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Chapter 4. Time Response I may not have gone where I intended to go, but I think I have ended up where I needed to be. Pusan National University Intelligent Robot Laboratory
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Table of Contents Introduction Poles, Zeros and System Response
First-order System Second-order System: Introduction General Second-order System Underdamped Second-order System System Response with Additional Poles System Response with Zeros Effects of Nonlinearities upon Time Response Laplace Transform Solution of State Equations Time Domain Solution of State Equations
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Introduction Control System Definition
Analysis of system transient response Step response of the first and second-order systems Poles and zeros of a transfer function
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Poles, Zeros and System Response
Output response of a system Forced response (steady-state response, particular solution) Natural response (homogeneous solution) Forced response + natural response Poles of a transfer function S that causes the transfer function to become infinite. Zeros of a transfer function S that causes the transfer function to become zero.
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Poles, Zeros and System Response
Poles and zeros of a first-order system : An example Figure 4.1 a. System showing input and output; b. pole-zero plot of the system
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Poles, Zeros and System Response
Poles and zeros of a first-order system: An example Figure 4.1 c. Evolution of system response A pole of the input function -> forced response A pole of the transfer function -> natural response A pole on the real axis -> exponential response Zeros and poles -> amplitudes.
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Poles, Zeros and System Response
Example 4.1 Given the system of Figure 4.2, write the output, , in general terms. Specify the forced and natural parts of the solution Figure 4.2 System for Example 4.1
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First-Order System A first-order system without zeros
(b) Figure 4.3 a. First-order system b. pole plot
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First-order System Time constant, Rise time, Tr Settling time, Ts
Time to decay to 37 % of its initial value or to rise to 63 % of its final value. : the exponential frequency Rise time, Tr Time for the waveform to go from 0.1 to 0.9 of its final value Settling time, Ts Time for the response to reach, and stay within, 2 % of its final value Figure 4.4 First-order system response to a unit step
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First-Order System First-order transfer function via testing
No overshoot Nonzero initial slope Figure 4.5 Laboratory results of a system step response test
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Second-order System: Summary
Figure 4.6 Second order system, pole plots, and step responses
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Second-order System: Overdamped
Over damped response Two system real poles An exponential natural response whose exponential frequency is equal to the pole location
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Second-order System: Underdamped
Under damped response Two system complex poles Real part: exponential decay of the sinusoid's amplitude Imaginary part: the frequency of the sinusoidal oscillation
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Second-order System: Envelope
Under damped response Second-order step response components generated by complex poles Sinusoidal Freq.: damped frequency of oscillation, Figure 4.7 Second-order step response components generated by complex poles
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Second-order System: Example 4.2
By inspection, write the form of the step response of the system in Figure 4.8 Factoring the denominator of the transfer function in Figure 4.8, We find the poles to be Real part, -5, is the exponential frequency for the damping. Imaginary part, 13.23, is the radian frequency for the sinusoidal oscillations is a constant plus an exponentially damped sinusoid. Figure 4.8 System for Example 4.2
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Second-order System: Undamped
Un-damped response Two system imaginary poles A sinusoidal natural response whose frequency is equal to the location of the imaginary poles. Not decay
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Second-order S.: Critically damped
Critically damped response Two system multiple real poles An exponential and an exponential multiplied by time.
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Second-order System: Summary
Figure 4.9 Step response for second-order system damping cases Over damped responses Poles: Natural response: Under damped responses Poles: Natural response: Un-damped responses Poles: Natural response: Critically damped responses Poles: Natural response:
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The general Second-order System
Natural frequency, Frequency of oscillation of the system without damping. Damping ratio, General second-order system Without damping,
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The general Second-order System
By definition, the natural frequency, is the oscillation of this system. Poles of this system are on the axis at Assuming an under damped system, the complex poles have a real part, , equal to –a/2 From which Second-order transfer function finally looks like this:
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The general Second-order System
Example 4.3 Given the transfer function of find and General second-order transfer function , from which Also, Substituting the value of
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The general Second-order System
Figure Second-order response as a function of damping ratio
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The general Second-order System
Example 4.4 For each of the system shown in Figure 4.11, find the value of Report the kind of response expected Figure System for Example 4.4
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The general Second-order System
Using the values of a and b from each of the system of Figure 4.11, we find for system , which is thus over damped. Since for system , which is thus critically damped And for system , which is thus under-damped, since
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Underdamped Second-order System
The transform of the response, , is the transform of the input times the transfer function where it is assumed that (the under damped case)
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Underdamped Second-order System
Taking the inverse Laplace transform Figure Second-order under damped response for damping ratio values
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Underdamped Second-order System
Figure Second-order under damped response specifications
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Underdamped Second-order System
Second-order under damped response specifications Rise time, : the time required for the waveform to go from 0.1 of the final value to 0.9 of the final value Peak time, : the time required to reach the first or maximum, peak Percent overshoot, : the amount that the waveform overshoots the steady-state, or final, value at the peak time, expressed as a percentage of the steady-stat value Settling time, : the time required for the transient’s damped oscillations to reach and stay within of the steady-state value
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Underdamped Second-order System
Evaluation of Assuming zero initial conditions Completing squares in the denominator Therefore Settling the derivative equal to zero yields
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Underdamped Second-order System
Evaluation of From Figure 4.13 the percent overshoot is given by The term is found by evaluating at the peak time, Figure 4.13,
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Underdamped Second-order System
For the unit step, Figure percent overshoot versus damping ratio
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Underdamped Second-order System
Evaluation of Stays within of the steady-state value, Assuming that at the settling time Damping ratio varies from 0 to 0.9 => the numerator varies from 3.91 to 4.74
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Underdamped Second-order System
Evaluation of Stays within of the steady-state value, Figure Normalized rise time versus damping ratio for a second-order underdamped response
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Underdamped Second-order System
Example 4.5 Given the transfer function Find is 0.475 is 2.838
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Underdamped Second-order System
is second Using the table The normalized rise time is approximately 2.3 seconds Dividing by yields second
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Underdamped Second-order System
The pole plot for a general, under damped second-order system The radial distance from the origin to the pole is the natural frequency, and the is the imaginary part of the part and is called the damped frequency of oscillation is the magnitude of the real part of the pole and is the exponential damping frequency Figure 4.16 Pole plot for an under damped second-order system
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Underdamped Second-order System
Note : Figure 4.17 Lines of constant peak time, , settling time, , and percent overshoot,
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Underdamped Second-order System
Step responses of second-order under damped system as poles move Figure 4.18 With constant real part (constant ) Figure 4.19 With constant imaginary part (constant )
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Underdamped Second-order System
Step responses of second-order under damped system as pole move Figure 4.20 With constant real part (constant )
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Underdamped Second-order System
Example 4.6 Given the pole plot shown in Figure 4.21, find The damping ratio is given by The natural frequency, , is the radial distance from the origin to the pole, or The peak time is The percent overshoot is The approximate settling time is Figure 4.21 Pole plot for Example 4.6
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Underdamped Second-order System
Example 4.7 Given the system shown in Figure 4.22, find J and D to yield 20% overshoot and a settling time of 2 seconds for a step input of torque The transfer function for the system is From the transfer function, Figure 4.22 Rotational mechanical system for Example 4.7
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Underdamped Second-order System
From the problem statement, Or Hence, , 20% overshoot implies From the problem statement, K=5 N-m/rad, D=1.04 N-m/rad, J=0.26kg-m^2
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System Response with Additional Poles
Dominant pole Under certain conditions, a system with more than two pole or with zeros can be approximated as a second-order system that has just two complex dominant pole Assuming that the complex poles are at the real pole is at In the time domain,
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System Response with Additional Poles
Case , and is not much larger than Case , and is much larger than
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System Response with Additional Poles
Case , Figure 4.23 Component responses of a three-pole system : component responses
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System Response with Additional Poles
Assume a step response, , of a three-pole system where we assume that the non-dominant pole is located at -c on the real axis The steady-state response approaches unity As the non-dominant pole approaches , or , Thus, for this example, D, the residue of the non-dominant pole and its response, becomes zero as the non-dominant pole approaches infinity.
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System Response with Additional Poles
Example 4.8 Find the step response of each of the transfer functions shown in through and compare them. The step response, , for the transfer function, , can be found by multiplying the transfer function by 1/s, a step input. Figure 4.24 Step responses of system
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System Response with Zeros
Effect of adding a zero to a two-pole system The closer the zero is to the dominant poles, the greater its effect on the transient response As the zero moves away from the dominant poles, the response approaches that of the two-pole system Figure 4.25 Effect of adding a zero to a two-pole system
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System Response with Zeros
The residue of each pole will be affected the same by the zero The relative amplitudes remain appreciably the same If the zero is far from the poles, then a is large compared to b and c
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System Response with Zeros
The zero does not change the relative amplitudes of the components of the response If we add a zero to the transfer function, yielding Notice that the response begins to turn toward the negative direction even though the final value is positive non-minimum-phase system Figure 4.26 Step response of a non-minimum-phase system
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System Response with Zeros
Example 4.9 a. Find the transfer function, , for the operational amplifier circuit shown in Figure 4.27 The current, , through , is the same Figure 4.27 Non-minimum-phase electric circuit
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System Response with Zeros
But Substituting into , Using voltage division, Substituting and into Let A approach infinity
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System Response with Zeros
b. Let , and find the step response of the filter Letting Expanding into partial fractions,
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System Response with Zeros
Or the response with a zero is Also, from Or the response without a zero is Figure 4.28 Step response of the non-minimum-phase network of Figure and normalized step response of an equivalent network without the zero
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System Response with Zeros
Example 4.10 , determine whether there is cancellation between the zero and the pole closest to the zero And for any function for which pole-zero cancellation is valid, find the approximate response The partial-fraction expansion of is The residue of the pole at -3.5, which is closest to the zero at -4, is equal to 1 and is not negligible compared to the other residues Thus, a second-order step response approximation cannot be made for
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System Response with Zeros
The partial-fraction expansion for is The residue of the pole at -4.01, which is closest to the zero at -4, is equal to 0.033, about two order of magnitude below any of the other residues The response is approximately
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Effects of Nonlinearities upon Time Response
Chapter 2 and look at the load angular velocity, , where Figure 4.29 a. Effect of amplifier saturation on load angular velocity response; b. Simulink block diagram
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Effects of Nonlinearities upon Time Response
Figure 4.30 a. Effect of deadzone on load angular displacement; b. Simulink block diagram
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Effects of Nonlinearities upon Time Response
Figure 4.31 a. Effect of backlash on load angular displacement response; b. Simulink block diagram
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Laplace Transform Solution of State Equations
Consider the state equation and the output equation Taking the Laplace transform of both sides of the state equation yields Combining all of the terms The final solution for is Taking the Laplace transform of the output equation yields
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Laplace Transform Solution of State Equations
Eigen values and transfer function poles If a system is represented in state-space We can find the poles from
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Laplace Transform Solution of State Equations
Example 4.11 Given the system represented in state space by Do following: a. Solve the preceding state equation and obtain the output for the given exponential input b. Find the eigen values and the system poles
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Laplace Transform Solution of State Equations
a. We will solve the problem by finding the component parts of , followed by substitution into
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Laplace Transform Solution of State Equations
Since (the Laplace transform for ) is can be calculated And using and from , respectively, we get
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Laplace Transform Solution of State Equations
The output equation is found from or where the pole at -1 canceled a zero at -1. Taking the inverse Laplace transform, b. The denominator of , which is , is also the denominator of the system’s transfer function Thus, furnishes both the poles of the system and the eigenvalues -2,-3, and -4
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Time Domain Solution of State Equations
The solution in the time domain is given directly by : the state-transition matrix : zero input response, : zero state response Thus, for the unforced system is the Laplace transform of the state-transition matrix
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Time Domain Solution of State Equations
Example 4.12 For the state equation and initial state vector shown in , where is a unit step, find the state-transition matrix and then solve for Since the state equation is in the from Find the eigenvalues using Hence, from which Since each term of the state-transition matrix is the sum of response generated by the poles
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Time Domain Solution of State Equations
Since each term of the state-transition matrix is the sum of response generated by the poles And since then
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Time Domain Solution of State Equations
The constants are solved by taking two simultaneous equations four time Also, Hence, the first term
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Time Domain Solution of State Equations
The last term The final result is found
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Time Domain Solution of State Equations
Example 4.13 Find the state-transition matrix of Example 4.12, using We use the fact that is the inverse Laplace transform of Then, first find as From which Finally, taking the inverse Laplace transform of each term, we obtain
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Intelligent Robotics Laboratory
H A N K Y O U Homework: divide by 9 Intelligent Robotics Laboratory
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