Project!
Model-based Investigation of the Effect of Tuning Parameters on a Servo-Motor Response and Mode Transition
1.0 Project Objective: The objective of this project is to familiarize the students with the use and limitations of models in understanding the response of a practical servo-system and evaluating its usefulness. It introduces system modes as a tool of evaluating the quality of a system’s output. It also explores the ability of a controller’s tuning parameters to affect the system’s behavior and cause it to shift from one mode to another. 2.0 Equipment: Matlab, Simulink and the EE380 textbook.
3.0 Background: This section provides a brief background about the tools and concepts needed to understand the role of mathematical constructs in modeling, predicting and tuning the behavior of a system. 3.1: System Modes and the Quality of its Response Control systems are enablers whose objective is to make a servo-process (SP) yield to the commands of an operator and provide him with useful work. When the operator issues a command, the SP can only respond by being in one of the following modes:
o If the SP complies with the command of the operator it is in a stable mode o If it does not comply with the command, the mode is called unstable
Figure-1: Stability does not imply useful outcome from the servo-process
Obviously, the minimum expectation of the operator from the servo-process is to be in a stable mode. However, the system being in a stable mode need not necessarily mean that it is providing the operator with useful work. The quality of the system response could be too poor and practically useless. Take for example a car as a servo-process (figure-1).
If the car fails to reach the destination, one may call the process unstable. However, if the process is stable and the car is able to reach the destination but the path it took is too long, rough, consumes too much fuel and contains many detours, the effort derived from the car cannot be called useful. Measuring the usefulness of the outcome from a servo-process is important since a response that is not useful defeats the purpose of control. One way to assess the quality of a system’s response is though the use of performance measures such as overshoot, settling time, rise time and steady state error. A more general way of describing the quality of system behavior is through using system modes. System modes may be used to qualitatively describe the whole state of the response not particular aspects of it, as in performance measures. In industrial applications, six system modes are used to describe the response of a practical servo-process. They are unstable, over-damped, critically-damped, under-damped, oscillatory and chattering. Their description and profile are shown in table-1.
Mode Description Profile
1 Unstable Instability causes the position to diverge from the reference position in either an oscillatory or an exponential manner.
2 Over- damped
A steady and slow motion towards the reference position with no oscillation
3 Critically- damped
Fast, steady and oscillation-free motion towards the reference position.
4 Under- damped
Oscillations whose strength decays with time are present in motion as it approach the reference position.
5 Oscillatory Sustained position oscillations of equal magnitude where motion does not settle at the reference location.
6 Chattering Audible high-frequency, low-magnitude, sustained
oscillations around the reference position are present in motion
Table-1: System modes 3.2: The Tuning Parameters of a System. Assume that the quality of a system’s response was assessed. If the quality is satisfactory, the operator may use the system. On the other hand, if the quality is not satisfactory, the operator has to either replace the system or adjust its performance to meet the needed quality level. Being able to tune a servo-system requires the system to have a second input port besides the command port the operator uses to drive the output to the desired value (figure-2). Through the extra port, a set of variables called the tuning parameters set (β) affects the behavior of the system. In effect, this makes the transfer function of the system a function of both the complex frequency S and the set of tuning
parameters (equation-1). Changing the tuning parameters will simply change the transfer function of the process.
)H(S, X Y
β= (1) Such a situation is mostly encountered when feedback control is used to adjust the behavior of a servo-process. For example, in figure-3 position and velocity feedback are used with a servo-motor, β = {Kp, Kv}, where Kp is the forward position gain and Kv is the velocity feedback gain.
Figure-2: Servo-system with and without tuning parameters (β).
Figure-3: A servo-system with two tuning parameters Kp and Kv.
3.3: Models and Virtual Experiments. One can experiment with different values of the tuning parameters and select the one’s that provide the best output quality. This approach may be undesirable since some values of the tuning parameters could cause instability and damage the servo-process. In addition, physical experiments are usually costly and although a must, they should be used sparingly in tuning a system. One way to lower the burden of physical experimentation is to use simulation. The core of a simulation experiment is a mathematical model of the servo-process. A model is a device or a process used by an engineer to predict the future output of a system in response to an action applied on the system in the present. A Model is constructed by studying the components of a system and mathematically capturing their input-output relation. An aggregator is then used to combine the individual behaviors of the components to yield the system behavior. For example, ohm’s law, which is an empirical law, is used to model circuit components (figure-4). Kirchhoff’s laws are then used to aggregate the individual relations to yield the equation that ties the input of the electric circuit to its output (figure-5).
Figure-4: Ohm’s law used to model individual circuit components
Figure-5: Kirchhoff’s laws used to construct the overall circuit (system) model.
The ability of a model to emulate a system’s response and predict its behavior depends on using full information about how the system works. This is usually not possible; there will always be missing information about the forces that act on a system. As a result, the ability of a model to predict the behavior of a system will always be limited. This does not mean that the model is useless; it only means that it cannot predict everything about the behavior of the system. In any case, some applications do not require high accuracy and full behavior spectrum prediction while other applications do. The more information used in constructing a model, the better is its ability to predict a system’s behavior.
3.4: Analysis Tools: System Poles , Rootlocus and Routh-Horowitz Using a model of a physical system to assist in predicting its behavior requires the identification of the information bearing elements of the model and the development of tools that can extract the needed information from these elements.
Figure-6: System behavior versus pole location.
Poles and zeros, especially poles, seem to be the most important element that bears information about the behavior of linear systems and the modes it can assume (figure-6). The most important behavior about a system is to know whether it is stable or unstable. In control system theory, the Routh–Hurwitz stability criterion is a mathematical test that
is a necessary and sufficient condition for the stability of a linear time invariant (LTI) control system. The Routh test is an efficient recursive algorithm to determine whether all the roots of the characteristic polynomial of a linear system have negative real parts (i.e. the system is stable). It operates on the denominator of the system transfer function (2). It begins by constructing the Routh-Horowitz matrix (3) then checking the signs of its first column. If all the sign are the same, the poles of H(S) (zeros of D(S)) are all in the right hand side and the system is stable.
D(S) N(S)
H(S)_ =
(2)
(3) While the location of the system poles provides information about system behavior, the trajectories of the poles (figure-7) provide a good idea about how the system modes change.
Figure-7: Pole trajectory versus system response
Studying the trajectories the system poles as a function of the tuning parameters used to modify the behavior of the system is important in understanding the effect of these parameters on the system. It is also an important aid for the designer that allows him to determine the value or set of parameter that place the system in a desired mode. In control theory and stability theory, root locus analysis (figure-8) is a graphical method for examining how the roots of a system change with variation of a certain system parameter. The root locus plots the poles of the closed loop transfer function in the complex s-plane as a function of a gain parameter.
Figure-8: A Root-locus Plot.
In this project, the effect of changing the tuning parameters of a controlled servo-motor on its mode changes is investigated. Three different models of the position servo with different accuracies are used:
1- a second order linear model of position obtained by neglecting the small electric time constant of the motor and considering only the large mechanical time constant.
2- A third order linear model of position that includes both electrical and mechanical time constants of the motor.
3- A third order model of position that includes nonlinearities such as servo- amplifier saturation and deadzone nonlinearities induced by phenomenon such as static friction.
A single loop, position feedback control is used with a single tuning parameter, the forward error proportional gain Kp.
4.0 Second order motor model-based behavior prediction: A motor is an electromechanical device with an electrical input component connected to a mechanical output component (figure-9). The system is assumed linear and can be modeled by an integrator and first order transfer function for the mechanical and electrical components with time constants τ m and τ e respectively. Furthermore, the electric time constant is assumed to be much lower than the mechanical one and is ignored (τ m >>>τ e). In this case, the position transfer function of the motor reduces to the second order transfer function in (4).
1S
K S K
1)S(S K
S)(H m
21m
+⋅ +=
+⋅ =
ττ m (4)
Figure-9: An electromechanical machine in a motor mode.
A single loop position feedback is used to control the motor (figure-10). The only tuning parameter used to adjust the quality of system behavior is the error proportional gain (Kp).
Figure-10: Servo-loop, 2nd order linear servo-motor
4.1 Mode transition for 2nd order system:
1- Choose τ m =0.5 sec and Km=5. 2- Derive the the tuning parameter (Kp) dependant transfer function of the system in
figure-10. 3- Use matlab to draw the rootlocus of the closed loop system with Kp as the free
variable. 4- From the root-locus plot, determine the modes of the servo-motor which the second
order linear model can predict. Also, determine the region of values of Kp that would place the system in each one of these modes.
5- Select specific values for Kp that would place the feedback system in each of these modes and use matlab to plot on the same graph the step response of the system for all the values of the selected Kps.
5.0 Third order motor model-based behavior prediction: To improve the accuracy of the servo-motor model, the electric time (τ e) constant is not neglected. In this case, the transfer function of the motor becomes third order (5).
1S K
1S K
S K
1)S(1)S(S K
S)(H 3 e
21m
+⋅ +
+⋅ +=
+⋅+⋅ =
mem ττττ (5)
Similar to the second order model, the servo-motor is placed in a position feedback configuration with tuning parameter Kp (figure-11).
Figure-11: Servo-loop, 3rd order linear servomotor
5.1 Sensitivity to the electric time constant: In this section, the effect of the electric time constant (τ e) on the ability of the second order model of the motor to predict the behavior of the third order model (figure-11) is examined.
1- Obtain the transfer function of the feedback system in figure-11. 2- Select Km=5, τ m =0.5 and Kp as the critically damped value you obtained
previously for the second order model (section 4.1). 3- Plot on the same graph the step response the following values of ( τ e
=1, .5, .4, .3, .2, .1). Record your observations. 4- Plot the maximum overshoot from the step response versus the electric time
constant for the following values of ( τ e =1, .5, .4, .3, .2, .1). Record your observations.
5- Use the transfer function to derive the rootlocus equation of the system with the electrical time constant (τ e) as the free variable (6)
0 D(S) N(S)
1 e =⋅+τ (6)
6- For the three values of Kp representing the other modes in the 2nd order model obtained in section 4.1,
7- Use the Routh-Horowtiz criterion to determine the range of τ e for which the system is stable,
8- use the root-locus plot to determine the range of values of τ e where the third order system may be approximated by a second order system with two dominant poles. This can be done by determining the range of τ e for which the quickly fading pole is about 10 times the real part of the two dominant complex poles.
9- Is it always possible to find a value of τ e where approximating a 3rd order system with a 2nd order system is possible? If you find more than one solution for τ e that satisfies the quickly fading condition, which one to keep and which one to reject?
10- Draw the 3 rootloci for each Kp on the same graph. 5.2 Mode transitions as a function of Kp:
1- Choose the electric time constant equal to τ e =0.05 and τ e =.5. 2- What are the range of values for Kp for which the system is stale? 3- Derive the root-locus equation of the system in figure-11 with Kp as the free
parameter. 4- Use the root-locus to determine the modes that the system response can assume.
Also, determine the range of values of Kp that will place the system in each one of these modes.
5- Select a point from each range a mode can assume and plot the step response on the same graph for the 3rd order system.
6- Also, for the second order model, plot the step response for the three values of Kp on the same graph. Compare the two cases.
6.0 Third order motor, Nonlinear, model-based behavior: Servo-systems, in particular motors, do experience a multitude of forces that cannot be expressed using linear models. For example, saturation in the servo-amplifier, dead-zone caused by static friction or semiconductor nonlinearity and hysteresis caused by mechanical couplings. This subsection will explore the effect of saturation and dead zone nonlinearities (figure-12) on the response of the servo-motor. Saturation affects the high magnitude of the motor actuating signal and causes a slowdown in the system. Dead zone affect the low magnitude of actuation and causes mainly steady state error.
Figure-12: Saturation and dead zone nonlinearities affecting the actuation signal
The presence of such nonlinearities can seriously alter the behavior of the system and disrupt the ability of linear models to predict the motor’s behavior. For the model to better predict the experiment, the effect of these nonlinearities is usually introduced in series with the motor (figure-13).
Figure-13: Servo-loop, 3rd order linear servomotor
Figure-14: Servo-loop, 3rd order linear servomotor – Simulink realization
6.1 Effect of amplifier saturation on system response: 1- Use simulink (figure-14) to construct the system in figure-13 and choose the
saturation nonlinearity from the discontinuous components menu. 2- Set Km=5, τ e =.05, τ m= .5 3- Select a step input from the sources block set of simulink and the “to workspace’
block to export the output to matalb, 4- Repeat the following steps for the three values of Kp representing each mode of the
third order linear model (section 5.2)
5- Assume that the saturation nonlinearity does not exist (you may do so by setting the saturation block level to a very high value), determine the maximum absolute value the input to the motor can assume when the system is subjected to a unit step input.
6- On the same graph, plot on the same graph the step response for the following saturation levels: 90%, 70%, 40%, 15% of the maximum absolute value of the input to the saturation-free case
7- Record your observation about the effect of saturation on the step response of the system
6.2 Mode transition as a function of the tuning parameters - saturation: 1- Select a 70% saturation level of the maximum absolute value 2- Use simulink to explore the effect of Kp on the modes that the system can assume. 3- Determine the range of values of Kp that will place the system in each one of these
modes. 4- Select a point from each range a mode can assume and plot the step response, 5- Repeat the above for a 40% and 10% saturation levels, 6- Record your observations
6.3 Effect of dead zone nonlinearity on system response:
1- Use simulink (figure-14) to construct the system in figure-13 and choose the dead zone nonlinearity from the discontinuous components menu.
2- Set Km=5, τ e =.05, τ m= .5 3- Select a step input from the sources and the “to workspace’ block to export the
output to matalb, 4- Repeat the following steps for the three values of Kp representing each mode of the
third order linear model (section 5.2) 5- Assume that the dead zone nonlinearity does not exist (you may do so by setting
the dead zone block level to a zero value), determine the maximum absolute value the input to the motor can assume when the system is subjected to a unit step input.
6- On the same graph, plot the step response for the following saturation levels: 30%, 20%, 10%, 5% of the maximum absolute value of the input to the dead zone-free case
7- Record your observation about the effect of dead zone on the step response of the system
6.4 Mode transition as a function of the tuning parameters – Dead Zone:
1- Select a 30% Dead Zone level of the maximum absolute value 2- Use simulink to explore the effect of Kp on the modes that the system can
assume. 3- Determine the range of values of Kp that will place the system in each one of
these modes. 4- Select a point from each range a mode can assume and plot the step response, 5- Repeat the above for a 20% and 10% saturation levels, 6- Record your observations
7.0 Instructions for writing the report: 1- The report must be typed, well-organized and contain all formal components of
objective, introduction, data section, data analysis section, conclusions and references
2- Details of the mathematical derivations has to be shown in full and all used equations must be numbered
3- All figures and tables must be numbered and provided with small descriptive captions
4- You have to comment on the results obtained. Figures, equations etc. that are merely stated with no comments will not be considered
5- You have to provide a meaningful conclusion section 6- You may explore the topic more. Any meaningful and well-documented extra work
will be rewarded with extra credit. 7- Any constructive suggestion about how to enhance the project will be taken into
consideration in marking the report.