Active Disturbance Rejection Control For An Electric Power .
INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMSVOL.15,NO.1,MARCH 2010,18-24Active Disturbance Rejection Control for an Electric Power AssistSteering SystemLili DONG1, *, Member, IEEE, Prasanth KANDULA1, Zhiqiang GAO1, Member, IEEE,Dexin WANG2Abstract—This paper presents the application of an active disturbancerejection controller (ADRC) to an electrical power assist steering system(EPAS) in automobiles. The control objective is to reduce the steeringtorque exerted by a driver so as to achieve good steering feel in thepresence of external disturbances and system uncertainties. With theproposed ADRC, the driver can turn the steering wheel with the desiredsteering torque which is independent of load torques that tend to varydepending on driving conditions. The ADRC is constructed on a columntype EPAS model. The computer simulation and frequency-domainanalyses verify the robustness and stability of the ADRC controlledsystem.Index Terms—EPAS, active disturbance rejection control, robustness,system uncertainty, external disturbance, automobile.I. INTRODUCTIONAsteering system is one of major subsystems for vehicleoperation [1]. The most conventional steeringarrangement is to turn the front wheel using a hand–operatedsteering wheel which is positioned in front of the driver, viathe steering column. However, as vehicles have becomeheavier and switched to front wheel drive, the effort to turn thesteering wheel manually will be greatly increased, often to thepoint where major physical exertion is required. To alleviatethis, automobile manufacturers have developed power steeringsystems. In modern world, there are two major types of powersteering systems: hydraulic and electric. A hydraulic powersteering (HPS) system uses hydraulic pressure supplied by anengine-driven pump to assist the motion of turning the steeringwheel. Electric power assist steering (EPAS) is more efficientthan the hydraulic power steering, since the electric powersteering motor only needs to provide assistance when thesteering wheel is turned, whereas the hydraulic pump must runconstantly. In addition, the EPAS system has the advantagesof fuel economy, space efficiency, and environmentalprotection [2, 3] over the HPS system. The EPAS system hasbegun replacing the HPS system in some advanced smallvehicles. The utilization of the EPAS to all types of vehicles isexpected to be expanded exponentially [4] in the future.The essence of an EPAS system is an electronically1The authors are with the Department of Electrical & ComputerEngineering, Cleveland State University, Cleveland, OH 44115 USA.2The author is with Ford Motor Corp., Dearborn, MI 48121, USA.*Corresponding author. Email: L.Dong34@csuohio.edu.controlled assist motor [5] that can be taken as a smartactuator. The EPAS is a classical example of the actuatoroperating under feedback control [6]. When an appropriateassist torque from the assist motor is applied in the samedirection as the driver’s steering direction, the amount ofsteering torque required by the driver for steering can beconsiderably relieved. Therefore the controller in EPAS aimsat driving the motor to output the assist torque so as toimprove the steering feel of the driver. The high performancerequirements of the EPAS in automobile industry also requirethe control system to be robust against modeling uncertaintiesand unknown disturbances. Specifically, a controller is crucialfor the EPAS system to stabilize the feedback system withtorque sensor and actuator (motor), to reduce the steeringtorque, and to improve the steering feel of the driver.There has been much research and development performedon EPAS systems though detailed information on their controlstrategies has not been extensively released in the literature [7].Most of the current research [1-6] has successfully employedhigh-order lead-lag compensator or Proportional IntegralDerivative (PID) controller to reach the control objectivegiven above. However, the multiple tuning parameters of thecompensator (with 4 or 5 tuning parameters) and PIDcontroller (with at least 3 tuning parameters) make themdifficult to adjust and implement in the real world. In addition,the compensator and PID controller are sensitive to bothunknown external disturbances and parameter variations.Their performances are greatly degraded in the presence of thedisturbance and structural uncertainties that are common in theEPAS system. Hence dealing with the uncertain dynamics inEPAS systems makes the control problem extremelychallenging and critically important. A limited amount ofresearch is reported using the advanced controller such as H2norm or H approach [8, 9] to solve the robustness problem.But the implementations of such advanced controllers inpractice are also restricted by their complex configurationsand multiple tuning parameters. Therefore, finding a simpleto-implement control solution being robust againstuncertainties and disturbances will have wide impacts on thedeployment of the EPAS in the next generation of vehicles.The paper presents the employment of an emergingpractical control strategy that is Active Disturbance RejectionControl (ADRC) [10-14] to the EPAS system. The controllertreats the discrepancy between the real EPAS system and itsmathematical model as the generalized disturbance andactively estimates and rejects the disturbance in real time,
Dong et al: Active Disturbance Rejection Control for an Electric Power Assist Steering System 19hence the name ADRC. Since the controller design is notbased on the accurate mathematical model of a system, it isvery effective in dealing with a large amount of uncertaintiesand external disturbances, the main challenges pertaining toEPAS systems. For the first time, we modify the controllerand extend its use to the EPAS systems.The rest of this paper is organized as follows. The dynamicmodeling of an EPAS system is given in section II. Thedesign of ADRC is presented in section III. The stability androbust analyses are included in section IV. The simulationresults are shown in section V. Section VI gives concludingremarks and suggests future research.II. DYNAMIC MODEL OF AN EPAS SYSTEMWe choose a column-type EPAS system to design itscontroller. The plant includes the steering wheel-columntorque sensor assembly, the steering rack shaft, and theelectric motor-gear box assembly. The mechanical model isextracted from [1] and shown in Fig.1, where Ts denotessteering torque, Tm motor torque, Ta the assist torque deliveredto the shaft, Tl load torque, θm motor angle, θss the angularposition at the steering shaft, and θsw the angular position atthe steering wheel.A major function of the EPAS system is reduction insteering torque for improving the steering feel of the driver. Inorder to realize the function, a proper amount of assist torqueshould be provided by the assist motor to reduce the driver’ssteering torque. So a steer controller is applied to drive theassist motor to output the assist torque. A block diagram of theEPAS control system is shown in Fig. 2, in which Trs is areference steering torque, Fd is an external disturbance, and uis the control input to the assist motor. The reference torque isalso taken as the desired assist torque. It can be determinedbased on the information of vehicle speed and steering wheelangle θsw [1].In Fig.2, the dynamics of the sensor and the motor can bemodeled as(1)PtPaP( s) s Pt s Pa,where -Pt and -Pa are the poles of torque sensor and actuator.The transfer function of steering column and rack system isrepresented by(2)KJ s2 b sG (s) sssRs2 ( J s s 2 bs s K s )(me s 2 be s K e ),where me and be are effective rack mass and dampingcoefficient; Js, bs, and Ks are moment of inertia, dampingcoefficient and stiffness of the steering column. Theparameters values in (1) and (2) are listed in Table II inAppendix. From (1) and (2), the transfer functionrepresentation of the block diagram in Fig. 2 can be illustratedas Fig. 3, where P(s) represents the dynamics of the sensor andthe actuator (motor), Ka is a feed-forward gain from theactuator, G(s) is the transfer function of the steering columnand rack system, Ts is a measured steering torque, and C(s)represents the controller to be designed.Fig.1: The dynamic model of an EPAS system orTaSteeringmechanismFig. 2: Block diagram of the steering control systemFdTru C(s)KaP(s)G(s)Ts-Fig. 3: Transfer function representation of EPAS controlsystemIII. CONTROLLER DESIGNDefine y Ts. The open-loop transfer function of the plant inFig. 3 can be represented byy(s) K a P( s )G ( s )u ( s ) Fd ( s ) Ka Kspt p a ( J s s 2 bs s )Rs2 ( s Pt )( s Pa )( J s s 2 bs s K s )(me s 2 be s K e )(3)Since the product of P(s) and G(s) is a sixth-order system, thetransfer function (3) can be rewritten asa 2 s 2 a1 sy(s) u ( s ) Fd ( s ) b6 s 6 b5 s 5 b4 s 4 b3 s 3 b2 s 2 b1 s b0(4)where the parameters ai and bi are the coefficients of thenumerator and denominator of the transfer function (4). Thecoefficients can be obtained through comparing the right sidesof equations (3) and (4). Converting the transfer function (4)to an ODE form yieldsb6 y (6) b5 y (5) b4 y ( 4) b3 y (3) b2 y" b1 y ' b0 y a2u" a1 u ' a1Fd" a1Fd' .(5)
20 INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS,VOL.15,NO.1,MARCH 2010Integrating both sides of (5) double times, we will havey ( 4 ) f ( y (3) , y" , y ' , y, y, y, u , Fd , Fd ) a2ub6(6)where f ( y (3) , y", y ' , y, y, y, u, F , F ) , also denoted as f, is d dtaken as a generalized disturbance including internal dynamicsand unknown external disturbances. The basic idea of theADRC is to estimate the generalized disturbance using anExtended State Observer (ESO), and actively cancel it in realtime to reduce the original system model (6) to a multipleintegrator. Then a traditional proportional derivative controllerwill be used to drive the measured torque output y to a desiredvalue. Following the analysis above, we rewrite (6) as:(7)y ( 4) f β u ,where β a2/b6 is a controller gain.We suppose fˆ is anestimated f through the ESO. Then the controller:u controller and observer respectively. The details about how totune the parameters of the ADRC are explained in [16]. Fortuning simplicity, we generally choose ωo 5ωc [16]. Soeventually there is just one tuning parameter, which is ωc, forthe ADRC controller.In summary, the proposed ADRC design proves to be agood fit for an EPAS system for three reasons: 1) it requiresminimal a priori information of the plant (just the relativeorder of the plant and its highest-order gain [11]), 2) itactively estimates and compensates the unknown dynamicsand disturbances, 3) the controller only has one tuningparameter, which is easy to be implemented and tuned,compared to other methods.(8)1 ˆ( f u0 ),βreduces (7) to a merely multiple integral plant of the formy ( 4) f fˆ u0 u0 ,(9)which can be readily driven to the desired torque by aproportional derivative (PD) controller. Let r denote thereference torque and r Trs. So the tracking error e r-y. Thenthe PD controller can be represented byu 0 k p e k d1e k d 2 e k d 3 e ,(10)where the controller gains are selected asIV. PERFORMANCE AND ROBUSTNESS ANALYSESIn this section, the steady-state performance of the closedloop control system is discussed. In addition, the robustness ofADRC against parameter variations is analyzed throughfrequency responses. The frequency-domain analyses ofADRC are initially reported in [16]. This paper originallyemployed the frequency-domain analyses to the EPAS system.A. Transfer function expression of an EPAS control systemThe Lapalce transform of the ESO given by (12) issZ ( s ) ( A LC ) Z ( s ) LY ( s ) BU ( s ) ,U (s) k di C 4i ωc4 i , i 0,1,2,3 .1β[kpkd 1 kd 2where[0100000100L 5ω o 10ω o2 10ω o3000100 yˆ 0 y ˆ 0 0 0 , B 0 , C [1 0 0 0 0], z yˆ , 0 yˆ β ˆ 0 0 f 5ω o4ω o5 ]. [l1 l 2l3l4l5 ], and z is theestimated state vector, in which ŷ is the estimated y, y ̂ theestimated y , yˆ the estimated y , and y ˆ the estimated y . Thevector L is denoted as observer gain vector and ωo is observerbandwidth. Similarly, the observer gains of the ESO have beenchosen such that all the eigenvalues of the ESO equal - ωo.From (11) and (12), we can see that in order to reach thecontrol objective, only two tuning parameters are needed forthe ADRC. They are the bandwidths (ωo and ωc) of the R( s) sR ( s ) 1 k s 2 R( s) β p 3 s R( s) ][kd 1 kd 2]kd 3 1 Z ( s)(14)Substituting (14) into (13) yieldsU ( s ) H ( s )Gc ( s ) R ( s ) Gc ( s ) Z ( s ).(15)In (15),H (s) ( s 5 l1 s 4 l 2 s 3 l3 s 2 l 4 s l5 )(k p k d 1 s k d 2 s 2 k d 3 s 3 s 4 )µ1 s 4 µ 2 s 3 µ 3 s 2 µ 4 s µ 5,(16)where μ1 l1kp l2kd1 l3kd2 l4kd3 l5, μ2 l2kp l3kd1 k4kd2 l5kd3,μ3 l3kp l4kd1 l5kd2, μ4 l4kp l5kd1, μ5 l5kp, andyˆ Cz 1 0 A 0 0 0kd 3(11)In this way, the only tuning parameter of the controller is thecontroller bandwidth ωc and all the closed-loop poles are set to–ωc. As analyzed above, the performance of the ADRC isrigorously dependent on the accurate estimation of the ESOwhose design is stated as follows.We suppose function f is continuous and its derivative isbounded. Then the ESO including one augmented state f canbe represented by(12)z Az Bu L( y yˆ ) ,(13)where Z(s) is the Laplace transform of the estimated statevector. The Laplace transform of the controller represented by(8) and (10) isGc ( s ) µ1s 4 µ 2 s 3 µ3 s 2 µ 4 s µ5,β ( s 5 λ1s 4 λ2 s 3 λ3 s 2 λ4 s )(17)where λ1 kd3 l1, λ2 kd2 kd3l1 l2, λ3 kd1 kd2l1 kd3l2 l3, andλ4 kp kd1l1 kd2l2 kd3l3 l4.According to (15), (16) and (17), the block diagram of theclosed-loop control system can be represented by Fig. 4, wherethe plant Gp(s) is equal to KaP(s)G(s) as given in (3).D(s)R(s)U(s) Gc(s)H(s)Z(s) Gp(s)Fig.4: Block diagram of the closed-loop control system
Dong et al: Active Disturbance Rejection Control for an Electric Power Assist Steering System 21From Fig.4, the open-loop transfer function isGo ( s ) Gc ( s )G p ( s ) .(18)G d (s) β s(a 2 s 5 (a 2 λ1 a1 ) s 4 (a 2 λ 2 a1 λ1 ) s 3 (a1 λ 2 a 2 λ3 ) s 2 (a 2 λ 4 a1 λ3 ) s a1 λ 4 ),Ad ( s)(23)where Ad(s), given in Appendix, is a tenth-order polynomial(19) including non-zero constant term. From (23), we can see thatZ ( s ) H ( s )Gc ( s )G p ( s ) .Gcl ( s ) as the input frequency ω goes to zero or infinite, Gd(jω) willR( s)1 Gc ( s )G p ( s )convergeto zero. This suggests that the disturbance will beIn addition, the transfer function from the disturbance input toattenuatedto zero with the increase of system bandwidth. Thethe torque output isBodeplotsof (23) are shown in Fig. 6, in which the system.(20)G p ( s)Z ( s) Gd ( s ) parameters(Js, bs, Ks, Rs, m, Jm, Rm, bm and b) are varying fromD( s ) 1 Gc ( s )G p ( s )the nominal values listed in Table II in Appendix to 20% oftheir nominal values. The figure demonstrated a desirableB. Convergence of tracking errorIn Fig. 4, the reference signal R(s) (or desired assist steering disturbance rejection property in the presence of the variationstorque) for the EPAS system is usually a sinusoidal signal [8] of the system parameters.Bode Diagramand its expression is R(s) Asin(ωt). According to (19), the-50steady-state output of the EPAS system is0% variationThe closed-loop transfer function is(21)xss A Gcl ( jω ) sin(ωt φ ),where the phase shift is(22)Im(Gcl ( jω )).Re(Gcl ( jω ))Define the magnitude error between the steady-state output ofthe EPAS and the reference signal as em A-A Gcl(jω) . The emand Φ versus the controller gain ωc are shown in Fig. 5, whereboth the magnitude error and the phase shift of the stead-stateoutput of the EPAS system are converging to zeros with theincrease of the controller bandwidth ωc. According to Fig. 5,we choose ωc 5 103 rad/s, for which em is about 0.6% of thereference magnitude and Φ -6 10-5rad, in the followingcomputer simulations of the ADRC.10% variation20% variation-150-200-250-300900Phase (deg)φ Gcl ( jω ) tan 1Magnitude ncy (rad/sec)Fig.6: the Bode diagrams of Gd(s) for varying systemparameterssteady state Magnitude error and phase shiftmagnitude error0.1D. Robustness and stability marginsThe Bode diagram of the loop gain transfer function (18)with varying system parameters (Js, bs, Ks, Rs, m, Jm, Rm, bmand b) is shown in Fig. 7. The stability margins of the systemwith the variant system parameters are shown in Table I.0.050-0.05-0.10.501.51frequency(wc rad/sec)Bode Diagram2.522004x 10-3Magnitude (dB)x 10phase shift210100-20% variation-10% variation00% variation10% variation20% variation-100-1-200180-20.51.51frequency(wc rad/sec)25x 10Fig.5: The steady-state magnitude error and phase shiftC. External disturbance rejectionThis section will show how the external disturbance isrejected by the ADRC in the presence of structuraluncertainties of the EPAS system. Substituting Gp(s) in (3) andGc(s) in (17) into (20), we have0Phase (deg)0-180-360-540-410-210010210410Frequency (rad/sec)Fig. 7: Bode diagrams of Go(s) with varying systemparameters610
22 INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS,VOL.15,NO.1,MARCH 2010TABLE I: STABILITY MARGINS WITH DIFFERENTSYSTEM PARAMETERSChanges in parameters-20%-10%0%10%20%Gain margin(dB)8.988.6310.778.188.69Trs. As introduced in [8], the reference steering torque Trs canbe determined by a torque map based on the drivingconditions (vehicle speed and steering wheel angle). In oursimulation, we take Trs as a step input signal with magnitudeof 5Nm, and the square-wave and sinusoidal signals with theamplitudes of 5Nm and frequencies of 0.25Hz respectively.The simulation results are shown in Figs 9, 10, and 11, fromwhich we can see that under the control of ADRC, the steeringtorques follows the reference steering torques very well. As adisturbance force (with the magnitude of 2Nm) is added to theEPAS system and the values of the parameters such aseffective rack mass, damping and moment of inertia of thesteering column are varying within 20% of the original values,the steering torque output still tracks the reference steeringtorque as well as the ones shown in Figs 8 through 10. Fig. 11shows the external disturbance added to the system. Fig. 12shows the ESO-estimated torque output approximates the realtorque output very well in the presence of the disturbance.The simulation results have validated the good trackingperformance and the robustness of the ADRC against thedisturbances and parameter variations successfully.Phase Margin(deg)7.137.1410.167.527.07From Fig. 7, we can see that as the system parameters arechanging from -20% to 20% of their nominal values, the Bodediagrams are almost unchanged. Table I shows positivestability margins of the ADRC controlled EPAS system. Thefigure and table demonstrate the stability of the system in thepresence of system uncertainties. The stability of the ADRCfor a general n-th order physical plant is theoretically provedin [18]. The frequency-domain analyses introduced in thissection confirmed the stability and robustness of the ADRCfor this specific EPAS system.E. Frequency responses with varying loop gainThe assist gain Ka in Fig.3 determines the driving andparking modes of a vehicle. As Ka 40, it represents parkingmode where a large assistance for steering wheel is required.As Ka 1, it represents high-speed driving mode under whichthe assistant torque is much smaller than the one for parkingmode. As Ka is ranging from 1 through 40, the driving speedis decreasing with the increase of Ka. The Bode diagrams ofloop gain transfer function with varying assist gain Ka aregiven in Fig. 8.VI. CONCLUDING REMARKSBode DiagramMagnitude (dB)200100Ka 1Ka 100Ka
practical control strategy that is Active Disturbance Rejection Control (ADRC) [10-14] to the EPAS system. The controller treats the discrepancy between the real EPAS system and its mathematical model as the generalized disturbance and actively estimates and rejects in real time, the disturbance
ON ACTIVE DISTURBANCE REJECTION CONTROL: STABILITY ANALYSIS AND APPLICATIONS IN DISTURBANCE DECOUPLING CONTROL QING ZHENG ABSTRACT One main contribution of this dissertation is to analyze the stability char-acteristics of extended state observer (ESO) and active disturbance rejection control (ADRC).
4. Active Disturbance Rejection Control . The active disturbance rejection control was proposed by Han [9][10][11][12]. It is designed to deal with systems having a large amount of uncertainty in both the internal dynamics and external disturbances. The particularity of the ADRC design is that the total disturbance is defined as
ing control technology, known as active disturbance rejection con-trol to this day. Three cornerstones toward building the founda-tion of active disturbance rejection control, namely, the track-ing di erentiator, the extended state observer, and the extended state observer based feedback are expounded separately. The paper
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shift. The active disturbance rejection concept, introduced in section III, could very well serve as the basis of the new paradigm, which is characterized in section IV. Also included in section III is the demonstration of the broad range applications of the active disturbance concept. Finally, the paper is concluded with a few remarks in .
The Active Disturbance Rejection Controller (ADRC), invented by Han Jing-Qing who serves in Chinese Academy of Science, is a new type of nonlinear controller which is based on active disturbance rejection control technology. It does not depend on system model andcan estimate and compensate the influences of all the
e recently developed active disturbance rejection con-trol (ADRC) is an e cient nonlinear digital control strategy which regards the unmodeled part and external disturbance as overall disturbance of the controlled system [ ] . e ADRC mainly consists of a tracking di erentiator (TD), an extended state observer (ESO), and a nonlinear state error
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