1 Introduction

Intelligent vehicles have attracted more and more attention due to their great advancements in enhancing vehicle safety and performance as well as traffic efficiency in the recent years [13, 14, 25, 43, 46, 51]. On the other hand, the electric vehicle driven by four in-wheels motors are easy to be intelligentized because of their features [7, 30, 31]. Therefore, the research about the intelligent electric vehicles driven by four in-wheel motors (IEV-DFIM) will attract more and more researchers [19, 20, 36]. Generally speaking, control methods are the keys of the IEV-DFIM.

The suspension system is very important for the riding comfort and maneuverability of the IEV-DFIM. For the IEV-DEIM, the in-wheel motor increases the unsprung mass that influences the riding comfort and maneuverability of the vehicle, and also causes the nonlinearity and uncertainty. In order to improve the riding comfort and maneuverability of the vehicle, the active suspension is necessary. For the active suspension, the control method is crucial, which decides the performance of the suspension directly. Many researchers have studied the issue widely, and presented some control methods via modern control theory. Now the active suspension control systems mainly adopt H \(\infty \) control method [3, 49, 53], explicit MPC-based RBF neural network method [6], sliding mode control method [24, 26, 34], rule-based control method [44], bioinspired dynamics-based adaptive tracking control method [35], robust and linear-parameter varying methods [33], neural network control method [22, 27], and adaptive fuzzy control method [21, 23], et. al. New control methods such as sampled-data control method [29] and homogeneous control method [28] were also developed for the active suspension.

The proposed control methods in the aforementioned literature need at least two sensors on each actuator to obtain the kinematic parameters of the active suspension. It means that the most active suspension control methods are state feedback control. But more sensors increase the cost of the active suspension which limits the application widely in practice. In order to reduce the cost, i.e. to reduce the number of sensors in the active suspension, it is necessary to design new control methods depending on less sensors while ensuring the riding comfort and maneuverability. This kind control method can be realized via output feedback control. Researchers have proposed some effective methods in this respect such as \(H_\infty \) output feedback control methods [45], sliding mode control method [8, 42, 56], and other new method [1, 12, 47]. It should be noted that the parameters of the active suspension models used in these control methods are dealt as constants for controlling easily. In fact, the sprung mass, stiffness coefficients and damping coefficients are time-varying.

Generally speaking, the most control methods of the active suspension are developed based on the linear growth condition. In engineering, the active suspension control system can not meet the linear growth condition strictly. In the decades years, a homogeneous domination approach is presented and developed rapidly which can solve this problem. The approach has strong robustness, and can handle the linear or nonlinear terms of control systems which do not require the restrict linear growth condition [10, 32, 37, 39, 41]. Some researchers have studied the homogeneous control method in theory. For example, an output feedback controller based on homogeneous domination was designed for upper-triangular systems via adding a power integrator method [55]. Furthermore, a new design method based on adding a power integrator and the interval homogeneous domination approach for a kind of nonlinear systems with time-varying powers was proposed in [5]. And a homogeneous domination idea and recursive design were used to design a continuous adaptive state-feedback controller to ensure the time-delay systems being globally stable [48]. A Lyapunov-based homogeneous controller was designed to stabilize a perturbed chain of integrators of arbitrary order [16]. Based on the aforementioned researches, the homogeneous control method was developed further. In paper [52], a generalized homogeneous adaptive stabilization method based on a general approximated homogeneous function restraining hypothesis was developed for a class of high order nonlinear systems without controllable/observable linearization. A state feedback control strategy is presented based on homogeneous domination for a kind of high-order uncertain nonlinear systems with multiple time-delays in [54]. In [4], a homogeneous state feedback controller with an adaptive strategy was constructed. A fixed-time synchronization control method based on homogeneous system theory was proposed for master-slave coupling systems in [50]. The paper [2] studied the second order practical consensus of homogeneous sampled-data multi-agent systems, and proposed a nonlinear emulation method based on homogeneity. In [11], a homogeneous version of Gronwall-Bellman inequality is introduced into the sampled-data control systems to solve the systems’ stabilization which is uncontrollable and unobservable around the origin.

But uncertain or nonlinear terms of the present methods must to be triangular and satisfy higher-order growth condition. Unfortunately, the active suspension dynamic model of our paper does not meet the aforementioned condition. Considering the aforementioned issues of the active suspension, in order to develop the homogeneous control method application for the active suspension of IEV-DFIM in engineering, we design a novel homogeneous output feedback controller to address this issues. Our contributions of this paper are following.

  • An active suspension model which considers the time-varying sprung and unsprung masses, stiffness coefficient and damping coefficient is built compared with other active suspension models which consider the sprung mass, stiffness coefficient and damping coefficient as constants.

  • A homogeneous observer is constructed to estimate the unmeasurable road profile that can help to improve the vehicle ride comfort.

  • A homogeneous domination method is developed to design an output feedback controller to solve the problem which the uncertain and nonlinear terms are not higher-order or linear growth in the active suspension control system.

The paper is organized as follows. The active suspension control system model based on the general active suspension dynamic model is given in Sect. 2. In Sect. 3, a novel active suspension control method based on homogeneous output feedback approach is proposed for IEV-DFIM. Simulation and test are carried out to verify the effectiveness of the designed homogeneous output feedback controller compared with the sliding mode controller and passive suspension in Sect. 4. And Conclusions are given in Sect. 5.

2 Active suspension control system modeling with time-varying coefficients

When an IEV-DFIM runs on road, the sprung mass \(m_s\) will transfer between the front axle and rear axle. And the unsprung mass (tyre mass) \(m_u\) will reduce because of wear. Therefore, the sprung mass \(m_s\) and unsprung mass \(m_u\) are time-varying. On the other hand, the actuator is also nonlinear, i.e., the stiffness coefficient \(k_s\) and damping coefficient of the actuator \(c_s\) are time-varying. Of course, the stiffness coefficient \(k_t\) and damping coefficient of the tyre \(c_t\) are also time-varying with tyre pressure and load. These time-varying parameters influence the performance of the active suspension greatly. But in the existing active suspension models, these parameters are considered as constants for controlling easily. In this section, we will build an active suspension dynamic model of an IEV-DFIM considering time-varying parameters. Then the active suspension control system is presented based on the dynamic model. In fact, the active suspension used in the IEV-DFIM is similar with the traditional vehicle. Therefore, we build the active suspension dynamic model referring the paper [28]. The dynamic model is shown in Fig. 1, and described as

Fig. 1
figure 1

The quarter-car active suspension model of an intelligent electric vehicles driven by four in-wheel motors

Full size image
$$\begin{aligned} \begin{aligned}&m_s(t)\ddot{x}_s(t)+k_s(t)(x_s(t)-x_u(t))\\&\qquad +c_s(t)({\dot{x}}_s(t)-{\dot{x}}_u(t))\\&\qquad +v(t)=0 \\&\qquad m_u(t)\ddot{x}_u(t)-k_s(t)(x_s(t)\\&\qquad -x_u(t))-c_s(t)({\dot{x}}_s(t)-{\dot{x}}_u(t)) \\&\qquad +k_t(x_u(t)-x_g(t))\\&\qquad +c_t({\dot{x}}_u(t)-{\dot{x}}_g(t))\\ {}&~~-v(t)=0, \end{aligned} \end{aligned}$$
(1)

where \(m_s(t)\) and \(m_u(t)\) are the sprung and unsprung mass respectively. \(x_s(t)\), \(x_u(t)\) and \(x_g(t)\) are the vertical displacements of the sprung mass, unsprung mass and vertical road profile respectively. \(k_s(t)\) and \(c_s(t)\) are the stiffness and damping coefficients of the suspension between the body and tyre respectively. \(k_t\) and \(c_t\) are the stiffness and damping coefficients of a tyre respectively.

The dynamic Eq. (1) can be rewritten as

$$\begin{aligned} m_s(t){\ddot{x}_s}(t)+m_u(t){\ddot{x}_u}(t)=&-k_t(t)(x_u(t)-x_g(t))\nonumber \\&-c_t(t)({\dot{x}}_u(t)-{\dot{x}}_g(t)). \end{aligned}$$
(2)

Defining

$$\begin{aligned} x_1&=m_s(t) x_s(t)+m_u(t) x_u(t), \\ x_2&=m_s(t){\dot{x}}_s(t)+m_u(t) {\dot{x}}_u(t),\\ x_3&=-k_t(t)(x_u(t)-x_g(t)), x_4=-k_t(t)({\dot{x}}_u(t)-{\dot{x}}_g(t)), \\ v&=m_u(t)\ddot{x}_u(t). \end{aligned}$$

The dynamic Eq. (2) is transferred to

$$\begin{aligned} \left\{ \begin{aligned}&{\dot{x}}_1=x_2 +\phi _1(t)\\&{\dot{x}}_2=x_3+\phi _2(t)\\&{\dot{x}}_3=x_4+\phi _3(t) \\&{\dot{x}}_4=u+\phi _4(t), \end{aligned} \right. \end{aligned}$$
(3)

where \(\phi _1(t)=0\), \(\phi _2(t)=\frac{c_t(t)}{k_t(t)}x_4\), \(\phi _3(t)=0\), \(\phi _4(t)=k_t(t)\ddot{x}_g(t)\), \(u=-\frac{k_t(t)}{m_u(t)}v\).

3 Homogeneous active suspension control method base on homogeneous output feedback approach

3.1 Problem statement

Before designing the homogeneous output feedback controller of the active suspension for the IEV-DFIM, we firstly give some important definitions and lemmas which have been proposed in literature.

Definition 1

[17]: Weighted Homogenity: For fixed coordinates \((x_1,x_2,\cdots , x_n) \in {\mathbb {R}}^n\) and real numbers \(r_i >0, i=1,2,\cdots ,n\), the dilation \(\varDelta _\varepsilon (x)\) is defined as \(\varDelta _\varepsilon (x)=(\varepsilon ^{r_1}x_1,\varepsilon ^{r_2}x_2,\cdots , \varepsilon ^{r_n}x_n), \forall \varepsilon >0\), in which \(r_i\) are called as the weights of the coordination. A function \(V \in C({\mathbb {R}}^n, {\mathbb {R}})\) is homogeneous of degree \(\tau \) where \(\tau \in {\mathbb {R}}\), \(\forall x \in {\mathbb {R}}^n \setminus {0}, \varepsilon >0, V(\varDelta _{\varepsilon }(x))=\varepsilon ^\tau V(x_1, x_2, \cdots , x_n)\). A vector field \(f \in C({\mathbb {R}}^n, {\mathbb {R}}^n)\) is homogeneous of degree of \(\tau \) where \(\tau \in {\mathbb {R}}\), \(\forall x \in {\mathbb {R}}^n \setminus {0}, \varepsilon >0, f_i(\varDelta _{\varepsilon }(x))=\varepsilon ^{\tau +r_i} f_i(x)\). And a homogeneous p-norm is defined as \(\Vert x\Vert _{\varDelta ,p}=\left( \sum _{i=1}^n|x_i|^{p/r_i}\right) ^{1/p}, \forall x \in {\mathbb {R}}^n\) where \(p \ge 1\).

Lemma 1

[17]: \(V_1\) and \(V_2\) are homogeneous functions of degree \(\tau _1\) and \(\tau _2\) with respect to a same dilation weight respectively. Then \(V_1\cdot V_2\) is also homogeneous with respect to the same dilation weight and the homogeneous degree is \(\tau _1+\tau _2\).

Lemma 2

[17]: \(V: {\mathbb {R}}^n \longrightarrow {\mathbb {R}}\) is a homogeneous function of degree \(\tau \) with respect to a given dilation weight \(\varDelta =(r_1,\cdots ,r_n)\). Then the following items hold.

  • \(\partial V / \partial x_i\) is homogeneous of degree \(\tau -r_i\) with respect to the dilation weight \(\varDelta \).

  • There is a positive constant \(c_1\) such that \(V(x) \le c_1 \Vert x\Vert ^\tau _\varDelta \).

  • If V(x) is positive definite, there is a positive constant \(c_2\) such that \(c_2\Vert x\Vert ^\tau _\varDelta \le V(x)\).

Lemma 3

[40]: When c and d are positive constants, the following inequality holds,

$$\begin{aligned} |x|^c|y|^d \le \frac{c}{c+d} \gamma |x|^{c+d}+\frac{d}{c+d}\gamma ^{-c/d}|y|^{c+d}. \end{aligned}$$
(4)

Lemma 4

[40]: For a positive odd integer \(p \ge 1\), the following inequality holds,

$$\begin{aligned} |x^p-y^p| \le p|x-y|(|x|^{p-1}+|y|^{p-1}). \end{aligned}$$
(5)

Lemma 5

[15]: For any \(x_i \in {\mathbb {R}},i=1,\cdots ,n\) and a real number \(p \ge 1\), the following inequalities hold,

$$\begin{aligned}&(|x_1|+\cdots +|x_n|)^p \le n^{p-1}(|x_1|^p+\cdots +|x_n|^p),\nonumber \\&\quad (|x_1|+\cdots +|x_n|)^{1/p} \le |x_1|^{1/p}+\cdots +|x_n|^{1/p}. \end{aligned}$$
(6)

The purpose of this paper is to design a homogeneous output feedback controller of the active suspension for the IEV-DFIM. For the purpose, a homogeneous state feedback controller will be firstly designed, and based on which a homogeneous output feedback controller will be designed. Firstly, we give a definition and an assumption for the designed controller.

Definition 2

Denote \(\tau =q/p\), where q is even and p is odd, then there is

$$\begin{aligned} r_i=(i-1)\tau +1, i=1,2,3,4 \end{aligned}$$
(7)

Assumption 1

There are constants \(\tau \ge 0\) and \(c \ge 0\) leading to

$$\begin{aligned} |\phi _i| \le c(|x_1|^{i\tau +1}+\cdots +|x_i|^{p_m}),i=1,2,3,4 \end{aligned}$$
(8)

where \(p_m=\frac{i\tau +1}{(i-1)\tau +1}\).

3.2 Design of the homogeneous state feedback controller

Next, we present a theorem to describe the main contribution of this paper.

Theorem 1

Based on the Assumption 1, the active suspension system (3) can be asymptotically stabilized by a homogeneous domination output feedback controller to improve the ride comfort and maneuverability of the IEV-DFIM.

$$\begin{aligned} u({\hat{x}})=-\beta _i \xi _4^{\frac{r_i+\tau }{r_i}},i=1, 2, 3, 4. \end{aligned}$$
(9)

where \(\beta _i\) is constant, \(\xi _4\) is an expression including the homogeneous state observer \({\hat{x}}\) which has the following form

$$\begin{aligned} \left\{ \begin{aligned}&{\hat{x}}_1=x_1\\&{\dot{\eta }}_j=f_{j} (x_1, \eta _2, \cdots , \eta _j)\\&{\hat{x}}_{j+1}=(\eta _{j}+k_{j} {\hat{x}}_{j})^{p_{j}}, k_{j}>0, p_{j}>0,\\ \end{aligned} \right. \end{aligned}$$
(10)

where \(j=1,2,3\).

Proof

We will firstly design an active suspension homogeneous state feedback controller according to Theorem 1.

$$\begin{aligned} u(x)=-\beta _4(x_4+\beta _3(x_3+\beta _2(x_2+\beta _1x_1^{p_1})^{p_2})^{p_3})^{p_4}, \end{aligned}$$
(11)

where \(\beta _i\) is constant, \(p_1=r_2/r_1,p_2=r_3/r_2,p_3=r_4/r_3,p_4=(r_4+\tau )/r_4\). \(\square \)

Next, for the following proving convenience, we construct a simple system (12) similar with the system (3), and prove that the controller (11) can stabilize the system (12).

$$\begin{aligned} \left\{ \begin{aligned}&{\dot{x}}_i=x_{i+1} \\&{\dot{x}}_4=u,\\ \end{aligned} \right. \end{aligned}$$
(12)

where \(i=1, 2, 3\), \(u:=x_{5} \in {\mathbb {R}}\) and \(y:=x_{1} \in {\mathbb {R}}\).

We select a Lyapunov function

$$\begin{aligned} V_1(x_1)=\frac{r_1}{2r_4-\tau } {x_1}^{\frac{2r_4-\tau }{r_1}}, \end{aligned}$$
(13)

The time derivative of the \(V_1\) along the controller (11) is

$$\begin{aligned} \begin{aligned} {\dot{V}}_1&={x_1}^{\frac{2r_4-\tau }{r_1}-1} x_2 \le {x_1}^{\frac{2r_4-\tau }{r_1}-1}(x_2+e{x_1}^{\frac{r2}{r1}})\\&~~ \le {x_1}^{\frac{2r_4-\tau }{r_1}-1}(x_2-{x_2}^{*}+{x_2}^{*}+e{x_1}^{\frac{r2}{r1}}), \end{aligned} \end{aligned}$$
(14)

where \(x^*_2\) is a virtual controller, and \( {x_2}^{*}=-\beta _1{x_1}^{r2/r1}\), \(\beta _1=4+e \). Then Eq. (14) is rewritten as

$$\begin{aligned} {\dot{V}}_1 \le -4{x_1}^{\frac{2r_4}{r_1}}+{x_1}^{\frac{2r_4-\tau }{r_1-1}}({x_2}-{x_2}^{*}). \end{aligned}$$
(15)

Suppose that there exists a positive definite and homogeneous \(C^1\) Lyapunov function \( V_{k-1}:{\mathbb {R}}^{k-1}\rightarrow {\mathbb {R}} \) with respect to Eq. (7) at step \(k-1,k=2,3,4\). And a set of \(C^1\) virtual controllers \( {x_1}^{*}, {x_2}^{*}, {x_3}^{*}, {x_4}^{*} \) are defined as

$$\begin{aligned} \begin{aligned}&{x_1}^{*}=0, \xi _1={x_1}-{x_1}^{*},\\&\quad {x_j}^{*}=-\beta _{j-1}{\xi _{j-1}}^{\frac{r_j}{{r_j}-1}} , \xi _j={x_j}-{x_j}^{*}, j=2, 3, 4, \end{aligned} \end{aligned}$$
(16)

where \(\beta _1>0, \beta _j >0 \). Therefore,

$$\begin{aligned}&{\dot{V}}_{k-1} \le -(6-k)\sum _{j=1}^{k-1} \xi _{k-1}^{(2r_4-\tau )/r_{k-1}-1}\nonumber \\&\quad +\xi _j^{2r_4/r_j}({x_k}-{x_k}^{*}). \end{aligned}$$
(17)

We claim that Eq. (17) also holds at step k. In order to prove the claim, we construct the Lyapunov function

$$\begin{aligned} V_k(x_1, \cdots , x_k)=V_{k-1}+\frac{r_k}{2r_4-\tau } \xi _k^{\frac{2r_4-\tau }{r_k}}. \end{aligned}$$
(18)

Then the derivative of the \(V_k(x_1, \cdots , x_k)\) is

$$\begin{aligned} \begin{aligned}&\dot{V_k}(x_1, \cdots , x_k) \le -(6-k)\sum _{j=1}^{k-1} \xi _j^{2r_4/r_j}+\xi _{k-1}^{\frac{2r_4-\tau }{r_k-1}-1} \xi _{k}\\&\quad +\xi _{k}^{\frac{2r_4-\tau }{r_k}-1} (x_{k+1}+\phi _k(\cdot )-\sum _{l=1}^{k-1} \frac{\partial {x_k}^{*}}{\partial x_l} {\dot{x}}_l). \end{aligned} \end{aligned}$$
(19)

Next, we will estimate each term in the right hand side of Eq. (19). By Yang Inequality (\(p=2r_4-\tau -r_{k-1}=2r_4-r_k, q=r_k \)), and denoting \(c_x= 2r_4 - \tau - r_{k-1}, d_x=r_k, \gamma =r_4/c_x, c_x+d_x=2r_4\), for a onstant \(c_k>0\), there is

$$\begin{aligned} \begin{aligned} {\xi _{k-1}}^{\frac{2r_4-\tau }{r_k-1}-1} {\xi _{k}}&\le \frac{c_x}{2r_4}\gamma \xi _{k-1}^{\frac{2r_4}{r_{k-1}}}+\frac{d_x}{2r_4} \gamma ^{-\frac{c_x}{d_x}} \xi _{k-1}^{\frac{2r_4}{r_{k}}}\\&=\frac{1}{2} \xi _{k-1}^{\frac{2r_4}{r_{k-1}}}+c_k \xi _{k}^{\frac{2r_4}{r_{k}}}. \end{aligned} \end{aligned}$$
(20)

According to Eq. (7) and Assumption 1, one obtains

$$\begin{aligned} \begin{aligned} \frac{r_k+\tau }{r_j}&=\frac{k \tau +1}{(j-1)\tau +1}&=\frac{kq+p}{(j-1)q+p}\ge 1 \end{aligned} \end{aligned}$$
(21)

is an odd function.

According to Lemma (4) and Eq. (16), there exists

$$\begin{aligned} \begin{aligned} |x_j|^{\frac{k \tau +1}{(j-1)\tau +1}}&= |\xi _j-\xi _{j-1}^{\frac{r_j}{r_{j-1}}} \beta _{j-1}| \\&\left. \le 2^{\frac{k \tau +1}{r_j}-1}| \xi _j^{\frac{r_k+\tau }{r_j}}-\beta _{j-1}^{\frac{r_k+\tau }{r_j}} \xi _{j-1}^{\frac{r_j}{r_{j-1}} \frac{r_k+\tau }{r_j}}\right. |\\&\le 2^{\frac{k \tau +1}{r_j}-1}(|{\xi _j}^{\frac{r_k+\tau }{r_j}}|+\beta _{j-1}^{\frac{r_k+\tau }{r_j}} |\xi _{j-1}^{\frac{r_k+\tau }{r_{j-1}}}|). \end{aligned} \end{aligned}$$
(22)

Combining Eq. (22) with Assumption (1), one obtains

$$\begin{aligned} \begin{aligned} |\phi _k|&\le c \sum _{j=1}^{k} |\xi _j-{\xi _{j-1}}^{\frac{r_j}{r_{j-1}}} \beta _{j-1}|\\ {}&\le c \sum _{j=1}^{k} 2^{\frac{k \tau +1}{r_j}-1}(|\xi _j^{\frac{r_k+\tau }{r_j}}|+\beta _{j-1}^{\frac{r_k+\tau }{r_j}} |\xi _{j-1}^{\frac{r_k+\tau }{r_{j-1}}}|)\\&={\bar{c}}_k \sum _{j=1}^{k} |{\xi _j}|^{\frac{r_k+\tau }{r_j}}. \end{aligned} \end{aligned}$$
(23)

The last right hand side of Eq. (19) can be estimated as the following proposition that has been proved in the paper [38].

Proposition 1

There is a constant \({\tilde{c}}_k\) such that

$$\begin{aligned} \sum _{l=1}^{k-1} \frac{\partial {x_k}^{*}}{\partial x_l} {\dot{x}}_l \le {\tilde{c}}_k(|\xi _1|^{\frac{r_k+\tau }{r_1}}+\cdots +|\xi _k|^{\frac{r_k+\tau }{r_k}}). \end{aligned}$$
(24)

By Lemma 3, combining Eqs. (23) and (24) yields

$$\begin{aligned} \begin{aligned}&\xi _{k}^{\frac{2r_4-\tau }{r_k}-1} (\phi _k(\cdot )-\sum _{l=1}^{k-1} \frac{\partial {x_k}^{*}}{\partial x_l} {\dot{x}}_l) \\&\quad \le x_{k}^{\frac{2r_4-\tau }{r_k}-1} ({\bar{c}}_k+{\tilde{c}}_k) \sum _{l=1}^{k} |\xi _l|^{\frac{r_k+\tau }{r_l}}\\&\quad \le ({\bar{c}}_k+{\tilde{c}}_k) \sum _{l=1}^{k-1} \left( \frac{c}{c+d}\gamma |\xi _{k}^{\frac{r_l}{r_k}}|^{\frac{2r_4}{r_l}}\right. \\&\qquad \left. +\frac{d}{c+d}\gamma ^{-\frac{c}{d}} |\xi _l|^{\frac{2r_4}{r_l}}\right) \\&\qquad + ({\bar{c}}_k+{\tilde{c}}_k) \xi _l^{\frac{2r_4}{r_k}} \\&\quad =\frac{1}{2} \sum _{l=1}^{k-1} \xi _l^{\frac{2r_4}{r_l}}\\&\qquad +{\hat{c}}_k \xi _l^{\frac{2r_4}{r_k}}, \end{aligned} \end{aligned}$$
(25)

where \(c={\frac{2r_4-\tau -r_k}{r_l}}, d=\frac{\tau +r_k}{r_l}, \frac{d}{c+d}\gamma ^{-\frac{c}{d}} ({\bar{c}}_k+{\tilde{c}}_k)= \frac{1}{2}\) and \( {\hat{c}}_k \) is a constant.

Substituting Eqs. (20) and (25) into Eq. (19) generates

$$\begin{aligned} \begin{aligned} {\dot{V}}_k&\le -(6-k) \sum _{j=1}^{k-1} \xi _j^{\frac{2r_4}{r_j}}+\frac{1}{2}\xi _{k-1}^{\frac{2r_4}{r_{k-1}}} +c_k\xi _k^{\frac{2r_4}{r_k}}\\&~~+\xi _k^{\frac{2r_4-\tau }{r_k}-1}x_{k+1} +\frac{1}{2} \sum _{l=1}^{k-1} \xi _l^{\frac{2r_4}{r_l}}+{\hat{c}}_k \xi _k^{\frac{2r_4}{r_k}}\\&\le -(6-k) \sum _{j=1}^{k-1} \xi _j^{\frac{2r_4}{r_j}}+\sum _{l=1}^{k-1} \xi _l^{\frac{2r_4}{r_l}}+\xi _k^{\frac{2r_4-\tau }{r_k}-1}x_{k+1}\\ {}&~~+(c_k+{\hat{c}}_k) \xi _k^{\frac{2r_4-\tau }{r_k}-1} \xi _k^{\frac{2r_k+\tau }{r_k}}\\&=-(5-k) \sum _{j=1}^{k-1} \xi _j^{\frac{2r_4}{r_j}}\\ {}&~~+\xi _k^{\frac{2r_4-\tau }{r_k}-1} (x_{k+1}+(c_k+{\hat{c}}_k) \xi _k^{\frac{r_k+1}{r_k}}). \end{aligned}\nonumber \\ \end{aligned}$$
(26)

According to the defined virtual controller, there exists

$$\begin{aligned} x_{k+1}^{*}=-\beta _k \xi _k^{\frac{r_k+1}{r_k}}=-(5-k+c_k+{\hat{c}}_k) \xi _k^{\frac{r_k+1}{r_k}}. \end{aligned}$$
(27)

Then

$$\begin{aligned} \begin{aligned} {\dot{V}}_k&\le -(5-k) \sum _{j=1}^{k-1} \xi _j^{\frac{2r_4}{r_j}}\\&\quad +\xi _k^{\frac{2r_4-\tau }{r_k}-1} (x_{k+1}- x_{k+1}^{*}-(5-k) \xi _k^{\frac{r_k+1}{r_k}})\\&=-(5-k) \sum _{j=1}^{k-1} \xi _j^{\frac{2r_4}{r_j}}-(5-k) \xi _j^{\frac{2r_4}{r_k}} \\&\quad + \xi _k^{\frac{2r_4-\tau }{r_k}-1} (x_{k+1}- x_{k+1}^{*})\\&=-(5-k) \sum _{j=1}^{k} \xi _j^{\frac{2r_4}{r_j}}+\xi _k^{\frac{2r_4-\tau }{r_k}-1} (x_{k+1}- x_{k+1}^{*}). \end{aligned} \end{aligned}$$
(28)

Then the inductive proof ends. The inductive argument shows that Eq. (16) holds for \(k=5 \). Therefore, the state feedback controller can be designed as

$$\begin{aligned} u=x_5^{*}=-\beta _4 \xi _4^{\frac{r_4+\tau }{r_4}}, \end{aligned}$$
(29)

which insures \({\dot{V}}_4 \le -(\xi _1^{2r_4/r_1}+\xi _2^{2r_4/r_2}+\xi _3^{2r_4/r_3}+\xi _4^{2r_4/r_4}) < 0\), i.e., system (12) can be globally asymptotically stabilized under the controller (11) or (29).

Next, we will prove that the state feedback controller (29) can stabilize the system (3). According to Eq. (29), one obtains

$$\begin{aligned} {\dot{V}}_4 \le -\sum _{j=1}^{4} \xi _j^{2r_4/r_j} + \sum _{j=1}^{4} \frac{\partial V_4(x_4)}{\partial x_j} \phi _j(\cdot ). \end{aligned}$$
(30)

According to Assumption 1, there exists

$$\begin{aligned} \begin{aligned}&\phi _j(t, x_{i+1},\cdots ,x_4) \le \kappa (|x_1| ^{p_1} + |x_2| ^{p_2} + |x_3| ^{p_3} + |x_4| ^{p_4} ) \\&\quad \times (|x_1| ^{p_{m1}-p_1} + |x_2| ^{p_{m2}-p_2} + |x_3| ^{p_{m3}-p_3} + |x_4| ^{p_{m4}-p_4}). \end{aligned} \end{aligned}$$
(31)

Then based on the aforementioned homogeneity definition, one can prove that \(\sum _{j=1}^{4} \xi _j^{2r_4/r_j}\) and

\(\sum _{j=1}^{4} \frac{\partial V_4(x_4)}{\partial x_j}(|x_1| ^{p_1} + |x_2|^{p_2}+|x_3|^{p_3}+|x_4|^{p_4})\) have the same homogeneity. According to the conclusion in the paper [9], there exists

$$\begin{aligned} \begin{aligned} {\dot{V}}_4&\le -\sum _{j=1}^{4} \xi _j^{2r_4/r_j}+ {\overline{\kappa }} \sum _{j=1}^{4} \xi _j^{2r_4/r_j}\times (|x_1| ^{p_{m1}-p_1}\\&\quad + |x_2| ^{p_{m2}-p_2} + |x_3| ^{p_{m3}-p_3} + |x_4| ^{p_{m4}-p_4} ), \end{aligned} \end{aligned}$$
(32)

when \({\overline{\kappa }} > 0 \).

According to Assumption (1) and (11), one knows that \( p_{mi}=\frac{i\tau +1}{(i-1)\tau +1}\), \( p_1=r2/r1, p_2=r3/r2, p_3=r4/r3, p_4=(r4+\tau )/r4 \), i.e., \(p_{m1}=p_1,p_{m2}=p_2, p_{m3}=p_3,p_{m4}=p_4\). Then there is \( (|x_1| ^{p_{m1}-p_1} + |x_2| ^{p_{m2}-p_2} + |x_3| ^{p_{m3}-p_3} + |x_4| ^{p_{m4}-p_4} )=4 \). Therefore,

$$\begin{aligned}&{\dot{V}}_4 \le -\sum _{j=1}^{4} \xi _j^{2r_4/r_j}\nonumber \\&\quad + 4 {\overline{\kappa }} \sum _{j=1}^{4} \xi _j^{2r_4/r_j} \le \sum _{j=1}^{4} \xi _j^{2r_4/r_j} (1-4 {\overline{\kappa }}). \end{aligned}$$
(33)

That is to say, \({\dot{V}}_4 < 0 \) if one selects suitable \({\overline{\kappa }} \). Therefore, The designed controller (11) can globally asymptotically stabilize the system (3).

Next, we will design the homogeneous observer and controller based on the designed state feedback controller.

3.3 Design of homogeneous state observer and controller

We construct a homogeneous observer inspired by the one used in [38] as follows

$$\begin{aligned} \left\{ \begin{aligned}&{\hat{x}}_1=x_1\\&{\dot{\eta }}_i=-Ll_{i} {\hat{x}}_{i}\\&{\hat{x}}_{i+1}=(\eta _{i}+k_{i} {\hat{x}}_{i})^{p_{i}}, \end{aligned} \right. \end{aligned}$$
(34)

where \(L\ge 1, l_i>0, k_i>0,i=1,2, 3\).

Based on the designed homogeneous state feedback controller, the homogeneous output feedback controller is designed as

$$\begin{aligned} u({\hat{x}})=-\beta _4({\hat{x}}_4+\beta _3({\hat{x}}_3+\beta _2({\hat{x}}_2+\beta _1{\hat{x}}_1^{p_1})^{p_2})^{p_3})^{p_4}, \end{aligned}$$
(35)

i.e.,

$$\begin{aligned} u({\hat{x}})=-\beta _i \xi _4^{\frac{r_i+\tau }{r_i}},i=1, 2, 3, 4. \end{aligned}$$
(36)

We construct

$$\begin{aligned} U_i=\int _{{\eta _i+l_{i-1} x_{i-1}}^{\frac{2r_4-r_i}{r_{i-1}}}}^{x_i^{\frac{2r_4-r_i}{r_{i}}}} (s^{\frac{r_{i-1}}{2r_4-r_i}}-(\eta _i+l_{i-1} x_{i-1})) \mathrm{d}s, \end{aligned}$$
(37)

where \(i=2,3,4\), and \(U_i\) is \(C^1\). One obtains

$$\begin{aligned} \begin{aligned}&\frac{\partial U_i}{\partial x_i}=\frac{2r_4-r_i}{r_i} x_i^{\frac{2r_4-2r_i}{r_i}}(x_i^{\frac{r_{i-1}}{r_i}}-(\eta _i+l_{i-1} x_{i-1})),\\&\quad \frac{\partial U_i}{\partial \eta _i}=-(x_i^{\frac{2r_4-2r_i}{r_i}}-(\eta _i+l_{i-1} x_{i-1})^{\frac{2r_4-2r_i}{r_{i-1}}}),\\&\quad \frac{\partial U_i}{\partial x_{i-1}}=-l_{i-1}(x_i^{\frac{2r_4-2r_i}{r_i}}-(\eta _i+l_{i-1} x_{i-1})^{\frac{2r_4-2r_i}{r_{i-1}}}). \end{aligned} \end{aligned}$$
(38)

Then the derivative of \(U_i\) along Eqs. (11), (12), (16), (34) is

$$\begin{aligned} \begin{aligned} {\dot{U}}_i&=\frac{2r_4-r_i}{r_i}x_{i+1} x_i^{\frac{2r_4-2r_i}{r_i}}(x_i^{\frac{r_{i-1}}{r_i}}-(\eta _i+l_{i-1} x_{i-1}))\\&\quad -l_{i-1} e_i (x_{i}^{\frac{2r_4-r_i}{r_i}}-{\hat{x}}_{i}^{\frac{2r_4-r_i}{r_i}})\\&\quad -l_{i-1} e_i ({\hat{x}}_i^{\frac{2r_4-2r_i}{r_i}}-(\eta _i+l_{i-1} x_{i-1})^{\frac{2r_4-2r_i}{r_{i-1}}}), \end{aligned} \end{aligned}$$
(39)

where \(e_i=x_i-{\hat{x}}_i, i=2, 3, 4\), and \(x_5=u({\hat{x}})\).

Next, we will estimate each term of the right hand side of Eq. (39). By Lemma 4, one obtains

$$\begin{aligned} -l_{i-1} e_i (x_{i}^{\frac{2r_4-r_i}{r_i}}-{\hat{x}}_{i}^{\frac{2r_4-r_i}{r_i}}) \le -l_{i-1} e_i^{\frac{2r_4}{r_i}} 2^{\frac{2r_i-2r_4}{r_i}}. \end{aligned}$$
(40)

The other terms can be verified by the following propositions which have been proved in [38].

Proposition 2

For \(i=2,3\), there exists

$$\begin{aligned} \begin{aligned}&\frac{2r_4-r_i}{r_i}x_{i+1} x_i^{\frac{2r_4-2r_i}{r_i}}(x_i^{\frac{r_{i-1}}{r_i}}-(\eta _i+l_{i-1} x_{i-1})) \\&\quad \le \frac{1}{12}(\xi _{i-1}^{\frac{2r_4}{r_{i-1}}}+ \xi _{i}^{\frac{2r_4}{r_{i}}}+\xi _{i+1}^{\frac{2r_4}{r_{i+1}}})\\&\qquad +\gamma _i e_i^{\frac{2r_4}{r_{i}}}+g_i(l_{i-1}) e_{i-1}^{\frac{2r_4}{r_{i-1}}}, \end{aligned} \end{aligned}$$
(41)

where \(\alpha _i\) is a constant, \(g_i\) is a continuous function of \(l_{i-1}\) with \(g_2(\cdot )=0\).

Proposition 3

For controller (36), there exists

$$\begin{aligned} \begin{aligned}&r_4 u({\hat{x}})(x_4^{\frac{2r_3}{r_4}}-(\eta _4+l_3 z_3)) \\&\quad \le c (||\xi ||_{\varDelta _x}^{r_4+\tau } +||e||_{\varDelta _x}^{r_4+\tau })(c |e_4|^{\frac{r_3}{r_4}}+l_3 |e_3|)\\&\quad \le \frac{1}{8}\sum _{i=1}^{4}\xi _i^{2r_4/r_i}+ {\bar{\gamma }}\sum _{i=1}^{4}e_i^{2r_4/r_i}+g_4(l_3)e_3^{2r_4/r_3}. \end{aligned} \end{aligned}$$
(42)

Proposition 4

For \(i=3,4\), there exists

$$\begin{aligned} \begin{aligned} -l_{i-1}&e_i({\hat{x}}_i^{\frac{2r_4-r_i}{r_i}}-(\eta _i+l_{i-1} x_{i-1})^{\frac{2r_4-r_i}{r_{i-1}}}) \\&\le e_i^{\frac{2r_4}{r_i}}+\frac{1}{16}(\xi _i^{\frac{2r_4}{r_i}}+ \xi _{i-1}^{\frac{2r_4}{r_{i-1}}})+h_i(l_{i-1})e_{i-1}^{\frac{2r_4}{r_i}}, \end{aligned} \end{aligned}$$
(43)

where \(h_i(l_{i-1})\) is a polynomial function.

With the help of Proposition 2–4, we can obtain the derivative of \(U=\sum _{i=2}^{4} U_i\)

$$\begin{aligned} \begin{aligned} {\dot{U}}&\le \frac{1}{2} \sum _{i=1}^{4} \xi _{i}^{\frac{2r_4}{r_{i}}}-(l_1 2^{\frac{2r_1-2r_4}{r_1}}-\gamma _2-{\bar{\gamma }}-g_3(l_2)\\&-h_3(l_2))e_2^{\frac{2r_4}{r_2}}-(l_{2} 2^{\frac{2r_3-2r_4}{r_{3}}}-\gamma _3-1-{\bar{\gamma }}-g_4(l_3)\\&-h_4(l_3))e_3^{\frac{2r_4}{r_3}} -(l_3-1-{\widetilde{\gamma }}-{\bar{\gamma }})e_4^{2}. \end{aligned} \end{aligned}$$
(44)

Now we will discuss how to determine \(l_i\). Because the states \(x_2,x_3,x_4\) are not measurable, system (29) generates a redundant term \(\xi _4^{2r_4-\tau /r_4-1}(u-u^*)\) under controller (35). We use Proposition 5 to solve this issue.

Proposition 5

For a constant \({\bar{\gamma }} \ge 0\) , there exists

$$\begin{aligned} u({\hat{x}})-u^{*}(x)=\sum _{i=2}^{4} e_i \int _{0}^{1} \frac{\partial ({\mathcal {X}})}{\partial {\mathcal {X}}_i} \mathrm{d} \lambda , \end{aligned}$$
(45)

where \(\mathbb {{\mathcal {X}}}=x-\lambda e\).

According to the homogeneity of \(u^*(x)\) whose degree is \(r_4+\tau \), the degree of \(\frac{\partial ({\mathcal {X}})}{\partial {\mathcal {X}}_i}\) is \(r_4+\tau -r_i\). According to the definition of homogeneous norm, there exists

$$\begin{aligned} \frac{\partial ({\mathcal {X}})}{\partial {\mathcal {X}}_i}\le c||\xi ||_{\varDelta _x}^{r_4+\tau -r_i}+ c||e||_{\varDelta _x}^{r_4+\tau -r_i}. \end{aligned}$$

By Yong’s Inequality, there exists

$$\begin{aligned} \begin{aligned}&\xi _4^{\frac{2r_4-\tau }{r_4}-1}(u^{*}({\hat{x}})-u(x)) \\&\quad \le c |\xi _4^{\frac{2r_4-\tau }{r_4}-1}| \sum _{i=2}^{4} |e_i| \times (||\xi ||_{\varDelta _x}^{r_4+\tau -r_i}\\&\qquad +||e||_{\varDelta _x}^{r_4+\tau -r_i})\\&\quad \le \sum _{i=1}^{4} \frac{\xi _i^{\frac{2r_4}{r_i}}}{4}+{\tilde{\gamma }} \sum _{i=2}^{4}e_i^{\frac{2r_4}{r_i}}, \end{aligned} \end{aligned}$$
(46)

for a constant \({\tilde{\gamma }} \ge 0\).

We construct a Lyapunov function

$$\begin{aligned} W=U+V_4. \end{aligned}$$
(47)

It’s derivative is

$$\begin{aligned} \begin{aligned}&{\dot{W}} \le \frac{1}{4} \sum _{i=1}^{4} \xi _{i}^{\frac{2r_4}{r_{i}}}-(l_1 2^{\frac{2r_1-2r_4}{r_1}}\\&\qquad -\gamma _2 -{\tilde{\gamma }} -{\bar{\gamma }}-g_3(l_2)\\&\qquad -h_3(l_2))e_2^{\frac{2r_4}{r_2}}-(l_{2} 2^{\frac{2r_3-2r_4}{r_{3}}}\\&\qquad -\gamma _3-1 -{\tilde{\gamma }} -{\bar{\gamma }}-g_4(l_3)\\&\qquad -h_4(l_3))e_3^{\frac{2r_4}{r_3}} -(l_3-1 \\&\qquad -{\tilde{\gamma }} -{\bar{\gamma }})e_4^{3}. \end{aligned} \end{aligned}$$
(48)

Obviously, if one selects

$$\begin{aligned} \begin{aligned} l_1&= 2^{1+\frac{r_1}{2r_4-2r_1}}(\frac{1}{4}+ \gamma _2\\&\qquad +{\tilde{\gamma }}+{\bar{\gamma }}+g_3(l_2)+h_3(l_2))\\ l_{2}&=2^{1+\frac{r_3}{2r_4-2r_3}}(\frac{5}{4}+ \gamma _3+{\tilde{\gamma }}\\&\qquad +{\bar{\gamma }}+g_{4}(l_3)+h_{4}(l_3))\\ l_3&=\frac{5}{4}+{\tilde{\gamma }}+{\bar{\gamma }}, \end{aligned} \end{aligned}$$
(49)

Equation (48) can be rewritten as

$$\begin{aligned} {\dot{W}} \le -\frac{1}{4}(\xi _1^{2r_4/r_1}+\sum _{j=2}^{4}(\xi _j^{2r_4/r_j}+e_j^{2r_4/r_1})). \end{aligned}$$
(50)

According to the construction of W, it can be proved that W is positive definite. And according to the constructions of \(\xi _i\) and \(e_i\), \({\dot{W}}\) is negative definite. Therefore, the closed-loop system 12 is globally asymptotically stable.

We rewrite system (12), (34) and (35) as

$$\begin{aligned} {\dot{X}}=F(X)=(x_2, x_3, x_4,u({\hat{x}}), F_5, F_6, F_7)^T. \end{aligned}$$
(51)

System (51) is homogeneous, and the dilations can be selected as

$$\begin{aligned} \varDelta =(\varDelta _x, \varDelta _\eta )=(r_1, r_2, r_3, r_4, r_5, r_6, r_7), \end{aligned}$$
(52)

where \(\varDelta _x=(1, 1+\tau , 1+2\tau , 1+3\tau )\), \(\varDelta _\eta =(1, 1+\tau , 1+2\tau )\). Therefore, system (51) is homogeneous of degree of \(\tau \). W is homogeneous of degree of \(2r_4-\tau \), and \({\dot{W}}\) is homogeneous of degree of \(2r_4\).

According to Lemma 2, Eq. (48) is rewritten as

$$\begin{aligned} \frac{\partial W(X)}{\partial X} \le -c_1 ||X||_\varDelta ^{2r_4}, \end{aligned}$$
(53)

where

$$\begin{aligned} ||X||_\varDelta =\sqrt{\sum ||X_i||^{2/r_i}},i=1,\cdots ,7. \end{aligned}$$

Next, we will prove that system (3) is stable under observer (34) and controller (35). A coordinate change is introduced firstly. Let \(z_1=x_1 \), \(z_2 = \frac{x_2}{L} \), \( z_3 = \frac{x_3}{L^2} \), \( z_4 = \frac{x_4}{L^3} \), \(u({\hat{z}}) = \frac{u({\hat{x}})}{L^4} \), system 3 is changed as

$$\begin{aligned} \begin{aligned} {\dot{z}}_1&= L z_2 + \phi _1(t, z)\\ {\dot{z}}_2&= Lz_3+ \frac{\phi _2(t, z)}{L}\\ {\dot{z}}_3&= L z_4 + \frac{\phi _3(t, z)}{L^2}\\ {\dot{z}}_4&= L u({\hat{z}})+ \frac{\phi _4(t, z)}{L^3}. \end{aligned} \end{aligned}$$
(54)

Combining with Eqs. (48), (50) and (53), the derivative of the Lyapunov function for system (54) is

$$\begin{aligned} {\dot{W}}_f\le & {} -\frac{1}{4}(\xi _1^{2r_4/r_1}+\sum _{j=2}^{4}(\xi _j^{2r_4/r_j}+e_j^{2r_4/r_j})) \nonumber \\&+\frac{\partial W_f(Z)}{\partial Z} \left[ \phi _1 (\cdot ) \cdot \frac{\phi _2 (\cdot )}{L} \cdot \frac{\phi _3 (\cdot )}{L^2} \cdot \frac{\phi _4 (\cdot )}{L^3} \right] ^T\nonumber \\\le & {} -L c_1 ||Z||_\varDelta ^{2r_4} \nonumber \\&+\frac{\partial W_f(Z)}{\partial Z} \left[ \phi _1 (\cdot ) \cdot \frac{\phi _2 (\cdot )}{L} \cdot \frac{\phi _3 (\cdot )}{L^2} \cdot \frac{\phi _4 (\cdot )}{L^3} \right] ^T,\nonumber \\ \end{aligned}$$
(55)

where \(Z = [z_1 \quad z_2 \quad z_3 \quad z_4]^T \).

Considering the characteristics of the active suspension, there exists

$$\begin{aligned} \begin{aligned} |\phi _1(t, z)|&= 0 \le c|z_1|^{\tau +1}\\ |\phi _2(t, z)|&=\frac{c_t}{k_t}x_4\le c (|z_1|^{\tau +1}+|L z_{2}|^{\frac{2\tau +1}{\tau +1}})\\ |\phi _3(t, z)|&= 0 \le c (|z_1|^{\tau +1}+|L z_{2}|^{\frac{2\tau +1}{\tau +1}}+|L^2 z_{3}|^{\frac{3\tau +1}{2 \tau +1}})\\ |\phi _4(t, z)|&=k_t\ddot{x}_g(t) \le c (|z_1|^{\tau +1}+|L z_{2}|^{\frac{2\tau +1}{\tau +1}}\\&\quad +|L^2 z_{3}|^{\frac{3\tau +1}{2 \tau +1}}+ |L^3 z_{4}|^{\frac{4\tau +1}{3\tau +1}}). \end{aligned} \end{aligned}$$
(56)

Because \(L \ge 1\),

$$\begin{aligned} \begin{aligned} |\phi _1(t, z)|&=0\\ |\frac{\phi _2(t, z)}{L}|&\le c L^{1-\frac{1}{\tau +1}} (|z_1|^{\frac{r_2+\tau }{r_1}}+|z_2|^{\frac{r_2+\tau }{r_2}})\\ |\frac{\phi _3(t, z)}{L^2}|&=0\\ |\frac{\phi _4(t, z)}{L^3}|&\le c L^{1-\frac{1}{2 \tau +1}} (|z_1|^{\frac{r_4+\tau }{r_1}}+|z_2|^{\frac{r_4+\tau }{r_2}}\\&\quad +|z_3|^{\frac{r_4+\tau }{r_3}}+z_4|^{\frac{r_4+\tau }{r_4}}). \end{aligned} \end{aligned}$$
(57)

According to the definition of homogeneous function, \(\frac{\partial W(Z)}{\partial Z_{i}}\) is homogeneous of degree of \(2r_4-\tau -r_i\). Based on Lemma 1,

$$\begin{aligned} \begin{aligned}&\left| \frac{\partial W_f(Z)}{\partial Z_1}\right| (|z_{1}|^{\frac{r_1+\tau }{r_1}}),\\&\quad \left| \frac{\partial W_f(Z)}{\partial Z_2}\right| (|z_{1}|^{\frac{r_2+\tau }{r_1}}+|z_{2}|^{\frac{r_2+\tau }{r_2}}),\\&\quad \left| \frac{\partial W_f(Z)}{\partial Z_3}\right| (|z_{1}|^{\frac{r_3+\tau }{r_1}}+|z_{2}|^{\frac{r_3+\tau }{r_2}}\\&\quad +|z_{3}|^{\frac{r_3+\tau }{r_3}}),\\&\left| \frac{\partial W_f(Z)}{\partial Z_4}\right| (|z_{1}|^{\frac{r_4+\tau }{r_1}}+|z_{2}|^{\frac{r_4+\tau }{r_2}}\\&\qquad +|z_{3}|^{\frac{r_4+\tau }{r_3}}+|z_{4}|^{\frac{r_4+\tau }{r_4}}) \end{aligned} \end{aligned}$$
(58)

are homogeneous of degree of \(2r_4\). Therefore, we can find a constant \(\chi _i>0 \) in the Eqs. (57) and (58) to get

$$\begin{aligned} \frac{\partial W_f(Z)}{\partial Z_i} \frac{\phi _i(\cdot )}{L^{i-1}} \le \chi _i L ^{1-\frac{1}{(i-1) \tau +1}} ||Z||_{\varDelta }^{2r_4}. \end{aligned}$$
(59)

Substituting Eq. (59) into (50) leads to

$$\begin{aligned} {\dot{W}}_f \le -L(c_1- \chi _2 L ^{-\frac{1}{\tau +1}} - \chi _4 L ^{-\frac{1}{3 \tau +1}} )||Z||_\varDelta ^{2r_4}. \end{aligned}$$
(60)

Therefore, as long as we select a suitable L, there exists \({\dot{W}}_f < 0 \) which ensures \({\dot{W}}_f < 0 \), i.e., the active suspension control system (3) is asymptotically stable under observer (34) and controller (35).

The proof of the Theorem 1 ends.

4 Simulation and test

We firstly numerically simulate the designed homogeneous output feedback controller Eq. (35) with the homogeneous observer Eq. (34) for the active suspension compared with the Sliding mode controller. The sliding mode controller is designed inspired by Reference [18]. We analyze the effectiveness of the designed homogeneous output feedback controller about the sprung mass vertical displacement and unsprung mass vertical displacement according to the suspension evaluation index. In the simulation, the parameters of the active suspension and the designed homogeneous output feedback controller are shown in Tables 1 and 2 respectively.

Table 1 The parameters of an active suspension of an IEV-DFIM
Full size table
Table 2 The parameters of the designed homogeneous output feedback controller
Full size table

We select three classic road profiles: sinusoidal road profile, convergent sinusoidal road profile and step road profile, to verify the designed controller. Figures 2, 3, 4 are the selected three road profiles.

Fig. 2
figure 2

The disturbance of \(0.01\sin (2\pi t)\)m road profile

Full size image
Fig. 3
figure 3

The disturbance of \(0.01\sin (2\pi t)/(1+t^3)\)m road profile

Full size image
Fig. 4
figure 4

The disturbance of step road profile

Full size image

Figures 5, 6, 7 are the vertical displacement of the body under different controllers and passive suspension suffering different road profiles. The homogeneous controller can decrease the vertical displacement of the sprung mass under sinusoidal road profile better than the sliding mode controller as shown in Fig. 5. The vertical displacements of the sprung mass under different controllers are in Table 3. Figure 6 shows that the vertical displacement of the sprung mass is more smoother with the homogeneous controller than the sliding mode controller under \(0.01\sin (2\pi t)/(1+t^3)\)m road profile. Figure 7 shows that the homogeneous controller can stabilize the vertical displacement of the sprung mass smoothly under step road profile. The sliding mode controller is better than the homogeneous controller on decreasing the vertical displacement of the sprung mass, but worse on smoothness. Figures 8, 9, 10 are the outputs of the homogeneous controller and sliding mode controller under different road profiles, from which we can obtain that the outputs of the two controllers are similar under sinusoidal and \(0.01\sin (2\pi t)/(1+t^3)\)m road profiles. But the output of the homogeneous controller is greatly smaller than the sliding mode controller under step road profile as shown in Fig. 10, which means the homogeneous controller is more practical than the sliding mode controller.

Fig. 5
figure 5

The vertical displacement of the body under different controllers suffering sinusoidal road profile

Full size image
Table 3 The vertical displacements of the sprung mass under different controllers suffering sinusoidal road profile
Full size table
Fig. 6
figure 6

The vertical displacement of the body under different controllers suffering \(0.01\sin (2\pi t)/(1+t^3)\)m road profile

Full size image
Fig. 7
figure 7

The vertical displacement of the body under different controllers suffering step road profile

Full size image
Fig. 8
figure 8

The outputs of different controllers suffering sinusoidal road profile

Full size image
Fig. 9
figure 9

The outputs of different controllers suffering \(0.01\sin (2\pi t)/(1+t^3)\)m road profile

Full size image
Fig. 10
figure 10

The outputs of different controllers suffering step road profile

Full size image

In order to further verify the effectiveness of the designed homogeneous controller in engineering, we test the controller compared with the sliding mode controller via an active suspension testbed shown in Fig. 11. In the test, the parameters of the vehicle and the designed homogeneous controller are shown in Tables 4 and 5 respectively.

Fig. 11
figure 11

The active suspension testbed

Full size image
Table 4 The parameters of an IEV-DFIM used in the test
Full size table
Table 5 The parameters of the designed homogeneous output feedback controller used in the test
Full size table
Fig. 12
figure 12

The outputs of different controllers suffering \(\sin (2\pi t)\)m road profile in the test

Full size image
Fig. 13
figure 13

The outputs of different controllers suffering \(\sin (2\pi t)/(1+t^3)\) road profile in the test

Full size image
Fig. 14
figure 14

The outputs of different controllers suffering step road profile in the test

Full size image

In the test, we use the vertical acceleration and dynamic travel of the spruang mass to verify the performance of the active suspension furtherly. The test is carried out with the three road profiles used in the Sect. 4 under 40 km/h. Figures 12, 13, 14 are the outputs of the homogeneous controller and sliding mode controller under sinusoidal, \(0.01\sin (2\pi t)/(1+t^3)\)m and step road profiles. Figures 15, 16, 17 are the vertical accelerations of the spruang mass under sinusoidal, \(0.01\sin (2\pi t)/(1+t^3)\)m and step road profiles with the homogeneous controller and sliding mode controller respectively. Figure 15 show that the homogeneous controller and sliding mode controller both decrease the vertical acceleration of the body to small values suffering \(0.01\sin (2\pi t)\)m road profile, but the homogeneous controller has smaller values. In Fig. 16, the homogeneous controller stabilizes the vertical acceleration to zero within 6 seconds which is faster than the sliding mode controller suffering

Fig. 15
figure 15

The vertical acceleration of the body under different controllers suffering \(\sin (2\pi t)\) road profile in the test

Full size image
Fig. 16
figure 16

The vertical acceleration of the body under different controllers suffering \(\sin (2\pi t)/(1+t^3)\) road profile in the test

Full size image

\(0.01\sin (2\pi t)/(1+t^3)\)m road profile. As shown in Fig. 17, the two controllers stabilize the vertical acceleration to zero within almost the same time, but the homogeneous controller obtains smaller values. Figures 18, 19, 20 show the dynamic travels of the suspension have the same trend with the vertical accelerations suffering three road profiles. In conclusion, the test results show that the designed homogeneous controller has better effectiveness than the sliding mode controller in practice.

Fig. 17
figure 17

The vertical acceleration of the body under different controllers suffering step road profile in the test

Full size image
Fig. 18
figure 18

The dynamic travel of the front-left suspension under different controllers suffering \(\sin (2\pi t)\) road profile in the test

Full size image

5 Conclusion

Fig. 19
figure 19

The dynamic travel of the front-left suspension under different controllers suffering \(\sin (2\pi t)/(1+t^3)\) road profile in the test

Full size image
Fig. 20
figure 20

The dynamic travel of the front-left suspension under different controllers suffering step road profile in the test

Full size image

In this paper, we build an active suspension dynamic model of an IEV-DFIM which considers the coefficients’ time-varying in practice. Based on the dynamic model, the active suspension control system is proposed for the IEV-DFIM. A homogeneous output feedback control method for the active suspension of the IEV-DFIM is presented. And we prove that the designed homogeneous controller can stabilize the system by Lyapunov method. The numerical simulation results show that the homogeneous controller is better than the sliding model controller to decrease the vertical displacement of the sprung mass under different road profiles. The test results verify that the designed homogeneous controller also has better effectiveness on vertical acceleration and dynamic travel than the sliding mode controller under different road profiles in practice.