Review last lectures

procedures for drawing block diagram Write the equations that describe the dynamic behavior for each component. Take Laplace transform of these equations, assuming zero initial conditions. Represent each Laplace-transformed equation individually in block form. Assembly the elements into a complete block diagram.

block diagram: example ei eo i Let consider the RC circuit: The equations for this circuit are:

block diagram: example take Laplace transform:

block diagram: example block representations for Laplace transforms: _ +

block diagram: example Assembly the elements into a complete block diagram. _ +

block diagram reduction Rules for reduction of the block diagram: Any number of cascaded blocks can be reduced by a single block representing transfer function being a product of transfer functions of all cascaded blocks. The product of the transfer functions in the feedforward direction must remain the same. The product of the transfer functions around the loop mast remain the same.

block diagram: reduction example _ + _ + +

block diagram: reduction example _ _ + + +

block diagram: reduction example _ _ + + +

block diagram: reduction example _ _ + + + +

block diagram: reduction example _ _ + +

block diagram: reduction example _ _ + +

block diagram: reduction example _ +

block diagram: reduction example

block diagram: reduction example NOTICE: Numerator of the closed-loop transfer function C(s)/R(s) is the product of the transfer functions of the feedforward path. The denominator of the closed-loop transfer function C(s)/R(s) is equal to: 1-Σ( product of the transfer functions around each loop) The positive feedback loop yields a negative term in the denominator.

signal flow graph input node (source) transmittance branch mixed node path forward path loop input node (source) output node (sink)

flow graphs of control systems block diagram: signal flow graph:

flow graphs of control systems _ + block diagram: signal flow graph:

flow graphs of control systems + _ + + block diagram: signal flow graph:

signal flow graph algebra

signal flow graph algebra

signal flow graph algebra

flow graphs of linear systems

Transient and steady state response analyses Input signal not known ahead of time but is random in nature! In analyzing and designing we need basic comparison of performance so we need input signal by specifying particular test input signals.

Typical Test Signals Step functions Ramp functions Acceleration (Parabolic) functions Impulse functions Sinusoidal functions Polynomial functions Note: which input signals we must use? Depend on the system normal operation inputs

Test Input Signals The impulse input is useful when we consider the convolution integral for the output y(t) in terms of an input r(t): This relationship is shown in the block diagram: If the input is a unit impulse function then

Transient response and Steady- State response Time response parts Steady state Output which behaves as t approach infinity Transient Which Goes from initial state to the final state Or output minus steady state output!

First order systems response

Final value Theorem for steady state response

Second order systems response (steady state part)

Second order systems types Over damped real poles Critically damped Real equal poles Under damped Imaginary poles

Over damped response

Critically damped

Under damped

Under damped 2

Performance of a Second-Order System Consider the system: The closed loop output is:

Stable Unstable

Performance of a Second-Order System The transient response of this second-order system is shown below. As ζ decreases, the closed-loop roots approach the imaginary axis, and the response becomes oscillatory.

Performance of a Second-Order System For the unit impulse (R(s)=1) the output is: The transient response is: The impulse response of the second order system is shown here

Performance of a Second-Order System Standard performances are usually defined in terms of the step response of a system as shown below: The swiftness of the response is measured by the rise time (Tr), and the peak time (Tp).

Performance of a Second-Order System The percent overshoot for the unit step input is defined as: Mpt: is the peak value of the time response fv: is the final value of the response Normally fv is the magnitude of the input, but many systems have a final value that is different from the desired input magnitude.

Performance of a Second-Order System Settling time (Ts): the time required for the system to settle within a certain percentage, δ, of the input amplitude. The settling time is four time constants (τ=1/ζωn) of the dominant roots of the characteristic equation. The steady-state error of the system may be measured on the step response of the system as shown in the previous figure.

Performance of a Second-Order System Consider the second order system with closed-loop damping constant ζωn and a response described by We seek to determine the time, Ts, for which the response remains within 2% of the final value.

Performance of a Second-Order System The transient response of the system may be described in terms of two factors: The swiftness of the response, as represented by the rise time and the peak time The closeness of the response to the desired response, as represented by the overshoot and settling time

Performance of a Second-Order System The peak time relationship for this second-order system is: The peak response is: The percent overshoot is:

Performance of a Second-Order System The swiftness of step response can be measured as the time it takes to rise from 10% to 90% of the magnitude of the step input. This is the rise time (Tr). The swiftness of a response to a step input is dependent of ζ and ωn. For a given ζ, the response is faster for larger ωn. The overshoot is independent of ωn. For a give ωn, the response is faster for lower ζ. The swiftness of the response will be limited by the overshoot that can be accepted

Quiz and Home work Don’t forget quiz in next time Home works of chapter 1 No: 1, 2, 3, 4, 5, 6, 7, 8, 13.