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LAB3: Study the Effects of Steady-State Error for a Physical System
Objective:
To study the effects of steady-state errors by varying system type, input waveforms and loop gains.
Description of system: Modern robotic manipulators that act directly upon their target environments must be controlled so that impact forces as well as steady-state forces do not damage the targets. At the same time, the manipulator must provide sufficient force to perform the task. In order to develop a control system to regulate these forces, the robotic manipulator and target environment must be modeled. Assuming the model shown in Figure (1) (Chiu, 1997)
Pre lab: (by hand)
Represent in transfer function of the manipulator and its environment under the following conditions
- The manipulator is not in contact with its target environment.
- The manipulator is in constant contact with its target environment.
Background:
Useful Matlab commands to check your prelab work
‘sys (‘s’); [B, A] = tf (H(s))’ returns the vector of numerator coefficients, B, and the vector of denominators, A, for the equivalent transfer function.
Figure: 1 Robotic manipulator and target environment (© 1997 IEEE)
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Steady-State Error (http://ctms.engin.umich.edu/CTMS/index.php?aux=Extras_Ess)
Steady-state error is defined as the difference between the input (command) and the output of a system in the limit as time goes to infinity (i.e. when the response has reached steady state). The steady-state error will depend on the type of input (step, ramp, etc.) as well as the system type (0, I, or II).
Calculating steady-state errors
Steady-state error can be calculated from the open- or closed-loop transfer function for unity feedback systems. For example, let's say that we have the system given below.
This is equivalent to the following system, where T(s) is the closed-loop transfer function.
Figure: 2 General negative feedback system with input x (t) and output y (t)
Figure: 3 closed loop system with input r (t) and output y (t)
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Therefore, a system can be type 0, type 1, etc. The following tables summarize how steady- state error varies with system type.
Type 0 system Step Input Ramp Input Parabolic Input
Steady-State Error Formula 1/(1+Kp) 1/Kv 1/Ka
Static Error Constant Kp = constant Kv = 0 Ka = 0
Error 1/(1+Kp) infinity infinity
Type 1 system Step Input Ramp Input Parabolic Input
Steady-State Error Formula 1/(1+Kp) 1/Kv 1/Ka
Static Error Constant Kp = infinity Kv = constant Ka = 0
Error 0 1/Kv infinity
Type 2 system Step Input Ramp Input Parabolic Input
Steady-State Error Formula 1/(1+Kp) 1/Kv 1/Ka
Static Error Constant Kp = infinity Kv = infinity Ka = constant
Error 0 0 1/Ka
Figure: 5 General closed loop transfer function with input r (t) and output y (t)
Table: 1 Relationships between input, system type, static error constants and steady-state errors
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Lab: (Using Simulink)
- Set up the negative feedback system of and H(s) =1. Plot on one graph the error signal of the system for an input of 3u (t) and K= 10, 100, 1000, 5000. Repeat for inputs 3tu (t) and 3t^2 u (t).
- Set up the negative feedback system of and H(s) =1. Plot on one graph the error signal of the system for an input of 3u (t) and K= 10, 100, 1000, 5000. Repeat for inputs 3tu (t) and 3t^2 u (t).
- Set up the negative feedback system of and H(s) =1. Plot on one graph the error signal of the system for an input of 3u (t) and K= 10, 100, 1000, 5000. Repeat for inputs 3tu (t) and 3t^2 u (t).
Lab Procedure:
- Check your prelab work using matlab commands given in background section.
- Get the Simulink block diagram for the case “ manipulator is not in constant contact with target environment ” and simulate the system with u (t) =3 for 10 secs. Discuss your output.
- Get the Simulink block diagram for the case “ manipulator is in constant contact with target environment ” and simulate the system with u (t) =3 for 10 secs. Discuss your output.
- Calculate the steady-state errors by hand for each case described in Lab section.
- Build the simulink diagram and outputs for each case described in Lab section. Note your observations and discuss.
- For Each case in Lab section compare your simulink result with calculation by hand. Explain the reasons for any discrepancies.
- Open the simulink ‘Library Browser’ by clicking on Start → Simulink. Open a new model file (.mdl). Build a model of the above closed loop system by using the respective blocks.
- Double-click on the blocks to open its property editor. This will give you options of changing the parameters of the blocks.
- Arrange the blocks in the proper order, and connect them. 10)Also save your workspace to matlab.
- Go to Simulation → Configuration Parameters. NOTE: Do not forget to save your work
Post Lab: Write a report in abstract, objective, theory, procedure, results, conclusion and appendices format. All steps should be clearly mentioned. Include all plots and results.
