# Dynamic event-triggered practical stabilization of random suspension system based on immersion and invariance

*Complex Eng Syst*2023;3:17.

## Abstract

This article investigates the practical stabilization problem of random quarter-car active suspension systems. An adaptive dynamic event-trigger strategy is proposed to stabilize the states of vehicle suspension in response to system uncertainty and controller area network resource constraints. Moreover, the model of random active suspension systems is extended to the general random robot systems; the controller is developed with the aid of a double dynamic surface filter, immersion and invariance (I&I) techniques, and event-triggered mechanisms. The results show that the semi-global stability of error systems is achieved, and there are some improvements in triggering times and adaptive estimation performance under the control framework. Finally, simulation comparison results are provided to prove the advantages of the proposed scheme.

## Keywords

*,*immersion and invariance

*,*double dynamic surface filter

*,*dynamic event-triggered control

## 1. INTRODUCTION

With the rapid development of science and technology, vehicles have become a commonly used means of transportation. The suspension system is a force transmission connection device between the vehicle body and the wheels. Due to their ability to effectively alleviate impacts and body vibrations caused by uneven road surfaces to ensure ride comfort, the suspension systems have received great attention from researchers ^{[1,2]}. As a type of suspension system, active suspension systems (ASS) were often used to improve vehicle damping characteristics. Subsequently, some novel controllers were designed to achieve the expected performance of ASS, such as robust sampled-data ^{[3]}, adaptive fuzzy sliding mode control ^{[4]}, saturated control ^{[5]}, fault-tolerant control ^{[6,7]}, performance constraint control ^{[8–10]}. The adaptive finite time filtering control problem of a nonlinear quarter ASS subjected to actuator failure was studied in ^{[7]}. A new integral barrier Lyapunov function is proposed in ^{[10]}; on the one hand, it satisfies the vertical displacement constraint condition, and on the other hand, it stabilizes the suspension position within the neighborhood of the expected position in a finite time.

Due to the lack of accurate identification of model parameters and partial measurement of the system states, nonlinear adaptive control must cope with high levels of uncertainty. For ASS with uncertain parameters, adaptive schemes are proposed in ^{[8,11,12]}. As is well known, the two basic methods for dealing with nonlinearity are certain equivalence (CE) ^{[13]} and immersion and invariance (I&I) ^{[14]}. In principle, CE is to design Lyapunov functions for the error dynamic equation and obtain the update law of parameter estimation, which takes the form of an error nonlinear integrator, while I&I indirectly introduced unknown parameters into the estimation with the aid of state correction terms, which means incorporating system dynamics into lower order expected behavior to achieve control objectives ^{[15]}. Whereafter, the I&I technology has been verified in practical robot systems, such as quadrotors ^{[16–18]}, balls, beam systems ^{[19]}, and so on. In addition, as a typical nonlinear system, ASS inevitably suffers from the problem of explosion of terms caused by the analytic calculation on the command derivative of stabilizing function. To overcome this difficulty, the command derivative was approximated by a dynamic surface filter (DSF) ^{[20]}, and then a command filter with compensated dynamics was proposed by ^{[21]}, which revealed the relation of command filter control with traditional backstepping approaches. This technology is further applied to theoretical development ^{[22–24]} and practical systems ^{[25–27]}.

With the continuous deepening of research on network systems and electronic control components, saving limited bandwidth resources has become an important topic. Usually, it is required that the sensors, controllers, and actuators of the suspension system can continuously obtain information from each other. It is worth mentioning that event-triggered control (ETC) is an effective way to reduce the communication burden on the controller area network ^{[28]} proposed a new adaptive event-triggered tracking scheme that can not only offset severe uncertainty but also ensure any pre-set tracking accuracy. Nowadays, a number of results have been presented on ASS under the ETC condition ^{[29–31]}.

In addition, the event-triggered communication mechanism is widely used in multi-agent systems ^{[32–35]}. Only when the event-triggering conditions are satisfied the information of each agent will propagate to adjacent agents, greatly reducing the communication burden ^{[34]} proposed two different position controllers to handle the time-varying formation problem of multi-rotor systems based on an event-triggered integral sliding mode method. Further consideration still needs to be given to the collaborative control problem of unmanned aerial vehicle systems in the case of hybrid active interactions between humans ^{[36]}. This is a question worth exploring, once again discussing the suspension system.

In real life, the impact of rough roads on vehicles cannot be ignored. In order to improve passenger comfort, the suspension system must absorb road vibrations and prevent them from being transmitted to the vehicle body. Therefore, there are sufficient reasons to consider the control of random nonlinear ASS on rough road surfaces. The random model of ASS was given in ^{[37]}; however, it does not consider the issues of unknown parameters and reduced system signal transmission frequency. Motivated by the aforementioned papers, the dynamic event-triggered practical stabilization of uncertain random quarter-car ASS is devoted and even extended to general random nonlinear systems to deal with a class of robot control problems.

The main contributions of this paper are as follows: First, the application of I&I techniques and dynamic event-trigger mechanisms (DETM) to random nonlinear systems has achieved practical stabilization of random ASS.

Compared with ^{[11]}, it is not necessary to invoke CE and Lyapunov functions. I&I indirectly introduced unknown parameters into the estimation by state correction terms, avoiding the coupling between estimation law and error terms from the perspective of nonlinear regulation, which, to some extent, improves estimation performance.

Second, the double DSF proposed in this paper removes the compensation signals and achieves awesome properties calculated by double integration. Another advantage of DSF is that compared to ^{[38,39]}, it eliminates the boundedness assumption of prior filter errors and provides a reasonable stability analysis process.

The paper is divided into six parts. The first section is the introduction of relevant background knowledge of the research content. In the second section, the random suspension system and problem formulation are presented. The adaptive dynamic event-trigger controller is designed in the third section, and the performance analysis is discussed in the fourth section. The simulation results are provided in the fifth section, and the last section is a conclusion.

Notations:

## 2. MODEL DESCRIPTION AND PROBLEM SETUP

### 2.1. Random active suspension model

As presented in Figure 1, the quarter-car active suspension in a random environment is shown in this paper. The masses of the wheel and car body are

Borrowed from ^{[37]}, the random model of ASS by the aid of Lagrangian principles and relative motions is constructed as

Let

Then, the random model (1) with an unknown damping coefficient is shown as

where

### 2.2. Problem setup

In response to the problems of unknown parameters and high communication requirements in traditional robot control systems, the objective of this paper is to design an adaptive DETM for random suspension systems (2) to achieve semi-global practical stabilization. In order to solve a type of control problem similar to a random suspension system, system (2) is organized into the following general forms of random systems for controller design, that is

where

**Assumption 1** For any

**Assumption 2** There exist constants

The purpose of introducing (4) is to consider that the disturbance energy is bounded in practical situations ^{[40]}. In order to better select parameters in stability analysis, (5) was cited, and similar considerations were also used in coordinated control systems ^{[41]}. To facilitate the controller design, the following inequalities are presented.

**Lemma 1**^{[32]} For any vectors

**Lemma 2**^{[10]} For all

## 3. ADAPTIVE EVENT-TRIGGERED CONTROLLER DESIGN

Compared with traditional adaptive laws, adaptive techniques based on I&I do not require linear parameterization conditions or CE and can better compensate for parameter uncertainty. In order to avoid explosive terms generated by recursive processes, inspired by ^{[20,21]}, a novel double DSF is proposed:

with filter time constant

where

where

**Step 1.** From (3), (7), and (8), the dynamic of

In light of (3) and (8), one obtains

Select the fictitious control and adaptive law as

where

Which used Young inequality (see Lemma 1) and (5):

**Step i.**

where

**Step n.** Construct the Lyapunov function

where

Naturally, the adaptive law is given as

where ^{[10,42]}, the following DETM is given

where

With the aid of (17), it can come to a conclusion that

where

where

note that the item based on Lemma 2 satisfies

and

**Remark 1** The DETM (17) adjusts the update interval and control accuracy based on a control signal. Compared with the static event-triggered mechanism (SETM), the proposed controller has a dynamically adjusted trigger threshold and a shorter execution interval ^{[42]}. Due to the presence of the internal dynamic variable

## 4. PERFORMANCE ANALYSIS

The stability properties of the random nonlinear system (3) with adaptive DETM controllers (17) and DSF techniques are summarized in this section. Let us make some preparations for stability analysis. For

Denote

Unfolding

where

For the

which satisfies

For any constant

where

where

where

Let

where

The error system is summarized as follows

Based on the above argument, we intend to summarize the following results.

**Theorem 1** Consider the random ASS system described by (3), under Assumption 1 and Assumption 2, with fictitious control law and adaptive update law in (9), (12), (16), the event-triggering rule (17), satisfying parameter requirements (27), the closed-loop system (29) has the following performance:

(1). the system (29) is semi-globally noise to state practically stable in probability (SGNSpS-P);

(2). all signals in (29) are bounded in probability;

(3). the desired performance of the stabilization error

(4). the inter-execution intervals

**Proof** It is clear that the closed-loop system is SGNSpES-P along (24) and (28), which means that

Regarding (28), with the aid of Gronwall inequality, we arrive at

for any

In the following, it needs to be demonstrated that the designed trigger control can avoid the Zeno phenomenon. The derivative of (19) becomes

which is bounded in the compact set

**Remark 2** A command filter proposed in ^{[21]} is replaced with a double DSF (6) in this paper. This results in a simpler form of Lyapunov functions used in

## 5. SIMULATION

For the random ASS (1), following the previous DETM controller design with I&I in section 3, we can arrive at

where

the dynamic of (31), along with the design of controllers (30) and the handling of inequalities, leads to

where

**Theorem 2** For random suspension systems (1) with control law (30) satisfying parameter requirements (32), the closed-loop system is SGNSpS-P; the stabilization error

Following Theorem 1, the previous properties can be obtained directly; the specific proof process was omitted.

Next, the practical stabilization problem is simulated to demonstrate the merit of the obtained feedback controller (30). Choose system parameters as

Followed by section Ⅵ of ^{[40]}, the disturbances

In addition, the parameters in controller are

The continuous control

Below, we will conduct a simulation comparison between DETM and SETM under I&I and DSF. As shown in Figure 5, the relative threshold triggering strategy in ^{[43,44]} is demonstrated, that is

where

In order to illustrate the difference in adaptive effects between I&I and CE under DETM and DSF, following ^{[11,12]}, an adaptive law is designed as

where

## 6. CONCLUSION

In this work, an adaptive ETC method with double DSF has been presented for random quarter-car active suspension models. Compared with static event-trigger, the designed dynamic event-triggered strategy can effectively reduce communication burden and save network resources. Not only automotive suspension systems but also the research on practical stabilization problems of general random nonlinear systems have also been provided. More importantly, for general random nonlinear systems, tracking controllers can also be designed to achieve the tracking goals. Future work may include vehicle network control issues under network attacks or adaptive ETC issues for stochastic under-actuated systems. The safety issues of multi-agent under-actuated systems, such as unmanned aerial vehicles and surface vessels, are also worth further investigation.

## DECLARATIONS

### Authors’ contributions

Made significant contributions to the conception: Wu Z

Made significant contributions to the writing: Yang C

Made significant contributions to the revision: Feng L

### Availability of data and materials

Not applicable.

### Financial support and sponsorship

This work is supported by the National Science Natural Foundation of China under Grant (No. 62073075).

### Conflicts of interest

All authors declared that there are no conflicts of interest.

### Consent for publication

Not applicable.

### Ethical approval and consent to participate

Not applicable.

### Copyright

© The Author(s) 2023.

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## Cite This Article

## How to Cite

Yang, C.; Wu Z.; Feng L. Dynamic event-triggered practical stabilization of random suspension system based on immersion and invariance. *Complex Eng. Syst..* **2023**, *3*, 17. http://dx.doi.org/10.20517/ces.2023.25

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