Analysis of Dynamic Process Models

1. Analysis ofDynamic Process Models C13

2. Overview of Dynamic Analysis • Controllability and observability • Stability • Structural control properties • Model structure simplification • Model reduction

3. State Controllability A system is said to be “(state) controllable” if for any t0 and any initial state x(t0)= x0 and any final state xf, there exists a finite time t1> t0and control u(t), such that x(t1)= xf

4. State Observability A system is said to be “(state) observable” if for any t0 and any initial state x(t0)= x0 there exists a finite time t1> t0such that knowledge of u(t) and y(t) for t0tt1 suffices to determine x0

5. MATLAB functions (V4.2) • Controllability • Observability

6. Example Model equations Controllability Observability

7. 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. 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. 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. 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. MATLAB Function and Example Model equations Analysis Stable!

12. 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. 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. 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. 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. 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. Example: Heat exchanger modelled by 3 connected lumped volumes Structure graph [A]is of full structural rank(because of self loops)

18. Example: a heat exchanger network Condensed structure graph: strong components collapsed into a single node Identical to the equipment flowsheet

19. 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. 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. Structural Stability • Method of circle familiesconditions 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. 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. Model Structure Simplification Elementary simplification steps • variable removal: steady state assumption on a state variableremoves the vertex and all adjacent edgesand conserves the paths. • variable lumping:for a vertex pair with similar dynamics, it lumps the two vertices together, unites adjacent edgesand conserves the paths.

24. Example: A heat exchanger 1. Variable removal Steady-state variables: cold side temperatures

25. Example: A heat exchanger1. Variable lumping Lumped variables: cold side temperatures hot side temperatures

26. 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. 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. Model Reduction • Use Grammian information for reduction • eliminate states where g(i)<g(1)/10 • Model reduction of states x(ie1),…, x(ie1) done using (Matlab 4.2):