A Dual-Observer Design for Nonlinear Suspension System Based on Feedback Linearization

Liu, Zheng; Zhao, Yuzhuang; Chen, Sizhong

doi:https://doi.org/10.1155/2018/2485263

Mathematical Problems in Engineering

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Abstract Introduction Analysis Conclusion Appendix Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2018 | Article ID 2485263 | https://doi.org/10.1155/2018/2485263

A Dual-Observer Design for Nonlinear Suspension System Based on Feedback Linearization

Zheng Liu,¹Yuzhuang Zhao,¹and Sizhong Chen¹

Academic Editor: Nicola Caterino

Received25 Sept 2018

Revised23 Nov 2018

Accepted13 Dec 2018

Published26 Dec 2018

Abstract

A novel approach to estimate suspension state information and payload condition was developed in this article. A nonlinear quarter car model with air spring and damper was built. After verification of system observability and solvability, a certain coordinate transform was built to transform the nonlinear system into a linear one. Then a Kalman filter observer was applied. A sprung mass observer, which works cooperatively with suspension state information observer, was also designed. Designed dual-observer was verified under typical road profile and sprung mass disturbance. Compared with extended Kalman filter, the dual-observer showed better accuracy and robustness.

1. Introduction

1.1. Background

The function of the suspension system is to isolate the passenger and payload from road disturbance and maintain contact between the wheel and the road at all times, which is called “ride comfort” and “ride holding”, respectively [1, 2]. Conventionally, these conflicting objectives are achieved by designing a passive suspension whose damping coefficient and stiffness curves are carefully selected for a compromised solution [3], while in recent years, there have been growing interests in the control strategy of semiactive and active suspension systems [4, 5]. Semiactive and active suspension systems can have better performance compared to conventional suspension systems due to their ability to change damping coefficient or stiffness and damping coefficient at the same time [6–8]. The performance of these control strategies strongly depends on the accuracy of vehicle parameters and state information [9], while due to system complexity, expenses, and technical limits, not all the state information can be measured directly by sensors [10] and online measurements of all the state variables of a process are rarely available. Now in industrial application, common technique to obtain state information which can not be measured directly from sensors such as vehicle body velocity is to integrate signals from accelerometers. However, such technique suffers a lot from errors caused by drift [11, 12].

1.2. Formulation of the Problem of Interest of This Investigation

In such cases, reliable information of unmeasurable variables such as vertical velocity of spring mass is obtained by applying a state observer. State observers are systems designed based on a mathematical model and are capable of reconstructing the inaccessible variables from easily available measurements [13–15]. Now most mutual observers designed for vehicle vibration state such as Kalman filter (KF) are based on linear system, while in industrial practice, most of the vehicle system has nonlinear character. Simply neglecting those nonlinear character and treating the vehicle system as a linear one can cause serious problems like the deterioration of estimation accuracy which will finally lead to poor control effect of active/semiactive suspension system.

1.3. Literature Survey

Kalman filter (KF) is an algorithm that uses a series of measurements in time domain and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone are. It is the optimal estimate for linear system models [9, 10, 16]. To apply this filtering method to nonlinear systems, many efforts are made by former researchers. Extended Kalman filter (EKF) is applied to vehicle system by Rigatos and Jurkiewicz [17, 18]. The EKF uses first order Taylor expansion to approximate the nonlinear system [19]. However, the linear system obtained through this method can only reflect the system character around the equilibrium position. When the system is severely nonlinear, it may lead to deteriorated estimation result [20]. Then Unscented Kalman filter is developed by Jeffrey Uhlmann. Researchers like Anronov and Hedrick applied UKF to vehicle system and accomplished better results compared to EKF [21, 22]. The UKF performs a stochastic linearization by performing unscented transform which uses a bunch of sigma points to approximate the mean and the variance of the system. So UKF is approved to have higher order accuracy then EKF [20]. Sliding mode theory is also used in the design of state observers [23, 24]. The sliding mode observer uses the error between the estimated state and real state to design the sliding mode surface, which can ensure that the estimated states can follow the real states [23, 25, 26]. Other newly developed observers like fuzzy observer and neural network observer do not rely on system dynamic model and their accuracy rely on large amount of training data, which sometimes is not easy to get [24, 27–30]. System identification is also considered in modern observer design. Gordon implemented online mass and stiffness identification by using recursive least-square [31]. An adaptive observer design, which was used for observer-based parameter identification in the active suspension system of an automobile, was implemented by Hedrick [32].

1.4. Scope and Contribution of This Study

The main motivation of this study is to apply a new approach of observer design to suspension system. Instead of trying to get approximation of the nonlinear system, feedback linearization transforms the nonlinear system to an observable linear one. Theoretically, the process does not affect the accuracy of the designed observer [33]. Moreover, the linearized form is easy to dealt with, which leaves much possibility for future work of implementing some mutual linear observer design technique such as Adaptive Kalman filter to nonlinear system.Then dual-observer design is introduced to the system which identify the vehicle sprung mass in real time.

1.5. Organization of the Paper

In this paper, a feedback linearization method has been implemented, which transform the original nonlinear system into an observable one by finding a specific transforming coordinate. Then the traditional linear Kalman filter algorithm can be applied in this transformed system. After each Kalman filter iteration, the linear system can be transformed back into the original system through inverse transformation, then the state information can be obtained. Payload variation is also considered in this paper. Since vehicle mass can vary significantly from one loading condition to another [34]. So we designed another Kalman filter to identify the vehicle payload by measuring the sprung mass vertical acceleration. The robustness of the system is significantly improved by exchanging the estimated results between these two observers in real time.

The paper is organized as follows: a nonlinear quarter car model is explained in Section 2; feedback linearization for above nonlinear model is introduced in the subsequent section; Kalman filter for state information observation and sprung mass identification is designed in Section 4; simulation tests are done in Section 5 and performance of above algorithm is illustrated in the same section; conclusions and future work are explained in the last section.

2. Nonlinear Quarter Car Model

Linear model can replace the nonlinear model around the operating point, out of the range, deviation cannot be ignored and the linear one is not valid. Therefore, a quarter car nonlinear model is established as shown in Figure 1.

There are two degrees of freedom of vibration in this model, vertical movement of sprung and unsprung mass. In Figure 1, is sprung mass, which varies when payload changes. Sprung mass is supported by air spring whose stiffness coefficient is and damping shock absorber whose damping coefficient is . Unsprung mass, shown in Figure 1 as , is supported by tire, which is simplified as linear model with constant stiffness coefficient . and are displacement of sprung and unsprung mass, respectively, and road profile is denoted as .

The dynamic function of this model can be described aswhere is the spring force, which is the function of spring stroke, and is damping force which is the function of damper stroke and stroke rate.

In our system, the suspension system shows nonlinear character because of the nonlinear force generated by air spring and the damper. So models of air spring and damper are also built in this paper.

The nonlinear spring force can be described aswhere r is gas polytrophic exponent, is effective section area of the piston, is the static equilibrium pressure of the energy accumulator, is the spring stroke, and is volume of the air bag.

Stiffness characteristic of air suspension is shown in the Figure 2.

Piecewise function of damper property is as follows; coefficients are acquired by data fitting of experimental data,where is the throttle opening rate in extending stroke, is the throttle opening rate in compressing stroke, and are damping coefficient in extending stroke, and and are damping coefficient in compressing stroke.

Characteristic of the damping force compared with experiment value is shown in Figure 3.

All the parameters used in this chapter are listed in the Table 1.

3. Feedback Linearization of Nonlinear Suspension System

The basic idea for feedback linearization is to find a proper nonlinear coordinate transformation and turn a nonlinear system into a linear system. Then traditional and well-developed linear observer algorithm can be applied to the system. Our system is a no input multioutput stochastic system and the feedback linearization method is based on differential geometry [33].

Take the following system, for example:A linearized form can be obtained through coordinate transfer x = X(ξ).where (A, C) is the observation matrix pair. ξ is the state vector of the transformed system.

For our certain system, considering system process noise, and measurement noise we can build the state space model for our system. Define state vectordefine measurement variable,the reason for such measurement variable selection will be explained later in our article, and the state space model can be expressed aswhere is the process noise and v is the measurement noise.

3.1. System Observability Verification

The system mentioned above is a 4-dimension system with a 2-dimension output. Make n=4 and m=2, where n is the dimension of the system and m is the dimension of the output function. Define observability index , where L is a K dimension real number sequence and meetsAccording to differential geometry theory [33], the necessary and sufficient condition for the system observability is to find a certain observability index L which can make observability matrix Q nonsingular, whereTo find out whether , we need to calculate the partial derivatives and Lie derivatives of the output function.

For output function where detailed information about (13) to (15) can be found in Appendix, to .

For output function With above partial derivatives and Lie derivatives, it can be easily verified that when observability index, and our nonlinear suspension system is observable.

3.2. Feedback Linearization Transfer Solving

Define,Again, according to differential geometry theory [33], the nonlinear feedback is solvable if and only if (1)(2)observation matrix pair(A,C) is in condensed dual Brunovský form listed below

where C₁ is a nonsingular and square matrix and is the ith column of the -dimensional unit matrix.

Carry output functions and Lie derivatives of output functions into above defined equations; we can easily verify that (21) is met.

And there exists a observer matrix pair (A,C) which meets (22), whereFor nonlinear coordinate transfer x = X(ξ), define ∂X/∂ =G whereThen following equation must be satisfied:whereAccording to the definition of generalized inverse matrix, (25) equals where is an arbitrary - inverse matrix of matrix and is an arbitrary matrix which has the same dimension as .

Then we can getIntegrate above equation; we can get coordinate transferThen we can get inverse coordinate transfer

3.3. Sensor Configuration Selection

Limited by technical feasibility, there are at least 10 possible sensor configurations for our system which is shown in Table 2.

In above two chapters, we choose configuration ⑥, verified its observability and solvability, and deduced the nonlinear feedback coordinate transfer.

Repeat the process of Sections 3.1 and 3.2; we can find out that though all of above sensor configurations are observable, not all of the sensor configurations are solvable. Some are not solvable at all, some are only solvable under certain limits, as shown in chart in Table 3.

As shown in Table 3, configuration ⑥ is observable and solvable when both spring force and damping force are nonlinear. That is why we choose suspension deflection and suspension deflection velocity as our measurement variable.

3.4. Kalman Filter Design

Apply the coordinate transfer to our system; we can getConsider process noise and measurement noise; the system state form iswhere is process noise and is measurement noise.

Compared with (8), the system measurement noise is determined by the measurement system, which makesTo find out process noise and process noise coefficient matrix B, first of all, we can assume there is no error in the measurement variables, then , which makesCompared to the coefficient of , we can get coefficient matrix

Now considering the process noise in the system, .

According to first order Taylor expansion,where DefineThen,where is the rigid coefficient and is the damping coefficient.

Then the linear Kalman filter algorithm can be applied to design the observer, which can estimate the state of .

Discretize the linearized system with sample time , and define , and we can get the discrete form of (32)where , and C are matrixes with constant coefficients and is the mapping of output function.

Then the structure of feedback linearization Kalman estimator algorithm is shown in Figure 4.

4. Sprung Mass Identification

Different driving conditions cause change of sprung mass, which leads to deteriorating estimator outcome if no corresponding change is made to the estimator parameters. Therefore, we designed a sprung mass identification estimator based on linear Kalman filter, which is combined with the estimator designed in last section. Those two estimators change results in real time. Assuming the sprung mass does not change in a short time. We chose as system state and sprung mass acceleration as measurement. The system function then iswhere H is the suspension force and and are process noise and measurement noise, respectively.

Figure 5 is the mass estimation estimating result, and we can see that this method is working pretty well in our system.

5. Simulation and Analysis

A simulation based on MATLAB/Simulink is implemented in this section under typical road profile. To verify the accuracy and efficiency of our designed feedback linearize Kalman filter estimator (FL-KF), a comparison is made between FL-KF and extended Kalman filter (EKF).

The road profile is grade-B random road excitation determined in ISO; vehicle speed is kept at 10m/s.

Figures 6–10 show the comparison of estimation result of FL-KF and EKF. To better evaluate the accuracy of the designed estimator, an accuracy index is introduced,where x is the system state, is the estimated system state, and N is the quantity of the number of the sample. The closer the index gets to the value of 1, the more precise the estimation result is.

Error analysis is plotted in Figure 11, where V_ms is the sprung mass vertical velocity and V_mu is the unsprung mass vertical velocity. It can be seen that the results of FL-KF show better accuracy compared with the results of EKF.

From Figures – and Table 4 we can find out that FL-KF has better accuracy than EKF in the estimation of all five parameters listed in the table. Differences are more obvious in the estimation results of vertical velocity of sprung/unsprung mass. That is because other three parameters are either part of the measurement vector or derivative of the measurement vector.

To better verify the success of the observer, simulation under ISO grad A and grade C road is also implemented, accuracy index of which compared with the result of grad B road is listed in the Table 5. The vehicle speed is kept at 10m/s in all sets of simulations.

From data in Table 5. We can find out that the designed observer is valid under different road level excitations ranging from grade-A level to grade C level. The difference between EKF and FL-KF observers gets larger when random road excitation gets severer, that is because the nonlinear character of the suspension system is more obvious when the system is far from the equilibrium position.

To verify the efficiency of our designed estimator under different vehicle load conditions, a sprung mass disturbance (plus 20%, which is 42kg) is given to the initial sprung mass value of the estimator. Road level of the simulation is ISO grade-B level and speed of the vehicle is kept at 10m/s. The outcome of our designed estimator is compared with the outcome of a feedback linearize Kalman filter without sprung mass estimator.

Figures 12–17 and Table 6 show the result of estimators with and without sprung mass identification filter. FL-KL is the result of our designed feedback linearization dual Kalman filter and FL-KF-NoME is the result of a feedback linearization single Kalman filter without sprung mass identification estimator. Results show that our designed estimator is much more accurate for sprung mass and unsprung mass velocity estimating when the payload changes, while feedback linearization Kalman filter with and without sprung mass identification estimator does not show much accuracy difference when estimating suspension deflection, suspension deflection velocity, and sprung mass acceleration. That is because suspension deflection and suspension deflection velocity are part of the measurement vector; sprung mass acceleration is the derivative of sprung mass velocity; sprung mass velocity of two estimators shares almost the same trend.

6. Conclusion

In this paper, a feedback linearization observer with sprung mass estimator is designed based on differential geometry. A nonlinear suspension system model is built and transformed into linear observable form, and then linear Kalman filter algorithm is applied to this transformed linear model. Efficiency and accuracy of feedback linearization observer are verified under typical road profile. Compared with EKF observer, feedback linearization observer has better performance. Sprung mass estimator is designed and its performance is verified; its necessity is verified by comparing estimation result of feedback linearization observer with and without sprung mass estimator under typical road profile. Results show that the sprung mass estimator can improve the accuracy of the observer significantly.

There are still some important issues that deserve future research; state observer should be designed for full vehicle suspension system; control algorithm for semiactive suspension system combined with state observer should be developed.

Appendix

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work is supported by the National Nature Science Foundation of China (Grants nos. 51375046 and 51205021).

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Copyright © 2018 Zheng Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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