A Generalized Human-In-The-Loop Stability Analysis in the Presence of Uncertain and Redundant Actuator Dynamics

[Audio] Hello everyone, In this presentation, I would like to talk about human-in-the-loop stability analysis in the presence of uncertain and redundant actuator dynamics

[Audio] Overview Adaptive control allocation Controller Human-In-The-Loop stability analysis Simulation results 1 / 15.

[Audio] Problem statement 2 / 15. . . Problem statement.

[Audio] Control allocation S. S. Tohidi, Y. Yildiz, and I. Kolmanovsky, " Adaptive Control Allocation for Constrained Systems," Automatica, 2020. 3 / 15.

[Audio] Adaptive control allocation 4 / 15. . . Adaptive control allocation.

[Audio] Adaptive control allocation ▶ Virtual dynamics ˙y = Amy + BΛu − v 4 / 15.

[Audio] Adaptive control allocation ▶ Virtual dynamics ˙y = Amy + BΛu − v ▶ Reference model ˙ym = Amym 4 / 15.

[Audio] Adaptive control allocation ▶ Virtual dynamics ˙y = Amy + BΛu − v ▶ Reference model ˙ym = Amym ▶ Adaptive law ˙ θv = ΓθProj � θv, −veT PB � 4 / 15.

[Audio] Adaptive control allocation ▶ Virtual dynamics ˙y = Amy + BΛu − v ▶ Reference model ˙ym = Amym ▶ Adaptive law ˙ θv = ΓθProj � θv, − veT PB � ▶ Control input u = θT v v 4 / 15.

[Audio] Controller ▶ Consider the overacuated plant to be in the following form: � ˙x( 1) ˙x(2) � = �A1,1 A1,2 A2,1 A2, 2 � �x(1) x(2) � +Bvv, y = [ 0r×(n−r) Ir] �x(1) x(2) � 5 / 15.

[Audio] Sliding mode controller ▶ Sliding surface s(x(2)(t), x( 2)(t0), t) = x(2)(t) − x(2)(t0)e−¯λ(t−t0) − 2 πr(t)tan− 1�¯λ(t − t0) � = 0 S. S. Tohidi, Y. Yildiz, and I. Kolmanovsky, " Sliding mode control for over-actuated systems with adaptive control allocation and its applications to flight control," IEEE Conference on Control Technology and Applications ( CCTA), 2021. 6 / 15.

[Audio] Sliding mode controller ▶ Sliding surface s(x( 2)(t), x(2)(t0), t) = x(2)(t) − x(2)(t0)e−¯λ(t−t0) − 2 πr(t)tan−1�¯λ(t − t0) � = 0 ▶ Sliding mode controller v(t) = − A2, 1x(1)(t) − A2,2x(2)(t) − ¯λx(2)(0)e−¯λt + 2 π ˙r(t)tan−1(¯λt) + 2 πr(t) ¯λ 1 + ¯λ2t2 − signv(s(x(2)(t), x(2)(0), t)) ρ, S. S. Tohidi, Y. Yildiz, and I. Kolmanovsky, " Sliding mode control for over-actuated systems with adaptive control allocation and its applications to flight control," IEEE Conference on Control Technology and Applications ( CCTA), 2021. 6 / 15.

[Audio] ADMIRE model x = [α β p q r]T y = [p q r]T u = [ uc ure ule ur]T uc ∈ [ −55, 25] × π 180rad, ure, ule, ur ∈ [ −30, 30] × π 180rad ˙x = Ax + BuΛu = Ax + Bvv v = BΛu, Bu = BvB, Bv = � 02× 3 I3×3 � 7 / 15.

[Audio] Closed loop dynamics 8 / 15. . . Closed loop dynamics.

[Audio] Closed loop dynamics ▶ � ˙α(t) ˙β(t) � = A1, 1 �α(t) β(t) � + A1, 2   p(t) q(t) r(t)   8 / 15.

[Audio] Closed loop dynamics ▶ � ˙α(t) ˙β(t) � = A1,1 �α(t) β(t) � + A1,2   p(t) q(t) r(t)   ▶   ˙p(t) ˙q(t) ˙r(t)   = −¯λe−¯ λt   p(0) q( 0) r(0)   + 2 πtan− 1(¯λt)   ˙pd(t) ˙qd(t) ˙rd(t)   + 2 π ¯λ 1 + ¯ λ2t2   pd(t) qd(t) rd(t)   − signv(s)ρ 8 / 15.

[Audio] Stability analysis: transfer function model ▶ qd(s) θd(s) − θ(s) = bms ˆm + bm−1s ˆm− 1 + ... + b0 sˆn + an−1sˆn−1 + ... + a0 9 / 15.

[Audio] Stability analysis: transfer function model ▶ qd(s) θd(s) − θ(s) = bms ˆm + bm−1s ˆm− 1 + ... + b0 sˆn + an−1sˆn−1 + ... + a0 ▶ ˙xh(t) = Ahxh(t) + Bh (θd(t) − θ(t)) qd(t) = Chxh(t) + Dh (θd(t) − θ(t)) 9 / 15.

[Audio] Stability analysis: transfer function model ▶ qd(s) θd(s) − θ(s) = bms ˆm + bm−1s ˆm−1 + ... + b0 sˆn + an−1sˆn− 1 + ... + a0 ▶ ˙xh(t) = Ahxh(t) + Bh (θd(t) − θ(t)) qd(t) = Chxh(t) + Dh (θd(t) − θ(t)) ▶ q(t) = q( 0)e−¯λt + 2 πtan−1(¯λt)qd(t) 9 / 15.

[Audio] Stability analysis: transfer function model ▶ qd(s) θd(s) − θ(s) = bms ˆm + bm−1s ˆm−1 + ... + b0 sˆn + an−1sˆn−1 + ... + a0 ▶ ˙xh(t) = Ahxh(t) + Bh (θd(t) − θ(t)) qd(t) = Chxh(t) + Dh (θd(t) − θ(t)) ▶ q(t) = q( 0)e−¯λt + 2 πtan− 1(¯λt)qd(t) ▶ ˙θ(t) = q(t) 9 / 15.

[Audio] Stability analysis: transfer function model ▶ � ˙θ(t) ˙xh(t) � = �− 2 πtan−1(¯λt)Dh 2 πtan− 1( ¯λt)Ch −Bh Ah � � �� � ¯ A(t) � θ(t) xh(t) � � �� � ¯x(t) + � 2 πtan−1(¯λt)Dh Bh � � �� � ¯B θd(t) + � q( 0)e−¯λt 0 � � �� � ω(t) 10 / 15.

[Audio] Stability analysis: transfer function model ▶ � ˙θ(t) ˙xh(t) � = �− 2 πtan− 1(¯λt)Dh 2 πtan−1( ¯λt)Ch −Bh Ah � � �� � ¯ A(t) � θ(t) xh(t) � � �� � ¯x(t) + � 2 πtan−1(¯λt)Dh Bh � � �� � ¯B θd(t) + � q( 0)e−¯λt 0 � � �� � ω(t) ▶ ˙¯x(t) = ¯A(t)¯x(t) + ¯Bθd(t) + ω(t) 10 / 15.

[Audio] Human-In-The-Loop stability analysis: TF model Theorem (1) The time-varying system ˙¯x(t) = ¯A(t)¯x(t), is uniformly exponentially stable if λi ��− Dh Ch −Bh Ah �� ∈ C−, i = 1, ..., ˆn + 1, where λi(.) provides ith eigenvalue of the matrix. 11 / 15.

[Audio] Human-In-The-Loop stability analysis: TF model Theorem ( 2) The solution of the linear time-varying system ˙ ˆx(t) = ˆA(t)ˆx(t) + ˆBθd(t) + ω(t), is bounded if λi ��− Dh Ch −Bh Ah �� ∈ C−, i = 1, ..., ˆn + 1. 12 / 15.

[Audio] Simulation results Human transfer function λi       − Dh Ch −Bh Ah       TF1 10 s+10 −1.127, −8.873 TF2 s+1 (s+2)(s+ 3) −0.16, −2.4 ± 0.6j TF3 s+ 1 (s−2)(s+3) 1.655, 0.21, −2.86 TF4 s−1 (s+2)(s+3) 0.13, −2.5 ± 1.04j TF5 (s+1)(s+ 2) (s+3)(s+ 4) −0.144, −2.678, −5.177 TF6 (s+1)(s+2) (s+3)(s+4)(s+ 5) −0.03, −2.8, −4.6 ± 0.96j 13 / 15.

[Audio] Simulation results 14 / 15. . . Simulation results.

[Audio] Future works ▶ Human-In-The-Loop stability analysis considering time-delay, plant non-linearity. 15 / 15.

[Audio] Future works ▶ Human-In-The-Loop stability analysis considering time-delay, plant non-linearity. ▶ Experimental results. 15 / 15.

[Audio] Main references [ 1] S. S. Tohidi, Y. Yildiz, and I. Kolmanovsky, " Adaptive Control Allocation for Constrained Systems," Automatica, 2020. [ 2] S. S. Tohidi, Y. Yildiz, and, I. Kolmanovsky, " Fault Tolerant Control for Over-Actuated Systems: An Adaptive Correction Approach," American Control Conference, 2016. [ 3] S. S. Tohidi, and Y. Yildiz, " Adaptive Control Allocation: A Human-In-The-Loop Stability Analysis," International Federation of Automatic Control ( IFAC), Berlin, Germany, 2020. [ 4] S. S. Tohidi, and Y. Yildiz, " A Control Theoretical Adaptive Human Pilot Model: Theory and Experimental Validation," IEEE Transaction on Control Systems Technology, 2020. [ 5] S. S. Tohidi, and Y. Yildiz, "Adaptive human pilot model for uncertain systems," European Control Conference ( ECC), 2019. [ 6] D. T. Mcruer, and H. Jex, "A review of quasi-linear pilot models," IEEE Transactions on human Factors in Electronics, vol. 8, pp. 231- 249, 1967..

[Audio] Thank you.. . . Thank you..