Youngjune, Han young@ssu.ac.kr Chapter 4 Time Response Youngjune, Han young@ssu.ac.kr

Chapter Objectives How to find the time response from the transfer function How to use poles and zeros to determine the response of a control system How to describe quantitatively the transient response of first- and second- order system How to approximate higher-order systems as first or second order How to view the effects of nonlinearities on the system time response How to find the time response from the state-space representation

Introduction In chapter 2, how transfer function can represent linear, time-invariant system In chapter 3, systems were represented directly in the time domain via the state and output equation Chapter 4 is devoted to the analysis of system transient response

Poles, zeros, and System response Output response of a system = the forced response + natural response Solving a differential equation Taking the inverse Laplace transform Laborious and time-consuming Using the attribute qualitative to describe the method Using poles and zeros, and their relationship to the time response of a system

Poles, zeros, and System response Poles of a Transfer Function Values of the Laplace transform variable, s, that cause the transfer function to become infinite Any roots of the denominator of the transfer function that are common to roots of the numerator Zeros of a Transfer Function Values of the Laplace transform variable, s, that cause the transfer function to become zero Any roots of the numerator of the transfer function that are common to roots of denominator

Poles, zeros, and System response Pole and zeros of a First-Order System: An Example Poles and Zeros are plotted on the complex s-place Using an X for the poles Using a O for the zeros Finding the unit step response of the system

Poles, zeros, and System response A pole of the input function  generating the form of the forced response A pole of the transfer function  generating the form of the natural response A pole on the real axis  generating an exponential response of the form e-at (where –a is the pole location on the real axis) The zeros and poles  generating the amplitudes for both the forced and natural response

First-Order System First-order system without zeros  Laplace transform of step response Pole at –a generates response Ke-at

First-Order System Significance of parameter a, exponential frequency When t=1/a, e-at|t=1/a = e-1 =0.37 c(t)|t=1/a =1-e-at|t=1/a = 1- 0.37= 0.63

First-Order System Three transient response performance specifications Time Constant, 1/a the time for e-at to decay to 37% of its initial vaue the time it takes for the step response to rise to 63% of the system response Rise Time, Tr the time for the waveform to go from 0.1 to 0.9 of its final value Settling Time, Ts the Time for the response to reach, and stay with, 2% of its final value

First-Order System First-order Transfer Functions via Testing  

Second-Order System Introduction

Second-Order System Overdamped Response, 4.7(a)  A pole at the original that comes from the unit step input: generating the constant forced response Two system poles on real axis: generating an exponential natural response whose exponential frequency is equal to the pole location 

Second-Order System Underdamped Response4.7(c)  A pole at the original that comes from the unit step input: generating the constant forced response Two complex poles: generating an exponential natural response An exponentially decaying amplitude, time constant: the reciprocal of the real part of the system pole A sinusoidal waveform, frequency: the imaginary part of the system pole 

Second-Order System Underdamped Response, 4.7(c)

Second-Order System Undamped Response4.7(c)  A pole at the original that comes from the unit step input: generating the constant forced response Two complex poles at imaginary axis: generating an exponential natural response zero decaying amplitude a sinusoidal waveform, frequency: the imaginary part of the system pole 

Second-Order System Critically damped Response4.7(c)  A pole at the original that comes from the unit step input: generating the constant forced response Two multiple real poles at real axis: generating an exponential natural response An exponential and an exponential multiplied by time Exponential frequency is equal to the location of the real poles 

The General Second-Order System Two physically meaningful specification: natural frequency and damping ratio Natural Frequency, ωn The frequency of oscillation of the system without damping For exampling, RLC circuit with the resistance shorted Damping ratio,ζ The ratio of the exponential decay frequency of the envelope to the natural frequency

The General Second-Order System Consider the general system, Without damping,  the natural frequency  Assuming an underdamped system, General second-order transfer function A real part, σ, equal to –a/2

The General Second-Order System Second-order response as a function of damping ratio

Underdamped Second-Order System A detailed description of the underdamped response is necessary for both analysis and design. Finding the step response for the general second-order system Expanding by partial fraction

Underdamped Second-Order System Taking the inverse Laplace transform

Underdamped Second-Order System Other parameters associated with the underdamped response: Peak time : the time required to reach the first, or maximum, peak Percent overshoot (%OS) : The amount that the waveform overshoots the steady-state, or final, value at peak time, expressed as a percentage of the steady-state value Settling time : the time required for the transient’s damped oscillations to reach and stay within ±2% of the steady-state value Rise time : the time required for the waveform to go 0.1 of the final value to 0.9 of the final value

Underdamped Second-Order System Other parameters associated with the underdamped response

Underdamped Second-Order System Evaluation of Tp Assuming zero initial conditions Completing squares in the denominator Setting the derivative equal to zero,  Peak Time(Tp):

Underdamped Second-Order System Evaluation of %OS Given by Cmax is found by evaluating c(t) at the peak time,  Percent overshoot, %OS for the unit step( cfinal = 1 ):

Underdamped Second-Order System Evaluation of Ts  Settling Time (Ts):  an approximation of Settling Time(Ts):

Underdamped Second-Order System Evaluation of Tr

Underdamped Second-Order System Relating peak time, percent overshoot, and settling time to the location of the poles  the radial distance from the original to the pole = natural frequency( ) 

Underdamped Second-Order System Tp: Inversely proportional to the imaginary part of the pole  Horizontal lines: lines of constant peak time Ts: Inversely proportional to the real part of the pole  Vertial lines: lines of constant settling time Since , radial line = lines of constant Since percent overshoot is only a function of , radial line = lines of constant percent overshoot

Underdamped Second-Order System Lines of constant peak time, settling time, percent overshoot

Underdamped Second-Order System Lines of constant peak time, settling time, percent overshoot

Example Transient Response through component design The transfer function From the transfer function, From the problem statement, 

Example Transient Response through component design A 20% overshoot  From the problem statement, K= 5N-m/rad  D=1.04 N-m-s/rad, and J=0.26kg-m2

System Response with additional Poles A system with more than two poles or zeros Can be approximated as a second-order system that has just two complex dominant poles Consider a three-pole system with complex poles and a third poles on the real axis Complex poles: Real pole: The output transform:  In time domain,

System Response with additional Poles Figure 4.23 Component responses of a three-pole system: a. pole plot; b. component responses: non-dominant pole is near dominant second-order pair (Case I), far from the pair (Case II), and at infinity (Case III)

System Response with additional Poles Case 1: αr = αr1 and is not much larger than ζωn The real pole’s transient response will not decay to insignificance at the peak time or settling time generated by the second-order pair.  Cannot be represented as a second-order system Case 2, 3: αr ≫ζωn The pure exponential will die out much rapidly than the second-order underdamped step response  parameters as percent overshoot, settling time, and peak time will be generated by the second-order underdamped step response component Can be represented as a pure second-order system If the real pole is five times farther to the left than the dominant poles, the system can be represented by its dominant second-order pair of poles

System Response with zeros Considering the general effect of a zero Adding a zero to the transfer function The response: the derivative of the original response and a scaled version of the original response If a , the negative of the zero, is very large, the response is approximately aC(s). As a become smaller, the derivative term contributes more to the response and has a greater effect. If a is negative, placing the zero in the right half-plane, the response may begin to turn toward the negative direction  nonminimum-phase system

System Response with zeros A two-pole system with a poles at (-1+j2.828) Figure 4.25 Effect of adding a zero to a two-pole system Figure 4.26 Step response of a nonminimum-phase system

4.10 Laplace Transform Solution of State Equation

4.10 Time Domain Solution of State Equation Assuming a homogeneous state equation of the form Assuming a series solution Substituting X(t) into homogeneous state eq.

4.10 Time Domain Solution of State Equation Equating like coefficients yields

4.10 Time Domain Solution of State Equation From previous eq. X(0) = bo Therefore, where eAt is state-transition matrix where Φ(t)= eAt

4.10 Time Domain Solution of State Equation Let us now solve the forced, or nonhomo-geneous, problem. Rearrange and multiply both sides by eAt

4.11 Time Domain Solution of State Equation The solution in the time domain The state-transition matrix

Case Studies: Antenna Control: Open-Loop Response Figure 4.32 Antenna azimuth position control system for angular velocity: a. forward path; b. equivalent forward path

Case Studies: Antenna Control: Open-Loop Response (a) Predicting the nature of the unit step response (b) the damping ratio and natural frequency  and (c) the angular velocity response to a step input  

Case Studies: Antenna Control: Open-Loop Response (d) converting the transfer function to state space The transfer function Taking the inverse Laplace transform Defining the phase variable as State equation and output equation