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CONTROL SYSTEM DESIGN FOR AN AUTONOMOUS MOBILE ROBOT

A thesis submitted to the Division of Graduate Studies and Advanced Research of the University of Cincinnati

in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

In the Department of Mechanical, Industrial and Nuclear Engineering of the College of Engineering 2000

by

Sathish K. Shanmugasundaram B.E (Mechanical Engineering), PSG College of Technology Coimbatore, India

Thesis Advisor and Committee Chair: Dr. Ernest L. Hall

Abstract

A good control system is an absolute requirement for a mobile robot or, for any system that has to be automated. This thesis deals mainly with designing an optimal, stable control system for an autonomous robot. The robot is Bearcat II. Bearcat II is being developed as an automated guided vehicle to participate in the annual International Ground Robotics Competition automated vehicles contest. A closed loop control system called as proportional, integral, and derivate control system (PID) is used currently on the robot. It uses an elaborate algorithm to manipulate the parameters. Tuning of the controller is the real tricky part of these types of controllers. Various tuning methods currently in use are discussed. The difficulties in using these are also highlighted. The model of the control system is reviewed and the system response studied. A general model of a motor and load system is also developed. Based on the model, a new technique using optimization theory is developed. This theory concentrates on reducing the error between output and input at multiple instants of time. This is a multi-criteria optimization formulation of the problem. Methods used in the formulation are discussed. Finally, an optimized set of tuning constants is determined. The developed method is more logical, scientific and better in terms of minimal mean square error than previous methods. It also reduces the number of iterations required for tuning the controller.

Acknowledgements

First, I would like to sincerely thank Dr. Ernest Hall my advisor, for having pointed me to this direction and, for his enthusiastic and energetic guidance throughout my career at the University of Cincinnati. He has been a constant source of information and support for this thesis.

A special word of thanks should go the robotic team members, past and present for having supported in my work. They have helped me in all possible ways to make this process smooth. I would also like to add a note of thanks and special appreciation to the staff of the MINE department for having supported the team and me in all possible ways.

My committee members, Dr. Richard L. Shell and Dr. Ronald L. Huston, should be thanked for looking through my work and making valid suggestion. Their suggestions have made this work more focused.

Last but not least, my family should be thanked. They have stood by me and watched my career evolve for the better. They have also been a source of constant support and encouragement. I love you all very much.

Table of Contents

List of Figures………………………………………………………. 1. Introduction………………………………………………………….. 1.1 Introduction…………………………………………….. 1.2 Motivation and problem statement…………………….. 1.3 Objective………………………………………………. 1.4 Acknowledgement……………………………………… 1.5 Organization…………………………………………… 2. Background work and literature search…………………………… 2.1 Introduction…………………………………………….. 2.2 Linear and non-linear controller design ……………… 2.3 Parallel linkage steering for an AGV………………... 2.4 Modeling and control of an automated vehicle……… 3. Control System theory……………………………………………. 3.1 Introduction…………………………………………… 3.2 The basic control system……………………………… 3.2.1 3.2.2 The control problem…………………… Description of the input and output… 4 6 6 7 7 7 8 9 9 13 14 17 20 20 21 22 23 24 29

3.3 Feedback and feed forward control…………………… 3.4 Linear control systems…………………………………

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3.5 Control of non-linear systems………………………… 3.6 The design process……………………………………..

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4. PID and the Galil DMC1000 controller. …………………………. 4.1 Digital control systems……………………………….. 4.2 PID controller…………………………………………. 4.2.1 Proportional band.……………………………… 4.2.2 Integral.………………………………………… 4.2.3 Derivative.……………………………………… 4.3 Control loop tuning…………………………………… 4.4 Galil DMC 1000……………………………………… 4.5 Microcomputer section……………………………… 4.6 Motor Interface……………………………………… 4.7 Communication……………………………………… 4.8 General I/O.………………………………………… 4.9 Amplifier.…………………………………………… 4.10 Encoder…………………………………………… 5. Control parameter determination for the current robot……. 6. Simulation and optimizing the parameters………..………. 6.1 Introduction.………………………………………. 6.2 Simulation on SIMULINK.…………………………

38 38 44 46 47 48 50 55 57 57 58 58 58 59 61 65 65 65

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6.3 SIMULINK model…………………………………. 6.4 Optimizing the control system……………………… 6.5 The lsqnonlin method………………………………. 7. Results and summary…………………………………………… 8. Recommendations……………………………………………… 9. References ...……………………………………………………. 10. Appendix…………………………………………………………

68 69 70 74 75 76 83

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List of figures

Page

Chapter 2

2.1 2.2 2.3 2.4 Lane following feedback structure…………………………. Design of experimental automated guided vehicle………… Control system……………………………………………… Closed loop system performance……………………………. 7 11 13 14

Chapter 3

3.1 3.2 3.3 3.4 3.5 Simplified description of a control system…………………… Input output for the elevator system…………………………. A typical feedback control system……………………………. An antenna azimuth position control system………………… Functional block diagram of a position control system……… 17 19 21 28 29

Chapter 4

4.1 4.2 4.3 4.4 An example of a 3-bit quantized signal………………………… A control system……………………………………………….. A digital control system controlling a controlling a cont. plant… PID equations…………………………………………………… 36 37 38 41

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4.5 4.6 4.7

A time response plot showing controller action…………………. Controller frequency response plot………………………………. A cruise controller…………………………………………………..

43 45 48

Chapter 5

5.1 5.2 5.3 Start-up menu of Galil motion control…………………………….. Tuning methods menu of the Galil motion control………………. Step response plot from the Galil servo design kit………………. 57 58 59

Chapter 6

6.1 6.2 6.3 6.4 6.5 Simulink model of the system………………………………………. Step response for the system………………………………………. General motor and load control system…………………………….. Step response for the general system………………………………. Error diagram…………………………………………………………… 62 63 64 65 66

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Chapter 1

Introduction

1.1 Introduction Control systems are the very heart of mobile robots. Problems such as coordination of manipulators, motion planning and coordination of mobile robots require a central controller. Motion control is one of the technological foundations of automation. Whether the motion of a product, path of a cutting tool, motion of a robot arm, the control of motion is a fundamental concern. To be able to control a motion process, the precise position of objects needs to be measured. Then a feedback comparison of the target and actual positions is then a natural step in implementing a motion control system. This comparison generates an error signal that may be used to correct the system. If the error is small, the system performance is repeatable and accurate. However, the use of feedback can lead to an unstable system whose output can go to infinity with a small input signal. This shows the importance of a good control system design. Many feedback controllers use a proportional-derivative-integral algorithm to manipulate the error signal and apply a corrective effort to the process.

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1.2 Motivation and problem statement The main motivation of this thesis is the mobile robot Bearcat II that we are developing and continually improving. The robot uses a PID controller (Reference). Though this type of controller is common it is very hard to tune and attain stable PID parameters. The team has tried many different methods to tune the controller. A few of them are iterative trial and error and, brute force. Though these work for most situations, the stability of the system is lost at certain times. For example using the largest possible values could make the controller very aggressive and cause amplification of errors. On the other hand if the tuning it too conservative does not eliminate the errors and the system appears sluggish. Hence this is a difficult adaptive tuning process.

1.3 Objective To solve this problem a new technique using optimization theory is proposed in this thesis. It uses both simulation and optimization. Because it uses an objective criteria function, minimum mean square error, it is a more scientific and logical approach to the problem. Although it is not a perfect and final solution, it is a definite step toward reaching the perfect and one solution for these adaptive controllers.

1.4 Acknowledgement This thesis is an extension to the work done in this area especially by two previous team members, Kalyan Kolli and Krishna Mohan Kola. Some of the ideas and topics have been

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derived from their work and have been worked on and extended. We acknowledge them for their good and fascinating work. 1.4 Organization Chapter 2 deals with the introduction and survey of automated guided vehicles. Chapter 3 gives an introduction to control system theory and the various types of control techniques used. Chapter 4 looks at the PID controller used in the Bearcat II, Galil DMC 1000 controller. It also explains the purpose and meaning of the various parameters in the PID controller. Chapter 5 details the current tuning procedure for the Bearcat II. The software used to assist the tuning is described in detail. Chapter 6 models the control system and brings out the new optimization technique for parameter determination. Finally the results and recommendations for future work are discussed in Chapter 7. The Appendix shows the actual Matlab and Simulink code used to obtain the results.

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Chapter 2 Background work and literature search

2.1

Introduction

Automated guided vehicle systems (AGVS) are commonly used for transporting material within a manufacturing, warehousing, or distribution system. These systems provide for asynchronous movement of material through the system and are used in a wide variety of applications. They offer many advantages relative to other types of material handling systems, including reliable, automatic operation, flexibility to changes in the material handling requirements, improved positioning accuracy, reduced handling damage, easily expandable layout and system capacity, and automated interfaces with other systems (Miller, 1987)(References should be numbered and listed in the Reference List in order they are encountered in the text).

2.1.1

Classification of AGV’s – Following are the ways AGV’s are classified

1. Guide path determination

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1. Static path 1. Unidirectional 2. Bi-directional 2. Dynamic path 2. Vehicle capacity 1. Single unit load 2. Multiple loads 3. Vehicle addressing mechanism 1. Direct address 2. Indirect address

Most AGVs have a specification of maximum speed about 1 m/sec (3.28 ft/sec, 2.24 mph) although, in practice, they are usually operated at half that speed. In a fully automated manufacturing environment, there should be few personnel in the area; therefore, the AGV should be able to run at full speed. As the speed of the AGV increases, so does the difficulty in designing stable and smooth tracking controls.

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Consequently, it becomes important to understand the dynamic effects of various design parameters on the steering action of the AGV.

Work has been done in the past to analyze various aspects of steering control in different situations for automobile steering or mobile robots. Some deals with the kinematics and dynamics involved (Tagawa et al [11]; and others have dealt with the controller designs for a linear or non-linear control (Hattori et al [19]).

Ozguner, et al [3] described an analytical study of vehicle steering control. They considered the design and stability analysis of a steering controller. The objective of the controller was to steer a ground vehicle along a reference line located in the middle of the lane. An arbitrary look-ahead point was located on the local longitudinal axis of the vehicle. The displacement between the look-ahead point and the reference point was called the look-ahead offset. During perfect lane tracking, the ratio of the steering angle to the look-ahead offset was independent of the curve radius under reasonable approximations. That ratio was computed in terms of the vehicle speed and various vehicle parameters. Then, a constant controller was designed to achieve that ratio at steady state. The controller was updated as a function of the vehicle speed. The only information processed by the controller was the look-ahead offset, which was measured using a radar-based or vision-based sensor. Using Routh-Hurwitz analysis, it was analytically proven that the closed loop system is stable. Given any range of longitudinal

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speeds, there exists a sufficiently large look-ahead distance ensuring the closed loop closed-loop stability for all speeds in that speed range. The look-ahead information was a set of geometrical information, including road feedback topology ahead of the vehicle center of gravity and the vehicle’s orientation and position relative to the topology. In this paper the objective was to develop a new control law for automatic steering based on specific look-ahead information. The paper considered the vehicle model and discussed the control design and the stability analysis of the controller. A simulation example was presented. The mathematical block diagram of the proposed model was a shown in the Figure 2.1 below.

Sensor Look-ahead offset

Constant Controller

Steer angle Command

Vehicle dynamics & Vehicle-road interaction

Longitudinal Speed

Figure 2.1 Lane following feedback structure (Ozguner, et al. [3]) The authors conclude that the controller was successfully tested at the Transportation Research Center, Marysville, OH.

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2.2

Linear and non-linear controller design for robust automatic steering

Ackerman, et al4 proposed a “linear and non-linear controller design for robust steering.” The research work presented by them concentrated on the automatic steering of vehicles, which they described as part of an integrated system of integrated highway system of the future. The primary task of an automated steering is to track a reference path, where the displacement sensor measures the displacement from the guide path. The reference may consist of a magnetic field of an electrically supplied wire or permanent magnets in the road. The sensor is mounted in the center of the front end of the vehicle. The controller output acts on the front wheel steering angle.

The design of an automatic steering system is a robustness problem in view of the large variations in velocity and mass of the vehicle and contact between tire and road surface. In the proposed research the authors model the data and specifications for a city bus. A comparison was then made between linear and non-linear controller concepts.

For non-linear control it was investigated that the track accuracy was improved by additional feedback of the yaw rate, which could be measured by a gyro. Therefore the automatic control problem became much less dependent on the uncertain operating conditions of velocity, mass, and road-tire contact. The study showed a significant reduction in the displacement from the guideline for all maneuvers and operating

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conditions. In their study, the design method used the parameter space approach, which was further exploited to explore extreme design directions. The resulting controller with fixed gains achieved good performance for a wide range of uncertainty in the operating conditions.

Additionally, in their research a non-linear controller structure was designed in an effort to further improve the performance of the automatic steering system. The non-linear controller was based on sliding mode control and included dynamic adaptation to changing operating conditions via an estimator-like observer. The advantages and drawbacks of the two approaches were contrasted in simulation studies. Finally, controller parameters of both the linear and non-linear controller were tuned automatically by optimizing a vector performance for typical maneuvers.

2.3

Parallel Linkage Steering for an Automated Guided Vehicle.

Sung, et al [5] have considered in their work a parallel linkage steering for an automated guided vehicle. The paper described the computer-aided design of suitable control strategies for parallel linkage steering of an experimental automated guided vehicle. The microprocessor-based steering control system was modeled, and stability characteristics,

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using proportional control, were investigated via describing functions. Although the approximation is coarse, the analysis indicates the upper bound for gain and suggests derivative action for improved control. For better accuracy, a flexible and fast simulation program was developed that enabled an efficient search for “optimal” control laws for operation of the vehicle under the constant speed mode and the slow-down mode. Timeindependent control laws were investigated, and the solutions found gave tolerable tracking over 90-deg arcs of 0.5-m radius. The simulation results were shown in which the time dependent control laws were proved to be essential for improved, nonoscillatory tracking. The experimental AGV has a differential gear drive and parallel linkage steering as shown in Figure 2.4.

Figure 2.2 Design of experimental automated guided vehicle (Sung, et al. [5] ) Six infrared sensors [light emitting diodes (LEDs)] were used on the forward and reverse guidance systems. Because of the designed configuration of the wheel positions, the AGV was symmetrical in the forward and reverse directions, which simplified bi-

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directional control. A DC servomotor was used for the drive; a stepper actuated the steering wheels. The steering loop was closed by a microprocessor-based-controller that computed the steering control signal to the stepper motor based on the derivation of the AGV from the track. The track contained reflective tape, and the derivation was measured by discrete number of LEDs “out of track”. This coarse sensing method was selected over more sophisticated continuous sensors for cost reasons but at the expense of the anticipated poorer tracking response.

The critical parameters affecting the steering action of the AGV were identified as follows: 1. Method of sensing the deviation from the guide path 2. Geometrical configuration of the AGV’s wheels 3. Speed of the AGV and drive dynamics in case of variable speed operation 4. Steering angle as a function of the deviation, i.e., the control strategy or control law.

Parameters 1 and 2 above pertained to the AGV firmware, whereas items 3 and 4 are required to be designed on the basis of firmware. It was desirable to operate the AGV at its maximum speed (constant) all the time but as was seen in the research study that was not possible when the AGV was negotiating a curve. Thus, the computer-aided design focused on finding:

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1. The maximum speed possible in the constant speed mode of operation and the associated optimal control law 2. A suitable slow-down mode of operation for the AGV at 1m/sec and the associated optimal control law

2.4

Modeling and Control of an Automated vehicle

Will and Zak, [6] present a vehicle model that includes the vehicle dynamics and vehicle dynamic model. The model developed was then used for conducting steering analysis of an automated vehicle. They tested the developed model on a step lane change maneuver and proposed a model-reference-based controller for the remote control of a vehicle. Stability analysis of the closed-loop system using the Lyapunov approach was included.

One of the main objectives of their research was to develop a vehicle model that could be used to predict the dynamics for steering and braking maneuvers. They proposed a model that is a spring mass system acting under the influence of tire forces developed by three driver inputs. Those inputs were the front and rear wheel angles and the brake force. Another objective of their study was to develop design control strategies for automatically guided vehicles. They proposed a model-reference tracking controller design. Figure 2.6 shows the model-reference control system block diagram.

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Figure 2.3 Model-reference control system block diagram with input scaling (Will and Zak [6] )

Figure 2.4. Not Closed-loop system performance with model-reference tracking controller (Will and Zak [6] )

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Figure 2.6 above shows the responses of the truth model to the same input signal. The truth model was incapable of tracking the desired path when using an open loop control.

To improve the tracking performance, they proposed a closed-loop tracking controller. The proposed model-reference-based controller required the availability of state variables. They proposed to use combined controller state estimator compensators that require only the input and output signals in their implementations. The model was also discontinuous in the state, which resulted in high activity, referred to as chattering. Chattering could be reduced in a continuous version of the controller proposed. The continuous controller was obtained by introducing a so-called boundary layer that smoothed out the discontinuous controller. However, instead of asymptotic stability, only a uniform ultimate bounded ness can be guaranteed.

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Chapter 3

Control System Theory

3.1

Introduction

This chapter presents an introduction to basic control theory and the Galil motion controller. Control systems are an integral part of modern society. Control systems find widespread application in the steering of missiles, planes, spacecraft and ships at sea. With control systems we can move large equipment with a precision that would otherwise be impossible. We can point large antennas toward the farthest reaches of the universe to pick up faint radio signals. Moving the antennas so precisely by hand would be impossible. The home is not without its own control systems. In a videodisc or compact disc machine, microscopic pits representing the information are cut into a disc by a laser during the recording process. During playback, a reflected laser beam, focused on the pits, changes intensity. The changes in light intensity are then converted into an electrical signal and processes as sound or picture. A control system keeps the laser beam positioned on the pits, which are cut as concentric circles. A home heating system is a simple control system consisting of a thermostat or bimetallic material that expands or contracts with changing temperature. This expansion or contraction moves a vial of

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mercury that acts as a switch, turning the heater on and off. The amount of expansion or contraction required to move the mercury switch is determined by the temperature setting.

3.2

The basic control system

A control system consists of subsystems and processes (or plants) assembled for the purposes of controlling the output of processes. For example, a furnace is a process that produces heat as a result of the flow of fuel. This process, assembled with subsystems called fuel valves and fuel-valve actuators, regulates the temperature of the room by controlling the heat output from the furnace. Other subsystems, such as thermostats, which act as sensors, measure the room temperature. In the simplest form, a control system provides an output response for a given input stimulus as shown in block diagram form in Figure 3.1.

Input; stimulus Desired response

Control system

Output; response Actual response

Figure 3.1 Simplified description of a control system (Nise, Norman [29])

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3.2.1 The control problem

A generic closed-loop system consists of : ? ? ? ? input; reference input gives the desired output (usually called a setpoint) controller plant, a system to be controlled measurement device; this allows the current state of the system to be assessed and generation of an appropriate error signal ? ? output; the controlled output actually generated by the closed loop system reward function; in reinforcement problems, we do not know what the set points are; instead we get evaluative feedback about how effective our control law is.

There may also be a ? Plant model; having a sufficiently accurate model helps us how to build an optimal controller. In general, the plant will be a dynamical system, that is, a system describable by a set of differential (or difference) equations. The distinction between continuous and discrete dynamical systems can be significant; in the reinforcement literature, several important results have been found for the control of discrete systems that do not apply in the continuous case. First, the difference between feedback and feed forward control needs to emphasized.

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3.2.2 Description of the Input and output

The input represents a desired response; the output is the actual response. For example, when the fourth-floor button of an elevator is pushed on the ground floor, the elevator rises to the fourth floor with a speed and floor-leveling accuracy designed for passenger comfort. Figure 3.2 shows the input and output for the elevator system.

Figure 3.2 Input output for the elevator system (Nise, Norman [29]) The push of the fourth floor button is the input and is represented by a step command. The input represents what the desired output should be after the elevator has stopped; the elevator itself follows the displacement described by the curve marked elevator response. Two factors make the output different from the input. The first factor is the instantaneous change of the input against the gradual change of the output. Figure 3.2 shows that the physical entities cannot change their state instantaneously. The state changes through a path that is related to the physical device and the way it acquires or dissipates energy. This is called the transient response of the system.

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After the transient response, the physical system approaches its steady-state response, which is its approximation to the commanded or desired response. The accuracy of the elevator’s final leveling with the floor is a second factor that could make the output different from the input. This difference is called the steady-state error.

3.3

Feedback and feed forward control

A feedback controller is designed to generate an output that causes some corrective effort to be applied to a process so as to drive a measurable process variable towards a desired value known as the set point. The controller uses an actuator to affect the process and a sensor to measure the results. Figure 3.1 shows a typical feedback control system with blocks representing the dynamic elements of the system and arrows representing the flow of information, generally in the form of electrical signals. Feedback control is an errordriven strategy; corrections are made on the basis of a difference between the system's current state, and the desired state. In the simplest case of a linear feedback control, the corrections are proportion to the magnitude of the difference, or error. This may not work very well for nonlinear plants, so it is common to multiply these control inputs by a gain matrix. The desired state acts as an attractor, and the system behaves as a simple spring.

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Figure 3.3 Most feedback controllers for continuous processes use the proportionalderivative-integral (PID) algorithm to manipulate the process variable by applying a corrective effort to the process. (VanDoren Vance J [58]). Springs are notorious for exhibiting oscillations, so it is common to include some damping terms. e.g. r(t) = Kp( x d(t) – x (t)) + Kv (d x(t)/dt); (3.1)

where r(t) is the control input, x (t) the state of the system, x

d(t)

the target state, Kp the

proportional constant or the damping coefficient, Kv the derivative constant or spring constant in this case and dx(t)/dt the rate of change of the state of the system.

The damping coefficient, Kp, can be set to give critical damping (a state in which there is just the right amount of damping to bring the system to the desired output level). Note that this is a completely model-free approach; the complex dynamical interactions of the system are considered as errors to be eliminated by feedback. As it stands, Equation (3.1) is not quite right; constant disturbances like gravity will leave a steady state offset in

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position. We can never get rid of this (unless we have an infinite value of Kp, so we need an extra integrative term:

r(t) = Kp( x d(t) – x (t)) + Kv (d x(t)/dt) + KI ! ( x d(t) – x (t)) dt

(3.2)

where KI is the integral constant. So now there are three terms in the controller, it is seen that feedback loops like this are described as a three-term controller, or a proportional, integrative, derivative (PID) controller.

Virtually all feedback controllers determine their output by observing the error between the set point and a measurement of the process variable. Errors occur when an operator changes the set point intentionally or when a process load changes the process variable accidentally. Feedback control is, by definition, an error-driven process, so there need to be errors for it to be doing anything. This means that it is very likely that there will be a path that lags continuously behind the desired path. Also, feedback systems usually take a finite amount of time to respond to errors, so perturbing oscillations above a certain frequency will not by corrected. Finally, it should be noted that feedback control might not be limited to the position of the system; and that feedback control can be defined at any level. Thus, it is perfectly possible to process feedback error signals at a task level, as long as we can find a reliable way of decreasing the error.

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Feed forward (model-based, indirect) control takes an alternative approach; a model of the dynamics of the system is built, and the inverse dynamics are solved for input torque. This method has the potential for very accurate control, but depends critically, on the integrity of the model. If we denote the plant dynamics by x(t) = R(r(t), x(0)) then

r(t) = R-1( xd(t))

(3.3)

This is the inverse dynamics of the plant; given the desired state of the system, this tells us the control inputs required to achieve that state (assuming such control inputs actually exists). If, however, we only have a model of the plant, say R^, then the path of the system will be described by then,

θ(t) = R(R^-1( xd(t)))

(3.4)

There could be several reasons for feed forward control to be unsatisfactory:

?

It may be impossible to get a sufficiently accurate model of the system, and the deviation of the produced path from the target may be a rapidly increasingly

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function of time (due to non-linearities or non- stationarities in the dynamics of the system). ? Even assuming that our model it perfectly accurate, we may have an error in the measurement of the initial state of the system, i.e. at t = t0. ? Any external objects that cause the system to deviate from its path will not be corrected; simply brushing against a surface may introduce errors that cannot be eliminated. ? It would seem unlikely that the control system is capable of computing inverse dynamics fast enough, even given a perfect model. This may not be a problem for dedicated hardware controlling robot arms, but if were concerns about biological plausibility, then a system using purely feed forward control appears unrealistic.

It can be useful to think of control problems in a dynamical systems context. If we consider the plant and the controller as one system, then the control problem can be construed as placing demands on the dynamics of this combined system. In stable control, we require that the set point of the plant be a fixed point of the system. We would also usually require this attractor to be stable: small perturbations from the goal state should cause the system to converge back to the attractor. We may also be concerned with the flow field surrounding the fixed point, particularly if we want the system to collapse onto the goal state as quickly as possible, but without overshooting. The

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reference model mentioned above is a description of the desired behavior of the combined system. For example, in designing the control system of an aircraft, we don't have any desired states, but we will have some idea about how we wish the combined system to handle.

Most control architectures can be categorized as direct or indirect. Indirect control makes use of a model of the plant in designing the controller; direct control tries to optimize the outputs of the plant directly. In the case of indirect control, a further distinction can be made between on-line and off-line optimization. Off-line optimization allows us to learn a plant model by observing the plant’s behavior under different conditions (system identification), and subsequently uses that model to design a controller. In on-line optimization, the plant has to be controlled all the time, while the plant model is being learned. This is obviously a much more difficult problem, as the combined system is now time-variant.

3.4

Linear control systems

Most physical systems have non-linear elements, but in some circumstances it may be possible to treat them as linear. Then the edifice of linear mathematics, which is very highly developed, can be employed to yield solutions. If a system only operates over a small range of input values, then the non-linearities can often be effectively approximated by linear functions. This may be referred to as operation about some reference point or

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nominal trajectory; if the non-linear equations are known, then the linearized form of these equations are often called small perturbation equations. If the non-linearities are severe enough to render the linearization approach invalid, then there is no general theory of non-linear control available at present only specific methods. Methods for control in such circumstances, based on artificial neural and fuzzy logic can be found in various research works. The methods for control of linear, time-invariant systems are very well known. The only difficulty is that it requires a moderate amount of linear algebra, which can at first be intimidating for the uninitiated and often a large amount of computation that can be intimidating for the fastest computers.

3.5

Control of non-linear systems

Non-linear systems are much more difficult to control than linear systems, mainly because the system equations are not necessarily solvable. Remembering that solving a set of differential equations means writing down a closed-form equation describing the behavior of the system under a whole variety of boundary conditions. Linear, time invariant (LTI) control is concerned with systems whose behavior we can completely specify; this means that the addition of carefully designed feedback produces a system that we know will behave in the desired way. When we can't solve a set of differential equations analytically, we don't know how the system will behave under a set of boundary conditions, so they need to treated on a case by case basis.

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Obviously, controlling such a system is going to be difficult, as we can make no claims about the response of the system to a control input. Except for a few special cases, we will not be able to guarantee that the combined system will even be stable, let alone optimal. These two issues, stability and convergence, are much more difficult in the nonlinear case. Linear systems theory is extremely well developed, and it is often the case that convergence and stability for an adaptive controller can be proven. If we're trying to control, for example, a power plant, it may be that the consequences of the system becoming unstable are disastrous. This is why control engineers are so concerned about being able to prove stability, and it's also why they try to linearise systems that are actually nonlinear. Of course, the other side to this coin is the control problem in biology where the systems are inherently nonlinear. If we're interested in understanding how biological control systems work, it seems natural to borrow concepts from engineering. However, this can lead to us viewing these problems from a rather strange perspective. Biological systems routinely control highly nonlinear plants, for example, flapping wing systems, which just could not be controlled with current engineering technology. How can this be?

The point about biological control systems is that they tend to be rather small, so it seems implausible that they could possibly model the dynamics of a set of four or six flexible flapping wings. An optimization method such as evolution has no reason to be concerned with controllers that are provably stable; it's much more likely to go for the 'quick hack'

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approach, and let all the unsuccessful designs die. This means that it may be very difficult to get the data on biological control into some kind of unified framework. Control theory certainly gives us a way of understanding the nature of biological control problems, but understanding how the controllers actually work might mean a reinterpretation of control theory concepts.

3.6

The design process

The design of a control system involves a step-by-step but iterative process. The following section example of an antenna azimuth position control system is presented. The azimuth angle output of the antenna, θo (t), follows the input angle of the potentiometer, θi (t). Figure 3.3 shows the diagram of the system and Figure 3.4 shows the functional block diagram of the system. The functions are shown above the blocks, and the required hardware is indicated inside the blocks.

Figure 3.4 An antenna azimuth position control system (Nise, Norman [29] )

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Figure 3.5 Functional Block Diagram of a position control system (Nise, Norman [29])

Step 1 : Transform the requirements into a physical system

Antenna example: To design a control system for the position control of the azimuth of the antenna.

Requirements: To position the antenna from a remote location and describe features such as weight, and the physical dimensions.

Using these requirements the desired transient response and steady state accuracy are determined.

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Step 2: Draw a functional block diagram

Involves the translation of the qualitative description of the system into a functional block diagram that describes the component parts of the system and shows their interconnections.

In the antenna example: The bBlock diagram indicates functions as input transducer, the controller, and relevant descriptions of the amplifiers and the motors.

Step 3: Create the schematic

Process of converting the physical system into a schematic diagram.Relevant approximations of the system should be made and neglect certain phenomena or else making it difficult to extract information for the mathematical model.

After a single loop of design involving the analysis and interpretation of the system, decisions have to be made as to whether or not reasonable approximations were made.

If the designer feels that the system was not described fully, additional parameters are built into the design schematic.

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Example: Potentiometers made like neglecting the friction or inertia, although these mechanical characteristics yield a dynamic response rather than an instantaneous response.

Implications of the assumptions are that the mechanical effects are neglected and the voltage across the potentiometer changes instantaneously as the potentiometer shaft turns.

Differential and power amplification: Assuming that the dynamics of the amplifier are rapid compared to the response time of the motors, hence we model it as pure gain K.

Dc motor: The output speed of the motor is proportional to the voltage applied to the motor’s armature. Armature consists of both the inductive and resistive effects and we assume that the inductance is negligible for the DC motor.

Load: The load consists of rotating mass and bearing friction. Hence the model consists of inertia and viscous damping whose resistive torque increases with speed like the automobile’s shock absorber or the screen door damper.

Step 4 : Develop a mathematical model (block diagram)

From the schematic, the physical laws such as the Kirchhoff’s laws and Newton’s laws are used with modifying assumptions.

Kirchoff’s voltage law: The sum of voltages around a closed path equals zero.

Kirchoff’s current law: The sum of electric currents flowing from a node equals zero.

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Newton’s laws: The sum of forces on a body equals zero, the sum of moments on a body equals zero.

These laws lead to mathematical models that describe the relationship between input and output of a dynamic system. Model 1 : One such model is also a linear time invariant differential equation. Model 2: Transfer function is another way of modeling a system. This is obtained from linear time invariant differential equation using what is called the Lap lace transform. Lap lace transform can be used for linear systems, but yields more intuitive information than the differential equations. Ability to change system parameters and rapidly sense the effect of these changes on the system’s response. Model 3: State space methods: Advantage of modeling a system in state space is that they can be used for systems that cannot be described by differential equations. These methods are also used to model systems for simulation the digital computer. This representation turns an n-th order differential equation into n simultaneous first-order differential equations.

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Step 5: Reduce the block diagram. Subsystem models are interconnected to form a block diagram of a larger system where each block has a mathematical description. The step involves the reduction of a large number of subsystems into a large system single block, with a mathematical description that represents the system from its input to its output. Step 6: Analyze and design Analyze the system to see if the response specifications and performance requirements can be met by simple adjustments of the system parameters. If the specifications are not yet met the designer then designs additional hardware in order to affect a desired performance.

Test inputs are signals that are used analytically and during testing to verify the design. Some of the standard input signals are impulses, steps, ramps, parabolas, and sinusoids. These steps were followed in the following analysis and synthesis described in the next chapter.

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Chapter 4

The Galil DMC 1000 Controller

4.1

Digital control systems

The rapid development of digital technology has radically changed the boundaries of practical control system design options. It is now routinely feasible to employ very complicated, high-order digital controllers and to carry out the extensive calculations required for their design. These advantages in implementation and design capability can be achieved at low cost because of the widespread availability of inexpensive, powerful digital computers and related devices.

A digital control system uses digital hardware, usually in the form of a programmed digital computer, as the heart of the controller. In contrast, the controller, in an analog control system is composed of analog hardware, typically analog electronic, mechanical, electro-mechanical, and hydraulic devices. Digital controllers normally have analog elements at their periphery to interface with the plant; it is the internal workings of the controller that distinguish digital from analog control.

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Digital control systems offer many advantages over their analog counterparts. Among these advantages are the following: 1. Low susceptibility to environment conditions, such as temperature, humidity, and component aging. 2. The cost reduction and interference rejection associated with digital signal transmission. 3. Zero “drift” of parameters. 4. High potential reliability. 5. The ability to perform highly complex tasks at low cost. 6. The potential flexibility of easily making changes in software. 7. Relatively simple interfaces with other digital systems, such as those for accounting, forecasting, and data collection.

Among the disadvantages are: 1. The introduction of errors (or noise) due to the finite precision of digital computations and the abrupt changes due to the discrete time nature of digital control. 2. The need for more sophisticated engineering in order to take advantage of higher-performance control algorithms. 3. Greater limitations on speed of operation. 4. Greater potential for catastrophic failure.

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The signals used in the description of control systems are classified as continuous-time or discrete-time. Continuous-time signals are functions of continuous variable, whereas discrete-time signals are defined only for discrete values of the variable, usually with evenly spaced time steps. Discrete-time signals and their manipulators are inherently well suited to digital computation and are used in describing the digital portions of a control system. Most often, continuous-time signals are involved in describing the plant and the interfaces between a controller and the plant it controls. Signals are further classified as being of continuous amplitude or discrete amplitude. Discrete-amplitude (or quantized ) signals can attain only discrete values, usually evenly spaced. The digital values are also quantized in amplitude signal shown in Figure 4.1, represented by a 3-bit binary code at evenly spaced time instants.

Figure 4.1 An example of a 3-bit quantized signal (Santina, Stubberud, Hostetter [36]) In general, an n-bit binary code can represent only 2n different values. Because of the complexity of dealing with quantized signals, digital control system design proceeds as if

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computer-generated signals were not of discrete amplitude. If necessary, further analysis is then done to determine if a proposed level of quantization error is acceptable. A general control system diagram with a controller is show in Figure 4.2

Figure 4.2 A control system ( Santina, Stubberud, Hostetter [36] ) The plant is affected by input signals, some of which (the control inputs) are accessible to the controller, and some of which (the disturbance inputs) are not. Some of the plant signals (the tracking outputs) are to be controlled, and some of the plant signals (the measurement outputs) are available to the controller. A controller generates control inputs to the plant with the objective of having the tracking outputs closely approximate the reference inputs.

As described earlier, the systems and system components are classified by the nature of their mathematical model and termed continuous time or discrete time, according to the type of signals they involve. They are classified as linear if signal components in them can be superimposed. Any linear combination of signal components applied to input produces the same linear combination of corresponding output components; otherwise the system is non-linear. A continuous-time system or component is time invariant if its properties do not change with time. Any time shift of the input produces an equal time

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shift of every corresponding signal. If a continuous-time system is not time-invariant, then it is time varying. On the other hand, if the properties of a discrete-time system do not change with step, then it is called step-invariant [26]. And, if the discrete-time system is not step-invariant, then it is step varying. Figure 4.3 shows the example of a digital control system for a continuous-time plant.

Figure 4.3 A digital control system controlling a continuous-time plant ( Santina, Stubberud, Hostetter [36] ) The system has two reference inputs and five outputs, only two of which are measured by the sensors. The analog-to-digital converters (A/D) perform sampling of the sensor signals and produce binary representations of these sensor signals. The digital controller algorithm then modifies the sensor signals and generates digital control inputs u1(k) and u2(k). These control inputs are then converted to analog signals via digital-to-analog converters (D/A). This process of transforming digital codes to analog signals begins by converting the digital codes to signal samples and then producing step reconstruction

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from the signal samples by transforming the binary coded digital inputs to voltages. These voltages are held constant for a sampling period T until the next samples are available. The process of holding each sample to perform step reconstruction is termed sample and hold.

The system also usually consists of a real-time clock that synchronizes the actions of the A/D and D/A and the shift registers. The analog signals inputs u1(k) and u2(k) are applied to the plant actuators or control elements to control the plant’s behavior.

There are many variations on this theme, including situations in which the sampling period is not fixed, in which the A/D and D/A are not synchronized, in which the system has many controllers with different sampling periods, and in which sensor produce digital signals and the actuator accepts digital commands. Two important classes of control systems are the regulator and the tracking system (or servo system). In the former, the objective is to bring the system-tracking outputs near zero in an acceptable manner, often in the face of disturbances. For example, a regulator might be used to keep a motor-driven satellite dish antenna on a moving vehicle accurately pointed in a fixed direction, even when the antenna base is moving and vibrating and the antenna itself is buffeted by winds. In a tracking system, the objective is for system outputs to track, as nearly as possible, an equal number of reference input

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signals. Regulation is a special case of tracking, in which the desired system-tracking output is zero.

4.2

PID controller

PID controllers are designed to eliminate the need for continuous operator attention. Cruise control in a car and a house thermostat are common examples of how controllers are used to automatically adjust some variable to hold the measurement (or process variable) at the set point. A PID controller performs much the same function as a thermostat but with a more elaborate algorithm for determining its output. It looks at the current value of the error, the integral of the error over a recent time interval, and the current derivative of the error signal to determine not only how much of a correction to apply, but for how long. Those three quantities are each multiplied by a tuning constant and added together to produce the current controller output CO(t) as shown in Figure 4.4. In this equation, P is the proportional tuning constant, I is the integral tuning constant, D is the derivative tuning constant, and the error e(t) is the difference between the set point P(t) and the process variable PV(t) at time t. If the current error is large or the error has been sustained for some time or the error is changing rapidly, the controller will attempt to make a large correction by generating a large output. Conversely, if the process variable has matched the set point for some time, the controller will leave well enough alone. The set point is where one would like the measurement to be.

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Figure 4.4 Equations [1] and [6]--both forms of the PID algorithm generate an output CO(t) according to recent values of the sepoint SP(t), the process variable PV(t), and the error between them e(t)=SP(t) - PV(t). (VanDoren Vance J [58]) Error is defined as the difference between the set point and measurement values. E=S–M (4.1)

Where, E is the error, S is the set point and M is the measurement value. The variable being adjusted is called the manipulated variable that usually is equal to the output of the controller. The output of PID controllers will change in response to a change in measurement or set point. Manufacturers of PID controllers use different names to identify the three modes. These equations show the relationships:

P I

Proportional band = 100/gain Integral = 1/reset (time units)

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D

Derivative = rate = pre-act (1/units of time)

Where gain is the value of the amplification factor in the control loop, reset is the time taken for the controller to bring the output to the desired output state and the rate or preact is the actual rate of change of the system stability to achieve the desired state.

Depending on the manufacturer, integral or reset action is set in either time/repeat or repeat/time. One is simply, the reciprocal of the other. Note that manufacturers are not consistent and often use reset in units of time/repeat or integral in units of repeats/time. Derivative and rate are the same.

4.2.1 Proportional

With proportional band, the controller output is proportional to the error or a change in measurement (depending on the controller).

O = E*100/P

(4.2)

Where O is the output of the controller. With a proportional controller, offset (deviation from set-point) is present. Increasing the controller gain will make the loop become unstable. Integral action was included in controllers to eliminate this offset.

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4.2.2 Integral

With integral action, the controller output is proportional to the amount of time the error is present. Integral action eliminates offset. O= 1 ( e(t ) dt ) I ∫

(4.3)

Notice that the offset (deviation from set-point) in the time response plots is now gone. Integral action has eliminated the offset. The response is somewhat oscillatory and can be stabilized some by adding derivative action.

Figure 4.5: A Time Response plot showing controller action (Graphic courtesy of ExperTune Inc. Loop Simulator (Expertune Inc [59])

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Integral action gives the controller a large gain at low frequencies that results in eliminating offset and "beating down" load disturbances.

4.2.3 Derivative

With derivative action, the controller output is proportional to the rate of change of the measurement or error. The controller output is calculated by the rate of change of the measurement with time.

O = D (dm/dt)

(4.4)

Where m is the measurement at time t. Derivative action can compensate for a changing measurement. Thus, derivative takes action to inhibit more rapid changes of the measurement than proportional action. When a load or set point change occurs, the derivative action causes the controller gain to move the "wrong" way when the measurement gets near the set point. Derivative is often used to avoid overshoot.

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Figure 4.6: Controller frequency response plot (Graphic courtesy of ExperTune Inc. Loop Simulator (Expertune Inc [59]) Derivative action can stabilize loops since it adds phase lead. Generally, if one uses derivative action, more controller gain and reset can be used. With a PID controller the amplitude ratio now has a dip near the center of the frequency response. Integral action gives the controller high gain at low frequencies, and derivative action causes the gain to start rising after the "dip”. At higher frequencies the filter on derivative action limits the derivative action. At very high frequencies (above 314 radians/time; the Nyquist frequency) the controller phase and amplitude ratio increase and decrease quite a bit because of discrete sampling. If the controller had no filter the controller amplitude ratio would steadily increase at high frequencies up to the Nyquist frequency (1/2 the sampling

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frequency). The controller phase now has a hump due to the derivative lead action and filtering. The time response is less oscillatory than with the PI controller. Derivative action has helped stabilize the loop.

4.3

Control loop tuning

It is important to keep in mind that understanding the process is fundamental to getting a well-designed control loop. Conceptually, that's all there is to a PID controller. The tricky part is tuning it; i.e., setting the P, I, and D tuning constants appropriately. The idea is to weight the sum of the proportional, integral, and derivative terms so as to produce a controller output that steadily drives the process variable in the direction required to eliminate the error. The brute force solution to this problem would be to generate the largest possible output by using the largest possible tuning constants. A controller thus tuned would amplify every error and initiate extremely aggressive efforts to eliminate even the slightest discrepancy between the set point and the process variable. However, an overly aggressive controller can actually make matters worse by driving the process variable past the set point as it attempts to correct a recent error. In the worst case, the process variable will end up even further away from the set point than before and the output will go to infinity.

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On the other hand, a PID controller that is tuned to be too conservative may be unable to eliminate one error before the next one appears. The system could appear so sluggish that it doesn’t move at all. A well-tuned controller performs at a level somewhere between those two extremes. It works aggressively to eliminate an error quickly, but without over doing it. How to best tune a PID controller depends upon how the process responds to the controller's corrective efforts. Processes that react instantly and predictably don't really require feedback at all. A car's headlights, for example, apparently come on as soon as the driver hits the switch. No subsequent corrections are required to achieve the desired illumination. On the other hand, the car's cruise controller cannot accelerate the car to the desired cruising speed so quickly. Because of friction and the car's inertia, there is always a delay between the time that the cruise controller activates the accelerator and the time that the car's speed reaches the set point. A PID controller must be tuned to account for such lags.

4.3.1 PID in action

Consider a sluggish process with a relatively long lag--an overloaded car with an undersized engine, for example. Such a process tends to respond slowly to the controller's efforts. If the process variable should suddenly begin to differ from the set point, the

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controller's immediate reaction will be determined primarily by the actions of the derivative term. This will cause the controller to initiate a burst of corrective efforts the instant the error changes from zero. A cruise controller with derivative action would kick in when the car encounters an uphill climb and suddenly begins to slow down. The change in speed would also initiate the proportional action that keeps the controller's output going until the error is eliminated. After a while, the integral term will also begin to contribute to the controller's output as the error accumulates over time. In fact, the integral action will eventually come to dominate the output signal since the error decreases so slowly in a sluggish process. Even after the error has been eliminated, the controller will continue to generate an output based on the history of errors that have been accumulating in the controller's integrator. The process variable may then overshoot the set point, causing an error in the opposite direction.

Figure 4.7 A cruise controller attempts to minimize errors between the desired speed set by the driver and the car's actual speed measured by the speedometer. The

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controller detects a speed error when the desired speed is increased or when an added load (such as an uphill climb) slows the car (VanDoren Vance J [58]). If the integral tuning constant is not too large, this subsequent error will be smaller than the original, and the integral action will begin to diminish as negative errors are added to the history of positive ones. This whole operation may then repeat several times until both the error and the accumulated error are eliminated. Meanwhile, the derivative term will continue to add its share to the controller output based on the derivative of the oscillating error signal. The proportional action, too, will come and go as the error waxes and wanes. Now suppose the process has very little lag so that it responds quickly to the controller's efforts. The integral will not play as dominant a role in the controller's output since the errors will be so short lived. On the other hand, the derivative action will tend to be larger since the error changes rapidly in the absence of long lags. Clearly, the relative importance of each term in the controller's output depends on the behavior of the controlled process. Determining the best mix suitable for a particular application is the essence of controller tuning. For the sluggish process, a large value for the derivative tuning constant D might be advisable to accelerate the controller's reaction to an error that appears suddenly. For the fast-acting process, however, an equally large value for D might cause the controller's

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output to fluctuate wildly as every change in the error (including extraneous changes caused by measurement noise) is amplified by the controller's derivative action.

4.3.2 Tuning methods

There are basically three schools of thought on how to select P, I, and D values to achieve an acceptable level of controller performance. The first method is simple trial-and-error-tweak the tuning constants and watch the controller handle the next error. If it can eliminate the error in a timely fashion, quit. If it proves to be too conservative or too aggressive, increase or decrease one or more of the tuning constants. Experienced control engineers seem to know just how much proportional, integral, and derivative action to add or subtract to correct the performance of a poorly tuned controller. Unfortunately, intuitive tuning procedures can be difficult to develop since a change in one tuning constant tends to affect the performance of all three terms in the controller's output. For example, turning down the integral action reduces overshoot. This in turn slows the rate of change of the error and thus reduces the derivative action as well. The analytical approach to the tuning problem is more rigorous. It involves a mathematical model of the process that relates the current value of the process variable to its current rate of change plus a history of the controller's output. The third approach to the tuning problem is something of a compromise between purely heuristic trial-and-error techniques and the more rigorous analytical techniques. It was originally proposed in

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1942 by John G. Ziegler and Nathaniel B. Nichols of Taylor Instruments and remains popular today because of its simplicity and its applicability to any process governed by a model in the form of equation [2]. The Ziegler-Nichols tuning technique will be the subject of "Back to Basics" (CE, Aug. 1998). For the Galil DMC-1000 controller that was used the user manual [49], describes a set of tuning procedure to be followed.

4.4

Galil DMC 1000

The DMC-1000 series motion controller is a state-of the-art motion controller that plugs into the PC bus. Extended performance capability over the previous generation of controllers include: 8 MHz encoder input frequency, 16-bit motor command output DAC, +/- 2 billion counts total travel per move, faster sample rate, bus interrupts and nonvolatile memory for parameter storage. The controllers provide high performance and flexibility while maintaining ease-of-use and low cost.

Designed for maximum system flexibility, the DMC-1000 is available for one, two, three or four axes per card (add on cards are available for control of five, six, seven, or eight axes). The DMC-1000 can be interfaced to a variety of motors and drives, including stepper motors, servomotors and hydraulic actuators.

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Each axis accepts feedback from a quadrature linear or rotary encoder with input frequencies up to 8 million quadrature counts per second. For dual-loop applications in which an encoder is required on both the motor and the load, auxiliary encoder inputs are included for each axis.

The DMC-1000 provides many modes of motion, including jogging, point-to-point positioning, linear and circular interpolation, electronic gearing and user-defined path following. Several motion parameters can be specified, including acceleration and deceleration rates, velocity and slew speed. The DMC-1000 also provides motion smoothing to eliminate jerk.

For synchronizing motion with external events, the DMC-1000 includes 8 optoisolated inputs, 8 programmable outputs and 7 analog inputs. I/O expansion boards provide additional inputs and outputs or interface to OPTO 22 racks. Event triggers can automatically check for the elapsed time, distance and motion to be complete. Despite its full range of sophisticated features, the DMC-1000 is easy to program. Instruction are represented by two letter commands such as BG for begin and SP for speed. Conditional instructions, jump statements, and arithmetic functions are included for writing self-contained applications programs. An internal editor allows programs to be quickly entered and edited, and support software such as the SDK (servo design kit) allows quick system set-up and tuning.

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To prevent system damage during machine operation, the DMC-1000 provides several error handling features. These include software and hardware limits, automatic shut-off on excessive error, abort input, and user-definable error and limit routines.

4.5 Microcomputer section

The main processing unit of the DMC-1000 is a specialist 32-bit Motorola 68331 series microcomputer with 64K RAM (256K available as an option), 64K EPROM and 256 bytes EEPROM. The RAM provides memory for variables, array elements and application programs. The EPROM stores the firmware of the DMC-1000. The EEPROM allows certain parameters to be saved in non-volatile memory upon power down.

4.6 Motor Interface

For each axis, a GL-1800 custom, sub-micron gate array performs quadrature decoding of the encoders at up to 8 MHz, generates a +/- 10 Volt analog signal (16 Bit D-to-A) for input to a servo amplifier, and generates step and direction signal for step motor drives.

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4.7 Communication

The communication interface with the host PC uses a bi-directional FIFO (AM470) and includes PC-interrupt handling circuitry.

4.8

General I/O

The DMC-1000 provides interface circuitry for eight optoisolated inputs, eight general outputs and seven analog inputs (12-bit ADC). Controllers with 5 or more axes provide 24 inputs and 16 outputs. Controllers with 1 to 4 axes can add additional I/O with an auxiliary board, the DB-10096 or DB-10072. The DB-10096 provides 96 additional I/O. The DB-10072 provides interface to up to three OPTO 22 racks with 24 I/O modules each.

4.9

Amplifier

For each axis, the power amplifier converts a +/- 10 -Volt signal from the controller into current to drive the motor. (For stepper motors, the amplifier converts step and direction signals into current). The amplifier should be sized properly to meet the power requirements of the motor. For brush less motors, an amplifier that provides electronic computation is required. The amplifiers may either be pulse-width-modulated (PWM) or linear. They may also be configured for operation with or without a tachometer. For current amplifiers, the amplifier gain should be set such that a 10 Volt command generates the maximum required current. For example, if the motor peak current is 10 A,

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the amplifier gain should be 1 A/V. For velocity mode amplifiers, 10 Volts should run the motor at the maximum speed.

4.10 Encoder

An encoder translates motion into electrical pulses, which are fed back into the controller. The DMC-1000 accepts feedback from either a rotary or linear encoder. Typical encoders provide two channels in quadrature, known as CHA and CHB. This type of encoder is known as quadrature encoder. Quadrature encoders may either be single-ended (CHA and CHB) or differential (CHA, CHA-, CHB, CHB-). The DMC-1000 decodes either type into quadrature states or four times the number of cycles. Encoders may also have a third channel (or index) for synchronization. For stepper motors, the DMC-1000 can also interface to encoders with pulse and direction signals . There is no limit on encoder line density; however, the input frequency to the controller must not exceed 2,000,000 full encoder cycles/second (8,000,000 quadrature counts/sec). For example, if the encoder line density is 10000 cycles per inch, the maximum speed should be 200 inches/second.

The standard voltage level is TTL (zero to five volts); however, voltage levels up to 12 volts are acceptable. (If using differential signals, 12 volts can be input directly to DMC-

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1000. Single-ended 12-volt signals require a bias voltage input to the complementary inputs).

To interface with other types of position sensors, such as resolvers or absolute encoders, Galil offers the DB-10096 auxiliary card, which can be customized for a particular sensor.

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Chapter 5 Control parameter determination for the Bearcat

Presently the control parameters i.e., PID values for the Galil controller are determined using a software tool supplied by Galil Motion Inc called Windows Servo Design Kit (WSDK 1000). This software takes into account the frictional losses in the gear mesh, motors and the drive train. The WSDK kit is a menu driven graphical user interface that allows tuning of the system. The following two figures show the start-up and tuning methods menu of the Galil Motion Control – Servo Design Kit Version 4.04

Figure 5.1. Start-up menu of Galil motion control - servo design kit, Ver. 4.04

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Figure 5.2 Tuning methods menu of the Galil motion control - servo design kit, Ver. 4.04 The system elements can be identified and various menu options are provided to make sure that the elements are connected. A closed loop test is done to see if the system is stable. If stable values are obtained then PID parameters can be tuned. The actual tests were made in three conditions: wheels off the ground, wheels on the ground with the robot stationary, and with the robot moving. Care was taken during this process because it causes violent shaking of the robot. To get a better idea of the parameters this process can be iterated on various surfaces. The different surfaces that the robot would encounter are grass, asphalt, wood, tarp, sand, and highly polished surfaces like granite as well as concrete. Based on the various values obtained a representative value can be chosen or, a

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value based on the roughest surface can be chosen. Figure 5.3 shows the step response plot evaluation after the values have been analyzed.

Figure 5.3: Step Response Plot from the Galil servo design kit.

First, a sinusoidal input is supplied to the motors. This causes the wheels to rotate back and forth. During this period of time the system response magnitude and phase frequency response is measured. The software allows individual tuning of up to four axes. Using these values a Bode plot frequency, which is the open loop frequency transfer function response, is plotted to determine to see if the overshoot is within satisfactory limits.

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The tuning process is an iterative process. Each of the parameter slides can be changed to adjust the PID values. First, a starting value is chosen. Using the starting value the WSDK program plots the Bode Frequency plot( step response plot). The overshoot is observed by visually observing the plot. Based on the plot the sliders adjusted to put in a different set of PID values. Again the program is run. This process may take a long time to arrive at a satisfactory PID values. Once satisfactory values are obtained then they can be used in the control program. If there is a slight change in the drive system hardware the PID values will have to be recalculated if proper control of the system is required. The interface for the system is implemented using a Galil 1030 motion control computer interface board. A Galil breakout board permits the amplifier and encoder to be easily connected. As mentioned this process may take many iterations and a long time to finish. To deal with this drawback a new optimized tuning method is proposed in the next chapter. But the PID values obtained with new process can always be tested and fine-tuned using the WSDK kit.

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Chapter 6 Simulation and optimizing the parameters

6.1 Introduction

This chapter presents the main theme of this thesis. It deals with the modeling, simulation and optimizing of the control system. The basic model of the control system was setupusing MATLAB. For this purpose the SIMULINK toolbox was used. This toolbox allows the modeling of the various systems on the control system. Optimization was done using the OPTIMIZATION toolbox. This invokes the basic SIMULINK model and optimizes the parameters based on the optimizing routine used.

6.2

Simulation on SIMULINK

The objective of the model was to attain a stable control system. One of the main requirements was that the phase margin should be less than 45 degrees and a gain margin more than 10 decibels with the percentage overshoot not exceeding 20%. The Galil DMC 1000 controller has a proportional integral and derivative controller to provide the necessary compensation. The simulation involves three steps. 1. A Matlab file that has the model of the transfer function. 2. A second Matlab source file converts the digital gains to analog gains.

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3. A Simulink graphics model which takes the analog values of the gains and simulates the system step response The actual setup that models the control system on the robot is shown below

Step Input

+ Sum

gs 2+(g*a+f)s+f*b s 2+bs The PID controller

1 .0005s+1 Zero Order Hold

0.00122 DAC

2 AMP

.54 .0003773s 2 Motor and Load

-KGain2

Auto-Scale Graph

Auto-Scale Graph1

Figure 6.1: Simulink model of the system (Kalyan [54])

The model consists of a step input signal fed to a summation block. The constant values to be used in the PID block are calculated using a separate M-file. These values calculate the analog gains for the various digital gains. The calculated analog values are stored in the Matlab kernel and, are read automatically when the model file is run. Analog values in the PID controller adjust the input signal and feed it to the Zero order hold. The zero order hold holds the input level until the next input is given in order to smoothen the input wave. Sampling time for this is modeled in the block. The signal is then fed into the

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DAC (digital to analog converter) and then to the amplifier. The amplifier gain is set at 2 but could be changed on the actual device to suit the amplification needed. The amplified signal is fed to the load which in our case is the overall system including the motor and drive train. The motors have encoders that give the position feed back signal, which is fed to a summation block for correction. Figure 6.2 shows the step response of the system.

0.12 0.1 0.08 0.06 0.04 0.02 0 0 0.02 0.04 0.06 0.08 0.1 0.12

Figure 6.2 Step response for the system.

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The unit step response modeled in Matlab is shown above. From the simulation it was found that the phase margin was within tolerable limits and the overshoot was less than 15%. These are observed in the magnitude and phase of the Bode plots.

6.3

Simulink Model of the general system to be optimized.

The actual model of the control system is shown above. But, it is not general to any motor and load system. The model being dealt with in the subsequent sections is a generalized model that is representative of any motor and load system. Given below is the model.

Figure 6.3 General motor and load control system.

The step response after giving an input is shown in the figure below. The ideal case would be to have the response rise immediately after the signal input is given. That is

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what one would like to achieve ideally. But depends of the components and parameters of the system.

Time (seconds)

Figure 6.4 Step response of the system

6.4

Optimizing the control system to obtain PID values

The control parameters in the model above are optimized using a solver. This is a nonlinear model to be optimized. This would enable the system to track a unit step input to the system with minimum error. MATLAB’s optimization toolbox is used to solve the non-linear problem. The way this problem is tackled is in two steps. 1. The first method minimizes the error between the output and the input signal using ‘lsqnonlin’ function 2. The second method uses the ‘fminimax’ function. In this case, rather than minimizing the error between the output and input the function minimizes the maximum value of the output at any time. The variables are the parameters of the PID controller. If the error

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needs to be minimized only at one time, it would be a single objective function. But, we need to minimize the error for all time steps so it becomes a multiobjective function. 6.5 The lsqnonlin method

The routine lsqnonlin is used to perform a least square fit on the tracking of the output. The following figure shows the error between the expected and the attained values at several times.

Figure 6.5 Errors between desired and actual response at several times.

In the above figure E1,..E6 are the errors or difference between the input and output. Ei = Input – Output The objective would be to minimize the mean squared error , i.e.

70

Minimize E = 1 N

∑E

i =1

N

2 i

The routine lsqnonlin is used to perform a least squares fit on the tracking of the output. This is defined via a function in a separate M-file called ‘lsq.m’. It defines the error signal. The function ‘lsq’ runs the simulation. Shown below is the file Step 1: File lsq.m

function F = lsq(pid, a1, a2) Kp = pid(1); % Move variables into model parameters Ki = pid(2); Kd = pid(3); % choose solver and set model workspace to this function opt = simset('solver', 'ode5', 'SrcWorkspace', 'Current'); [tout, xout, yout] = sim('optsim', [0 100], opt); F = yout-1; % compute error signal

The simulink model file is specified in another M-file called the ‘optroutine’. The model is the one shown above. Step 2: File optroutine.m

kolmodel % loads the model pid0 = [1 10 12] % setting initial values a1 = 0; a2 = 0.0005; options = optimset('LargeScale', 'off', 'Display', 'iter', 'TolX', 0.001, 'TolFun', 0.001); pid = lsqnonlin('lsq', pid0, [], [], options, a1, a2) % put variables back into the base workspace Kp = pid(1); Ki = pid(2); Kd = pid(3);

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First the variables Kp, Kd, Ki are all defined and initialized before calling ‘lsq’. When the ‘optroutine’ file is run the model is loaded and function ‘lsq’ does the optimization based on the initial values of specified for the parameters and, constraints. A solver called simset is chosen and the simulation is run using sim.

Finally the optimization gives the solution for the Proportinal, Integral, and Derivative (Kp, Ki, Kd) gains of the controller after 55 function evaluations. The result from Matlab is shown below.

pid0 = 1 5 10 Directional Iteration 1 2 3 4 5 6 7 Func-count 3 10 17 24 31 39 55 Residual 4.76719e+012 4.70723e+012 4.65431e+012 4.58281e+012 4.56367e+012 4.53102e+012 4.53102e+012 Step-size 1 0.375 0.253 0.293 0.0868 0.119 derivative -4.91e+012 -1.51e+011 -2.33e+011 -8.38e+010 -8.32e+010 -1.63e+011 1.32032e+012 8.77204e+011 1.68981e+012 2.02202e+012 2.60576e+012 1.07078e+018 Lambda

3.28e-007 -3.27e+011

Optimization terminated successfully:

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Search direction less than tolX

pid = 0.7234 4.9399 10.0233

The final optimized parameter values are, Kp = 0.7235; Ki = 4.9399; Kd = 10.0233

Now, these optimized values are put into the PID block and the simulation is run. The step response obtained is shown below.

Time (seconds) Figure 6.6 Step response for using the optimized values. The results are discussed in the subsequent chapter.

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Chapter 7 Results and summary

The PID values obtained through the optimization method are the minimum mean square error solution needed for the time response of the control system. This is because it tries to minimize the error between the input and output. The step response plot using the optimized value is also plotted. Optimized PID values are better as seen by comparing both the plots. In the graph using optimized values the signal rises to the required level faster (at time t = 200 using the optimized PID values, whereas at time t = 300 for starting values). Hence the optimized values are definitely better than the previous values. It is also a better alternative to the trial and error method used presently on the current robot. The method is also more scientific and logical approach to a difficult problem of tuning the PID controller.

The obtained values can be compared directly to the results obtained with the WSDK kit to see the response of the system rather than trying iterative values. But, the kit can be used to fine-tune the PID values after we arrive at the optimized value.

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Chapter 8 Recommendations

Due the complex nature of framing the model, a generalized model was taken for optimization. In future methods more work could be done to model actual control systems for individual applications and be solved using the optimization routine. Even though the iterative steps are eliminated, a better method for choosing the intial parameter values should be developed. This thesis deals with only one of the optimization methods but there are many more methods to solve multi objective nonlinear problems. One of them is using a minmax approach. More work could be directed toward that area where many routines are used on the same problem and the best control parameters chosen. Based on those the software kit can be used to fine-tune the values. A stable testing platform, Bearcat II is available for testing. Various values can be used in the program and the performance evaluated.

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Appendix

MATLAB code for the calculation of digital gains Section of code which takes in digital gains and converts them into analog gains

GN=800 ZR=0.9 PL=0 KI=0

% Given GN...solve for K,A,B,C K=GN; A=ZR; B=PL/256; C=KI/8;

% solve for a,b,g,f

a=(2000-2000*A)/(1+A) b=(2000-2000*B)/(1+B) f=1000*C g=K*(1+A)/(1+B)

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