Experimental verification of active oscillation controller for vehicle drivetrain with backlash nonlinearity based on norm-limited SPSA

Abstract

To address vehicle drivetrain vibrations that cause discomfort and poor drivability, this study proposes a new active damping strategy with simple backlash compensation based on the simultaneous perturbation stochastic approximation (SPSA) with norm-limited update vector. First, an experimental device developed for a simplified drivetrain mechanism is demonstrated. A mechanism for reproducing both the contact mode and the backlash mode is included in the device. For the contact mode, a model-based $H_{2}$ controller is employed as the baseline damping strategy. Further, to mitigate the backlash effect, a simple algorithm based on mode-switching-based compensation is used with the $H_{2}$ controller. In particular, for the critical controller parameters, this article presents a systematic design approach to search for their optimal values. The key parameters, which are needed for the backlash and contact mode controllers, are simultaneously auto-tuned using norm-limited update vector-based SPSA, which ensures the stability in the iterative tuning. The novelty of this study is that both the backlash mode controller and the contact mode controller are simultaneously optimized by the improved version of SPSA, thus realizing a comprehensive auto-tuning design of an active drivetrain damping system. Finally, the active controller is experimentally verified using the actual test device. Comparative studies show that the proposed approach significantly reduces drivetrain vibrations and is robust against fluctuations in the backlash.

Keywords

Powertrain active vibration control drivetrain backlash optimization simultaneous perturbation stochastic approximation vibration vehicle system

Introduction

Vibrations cause noise, fatigue failure, and discomfort, providing an adverse influence for mechanical systems. The need for vibration suppression technology has grown as a result of the current movement toward downsizing, greater comfort, weight reduction, and superior performance.^1–3 Due to their potential to deliver strong dampening effects for intricate mechanical systems, active vibration damping has received a lot of research attention.

Vehicle drivetrains also require vibration control as one of the essential important technologies to maintain their functions. Vibrations induced by a sudden driving force change in the drivetrain⁴ of a vehicle persist as a serious problem. These vibrations not only degrade the comfort and drivability but also reduce the durability of the drivetrain components. As one of the promising countermeasures, coupling a nonlinear dynamic vibration absorber in parallel to a drivetrain mechanism can be considered. For example, a nonlinear energy sink (NES)^5,6 can efficiently dissipate vibrational energy through structural damping by redistributing the vibration energy within the modal space of the primary structure or passively absorb a part of this energy from the drivetrain in a nearly irreversible form. The advantage is that it can operate over a broad range of frequencies. In the successful application to attenuation of drivetrain vibrations,⁵ three types of NESs were investigated: a third-order NES stiffness, a vibro-impact nonsmooth NES and a NES with fifth-order smooth nonlinear stiffness.

The use of NESs is categorized as passive vibration control, whereas this study focuses on developing the active vibration control method for drivetrain vibrations. NES is a lightweight absorber effective over a broad range of frequencies; however, active vibration control is also needed in order to obtain higher vibration suppression performances without increasing the weight of the driveline at all. Compared to passive control such as NESs, active control can provide stronger damping effects based on control algorithms by using only an existing actuator and do not need to incorporate additional palliative devices, thus reducing the associated costs. For the active vibration control of the vehicle drivetrain, some advanced control strategies have been presented, such as compensation for a disturbance based on an adaptive observer⁷ for damping control of internal combustion engine powered powertrains. Further, an adaptive active control scheme has been developed by combining model prediction and Butterworth filter control to solve the torsional vibration problem.⁸ In addition, the filtered-x least mean square algorithm⁹ and a model reference controller have been implemented.¹⁰ Moreover, a previous work has improved vehicle drivability by using a linear quadratic tracking controller with a model reference adaptive technique.¹¹

However, the aforementioned studies have not considered the nonlinear backlash existing in the gears of drivetrains. Backlash in a drivetrain leads to an undesirable phenomenon, which significantly increases the vibration amplitude.^12,13 The drivetrain dynamics switch between two conditions: “contact mode,” during which mechanical contact occurs, and “backlash mode.” Backlash mode disconnects the output side (wheel side) from the input side (actuator side) owing to the traversal in backlash. This phenomenon, that is, switching of the dynamics modes, also leads to higher complexity of control system structures and increases the designer's burden in terms of tuning tasks.

For a drivetrain with backlash, a hybrid strategy of a feedforward-feedback controller and a robust disturbance observer has been introduced to compensate for the backlash.¹⁴ Sliding mode control-based approaches have also been investigated for a drivetrain with nonlinearity.^15,16 Another study has adopted a combination of linear quadratic regulator (LQR) and a notch filter for the contact mode while dealing with the backlash mode using a proportional integral differential (PID) controller.¹⁷

However, in the existing studies, the following shortcomings and rooms for improvement remain as:

With respect to controller parameters that must be determined by designers, there has been no attempt to adopt efficient optimization for auto-tuning of the parameters.

The mechanism of compensation for backlash can be improved by making it simpler with a lower online computational load. Moreover, versatility is desired to facilitate its use with a wide variety of model-based linear controllers for satisfactory vibration reduction in the contact mode.

The optimization design is closely related to the complicated powertrain dynamics.^18,19 Some researchers have employed efficient optimization algorithms for their applications, such as a gear shifting fuzzy controller and an engine management approach optimized using an advanced genetic algorithm (GA) with interactive adaptive-weight,^20,21 multi-objective optimization,¹⁸ lookup-table-based torque control,²² and design of a torsional vibration damper (a dual mass flywheel with an internal damper).²³ The aforementioned works suggest the following important aspects: (1) effectiveness of iterative offline simulations based on powertrain dynamics models; (2) necessity of application of efficient optimization algorithms (rather than the designer's subjective selection) to automatically search for the controller parameters.¹⁹

Focusing on active vibration controls of a drivetrain, their controller design can be framed as optimization problems. For the backlash mode, a previous study has developed a new clunk controller with model-based torque shaping.¹² It consists of a soft-landing reference governor and a proportional-derivative (PD) baseline controller to track the modified reference. The ingenious point is that the competing design requirements, that is, reducing the clunk when the backlash traversal finishes and allowing the backlash to be crossed rapidly, are defined as an optimal control problem. However, a contact mode controller^12,24 and the aforementioned PD gains have not been included in auto-tuning optimization, although they are critical for addressing transient oscillations in a drivetrain. In addition, the optimization solution can be obtained more conveniently by introducing an efficient optimization algorithm. For the minimization of an objective function based on the root mean square (RMS) of drivetrain vibration responses, the LQR controller has been optimized offline.²⁵ The selection of the critical weighting matrices has been optimized with the GA. However, this work does not consider the backlash nonlinearity in a drivetrain. Another study has proposed a safe calibration framework based on Bayesian optimization to obtain the optimal controller gain.²⁶ However, the backlash has not been explicitly addressed in the controller. Among previous approaches, the most popular choice for powertrain dynamics control is model predictive control (MPC).²⁷ To address undesired oscillations due to backlash, the online MPC, which is a quadratic form-defined optimal control in a receding horizon, has been implemented.²⁸ In the literatures,^29,30 switching strategies for multiple predictive controllers have addressed the nonlinearity in a powertrain. The MPC controller can produce ideal responses including little error states such as excessive jerks and transient oscillations.³¹ Although MPC has achieved promising results, one of its main drawbacks is its high computational burden online. This makes it difficult to implement MPC on real-time applications without sacrificing the control performance.¹² Furthermore, the need to implement many different controllers makes the control structure more complicated, involving high requirements for the control module memory.

In summary, previous studies have not applied an optimization process for the contact mode controller or the backlash mode controller, that is, auto-tuning of the controller parameters is insufficient. In addition, few studies have focused on developing a simple and versatile backlash compensation strategy with lower online computational loads. Therefore, the novelty and principal contributions of our work are to bridge these gaps.

Based on our previous works,^13,32 this study proposes an improved version of active drivetrain vibration control with compensation for backlash. In this article, the active controller is verified using an actual test device of a simplified drivetrain mechanism with backlash. We employ the reduced drivetrain configuration similar to the existing work⁵: this is a reduced lumped parameter model composed of a few inertias connected via stiffness. The model in the work⁵ incorporates all the nonlinearities in the equation of motion such as the nonlinear stiffness of NESs, the aerodynamic drag torque and the tire rolling resistance, whereas this study only considers the dead zone property due to the backlash nonlinearity. This is because we need to focus on the effect of backlash and make it easier to evaluate the efficacy of the proposed backlash compensation. For considering the backlash and contact modes, the proposed control system is constructed as a simple compensation algorithm based on switching of the control modes, which can be applicable for a single model-based vibration controller. It can reduce the shock when a drivetrain is changed from the backlash mode to the contact mode while allowing the backlash to be traversed rapidly. Even though its versatility has been proven through some application examples with various model-based linear controllers,^13,32 its critical controller parameters require manual setting based on trial-and-error tuning by the designer.³² In other words, an auto-tuning design based on an optimization algorithm needs to be newly introduced. The contributions of this article can be summarized as follows.

As a novelty of this study compared to previous studies, both the backlash mode controller (i.e. a soft-landing compensator) and the contact mode controller (i.e. a baseline $H_{2}$ vibration suppression) are simultaneously optimized, resulting in the realization of an auto-tuning design. This auto-tuning optimization reduces the designer's burden as well as the cost of trial-and-error procedures.

As a tuning algorithm, the simultaneous perturbation stochastic approximation with a norm-limited update vector (hereafter referred to as “NLSPSA” in this article) is employed. This work is the first to propose the application of NLSPSA for active damping of a drivetrain. This enables a high calculation efficiency while guaranteeing the stability of the iterative tuning process.

To realize compensation for backlash, a simpler algorithm based on switching of the control modes is combined with a single output feedback $H_{2}$ controller. Its advantages include not only lower online computational requirements, but also reduction in complexity of the control structure.

Compared to a previous study that has only performed simulation tests,³³ this study experimentally confirms the effectiveness of the proposed method using an actual test device of a simplified drivetrain mechanism with backlash.

Basic experimental device

Simplified drivetrain with backlash

As the controlled object, this study considers a basic experimental device. By simplifying the actual driveline mechanism, this allows us to concentrate on the oscillation phenomena brought on by backlash effects and an abrupt shift in driving force. The device specifications are summarized in Table 1. The mechanical model and the actual test rig are given in Figure 1(a) and (b), respectively. The structure is a three-degree-of-freedom translational vibration system that is equivalent to a simplified drivetrain.^13,34 The three mass points ( $M_{E}$ , $m_{G}$ , $M_{B}$ ) are connected via springs and dampers. Backlash is produced as a dead-zone band by providing a large adjustable space between the plate springs on both sides of $m_{G}$ . Further details on the experimental device, such as the simplifications, can be found in references.^13,35 The motor acts as the actuator to realize the control input.

Figure 1.

Basic experimental device: (a) drivetrain model; (b) real test device; (c) contact mode; and (d) backlash mode.

Table 1.

Specifications of the test device.

Parameter	Value	Unit
$K_{D}$	$2.2 \times 10^{4}$	N/m
$K_{C}$	$660.0$	N/m
$K_{G}$	$5.3 \times 10^{4}$	N/m
$m_{G}$	$0.039$	Kg
$M_{B}$	$0.232$	Kg
$M_{E}$	$1.04$	Kg
$C_{C}$	$0.1$	Ns/m
$C_{c l}$	$1.5$	Ns/m
$C_{G}$	$36.0$	Ns/m
$C_{D}$	$12.5$	Ns/m

Figure 1(c) and (d) show the contact mode and the backlash mode, respectively. In the contact mode, the mechanical connection is established between the actuator and the vehicle body side. By contrast, the backlash mode is a disconnected situation during the backlash traversal.

State-space representation for modeling

For the purpose of deriving a baseline vibration controller used in the contact mode, we need a linearized plant model because the experimental device involves nonlinearities such as backlash. As indicated below, the previous works have already derived the linear state-space representation of the plant dynamics in the form of the time-varying state equation.^13,35

{\dot{x}}_{p} = A_{p} x_{p} + B_{p 1} w_{p} + B_{p 2} u

(1)

y_{p} = C_{p} x_{p} + D_{p 1} w_{p} + D_{p 2} u

(2)

Each coefficient matrix and the state vector are expressed as follows:

\begin{aligned} A_{p} = [\begin{matrix} 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ - \frac{(K_{D} + K_{C})}{M_{B}} & \frac{K_{D}}{M_{B}} & 0 & - \frac{(C_{D} + C_{C})}{M_{B}} & \frac{C_{D}}{M_{B}} & 0 \\ \frac{K_{D}}{m_{G}} & - \frac{(S w K_{G} + K_{D})}{m_{G}} & \frac{S w K_{G}}{m_{G}} & \frac{C_{D}}{m_{G}} & - \frac{(S w C_{G} + C_{D})}{m_{G}} & \frac{S w C_{G}}{m_{G}} \\ 0 & \frac{S w K_{G}}{M_{E}} & - \frac{S w K_{G}}{M_{E}} & 0 & \frac{S w C_{G}}{M_{E}} & - \frac{(S w C_{G} + C_{c l})}{M_{E}} \end{matrix}] \\ B_{p 1} = [\begin{matrix} 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & \frac{1}{M_{B}} \\ \frac{1}{m_{G}} & 0 \\ - \frac{1}{M_{E}} & 0 \end{matrix}], B_{p 2} = [\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ \frac{1}{M_{E}} \end{matrix}], C_{p} = [\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 \end{matrix}], D_{p 1} = 0, D_{p 2} = 0 \end{aligned}

(3)

x_{p} = {[\begin{matrix} X_{B} & x_{G} & X_{E} & {\dot{X}}_{B} & {\dot{x}}_{G} & {\dot{X}}_{E} \end{matrix}]}^{T}

(4)

In equations (1) and (2), u is the control input and

w_{p}

is the disturbance including the force due to the backlash. By switching the parameter

S w

, the dead-zone characteristic of the backlash can be expressed. The switching rule is shown below.^36,37 Here, a force, which originates from the spring

K_{G}

, is denoted as F. The width of a dead zone by the backlash effect is denoted as

| b l |

S w

is set to 1.0 in design of the baseline controller.

\begin{aligned} F = O K G + S w \cdot K_{G} \cdot Δ X Δ X = X_{E} - x_{G} \\ S w = {\begin{matrix} 1, Δ X > | b l | \\ 1, Δ X < - | b l | \\ 0, | Δ X | \leq | b l | \end{matrix} \\ O K G = {\begin{matrix} 0, | Δ X | \leq | b l | \\ - | K_{G} \times | δ | |, Δ X > | b l | \\ | K_{G} \times | δ | |, Δ X < - | b l | \end{matrix} \end{aligned}

(5)

The experimental device, which involves nonlinear characteristics such as backlash, has been modeled by the time-varying linear state equation, which is more detailed in the previous study.¹³

Active vibration control with compensation for backlash

The entire control structure consists of two main parts: the contact mode control (baseline vibration suppression) and the backlash mode control (simple mode-switching-based compensation). The control structure is shown in Figure 2.

Figure 2.

Active damping strategy composed of backlash and contact mode controls.

Model-based $H_{2}$ vibration controller

As the baseline controller, a model-based linear $H_{2}$ controller is designed to achieve high damping effects and excellent transient responses.^38,39

Figure 3 shows the block diagram used to design the $H_{2}$ baseline controller. Here, $z_{y}$ and $z_{u}$ denote the controlled variables with respect to transient response of the oscillation of the vehicle body and penalty on the amount of the control input (energy saving), respectively. The vehicle vibration y indicates the observed output. u and w denote the control input and disturbance containing the force owing to the effect of backlash, respectively. $P_{c}$ , $F_{H i g h p a s s}$ , and $F_{L o w p a s s}$ denote the controlled object including an integrator, a high-pass filter, and a frequency transfer function of a low-pass filter including a first-order lag element, respectively.^13,32 The expanded controlled object $G (s)$ is constructed to eliminate unmodeled uncertain dynamics in a high-frequency band.¹³ As discussed later, $Q_{k}$ and $R_{k}$ are weighting constants, that is, critical tuning parameters, for the controller design. k is the number of iterations for the optimization, which is explained later.

Figure 3.

Generalized plant used to design the contact mode controller.

From the external input to $z = {[\begin{matrix} z_{y} & z_{u} \end{matrix}]}^{T}$ , a transfer function $T_{z w} (s)$ can be defined. Therefore, the baseline vibration controller is designed such that the $H_{2}$ norm $‖ T_{z w} (s) ‖_{2}^{2}$ is minimized.³⁸ The design approach is implemented utilizing the Robust Control Toolbox and Control System Toolbox, which are installed n MATLAB.

When evaluating the $H_{2}$ -norm criterion, $z_{u}$ and $z_{y}$ are defined as the multiplication of the control input $u_{G}$ and the vehicle vibration $y_{G}$ from the expanded controlled object by the evaluation weights $\sqrt{R_{k}}$ and $\sqrt{Q_{k}}$ , respectively, as follows:

z_{y} = \sqrt{Q_{k}} \cdot y_{G}

(6)

z_{u} = \sqrt{R_{k}} \cdot u_{G}

(7)

Equations (6) and (7) imply that

Q_{k}

and

R_{k}

are the critical parameters affecting the vibration attenuation performance. Because they amplify

y_{G}

and

u_{G}

via the multiplication, how much to highlight each of the vibration response and the control input limitation is directly determined in the

H_{2}

-norm minimization. Consequently, trial-and-error adjustments for effective balancing between

Q_{k}

and

R_{k}

have been imposed in our previous studies.³²

Backlash compensation by simple control mode switching algorithm

During the backlash mode, the aims of our backlash compensation strategy are as follows:

to push the backlash to travel from negative contact toward positive contact as soon as possible for good responsiveness;

to reduce the impact (shock) when the drivetrain is transferred from the backlash mode to the contact mode and the backlash traversal finishes by soft landing.

To meet these competing requirements, the compensation algorithm based on switching of the control modes is applied to the baseline

H_{2}

vibration controller.^13,16 This is outlined in Figure 4. The backlash can be traversed quickly yet smoothly with the help of this compensation, which also eliminates the accumulation (buildup) of control errors in the backlash.

Figure 4.

Simple control mode switching algorithm.

In equations (8) to (10) and Figure 4, as the contact mode controller that operates during the mechanical contact, the normal $H_{2}$ vibration control is performed. The low-frequency resonance should be attenuated during the contact mode.

In equations (11) to (15), the compensation during the backlash traversal is performed by the backlash mode controller. Here, by switching the target signal $r (n)$ to a small positive value $r_{k}^{B L}$ that is a key compensation reference signal (parameter) for realizing soft landing, the backlash can be traversed gently and rapidly. This is because the control error $e_{B L} (n)$ originating from $r_{k}^{B L}$ makes the $H_{2}$ controller compute the moderate input command for the backlash traversal. The dead-zone effect due to the backlash causes excessive control errors to accumulate in the controller, resulting in a shock with the calculation of a large control command. Therefore, anti-windup needs to be introduced. Specifically, for the discretized controller $(A_{c k}, B_{c k}, C_{c k}, D_{c k})$ , the updating of the state vector $x_{c} (n)$ is temporarily stopped by anti-windup. This processing avoids the undesired effect caused by the dead zone in the backlash, that is, the cumulation (buildup) of the control errors. The controller with a frozen state vector no longer unnecessarily accelerates the actuator during the backlash traversal.

Due to this compensation, we can prevent unnecessary excessive control inputs during the backlash traversal, resulting in a considerably reduced shock with a collision in the backlash. Refer the previous studies^13,32 for further details on the basic idea of this compensation.

Note that the same single $H_{2}$ controller is commonly used in both the backlash mode and the contact mode. Its discretized state space representation is shown as $(A_{c k}, B_{c k}, C_{c k}, D_{c k})$ . This simple mechanism means that multiple controllers according to the mode switching do not need to be prepared.

(Contact mode control):

x_{c} (n + 1) = A_{c k} x_{c} (n) + B_{c k} e (n)

(8)

u_{C O} (n) = C_{c k} x_{c} (n) + D_{c k} e (n) + K_{C} r (n)

(9)

e (n) = r (n) - X_{B} (n)

(10)

(Backlash mode control):

x_{A W} (n + 1) = A_{c k} x_{c} (n) + B_{c k} e_{B L} (n)

(11)

x_{A W} (n) = z^{- 1} [x_{A W} (n + 1)] (x_{A W} (n) = x_{c} (n))

(12)

x_{c} (n + 1) = x_{A W} (n) i . e ., Substitute x_{A W} (n) for x_{c} (n + 1)

(13)

u_{B L} (n) = C_{c k} x_{c} (n) + D_{c k} e_{B L} (n) + K_{C} r_{k}^{B L}

(14)

e_{B L} (n) = r_{k}^{B L} (n) - X_{B} (n) = r_{k}^{B L} - X_{B} (n)

(15)

To identify the moment at which the drivetrain is transferred from the backlash mode to the contact mode and the backlash traversal finishes, a vehicle jerk can be utilized. The vehicle body acceleration with regard to time is differentiated to produce a jerk. Specifically, we need a threshold condition as follows:

| J e r k (n) | \geq T h r e s h o l d

(16)

With discontinuous change of the vehicle acceleration owing to the mode switching, the steep jerk value is well suited for the judgment because it can be observed independently of the backlash length fluctuation. Moreover, the use of the vehicle body jerk contributes to simplification of the compensation structure.

The soft-landing reference $r_{k}^{B L}$ during the backlash mode significantly affects the vibration control performance. If this value is too large, unsuccessful soft landing will cause an impact (shock) because an excessive control input command will be given to the actuator. Meanwhile, if $r_{k}^{B L}$ is too small, the responsiveness will be poor. Achieving an effective balance is a challenging task, and it has been manually tuned with trial-and-error adjustments in our previous studies.^13,32

Auto-optimization scheme for the backlash and contact mode controllers

SPSA with norm-limited update vector

The proposed tuning approach employs SPSA because of its calculation efficiency. The algorithm of SPSA updates a design variable $θ \in R^{p}$ so that a non-negative scalar loss function $L (θ)$ is minimized. At the $k$ th iteration, the update rule is given as shown in equation (17).⁴⁰ The gradient $g (θ) = \partial L / \partial θ = \nabla L (θ)$ is stochastically estimated. The estimated gradient is denoted as ${\hat{g}}_{k} (θ_{k}) \in R^{p}$ in the $k$ th iteration. The $i$ th element of ${\hat{g}}_{k}$ is ${\hat{g}}_{k, i}$ . $Δ_{k} \in R^{p}$ is a random perturbation vector, which contains each component $Δ_{k, i}$ . A typical selection of $Δ_{k} \in R^{p}$ is Bernoulli distribution, which generates $\pm 1$ .⁴¹ For all $k, i$ , $Δ_{k, i}$ with finite inverse moments $E [| Δ_{k, i}^{- 1} |]$ should be symmetrically distributed about 0 and independent.^40,42

θ_{k + 1} = θ_{k} - a_{k} {\hat{g}}_{k} (θ_{k})

(17)

{\hat{g}}_{k, i} (θ_{k}) = \frac{L (θ_{k} + c_{k} Δ_{k}) - L (θ_{k} - c_{k} Δ_{k})}{2 c_{k} Δ_{k, i}}

(18)

Δ_{k} = {[\begin{matrix} Δ_{k, 1} & Δ_{k, 2} & \dots & Δ_{k, p} \end{matrix}]}^{T}

(19)

Equations (20) to (23) are the conditions to be satisfied in selections of the gain sequences

a_{k}

and

c_{k}

.⁴⁰ With positive constants a, A, c,

α

, and

γ

a_{k}

and

c_{k}

are typically set, as indicated in equations (24) and (25).^40,42

a_{k} > 0, c_{k} > 0

(20)

lim_{k \to \infty} a_{k} \to 0, lim_{k \to \infty} c_{k} \to 0

(21)

\sum_{k = 1}^{\infty} a_{k} = \infty

(22)

\sum_{k = 1}^{\infty} \frac{a_{k}^{2}}{c_{k}^{2}} < \infty

(23)

a_{k} = \frac{a}{{(A + k)}^{α}}

(24)

c_{k} = \frac{c}{k^{γ}}

(25)

High calculating efficiency is one of the SPSA's benefits. According to equation (18),

{\hat{g}}_{k}

can be estimated using just two L computations (i.e. two loss function measurements), and it is not reliant on the design variable

θ_{k}

's size p.⁴⁰ Calculating the loss function value is typically the most challenging work (time and cost consuming step) in many engineering optimizations.^43,44 Hence, especially in multi-dimensional optimizations, SPSA is highly efficient because few loss function computations are needed at each iteration. This benefit has resulted in practical controller tuning applications.⁴⁴

However, several researchers have indicated that the computation stability should be improved to apply SPSA for practical applications.⁴⁵ An effective approach is to explicitly limit the updating amount of design variable.⁴⁵ Therefore, the term $a_{k} {\hat{g}}_{k} (θ_{k})$ is saturated as follows:

θ_{k + 1} = θ_{k} - δ (a_{k} {\hat{g}}_{k} (θ_{k}))

(26)

δ (a_{k} {\hat{g}}_{k} (θ_{k})) = [\begin{matrix} sgn (a_{k} {\hat{g}}_{k, 1}) min {| a_{k} {\hat{g}}_{k, 1} |, d} \\ sgn (a_{k} {\hat{g}}_{k, 2}) min {| a_{k} {\hat{g}}_{k, 2} |, d} \\ ⋮ \\ sgn (a_{k} {\hat{g}}_{k, p}) min {| a_{k} {\hat{g}}_{k, p} |, d} \end{matrix}]

(27)

In addition, by restricting

Δ_{k} \in R^{p}

to a Bernoulli distribution, equation (27) can be simply rewritten as follows:

δ (a_{k} {\hat{g}}_{k} (θ_{k})) = min {| \frac{a_{k} Δ L}{2 c_{k}} |, d} [\begin{matrix} sgn (Δ L / Δ_{k, 1}) \\ sgn (Δ L / Δ_{k, 2}) \\ ⋮ \\ sgn (Δ L / Δ_{k, p}) \end{matrix}]

(28)

Δ L = L (θ_{k} + c_{k} Δ_{k}) - L (θ_{k} - c_{k} Δ_{k})

(29)

This improved version is referred to as SPSA with a norm-limited update vector (NLSPSA). Here,

δ : R^{p} \to R^{p}

in equation (26) shows the saturation function, where

d > 0

denotes the scalar-valued constant. The remarkable point of this modified algorithm is that a theoretical analysis of the convergence of equation (26) has been provided.⁴⁵ Note that each element of

δ (a_{k} {\hat{g}}_{k} (θ_{k}))

is limited within

[- d, d]

; thus, the instability of the iterative optimization process is avoided. This article is the first to demonstrate the application of NLSPSA for controlling a drivetrain with backlash, to the best of our knowledge.

Norm-limited SPSA-based parameter tuning scheme

For the backlash mode and the contact mode, a parameter tuning scheme is newly proposed on the basis of NLSPSA. The tuning scheme is given in Figure 5. In the following, “design variables ( $θ_{k} \in R^{p}$ )” refers to the parameters that NLSPSA directly updates in equation (26). “Tuning parameters” refers to the parameters that designers must determine to derive a controller. For evaluating the active damping performance, we need to set a loss function $J (θ_{k})$ , which takes non-negative values. In Figure 5, NLSPSA searches for the $θ_{k}$ that minimizes $J (θ_{k})$ . $C_{s t o p} \in N$ indicates a stop criterion, that is, the total number of iterations.

Figure 5.

Tuning scheme for the controller design parameters using SPSA with norm-limited update vector. SPSA: simultaneous perturbation stochastic approximation.

An advantage of the proposed method is that both the contact mode controller (i.e., the $H_{2}$ baseline controller) and the backlash mode controller (i.e. the soft-landing compensator) can be simultaneously and automatically optimized. The proposed approach can reduce not only the subjective selection of the critical controller parameters by the designer, but also the load on the designer caused by laborious trial-and-error processes.

Tuning parameter expression

The proposed method automatically tunes all of $Q_{k}$ , $R_{k}$ , and $r_{k}^{B L}$ in equations (6), (7), and (15), resulting in the simultaneous optimization for the compensator in the backlash mode and the $H_{2}$ controller in the contact mode. That is, the tuning parameters are $Q_{k}$ , $R_{k}$ , and $r_{k}^{B L}$ in this research. Equations (30) to (32) express $Q_{k}$ , $R_{k}$ , and $r_{k}^{B L}$ using the design variable $θ_{k}$ .

Q_{k} = | θ_{k, 1} | Q_{0}

(30)

R_{k} = | θ_{k, 2} | R_{0}

(31)

r_{k}^{B L} = | θ_{k, 3} | r_{0}^{B L}

(32)

θ_{k, i}

represents the

i

th component of

θ_{k}

;

Q_{0}

is the initial value of

Q_{k}

, and

R_{0}

is that of

R_{k}

. Similarly,

r_{0}^{B L}

is the initial value of

r_{k}^{B L}

. Therefore,

θ_{k}

is a three-dimensional variable. For the backlash and contact modes, all the critical design variables are stored in

θ_{k}

. The initial values of those are set as

Q_{0} = 1.0

R_{0} = 1.0

, and

r_{0}^{B L} = 1.0 \times 10^{- 1}

in this research.

Loss function in optimization

The proposed tuning design uses the loss function $J (θ_{k})$ shown in equation (33).

J (θ_{k}) = J_{k} = \frac{1}{N} {W_{y} (\sum_{n = 1}^{N} {\tilde{y}}_{k, n}^{2}) + W_{u} (\sum_{n = 1}^{N} u_{k, n}^{2})}

(33)

Here,

{\tilde{y}}_{k, n}

is the control error between an ideal target response and the vibration response (observed output) at the

k

th iteration at the

n

th sampling step.

u_{k, n}

indicates the control input component for vibration suppression at the

k

th iteration at the

n

th sampling step. In each simulation, the number of sampling points is denoted as N. The scaling factors on

{\tilde{y}}_{k, n}

and

u_{k, n}

are

W_{y}

and

W_{u}

, respectively. By taking into account the scale gap between the two terms in equation (33), they are set to

W_{y} = 10^{3}

and

W_{u} = 1

J (θ_{k})

is therefore defined as the mean square of

{\tilde{y}}_{k, n}

plus that of

u_{k, n}

derived from the vibration suppression controller with

θ_{k}

Parameter tuning condition

Table 2 lists the conditions for NLSPSA shown in equations (17) to (29). They are determined according to the guideline on SPSA implementation.⁴¹ The simulation time in each iteration is 4.0 s.

Table 2.

Conditions for NLSPSA in the proposed method.

Properties	Value
Number of iterations	$100$
$α$	$0.602$
$θ_{0}$	${[\begin{matrix} 1 & 1 & 1 \end{matrix}]}^{T}$
$c$	$1.0 \times 10^{- 1}$
$a$	$7.0 \times 10^{- 4}$
$A$	$1.0 \times 10^{0}$
$d$	$40$
$γ$	$0.101$
$Δ_{k}$	Bernoulli distribution with probability of 1/2 for each $\pm 1$

NLSPSA: Norm-limited update vector-based simultaneous perturbation stochastic approximation.

Tuning results

Here, the tuning results obtained by the proposed method are discussed. Figures 6 and 7 show the tuning histories of each tuning parameter and the value of $J (θ_{k})$ by iteration, respectively. Note that Figure 6(c) shows that $r_{k}^{B L}$ converges to a small non-zero positive value.

Figure 6.

Tuning history of each tuning parameter: (a) Q_k; (b) R_k; and (c) $r_{k}^{B L}$ .

Figure 7.

Tuning history of the loss function.

Figure 7 means that the auto-tuned (resultant) controller suppresses vibrations sufficiently. The successful tuning history is observed in Figure 7, where the initial large value of $J (θ_{k})$ eventually converges to a small positive value. Table 3 summarizes the tuning results shown in Figures 6 and 7. These results quantitatively reflect the improvement of the control performance by iteration.

Table 3.

Tuning result of the CO and BL controller parameters (mean of 5 trials).

Properties	Value
$Q_{100}$	$20.9217$
$R_{100}$	$31.4235$
$r_{100}^{B L}$	$0.008371397$
$J_{0}$	$2.0134 \times 10^{4}$
$J_{100}$	$4.197$

Experimental verifications

Setting of experimental verifications

In this study, the active damping system is implemented on the experimental system, and the effectiveness is validated using the actual test device. Figure 8 shows the closed-loop diagram of the experimental verifications. The aim of the controllers is to damp the vehicle vibration $X_{B}$ and thus realize an ideal target response. The vehicle body vibration is measured with a laser sensor (KEYENCE, IL-300). The measured signal is amplified by a signal amplifier and is subsequently sent to a real-time controller. A digital signal processor (DSP: iBIS DSP7101A) calculates the control input commands based on the measured output. The control commands output from DSP is delivered to a servo amplifier to drive the actuator (i.e. motor). In addition, the measured vibration signals are recorded by a data-collecting device (CATEC INC, Cat System). Because the vehicle vibration frequency to be suppressed is about 4 Hz in this study, a sampling frequency of the experimental system was set as about 3 kHz. Therefore, the sampling frequency has sufficient bandwidth for the target resonance to be controlled and realizes a fast control cycle.

Figure 8.

Closed-loop diagram of the experimental system.

Experimental results and discussion

Figure 9 shows the experimental results. The upper and lower graphs show the time responses of the vehicle body vibrations and the control inputs, respectively. The black line represents the response without any active vibration control. The green line represents the ideal response to be realized. The red line represents the experimental result obtained by the proposed method.

Figure 9.

Vehicle body vibration response and control input measured by the experimental verification.

Figure 9 clearly demonstrates the high control performance achieved by the proposed active damping strategy. Compared to the open-loop response represented by the black line, the excellent transient characteristic is realized as indicated by the red line, making the controlled result close to the ideal target response. The vibration appearing after 2.0 s indicated by the black line is hardly suppressed, whereas that indicated by the red line is considerably attenuated.

According to the control input values near 2.0 s during the backlash traversal, we can see that the controller successfully switches from the backlash mode to the contact mode.

Comparison with the case without backlash compensation

For comparative investigations, only a $H_{2}$ controller that does not involve any backlash compensation is also implemented. Figure 10 shows the comparison result. The response in the case without backlash compensation is indicated by the blue line.

Figure 10.

Comparison between the proposed system and case without compensation for backlash.

The blue line shows a large overshoot as well as residual vibrations between 2.0 s and 3.0 s. This poor performance is due to the absence of backlash compensation.

Meanwhile, the proposed method, represented by the red line, improves the performance by suppressing the overshoot and residual vibration amplitude. This result proves that the aim of our backlash compensation strategy is achieved by $r_{k}^{B L}$ with the properly tuned value. Note that this key parameter was simultaneously auto-tuned together with the contact mode $H_{2}$ controller. The proposed tuning approach reduces the burden of designers in terms of subjective tuning tasks.

Table 4 summarizes the quantitative comparison between the experimental results presented above. The 2-norm of the control error between the ideal response and each vibration response is shown. The notable point is that the smallest value is given by the proposed approach. The 2-norm by the proposed approach is reduced by 51.1688% and 74.4953% compared to “Without compensation for backlash” and “Without control”, respectively.

Table 4.

Quantitative performance comparison based on 2-norm of the control error between the ideal response and each vibration response.

Each control result	Proposed method	Without compensation for backlash	Without control (open loop response)
Values of the 2-norm	$0.1868$	$0.3826$	$0.7326$

Robustness against plant variation

The robustness of the controller tuned by the proposed approach is verified in this section. For this purpose, a fluctuation of $- 50 %$ was newly set for the length of the backlash of the actual test device. Through the application of the same controller as that used in the previous sections to this fluctuated device, the robustness against changes in the backlash can be validated.

Figure 11 shows the comparison between the proposed approach and the open-loop response under the fluctuation in the actual backlash. Despite the fluctuation, the high vibration suppression is retained as indicated by the red line.

Figure 11.

Vehicle body vibration response and control input with the fluctuation of the backlash.

Compared to the case without backlash compensation as shown in Figure 12, the importance of addressing backlash is also observed. The reduced overshoot as well as the lower residual vibrations reflect the robustness of the backlash compensation. Even though the backlash is changed, the jerk can be observed independently of the fluctuation. Therefore, it allows switching of the control modes, leading to robust backlash compensation.

Figure 12.

Comparison between the proposed approach and the case without backlash compensation when there is a fluctuation in the backlash.

Table 5 summarizes the 2-norm of each vibration response when the backlash is changed. The 2-norm by the proposed approach is reduced by 60.6790% and 77.5419% compared to “Without compensation for backlash” and “Without control”, respectively. These results quantitatively demonstrate the robust performance of the active vibration controller.

Table 5.

2-norm of each experimental result with the fluctuation in the backlash.

Each control result	Proposed method	Without compensation for backlash	Without control (open loop response)
Values of the 2-norm	$0.1475$	$0.3751$	$0.6568$

The limitation of this study is that it is necessary to further verify the robustness of the controller tuned by the proposed approach. Severer changes should be considered under additional conditions of the drivetrain dynamics.

Conclusion

For a drivetrain mechanism with nonlinear backlash, this study proposed a norm-limited SPSA-based optimized active vibration controller. First, a basic experimental device reflecting a simplified drivetrain was demonstrated. The mechanism reproduces both the contact mode and the backlash mode. For the contact mode, a model-based $H_{2}$ controller was employed as the baseline damping strategy. For backlash compensation, a simple control mode switching algorithm was applied to the $H_{2}$ controller. This reduces the shock when the drivetrain is transferred from the backlash mode to the contact mode while allowing the backlash to be traversed rapidly. In our novel systematic design algorithm, the key controller parameters that influence the transient vibration attenuation and the compensation for backlash were comprehensively auto-tuned with reduced burden. This was realized on the basis of NLSPSA with high calculation efficiency and stability of the iterative tuning process. Finally, the controller tuned by the proposed approach was experimentally verified with the actual test device. The experimental results demonstrated that the proposed damping strategy significantly improves the transient response. Moreover, the robustness of the controller against changes in the backlash was experimentally confirmed.

In the future, severer fluctuations will be tested for additional conditions of the drivetrain dynamics. In addition, a more robust controller will be developed by introducing an SPSA-based online tuning mechanism into the active damping strategy.

Footnotes

Declaration of conflicting interests

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

Funding

The authors disclosed receipt of the following financial supports for the research, authorship, and/or publication of this article: This work was partly supported by the Transmission Research Association for Mobility Innovation (TRAMI) (23B3-01) and the Japan Society for the Promotion of Science (JSPS) KAKENHI (grant numbers 22K20396; 23K13273).

ORCID iDs

Heisei Yonezawa

Ansei Yonezawa

Itsuro Kajiwara

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Experimental verification of active oscillation controller for vehicle drivetrain with backlash nonlinearity based on norm-limited SPSA

Abstract

Keywords

Introduction

Basic experimental device

Simplified drivetrain with backlash

State-space representation for modeling

Active vibration control with compensation for backlash

Model-based H 2 vibration controller

Backlash compensation by simple control mode switching algorithm

Auto-optimization scheme for the backlash and contact mode controllers

SPSA with norm-limited update vector

Norm-limited SPSA-based parameter tuning scheme

Tuning parameter expression

Loss function in optimization

Parameter tuning condition

Tuning results

Experimental verifications

Setting of experimental verifications

Experimental results and discussion

Comparison with the case without backlash compensation

Robustness against plant variation

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

Appendix

References

Model-based $H_{2}$ vibration controller