Nonlinear observer design
1 like1,440 views
The document discusses the Frobenius theorem related to state observability and observer design, detailing concepts such as distributions, integrability, and observer linearization. It includes definitions, theorems, and examples of mathematical structures involved in control systems design. Key notions include nonsingular distributions, codistributions, and local coordinate transformations essential for observer design in dynamic systems.
![The Frobenius Theorem
Introduction to Distributions
State Observability
Duality and Integrability
Analysis and Design
The Frobenius Theorem
Literature
Lie Algebras on Mn ( ) and C ∞ ( n
).
A Lie Algebra is a linear vector space L over a field F endowed
with a bilinear operation [·, ·] : L × L → L, such that:
1. [x, x] = 0 for every x ∈ L
2. [x[yz]] + [y [zx]] + [z[xy ]] = 0 for every x, y , z ∈ L
This operator is known as Lie Bracket or Commutator.
1. If L = Mn ( ) then the commutator is (usually) defiend to be
[A, B] = AB − BA for every A, B ∈ Mn ( ).
2. If L = C inf ( n) then [f , g ] = g ·f − f ·g
Sopasakis Pantelis State Observer Design](https://image.slidesharecdn.com/nonlinearobserver-121229075729-phpapp02/85/Nonlinear-observer-design-5-320.jpg)
![The Frobenius Theorem
Introduction to Distributions
State Observability
Duality and Integrability
Analysis and Design
The Frobenius Theorem
Literature
Involutive Distributions
A distribution D ∈ D(U) is called involutive if [f , g ] ∈ D for every
f , g ∈ D.
Sopasakis Pantelis State Observer Design](https://image.slidesharecdn.com/nonlinearobserver-121229075729-phpapp02/85/Nonlinear-observer-design-6-320.jpg)
![The Frobenius Theorem
State Observability Local Decompositions of Control Systems
Analysis and Design State Observability
Literature
f -invariant distributions
Let D ∈ D(U) and f : n → n . The distribution D is said to be
f -invariant if for every τ ∈ D : [f , τ ] ∈ D.
Sopasakis Pantelis State Observer Design](https://image.slidesharecdn.com/nonlinearobserver-121229075729-phpapp02/85/Nonlinear-observer-design-11-320.jpg)
![State Observers
The Frobenius Theorem
Observer Linearization Problem
State Observability
Observer Canonical Form
Analysis and Design
Observer Design
Literature
Extended Linearization Design Method
Observer Error Dynamics
˙
Let us define e = x − ξ, with ξ = f (ξ, u) + g (y ) − g (ˆ ). Then:
y
˙ ˙ ˙
e = x − ξ = f (x, u) − f (x − e, u) − g (y ) + g (ˆ )
y
for (x, u) close to (xε , ε) we have:
e = [D1 f (xε , ε) − Dg (yε ) Dh (xε )] e = Ye
˙
The aim of the design consists in determining proper analytic
function g such that Y perserves constant stable eigenvalues -
independent of ε!
Sopasakis Pantelis State Observer Design](https://image.slidesharecdn.com/nonlinearobserver-121229075729-phpapp02/85/Nonlinear-observer-design-24-320.jpg)
![State Observers
The Frobenius Theorem
Observer Linearization Problem
State Observability
Observer Canonical Form
Analysis and Design
Observer Design
Literature
Extended Linearization Design Method
Assumptions
We assume that the following hold:
1. D1 f (0, 0)−1 exists
2. (D1 f (0, 0) , Dh (0)) is observable
∂yε
3. ∂ε |ε=0= Dyε |ε=0 = Dh (0) Dxε (0) =
−Dh (0) [D1 f (0, 0)]−1 = 0
Sopasakis Pantelis State Observer Design](https://image.slidesharecdn.com/nonlinearobserver-121229075729-phpapp02/85/Nonlinear-observer-design-25-320.jpg)
Nonlinear observer design
- 1. The Frobenius Theorem State Observability Analysis and Design Literature State Observer Design Sopasakis Pantelis October 5, 2012 Sopasakis Pantelis State Observer Design
- 2. The Frobenius Theorem Introduction to Distributions State Observability Duality and Integrability Analysis and Design The Frobenius Theorem Literature What is a Distribution A distribution is a mapping D : n → V( n ) which maps every − x ∈ n to a linear subspace of n according to the formula D(x) = span{f1 (x) , f2 (x) , . . . , fp (x)}, where fi : n → n are the − generator vector fields. The set of all distributions defined U ⊂ n will be denoted as D(U). Sopasakis Pantelis State Observer Design
- 3. The Frobenius Theorem Introduction to Distributions State Observability Duality and Integrability Analysis and Design The Frobenius Theorem Literature Some Definitions A distribution is called nonsingular or a distribution of constant-degree k if dimD(x) = k for every x ∈ n . Let D ∈ D(U). A x0 ∈ U is said to be a regular point of D if there exists an open neighborhood U0 of x0 such that D is nonsingular in U0 . A distribution D is called smooth if there exist vector fields {fi }i∈F that span D. Sopasakis Pantelis State Observer Design
- 4. The Frobenius Theorem Introduction to Distributions State Observability Duality and Integrability Analysis and Design The Frobenius Theorem Literature A fundamental result on smooth distributions Let D ∈ D(U) be a nonsingular smooth distribution of constant degree k and Y : U → n a smooth vector-valued function in D, − i.e Y (x) ∈ D(x) for every x ∈ U. Then there exist k smooth real-valued functions mj : U → such that k Y (x) = mj (x)Xj (x) j=1 where Xj ∈ D. Sopasakis Pantelis State Observer Design
- 5. The Frobenius Theorem Introduction to Distributions State Observability Duality and Integrability Analysis and Design The Frobenius Theorem Literature Lie Algebras on Mn ( ) and C ∞ ( n ). A Lie Algebra is a linear vector space L over a field F endowed with a bilinear operation [·, ·] : L × L → L, such that: 1. [x, x] = 0 for every x ∈ L 2. [x[yz]] + [y [zx]] + [z[xy ]] = 0 for every x, y , z ∈ L This operator is known as Lie Bracket or Commutator. 1. If L = Mn ( ) then the commutator is (usually) defiend to be [A, B] = AB − BA for every A, B ∈ Mn ( ). 2. If L = C inf ( n) then [f , g ] = g ·f − f ·g Sopasakis Pantelis State Observer Design
- 6. The Frobenius Theorem Introduction to Distributions State Observability Duality and Integrability Analysis and Design The Frobenius Theorem Literature Involutive Distributions A distribution D ∈ D(U) is called involutive if [f , g ] ∈ D for every f , g ∈ D. Sopasakis Pantelis State Observer Design
- 7. The Frobenius Theorem Introduction to Distributions State Observability Duality and Integrability Analysis and Design The Frobenius Theorem Literature Duality A codistribution is a mapping W : n → V(( n ) ) which maps every x ∈ n to a linear subspace of ( n ) . The set of codistributions defined on a subset U ⊂ n is denoted by D (U). The annihilator of a distribution D ∈ D(U) is a codistribution D⊥ ∈ D (U) defined as D⊥ (x) = {w ∈ ( n ) : w , u = 0, ∀u ∈ D(x)} It is remarkable that dim(D) + dim(D ) = n. Sopasakis Pantelis State Observer Design
- 8. The Frobenius Theorem Introduction to Distributions State Observability Duality and Integrability Analysis and Design The Frobenius Theorem Literature The Distribution Integrability Problem Problem Formulation: Given a distribution DF ∈ D(U) of constant degree p which is spanned by the k ≥ p columns of a mapping F : n → Mn×k ( ) specify necessary and sufficient conditions such that there exist n − p vector fields λ1 , λ2 , . . . , λp : U → n that their derivatives annigilate the vector fields in D, i.e. dλj · F (x) = 0 Sopasakis Pantelis State Observer Design
- 9. The Frobenius Theorem Introduction to Distributions State Observability Duality and Integrability Analysis and Design The Frobenius Theorem Literature Complete Integrability Let D be a nonsingular distribution of constant degree d defined on an open set U ⊂ n and U = n . The distribution D is called completely integrable if for each x0 ∈ U there exists an open neighborhood U0 of x0 and n − d real-valued functions λ1 , λ2 , . . . , λn−d , defined on U0 such that dλ1 , dλ2 , . . . , dλn−d span the annihilator of D. Sopasakis Pantelis State Observer Design
- 10. The Frobenius Theorem Introduction to Distributions State Observability Duality and Integrability Analysis and Design The Frobenius Theorem Literature The Frobenius Theorem Let D ∈ D(U) be a nonsingular distribution of constant degree d and U = n . The following are equivalent: 1. D is completely integrable 2. D is involutive Sopasakis Pantelis State Observer Design
- 11. The Frobenius Theorem State Observability Local Decompositions of Control Systems Analysis and Design State Observability Literature f -invariant distributions Let D ∈ D(U) and f : n → n . The distribution D is said to be f -invariant if for every τ ∈ D : [f , τ ] ∈ D. Sopasakis Pantelis State Observer Design
- 12. The Frobenius Theorem State Observability Local Decompositions of Control Systems Analysis and Design State Observability Literature ∗ Local Inner Triangular Decomposition Let D be a distribution posessing the following properties: 1. D is nonsingular, of constant degree d and involutive. 2. D is f -invariant for some f : n → n Then for every x0 ∈ U there exists a neighborhood U0 x0 and a coordinate transormation z = Φ(x); x ∈ U, such that: f1 (z1 , z2 , . . . , zd , zd+1 , . . . , zn ) f2 (z1 , z2 , . . . , zd , zd+1 , . . . , zn ) .. ˆ(z) = f (Φ−1 (x)) = . f fd (zd , zd+1 , . . . , zn ) .. . fn (zd , zd+1 , . . . , zn ) A.J.Kerner, Normal Forms for linear and nonlinear systems, Contemp. Math. 68, 157-189, 1987 Sopasakis Pantelis State Observer Design
- 13. The Frobenius Theorem State Observability Local Decompositions of Control Systems Analysis and Design State Observability Literature ∗∗ How much state can we know? For a dynamic system x = f (x) + m gi (x)ui ˙ i=1 y = h(x), y ∈ p Let its triangular representation be: ζ1 = θ1 (ζ1 , ζ2 ) + m γ1,i (ζ1 , ζ2 ) ui ˙ i=1 ζ2 = θ2 (ζ2 ) + m γ2,i (ζ2 ) ui ˙ i=1 yi = hi (ζ2 ) for x ∈ U0 x0 . The “unobservable” manifold of the system is the slice: Sx = {υ ∈ U0 : ζ2 (υ) = ζ2 (x)} S.R.Kou, D.L.Eliot and T.J.Tarn, Observability of Nonlinear Systems, Information and Control 22(1), 89-99, 1972 Sopasakis Pantelis State Observer Design
- 14. State Observers The Frobenius Theorem Observer Linearization Problem State Observability Observer Canonical Form Analysis and Design Observer Design Literature Extended Linearization Design Method Observers An observer is a dynamic system such that its output converges to the state of a given system as t → ∞, that is t→∞ ξ (t) − x (t) − − 0 −→ Sopasakis Pantelis State Observer Design
- 15. State Observers The Frobenius Theorem Observer Linearization Problem State Observability Observer Canonical Form Analysis and Design Observer Design Literature Extended Linearization Design Method ∗∗∗ The Observer Linearization Problem Given a dynamical system x = f (x), y = h(x) with scalar output y ˙ and an initial state x0 , specify a neighborhood U0 x0 and a local coord. transf. z = Φ(x) and a mapping k : h(U0 ) → n such that: ∂Φ ∂x f (x) x=Φ−1 (z) = Az + k(Cz) h(Φ−1 (z)) = Cz for z ∈ Φ(U0 ) and A ∈ Mn ( ) and C T ∈ n such that: C CA rank = n ⇔ (C , A) is observable . . . CAn−1 Sopasakis Pantelis State Observer Design
- 16. State Observers The Frobenius Theorem Observer Linearization Problem State Observability Observer Canonical Form Analysis and Design Observer Design Literature Extended Linearization Design Method ∗∗∗∗ Solvability of the Observer Linearization Problem The OLP is solvable at x0 only if the following conditions holds: dim span dh|x0 , d (fh) |x0 , d f 2 h |x0 , . . . , d f n−1 h |x0 = n where fh = f (h) = dh, f = Lf h and f k h = Lk h f A. Isidori, Nonlinear Control Systems - An Introduction, Springer-Verlag editions, 2ns ed, 1989, pp.217-226 Sopasakis Pantelis State Observer Design
- 17. State Observers The Frobenius Theorem Observer Linearization Problem State Observability Observer Canonical Form Analysis and Design Observer Design Literature Extended Linearization Design Method Solvability of the Observer Linearization Problem The OLP is solvable at x0 iff the following conditions hold: 1. dim span dh|x0 , d (fh) |x0 , d f 2 h |x0 , . . . , d f n−1 h |x0 = n 2. The unique vector field τ which satisfies dh|x0 0 d (fh) |x 0 0 d f 2 h |x . ·τ = . . 0 . . . 0 d f n−1 h |x0 1 is such that adfi τ, adfj τ = 0 for every 0 ≤ i and j ≤ n − 1 Sopasakis Pantelis State Observer Design
- 18. State Observers The Frobenius Theorem Observer Linearization Problem State Observability Observer Canonical Form Analysis and Design Observer Design Literature Extended Linearization Design Method Observer Canonical Form Given an observable system x = f (x) with x ∈ n and y = h(x) ˙ with y ∈ find a local coord. trans x = T (x ∗ ) s.t. 0 ··· 0 ∗ ∗ f0 (xn ) . . f ∗ (x ∗ ) 1 . ∗ 1 n x∗ = ˙ .. . x − . = f ∗ (x ∗ ) . . . . . ∗ (x ∗ ) fn−1 n 1 0 y= 0 ... 0 1 x ∗ = ηT x ∗ X.H. Xia, W.B. Gao, Nonlinear observer design by observer canonical forms, Int. J. contr. 47(4) 2988, 1081-1100 Sopasakis Pantelis State Observer Design
- 19. State Observers The Frobenius Theorem Observer Linearization Problem State Observability Observer Canonical Form Analysis and Design Observer Design Literature Extended Linearization Design Method ∗∗∗∗∗ Inverted Pendulum - OCF Suppose of the system: x1 ˙ x2 x2 = sinx1 + x3 = f (x) ˙ x3 ˙ x2 + x3 y = x1 = h (x) Hint: O(x) · ∂T ∗ ∂x1 = η and ∂T ∂x ∗ = ∂T adf0 ∂x ∗ · · · adfn−1 ∂x ∗ ∂T 1 1 Sopasakis Pantelis State Observer Design
- 20. State Observers The Frobenius Theorem Observer Linearization Problem State Observability Observer Canonical Form Analysis and Design Observer Design Literature Extended Linearization Design Method Inverted Pendulum - OCF The OCF of the inverted pendulum nonlinear system is: ∗ 0 0 0 sinx3 ∗ ∗ x ∗ = 1 0 0 x ∗ + x3 + sinx3 = f ∗ (x ∗ ) ˙ 0 1 0 ∗ x3 ∗ y = x3 Sopasakis Pantelis State Observer Design
- 21. State Observers The Frobenius Theorem Observer Linearization Problem State Observability Observer Canonical Form Analysis and Design Observer Design Literature Extended Linearization Design Method Observer Design Based on the OCF With every observable system in the OCF we associate the following observer: 0 0 0 ··· 0 f0∗ (xn ) ∗ 1 0 0 ··· 0 ∗ (x ∗ ) f1 n ξ = 0 1 0 ··· ˙ 0 − − KTe . . . . .. . . . . . . . . . ∗ ∗ fn−1 (xn ) 0 0 ··· 1 0 where e = ξ − x ∗ and K T = k0 k1 · · · kn−1 ∈ n Sopasakis Pantelis State Observer Design
- 22. State Observers The Frobenius Theorem Observer Linearization Problem State Observability Observer Canonical Form Analysis and Design Observer Design Literature Extended Linearization Design Method Error Dynamics The error e ∗ = ξ − x ∗ evolves with respect to the linear dynamics: 0 0 0 ··· −k0 1 0 0 ··· −k1 de 0 1 0 ··· −k2 e = Y · e dt = . . .. . . . . . . . . 0 0 · · · 1 −kn−1 The characteristic polynomial of the matrix Y is χ (Y ) (s) = k0 + k1 s + . . . + kn−1 s n−1 + s n Sopasakis Pantelis State Observer Design
- 23. State Observers The Frobenius Theorem Observer Linearization Problem State Observability Observer Canonical Form Analysis and Design Observer Design Literature Extended Linearization Design Method Introduction to Extended Linearization Extended Linearization is a method to tackle the observer design problem for control systems x = f (x, u) ˙ y = h (x) x ∈ n, y ∈ p , u ∈ f (0, 0) = 0, h (0) = 0 Let {u = ε, x = xε , f (xε , ) = 0} be a collection of equillibrium points. We assume that the observer admits the representation: ˙ ξ = f (ξ, u) + g (y ) − g (ˆ ) y B.Walcott et al., Comparative study of nonlinear state observation techniques, Int. J. Contr. 45(6) 1987, 2109-32 F.Thau, Observing the state of nonlinear dynamic systems,Int. J. Contrl 17(3), 1973, 471-9 Sopasakis Pantelis State Observer Design
- 24. State Observers The Frobenius Theorem Observer Linearization Problem State Observability Observer Canonical Form Analysis and Design Observer Design Literature Extended Linearization Design Method Observer Error Dynamics ˙ Let us define e = x − ξ, with ξ = f (ξ, u) + g (y ) − g (ˆ ). Then: y ˙ ˙ ˙ e = x − ξ = f (x, u) − f (x − e, u) − g (y ) + g (ˆ ) y for (x, u) close to (xε , ε) we have: e = [D1 f (xε , ε) − Dg (yε ) Dh (xε )] e = Ye ˙ The aim of the design consists in determining proper analytic function g such that Y perserves constant stable eigenvalues - independent of ε! Sopasakis Pantelis State Observer Design
- 25. State Observers The Frobenius Theorem Observer Linearization Problem State Observability Observer Canonical Form Analysis and Design Observer Design Literature Extended Linearization Design Method Assumptions We assume that the following hold: 1. D1 f (0, 0)−1 exists 2. (D1 f (0, 0) , Dh (0)) is observable ∂yε 3. ∂ε |ε=0= Dyε |ε=0 = Dh (0) Dxε (0) = −Dh (0) [D1 f (0, 0)]−1 = 0 Sopasakis Pantelis State Observer Design
- 26. State Observers The Frobenius Theorem Observer Linearization Problem State Observability Observer Canonical Form Analysis and Design Observer Design Literature Extended Linearization Design Method Since (D1 f (0, 0) , Dh (0)) is observable, D1 f (0, 0)T , Dh (0)T is controllable. Hence we may use the Ackermann’s synthesis formula to determine a C : → n such that D1 f (xε , ε)T − Dh (xε )T C (ε) has a prespecified desired spectrum. Have we found C , g is given by: Dg (yε )T = C (ε) More Details on the whiteboard... Sopasakis Pantelis State Observer Design
- 27. State Observers The Frobenius Theorem Observer Linearization Problem State Observability Observer Canonical Form Analysis and Design Observer Design Literature Extended Linearization Design Method Fin! Thank you for your attention! Sopasakis Pantelis State Observer Design
- 28. The Frobenius Theorem State Observability Analysis and Design Literature Important References 1. A. Isidori, Nonlinear Control Systems - An Introduction, Springer-Verlag editions, 2ns ed, 1989, pp.217-226 2. A.J.Kerner, Normal Forms for linear and nonlinear systems, Contemp. Math. 68, 157-189, 1987 3. S.R.Kou, D.L.Eliot and T.J.Tarn, Observability of Nonlinear Systems, Information and Control 22(1), 89-99, 1972 4. X.H. Xia, W.B. Gao, Nonlinear observer design by observer canonical forms, Int. J. contr. 47(4) 2988, 1081-1100 5. B.Walcott et al., Comparative study of nonlinear state observation techniques, Int. J. Contr. 45(6) 1987, 2109-32 6. F.Thau, Observing the state of nonlinear dynamic systems,Int. J. Contrl 17(3), 1973, 471-9 Sopasakis Pantelis State Observer Design