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Modelling: State Space Representation

Modelling procedure: a step by step approach What is a state, an input, an output Model in state space notation

H Preisig 2006: Prosessregulering H Preisig 2006: Prosessregulering

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Modelling step by step (1)

Problem formulation Sketch process List what is known what one looks for primary assumptions Abstraction Represent plant as a network of capacities & flows of extensive quantities -- a directed graph Label systems and flows Indicate what you know and what you look for This graph should contain ALL information Establish basic dynamic model: Components mass balances (see separate discussion) Energy balances Second set of assumptions example constant pressure: internal energy enthalpy system does not move in space etc. Defines primary state (see below)

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Modelling step by step (2)

Detailed description of Flows component mass convective

diffusive enthalpy due to mass: heat shaft work, friction... Reactions & phase changes reactions occur inside capacities whilst phase changes can be thought of occurring inside the differential volume of the boundary State variable transformation transfer laws and reactions | phase changes define new quantities, which must be represented as a function of the fundamental state, namely component mass and enthalpy. Typical new variables are: concentration: from mass and volume volume from mass and density pressure given through fast hydraulic (mostly (implicitly) given) surface and other geometrical quantities (given)

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Model in state-space notation

State: primary: what is differentiated with respect to time in the balances the variables in the accumulation term for an arbitrary lumped system a: for the whole plant P consisting of lumped system a,b,...s, stack them simply up:

Note that the momentum is not considered here. In most application outside fluid mechanics, the momentum dissipation is assumed fast, because pressure changes propagate at speed of sound. Thus common assumption here is event dynamic: things happen instantaneously. secondary: all state-derived quantities such as concentrations, volumes, chemical potential, temperature, pressure etc.

Conditions (z): quantities that are given and belong to the class of states, we call conditions. Typical examples are states of feed systems, such as concentrations, temperature. These can only be set for feed systems, represented as reservoirs or combinations thereof, because the states are otherwise defined by the conservation principles or systems for which the ASSUMPTION event-dynamic is made, thus one assumes zero capacity effect. An example is the momentum and its derived quantities, mainly the pressure. Other such quantities are often associated with the geometry (volume, areas,...) Parameters ( ): This class of variables essentially only appear as quantities that were fitted to empirical model components, such as kinetic constants, coefficients in polynomial approximations for physical properties or the like.

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Global state-space representation

dynamics: ODEs

in words: the change of the state of the system a is given by the flows of extensive quantities it exchanges with the environment e and the internal conversion of extensive quantities. transfer: algebraic connecting systems thus coupling ODEs:

reactions (transposition): algebraic, internal conversion:

transformations: algebraic, link secondary states with primary states

these equations can be implicit !

"parameters" : the quotes have been added because these quantities may be yet a function of the state, thus not really constant, but a low-level model component itself:

Thus this is a representation in the form of a set of ordinary differential equations augmented with a set of algebraic equations. This one calls a set of Differential Algebraic Equations (DAEs).

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Common substitution of secondary state for primary state

Solving the representation as it was given above is usually not done, though it is clear that the new software development will be using the DAE representation. Solving things by hand ask for a transformation of the DAEs into ODEs by implementing a state space transformation, in this case from the state space spanned by the primary state into the state space spanned by the secondary state. For the whole plant we write:

The respective matrices are now block matrices and the vectors are stacks of vectors. By differentiating the secondary state variables we find:

If we deal with constant "parameters" things are relatively simple, as the derivatives of the parameters with respect to time are zero and the above equation. Thus:

Substitution leads to the desired ODE in the secondary state.

If you analyse on what is being taught in chemical engineering, then it is the these equations that are derived, not the former, though it is the representation in the primary state variables that is the fundamental one!

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