Challenges and Paradigms in Applied Robust Control Part 8 pdf

Challenges and Paradigms in Applied Robust Control 200 Figure 5 shows the tracking errors of the BSC and ABSC systems having perturbations of the system parameters such as the Coulomb fr

Robust Control of Electro-Hydraulic Actuator Systems

Fig 4 Estimated value for the system uncertainties of the ABSC system for the sinusoidal reference input

Fig 5 Tracking errors of the BSC and ABSC systems with the perturbation of the system parameters

Challenges and Paradigms in Applied Robust Control

200

Figure 5 shows the tracking errors of the BSC and ABSC systems having perturbations of the system parameters such as the Coulomb friction, viscous friction and pump leakage coefficient in the EHA system for the sinusoidal reference input It was assumed that the system parameters have a perturbation of 50% From Fig 5, it was found that the perturbations of the system parameters of the EHA system are closely related with the tracking performance of the EHA system Table 2 shows the tracking RMS errors of the BSC and ABSC systems according to the perturbation of the system parameters The variations of the tracking RMS errors due to the 50% perturbation of the system parameters for the BSC and ABSC systems are 17.6% and 3.02%, respectively These results show that the proposed position control scheme has the desired robustness to system uncertainties such as the perturbation of the viscous friction, Coulomb friction and pump leakage coefficient

Control scheme Perturbation ratio RMS value

Table 2 Tracking RMS errors of the BSC and ABSC systems according to the perturbations

of the system parameters

5 Experimental results and discussion

Figure 6 shows the experimental setup of the EHA system To evaluate the effectiveness of the proposed control system, the PCM-3350(AMD Geode processor, 300MHz) was used The control algorithms were programmed by Turbo-C++ language on MS-DOS, in order to directly handle the PCM-3718 as a data acquisition board The PCM-3718 is a fully multifunctional card with PC/104 interface In addition, to measure the position of the piston, an LVDT(linear variable differential transformer) sensor was used The sampling rate was set to 1 kHz

Fig 6 Experimental setup of the EHA system

Robust Control of Electro-Hydraulic Actuator Systems

Figure 7 shows the tracking errors of the BSC and ABSC systems for the sinusoidal reference input, which was used in the computer simulation The tracking error of the BSC system is relatively large when the direction of the piston is changed because the BSC system cannot compensate the friction of the EHA system In addition, the tracking error of the BSC varies according to the direction of the piston because of the system uncertainties of the EHA system However, the ABSC system has better tracking performance than the BSC system because the ABSC system can effectively compensate the system uncertainties as well as the nonlinear friction effects by using the estimated value ˆf , which is shown in Fig 8

Figure 9 shows the speed of the motor as the control input for the sinusoidal reference input Figure 10 shows the tracking errors of the BSC and ABSC systems for the square wave type reference input The characteristics of the transient responses of the BSC and the ABSC systems are almost same In the BSC system, however, steady-state error occurs relatively large in the backward direction This shows that the BSC system cannot compensate the system uncertainties of the EHA system But we can show that the ABSC system can effectively compensate the system uncertainties regardless of the piston direction Figure 11 shows the estimated value ˆf for the system uncertainties of the ABSC system for the square

wave type reference input The estimated value ˆf for the system uncertainties makes the

desired tracking performance and robustness to the EHA system with system uncertainties Figure 12 shows the speed of the motor as the control input for the square wave type reference input

Fig 7 Tracking errors of the BSC and ABSC systems for the sinusoidal reference input

Challenges and Paradigms in Applied Robust Control

Robust Control of Electro-Hydraulic Actuator Systems

Fig 10 Tracking errors of the BSC and ABSC systems for the square wave type reference input

Fig 11 Estimated value for the system uncertainties of the ABSC system for the square wave type reference input

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Fig 12 Speed of the motor as the control input for the square wave type reference input

Robust Control of Electro-Hydraulic Actuator Systems

Table 3 shows the tracking RMS errors of the BSC and ABSC systems for the sinusoidal reference input and the square wave type reference input at steady-state From Table 3, it was found that using the ABSC system instead of the BSC system yields about 5 times improvement in the tracking performance of the EHA position control system

Control system Sinusoidal reference input

Square wave type reference

7 References

Y Chinniah, R Burton and S Habibi (2006), Failure monitoring in a high performance

hydrostatic actuation system using the extended kalman filter, Int J Mechatronics 16(10) , pp 643-653

J J Choi, J S Kim and S I Han (2004), Pre-sliding friction control using the sliding mode

controller with hysteresis friction compensator, KSME Int’l J 18(10), pp 1755-1762

S Habibi and A Goldenberg (2000), Design of a new high-performance electro-hydraulic

actuator, IEEE Trans Mechatronics 5(2), pp 158-164

L Jun, F Yongling, Z Guiying, G Bo and M Jiming (2004), Research on fast response and

high accuracy control of an airborne brushless DC motor, Proc 2004 IEEE Int Conf Robotics and Biomimetics, Shenyang, China, pp 807-810

C Kaddissi, J P Kenne and M Saad (2006), Indirect adaptive control of an electro-hydraulic

servo system based on nonlinear backstepping, IEEE Int Symposium Ind Electron,

Montreal, Quebec, Canada, pp 3147-3153

V V Kokotovic, J Grabowski, V Amin and J Lee (1999), Electro hydraulic power steering

system, Int Congress & Exposition, Detroit, Michigan, USA, pp 1-4

M Krstic, I Kanellakopoulos and P Kokotovic (1995), Nonlinear and Adaptive Control

Design, Wiley Interscience, New York, USA

Challenges and Paradigms in Applied Robust Control

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K J Lee, H M Kim and J S Kim (2004), Design of a chattering-free sliding mode controller

with a friction compensator for motion control of a ball-screw system, IMechE J of Systems and Control Engineering, 218, pp 369-380

H E Merritt (1967), Hydrostatic Control Systems, Wiley, New York, USA

M Perron, J de Lafontaine and Y Desjardins (2005), Sliding-mode control of a

servomotor-pump in a position control application, IEEE Conf Electrical and Computer Eng,

Saskatoon, Canada, pp 1287-1291

J J Slotine and W Li (1991), Applied Nonlinear Control, Pearson Education, New Jersey,

USA

S Wang, R Burton and S Habibi (2005), Sliding mode controller and filter applied to a

model of an electro-hydrostatic actuator system, ASME Int Mechanical Engineering Congress & Exposition, Orlando, Florida, USA, pp 1-10

Discussion on Robust Control Applied to Active

Magnetic Bearing Rotor System

Rafal P Jastrzebski1, Alexander Smirnov1, Olli Pyrhönen1

and Adam K Piłat2

1Dept of Electrical Engineering, LUT Energy, Lappeenranta University of Technology

2Dept of Automatics, AGH University of Science and Technology, Krakow

AMBs are employed in high-speed rotating machines such as turbo compressors, flywheels,machine tools, molecular pumps, and others (Schweitzer & Maslen, 2009) The support ofrotors using an active magnetic field instead of mechanical forces of the fluid film, contactrolling element, or ball bearings enables high-speed operation and lower friction losses.Other major advantages of AMBs include no lubrication, long life, programmable stiffnessand damping, built-in monitoring and diagnostics, and availability of automatic balancing.However, AMB rotor system forms an open-loop unstable, multiple-input multiple-output(MIMO) coupled plant with uncertain dynamics that can change over time and that can varysignificantly at different rotational speeds In practical systems, the sensors are not collocatedwith the actuators, and therefore, the plant cannot always be easily decoupled Additionally,the control systems face a plethora of external disturbances

The major drawback of an AMB technology is a difficulty in designing a high-performancereliable control and its implementation For such systems, theμ and H∞control approachesoffer useful tools for designing a robust control (Moser, 1993; Zhou et al., 1996)

The high-performance and high-precision control for the nominal plant without uncertaintiescan be realized by using model-based, high-order controllers In the case of control synthesis,which is based on the uncertain plant model, there is a tradeoff between the nominalperformance (time- and frequency-domain specifications) and the robustness The modeleduncertainties cannot be too conservative or otherwise obtaining practical controllers might

be not feasible (Sawicki & Maslen, 2008) Moreover, too complex uncertainty models lead toincreased numerical complexity in the control synthesis The models applied for the control

10

2 Will-be-set-by-IN-TECHsynthesis of AMBs can vary from a point mass (Oliveira et al., 2006) to very complex MIMOplants (Li, Lin, Allaire & Luo, 2006).

The literature presents different weighting or interconnection design schemes Each of theschemes has its contradictive objectives and tradeoffs For the point mass levitated systems,the load uncertainty is typically applied As an example, Li, Lin, Allaire & Luo (2006)present anS/T/KS scheme, where the S, T, K, and G are the sensitivity, complementary

sensitivity, controller, and plant transfer functions The corresponding weights are tunedusing engineering judgment and manual trial and error simulations Losch (2002) splits theavailable design schemes to signal-based and the loop-shaping schemes The signal-basedschemes are considered to be more complex and conservative The loop-shaping schemes,for example, discussed by Losch (2002) includeKS/SG/T for the control of the rigid rotor

and KS/SG/T/S for the control of the flexible rotor Another loop-shaping procedure

is developed by Glover & McFarlane (1989) It applies robust stabilization of normalizedcoprime factorization of the plant using two weights: pre- and post-compensators Skogestad

& Postlethwaite (2005) give a general recommendation on the selection of these weights.This chapter reviews different weighting schemes for building the robust control of AMBsystems The presentation starts with the point mass levitation and then undertakesnon-gyroscopic and gyroscopic coupled AMB rotor systems The aim of the robust control is

to stabilize the rotor suspension independently to the assumed uncertainties The robustnessmust be satisfied in the full range of the operating frequencies and for the selected range ofthe state variables The work studies how to select the optimal control weighting functionsfor selected schemes based on genetic algorithms and experimental data obtained from thetest rig The Linear Parameter-Varying (LPV) technique is applied to suppress the influence

of the variable rotational speed on the plant dynamics, thus reducing the uncertainty set.The real-time controller operating conditions are considered The nonlinear simulations ofthe synthesized controllers and the accurate plant models in Simulink are compared withexperimental results

2 Suspension of the point mass

2.1 Introduction

The main component of the AMB system is an electromagnet that is used for the levitationpurposes to keep the ferromagnetic object (e.g rotor) levitated The electromagnetic forcevalue is controlled by the coil current steered by the external regulator The introduction tothe robust control is described by the example of Active Magnetic Suspension (AMS), which

is also referred as Active Magnetic Levitation (AML) The robust approach can be applied tothe uncertainty of the electromagnetic actuator and the levitated object mass The controllersynthesis and experiments are devoted to the MLS2EM (InTeCo, 2008) system (see Fig 1) thatextends the standard single electromagnet AML and represents one axis of the typical fourhorse-shoe AMB configuration

2.2 Why robust control is required

In the classical state-feedback control approach for locally linearized AML model (Pilat,2002) the mass uncertainty affects the control quality and object position For the designedstate-feedback controller with different closed-loop properties the 90 % mass perturbation hasbeen introduced and presented with Bode diagrams in Fig 2 One can find the influence ofthe mass change on the phase and amplitude depending on the designed controller properties

208 Challenges and Paradigms in Applied Robust Control

Discussion on Robust Control Applied to Active Magnetic Bearing Rotor System 3

Fig 1 Dual electromagnet Active Magnetic Levitation System - concept and test-rig

The closed-loop characteristics remain unchanged due to the fixed and non-robust structure

Frequency [Hz]

m m+

Frequency [Hz]

m m+

m−

(b)Fig 2 Influence on the mass perturbation for the state feedback controller: a) for k = 250

4 Will-be-set-by-IN-TECH

at the modelling and simulation stage, or by the application of an on-line adopted neuralnetwork (Pilat & Turnau, 2009), where the weights and biases are updated while the real-timecontrol is pending Another approach is based on the linear control theory and parameteruncertainty Some applications to the magnetic levitation and bearing systems can be found

in (Fujita et al., 1995; Gosiewski & Mystkowski, 2008; Mystkowski & Gosiewski, 2009) Thefollowing section will present a robust controller design to stabilize the levitated objectindependently to its mass uncertainty More detailed, simulation results and comparison tothe state feedback controller can be found in (Pilat, 2010)

2.3 AML modelling and control

2.3.1 Nonlinear and linear AML model

The open loop structurally unstable model of the current driven single electromagnet AML(Pilat, 2009) is given by Equation (1)

¨

x1= − Kem (i0+i)2

where: x1- object displacement with respect to the x10[m], x10- nominal object distance from

the electromagnet surface [m] (x1>0), x2 - object velocity [m s−1 ], m object mass [kg], g

-gravity acceleration [m s−2 ], Kem- actuator constant describing its construction [N m2A−2 ], i

- coil current [A], i0nominal coil current for the object distance x10 This research will use the

laboratory setup (Fig 1b) characterized by the following parameter values: m = 0.056 kg, Kem

= 5.594·10−5N m2A−2 By analyzing the nonlinear model one can observe that the variablemass affects the system dynamics so that heavier objects require an increase in the coil currentwhen the actuator construction remains the same It means that the controller should react tothe variable load using the robustness property The steady-state coil current depends on thenominal object distance and the levitated object mass and the actuator design 2

0

m −1 β0



(3)with:α0=2Kemi2x10−3kg s−2,β0= − 2Kemi0x10−2kg m A−1s−2

2.3.2 Robust controller design

TheH2,H∞andμ-synthesis theory allows to perform an analysis and synthesis of the robust

control systems (Battachatyya et al., 1995; Gu et al., 2005a; Kwakernaak, 1993; 2002) in thecase of model-system uncertainties and perturbations In the AML, the exact physical value

of the levitated object mass is not known, but can be measured before an experiment Whenapplying the AML in real applications the mass value can vary It can be assumed that themass value is known with a certain, known interval Thus, we can represent the mass asfollows:

210 Challenges and Paradigms in Applied Robust Control

Discussion on Robust Control Applied to Active Magnetic Bearing Rotor System 5

0

− pm

,B2=

0

Fig 3 AML closed loop system with an uncertain mass

Thus, a key point in the controller design is to develop the sensitivity function to satisfy therequired closed-loop performance over a specified frequency range There are many possibleapproaches to propose the weighting functions, for example they can be chosen as follows:

211

Discussion on Robust Control Applied to Active Magnetic Bearing Rotor System

6 Will-be-set-by-IN-TECH

W p(s) = wn0

The control weighting function W u(s) is chosen as a scalar value of 10−3 By adjusting the

values of wn0, wd1, and wd0the performance of the robust controller could be tuned up.The robustμ-synthesis based on the D-K iteration procedure involving a set of optimizations

produces the controller in a continuous form The resulting controller order can be high anddepend on the mass perturbation, formulation of the weighting function, and the number

of iterations executed to find the optimal controller The obtained 3rd order controller has

the following parameters: a2=-1.473·106, a1=-8.457·107, a0= -8.552·108, b2 = 1.648·103, b1 =3.014·105, b0= 3.012·104and it is given by equation (11)

MATLAB/Simulink via RTWT at a sampling frequency of F S = 4 kHz The extra forcegenerated in the programmable way and produced by the lower electromagnet was attractingthe levitated object and therefore simulating mass uncertainty To show the performance ofthe robust controller, the experimental data has been filtered to remove the high frequenciesfrom the measured signals

In the case of a step-type load representing a narrow mass change of 15 % the object is broughtdown to the desired level in 100 ms The maximal overshoot versus desired object position isequal to 317 μm while for the triangular load corresponding to the low-frequency mass change

of 33% is equal to 237 μm

2.3.4 Conclusions to AML robust control design

The analytical robust control approach requires a good model of the system at the operatingpoint The parameter uncertainty does not cancel the structural nonlinearities, but issatisfactory for the required control performance In some cases, the obtained high-ordercontroller structure could not be realized by the hardware resources In this case, the orderreduction under special attendance of the controller quality is required

3 Modelling of the AMB rotor systems

The second case study plant is a laboratory test stand with an AMB-supported custom rotor.The machine was originally a solid rotor induction motor for general industrial high-speedapplications with the rated speed 12000 rpm The original machine was produced by RotatekFinland Oy The AMB setup consist of two radial actuators and one axial actuator The control

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Discussion on Robust Control Applied to Active Magnetic Bearing Rotor System 7

0.0095

0.0098

0.01 0.0103

δm

0.8 0.85 0.9 0.95 1 1.05 1.1

δm

0.8 0.85 0.9 0.95 1 1.05 1.1

Time [s]

(b)

Fig 4 Real-time experimens: a) narrow load change, b) slow load change

layout comprises the inner current control loop and the outer position control loop Thissection focuses on the radial suspension

The studied AMB system is non-symmetric and non-collocated The rotor is of a long rotortype without a significant gyroscopic effect The machine is subcritical, that is, the maximumrotational speed is below the first flexible bending mode From the radial position controlpoint of view, the measured outputs are rotor displacements in two axes in two sensor planesand the applied control signals are four control currents of two radial eight-pole magneticbearings The system parameters are presented in Table 1

Table 1 Key AMB system parameters and their nominal values

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Discussion on Robust Control Applied to Active Magnetic Bearing Rotor System