Survey of state estimation for (bio)chemical systems – A personal perspective

1. Survey of state estimation for (bio)chemical systems –A personal perspective Denis Dochain

2. My talk… • Central motivation : how to provide reliable software measurements in presence of uncertainty • 2 parts • classical observers • 5 (selected) alternatives • No detailed proof • One leading example : simple microbial growth

3. Menu • Basic observer structure • Extended Luenberger observer • Extended Kalman observer • Limitations of classical observers • Some alternatives Asymptotic observer Observer with model parameters as design parameters Interval observer Finite time converging observer Robust asymptotic observer for systems unobservable on their boundary

4. Menu • Basic observer structure • Extended Luenberger observer • Extended Kalman observer • Limitations of classical observers • Some alternatives Asymptotic observer Observer with model parameters as design parameters Interval observer Finite time converging observer Robust asymptotic observer for systems unobservable on their boundary

5. Basic observer structure • Consider the dynamical system : • On-line measured variables :y(t) = h(x) • State observer :

6. Basic observer structure • Consider the dynamical system : • On-line measured variables :y(t) = h(x) • State observer : • If the system is observable, then it is possible to find an observer that will reconstruct the unmeasured state with an arbitrary convergence rate (in absence of model uncertainty) state estimate observer gain

7. State observer for reaction systems • Starting point : the general dynamical model (N components, M reactions) • Measurements : p measured components :y(t) = L x(t) withLa pxN matrix with 1’s and 0’s (p ≥ N - M) • State observer :

8. Menu • Basic observer structure • Extended Luenberger observer • Extended Kalman observer • Limitations of classical observers • Some alternatives Asymptotic observer Observer with model parameters as design parameters Interval observer Finite time converging observer Robust asymptotic observer for systems unobservable on their boundary

9. Extended Luenberger observer Design rule : such that

10. Menu • Basic observer structure • Extended Luenberger observer • Extended Kalman observer • Limitations of classical observers • Some alternatives Asymptotic observer Observer with model parameters as design parameters Interval observer Finite time converging observer Robust asymptotic observer for systems unobservable on their boundary

11. Extended Kalman observer • Solution of a minimization problem : with • with R a NxN symmetric matrix solution of the Riccati equation :

12. Example State observer

13. Luenberger observer • to assign the observer error dynamics • characteristic polynomial det(lI – A) = (l + l1)(l + l2)

15. Kalman observer

17. Menu • Basic observer structure • Extended Luenberger observer • Extended Kalman observer • Limitations of classical observers • Some alternatives Asymptotic observer Observer with model parameters as design parameters Interval observer Finite time converging observer Robust asymptotic observer for systems unobservable on their boundary

18. Limitations of the classical observers • Sensitivity to model uncertainty • Influence of the zeros dynamics on the rate of convergence

19. Sensitivity to model uncertainty Illustration : Estimation of X from measurements of S in a bioreactor • S X ; Monod kinetics • extendedLuenberger observer (knows the kinetics model) Simulation conditions : • observer gains : assign the observer dynamics (2 choices : slow - fast) • error on the values of the parameters of the Monod model in the observer

22. Influence of the zero dynamics

24. • Observation error dynamics (2nd order system) : • Transfer function between the unmeasured outputs and the initial values of the state variables : • e1(0) is unknown…

25. Solution: usee2(0)as an extra design parameter e1(t) ___ : e2(0) = 1 - - : e2(0) = 0.5 _ . : e2(0) = -1

26. Menu • Basic observer structure • Extended Luenberger observer • Extended Kalman observer • Limitations of classical observers • Some alternatives Asymptotic observer Observer with model parameters as design parameters Interval observer Finite time converging observer Robust asymptotic observer for systems unobservable on their boundary

27. Asymptotic observer • Basis : reaction invariants • Assumptions A1. The stochiometric coefficients K are known A2. The reaction rate vector r is unknown A3. q components are measured on-line (q ≥ rank (K)) A4. The vectors FandQare known • State partition (arbitrary) : Z = A0 xa + xb •Measured components x1 : Z = A1 x1 + A2 x2

28. Observer design • Asymptotic observer : with a left inverse of A2 Remarks : • K full rank if independent reactions (reversible reaction = one reaction) • submatrix K1 (→ measured components) : full rank i.e. q“independent” measured variables)

29. Convergence • Estimation error : • Error dynamics :  “persistence of excitation” Comments • no requirement on the knowledge of the kinetics • no correction term  the error dynamics depend on D

30. Example S1 X1 + S2 S2  X2 Possible state transformation

31. Some possible cases Case #1 Case #2

32. Application: production of antibiotics

33. Menu • Basic observer structure • Extended Luenberger observer • Extended Kalman observer • Limitations of classical observers • Some alternatives Asymptotic observer Observer with model parameters as design parameters Interval observer Finite time converging observer Robust asymptotic observer for systems unobservable on their boundary

34. Observer with the model parameters as design parameters Example : Estimation of the biomass concentration X frommeasurements of the substrate concentration S in a simple microbialgrowthprocesswith Blackmankinetics : m = aS Objective : select such that the estimation error on X is equal to zero in steady-state

35. Model equations : Observer equations : --->3 “design” parameters : w1, w2,

36. Choice of the design parameters : 1) w1 and w2: to assign the observer dynamics (1 and 2 : poles of the observer dynamics) 2) : to handle the model uncertainty

37. Stability properties of the observer • Define the observation error : • Error dynamics : • A is asymptotically stable if : bounded and

38. Extensions (generalization) : e.g. 1) 2) other kinetic models : e.g. Monod kinetics :

40. Menu • Basic observer structure • Extended Luenberger observer • Extended Kalman observer • Limitations of classical observers • Some alternatives Asymptotic observer Observer with model parameters as design parameters Interval observer Finite time converging observer Robust asymptotic observer for systems unobservable on their boundary

41. Interval Observer • Assumption : bounded uncertainty for the parameters • Interval observer : provides bounds of the state estimation based on this uncertainty bounds • Key issue : cooperativity of the system observer : (off-diagonal entries of the Jacobian matrix > 0)

42. Design of the interval observer • Simple microbial growth process with m(S) bounded : m-(S) ≤ m(S) ≤ m+(S) • S is measured, X is not measured • The dynamical model in its original format is not a good candidate since the observer equations are not cooperative Jacobian: <0!

43. Design of the interval observer (continued) • State transformation: Z = X + S/k1 • Observer: Jacobian: cooperative if w1 > 0 and w2 < 0

44. Interval observer equations 1) 2)

45. Stability analysis If then with i.e. EX small if w2 large

46. Simulation results • Monod (m+(S)/m-(S)) or Monod (m+(S)) /Haldane (m-(S)) models • k1 = 2, mmax = 0.33 h-1, KS = 5 g/l, Sin = 5 g/l, D = 0.05 h-1 • 0.165 h-1 ≤ mmax ≤ 0.395 h-1, m0 = 0.99 x 0.33 h-1, KI = 25 g/l

47. w1 = q C1s, w2 = q2 C2s

48. Monod (m+(S)) /Haldane (m-(S)) models

49. Monod (m+(S)) /Haldane (m-(S)) models

50. Menu • Basic observer structure • Extended Luenberger observer • Extended Kalman observer • Limitations of classical observers • Some alternatives Asymptotic observer Observer with model parameters as design parameters Interval observer Finite time converging observer Robust asymptotic observer for systems unobservable on their boundary