The RFPT-based MRAC Controller for a 2 DoF TORA System

The “TORA” (Translational Oscillations with an Eccentric Rotational Proof Mass Actuator) system was considered as a simplified model of a dual-spin spacecraft with mass imbalance in the literature. It served as a “benchmark problem” for controller design in [66]. In [67] it was controlled by a cascade and a passivity based controller. In [68] for example the “Tensor Product Form” of the system model was applied to develop a model-based controller. For our purposes its 2 DoF variant was considered that consisted of a cart, a pendulum (practically a beam) and a dial that can be rotated around an axle attached to the end of the beam. Its equations of motion are given in (4.1). This seemingly 3 DoF system was made 2 DoF indirectly driven one by settingQ₃≡0and trying to control the motion of axlesq2[rad]andq3[m]by properly settingQ1[N×m]andQ2[N×m][A. 6].

with the dynamic parameters as follows: m= 20 [kg](the mass of the dial),M= 30 [kg](the mass of the body of the cart,L= 2 [m](the length of the beam), andΘ= 20 [kg×m²](the momentum of inertia of the dial with respect to its own mass center point). Utilizing thatQ₃ ≡0the last equation of (4.1) determinesq¨1in (4.2) [A. 6]

¨

q₁=mLsin(q1) ˙q12−(m+M) ¨q3

mLcos(q₁) (4.2)

that can be substituted into the first two equations of (4.1) to calculate the necessaryQ₁andQ₂torque components in (4.3).

that is considerably different to that of the directly driven Classical Mechanical systems: certain ele-ments of the “inertia matrix” and the additional terms are singular and may contain infinite eleele-ments if cos(q₁) = 0. The aim of the MRAC controller in our case was to simplify the dynamics given in (4.3)for an external control loopas a 2 DoFreference modeldescribed by (4.4) [A. 6].

Θ_refq¨₂=Q₂,−(

Mref +mref

)q¨₃=Q₁. (4.4)

with the parametersm_ref = 15 [kg],M_ref = 25 [kg], andΘ_ref = 16 [kg×m²]for nominal motion that avoids the configurations in whichcos(q1)approaches 0. The details of how to develop MRAC con-trollers by the use of RFPT were described in details e.g. in [38]. In the sequel we give only simulation results that exemplify how can the “precursor” oscillations be utilized for guaranteeing stable tuning for A_c[A. 6]. Λ= 10/s kinematic trackingwas prescribed fori= 2,3. The actively driven axleq1 started from the

“not dangerous”q₁ = 0initial position. A digital controller was assumed to yield constant torque for cycle time of∆t = 1ms. In the adaptive casethe adaptive control parameters were set as follows:

{Kc = −10⁴, Bc = 1} or{Kc = 10⁴, Bc =−1} for yielding the appropriate precursor phenomenon, and A_cini = 10⁻⁷were chosen. For tuningA_canasymmetric rulewas applied that cautiously increases its value if no “precursor oscillations” are observed but causes fast decrease if the oscillations appear as

A˙_c=

{ v₊if Fˆ_n−F_thr≥0andA_c≤A_cini

−cvv₊if Fˆ_n−F_thr<0orA_c≥A_cmax (4.5) withv₊= 10⁻⁴,c_v= 3,F_thr= 10⁻⁴, andA_cmax = 10A_cini. TheFˆ_nvalues were calculated by a “forgetting integral” that served as a model-independent observerin the following manner: the scalar products F(t_n) := [Q(t⃗ _n)−Q(t⃗ _n−∆t)]^T[Q(t⃗ _n−∆t)−Q(t⃗ _n−2∆t)]in general can be used formonitoring the precursor of chatteringsince its positive value pertains to definitemodification of the control force“approximately in the same direction” while negative value reveals a significant fluctuation in the direction of the force between the subsequent control cycles. (The scalar productQ(t⃗ _n)^TQ(t⃗ _n−∆t)could reveal only a rough chattering.) The forgetting integral with the outputFˆn:= (1−β)∑_n

s=0β^sFn−swithβ= 0.1and threshold valueF_thris able to filter out single changes in the direction of the forces that normally occur only in certain points of a smooth variation [A. 6].

4.1.1.2 Simulations for Big NegativeKcandBc= 1

In Fig. 4.5 the movement of the axles in the adaptive case are described for a nominal trajectory con-verging from zero to a third order spline function of time. In Fig. 4.6 the trajectory tracking errors are described fornon-adaptiveand theadaptivecontrollers [A. 6].

Figure 4.5: Trajectory tracking of theadaptive controller[q₂[rad]: black,q₃[m]: green,q^{N om}₂ [rad]: red, q₃^{N om}[m]: ochre lines] (LHS), and the motion of the actively driven axis [q₁: [rad]vs. time in[s]units (RHS)] [A. 6]

Figure 4.6: Trajectory tracking errors of thenon-adaptive controller(LHS) and theadaptive one(RHS) [q₂[rad]: black,q₃[m]: green lines vs. time in[s]units] [A. 6]

Figure 4.7 reveals the control forces that have the following definitions: the “desired forces” with super-script “Des” denote the force needs of thereference modelcalculated by the external control loop using thekinematically prescribed tracking error relaxationand thedynamic model of the reference system given in (4.4). The “exerted” forces are calculated by the internal loop of the MRAC controller that has double responsibility: a) guaranteeing precise trajectory tracking, and b) generating the illusion to the external loop that the reference model well describes the controlled system (the “MRAC illusion”). The

“recalculated forces” marked by the superscript “Rec” meansthe force need of the reference modelat the realized response of the controlled system in its actual state. It is evident that the controller well generated the MRAC illusion since the “recalculated” forces are in the vicinity of the “desired ones” while both of them considerably differ from the adaptively deformed, realized control forces actually exerted.

Figure 4.8 displays zoomed excerpts of the force diagram of the adaptive controller. In the diagram little fluctuations in the control signal in certain segments can well be observed. These fluctuations are related to the tuned value ofAcdepicted in Fig. 4.9. By the use of the time axis of the diagrams it can well be seen that the appearance of the the little fluctuations in the control signal are related to the too great values ofA_cand that the fast decrease inA_cmakes them cease quickly [A. 6].

Figure 4.7: Thegeneralized forcesfor thenon-adaptive(LHS) and theadaptive(RHS) cases versus time in[s]units [Q^Des₁ : black,Q^Exerted₁ : red,Q₁^Rec: brown, andQ^Des₂ : green,Q^Exerted₂ : ochre,Q^Rec₂ : dark blue lines in[N×m]units] [A. 6]

Figure 4.8: Detailed excerpts of thegeneralized forcesof theadaptive controllerversus time in[s]units [Q^Des₁ : black,Q^Exerted₁ : red,Q₁^Rec: brown, andQ₂^Des: green,Q^Exerted₂ : ochre,Q^Rec₂ : dark blue lines in[N×m]

units] [A. 6]

Figure 4.9: Variation ofAcvs. time in[s]units (LHS) and the time-dependence of the forgetting integral (RHS) [A. 6]

4.1.1.3 Simulations for Big PositiveK_candB_c=−1

These alternative settings also produce observable and useful “precursor oscillations” for the same nominal motion. Figure 4.10 reveals precise trajectory and smooth and precise phase trajectory track-ing. According to Fig. 4.11 it can be stated that similar and efficient parameter tuning happened as in the case of the original settings{Kc=−10⁴, B_c= 1}. In the diagram of the generalized forces (Fig. 4.12) similar precursor oscillations can be observed, too [A. 6].

Figure 4.10: The phase trajectories [q₂[rad]: black,q₃ [m]: green,q^{N om}₂ [rad]: red,q^{N om}₃ [m]: ochre lines] (LHS) and the trajectory tracking errors of theadaptive controllerusing the alternative parameter settings (RHS) [q₂[rad]: black,q₃[m]: green lines vs. time in[s]units] [A. 6]

Figure 4.11: Variation ofA_cvs. time in[s]units (LHS) and the time-dependence of the forgetting integral (RHS) using the alternative parameter settings [A. 6]

Figure 4.12: The chart of thegeneralized forcesof theadaptive controllerat the alternative parameter settings versus time in[s]units and its detailed excerpts [Q^Des₁ : black,Q^Exerted₁ : red,Q^Rec₁ : brown, and Q^Des₂ : green,Q₂^Exerted: ochre,Q^Rec₂ : dark blue lines in[N×m]units] [A. 6]