# Weakly Hard Real-Time Model for Control Systems: A Survey

^{*}

## Abstract

**:**

## 1. Introduction

- The weakly hard real-time models presented in the literature and their applications;
- The existing approaches for scheduling tasks under weakly hard real-time constraints;
- The existing methods for analyzing the schedulability of weakly hard real-time systems;
- Recent work on control and scheduling co-design, focusing on the approaches that rely on the weakly hard real-time model.

## 2. Preliminaries

#### 2.1. System Model

**Definition**

**1**

- $(m,k)$-firm model,
- Skip-over model,
- Generalized weakly hard real-time model.

#### 2.2. Motivation and Industrial Applications

**Example**

**1**

## 3. Overview of Weakly Hard Real-Time System Models

#### 3.1. Task Model with $(m,k)$-Firm Deadlines

#### 3.2. Skip-Over Model

#### 3.3. Generalized Weakly Hard Real-Time System Model

**Definition**

**2**

**Definition**

**3**

**Definition**

**4**

**Definition**

**5**

**Definition**

**6**

**Definition**

**7**

#### 3.4. Research Problems

- Specifying the weakly hard temporal constraints;
- Analyzing the schedulability of the system under weakly hard constraints;
- Implementation of the algorithms for scheduling systems under weakly hard constraints.

## 4. Overview of Scheduling Approaches

#### 4.1. Scheduling Approaches for $(m,k)$-Firm System Model

#### 4.2. Scheduling Approaches for the Skip-Over System Model

- If there are no blue jobs in the system, red jobs are scheduled as soon as possible according to the EDF;
- If blue jobs are present in the system, red jobs are processed as late as possible and blue jobs are processed in the idle time of red jobs.

#### 4.3. Scheduling Approaches for Generalized Weakly Hard Real-Time System Model

## 5. Schedulability Analysis for Weakly Hard Real-Time Systems

#### 5.1. Schedulability Analysis for $(m,k)$-Firm System Model

**Theorem**

**1.**

#### 5.2. Schedulability Analysis for the Skip-Over System Model

**Lemma**

**1.**

**Lemma**

**2.**

#### 5.3. Schedulability Analysis for the Generalized Weakly Hard Real-Time System Model

- Check whether the system is schedulable under the typical behavior using classical analysis;
- Check if the given $\left(\genfrac{}{}{0pt}{}{n}{m}\right)$ constraint is satisfied in overload conditions.

## 6. Applications of the Weakly Hard Real-Time Model in Control Systems’ Design

- Stability analysis: determining timing constraints for control tasks that ensure control loop stability;
- Optimal control system design: leveraging the weakly hard real-time constraints to reduce system utilization in overload conditions while aiming to maximize control performance.

#### 6.1. Overview of Basic Concepts from Control Theory

**Definition**

**8**.

- Integral squared error, ISE: ${\int}_{0}^{t}{e}^{2}\left(t\right)dt$;
- Integral absolute error, IAE: ${\int}_{0}^{t}\left|e\left(t\right)\right|dt$;
- Integral time-weighted absolute error, ITAE: ${\int}_{0}^{t}t\left|e\left(t\right)\right|dt$.

#### 6.2. Review of the Applications of the Weakly Hard Real-Time Model in Control Systems

- ${T}_{i}^{+}$ - period value beyond which the control system output response becomes unacceptable;
- ${T}_{i}^{-}$ - period value that ensures that the utilization of the task ${\tau}_{i}$ will not exceed the maximum allowed value.

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AMR | Autonomous Mobile Robot |

BMS | Battery Management System |

BM | Bi-Modal |

BWP | Blue When Possible |

CAN | Control Area Network |

CPU | Central Processing Unit |

CSA | Class Selection Algorithm |

D-CSA | Dynamic Class Selection Algorithm |

DBP | Distance-Based Priority |

DMAC | Deadline-Miss-Aware Controller |

DP | Dynamic Priority |

DPS | Dual Priority Scheduling |

DWCS | Dynamic Window-Constrained Scheduling |

EDBP | Enhanced Distance Based Priority |

EDF | Earliest Deadline First |

EDL | Earliest Deadline as Late as Possible |

FP | Fixed Priority |

GDPA | Guaranteed Dynamic Priority Assignment |

GDPA-S | Guaranteed Dynamic Priority Assignment-Simplified |

GEBS | Global Emergency-Based Scheduling |

IAE | Integral Absolute Error |

IDBP | Integrated Distance-Based Priority |

ISE | Integral Squared Error |

ITAE | Integral Time-Weighted Absolute Error |

JCLS | Job-Class-Level Scheduler |

LED | Light-Emitting Diode |

LTI | Linear Time Invariant |

MAA | Meet Any Algorithm |

MILP | Mixed-Integer Linear Programming |

MRA | Meet Row Algorithm |

PDS | Probability of Deadline Satisfaction |

QoS | Quality of Service |

RLP | Red Tasks as Late as Possible |

RLP/T | Red Tasks as Late as Possible with Blue Acceptance Test |

RMS | Rate Monotonic Scheduling |

RTO | Red Tasks Only |

S-CSA | Static Class Selection Algorithm |

TBS | Total Bandwidth Server |

ToF | Time of Flight |

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**Figure 2.**System model with $(m,k)$-firm constraints. Modified from [5]. The streams are denoted as ${\tau}_{i}$, while the j-th request of the i-th stream is denoted as ${J}_{i}^{j}$.

**Figure 3.**Transition diagram for a task with $\left(\genfrac{}{}{0pt}{}{2}{3}\right)$ constraint. Modified from [5].

**Figure 4.**System model for the CSA. Modified from [57].

**Figure 5.**A comparison of EDF, IDBP, and S-CSA with different class numbers. Data taken from [57].

**Figure 6.**A comparison of PDS obtained by the GDPA, GDPA-S, DBP, and EDF approaches. Data taken from [68].

**Figure 7.**A comparison of job-skipping algorithms according to QoS metric. Data taken from [73].

**Figure 8.**A comparison of job-skipping algorithms with respect to the dynamic overhead. Data taken from [73].

**Figure 9.**A comparison of the number of dynamic failures for DWCS, EDF, and BM scheduling algorithms. Data taken from [81].

**Figure 10.**Mean rate of met deadlines obtained by the MRA, MAA, and EDF algorithms. Data taken from [82].

**Figure 11.**An illustration of control loop timing characteristics. (

**a**) Input-output latency and sampling period. (

**b**) Sampling and latency jitter.

**Figure 12.**An illustration of performance metrics analyzed through the step response: delay time ${t}_{d}$, rise time ${t}_{r}$, overshoot ${M}_{p}$, and settling time ${t}_{s}$.

${\mathit{\tau}}_{\mathit{i}}$ | ${\mathit{T}}_{\mathit{i}}$ = ${\mathit{D}}_{\mathit{i}}$ (ms) | ${\mathit{C}}_{\mathit{i}}$ (ms) | Functionality | Constraint |
---|---|---|---|---|

${\tau}_{1}$ | 10 | 2 | localization | hard real-time |

${\tau}_{2}$ | 10 | 3 | navigation | hard real-time |

${\tau}_{3}$ | 10 | 1 | obstacle detection | hard real-time |

${\tau}_{4}$ | 20 | 2 | battery management | weakly hard real-time: ${s}_{i}=5$ |

${\tau}_{5}$ | 10 | 2 | motor control | hard real-time |

${\tau}_{6}$ | 10 | 1 | user signalization | weakly hard real-time: $(1,5)$ |

${\tau}_{7}$ | 10 | 1 | obstacle avoidance | hard real-time |

Met Deadlines | Missed Deadlines | |
---|---|---|

Any order | $\left(\genfrac{}{}{0pt}{}{n}{m}\right)$ | $\overline{\left(\genfrac{}{}{0pt}{}{n}{m}\right)}$ |

Consecutive | $\u2329\genfrac{}{}{0.0pt}{}{n}{m}\u232a$ | $\overline{\u2329\genfrac{}{}{0.0pt}{}{n}{m}\u232a}\equiv \overline{\langle n\rangle}$ |

Algorithm | Constraint | Guaranteed/Best-Effort | FP/DP | Complexity | Advantages |
---|---|---|---|---|---|

DBP [5] | $\left(\genfrac{}{}{0pt}{}{n}{m}\right)$ | Best-effort | DP | $O\left(N{M}_{\mathrm{max}}\right)$ | Simple implementation |

DWCS [61] | $\left(\genfrac{}{}{0pt}{}{n}{m}\right)$ | Guaranteed for a special case of ${D}_{i}={T}_{i}$ and equal loss tolerance | DP | $O\left(N\right)$ | Considers both deadlines and loss tolerance, low computational cost |

DPS [64] | $\left(\genfrac{}{}{0pt}{}{n}{m}\right)$ | Guaranteed | FP | $O\left(N{M}_{\mathrm{max}}\right)$ | Considers general process model |

GDPA [68] | $\left(\genfrac{}{}{0pt}{}{n}{m}\right)$ | Guaranteed | DP | $O\left({N}^{2}\right)$ $O\left(N\right)$ for GDPA-S | Maximizes the QoS |

CSA [57] | $\left(\genfrac{}{}{0pt}{}{n}{m}\right)$ | Guaranteed | DP | $O\left(N{M}_{\mathrm{max}}\right)$ for class assignment $O\left(C{M}_{\mathrm{max}}\right)$ for scheduling, C is the number of classes | Achieves a trade-off between QoS granularity and scalability |

JCLS [16] | $\left(\genfrac{}{}{0pt}{}{n}{m}\right)$ | Guaranteed | FP | $O\left(1\right)$ if $\frac{{n}_{i}}{{m}_{i}}\ge 0.5$, otherwise exponentially depends on ${m}_{i}$ | Used for scheduling sporadic tasks applicable to scenarios with jitter |

GEBS [69] | $\left(\genfrac{}{}{0pt}{}{n}{m}\right)$ | Guaranteed | FP | $O(N{m}^{2}log\left(N{m}^{2}\right))$ where $m=max\left\{{m}_{i}\right|\forall {\tau}_{i}\in \mathcal{T}\}$ | Global priority allocation scheme, takes into account emergency degrees of all tasks |

BWP [47] | $\u2329\genfrac{}{}{0.0pt}{}{m-1}{m}\u232a$ | Guaranteed | DP | $O\left(N\right)$ | Simple implementation and low computational cost |

RLP [72] | $\u2329\genfrac{}{}{0.0pt}{}{m-1}{m}\u232a$ | Guaranteed | DP | Determined by the complexity of the EDL: $O({D}_{\mathrm{max}}/{T}_{\mathrm{min}}\xb7N)$ | Implements a mechanism for stimulating the execution of blue jobs |

RLP/T [75] | $\u2329\genfrac{}{}{0.0pt}{}{m-1}{m}\u232a$ | Guaranteed | DP | Schedulability test runs in $O({D}_{\mathrm{max}}/{T}_{\mathrm{min}}\xb7N+\mathcal{B}\left(t\right))$ | Provides an acceptance test for blue jobs |

BM [81] | $\left(\genfrac{}{}{0pt}{}{n}{m}\right)$, $\u2329\genfrac{}{}{0.0pt}{}{n}{m}\u232a$ | Guaranteed | FP in panic mode, normal mode can use both FP and DP | Depends on the priority assignment in normal and panic modes, $O\left({N}^{2}\right)$ for optimal priority assignment | Considers general process model |

MAA [82] | $\left(\genfrac{}{}{0pt}{}{n}{m}\right)$ | Guaranteed | DP | $O\left(N{M}_{\mathrm{max}}\right)$ | Combines a scheduling policy that guarantees the weakly hard constraint with an arbitrary scheduling policy |

MRA [83] | $\u2329\genfrac{}{}{0.0pt}{}{n}{m}\u232a$ | Guaranteed | DP | $O\left(N\right)$ |

Co-Design Approach | Objective | Target Application | Features |
---|---|---|---|

[101] | performance index minimization | generalized control problem, expanded to linear-quadratic control problem | uses the notation of accelerable tasks that minimize performance index with every invocation |

[103] | stability analysis for nonlinear control systems | applicable to a wide class of nonlinear control systems | weakly hard real-time constraints derived from sufficient condition for asymptotic stability |

[106] | optimal controller design | generalized control problem | the effects of several strategies for handling deadline misses are discussed |

[107] | stability analysis | generalized control problem | considers $\overline{\langle n\rangle}$ constraint and several strategies for handling deadline misses |

[13] | stabilization of control systems | linear discrete-time systems | considers both time-triggered stabilization and event-triggered stabilization |

[114,115] | stability analysis | networked control systems | stability conditions for arbitrary state matrices and weakly hard constraints |

[117] | performance index minimization | cyber-physical systems | scheduling algorithm that skips jobs in order to reduce system utilization |

[33] | optimal task parameter assignment that maximizes the worst-case control performance while guaranteeing stability | cyber-physical systems | task model that captures the relation between sampling period and weakly hard real-time constraints |

[15] | bounding consecutive deadline misses and optimizing performance | cyber-physical systems | jobs are classified based on previous number of deadline misses; considers $\overline{\langle n\rangle}$ constraint |

[119] | optimizing resource efficiency | cyber-physical systems | introduces an additional metric–resource efficiency and a dual-mode task model |

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**MDPI and ACS Style**

Salamun, K.; Pavić, I.; Džapo, H.; Čuljak, I.
Weakly Hard Real-Time Model for Control Systems: A Survey. *Sensors* **2023**, *23*, 4652.
https://doi.org/10.3390/s23104652

**AMA Style**

Salamun K, Pavić I, Džapo H, Čuljak I.
Weakly Hard Real-Time Model for Control Systems: A Survey. *Sensors*. 2023; 23(10):4652.
https://doi.org/10.3390/s23104652

**Chicago/Turabian Style**

Salamun, Karla, Ivan Pavić, Hrvoje Džapo, and Ivana Čuljak.
2023. "Weakly Hard Real-Time Model for Control Systems: A Survey" *Sensors* 23, no. 10: 4652.
https://doi.org/10.3390/s23104652