State Space Models Sheffield University

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This chapter gives a summary of key methods and concepts around state space models. The content is primarily a targeted at students doing a single course in state space methods and hence does not dwell on some fine details which would be covered in a 2nd course or in research applications; one such example is nonsimple Jordan forms and another is finding approximate state space models by linearisation of 1st principles models. Once the principles are understood clearly, Students are encouraged to use tools like MATLAB for some of the number crunching as manipulation of state space models is not a paper and pen exercise in general.

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State Space Models 1  Introduction
Introduces the concept of taking first principles models for systems and converting them into state space form. Explains the key assumption in a state space model is that one can write an equation for all the key dynamics in terms of their 1st order derivatives. Gives simple examples from 1st order engineering systems (massdamper, resistorcapacitor, tank system).

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State Space Models 2  2nd Order ODEs
Extends the concept of taking first principles models for systems and converting them into state space form. Uses some 2nd and 3rd order examples (massspringdamper, RLC circuit, dc servo and pendulum) to demonstrate the process of constructing a statespace equivalent. Introduces the concept that state space descriptions for a given system are not unique as they depend on the selection and ordering of states.

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State Space Models 3  From a generic ODE
In some cases a system model is supplied solely as an ODE rather than separate 1st principles equations. This resource shows how an equivalent state space model can be derived from an ODE. It is assumed that the results are given in canonical forms but again emphasis is made on the state space matrices not being unique as they depend on the selection and ordering of states.

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State Space Models 4  Finding the output
State space models have numerous states but the user may only be interested in a subset of these. The selected states are denoted as outputs; outputs are only those states you want to measure. This resource shows how the extraction of outputs from a state space model leads to another matrix definition or set of equations which are therefore part of the state space model.

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State Space Models 5  Transfer function from state space
It is useful to understand the relationship between state space models and transfer function models. This resource shows how one can form an equivalent transfer function model from a state space model. Several numerical examples are given but it is emphasised that the process is numerically intensive and thus in general should be performed on a computer and not by hand. Tools like MATLAB are demonstrated. It is also noted that the system poles correspond to the eigenvalues of the A matrix.

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State Space Models 6  Transfer function to state space
This resource shows how one can form a state space model from a transfer function. The process is analogous to that used for ODEs but with the extra subtlety of allowing more complex numerators than a constant. The resource gives the controllable canonical form only as this can be constructed by inspection from the transfer function parameters.