Analysis and Design of Nonlinear Control Systems – London, 2008 1 / 19
A Taxonomy for Time-Varying Immersionsin
Periodic Internal-Model Control
Prof. Andrea Serrani
Department of Electrical and Computer Engineering
The Ohio State University
Outline of the Talk
Introduction
• Outline of the Talk• Milestones in OutputRegulation Theory
Problem Statement
Problem Solvability
Robust IM Design
Adaptive IM Design
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 2 / 19
1. Statement of the Problem
• Regulation in the Periodic Setup2. Solvability of the Problem
• Periodic Internal Model Principle• Periodic Immersions
3. Robust Internal Model Design
• Canonical Realizations• Regulator Structure
4. Adaptive Internal Model Design
• Canonical Parameterization• Adaptive Regulator Structure• Weak Immersions for Adaptive Design
5. Illustrative Example6. Conclusions
Milestones in Output Regulation Theory
Introduction
• Outline of the Talk• Milestones in OutputRegulation Theory
Problem Statement
Problem Solvability
Robust IM Design
Adaptive IM Design
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 3 / 19
1. Davison, E. J.; Goldenberg, A., “Robust control of a general
servomechanism problem: The servo compensator,”
Automatica, 1975.
2. Francis, B. A.; Wonham, W. M., “The internal model principle of
control theory,” Automatica, 1976.3. Francis, B. A., “The linear multivariable regulator problem,”
SIAM Journal on Control and Optimization, 1977.
4. Isidori, A.; Byrnes, C. I., “Output regulation of nonlinearsystems,” IEEE Transactions on Automatic Control, 1990.
5. Byrnes, C. I.; Isidori, A., “Limit sets, zero dynamics, and internalmodels in the problem of nonlinear output regulation”,
IEEE Transactions on Automatic Control, 2003.
6. Marconi, L.; Praly, L.; Isidori, A., “Output Stabilization via
Nonlinear Luenberger Observers”, SIAM Journal on Control and
Optimization, 2007.
Milestones in Output Regulation Theory
Introduction
• Outline of the Talk• Milestones in OutputRegulation Theory
Problem Statement
Problem Solvability
Robust IM Design
Adaptive IM Design
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 3 / 19
1. Davison, E. J.; Goldenberg, A., “Robust control of a general
servomechanism problem: The servo compensator,”
Automatica, 1975.
2. Francis, B. A.; Wonham, W. M., “The internal model principle of
control theory,” Automatica, 1976.3. Francis, B. A., “The linear multivariable regulator problem,”
SIAM Journal on Control and Optimization, 1977.
4. Isidori, A.; Byrnes, C. I., “Output regulation of nonlinearsystems,” IEEE Transactions on Automatic Control, 1990.
5. Byrnes, C. I.; Isidori, A., “Limit sets, zero dynamics, and internalmodels in the problem of nonlinear output regulation”,
IEEE Transactions on Automatic Control, 2003.
6. Marconi, L.; Praly, L.; Isidori, A., “Output Stabilization via
Nonlinear Luenberger Observers”, SIAM Journal on Control and
Optimization, 2007.
Milestones in Output Regulation Theory
Introduction
• Outline of the Talk• Milestones in OutputRegulation Theory
Problem Statement
Problem Solvability
Robust IM Design
Adaptive IM Design
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 3 / 19
1. Davison, E. J.; Goldenberg, A., “Robust control of a general
servomechanism problem: The servo compensator,”
Automatica, 1975.
2. Francis, B. A.; Wonham, W. M., “The internal model principle of
control theory,” Automatica, 1976.3. Francis, B. A., “The linear multivariable regulator problem,”
SIAM Journal on Control and Optimization, 1977.
4. Isidori, A.; Byrnes, C. I., “Output regulation of nonlinearsystems,” IEEE Transactions on Automatic Control, 1990.
5. Byrnes, C. I.; Isidori, A., “Limit sets, zero dynamics, and internalmodels in the problem of nonlinear output regulation”,
IEEE Transactions on Automatic Control, 2003.
6. Marconi, L.; Praly, L.; Isidori, A., “Output Stabilization via
Nonlinear Luenberger Observers”, SIAM Journal on Control and
Optimization, 2007.
Regulation in the Periodic Setup
Introduction
Problem Statement• Regulation in thePeriodic Setup
• Certainty EquivalenceControl
• Adaptive RegulationProblem Solvability
Robust IM Design
Adaptive IM Design
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 4 / 19
• Regulation of parametrized families of linear T -periodic systems:
ẇ = S(t, σ)wẋ = A(t, µ)x + B(t, µ)u + P (t, µ)we = C(t, µ)x + Q(t, µ)w , (1)
◦ exosystem state w ∈ Rnw , plant state x ∈ Rn◦ control input u ∈ R, and regulated error e ∈ R◦ parameter vectors (σ, µ) ∈ Kσ ×Kµ ⊂ Rs × Rp
Regulation in the Periodic Setup
Introduction
Problem Statement• Regulation in thePeriodic Setup
• Certainty EquivalenceControl
• Adaptive RegulationProblem Solvability
Robust IM Design
Adaptive IM Design
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 4 / 19
• Regulation of parametrized families of linear T -periodic systems:
ẇ = S(t, σ)wẋ = A(t, µ)x + B(t, µ)u + P (t, µ)we = C(t, µ)x + Q(t, µ)w , (1)
◦ exosystem state w ∈ Rnw , plant state x ∈ Rn◦ control input u ∈ R, and regulated error e ∈ R◦ parameter vectors (σ, µ) ∈ Kσ ×Kµ ⊂ Rs × Rp
• Look for a parameterized family of T -periodic controllers
ξ̇ = F (t, θ)ξ + G(t, θ)eu = H(t, θ)ξ + K(t, θ)e , (2)
with state ξ ∈ Rν and tunable parameter vector θ ∈ Kθ ⊂ Rρ.
Certainty Equivalence Control
Introduction
Problem Statement• Regulation in thePeriodic Setup
• Certainty EquivalenceControl
• Adaptive RegulationProblem Solvability
Robust IM Design
Adaptive IM Design
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 5 / 19
The controller (2) is a certainty equivalence controller if ∀µ ∈ Kµ:1. The unforced closed-loop system
ẋ =[A(t, µ) + B(t, µ)K(t, θ)C(t, µ)
]x + B(t, µ)H(t, θ)ξ
ξ̇ = F (t, θ)ξ + G(t, θ)C(t, µ)x
is uniformly asymptotically stable for all θ ∈ Kθ
Certainty Equivalence Control
Introduction
Problem Statement• Regulation in thePeriodic Setup
• Certainty EquivalenceControl
• Adaptive RegulationProblem Solvability
Robust IM Design
Adaptive IM Design
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 5 / 19
The controller (2) is a certainty equivalence controller if ∀µ ∈ Kµ:1. The unforced closed-loop system
ẋ =[A(t, µ) + B(t, µ)K(t, θ)C(t, µ)
]x + B(t, µ)H(t, θ)ξ
ξ̇ = F (t, θ)ξ + G(t, θ)C(t, µ)x
is uniformly asymptotically stable for all θ ∈ Kθ2. There exists a continuous assignment σ �→ θσ such that for any
given σ ∈ Kσ, the fixed controller
ξ̇ = F (t, θσ)ξ + G(t, θσ)eu = H(t, θσ)ξ + K(t, θσ)e
solves the robust output regulation problem for (1), i.e.,
boundedness of all trajectories, and limt→∞ e(t) = 0. �
Adaptive Regulation
Introduction
Problem Statement• Regulation in thePeriodic Setup
• Certainty EquivalenceControl
• Adaptive RegulationProblem Solvability
Robust IM Design
Adaptive IM Design
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 6 / 19
Once a certainty-equivalence has
been found, look for an update law
˙̂θ = ϕ(ξ, e)
to tune the family of controllers to
the one achieving regulation.
Note that adaptation is not used
for stabilization.
ξ̇ = F (t, θ̂)ξ + G(t, θ̂)e
u = H(t, θ̂)ξ + K(t, θ̂)e
ẇ = S(t, σ)w
ẋ = A(t, µ)x + B(t, µ)u + P (t, µ)we = C(t, µ)x + Q(t, µ)w
˙̂θ = ϕ(ξ, e)
ξ
u
w
e
θ̂
Adaptive Regulation
Introduction
Problem Statement• Regulation in thePeriodic Setup
• Certainty EquivalenceControl
• Adaptive RegulationProblem Solvability
Robust IM Design
Adaptive IM Design
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 6 / 19
Once a certainty-equivalence has
been found, look for an update law
˙̂θ = ϕ(ξ, e)
to tune the family of controllers to
the one achieving regulation.
Note that adaptation is not used
for stabilization.
ξ̇ = F (t, θ̂)ξ + G(t, θ̂)e
u = H(t, θ̂)ξ + K(t, θ̂)e
ẇ = S(t, σ)w
ẋ = A(t, µ)x + B(t, µ)u + P (t, µ)we = C(t, µ)x + Q(t, µ)w
˙̂θ = ϕ(ξ, e)
ξ
u
w
e
θ̂
Issues
• Find a characterization of all certainty-equivalence regulators(canonical realization)
Adaptive Regulation
Introduction
Problem Statement• Regulation in thePeriodic Setup
• Certainty EquivalenceControl
• Adaptive RegulationProblem Solvability
Robust IM Design
Adaptive IM Design
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 6 / 19
Once a certainty-equivalence has
been found, look for an update law
˙̂θ = ϕ(ξ, e)
to tune the family of controllers to
the one achieving regulation.
Note that adaptation is not used
for stabilization.
ξ̇ = F (t, θ̂)ξ + G(t, θ̂)e
u = H(t, θ̂)ξ + K(t, θ̂)e
ẇ = S(t, σ)w
ẋ = A(t, µ)x + B(t, µ)u + P (t, µ)we = C(t, µ)x + Q(t, µ)w
˙̂θ = ϕ(ξ, e)
ξ
u
w
e
θ̂
Issues
• Find a characterization of all certainty-equivalence regulators(canonical realization)
• Find a parameterization that is amenable to adaptive control(canonical parameterization)
Periodic Internal Model Principle
Introduction
Problem Statement
Problem Solvability• Periodic InternalModel Principle
• Periodic Immersion• ExamplesRobust IM Design
Adaptive IM Design
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 7 / 19
Assume σ fixed (look at robust regulation first).
Periodic Internal Model Principle
Introduction
Problem Statement
Problem Solvability• Periodic InternalModel Principle
• Periodic Immersion• ExamplesRobust IM Design
Adaptive IM Design
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 7 / 19
Assume σ fixed (look at robust regulation first).
A periodic stabilizing controller (F, G, H, K) is a robust regulatoriff there exist T -periodic mappings Π , Ξ and R solving the DAEs
Π̇(t, µ) + Π(t, µ)S(t) = A(t, µ)Π(t, µ) + B(t, µ)R(t, µ) + P (t, µ)0 = C(t, µ)Π(t, µ) + Q(t, µ)
Ξ̇(t, µ) + Ξ(t, µ)S(t) = F (t) Ξ(t, µ)R(t, µ) = H(t) Ξ(t, µ)
for all t ∈ [0, T ) and all µ ∈ Kµ.
Periodic Internal Model Principle
Introduction
Problem Statement
Problem Solvability• Periodic InternalModel Principle
• Periodic Immersion• ExamplesRobust IM Design
Adaptive IM Design
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 7 / 19
Assume σ fixed (look at robust regulation first).
A periodic stabilizing controller (F, G, H, K) is a robust regulatoriff there exist T -periodic mappings Π , Ξ and R solving the DAEs
Π̇(t, µ) + Π(t, µ)S(t) = A(t, µ)Π(t, µ) + B(t, µ)R(t, µ) + P (t, µ)0 = C(t, µ)Π(t, µ) + Q(t, µ)
Ξ̇(t, µ) + Ξ(t, µ)S(t) = F (t) Ξ(t, µ)R(t, µ) = H(t) Ξ(t, µ)
for all t ∈ [0, T ) and all µ ∈ Kµ.
The periodic feed-forward control
ẇ = S(t)wv = R(t, µ)w
must be embedded in the controller
+
Stabilizer
Internal Model
e u
Periodic Immersion
Introduction
Problem Statement
Problem Solvability• Periodic InternalModel Principle
• Periodic Immersion• ExamplesRobust IM Design
Adaptive IM Design
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 8 / 19
The parameterized family of periodic systems (S(t), R(t, µ))is immersed into (Φ(t), Γ (t)) if there exists a periodic map Υsuch that:
Υ̇ (t, µ) + Υ (t, µ)S(t) = Φ(t)Υ (t, µ)R(t, µ) = Γ (t)Υ (t, µ)
Υ←−{
ẇ = S(t)wv = R(t, µ)w
Periodic Immersion
Introduction
Problem Statement
Problem Solvability• Periodic InternalModel Principle
• Periodic Immersion• ExamplesRobust IM Design
Adaptive IM Design
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 8 / 19
The parameterized family of periodic systems (S(t), R(t, µ))is immersed into (Φ(t), Γ (t)) if there exists a periodic map Υsuch that:
Υ̇ (t, µ) + Υ (t, µ)S(t) = Φ(t)Υ (t, µ)R(t, µ) = Γ (t)Υ (t, µ)
Υ←−{
ẇ = S(t)wv = R(t, µ)w
Different observability properties characterize the immersion map
1. regular immersion, if (Φ(·), Γ (·)) is uniformly completelyobservable;
Periodic Immersion
Introduction
Problem Statement
Problem Solvability• Periodic InternalModel Principle
• Periodic Immersion• ExamplesRobust IM Design
Adaptive IM Design
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 8 / 19
The parameterized family of periodic systems (S(t), R(t, µ))is immersed into (Φ(t), Γ (t)) if there exists a periodic map Υsuch that:
Υ̇ (t, µ) + Υ (t, µ)S(t) = Φ(t)Υ (t, µ)R(t, µ) = Γ (t)Υ (t, µ)
Υ←−{
ẇ = S(t)wv = R(t, µ)w
Different observability properties characterize the immersion map
1. regular immersion, if (Φ(·), Γ (·)) is uniformly completelyobservable;
2. strong immersion, if (Φ(·), Γ (·)) is uniformly observable;
Periodic Immersion
Introduction
Problem Statement
Problem Solvability• Periodic InternalModel Principle
• Periodic Immersion• ExamplesRobust IM Design
Adaptive IM Design
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 8 / 19
The parameterized family of periodic systems (S(t), R(t, µ))is immersed into (Φ(t), Γ (t)) if there exists a periodic map Υsuch that:
Υ̇ (t, µ) + Υ (t, µ)S(t) = Φ(t)Υ (t, µ)R(t, µ) = Γ (t)Υ (t, µ)
Υ←−{
ẇ = S(t)wv = R(t, µ)w
Different observability properties characterize the immersion map
1. regular immersion, if (Φ(·), Γ (·)) is uniformly completelyobservable;
2. strong immersion, if (Φ(·), Γ (·)) is uniformly observable;3. weak immersion, if (Φ(·), Γ (·)) is detectable and
not completely observable.
Periodic Immersion
Introduction
Problem Statement
Problem Solvability• Periodic InternalModel Principle
• Periodic Immersion• ExamplesRobust IM Design
Adaptive IM Design
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 8 / 19
The parameterized family of periodic systems (S(t), R(t, µ))is immersed into (Φ(t), Γ (t)) if there exists a periodic map Υsuch that:
Υ̇ (t, µ) + Υ (t, µ)S(t) = Φ(t)Υ (t, µ)R(t, µ) = Γ (t)Υ (t, µ)
Υ←−{
ẇ = S(t)wv = R(t, µ)w
Different observability properties characterize the immersion map
1. regular immersion, if (Φ(·), Γ (·)) is uniformly completelyobservable;
2. strong immersion, if (Φ(·), Γ (·)) is uniformly observable;3. weak immersion, if (Φ(·), Γ (·)) is detectable and
not completely observable.
1. and 2. are not equivalent (2.⇒1.) even for periodic systems2.⇒ existence of observer and observability canonical forms3. is useful only for adaptive regulation
Examples
Introduction
Problem Statement
Problem Solvability• Periodic InternalModel Principle
• Periodic Immersion• ExamplesRobust IM Design
Adaptive IM Design
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 9 / 19
The periodic exosystem
S(t) =(
0 sin(t)− sin(t) 0
), R(µ) =
(µ1 µ2
), ‖µ‖2 = 1
• Is UCO, but not UO, since detO = sin(t)• Is immersed into a 3-dim UO system in observability form
Φ(t) =
0 1 00 0 1−3 sin(t) cos(t) −1− sin2(t) 0
, Γ = (1 0 0)
Examples
Introduction
Problem Statement
Problem Solvability• Periodic InternalModel Principle
• Periodic Immersion• ExamplesRobust IM Design
Adaptive IM Design
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 9 / 19
The periodic exosystem
S(t) =(
0 sin(t)− sin(t) 0
), R(µ) =
(µ1 µ2
), ‖µ‖2 = 1
• Is UCO, but not UO, since detO = sin(t)• Is immersed into a 3-dim UO system in observability form
Φ(t) =
0 1 00 0 1−3 sin(t) cos(t) −1− sin2(t) 0
, Γ = (1 0 0)
The same system, with R(t, µ) =(µ1 + µ2 cos(t) 0
)• Is UCO, but not UO• Admits an 8-dim regular immersion, but not a strong immersion
Examples
Introduction
Problem Statement
Problem Solvability• Periodic InternalModel Principle
• Periodic Immersion• ExamplesRobust IM Design
Adaptive IM Design
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 9 / 19
The periodic exosystem
S(t) =(
0 sin(t)− sin(t) 0
), R(µ) =
(µ1 µ2
), ‖µ‖2 = 1
• Is UCO, but not UO, since detO = sin(t)• Is immersed into a 3-dim UO system in observability form
Φ(t) =
0 1 00 0 1−3 sin(t) cos(t) −1− sin2(t) 0
, Γ = (1 0 0)
The same system, with R(t, µ) =(µ1 + µ2 cos(t) 0
)• Is UCO, but not UO• Admits an 8-dim regular immersion, but not a strong immersion
Assuming that a regular immersion exists, how does one constructand internal model?
Canonical Realization of the Internal Model
Introduction
Problem Statement
Problem Solvability
Robust IM Design
• Canonical Realizationof the Internal Model• Structure of theRobust Regulator
Adaptive IM Design
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 10 / 19
The regular internal-model pair (Φ(·), Γ (·)) is said to admita canonical realization if there exist a periodic map M(·) anda periodic system (Fim(·), Gim(·), Him(·)) such that:1. Fim(t) ∈ Rm×m has all characteristic multipliers in |λ| < 12. M(t) has constant rank, and satisfies for all t ∈ [0, T )
Ṁ(t) + M(t)Φ(t) = (Fim(t) + Gim(t)Him(t))M(t)Γ (t) = Him(t)M(t)
Canonical Realization of the Internal Model
Introduction
Problem Statement
Problem Solvability
Robust IM Design
• Canonical Realizationof the Internal Model• Structure of theRobust Regulator
Adaptive IM Design
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 10 / 19
The regular internal-model pair (Φ(·), Γ (·)) is said to admita canonical realization if there exist a periodic map M(·) anda periodic system (Fim(·), Gim(·), Him(·)) such that:1. Fim(t) ∈ Rm×m has all characteristic multipliers in |λ| < 12. M(t) has constant rank, and satisfies for all t ∈ [0, T )
Ṁ(t) + M(t)Φ(t) = (Fim(t) + Gim(t)Him(t))M(t)Γ (t) = Him(t)M(t)
Internal model unit
ξ̇ = Fim(t)ξ + Gim(t)uu = Him(t)ξ + ust ←− stabilizer
Canonical Realization of the Internal Model
Introduction
Problem Statement
Problem Solvability
Robust IM Design
• Canonical Realizationof the Internal Model• Structure of theRobust Regulator
Adaptive IM Design
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 10 / 19
The regular internal-model pair (Φ(·), Γ (·)) is said to admita canonical realization if there exist a periodic map M(·) anda periodic system (Fim(·), Gim(·), Him(·)) such that:1. Fim(t) ∈ Rm×m has all characteristic multipliers in |λ| < 12. M(t) has constant rank, and satisfies for all t ∈ [0, T )
Ṁ(t) + M(t)Φ(t) = (Fim(t) + Gim(t)Him(t))M(t)Γ (t) = Him(t)M(t)
Internal model unit
ξ̇ = Fim(t)ξ + Gim(t)uu = Him(t)ξ + ust ←− stabilizer
In a nutshell, the canonical realization yields for the IMU:
“zeros” in |λ| = 1 (characteristic multipliers of Fim + GimHim)“poles” in |λ| < 1 (characteristic multipliers of Fim).
Structure of the Robust Regulator
Introduction
Problem Statement
Problem Solvability
Robust IM Design
• Canonical Realizationof the Internal Model• Structure of theRobust Regulator
Adaptive IM Design
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 11 / 19
ζ̇ = Fst(t)ζ + Gst(t)e
ust = Hst(t)ζ + Kst(t)e
ξ̇ = Fim(t)ξ + Gim(t)u
uim = Him(t)ξ
e u+
Regular IM pair (Φ(·), Γ (·))
Fim(t) = −αI − Φ′(t), α > 0Gim(t) = Γ ′(t)
Him(t) = Γ (t)M−1(t)
Strong IM pair (Φo(·), Γo)
Fim(t) = Fim Hurwitz
Gim(t) = Q−1[φ1(t) + b
]Him(t) = Him observable pair
• For regular IM pairs, it is a passivity-based design◦ closed-loop eigenvalues not free◦ requires the computation of M−1(t)
• For strong IM pairs, eigenvalues are assigned via output injection
Canonical Parameterization of the Internal Model
Introduction
Problem Statement
Problem Solvability
Robust IM Design
Adaptive IM Design
• CanonicalParameterization of theInternal Model• Structure of theAdaptive Regulator
• How to Construct aCanonicalParameterization?• Non-minimal InternalModel Unit• Usefulness of WeakImmersions
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 12 / 19
The family of internal-model pairs(Φ(·, σ), Γ (·, σ)) is said to admit
a canonical parametrization in feedback form if there exist a family of
periodic maps M(·, θ) and a family of periodic systems(Fim(·), Gim(·), Him(·, θ)) such that:1. Fim(t) has all characteristic multipliers in |λ| < 12. Him(t, θ) is affine in θ3. there exists a continuous map σ �→ θσ such that the matrix
M(t, θσ) has constant rank ∀ t ∈ [0, T ) and ∀σ ∈ Kσ, and
Ṁ(t, θσ) + M(t, θσ)Φ(t, σ) = (Fim(t) + Gim(t)Him(t, θσ))M(t, θσ)Γ (t, σ) = Him(t, θσ)M(t, θσ) .
Canonical Parameterization of the Internal Model
Introduction
Problem Statement
Problem Solvability
Robust IM Design
Adaptive IM Design
• CanonicalParameterization of theInternal Model• Structure of theAdaptive Regulator
• How to Construct aCanonicalParameterization?• Non-minimal InternalModel Unit• Usefulness of WeakImmersions
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 12 / 19
The family of internal-model pairs(Φ(·, σ), Γ (·, σ)) is said to admit
a canonical parametrization in feedback form if there exist a family of
periodic maps M(·, θ) and a family of periodic systems(Fim(·), Gim(·), Him(·, θ)) such that:1. Fim(t) has all characteristic multipliers in |λ| < 12. Him(t, θ) is affine in θ3. there exists a continuous map σ �→ θσ such that the matrix
M(t, θσ) has constant rank ∀ t ∈ [0, T ) and ∀σ ∈ Kσ, and
Ṁ(t, θσ) + M(t, θσ)Φ(t, σ) = (Fim(t) + Gim(t)Him(t, θσ))M(t, θσ)Γ (t, σ) = Him(t, θσ)M(t, θσ) .
Note that:
• Fim(t) and Gim(t) are both independent of θ• Him(t, θ) = Him, 0(t) + θT Him, 1(t)
Structure of the Adaptive Regulator
Introduction
Problem Statement
Problem Solvability
Robust IM Design
Adaptive IM Design
• CanonicalParameterization of theInternal Model• Structure of theAdaptive Regulator
• How to Construct aCanonicalParameterization?• Non-minimal InternalModel Unit• Usefulness of WeakImmersions
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 13 / 19
Adaptive internal model unit (for relative-degree 1 minimum-phase
plants)ξ̇ = Fim(t)ξ + Gim(t)u˙̂θ = −γ Him,1(t)ξ e, γ > 0u = Him(t, θ̂)ξ + ust ←− stabilizer
ust = −ke
ξ̇ = Fim(t)ξ + Gim(t)u
uim = Him(t, θ̂)ξ
e u+
˙̂θ = −γ Him,1(t)ξ e
ξ
θ̂
How to Construct a Canonical Parameterization?
Introduction
Problem Statement
Problem Solvability
Robust IM Design
Adaptive IM Design
• CanonicalParameterization of theInternal Model• Structure of theAdaptive Regulator
• How to Construct aCanonicalParameterization?• Non-minimal InternalModel Unit• Usefulness of WeakImmersions
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 14 / 19
• Start from a strong immersion⇒ (Φo(·, σ), Γo) in observer form• Find a linear parameterization of the first column of Φo(·, σ)
Φo(t, σ) = Φb −Θβ(t)Γo
Φb =
0 1 · · · 0...
.... . .
...0 0 · · · 10 0 · · · 0
, Θ =
θTq−1...
θT1θT0
.
• Choose L0 such that F = Φb − L0Γo is Hurwitz• Let G(t, θ) = L0 −Θβ(t), H = Γo◦ The triplet
(F, G(·, θ), H) is not yet in feedback form.
◦ We need to “shift” the parameter θ from G to H◦ Impossible to do with a change of coordinates!
How to Construct a Canonical Parameterization?
Introduction
Problem Statement
Problem Solvability
Robust IM Design
Adaptive IM Design
• CanonicalParameterization of theInternal Model• Structure of theAdaptive Regulator
• How to Construct aCanonicalParameterization?• Non-minimal InternalModel Unit• Usefulness of WeakImmersions
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 14 / 19
• Start from a strong immersion⇒ (Φo(·, σ), Γo) in observer form• Find a linear parameterization of the first column of Φo(·, σ)
Φo(t, σ) = Φb −Θβ(t)Γo
Φb =
0 1 · · · 0...
.... . .
...0 0 · · · 10 0 · · · 0
, Θ =
θTq−1...
θT1θT0
.
• Choose L0 such that F = Φb − L0Γo is Hurwitz• Let G(t, θ) = L0 −Θβ(t), H = Γo◦ The triplet
(F, G(·, θ), H) is not yet in feedback form.
◦ We need to “shift” the parameter θ from G to H◦ Impossible to do with a change of coordinates!
How to Construct a Canonical Parameterization?
Introduction
Problem Statement
Problem Solvability
Robust IM Design
Adaptive IM Design
• CanonicalParameterization of theInternal Model• Structure of theAdaptive Regulator
• How to Construct aCanonicalParameterization?• Non-minimal InternalModel Unit• Usefulness of WeakImmersions
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 14 / 19
• Start from a strong immersion⇒ (Φo(·, σ), Γo) in observer form• Find a linear parameterization of the first column of Φo(·, σ)
Φo(t, σ) = Φb −Θβ(t)Γo
Φb =
0 1 · · · 0...
.... . .
...0 0 · · · 10 0 · · · 0
, Θ =
θTq−1...
θT1θT0
.
• Choose L0 such that F = Φb − L0Γo is Hurwitz• Let G(t, θ) = L0 −Θβ(t), H = Γo◦ The triplet
(F, G(·, θ), H) is not yet in feedback form.
◦ We need to “shift” the parameter θ from G to H◦ Impossible to do with a change of coordinates!
How to Construct a Canonical Parameterization?
Introduction
Problem Statement
Problem Solvability
Robust IM Design
Adaptive IM Design
• CanonicalParameterization of theInternal Model• Structure of theAdaptive Regulator
• How to Construct aCanonicalParameterization?• Non-minimal InternalModel Unit• Usefulness of WeakImmersions
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 14 / 19
• Start from a strong immersion⇒ (Φo(·, σ), Γo) in observer form• Find a linear parameterization of the first column of Φo(·, σ)
Φo(t, σ) = Φb −Θβ(t)Γo
Φb =
0 1 · · · 0...
.... . .
...0 0 · · · 10 0 · · · 0
, Θ =
θTq−1...
θT1θT0
.
• Choose L0 such that F = Φb − L0Γo is Hurwitz• Let G(t, θ) = L0 −Θβ(t), H = Γo◦ The triplet
(F, G(·, θ), H) is not yet in feedback form.
◦ We need to “shift” the parameter θ from G to H◦ Impossible to do with a change of coordinates!
How to Construct a Canonical Parameterization?
Introduction
Problem Statement
Problem Solvability
Robust IM Design
Adaptive IM Design
• CanonicalParameterization of theInternal Model• Structure of theAdaptive Regulator
• How to Construct aCanonicalParameterization?• Non-minimal InternalModel Unit• Usefulness of WeakImmersions
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 14 / 19
• Start from a strong immersion⇒ (Φo(·, σ), Γo) in observer form• Find a linear parameterization of the first column of Φo(·, σ)
Φo(t, σ) = Φb −Θβ(t)Γo
Φb =
0 1 · · · 0...
.... . .
...0 0 · · · 10 0 · · · 0
, Θ =
θTq−1...
θT1θT0
.
• Choose L0 such that F = Φb − L0Γo is Hurwitz• Let G(t, θ) = L0 −Θβ(t), H = Γo◦ The triplet
(F, G(·, θ), H) is not yet in feedback form.
◦ We need to “shift” the parameter θ from G to H◦ Impossible to do with a change of coordinates!
Non-minimal Internal Model Unit
Introduction
Problem Statement
Problem Solvability
Robust IM Design
Adaptive IM Design
• CanonicalParameterization of theInternal Model• Structure of theAdaptive Regulator
• How to Construct aCanonicalParameterization?• Non-minimal InternalModel Unit• Usefulness of WeakImmersions
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 15 / 19
The key is to look for a non-minimal periodic realization of the I/O
response of the IM unit
h(t, τ, θ) = HeF (t−τ)G(τ, θ)
Non-minimal Internal Model Unit
Introduction
Problem Statement
Problem Solvability
Robust IM Design
Adaptive IM Design
• CanonicalParameterization of theInternal Model• Structure of theAdaptive Regulator
• How to Construct aCanonicalParameterization?• Non-minimal InternalModel Unit• Usefulness of WeakImmersions
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 15 / 19
The key is to look for a non-minimal periodic realization of the I/O
response of the IM unit
h(t, τ, θ) = HeF (t−τ)G(τ, θ)= HeF (t−τ)L0︸ ︷︷ ︸
LTI
−HeF (t−τ)Θβ(τ)︸ ︷︷ ︸periodic
Non-minimal Internal Model Unit
Introduction
Problem Statement
Problem Solvability
Robust IM Design
Adaptive IM Design
• CanonicalParameterization of theInternal Model• Structure of theAdaptive Regulator
• How to Construct aCanonicalParameterization?• Non-minimal InternalModel Unit• Usefulness of WeakImmersions
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 15 / 19
The key is to look for a non-minimal periodic realization of the I/O
response of the IM unit
h(t, τ, θ) = HeF (t−τ)G(τ, θ)= HeF (t−τ)L0︸ ︷︷ ︸
LTI
−HeF (t−τ)Θβ(τ)︸ ︷︷ ︸periodic
F0 =
−lq−1 · · · −l1 −l0
1 · · · 0 0...
. . ....
...0 · · · 1 0
G0 =
10...0
H0 = L′0
F1 =
−lq−1Iρ · · · −l1Iρ −l0Iρ
Iρ · · · 0 0...
. . ....
...0 · · · Iρ 0
G1(t) =
β(t)0...0
H1(θ) = θ′
Usefulness of Weak Immersions
Introduction
Problem Statement
Problem Solvability
Robust IM Design
Adaptive IM Design
• CanonicalParameterization of theInternal Model• Structure of theAdaptive Regulator
• How to Construct aCanonicalParameterization?• Non-minimal InternalModel Unit• Usefulness of WeakImmersions
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 16 / 19
The triplet
Fim =(
F0 00 F1
), Gim(t) =
(G0
G1(t)
)
Him(θ) =(H0 −H1(θ)
)is a canonical parameterization in feedback form of the original
internal-model pair (Φo(t), Γo).
Usefulness of Weak Immersions
Introduction
Problem Statement
Problem Solvability
Robust IM Design
Adaptive IM Design
• CanonicalParameterization of theInternal Model• Structure of theAdaptive Regulator
• How to Construct aCanonicalParameterization?• Non-minimal InternalModel Unit• Usefulness of WeakImmersions
Illustrative Example
Analysis and Design of Nonlinear Control Systems – London, 2008 16 / 19
The triplet
Fim =(
F0 00 F1
), Gim(t) =
(G0
G1(t)
)
Him(θ) =(H0 −H1(θ)
)is a canonical parameterization in feedback form of the original
internal-model pair (Φo(t), Γo).
The canonical parameterization has been obtained via a weak
immersion of the original exosystem.
(S(t, σ), R(t, µ)
) strong−→ (Φo(t, θ), Γo) weak−→ (Fim, Gim(t), Him(θ))
Synchronization of a Pendulum with a Mathieu System
Introduction
Problem Statement
Problem Solvability
Robust IM Design
Adaptive IM Design
Illustrative Example
• Synchronization of aPendulum with aMathieu System
• Simulation Results• Conclusions
Analysis and Design of Nonlinear Control Systems – London, 2008 17 / 19
Controlled pendulum:
δ̈ = −aδ + bu
Vertically-oscillating pendulum:
δ̈1 = (−a− 2d cos(2t))δ1 m
δ1 l
d12 cos(2t)
δl
m
uO
Exosystem
S(t, σ) =(
0 1−a− 2d cos(2t) 0
), Rσ(t, µ) =
(2bd cos(2t) 0
)
strongly immersed in a 4-dim IM pair (Φo(t, θ), Γo) in observercanonical form with new parameter vector θ = (a, d, a2, ad)′.
Simulation Results
Introduction
Problem Statement
Problem Solvability
Robust IM Design
Adaptive IM Design
Illustrative Example
• Synchronization of aPendulum with aMathieu System
• Simulation Results• Conclusions
Analysis and Design of Nonlinear Control Systems – London, 2008 18 / 19
0 10 20 30 40 50 60 70−2
−1.5
−1
−0.5
0
0.5
Time [s]
e(t)
θ̃1(t)
θ̃2(t)Parameter change
0 5 10 15−2
−1.5
−1
−0.5
0
0.5
1
1.5
Time [s]
uim(t) = Him(θ̂(t))ξ(t)Rσ(t, µ)w(t)
Conclusions
Introduction
Problem Statement
Problem Solvability
Robust IM Design
Adaptive IM Design
Illustrative Example
• Synchronization of aPendulum with aMathieu System
• Simulation Results• Conclusions
Analysis and Design of Nonlinear Control Systems – London, 2008 19 / 19
• We have proposed a classification of the property of systemimmersion for periodic systems to underly the connections
between various non-equivalent definitions of systems
observability and the existence of robust internal model-based
controllers.
Conclusions
Introduction
Problem Statement
Problem Solvability
Robust IM Design
Adaptive IM Design
Illustrative Example
• Synchronization of aPendulum with aMathieu System
• Simulation Results• Conclusions
Analysis and Design of Nonlinear Control Systems – London, 2008 19 / 19
• We have proposed a classification of the property of systemimmersion for periodic systems to underly the connections
between various non-equivalent definitions of systems
observability and the existence of robust internal model-based
controllers.• Weaker detectability properties are related to the possibility of
obtaining canonical realizations of periodic internal models to be
used in certainty-equivalence design to deal with parameter
uncertainty on the exosystem model.
Conclusions
Introduction
Problem Statement
Problem Solvability
Robust IM Design
Adaptive IM Design
Illustrative Example
• Synchronization of aPendulum with aMathieu System
• Simulation Results• Conclusions
Analysis and Design of Nonlinear Control Systems – London, 2008 19 / 19
• We have proposed a classification of the property of systemimmersion for periodic systems to underly the connections
between various non-equivalent definitions of systems
observability and the existence of robust internal model-based
controllers.• Weaker detectability properties are related to the possibility of
obtaining canonical realizations of periodic internal models to be
used in certainty-equivalence design to deal with parameter
uncertainty on the exosystem model.
• Open problem: It is not clear whether coordinate-free conditionscan be found to check a priori the existence of a regular
immersion.