Observer-based motion axis control for hydraulic actuation systems

Xiaowei YANG, Yaowen GE, Wenxiang DENG, Jianyong YAO

School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China

KEYWORDS

Abstract Unknown dynamics including mismatched mechanical dynamics(i.e.,parametric uncertainties, unmodeled friction and external disturbances) and matched actuator dynamics (i.e., pressure and flow characteristic uncertainties) broadly exist in hydraulic actuation systems (HASs),which can hinder the achievement of high-precision motion axis control.To surmount the practical issue, an observer-based control framework with a simple structure and low computation is developed for HASs.First, a simple observer is utilized to estimate mismatched and matched unknown dynamics for feedforward compensation.Then combining the backstepping design and adaptive control, an appropriate observer-based composite controller is provided, in which nonlinear feedback terms with updated gains are adopted to further improve the tracking accuracy.Moreover,a smooth nonlinear filter is introduced to shun the ‘‘explosion of complexity” and attenuate the impact of sensor noise on control performance.As a result,this synthesized controller is more suitable for practical use.Stability analysis uncovers that the developed controller assures the asymptotic convergence of the tracking error.The merits of the proposed approach are validated via comparative experiment results applied in an HAS with an inertial load as well.

1.Introduction

Hydraulic actuation systems(HASs)are broadly utilized in the aerospace industry and other fields.1-6This reason is that HASs have prominent merits of large force/torque output capability, small size-to-power ratio and high response.Nonetheless, given the highly nonlinear dynamics and unknown dynamics including mismatched mechanical dynamics (i.e., parametric uncertainties, unmodeled friction and external disturbances) and matched actuator dynamics (i.e.,pressure and flow characteristic uncertainties) existing in HASs, the attainment of high-performance motion axis control is hindered.Therefore, to figure out this practical issue,it is essential to investigate advanced control strategies.

To date, plentiful efficient control approaches have been proposed to achieve high-precision tracking performance for HASs.Typically, feedback linearization control7was adopted to dispose of dynamic nonlinearities in a feedforward compensation way.To address parametric uncertainties,adaptive control was presented8, but could not restrain unmodeled dynamics9.Adaptive robust control9could dispose of both unmodeled dynamics and parametric uncertainties simultaneously by integrating the backstepping design and adaptive control,thus it is widely performed in the literature.10–12However,theoretical analysis of the control method8–12validates that only bounded convergence was pledged while time-variant uncertainties exist.Via combining an integral sliding mode control13and adaptive control, a composite adaptive position tracking controller was proposed for HASs,14,15where the asymptotic stability of the tracking error was attained while smooth enough disturbances exist.Moreover, owing to its strong robustness and simple structure, sliding mode control could be adopted to restrain strong disturbances and gain the improvement of tracking accuracy for hydraulic servomechanisms.16Unfortunately, as unknown dynamics increase heavily in HASs, a high-gain feedback way13–16will be utilized to assure the achievable tracking performance,which might arise high-frequency dynamics and let the system become unstable.

To relieve the influence of both mismatched and matched unknown dynamics on control performance and avoid the high-gain feedback, lots of disturbance observers have been presented for HASs.In the literature,17an active disturbance rejection adaptive controller was investigated for hydraulic actuators, in which an extended state observer (ESO) was utilized to estimate mismatched and matched disturbances for feedforward compensation.Subsequently, a lot of ESO-based robust controllers18–21of hydraulic systems were presented.However, the asymptotic convergence performance of the ESO-based controllers was destroyed while time-variant disturbances exist.A novel sliding mode observer (SMO)22was recently utilized to cope with force dynamics and pressure dynamics in hydraulic systems.The asymptotic tracking performance could be attained.Nonetheless, the SMO22needed to assume that the first-order derivatives of disturbances are bounded.This assumption is a little strong and limits the applied range of SMO.Also, the neural network was taken as an estimator23–26to achieve the estimations and compensation of unknown dynamics in servo systems.Whereas, to obtain accurate estimations of unknown dynamics,lots of neural network nodes and training data were needed,which results in a long online learning time and high computation.As a result, how to effectively deal with unknown dynamics by using a simple structure and obtain the high-accuracy asymptotic convergence performance for HASs is still challenging.

In this article,an observer-based robust tracking controller is developed for the motion axis of HASs with mechanical dynamics and actuator dynamics.Firstly, an observer with a simple structure is adopted to estimate mismatched and matched unknown dynamics for feedforward compensation.Then combining the backstepping design and unknown dynamics observers, a composite control framework is proposed, where nonlinear feedback terms with updated gains are utilized to further improve the tracking accuracy.Moreover, a smooth nonlinear filter is introduced to shun the ‘‘explosion of complexity” and attenuate the impact of sensor noise on control performance.Based on the stability analysis,the boundedness of all signals in the closed-loop system can be ensured and the tracking error can converge asymptotically.Contrastive experimental results of an HAS with an inertial load validate the preponderance of the proposed method.The main contributions of this article are summarized as: (1)An observer-based tracking control framework with a simple structure and low computation, is developed for HASs, which is more suitable for practical implementation.(2) Both the boundedness of all signals in the closed-loop system and asymptotic convergence of the tracking errors can be attained with the developed controller.(3) Compared to most existing approaches focusing on HASs, the presented observer does not need any prior assumption on unknown dynamics.This makes the applied range of the developed controller broadened considerably.

2.System description and modeling

The sketch of the considered hydraulic actuation system(HAS) is presented in Fig.1, in which a servo-valvecontrolled hydraulic cylinder actuates a load to move.Our objective is to design a controller to let the load well track the reference motion trajectory.

As shown in Fig.1, by using Newton’s second law, the dynamics of the load are depicted as.17,18,27

where m and y stand for the mass and displacement of the load;P1and P2stand for the pressure values of the two chambers; A stands for the effective piston area; B denotes the viscous friction coefficient; AfSfstands for the approximated Coulomb friction, where Afstands for the amplitude of Coulomb friction and Sfstands for a known function; f(t) stands for unknown mechanical dynamics containing parametric uncertainties, unmodeled friction and disturbances.

The pressure dynamics of the actuator are depicted as.17,18,27

where V1=V01+Ay and V2=V02-Ay represent volumes of two chambers,where V01and V02are the initial volumes of the two chambers; βestands for the oil bulk modulus; Ctdenotes internal leakage coefficient; q1is the supplied flow rate and q2is the return flow rate;⊿1and ⊿2denote unmodeled uncertainties caused by pressure and flow characteristics.

Fig.1 Sketch of considered HAS.

The impact of servo-valve dynamics on control performance has been investigated.21,23Whereas, an additional sensor is needed to gain the spool position, and only the minor precision enhancement is ensured for position tracking.Also,the valve dynamics can augment the system order and make the designed controller more complicated.As a result, the servo-valve dynamics can be ignored in lots of existing studies.1–12,14–20,22,24–26Hence,the control input u can be supposed to be linear with the spool displacement xv, thus q1and q2are reconstructed by17,18,27.

where kvrepresents the flow gain; Psdenotes the supply pressure and Prdenotes the return pressure; sign(*) is defined by.

Defining x =[x1;x2;x3]=[y; ˙y;A(P1-P2)/m], one has.

where

Remark 1.Indeed, the nominal parameter values of the HAS that are able to be gained by off-line identification, are employed in the subsequent designed observer and controller.The discrepancy between the identified and real values can be regarded as the unknown dynamics D1(t) and D2(t).The mismatched dynamics D1(t) and matched dynamics D2(t) are then observed and compensated with the aid of the developed unknown dynamics observer.

Some assumptions below are provided.

Assumption 1.The mismatched and matched unknown dynamics in Eq.(5) are bounded and satisfy.

with D-1and D-2being unknown positive constants.

Assumption 2.The desired trajectory xd(t) and its derivatives ˙xd(t) and ¨xd(t) are bounded.

Remark 2.It is noteworthy that Assumption 1 is more reasonable and relaxed for practical HASs when compared to the previous results.17–22Assumption 1 ensures the presented observer does not need any precondition on the unknown dynamics D1(t) and D2(t) while the existing SMO22and ESO17–21need to assume that their first-order derivatives are bounded.In addition, Assumption 2 indicates that only the first-order and second-order derivatives of xdare necessitated,which breaks down many limitations in comparison with lots of previous control methods8–26needing higher-order derivatives of xd.These make the use range of the presented controller broadened significantly.

3.Controller development

3.1.Observer design for system unknown dynamics

To estimate mismatched and matched unknown dynamics existing in the HAS, a set of auxiliary variables xai(i=2,3)are firstly constructed by.

and

where xri=xai-xiare known since xaiand xiare known; βidenote non-negative adjustable constants for i=2,3.

The estimations of xriare designed as.

where ˙xristand for the numerical differentiation of the known xri;^?stands for the estimation of the variable ?throughout this paper.

Then two unknown dynamics observers for D1(t)and D2(t)are depicted by.

and

Thus, the error dynamics of observers can be exhibited as.

and

in which ?～=^?-? stands for the estimation error of the variable.

Given the previous analysis, the following Theorem 1 holds.

Theorem 1.With the observer updated laws Eqs.(11)and(12),by selecting positive constant values β2and β3,it is inferred that the estimation errors D～iof the unknown dynamics Di(t) can achieve the asymptotic convergence.

Proof: Take a Lyapunov function as.

According to Eq.(13),the derivative of Vocan be presented by.

Hence,the asymptotic convergence of the estimation errors D～iis proved, which leads to Theorem 1.

Remark 3.It is obtained from Eq.(17) that the estimation errors for both mismatched and matched unknown dynamics can asymptotically converge, while most existing extended state observers17–21only assured the ultimate boundness of estimation errors.Moreover, from Eq.(17), the faster estimation rate of the developed observers can be acquired by choosing larger parameters β2and β3.

3.2.Observer-based controller design

The backstepping method28is a classic tool to dispose of unknown dynamics existing in the HAS.Nonetheless, considering that the order of the system in Eq.(5) is third-order, the traditional backstepping technique28brings the problem named ‘‘explosion of complexity” on account of the high computational burden resulting from the repeated derivatives of virtual controls.To overcome the problem, a modified backstepping design with nonlinear filters will be provided here.

Before accomplishing the controller design, define the following variables.

where e1stands for the tracking error;αi-1stand for the virtual controls at the ith step; α(i-1)fstand for the filtered signals of αi-1for i=2,3.

Notably, to get the filtered signals of α1and α2, a set of smooth nonlinear filters are designed by.

Step 1: Considering Eq.(5) and differentiating e1in Eq.(18), one has.

Design α1as.

where k1denotes a non-negative feedback gain.

Substituting Eqs.(21) into (20) results in.

Taking a Lyapunov function asyields.

Step 2: Differentiating e2in Eq.(18), it has.

Step 3: The derivative of e3in Eq.(18) is expressed by.

From Eq.(28), the control input u can be depicted by.

3.3.Stability analysis

Before demonstrating the main result, define.

Fig.2 Block diagram of developed controller.

with

Theorem 2.With the observer updated laws in Eqs.(11)-(12)and the control law in Eq.(29), by choosing appropriate controller parameters k1,k2,k3,β2,β3,l1,l2,τ1and τ2such that the matrix Λ defined in Eq.(33) is positive definite, then it is inferred that all signals in the closed-loop system are bounded.Furthermore,the tracking error e1can asymptotically converge.

Proof: Firstly, a new Lyapunov function by integrating tracking errors, observer errors and filter errors is constructed as.

Combining Eqs.(16), (19) and (32), one has.

Therefore,V ?L∞,Φ ?L2, and the boundness of all signals is attained on Ω1×Ω2.As a result,Φ is uniformly continuous.Via adopting the Barbalat’s lemma,29Φ →0 as t →∞on Ω1×Ω2.Hence,the output error e1is able to converge to zero asymptotically.Thus, Theorem 2 holds.

Remark 4.The objective of designing observer parameters β2and β3is to attain a faster estimation rate for mismatched and matched unknown dynamics.The nonlinear filter parameters l1,l2,τ1and τ2are used to get the filtered signals for the virtual control α1and α2, and further attenuate the effects of the sensor noise on tracking accuracy.Meanwhile,the objective of using k1,k2,k3, γ1and γ2is to realize better control performance.Moreover, the efforts of arbitrarily increasing k1,k2,k3,β2,β3,l1,l2,τ1,τ2,γ1and γ2can be performed to achieve better tracking accuracy.Whereas, from Eqs.(19), (21), (25)and (29), the increase of the above control parameters may result in the augmentation of the control input and even destabilize the system.Consequently, a tradeoff between the tracking accuracy and the control input is worth considering.

4.Experiment setup and results

To uncover the availability of the developed controller, a hydraulic actuation system (HAS) with an inertial load is set up, as shown in Fig.3.The components of the HAS are collected in Table 1.The sample time is 0.5 ms.

Five controllers can be done to validate the merits of the developed controller.

1) UDOC: This controller with two unknown dynamics observes(UDOC)is described in this paper.In the HAS,some system parameters are provided as:A=9?05×10-4m2,V01=V02=3?98×10-5m3.By off-line friction identification,the corresponding parameters are provided as B=4000N ?s/m,Af=200N,Sf=2 arctan(900x2)/π.Meanwhile the feedback gains are set as k1=1600,k2=350,k3=150, the nonlinear filter parameters are provided by τ1=τ2=1000,l1=l2=1, and the observer parameters are set as β2=100,β3=15000.The parameter adaption rates are set as γ1=0?1 and γ2=1.

2) FLC: This is a feedback linearization controller (FLC),which is the same as UDOC but without unknown dynamics compensation, i.e., β2=β3=0.The other control parameters are the same as UDOC.

Fig.3 HAS.

Table 1 Components of HAS.

3) PI: This is a proportional-integral controller (PI).The parameters are set as kP=10000 and kI=500, which stand for the P-gain and I-gain, respectively.

4) VPI: This is a velocity feedforward-based proportionalintegral controller (VPI).The parameters are set as kP=10000, kI=500 and kve=0?0281V ?s/mm, which stands for the open-loop velocity feedforward gain, respectively.

5) AESO: This adaptive robust controller with extended state observers(AESO)is presented.17The corresponding control gains are set as k1=1600, k2=350 and k3=150.The other parameters and initial values can be seen17.

To validate the merits of these controllers, three indices including maximum(Max),average(Ave)and standard deviation (Std) of e1are provided:

1) Max stands for.

where n is on behalf of the number of the recorded points.

2) Ave stands for.

3) Std stands for.

Three experiment cases are done.Case 1.First a common signal xd=10 arctan(sin(πt))[1-e-t]/0?7854 mm is carried out.The corresponding results are depicted in Figs.4-6 and Table 2, respectively.As shown,all the performance indices in Table 2 indicate that the proposed UDOC achieves better tracking performance than the other controllers, since two unknown dynamics observers and nonlinear filters are introduced into the designed controller.Concretely,the index Max of C1 is 0.0209 mm(i.e.,the accuracy at 0.5 Hz is 0.209 %), which decreases by about 98%,91%,59%and 42%when compared to those of FLC,PI, VPI and AESO severally.Though FLC utilizes the same model-based approach like UDOC,its control accuracy is still inequitable to that of UDOC for lack of the unknown dynamics compensation.This indicates the validity of the unknown dynamics compensation method presented in this paper, which can decrease the time-variant disturbances and further gain the attainment of high-precision tracking.From performance indices in Table 2,FLC and PI only have a bit of robustness against uncertainties and their tracking performance is worse than the other three controllers.Moreover,it is easy to find that the tracking error of PI is larger than that of FLC,which uncovers that feedback gains in UDOC,FLC and AESO are weaker than those of PI and VPI.Even though,UDOC and AESO can attain better control performance than PI.This verifies the merit of the uncertainty compensationbased controller design of UDOC and AESO.Furthermore,it is observed that the performance of VPI is superior to that of FLC and PI, which is mainly owing to both large feedback gains and velocity feedforward compensation in VPI to restrain parametric uncertainties and disturbances.Thus, the augmentation of control performance of VPI is attained.Nonetheless, VPI gains a worse control performance than UDOC.

Fig.4 Tracking errors in Case 1.

Fig.5 Control input of UDOC.

Fig.6 Tracking errors in Case 2.

Table 2 Performance indexes in Case 1.

Also, AESO17obtains a better control performance than FLC,PI and VPI.This reason is that parametric uncertainties and external disturbances can be disposed of by adaptive laws and the ESO technique severally in AESO.Whereas,considering that only ultimately bounded-error tracking performance is obtained in AESO,its performance indices are still inequitable to those of UDOC.As a result, the developed UDOC adopts the weakest feedback gains but attains the best control performance by virtue of actively compensating unknown disturbances.In addition, the control input of UDOC is seen from Fig.5.

Case 2.A slow trajectory xd=10 arctan(sin(0?4πt))[1-e-t]/0?7854 mm is performed to further attest to the merit of the developed approach.The control errors and indices of all the controllers can be thus found from Fig.6 and Table 3,respectively.

As shown,the index Max of C1 is 0.0074 mm(i.e.,the accuracy at 0.2 Hz is 0.074 %), which decreases by about 98 %,94 %, 75 % and 66 % when compared to those of FLC, PI,VPI and AESO severally.This uncovers that the control precision of the proposed UDOC is superior to those of FLC, PI,VPI and AESO once again.

Case 3.Finally, an input disturbance experiment is done by using the same reference trajectory xd=10 arctan(sin(0?4πt))[1-e-t]/0?7854 mm, where a similar technique17can be adopted to insert the input disturbance.The inserted disturbance signal 0?2 arctan(sin(0?4πt))[1-e-t]/0?7854 V is added to the control input directly.According to the Eq.(5), the disturbance D2(t) can be increased greatly, and the HAS willwork under such strong disturbance.As a result, the learning capability of the developed method can be verified.The tracking errors and indices of UDOC, FLC, PI, VPI and AESO are presented in Fig.7 and Table 4.It is easy to find that the control performance of the three controllers increases when compared to the former case.Nevertheless,in view of the merit of compensating for unknown disturbances,the tracking performance of the developed UDOC still surpasses those of FLC, PI, VPI and AESO as well.

Table 3 Performance indexes in Case 2.

Fig.7 Tracking errors in Case 3.

Table 4 Performance indexes in Case 3.

5.Conclusions

In this article, an observer-based backstepping tracking controller is developed for the motion axis of hydraulic actuation systems.Firstly,an observer with a simple structure is adopted to estimate mismatched and matched unknown dynamics for feedforward compensation.Then combining the backstepping design and unknown dynamics observers, a composite robust axis control framework is proposed,where nonlinear feedback terms with updated gains are utilized to further improve the tracking precision.Moreover,a smooth nonlinear filter is introduced into the control development process to shun the ‘‘explosion of complexity” and attenuate the impact of sensor noise on control performance.Furthermore,the control framework makes the assumptions on both unknown dynamics and the reference trajectory much relaxed.As a result, this makes the applied range of the developed control method extended.Based on the Lyapunov function, both the boundness of all signals in the closed-loop system and asymptotic convergence of the tracking errors can be attained.The results of the three experiment cases performed in a hydraulic actuation system with an inertial load validate the merits of the presented method.As our future work, it is worth studying how to attain the development of an output feedback-based controller for hydraulic actuation systems.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported in part by the National Key R&D Program of China (No.2021YFB2011300), the National Natural Science Foundation of China (No.52075262, 51905271,52275062), the Fok Ying-Tong Education Foundation of China(No.171044)and the Postgraduate Research&Practice Innovation Program of Jiangsu Province (No.KYCX22_0471).