Sliding Mode Control Part 12 potx

Studies of braking mechanisms of railway rolling stocks focus on the adhesion force, which is the tractive friction force that occurs between the rail and the wheel Kadowaki, 2004.. The

0 0.05 0.1 0.15 0.2 0.25

0 2 4 6 8

x 1050 500 1000 1500 2000

3 Control of the carriage position

The different sliding mode controllers for the carriage position are designed by exploitingthe differential flatness property of the system under consideration (Fliess et al (1995),

Sira-Ramirez & Llanes-Santiago (2000)) For the mechanical system the carriage position z C and the mean muscle pressure p M=0.5(p Ml+p Mr)are chosen as flat output candidates Thetrajectory control of the mean pressure allows for increasing stiffness concerning disturbanceforces acting on the carriage (Bindel et al (1999)) As the inner controls have been assigned

a high bandwidth, these underlying controlled muscle pressures can be considered as idealcontrol inputs of the outer control

y2=p M=0.5(p Ml+p Mr) (15)

The disturbance force F U is estimated by a disturbance observer and used for disturbancecompensation Due to the differential flatness of the system, the inverse dynamics can beobtained by solving the equations (14) and (15) for the input variables

3.1 Sliding mode control

Now, the tracking error e z = z Cd − z C can be stabilised by sliding mode control For this

purpose, the following sliding surface s zis defined for the outer control loop in the form

s z= ˙z Cd − ˙z C+α(z Cd − z C) (17)

At this, the coefficientα must be chosen positive in order to obtain a Hurwitz-polynomial The

convergence to the sliding surfaces in face of model uncertainty can be achieved by specifying

a discontinuous signum-function

˙s z = − W z · sign(s z) , W z >0 (18)

With a properly chosen positive coefficient W z dominating the corresponding model

uncertainties, the sliding surface s z = 0 is reached in finite time depending on the initialconditions This leads to the stabilising control law for each crank angle

υ z= ¨q id+α · ( ˙z Cd − ˙z C) +W z · sign(s z) (19)

Here, the carriage position z C , the carriage velocity ˙z C, the desired trajectory for the carriage

position z Cdand their first two time derivatives have to be provided For the second stabilisingcontrol input υ p , the desired trajectory for the mean pressure p Md is directly utilised in afeedforward manner, i.e.,υ p=p Md Inserting these new defined inputs into (16), the inversedynamics becomes

Having once reached the sliding surfaces, the final sliding mode is maintained during

trajectory tracking provided that the tracking error e z = z Cd − z C is governed by anasymptotically stable first-order error dynamics

3.2 Higher-order sliding mode control

An alternative method to reduce high frequency chattering effects is to employ higher-ordersliding mode techniques for control design, Levant (2008) For this approach the controlderivative is considered as a new control input Containing an integrator in the dynamicfeedback law, real discontinuities in the control input are avoided at higher-order slidingmode In this contribution a quasi-continuous second-order sliding mode controller asproposed in Levant (2005) is utilised Then the tracking error is stabilised by the followingcontrol law

For further reduction of the chattering phenomena, similar to the first-order sliding mode

control law (23) the discontinuous function sign(s z)in (25) can be replaced by the smoothfunction tanh s z



,  >0 Again, the new control inputυ zhas to be inserted in the inversedynamics (16), at which the second control inputυ premains the same

3.3 Proxy-based sliding mode control

Proxy-based sliding mode control is a modification of sliding mode control as well as anextension of PID-control, see Kikuuwe & Fujimoto (2006), Van-Damme et al (2007) Thebasic idea is to introduce a virtual carriage, called proxy, which is controlled using slidingmode techniques, whereas the proxy is connected to the real carriage by a PID-type couplingforce, see Fig 5 The goal of proxy-based sliding mode is to achieve precise tracking duringnormal operation and smooth, overdamped recovery in case of large position errors The

sliding mode control law for the virtual carriage results from equation (19) with z Sdenotingthe carriage position of the proxy

new variable a as integrated difference between the real and the virtual carriage position a=

The implementation of the control law is shown in the right part of Fig 5

2 + +

s

K s K s K

2 2 + +

a

Sliding Mode Control

High-Speed Linear Axis

[z z C C ] [z z z Cd Cd Cd ]

Inverse Dynamics

Figure 5 Coupling between virtual and real carriage (left) Implementation of the

proxy-based sliding mode control (right)

4 Control of internal muscle pressure

The internal pressures of the pneumatic muscles are controlled separately with high accuracy

in fast underlying control loops The pneumatic subsystem represents a differentially flatsystem with the internal muscle pressure as flat output, see Aschemann & Schindele (2008).Hence, equation (10) can be solved for the input variable

k ui(Δ Mi , p Mi)[˙p Mi+k pi Δ Mi,Δ˙ Mi , p Mi

The contraction length Δ Mi as well as its time derivative Δ˙ Mi can be considered as

scheduling parameters in a gain-scheduled adaptation of k ui and k pi With the internal

pressure as flat output, its first time derivative ˙p Mi =υ iis introduced as new control input

The error dynamics of each muscle pressure p Mi , i = { l, r }, can be asymptotically stabilised

by the following control law

υ i= ˙p Mid+a i · ( p Mid − p Mi) , (31)

where the constant a i is determined by pole placement By introducing the definition e i =

p Mid − p Mi for the control error w.r.t the internal muscle pressure, the corresponding errordynamics is governed by the following first order differential equation

5 Feedforward friction compensation

The main part of the friction is considered by a dynamical friction model in a feedforwardmanner For this purpose, the LuGre friction model, introduced by de Wit et al (1995), isemployed This friction model is capable of describing the Stribeck effect, hysteresis, stick-sliplimit cycling, presliding displacement as well as rising static friction

Here, the internal state variable z describes the deflection of the contact surfaces The model parameters are given by the static friction F S , the Coulomb friction F C and the Stribeck

velocity v S The parameterσ0is the stiffness coefficient,σ1the damping coefficient andσ2theviscous friction coefficient All parameters have been identified using nonlinear least squaretechniques

6 Reduced nonlinear disturbance observer

Disturbance behaviour and tracking accuracy in view of model uncertainties can besignificantly improved by introducing a compensating control action provided by a nonlinearreduced-order disturbance observer as described in Friedland (1996) The observer design isbased on the equation of motion The key idea for the observer design is to extend the stateequation with integrators as disturbance models

˙y=f(y, FU, u) ,

where y=q ˙qT

denotes the measurable state vector The estimated disturbance force ˆF U

is obtained from ˆF U=hTy+z with the chosen observer gain vector h T

The observer gain vector h and the nonlinear function Φ have to be chosen such that the

steady-state observer error e = F U − FˆU converges to zero Thus, the function Φ can bedetermined as follows

The linearised error dynamics ˙e has to be made asymptotically stable Accordingly, all

eigenvalues of the Jacobian

Je= ∂Φ(y, FU, u)

must be located in the left complex half-plane This can be achieved by proper choice of the

observer gain h1 The stability of the closed-loop control system has been investigated bythorough simulations

7 Control implementation

For the implementation at the test rig the control structure as depicted in Fig 6 has been used.Fast underlying pressure control loops achieve an accurate tracking behaviour for the desiredpressures stemming from the outer control loop The nonlinear valve characteristic (VC) hasbeen identified by measurements, see Aschemann & Schindele (2008), and is compensated by

Figure 6 Implementation of the cascaded control structure.

its approximated inverse valve characteristic (IVC) in each input channel For each pulleytackle one pneumatic muscle is equipped with a piezo-resistive pressure sensor mounted

at the connection flange that connects the muscle with the connection plate The carriage

position z C is obtained by a linear incremental encoder providing high resolution The

carriage velocity ˙z C is derived from the carriage position z Cby means of real differentiationusing a DT1-System with the corresponding transfer function G DT1(s) = s

T1s+1 The desired

value for the time derivative of the internal muscle pressure can be obtained either by real

differentiation of the corresponding control input p Miin (16) or by model-based calculationusing only desired values, i.e

0 5 10 15 20

−0.2 0 0.2

0 5 10 15 20

−1

−0.5 0 0.5 1

0 5 10 15 20

−6

−4

−2 0 2 4 6

0 5 10 15 20

−5 0

Figure 7 Desired values for the carriage position, velocity, and acceleration Corresponding

control error e z=z Cd − z Cfor standard sliding mode control

0 5 10 15 20 1

2 3 4 5 6 7

0 5 10 15 20 1

2 3 4 5 6 7

Figure 8 Comparison of desired and actual values for the left and right muscle pressure

−0.35 m and 0.35 m The maximum velocities are approximately 1.3 m/s and the maximumaccelerations are about 5 m/s2 The resulting tracking errors for the carriage e z = z Cd − z C

are shown in the right lower part of Fig 7 As for the carriage position, the maximumtracking error during the acceleration and deceleration intervals is approximately 3.5 mm Themaximum steady-state error is approximately 0.6 mm Fig 8 shows the corresponding desiredand actual values of the internal muscle pressure Obviously, the underlying fast controlloops achieve a precise tracking of the desired values, which stem from the outer decouplingcontrol loop Due to a time-optimal trajectory planning using desired ansatzfunctions withlimited jerk as described in Aschemann & Hofer (2005), the admissible range of the internalmuscle pressure is not exceeded In Fig 9 the different control approaches, introduced in

this contribution, are compared concerning the control error e z The higher-order slidingmode (HOSM) control approach results in a slightly larger maximum tracking error than

0 5 10 15 20 25

−6

−4

−2 0 2 4 6

8x 10

−3

PBSM HOSM SM

t in s

e z

Figure 9 Comparison of different control approaches concerning the corresponding control

errror e z: Proxy-based sliding mode control (PBSM), Higher-order sliding mode control(HOSM) and standard sliding mode control (SM)

with the standard sliding mode technique (SM) Nevertheless, the steady-state accuracy of theHOSM approach is superior to the other approaches As the chattering phenomena is reduced

by HOSM control the parameter  in equation (25) can be chosen very small, so that the

hyperbolic tangent function is very close to the ideal switching-function The parameter in

(23) have to be chosen about 100 times larger as compared to the value in HOSM, to avoid thehigh-frequency chattering, which is critical for the proportional valves and results in a reducedlifetime of the valves The largest tracking errors occur with proxy-based sliding mode (PBSM)control, which represents a PID-controller at normal operation The benefits of the PBSMcontrol are its high robustness and its slow and safe recovery from unexpected disturbancesand abnormal events, which leads to an inherent safety property In Fig 10 the impact ofthe feedforward friction compensation and the nonlinear reduced disturbance observer isdemonstrated Here the tracking errors of SM control with feedforward friction compensation(f.f.c.) and disturbance observer (d.o.), SM control only with f.f.c and SM control without f.f.c.and d.o are depicted As can be seen the tracking errors can be significantly reduced byemploying the proposed disturbance compensation strategy The sum of the feedforward

friction force F Fr and the disturbance force estimated by the disturbance observer ˆF U isdepicted in Fig 11 The robustness of the proposed solution is shown by a unmodelledadditional mass of 25 kg, which represents almost the double of the nominal value In thecorresponding force, the increase due to the higher inertial forces becomes obvious Thecorresponding tracking errors are shown in Fig 12 All three control approaches show similarresults Whereas the steady-state errors remain almost unchanged, the maximum trackingerrors are now approximately 8 mm due to the inertia forces during the acceleration anddeceleration phases The closed-loop stability is not affected by this parametric uncertainty

381

Sliding Mode Control Applied to a Novel Linear Axis Actuated by Pneumatic Muscles

without f.f.c and d.o.

200

mass mC+25kg mass m C

Figure 11 Estimated disturbance force with and without additional mass of 25 kg

8x 10

−3

PBSM HOSM SM

a compensation scheme consisting of a feedforward friction compensation and a nonlinearreduced disturbance observer Experimental results emphasise the excellent closed-loopperformance with maximum position errors of approximately 4 mm The robustness of theproposed control is shown by measurements with an almost doubled carriage mass

10 References

Aschemann, H & Hofer, E (2004) Flatness-based trajectory control of a pneumatically

driven carriage with uncertainties, Proceedings of NOLCOS 2004, Stuttgart, Germany

pp 239–244

383

Sliding Mode Control Applied to a Novel Linear Axis Actuated by Pneumatic Muscles

Aschemann, H & Hofer, E (2005) Flatness-based trajectory planning and control of a parallel

robot actuated by pneumatic muscles, CD-Proceedings of ECCOMAS Thematic Conf.

Multibody Dyn., Madrid, Spain

Aschemann, H & Schindele, D (2008) Sliding-mode control of a high-speed linear axis driven

by pneumatic muscle actuators, IEEE Trans Ind Electronics 55(11): 3855–3864.

Aschemann, H., Schindele, D & Hofer, E (2006) Nonlinear optimal control of a

mechatronic system with pneumatic muscle actuators, CD-Proceedings of MMAR

2006, Miedzyzdroje, Poland

Bindel, R., Nitsche, R., Rothfuß, R & Zeitz, M (1999) Flatness based control of two valve

hydraulic joint actuator of a large manipulator, CD-Proceedings of ECC 1999, Karlsruhe,

Germany

de Wit, C C., Olsson, H., Âström, K & Lischinsky, P (1995) A new model for control of

systems with friction, IEEE Transactions on Automatic Control 40(3): 419–425.

Fliess, M., Levine, J., Martin, P & Rouchon, P (1995) Flatness and defect of nonlinear systems:

Introductory theory and examples, Int J Control 61: 1327–1361.

Friedland, B (1996) Advanced Control System Design, Prentice-Hall.

Kikuuwe, R & Fujimoto, H (2006) Proxy-based sliding mode control for accurate and safe

position control, IEEE Trans on Industr Electr 53(5): 25–30.

Levant, A (2005) Quasi-continuous high-order sliding-mode controllers, IEEE Transactions on

Automatic Control 50(11): 1812–1816.

Levant, A (2008) Homogeneous high-order sliding modes, Proceedings of the 17th IFAC World

Congress, Seoul, Korea pp 3799–3810.

Lilly, J & Yang, L (2005) Sliding mode control tracking for pneumatic muscle actuators in

opposing pair configuration, IEEE Trans on Contr Syst Techn 13(4): 550–558.

Pukdeboon, C., Zinober, A S I & Thein, M.-W L (2010) Quasi-continuous higher

order sliding-mode controllers for spacecraft-attitude-tracking maneuvers, IEEE

Transactions on Industrial Electronics 57(4): 1436–1444.

Schindele, D & Aschemann, H (2010) Norm-optimal iterative learning control for a

high-speed linear axis with pneumatic muscles, Proc of NOLCOS 2010, Bologna, Italy.

to be published

Sira-Ramirez, H & Llanes-Santiago, O (2000) Sliding mode control of nonlinear mechanical

vibrations, J of Dyn Systems, Meas and Control 122(12): 674–678.

Smith, J., Ness, H V & Abott, M M (1996) Introduction to Chemical Engineering

Thermodynamics, McGraw-Hill, New York.

Van-Damme, M., Vanderborght, R., Ham, R V., Verrelst, B., Daerden, F & Lefeber, D

(2007) Proxy-based sliding-mode control of a manipulator actuated by pleated

pneumatic artificial muscles, Proc IEEE Int Conf on Robotics and Automation, Rome,

Italy pp 4355–4360.

Zhu, X., Tao, G., Yao, B & Cao, J (2008) Adaptive robust posture control of a parallel

manipulator driven by pneumatic muscles, Automatica 44(9): 2248–2257.

Studies of braking mechanisms of railway rolling stocks focus on the adhesion force, which

is the tractive friction force that occurs between the rail and the wheel (Kadowaki, 2004) During braking, the wheel always slips on the rail The adhesion force increases or decreases according to the slip ratio, which is the difference between the velocity of the rolling stocks and the tangential velocity of each wheel of the rolling stocks normalized with respect to the velocity of the rolling stocks A nonzero slip ratio always occurs when the brake caliper holds the brake disk, and thus the tangential velocity of the wheel so that the velocity of the wheel is lower than the velocity of the rolling stocks Unless an automobile is skidding, the slip ratio for an automobile is always zero In addition, the adhesion force decreases as the rail conditions change from dry to wet (Isaev, 1989) Furthermore, since it is impossible to directly measure the adhesion force, the characteristics of the adhesion force must be inferred based on experiments (Shirai, 1977)

To maximize the adhesion force, it is essential to operate at the slip ratio at which the adhesion force is maximized In addition, the slip ratio must not exceed a specified value determined to prevent too much wheel slip Therefore, it is necessary to characterize the adhesion force through precise modeling

To estimate the adhesion force, observer techniques are applied (Ohishi, 1998) In addition, based on the estimated value, wheel-slip brake control systems are designed (Watanabe, 2001) However, these control systems do not consider uncertainty such as randomness in the adhesion force between the rail and the wheel To address this problem, a reference slip ratio generation algorithm is developed by using a disturbance observer to determine the desired slip ratio for maximum adhesion force Since uncertainty in the traveling resistance and the mass of the rolling stocks is not considered, the reference slip ratio, at which adhesion force is maximized, cannot always guarantee the desired wheel slip for good braking performance

In this paper, two models are developed for the adhesion force in railway rolling stocks The first model is a static model based on a beam model, which is typically used to model automobile tires The second model is a dynamic model based on a bristle model, in which the friction interface between the rail and the wheel is modeled as contact between bristles (Canudas de Wit, 1995) The validity of the beam model and bristle model is verified through an adhesion test using a brake performance test rig

Sliding Mode Control

386

We also develop wheel-slip brake control systems based on each friction model One control

system is a conventional PI control scheme, while the other is an adaptive sliding mode

control (ASMC) scheme The controller design process considers system uncertainties such

as the traveling resistance, disturbance torque, and variation of the adhesion force according

to the slip ratio and rail conditions The mass of the rolling stocks is also considered as an

uncertain parameter, and the adaptive law is based on Lyapunov stability theory The

performance and robustness of the PI and adaptive sliding mode wheel-slip brake control

systems are evaluated through computer simulation

2 Wheel-slip mechanism for rolling stocks

To reduce braking distance, automobiles are fitted with an anti-lock braking system (ABS)

(Johansen, 2003) However, there is a relatively low adhesion force between the rail and the

wheel in railway rolling stocks compared with automobiles A wheel-slip control system,

which is similar to the ABS for automobiles, is currently used in the brake system for

railway rolling stocks

The braking mechanism of the rolling stocks can be modeled by

( )

a

v r v

ω

where F a is the adhesion force, ( )μ λ is the dimensionless adhesion coefficient, λ is the slip

ratio, N is the normal force, v is the velocity of the rolling stocks, and ω and r are the

angular velocity and radius of each wheel of the rolling stocks, respectively The velocity of

the rolling stocks can be measured (Basset, 1997) or estimated (Alvarez, 2005) The adhesion

force F is the friction force that is orthogonal to the normal force This force disturbs the a

motion of the rolling stocks desirably or undesirably according to the relative velocity

between the rail and the wheel The adhesion force F a changes according to the variation of

the adhesion coefficient ( )μ λ , which depends on the slip ratioλ, railway condition, axle

load, and initial braking velocity, that is, the velocity at which the brake is applied Figure 1

shows a typical shape of the adhesion coefficient ( )μ λ according to the slip ratioλ and rail

conditions

To design a wheel-slip control system, it is useful to simplify the dynamics of the rolling

stocks as a quarter model based on the assumption that the rolling stocks travel in the

longitudinal direction without lateral motion, as shown in Fig 2 the equations of motion for

the quarter model of the rolling stocks can be expressed as

T =rF and T b are the adhesion and brake torques, respectively, T d is the disturbance

torque due to the vibration of the brake caliper, J and r are the inertia and radius,

respectively, of each wheel of the rolling stocks, and M and F are the mass and traveling r

resistance force of the rolling stocks, respectively

Adaptive Sliding Mode Control of Adhesion Force in Railway Rolling Stocks 387

Fig 2 Quarter model of the rolling stocks

From (3) and (4), it can be seen that, in order to achieve sufficient adhesion force, a large brake torque T b must be applied When T b is increased, however, the slip ratio increases, which causes the wheel to slip When the wheel slips, it may develop a flat spot on the rolling surface This flat spot affects the stability of the rolling stocks, the comfort of the passengers, and the life cycle of the rail and the wheel To prevent this undesirable braking situation, a desired wheel-slip control is essential for the brake system of the rolling stocks

In addition, the adhesion force between the wheel and the contact surface is dominated by the initial braking velocity, as well as by the mass M and railway conditions In the case of

automobiles, which have rubber pneumatic tires, the maximum adhesion coefficient changes from 0.4 to 1 according to the road conditions and the materials of the contact surface (Yi, 2002) In the case of railway rolling stocks, where the contact between the wheel and the rail is that of steel on steel, the maximum adhesion force coefficient changes from approximately 0.1 to 0.4 according to the railway conditions and the materials of the contact surface (Kumar, 1996) Therefore, railway rolling stocks and automobiles have significantly different adhesion force coefficients because of different materials for the rolling and contact surfaces However, the brake characteristics of railway rolling stocks (Jin, 2004) and automobiles (Li, 2006) are similar

Sliding Mode Control

388

According to adhesion theory, the maximum adhesion force occurs when the slip ratio is approximately between 0.1 and 0.4 in railway rolling stocks Therefore, the slip ratio at which the maximum adhesion force is obtained is usually used as the reference slip ratio for the brake control system of the rolling stocks Figure 3 shows an example of a wheel-slip control mechanism based on the relationship between the slip ratio and braking performance

60 70 80 90 100 110

0 5 10 15 20

Fig 4 Simplified contact model for the rail and wheel

3 Static adhesion force model based on the beam model

To model the adhesion force as a function of the slip ratio, we consider the beam model, which reflects only the longitudinal adhesion force Figure 4 shows a simplified contact model for the rail and wheel, where the beam model treats the wheel as a circular beam

Adaptive Sliding Mode Control of Adhesion Force in Railway Rolling Stocks 389

supported by springs The contact footprint of an automobile tire is generally approximated

as a rectangle by the beam model (Sakai, 1987) In a similar manner, the contact footprint

between the rail and the wheel is approximated by a rectangle as shown in Fig 5

Fig 5 Contact footprint between the rail and the wheel

The contact pressure p between the rail and the wheel at the displacement x c from the tip

of the contact footprint in the longitudinal direction is given by (Sakai, 1987)

where N is the normal force, and l and w are the length and width of the contact

footprint, respectively Figure 6 shows a typical distribution of the tangential force

coefficient in a contact footprint (Kalker, 1989)

In Fig 6, the variable f , which is the derivative of the adhesion force x F with respect to the a

displacement x c from the tip of the contact footprint, is given by

≤ ≤

⎧

where C is the modulus of transverse elasticity, x l is the displacement from the tip of the h

contact footprint at which the adhesion-force derivativef x changes rapidly, and μd is the

dynamic friction coefficient In particular, μd is defined by

where μmax is the maximum adhesion coefficient, a is a constant that determines the

dynamic friction coefficient in the slipping regime, and l is expressed as (Sakai, 1987) h

max

13

Sliding Mode Control

390

where K x is the traveling stiffness calculated by

2

12

The wheel load, which is the normal force, is equal to the integrated value of the contact

pressure between the rail and the wheel over the contact footprint Therefore, the adhesion

force F a between the rail and the wheel can be calculated by integrating (6) over the length

of the contact footprint and substituting (7) and (8) into (6), which is expressed as

2 2

max

3 max

Fig 6 A typical distribution of the tangential force coefficient in a contact footprint

4 Dynamic adhesion force model based on bristle contact

As a dynamic adhesion force model, we consider the Dahl model given by (Dahl, 1976)

where z is the internal friction state, σ is the relative velocity, αis the stiffness coefficient,

and F and F c are the friction force and Coulomb friction force, respectively Since the

steady-state version of the Dahl model is equivalent to Coulomb friction, the Dahl model is a

generalized model for Coulomb friction However, the Dahl model does not capture either

the Stribeck effect or stick-slip effects In fact, the friction behavior of the adhesion force