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Abstract

This article focuses on the problem of optimal linear quadratic Gaussian control for networked control systems with multiple delays and packet dropouts. The main contributions are twofold. Firstly, based on the introduced maximum principle for linear quadratic Gaussian system with multiple input delays and packet dropouts, a nonhomogeneous relationship between the state and costate is obtained, which is the key technical tool to solve the problem. Secondly, a necessary and sufficient condition for the optimal networked control problem is given in virtue of the coupled Riccati equations, and the explicit expression of the optimal controller is presented. Numerical examples are shown to illustrate the proposed algorithm.

Keywords

Networked control systems LQG optimal control multiple delays packet dropouts

Introduction

Networked control systems (NCSs) are spatially distributed systems which transmit information between sensors, actuators, and controllers via a shared communication network.^1
–3 With the wide use of computer and the rapid development of network technology, NCSs are found in a wide range of applications in various fields, such as mobile sensor networks, building automation surveillance, and reduction of car wiring, which makes the control systems more complicated and has attracted considerable attentions.^4
–6 However, the situation of time delay and data loss often occurs in the process of communication. This motivated us to study the optimal control for NCSs with multiple delays and packet dropouts.

As for the optimal control for NCSs problem, many results have been surveyed.^7

–14 Cacace et al.⁷ studied the linear time invariant linear quadratic Gaussian (LQG) problems with a single input delay and instantaneous state feedback. Yang et al.¹⁰ presented a method for the compensation of a networked state-feedback control system, which possesses a randomly varying transmission delay and uncertain process parameters. Zhang et al.¹² studied the classic linear quadratic regulation (LQR) problem for both continuous-time and discrete-time systems with multiple input delays. Ma et al.¹³ designed a state-feedback controller for the closed-loop system with both discrete and distributed input time delays in terms of the solvability of certain Hamilton–Jacobi inequalities. And they¹⁴ studied the nonlinear Markovian jump systems subject to actuator failures and mixed time-delays and proposed the sufficient conditions for the existence of the desired controller. On the other hand, the problem on packet dropout occurred in NCSs is studied widely. Yu et al.¹⁵ obtained the sufficient conditions on the stability and stabilization of the NCSs with packet dropout. Gao et al.¹⁶ considered linear quadratic (LQ) optimal control problems based on state observer for a class of discrete-time system over lossy data network. Gupta et al.¹⁷ investigated the optimal LQG control by decomposing the system into a standard LQR state-feedback controller, applying the method of optimal encoder–decoder. Liang and Xu¹⁸ studied the packet dropout LQG control for NCSs with both local and remote controllers and gave the expression of the optimal controller. Moreover, the more applicability research on NCSs with both packet dropouts and time delay can be shown in the literature.^19
–21 Liang et al.¹⁹ obtained the necessary and sufficiency condition for the discrete stochastic LQG control system with multiplicative noises and input delay and presented the suboptimal controller when the separation principle did not hold. Zhang et al.²⁰ investigated the optimal results of LQR control system in both finite-horizon case and infinite-horizon case, where exist delay and multiplicative noise. What’s more, there are many references about the control of NCSs with multiple time delays and packet loss. Fischer et al.²² derived an optimal solution to the LQG control problem and proved that the separation principle holds, based on the network control system where control inputs and measurements are transmitted over transfer control protocol (TCP)-like network connections that are subject to random transmission delays and packet losses. Liang et al.²³ obtained the optimal LQR controller separately from the state estimator and derived a necessary and sufficient condition for the mean-square stabilization when there are both delays and packet dropouts in the NCSs.

However, there is no LQG control for NCSs with both multiple time delays and packet dropouts in the above literature. Inspired by the work of Liang and Xu¹⁸ and Liang et al.²³ we consider the NCSs with the assumption that one communication channel with large packet dropout probability, while the other communication channel with both small packet dropout probability and time delay. The case is shown in Figure 1, which is composed of a plant, a controller, and two actuators. In the closed loop, obviously, the state x_k is sent to the controller, and output as u_k . When the signal is transited to actuators over unreliable communication channel in the case of assumption, there will exist packet dropouts and time delays, that is, $γ_{k} (i) = 0$ signifies the packet is lost with the probability of packet dropout p_i , and $γ_{k} (i) = 1$ denotes the packet delivered successfully.

Figure 1.

Overview of NCSs with time delays and packet dropouts. NCS: networked control system.

Motivated by Li et al.,²⁴ we expend the results of optimal LQR control with multiplicative noises and multiple input delays into NCSs subject to multiple delays and packet dropouts. Moreover, we study the optimal LQG control of NCSs by adding the additive noise. Noting the approach of solving the forward and backward stochastic difference equations (FBSDEs),^19,20,24 this article studies the optimal LQG control problem for NCSs with both multiple time delays and packet dropout. Different from the literature,^13,14 the key technique of this article is to apply the maximum principle of multiple delays system to obtained the solution to FBSDEs, which is the nonhomogeneous relationship of the state and the costate. The main contributions of this article are summarized as follows: (1) A solution to the FBSDEs is obtained based on the derived maximum principle for NCSs with delays and packet loss. (2) In terms of the solution of the FBSDEs, a necessary and sufficient condition is given for the optimal LQG control.

Notation

Rⁿ denotes the n-dimensional real Euclidean space. I represents the unit matrix of appropriate dimension. The superscript $'$ denotes the transpose of the matrix. ${Ω, F, P, {F_{k}}_{k \geq 0}}$ denotes a complete probability space on which random variable $ν_{k}$ are defined such that ${F_{k}}_{k \geq 0}$ is the natural filtration generated by $ν_{k}$ , that is, $F_{k} = σ {ν_{0}, \dots, ν_{k}}$ , augmented by all the $P -null$ sets in $F$ . A symmetric $A > 0 (\geq 0)$ means that it is a positive definite (positive semi-definite) matrix. $Tr (A)$ represents the trace of matrix A.

Problem formulation

Consider the discrete stochastic LQG system with multiple input delays and packet dropouts, the corresponding plant in Figure 1 is given by

$x_{k + 1} = C x_{k} + γ_{k} (0) D_{0} u_{k} + γ_{k} (d) D_{d} u_{k - d} + ν_{k}$
1

for $k = 0, ..., N$ , where $x_{k} \in R^{n}$ is the state, $u_{k} \in R^{m}$ is the control input with a constant delay $d > 0$ , $γ_{k} (j)$ is an independent identically distributed Bernoulli random noise representing the packet dropouts with the probability of $p_{j} \in (0, 1)$ , that is,

$\begin{array}{l} P (γ_{k} (j) = 0) = p_{j}, with packet dropouts \\ P (γ_{k} (j) = 1) = 1 - p_{j}, without packet dropouts \end{array}$

And $ν_{k} \in R^{n}$ is a stochastic sequence, and $C, D_{0}$ , and D_d are deterministic matrices with compatible dimensions. Denote $w_{k} (j) = γ_{k} (j) - E [γ_{k} (j)] = γ_{k} (j) - (1 - p_{j})$ . Then the system (1) can be written as

$\begin{matrix} x_{k + 1} = C x_{k} + ((1 - p_{0}) + w_{k} (0)) D_{0} u_{k} \\ + ((1 - p_{d}) + w_{k} (d)) D_{d} u_{k - d} + ν_{k} \end{matrix}$
2

where $w_{k} (j)$ is a random variable. Besides, $w_{k} (j)$ and $ν_{k}$ are uncorrelated and satisfy

$\begin{array}{l} E [w_{k} (j)] = 0, E [w_{k} (j) w_{k} (j)'] = p_{j} (1 - p_{j}) \\ E [ν_{k}] = 0, E [ν_{k} ν_{k}^{'}] = Q_{ν_{k}}, E [w_{k} (j) ν_{k}] = 0 \\ E [w_{k} (j) w_{l} (j)'] = 0, E [ν_{k} ν_{l}^{'}] = 0, E [w_{k} (j) ν_{l}] = 0 \end{array}$

for $j = 0, d$ and $k \neq l,$ , with the initial values $x_{0}, u_{i}$ for $i = - d, \dots, - 1$ are known. The NCSs with delays and packet loss can be well described as $[C, D_{0}, D_{d}, p | d]$ conveniently.

The associated cost function for system (2) is given by

$J_{N} = E {\sum_{k = 0}^{N} (x_{k}' Q x_{k} + u_{k}' R u_{k}) + x_{N + 1}^{'} T_{N + 1} x_{N + 1}}$
3

where Q, R, and $T_{N + 1}$ are positive semi-definite constant matrices with appropriate dimensions, and N is the horizon length.

Problem 1

Find the unique $F_{k - 1}$ -measurable u_k , for $k = 0, \dots, N$ , to minimize the cost function (3) subject to equation (2).

Main results

By applying the Pontryagin’s maximum principle to the discrete LQG system (2) with the cost function (3) yields that

$ξ_{N} = T_{N + 1} x_{N + 1}$
4

$ξ_{k - 1} = E [C^{'} ξ_{k} | F_{k - 1}] + Q x_{k}$
5

$0 = E [D_{0}^{'} (k) ξ_{k} + D_{d}^{'} (k + d) ξ_{k + d} | F_{k - 1}] + R u_{k}$
6

where

$\begin{array}{l} D_{0} (k) = ((1 - p_{0}) + w_{k} (0)) D_{0} \\ D_{d} (k) = ((1 - p_{d}) + w_{k} (d)) D_{d} \end{array}$

for $k = 0, \dots, N$ . And $ξ_{k}$ is the costate with $ξ_{k} = 0$ for $k > N$ .

To make further study, we define the following coupled Riccati difference equations

$T_{k} = C^{'} T_{k + 1} C - L_{k}^{'} Ω_{k}^{- 1} L_{k} + Q$
7

where

$\begin{matrix} Ω_{k} = R + (1 - p_{0}) D_{0}^{'} T_{k + 1} D_{0} + (1 - p_{d}) D_{d}^{'} T_{k + d + 1} D_{d} \\ + (1 - p_{0}) (D_{0}^{'} T_{k + 1}^{d - 1} + (T_{k + 1}^{d - 1})' D_{0}) - \sum_{i = 1}^{d} (L_{k + i}^{d - i})' \\ \times Ω_{k + i}^{- 1} L_{k + i}^{d - i} \end{matrix}$
8

$L_{k} = (1 - p_{0}) D_{0}^{'} T_{k + 1} C + (T_{k + 1}^{d - 1})' C$
9

with

$L_{k}^{0} = (1 - p_{0}) (1 - p_{d}) D_{0}^{'} T_{k + 1} D_{d} + (1 - p_{d}) (T_{k + 1}^{d - 1})' D_{d}$
10

$\begin{matrix} L_{k}^{j} = (1 - p_{0}) D_{0}^{'} T_{k + 1}^{j - 1} + (1 - p_{d}) (T_{k + j + 1}^{d - j - 1})' D_{d} \\ - \sum_{i = 1}^{j} (L_{k + i}^{d - i})' Ω_{k + i}^{- 1} L_{k + i}^{j - i} \end{matrix}$
11

$T_{k}^{0} = (1 - p_{d}) C^{'} T_{k + 1} D_{d} - L_{k}^{'} Ω_{k}^{- 1} L_{k}^{0}$
12

$T_{k}^{j} = C^{'} T_{k + 1}^{j - 1} - L_{k}^{'} Ω_{k}^{- 1} L_{k}^{j}, j = 1, \dots, d - 1$
13

The terminal values are given by

$\begin{array}{l} T_{N + 1}, T_{N + i + 1} = 0, T_{N + i}^{j} = 0 \\ L_{N + i}^{j} = 0, Ω_{N + i} = I, i \geq 1, j = 0, \dots, d - 1 \end{array}$
14

It is stressed that the key to solve the optimal LQG control problem is to solve the FBSDEs (2) and (4)–(6). We now give the nonhomogeneous relationship of the optimal costate $ξ_{k - 1}$ and state x_k as follows.

Lemma 1

Supposing that $Ω_{k}$ are positive definite for $k = 0, \dots, N$ , the following equation

$ξ_{k - 1} = T_{k} x_{k} + \sum_{j = 0}^{d - 1} T_{k}^{j} u_{j + k - d}$
15

is the solution to FBSDEs (2) and (4)–(6). And $T_{k}$ , $T_{k}^{j}$ satisfy the coupled equations (12), (13) with the terminal value (14).

Proof

The proof of Lemma 1 is provided in Appendix 1.

Based on the maximum principle and the solution to the FBSDEs from Lemma 1, we can present the optimal control results of Problem 1.

Theorem 1

There exists a unique $F_{k - 1}$ -measurable u_k for Problem 2 if and only if $Ω_{k}, k = 0, \dots, N$ , are all invertible. In this context, the optimal controller u_k will be given by

$u_{k} = - Ω_{k}^{- 1} L_{k} x_{k} - Ω_{k}^{- 1} \sum_{j = 0}^{d - 1} L_{k}^{j} u_{j + k - d}$
16

The associated optimal performance index is given by

$\begin{matrix} J_{N}^{} = x_{0}^{'} T_{0} x_{0} + 2 x_{0}^{'} \sum_{j = 0}^{d - 1} T_{0}^{j} u_{j - d} + (1 - p_{d}) \sum_{j = 0}^{d - 1} u_{j - d}^{'} \\ \times (D_{d}^{'} T_{j + 1} D_{d}) u_{j - d} + 2 (1 - p_{d}) \sum_{j = 0}^{d - 1} \sum_{i = 0}^{d - 1} u_{j - d}^{'} D_{d}^{'} \\ \times T_{j + 1}^{i - j - 1} u_{i - d} - \sum_{j = 0}^{d - 1} \sum_{i = 0}^{d - 1} \sum_{m = 0}^{d - 1} u_{j - d}^{'} (L_{m}^{j - m})' Ω_{m}^{- 1} \\ \times L_{m}^{i - m} u_{i - d} + \sum_{k = 0}^{N} Tr [T_{k + 1} Q_{ν_{k}}] \end{matrix}$
17

where $Ω_{k}$ , L_k* , $L_{k}^{j}$ , $T_{k}$ , and $T_{k}^{j}$ satisfy the coupled equations (7) –(13) with the terminal value (14).

Proof

The proof of Theorem 1 is provided in Appendix 2.

Now we obtained the special case of NCSs with multiple delays and packet dropouts which is contained with only u_k and $u_{k - d}$ .

Remark 1

For stochastic NCSs with multiple delays and packet dropouts, the corresponding plant is given by

$x_{k + 1} = C x_{k} + \sum_{i = 0}^{d} γ_{k} (i) D_{i} u_{k - i} + ν_{k}$
18

for $k = 0, ..., N$ , with the initial values $x_{0}, u_{i}$ for $i = - d, \dots, - 1$ .

Besides, the Riccati equations (8), (11), and (13) will be as follows:

$\begin{matrix} Ω_{k} = R + \sum_{i = 0}^{d} (1 - p_{i}) D_{i}^{'} T_{k + i + 1} D_{i} + \sum_{i = 0}^{d - 1} (1 - p_{i}) (D_{i}^{'} \\ \times T_{k + i + 1}^{d - i - 1} + (T_{k + i + 1}^{d - i - 1})' D_{i}) - \sum_{i = 1}^{d} (L_{k + i}^{d - i})' Ω_{k + i}^{- 1} L_{k + i}^{d - i} \end{matrix}$
19

$\begin{matrix} L_{k}^{j} = \sum_{i = i}^{j} (1 - p_{i}) (1 - p_{i - j + d}) D_{i}^{'} T_{k + i + 1} D_{i - j + d} \\ + \sum_{i = 0}^{j - 1} (1 - p_{i}) D_{i}^{'} T_{k + i + 1}^{j - i - 1} + \sum_{i = 0}^{j} (1 - p_{i - j - d}) (T_{k + i + 1}^{d - i - 1})' \\ \times D_{i - j + d} - \sum_{i = 1}^{j} (L_{k + i}^{d - i})' Ω_{k + i}^{- 1} L_{k + i}^{j - i} \end{matrix}$
20

$T_{k}^{j} = (1 - p_{d - j}) C^{'} T_{k + 1} D_{d - j} + C^{'} T_{k + 1}^{j - 1} - L_{k}^{'} Ω_{k}^{- 1} L_{k}^{j}$
21

for $j = 0, \dots, d - 1$ , where the terminal value is as (14).

In this context, there exists a unique $F_{k - 1}$ -measurable u_k for system (18) subject to the cost function (3) if and only if $Ω_{k}, k = 0, \dots, N$ , are all invertible. The optimal controller u_k will be given by

$u_{k} = - Ω_{k}^{- 1} L_{k} x_{k} - Ω_{k}^{- 1} \sum_{j = 0}^{d - 1} L_{k}^{j} u_{j + k - d}$
22

with the associated optimal cost function

$\begin{matrix} J_{N}^{} = x_{0}^{'} T_{0} x_{0} + 2 x_{0}^{'} \sum_{j = 0}^{d - 1} T_{0}^{j} u_{j - d} + \sum_{i = 0}^{d - 1} \sum_{j = 0}^{d - 1} (1 - p_{j + d - i}) u_{i - d}^{'} \\ \times [D_{j + d - i}^{'} T_{j + 1} D_{j + d - i} + (D_{j + d - i}^{'} T_{j + 1}^{i - j - 1} + (T_{j + 1}^{i - j - 1})' \\ \times D_{j + d - i}) - (L_{j}^{i - j})' Ω_{j}^{- 1} L_{j}^{i - j}] u_{i - d} + \sum_{k = 0}^{d} Tr [T_{k + 1} Q_{ν_{k}}] \end{matrix}$
23

In addition, the relationship of the optimal costate $ξ_{k - 1}$ and state x_k* is as (15).

Numerical examples

Example 1

Consider the scalar case of LQG control system with multiple delays and packet dropouts (2), and the cost function (3). As the coefficients with

$\begin{array}{l} C = 1.1, D_{0} = 0.6, D_{d} = 0.7 \\ d = 1, p_{0} = p_{d} = 0.6, Q_{ν_{k}} = 1, N = 10, x_{0} = 0 \\ u_{- 1} = - 0.7, Q = 1, R = 1, T_{N + 1} = 0 \end{array}$

By applying Theorem 2 and equations (8) –(15), it yields that

$\begin{array}{l} T_{1} = 2.3087, T_{2} = 2.3086, T_{3} = 2.3085, T_{4} = 2.3080 \\ T_{5} = 2.3063, T_{6} = 2.3000, T_{7} = 2.2770, T_{8} = 2.1936 \\ T_{9} = 1.8897, T_{10} = 1 \end{array}$

$\begin{array}{l} Ω_{1} = 2.4061, Ω_{2} = 2.4060, Ω_{3} = 2.4056, Ω_{4} = 2.4041 \\ Ω_{5} = 2.3987, Ω_{6} = 2.3986, Ω_{7} = 2.3058, Ω_{8} = 2.0489 \\ Ω_{9} = 1.3600, Ω_{10} = 1 \end{array}$

$\begin{array}{l} L_{1} = 1.8901, L_{2} = 1.8900, L_{3} = 1.8896, L_{4} = 1.8879 \\ L_{5} = 1.8820, L_{6} = 1.8604, L_{7} = 1.7821, L_{8} = 1.4963 \\ L_{9} = 0.6600, L_{10} = 0 \end{array}$

Obviously, $Ω_{k} > 0$ , thus, there is an optimal solution to the NCSs with multiple delays and packet dropouts from Theorem 2. The optimal controller is shown in Figure 2.

Figure 2.
The optimal controller u_k .

Accordingly, the optimal value of equation (18) is $J_{N}^{} = 0.6788$ .

Example 2

Consider the LQG control system with multiple delays and packet dropouts (2) with $x_{k} \in R^{2}$ , $u_{k} \in R^{2}$ , and the cost function (3). As the coefficients with

$\begin{array}{l} C = [\begin{matrix} 1 & 2 \\ 2 & 1 \end{matrix}], D_{0} = [\begin{matrix} - 1 & 1 \\ 1 & 0 \end{matrix}], D_{d} = [\begin{matrix} 0 & 2 \\ 1 & 1 \end{matrix}] \\ x_{0} = [\begin{matrix} 1 \\ - 1 \end{matrix}], u_{- 1} = [\begin{matrix} - 1 \\ 0 \end{matrix}], Q_{ν_{k}} = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}], T_{N + 1} = 0 \\ d = 1, p_{0} = 0.8, p_{d} = 0.4, N = 3, Q = R = I \end{array}$

By applying Theorem 2 and equations (8) –(15), it yields that

$\begin{array}{l} T_{1} = [\begin{matrix} 12.02 & 9.89 \\ 9.96 & 13.67 \end{matrix}], T_{2} = [\begin{matrix} 5.92 & 3.95 \\ 3.95 & 5.84 \end{matrix}], T_{3} = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] \\ Ω_{1} = [\begin{matrix} 1.97 & 0.28 \\ 0.64 & 3.75 \end{matrix}], Ω_{2} = [\begin{matrix} 1.32 & - 0.16 \\ - 0.16 & 1.16 \end{matrix}], Ω_{3} = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] \\ L_{1} = [\begin{matrix} 2.76 & 2.59 \\ 11.16 & 10.96 \end{matrix}], L_{2} = [\begin{matrix} 0.20 & - 0.20 \\ 0.20 & 0.40 \end{matrix}], L_{3} = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}] \end{array}$

Obviously, for $i = 1, 2, 3$ , $Ω_{k} > 0$ , thus, there is an optimal solution to the NCSs with multiple delays and packet dropouts from Theorem 2. The optimal controller can be calculated as

$u_{0} = [\begin{matrix} 0.45 \\ 1.63 \end{matrix}], u_{1} = [\begin{matrix} - 0.04 \\ 0.32 \end{matrix}], u_{2} = [\begin{matrix} 0 \\ 0 \end{matrix}]$

Accordingly, the optimal value of equation (18) is $J_{N}^{} = 10.9390$ .

Conclusion

In this article, the optimal LQG control of NCSs with multiple delays and packet dropouts has been studied. A necessary and sufficient condition for the existence of unique optimal controller to the problem is given, which is based on the obtained maximum principle and the solution to the introduced coupled difference equations. Under this condition, the optimal controller and the minimized performance index are represented. In the future works, we expect that the results in this article pave new ways for the stabilization of the NCSs with multiple delays and packet dropouts. Moreover, the optimal output-feedback control for the NCSs with delays and packet dropouts will be studied in our further research.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (61903233), the “Special Funding for Top Talents” of Shandong Province (61773245, 61603068, 61806113, 61922051, 61933006, 91848206, 61733017, 61873179), Shandong Province Natural Science Foundation (ZR2018ZC0436, ZR2018PF011, ZR2018MF019), Key Research and Development Program of Shandong Province (2018GGX101053), and Taishan Scholarship Construction Engineering.

ORCID iDs

Qiyan Zhang

Chunyang Sheng

Zhiguo Zhang

Appendix 1

Appendix 2

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Optimal control for networked control systems with multiple delays and packet dropouts

Abstract

Keywords

Introduction

Notation

Problem formulation

Problem 1

Main results

Lemma 1

Proof

Theorem 1

Proof

Remark 1

Numerical examples

Example 1

Example 2

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

Appendix 1

Appendix 2

References