Modeling This lecture we will consentrate on how to do system modeling based on two commonly used techniques In frequency domain using Transfer Function (TF) representation In time domain via using State Space representation Transition between the TF to SS and SS to TF will also be discussed.
Transfer Function Representation Transfer functions is an Input/Output approach for system modelling In Laplace Domain this becomes where Relating the output to the input is called the transfer function of the system
Transfer Function (TF) For the differential equation of the form the transfer function is Note that transfer function is obtained by assuming that all the initial conditions are zero Roots of the numerator of Roots of the denominator of are the zeros of are the poles of
TF Models of Physical Systems Electrical Systems
Electrical Systems Back to the basic example of RLC circuit Time Domain Laplace Domain
Electrical System After some mathematical manipulatons That is
2-Loop Electrical System Find the relation between the input voltage and the voltage across the capacitor
2-Loop Electrical System System in Laplace domain (1) (2)
2-Loop Electrical System Solve for with respect to from the mesh equation (1) and replace it in the output equation (2) Transfer function is then
Important Note Becarefull The approach : Loop1 Loop2 will not work... WHY??
3-loop example Find the input output realtion for the given circuit
3-loop electrical system Mesh equation : Output equation : Solve the mesh equation for equation then insert it in the output
Mechanical Systems
Mechanical Systems
Mechanical System Example An easy mass spring damper system Time domain Laplace (frequency) domain
Mechanical Systems Find the input output relationship of the system EOM (equations of motion) in s-domain (1) (2)
Mechanical Systems Use (1) to find a realtionship between and Then insert this into (2) to obtain The transfer function is then
Rotating Drive System Relationship between and EOM in s-domain The rest is similar to our previous electrical system example
Electromechanical System DC motor, relationship between input voltage and motor's rotational position Note that:
Electromechanical System The EOM of the system in s-domain inserting for the voltage across the motor and the motor torque Rest is mathematical manipulations
State Space Representation We start with a similar example The differential equation representing the system is and the system output is the position
State Space Representation Define the states as Taking the time derivative of system equation yields and inserting for the which can be written in vector form as
State Space Representation In a more general form For our special case
So What? What have we gained? We were able to represent a high order scalar differential equation with a first order, but in matrix form, differential eqaution. Solution should be much more easier but requires linear algebra knowledge as well. Can we somehow go back to Transfer function representation? Lets give a try
Converting from SS to TF Start from Apply Laplace Trasfromation, omit initial conditions, to obtain Now rearrange since The transfer function representation can be obtained as
Back to our Example For our special case
Converting from TF to SS * Consider the transfer function of the form which is assumed to be strictly proper. That is Rearrange the transfer function to have the following form V 's are the states where : State Eq : Output Eq
Converting from TF to SS This enables us to write the matrix eqaution and the output equation as
Example Consider the case n = 3 then In matrix form
Block Diagram Representation Note that the block diagram representation of the previous example which was converted to aside from conversion to SS this also enables us to form the following equivalent block diagram representation
Block Diagram Representation
Solution of State Equations (1) Solution by Laplace Transform start from Using Laplace Transform obtain where is called Resolvent Matrix is the State Transition Matrix
Solution of State Equations Observe that and since we can rewrite the solution for as follows
Example Consider the system of the form calculate the output response to unit step input when the initial conditions are given as
Example Solution : start from then solution of the states are
Example (cont.) Taking inverse Laplace Transform yields Alternatively, Then we can directly calculate
Example (still cont..) First simplify the evaluate the integral the result is then
Solution of State Equations (2) Formal (Classical) Solution of State Equation Take the scalar case rearrange to have multiply both sides by integrate both sides
Solution of State Equations Multiply each side by to obtain which in matrix form would have been or
State Transition Matrix Some of the usefull Properties of the State Transition Matrix 1. 2. 3. as as