Download presentation

Presentation is loading. Please wait.

Published byDamaris Milem Modified over 2 years ago

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)

Similar presentations

Presentation is loading. Please wait....

OK

Chapter 1 Systems of Linear Equations Linear Algebra.

Chapter 1 Systems of Linear Equations Linear Algebra.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on money and credit download Ppt on assembly line balancing Ppt on acid-base titration calculations Ppt on international stock exchange Ppt on cost center accounting Ppt on ways to improve communication skills Ppt on computer languages wiki Ping pay ppt online Design for ppt on deforestation Ppt on non biodegradable wastes