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Nonlinear observer design

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Design of observers for nonlinear systems using the Frobenius theorem. Presentation for the defense of my MSc Thesis at the School of Applied Mathematics, NTU Athens.

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Nonlinear observer design

  1. 1. The Frobenius Theorem State Observability Analysis and Design Literature State Observer Design Sopasakis Pantelis October 5, 2012 Sopasakis Pantelis State Observer Design
  2. 2. The Frobenius Theorem Introduction to Distributions State Observability Duality and Integrability Analysis and Design The Frobenius Theorem LiteratureWhat 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. 3. The Frobenius Theorem Introduction to Distributions State Observability Duality and Integrability Analysis and Design The Frobenius Theorem LiteratureSome 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. 4. The Frobenius Theorem Introduction to Distributions State Observability Duality and Integrability Analysis and Design The Frobenius Theorem LiteratureA 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. 5. The Frobenius Theorem Introduction to Distributions State Observability Duality and Integrability Analysis and Design The Frobenius Theorem LiteratureLie 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. 6. The Frobenius Theorem Introduction to Distributions State Observability Duality and Integrability Analysis and Design The Frobenius Theorem LiteratureInvolutive Distributions A distribution D ∈ D(U) is called involutive if [f , g ] ∈ D for every f , g ∈ D. Sopasakis Pantelis State Observer Design
  7. 7. The Frobenius Theorem Introduction to Distributions State Observability Duality and Integrability Analysis and Design The Frobenius Theorem LiteratureDuality 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. 8. The Frobenius Theorem Introduction to Distributions State Observability Duality and Integrability Analysis and Design The Frobenius Theorem LiteratureThe 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. 9. The Frobenius Theorem Introduction to Distributions State Observability Duality and Integrability Analysis and Design The Frobenius Theorem LiteratureComplete 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. 10. The Frobenius Theorem Introduction to Distributions State Observability Duality and Integrability Analysis and Design The Frobenius Theorem LiteratureThe 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. 11. The Frobenius Theorem State Observability Local Decompositions of Control Systems Analysis and Design State Observability Literaturef -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. 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. 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. 14. State Observers The Frobenius Theorem Observer Linearization Problem State Observability Observer Canonical Form Analysis and Design Observer Design Literature Extended Linearization Design MethodObservers 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. 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. 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. 17. State Observers The Frobenius Theorem Observer Linearization Problem State Observability Observer Canonical Form Analysis and Design Observer Design Literature Extended Linearization Design MethodSolvability 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. 18. State Observers The Frobenius Theorem Observer Linearization Problem State Observability Observer Canonical Form Analysis and Design Observer Design Literature Extended Linearization Design MethodObserver 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. 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. 20. State Observers The Frobenius Theorem Observer Linearization Problem State Observability Observer Canonical Form Analysis and Design Observer Design Literature Extended Linearization Design MethodInverted 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. 21. State Observers The Frobenius Theorem Observer Linearization Problem State Observability Observer Canonical Form Analysis and Design Observer Design Literature Extended Linearization Design MethodObserver 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. 22. State Observers The Frobenius Theorem Observer Linearization Problem State Observability Observer Canonical Form Analysis and Design Observer Design Literature Extended Linearization Design MethodError 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. 23. State Observers The Frobenius Theorem Observer Linearization Problem State Observability Observer Canonical Form Analysis and Design Observer Design Literature Extended Linearization Design MethodIntroduction 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. 24. State Observers The Frobenius Theorem Observer Linearization Problem State Observability Observer Canonical Form Analysis and Design Observer Design Literature Extended Linearization Design MethodObserver 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. 25. State Observers The Frobenius Theorem Observer Linearization Problem State Observability Observer Canonical Form Analysis and Design Observer Design Literature Extended Linearization Design MethodAssumptions 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. 26. State Observers The Frobenius Theorem Observer Linearization Problem State Observability Observer Canonical Form Analysis and Design Observer Design Literature Extended Linearization Design MethodSince (D1 f (0, 0) , Dh (0)) is observable, D1 f (0, 0)T , Dh (0)T iscontrollable. Hence we may use the Ackermann’s synthesis formulato 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 givenby: Dg (yε )T = C (ε)More Details on the whiteboard... Sopasakis Pantelis State Observer Design
  27. 27. State Observers The Frobenius Theorem Observer Linearization Problem State Observability Observer Canonical Form Analysis and Design Observer Design Literature Extended Linearization Design MethodFin! Thank you for your attention! Sopasakis Pantelis State Observer Design
  28. 28. The Frobenius Theorem State Observability Analysis and Design LiteratureImportant 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

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