2. 火灾科学国家重点实验室 中国科学技术大学先进技术研究院合肥 230027;
3. 中国科学技术大学先进技术研究院 合肥 230027;
4. 中国科学院空间信息处理与应用系统技术重点实验室 北京 100190;
5. 安庆师范大学物理与电子信息学院 安庆 246011
2. State Key Laboratory of Fire Science, Department of Automation, Institute of Advanced Technology, University of Science and Technology of China, Hefei 230027, China, and also with the Key Laboratory of Technology in GeoSpatial Information Processing and Application System, Chinese Academy of Sciences, Beijing 100190, China;
3. Physics and Electronic Engineering, Anqing Normal University, Anqing 246011, China
Networked control systems (NCSs) are spatially distributed control systems, where controllers, sensors and actuators are connected via a limited bandwidth communication network. There are many advantages of NCSs, such as flexible architectures, low installation costs and high reliability. However, the insertion of the communication network will inevitably lead to communication delay, data packets dropout and disorder [1], which usually result in performance degradation or even instability of systems [2], [3].
Focusing on these issues, various approaches have been developed, for example, stochastic control approach [4], switched system approach [5], queuing approach [6], robust control approach [7] and so on. However, these methods are proposed based on some restrictive assumptions on NCSs which ignore the features of NCSs. In recent years, a new approach called the networked predictive control is proposed by taking full advantage of the specialities of NCSs, such as timestamp technique, packetbased transmission and caching technology [8], [9]. As shown in Fig. 1, the networked predictive control system mainly consists of control prediction generator (CPG) and network delay compensator (NDC), where a set of future control predictions are generated by CPG and the appropriate control value in the latest control prediction sequence is chosen to compensate the networkinduced issues actively by NDC. In addition, two data buffers are located at the controller and actuator to reorder and store the received data from networks. Based on the networked predictive control scheme, many results have been obtained for networked linear control systems. The design, analysis and implementation of the networked predictive control scheme have been widely studied [10][12]. The stability and robustness criteria have been derived when uncertainties exist in NCSs [13], [14].
Following the predictive control based approach of linear systems in [8], we aim to study the networked predictive control for bilinear systems in this paper. As a kind of important nonlinear system, the bilinear systems are applied widely in engineering, economics, biology and other areas [15]. Further, the bilinear systems can approach other nonlinear systems with higher precision than any other traditional linear systems [16]. Therefore, it is of both theoretical and practical interest to explore the networked predictive control of bilinear systems. Actually, the networked predictive control scheme has been extended to some nonlinear systems [17][21]. However, the analytical methods are restricted to some specific nonlinear systems, for instance, hammerstein systems [17] and wiener systems [18], which are not applicable to bilinear systems. The networked predictive control schemes for general nonlinear systems are proposed in [19][21]. However, the linearization theory of nonlinear systems adopted in [19] is always accompanied by strong conservatism. The system states and control variables in [20] are subject to some constraints, which will reduce the versatility of system. The nonlinear dynamical controller is given in [21] in advance, which is always difficult to be obtained in reality.
Different from most of the existing literatures which state the form of controller firstly, a constrained optimization problem consisted of system states and inputs is considered in this paper to obtain the future control inputs. However, the system predictive states will present a highly coupled form because of the nonlinearity inherent in bilinear model, which makes the optimization problem to be nonconvex. A general method for solving such an optimization problem is the sequential quadratic programming (SQP) [22], where a quadratic program is solved to determine the search direction and step size. However, if the approximation of SQP is poor, the search direction derived would also be poor. Another effective method is linearizing the nonlinear model about the input trajectory [23], [24]. However, the linearization method always impose strong constraints which subsequently leads to strong conservatism. Different from approximating the original problem to a quadratic program by SQP or linearization, the special structure of bilinear model will be fully exploited in this paper, and the original nonconvex optimization problem will be transformed into a quadratic program subproblem. Based on the above analysis, two algorithms, called the stepwise iterative (SI) algorithm and approximate forward stagewise (AFS) algorithm, will be proposed to calculate the predictive control sequence of the networked bilinear systems. Subsequently, the future inputs will be transmitted to the actuator via the forward channel.
The remainder of this paper is organized as follows: In Section 2, some assumptions and problem formulations are introduced. Further, the networked predictive control scheme is implemented and the related issues are converted to a nonconvex optimization problem. Two graduallyoptimized algorithms are given to solve the proposed optimization problem in Section 3. The numerical simulation is presented and the conclusions are summarized in Sections 4 and 5, respectively.
2 Design of Networked Predictive Control System for Networked Bilinear SystemsConsider the following discretetime nonlinear system
${x_{t + 1}} = A{x_t} + {u_t}N{x_t}$  (1) 
where
To simplify the presentation of the networked bilinear predictive control scheme, some assumptions are made as follows.
Assumption 1: The communication delay
Assumption 2: The number of consecutive data dropouts in the forward channel and feedback channel are upper bounded by
Assumption 3: The sensor, controller and actuator considered in the networked predictive control system (shown in Fig. 1) are timesynchronized.
Assumption 4: All the data packets sent from both the sensor and the controller are timestamped to notify when they are sent.
Remark 1: From Assumptions 1 and 2, we can know that the sum of consecutive data dropouts and communication delay in the forward channel and feedback channel are all upper bounded, i.e.,
Remark 2: In practical NCSs, the so called data dropout can be divided into three situations: 1) the packet fails to arrive at the destination in a certain transmission period, it will be treated as dropout based on the network protocols although it arrives in the future; 2) the "first sent later arrival", i.e., packet A is sent earlier than packet B, but it arrives later than packet B, then packet A would be treated as dropout; 3) the real data loss, i.e., the packet never reached. Additionally, the number of consecutive data dropouts should be finite in order to avoid the system becoming open loop. Thus, the Assumption 2 is reasonable.
Remark 3: From Assumptions 3 and 4, the networkinduced delays in the feedback channel and forward channel can be determined, respectively, according to the time when the data packets arrive at the controller and actuator.
Remark 4: In the networked predictive control scheme, when the data dropouts happen in the forward channel within the current sampling period, the
Based on the aforementioned assumptions, the following networked predictive control method for networked bilinear systems is proposed.
When the communication delay in the feedback channel is
$\begin{array}{l} {{\hat x}_{t  {k_t} + 1t  {k_t}}} = A{x_{t  {k_t}}} + {u_{t  {k_t}}}N{x_{t  {k_t}}}\\ \quad \quad \quad \quad = \left[ {A + {u_{t  {k_t}}}N} \right]{x_{t  {k_t}}}. \end{array}$  (2) 
Following by (2), the state predictions from time
$\begin{array}{l} {{\hat x}_{t  {k_t} + 2t  {k_t}}} = A{{\hat x}_{t  {k_t} + 1t  {k_t}}} + {u_{t  {k_t} + 1}}N{{\hat x}_{t  {k_t} + 1t  {k_t}}}\\ \quad \quad \quad \quad = \left[ {A + {u_{t  {k_t} + 1}}N} \right]{{\hat x}_{t  {k_t} + 1t  {k_t}}}\\ \quad \quad \quad \quad \quad \vdots \\ \quad \quad {{\hat x}_{tt  {k_t}}} = A{{\hat x}_{t  1t  {k_t}}} + {u_{t  1}}N{{\hat x}_{t  1t  {k_t}}}\\ \quad \quad \quad \quad \quad = [A + {u_{t  1}}N]{{\hat x}_{t  1t  {k_t}}} \end{array}$  (3) 
which results in
${\hat x_{tt  {k_t}}} = \prod\limits_{j = 1}^{{k_t}} {\left[ {A + {u_{t  j}}N} \right]} {x_{t  {k_t}}}.$  (4) 
Moreover, when the communication delay
$\begin{array}{l} {{\hat x}_{t + 1t  {k_t}}} = A{{\hat x}_{tt  {k_t}}} + {u_{tt  {k_t}}}N{{\hat x}_{tt  {k_t}}}\\ \quad \quad \quad = \left[ {A + {u_{tt  {k_t}}}N} \right]{{\hat x}_{tt  {k_t}}}\\ {{\hat x}_{t + 2t  {k_t}}} = A{{\hat x}_{t + 1t  {k_t}}} + {u_{t + 1t  {k_t}}}N{{\hat x}_{t + 1t  {k_t}}}\\ \quad \quad \quad = \left[ {A + {u_{t + 1t  {k_t}}}N} \right]{{\hat x}_{t + 1t  {k_t}}}\\ \quad \quad \quad \quad \vdots \\ {{\hat x}_{t + {i_t}t  {k_t}}} = A{{\hat x}_{t + {i_t}  1t  {k_t}}} + {u_{t + {i_t}  1t  {k_t}}}N{{\hat x}_{t + {i_t}  1t  {k_t}}}\\ \quad \quad \quad = \left[ {A + {u_{t + {i_t}  1t  {k_t}}}N} \right]{{\hat x}_{t + {i_t}  1t  {k_t}}}{\rm{ }} \end{array}$  (5) 
which results in
${\hat x_{t + {i_t}t  {k_t}}} = \prod\limits_{l = 1}^{{i_t}} {\left[ {A + {u_{t + {i_t}  lt  {k_t}}}N} \right]} {\hat x_{tt  {k_t}}}.$  (6) 
Combining with (4), the predictive state of time
${\hat x_{t + {i_t}t  {k_t}}} = \prod\limits_{l = 1}^{{i_t}} {\left[ {A + {u_{t + {i_t}  lt  {k_t}}}N} \right]} \prod\limits_{j = 1}^{{k_t}} {\left[ {A + {u_{t  j}}N} \right]} {x_{t  {k_t}}}.$  (7) 
Remark 5: From (7), it is clear that the predictive state of time
Considering the optimization problems depend on state vector and control vector
$\begin{align}\label{eq10} & \min\ J(k)=\overline{X}^TQ_x\overline{X}+\overline{U}^TR_u\overline{U}\notag\\ & \begin{cases} x_{t+1}=Ax_t+u_tNx_t &\\ \ u_{t+i}\ \le \theta_i,&i\in \{0,1,2,\ldots,\overline{M}1\}\\ \end{cases} \end{align}$  (8) 
where
Obviously, (8) is a nonconvex optimization problem because of the nonlinear form of system (1) and predictive state (7). Therefore, setting the derivative of performance index with respect to the control vector be zero directly did not obtain the desired control values. It is the main difficulty in dealing with networked predictive control of nonlinear system, which prompts us to explore new algorithms.
3 Two Algorithms for Solving the Nonconvex Optimization ProblemObserve the system predictive state (7), we can find that the order of each control input in the expression is up to one order. It means that the system state can be represented as the first order polynomial of one control input when the other inputs are known. This characteristic is decided by the special structure of bilinear system (1), which can be used as the breakthrough point to solve the nonconvex optimization problem (8).
Based on the above observation, two algorithms are proposed as follows. Algorithm 1, which is called the stepwise iterative algorithm, begins with a given initial sequence of control inputs. Only one control input is calculated every time and the results obtained previously will be used to calculate the latter ones. Substituting the obtained control inputs into performance index, and iterating the whole process until the variations of index meets the given threshold value
An initial sequence of control inputs is given firstly
${{\bar{U}}_{0}}={{\left[ {{u}_{tt{{k}_{t}}}},{{u}_{t+1t{{k}_{t}}}},\ldots ,{{u}_{t+\bar{M}t{{k}_{t}}}} \right]}^{T}}.$ 
Now, assume that the communication delay in the feedback channel is
Let
${{\bar{X}}_{\bar{N},\bar{M}}}={{\left[ \hat{x}_{t+1t{{k}_{t}}}^{T},\hat{x}_{t+2t{{k}_{t}}}^{T},\ldots ,\hat{x}_{t+\bar{M}+1t{{k}_{t}}}^{T} \right]}^{T}}.$  (9) 
According to (7), we have known that the state prediction
$\begin{align} &{{{\hat{x}}}_{t+{{i}_{t}}t{{k}_{t}}}}={{K}_{{{i}_{t}},i}}+{{V}_{{{i}_{t}},i}}{{u}_{t+it{{k}_{t}}}} \\ &\qquad \qquad \qquad \qquad \qquad {{i}_{t}}=1,2,\ldots ,\bar{M}+1 \\ \end{align}$  (10) 
where
When it is time to calculate
$\begin{align} &\bar{U}_{t+i1}^{\star }=\ [u_{tt{{k}_{t}}}^{\star },\ldots ,u_{t+i1t{{k}_{t}}}^{\star },{{u}_{t+it{{k}_{t}}}},\ldots , \\ &\quad \quad \quad {{u}_{t+{{i}_{t}}t{{k}_{t}}}},\ldots ,{{u}_{t+\bar{M}t{{k}_{t}}}}{{]}^{T}}. \\ \end{align}$ 
Further according to (7), the
$\begin{align} &{{{\hat{x}}}_{t+{{i}_{t}}t{{k}_{t}}}}=\prod\limits_{l=1}^{{{i}_{t}}i1}{\left[ A+{{u}_{t+{{i}_{t}}lt{{k}_{t}}}}N \right]}\left[ A+{{u}_{t+it{{k}_{t}}}}N \right] \\ &\ \quad \ \quad \ \quad \times \prod\limits_{m={{i}_{t}}i+1}^{{{i}_{t}}}{\left[ A+u_{t+{{i}_{t}}mt{{k}_{t}}}^{\star }N \right]} \\ &\ \quad \ \quad \ \quad \ \times \prod\limits_{n=1}^{{{k}_{t}}}{\left[ A+{{u}_{tn}}N \right]}{{x}_{t{{k}_{t}}}}. \\ \end{align}$  (11) 
Setting
$\begin{align} &{{K}_{{{i}_{t}},i}}=\ {{{\hat{x}}}_{t+{{i}_{t}}t{{k}_{t}}}}[{{u}_{t+it{{k}_{t}}}}=0] \\ &\quad \quad =\prod\limits_{l=1}^{{{i}_{t}}i1}{\left[ A+{{u}_{t+{{i}_{t}}lt{{k}_{t}}}}N \right]}A \\ &\ \quad \quad \quad \times \prod\limits_{m={{i}_{t}}i+1}^{{{i}_{t}}}{\left[ A+u_{t+{{i}_{t}}mt{{k}_{t}}}^{\star }N \right]} \\ &\ \quad \quad \quad \times \prod\limits_{n=1}^{{{k}_{t}}}{\left[ A+{{u}_{tn}}N \right]}{{x}_{t{{k}_{t}}}}. \\ \end{align}$  (12) 
And setting
$\begin{align} &{{V}_{{{i}_{t}},i}}={{{\hat{x}}}_{t+{{i}_{t}}t{{k}_{t}}}}[{{u}_{t+it{{k}_{t}}}}=1]{{K}_{{{i}_{t}},i}} \\ &\quad \quad =\prod\limits_{l=1}^{{{i}_{t}}i1}{\left[ A+{{u}_{t+{{i}_{t}}lt{{k}_{t}}}}N \right]}N \\ &\times \prod\limits_{m={{i}_{t}}i+1}^{{{i}_{t}}}{\left[ A+u_{t+{{i}_{t}}mt{{k}_{t}}}^{\star }N \right]}\prod\limits_{n=1}^{{{k}_{t}}}{\left[ A+{{u}_{tn}}N \right]}{{x}_{t{{k}_{t}}}}. \\ \end{align}$  (13) 
Remark 6: It is noticed that the
${{\hat{x}}_{t+{{i}_{t}}t{{k}_{t}}}}=\prod\limits_{l=1}^{{{i}_{t}}}{\left[ A+u_{t+{{i}_{t}}lt{{k}_{t}}}^{\star }N \right]}\prod\limits_{j=1}^{{{k}_{t}}}{\left[ A+{{u}_{tj}}N \right]}{{x}_{t{{k}_{t}}}}$ 
and then we hav
${{K}_{{{i}_{t}},i}}={{\hat{x}}_{t+{{i}_{t}}t{{k}_{t}}}},\ {{V}_{{{i}_{t}},i}}=0,\quad \quad {{i}_{t}}=1,2,\ldots ,i.$ 
Based on the above discussion, the system state vector
$\begin{align} &{{{\bar{X}}}_{\bar{N},\bar{M},i}}=\left[ \begin{matrix} {{K}_{1,i}}+{{V}_{1,i}}{{u}_{t+it{{k}_{t}}}} \\ \vdots \\ {{K}_{\bar{M}+1,i}}+{{V}_{\bar{M}+1,i}}{{u}_{t+it{{k}_{t}}}} \\ \end{matrix} \right] \\ &\quad \quad \quad ={{K}_{i}}+{{V}_{i}}{{u}_{t+it{{k}_{t}}}} \\ \end{align}$ 
where
${{K}_{i}}=\left[ \begin{matrix} {{K}_{1,i}} \\ \vdots \\ {{K}_{\bar{M}+1,i}} \\ \end{matrix} \right],\ {{V}_{i}}=\left[ \begin{matrix} {{V}_{1,i}} \\ \vdots \\ {{V}_{\bar{M}+1,i}} \\ \end{matrix} \right].$ 
Consider the following performance index
$\begin{align} &{{J}_{\bar{N},\bar{M},i}}=\ \bar{X}_{\bar{N},\bar{M},i}^{T}{{Q}_{\bar{N},\bar{M}}}{{{\bar{X}}}_{\bar{N},\bar{M},i}}+\bar{U}_{\bar{N},\bar{M},i}^{T}{{R}_{\bar{N},\bar{M}}}{{{\bar{U}}}_{\bar{N},\bar{M},i}} \\ &\quad \quad \quad =\ {{[{{K}_{i}}+{{V}_{i}}{{u}_{t+it{{k}_{t}}}}]}^{T}}{{Q}_{\bar{N},\bar{M}}}[{{K}_{i}}+{{V}_{i}}{{u}_{t+it{{k}_{t}}}}] \\ &\quad \quad \quad +\bar{U}_{\bar{N},\bar{M},i}^{T}{{R}_{\bar{N},\bar{M}}}{{{\bar{U}}}_{\bar{N},\bar{M},i}} \\ \end{align}$  (14) 
where
Now, let the partial derivative of
$u_{t+it{{k}_{t}}}^{\star }=K_{i}^{T}{{Q}_{\bar{N},\bar{M}}}{{V}_{i}}{{\left[ V_{i}^{T}{{Q}_{\bar{N},\bar{M}}}{{V}_{i}}+\sum\limits_{j=1}^{\bar{M}+1}{{{R}_{i,j}}} \right]}^{1}}.$  (15) 
Updating
$\begin{align} &\bar{U}_{t+i}^{\star }=\ [u_{tt{{k}_{t}}}^{\star },\ldots ,u_{t+i1t{{k}_{t}}}^{\star }, \\ &\quad \ \quad \ u_{t+it{{k}_{t}}}^{\star },{{u}_{t+i+1t{{k}_{t}}}},\ldots ,{{u}_{t+\bar{M}t{{k}_{t}}}}{{]}^{T}}. \\ \end{align}$ 
Let
$\begin{array}{*{35}{l}} \bar{U}_{t+\bar{M}}^{\star }=[u_{tt{{k}_{t}}}^{\star },&\ldots ,u_{t+{{i}_{t}}t{{k}_{t}}}^{\star },\ldots ,u_{t+\bar{M}t{{k}_{t}}}^{\star }{{]}^{T}}. \\ \end{array}$ 
After the predictive control sequence
$\begin{align} &\bar{U}_{t+\bar{M}{{i}_{t}}}^{\star } \\ &\quad ={{[u_{t{{i}_{t}}t{{i}_{t}}{{k}_{t}}}^{\star },\ldots ,u_{tt{{i}_{t}}{{k}_{t}}}^{\star },\ldots ,u_{t+\bar{M}{{i}_{t}}t{{i}_{t}}{{k}_{t}}}^{\star }]}^{T}}. \\ \end{align}$ 
To compensate for the communication delay
The detailed steps of Algorithm 1 are given in the following.
Algorithm 1 Stepwise iterative algorithm 
Step 1. Given an initial predictive control sequence 
Step 2. Calculate 
Step 3. Determine the communication delay 
Step 3.1 Calculate 
Step 3.2 Calculate 
Step 3.3 Calculate 
Step 4. Calculate 
Step 5. If 
Each control input in the predictive control sequence calculated by Algorithm 1 will make the performance index to be minimal, which leads the final optimal predictive control sequence to approximate the optimal solution with any small error after several iterations. However, it should be noted that Algorithm 1 will cost lots of computation time if the initial sequence
Remark 7: As the system state converge asymptotically to the equilibrium state over time, the control inputs should be reduced to avoid system oscillation. Therefore, before calculating a new control sequence for the next sampling period, some processing should be done to the initial sequence
Algorithm 1 usually requires many iterations, which result lots of computation and thus it is time consuming. For higher calculation efficiency, Algorithm 2 is proposed as follows.
For convenience, we consider that the forward communication delay is
$\bar{U}_{{{k}_{t}},{{i}_{t}}2}^{\star }={{\left[ u_{tt{{k}_{t}}}^{\star },u_{t+1t{{k}_{t}}}^{\star },\ldots ,u_{t+{{i}_{t}}2t{{k}_{t}}}^{\star } \right]}^{T}}.$ 
According to (6), we have
$\begin{align} &{{{\hat{x}}}_{t+{{i}_{t}}t{{k}_{t}}}} \\ &\quad =\left[ A+{{u}_{t+{{i}_{t}}1t{{k}_{t}}}}N \right]\prod\limits_{j=2}^{{{i}_{t}}}{\left[ A+u_{t+{{i}_{t}}jt{{k}_{t}}}^{\star }N \right]}{{{\hat{x}}}_{tt{{k}_{t}}}} \\ \end{align}$ 
where only
Further it is easy to find that
${{\hat{x}}_{t+{{i}_{t}}t{{k}_{t}}}}={{K}_{{{i}_{t}}}}+{{V}_{{{i}_{t}}}}{{u}_{t+{{i}_{t}}1t{{k}_{t}}}}$  (16) 
where
$\begin{align} &{{K}_{{{i}_{t}}}}={{{\hat{x}}}_{t+{{i}_{t}}t{{k}_{t}}}}[{{u}_{t+{{i}_{t}}1t{{k}_{t}}}}=0] \\ &\quad =A\prod\limits_{j=2}^{{{i}_{t}}}{\left[ A+u_{t+{{i}_{t}}jt{{k}_{t}}}^{\star }N \right]}{{{\hat{x}}}_{tt{{k}_{t}}}} \\ \end{align}$  (17) 
$\begin{align} &{{V}_{{{i}_{t}}}}={{{\hat{x}}}_{t+{{i}_{t}}t{{k}_{t}}}}[{{u}_{t+{{i}_{t}}1t{{k}_{t}}}}=1]{{K}_{{{i}_{t}}}} \\ &\quad =N\prod\limits_{j=2}^{{{i}_{t}}}{\left[ A+u_{t+{{i}_{t}}jt{{k}_{t}}}^{\star }N \right]}{{{\hat{x}}}_{tt{{k}_{t}}}}. \\ \end{align}$  (18) 
Obviously,
Let
$\begin{align} &{{{\bar{U}}}_{{{k}_{t}},{{i}_{t}}1}}={{\left[ {{u}_{tt{{k}_{t}}}},{{u}_{t+1t{{k}_{t}}}},\ldots ,{{u}_{t+{{i}_{t}}1t{{k}_{t}}}} \right]}^{T}} \\ &{{{\bar{X}}}_{{{k}_{t}},{{i}_{t}}}}={{\left[ \hat{x}_{t+1t{{k}_{t}}}^{T},\hat{x}_{t+2t{{k}_{t}}}^{T},\ldots ,\hat{x}_{t+{{i}_{t}}t{{k}_{t}}}^{T} \right]}^{T}} \\ \end{align}$ 
where
Based on the discussion above, we have
${{\bar{X}}_{{{k}_{t}},{{i}_{t}}}}={{\bar{K}}_{{{i}_{t}}}}+{{\bar{V}}_{{{i}_{t}}}}{{\bar{U}}_{{{k}_{t}},{{i}_{t}}1}}$  (19) 
where
${{\bar{K}}_{{{i}_{t}}}}=\left[ \begin{matrix} {{K}_{1}} \\ \vdots \\ {{K}_{{{i}_{t}}}} \\ \end{matrix} \right],\ \ {{\bar{V}}_{{{i}_{t}}}}=\left[ \begin{matrix} {{V}_{1}}&\ldots &0 \\ \vdots &\ddots &\vdots \\ 0&\ldots &{{V}_{{{i}_{t}}}} \\ \end{matrix} \right]$ 
Defining the following performance index
${{J}_{{{k}_{t}},{{i}_{t}}1}}=\bar{X}_{{{k}_{t}},{{i}_{t}}}^{T}{{Q}_{{{k}_{t}},{{i}_{t}}}}{{\bar{X}}_{{{k}_{t}},{{i}_{t}}}}+\bar{U}_{{{k}_{t}},{{i}_{t}}1}^{T}{{R}_{{{k}_{t}},{{i}_{t}}}}{{\bar{U}}_{{{k}_{t}},{{i}_{t}}1}}$ 
where
Let the partial derivative of
$\bar{U}_{{{k}_{t}},{{i}_{t}}1}^{\star T}=\bar{K}_{{{i}_{t}}}^{T}{{Q}_{{{k}_{t}},{{i}_{t}}}}{{\bar{V}}_{{{i}_{t}}}}{{\left[ \bar{V}_{{{i}_{t}}}^{T}{{Q}_{{{k}_{t}},{{i}_{t}}}}{{{\bar{V}}}_{{{i}_{t}}}}+{{R}_{{{k}_{t}},{{i}_{t}}}} \right]}^{1}}.$  (20) 
Remark 8: For the case
${{\hat{x}}_{t+1t{{k}_{t}}}}=[A+{{u}_{tt{{k}_{t}}}}N]{{\hat{x}}_{tt{{k}_{t}}}}$ 
and the performance index
${{J}_{{{k}_{t}},0}}=\hat{x}_{t+1t{{k}_{t}}}^{T}{{Q}_{{{k}_{t}},1}}{{\hat{x}}_{t+1t{{k}_{t}}}}+u_{tt{{k}_{t}}}^{T}{{R}_{{{k}_{t}},1}}{{u}_{tt{{k}_{t}}}}.$ 
Let the partial derivative of
Assume that the communication delay
$\bar{U}_{\bar{N},\bar{M}}^{\star }={{\left[ u_{tt\bar{N}}^{\star },u_{t+1t\bar{N}}^{\star },\ldots ,u_{t+\bar{M}1t\bar{N}}^{\star },u_{t+\bar{M}t\bar{N}}^{\star } \right]}^{T}}.$ 
Next, the predictive control inputs
Remark 9: After obtaining
The detailed steps of Algorithm 2 are given in the following.
Algorithm 2 Approximate forward stagewise algorithm 
Step 1. Calculate 
Step 2. For the forward communication delay 
Step 2.1 Add 
Step 2.2 Add 
Step 2.3 Extend matrixes 
Step 2.4 Calculate 
Step 2.5 Obtain 
Step 3. Take 
Analyzing the process of Algorithm 2, we can know that
In this section, an illustrative example is provided to verify the design scheme and algorithms developed in this paper.
Consider the following discretetime bilinear system
${{x}_{t+1}}=A{{x}_{t}}+B{{u}_{t}}+{{u}_{t}}N{{x}_{t}}$  (21) 
with the system matrixes
$\begin{align} &A=\left[ \begin{array}{*{35}{r}} 0.65&0.24 \\ 0.59&0.27 \\ \end{array} \right] \\ &B=\left[ \begin{array}{*{35}{r}} 0.43 \\ 0.35 \\ \end{array} \right] \\ &N=\left[ \begin{array}{*{35}{r}} 1.49&0.34 \\ 1.08&0.41 \\ \end{array} \right]. \\ \end{align}$ 
The initial state of the system is chosen as
To illustrate the effectiveness of the networked predictive control scheme proposed in this paper, we consider the following two cases.
Case Ⅰ: Local control (offline) method presented in [25]. Here, the linear statefeedback controller is designed as
Case Ⅱ: Networked predictive control scheme proposed in this paper. Applying the networked predictive control scheme proposed in Section Ⅱ and solving the predictive control sequence by using Algorithm 1 and Algorithm 2 presented in Section Ⅲ. The constant matrixes R and Q in performance index are unit matrixes with appropriate dimensions. The initial control inputs in
The system (21) is controlled by the two kinds of controllers stated above, respectively. The state curves of
The simulation results show that the predictive control sequences, calculated by Algorithm 1 and Algorithm 2, can guarantee the stability of the networked bilinear system while random time delays exist both in the forward channel and feedback channel. Meanwhile, the state curves under Algorithm 1 and Algorithm 2 have the similar trend with the local linear statefeedback controller [25], which shows the effectiveness of the proposed algorithms. Further according to Figs. 2 and 3, we can find that the system state under Algorithm 2 has the faster convergence speed and better control performance compared with Algorithm 1. In addition, the time spent on Algorithm 1 and Algorithm 2 are 0.159 s and 0.067 s, respectively, which shows that Algorithm 2 has lower computational complexity.
5 ConclusionIn this paper, the networked predictive control of bilinear systems is investigated. Two algorithms, called the stepwise iterative (SI) algorithm and approximate forward stagewise (AFS) algorithm, are proposed to solve a nonconvex optimization problem to obtain the predictive control sequence. The SI algorithm begins with a given initial sequence of control inputs, and only one control input is calculated every time. Keeping on the iterative process until the variations of performance index meets the given threshold value
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