Simulation Results for Section “7.3. Simulation Example for Potential

At first the paradigm used for the investigations is described mathematically, i.e. two coupled cart plus double pendulum systems.

m ₁ ,L ₁ ,q ₁

M

m ₂ ,L ₂ ,q ₂

q ₃

Figure A.5.1. The cart plus double pendulum system

Each cart under consideration consists of a body of considerable mass and wheels of negligible masses and momentums. The overall cart-masses are M^A=4 [kg], and M^B=4 [kg]. The pendulums are assembled on the cart by parallel shafts and arms of negligible masses and lengths L1A=2 and L2A=2 [m], L1B=1.5 and L2B=1.5 [m], respectively. At the end of each arm a ball of negligible size and considerable mass (m1A=10 and m2A=10, m1B=8 and m2B=7) [kg] are attached, respectively [Fig. A.5.1.].

The Euler-Lagrange equations of motion of a single cart are given as follows (A.5.4):

( )

[N×m] denote the driving torque at shaft 1 and 2, respectively, and Q3 [N] stands for the force moving the cart in the horizontal direction. The appropriate rotational angles are q1 and q2 [rad], and the linear degree of freedom belongs to q3 [m]. The 1st rotational and the linear degrees of freedom were the controlled and actuated ones, while the second rotary axis is without observation, control, and actuation that means that Q₂ takes the constant value zero. Furthermore, two pieces of the above described subsystems are coupled along their linear direction of motion by the forces Q3A = –Q3B given in [N] as spring force length.

Figure A.5.2. Simulation results for the non-adaptive (LHS) and the “centralized adaptive” (RHS) control approaches (the notation “β=1” refers to the occurrence of

integer order derivatives in the symplectic matrices)

To model the buffers two non-linear terms are applied that are very sharp near the 0.5×L0 and 1.5×L0 distances, while in the “internal points” they are very flat. They are described by two parameters, namely by the “strength” A=1000 [N×m²], and a small parameter εbump=10^-3 [m] determining the “nearness” of the singularity of these coupling forces. In the simulation the rough initial system model for both carts was

]T 1 , 1 , 1 [ 10 10 +

= q

Q && instead of (A.5.4). A PID-type kinematic trajectory tracking

strategy was prescribed for the relaxation of the tracking error h=q^N-q according to

three oscillation-free (real) time-constants α1=α, α2=0.9×α ,α3=0.8×α with α=20 [1/s]:

( ) _{1 2 2 3 3 1 1 2 3 1 2 3}

0

, + + ,

+ +

, αα α α α α α α α αα α

τ

τ = = =

−

−

−

= ^{P D I}∫^{t d P D I}

Des h h h

h& &

& . (A.5.6)

Figure A.5.3. Simulation results for the non-adaptive (LHS) and the “centralized adaptive” (RHS) control approaches [continued] (the notation “β=1” refers to the

occurrence of integer order derivatives in the symplectic matrices)

In the control the “Regulating Factor” was calculated according to (7.2.19) with υ=0.5, ε1=0.2, ε2=10^-5. The finite element time-resolution for the control was δt=10

-3 [s], the numerical integration happened according to the Euler formula with step length δt/10. For the “centralized” approach the special symplectic matrices of size 12×12 defined in (7.1.10) were applied with the “dummy parameter” d=80. The first column in the upper half of these matrices were defined as

contributions” in each control cycle “i” as

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

The function of this weighting is maintaining a proper relationship between b and the norm of the appropriate accelerations: the physically interpreted part of the column remains commensurate with b. (The former solutions that applied the accelerations without weighting with fixed dummy parameter b were found to be less precise.)

The simulation results obtained for the non-adaptive and the “centralized adaptive” approaches are given in Figs. A.5.2., A.5.3., A.5.4., and A.5.5. It is clear that both the trajectory and the phase trajectory tracking accuracy have been considerably improved by switching on the adaptive law. Following a sharp transient section the driving forces applied at both subsystems have been well stabilized. The same statement can be done in connection with the “weighting factor” in the case of the adaptive control. Figure A.5.5. also reveals that the variation of the “regulating factor” became “canonical”, and that the symplectic matrices applied by the control were really in the vicinity of the unit matrix.

Figure A.5.4. Simulation results for the non-adaptive (LHS) and the “centralized adaptive” (RHS) control approaches [continued] (the notation “β=1” refers to the

occurrence of integer order derivatives in the symplectic matrices); (these factors are not in use in the non-adaptive case)

The simulation results pertaining to the “distributed approach” are described in Figs. A.5.7.-A.5.11. It used two smaller symplectic matrices with the first columns in the upper half as

[

]

Figure A.5.5. Simulation results for the “centralized adaptive” control approach [continued] (the notation “β=1” refers to the occurrence of integer order derivatives

in the symplectic matrices)

Switching on the adaptive law again considerably improved the precision of the trajectory- and phase-trajectory tracking. As in the case of the “centralized approach” the existence of a sharp initial transient section can be observed in which the appropriate symplectic matrices are not in the close vicinity of the unit matrix. In these regimes the use of the “regulating factor” plays important role in guaranteeing the convergence of the method. Following this transient phase stable control can be observed in which the transformation matrices remain in the close vicinity of the unit matrix, the weighting and regulating factors as well as the control forces and torques vary “regularly”.

Figure A.5.6. Simulation results for the non-adaptive (LHS) and the “distributed adaptive” (RHS) control approaches (the notation “β=1” refers to the occurrence of

integer order derivatives in the symplectic matrices)

Figure A.5.7. (continued)Simulation results for the non-adaptive (LHS) and the

“distributed adaptive” (RHS) control approaches (the notation “β=1” refers to the occurrence of integer order derivatives in the symplectic matrices)

Figure A.5.8. Simulation results for the non-adaptive (LHS) and the “distributed adaptive” (RHS) control approaches [continued] (the notation “β=1” refers to the

occurrence of integer order derivatives in the symplectic matrices)

Figure A.5.9. (continued) Simulation results for the non-adaptive (LHS) and the

“distributed adaptive” (RHS) control approaches [continued] (the notation “β=1”

refers to the occurrence of integer order derivatives in the symplectic matrices)

Figure A.5.10. Simulation results for the non-adaptive (LHS) and the “distributed adaptive” (RHS) control approaches [continued] (the notation “β=1” refers to the

occurrence of integer order derivatives in the symplectic matrices); (these factors are not in use in the non-adaptive case)

Figure A.5.11. Simulation results for the “distributed adaptive” control approach [continued] (the notation “β=1” refers to the occurrence of integer order derivatives

in the symplectic matrices)

A.6. Illustrative Figures for Section “8.1. Fixed Point Transformations with a