Further Details Belonging to Subsection “8.1.1. A Higher Order

x^d y=f(x)

x₀ x₁

0>f’(x)

( )

− +

− +

−

∆

= −

−

∆

−

D x

x f D x x^d

0 0 1

Figure A.6.6. (continued) Two parametric transformations for sequences for

“decreasing system”

A.6.1. Further Details Belonging to Subsection “8.1.1. A Higher Order Application Example for Fixed Point Transformations of a Few Parameters”

In the control of this system a ball or cylinder can roll on the surface of a beam the tilting angle of which is driven by some actuator. The motion of the ball essentially is determined by the tilting angle and the force of gravitation. This means that even if we are in the possession of a very strong actuator, the acceleration of the ball along the beam is limited by the above two factors. Since the directly controllable quantity is the torque determining the 2^nd time-derivative of the angle tilting the beam, this system acts as a 4^th order one in the sense that the 4th time-derivative of the ball’s position along the beam is determined by the tilting torque. It has the following parameters: the momentum of the beam Θ_Beam=2 (kg×m²), the mass of the ball mBall=2 (kg), the radius of the ball r=0.05 (m), and the gravitational acceleration is g=9.81 (m/s²). Via introducing the quantities A=ΘBeam, and B=ΘBall/r²+mBall, the following equations of motion are obtained as given in Fig. A.6.1.1. in which variable ϕ (rad) describes the rotation of the beam counter-clockwisely with respect to the horizontal position, and x (m) denotes the distance of the ball from the center of the beam where it is supported. Variable Q (N×m) describes the torque at the axis rotating the beam. This quantity consists of two different components: the torque directly exerted by the drive and the contribution by the friction forces acting at the surface of the axle. In the present investigations this latter component is unknown by the controller, only the consequences of its existence in the trajectory tracking can be observed. It is evident that only the 4^th time-derivative of x can be related to the 2^nd time-derivative of the tilting angle of the beam that is in direct relationship with the rotating torque taking part in tilting this angle. For making the model more realistic in the simulations it was assumed that the axle of the beam has considerable dynamic friction approximated by the LuGre model as follows.

r

directly be set

by the torque tilting the beam

(Q).

A 4th order system as investigated paradigm:

The Ball-Beam System.

Dynamic friction at the axle (unknown by the controller).

2

Figure A.6.1.1. The Ball-Beam System

Instead of the dubious “velocity limit” normally applied in simulations with static friction models (i.e. the limit value at which the relative motion of the contacted surfaces is practically zero) to describe the “stick-slip phenomenon” an “internal degree of freedom”, z is introduced with the appropriate equations of motion as

( )

describes the rotational speed characteristic to the surfaces in contact at the axle. The kinematic tracking requirements were set by (A.6.1.2) with the order of from which the desired 4^th time-derivative ( )Des

x ⁴ can be computed. Since normally the beam must be in an almost horizontal position for stabilizing purposes it was expedient to limit its allowable rotational angle, and angular speed. For this purpose potential-like limiting terms were introduced in the calculation of the desired ϕ&&^Des as follows:

( )



 





∂ Γ ∂

−

−



 





∂ Γ ∂

−

− +

=

cosh 3

5 . cosh 1 cos tan

~

~ 4 2

ϕ β ϕ

ϕ β ϕ ϕ

ϕ ϕ ϕ

ϕ

ϕ

&

&

&

&

&

& ^Pot

Pot Ball

Des Des

g m

x B

(A.6.1.3)

with Γ_ϕ =Γ_ϕ& =3 and β_Pot =5 that worked well setting limitation to the angle at 1.5 (rad) and angular velocity of 3 (rad/s). The parameters B~

and m~_Ball mean the estimated model values. The effects of the rough dynamic model data and the friction that were unknown by the controller could manifest themselves in the low accuracy of the non-adaptive control. The significance of adaptation can be measured by observing the improved tracking accuracy of the adaptive controller. In the control approach applied for negative realized x( )4 the g(x|x^d,D-,∆+) function, for positive realized values the h(x|x^d,D-,∆-) functions were used.

Figure A.6.1.2. The phase space of the tilting angle ϕ& vsϕ: non-adaptive (LHS) and adaptive (RHS) solutions

According to Fig. A.6.1.2. adaptivity considerably “regularizes” the motion of the beam.

Figure A.6.1.3. The tilting angle ϕ vs time: non-adaptive (LHS) and adaptive (RHS) solutions

Figure A.6.1.4. The phase space and time dependence of the displacement of the cylinder along the beam: non-adaptive (LHS) and adaptive (RHS) solutions

Figure A.6.1.5. The tracking error vs. time: non-adaptive (LHS) and adaptive (RHS) solutions

Similar can be stated for the displacement of the ball (cylinder) along the beam (Fig. A.6.1.4.). Adaptivity drastically improved the tracking accuracy (Fig. A.6.1.5.).

In both the adaptive and the non-adaptive cases the effort of the feedback exerted for the compensation of the friction torque can be traced. In these figures the adaptive and the non-adaptive solutions show differences only nuances (Fig. A.6.1.6.), however, due to the integration according to time these nuances have significant effect on the tracking accuracy. Figure A.6.1.7. well reveals the essence of the adaptive method that realizes precise 4^th time-derivative of the coordinate x. To study the operation of the adaptive control further charts were made (Fig. A.6.1.8.) that displays when the functions g or h were used for realizing adaptivity.

Figure A.6.1.6. Compensation of the friction torques: non-adaptive (LHS) and adaptive (RHS) solutions

Figure A.6.1.7. Desired and realized 4^th time-derivative of “x”: non-adaptive (LHS) and adaptive (RHS) solutions

Figure A.6.1.8. The use of functions “g” and “h” versus time, and the “cumulative deformation factor” vs. time in the case of the adaptive control

According to (8.1.1) and (8.1.2) a “cumulative deformation factor” can be defined for functions h and g as follows:

( ) ∏ ( ( ) ) ^{( ( ( ) ) )}

= −∆+ −∆+

= ^k

i

i d i

k x t f x t

t s

0

: and

( ) ∏ ( ( ( ) ) ) ( ( ) )

= −∆− −∆−

= ^k

i

d i i

k f x t x t

t s

0

: that somehow are characteristic to the control

(Fig. A.6.1.8.).

The consequences of the strongly nonlinear nature of the friction model applied can well be traced in Fig. A.6.1.8.

In document Adaptive Control of Smooth Nonlinear Systems Based on Lucid Geometric Interpretation (Pldal 123-128)