Discontinuous Observers with strong convergence properties and some - - PowerPoint PPT Presentation

Discontinuous Observers with strong convergence properties and some applications

Jaime A. Moreno JMorenoP@ii.unam.mx

Instituto de Ingenier´ ıa Universidad Nacional Aut´

• noma de M´

exico 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 Observer Generalized Super-Twisting Observers

3 Lyapunov Approach for Second-Order Sliding Modes

GSTA without perturbations: ALE GSTA with perturbations: ARI

4 Optimality of the ST with noise 5 A Recursive Finite-Time Convergent Parameter Estimation Algorithm

The Classical Algorithm The 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 Observer Generalized Super-Twisting Observers

3 Lyapunov Approach for Second-Order Sliding Modes

GSTA without perturbations: ALE GSTA with perturbations: ARI

4 Optimality of the ST with noise 5 A Recursive Finite-Time Convergent Parameter Estimation Algorithm

The Classical Algorithm The 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, practical

• bservers, 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 dt h (x (t)) = ∂h (x) ∂x ˙ x (t) = ∂h (x) ∂x f (x) := Lf h (x) ¨ y (t) = ∂Lf h (x) ∂x ˙ x (t) = ∂Lf h (x) ∂x f (x) := L2

f h (x)

. . . y (n−1) (t) = ∂Ln−2

f

h (x) ∂x ˙ x (t) = ∂Ln−2

f

h (x) ∂x f (x) := Ln−1

f

h (x) where Lk

f h (x) are Lie’s derivatives of h along f .

24 April 2012, Jaime A. Moreno Discontinuous Observers 3 / 59

Evaluating at t = 0        y (0) ˙ y (0) ¨ y (0) . . . y (k) (0)        =        h (x0) Lf h (x0) L2

f h (x0)

. . . Lk

f 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−1

n

(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: ˙ ˆ x1 = −l1 (ˆ x1 − x1) + ˆ x2 , ˙ ˆ x2 = −l2 (ˆ x1 − x1) Estimation Error: e1 = ˆ x1 − x1, e2 = ˆ 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

20 40 60 10 20 30 40 50 60

Time (sec) State x1

20 40 60 0.5 1 1.5 2

Time (sec) Statet x2

20 40 60 −1 1 2 3 4

Time (sec) Estimation error e1

20 40 60 −1.5 −1 −0.5 0.5 1 1.5 2

Time (sec) Estimation error e2

Linear Observer Linear Observer

Figure: Behavior of Plant and the Linear Observer without unknown input.

24 April 2012, Jaime A. Moreno Discontinuous Observers 7 / 59

20 40 60 20 40 60 80 100 120 140

Time (sec) State x1

20 40 60 0.5 1 1.5 2 2.5 3 3.5 4

Time (sec) Statet x2

20 40 60 −2 −1 1 2 3 4

Time (sec) Estimation error e1

20 40 60 −1.5 −1 −0.5 0.5 1 1.5 2

Time (sec) Estimation error e2

Linear Observer Linear Observer

Figure: Behavior of Plant and the Linear Observer with unknown input.

24 April 2012, Jaime A. Moreno Discontinuous Observers 8 / 59

20 40 60 10 20 30 40 50 60

Time (sec) State x1

20 40 60 0.5 1 1.5 2

Time (sec) Statet x2

20 40 60 −1 1 2 3 4

Time (sec) Estimation error e1

20 40 60 −1.5 −1 −0.5 0.5 1 1.5 2

Time (sec) Estimation error e2

Linear Observer Linear Observer

Figure: Behavior of Plant and the Linear Observer without UI with large initial conditions.

24 April 2012, Jaime A. Moreno Discontinuous Observers 9 / 59

20 40 60 10 20 30 40 50 60

Time (sec) State x1

20 40 60 0.5 1 1.5 2

Time (sec) Statet x2

20 40 60 −1 1 2 3 4

Time (sec) Estimation error e1

20 40 60 −1.5 −1 −0.5 0.5 1 1.5 2

Time (sec) Estimation error e2

Linear Observer Linear Observer

Figure: Behavior of Plant and the Linear Observer without UI with very large initial conditions.

24 April 2012, Jaime A. Moreno Discontinuous Observers 10 / 59

Recapitulation.

Linear Observer for Linear Plant If no unknown inputs/Uncertainties: it converges exponentially fast.

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 best bounded error.

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 best bounded error. Convergence time depends on the initial conditions of the observer

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 best bounded 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 = 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 is discontinuous. After arriving at σ = 0, sliding on σ ≡ 0.

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −1 −0.5 0.5 1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.4 0.6 0.8 1 1.2

σ u ¯ u Finite-Time convergence. Differential Inclusion. ˙ σ ∈ [−α, α] − sign(σ)

24 April 2012, Jaime A. Moreno Discontinuous Observers 13 / 59

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.

24 April 2012, Jaime A. Moreno Discontinuous Observers 14 / 59

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

20 40 60 10 20 30 40 50 60

Time (sec) State x1

20 40 60 0.5 1 1.5 2

Time (sec) Statet x2

20 40 60 −1 1 2 3 4

Time (sec) Estimation error e1

20 40 60 −1.5 −1 −0.5 0.5 1 1.5 2

Time (sec) Estimation error e2

Linear Observer Nonlinear Observer Linear Observer Nonlinear Observer

Figure: Behavior of Plant and the SM Observer without unknown input.

24 April 2012, Jaime A. Moreno Discontinuous Observers 16 / 59

20 40 60 20 40 60 80 100 120 140

Time (sec) State x1

20 40 60 0.5 1 1.5 2 2.5 3 3.5 4

Time (sec) Statet x2

20 40 60 −1 1 2 3 4

Time (sec) Estimation error e1

20 40 60 −1.5 −1 −0.5 0.5 1 1.5 2

Time (sec) Estimation error e2

Linear Observer Nonlinear Observer Linear Observer Nonlinear Observer

Figure: Behavior of Plant and the SM Observer with unknown input.

24 April 2012, Jaime A. Moreno Discontinuous Observers 17 / 59

Recapitulation.

Sliding Mode Observer for Linear Plant If no unknown inputs/Uncertainties: e1 converges in finite time, and e2 converges exponentially fast.

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, and e2 converges exponentially fast. If there are unknown inputs/Uncertainties: no convergence. At best bounded error. Only e1 converges in finite time!

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, and e2 converges exponentially fast. If there are unknown inputs/Uncertainties: no convergence. At best bounded error. Only e1 converges in finite time! Convergence time depends on the initial conditions of the observer

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, and e2 converges exponentially fast. If there are unknown inputs/Uncertainties: no convergence. At best bounded 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 been achieved!

24 April 2012, Jaime A. Moreno Discontinuous Observers 18 / 59

Overview

1 Introduction 2 Observers a la Second Order Sliding Modes (SOSM)

Super-Twisting Observer Generalized Super-Twisting Observers

3 Lyapunov Approach for Second-Order Sliding Modes

GSTA without perturbations: ALE GSTA with perturbations: ARI

4 Optimality of the ST with noise 5 A Recursive Finite-Time Convergent Parameter Estimation Algorithm

The Classical Algorithm The 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 Observer Generalized Super-Twisting Observers

3 Lyapunov Approach for Second-Order Sliding Modes

GSTA without perturbations: ALE GSTA with perturbations: ARI

4 Optimality of the ST with noise 5 A Recursive Finite-Time Convergent Parameter Estimation Algorithm

The Classical Algorithm The 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: ˙ ˆ x1 = −l1 |e1|

1 2 sign (e1) + ˆ

x2 , ˙ ˆ x2 = −l2 sign (e1) Estimation Error: e1 = ˆ x1 − x1, e2 = ˆ x2 − x2 ˙ e1 = −l1 |e1|

1 2 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.

24 April 2012, Jaime A. Moreno Discontinuous Observers 20 / 59

20 40 60 10 20 30 40 50 60

Time (sec) State x1

20 40 60 0.5 1 1.5 2

Time (sec) Statet x2

20 40 60 −1 1 2 3 4

Time (sec) Estimation error e1

20 40 60 −1.5 −1 −0.5 0.5 1 1.5 2

Time (sec) Estimation error e2

Linear Observer Nonlinear Observer Linear Observer Nonlinear Observer

Figure: Behavior of Plant and the Non Linear Observer without unknown input.

24 April 2012, Jaime A. Moreno Discontinuous Observers 21 / 59

20 40 60 20 40 60 80 100 120 140

Time (sec) State x1

20 40 60 0.5 1 1.5 2 2.5 3 3.5 4

Time (sec) Statet x2

20 40 60 −1 1 2 3 4

Time (sec) Estimation error e1

20 40 60 −1.5 −1 −0.5 0.5 1 1.5 2

Time (sec) Estimation error e2

Linear Observer Nonlinear Observer 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

20 40 60 10 20 30 40 50 60

Time (sec) State x1

20 40 60 0.5 1 1.5 2

Time (sec) Statet x2

20 40 60 −1 1 2 3 4

Time (sec) Estimation error e1

20 40 60 −1.5 −1 −0.5 0.5 1 1.5 2

Time (sec) Estimation error e2

Linear Observer Nonlinear Observer Linear Observer Nonlinear Observer

Figure: Behavior of Plant and the Non Linear Observer without UI with large initial 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!

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 in finite-time! Observer is insensitive to perturbation/uncertainty!

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 in finite-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 Observer Generalized Super-Twisting Observers

3 Lyapunov Approach for Second-Order Sliding Modes

GSTA without perturbations: ALE GSTA with perturbations: ARI

4 Optimality of the ST with noise 5 A Recursive Finite-Time Convergent Parameter Estimation Algorithm

The Classical Algorithm The 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: ˙ ˆ x1 = −l1φ1 (e1) + ˆ x2 , ˙ ˆ x2 = −l2φ2 (e1) Estimation Error: e1 = ˆ x1 − x1, e2 = ˆ x2 − x2 ˙ e1 = −l1φ1 (e1) + e2 ˙ e2 = −l2φ2 (e1) − w (t) , Solutions in the sense of Filippov. φ1 (e1) = µ1 |e1|

1 2 sign (e1) + µ2 |e1| 3 2 sign (e1) , µ1 , µ2 ≥ 0 ,

φ2 (e1) = µ2

1

2 sign (e1) + 2µ1µ2e1 + 3 2µ2

2 |e1|2 sign (e1) ,

24 April 2012, Jaime A. Moreno Discontinuous Observers 25 / 59

Figure: Linear Plant with an unknown input and a Non Linear Observer.

24 April 2012, Jaime A. Moreno Discontinuous Observers 26 / 59

20 40 60 10 20 30 40 50 60

Time (sec) State x1

20 40 60 0.5 1 1.5 2

Time (sec) Statet x2

20 40 60 −1 1 2 3 4

Time (sec) Estimation error e1

20 40 60 −1.5 −1 −0.5 0.5 1 1.5 2

Time (sec) Estimation error e2

Linear Observer Nonlinear Observer Linear Observer Nonlinear Observer

Figure: Behavior of Plant and the Non Linear Observer without unknown input and large initial conditions.

24 April 2012, Jaime A. Moreno Discontinuous Observers 27 / 59

20 40 60 10 20 30 40 50 60

Time (sec) State x1

20 40 60 0.5 1 1.5 2

Time (sec) Statet x2

20 40 60 −1 1 2 3 4

Time (sec) Estimation error e1

20 40 60 −1.5 −1 −0.5 0.5 1 1.5 2

Time (sec) Estimation error e2

Linear Observer Nonlinear Observer Linear Observer Nonlinear Observer

Figure: Behavior of Plant and the Non Linear Observer without unknown input and very large initial conditions.

24 April 2012, Jaime A. Moreno Discontinuous Observers 28 / 59

20 40 60 20 40 60 80 100 120 140

Time (sec) State x1

20 40 60 0.5 1 1.5 2 2.5 3 3.5 4

Time (sec) Statet x2

20 40 60 −1 1 2 3 4

Time (sec) Estimation error e1

20 40 60 −1.5 −1 −0.5 0.5 1 1.5 2

Time (sec) Estimation error e2

Linear Observer Nonlinear Observer Linear Observer Nonlinear Observer

Figure: Behavior of Plant and the Non Linear Observer with UI with large initial conditions.

24 April 2012, Jaime A. Moreno Discontinuous Observers 29 / 59

Effect: Convergence time independent of I.C.

10

1

10

2

10

3

10

4

2 4 6 8 10 12 14 16 norm of the initial condition ||x(0)|| (logaritmic scale) Convergence Time T NSOSMO GSTA with linear term STO

Figure: Convergence time when the initial condition grows.

24 April 2012, Jaime A. Moreno Discontinuous Observers 30 / 59

Recapitulation.

Generalized Super-Twisting Observer for Linear Plant If no unknown inputs/Uncertainties: e1 and e2 converge in finite-time!

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 in finite-time! Observer is insensitive to perturbation/uncertainty!

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 in finite-time! Observer is insensitive to perturbation/uncertainty! Convergence time is independent of the initial conditions of the

• bserver!.

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 in finite-time! Observer is insensitive to perturbation/uncertainty! Convergence time is independent of the initial conditions of the

• bserver!.

All objectives were achieved!

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 in finite-time! Observer is insensitive to perturbation/uncertainty! Convergence time is independent of the initial conditions of the

• bserver!.

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 initial condition. 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 Observer Generalized Super-Twisting Observers

3 Lyapunov Approach for Second-Order Sliding Modes

GSTA without perturbations: ALE GSTA with perturbations: ARI

4 Optimality of the ST with noise 5 A Recursive Finite-Time Convergent Parameter Estimation Algorithm

The Classical Algorithm The 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 are

Quadratic-like

2 This family allows the estimation of convergence time, 3 It allows to study the robustness of the algorithm to different kinds of

perturbations,

4 All results are obtained in a Linear-Like framework, known from

classical control,

5 The analysis can be obtained in the same manner for a linear

algorithm, the classical ST algorithm and a combination of both algorithms (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 Observer Generalized Super-Twisting Observers

3 Lyapunov Approach for Second-Order Sliding Modes

GSTA without perturbations: ALE GSTA with perturbations: ARI

4 Optimality of the ST with noise 5 A Recursive Finite-Time Convergent Parameter Estimation Algorithm

The Classical Algorithm The 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|

1 2 sign (e1) + µ2 |e1|q sign (e1) , µ1 , µ2 ≥ 0 , q ≥ 1 ,

φ2 (e1) = µ2

1

2 sign (e1) +

• q + 1

2

• µ1µ2 |e1|q− 1

2 sign (e1) +

+ qµ2

2 |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

• .

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

Figure: The Lyapunov function.

24 April 2012, Jaime A. Moreno Discontinuous Observers 36 / 59

Lyapunov Function

Proposition

If A Hurwitz then x = 0 Finite-Time stable (if µ1 = 1) and for every Q = QT > 0, V (x) = ζTPζ is a global, strong Lyapunov function, with P = PT > 0 solution of the ALE, and ˙ V ≤ −γ1 (Q, µ1) V

1 2 (x) − γ2 (Q, µ2) V (x) ,

where γ1 (Q, µ1) µ1 λmin {Q} λ

1 2

min{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 at x0 ∈ R2 converges to the origin in finite time if µ1 = 1, and it reaches that point at most after a time T =   

2 γ1(Q,µ1)V

1 2 (x0)

if µ2 = 0

2 γ2(Q,µ2) ln

• γ2(Q,µ2)

γ1(Q,µ1)V

1 2 (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 Observer Generalized Super-Twisting Observers

3 Lyapunov Approach for Second-Order Sliding Modes

GSTA without perturbations: ALE GSTA with perturbations: ARI

4 Optimality of the ST with noise 5 A Recursive Finite-Time Convergent Parameter Estimation Algorithm

The Classical Algorithm The 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) . Assume 2 |ρ (t, x)| ≤ δ Analysis: The construction of Robust Lyapunov Functions can be done with the classical method of solving an Algebraic Ricatti Inequality (ARI),

• r equivalently, solving the LMI

ATP + PA + ǫP + δ2C TC PB BTP −1

• ≤ 0 ,

where A = −k1 1 −k2

• , C =
• 1
• , B =

1

• .

24 April 2012, Jaime A. Moreno Discontinuous Observers 39 / 59

Overview

1 Introduction 2 Observers a la Second Order Sliding Modes (SOSM)

Super-Twisting Observer Generalized Super-Twisting Observers

3 Lyapunov Approach for Second-Order Sliding Modes

GSTA without perturbations: ALE GSTA with perturbations: ARI

4 Optimality of the ST with noise 5 A Recursive Finite-Time Convergent Parameter Estimation Algorithm

The Classical Algorithm The 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.

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: ˙ ˆ x1 = − α1

ε φ1(ˆ

x1 − y) + ˆ x2, ˙ ˆ x2 = − α2

ε2 φ2(ˆ

x1 − y), where φ1(x) = µ1|x|

1 2 sign(x) + µ2x,

µ1, µ2 ≥ 0 φ2(x) = 1

2µ2 1sign(x) + 3 2µ1µ2|x|

1 2 sign(x) + µ2

2x,

24 April 2012, Jaime A. Moreno Discontinuous Observers 40 / 59

µ1 = 0: High Gain Observer

24 April 2012, Jaime A. Moreno Discontinuous Observers 41 / 59

µ1 = 0: High Gain Observer µ2 = 0: Super-Twisting differentiator.

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.

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” of the differentiation error ˆ x2 − x2.

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” of the differentiation error ˆ 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, unless

• therwise stated.

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” of the differentiation error ˆ 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, unless

• therwise stated.

Differentiation error ˜ x = ˆ x − x ˙ ˜ x1 = − α1

ε φ1(˜

x1 − η) + ˜ x2, ˙ ˜ x2 = − α2

ε2 φ2(˜

x1 − η) + ρ,

24 April 2012, Jaime A. Moreno Discontinuous Observers 41 / 59

The performance of linear and ST differentiators.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6

gain ε differentiation error

0.5 1 1.5 2 2.5 3 3.5 4 1 2 3 4 5 6 7

gain ε differentiation error

1 2 3 4 5 6 7 8 0.5 1 1.5 2 2.5 3

gain ε differentiation error

Figure: Ultimate bound of the differentiation error as a function of ε. Solid: linear case µ1 = 0, µ2 = 1; dashed: pure ST case µ1 = 1, µ2 = 0. Left: parameters L = 1, δ = 0.01; middle: parameters L = 1, δ = 1; right: parameters L = 0.1, δ = 1.

24 April 2012, Jaime A. Moreno Discontinuous Observers 42 / 59

1 2 3 4 5 6 7 8 0.1 0.2 0.3 0.4 0.5 0.6

gain ε

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

gain ε

Figure: Steady state error in the differentiation versus the gain from L = 1, δ = 0.01 to L = 0.001, δ = 0.01. The figure on the right is a zoom of the left figure. Solid: linear case µ1 = 0, µ2 = 1; dashed: pure ST case µ1 = 1, µ2 = 0.

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Advantages of the GST

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 0.7 0.8 0.9

gain ε differentiation error

Figure: Ultimate bound of the differentiation error as a function of ε, for L = 1 and δ = 0.01. Solid: linear case µ1 = 0, µ2 = 1; dashed: pure ST case µ1 = 1, µ2 = 0; dotted: two experiments µ1 = 0.8, µ2 = 0.2 and µ1 = 0.5, µ2 = 0.5 (with circles).

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The sensitivity to variations in the noise amplitude

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.5 1 1.5 2 2.5

noise amplitude δ differentiation error

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.5 1 1.5

noise amplitude δ differentiation error

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: GST case with µ1 = µ2 = 0.5. Dash-dot: the nominal noise 0.01 for which the

• ptimal gains are selected. Left: for L = 1, Right: for L = 0.

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Sensitivity to variations in the perturbation amplitude.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

perturbation amplitude L differentiation error

Figure: The differentiation error as a function of the perturbation amplitude, for δ = 0.01. Solid: linear µ1 = 0, µ2 = 1; dashed: ST µ1 = 1, µ2 = 0; dotted: GST µ1 = µ2 = 0.5. Dash-dotted: the nominal perturbation L = 1 for which the

• ptimal gains are selected.

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5 10 15 20 25 30 35 40 45 50 1 1.5 2 2.5 3 3.5 4 4.5 5

Time (sec) Statet x2

5 10 15 20 25 30 35 40 45 50 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5

Time (sec) Estimation error e2

Linear Observer Nonlinear Observer

Figure: Behavior of Linear and ST Observers under noise and perturbation.

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Overview

1 Introduction 2 Observers a la Second Order Sliding Modes (SOSM)

Super-Twisting Observer Generalized Super-Twisting Observers

3 Lyapunov Approach for Second-Order Sliding Modes

GSTA without perturbations: ALE GSTA with perturbations: ARI

4 Optimality of the ST with noise 5 A Recursive Finite-Time Convergent Parameter Estimation Algorithm

The Classical Algorithm The 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 and parameter 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 Observer Generalized Super-Twisting Observers

3 Lyapunov Approach for Second-Order Sliding Modes

GSTA without perturbations: ALE GSTA with perturbations: ARI

4 Optimality of the ST with noise 5 A Recursive Finite-Time Convergent Parameter Estimation Algorithm

The Classical Algorithm The 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 piecewise continuous 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)

• is

uniformly bounded, there exist T > 0, ǫ > 0 such that for any unit vector w ∈ Rn2

t+T

t

B(τ)w dτ ≥ ǫ (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 ˙ ˆ y = α(x, t) + Γ(t) ˆ θ − kyey ˙ ˆ θ = −kθΓT(t)ey Defining errors as ey = ˆ y − y and eθ = ˆ θ − θ. Error Dynamics ˙ ey = −kyey + Γ(t)eθ ˙ ˆ 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 Observer Generalized Super-Twisting Observers

3 Lyapunov Approach for Second-Order Sliding Modes

GSTA without perturbations: ALE GSTA with perturbations: ARI

4 Optimality of the ST with noise 5 A Recursive Finite-Time Convergent Parameter Estimation Algorithm

The Classical Algorithm The Proposed Algorithm

6 Conclusions

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˙ 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

1

2 sign (x1) + 2 3µ1µ2 |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, ǫ0, δ0, with t2 ∈ [t, t + T] such that for any unit vector w ∈ Rn2

• 1

T0

t2+δ0

t2

B(τ)w dτ

• ≥ ǫ0

(5)

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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.

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Case (µ1, µ2, p, q) Linear (0,1,–,1) STA (1,0,1/2,–) GSTA (1,1,1/2,q)

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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

ζT

2

=

• 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, ˙ ˆ y = −k1φ1(ey) + Γ (t) ˆ θ ˙ ˆ θ = −k2φ2(ey)ΓT (t) (7) φ1(ey) = µ1 |ey|1/2 sign (ey) + µ2ey, µ1, µ2 > 0 (8) φ2(ey) = µ2

1

2 sign (ey) + 3 2µ1µ2 |ex|1/2 sign (ey) + µ2

2ey

(9) Estimation Error Dynamics, where ey = ˆ y − y, eθ = ˆ θ − θ ˙ ey = −k1φ1(ey) + Γ (t) eθ (10a) ˙ eθ = −k2φ2(ey)ΓT (t) (10b)

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Example: A simple pendulum

˙ x1 = x2 (11a) ˙ x2 = 1 J u − Mgl 2J sin x1 − Vs J x2 (11b) x1 angular position, x2 angular velocity, M mass, g gravity, L rope’s length mass and J = ML2 mass inertia. Parameter’s estimator ˙ ˆ x2 = −k1φ1(ex2) + Γ ˆ θ (12a) ˙ ˆ θ = −k2φ2(ex2)ΓT (12b) ex2 = ˆ x2 − x2 velocity’s error, Γ =

• x2

sin x1 u

• regressor,

ˆ θ =

• − ˆ

Vs ˆ J

−

ˆ M ˆ g ˆ L ˆ J 1 ˆ J

• vector of unknown parameters, φ1(·) and φ2(·)

are given as in the GSTA.

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Parameter’s estimation

10 20 30 40 50 −2.5 −2 −1.5 −1 −0.5 0.5

Time (sec)

θ1 and ˆ θ1 θ1 ˆ θ1 with linear algorithm ˆ θ1 with nonlinear algorithm 10 20 30 40 50 −12 −10 −8 −6 −4 −2

Time (sec)

θ2 and ˆ θ2 θ2 ˆ θ2 with linear algorithm ˆ θ2 with nonlinear algorithm 10 20 30 40 50 −1 −0.5 0.5 1 1.5 2 2.5

Time (sec)

θ3 and ˆ θ3 θ3 ˆ θ3 with linear algorithm ˆ θ3 with nonlinear algorithm

Figure: Parameter estimation of θ1, θ2 and θ3 with GSTA and with linear algorithm.

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Parameter’s estimation, with perturbation in θ1 = −0.2 + 0.2sin(t)

10 20 30 40 50 −2 −1.5 −1 −0.5 0.5

Time (sec)

θ1 and ˆ θ1 θ1 ˆ θ1 with linear algorithm ˆ θ1 with nonlinear algorithm 10 20 30 40 50 −12 −10 −8 −6 −4 −2 2

Time (sec)

θ2 and ˆ θ2 θ2 ˆ θ2 with linear algorithm ˆ θ2 with nonlinear algorithm 10 20 30 40 50 −1 −0.5 0.5 1 1.5 2

Time (sec)

θ3 and ˆ θ3 θ3 ˆ θ3 with linear algorithm ˆ θ3 with nonlinear algorithm

Figure: Parameter estimation of θ1, θ2 and θ3 with GSTA and with linear algorithm.

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Overview

1 Introduction 2 Observers a la Second Order Sliding Modes (SOSM)

Super-Twisting Observer Generalized Super-Twisting Observers

3 Lyapunov Approach for Second-Order Sliding Modes

GSTA without perturbations: ALE GSTA with perturbations: ARI

4 Optimality of the ST with noise 5 A Recursive Finite-Time Convergent Parameter Estimation Algorithm

The Classical Algorithm The Proposed Algorithm

6 Conclusions

24 April 2012, Jaime A. Moreno Discontinuous Observers 58 / 59

Conclusions

1 The GSTO is an observer that is

Robust: it converges despite of unknown inputs/uncertainties Exact: it converges in finite-time Preassigned Convergence time for any initial condition. But there is no free lunch!

2 It can be extended to estimate parameters in finite time with

applications in Adaptive Control

3 Applications for Bioreactors seem to be attractive due to:

Robustness against uncertainties Possibility 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, ...

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Thank you!

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Overview

7 Some Applications

Finite-Time, robust MRAC

24 April 2012, Jaime A. Moreno Discontinuous Observers 59 / 59

Overview

7 Some Applications

Finite-Time, robust MRAC

24 April 2012, Jaime A. Moreno Discontinuous Observers 59 / 59

Introduction

Direct Model Reference Adaptive Control (MRAC) is a well-known approach for adaptive control of linear and some nonlinear systems.

24 April 2012, Jaime A. Moreno Discontinuous Observers 60 / 59

Introduction

Direct Model Reference Adaptive Control (MRAC) is a well-known approach for adaptive control of linear and some nonlinear systems. If Plant has relative degree n∗ = 1 and the reference model is Strictly Positive Real (SPR), the controller is particularly simple to implement, and to design.

24 April 2012, Jaime A. Moreno Discontinuous Observers 60 / 59

Introduction

Direct Model Reference Adaptive Control (MRAC) is a well-known approach for adaptive control of linear and some nonlinear systems. If Plant has relative degree n∗ = 1 and the reference model is Strictly Positive Real (SPR), the controller is particularly simple to implement, and to design. The Adjustment Mechanism of the MRAC is basically a parameter estimation algorithm.

24 April 2012, Jaime A. Moreno Discontinuous Observers 60 / 59

Introduction

Direct Model Reference Adaptive Control (MRAC) is a well-known approach for adaptive control of linear and some nonlinear systems. If Plant has relative degree n∗ = 1 and the reference model is Strictly Positive Real (SPR), the controller is particularly simple to implement, and to design. The Adjustment Mechanism of the MRAC is basically a parameter estimation algorithm. Our objective: Modify the Adjustment Mechanism of the classical Direct MRAC by adding the Super-Twisting-Like nonlinearities, so as to achieve the finite-time convergence and robustness properties.

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MRAC with relative degree n∗ = 1

Figure: General structure of MRAC scheme.

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SISO Plant: yp = Gp(s)up = kp

Zp(s) Rp(s), Relative degree n∗ = 1.

Reference model: ym = Wm(s)r = km

Zm(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).

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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

wT

2

yp r T , θ∗ =

• θ∗T

1

θ∗T

2

θ∗T

3

c∗ 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 kp km

• , Γ = ΓT > 0 .

(17)

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Classical MRAC

Theorem

The MRAC scheme guarantees that:

1 All signals in the closed-loop plant are bounded and the tracking error

e1 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) are relatively coprime, then the parameter error | ˜ θ| = |θ − θ∗| and the tracking error e1 converge to zero exponentially fast.

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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

1

2 sign (e1) + 3 2µ1µ2 |e1|1/2 sign (e1) + µ2 2e1

(18) and µ1 > 0, µ2 > 0.

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Theorem

The modified MRAC scheme guarantees that:

1 All signals in the closed-loop plant are bounded and the tracking error

e1 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) are relatively coprime, then the parameter error | ˜ θ| = |θ − θ∗| and the tracking 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+1up .

24 April 2012, Jaime A. Moreno Discontinuous Observers 67 / 59

Example

Second order plant yp =

(s+1) s2−3s+1up .

Reference model ym =

1 s+1r .

24 April 2012, Jaime A. Moreno Discontinuous Observers 67 / 59

Example

Second order plant yp =

(s+1) s2−3s+1up .

Reference model ym =

1 s+1r .

Nominal controller ˙ w1 = −2w1 + up, w1(0) = 0 ˙ w2 = −2w2 + yp, w2(0) = 0 up = θ1w1 + θ2w2 + θ3yp + c0r , (19) θ =

• θ1

θ2 θ3 c0 T, w =

• w1

w2 yp r T, and

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Example

Second order plant yp =

(s+1) s2−3s+1up .

Reference model ym =

1 s+1r .

Nominal controller ˙ w1 = −2w1 + up, w1(0) = 0 ˙ w2 = −2w2 + yp, w2(0) = 0 up = 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∗ T =

• 0 ,

6 , −5 , 1 T.

24 April 2012, Jaime A. Moreno Discontinuous Observers 67 / 59

Example

Second order plant yp =

(s+1) s2−3s+1up .

Reference model ym =

1 s+1r .

Nominal controller ˙ w1 = −2w1 + up, w1(0) = 0 ˙ w2 = −2w2 + yp, w2(0) = 0 up = 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∗ 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

5 10 15 20 25 −6 −4 −2 2 4 6 8

Time (sec) ym and yp

Model reference output Plant output

Figure: Model ym (continuous line) and the Plant’s output yp (dotted line) with the classical MRAC scheme with reference signal r = 5 cos (t) + 10 cos (5t).

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5 10 15 20 25 −6 −4 −2 2 4 6

Time (sec) yp and ym

Model reference output Plant output

Figure: Model ym (continuous line) and the Plant’s output yp (dotted line) with the nonlinear MRAC scheme with reference signal r = 5 cos (t) + 10 cos (5t).

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5 10 15 20 25 −4 −3 −2 −1 1 2 3 4

Time (sec) Error

5 10 15 20 25 −4 −3 −2 −1 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).

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5 10 15 20 25 −40 −30 −20 −10 10 20 30

Time (sec) Control

Linear control Nonlinear control Figure: Control variable up for the classical MRAC (continuous line) and the proposed nonlinear MRAC (dotted line) with reference signal r = 5 cos (t) + 10 cos (5t).

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5 10 15 20 25 30 35 40 −10 10 Time (sec) A) 5 10 15 20 25 30 35 40 −10 10 Time (sec) B) 5 10 15 20 25 30 35 40 −10 10 Time (sec) C) 5 10 15 20 25 30 35 40 −5 5 Time (sec) D)

linear estimation nonlinear estimation

• riginal value

Figure: Parameter convergence to the real values with reference signal r = 5 sin (t) + 10 sin (5t). A) θ∗

1 = 0, B) θ∗ 2 = 6, C) θ∗ 3 = −5 and D) c∗ 0 = 1

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Persistence of Excitation conditions, with perturbations

5 10 15 20 25 −4 −3 −2 −1 1 2 3 4

Time (sec)

Error 5 10 15 20 25 30 35 40 −4 −3 −2 −1 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), with perturbation p (t) = 5 sin (6t).

24 April 2012, Jaime A. Moreno Discontinuous Observers 73 / 59

5 10 15 20 25 30 35 40 −10 10 Time (sec) 5 10 15 20 25 30 35 40 −10 10 Time (sec) 5 10 15 20 25 30 35 40 −20 20 Time (sec) 5 10 15 20 25 30 35 40 −10 10 Time (sec)

Linear estimation Nonlinear estimation Real value

Figure: Parameter convergence to the real values with reference signal r = 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.

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Lack of Persistence of Excitation conditions

5 10 15 −2 −1 1 2 3 4 5 6 7

Time (sec)

yp and ym

Model reference output Plant output

Figure: Model ym (continuous line) and the Plant’s output yp (dotted line) with the classical MRAC scheme with constant reference signal r = 6.

24 April 2012, Jaime A. Moreno Discontinuous Observers 75 / 59

5 10 15 −1 1 2 3 4 5 6

Time (sec) yp and ym Model reference output Plant output Figure: Model ym (continuous line) and the Plant’s output yp (dotted line) with the nonlinear MRAC scheme with reference signal r = 6.

24 April 2012, Jaime A. Moreno Discontinuous Observers 76 / 59

5 10 15 −15 −10 −5 5 10 15 20

Time (sec) Control Linear control Nonlinear control Figure: Control variable up for the classical MRAC (continuous line) and the proposed nonlinear MRAC (dotted line) with reference signal r = 6.

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Adetola, V. and Guay, M. (2008). Finite-time parameter estimation in adaptive control of nonlinear systems. IEEE Transactions on Automatic Control, 53(3):pp. 807–811. Davila, J., Fridman, L., and Levant, A. (2005). Second order sliding-modes observer for mechanical systems. IEEE Transactions on Automatic Control, 50(11):pp. 2292–2299. Davila, J., Fridman, L., and Poznyak, A. (2006). Observation and identification of mechanical systems via second order sliding modes. Journal of Control., 79(10):pp. 1251–1262. Ioannou, P. A. and Findan, B. (2006). Adaptive Control Tutorial. Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia.

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THANKS

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