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

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

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

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

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

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

State Space 1 - Hany Ferdinando7 Example i(t)

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

State Space 1 - Hany Ferdinando9 Example Matrix AMatrix B

State Space 1 - Hany Ferdinando10 General Form of SS

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

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

State Space 1 - Hany Ferdinando13 Example

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

State Space 1 - Hany Ferdinando15 Example

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

State Space 1 - Hany Ferdinando17 Example

State Space 1 - Hany Ferdinando18 TF from SS

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