Modern Control System EKT 308 Transfer Function Poles and Zeros

Transfer Function A system can be represented, in s-domain, using the following block diagram. Transfer FunctionInput Output For a linear, time-invariant system, the transfer function is given by,

Transfer Function (contd…) Consider the following linear time invariant (LTI) system With zero initial conditions, taking Laplace transform on both sides

Transfer Function (contd…) Rearranging, we get

Impulse Response Suppose, input to a LTI system is unit impulse. We get, Inverse Laplace Transform of this output gives the impulse response of the system. I.e. impulse response of the system is given by, Given g(t), input-output relationship in t-domain is given by the following convolution

Analisys of Transfer Function Consider the transfer function, If the denominator polynomialis set to 0, the resulting equation is called the characteristic equation The roots of the characteristic equation are called poles. zeros.

Example of Poles and Zeros Suppose the following transfer function Note: This is only for illustration. Positive real poles lead instability. Characteristic equation

Example of Poles and Zeros

Poles and Zeros plot Suppose, the following transfer function Zeros are represented by circles (O) and poles by cross (x).

Block Diagram representation Transfer FunctionInput Output

Closed-Loop System Block Diagram + -

Closed Loop Transfer function

Closed Loop Transfer function (contd…) + -

Block Diagram Transfer FunctionInput Output

Block Diagram Transformation

Block Diagram Reduction

Moving a pickoff point behind a block Eliminating feedback loop

SIGNAL FLOW GRAPH MODEL Nodes which are connected by several directed branches Graphical representation of a set of linear relation. Basic element is unidirectional path segment called a branch. The branch relates the dependency of input/output variable in a manner equivalent to a block of block diagram.

SIGNAL FLOW GRAPH MODEL Variables are reperesented as nodes. Transmittence with directed branch. Source node: node that has only outgoing branches. Sink node: node that has only incoming branches. R(s)R(s) G(s) Y(s) Y(s)Y(s) B(s)B(s) E(s)E(s) D(s) F(s)F(s) As signal flow graph D(s) R(s) E(s) 1 G(s)F(s) 1 Y(s) 1 H(s) B(s)

Cascade connection 

Parallel connection Two parallel branch

Mason rule where  Total transmittence for every single loop  Total transmittence for every 2 non-touching loops  Total transmittence for every 3 non-touching loops  Total transmittence for every m non-touching loops  Total transmittence for k paths from source to sink nodes. where:  Total transmittence for every single non-touching loop of ks’ paths  Total transmittence for every 2 non-touching loop of ks’ paths  Total transmittence for every 3 non-touching loop of ks’ paths.  Total transmittence for every n non-touching loop of ks’ paths.

Example: Determine the transfer function of the following block diagram. P Q H RY and. etc. Transfer function

Example: Determine. A B D C E RY and