Discontinuous Observers with strong convergence properties ...
Transcript of Discontinuous Observers with strong convergence properties ...
Discontinuous Observers with strong convergenceproperties and some applications
Jaime A. [email protected]
Instituto de IngenierıaUniversidad Nacional Autonoma de Mexico
Mexico City, Mexico
Seminars in Systems and Control, CESAME, UCL, Belgium
24 April 2012
24 April 2012, Jaime A. Moreno Discontinuous Observers 1 / 59
Overview
1 Introduction
2 Observers a la Second Order Sliding Modes (SOSM)Super-Twisting ObserverGeneralized Super-Twisting Observers
3 Lyapunov Approach for Second-Order Sliding ModesGSTA without perturbations: ALEGSTA with perturbations: ARI
4 Optimality of the ST with noise
5 A Recursive Finite-Time Convergent Parameter Estimation AlgorithmThe Classical AlgorithmThe Proposed Algorithm
6 Conclusions
24 April 2012, Jaime A. Moreno Discontinuous Observers 1 / 59
Overview
1 Introduction
2 Observers a la Second Order Sliding Modes (SOSM)Super-Twisting ObserverGeneralized Super-Twisting Observers
3 Lyapunov Approach for Second-Order Sliding ModesGSTA without perturbations: ALEGSTA with perturbations: ARI
4 Optimality of the ST with noise
5 A Recursive Finite-Time Convergent Parameter Estimation AlgorithmThe Classical AlgorithmThe Proposed Algorithm
6 Conclusions
24 April 2012, Jaime A. Moreno Discontinuous Observers 1 / 59
Basic Observation Problem
Variations of the observation problem: with unknown inputs, practicalobservers, robust observers, stochastic framework to deal with noises, ....
24 April 2012, Jaime A. Moreno Discontinuous Observers 2 / 59
An important Property: Observability
Consider a nonlinear system without inputs, x ∈ Rn, y ∈ R
x (t) = f (x (t)) , x (t0) = x0
y (t) = h (x (t))
Differentiating the output
y (t) = h (x (t))
y (t) =d
dth (x (t)) =
∂h (x)
∂xx (t) =
∂h (x)
∂xf (x) := Lf h (x)
y (t) =∂Lf h (x)
∂xx (t) =
∂Lf h (x)
∂xf (x) := L2
f h (x)
...
y (n−1) (t) =∂Ln−2
f h (x)
∂xx (t) =
∂Ln−2f h (x)
∂xf (x) := Ln−1
f h (x)
where Lkf h (x) are Lie’s derivatives of h along f .
24 April 2012, Jaime A. Moreno Discontinuous Observers 3 / 59
Evaluating at t = 0y (0)y (0)y (0)...
y (k) (0)
=
h (x0)Lf h (x0)L2f h (x0)
...Lkf h (x0)
:= On (x0)
On (x): Observability map
Theorem
If On (x) is injective (invertible) → The NL system is observable.
24 April 2012, Jaime A. Moreno Discontinuous Observers 4 / 59
Observability Form
In the coordinates of the output and its derivatives
z = On (x) , x = O−1n (z)
the system takes the (observability) form
z1 = z2
z2 = z3...
zn = φ (z1, z2, . . . , zn)y = z1
So we can consider a system in this form as a basic structure.
24 April 2012, Jaime A. Moreno Discontinuous Observers 5 / 59
A Simple Observer and its Properties
Plant: x1 = x2 , x2 = w(t)Observer: ˙x1 = −l1 (x1 − x1) + x2 , ˙x2 = −l2 (x1 − x1)Estimation Error: e1 = x1 − x1, e2 = x2 − x2
e1 = −l1e1 + e2 , e2 = −l2e1 − w (t)
Figure: Linear Plant with an unknown input and a Linear Observer.
24 April 2012, Jaime A. Moreno Discontinuous Observers 6 / 59
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Linear Observer Linear Observer
Figure: Behavior of Plant and the Linear Observer without unknown input.
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Linear Observer Linear Observer
Figure: Behavior of Plant and the Linear Observer with unknown input.
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Linear Observer Linear Observer
Figure: Behavior of Plant and the Linear Observer without UI with large initialconditions.
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Linear Observer Linear Observer
Figure: Behavior of Plant and the Linear Observer without UI with very largeinitial conditions.
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Recapitulation.
Linear Observer for Linear Plant
If no unknown inputs/Uncertainties: it converges exponentially fast.
If there are unknown inputs/Uncertainties: no convergence. At bestbounded error.
Convergence time depends on the initial conditions of the observer
Is it possible to alleviate these drawbacks?
24 April 2012, Jaime A. Moreno Discontinuous Observers 11 / 59
Recapitulation.
Linear Observer for Linear Plant
If no unknown inputs/Uncertainties: it converges exponentially fast.
If there are unknown inputs/Uncertainties: no convergence. At bestbounded error.
Convergence time depends on the initial conditions of the observer
Is it possible to alleviate these drawbacks?
24 April 2012, Jaime A. Moreno Discontinuous Observers 11 / 59
Recapitulation.
Linear Observer for Linear Plant
If no unknown inputs/Uncertainties: it converges exponentially fast.
If there are unknown inputs/Uncertainties: no convergence. At bestbounded error.
Convergence time depends on the initial conditions of the observer
Is it possible to alleviate these drawbacks?
24 April 2012, Jaime A. Moreno Discontinuous Observers 11 / 59
Recapitulation.
Linear Observer for Linear Plant
If no unknown inputs/Uncertainties: it converges exponentially fast.
If there are unknown inputs/Uncertainties: no convergence. At bestbounded error.
Convergence time depends on the initial conditions of the observer
Is it possible to alleviate these drawbacks?
24 April 2012, Jaime A. Moreno Discontinuous Observers 11 / 59
Recalling Sliding Mode Control
Consider a plant
σ = α + u, σ(0) = 1
where α ∈ (−1, 1) is a perturbation.Continuous (linear) Control
σ = α− kσ, k > 0, σ(0) = 1
Comments:
RHS of DE continuous (linear).
If α = 0 exponential (asymptotic) convergence to σ = 0.
If α 6= 0 practical convergence.
24 April 2012, Jaime A. Moreno Discontinuous Observers 12 / 59
Discontinuous Control
σ = α− sign(σ), σ(0) = 1
with α ∈ (−1, 1).
σ > 0⇒ σ =< 0
σ < 0⇒ σ => 0
y σ(t) ≡ 0, ∀t ≥ T .
Comments:
¿0 = α− sign(0)?
RHS of DE isdiscontinuous.
After arriving at σ = 0,sliding on σ ≡ 0.
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Finite-Time convergence.
Differential Inclusion.
σ ∈ [−α, α]− sign(σ)
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Why sliding-mode (discontinuous) control?
Precise.
More than robust (insensitive) against a class of perturbations.
When to use sliding-mode control?
Unaccounted dynamics.
Parametric uncertainty.
Severe external perturbations.
Some applications
Robotics.
Aerospace vehicles.
Electric systems (motors, generators, etc).
Automobiles.
Biomedical.
etc.
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Sliding Mode Observer (SMO)
Figure: Linear Plant with an unknown input and a SM Observer.24 April 2012, Jaime A. Moreno Discontinuous Observers 15 / 59
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Linear Observer
Nonlinear ObserverLinear Observer
Nonlinear Observer
Figure: Behavior of Plant and the SM Observer without unknown input.
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Linear Observer
Nonlinear ObserverLinear Observer
Nonlinear Observer
Figure: Behavior of Plant and the SM Observer with unknown input.
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Recapitulation.
Sliding Mode Observer for Linear Plant
If no unknown inputs/Uncertainties: e1 converges in finite time, ande2 converges exponentially fast.
If there are unknown inputs/Uncertainties: no convergence. At bestbounded error. Only e1 converges in finite time!
Convergence time depends on the initial conditions of the observer
It is not the solution we expected! None of the objectives has beenachieved!
24 April 2012, Jaime A. Moreno Discontinuous Observers 18 / 59
Recapitulation.
Sliding Mode Observer for Linear Plant
If no unknown inputs/Uncertainties: e1 converges in finite time, ande2 converges exponentially fast.
If there are unknown inputs/Uncertainties: no convergence. At bestbounded error. Only e1 converges in finite time!
Convergence time depends on the initial conditions of the observer
It is not the solution we expected! None of the objectives has beenachieved!
24 April 2012, Jaime A. Moreno Discontinuous Observers 18 / 59
Recapitulation.
Sliding Mode Observer for Linear Plant
If no unknown inputs/Uncertainties: e1 converges in finite time, ande2 converges exponentially fast.
If there are unknown inputs/Uncertainties: no convergence. At bestbounded error. Only e1 converges in finite time!
Convergence time depends on the initial conditions of the observer
It is not the solution we expected! None of the objectives has beenachieved!
24 April 2012, Jaime A. Moreno Discontinuous Observers 18 / 59
Recapitulation.
Sliding Mode Observer for Linear Plant
If no unknown inputs/Uncertainties: e1 converges in finite time, ande2 converges exponentially fast.
If there are unknown inputs/Uncertainties: no convergence. At bestbounded error. Only e1 converges in finite time!
Convergence time depends on the initial conditions of the observer
It is not the solution we expected! None of the objectives has beenachieved!
24 April 2012, Jaime A. Moreno Discontinuous Observers 18 / 59
Overview
1 Introduction
2 Observers a la Second Order Sliding Modes (SOSM)Super-Twisting ObserverGeneralized Super-Twisting Observers
3 Lyapunov Approach for Second-Order Sliding ModesGSTA without perturbations: ALEGSTA with perturbations: ARI
4 Optimality of the ST with noise
5 A Recursive Finite-Time Convergent Parameter Estimation AlgorithmThe Classical AlgorithmThe Proposed Algorithm
6 Conclusions
24 April 2012, Jaime A. Moreno Discontinuous Observers 18 / 59
Overview
1 Introduction
2 Observers a la Second Order Sliding Modes (SOSM)Super-Twisting ObserverGeneralized Super-Twisting Observers
3 Lyapunov Approach for Second-Order Sliding ModesGSTA without perturbations: ALEGSTA with perturbations: ARI
4 Optimality of the ST with noise
5 A Recursive Finite-Time Convergent Parameter Estimation AlgorithmThe Classical AlgorithmThe Proposed Algorithm
6 Conclusions
24 April 2012, Jaime A. Moreno Discontinuous Observers 18 / 59
Super-Twisting Algorithm (STA)
Plant:x1 = x2 ,x2 = w(t)
Observer:˙x1 = −l1 |e1|
12 sign (e1) + x2 ,
˙x2 = −l2 sign (e1)
Estimation Error: e1 = x1 − x1, e2 = x2 − x2
e1 = −l1 |e1|12 sign (e1) + e2
e2 = −l2 sign (e1)− w (t) ,
Solutions in the sense of Filippov.
24 April 2012, Jaime A. Moreno Discontinuous Observers 19 / 59
Figure: Linear Plant with an unknown input and a SOSM Observer.
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Linear Observer
Nonlinear Observer
Linear Observer
Nonlinear Observer
Figure: Behavior of Plant and the Non Linear Observer without unknown input.
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Linear Observer
Nonlinear Observer
Figure: Behavior of Plant and the Non Linear Observer with unknown input.
24 April 2012, Jaime A. Moreno Discontinuous Observers 22 / 59
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Linear Observer
Nonlinear Observer
Linear Observer
Nonlinear Observer
Figure: Behavior of Plant and the Non Linear Observer without UI with largeinitial conditions.
24 April 2012, Jaime A. Moreno Discontinuous Observers 23 / 59
Recapitulation.
Super-Twisting Observer for Linear Plant
If no unknown inputs/Uncertainties: e1 and e2 converge in finite-time!
If there are unknown inputs/Uncertainties: e1 and e2 converge infinite-time! Observer is insensitive to perturbation/uncertainty!
Convergence time depends on the initial conditions of the observer.This objective is not achieved!
24 April 2012, Jaime A. Moreno Discontinuous Observers 24 / 59
Recapitulation.
Super-Twisting Observer for Linear Plant
If no unknown inputs/Uncertainties: e1 and e2 converge in finite-time!
If there are unknown inputs/Uncertainties: e1 and e2 converge infinite-time! Observer is insensitive to perturbation/uncertainty!
Convergence time depends on the initial conditions of the observer.This objective is not achieved!
24 April 2012, Jaime A. Moreno Discontinuous Observers 24 / 59
Recapitulation.
Super-Twisting Observer for Linear Plant
If no unknown inputs/Uncertainties: e1 and e2 converge in finite-time!
If there are unknown inputs/Uncertainties: e1 and e2 converge infinite-time! Observer is insensitive to perturbation/uncertainty!
Convergence time depends on the initial conditions of the observer.This objective is not achieved!
24 April 2012, Jaime A. Moreno Discontinuous Observers 24 / 59
Overview
1 Introduction
2 Observers a la Second Order Sliding Modes (SOSM)Super-Twisting ObserverGeneralized Super-Twisting Observers
3 Lyapunov Approach for Second-Order Sliding ModesGSTA without perturbations: ALEGSTA with perturbations: ARI
4 Optimality of the ST with noise
5 A Recursive Finite-Time Convergent Parameter Estimation AlgorithmThe Classical AlgorithmThe Proposed Algorithm
6 Conclusions
24 April 2012, Jaime A. Moreno Discontinuous Observers 24 / 59
Generalized Super-Twisting Algorithm (GSTA)
Plant:x1 = x2 ,x2 = w(t)
Observer:˙x1 = −l1φ1 (e1) + x2 ,˙x2 = −l2φ2 (e1)
Estimation Error: e1 = x1 − x1, e2 = x2 − x2
e1 = −l1φ1 (e1) + e2
e2 = −l2φ2 (e1)− w (t) ,
Solutions in the sense of Filippov.
φ1 (e1) = µ1 |e1|12 sign (e1) + µ2 |e1|
32 sign (e1) , µ1 , µ2 ≥ 0 ,
φ2 (e1) =µ2
1
2sign (e1) + 2µ1µ2e1 +
3
2µ2
2 |e1|2 sign (e1) ,
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Figure: Linear Plant with an unknown input and a Non Linear Observer.
24 April 2012, Jaime A. Moreno Discontinuous Observers 26 / 59
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Linear Observer
Nonlinear Observer
Linear Observer
Nonlinear Observer
Figure: Behavior of Plant and the Non Linear Observer without unknown inputand large initial conditions.
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Linear Observer
Nonlinear Observer
Linear Observer
Nonlinear Observer
Figure: Behavior of Plant and the Non Linear Observer without unknown inputand very large initial conditions.
24 April 2012, Jaime A. Moreno Discontinuous Observers 28 / 59
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Linear Observer
Nonlinear Observer
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Nonlinear Observer
Figure: Behavior of Plant and the Non Linear Observer with UI with large initialconditions.
24 April 2012, Jaime A. Moreno Discontinuous Observers 29 / 59
Effect: Convergence time independent of I.C.
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norm of the initial condition ||x(0)|| (logaritmic scale)
Co
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Tim
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NSOSMO
GSTA with linear term
STO
Figure: Convergence time when the initial condition grows.
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Recapitulation.
Generalized Super-Twisting Observer for Linear Plant
If no unknown inputs/Uncertainties: e1 and e2 converge in finite-time!
If there are unknown inputs/Uncertainties: e1 and e2 converge infinite-time! Observer is insensitive to perturbation/uncertainty!
Convergence time is independent of the initial conditions of theobserver!.
All objectives were achieved!
How to proof these properties?
24 April 2012, Jaime A. Moreno Discontinuous Observers 31 / 59
Recapitulation.
Generalized Super-Twisting Observer for Linear Plant
If no unknown inputs/Uncertainties: e1 and e2 converge in finite-time!
If there are unknown inputs/Uncertainties: e1 and e2 converge infinite-time! Observer is insensitive to perturbation/uncertainty!
Convergence time is independent of the initial conditions of theobserver!.
All objectives were achieved!
How to proof these properties?
24 April 2012, Jaime A. Moreno Discontinuous Observers 31 / 59
Recapitulation.
Generalized Super-Twisting Observer for Linear Plant
If no unknown inputs/Uncertainties: e1 and e2 converge in finite-time!
If there are unknown inputs/Uncertainties: e1 and e2 converge infinite-time! Observer is insensitive to perturbation/uncertainty!
Convergence time is independent of the initial conditions of theobserver!.
All objectives were achieved!
How to proof these properties?
24 April 2012, Jaime A. Moreno Discontinuous Observers 31 / 59
Recapitulation.
Generalized Super-Twisting Observer for Linear Plant
If no unknown inputs/Uncertainties: e1 and e2 converge in finite-time!
If there are unknown inputs/Uncertainties: e1 and e2 converge infinite-time! Observer is insensitive to perturbation/uncertainty!
Convergence time is independent of the initial conditions of theobserver!.
All objectives were achieved!
How to proof these properties?
24 April 2012, Jaime A. Moreno Discontinuous Observers 31 / 59
Recapitulation.
Generalized Super-Twisting Observer for Linear Plant
If no unknown inputs/Uncertainties: e1 and e2 converge in finite-time!
If there are unknown inputs/Uncertainties: e1 and e2 converge infinite-time! Observer is insensitive to perturbation/uncertainty!
Convergence time is independent of the initial conditions of theobserver!.
All objectives were achieved!
How to proof these properties?
24 April 2012, Jaime A. Moreno Discontinuous Observers 31 / 59
What have we achieved?
An algorithm
Robust: it converges despite of unknown inputs/uncertainties
Exact: it converges in finite-time
The convergence time can be preassigned for any arbitrary initialcondition.
But there is no free lunch!
It is useful for
Observation
Estimation of perturbations/uncertainties
Control: Nonlinear PI-Control
in practice?
Some Generalizations are available but Still a lot is missing
24 April 2012, Jaime A. Moreno Discontinuous Observers 32 / 59
Overview
1 Introduction
2 Observers a la Second Order Sliding Modes (SOSM)Super-Twisting ObserverGeneralized Super-Twisting Observers
3 Lyapunov Approach for Second-Order Sliding ModesGSTA without perturbations: ALEGSTA with perturbations: ARI
4 Optimality of the ST with noise
5 A Recursive Finite-Time Convergent Parameter Estimation AlgorithmThe Classical AlgorithmThe Proposed Algorithm
6 Conclusions
24 April 2012, Jaime A. Moreno Discontinuous Observers 32 / 59
Lyapunov functions:
1 We propose a Family of strong Lyapunov functions, that areQuadratic-like
2 This family allows the estimation of convergence time,
3 It allows to study the robustness of the algorithm to different kinds ofperturbations,
4 All results are obtained in a Linear-Like framework, known fromclassical control,
5 The analysis can be obtained in the same manner for a linearalgorithm, the classical ST algorithm and a combination of bothalgorithms (GSTA), that is non homogeneous.
24 April 2012, Jaime A. Moreno Discontinuous Observers 33 / 59
Overview
1 Introduction
2 Observers a la Second Order Sliding Modes (SOSM)Super-Twisting ObserverGeneralized Super-Twisting Observers
3 Lyapunov Approach for Second-Order Sliding ModesGSTA without perturbations: ALEGSTA with perturbations: ARI
4 Optimality of the ST with noise
5 A Recursive Finite-Time Convergent Parameter Estimation AlgorithmThe Classical AlgorithmThe Proposed Algorithm
6 Conclusions
24 April 2012, Jaime A. Moreno Discontinuous Observers 33 / 59
Generalized STA
x1 = −k1φ1 (x1) + x2
x2 = −k2φ2 (x1) ,(1)
Solutions in the sense of Filippov.
φ1 (e1) = µ1 |e1|12 sign (e1) + µ2 |e1|q sign (e1) , µ1 , µ2 ≥ 0 , q ≥ 1 ,
φ2 (e1) =µ2
1
2sign (e1) +
(q +
1
2
)µ1µ2 |e1|q−
12 sign (e1) +
+ qµ22 |e1|2q−1 sign (e1) ,
Standard STA: µ1 = 1, µ2 = 0
Linear Algorithm: µ1 = 0, µ2 > 0, q = 1.
GSTA: µ1 = 1, µ2 > 0, q > 1.
24 April 2012, Jaime A. Moreno Discontinuous Observers 34 / 59
Quadratic-like Lyapunov Functions
System can be written as:
ζ = φ′1 (x1)Aζ , ζ =
[φ1 (x1)
x2
], A =
[−k1 1−k2 0
].
Family of strong Lyapunov Functions:
V (x) = ζTPζ , P = PT > 0 .
Time derivative of Lyapunov Function:
V (x) = φ′1 (x1) ζT(
ATP + PA)
ζ = −φ′1 (x1) ζTQζ
Algebraic Lyapunov Equation (ALE):
ATP + PA = −Q
24 April 2012, Jaime A. Moreno Discontinuous Observers 35 / 59
Lyapunov Function
Proposition
If A Hurwitz then x = 0 Finite-Time stable (if µ1 = 1) and for everyQ = QT > 0, V (x) = ζTPζ is a global, strong Lyapunov function,with P = PT > 0 solution of the ALE, and
V ≤ −γ1 (Q, µ1)V12 (x)− γ2 (Q, µ2)V (x) ,
where
γ1 (Q, µ1) , µ1λmin {Q} λ
12min{P}
2λmax {P}, γ2 (Q, µ2) , µ2
λmin {Q}λmax {P}
If A is not Hurwitz then x = 0 unstable.
24 April 2012, Jaime A. Moreno Discontinuous Observers 37 / 59
Convergence Time
Proposition
If k1 > 0 , k2 > 0, and µ2 ≥ 0 a trajectory of the GSTA starting atx0 ∈ R2 converges to the origin in finite time if µ1 = 1, and it reachesthat point at most after a time
T =
2
γ1(Q,µ1)V
12 (x0) if µ2 = 0
2γ2(Q,µ2)
ln(
γ2(Q,µ2)γ1(Q,µ1)
V12 (x0) + 1
)if µ2 > 0
,
When µ1 = 0 the convergence is exponential.
For Design: T depends on the gains!
24 April 2012, Jaime A. Moreno Discontinuous Observers 38 / 59
Overview
1 Introduction
2 Observers a la Second Order Sliding Modes (SOSM)Super-Twisting ObserverGeneralized Super-Twisting Observers
3 Lyapunov Approach for Second-Order Sliding ModesGSTA without perturbations: ALEGSTA with perturbations: ARI
4 Optimality of the ST with noise
5 A Recursive Finite-Time Convergent Parameter Estimation AlgorithmThe Classical AlgorithmThe Proposed Algorithm
6 Conclusions
24 April 2012, Jaime A. Moreno Discontinuous Observers 38 / 59
GSTA with perturbations: ARI
GSTA with time-varying and/or nonlinear perturbations
x1 = −k1φ1 (x1) + x2
x2 = −k2φ2 (x1) + ρ (t, x) .
Assume2 |ρ (t, x)| ≤ δ
Analysis: The construction of Robust Lyapunov Functions can be donewith the classical method of solving an Algebraic Ricatti Inequality (ARI),or equivalently, solving the LMI[
ATP + PA + εP + δ2CTC PBBTP −1
]≤ 0 ,
where
A =
[−k1 1−k2 0
], C =
[1 0
], B =
[01
].
24 April 2012, Jaime A. Moreno Discontinuous Observers 39 / 59
Overview
1 Introduction
2 Observers a la Second Order Sliding Modes (SOSM)Super-Twisting ObserverGeneralized Super-Twisting Observers
3 Lyapunov Approach for Second-Order Sliding ModesGSTA without perturbations: ALEGSTA with perturbations: ARI
4 Optimality of the ST with noise
5 A Recursive Finite-Time Convergent Parameter Estimation AlgorithmThe Classical AlgorithmThe Proposed Algorithm
6 Conclusions
24 April 2012, Jaime A. Moreno Discontinuous Observers 39 / 59
Derivation of a signal under noise
Signal model: x1 := σ and x2 := σ
x1(t) = x2(t),x2(t) = ρ(t) = −σ(t),y(t) = x1(t) + η(t),
where|ρ(t)| ≤ L, |η(t)| ≤ δ, ∀t ≥ 0.
The estimator (differentiator/observer) proposed is:
˙x1 = − α1ε φ1(x1 − y) + x2,
˙x2 = − α2ε2 φ2(x1 − y),
where
φ1(x) = µ1|x |12 sign(x) + µ2x , µ1, µ2 ≥ 0
φ2(x) =12 µ2
1sign(x) + 32 µ1µ2|x |
12 sign(x) + µ2
2x ,
24 April 2012, Jaime A. Moreno Discontinuous Observers 40 / 59
Derivation of a signal under noise
Signal model: x1 := σ and x2 := σ
x1(t) = x2(t),x2(t) = ρ(t) = −σ(t),y(t) = x1(t) + η(t),
where|ρ(t)| ≤ L, |η(t)| ≤ δ, ∀t ≥ 0.
The estimator (differentiator/observer) proposed is:
˙x1 = − α1ε φ1(x1 − y) + x2,
˙x2 = − α2ε2 φ2(x1 − y),
where
φ1(x) = µ1|x |12 sign(x) + µ2x , µ1, µ2 ≥ 0
φ2(x) =12 µ2
1sign(x) + 32 µ1µ2|x |
12 sign(x) + µ2
2x ,
24 April 2012, Jaime A. Moreno Discontinuous Observers 40 / 59
µ1 = 0: High Gain Observer
µ2 = 0: Super-Twisting differentiator.
µ1 > 0 and µ2 > 0: combined actions.
Performance of the differentiator: “global uniform ultimate bound” ofthe differentiation error x2 − x2.
Remark about the Figures. In all figures and examples that follow,the parameters are set as α1 = 1.5, α2 = 1.1, L = 1, δ = 0.01, unlessotherwise stated.
Differentiation error x = x − x
˙x1 = − α1ε φ1(x1 − η) + x2,
˙x2 = − α2ε2 φ2(x1 − η) + ρ,
24 April 2012, Jaime A. Moreno Discontinuous Observers 41 / 59
µ1 = 0: High Gain Observer
µ2 = 0: Super-Twisting differentiator.
µ1 > 0 and µ2 > 0: combined actions.
Performance of the differentiator: “global uniform ultimate bound” ofthe differentiation error x2 − x2.
Remark about the Figures. In all figures and examples that follow,the parameters are set as α1 = 1.5, α2 = 1.1, L = 1, δ = 0.01, unlessotherwise stated.
Differentiation error x = x − x
˙x1 = − α1ε φ1(x1 − η) + x2,
˙x2 = − α2ε2 φ2(x1 − η) + ρ,
24 April 2012, Jaime A. Moreno Discontinuous Observers 41 / 59
µ1 = 0: High Gain Observer
µ2 = 0: Super-Twisting differentiator.
µ1 > 0 and µ2 > 0: combined actions.
Performance of the differentiator: “global uniform ultimate bound” ofthe differentiation error x2 − x2.
Remark about the Figures. In all figures and examples that follow,the parameters are set as α1 = 1.5, α2 = 1.1, L = 1, δ = 0.01, unlessotherwise stated.
Differentiation error x = x − x
˙x1 = − α1ε φ1(x1 − η) + x2,
˙x2 = − α2ε2 φ2(x1 − η) + ρ,
24 April 2012, Jaime A. Moreno Discontinuous Observers 41 / 59
µ1 = 0: High Gain Observer
µ2 = 0: Super-Twisting differentiator.
µ1 > 0 and µ2 > 0: combined actions.
Performance of the differentiator: “global uniform ultimate bound” ofthe differentiation error x2 − x2.
Remark about the Figures. In all figures and examples that follow,the parameters are set as α1 = 1.5, α2 = 1.1, L = 1, δ = 0.01, unlessotherwise stated.
Differentiation error x = x − x
˙x1 = − α1ε φ1(x1 − η) + x2,
˙x2 = − α2ε2 φ2(x1 − η) + ρ,
24 April 2012, Jaime A. Moreno Discontinuous Observers 41 / 59
µ1 = 0: High Gain Observer
µ2 = 0: Super-Twisting differentiator.
µ1 > 0 and µ2 > 0: combined actions.
Performance of the differentiator: “global uniform ultimate bound” ofthe differentiation error x2 − x2.
Remark about the Figures. In all figures and examples that follow,the parameters are set as α1 = 1.5, α2 = 1.1, L = 1, δ = 0.01, unlessotherwise stated.
Differentiation error x = x − x
˙x1 = − α1ε φ1(x1 − η) + x2,
˙x2 = − α2ε2 φ2(x1 − η) + ρ,
24 April 2012, Jaime A. Moreno Discontinuous Observers 41 / 59
µ1 = 0: High Gain Observer
µ2 = 0: Super-Twisting differentiator.
µ1 > 0 and µ2 > 0: combined actions.
Performance of the differentiator: “global uniform ultimate bound” ofthe differentiation error x2 − x2.
Remark about the Figures. In all figures and examples that follow,the parameters are set as α1 = 1.5, α2 = 1.1, L = 1, δ = 0.01, unlessotherwise stated.
Differentiation error x = x − x
˙x1 = − α1ε φ1(x1 − η) + x2,
˙x2 = − α2ε2 φ2(x1 − η) + ρ,
24 April 2012, Jaime A. Moreno Discontinuous Observers 41 / 59
The performance of linear and ST differentiators.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
0.1
0.2
0.3
0.4
0.5
0.6
gain ε
diffe
ren
tia
tio
n e
rro
r
0 0.5 1 1.5 2 2.5 3 3.5 40
1
2
3
4
5
6
7
gain ε
diffe
rentiation e
rror
0 1 2 3 4 5 6 7 80
0.5
1
1.5
2
2.5
3
gain ε
diffe
rentiation e
rror
Figure: Ultimate bound of the differentiation error as a function of ε. Solid: linearcase µ1 = 0, µ2 = 1; dashed: pure ST case µ1 = 1, µ2 = 0. Left: parametersL = 1, δ = 0.01; middle: parameters L = 1, δ = 1; right: parametersL = 0.1, δ = 1.
24 April 2012, Jaime A. Moreno Discontinuous Observers 42 / 59
0 1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
gain ε
steady
state
err
or
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
gain ε
ste
ad
y st
ate
err
or
Figure: Steady state error in the differentiation versus the gain fromL = 1, δ = 0.01 to L = 0.001, δ = 0.01. The figure on the right is a zoom of theleft figure. Solid: linear case µ1 = 0, µ2 = 1; dashed: pure ST caseµ1 = 1, µ2 = 0.
24 April 2012, Jaime A. Moreno Discontinuous Observers 43 / 59
Advantages of the GST
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.3
0.4
0.5
0.6
0.7
0.8
0.9
gain ε
diffe
ren
tia
tio
n e
rro
r
Figure: Ultimate bound of the differentiation error as a function of ε, for L = 1and δ = 0.01. Solid: linear case µ1 = 0, µ2 = 1; dashed: pure ST caseµ1 = 1, µ2 = 0; dotted: two experiments µ1 = 0.8, µ2 = 0.2 andµ1 = 0.5, µ2 = 0.5 (with circles).24 April 2012, Jaime A. Moreno Discontinuous Observers 44 / 59
The sensitivity to variations in the noise amplitude
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
0.5
1
1.5
2
2.5
noise amplitude δ
diffe
rentiation e
rror
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
0.5
1
1.5
noise amplitude δ
diffe
rentiation e
rror
Figure: The differentiation error as a function of the noise amplitude. Solid:linear case µ1 = 0, µ2 = 1; dashed: pure ST case µ1 = 1, µ2 = 0; dotted: GSTcase with µ1 = µ2 = 0.5. Dash-dot: the nominal noise 0.01 for which theoptimal gains are selected. Left: for L = 1, Right: for L = 0.
24 April 2012, Jaime A. Moreno Discontinuous Observers 45 / 59
Sensitivity to variations in the perturbation amplitude.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
perturbation amplitude L
diffe
rentiation e
rror
Figure: The differentiation error as a function of the perturbation amplitude, forδ = 0.01. Solid: linear µ1 = 0, µ2 = 1; dashed: ST µ1 = 1, µ2 = 0; dotted: GSTµ1 = µ2 = 0.5. Dash-dotted: the nominal perturbation L = 1 for which theoptimal gains are selected.24 April 2012, Jaime A. Moreno Discontinuous Observers 46 / 59
0 5 10 15 20 25 30 35 40 45 501
1.5
2
2.5
3
3.5
4
4.5
5
Time (sec)
Sta
tet x
2
0 5 10 15 20 25 30 35 40 45 50−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Time (sec)E
stim
ation e
rror
e2
Linear Observer
Nonlinear Observer
Figure: Behavior of Linear and ST Observers under noise and perturbation.
24 April 2012, Jaime A. Moreno Discontinuous Observers 47 / 59
Overview
1 Introduction
2 Observers a la Second Order Sliding Modes (SOSM)Super-Twisting ObserverGeneralized Super-Twisting Observers
3 Lyapunov Approach for Second-Order Sliding ModesGSTA without perturbations: ALEGSTA with perturbations: ARI
4 Optimality of the ST with noise
5 A Recursive Finite-Time Convergent Parameter Estimation AlgorithmThe Classical AlgorithmThe Proposed Algorithm
6 Conclusions
24 April 2012, Jaime A. Moreno Discontinuous Observers 47 / 59
Background
Super Twisting-Algorithm
Applications: Differentiator [Levant, 1998], [Levant, 2003], Controller[Levant, 2003], Observer [Davila et al., 2005] and as an observer andparameter estimator [Davila et al., 2006].
[Moreno, 2009], (Generalized Super-Twisting Algorithm).
Adaptive Systems
Some documents that summarize several techniques are[Narendra and Annaswamy, 1989], [Sastry and Bodson, 1989],[Ioannou and Sun, 1996] and [Ioannou and Findan, 2006] among others.
Finite-Time Parameter Estimation
Finite-Time Parameter Estimator ([Adetola and Guay, 2008])
24 April 2012, Jaime A. Moreno Discontinuous Observers 48 / 59
Overview
1 Introduction
2 Observers a la Second Order Sliding Modes (SOSM)Super-Twisting ObserverGeneralized Super-Twisting Observers
3 Lyapunov Approach for Second-Order Sliding ModesGSTA without perturbations: ALEGSTA with perturbations: ARI
4 Optimality of the ST with noise
5 A Recursive Finite-Time Convergent Parameter Estimation AlgorithmThe Classical AlgorithmThe Proposed Algorithm
6 Conclusions
24 April 2012, Jaime A. Moreno Discontinuous Observers 48 / 59
[Morgan and Narendra, 1977]
x1 = A(t)x1 + B(t)x2
x2 = −BT (t)P(t)x1(2)
where xi ∈ Rni , i = 1, 2, A(t), B(t) are matrices of bounded piecewisecontinuous functions.
There exist a symmetric, positive definite matrix P(t), which satisfies
P(t) + AT (t)P(t) + P(t)A(t) = −Q(t)
Persistence of Excitation Conditions: B(t) is smooth,∣∣B(t)
∣∣ isuniformly bounded, there exist T > 0, ε > 0 such that for any unitvector w ∈ Rn2 ∫ t+T
t‖B(τ)w‖ dτ ≥ ε (3)
Then x(t)→ 0 exponentially.
24 April 2012, Jaime A. Moreno Discontinuous Observers 49 / 59
Application: Parameter Estimation
Linearly parametrized System
y = α(x , t) + Γ(t)θ
Parameter Estimation Algorithm
˙y = α(x , t) + Γ(t)θ − kyey˙θ = −kθΓT (t)ey
Defining errors as ey = y − y and eθ = θ − θ.Error Dynamics
ey = −kyey + Γ(t)eθ˙θ = −kθΓT (t)ey
24 April 2012, Jaime A. Moreno Discontinuous Observers 50 / 59
Overview
1 Introduction
2 Observers a la Second Order Sliding Modes (SOSM)Super-Twisting ObserverGeneralized Super-Twisting Observers
3 Lyapunov Approach for Second-Order Sliding ModesGSTA without perturbations: ALEGSTA with perturbations: ARI
4 Optimality of the ST with noise
5 A Recursive Finite-Time Convergent Parameter Estimation AlgorithmThe Classical AlgorithmThe Proposed Algorithm
6 Conclusions
24 April 2012, Jaime A. Moreno Discontinuous Observers 50 / 59
x1 = A(t)φ1(x1) + B(t)x2
x2 = −BT (t)P(t)φ2(x1)+δ (t)(4)
where
φ1(x1) = µ1 |x1|1/2 sign (x1) + µ2x1, µ1, µ2 > 0
φ2(x1) =µ2
12 sign (x1) +
23 µ1µ2 |x1|1/2 sign (x1) + µ2
2x1
where x1 ∈ Rn1 , x2 ∈ Rn2 , n1 = 1, n2 ≥ 1, µ1 > 0 y µ2 ≥ 0
There exist a symmetric, positive definite matrix P(t), which satisfies
P(t) + AT (t)P(t) + P(t)A(t) = −Q(t)
Persistence of Excitation Conditions (PE): There exist T0, ε0, δ0,with t2 ∈ [t, t + T ] such that for any unit vector w ∈ Rn2∥∥∥∥ 1
T0
∫ t2+δ0
t2
B(τ)w dτ
∥∥∥∥ ≥ ε0 (5)
24 April 2012, Jaime A. Moreno Discontinuous Observers 51 / 59
Finite-Time Convergence
Under PE conditions, if δ(t) = 0 then x(t)→ 0 in Finite-Time.
Robustness
Under PE conditions, if δ(t) is bounded then x(t) is bounded (ISS).Under some extra conditions it converges in finite time.
24 April 2012, Jaime A. Moreno Discontinuous Observers 52 / 59
Case (µ1, µ2, p, q)
Linear (0,1,–,1)
STA (1,0,1/2,–)
GSTA (1,1,1/2,q)
24 April 2012, Jaime A. Moreno Discontinuous Observers 53 / 59
Idea of the Proof
Under PE conditions it is possible to construct a strict Lyapunov function[Moreno, 2009]
V (t, x) = ζT Π(t)ζ
ζT =[ζ1 ζT2
]=[φ1(x1) xT
2
]V ≤ −γ1V 1/2 − γ2V
24 April 2012, Jaime A. Moreno Discontinuous Observers 54 / 59
Application: Parameter Estimation
System representation,y = Γ(t)θ (6)
Finite-time parameter estimator,
˙y = −k1φ1(ey ) + Γ (t) θ˙θ = −k2φ2(ey )ΓT (t)
(7)
φ1(ey ) = µ1 |ey |1/2 sign (ey ) + µ2ey , µ1, µ2 > 0 (8)
φ2(ey ) =µ2
1
2sign (ey ) +
3
2µ1µ2 |ex |1/2 sign (ey ) + µ2
2ey (9)
Estimation Error Dynamics, where ey = y − y , eθ = θ − θ
ey = −k1φ1(ey ) + Γ (t) eθ (10a)
eθ = −k2φ2(ey )ΓT (t) (10b)
24 April 2012, Jaime A. Moreno Discontinuous Observers 55 / 59
Example: A simple pendulum
x1 = x2 (11a)
x2 =1
Ju − Mgl
2Jsin x1 −
Vs
Jx2 (11b)
x1 angular position, x2 angular velocity, M mass, g gravity, L rope’slength mass and J = ML2 mass inertia.Parameter’s estimator
˙x2 = −k1φ1(ex2) + Γθ (12a)
˙θ = −k2φ2(ex2)ΓT (12b)
ex2 = x2 − x2 velocity’s error, Γ =[x2 sin x1 u
]regressor,
θ =[− Vs
J− Mg L
J1J
]vector of unknown parameters, φ1(·) and φ2(·)
are given as in the GSTA.24 April 2012, Jaime A. Moreno Discontinuous Observers 56 / 59
Parameter’s estimation
0 10 20 30 40 50−2.5
−2
−1.5
−1
−0.5
0
0.5
Time (sec)θ 1
andθ 1
θ1
θ1 with linear algorithm
θ1 with nonlinear algorithm
0 10 20 30 40 50−12
−10
−8
−6
−4
−2
0
Time (sec)
θ 2an
dθ 2
θ2θ2 with linear algorithmθ2 with nonlinear algorithm
0 10 20 30 40 50−1
−0.5
0
0.5
1
1.5
2
2.5
Time (sec)
θ 3an
dθ 3
θ3
θ3 with linear algorithm
θ3 with nonlinear algorithm
Figure: Parameter estimation of θ1, θ2 and θ3 with GSTA and with linearalgorithm.
24 April 2012, Jaime A. Moreno Discontinuous Observers 57 / 59
Parameter’s estimation, with perturbation in θ1 = −0.2 + 0.2sin(t)
0 10 20 30 40 50−2
−1.5
−1
−0.5
0
0.5
Time (sec)θ 1
andθ 1
θ1
θ1 with linear algorithm
θ1 with nonlinear algorithm
0 10 20 30 40 50−12
−10
−8
−6
−4
−2
0
2
Time (sec)
θ 2an
dθ 2
θ2θ2 with linear algorithmθ2 with nonlinear algorithm
0 10 20 30 40 50−1
−0.5
0
0.5
1
1.5
2
Time (sec)
θ 3an
dθ 3
θ3
θ3 with linear algorithm
θ3 with nonlinear algorithm
Figure: Parameter estimation of θ1, θ2 and θ3 with GSTA and with linearalgorithm.
24 April 2012, Jaime A. Moreno Discontinuous Observers 58 / 59
Overview
1 Introduction
2 Observers a la Second Order Sliding Modes (SOSM)Super-Twisting ObserverGeneralized Super-Twisting Observers
3 Lyapunov Approach for Second-Order Sliding ModesGSTA without perturbations: ALEGSTA with perturbations: ARI
4 Optimality of the ST with noise
5 A Recursive Finite-Time Convergent Parameter Estimation AlgorithmThe Classical AlgorithmThe Proposed Algorithm
6 Conclusions
24 April 2012, Jaime A. Moreno Discontinuous Observers 58 / 59
Conclusions
1 The GSTO is an observer that isRobust: it converges despite of unknown inputs/uncertaintiesExact: it converges in finite-timePreassigned Convergence time for any initial condition.But there is no free lunch!
2 It can be extended to estimate parameters in finite time withapplications in Adaptive Control
3 Applications for Bioreactors seem to be attractive due to:Robustness against uncertaintiesPossibility of reconstructing uncertainties, e.g. reaction rates
4 Lyapunov functions for Higher Order algorithms is an ongoing work.
5 Useful for control, reaction rate parameters (functional form)estimation, ...
24 April 2012, Jaime A. Moreno Discontinuous Observers 59 / 59
Overview
7 Some ApplicationsFinite-Time, robust MRAC
24 April 2012, Jaime A. Moreno Discontinuous Observers 59 / 59
Overview
7 Some ApplicationsFinite-Time, robust MRAC
24 April 2012, Jaime A. Moreno Discontinuous Observers 59 / 59
Introduction
Direct Model Reference Adaptive Control (MRAC) is a well-knownapproach for adaptive control of linear and some nonlinear systems.
If Plant has relative degree n∗ = 1 and the reference model is StrictlyPositive Real (SPR), the controller is particularly simple toimplement, and to design.
The Adjustment Mechanism of the MRAC is basically a parameterestimation algorithm.
Our objective: Modify the Adjustment Mechanism of the classicalDirect MRAC by adding the Super-Twisting-Like nonlinearities, so asto achieve the finite-time convergence and robustness properties.
24 April 2012, Jaime A. Moreno Discontinuous Observers 60 / 59
Introduction
Direct Model Reference Adaptive Control (MRAC) is a well-knownapproach for adaptive control of linear and some nonlinear systems.
If Plant has relative degree n∗ = 1 and the reference model is StrictlyPositive Real (SPR), the controller is particularly simple toimplement, and to design.
The Adjustment Mechanism of the MRAC is basically a parameterestimation algorithm.
Our objective: Modify the Adjustment Mechanism of the classicalDirect MRAC by adding the Super-Twisting-Like nonlinearities, so asto achieve the finite-time convergence and robustness properties.
24 April 2012, Jaime A. Moreno Discontinuous Observers 60 / 59
Introduction
Direct Model Reference Adaptive Control (MRAC) is a well-knownapproach for adaptive control of linear and some nonlinear systems.
If Plant has relative degree n∗ = 1 and the reference model is StrictlyPositive Real (SPR), the controller is particularly simple toimplement, and to design.
The Adjustment Mechanism of the MRAC is basically a parameterestimation algorithm.
Our objective: Modify the Adjustment Mechanism of the classicalDirect MRAC by adding the Super-Twisting-Like nonlinearities, so asto achieve the finite-time convergence and robustness properties.
24 April 2012, Jaime A. Moreno Discontinuous Observers 60 / 59
Introduction
Direct Model Reference Adaptive Control (MRAC) is a well-knownapproach for adaptive control of linear and some nonlinear systems.
If Plant has relative degree n∗ = 1 and the reference model is StrictlyPositive Real (SPR), the controller is particularly simple toimplement, and to design.
The Adjustment Mechanism of the MRAC is basically a parameterestimation algorithm.
Our objective: Modify the Adjustment Mechanism of the classicalDirect MRAC by adding the Super-Twisting-Like nonlinearities, so asto achieve the finite-time convergence and robustness properties.
24 April 2012, Jaime A. Moreno Discontinuous Observers 60 / 59
MRAC with relative degree n∗ = 1
Figure: General structure of MRAC scheme.
24 April 2012, Jaime A. Moreno Discontinuous Observers 61 / 59
SISO Plant: yp = Gp(s)up = kpZp(s)Rp(s)
, Relative degree n∗ = 1.
Reference model: ym = Wm(s)r = kmZm(s)Rm(s)
Hypothesis:
A1 An upper bound n of the degree np of Rp(s) is known.A2 The relative degree n∗ = np −mp of Gp(s) is one, i.e.
n∗ = 1.A3 Zp(s) is a monic Hurwitz polynomial of degree
mp = np − 1.A4 The sign of the high frequency gain kp is known.B1 Zm(s), Rm(s) are monic Hurwitz polynomials of degree
qm, pm, respectively, where pm ≤ n.B2 The relative degree n∗m = pm − qm of Wm is the same
as that of Gp(s), i.e., n∗m = n∗ = 1.B3 Wm(s) is designed to be Strictly Positive Real (SPR).
24 April 2012, Jaime A. Moreno Discontinuous Observers 62 / 59
Solution of the MRC problem for known parameters:
w1 = Fw1 + gup, w1(0) = 0 (13a)
w2 = Fw2 + gyp, w2(0) = 0 (13b)
up = θ∗Tw (13c)
w =[wT
1 wT2 yp r
]T, θ∗ =
[θ∗T1 θ∗T2 θ∗T3 c∗0
]T(14)
For unknown parameters the control law is
up = θT (t)w (15)
θ (t) = −Γe1w sign (ρ∗) , (16)
e1 = yp − ym , sign (ρ∗) = sign
(kpkm
), Γ = ΓT > 0 . (17)
24 April 2012, Jaime A. Moreno Discontinuous Observers 63 / 59
Classical MRAC
Theorem
The MRAC scheme guarantees that:
1 All signals in the closed-loop plant are bounded and the tracking errore1 converges to zero asymptotically for any reference input r ∈ L∞ .
2 If r is sufficiently rich of order 2n, r ∈ L∞ and Zp(s), Rp(s) arerelatively coprime, then the parameter error |θ| = |θ − θ∗| and thetracking error e1 converge to zero exponentially fast.
24 April 2012, Jaime A. Moreno Discontinuous Observers 64 / 59
Modified MRAC
Modified adaptive control law
up =θT (t)w − k1φ1 (e1) sign (ρ∗)
θ (t) =− Γφ2 (e1)w sign (ρ∗) ,
φ1 (e1) = µ1 |e1|1/2 sign (e1) + µ2e1
φ2 (e1) =µ2
12 sign (e1) +
32 µ1µ2 |e1|1/2 sign (e1) + µ2
2e1
(18)
and µ1 > 0, µ2 > 0.
24 April 2012, Jaime A. Moreno Discontinuous Observers 65 / 59
Theorem
The modified MRAC scheme guarantees that:
1 All signals in the closed-loop plant are bounded and the tracking errore1 converges to zero asymptotically for any reference input r ∈ L∞ .
2 If r is sufficiently rich of order 2n, r ∈ L∞ and Zp(s), Rp(s) arerelatively coprime, then the parameter error |θ| = |θ − θ∗| and thetracking error e1 converge to zero in finite time.
3 If PE is satisfied then algorithm is robust (ISS)
24 April 2012, Jaime A. Moreno Discontinuous Observers 66 / 59
Example
Second order plant yp = (s+1)s2−3s+1
up .
Reference model ym = 1s+1 r .
Nominal controller
w1 = −2w1 + up, w1(0) = 0w2 = −2w2 + yp, w2(0) = 0up = θ1w1 + θ2w2 + θ3yp + c0r ,
(19)
θ =[θ1 θ2 θ3 c0
]T, w =
[w1 w2 yp r
]T, and
Nominal parameter values:
θ∗ =[θ∗1 θ∗2 θ∗3 c∗0
]T=[0 , 6 , −5 , 1
]T.
Three simulation scenarios will be presented.
24 April 2012, Jaime A. Moreno Discontinuous Observers 67 / 59
Example
Second order plant yp = (s+1)s2−3s+1
up .
Reference model ym = 1s+1 r .
Nominal controller
w1 = −2w1 + up, w1(0) = 0w2 = −2w2 + yp, w2(0) = 0up = θ1w1 + θ2w2 + θ3yp + c0r ,
(19)
θ =[θ1 θ2 θ3 c0
]T, w =
[w1 w2 yp r
]T, and
Nominal parameter values:
θ∗ =[θ∗1 θ∗2 θ∗3 c∗0
]T=[0 , 6 , −5 , 1
]T.
Three simulation scenarios will be presented.
24 April 2012, Jaime A. Moreno Discontinuous Observers 67 / 59
Example
Second order plant yp = (s+1)s2−3s+1
up .
Reference model ym = 1s+1 r .
Nominal controller
w1 = −2w1 + up, w1(0) = 0w2 = −2w2 + yp, w2(0) = 0up = θ1w1 + θ2w2 + θ3yp + c0r ,
(19)
θ =[θ1 θ2 θ3 c0
]T, w =
[w1 w2 yp r
]T, and
Nominal parameter values:
θ∗ =[θ∗1 θ∗2 θ∗3 c∗0
]T=[0 , 6 , −5 , 1
]T.
Three simulation scenarios will be presented.
24 April 2012, Jaime A. Moreno Discontinuous Observers 67 / 59
Example
Second order plant yp = (s+1)s2−3s+1
up .
Reference model ym = 1s+1 r .
Nominal controller
w1 = −2w1 + up, w1(0) = 0w2 = −2w2 + yp, w2(0) = 0up = θ1w1 + θ2w2 + θ3yp + c0r ,
(19)
θ =[θ1 θ2 θ3 c0
]T, w =
[w1 w2 yp r
]T, and
Nominal parameter values:
θ∗ =[θ∗1 θ∗2 θ∗3 c∗0
]T=[0 , 6 , −5 , 1
]T.
Three simulation scenarios will be presented.
24 April 2012, Jaime A. Moreno Discontinuous Observers 67 / 59
Example
Second order plant yp = (s+1)s2−3s+1
up .
Reference model ym = 1s+1 r .
Nominal controller
w1 = −2w1 + up, w1(0) = 0w2 = −2w2 + yp, w2(0) = 0up = θ1w1 + θ2w2 + θ3yp + c0r ,
(19)
θ =[θ1 θ2 θ3 c0
]T, w =
[w1 w2 yp r
]T, and
Nominal parameter values:
θ∗ =[θ∗1 θ∗2 θ∗3 c∗0
]T=[0 , 6 , −5 , 1
]T.
Three simulation scenarios will be presented.
24 April 2012, Jaime A. Moreno Discontinuous Observers 67 / 59
Persistence of Excitation conditions, without perturbations
0 5 10 15 20 25−6
−4
−2
0
2
4
6
8
Time (sec)
y mandy p
Model reference output
Plant output
Figure: Model ym (continuous line) and the Plant’s output yp (dotted line) withthe classical MRAC scheme with reference signal r = 5 cos (t) + 10 cos (5t).
24 April 2012, Jaime A. Moreno Discontinuous Observers 68 / 59
0 5 10 15 20 25−6
−4
−2
0
2
4
6
Time (sec)
y pandy m
Model reference output
Plant output
Figure: Model ym (continuous line) and the Plant’s output yp (dotted line) withthe nonlinear MRAC scheme with reference signal r = 5 cos (t) + 10 cos (5t).
24 April 2012, Jaime A. Moreno Discontinuous Observers 69 / 59
0 5 10 15 20 25−4
−3
−2
−1
0
1
2
3
4
Time (sec)
Error
0 5 10 15 20 25−4
−3
−2
−1
0
1
2
3
4
Time (sec)
Error
Figure: Tracking error e1 = yp − ym for the classical (left) and the proposed(right) MRAC schemes with reference signal r = 5 cos (t) + 10 cos (5t).
24 April 2012, Jaime A. Moreno Discontinuous Observers 70 / 59
0 5 10 15 20 25−40
−30
−20
−10
0
10
20
30
Time (sec)
Control
Linear control
Nonlinear control
Figure: Control variable up for the classical MRAC (continuous line) and theproposed nonlinear MRAC (dotted line) with reference signalr = 5 cos (t) + 10 cos (5t).
24 April 2012, Jaime A. Moreno Discontinuous Observers 71 / 59
0 5 10 15 20 25 30 35 40−10
0
10
Time (sec)
A)
0 5 10 15 20 25 30 35 40−10
0
10
Time (sec)
B)
0 5 10 15 20 25 30 35 40−10
0
10
Time (sec)
C)
0 5 10 15 20 25 30 35 40−5
0
5
Time (sec)
D)
linear estimation
nonlinear estimation
original value
Figure: Parameter convergence to the real values with reference signalr = 5 sin (t) + 10 sin (5t). A) θ∗1 = 0, B) θ∗2 = 6, C) θ∗3 = −5 and D) c∗0 = 1
24 April 2012, Jaime A. Moreno Discontinuous Observers 72 / 59
Persistence of Excitation conditions, with perturbations
0 5 10 15 20 25−4
−3
−2
−1
0
1
2
3
4
Time (sec)
Err
or
0 5 10 15 20 25 30 35 40−4
−3
−2
−1
0
1
2
3
4
Time (sec)
Error
Figure: Tracking error e1 = yp − ym for the classical (left) and the proposed(right) MRAC schemes with reference signal r = 5 cos (t) + 10 cos (5t), withperturbation p (t) = 5 sin (6t).
24 April 2012, Jaime A. Moreno Discontinuous Observers 73 / 59
0 5 10 15 20 25 30 35 40−10
0
10
Time (sec)
0 5 10 15 20 25 30 35 40−10
0
10
Time (sec)
0 5 10 15 20 25 30 35 40−20
0
20
Time (sec)
0 5 10 15 20 25 30 35 40−10
0
10
Time (sec)
Linear estimation
Nonlinear estimation
Real value
Figure: Parameter convergence to the real values with reference signalr = 5 sin (t) + 10 sin (5t), and perturbation p (t) = 5 sin (6t). A) θ∗1 = 0, B)θ∗2 = 6, C) θ∗3 = −5 and D) c∗0 = 1.
24 April 2012, Jaime A. Moreno Discontinuous Observers 74 / 59
Lack of Persistence of Excitation conditions
0 5 10 15−2
−1
0
1
2
3
4
5
6
7
Time (sec)
ypan
dy m
Model reference output
Plant output
Figure: Model ym (continuous line) and the Plant’s output yp (dotted line) withthe classical MRAC scheme with constant reference signal r = 6.
24 April 2012, Jaime A. Moreno Discontinuous Observers 75 / 59
0 5 10 15−1
0
1
2
3
4
5
6
Time (sec)
y pandy m
Model reference output
Plant output
Figure: Model ym (continuous line) and the Plant’s output yp (dotted line) withthe nonlinear MRAC scheme with reference signal r = 6.
24 April 2012, Jaime A. Moreno Discontinuous Observers 76 / 59
0 5 10 15−15
−10
−5
0
5
10
15
20
Time (sec)
Control
Linear control
Nonlinear control
Figure: Control variable up for the classical MRAC (continuous line) and theproposed nonlinear MRAC (dotted line) with reference signal r = 6.
24 April 2012, Jaime A. Moreno Discontinuous Observers 77 / 59
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Ioannou, P. A. and Findan, B. (2006).Adaptive Control Tutorial.Society for Industrial and Applied Mathematics, 3600 University CityScience Center, Philadelphia.
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Ioannou, P. A. and Sun, J. (1996).Robust Adaptive Control.Upper Saddle River, NJ.
Levant, A. (1998).Robust exact differentiation via sliding mode technique.Automatica, 34(3):379–384.
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24 April 2012, Jaime A. Moreno Discontinuous Observers 79 / 59
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Narendra, K. S. and Annaswamy, A. (1989).Stable Adaptive Systems.Prentice Hall, Englewood Cliffs, NJ.
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24 April 2012, Jaime A. Moreno Discontinuous Observers 80 / 59
G. Bastin and D. DochainOn-line estimation and adaptive control of bioreactors.Elsevier, 1990.
Moreno, J.A. and Alvarez, J. and Rocha-Cozatl, E. and Diaz-Salgado,J.Super-Twisting Observer-Based Output Feedback Control of a Classof Continuous Exothermic Chemical ReactorsProceedings of the 9th International Symposium on Dynamics andControl of Process Systems (DYCOPS 2010),Leuven, Belgium, July5-7, 2010,
Farza, M. and Busawon, K. and Hammouri, H.Simple nonlinear observers for on-line estimation of kinetic rates inbioreactorsAutomatica,Vol.34, Num.3, 1998, Elsevier ,
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