NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...
NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ... NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...
Figure 2.3: Torsional Spring Model of FBS Here, Lp is the Lagrangian (the kinetic energy minus the potential energy) and Mup is avector representing the external forces applied to the system. Using these equations, we obtain a model that is unwieldy and di cult to use in designing control laws, therefore an even simpler model is desired. If one assumes that the free end of the exible beam undergoes only small de ections, then it can be assumed to follow a linear path, and its de ection can be measured as a length rather than as an angle. We can then model the bending dynamics of the beam as a simple linear spring with spring constant k. The system can then be thought of as a rotational actuator that either induces or attenuates translational oscillations. Figure 2.4 shows a simple diagram of this concept. While this model ignores all of the higher order modes and dynamics of the beam as well as the nonlinear motion of the free end, it does yield a su ciently simple mathematical representation. The linear displacement of the motor xture at the free end of the exible beam is represented by y, is the angle made by the rigid proof mass, k is the equivalent linear sti ness of the exible beam structure, m is the mass at the end of the proof mass, M is the total mass of the motor xture, c is the length of the rigid proof mass, Rm is the motor's 10
Figure 2.4: Translational Oscillation Model for FBS electrical resistance, and I is the inertia of the proof mass. Letting x =(y _y _ ) T , and using the voltage applied to the motor as the input to the system, the resulting state space equations are as follows: 0 1 0 _y B C B B C B ByC B B C B _C = B C B @ A @ _y _2 Rm sin( )(m 2 c 3 +Imc)+mc cos( ) _ K 2 m K 2 g;ykRm(mc 2 +I) Rm[(I+mc 2 )(M +m);m 2 c 2 cos 2 ( )] _ ;m2c2Rm cos( )sin( ) _2 +yRmkmc cos( );y(m+M )Rmk Rm[(I+mc2 )(M +m);m2c2 cos2 ( )] 0 B + B @ 1 C A 0 ;mcKmKg cos( ) Rm[(I+mc2 )(M +m);m2c2 cos2 ( )] 0 KmKg(m+M ) Rm[(I+mc 2 )(M +m);m 2 c 2 cos 2 ( )] 1 C u(x) (2.4) C A Again, this state-space model was derived using the EL equations. One mightwonder at the validity of the simplifying assumptions necessary to derive this model. For the purpose of this thesis, modeling inaccuracy is highly desirable { the control strategies studied in this paper are purportedly \robust" in some sense. One would hope that 11
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- Page 7 and 8: Contents Acknowledgments vi List of
- Page 9: List of Tables 4.1 Tabular Comparis
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- Page 14 and 15: Figure 1.1: TORA System One of the
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- Page 25 and 26: 2.3 Software Set-up All of the cont
- 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
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- 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
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Figure 2.3: Torsional Spring Model of FBS<br />
Here, Lp is the Lagrangian (the kinetic energy minus the potential energy) and Mup<br />
is avector representing the external forces applied to the system.<br />
Using these equations, we obtain a model that is unwieldy and di cult to use<br />
in designing control laws, therefore an even simpler model is desired. If one assumes<br />
that the free end of the exible beam undergoes only small de ections, then it can<br />
be assumed to follow a linear path, and its de ection can be measured as a length<br />
rather than as an angle. We can then model the bending dynamics of the beam as<br />
a simple linear spring with spring constant k. The system can then be thought of<br />
as a rotational actuator that either induces or attenuates translational oscillations.<br />
Figure 2.4 shows a simple diagram of this concept. While this model ignores all of the<br />
higher order modes and dynamics of the beam as well as the nonlinear motion of the<br />
free end, it does yield a su ciently simple mathematical representation. The linear<br />
displacement of the motor xture at the free end of the exible beam is represented<br />
by y, is the angle made by the rigid proof mass, k is the equivalent linear sti ness of<br />
the exible beam structure, m is the mass at the end of the proof mass, M is the total<br />
mass of the motor xture, c is the length of the rigid proof mass, Rm is the motor's<br />
10
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