1. PROCESS MODELLING AND MODEL ANALYSIS © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Analysis of Dynamic Process Models C13

2. PROCESS MODELLING AND MODEL ANALYSIS 2 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Overview of Dynamic Analysis   Controllability and observability   Stability   Structural control properties   Model structure simplification   Model reduction

3. PROCESS MODELLING AND MODEL ANALYSIS 3 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences State Controllability A system is said to be “(state) controllable” if for any t 0 and any initial state x(t 0 )= x 0 and any final state x f, there exists a finite time t 1 > t 0 and control u(t), such that x(t 1 )= x f

4. PROCESS MODELLING AND MODEL ANALYSIS 4 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences State Observability A system is said to be “(state) observable” if for any t 0 and any initial state x(t 0 )= x 0 there exists a finite time t 1 > t 0 such that knowledge of u(t) and y(t) for t 0  t  t 1 suffices to determine x 0

5. PROCESS MODELLING AND MODEL ANALYSIS 5 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences MATLAB functions (V4.2)   Controllability   Observability

6. PROCESS MODELLING AND MODEL ANALYSIS 6 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Example Model equations Controllability Observability

7. PROCESS MODELLING AND MODEL ANALYSIS 7 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Stability of systems - overview   Two stability notions - bounded input bounded output (BIBO) - asymptotic stability   Testing asymptotic stability of LTI systems   MATLAB functions (e.g. eig(A))   Stability of nonlinear process systems - Lyapunov’s principle

8. PROCESS MODELLING AND MODEL ANALYSIS 8 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences BIBO Stability A system is said to be “bounded input, bounded output (BIBO) stable” if it responds with a bounded output signal to any bounded input signal, i.e. BIBO stability is external stability

9. PROCESS MODELLING AND MODEL ANALYSIS 9 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Asymptotic Stability A system is said to be “asymptotically stable” if for a “small” deviation in the initial state the resulting “perturbed” solution goes to the original solution in the limit, i.e. asymptotic stability is internal stability

10. PROCESS MODELLING AND MODEL ANALYSIS 10 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Asymptotic Stability of LTI Systems A LTI system with state space realization matrices (A,B,C) is asymptotically stable if and only if all the eigenvalues of the state matrix A have negative real parts, i.e. asymptotic stability is a system property

11. PROCESS MODELLING AND MODEL ANALYSIS 11 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences MATLAB Function and Example Model equations Analysis Stable!

12. PROCESS MODELLING AND MODEL ANALYSIS 12 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Asymptotic Stability of Nonlinear Systems Lyapunov principle: construct a generalized energy function V for the system, such that: If such a V exists then the system is asymptotically stable

13. PROCESS MODELLING AND MODEL ANALYSIS 13 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Structural properties of systems A dynamic system possesses a structural property if “almost every” system with the same structure has this property (“same structure” = identical structure graph) Properties include:   Structural controllability   Structural observability   Structural stability

14. PROCESS MODELLING AND MODEL ANALYSIS 14 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Structural Rank The structural rank (s-rank) of a structure matrix [Q] is its maximal possible rank when its structurally non 0 elements get numerical values

15. PROCESS MODELLING AND MODEL ANALYSIS 15 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Structural Controllability A system is structurally controllable if the structural rank (s-rank) of the block structure matrix [A,B] is equal to the number of state variables n

16. PROCESS MODELLING AND MODEL ANALYSIS 16 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Structural Controllability A system is structurally controllable if:   the state structure matrix [A] is of full structural rank.   the structure graph of the state space realization ([A],[B],[C],[D]) is input connectable. Structural rank: pairing of columns and rows. Input connectable: path to every state vertex from at least one input vertex.

17. PROCESS MODELLING AND MODEL ANALYSIS 17 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Example: Heat exchanger modelled by 3 connected lumped volumes [A] is of full structural rank (because of self loops) Structure graph

18. PROCESS MODELLING AND MODEL ANALYSIS 18 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Example: a heat exchanger network Identical to the equipment flowsheet Condensed structure graph: strong components collapsed into a single node

19. PROCESS MODELLING AND MODEL ANALYSIS 19 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Structural Observability A system is structurally observable if the structural rank (s-rank) of the block structure matrix [C,A] T is equal to the number of state variables n

20. PROCESS MODELLING AND MODEL ANALYSIS 20 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Structural Observability A system is structurally observable if:   the state structure matrix [A] is of full structural rank.   the structure graph of the state space realization ([A],[B],[C],[D]) is output connectable. Structural rank: pairing of columns and rows. Output connectable: path from every state vertex to at least one output vertex.

21. PROCESS MODELLING AND MODEL ANALYSIS 21 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Structural Stability   Method of circle families conditions depending on the sign of non- touching circle families (computationally hard)   Method of conservation matrices If the state matrix A is a conservation matrix then the system is structurally stable.

22. PROCESS MODELLING AND MODEL ANALYSIS 22 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Model Simplification and Reduction LTI models with state space representation States can be classified into:   slow modes (“small” negative eigenvalues) states essentially constant   fast modes (“large” negative eigenvalues) go to steady state rapidly   medium modes

23. PROCESS MODELLING AND MODEL ANALYSIS 23 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Model Structure Simplification Elementary simplification steps   variable removal: steady state assumption on a state variable removes the vertex and all adjacent edges and conserves the paths.   variable lumping: for a vertex pair with similar dynamics, it lumps the two vertices together, unites adjacent edges and conserves the paths.

24. PROCESS MODELLING AND MODEL ANALYSIS 24 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Example: A heat exchanger 1. Variable removal Steady-state variables: cold side temperatures

25. PROCESS MODELLING AND MODEL ANALYSIS 25 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Example: A heat exchanger 1. Variable lumping Lumped variables: cold side temperatures hot side temperatures

26. PROCESS MODELLING AND MODEL ANALYSIS 26 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Equivalent State Space Models Two state space models are equivalent if they give rise to the same input-output model. Equivalence transformation of state space models of LTI systems are:

27. PROCESS MODELLING AND MODEL ANALYSIS 27 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Model Reduction Balanced state-space realizations:   takes original A, B and C   returns new “balanced” AA, BB and CC   new LTI has equal controllability and observability Grammians   returns the Grammian vector G contains the contribution of the states to the controllability and observability Matlab 4.2

28. PROCESS MODELLING AND MODEL ANALYSIS 28 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Model Reduction   Use Grammian information for reduction eliminate states where g(i)<g(1)/10   Model reduction of states x(i e1 ),…, x(i e1 ) done using (Matlab 4.2):