ : ır kjr þ 1 ¼ 1;

8 xðır þ 1 Þ ¼ Axðır Þ þ BKxðkjr Þ > > > > < xðır þ 1 Þ ¼ A2 xðır Þ þ B2 Kxðkjr Þ >⋮ > > > 2d1 þ d2 2 : xðı xðır Þ þ B2d1 þ d2 2 Kxðkır Þ r þ 1Þ ¼ A

if ır þ 1 ır ¼ 1; if ır þ 1 ır ¼ 2; if ır þ 1 ır ¼ 2d1 þ d2 2:

Case ð2Þ : ır kjr þ 1 ¼ 2;

8 xðır þ 1 Þ ¼ Axðır Þ þ BKxðkjr Þ > > > > < xðır þ 1 Þ ¼ A2 xðır Þ þ B2 Kxðkjr Þ > ⋮ > > > 2d1 þ d2 3 : xðı xðır Þ þ B2d1 þ d2 3 Kxðkjr Þ r þ 1Þ ¼ A

if ır þ 1 ır ¼ 1; if ır þ 1 ır ¼ 2; if ır þ 1 ır ¼ 2d1 þ d2 3:

⋮ Case ðd1 þd2 1Þ : ır kjr þ 1 ¼ d1 þ d2 1; 8 if ır þ 1 ır ¼ 1; xðır þ 1 Þ ¼ Axðır Þ þ BKxðkjr Þ > > > > 2 < xðı if ır þ 1 ır ¼ 2; r þ 1 Þ ¼ A xðır Þ þ B2 Kxðkjr Þ > ⋮ > > > : xðı Þ ¼ Ad1 xðı Þ þ B Kxðk Þ if ı ı ¼ d : rþ1

r

d1

jr

rþ1

r

1

2 2 1 2 ð3d1 þ d2 þ4d1 d2 3d1 3d2 þ 2Þ

It is shown that different choices may arise. Similar to [36], by defining zðkÞ ¼ ½xðkÞT xðk 1ÞT x ðk 2ÞT ⋯xðk d1 d2 þ 2ÞT T and taking into account all the possibilities, (6) is rewritten as zðır þ 1 Þ ¼ Φðır Þzðır Þ Φðır Þ ¼ ∑ ∑ ωp ðır Þκ t ðır ÞΦpt p A DðtÞt A D3



4

ωp ðır Þ ¼ κ t ðır Þ ¼

1;

ır þ 1 ır ¼ p

definite matrices P such that

0

otherwise

1; 0

ır kjr þ 1 ¼ t otherwise

ΦTpt P Φpt P o 0; t A D3 ; p A DðtÞ: ð7Þ Proof. Choose the Lyapunov function as

where DðtÞ ¼ f1; 2; …; 2d1 þ d2 1 tg Φpt A R " #

Φpt ¼

ϕpt

½ I 0

ϕpt A RðpnÞðd1 þ d2 1Þn ,

ϕspt A RðpnÞn , s A D3 ; ϕsp1 ¼ 0, s a 1 and 2

ϕ

1 p1

6 6 ¼6 6 4

ð8Þ

Ap þ Bp K A p 1 þ Bp 1 K ⋮

ðd1 þ d2 1Þnðd1 þ d2 1Þn

h

d1 þ d2 1 ϕpt ¼ ϕ1pt ϕ2pt ⋯ϕpt

, i

V ðzðır ÞÞ ¼ zT ðır ÞPzðır Þ: Applying (7) and (9) yields V ðzðır þ 1 ÞÞ ¼ zT ðır ÞΦ ðır ÞP Φðır Þzðır Þ: T

3T

Hence,

7 7 7 for t a 1, ϕs ¼ 0, pt 7 5

V ðzðır þ 1 ÞÞ Vðzðır ÞÞ

Amaxf1;p d1 d2 þ 2g þ Bmaxf1;p d1 d2 þ 2g K 2 3 Ap Bp K 6 7 p 1 h i 6 7 A Bp 1 K 1 t 7. s a f1; tg and ϕpt ϕpt ¼ 6 6 7 ⋮ ⋮ 4 5 max f1;p d1 d2 þ 2g Bmaxf1;p d1 d2 þ 2g K A

Remark 1. According to Definitions 1 and 2 and Proposition 1, in NCS model (7), ır þ 1 ır ¼ p; ır kjr þ 1 ¼ t; t A D3 ; p A DðtÞ describe all the possibilities of network-induced delays. Further, it is easy to show that the existence of packet discards in buffer 1 and buffer 2 amounts to the effect of packet loss. Hence, the model in this paper is appropriate when network-induced delays and packet loss are all explicitly considered. The closed-loop model (6) is obtained by generalizing the procedure in [36] where NCS with packet loss and quantization in both links is considered. The difference is caused by the different properties between packet losses and network-induced delays. For packet loss process, once the controller receives new data at jl, it knows that the data is the current ^ l Þ ¼ xðjl Þ. However, for network-induced state of the plant, i.e. xðj delay process, it becomes 8 xðjl Þ; jl kjl ¼ 0 > > > > < xðjl 1Þ; jl kjl ¼ 1 ^ lÞ ¼ xðj ⋮ > > > > : xðjl d1 þ 1Þ; jl kj ¼ d1 1: l This difference gives rise to the major difficulty overcome when obtaining the synthesis approach of MPC in this paper.

ð9Þ

¼ zT ðır ÞðΦ ðır ÞP Φðır Þ PÞzðır Þ T

¼ zT ðır Þ

∑

!

∑ ωp ðır Þκ t ðır ÞðΦpt P Φpt PÞ zðır Þ: T

p A DðtÞt A D3

V ðzðır þ 1 ÞÞ Vðzðır ÞÞ o 0 for all zðır Þ a 0, which is guaranteed by (8), means asymptotic stability of (7). This completes the proof. □ Denote Q ¼ P 1 , 2 G1;1 G1;2 pt pt 6 2;1 6 Gpt G2;2 6 pt Gpt ¼ 6 6 ⋮ ⋮ 4 d1 þ d2 1;2 Gdpt1 þ d2 1;1 Gpt

⋯

1;d1 þ d2 1 Gpt

⋯

2;d1 þ d2 1 Gpt

⋱ ⋯

⋮ d1 þ d2 1;d1 þ d2 1 Gpt

3. Stability analysis and design of state feedback controller for NCS In this section, we analyze the stability property and design the state feedback controller of NCS (7). For NCS with bounded delay process, a sufficient condition is derived based on the Lyapunov approach. Then, with the stability result, a controller design technique is provided. Theorem 1. Consider the discrete-time system in (1), where bounded ^ delay occurs. By applying state feedback control defined by u^ ¼ K x, the closed-loop system is asymptotically stable if there exist positive-

7 7 7 7; 7 5

Gt;s pt ¼ G; s A D3 ; where each block in Gpt is of the same dimension. Then, with the stability condition in Theorem 1, a state feedback is derived in terms of a set of LMIs. Theorem 2. Consider the discrete-time system in (1), where bounded delay occurs. There exists a state feedback controller u^ ¼ K x^ such that the closed-loop system in (7) is asymptotically stable if and only if there exist matrices Y, Gpt , G, and Q such that " T # T Gpt þ Gpt Q GTpt Φpt 40; t A D3 ; p A DðtÞ ð10Þ Φpt Gpt Q where KG in Φpt Gpt yields Y. In this case, the state feedback gain is given by K ¼ YG 1 . Proof. By using a congruence transformation on (8) by Gpt, we have GTpt Φpt P Φpt Gpt GTpt PGpt o 0; t A D3 ; p A DðtÞ: T

Remark 2. The NCS model (7) is established on the standpoint of robustness analysis. The controller enables (7) to be applied to time-varying control law. This is the major difference between NCS model (7) and the other NCS models. Since both this paper and [35] introduce the augmented state space technique to study NCS, the NCS model (7) has similar form as in [35]. However, it is worth pointing out that the vertices of the model in (7) is distinct from that in [35], and the physical meanings of the two models are completely different.

3

ð11Þ

1

and the Schur complement, (11) becomes By applying Q ¼ P " T 1 # T T Gpt Q Gpt Gpt Φpt ð12Þ 4 0; t A D3 ; p A DðtÞ: Φpt Gpt Q Applying the dilation trick GTpt Q 1 Gpt Z GTpt þ Gpt Q , (12) is guaranteed by (10). This completes the proof. □ 4. Stabilization via model predictive control In Sections 2 and 3, the NCS model which can be used to timevarying control law, and the stability result for the resulting closedloop model, are provided. This section will apply the stability result for this system to MPC that explicitly considers the input and state constraints. The input and state constraints considered in this paper are u r uðı1 þhÞ r u; ψ r Ψ xðı1 þh þ 1Þ r ψ ;

8hZ0

ð13Þ

where u≔½u 1 ; u 2 ; …; u m , u j 4 0, jA m≔f1; 2; …; mg; ψ ≔½ψ 1 ; ψ 2 ; …; T

ψ q T , ψ s 4 0, s A q≔f1; 2; …; qg; Ψ A Rqn .

The purpose is to design a networked model predictive controller that drives the system (7) and (13) to the equilibrium point.



For our synthesis approach of off-line networked MPC, we first solve the optimization problem for MPC off-line, such that a sequence of state feedback gains K ℓ ’s, corresponding to a sequence of nested asymptotically invariant ellipsoids, are constructed. Then, in on-line implementing our networked MPC, at each jl, the controller performs a search to find a state feedback gain Kðjl Þ. ^ ^ l Þ ¼ Kðjl Þxðj ^ l Þ is sent into the network. This For jl rk ojl þ 1 , uðkÞ ¼ uðj off-line networked MPC method is a generalization of [29], where the non-networked counterpart of this off-line method is provided.

4.1. Optimization problem for networked MPC The off-line optimized state feedback gains are based on a sequence of pre-specified augmented states zℓ ’s. For each zℓ , as if it is at jl, the following optimization is solved: min

γ ;Q ;Kðjl Þ

s:t:

maxγ

ð14Þ

Ω

ψ r Ψ xðk þ 1jjl Þ r ψ ; u r uðkjjl Þ r u; k Z ısl

^ s þ τ jjl Þ ¼ Kðjl Þxðkjs uðısl þ τ jjl Þ ¼ Kðjl Þxðj l

l þτ

xðk þ 1jjl Þ ¼ Ak ısl þ τ þ 1 xðısl þ τ jjl Þ þ Bk ıs

ð15Þ

jjl Þ

l þτ

þ 1 Kðjl Þxðkjs

l þτ

jjl Þ

ısl þ τ r k o ısl þ τ þ 1 ; τ Z 0 ‖zðısl jjl Þ‖2Q 1 r 1

ð16Þ ð17Þ

‖zðısl þ τ þ 1 jjl Þ‖2Q 1 ‖zðısl þ τ jjl Þ‖2Q 1 r 1=γ ‖xðısl þ τ jjl Þ‖2Q 1=mbi; γ‖uðısl þ τ jjl Þ‖2R ; 8 τ Z 0 zðısl þ τ þ 1 jjl Þ ¼ Φðısl þ τ jjl Þzðısl þ τ jjl Þ; zðısl jjl Þ ¼ zℓ

ð18Þ ð19Þ

þ1 A D1 ; ısl þ τ jsl þ τ þ 1 A D2 ; ısl þ τ þ 1 where Ω ¼ fjsl þ τ kjs þ τ l kjs þ τ A D; τ Z 0g, Q and R are positive-definite weighting matrices. l In problem (14)–(19), the performance cost γ, the real-time condition on the augmented state (17), and the stability constraint (18), are provided by generalizing those in [29]. (16) is the closed-loop state prediction which is obtained by taking (6) as the prediction model. The purpose of this problem is to find fKðjl Þ; Q g such that (7) is asymptotically stable with a common Lyapunov matrix Q 1 . Note that the controller does not know a priori whether or not the current jl will be equal to jr , and hence, it assumes that jl ¼ jsl . It is not necessary to know the value of sl Z 1. Then, we convert problem (14)–(19) to LMI optimization problem. By applying Schur complement, (17) is equivalent to " # 1 n Z 0: ð20Þ zðısl jjl Þ Q Similar to Theorem 2, denote Kðjl Þ ¼ YG 1 and G1pt the first row of Gpt . By taking a procedure the same as in [36], (18) is guaranteed by 2 T 3 Gpt þ Gpt Q n n n 6 7 6 Φpt Gpt Q n n7 6 7 ð21Þ 6 7 Z 0; t A D3 ; p A DðtÞ 6 W 1=2 G1pt 0 γI n 7 4 5 R1=2 ½Y Y⋯Y 0 0 γ I In order to tackle physical constraints, partition symmetric matrix Q into ðd1 þ d2 1Þ ðd1 þd2 1Þ blocks of the same

dimensions, i.e. 2 Q 1;1 6 6 Q 2;1 Q ¼6 6 ⋮ 4 Q d1 þ d2 1;1

5

ðQ 2;1 ÞT

⋯

ðQ d1 þ d2 1;1 ÞT

Q 2;2 ⋮

⋯ ⋱

ðQ d1 þ d2 1;2 ÞT ⋮

Q d1 þ d2 1;2

⋯

Q d1 þ d2 1;d1 þ d2 1

3 7 7 7; 7 5

By generalizing “Lemma 10” in [35], we have the following Lemma. Lemma. Suppose there exist a scalar γ, symmetric matrices Γ , Z, Q, and any matrix Kðjl Þ such that (20) and (21) and the following inequalities are satisfied: " # GT þ G Q t;t n ð22Þ Z 0; t A D3 Y Γ "

GT þ G Q 1;1

n

#

Ψ ðAp G þ Bp YÞ Z

Z 0;

pAD

2

1;1 1;1 T ðG1;1 pt Þ þ Gpt Q 6 1;t t;1 6 ðG ÞT þ G Q 6 pt 4 Ψ ðAp G1;1 pt þ Bp YÞ

n

GT þ G Q t;t

ð23Þ

n

3 7

n7 7Z0

Ψ ðAp G1;t pt þ Bp YÞ Z

5

ð24Þ

t A f2; 3; …; d1 þ d2 1g; p A DðtÞ

ð25Þ

Γ ss ru 2s ; s A m; Z ss r ψ 2s ; s A q

ð26Þ

where Z ss ðΓ ss Þ is the s-th diagonal element of ZðΓ Þ. Then, the input and state constraints in (15) are satisfied. Considering the above derivations, the optimization problem (14)–(19) is approximately transformed into the following LMI optimization problem: min

γ ;Q ;Gpt ;Y;Z;Γ

γ;

s:t: ð20Þ ð25Þ:

ð27Þ

The optimization problem (27) is extended from problem (21) in [36]. Due to the unique properties of network-induced delays, the major difficulty for solving optimization problem (27) comes from dealing with the initial condition on the augmented state (20), which renders one of the contributions of this paper. In (20), zðısl jjl Þ ¼ ½xðısl jjl ÞT xðısl 1jjl ÞT ⋯xðjl jjl ÞT xðjl 1jjl ÞT ⋯xðısl d1 d2 þ 2jjl ÞT T , which cannot be obtained by the same method as in [36], since the current state x(k) and the historical control input are unknown to the controller. Here, we provide a new approach to calculate zðısl jjl Þ of (20), ad hoc for network-induced delay case. For ısl jl þ 1 ¼ ϑ and jl kjl þ 1 ¼ β, denote ϖ ¼ min fϱjϑ þ jl jl ϱ Z d1 þ d2 1g and h iT xðjl jjl Þ ¼ xðjl þ ϑ 1jjl ÞT ⋯ xðjl þ 1jjl ÞT xðjl jjl ÞT h iT xðjl0 1 jjl Þ ¼ xðjl0 1jjl ÞT ⋯ xðjl0 1 þ1jjl ÞT xðjl0 1 jjl ÞT 0

l A fl; l 1; …; l ϖ þ 1g: Then the following Proposition can be obtained: Proposition 2. zðısl jjl Þ consists of the first d1 þ d2 1 state values in h iT Zðjl Þ ¼ xðjl jjl ÞT xðjl 1 jjl ÞT ⋯xðjl ϖ jjl ÞT : Proof. See Appendix A. 4.2. Off-line formulation of networked MPC The on-line approach, based on receding-horizon solution of (27), will involve heavy on-line computational burden. This is not allowed for real applications. In this part, the algorithm and the stability of the off-line networked MPC are investigated.



(a) choose states zν , ν A f1; 2; …; Ng. For each ν, substitute zðısl jjl Þ with zν in (20) and solve (27) to obtain the corresponding matrices π ν ≔fγ ν ; Q ν ; Y ν ; Gν ; Z ν ; Γ ν g. (b) for all ν a s, check

ΦTpt P ν Φpt P s r 0; t A D3 ; p A DðtÞ:

ð28Þ

If (28) is satisfied, then the corresponding π ν is denoted as π 0μ ≔fγ μ ; Q μ ; Y μ ; Gμ ; Z μ ; Γ μ g, μ A f1; 2; …; N μ g with Nμ r N. (c) by adjusting the index of π 0μ , according to the value of γ μ , Π θ ≔fγ θ ; Q θ ; Y θ ; Gθ ; Z θ ; Γ θ g, θ A f1; 2; …; Nμ g, which satisfies γ 1 4 γ 2 4 ⋯ 4 γ Nμ , are obtained. On-line: (e) at each jl, calculate all the possible values of zðısl jjl Þ according to Proposition 2; then, perform a search over Q θ to find the largest index θ such that zðısl jjl Þ A εQ 1 ¼ fz A Rðd1 þ d2 1Þn jzT Q θ 1 z r1g, θ and adopt the following control law: 1 ^ l Þ: ^ l Þ ¼ Kðjl Þxðj ^ l Þ ¼ Y θ Gθ xðj uðj

links are d1 ¼ 3, d2 ¼ 2, respectively; (2) the bounded packet losses addressed in [35]. The bounds of packet losses in S–C and C–A links are selected as d1 ¼ 3, d2 ¼ 2, respectively; (3) packet losses and

3 S−C link

Algorithm. Off-line:

where θ ðradÞ and θ_ ðrad s 1 Þ are the angular position and the velocity of the antenna, respectively. ϵ ðs 1 Þ is proportional to the coefficient of viscous friction in the rotating parts of the antenna and satisfies 0:1 s 1 r ϵ r 10 s 1 . uðVÞ is the input voltage, subjected to 1 juj r1 V. By taking ϵ ¼ 0:1 s 1 and κ ¼ 0:787 rad V 1 s 2 , we get the values of the matrices A and B 1 0:1 0 A¼ ; B¼ 0 0:99 0:0787 For networked angular positioning system, Ethernet-like networks are deployed on both S–C and C–A transmission links. For comparison, we consider four kinds of network transmission conditions: (1) the bounded arbitrary network-induced delays investigated in this paper. The bounds of delays in S–C and C–A

60

80

100

120

80

100

120

1

0 0

20

40

60 k

Fig. 3. The network transmission status in this paper.

S−C link

1.5 1 0.5 0 −0.5

0

20

40

60

80

100

120

80

100

120

k 1.5

C−A link

1 0.5 0 −0.5

0

20

40

60 k

Fig. 4. The network transmission status in [35].

1.5 1 S−C link

9 AxðkÞ þ BuðkÞ

40

2

0.5 0 −0.5

0

20

40

60

80

100

120

80

100

120

k 1.5 1 C−A link

In this section, a numerical example is given to illustrate the developed off-line networked MPC for NCS with network-induced delays. Furthermore, some comparisons are carried out to demonstrate the differences between this paper and [35,36]. Consider the control problem for angular positioning system deployed in a networked environment (see Fig. 1), which has been studied in [35,36]. The controlled plant is the angular positioning system which is consisted of a rotating antenna and an electric motor. The target is to regulate the input voltage to the motor and rotate the antenna so that it always points in the direction of a moving object. The dynamic model of the angular positioning system can be expressed as the following discrete time system by discretization, using a sampling time of 0.1 s and Euler's approximation: " # θðk þ 1Þ 0 1 0:1 xðk þ 1Þ ¼ _ xðkÞ þ uðkÞ ¼ θ ðk þ 1Þ 0:1κ 0 1 0:1ϵ

20

k

Theorem 3. Consider the discrete-time system in (1), where bounded delay occurs. Algorithm 1 is applied. If ‖zðj1 Þ‖2Q 1 r 1, then the closed1 loop system is robustly asymptotically stable, and the input and state constraints are satisfied.

5. Numerical example

1

0

N ^ l Þ ¼ Kðjl 1 Þxðj ^ l Þ is sent (f) if zðısl jjl Þ A ⋃i ¼μ 1 εQ 1 , then the above uðj θ into the network.

Proof. See Appendix B.

2

0

C−A link

6

0.5 0 −0.5

0

20

40

60 k

Fig. 5. The network transmission status in [36].



7

Table 1 Computational results for off-line stage of presented MPC. zν

Y

G

z1

[ 0.8641 0.3781]

z2 z3 z4 z5

γ

4:5875 3:5510 3:5067

[ 0.1918 0.3543]

208.7389

[ 0.7207 0.3657]

11:7910 3:9448 8:6936

[ 0.2170 0.3917]

151.7446

[ 0.5824 0.3541]

2:9751 6:2440

2:8759 2:6270

[ 0.2476 0.4387]

107.1232

[ 0.4505 0.3424]

2:1970 4:3452

2:2900 1:9199

[ 0.2854 0.4992]

72.9930

[ 0.3228 0.3320]

1:5816 2:9231

1:7835 1:3700

[ 0.3313 0.5793]

47.6516

1:3564 0:9489 0:9988 0:6307

[ 0.3854 0.6890]

29.4647

[ 0.4378 0.8418]

17.0107

[ 0.4698 0.9773]

9.2222

[ 0.4701 0.9772]

4.0995

[ 0.4699 0.9772]

1.0247

z6

[ 0.1994 0.3225]

z7

[ 0.0731 0.3162]

1:1142 1:8890 0:7673 1:1670

[ 0.0037 0.2460]

0:5201 0:6521

0:7036 0:3636

0:3097 0:2897 0:1375 0:0725

0:4265 0:1615 0:1895 0:0404

z8

F

z9

[ 0.0018 0.1092]

z10

[ 0.0004 0.0273]

0:0344

0:0474

1.2

0.05 MPC with delays in this paper MPC with packet loss in [35] MPC with packet loss and quantization in [36] MPC with perfect networks

1

0

−0.05

0.8

−0.1

x2

x1

0.6

−0.15

0.4

−0.2 0.2

−0.25 0

−0.2

MPC with delays in this paper MPC with packet loss in [35] MPC with packet loss and quantization in [36] MPC with perfect networks

−0.3

0

20

40

60

80

100

120

k

−0.35

0

20

40

60

80

100

120

k

Fig. 6. The state responses (x1).

Fig. 7. The state responses (x2).

quantization are coexisted which is studied in [36]. The bounds of packet losses take d1 ¼ 3, d2 ¼ 2, and the quantization parameters are δf ¼ δg ¼ 0:1909; and (4) perfect networks (i.e. no networkinduced delays, no packet losses and quantization occur in both links). In the simulation, the network-induced delays and packet losses are generated randomly, and the detailed transmission sequences of network conditions (1), (2) and (3) are depicted in Figs. 3–5, respectively. For the off-line stage of the Algorithm presented in this paper, ten augmented states zν (ν ¼ 1; 2; …; 10) are chosen as zν ¼ βν zmax , where β 1 ¼ 1, β 2 ¼ 0:9, β 3 ¼ 0:8, β 4 ¼ 0:7, β 5 ¼ 0:6, β 6 ¼ 0:5, β7 ¼ 0:4, β8 ¼ 0:3, β9 ¼ 0:2, β10 ¼ 0:1, and zmax ¼ ½1 1 1 1 1 1 11T . By substituting zðısl jjl Þ with zν in (20), taking R ¼ 1, Q ¼ I (the comparison simulations share the same parameters) and solving the optimization problem (27), the corresponding matrices fγ θ ; Q θ ; Y θ ; Gθ ; Z θ ; Γ θ g are obtained, which are partially shown in Table 1. For on-line stage of the presented networked MPC, the

controller only needs to perform a search to find the appropriate state feedback F θ ¼ Y θ Gθ 1 according to the values of the current augmented state. Given an initial state xð0Þ ¼ ½1 0T , four groups of simulation tests are conducted: (a) MPC for networked angular positioning system subject to bounded arbitrary network-induced delays; (b) MPC for networked angular positioning system subject to bounded packet losses in [35]; (c) MPC for networked angular positioning system subject to both packet losses and quantization in [36]; and (d) MPC for the same system with perfect networks. The simulations are performed on our laptop, and the results are reported as follows. Figs. 6 and 7 show the two states x1 , x2 of closed-loop systems by applying four different methods, respectively. Fig. 8 compares the state trajectories of closed-loop system corresponding to different algorithms. The control inputs of four different tests are demonstrated in Fig. 9. The upper bounds of cost functions of the compared algorithms are depicted in Fig. 10.



8

160

0.05 MPC with delays in this paper MPC with packet loss in [35] MPC with packet loss and quantization in [36] MPC with perfect networks

0


140

−0.05

120

−0.1

100

−0.15

80

10 5 γ

γ

x2

15

0 60

−0.2

−5

−0.25

40

−0.3

20

−0.35

0

0.2

0.4

0.6

0.8

1

0

−10 35

0

20

40

40

60

45 k

50

80

55

100

120

k

x1

Fig. 10. The evolutions of γ.

Fig. 8. The comparison of state trajectories.

0.2

Table 2 Comparison of control performance and computation burden.

0.1 0 −0.1

J true

MPC with delays in this paper

32.950 203 (on-line)/0.03 (offline) 20.585 0.5 21.428 9.6

MPC with packet loss in [35] MPC with packet loss and quantization in [36] MPC with perfect networks

u

−0.2

T ave ðsÞ

Different cases

20.562 0.4

−0.3 −0.4

6. Conclusions

−0.5


−0.6 −0.7

0

20

40

60

80

100

120

k Fig. 9. The comparison of control inputs.

Table 2 shows the control performance and computation burden of T 2 different tests, where J true ¼ ∑1 k ¼ 0 ½xðkÞ xðkÞ þ uðkÞ and T ave denotes the average computation time for each jl. Clearly, from the simulation results, the following can be observed: (1) the networked angular positioning system controlled by the presented off-line MPC is stable and the input constraint is satisfied; (2) due to different physical meanings between networkinduced delay and packet loss, quantization, the closed-loop model in this paper is distinct from that in [35,36], and obtaining the real-time condition on the augmented state is more complex, which results in higher computation complexity; and (3) the control performance using the presented MPC is slightly decreased compared to the results of perfect networks, but the computation time is sharply decreased. The average time for the off-line computing a feedback gain is 203 s, which is about 6767 times faster than the 0.03 s it takes for on-line search. These aforementioned observations imply that the presented off-line networked MPC is effective and valid.

In this paper, an off-line synthesis approach of networked MPC has been introduced to deal with the problem of NCS with bounded deterministic network-induced delays. By respectively analyzing the effects of S–C and C–A delays on the closed-loop system, a novel model which can be used to tackle the problem of time-varying control law in NCSs, is established. A synthesis approach of MPC based on the new model that explicitly considers the satisfaction of input and state constraints is addressed. Only off-line approach is specified in this paper, since the on-line approach is computationally impractical.

Acknowledgments This work is supported by the National Natural Science Foundation of China (61403055, 61102145, 61374093, 61304197, 61301033), the Research Project of Chongqing Science & Technology Commission (cstc2014jcyjA40005, cstc2014jcyjA40025), the Doctoral Start-Up Fund of Chongqing University of Posts and Telecommunications (A2013-14), and the Youth Natural Science Foundation of Chongqing University of Posts and Telecommunications (A2013-26).

Appendix A. Proof of Proposition 2 In order to calculate zðısl jjl Þ, we need to show how to calculate Zðjl Þ. First, we focus on the calculation of xðjl jjl Þ in Zðjl Þ. xðjl jjl Þ is



composed of the values of x at the future time predicted at jl. Since ^ the current state xðjl Þ is unknown, we use the received data xðkÞ to construct the possible values of current state xðjl jjl Þ, which is expressed as 8 ^ lÞ xðj β¼1 > > > > > ^ Þ þ Buðj 1Þ β¼2 A xðj > l l < ⋮ ðA:1Þ xðjl jjl Þ ¼ > > β1 > > β1 ^ βN 1 > Buðjl β þ Njjl Þ β ¼ d1 : xðjl Þ þ ∑ A > :A N¼1

we get

(

^ l Þ þ Buðjl jjl Þ; Axðj

xðjl jjl Þ ¼

AX ðjl Þ þ Buðjl jjl Þ;

X ðjl Þ ¼

A

β1

xðjl Þ þ ∑ A

ðA:2Þ

βðjℓ Þ 1

∑

ðB:1Þ

The satisfaction of (21)–(25) for fQ θ ; Y θ ; Gθ ; Z θ ; Γ θ g yields 2 T K θ Q t;t θ K θ r Γ θ ; ðΓ θ Þss r u s ; s A m

ðB:2Þ

p T T Ψ ðAp þ Bp K θ ÞQ 1;1 θ ðA þ Bp K θ Þ Ψ rZ θ ; p A D

ðB:3Þ

Q 1;t θ Q t;t θ

3 5½Ap Bp K θ T Ψ T r Z θ

ðB:4Þ

ðB:5Þ

^ l Þ ¼ K θ xðj ^ l Þ will keep Eqs. (A.1)–(A.4) mean that the control law uðj the augmented state within εθ and drive it into εθ þ 1 , with input ^ l Þ ¼ K Nμ xðj ^ lÞ and state constraints (15) satisfied. The control law uðj will keep the augmented state within εNμ and drive it to the origin. This completes the proof. □

! βN1

References

Buðjl β þ Njjl Þ :

where xðjℓ Þ þ

t A D3 ; p A DðtÞ:

ðZ θ Þss r ψ 2s ; s A q; t A f2; 3; …; d1 þ d2 1gp A DðtÞ:

Second, we calculate xðjl 1 jjl Þ; xðjl 2 jjl Þ; ⋯xðjl ϖ jjl Þ in Zðjl Þ. Denote 0 ^ ℓ Þ to ℓ≔l 1 and β ðjℓ Þ≔jℓ kjℓ þ 1. Similar to (A.1), we take xðj construct all the possible values of system state xðjℓ jjl Þ, 8 ^ Þ xðj βðjℓ Þ ¼ 1 > > > ℓ > < Axðj ^ ℓ Þ þ Buðjℓ 1Þ βðjℓ Þ ¼ 2 xðjℓ jjl Þ ¼ ⋮ > > > > : X ðj Þ βðjℓ Þ ¼ d1 ℓ

X ðjℓ Þ ¼ A

T

Q 1;1 θ Ψ ½A Bp K θ 4 t;1 Qθ

ϑ A D2 ; β ¼ 1 ϑ A D2 ; β A f2; 3; …; d1 g

N¼1

βðjℓ Þ 1 ^

Q θ 1 Φpt Q θ 1 Φpt Z0;

p

where β1 ^

For zðısl jjl Þ A εθ , since fQ θ ; Y θ ; Gθ ; Z θ ; Γ θ g satisfy (20)–(25), K θ is feasible and stabilizing. The satisfaction of (21) for fQ θ ; Y θ ; Gθ g ensures that

2

By denoting h iT A ¼ ðAϑ 1 ÞT ðAϑ 2 ÞT ⋯ AT I h iT B ¼ ðBϑ 1 ÞT ðBϑ 2 ÞT ⋯ BT 0

9

A

βðjℓ Þ 1 N

N¼1

Bu jl

l

∑ 0

s ¼ l 1

!

βðjs Þ þ Njjl :

By denoting h iT Aðjℓ Þ ¼ ðAjl0 jℓ 1 ÞT ðAjl0 jℓ 2 ÞT ⋯ AT I 2 3 B AB ⋯ Ajl0 jℓ 2 B 6 jl0 jℓ 3 7 60 B ⋯ A B7 6 7 7 Bðjℓ Þ ¼ 6 ⋮ ⋱ ⋱ ⋮ 6 7 6 7 40 ⋯ 0 5 B 0 0 0 0 h iT Uðjℓ Þ ¼ uðjl0 2jjl ÞT ⋯ uðjℓ þ 1jjl ÞT uðjℓ jjl ÞT 0

xðjℓ jjl Þ for all l A fl; l 1; …; l ϖ þ 1g can be calculated by ( ^ ℓ Þ þ Bðjℓ ÞUðjℓ Þ; βðjℓ Þ ¼ 1; Aðjℓ Þxðj xðjℓ jjl Þ ¼ Aðjℓ ÞX ðjℓ Þ þ Bðjℓ ÞUðjℓ Þ; βðjℓ Þ A f2; 3; …; d1 g

ðA:3Þ

^ l h ch Þ, h, ch A f0; 1; …; d1 þ d2 2g, ch Z ch 1 . where uðjl hjjl Þ ¼ uðj The ranges of h and ch are obtained due to Definitions 1 and 2. Note that ∑ls ¼ l ϖ βðjs Þ r d1 þ d2 1 in (A.3). Finally, according to the above derivations, Zðjl Þ contains all the possible values of zðısl jjl Þ. Therefore, the conclusion holds. Appendix B. Proof of Theorem 3 This mimics the procedure in [29]. Accordingly, Algorithm 1 is recursive feasible.

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Variable sampling approach to mitigate instability in networked control systems with delays.

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Distributed model predictive control for constrained nonlinear systems with decoupled local dynamics.

Comprehensive control of networked control systems with multistep delay.

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Aperiodic linear networked control considering variable channel delays: application to robots coordination.

Adaptive sampling rate control for networked systems based on statistical characteristics of packet disordering.

An Improved Formulation of Hybrid Model Predictive Control With Application to Production-Inventory Systems.

Computation and Communication Evaluation of an Authentication Mechanism for Time-Triggered Networked Control Systems.

$H_\infty $ Control for 2-D Fuzzy Systems With Interval Time-Varying Delays and Missing Measurements.

Consensus of Euler-Lagrange systems networked by sampled-data information with probabilistic time delays.

Robust model predictive control for optimal continuous drug administration.

Choosing the appropriate forecasting model for predictive parameter control.

A systems approach to air pollution control.

A systems model of blood glucose control.

Tracking control for nonaffine systems: a self-organizing approximation approach.