How to simulate in Simulink: DAE, DAE-EKF, MPC & MHE

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1 How to simulate in Simulink: DAE, DAE-EKF, MPC & MHE Tamal Das PhD Candidate IKP, NTNU 24th May 2017

2 Outline 1 Differential algebraic equations (DAE) 2 Extended Kalman filter (EKF) for DAE 3 Moving horizon estimation (MHE) Tamal Das How to simulate in Simulink: DAE, DAE-EKF, MPC & MHE 1 / 24

3 Differential algebraic equations (DAE) Tamal Das How to simulate in Simulink: DAE, DAE-EKF, MPC & MHE 2 / 24

4 Types of DAEs: Fully implicit and Semi-explicit Fully implicit DAE Notation x : Differential variables z : Algebraic variables y = [ x T z T ] F ( y, dy ) = 0 dt Semi-explicit DAE y (0) = y 0 dx = f (x, z) dt 0 = g (x, z) Tamal Das How to simulate in Simulink: DAE, DAE-EKF, MPC & MHE 3 / 24

5 DAEs in Chemical Engineering In ChemEng, mathematical models consist of the following: Dynamic conservation laws [Differential equations] Mass balance Energy balance Static conservation laws [Algebraic equations] Constitutive equations [Algebraic equations] Equation of state Pressure drop equation Heat transfer equation Physical and operating constraints [Algebraic equations] Desired operation Design constraints Tamal Das How to simulate in Simulink: DAE, DAE-EKF, MPC & MHE 4 / 24

6 Index of a DAE Index 1-DAE g z In ChemEng, commonly encountered DAEs are semi-explicit dx dt = f (x, z) ; 0 = g (x, z) Index of DAEs determines how hard it is to solve them is non-singular ) det 0 g z ( g z is full rank matrix Higher index (2,3,... -) DAEs g z is singular ) det = 0 g z ( g z is not full rank matrix Tamal Das How to simulate in Simulink: DAE, DAE-EKF, MPC & MHE 5 / 24

7 Examples of DAE indices F, c 0 dc dt = (c 0 c) τ Index 1-DAE c 0 = γ (t) is specified; g = c 0 γ (t) g z = g c 0 = 1 (Full rank) CSTR with volume V Residence time τ = V F Differential variable (x) : c Algebraic variable (z) : c 0 F, c Higher index (2,3,... -) DAEs c = γ (t) is specified; g = c γ (t) g z = g c 0 = 0 (not full rank) Tamal Das How to simulate in Simulink: DAE, DAE-EKF, MPC & MHE 6 / 24

8 How to determine DAE index? Index = Number of differentiations of g (x, z) w.r.t time necessary to get a differential equation for each algebraic variable Index 1-DAE c 0 = γ (t) is specified Derivative 1: dc 0 dt Index = 1 = γ (t) Higher index (2,3,... -) DAEs c = γ (t) is specified Derivative 1: dc Rearrange: c 0 c τ dt = γ (t) = γ (t) Derivative 2: dc 0 dt dc dt = τγ Rearrange: dc 0 dt = γ + τγ Index = 2 Tamal Das How to simulate in Simulink: DAE, DAE-EKF, MPC & MHE 7 / 24

9 Why is Index of a DAE important? Index 1-DAEs are easy whereas higher index-daes are hard Higher index DAEs are hard due to initialization of variables (among others) In ChemEng, DAEs of both kinds are encountered dx = f (x, z) ; 0 = g (x, z) dt ( ) g det 0 (Index 1-DAE), or z ( ) g det = 0 (Higher index DAE) z Where does one encounter higher index DAEs in ChemEng Making simplifications in process models Converting simulation problems into control problems Tamal Das How to simulate in Simulink: DAE, DAE-EKF, MPC & MHE 8 / 24

10 Initialization issues of DAEs Index 1-DAE (Easy) Initialize differential variables x (0) Solve g (x (0), z (0)) = 0 for algebraic variables z (0) Higher index DAEs (Hard) x (0) and z (0) are interdependent x (0) and z (0) cannot be specified independently If specified independently, leads to inconsistent initial conditions DAE solver will crash Tamal Das How to simulate in Simulink: DAE, DAE-EKF, MPC & MHE 9 / 24

11 DAE example: Two component dynamic flash calculation F, z 1, z 2 P, T M V M L V, y 1, y 2 P out Volume V L, x 1, x 2 M T = F V L M 1 = Fz 1 Vy 1 Lx 1 M 2 = Fz 2 Vy 2 Lx 2 M T = M L + M V M 1 = M L x 1 + M V y 1 M 2 = M L x 2 + M V y 2 x 1 + x 2 = y 1 + y 2 7 Conservation equations Tamal Das How to simulate in Simulink: DAE, DAE-EKF, MPC & MHE 10 / 24

12 DAE example: Two component dynamic flash calculation F, z 1, z 2 P, T M V M L V, y 1, y 2 P out Volume V L, x 1, x 2 y 1 = K 1 x 1 y 2 = K 2 x 2 K 1 = f K 1 (T, P) K 2 = f K 2 (T, P) ρ L = f L (x 1 ) ρ V = f V (T, P) L = h L (M L, ρ L ) V = h V (P P out ) V = M L ρ L + M V ρ V } 8 Constitutive equations Design Tamal Das How to simulate in Simulink: DAE, DAE-EKF, MPC & MHE 11 / 24

13 DAE example: Two component dynamic flash calculation P, T V, y 1, y 2 P out 3 differential equations 3 differential variables M V 13 algebraic equations F, z 1, z 2 M L Volume V 13 algebraic variables We can find an output set for all the algebraic and differential variables L, x 1, x 2 Index 1 Tamal Das How to simulate in Simulink: DAE, DAE-EKF, MPC & MHE 12 / 24

14 DAE example: Modified two component dynamic flash F, z 1, z 2 P, T M L V, y 1, y 2 M L = F V L M 1 = Fz 1 Vy 1 Lx 1 M 2 = Fz 2 Vy 2 Lx 2 M 1 = M L x 1 M 2 = M L x 2 x 1 + x 2 = y 1 + y 2 6 Conservation equations L, x 1, x 2 Tamal Das How to simulate in Simulink: DAE, DAE-EKF, MPC & MHE 13 / 24

15 DAE example: Modified two component dynamic flash F, z 1, z 2 P, T M L V, y 1, y 2 y 1 = K 1 x 1 y 2 = K 2 x 2 K 1 = f K 1 (T, P) K 2 = f K 2 (T, P) L = h L (M L, ρ L ) ρ L = f L (x 1 ) 6 Constitutive equations L, x 1, x 2 Tamal Das How to simulate in Simulink: DAE, DAE-EKF, MPC & MHE 14 / 24

16 DAE example: Modified two component dynamic flash F, z 1, z 2 V, y 1, y 2 P, T 3 differential equations M L L, x 1, x 2 3 differential variables 9 algebraic equations 9 algebraic variables We cannot find an output set for algebraic variable V. We need to differentiate twice to get there Index 2 Tamal Das How to simulate in Simulink: DAE, DAE-EKF, MPC & MHE 15 / 24

17 Solving higher index DAE problems Reduce index to 1 or 0 Find consistent initial conditions for all the variables of the new system Solve the new system as a Index 1 DAE system or 0 index system (ODE system) The procedure 1 is explained on the MATLAB website with added support functions: Index / order reduction ( reducedaeindex ) Finding consistent initial guesses ( decic ) DAE solvers: ode15i, ode15s, or ode23t 1 Tamal Das How to simulate in Simulink: DAE, DAE-EKF, MPC & MHE 16 / 24

18 Solving index 1-DAE: Robertson DAE in Simulink A B + C B C dc A = 0.04C A C B C C dt dc B = 0.04C A 10 4 C B C C CB 2 dt 0 = 1 (C A + C B + C C ) Tamal Das How to simulate in Simulink: DAE, DAE-EKF, MPC & MHE 17 / 24

19 Solving index 1-DAE problem in Simulink: Implementation Interpreted MATLAB Fcn derivatives 1 s Integrator z Solve f(z) = 0 f(z) Algebraic Constraint Interpreted MATLAB Fcn Algebraics Differential states Algebraic states Tamal Das How to simulate in Simulink: DAE, DAE-EKF, MPC & MHE 18 / 24

20 Solving index 1-DAE problem in Simulink: Result Robertson DAE problem with a Conservation Law, solved by ODE15S C A C C Concentration [-] C B t (sec) Tamal Das How to simulate in Simulink: DAE, DAE-EKF, MPC & MHE 19 / 24

21 Extended Kalman filter (EKF) for DAE Tamal Das How to simulate in Simulink: DAE, DAE-EKF, MPC & MHE 20 / 24

22 Continuous-discrete EKF algorithm for DAE dx = f (x, z) + w dt x aug = 0 = g (x, z) ; y = h (x, z) + v [ ] d ˆx dt = f (ˆx) ; L = I ( ) 1 ( ) g g z x ( ) ( ) dp f aug f aug T dt = x aug P + P x aug + LQL T ˆx aug,+ k K k = P k ( h x aug ) ( T ( h R k + = ˆx aug, k + K k ( y k h k ( P + k = ( I K k ( h x aug )) P k x aug ˆx aug, k )) ) P k ( I K k ( [ ] [ ] x ; f aug f = z g ( h ) ) T 1 x aug h x aug )) T + K k R k K T k Tamal Das How to simulate in Simulink: DAE, DAE-EKF, MPC & MHE 21 / 24

23 Moving horizon estimation (MHE) Tamal Das How to simulate in Simulink: DAE, DAE-EKF, MPC & MHE 22 / 24

24 MHE is a dynamic optimization problem At each sample time, a set of past few measurements are used to provide a state estimate This problem is discretized to a nonlinear program (NLP) [for nonlinear estimation] NLP is solved using IPOPT solver in CasADi CasADi is interfaced with Simulink using system block to provide online state estimation Tamal Das How to simulate in Simulink: DAE, DAE-EKF, MPC & MHE 23 / 24

25 Conclusions DAE systems are hard: particularly higher index DAEs Two approaches: simultaneous (Mass matrix solver) sequential (Simulink with Algebraic constraint block) Estimation for DAE systems: CD-DAE-EKF in Simulink MPC in Simulink using system block and CasADi MHE in Simulink using system block and CasADi For simulation files and this presentation, follow 2 Thank you for your attention 2 folk.ntnu.no/tamald Tamal Das How to simulate in Simulink: DAE, DAE-EKF, MPC & MHE 24 / 24

10.34: Numerical Methods Applied to Chemical Engineering. Lecture 19: Differential Algebraic Equations

10.34: Numerical Methods Applied to Chemical Engineering Lecture 19: Differential Algebraic Equations 1 Recap Differential algebraic equations Semi-explicit Fully implicit Simulation via backward difference