# Where Freshness Matters in the Control Loop: Mixed Age-of-Information and Event-Based Co-Design for Multi-Loop Networked Control Systems

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## Abstract

**:**

## 1. Introduction

#### 1.1. Contributions

#### 1.2. Related Works

#### 1.3. Outline

#### 1.4. Notations

## 2. NCS Model and Problem Description

#### 2.1. NCS Model

**Definition**

**1.**

**Remark**

**1.**

#### 2.2. Problem Description

## 3. NCS Design

#### 3.1. CE Control Law

**Assumption**

**1.**

**Theorem**

**1.**

**Proof.**

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

#### 3.2. Co-Design of Sampling and Scheduling Laws

## 4. Performance Analysis of the Joint Design

#### 4.1. AoI Sampling and Scheduling Co-Design

**Proposition**

**1.**

- each sub-system is scheduled a transmission once every N time-steps.
- stable and unstable sub-systems send, respectively, $N-{\lambda}^{i}$ and $N-{\lambda}^{j}$ fresh samples to the buffer during every N time-steps.
- if ${\lambda}^{i}={\lambda}^{j}=N-1$, then the AoI sampling is equivalent to the time-triggered sampling.

#### 4.2. ET Sampling and AoI Scheduling Co-Design

**Lemma**

**1.**

**Proof.**

#### 4.3. Performance Comparisons

**Definition**

**2.**

**Theorem**

**2.**

**Proof.**

**Remark**

**6.**

**Remark**

**7.**

**Corollary**

**1.**

**Proof.**

## 5. Numerical Evaluations

#### Sampling Strategies

- Event triggering: The sampler (sensor) samples the plant in each time-step, and if the value of the estimation error is greater than a threshold, then the sample is forwarded to the queue. The threshold is generated from an exponential distribution, and the mean of the distribution is chosen from the set $\{0,0.1,0.5,1,2:2:30\}$, where 2:2:30 are integer values in $[2,30]$ that are divisible by two. We use a default setting where each sampler uses the same mean threshold.
- Period-n sampling: Each sampler samples the plant periodically with period n.
- AoI sampling (n): Each sampler samples the plant whenever the AoI at the sampler exceeds n. The AoI at the sampler is equal to the AoI at the respective estimator from the previous time-step plus one. The default value for n is $N-1$, and for brevity we use AoI sampling to refer to this case.

#### Network Scheduling

- Highest AoI: Under this scheduling policy, the network chooses the plant that has a packet in the queue and the maximum AoI at the estimator, which is the “highest-age-first” prioritization.
- Randomized: The network scheduler chooses a packet from the queue in the uniform random fashion.
- Prioritizing unstable (PU) highest AoI: The scheduler selects the unstable plant which has a packet in the queue and has maximum AoI. If no packet belonging to an unstable plant exists in the queue, then the scheduler selects the stable plant with a packet in the queue with highest AoI.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

NCS | Networked Control System |

AoI | Age-of-Information |

ET | Event-Triggered |

QoS | Quality-of-Service |

QoC | Quality-of-Control |

LQG | Linear-Quadratic Gaussian |

CE | Certainty Equivalence |

## Appendix A. Proof of Theorem 1

## Appendix B. Proof of Theorem 2

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**Figure 2.**Sampling and scheduling patterns for an illustrative heterogeneous NCS of three Stable (SS1, SS2, SS3) and three Unstable Sub-systems (US1, US2, US3) with the AoI/AoI co-design architecture.

**Figure 3.**Average estimation error variance vs. the mean threshold under ET and highest AoI scheduling for different numbers of sub-systems N.

**Figure 5.**Comparison of estimation error variance for various sampling policies and PU highest AoI scheduling policy.

**Table 1.**Considered combinations of sampling/scheduling policies. The combinations designated with * are discussed either analytically or in simulations.

Scheduling | |||||
---|---|---|---|---|---|

ET | AoI | R | P | ||

Sampling | ET | * | * | ||

AoI | * | * | |||

R | |||||

P | * | * |

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## Share and Cite

**MDPI and ACS Style**

Mamduhi, M.H.; Champati, J.P.; Gross, J.; Johansson, K.H.
Where Freshness Matters in the Control Loop: Mixed Age-of-Information and Event-Based Co-Design for Multi-Loop Networked Control Systems. *J. Sens. Actuator Netw.* **2020**, *9*, 43.
https://doi.org/10.3390/jsan9030043

**AMA Style**

Mamduhi MH, Champati JP, Gross J, Johansson KH.
Where Freshness Matters in the Control Loop: Mixed Age-of-Information and Event-Based Co-Design for Multi-Loop Networked Control Systems. *Journal of Sensor and Actuator Networks*. 2020; 9(3):43.
https://doi.org/10.3390/jsan9030043

**Chicago/Turabian Style**

Mamduhi, Mohammad H., Jaya Prakash Champati, James Gross, and Karl H. Johansson.
2020. "Where Freshness Matters in the Control Loop: Mixed Age-of-Information and Event-Based Co-Design for Multi-Loop Networked Control Systems" *Journal of Sensor and Actuator Networks* 9, no. 3: 43.
https://doi.org/10.3390/jsan9030043