mynotes2 - Part 2 State Space Representation Simulation...

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Part 2 - 1 Part 2 – State Space Representation, Simulation Diagrams So far, the end result of modeling a system has always been a single differential equation. There are, however, other possible ways of representing systems. Consider the system below: The current through inductor L 1 , the charge on capacitor C , and the current through inductor L 2 are obviously pretty key system quantities. Together they represent the complete state of the system. If we know these three quantities, we have a complete handle on what is happening inside the system. We will take i 1 , q , and i 2 as our state variables . Analysis of the systems produces the following basic equations: 0 ) / 1 ( 1 1 = + + q C L i V IN (sum of voltages around the left loop) 2 1 i i q = (current through the capacitor) 0 ) / 1 ( 2 2 2 = + q C R i L i (sum of voltages around the right loop) R i V OUT 2 = A bit of rearranging gives the equations below: q C L L V i IN ) / 1 ( / 1 1 1 = 2 1 i i q = 2 2 2 2 / ) / 1 ( L R i q C L i = R i V OUT 2 = The first three of these equations give the first derivatives of our state variables in terms of the state variables themselves and the system input. The last gives the system output in terms of the state variables. If we look at things in matrix terms, and take x to be a vector containing our state variables (the state vector ), u to be a vector containing our input (the
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Part 2 - 2 input vector ), and y to be vector containing our output (the output vector ), we get the following: [] [ ] [] [ ] u x y u x x y u x 0 0 0 0 0 / 1 / / 1 0 1 0 1 0 / 1 0 1 2 2 1 2 1 + = + = = = = R L L R C L C L V V i q i OUT IN In our case there is only input and one output, but the state-space approach to system modeling permits any number of inputs and outputs. In general, and assuming n state variables, r inputs, and m outputs, we have Du Cx y Bu Ax x + = + = where A is an n by n state matrix B is an n by r input matrix C is an m by n output matrix D is an m by r direct transmission matrix Taken together, matrices, A , B , C , and D completely define a system, and this representation of a system is a valid as reducing a system to a single differential equation. Matlab allows systems to be entered in this form (i.e. one can define a system by entering A , B , C , and D ), and it is also possible to analyze a system by manipulating its defining matrices. We will not pursue these possibilities beyond (in part 6) using matrix notation and Matlab as a means of obtaining a system&s transfer function. Instead we will largely restrict ourselves to viewing matrix notation as a kind of shorthand (i.e. as a convenient way of representing systems). We will also restrict ourselves to systems having a single input.
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This note was uploaded on 07/16/2009 for the course SYSC 3600 taught by Professor John bryant during the Winter '08 term at Carleton CA.

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mynotes2 - Part 2 State Space Representation Simulation...

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