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

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 $$H_{\mathrm{\infty }}$$ Control ^[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: $${\mathbb{R}}^n$$ denotes the real $$n$$-dimensional space. $$|x|=(\sum _i{x}_i^2{)}^{\frac{1}{2}}$$ is the distance norm of vector $$x$$. The partial differentiation of the variable is $$\frac{\partial y}{\partial x}$$. $$X^T$$ represents the transposition of a vector or matrix. $$B(x_0,r)=\{x\in {\mathbb{R}}^n:|x-x_0|\le r\}$$ stands for a ball with $$x_0$$ as the center and $$r$$ as radius.

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 $$m_c$$ and $$m_b$$, respectively, with displacements of $$q_1$$ and $$q_2$$. The hydraulic controller between the wheel and the body is $$u$$, and the linear spring and nonlinear damper are in parallel, where the spring coefficient is $$K_a$$ and the damping coefficient is $$C_a$$. The wheel is regarded as a linear spring with a spring coefficient of $$K_t$$.

Figure 1. Quarter active suspension model.

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

(1) $$\begin{array}{l}{m}_c{\ddot{q}}_1-K_a(q_2-q_1)+K_t{q}_1=C_a({\dot{q}}_2-{\dot{q}}_1)-u-m_c{\xi }_1,\\ m_b{\ddot{q}}_2+K_a(q_2-q_1)=-C_a({\dot{q}}_2-{\dot{q}}_1)+u-m_b{\xi }_2.\end{array}$$

Let $$x_1=m_c{q}_1+m_b{q}_2$$, $$x_2={\dot{x}}_1$$, $$x_3=q_2-q_1$$, $$x_4={\dot{x}}_3$$, note that $$C_a$$ is an unknown coefficient, denoted as $$\theta =C_a$$. The impact of rough road surface on the wheels and body is considered as a stationary process $${\xi }_1$$ and $${\xi }_2$$, respectively.

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

(2) $$\begin{array}{rl}{\dot{x}}_1=& x_2,\\ {\dot{x}}_2=& -\frac{K_t}{m_c+m_b}{x}_1+\frac{K_t{m}_b}{m_c+m_b}{x}_3-m_c{\xi }_1-m_b{\xi }_2,\\ {\dot{x}}_3=& x_4,\\ {\dot{x}}_4=& f_4+\varphi \theta +(\frac{1}{m_c}+\frac{1}{m_b})u-{\xi }_2+{\xi }_1,\end{array}$$

where $$f_4=-K_a\frac{m_b+m_c}{m_b{m}_c}{x}_3+K_t\frac{(x_1-m_b{x}_3)}{m_c(m_b+m_c)}$$, $$\varphi =-\frac{m_b+m_c}{m_b{m}_c}{x}_4$$.

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 $$k_i>0$$, which means that $${\dot{\alpha }}_i^c=k_i({\eta }_{i2}-{\eta }_{i1})$$ can be served as a substitute of $${\dot{\alpha }}_i$$. Referring to coordinate transformation, the tracking error can be rewritten as

where $${\alpha }_0^c=0$$, $${\mathrm{\Delta }}_i={\alpha }_i-{\alpha }_i^c$$, and $${\mathrm{\Delta }}_0=0$$ are specified for the convenience of subsequent discussions. In order to deal with unknown items, we define estimation errors based on I&I as

where $${\rho }_i(x_i)$$ is a smooth function that will be designed later.

Step 1. From (3), (7), and (8), the dynamic of $$z_1$$-subsystem is expressed as

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

Select the fictitious control and adaptive law as

where $${\rho }_1=\frac{{\lambda }_1}{d_o}{\int }_0^{x_1}{\varphi }_1(s)\mathrm{d}s$$, and $${\delta }_1,c_1,d_o,{\lambda }_1>0$$. Substituting (9) into the derivative of the Lyapunov function $$V_1=\frac{1}{2}{z}_1^2+\frac{1}{2{\lambda }_1}{\tilde{\theta }}_1^2$$ yields:

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

Step i.$$(i=2,\cdots ,n-1)$$ Following the above design process, we can get the virtual control and adaptive law

where $${\rho }_i=\frac{{\lambda }_i}{d_o}{\int }_0^{x_i}{\varphi }_i(s)\mathrm{d}s$$ and $${\delta }_i,c_i,{\lambda }_i>0$$. Together with (12) and Lyapunov function $$V_i=V_{i-1}+\frac{1}{2}{z}_i^2+\frac{1}{2{\lambda }_i}{\tilde{\theta }}_i^2$$, similar to (10), the dynamic of $${\dot{V}}_i$$ can be formulated as

Step n. Construct the Lyapunov function

where $${\lambda }_n>0$$, whose derivative can be organized as

Naturally, the adaptive law is given as

where $${\rho }_n=\frac{{\lambda }_n}{d_o}{\int }_0^{x_n}{\varphi }_n(s)\mathrm{d}s$$ and $${\delta }_n>0$$. Drawing on the idea in ^[10,42], the following DETM is given

where $$\nu (t)$$ is the control auxiliary function to be given later, and $$\beta (t)$$ is an internal dynamic variable. $$h(t)=\nu (t)-u(t)$$ is measurement errors between the sampling control and current control, the parameters $$ϵ,b>0$$. Once the mechanism (17) is triggered, the control is updated to $$\nu (t_{k+1})$$; otherwise, it will remain at a constant value $$\nu (t_k)$$.

With the aid of (17), it can come to a conclusion that $$|\nu (t)-u(t)|\le \beta (t)+ϵ$$ is always held on intervals $$[t_k,t_{k+1})$$, it is easy to find two continuous functions $$\omega (t)\in [-1,1]$$ to satisfy

where $$\omega (t_k)=0,\omega (t_{k+1})=\pm 1$$. The auxiliary control input function is selected as

where $${\alpha }_n=-c_n{z}_n-z_{n-1}-f_n-$$$${\varphi }_n({\hat{\theta }}_n+{\rho }_n)+{\dot{\alpha }}_{n-1}^c$$, parameters $$c_n,\sigma >0$$ and $$\overline{ϵ}>ϵ$$. Replacing (16)-(19) in (15) yields

note that the item based on Lemma 2 satisfies

and $$|z_n(\beta (t)+\overline{ϵ})|>|z_n\omega (t)(\beta (t)+ϵ)|$$. Following the calculation process in (11) and combining (13), the dynamic of $$V_{z\theta }$$ turns to

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 $$\beta (t)$$, triggers will not infinitely activate at $$u=0$$.

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 $$i=1,\cdots ,n-1$$, define the transform

Denote $$e=[e_1^T,\cdots ,e_{n-1}^T{]}^T$$, and $$e_i=[e_{i1},e_{i2}{]}^T$$. From (6) and (21), we have

Unfolding $${\dot{\alpha }}_i$$ gives

where $$z_0=0$$, $${\dot{z}}_i=-c_i{z}_i-z_{i-1}+z_{i+1}+e_{i1}+g_i{\xi }_i-{\varphi }_i{\tilde{\theta }}_i$$, $${\dot{x}}_j=-c_j{z}_j-z_{j-1}+z_{j+1}+e_{j1}$$$$-{\varphi }_j{\tilde{\theta }}_j+g_j{\xi }_j+$$$$k_{j-1}(e_{j-1,2}-e_{j-1,1})$$, $${\dot{\hat{\theta }}}_i=\frac{{\lambda }_i}{d_o}{\varphi }_i(f_i+x_{i+1}$$$$+{\varphi }_i({\hat{\theta }}_i+{\rho }_i)-{\lambda }_i{\delta }_i({\hat{\theta }}_i+{\rho }_i)$$, $${\dot{\rho }}_i=\frac{{\lambda }_i}{d_o}{\varphi }_i(x_i){\dot{x}}_i$$, and

For the $$e$$-subsystem, select Lyapunov function $$V_e=\frac{1}{2}\sum _{i=1}^{n-1}{e}_i^T{e}_i$$, combining with (14), the following whole Lyapunov function is obtained:

which satisfies $$a_1|\chi {|}^2\le V(\chi )\le a_2|\chi {|}^2$$ with $$\chi =[z^T,e^T,\tilde{\theta }{]}^T$$, $$a_1=\frac{1}{2}min\{1,\frac{1}{{\lambda }_i}\}$$, and $$a_2=\frac{1}{2}max\{1,\frac{1}{{\lambda }_i}\}$$.

For any constant $$v_0>0$$, define $$r_0=\sqrt{\frac{v_0}{a_2}}$$, the initial value contained within a compact set $$B(0,r_0)=\{\chi :|\chi |\le r_0\}$$. Since $$f_i,{\varphi }_i,{\rho }_i$$ are smooth, then $$\frac{\partial f_i}{\partial x_j},\frac{\partial {\varphi }_i}{\partial x_j},\frac{\partial {\rho }_i}{\partial x_j}$$ are bounded in the ball $$B(0,r_0)$$, which, along with (23), implies that constants $$b_{ij}>0(j=1,\cdots ,4)$$ can always be found such that

where $$b_{ij}(j=1,\cdots ,4)$$ depends on $$r_0$$, and only $$b_{i1}$$ depends on $${\overline{k}}_{i-1}$$. With the aid of Young inequality, for any constant $$d_o>0$$, there holds

where $${\kappa }_i=d_o(b_{i1}^2+b_{i2}^2+b_{i3}^2+b_{i4}^2)$$, which results in

where $${\kappa }_i$$, depending on $${\overline{k}}_{i-1}$$, is independent of $$k_i$$. Substituting (20) and (25) into the derivative of (24) yields that for $$V\le v_0$$, one has

Let $$c_i=c_{i1}+c_{i2}$$, $$k_i=k_{i1}+k_{i2}$$, $${\delta }_i={\delta }_{i1}+{\delta }_{i2}$$, and define $$c=min\{2c_{i1},k_{i1},{\delta }_{i1}\}$$ and $$\overline{c}=2min\{c_{i2}-\frac{d_o}{2}-\frac{n-1}{d_o}-\frac{d_o{d}_g}{4},$$$$\frac{1}{2}{k}_{i2}-{\kappa }_i-\frac{n}{d_o},\frac{1}{2}{\delta }_{i2}-\frac{d_{\varphi }{d}_g}{4d_o}-\frac{n-1}{d_o}\}$$. Denote $${\zeta }_1=\frac{n+1}{d_o}$$ and $${\zeta }_2=\sum _{i=1}^n\frac{1}{2}{\delta }_i{\theta }_i^2+0.2785\sigma$$. To let $$c>0,\overline{c}\ge 0$$, requirements on parameters are

where $${\kappa }_i$$ depends on $$b_{ij}(j=1,\cdots ,5)$$, and $$b_{ij}$$ depends on $$v_0$$. When $$c>0$$ and $$\overline{c}\ge 0$$ are required, it is concluded from (26) that

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 $$z_1$$ can be made arbitrarily small by adjusting parameters;

(4). the inter-execution intervals $$(t_{k+1}-t_k)$$ have lower bounds.

Proof It is clear that the closed-loop system is SGNSpES-P along (24) and (28), which means that $$z$$, $$e$$, and $$\tilde{\theta }$$ are all bounded in probability. From (7) and (9), the signals $$x_1,{\alpha }_1,{\hat{\theta }}_1$$ are bounded in probability. Recalling $$x_i=z_i+{\alpha }_{i-1}+e_{i-1,1}$$ and (12), (16), the signals $$x_i,{\alpha }_i,{\hat{\theta }}_i$$ are got. Based on (17) and (19), the boundedness in probability of the final control $$u$$ is obtained.

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

for any $$\epsilon >0$$, where $$d_M=\frac{1}{c}({\zeta }_{1}M+{\zeta }_2)$$ can be adjusted arbitrarily for any $$\epsilon$$ by tuning $$d_o,c$$ large enough.

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 $$B(0,r_0)$$, thus, $$\dot{\nu }(t)\le \overline{\nu }$$ with $$\overline{\nu }>0$$ is established. In time interval $$[t_k,t_{k+1})$$ of (17), it can obtain that $$|h(t)|=|u(t)-\nu (t)|\le \beta (t)+ϵ\le \overline{h}$$. Further, the derivative of $$h^2(t)$$ becomes $$|(h^2(t){)}^{\prime }|=2|h(t)|\cdot |\dot{\nu }(t)|\le 2\overline{h}\overline{\nu }$$. When $$t\in [t_k,t_{k+1}]$$, it can be asserted that $$(u(t_k^{+})-\nu (t_k^{+}){)}^2=0$$, $$(u(t_{k+1}^{-})-\nu (t_{k+1}^{-}){)}^2\ge {ϵ}^2$$, therefore, $${ϵ}^2\le (u(t_{k+1}^{-})-v(t_{k+1}^{-}){)}^2-(u(t_k^{+})-v(t_k^{+}){)}^2=$$$${\int }_{t_k^{+}}^{t_{k+1}^{-}}$$$$|h(s{)}^2|\mathrm{d}s\le 2\overline{h}\overline{\nu }(t_{k+1}-t_k)$$, it can come to a conclusion that $$t_{k+1}-t_k\ge \frac{{ϵ}^2}{2\overline{h}\overline{\nu }}$$, which implies that the inter-execution intervals have lower bounds.