Keywords

1 Introduction

For an optimal experience and maximum safety while driving a vehicle, comfort and road holding are important factors. The suspension has a large impact on both comfort and road holding [2]. In a regular passenger car the suspension typically consists of a spring and a damper. By adding an actuator to the suspension (active suspension), comfort and road holding can be improved even further [5]. An active suspension typically acts upon the measured sprung acceleration and suspension travel and is therefore reactive [7]. To improve performance, a preview controller can be used that uses information about the road ahead.

Little research has been done on active tire control. D’Ambrosio [1] introduces Active Tire Pressure Control (ATPC), aimed at improving the fuel consumption, safety, and drivability of a car. However, the system is rather slow. Nandikolla [6] introduces a deformable smart tire with the use of SMA springs. By heating and cooling the SMA springs, the shape of the smart tire switches between two states: circular and square. Even though not a road vehicles, Maglev trains can be modeled in a similar way as a car. Maglev trains offer the opportunity to actively control the interaction between track and unsprung mass of the train, thereby adding the equivalent of a tire actuator. This leads to the open research question of determining the performance gain in terms of handling and comfort achievable with a dual actuator system with preview control over conventional and active suspension.

The outline of this paper is as follows, in Sect. 2 the quarter car model is introduced. Using this model, the controller is derived in Sect. 3. The effects of preview time and a comparison with existing controllers is shown in Sect. 4. Finally, conclusions are drawn in Sect. 5.

2 Vehicle Model

In this research the standard quarter car model [7], shown in Fig. 1, is used. Here \(m_s\) represents the sprung mass and \(m_u\) unsprung mass, which are connected via a spring (\(k_s\)) and damper (\(d_s\)). Furthermore, the tire is modeled as a spring with stiffness \(k_t\). The actuator force between the sprung and unsprung mass is represented by \(F_s\) and the tire actuator by \(F_t\). Finally, the preview time is indicated by \(t_p\). The parameters of the quarter car model are given in Table 1. The equations of motion are given as

$$\begin{aligned} m_s\ddot{z}_s &= -k_s(z_s-z_u)-d_s(\dot{z}_s-\dot{z}_u)-F_{s} \end{aligned}$$
(1)
$$\begin{aligned} m_u\ddot{z}_u &= k_s(z_s-z_u)+d_s(\dot{z}_s-\dot{z}_u)-k_t(z_u-z_r)+F_{s}-F_{t}. \end{aligned}$$
(2)

For controller implementation, these equations of motion have been put into a linear state-space form

$$\begin{aligned} \dot{x}&=\textbf{A}x+\textbf{B}_{\textbf{1}}\textbf{u}_{\textbf{1}}+\textbf{B}_{\textbf{2}}z_r \end{aligned}$$
(3)
$$\begin{aligned} y&=\textbf{C}x+\textbf{D}_{\textbf{1}}\textbf{u}_{\textbf{1}} + \textbf{D}_{\textbf{2}}z_r \end{aligned}$$
(4)

with state space matrices \(\textbf{A}\), \(\textbf{B}_{\textbf{1}}\), \(\textbf{B}_{\textbf{2}}\), \(\textbf{C}\), \(\textbf{D}_{\textbf{1}}\), and \(\textbf{D}_{\textbf{2}}\) denoting the state transition matrix, actuator force and disturbance input matrix, state to output matrix, force input to output matrix and disturbance to output matrix respectively.

The main purpose of a suspension is to provide comfort to the driver and keep the tires in contact with the road, which is important for vehicle handling. These objectives can be captured in the vertical acceleration of the sprung mass, \(\ddot{z}_s\), which is a good measure of comfort, and in the dynamic tire compression, \(z_u-z_r\), which is a good measure of handling [7]. For both quantities, the RMS value is used. In addition to that, suspension travel, \(z_s-z_u\), is typically limited.

Fig. 1.
figure 1

Schematic representation of the quarter car model with suspension actuator, tire actuator, and preview control, adapted from [7]

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Table 1. System Parameters
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3 Controller Design

Given the system dynamics with multiple control objectives and two actuators, a control method that can generate the optimal input for both the actuators has to be designed. Assuming that the full state can be measured, an LQR is a suitable candidate. The cost function to be minimized is defined as

$$\begin{aligned} J=\frac{1}{2}\int _{t}^{t+t_p}(\textbf{y}^T\textbf{Q}\textbf{y} + \textbf{u}_{\textbf{1}}^T\textbf{R}\textbf{u}_{\textbf{1}})d\tau ; \\ \textbf{Q} = \textbf{Q}^T; \quad \textbf{Q} \ge 0; \quad \textbf{R} = \textbf{R}^T; \quad \textbf{R} > 0 \nonumber , \end{aligned}$$
(5)

where \(\textbf{Q}\) is a positive semidefinite weighting matrix on the output vector and \(\textbf{R}\) is a positive definite matrix on the force inputs. They can both be chosen such that the aforementioned performance metrics can be individually emphasized (comfort or handling), while taking into account suspension travel and actuator load. The input \(\textbf{u}_{\textbf{1}}\) that minimizes J is determined as

$$\begin{aligned} \textbf{u}_{\textbf{1}} = -\textbf{V}^{-1}(\textbf{B}_{\textbf{1}}^T\textbf{P} + \textbf{D}_{\textbf{1}}^T\textbf{Q}\textbf{C})\textbf{x} + \textbf{V}^{-1}\textbf{B}_{\textbf{1}}^T\textbf{r}, \end{aligned}$$
(6)

which has both a state feedback part and a feedforward part based on the future response of the system \(\textbf{r}\). The matrix \(\textbf{V}\) is defined as

$$\begin{aligned} \textbf{V} = \textbf{D}_{\textbf{1}}^T\textbf{Q}\textbf{D}_{\textbf{1}} + \textbf{R}; \quad \textbf{V} = \textbf{V}^T; \quad \textbf{V} > 0, \end{aligned}$$
(7)

and \(\textbf{P}\) is the solution of the Algebraic Ricatti Equation (ARE) [4]

$$\begin{aligned} \textbf{F}^T\textbf{P} + \textbf{P}\textbf{F} + \textbf{G} - \textbf{P}\textbf{H}\textbf{P} = \textbf{0}. \end{aligned}$$
(8)

In the Ricatti equation, the following matrices are defined

$$\begin{aligned} \textbf{F} = \textbf{A} - \textbf{B}_{\textbf{1}}\textbf{V}^{-1}\textbf{D}_{\textbf{1}}^T\textbf{Q}\textbf{C}, \end{aligned}$$
(9)
$$\begin{aligned} \textbf{G} = \textbf{C}^T(\textbf{Q} - \textbf{Q}\textbf{D}_{\textbf{1}}\textbf{V}^{-1}\textbf{D}_{\textbf{1}}^T\textbf{Q})\textbf{C}, \end{aligned}$$
(10)
$$\begin{aligned} \textbf{H} = \textbf{B}_{\textbf{1}}\textbf{V}^{-1}\textbf{B}_{\textbf{1}}^T. \end{aligned}$$
(11)

Road preview requires knowledge of the road disturbances up ahead. For now, it is assumed that this knowledge is available, either through, for example, detailed maps or a sensor on the vehicle that can measure the height of the road disturbance up ahead. In (6), the future response of the system, \(\textbf{r}\), was introduced. The upcoming section will discuss how the upcoming road disturbance is captured in \(\textbf{r}\).

First, the states of the system at the end of the preview time \(t_p\), \(\textbf{x}(t+t_p)\), are determined, which will be used as a boundary condition in the next step. The calculation strategy for \(\textbf{x}(t+t_p)\) is suggested by Huisman [3]. Secondly, the response of the system over the preview distance is determined by backward integrating of

$$\begin{aligned} \dot{\textbf{r}}(t^*) = -\mathbf {A_g}^T\textbf{r}(t^*) + \textbf{P}\textbf{h}(t^*); \quad \textbf{r}(t + t_p) = \textbf{P} \textbf{x}(t+t_p), \end{aligned}$$
(12)

over the preview time \(t_p\). Here \(\textbf{h}(t^*)\) defined as

$$\begin{aligned} \textbf{h} = \textbf{B}_{\textbf{2}}\textbf{u}_{\textbf{2}}, \end{aligned}$$
(13)

and \(\mathbf {A_g}\) as

$$\begin{aligned} \mathbf {A_g} = \textbf{F} - \textbf{H}\textbf{P}. \end{aligned}$$
(14)

To perform the backwards integration, the approach of [4] is followed, where the substitution \(\textbf{r}(t)=\textbf{s}(t+t_p-t^*)\) is made, such that

$$\begin{aligned} \dot{\textbf{s}}(t) = \mathbf {A_g}^T\textbf{s}(t) + \textbf{P}\textbf{B}_{\textbf{2}}\mathbf {z_r}(t+t_p-t); \quad \textbf{s}(0) = \textbf{r}(t+t_p). \end{aligned}$$
(15)

The states of the system, \(\textbf{x}(t)\), can be determined as

$$\begin{aligned} \dot{\textbf{x}}(t) = \mathbf {A_g}\textbf{x}(t) + \textbf{H}\textbf{r}(t) + \textbf{h}(t); \quad \textbf{x}(t) = \textbf{x}(t-\varDelta t). \end{aligned}$$
(16)

Finally, the optimal input \(\textbf{u}_{\textbf{1}}\) over the interval [\(t,t+t_p\)] can be determined by substituting \(\textbf{x}(t)\) and \(\textbf{r}(t)\) in (6).

4 Results

To achieve good comfort, the weighting matrices introduced in (5) have been chosen as

$$\begin{aligned} \textbf{Q}_{comfort}=\begin{bmatrix} 22 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} ;\quad \textbf{R}_{comfort}=\begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix}, \end{aligned}$$
(17)

thereby emphasizing the sprung acceleration, \(\ddot{z}_s\), over the dynamic tire compression and suspension travel. Below, first the influence of preview time on the performance of the dual actuator system is shown. Secondly, the frequency response of the proposed controller will be compared to a one actuator system with and without preview as well as the two actuator case without preview.

4.1 Influence of Preview Time

In this section, the effect of preview time for the dual actuator system is investigated. As input a 0.05 m step input is used. The preview times shown range from 0.1 to 1 s, with a step of 0.1 s. In Fig. 2, the results of the look-ahead time for a comfort-oriented controller are shown. The handling-oriented controller results are not shown here but show similar results in terms of the influence of the look-ahead time. From Fig. 2, it can be noted that the using road preview has the largest impact on the sprung acceleration and the suspension travel. Furthermore, the lines are converging with increasing preview times, which indicates the presence of an asymptote in performance. This is even more apparent from the fact that the controllers with larger preview time hardly show any movement of the suspension travel before 0.7 s. Changing the preview time shows little effect on the tire compression when the controller is tuned for comfort. In the next section the dual actuator controller is compared to other controllers, for this, a look-ahead time of 0.125 s will be used.

Fig. 2.
figure 2

Influence preview time on a dual actuator suspension with preview and comfort-oriented control with a step input of 0.05 m

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4.2 Frequency Response

In this section, the frequency response of the two actuator controller with preview is compared with various other controllers. Figure 3 shows that the two actuator solution shows better comfort over the whole frequency range compared to the passive system and also the system with a single actuator (both with and without preview). In addition to that, it also shows that the two actuator solution with preview performs better than a dual actuator without preview. Both dual actuator systems also show that the comfort invariant point no longer exists for this setup.

Fig. 3.
figure 3

Frequency response of systems with a comfort-oriented controller on a road disturbance

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5 Conclusion

A quarter car model with an LQR and a preview strategy is developed to determine the comfort gain achievable with a dual actuator suspension with preview control. The performance gain is measured by three metrics: sprung acceleration, tire compression, and suspension travel. The weighting matrices are tuned so that maximum comfort or handling can be achieved within the limits of the constraints. The simulation results show that an improvement of 182.3% in sprung acceleration can be achieved on a 0.05 m step input.