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Abstract

The present paper proposes an optimal polynomial control strategy for a quarter-vehicle suspension system based on dynamic programming. The optimal control objective is to decrease the responses of sprung mass acceleration, suspension deformation and road excitation, which are reflected in the performance index. The optimal control force is obtained through the principle of dynamic programming, and the form of the optimal control force mainly depends on the form of the cost function. The optimal nonlinear polynomial control (NPC) force is derived by selecting the cost function in polynomial form. The linear feedback matrix and nonlinear feedback matrix in the NPC force are determined by the Riccati equation and Lyapunov equation, respectively. The root mean square (RMS) responses of the sprung mass acceleration, suspension deformation and road load of driving vehicles under deterministic and random road surfaces are calculated. The numerical results showed that the proposed NPC strategy has a better control effect over the traditional linear quadratic regulator (LQR), especially on the peak reduction of sprung mass acceleration. Finally, the optimal control force is divided into active control force and passive control force. The energy input of the optimal control force can be reduced by only inputting the active control force.

Keywords

Nonlinear polynomial control dynamic programming optimal control active suspension peak reduction

Introduction

The suspension system has an important impact on the safety, ride comfort and handling stability of vehicles,¹ especially in some special vehicles, such as oil tank vehicles, signal vehicles and so on. The main purpose of the suspension system is to absorb the vibration and impact transmitted by the road surface.² Over the past decades, passive suspension has been widely used in traditional vehicles, such as tuned mass dampers,³ viscous dampers⁴ and dry friction dampers.⁵ However, it can only passively dissipate and store external input vibration. Thus, this kind of suspension cannot effectively improve the adaptability and driving performance in the complex external environment.

To solve these shortcomings, the active suspension was proposed by scholars in the 1950s, and a series of studies were subsequently carried out.⁶ Active suspension can continuously provide and adjust the control force with the variation of external loads. Thus, it can adapt to the complex environment. The disadvantages of using active suspension are the requirement of additional energy input and complex design of the control force. In recent years, research on active suspension has become more and more popular. The traditional control technologies used in the active suspension are PID control,⁷ $H_{\infty}$ control⁸ and Fuzzy control.⁹ Hrovat has reviewed different types of optimal control strategies in the design of active suspension for various models of dynamical models in the last century.¹⁰ However, the most widely used optimal control strategies are still the linear quadratic regulator (LQR) and its improved forms,^11–13 which can adjust the control effect by carefully selecting the performance index. In recent years, many creative control strategies in active suspension design are developed. To solve the energy-consuming problem in active suspension design, Ding et al. proposed a modified skyhook control by using the electromagnetic actuator.¹⁴ Using the state-dependent Riccati equation, Sinan studied the dynamics of active suspension with nonlinear actuator and compared the results with a passive suspension system.¹⁵ Based on the sliding mode control, Dou et al. considered the dynamics of a full-vehicle model with air suspension.¹⁶ Chen et al studied the chaotic behaviour of a fractional suspension system with nonlinear terms both in the conservative and the damping force,¹⁷ and the influence of the coefficients on the dynamical behaviour is analyzed. Tuwa et al studied the dynamics of a quarter car suspension with fractional order derivative by using multi-scale method,¹⁸ and the bifurcations and the chaotic behaviour of the system are investigated. To make a balance between the cost and the performance of the suspension, the theories and control algorithms for the semi-active suspension have been proposed and were well summarized in Ref. ¹⁹

The widely used control strategies in the actual engineering application are still based on linear system and deterministic excitation. In recent years, some scholars have proposed nonlinear control strategy,^20,21 which has a good effect on reducing the peak response of the system and high control efficiency. The theory of dynamic programming is first proposed by Bellman and further developed by Crandall by proposing the viscosity solutions,²² which has been widely used in optimal control problems. Based on the dynamic programming method, the optimal control force can be derived by minimizing the optimal control conditions and solving the related dynamical programming equation. The advantages of linear feedback control force are simple and easy to realize. However, in practical vibration control problems, the effective control force is to apply the increased control force when the response of the system is large and to apply the small control force when the response of the system is small. The control efficiency will be better when the control force changes in unequal proportion. According to this idea, Agrawal proposed optimal polynomial control for linear stochastic systems based on the stochastic dynamic programming method²³ and then extended this control strategy to nonlinear systems.²⁴ The results for various systems verified the effectiveness of nonlinear control strategies.

This paper proposes an optimal NPC strategy for the quarter-vehicle suspension system based on the principle of dynamic programming. Firstly, the vibration model of the quarter-vehicle suspension system is established. The responses of the sprung mass acceleration, suspension deformation and road load are taken as the control objectives to give the performance index. The polynomial form of optimal control force is obtained through appropriate selection of the cost function. Then, the responses of the sprung mass acceleration, suspension deformation and the change of road load under harmonic and random road excitation are calculated to show the control effect. For comparison, the results obtained by the traditional LQR are also presented. To reduce the input energy due to the active control force, the obtained optimal control force is decomposed into active and passive control forces. Finally, the numerical results are given to show the advantages of the proposed NPC.

Dynamic model of the quarter-vehicle suspension system

In this paper, the optimal control of a quarter-vehicle suspension system is studied. The quarter-vehicle model is shown in Figure 1.

Figure 1.

Model of a quarter-vehicle suspension system with active control force.

According to Newton's law of motion, the vibration equations of the suspension system are established

\begin{array}{l} m_{2} {\ddot{z}}_{2} + c ({\dot{z}}_{2} - {\dot{z}}_{1}) + k (z_{2} - z_{1}) - u = 0 \\ m_{1} {\ddot{z}}_{1} + c ({\dot{z}}_{1} - {\dot{z}}_{2}) + k (z_{1} - z_{2}) + k_{t} (z_{1} - z_{g}) + u = 0 \end{array}

(1)

where

m_{1}

is the unsprung mass (including wheels and axles), and

m_{2}

is the sprung mass (including body, etc.).

z_{1}

is the unsprung mass displacement,

z_{2}

is the sprung mass displacement,

z_{g}

is the road displacement excitation,

c

is the damping coefficient of the suspension,

k

is the suspension stiffness,

k_{t}

is the tyre stiffness and

u

is the undetermined control force. The model in equation (1) has been used extensively in the studies of existing suspension systems, which describe the main characteristics of the quarter-vehicle system.

By introducing $x = {z_{2} - z_{1}, {\dot{z}}_{2}, z_{1} - z_{g}, {\dot{z}}_{1}}^{T}$ as the new system state vector and $ξ (t) = {\dot{z}}_{g} (t)$ , the vibration equation of the quarter-vehicle suspension system can be written in matrix form

\dot{x} = f (x, u) = A x + B u + L ξ (t)

(2)

where

A = [\begin{array}{c} 0 & 1 & 0 & - 1 \\ - \frac{k}{m_{2}} & - \frac{c}{m_{2}} & 0 & \frac{c}{m_{2}} \\ 0 & 0 & 0 & 1 \\ \frac{k}{m_{1}} & \frac{c}{m_{1}} & - \frac{k_{t}}{m_{1}} & - \frac{c}{m_{1}} \end{array}]; B = [\begin{array}{c} 0 \\ \frac{1}{m_{2}} \\ 0 \\ - \frac{1}{m_{1}} \end{array}]; L = [\begin{array}{l} 0 \\ 0 \\ - 1 \\ 0 \end{array}]

Determinization and decomposition of the optimal NPC force

Formulation of the optimal control problem

Most of the previous suspension system control strategies are mainly based on the LQR control strategy, and the derived feedback control force is proportional to the state variables. However, the existing research showed that the nonlinear control strategy could adjust the control force nonlinearly according to the amplitude of the system response. Compared with linear control strategies, it can reduce the peak response of the system and has high control efficiency. In this section, we will establish an optimal polynomial control strategy based on the principle of dynamic programming.

The performance index $J$ can be expressed as

J (u (•)) = \int_{0}^{T} \bar{L} (x, u) d t + ψ (x (T))

(3)

where

x (T)

is the terminal state,

ψ (x (T))

represents the terminal cost and

\bar{L} (x, u)

is a nonnegative cost function. The optimal control problem is to minimize the performance index described in equation (3). Based on the principle of dynamic programming, the following Hamilton–Jacobi–Bellman (HJB) equation is derived

\frac{\partial V}{\partial t} = - \min_{u \in U} [H (x, u, V^{'}, t)]

(4)

where

V = V (x)

is the value function, and

V^{'} = \partial V / \partial x

is a vector.

H (x, u, V^{'}, t)

is the Hamiltonian function defined by the following equation

H (x, u, V^{'}, t) = {[V^{'}]}^{T} f + \bar{L} (x, u)

(5)

The optimal control force can be derived by minimizing $H (x, u, V^{'}, t)$ . Thus, the form of the optimal control force mainly depends on the expressions of $\bar{L} (x, u)$ .

Traditional LQR

Traditionally, the control objectives to be considered in vehicle suspension are sprung acceleration, suspension deformation and road excitation. Therefore, these quantities will eventually be used as control targets

\bar{L} = w_{1} {\ddot{z}}_{2}^{2} + w_{2} {(z_{2} - z_{1})}^{2} + w_{3} {(z_{1} - z_{g})}^{2}

where

w_{1}, w_{2}

and

w_{3}

are weighting coefficients. By using equations (1) and (2), the following cost function is established with transformed expressions of

x

\bar{L} (x, u) = x^{T} Q x + 2 x^{T} N u + u^{T} R u

(6)

where

Q = [\begin{array}{c} w_{1} \frac{k^{2}}{m_{2}^{2}} + w_{2} & w_{1} \frac{c k}{m_{2}^{2}} & 0 & - w_{1} \frac{c k}{m_{2}^{2}} \\ w_{1} \frac{c k}{m_{2}^{2}} & w_{1} \frac{c^{2}}{m_{2}^{2}} & 0 & - w_{1} \frac{c^{2}}{m_{2}^{2}} \\ 0 & 0 & w_{3} & 0 \\ - w_{1} \frac{c k}{m_{2}^{2}} & - w_{1} \frac{c^{2}}{m_{2}^{2}} & 0 & w_{1} \frac{c^{2}}{m_{2}^{2}} \end{array}], N = [\begin{array}{c} w_{1} \frac{k}{m_{2}^{2}} \\ w_{1} \frac{c}{m_{2}^{2}} \\ 0 \\ - w_{1} \frac{c}{m_{2}^{2}} \end{array}], R = \frac{w_{1}}{m_{2}^{2}}

Substituting equation (6) into equation (4) and according to the minimization condition, the linear optimal control force is derived with the following form

u = - R^{- 1} (B^{T} P + N) x

(7)

where matrix

P

satisfied the following equation

\dot{P} = Q - (P B + N) R^{- 1} (B^{T} P + N^{T}) + P A + A^{T} P

(8)

Considering the steady case, matrix $P$ satisfies the following algebraic Ricatti equation

Q - (P B + N) R^{- 1} (B^{T} P + N^{T}) + P A + A^{T} P = 0

(9)

Optimal nonlinear polynomial control

The optimal control force depends mainly on the expressions of $L (x, u)$ . Here, we can introduce a higher-order function to obtain the nonlinear optimal control force

\bar{L} (x, u) = x^{T} Q x + l (x) + 2 x^{T} N u + u^{T} R u

(10)

where

l (x)

is an undetermined function. By substituting equation (10) into equation (4), the value function

V (x)

satisfies

\frac{\partial V}{\partial t} = \min_{u \in U} {V^{'} f + \bar{L} (x, u)}

(11)

Assuming that $l (x)$ is a homogeneous polynomial higher than order 2, $V (x)$ will also be in the form of a polynomial. Supposing that the solution $V (x)$ has the following form

V (x) = x^{T} P x + g (x)

(12)

where

g (x)

is a polynomial to be determined. Substituting equation (12) into equation (5), we derive

\begin{array}{l} H (x, u, V^{'}, t) = V^{'} f + \bar{L} (x, u) \\ = x^{T} Q x + l (x) + 2 x^{T} N u + u^{T} R u + (2 x^{T} P + g^{'} {(x)}^{T}) (A x + B u) \end{array}

(13)

According to the minimization conditions, the control force can be obtained in the following form

u = - R^{- 1} (B^{T} P + N) x - R^{- 1} B^{T} g^{'} (x) / 2

(14)

By substituting the control force in equation (14) and value function in equation (12) into the dynamic programming equation, the following equation is derived

\begin{array}{l} - x^{T} \dot{P} x - \frac{\partial g (x)}{\partial t} = x^{T} Q x + l (x) - \frac{1}{4} g^{'} {(x)}^{T} B R^{- 1} B^{T} g^{'} (x) - g^{'} {(x)}^{T} B R^{- 1} (B^{T} P + N^{T}) x \\ - x^{T} (P B + N) R^{- 1} (B^{T} P + N^{T}) x + 2 x^{T} P A x + g^{'} {(x)}^{T} A x \end{array}

(15)

According to the order of $x$ in the equation, the quadratic term and higher-order term are separated, correspondingly

\begin{array}{l} - x^{T} \dot{P} x = x^{T} Q x - x^{T} (P B + N) R^{- 1} (B^{T} P + N^{T}) P x + 2 x^{T} P A x \\ - \frac{\partial g (x)}{\partial t} = l (x) - \frac{1}{4} g^{'} {(x)}^{T} B R^{- 1} B^{T} g^{'} (x) - g^{'} {(x)}^{T} B R^{- 1} (B^{T} P + N^{T}) x + g^{'} {(x)}^{T} A x \end{array}

(16)

From the first formula in equation (16), matrix

P

should satisfy the following equation

\dot{P} = Q - (P B + N) R^{- 1} (B^{T} P + N^{T}) + P A + A^{T} P

(17)

According to the expression of the control force in equation (14), the nonlinear expression of the control force $u$ mainly depends on the expressions of function $g (x)$ . Here, the function $g (x)$ is selected with the following form of a higher-order polynomial

g (x) = {(x^{T} P_{1} x)}^{2} / 2

(18)

where

P_{1}

is the undetermined matrix. Substituting

g (x)

in equation (18) into the second equation in equation (16) yields

\begin{array}{l} (x^{T} P_{1} x) x^{T} {\dot{P}}_{1} x \\ = l (x) - {(x^{T} P_{1} x)}^{2} x^{T} P_{1} B R^{- 1} B^{T} x^{T} P_{1} x + 2 (x^{T} P_{1} x) x^{T} P_{1} (A - B R^{- 1} (B^{T} P + N^{T})) x \end{array}

(19)

To eliminate the functions of $x$ higher than the fourth order, the following form of $l (x)$ is adopted

l (x) = (x^{T} P_{1} x) (x^{T} Q_{1} x) + {(x^{T} P_{1} x)}^{2} x^{T} P_{1} B R^{- 1} B^{T} x^{T} P_{1} x

(20)

where

Q_{1}

represents the nonlinear weighting matrix in the cost function. By substituting equation (20) into equation (19), matrix

P_{1}

should satisfy the following equation

{\dot{P}}_{1} = P_{1} (A - B R^{- 1} (B^{T} P + N^{T})) + {(A - B R^{- 1} (B^{T} P + N^{T}))}^{T} P_{1} + Q_{1}

(21)

Therefore, if the matrix $Q_{1}$ is given, matrix $P_{1}$ can be determined by equation (21). Finally, the optimal control force can be determined by equation (14).

Considering the steady state of the system, the equation for $P_{1}$ will meet the following Lyapunov function

P_{1} (A - B R^{- 1} (B^{T} P + N^{T})) + {(A - B R^{- 1} (B^{T} P + N^{T}))}^{T} P_{1} + Q_{1} = 0

(22)

where equation (22) is an algebraic equation. The final optimal nonlinear control force is of the following form

u = - R^{- 1} (B^{T} P + N) x - R^{- 1} B^{T} (x^{T} P_{1} x) P_{1} x

(23)

which is a polynomial function of the state variable.

Decomposition of optimal control force

In the realization of the control process, additional active control force means more energy consumption, and the power consumption of control force is strictly limited in many vehicle systems. To reduce the consumption of the active control force, the control force is decomposed into active control force and passive control force with the following form

u = - R^{- 1} (B^{T} P + N) x - R^{- 1} B^{T} g^{'} (x) = u_{a} + u_{p}

(24)

where

u_{a}

and

u_{p}

represent the optimal active control force and passive control force, respectively. The active control force

u_{a}

requires additional power input. The optimal passive control force

u_{p}

can be realized by adjusting the passive stiffness and damping coefficients, which means no extra control force is required for the passive control force. As it is not easy to realize nonlinear damping in engineering practice, the optimal passive control force containing

z_{2} - z_{1}

{\dot{z}}_{2} - {\dot{z}}_{1}

and

{(z_{2} - z_{1})}^{3}

is adopted, that is

u_{p} = c o e_{1} (z_{2} - z_{1}) + c o e_{2} ({\dot{z}}_{2} - {\dot{z}}_{1}) + c o e_{3} {(z_{2} - z_{1})}^{3}

(25)

where

c o e_{1}, c o e_{2}

and

c o e_{3}

are the optimal constant parameters of the passive control force

u_{p}

. It can be seen that the smaller the difference

| u - u_{p} |

, the smaller the active control force will be. Thus, the coefficients

c o e_{i}

can be selected to minimize the following conditions based on least square algorithm

\min E (u_{a}^{2}) = \min E {((u - u_{p})}^{2})

(26)

However, it is seen from equation (23) that the expressions of the optimal control force happen to be in the polynomial form of $(z_{2} - z_{1})$ . Therefore, there is a simple way to determine $c o e_{i}$ . That is, $c o e_{1}, c o e_{3}$ are selected as the coefficients of the linear and nonlinear conservative force in equation (23). $c o e_{2}$ can be selected as the coefficients of ${\dot{z}}_{2}, {\dot{z}}_{1}$ with the smallest absolute value.

Numerical results

The parameters of the quarter-vehicle suspension system used in the present paper are given in Table 1.

Table 1.

Values of parameters in a quarter car system.

Parameters	Values
Sprung mass $m_{2}$	400kg
Unsprung mass $m_{1}$	50kg
Damping coefficient ( $c$ )	1500Ns/m
Spring constant tyre ( $k_{t}$ )	240000N/m
Spring constant suspension ( $k$ )	20000N/m

In the LQR control strategy, the weighting coefficients are selected as $w_{1} = 2.0 \times 10^{6}$ , $w_{2} = 2.5 \times 10^{9}, w_{3} = 2.0 \times 10^{10}$ . Matrices $Q, N$ and $R$ can be obtained from equation (6). In order to make full use of the information of matrix A, matrix $Q_{1}$ can be selected as $Q_{1} = λ Q, λ = 7.5 \times 10^{- 4}$ , otherwise mentioned. Then, matrixes $P$ and $P_{1}$ can be derived by calculating the algebraic equations (9) and (22), respectively. Thus, the optimal NPC force is finally obtained by equation (23).

Harmonic excitation

First, we consider the case that the vehicle is driven on the road with a sinusoidal surface; In this case, the excitation will be harmonic. The excitation force is

z_{g} = f_{0} \sin (2 π f t)

(27)

where

f_{0}

and

f

are the amplitude and frequency of the excitation, respectively. In the following calculation, the value of the excitation amplitude is selected with

f_{0} = 0.003

Among all control objectives, the response of the sprung mass acceleration is related to the ride comfort of passengers and it is often used as the foremost objective in vehicle control. Under the given parameters, the peak value (amplitude) of the sprung mass acceleration under the proposed NPC is compared with the peak value of LQR and the uncontrolled system in Table 2. In Figure 2, the frequency-amplitude curves of the proposed NPC, traditional LQR and uncontrolled system under different values of frequencies are shown. It is seen that the conventional LQR and the proposed NPC control strategies both can reduce the response amplitude. But the proposed NPC control strategy has a better control effect in the peak reduction of the response amplitude, especially in the resonance interval of excitation frequency near

f = 11 Hz

Table 2.

The amplitude of the acceleration under different excitation frequencies.

Excitation frequency	Suspension type	Values
$f = 8$	Original system LQR control NPC control	0.72 0.493 0.49
$f = 11$	Original system LQR control NPC control	2.34 2.01 1.89
$f = 14$	Original system LQR control NPC control	2.01 1.57 1.47

Figure 2.

The frequency-amplitude curve under different values of frequencies.

To give a more intuitive exhibition, the typical time histories of the sprung mass acceleration are shown in Figure 3 with excitation frequency $f = 10 Hz$ . It can be seen that the peak value of the acceleration of the sprung mass under NPC reduces a lot compared to those under LQR control.

Figure 3.

Comparison of sprung mass acceleration under different control strategies with excitation frequency $f = 10 Hz$ .

Bumped road surface excitation

Then, we consider the case that the vehicle is driven over a bumped road surface with

\begin{array}{l} z_{g} = {\begin{cases} \frac{h}{2} (1 - \cos (\frac{2 π v}{l_{r}} t)), i f 0 \leq t \leq l_{r} / v \\ 0, i f l_{r} / v \leq t \end{cases} \end{array}

(28)

where

h

and

l_{r}

are the height and length of the bumped road surface, respectively,

v

is the driving speed.

In calculation, the parameters are selected with $v = 22 m / s, l_{r} = 2 m, h = 0.012 m$ . The time histories of the sprung mass acceleration are shown in Figure 4. Comparing the values of sprung mass acceleration, it can be seen that both LQR and NPC control strategies can reduce the peak value of the response, and the NPC control strategy will make the peak response smaller. This is because the NPC control strategy will provide greater feedback control force within the region of the peak as shown in Figure 5. It is seen that sprung mass acceleration will have bigger oscillation due to the optimal control force. This is because more energy of the system is saved in the form of restoring force.

Figure 4.

Comparison of sprung mass acceleration under different control strategies with bumped road surface excitation.

Figure 5.

Time histories of the optimal control force.

Random disturbance

The following power spectrum density of road surface is often used

S (n) = G_{0} n^{- p}

(29)

where

n

is the spatial frequency and the unit is

m^{- 1}

, which is the reciprocal of wavelength

λ

and represents the number of wavelengths contained in a unit length.

G_{0}

is the roughness coefficient of pavement and the unit is

m^{3}

; Its size increases with the increase of pavement roughness.

p

is the frequency index

In the actual calculation, the following model is often used for the pavement excitation model

{\dot{z}}_{g} (t) = - 2 π f_{0} z_{g} (t) + 2 π n_{0} \sqrt{G_{0} v_{0}} ς (t)

(30)

where

f_{0} = 0.1 m

is the lower spatial cut-off frequency,

v_{0}

is the vehicle speed and

n_{0} = 0.1

is the reference spatial frequency. In this paper, the road grade is chosen as B-class and the pavement roughness coefficients

G_{0} = 64 \times 10^{- 6}

, and

ς (t)

is the Gaussian white noise with the noise power 0.5.

Firstly, the road input excitation is simulated by equation (30), and the road unevenness is related to the vehicle speed. Generally, the greater the speed, the greater the excitation. As road excitation is a random process, the system’s responses are also random processes. Therefore, we can use the root mean square (RMS) response of the system to evaluate the control strategy

R M S (X) = \sqrt{E (X^{2})}

(31)

X

can be sprung acceleration, road excitation or suspension deformation. Under the given values of parameters, the responses can be calculated.

The RMS of the sprung mass acceleration under different driving speeds is shown in Figure 6. From Figure 6, it is seen that the RMS of the sprung mass acceleration decreased a lot by both the LQR and the proposed NPC strategies. In all the speed ranges investigated, the reduction rate of the proposed optimal control strategy is higher than that of the traditional LQR. Combined with the time histories of sprung mass acceleration shown in Figure 7, it is seen that the response of the system with nonlinear control force has smaller peak response, which also shows that the proposed NPC control strategy has a better peak reduction effect.

Figure 6.

The RMS of the sprung mass acceleration under different driving speeds with B-class road ( $G_{0} = 64 \times 10^{- 6}$ ).

Figure 7.

Time histories of sprung mass acceleration under driving speed $v = 15 m / s$ .

In order to provide a more intuitive view of the effects of the linear and nonlinear control forces, the time histories of the total control force $u (t)$ , the linear control force $u_{l} (t)$ and the nonlinear control force $u_{n} (t)$ are shown in Figure 8. It is seen that the nonlinear control force is much smaller than the linear control force and only has a larger value at the peak of the system response, which demonstrates the mechanism of the nonlinear control force. The RMS of the sprung mass acceleration under A-Class road with $G_{0} = 32 \times 10^{- 6}$ is also shown in Figure 9, which has a similar effect.

Figure 8.

Time histories of the control force $u (t)$ and its decomposition $u_{l} (t)$ and $u_{n} (t)$

Figure 9.

The RMS of the sprung mass acceleration under different driving speeds with A-class road ( $G_{0} = 32 \times 10^{- 6}$ ).

In Figures 10 and 11, the suspension deformations under different driving speeds are exhibited under different road conditions. Compared with the uncontrolled system, it is seen that the deformations of suspension in LQR and NPC strategies have decreased a lot. The suspension deformation of NPC strategy is almost the same as that of LQR for different driving speeds. Figure 12 shows the time histories of the suspension deformation under B-class road with driving speed $v = 15 m / s$ , from which we can also see the control effect under the proposed NPC and the traditional LQR control strategies intuitively.

Figure 10.

The RMS of the suspension deformation under different driving speeds with B-class road ( $G_{0} = 64 \times 10^{- 6}$ ).

Figure 11.

The RMS of the suspension deformation under different driving speeds with A-class road ( $G_{0} = 32 \times 10^{- 6}$ ).

Figure 12.

Time histories of suspension deformation under driving speed $v = 15 m / s$ .

As it is shown in Figures 13 and 14, the road load of the system under the proposed NPC and the traditional LQR control strategy increases compared with that of the uncontrolled system. This is because the control force can only be applied between the sprung mass and the unsprung mass in the control of the vehicle system. However, the acceleration of the sprung mass, road excitation and suspension deformation of the control target affect each other. Under the given control requirements, it is difficult to reduce all the indexes simultaneously. Generally, the usual treatment is to decrease the acceleration of the sprung mass and ensure that road excitation and suspension deformation are within a reasonable range. Figure 15 shows the time histories of the road load driving speed $v = 15 m / s$ . It is seen that the road excitation of NPC strategy is less than that of LQR. Thus, the proposed nonlinear control strategy is more effective in reducing the input excitation force than LQR control strategy.

Figure 13.

The RMS of the road load under different driving speeds with B-class road ( $G_{0} = 64 \times 10^{- 6}$ ).

Figure 14.

The RMS of the road load under different driving speeds with A-class road ( $G_{0} = 32 \times 10^{- 6}$ ).

Figure 15.

Time histories of the road load under driving speed $v = 15 m / s$ .

According to the discussion above, the proposed NPC control strategy has a better performance in the control target of the sprung mass acceleration and the road load and has almost the same performance in the control of suspension deformation. In most cases, the sprung mass acceleration affects the ride comfort and the safety of the loaded goods. Therefore, it is the primary control target. From this point of view, the proposed nonlinear control strategy has advantages.

Figure 16 shows the mean square values of the optimal control force and passive control force in the optimal control force at different driving speeds. The time histories of the total control force $u (t)$ , the passive control force $u_{p} (t)$ and the active control force $u_{a} (t)$ are shown in Figure 17. It can be seen that by decomposing the optimal control force, the energy consumption to be provided will be reduced by about 30%, which means the energy input due to the optimal control force can be reduced a lot.

Figure 16.

The RMS of the optimal control force and the passive control force under different driving speeds.

Figure 17.

Time histories of the control force $u (t)$ and its decomposition $u_{p} (t)$ and $u_{a} (t)$ .

Conclusion

In this paper, the vibration control problem of the quarter-vehicle system is studied based on the dynamic programming method. By reasonably selecting the index function, an optimal NPC strategy is developed. The derived optimal control force will be adjusted nonlinearly according to the vibration responses of the vehicle system, which can provide cross large control force when the vehicle vibration is large. Compared with the LQR control strategy, the proposed control strategy has good performance in the main control indexes and effectively reduces the peak response of the system. Finally, the optimal control force is decomposed into active and passive control force, and only the active control force needs additional energy input, which can effectively reduce the energy consumption in actual control execution. The developed optimal NPC strategy in this paper has good application in the design of active vehicle suspensions.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work in the present paper is finished under financial support from the National Nature Science Foundation of China under grant No. 11872307, No. 11972293 and the National College Student Innovation and Entrepreneurship Training Program under grant No. 202110699167.

ORCID iD

Xudong Gu

Data Availability Statement

Not applicable. The numerical results in the paper are all calculated by the authors.

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Optimal nonlinear polynomial control of a quarter-vehicle suspension system under harmonic and random road excitations

Abstract

Keywords

Introduction

Dynamic model of the quarter-vehicle suspension system

Determinization and decomposition of the optimal NPC force

Formulation of the optimal control problem

Traditional LQR

Optimal nonlinear polynomial control

Decomposition of optimal control force

Numerical results

Harmonic excitation

Bumped road surface excitation

Random disturbance

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

Data Availability Statement

References