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State Space Representation Hany Ferdinando Dept. of Electrical Engineering Petra Christian University

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State Space 1 - Hany Ferdinando2 Overview Introduction State variable Signal Flow Graph (revisited) State Space (SS) matrix Transfer function from SS SS in Matlab

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State Space 1 - Hany Ferdinando3 Introduction State Space (SS) represents a dynamic system in matrices It is MIMO (multi input multi output) system (transfer function is SISO – single input single output) With SS, one can get future condition of a dynamic system

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State Space 1 - Hany Ferdinando4 Examples On-off switch: There are only two positions The state of the switch can assume one of two possible states if the present state is ‘on’, then one can know the future state

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State Space 1 - Hany Ferdinando5 State Variable Definition: is a set of variables such that the knowledge of these variables and the input functions will, with the equations describing the dynamics, provide the future state and output of the system

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State Space 1 - Hany Ferdinando6 Model of A Dynamic System Model of a dynamic system is in n th - order differential equation The order depends on the number of storage element in that system SS will change one n th -order differential equation into n first-order differential equations

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State Space 1 - Hany Ferdinando7 Example i(t)

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State Space 1 - Hany Ferdinando8 Example If x 1 = v C and x 2 = i L then

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State Space 1 - Hany Ferdinando9 Example Matrix AMatrix B

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State Space 1 - Hany Ferdinando10 General Form of SS

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State Space 1 - Hany Ferdinando11 Signal Flow Graph (revisited) Sum of the forward-path factors Sum of the feedback loop factors

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State Space 1 - Hany Ferdinando12 Signal Flow Graph State From G(s) get the order of the equation Make the denominator of G(s) in the form of 1-(sum of the feedback loop) by dividing G(s) with s n (n is the order of G(s)) Draw the new SFG

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State Space 1 - Hany Ferdinando13 Example

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State Space 1 - Hany Ferdinando14 Example -a3 -a2 -a1 -a0 b1 b2 b3 1/s b0 x1x2x3 x4 U(s) Y(s) Phase Variable Format

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State Space 1 - Hany Ferdinando15 Example

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State Space 1 - Hany Ferdinando16 Example -a3 -a2 -a1 -a0 b1 b2 b3 1/s b0 x1x2x3 x4 U(s) Y(s) Input Feedforward Format

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State Space 1 - Hany Ferdinando17 Example

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State Space 1 - Hany Ferdinando18 TF from SS

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State Space 1 - Hany Ferdinando19 SS in Matlab (basic) Use two functions: tf2ss to convert transfer function model into state space model ss2tf to convert state space model into transfer function model

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