Adaptive Tracking and Estimation for Nonlinear Control Systems

Adaptive Tracking and Estimation for Nonlinear Control Systems Michael Malisoff, Louisiana State University Joint with Frédéric Mazenc and Marcio de Queiroz Sponsored by NSF/DMS Grant 0708084 AMS-SIAM Special Session on Control and Inverse Problems for Partial Differential Equations 2011 Joint Mathematics Meetings, New Orleans

Adaptive Tracking and Estimation Problem

Adaptive Tracking and Estimation Problem Consider a suitably regular nonlinear system ξ = J (t, ξ, Ψ, u) (1) with a smooth reference trajectory ξ R and a vector Ψ of uncertain constant parameters.

Adaptive Tracking and Estimation Problem Consider a suitably regular nonlinear system ξ = J (t, ξ, Ψ, u) (1) with a smooth reference trajectory ξ R and a vector Ψ of uncertain constant parameters. ξ R = J (t, ξ R, Ψ, u R ).

Adaptive Tracking and Estimation Problem Consider a suitably regular nonlinear system ξ = J (t, ξ, Ψ, u) (1) with a smooth reference trajectory ξ R and a vector Ψ of uncertain constant parameters. ξ R = J (t, ξ R, Ψ, u R ). Problem:

Adaptive Tracking and Estimation Problem Consider a suitably regular nonlinear system ξ = J (t, ξ, Ψ, u) (1) with a smooth reference trajectory ξ R and a vector Ψ of uncertain constant parameters. ξ R = J (t, ξ R, Ψ, u R ). Problem: Design a dynamic feedback with estimator u = u(t, ξ, ˆΨ), ˆΨ = τ(t, ξ, ˆΨ) (2) such that the error Y = ( Ψ, ξ) = (Ψ ˆΨ, ξ ξ R ) 0.

Adaptive Tracking and Estimation Problem Consider a suitably regular nonlinear system ξ = J (t, ξ, Ψ, u) (1) with a smooth reference trajectory ξ R and a vector Ψ of uncertain constant parameters. ξ R = J (t, ξ R, Ψ, u R ). Problem: Design a dynamic feedback with estimator u = u(t, ξ, ˆΨ), ˆΨ = τ(t, ξ, ˆΨ) (2) such that the error Y = ( Ψ, ξ) = (Ψ ˆΨ, ξ ξ R ) 0. This is a central problem with numerous applications in flight control and electrical and mechanical engineering.

Adaptive Tracking and Estimation Problem Consider a suitably regular nonlinear system ξ = J (t, ξ, Ψ, u) (1) with a smooth reference trajectory ξ R and a vector Ψ of uncertain constant parameters. ξ R = J (t, ξ R, Ψ, u R ). Problem: Design a dynamic feedback with estimator u = u(t, ξ, ˆΨ), ˆΨ = τ(t, ξ, ˆΨ) (2) such that the error Y = ( Ψ, ξ) = (Ψ ˆΨ, ξ ξ R ) 0. This is a central problem with numerous applications in flight control and electrical and mechanical engineering. Persistent excitation.

Adaptive Tracking and Estimation Problem Consider a suitably regular nonlinear system ξ = J (t, ξ, Ψ, u) (1) with a smooth reference trajectory ξ R and a vector Ψ of uncertain constant parameters. ξ R = J (t, ξ R, Ψ, u R ). Problem: Design a dynamic feedback with estimator u = u(t, ξ, ˆΨ), ˆΨ = τ(t, ξ, ˆΨ) (2) such that the error Y = ( Ψ, ξ) = (Ψ ˆΨ, ξ ξ R ) 0. This is a central problem with numerous applications in flight control and electrical and mechanical engineering. Persistent excitation. Annaswamy, Astolfi, Narendra, Teel..

First-Order Case (FM-MdQ-MM-TAC 09)

First-Order Case (FM-MdQ-MM-TAC 09) In 2009, we gave a solution for the special case ẋ = ω(x)ψ + u. (3)

First-Order Case (FM-MdQ-MM-TAC 09) In 2009, we gave a solution for the special case ẋ = ω(x)ψ + u. (3) We used adaptive controllers of the form u s = ẋ R (t) ω(x) ˆΨ+K (x R (t) x), ˆΨ = ω(x) (x R (t) x).

First-Order Case (FM-MdQ-MM-TAC 09) In 2009, we gave a solution for the special case ẋ = ω(x)ψ + u. (3) We used adaptive controllers of the form u s = ẋ R (t) ω(x) ˆΨ+K (x R (t) x), ˆΨ = ω(x) (x R (t) x). We used the classical PE assumption: constant µ > 0 s.t. µi p t t T ω(x R(l)) ω(x R (l)) dl for all t R. (4)

First-Order Case (FM-MdQ-MM-TAC 09) In 2009, we gave a solution for the special case ẋ = ω(x)ψ + u. (3) We used adaptive controllers of the form u s = ẋ R (t) ω(x) ˆΨ+K (x R (t) x), ˆΨ = ω(x) (x R (t) x). We used the classical PE assumption: constant µ > 0 s.t. Novelty: µi p t t T ω(x R(l)) ω(x R (l)) dl for all t R. (4)

First-Order Case (FM-MdQ-MM-TAC 09) In 2009, we gave a solution for the special case ẋ = ω(x)ψ + u. (3) We used adaptive controllers of the form u s = ẋ R (t) ω(x) ˆΨ+K (x R (t) x), ˆΨ = ω(x) (x R (t) x). We used the classical PE assumption: constant µ > 0 s.t. µi p t t T ω(x R(l)) ω(x R (l)) dl for all t R. (4) Novelty: Our explicit global strict Lyapunov function for the Y = (Ψ ˆΨ, x x R ) dynamics.

First-Order Case (FM-MdQ-MM-TAC 09) In 2009, we gave a solution for the special case ẋ = ω(x)ψ + u. (3) We used adaptive controllers of the form u s = ẋ R (t) ω(x) ˆΨ+K (x R (t) x), ˆΨ = ω(x) (x R (t) x). We used the classical PE assumption: constant µ > 0 s.t. µi p t t T ω(x R(l)) ω(x R (l)) dl for all t R. (4) Novelty: Our explicit global strict Lyapunov function for the Y = (Ψ ˆΨ, x x R ) dynamics. It gave input-to-state stability with respect to additive time-varying uncertainties δ on Ψ.

Input-to-State Stability (Sontag, TAC 89)

Input-to-State Stability (Sontag, TAC 89) This generalization of uniform global asymptotic stability (UGAS) applies to systems with disturbances δ, having the form Ẏ = G(t, Y, δ(t)). (5)

Input-to-State Stability (Sontag, TAC 89) This generalization of uniform global asymptotic stability (UGAS) applies to systems with disturbances δ, having the form Ẏ = G(t, Y, δ(t)). (5) It is the requirement that there exist functions γ i K such that the corresponding solutions of (5) all satisfy for all t t 0 0. Y (t) γ 1 ( e t 0 t γ 2 ( Y (t 0 ) ) ) + γ 3 ( δ [0,t] ) (6)

Input-to-State Stability (Sontag, TAC 89) This generalization of uniform global asymptotic stability (UGAS) applies to systems with disturbances δ, having the form Ẏ = G(t, Y, δ(t)). (5) It is the requirement that there exist functions γ i K such that the corresponding solutions of (5) all satisfy Y (t) γ 1 ( e t 0 t γ 2 ( Y (t 0 ) ) ) + γ 3 ( δ [0,t] ) (6) for all t t 0 0. UGAS is the special case where δ 0.

Input-to-State Stability (Sontag, TAC 89) This generalization of uniform global asymptotic stability (UGAS) applies to systems with disturbances δ, having the form Ẏ = G(t, Y, δ(t)). (5) It is the requirement that there exist functions γ i K such that the corresponding solutions of (5) all satisfy Y (t) γ 1 ( e t 0 t γ 2 ( Y (t 0 ) ) ) + γ 3 ( δ [0,t] ) (6) for all t t 0 0. UGAS is the special case where δ 0. Integral ISS is the same except with the decay condition γ 0 ( Y (t) ) γ 1 ( e t 0 t γ 2 ( Y (t 0 ) ) ) + t 0 γ 3( δ(r) )dr. (7)

Input-to-State Stability (Sontag, TAC 89) This generalization of uniform global asymptotic stability (UGAS) applies to systems with disturbances δ, having the form Ẏ = G(t, Y, δ(t)). (5) It is the requirement that there exist functions γ i K such that the corresponding solutions of (5) all satisfy Y (t) γ 1 ( e t 0 t γ 2 ( Y (t 0 ) ) ) + γ 3 ( δ [0,t] ) (6) for all t t 0 0. UGAS is the special case where δ 0. Integral ISS is the same except with the decay condition γ 0 ( Y (t) ) γ 1 ( e t 0 t γ 2 ( Y (t 0 ) ) ) + t 0 γ 3( δ(r) )dr. (7) Both are shown by constructing specific kinds of strict Lyapunov functions for Ẏ = G(t, Y, 0).

Higher-Order Case (FM-MM-MdQ, NATMA 11)

Higher-Order Case (FM-MM-MdQ, NATMA 11) We solved the adaptive tracking and estimation problem for { ẋ = f (ξ) ż i = g i (ξ) + k i (ξ) θ i + ψ i u i, i = 1, 2,..., s. (8)

Higher-Order Case (FM-MM-MdQ, NATMA 11) We solved the adaptive tracking and estimation problem for { ẋ = f (ξ) ż i = g i (ξ) + k i (ξ) θ i + ψ i u i, i = 1, 2,..., s. (8) Unknown constants ψ = (ψ 1,..., ψ s ) R s

Higher-Order Case (FM-MM-MdQ, NATMA 11) We solved the adaptive tracking and estimation problem for { ẋ = f (ξ) ż i = g i (ξ) + k i (ξ) θ i + ψ i u i, i = 1, 2,..., s. (8) Unknown constants ψ = (ψ 1,..., ψ s ) R s and constants θ = (θ 1,..., θ s ) R p 1+...+p s.

Higher-Order Case (FM-MM-MdQ, NATMA 11) We solved the adaptive tracking and estimation problem for { ẋ = f (ξ) ż i = g i (ξ) + k i (ξ) θ i + ψ i u i, i = 1, 2,..., s. (8) Unknown constants ψ = (ψ 1,..., ψ s ) R s and constants θ = (θ 1,..., θ s ) R p 1+...+p s. ξ = (x, z).

Higher-Order Case (FM-MM-MdQ, NATMA 11) We solved the adaptive tracking and estimation problem for { ẋ = f (ξ) ż i = g i (ξ) + k i (ξ) θ i + ψ i u i, i = 1, 2,..., s. (8) Unknown constants ψ = (ψ 1,..., ψ s ) R s and constants θ = (θ 1,..., θ s ) R p 1+...+p s. ξ = (x, z). Now Ψ = (θ, ψ).

Higher-Order Case (FM-MM-MdQ, NATMA 11) We solved the adaptive tracking and estimation problem for { ẋ = f (ξ) ż i = g i (ξ) + k i (ξ) θ i + ψ i u i, i = 1, 2,..., s. (8) Unknown constants ψ = (ψ 1,..., ψ s ) R s and constants θ = (θ 1,..., θ s ) R p 1+...+p s. ξ = (x, z). Now Ψ = (θ, ψ). The C 2 T -periodic reference trajectory ξ R = (x R, z R ) to be tracked must satisfy ẋ R (t) = f (ξ R (t)) everywhere.

Higher-Order Case (FM-MM-MdQ, NATMA 11) We solved the adaptive tracking and estimation problem for { ẋ = f (ξ) ż i = g i (ξ) + k i (ξ) θ i + ψ i u i, i = 1, 2,..., s. (8) Unknown constants ψ = (ψ 1,..., ψ s ) R s and constants θ = (θ 1,..., θ s ) R p 1+...+p s. ξ = (x, z). Now Ψ = (θ, ψ). The C 2 T -periodic reference trajectory ξ R = (x R, z R ) to be tracked must satisfy ẋ R (t) = f (ξ R (t)) everywhere. New PE condition:

Higher-Order Case (FM-MM-MdQ, NATMA 11) We solved the adaptive tracking and estimation problem for { ẋ = f (ξ) ż i = g i (ξ) + k i (ξ) θ i + ψ i u i, i = 1, 2,..., s. (8) Unknown constants ψ = (ψ 1,..., ψ s ) R s and constants θ = (θ 1,..., θ s ) R p 1+...+p s. ξ = (x, z). Now Ψ = (θ, ψ). The C 2 T -periodic reference trajectory ξ R = (x R, z R ) to be tracked must satisfy ẋ R (t) = f (ξ R (t)) everywhere. New PE condition: positive definiteness of the matrices P i def = T 0 λ i (t)λ i (t) dt R (p i +1) (p i +1), (9) where λ i (t) = (k i (ξ R (t)), ż R,i (t) g i (ξ R (t))) for each i.

Two Other Key Assumptions

Two Other Key Assumptions Set F(t, χ) = f (χ + ξ R (t)) f (ξ R (t)).

Two Other Key Assumptions Set F(t, χ) = f (χ + ξ R (t)) f (ξ R (t)). There is a feedback v f and a global strict Lyapunov function V for { Ẋ = F(t, X, Z ) Ż = v f (t, X, Z ) (10) so that V and V have positive definite quadratic lower bounds near 0

Two Other Key Assumptions Set F(t, χ) = f (χ + ξ R (t)) f (ξ R (t)). There is a feedback v f and a global strict Lyapunov function V for { Ẋ = F(t, X, Z ) Ż = v f (t, X, Z ) (10) so that V and V have positive definite quadratic lower bounds near 0 and V, and v f are T -periodic.

Two Other Key Assumptions Set F(t, χ) = f (χ + ξ R (t)) f (ξ R (t)). There is a feedback v f and a global strict Lyapunov function V for { Ẋ = F(t, X, Z ) Ż = v f (t, X, Z ) (10) so that V and V have positive definite quadratic lower bounds near 0 and V, and v f are T -periodic. Backstepping..

Two Other Key Assumptions Set F(t, χ) = f (χ + ξ R (t)) f (ξ R (t)). There is a feedback v f and a global strict Lyapunov function V for { Ẋ = F(t, X, Z ) Ż = v f (t, X, Z ) (10) so that V and V have positive definite quadratic lower bounds near 0 and V, and v f are T -periodic. Backstepping.. See Sontag text, Chap. 5.

Two Other Key Assumptions Set F(t, χ) = f (χ + ξ R (t)) f (ξ R (t)). There is a feedback v f and a global strict Lyapunov function V for { Ẋ = F(t, X, Z ) Ż = v f (t, X, Z ) (10) so that V and V have positive definite quadratic lower bounds near 0 and V, and v f are T -periodic. Backstepping.. See Sontag text, Chap. 5. There are known positive constants θ M, ψ and ψ such that for each i {1, 2,..., s}. ψ < ψ i < ψ and θ i < θ M (11)

Two Other Key Assumptions Set F(t, χ) = f (χ + ξ R (t)) f (ξ R (t)). There is a feedback v f and a global strict Lyapunov function V for { Ẋ = F(t, X, Z ) Ż = v f (t, X, Z ) (10) so that V and V have positive definite quadratic lower bounds near 0 and V, and v f are T -periodic. Backstepping.. See Sontag text, Chap. 5. There are known positive constants θ M, ψ and ψ such that ψ < ψ i < ψ and θ i < θ M (11) for each i {1, 2,..., s}. Known directions for the ψ i s.

Dynamic Feedback

Dynamic Feedback The estimator evolves on { s i=1 ( θ M, θ M ) p i } (ψ, ψ) s.

Dynamic Feedback The estimator evolves on { s i=1 ( θ M, θ M ) p i } (ψ, ψ) s. ˆθ i,j = (ˆθ i,j 2 θm 2 )ϖ i,j, 1 i s, 1 j p i ( ) ( ˆψ i = ˆψ i ψ ˆψ i ψ) i, 1 i s (12)

Dynamic Feedback The estimator evolves on { s i=1 ( θ M, θ M ) p i } (ψ, ψ) s. ˆθ i,j = (ˆθ i,j 2 θm 2 )ϖ i,j, 1 i s, 1 j p i ( ) ( ˆψ i = ˆψ i ψ ˆψ i ψ) i, 1 i s (12) Here ˆθ i = (ˆθ i,1,..., ˆθ i,pi ) for i = 1, 2,..., s

Dynamic Feedback The estimator evolves on { s i=1 ( θ M, θ M ) p i } (ψ, ψ) s. ˆθ i,j = (ˆθ i,j 2 θm 2 )ϖ i,j, 1 i s, 1 j p i ( ) ( ˆψ i = ˆψ i ψ ˆψ i ψ) i, 1 i s (12) Here ˆθ i = (ˆθ i,1,..., ˆθ i,pi ) for i = 1, 2,..., s, ϖ i,j = V z i (t, ξ)k i,j ( ξ + ξ R (t) )

Dynamic Feedback The estimator evolves on { s i=1 ( θ M, θ M ) p i } (ψ, ψ) s. ˆθ i,j = (ˆθ i,j 2 θm 2 )ϖ i,j, 1 i s, 1 j p i ( ) ( ˆψ i = ˆψ i ψ ˆψ i ψ) i, 1 i s (12) Here ˆθ i = (ˆθ i,1,..., ˆθ i,pi ) for i = 1, 2,..., s, ϖ i,j = V z i (t, ξ)k i,j ( ξ + ξ R (t) ) and i = V z i (t, ξ)u i (t, ξ, ˆθ, ˆψ). (13)

Dynamic Feedback The estimator evolves on { s i=1 ( θ M, θ M ) p i } (ψ, ψ) s. ˆθ i,j = (ˆθ i,j 2 θm 2 )ϖ i,j, 1 i s, 1 j p i ( ) ( ˆψ i = ˆψ i ψ ˆψ i ψ) i, 1 i s (12) Here ˆθ i = (ˆθ i,1,..., ˆθ i,pi ) for i = 1, 2,..., s, ϖ i,j = V z i (t, ξ)k i,j ( ξ + ξ R (t) ) and i = V z i (t, ξ)u i (t, ξ, ˆθ, ˆψ). (13) u i (t, ξ, ˆθ, ˆψ) = v f,i (t, ξ) g i (ξ) k i (ξ) ˆθ i +ż R,i (t) ˆψ i (14)

Dynamic Feedback The estimator evolves on { s i=1 ( θ M, θ M ) p i } (ψ, ψ) s. ˆθ i,j = (ˆθ i,j 2 θm 2 )ϖ i,j, 1 i s, 1 j p i ( ) ( ˆψ i = ˆψ i ψ ˆψ i ψ) i, 1 i s (12) Here ˆθ i = (ˆθ i,1,..., ˆθ i,pi ) for i = 1, 2,..., s, ϖ i,j = V z i (t, ξ)k i,j ( ξ + ξ R (t) ) and i = V z i (t, ξ)u i (t, ξ, ˆθ, ˆψ). (13) u i (t, ξ, ˆθ, ˆψ) = v f,i (t, ξ) g i (ξ) k i (ξ) ˆθ i +ż R,i (t) ˆψ i (14) The estimator and feedback can only depend on things we know.

Stabilization Analysis

Stabilization Analysis We build a global strict Lyapunov function for the Y = ( ξ, θ, ψ) = (ξ ξ R, θ ˆθ, ψ ˆψ) dynamics to prove the UGAS condition Y (t) γ 1 (e t 0 t γ 2 ( Y (t 0 ) )).

Stabilization Analysis We build a global strict Lyapunov function for the Y = ( ξ, θ, ψ) = (ξ ξ R, θ ˆθ, ψ ˆψ) dynamics to prove the UGAS condition Y (t) γ 1 (e t 0 t γ 2 ( Y (t 0 ) )). We start with the nonstrict Lyapunov function V 1 (t, ξ, θ, ψ) = V (t, ξ) + + s i=1 s p i i=1 j=1 ψi 0 θi,j 0 m θ 2 M (m θ i,j) 2 dm m (ψ i m ψ)(ψ ψ i + m) dm.

Stabilization Analysis We build a global strict Lyapunov function for the Y = ( ξ, θ, ψ) = (ξ ξ R, θ ˆθ, ψ ˆψ) dynamics to prove the UGAS condition Y (t) γ 1 (e t 0 t γ 2 ( Y (t 0 ) )). We start with the nonstrict Lyapunov function V 1 (t, ξ, θ, ψ) = V (t, ξ) + + s i=1 s p i i=1 j=1 ψi 0 θi,j 0 m θ 2 M (m θ i,j) 2 dm m (ψ i m ψ)(ψ ψ i + m) dm. It gives V 1 W ( ξ) for some positive definite function W.

Stabilization Analysis We build a global strict Lyapunov function for the Y = ( ξ, θ, ψ) = (ξ ξ R, θ ˆθ, ψ ˆψ) dynamics to prove the UGAS condition Y (t) γ 1 (e t 0 t γ 2 ( Y (t 0 ) )). We start with the nonstrict Lyapunov function V 1 (t, ξ, θ, ψ) = V (t, ξ) + + s i=1 s p i i=1 j=1 ψi 0 θi,j 0 m θ 2 M (m θ i,j) 2 dm m (ψ i m ψ)(ψ ψ i + m) dm. It gives V 1 W ( ξ) for some positive definite function W. This is insufficient for robustness analysis because V 1 could be zero outside 0.

Stabilization Analysis We build a global strict Lyapunov function for the Y = ( ξ, θ, ψ) = (ξ ξ R, θ ˆθ, ψ ˆψ) dynamics to prove the UGAS condition Y (t) γ 1 (e t 0 t γ 2 ( Y (t 0 ) )). We start with the nonstrict Lyapunov function V 1 (t, ξ, θ, ψ) = V (t, ξ) + + s i=1 s p i i=1 j=1 ψi 0 θi,j 0 m θ 2 M (m θ i,j) 2 dm m (ψ i m ψ)(ψ ψ i + m) dm. It gives V 1 W ( ξ) for some positive definite function W. This is insufficient for robustness analysis because V 1 could be zero outside 0. Therefore, we transform V 1.

Transformation (FM-MM-MdQ, NATMA 11)

Transformation (FM-MM-MdQ, NATMA 11) Theorem: We can construct K K C 1 such that V (t, ξ, θ, ψ) =. K ( V 1 (t, ξ, θ, ψ) ) s + Υ i (t, ξ, θ, ψ), (15) i=1 where Υ i (t, ξ, θ, ψ) = z i λ i (t)α i ( θ i, ψ i ) + 1 T ψ α i ( θ i, ψ i )Ω i (t)α i ( θ i, ψ i ), (16) λ i (t) = (k i (ξ R (t)), ż R,i (t) g i (ξ R (t))), (17) [ ] α i ( θ i, ψ θi ψ i ) = i θ i ψi, and ψ i Ω i (t) = (18) t t t T m λ i (s)λ i (s)ds dm, is a global strict Lyapunov function for the Y = ( ξ, θ, ψ) dynamics.

Transformation (FM-MM-MdQ, NATMA 11) Theorem: We can construct K K C 1 such that V (t, ξ, θ, ψ) =. K ( V 1 (t, ξ, θ, ψ) ) s + Υ i (t, ξ, θ, ψ), (15) i=1 where Υ i (t, ξ, θ, ψ) = z i λ i (t)α i ( θ i, ψ i ) + 1 T ψ α i ( θ i, ψ i )Ω i (t)α i ( θ i, ψ i ), (16) λ i (t) = (k i (ξ R (t)), ż R,i (t) g i (ξ R (t))), (17) [ ] α i ( θ i, ψ θi ψ i ) = i θ i ψi, and ψ i Ω i (t) = (18) t t t T m λ i (s)λ i (s)ds dm, is a global strict Lyapunov function for the Y = ( ξ, θ, ψ) dynamics. Hence, the dynamics are UGAS to 0.

Conclusions

Conclusions Adaptive tracking and estimation is a central problem with applications in many branches of engineering.

Conclusions Adaptive tracking and estimation is a central problem with applications in many branches of engineering. Standard adaptive control treatments based on nonstrict Lyapunov functions only give tracking and are not robust.

Conclusions Adaptive tracking and estimation is a central problem with applications in many branches of engineering. Standard adaptive control treatments based on nonstrict Lyapunov functions only give tracking and are not robust. Our strict Lyapunov functions gave robustness to additive uncertainties on the parameters using the ISS paradigm.

Conclusions Adaptive tracking and estimation is a central problem with applications in many branches of engineering. Standard adaptive control treatments based on nonstrict Lyapunov functions only give tracking and are not robust. Our strict Lyapunov functions gave robustness to additive uncertainties on the parameters using the ISS paradigm. We covered systems with unknown control gains including brushless DC motors turning mechanical loads.

Conclusions Adaptive tracking and estimation is a central problem with applications in many branches of engineering. Standard adaptive control treatments based on nonstrict Lyapunov functions only give tracking and are not robust. Our strict Lyapunov functions gave robustness to additive uncertainties on the parameters using the ISS paradigm. We covered systems with unknown control gains including brushless DC motors turning mechanical loads. It would be useful to extend to cover models that are not affine in θ, feedback delays, and output feedbacks.