control and robustness analysis for curve tracking with ...malisoff/papers/2013jmmslides.pdf · 1/9...
TRANSCRIPT
![Page 1: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/1.jpg)
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Control and Robustness Analysis forCurve Tracking with Unknown Control Gains
Michael Malisoff, Roy P. Daniels ProfessorLouisiana State University Department of Mathematics
Joint with Fumin Zhang from Georgia Tech School of ECESponsored by NSF/ECCS/EPAS Program
Summary of Forthcoming Paper in Automatica
2013 Joint Mathematics MeetingsSpecial Session on Theory and Interdisciplinary
Applications of Dynamical SystemsJanuary 9, 2013
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 2: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/2.jpg)
2/9
What Are Perturbed Control Systems ?
These are triply parameterized families of ODEs of the form
Y ′(t) = F(t,Y (t), u(t,Y (t)), Γ, δ(t)
), Y (t) ∈ Y. (1)
Y ⊆ Rn. We have freedom to choose the control function u.The functions δ : [0,∞)→ D represent uncertainty. D ⊆ Rm.The vector Γ is constant but unknown.
Specify u to get a doubly parameterized closed loop family
Y ′(t) = G(t,Y (t), Γ, δ(t)), Y (t) ∈ Y, (2)
where G(t,Y , Γ, d) = F(t,Y , u(t,Y ), Γ, d).
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective.
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 3: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/3.jpg)
2/9
What Are Perturbed Control Systems ?
These are triply parameterized families of ODEs of the form
Y ′(t) = F(t,Y (t), u(t,Y (t)), Γ, δ(t)
), Y (t) ∈ Y. (1)
Y ⊆ Rn. We have freedom to choose the control function u.The functions δ : [0,∞)→ D represent uncertainty. D ⊆ Rm.The vector Γ is constant but unknown.
Specify u to get a doubly parameterized closed loop family
Y ′(t) = G(t,Y (t), Γ, δ(t)), Y (t) ∈ Y, (2)
where G(t,Y , Γ, d) = F(t,Y , u(t,Y ), Γ, d).
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective.
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 4: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/4.jpg)
2/9
What Are Perturbed Control Systems ?
These are triply parameterized families of ODEs of the form
Y ′(t) = F(t,Y (t), u(t,Y (t)), Γ, δ(t)
), Y (t) ∈ Y. (1)
Y ⊆ Rn.
We have freedom to choose the control function u.The functions δ : [0,∞)→ D represent uncertainty. D ⊆ Rm.The vector Γ is constant but unknown.
Specify u to get a doubly parameterized closed loop family
Y ′(t) = G(t,Y (t), Γ, δ(t)), Y (t) ∈ Y, (2)
where G(t,Y , Γ, d) = F(t,Y , u(t,Y ), Γ, d).
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective.
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 5: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/5.jpg)
2/9
What Are Perturbed Control Systems ?
These are triply parameterized families of ODEs of the form
Y ′(t) = F(t,Y (t), u(t,Y (t)), Γ, δ(t)
), Y (t) ∈ Y. (1)
Y ⊆ Rn. We have freedom to choose the control function u.
The functions δ : [0,∞)→ D represent uncertainty. D ⊆ Rm.The vector Γ is constant but unknown.
Specify u to get a doubly parameterized closed loop family
Y ′(t) = G(t,Y (t), Γ, δ(t)), Y (t) ∈ Y, (2)
where G(t,Y , Γ, d) = F(t,Y , u(t,Y ), Γ, d).
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective.
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 6: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/6.jpg)
2/9
What Are Perturbed Control Systems ?
These are triply parameterized families of ODEs of the form
Y ′(t) = F(t,Y (t), u(t,Y (t)), Γ, δ(t)
), Y (t) ∈ Y. (1)
Y ⊆ Rn. We have freedom to choose the control function u.The functions δ : [0,∞)→ D represent uncertainty.
D ⊆ Rm.The vector Γ is constant but unknown.
Specify u to get a doubly parameterized closed loop family
Y ′(t) = G(t,Y (t), Γ, δ(t)), Y (t) ∈ Y, (2)
where G(t,Y , Γ, d) = F(t,Y , u(t,Y ), Γ, d).
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective.
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 7: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/7.jpg)
2/9
What Are Perturbed Control Systems ?
These are triply parameterized families of ODEs of the form
Y ′(t) = F(t,Y (t), u(t,Y (t)), Γ, δ(t)
), Y (t) ∈ Y. (1)
Y ⊆ Rn. We have freedom to choose the control function u.The functions δ : [0,∞)→ D represent uncertainty. D ⊆ Rm.
The vector Γ is constant but unknown.
Specify u to get a doubly parameterized closed loop family
Y ′(t) = G(t,Y (t), Γ, δ(t)), Y (t) ∈ Y, (2)
where G(t,Y , Γ, d) = F(t,Y , u(t,Y ), Γ, d).
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective.
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 8: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/8.jpg)
2/9
What Are Perturbed Control Systems ?
These are triply parameterized families of ODEs of the form
Y ′(t) = F(t,Y (t), u(t,Y (t)), Γ, δ(t)
), Y (t) ∈ Y. (1)
Y ⊆ Rn. We have freedom to choose the control function u.The functions δ : [0,∞)→ D represent uncertainty. D ⊆ Rm.The vector Γ is constant but unknown.
Specify u to get a doubly parameterized closed loop family
Y ′(t) = G(t,Y (t), Γ, δ(t)), Y (t) ∈ Y, (2)
where G(t,Y , Γ, d) = F(t,Y , u(t,Y ), Γ, d).
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective.
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 9: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/9.jpg)
2/9
What Are Perturbed Control Systems ?
These are triply parameterized families of ODEs of the form
Y ′(t) = F(t,Y (t), u(t,Y (t)), Γ, δ(t)
), Y (t) ∈ Y. (1)
Y ⊆ Rn. We have freedom to choose the control function u.The functions δ : [0,∞)→ D represent uncertainty. D ⊆ Rm.The vector Γ is constant but unknown.
Specify u to get a doubly parameterized closed loop family
Y ′(t) = G(t,Y (t), Γ, δ(t)), Y (t) ∈ Y, (2)
where G(t,Y , Γ, d) = F(t,Y , u(t,Y ), Γ, d).
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective.
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 10: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/10.jpg)
2/9
What Are Perturbed Control Systems ?
These are triply parameterized families of ODEs of the form
Y ′(t) = F(t,Y (t), u(t,Y (t)), Γ, δ(t)
), Y (t) ∈ Y. (1)
Y ⊆ Rn. We have freedom to choose the control function u.The functions δ : [0,∞)→ D represent uncertainty. D ⊆ Rm.The vector Γ is constant but unknown.
Specify u to get a doubly parameterized closed loop family
Y ′(t) = G(t,Y (t), Γ, δ(t)), Y (t) ∈ Y, (2)
where G(t,Y , Γ, d) = F(t,Y , u(t,Y ), Γ, d).
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective.
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 11: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/11.jpg)
3/9
What is One Possible Control Objective ?
Input-to-state stability generalizes global asymptotic stability.
Y ′(t) = G(t,Y (t), Γ), Y (t) ∈ Y (Σ)
|Y (t)| ≤ γ1
(et0−tγ2(|Y (t0)|)
)(UGAS)
Our γi ’s are 0 at 0, strictly increasing, and unbounded. γi ∈ K∞.
Y ′(t) = G(t,Y (t), Γ, δ(t)
), Y (t) ∈ Y (Σpert)
|Y (t)| ≤ γ1
(et0−tγ2(|Y (t0)|)
)+ γ3(|δ|[t0,t]) (ISS)
Find γi ’s by building special strict LFs for Y ′(t) = G(t,Y (t), Γ, 0).
Ex : Σpert is ISS iff it has an ISS Lyapunov function (Sontag-Wang)
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 12: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/12.jpg)
3/9
What is One Possible Control Objective ?
Input-to-state stability generalizes global asymptotic stability.
Y ′(t) = G(t,Y (t), Γ), Y (t) ∈ Y (Σ)
|Y (t)| ≤ γ1
(et0−tγ2(|Y (t0)|)
)(UGAS)
Our γi ’s are 0 at 0, strictly increasing, and unbounded. γi ∈ K∞.
Y ′(t) = G(t,Y (t), Γ, δ(t)
), Y (t) ∈ Y (Σpert)
|Y (t)| ≤ γ1
(et0−tγ2(|Y (t0)|)
)+ γ3(|δ|[t0,t]) (ISS)
Find γi ’s by building special strict LFs for Y ′(t) = G(t,Y (t), Γ, 0).
Ex : Σpert is ISS iff it has an ISS Lyapunov function (Sontag-Wang)
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 13: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/13.jpg)
3/9
What is One Possible Control Objective ?
Input-to-state stability generalizes global asymptotic stability.
Y ′(t) = G(t,Y (t), Γ), Y (t) ∈ Y (Σ)
|Y (t)| ≤ γ1
(et0−tγ2(|Y (t0)|)
)(UGAS)
Our γi ’s are 0 at 0, strictly increasing, and unbounded. γi ∈ K∞.
Y ′(t) = G(t,Y (t), Γ, δ(t)
), Y (t) ∈ Y (Σpert)
|Y (t)| ≤ γ1
(et0−tγ2(|Y (t0)|)
)+ γ3(|δ|[t0,t]) (ISS)
Find γi ’s by building special strict LFs for Y ′(t) = G(t,Y (t), Γ, 0).
Ex : Σpert is ISS iff it has an ISS Lyapunov function (Sontag-Wang)
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 14: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/14.jpg)
3/9
What is One Possible Control Objective ?
Input-to-state stability generalizes global asymptotic stability.
Y ′(t) = G(t,Y (t), Γ), Y (t) ∈ Y (Σ)
|Y (t)| ≤ γ1
(et0−tγ2(|Y (t0)|)
)(UGAS)
Our γi ’s are 0 at 0, strictly increasing, and unbounded. γi ∈ K∞.
Y ′(t) = G(t,Y (t), Γ, δ(t)
), Y (t) ∈ Y (Σpert)
|Y (t)| ≤ γ1
(et0−tγ2(|Y (t0)|)
)+ γ3(|δ|[t0,t]) (ISS)
Find γi ’s by building special strict LFs for Y ′(t) = G(t,Y (t), Γ, 0).
Ex : Σpert is ISS iff it has an ISS Lyapunov function (Sontag-Wang)
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 15: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/15.jpg)
3/9
What is One Possible Control Objective ?
Input-to-state stability generalizes global asymptotic stability.
Y ′(t) = G(t,Y (t), Γ), Y (t) ∈ Y (Σ)
|Y (t)| ≤ γ1
(et0−tγ2(|Y (t0)|)
)(UGAS)
Our γi ’s are 0 at 0, strictly increasing, and unbounded.
γi ∈ K∞.
Y ′(t) = G(t,Y (t), Γ, δ(t)
), Y (t) ∈ Y (Σpert)
|Y (t)| ≤ γ1
(et0−tγ2(|Y (t0)|)
)+ γ3(|δ|[t0,t]) (ISS)
Find γi ’s by building special strict LFs for Y ′(t) = G(t,Y (t), Γ, 0).
Ex : Σpert is ISS iff it has an ISS Lyapunov function (Sontag-Wang)
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 16: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/16.jpg)
3/9
What is One Possible Control Objective ?
Input-to-state stability generalizes global asymptotic stability.
Y ′(t) = G(t,Y (t), Γ), Y (t) ∈ Y (Σ)
|Y (t)| ≤ γ1
(et0−tγ2(|Y (t0)|)
)(UGAS)
Our γi ’s are 0 at 0, strictly increasing, and unbounded. γi ∈ K∞.
Y ′(t) = G(t,Y (t), Γ, δ(t)
), Y (t) ∈ Y (Σpert)
|Y (t)| ≤ γ1
(et0−tγ2(|Y (t0)|)
)+ γ3(|δ|[t0,t]) (ISS)
Find γi ’s by building special strict LFs for Y ′(t) = G(t,Y (t), Γ, 0).
Ex : Σpert is ISS iff it has an ISS Lyapunov function (Sontag-Wang)
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 17: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/17.jpg)
3/9
What is One Possible Control Objective ?
Input-to-state stability generalizes global asymptotic stability.
Y ′(t) = G(t,Y (t), Γ), Y (t) ∈ Y (Σ)
|Y (t)| ≤ γ1
(et0−tγ2(|Y (t0)|)
)(UGAS)
Our γi ’s are 0 at 0, strictly increasing, and unbounded. γi ∈ K∞.
Y ′(t) = G(t,Y (t), Γ, δ(t)
), Y (t) ∈ Y (Σpert)
|Y (t)| ≤ γ1
(et0−tγ2(|Y (t0)|)
)+ γ3(|δ|[t0,t]) (ISS)
Find γi ’s by building special strict LFs for Y ′(t) = G(t,Y (t), Γ, 0).
Ex : Σpert is ISS iff it has an ISS Lyapunov function (Sontag-Wang)
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 18: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/18.jpg)
3/9
What is One Possible Control Objective ?
Input-to-state stability generalizes global asymptotic stability.
Y ′(t) = G(t,Y (t), Γ), Y (t) ∈ Y (Σ)
|Y (t)| ≤ γ1
(et0−tγ2(|Y (t0)|)
)(UGAS)
Our γi ’s are 0 at 0, strictly increasing, and unbounded. γi ∈ K∞.
Y ′(t) = G(t,Y (t), Γ, δ(t)
), Y (t) ∈ Y (Σpert)
|Y (t)| ≤ γ1
(et0−tγ2(|Y (t0)|)
)+ γ3(|δ|[t0,t]) (ISS)
Find γi ’s by building special strict LFs for Y ′(t) = G(t,Y (t), Γ, 0).
Ex : Σpert is ISS iff it has an ISS Lyapunov function (Sontag-Wang)
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 19: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/19.jpg)
3/9
What is One Possible Control Objective ?
Input-to-state stability generalizes global asymptotic stability.
Y ′(t) = G(t,Y (t), Γ), Y (t) ∈ Y (Σ)
|Y (t)| ≤ γ1
(et0−tγ2(|Y (t0)|)
)(UGAS)
Our γi ’s are 0 at 0, strictly increasing, and unbounded. γi ∈ K∞.
Y ′(t) = G(t,Y (t), Γ, δ(t)
), Y (t) ∈ Y (Σpert)
|Y (t)| ≤ γ1
(et0−tγ2(|Y (t0)|)
)+ γ3(|δ|[t0,t]) (ISS)
Find γi ’s by building special strict LFs for Y ′(t) = G(t,Y (t), Γ, 0).
Ex : Σpert is ISS iff it has an ISS Lyapunov function (Sontag-Wang)
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 20: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/20.jpg)
3/9
What is One Possible Control Objective ?
Input-to-state stability generalizes global asymptotic stability.
Y ′(t) = G(t,Y (t), Γ), Y (t) ∈ Y (Σ)
|Y (t)| ≤ γ1
(et0−tγ2(|Y (t0)|)
)(UGAS)
Our γi ’s are 0 at 0, strictly increasing, and unbounded. γi ∈ K∞.
Y ′(t) = G(t,Y (t), Γ, δ(t)
), Y (t) ∈ Y (Σpert)
|Y (t)| ≤ γ1
(et0−tγ2(|Y (t0)|)
)+ γ3(|δ|[t0,t]) (ISS)
Find γi ’s by building special strict LFs for Y ′(t) = G(t,Y (t), Γ, 0).
Ex : Σpert is ISS iff it has an ISS Lyapunov function (Sontag-Wang)
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 21: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/21.jpg)
4/9
Adaptive Tracking and Parameter Identification
Consider a perturbed control system
ξ = J (t, ξ, Γ, u, δ) (3)
with a smooth reference trajectory ξR for a reference control uR .That means ξ′R(t) = J (t, ξR(t), Γ, uR(t), 0) ∀t ≥ 0.
Problem: Find a dynamic feedback and a parameter estimator
u(t, ξ, Γ) and·Γ = τ(t, ξ, Γ) (4)
that makes the Y = (ξ, Γ) = (ξ − ξR , Γ− Γ) dynamics ISS.
Flight control, electrical and mechanical engineering, etc.Persistent excitation. Annaswamy, Narendra, Teel,..
We proved a general theorem about how this can be solved.
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 22: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/22.jpg)
4/9
Adaptive Tracking and Parameter Identification
Consider a perturbed control system
ξ = J (t, ξ, Γ, u, δ) (3)
with a smooth reference trajectory ξR for a reference control uR .
That means ξ′R(t) = J (t, ξR(t), Γ, uR(t), 0) ∀t ≥ 0.
Problem: Find a dynamic feedback and a parameter estimator
u(t, ξ, Γ) and·Γ = τ(t, ξ, Γ) (4)
that makes the Y = (ξ, Γ) = (ξ − ξR , Γ− Γ) dynamics ISS.
Flight control, electrical and mechanical engineering, etc.Persistent excitation. Annaswamy, Narendra, Teel,..
We proved a general theorem about how this can be solved.
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 23: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/23.jpg)
4/9
Adaptive Tracking and Parameter Identification
Consider a perturbed control system
ξ = J (t, ξ, Γ, u, δ) (3)
with a smooth reference trajectory ξR for a reference control uR .That means ξ′R(t) = J (t, ξR(t), Γ, uR(t), 0) ∀t ≥ 0.
Problem: Find a dynamic feedback and a parameter estimator
u(t, ξ, Γ) and·Γ = τ(t, ξ, Γ) (4)
that makes the Y = (ξ, Γ) = (ξ − ξR , Γ− Γ) dynamics ISS.
Flight control, electrical and mechanical engineering, etc.Persistent excitation. Annaswamy, Narendra, Teel,..
We proved a general theorem about how this can be solved.
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 24: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/24.jpg)
4/9
Adaptive Tracking and Parameter Identification
Consider a perturbed control system
ξ = J (t, ξ, Γ, u, δ) (3)
with a smooth reference trajectory ξR for a reference control uR .That means ξ′R(t) = J (t, ξR(t), Γ, uR(t), 0) ∀t ≥ 0.
Problem:
Find a dynamic feedback and a parameter estimator
u(t, ξ, Γ) and·Γ = τ(t, ξ, Γ) (4)
that makes the Y = (ξ, Γ) = (ξ − ξR , Γ− Γ) dynamics ISS.
Flight control, electrical and mechanical engineering, etc.Persistent excitation. Annaswamy, Narendra, Teel,..
We proved a general theorem about how this can be solved.
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 25: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/25.jpg)
4/9
Adaptive Tracking and Parameter Identification
Consider a perturbed control system
ξ = J (t, ξ, Γ, u, δ) (3)
with a smooth reference trajectory ξR for a reference control uR .That means ξ′R(t) = J (t, ξR(t), Γ, uR(t), 0) ∀t ≥ 0.
Problem: Find a dynamic feedback and a parameter estimator
u(t, ξ, Γ) and·Γ = τ(t, ξ, Γ) (4)
that makes the Y = (ξ, Γ) = (ξ − ξR , Γ− Γ) dynamics ISS.
Flight control, electrical and mechanical engineering, etc.Persistent excitation. Annaswamy, Narendra, Teel,..
We proved a general theorem about how this can be solved.
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 26: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/26.jpg)
4/9
Adaptive Tracking and Parameter Identification
Consider a perturbed control system
ξ = J (t, ξ, Γ, u, δ) (3)
with a smooth reference trajectory ξR for a reference control uR .That means ξ′R(t) = J (t, ξR(t), Γ, uR(t), 0) ∀t ≥ 0.
Problem: Find a dynamic feedback and a parameter estimator
u(t, ξ, Γ) and·Γ = τ(t, ξ, Γ) (4)
that makes the Y = (ξ, Γ) = (ξ − ξR , Γ− Γ) dynamics ISS.
Flight control, electrical and mechanical engineering, etc.
Persistent excitation. Annaswamy, Narendra, Teel,..
We proved a general theorem about how this can be solved.
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 27: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/27.jpg)
4/9
Adaptive Tracking and Parameter Identification
Consider a perturbed control system
ξ = J (t, ξ, Γ, u, δ) (3)
with a smooth reference trajectory ξR for a reference control uR .That means ξ′R(t) = J (t, ξR(t), Γ, uR(t), 0) ∀t ≥ 0.
Problem: Find a dynamic feedback and a parameter estimator
u(t, ξ, Γ) and·Γ = τ(t, ξ, Γ) (4)
that makes the Y = (ξ, Γ) = (ξ − ξR , Γ− Γ) dynamics ISS.
Flight control, electrical and mechanical engineering, etc.Persistent excitation.
Annaswamy, Narendra, Teel,..
We proved a general theorem about how this can be solved.
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 28: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/28.jpg)
4/9
Adaptive Tracking and Parameter Identification
Consider a perturbed control system
ξ = J (t, ξ, Γ, u, δ) (3)
with a smooth reference trajectory ξR for a reference control uR .That means ξ′R(t) = J (t, ξR(t), Γ, uR(t), 0) ∀t ≥ 0.
Problem: Find a dynamic feedback and a parameter estimator
u(t, ξ, Γ) and·Γ = τ(t, ξ, Γ) (4)
that makes the Y = (ξ, Γ) = (ξ − ξR , Γ− Γ) dynamics ISS.
Flight control, electrical and mechanical engineering, etc.Persistent excitation. Annaswamy, Narendra, Teel,..
We proved a general theorem about how this can be solved.
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 29: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/29.jpg)
4/9
Adaptive Tracking and Parameter Identification
Consider a perturbed control system
ξ = J (t, ξ, Γ, u, δ) (3)
with a smooth reference trajectory ξR for a reference control uR .That means ξ′R(t) = J (t, ξR(t), Γ, uR(t), 0) ∀t ≥ 0.
Problem: Find a dynamic feedback and a parameter estimator
u(t, ξ, Γ) and·Γ = τ(t, ξ, Γ) (4)
that makes the Y = (ξ, Γ) = (ξ − ξR , Γ− Γ) dynamics ISS.
Flight control, electrical and mechanical engineering, etc.Persistent excitation. Annaswamy, Narendra, Teel,..
We proved a general theorem about how this can be solved.
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 30: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/30.jpg)
5/9
Application: 2D Curve Tracking for Marine Robots
Motivation: Search for pollutants from Deepwater Horizon disaster.
ρ = |r2 − r1|, φ = angle between x1 and x2, cos(φ) = x1 · x2
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 31: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/31.jpg)
5/9
Application: 2D Curve Tracking for Marine Robots
Motivation: Search for pollutants from Deepwater Horizon disaster.
ρ = |r2 − r1|, φ = angle between x1 and x2, cos(φ) = x1 · x2
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 32: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/32.jpg)
5/9
Application: 2D Curve Tracking for Marine Robots
Motivation: Search for pollutants from Deepwater Horizon disaster.
ρ = |r2 − r1|, φ = angle between x1 and x2, cos(φ) = x1 · x2
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 33: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/33.jpg)
5/9
Application: 2D Curve Tracking for Marine Robots
Motivation: Search for pollutants from Deepwater Horizon disaster.
ρ = |r2 − r1|, φ = angle between x1 and x2, cos(φ) = x1 · x2
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 34: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/34.jpg)
6/9
Adaptive Robust Curve Tracking
{ρ = − sin(φ)
φ = κ cos(φ)1+κρ + Γ[u + δ]
(ρ, φ) ∈state space︷ ︸︸ ︷
(0,∞)× (−π/2, π/2) (Σc)
h(ρ) = α{ρ+
ρ20ρ − 2ρ0
}, ρ0 = desired value for ρ (5)
Control : u(ρ, φ, Γ) = − 1Γ
(κ cos(φ)
1+κρ − h′(ρ) cos(φ) + µ sin(φ))
(6)
Estimator :˙Γ = (Γ− cmin)(cmax − Γ)∂V
](ρ,φ)∂φ u(ρ, φ, Γ) (7)
V ](ρ, φ) = − h′(ρ) sin(φ) +
∫ V (ρ,φ)
0γ(m)dm (8)
γ(q) = 1µ
(2
α2ρ40(q + 2αρ0)3 + 1
)+ µ
2 + 2 + 18αρ0
+ 576ρ4
0α2 q3 (9)
V (ρ, φ) = − ln(
cos(φ))
+ h(ρ) (10)
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 35: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/35.jpg)
6/9
Adaptive Robust Curve Tracking
{ρ = − sin(φ)
φ = κ cos(φ)1+κρ + Γ[u + δ]
(ρ, φ) ∈state space︷ ︸︸ ︷
(0,∞)× (−π/2, π/2) (Σc)
h(ρ) = α{ρ+
ρ20ρ − 2ρ0
}, ρ0 = desired value for ρ (5)
Control : u(ρ, φ, Γ) = − 1Γ
(κ cos(φ)
1+κρ − h′(ρ) cos(φ) + µ sin(φ))
(6)
Estimator :˙Γ = (Γ− cmin)(cmax − Γ)∂V
](ρ,φ)∂φ u(ρ, φ, Γ) (7)
V ](ρ, φ) = − h′(ρ) sin(φ) +
∫ V (ρ,φ)
0γ(m)dm (8)
γ(q) = 1µ
(2
α2ρ40(q + 2αρ0)3 + 1
)+ µ
2 + 2 + 18αρ0
+ 576ρ4
0α2 q3 (9)
V (ρ, φ) = − ln(
cos(φ))
+ h(ρ) (10)
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 36: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/36.jpg)
6/9
Adaptive Robust Curve Tracking
{ρ = − sin(φ)
φ = κ cos(φ)1+κρ + Γ[u + δ]
(ρ, φ) ∈state space︷ ︸︸ ︷
(0,∞)× (−π/2, π/2) (Σc)
h(ρ) = α{ρ+
ρ20ρ − 2ρ0
}, ρ0 = desired value for ρ (5)
Control : u(ρ, φ, Γ) = − 1Γ
(κ cos(φ)
1+κρ − h′(ρ) cos(φ) + µ sin(φ))
(6)
Estimator :˙Γ = (Γ− cmin)(cmax − Γ)∂V
](ρ,φ)∂φ u(ρ, φ, Γ) (7)
V ](ρ, φ) = − h′(ρ) sin(φ) +
∫ V (ρ,φ)
0γ(m)dm (8)
γ(q) = 1µ
(2
α2ρ40(q + 2αρ0)3 + 1
)+ µ
2 + 2 + 18αρ0
+ 576ρ4
0α2 q3 (9)
V (ρ, φ) = − ln(
cos(φ))
+ h(ρ) (10)
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 37: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/37.jpg)
6/9
Adaptive Robust Curve Tracking
{ρ = − sin(φ)
φ = κ cos(φ)1+κρ + Γ[u + δ]
(ρ, φ) ∈state space︷ ︸︸ ︷
(0,∞)× (−π/2, π/2) (Σc)
h(ρ) = α{ρ+
ρ20ρ − 2ρ0
}, ρ0 = desired value for ρ (5)
Control : u(ρ, φ, Γ) = − 1Γ
(κ cos(φ)
1+κρ − h′(ρ) cos(φ) + µ sin(φ))
(6)
Estimator :˙Γ = (Γ− cmin)(cmax − Γ)∂V
](ρ,φ)∂φ u(ρ, φ, Γ) (7)
V ](ρ, φ) = − h′(ρ) sin(φ) +
∫ V (ρ,φ)
0γ(m)dm (8)
γ(q) = 1µ
(2
α2ρ40(q + 2αρ0)3 + 1
)+ µ
2 + 2 + 18αρ0
+ 576ρ4
0α2 q3 (9)
V (ρ, φ) = − ln(
cos(φ))
+ h(ρ) (10)
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 38: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/38.jpg)
6/9
Adaptive Robust Curve Tracking
{ρ = − sin(φ)
φ = κ cos(φ)1+κρ + Γ[u + δ]
(ρ, φ) ∈state space︷ ︸︸ ︷
(0,∞)× (−π/2, π/2) (Σc)
h(ρ) = α{ρ+
ρ20ρ − 2ρ0
}, ρ0 = desired value for ρ (5)
Control : u(ρ, φ, Γ) = − 1Γ
(κ cos(φ)
1+κρ − h′(ρ) cos(φ) + µ sin(φ))
(6)
Estimator :˙Γ = (Γ− cmin)(cmax − Γ)∂V
](ρ,φ)∂φ u(ρ, φ, Γ) (7)
V ](ρ, φ) = − h′(ρ) sin(φ) +
∫ V (ρ,φ)
0γ(m)dm (8)
γ(q) = 1µ
(2
α2ρ40(q + 2αρ0)3 + 1
)+ µ
2 + 2 + 18αρ0
+ 576ρ4
0α2 q3 (9)
V (ρ, φ) = − ln(
cos(φ))
+ h(ρ) (10)
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 39: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/39.jpg)
7/9
Robustly Forwardly Invariant Hexagonal Regions
We must restrict the suprema of the perturbations δ(t) to keep(ρ, φ) from exiting the required state space (0,∞)× (−π/2, π/2).
View the state space (0,∞)× (−π/2, π/2)as a union of compact hexagonallyshaped regions H1 ⊆ H2 ⊆ . . . ⊆ Hi ⊆ . . ..For each i , all trajectories of (Σc) startingin Hi for all δ : [0,∞)→ [−δ∗i , δ∗i ] stayin Hi . The tilted legs have slope cminµ/cmax.
For each index i , we take δ∗i to be the largest allowabledisturbance bound to maintain forward invariance of Hi .
Then we can prove ISS of the tracking and parameter identificationdynamics for each set Hi for the disturbance set D = [−δ∗i , δ∗i ].
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 40: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/40.jpg)
7/9
Robustly Forwardly Invariant Hexagonal Regions
We must restrict the suprema of the perturbations δ(t) to keep(ρ, φ) from exiting the required state space (0,∞)× (−π/2, π/2).
View the state space (0,∞)× (−π/2, π/2)as a union of compact hexagonallyshaped regions H1 ⊆ H2 ⊆ . . . ⊆ Hi ⊆ . . ..For each i , all trajectories of (Σc) startingin Hi for all δ : [0,∞)→ [−δ∗i , δ∗i ] stayin Hi . The tilted legs have slope cminµ/cmax.
For each index i , we take δ∗i to be the largest allowabledisturbance bound to maintain forward invariance of Hi .
Then we can prove ISS of the tracking and parameter identificationdynamics for each set Hi for the disturbance set D = [−δ∗i , δ∗i ].
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 41: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/41.jpg)
7/9
Robustly Forwardly Invariant Hexagonal Regions
We must restrict the suprema of the perturbations δ(t) to keep(ρ, φ) from exiting the required state space (0,∞)× (−π/2, π/2).
View the state space (0,∞)× (−π/2, π/2)as a union of compact hexagonallyshaped regions H1 ⊆ H2 ⊆ . . . ⊆ Hi ⊆ . . ..
For each i , all trajectories of (Σc) startingin Hi for all δ : [0,∞)→ [−δ∗i , δ∗i ] stayin Hi . The tilted legs have slope cminµ/cmax.
For each index i , we take δ∗i to be the largest allowabledisturbance bound to maintain forward invariance of Hi .
Then we can prove ISS of the tracking and parameter identificationdynamics for each set Hi for the disturbance set D = [−δ∗i , δ∗i ].
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 42: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/42.jpg)
7/9
Robustly Forwardly Invariant Hexagonal Regions
We must restrict the suprema of the perturbations δ(t) to keep(ρ, φ) from exiting the required state space (0,∞)× (−π/2, π/2).
View the state space (0,∞)× (−π/2, π/2)as a union of compact hexagonallyshaped regions H1 ⊆ H2 ⊆ . . . ⊆ Hi ⊆ . . ..For each i , all trajectories of (Σc) startingin Hi for all δ : [0,∞)→ [−δ∗i , δ∗i ] stayin Hi . The tilted legs have slope cminµ/cmax.
For each index i , we take δ∗i to be the largest allowabledisturbance bound to maintain forward invariance of Hi .
Then we can prove ISS of the tracking and parameter identificationdynamics for each set Hi for the disturbance set D = [−δ∗i , δ∗i ].
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 43: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/43.jpg)
7/9
Robustly Forwardly Invariant Hexagonal Regions
We must restrict the suprema of the perturbations δ(t) to keep(ρ, φ) from exiting the required state space (0,∞)× (−π/2, π/2).
View the state space (0,∞)× (−π/2, π/2)as a union of compact hexagonallyshaped regions H1 ⊆ H2 ⊆ . . . ⊆ Hi ⊆ . . ..For each i , all trajectories of (Σc) startingin Hi for all δ : [0,∞)→ [−δ∗i , δ∗i ] stayin Hi . The tilted legs have slope cminµ/cmax.
For each index i , we take δ∗i to be the largest allowabledisturbance bound to maintain forward invariance of Hi .
Then we can prove ISS of the tracking and parameter identificationdynamics for each set Hi for the disturbance set D = [−δ∗i , δ∗i ].
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 44: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/44.jpg)
7/9
Robustly Forwardly Invariant Hexagonal Regions
We must restrict the suprema of the perturbations δ(t) to keep(ρ, φ) from exiting the required state space (0,∞)× (−π/2, π/2).
View the state space (0,∞)× (−π/2, π/2)as a union of compact hexagonallyshaped regions H1 ⊆ H2 ⊆ . . . ⊆ Hi ⊆ . . ..For each i , all trajectories of (Σc) startingin Hi for all δ : [0,∞)→ [−δ∗i , δ∗i ] stayin Hi . The tilted legs have slope cminµ/cmax.
For each index i , we take δ∗i to be the largest allowabledisturbance bound to maintain forward invariance of Hi .
Then we can prove ISS of the tracking and parameter identificationdynamics for each set Hi for the disturbance set D = [−δ∗i , δ∗i ].
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 45: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/45.jpg)
8/9
Summary (M. and Zhang, Automatica, 2013)
Theorem
Let κ > 0 and ρ0 > 0 be any constants. Let cmin and cmax be anypositive constants such that cmin < Γ < cmax. Let i ∈ N, and letHi and δ∗i satisfy the above requirements. Then the augmentedperturbed 2D tracking and parameter identification dynamics
Y1 = − sin(Y2)
Y2 = κ cos(Y2)1+κ(Y1+ρ0) + Γu
(Y1 + ρ0,Y2, Γ
)+ Γδ
˙Γ = u
(Y1 + ρ0,Y2, Γ
)(Γ− cmin
)(cmax − Γ
)∂V ](Y1+ρ0,Y2)∂Y2
for Y = (Y1,Y2, Γ) = (ρ− ρ0, φ, Γ− Γ) is ISS on the state spaceY = (Hi − {(ρ0, 0)})× (cmin − Γ, cmax − Γ) for D = [−δ∗i , δ∗i ].
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 46: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/46.jpg)
8/9
Summary (M. and Zhang, Automatica, 2013)
Theorem
Let κ > 0 and ρ0 > 0 be any constants. Let cmin and cmax be anypositive constants such that cmin < Γ < cmax. Let i ∈ N, and letHi and δ∗i satisfy the above requirements. Then the augmentedperturbed 2D tracking and parameter identification dynamics
Y1 = − sin(Y2)
Y2 = κ cos(Y2)1+κ(Y1+ρ0) + Γu
(Y1 + ρ0,Y2, Γ
)+ Γδ
˙Γ = u
(Y1 + ρ0,Y2, Γ
)(Γ− cmin
)(cmax − Γ
)∂V ](Y1+ρ0,Y2)∂Y2
for Y = (Y1,Y2, Γ) = (ρ− ρ0, φ, Γ− Γ) is ISS on the state spaceY = (Hi − {(ρ0, 0)})× (cmin − Γ, cmax − Γ) for D = [−δ∗i , δ∗i ].
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 47: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/47.jpg)
8/9
Summary (M. and Zhang, Automatica, 2013)
Theorem
Let κ > 0 and ρ0 > 0 be any constants.
Let cmin and cmax be anypositive constants such that cmin < Γ < cmax. Let i ∈ N, and letHi and δ∗i satisfy the above requirements. Then the augmentedperturbed 2D tracking and parameter identification dynamics
Y1 = − sin(Y2)
Y2 = κ cos(Y2)1+κ(Y1+ρ0) + Γu
(Y1 + ρ0,Y2, Γ
)+ Γδ
˙Γ = u
(Y1 + ρ0,Y2, Γ
)(Γ− cmin
)(cmax − Γ
)∂V ](Y1+ρ0,Y2)∂Y2
for Y = (Y1,Y2, Γ) = (ρ− ρ0, φ, Γ− Γ) is ISS on the state spaceY = (Hi − {(ρ0, 0)})× (cmin − Γ, cmax − Γ) for D = [−δ∗i , δ∗i ].
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 48: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/48.jpg)
8/9
Summary (M. and Zhang, Automatica, 2013)
Theorem
Let κ > 0 and ρ0 > 0 be any constants. Let cmin and cmax be anypositive constants such that cmin < Γ < cmax.
Let i ∈ N, and letHi and δ∗i satisfy the above requirements. Then the augmentedperturbed 2D tracking and parameter identification dynamics
Y1 = − sin(Y2)
Y2 = κ cos(Y2)1+κ(Y1+ρ0) + Γu
(Y1 + ρ0,Y2, Γ
)+ Γδ
˙Γ = u
(Y1 + ρ0,Y2, Γ
)(Γ− cmin
)(cmax − Γ
)∂V ](Y1+ρ0,Y2)∂Y2
for Y = (Y1,Y2, Γ) = (ρ− ρ0, φ, Γ− Γ) is ISS on the state spaceY = (Hi − {(ρ0, 0)})× (cmin − Γ, cmax − Γ) for D = [−δ∗i , δ∗i ].
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 49: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/49.jpg)
8/9
Summary (M. and Zhang, Automatica, 2013)
Theorem
Let κ > 0 and ρ0 > 0 be any constants. Let cmin and cmax be anypositive constants such that cmin < Γ < cmax. Let i ∈ N, and letHi and δ∗i satisfy the above requirements.
Then the augmentedperturbed 2D tracking and parameter identification dynamics
Y1 = − sin(Y2)
Y2 = κ cos(Y2)1+κ(Y1+ρ0) + Γu
(Y1 + ρ0,Y2, Γ
)+ Γδ
˙Γ = u
(Y1 + ρ0,Y2, Γ
)(Γ− cmin
)(cmax − Γ
)∂V ](Y1+ρ0,Y2)∂Y2
for Y = (Y1,Y2, Γ) = (ρ− ρ0, φ, Γ− Γ) is ISS on the state spaceY = (Hi − {(ρ0, 0)})× (cmin − Γ, cmax − Γ) for D = [−δ∗i , δ∗i ].
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 50: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/50.jpg)
8/9
Summary (M. and Zhang, Automatica, 2013)
Theorem
Let κ > 0 and ρ0 > 0 be any constants. Let cmin and cmax be anypositive constants such that cmin < Γ < cmax. Let i ∈ N, and letHi and δ∗i satisfy the above requirements. Then the augmentedperturbed 2D tracking and parameter identification dynamics
Y1 = − sin(Y2)
Y2 = κ cos(Y2)1+κ(Y1+ρ0) + Γu
(Y1 + ρ0,Y2, Γ
)+ Γδ
˙Γ = u
(Y1 + ρ0,Y2, Γ
)(Γ− cmin
)(cmax − Γ
)∂V ](Y1+ρ0,Y2)∂Y2
for Y = (Y1,Y2, Γ) = (ρ− ρ0, φ, Γ− Γ) is ISS on the state spaceY = (Hi − {(ρ0, 0)})× (cmin − Γ, cmax − Γ) for D = [−δ∗i , δ∗i ].
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 51: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/51.jpg)
8/9
Summary (M. and Zhang, Automatica, 2013)
Theorem
Let κ > 0 and ρ0 > 0 be any constants. Let cmin and cmax be anypositive constants such that cmin < Γ < cmax. Let i ∈ N, and letHi and δ∗i satisfy the above requirements. Then the augmentedperturbed 2D tracking and parameter identification dynamics
Y1 = − sin(Y2)
Y2 = κ cos(Y2)1+κ(Y1+ρ0) + Γu
(Y1 + ρ0,Y2, Γ
)+ Γδ
˙Γ = u
(Y1 + ρ0,Y2, Γ
)(Γ− cmin
)(cmax − Γ
)∂V ](Y1+ρ0,Y2)∂Y2
for Y = (Y1,Y2, Γ) = (ρ− ρ0, φ, Γ− Γ) is ISS on the state spaceY = (Hi − {(ρ0, 0)})× (cmin − Γ, cmax − Γ) for D = [−δ∗i , δ∗i ].
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 52: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/52.jpg)
9/9
Conclusions
It is important but complicated to design controllers whenthere are unknown parameters that we must identify.
We overcame this challenge for an interesting large class ofdynamics, including 2D curve tracking dynamics.
Our robust forward invariance approach leads to input-to-statestability under maximal perturbation bounds.
We can also cover time delayed perturbed systems whichmodel intermittent communication in marine environments.
We can generalize our work to 3D curve tracking and we planextensions to cases with other obstacles.
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 53: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/53.jpg)
9/9
Conclusions
It is important but complicated to design controllers whenthere are unknown parameters that we must identify.
We overcame this challenge for an interesting large class ofdynamics, including 2D curve tracking dynamics.
Our robust forward invariance approach leads to input-to-statestability under maximal perturbation bounds.
We can also cover time delayed perturbed systems whichmodel intermittent communication in marine environments.
We can generalize our work to 3D curve tracking and we planextensions to cases with other obstacles.
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 54: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/54.jpg)
9/9
Conclusions
It is important but complicated to design controllers whenthere are unknown parameters that we must identify.
We overcame this challenge for an interesting large class ofdynamics, including 2D curve tracking dynamics.
Our robust forward invariance approach leads to input-to-statestability under maximal perturbation bounds.
We can also cover time delayed perturbed systems whichmodel intermittent communication in marine environments.
We can generalize our work to 3D curve tracking and we planextensions to cases with other obstacles.
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 55: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/55.jpg)
9/9
Conclusions
It is important but complicated to design controllers whenthere are unknown parameters that we must identify.
We overcame this challenge for an interesting large class ofdynamics, including 2D curve tracking dynamics.
Our robust forward invariance approach leads to input-to-statestability under maximal perturbation bounds.
We can also cover time delayed perturbed systems whichmodel intermittent communication in marine environments.
We can generalize our work to 3D curve tracking and we planextensions to cases with other obstacles.
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 56: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/56.jpg)
9/9
Conclusions
It is important but complicated to design controllers whenthere are unknown parameters that we must identify.
We overcame this challenge for an interesting large class ofdynamics, including 2D curve tracking dynamics.
Our robust forward invariance approach leads to input-to-statestability under maximal perturbation bounds.
We can also cover time delayed perturbed systems whichmodel intermittent communication in marine environments.
We can generalize our work to 3D curve tracking and we planextensions to cases with other obstacles.
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains
![Page 57: Control and Robustness Analysis for Curve Tracking with ...malisoff/papers/2013JMMSlides.pdf · 1/9 Control and Robustness Analysis for Curve Tracking with Unknown Control Gains Michael](https://reader031.vdocuments.net/reader031/viewer/2022022703/5bc9938609d3f2aa798cc6bf/html5/thumbnails/57.jpg)
9/9
Conclusions
It is important but complicated to design controllers whenthere are unknown parameters that we must identify.
We overcame this challenge for an interesting large class ofdynamics, including 2D curve tracking dynamics.
Our robust forward invariance approach leads to input-to-statestability under maximal perturbation bounds.
We can also cover time delayed perturbed systems whichmodel intermittent communication in marine environments.
We can generalize our work to 3D curve tracking and we planextensions to cases with other obstacles.
Michael Malisoff (LSU) and Fumin Zhang (Georgia Tech) Control for Curve Tracking with Unknown Control Gains