Certainty-Equivalence Design for Output Regulation of a Nonlinear Benchmark System Fabio Celani Department of Computer and Systems Science Antonio Ruberti Sapienza University of Rome Italy
2009 American Control Conference St. Louis, USA June 12th , 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
N
I θ
disturbance rejection and output tracking I
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)
I
Zhao and Kanellakopoulos (IJRNC 1998) Jiang and Kanellakopoulos (TAC 2000)
xc
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Literature Review
I
stabilization I I I I
N
I θ
disturbance rejection and output tracking I
F
I xc
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) Zhao and Kanellakopoulos (IJRNC 1998) Jiang and Kanellakopoulos (TAC 2000)
asymptotic disturbance rejection (output regulation theory) I I
Huang and Hu (CTA 2004) Pavlov, Janssen, van de Wouw, and Nijmeijer (CST 2007)
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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 ω 2 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 ω 2 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 ω 2 w1 + 2 ω
˜ → 0 ⇒ e(t) → 0 |w1 (t)| ≤ ω 2 and ξ(t) ˜ w ) such that ξ(t) ˜ → 0 locally use backstepping → v˜ = φ(ξ, Lyapunov function → W × X estimate of region of attraction in original coordinates
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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. ↓ trajectories are bounded and e(t) → 0 ∀(w (0), x(0)) ∈ W × X
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Simulation Results = 0.2
ω=3
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Simulation Results = 0.2
ω=3
W = {w ∈ R2 |w | ≤ 0.01} X = {x ∈ R |x1 | ≤ 0.06, |x2 | ≤ 0.03, |x3 | ≤ 0.009, |x4 | ≤ 0.002} 4
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Simulation Results = 0.2
ω=3
W = {w ∈ R2 |w | ≤ 0.01} X = {x ∈ R |x1 | ≤ 0.06, |x2 | ≤ 0.03, |x3 | ≤ 0.009, |x4 | ≤ 0.002} 4
0.2
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x3 hat
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w1 hat
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w2 hat
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w2
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x4
x 0
−0.5
150 x4 hat
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Conclusions
I
certainty-equivalence design for asymptotic disturbance rejection for the TORA system via rotational-position feedback
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Conclusions
I
certainty-equivalence design for asymptotic disturbance rejection for the TORA system via rotational-position feedback
I
regulator is local
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Conclusions
I
certainty-equivalence design for asymptotic disturbance rejection for the TORA system via rotational-position feedback
I
regulator is local
I
lack of robustness with respect to either plant or exosystem uncertainties
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