Asymptotic Disturbance Rejection for the TORA System via Rotational-position Feedback Fabio Celani Department of Computer and Systems Science Antonio Ruberti Sapienza University of Rome Italy

IASTED Conference on Control and Applications 2009 Cambridge, UK July 13th , 2009

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Outline

I

TORA system

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Outline

I

TORA system

I

literature review

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Outline

I

TORA system

I

literature review

I

full-state regulator

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Outline

I

TORA system

I

literature review

I

full-state regulator

I

full-state observer

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Outline

I

TORA system

I

literature review

I

full-state regulator

I

full-state observer

I

certainty-equivalence regulator

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Translational Oscillator with Rotational Actuator (TORA)

N θ F

xc

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Translational Oscillator with Rotational Actuator (TORA)

N θ F

xc

x¨d + xd θ¨

=

(θ˙2 sin θ − θ¨ cos θ) + w1

=

u − ¨ xd cos θ

0<<1

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Translational Oscillator with Rotational Actuator (TORA)

N θ F

xc

x¨d + xd θ¨

=

(θ˙2 sin θ − θ¨ cos θ) + w1

=

u − ¨ xd cos θ

0<<1

˙ T x = (xd x˙ d θ θ)

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Translational Oscillator with Rotational Actuator (TORA)

N θ F

xc

x¨d + xd θ¨

=

(θ˙2 sin θ − θ¨ cos θ) + w1

=

u − ¨ xd cos θ

0<<1 0

˙ T x = (xd x˙ d θ θ) x˙

=

B B B B f (x) , B B B @

f (x) + g (x)u + p(x)w1

1 0 B − cos x3 C B C g (x) , A 0 1 − 2 cos2 x3 @ 1

1 − 2 cos2 x3 x4  cos x3 (x1 − x42 sin x3 )

1 C C C C C C C A

1 − 2 cos2 x3

0

1

x2 −x1 + x42 sin x3

0 1

B B p(x) , 1 − 2 cos2 x3 @

1 0 C 1 C A 0 − cos x3 3 / 11

Literature Review

I

stabilization I I I I

Wan, Bernstein, and Coppola (CDC 1994) Jankovic, Fontaine, and Kokotovic (CST 1996) Escobar, Ortega, and Sira-Ramirez (CST 1999) Karagiannis, Jiang, Ortega, and Astolfi (Automatica 2005)

N θ F

xc

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Literature Review

I

stabilization I I I I

I

disturbance rejection and output tracking I

N

I

θ F

Wan, Bernstein, and Coppola (CDC 1994) Jankovic, Fontaine, and Kokotovic (CST 1996) Escobar, Ortega, and Sira-Ramirez (CST 1999) Karagiannis, Jiang, Ortega, and Astolfi (Automatica 2005) Zhao and Kanellakopoulos (RNC 1998) Jiang and Kanellakopoulos (TAC 2000)

xc

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Literature Review

I

stabilization I I I I

I

disturbance rejection and output tracking I

N

I

θ F

I xc

Wan, Bernstein, and Coppola (CDC 1994) Jankovic, Fontaine, and Kokotovic (CST 1996) Escobar, Ortega, and Sira-Ramirez (CST 1999) Karagiannis, Jiang, Ortega, and Astolfi (Automatica 2005) Zhao and Kanellakopoulos (RNC 1998) Jiang and Kanellakopoulos (TAC 2000)

asymptotic disturbance rejection (output regulation theory)

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Literature Review

I

stabilization I I I I

I

disturbance rejection and output tracking I

N

I

θ F

I xc

Wan, Bernstein, and Coppola (CDC 1994) Jankovic, Fontaine, and Kokotovic (CST 1996) Escobar, Ortega, and Sira-Ramirez (CST 1999) Karagiannis, Jiang, Ortega, and Astolfi (Automatica 2005) Zhao and Kanellakopoulos (RNC 1998) Jiang and Kanellakopoulos (TAC 2000)

asymptotic disturbance rejection (output regulation theory) I

Huang and Hu (CTA 2004) local regulator - uses measurements of both translation and rotational positions

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Literature Review

I

stabilization I I I I

I

disturbance rejection and output tracking I

N

I

θ F

I xc

Wan, Bernstein, and Coppola (CDC 1994) Jankovic, Fontaine, and Kokotovic (CST 1996) Escobar, Ortega, and Sira-Ramirez (CST 1999) Karagiannis, Jiang, Ortega, and Astolfi (Automatica 2005) Zhao and Kanellakopoulos (RNC 1998) Jiang and Kanellakopoulos (TAC 2000)

asymptotic disturbance rejection (output regulation theory) I

I

Huang and Hu (CTA 2004) local regulator - uses measurements of both translation and rotational positions Celani (ACC 2009) local regulator - uses only measurements of rotational position

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Literature Review

I

stabilization I I I I

I

disturbance rejection and output tracking I

N

I

θ F

I xc

Wan, Bernstein, and Coppola (CDC 1994) Jankovic, Fontaine, and Kokotovic (CST 1996) Escobar, Ortega, and Sira-Ramirez (CST 1999) Karagiannis, Jiang, Ortega, and Astolfi (Automatica 2005) Zhao and Kanellakopoulos (RNC 1998) Jiang and Kanellakopoulos (TAC 2000)

asymptotic disturbance rejection (output regulation theory) I

I

Huang and Hu (CTA 2004) local regulator - uses measurements of both translation and rotational positions Celani (ACC 2009) local regulator - uses only measurements of rotational position

semiglobal regulator - uses only measurements of rotational position

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Asymptotic Disturbance Rejection via Rotational-position Feedback

exosystem w˙ = Sw

 

S = 



0 ω   −ω 0 

w χ˙ = ϕ(χ, y) u = ρ(χ, y) rotationalposition feedback

u

x˙ = f (x) + g(x)u + p(x)w1 e = x1 y = x3

N

e y

TORA

θ F

xc

design rotational-position feedback such that 1. trajectories are bounded 2. limt→∞ e(t) = 0

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Asymptotic Disturbance Rejection via Rotational-position Feedback

exosystem w˙ = Sw

 

S = 



0 ω   −ω 0 

w χ˙ = ϕ(χ, y) u = ρ(χ, y) rotationalposition feedback

u

x˙ = f (x) + g(x)u + p(x)w1 e = x1 y = x3

N

e y

TORA

θ F

xc

design rotational-position feedback such that 1. trajectories are bounded 2. limt→∞ e(t) = 0 regulated error e is not measured

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Certainty-equivalence Design 1. exosystem w˙ = Sw

 

S = 



0 ω   −ω 0 

w full-state feedback

u = u∗(x, w)

u

x˙ = f (x) + g(x)u + p(x)w1 e = x1 y = x3 x

e y

TORA

design memoryless full-state feedback such that I I

trajectories are bounded limt→∞ e(t) = 0

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Certainty-equivalence Design 1. exosystem

 

S = 

w˙ = Sw



0 ω   −ω 0 

w full-state feedback

u = u∗(x, w)

u

e

x˙ = f (x) + g(x)u + p(x)w1 e = x1 y = x3

y

TORA

x

design memoryless full-state feedback such that I I

trajectories are bounded limt→∞ e(t) = 0

2. exosystem w˙ = Sw w u

x˙ = f (x) + g(x)u + p(x)w1 e = x1 y = x3 TORA

e

     

y      



xˆ   wˆ 

xˆ˙   = F(ˆ ˆ y, u) x, w, wˆ˙  

full-state observer

design full-state observer such that „ « x(t) − xˆ(t) →0 w (t) − w ˆ (t) 6 / 11

Memoryless Full-state Feedback change of coordinates + memoryless feedback

w˙ x˙ e

= = =

Sw f (x) + g (x)u + p(x)w1 x1

w˙ ξ˜˙1

=

ξ˜˙2

=

ξ˜˙3

=

ξ˜4

ξ˜˙4

=

e

=

v˜ “ w ”” “ w1 1 ξ˜1 − 2 −  sin ξ˜3 − arcsin ω ω 2

=

→

Sw ξ˜2 “ “ w ”” 1 −ξ˜1 +  sin ξ˜3 − arcsin 2 ω w1 + 2 ω

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Memoryless Full-state Feedback change of coordinates + memoryless feedback

w˙ x˙ e

= = =

Sw f (x) + g (x)u + p(x)w1 x1

w˙ ξ˜˙1

=

ξ˜˙2

=

ξ˜˙3

=

ξ˜4

ξ˜˙4

=

e

=

v˜ “ w ”” “ w1 1 ξ˜1 − 2 −  sin ξ˜3 − arcsin ω ω 2

=

→

Sw ξ˜2 “ “ w ”” 1 −ξ˜1 +  sin ξ˜3 − arcsin 2 ω w1 + 2 ω

˜ → 0 ⇒ e(t) → 0 |w1 (t)| ≤ ω 2 and ξ(t)

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Memoryless Full-state Feedback change of coordinates + memoryless feedback

w˙ x˙ e

= = =

Sw f (x) + g (x)u + p(x)w1 x1

w˙ ξ˜˙1

=

ξ˜˙2

=

ξ˜˙3

=

ξ˜4

ξ˜˙4

=

e

=

v˜ “ w ”” “ w1 1 ξ˜1 − 2 −  sin ξ˜3 − arcsin ω ω 2

=

→

Sw ξ˜2 “ “ w ”” 1 −ξ˜1 +  sin ξ˜3 − arcsin 2 ω w1 + 2 ω

˜ → 0 ⇒ e(t) → 0 |w1 (t)| ≤ ω 2 and ξ(t) ˜ w ) such that ξ(t) ˜ → 0 globally w.r.t. ξ˜ use backstepping → v˜ = φ(ξ,

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Memoryless Full-state Feedback change of coordinates + memoryless feedback

w˙ x˙ e

= = =

Sw f (x) + g (x)u + p(x)w1 x1

w˙ ξ˜˙1

=

ξ˜˙2

=

ξ˜˙3

=

ξ˜4

ξ˜˙4

=

e

=

v˜ “ w ”” “ w1 1 ξ˜1 − 2 −  sin ξ˜3 − arcsin ω ω 2

=

→

Sw ξ˜2 “ “ w ”” 1 −ξ˜1 +  sin ξ˜3 − arcsin 2 ω w1 + 2 ω

˜ → 0 ⇒ e(t) → 0 |w1 (t)| ≤ ω 2 and ξ(t) ˜ w ) such that ξ(t) ˜ → 0 globally w.r.t. ξ˜ use backstepping → v˜ = φ(ξ, ↓ u=

u ∗ (x, w )

such that e(t) → 0 globally w.r.t. x

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Full-state Observer exosystem w˙ = Sw w u

x˙ = f (x) + g(x)u + p(x)w1 e = x1 y = x3

e y

TORA

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Full-state Observer exosystem w˙ = Sw w x˙ = f (x) + g(x)u + p(x)w1 e = x1 y = x3

u

e y

TORA

0 B B B z˙ = B B @

change of coordinates z = φ(x, w ) z ∈ R6 1 0 1 z˙ 1 z2 z˙ 2 C B ψ2 (z1 , z2 , u) + z3 C B C C z˙ 3 C B ψ3 (z1 , z2 , z3 , u) + z4 C C = F (z, u) C = B z˙ 4 C B ψ4 (z1 , z2 , z3 , z4 , u) + z5 C @ A z˙ 5 ψ5 (z1 , z2 , z3 , z4 , z5 , u) + z6 A z˙ 6 ψ6 (z1 , z2 , z3 , z4 , z5 , z6 , u) y

=

z1

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Full-state Observer exosystem w˙ = Sw w x˙ = f (x) + g(x)u + p(x)w1 e = x1 y = x3

u

e y

TORA

0 B B B z˙ = B B @

change of coordinates z = φ(x, w ) z ∈ R6 1 0 1 z˙ 1 z2 z˙ 2 C B ψ2 (z1 , z2 , u) + z3 C B C C z˙ 3 C B ψ3 (z1 , z2 , z3 , u) + z4 C C = F (z, u) C = B z˙ 4 C B ψ4 (z1 , z2 , z3 , z4 , u) + z5 C @ A z˙ 5 ψ5 (z1 , z2 , z3 , z4 , z5 , u) + z6 A z˙ 6 ψ6 (z1 , z2 , z3 , z4 , z5 , z6 , u) y

=

z1

bounded trajectories and bounded input → Gauthier-Kupka’s nonlinear observer zˆ˙ = F gl (ˆ z , u) + G (y − zˆ1 ) 8 / 11

Full-state Observer exosystem w˙ = Sw w x˙ = f (x) + g(x)u + p(x)w1 e = x1 y = x3

u

e y

TORA

0 B B B z˙ = B B @

change of coordinates z = φ(x, w ) z ∈ R6 1 0 1 z˙ 1 z2 z˙ 2 C B ψ2 (z1 , z2 , u) + z3 C B C C z˙ 3 C B ψ3 (z1 , z2 , z3 , u) + z4 C C = F (z, u) C = B z˙ 4 C B ψ4 (z1 , z2 , z3 , z4 , u) + z5 C @ A z˙ 5 ψ5 (z1 , z2 , z3 , z4 , z5 , u) + z6 A z˙ 6 ψ6 (z1 , z2 , z3 , z4 , z5 , z6 , u) y

=

z1

bounded trajectories and bounded input → Gauthier-Kupka’s nonlinear observer zˆ˙ = F gl (ˆ z , u) + G (y − zˆ1 ) (ˆ x, w ˆ)

=

φ−1 (ˆ z) 8 / 11

Rotational-position Feedback

exosystem w˙ = Sw

 

S = 



0 ω   −ω 0 

w zˆ˙ = F gl (ˆ z , u) + G(y − zˆ1) ˆ = φ−1(ˆ (ˆ x, w) z) full-state observer

(ˆ x, w) ˆ

ˆ u∗(ˆ x, w)

u∗

full-state feedback

σl (u∗) saturation

u

x˙ = f (x) + g(x)u + p(x)w1 e = x1 y = x3

e y

TORA

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Rotational-position Feedback

exosystem w˙ = Sw

 

S = 



0 ω   −ω 0 

w zˆ˙ = F gl (ˆ z , u) + G(y − zˆ1) ˆ = φ−1(ˆ (ˆ x, w) z) full-state observer

(ˆ x, w) ˆ

u∗

ˆ u∗(ˆ x, w)

full-state feedback

σl (u∗) saturation

u

x˙ = f (x) + g(x)u + p(x)w1 e = x1 y = x3

e y

TORA

F. Celani, Certainty-equivalence in nonlinear output regulation with unmeasurable regulated error, IFAC World Congress, 2008. ↓ X arbitrary compact set of initial states of the TORA |w (0)| < ω 2 and x(0) ∈ X ⇒ trajectories are bounded and e(t) → 0

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Simulation Results  = 0.2

ω=3

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Simulation Results  = 0.2

ω=3

X = {x ∈ R4 ||xi | ≤ 0.1 i = 1, . . . , 4} w1 (0) = 0.1 < ω 2 w2 (0) = 0

x1 (0) = x2 (0) = x3 (0) = x4 (0) = 0.1

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Simulation Results  = 0.2

ω=3

X = {x ∈ R4 ||xi | ≤ 0.1 i = 1, . . . , 4} w1 (0) = 0.1 < ω 2 w2 (0) = 0 2

x1 (0) = x2 (0) = x3 (0) = x4 (0) = 0.1 2

x1 hat

x2 hat

x

1 0 −1

x

1

1

2

0

0

50

100 t

0.2

150

200

x3 hat

−1

0

50

100 t

0.5

x3

150

200

x4 hat x4

0.1 0

0

−0.1 −0.2

0

50

100 t

0.5

150

200

−0.5

w1 hat

0.2

w1

0.1

0

0

50

100 t

150 200 w2 hat w2

0 −0.1

−0.5

0

50

100 t

150

200

−0.2

0

50

100 t

150

200

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Conclusions

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semiglobal regulator that achieves asymptotic disturbance rejection for the TORA system via rotational-position feedback

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Conclusions

I

semiglobal regulator that achieves asymptotic disturbance rejection for the TORA system via rotational-position feedback

I

lack of robustness with respect to either plant or exosystem uncertainties

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