NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...
NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ... NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...
a robust design strategy would, in practice, stabilize a physical plant that di ers substantially from its mathematical model. Thus the crudeness of our model only helps to demonstrate the robustness of the control laws that succeed in yielding adequate performance on the actual hardware system. Another advantage of this mechanical model is that it is identical, in form, to the system equations presented in [2] as a nonlinear benchmark problem. Since the object of this research is to compare nonlinear robust control strategies, this model is ideal in the sense that it is already widely used as a standard mathematical system for testing and comparing nonlinear control techniques. The mathematical model in (2.4) is slightly di erent than the standard model of the NLBP in that our input is voltage, not torque. This adds one extra term and a few constants in order to incorporate the voltage to torque transfer function into the model. Many of the papers involving the NLBP express the system equation in a non-dimensionalized form. Since some of the control strategies in this comparison make use of these dimensionless equations, and because they simplify the model by combining all of the physical parameters into a single value, we include them here: 0 1 _z B C B C BzC B C B _ C @ A = Where _ represents d z = y = t = d q M +m I+mc q k M +m 0 B @ v(x) =u(x) m+M k(I+mc2 ) mc = p (I+mc2 )(m+M ) _z ;z+ _2 sin( ); 2 cos( )_z K2 m K2 g Rmmc 1; 2 cos( ) 2 _ cos( )[z; 2 sin( )]+ K2 m K2 g Rmmc 1; 2 cos( ) 2 1 C A + 0 B @ 0 KmKg Rm cos( ) 1; 2 cos( ) 2 0 KmKg Rm 1; 2 cos( ) 2 and the following transformations are used: 12 1 C v(x): (2.5) C A
2.3 Software Set-up All of the control strategies presented in this paper were simulated in MAT- LAB's SIMULINK environment. They were then implemented by way of Quanser Consulting's Multiq3 I/O board which interfaced with the SIMULINK Real Time Workshop. Figure 2.5 shows the SIMULINK model used in the simulations and in the actual exible beam system. Their are two main blocks, one is the control block that takes the four states as inputs, as well as an initial open loop disturbance. The control has only one output, a voltage signal that is fed into the system block. The output of the strain gauge and the motor encoder were run through the Multiq3 board into SIMULINK. These provided a direct state measurement of the angle, , and the strain was a good estimator for the linear displacement, y. The state ve- locities in the system equations were numerically computed using lters of the form s s+ . Although this theoretically makes for a relatively crude observer, in practice these pseudo derivatives actually performed very well. Even though the assumption of full-state feedback is not strictly true, in practice a more complicated observer is unnecessary to achieve good performance by the control laws. The sample rate for in the SIMULINK model for both the simulation and the FBS was 1 kHz. All of the physical parameters for the Quanser system are noted here: k = 30N=m, m = :05Kg, M = :6Kg, c = :285m, I = :0030Kgm 2 , Km = :001Nm=A, Kg = 70, Rm =2:6 Ohms. 13
- Page 1 and 2: NONLINEAR CONTROLLER COMPARISON ON
- Page 3 and 4: BRIGHAM YOUNG UNIVERSITY GRADUATE C
- Page 5 and 6: ABSTRACT NONLINEAR CONTROLLER COMPA
- Page 7 and 8: Contents Acknowledgments vi List of
- Page 9: List of Tables 4.1 Tabular Comparis
- Page 12 and 13: 4.19 SGA: Nonlinear H1 vs. Linear O
- Page 14 and 15: Figure 1.1: TORA System One of the
- Page 16 and 17: 1.3 Literature Review In surveying
- Page 19 and 20: Chapter 2 Plant Speci cations and M
- Page 21 and 22: Figure 2.2: Mechanical Model of Fle
- Page 23: Figure 2.4: Translational Oscillati
- Page 27 and 28: Chapter 3 Overview of Control Strat
- Page 29 and 30: The optimal gain matrix was given b
- Page 31 and 32: 3.3 Passivity-Based Control Certain
- Page 33 and 34: limitations, k 1 and k 2 are tuning
- Page 35 and 36: Now we regard as the control variab
- Page 37 and 38: where V satis es the well known Ham
- Page 39 and 40: ecause one can quickly adjust the Q
- Page 41 and 42: Chapter 4 Simulation Results 4.1 Th
- Page 43 and 44: disturbances, whereas the hardware
- Page 45 and 46: Response in cm 4 3 2 1 0 −1 −2
- Page 47 and 48: Response in cm Response in cm 4 3 2
- Page 49 and 50: Response in cm Response in cm 5 4 3
- Page 51 and 52: Response in cm 4 3 2 1 0 −1 −2
- Page 53 and 54: Response in cm Response in cm 4 3 2
- Page 55 and 56: Response in cm Response in cm 4 3 2
- Page 57 and 58: It is seen in Figure 4.23 that the
- Page 59: all of the physical parameters exce
- Page 62 and 63: Response in m 0.05 0.04 0.03 0.02 0
- Page 64 and 65: Response in m 0.05 0.04 0.03 0.02 0
- Page 66 and 67: Response in m Response in m 0.05 0.
- Page 68 and 69: Response in m Response in m 0.05 0.
- Page 70 and 71: Response in m Response in m 0.05 0.
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2.3 Software Set-up<br />
All of the control strategies presented in this paper were simulated in MAT-<br />
LAB's SIMULINK environment. They were then implemented by way of Quanser<br />
Consulting's Multiq3 I/O board which interfaced with the SIMULINK Real Time<br />
Workshop. Figure 2.5 shows the SIMULINK model used in the simulations and in<br />
the actual exible beam system. Their are two main blocks, one is the control block<br />
that takes the four states as inputs, as well as an initial open loop disturbance. The<br />
control has only one output, a voltage signal that is fed into the system block. The<br />
output of the strain gauge and the motor encoder were run through the Multiq3<br />
board into SIMULINK. These provided a direct state measurement of the angle, ,<br />
and the strain was a good estimator for the linear displacement, y. The state ve-<br />
locities in the system equations were numerically computed using lters of the form<br />
s<br />
s+<br />
. Although this theoretically makes for a relatively crude observer, in practice<br />
these pseudo derivatives actually performed very well. Even though the assumption<br />
of full-state feedback is not strictly true, in practice a more complicated observer<br />
is unnecessary to achieve good performance by the control laws. The sample rate<br />
for in the SIMULINK model for both the simulation and the FBS was 1 kHz. All<br />
of the physical parameters for the Quanser system are noted here: k = 30N=m,<br />
m = :05Kg, M = :6Kg, c = :285m, I = :0030Kgm 2 , Km = :001Nm=A, Kg = 70,<br />
Rm =2:6 Ohms.<br />
13
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