Chapter 1 Introduction
1.1 Motivation
In order to improve the behavior or performance of physical systems we addressto control theory. The algorithm of control theory is simply to collect and use theinformation about physical systems. Most of time this information is introduced as amathematical model of the plant from which the controller design is derived. Our goalin designing a controller is to obtain some objectives such as stability or outputtracking of some reference signal.Typically for robustness reasons, the controller is realized as a dynamic feedbackof measured variables. From a system dynamics point of view, control theory can bedivided into two main classes, linear and nonlinear systems, the dynamic features ofwhich can be either dependent on time or time invariant. In this thesis, we focus onnonlinear system with input nonsmooth nonlinearity. Several designs have beensuggested to study robustness issues in the single loop control[5-9]and decentralizedcontrol of multi-loop systems[10-13]. Problems related to nonsmooth nonlinearitieshave also been well addressed in [14-17].Backstepping is a recursive Lyapunov-based scheme for parametric-strictfeedback systems to ensure tracking properties and global regulation. Backsteppingdesign method provides a step-by-step algorithm to design desirable controllers.Dead-zone nonsmooth nonlinearity[21-22]is common in industrial control systems.Dead-zone is a static input-output characteristic. Usually it appears in motors andactuators. Dead-zone varies with time and often limits system performance. Controlof systems with dead-zone nonlinearity is difficult. The control design should havean ability to adapt to system uncertainties. Different design approaches with differentcontrol objectives have been developed in theory and practice[26-27]. To cope withactuator dead-zone[28-32]adaptive control scheme was used. We can also mentionneural networks control in [22,34,35], fuzzy logic control in [21,36,37], variablestructure control in [23,30,38,39], pole placement control in [24,25,33] and recursiveleast-square algorithm in [40]. From the above discussion, it follows that the presenceof nonsmooth nonlinearity in actuators should be accounted for. Hence, an observableuncertainty should be taken into account in nonlinearities models parameters, in orderto match a spacious array of real situations. In this thesis, we will developbackstepping control for our proposed plant to get both perfect tracking and transientperformance.
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1.2 Review
Backstepping approach in this thesis is composed of simple steps. The techniqueis to step back toward the control input starting with scalar equation. After a br iefreview of DC Motors, backstepping control and mathematical preliminaries, thischapter introduces basic backstepping tools for systems with uncertainties and basicdescription of nonsmooth nonlinear characteristic dead-zone. Dead-zone is illustratedby an example in this chapter.Motors are devices which convert electrical energy into mechanical energy. In itsmost basic form, a motor consists of a loop of wire in a magnetic field to which currentis applied. The torque acting on the current carrying loop causes it to rotate. Attachingthe rotating armature to external devices can cause useful mechanical work. A DCmotor is motor in which the armature windings are on the rotor with current conductedfrom brushes. The rotor of a DC machine is often referred to as the armature. Thefield windings are located on the stator and are excited with help of direct current.DC motors are the most common choice when a controlled electrical drive operatingover a wide speed range is specified[50]. They have excellent operational propertiesand control characteristics[50].DC motors are classified as shunt, series or compound according to the methodof field connection. A discussion of these motors and their system models may befound in variety of sources, including [47,49,53,55]. In a series motor, the field circuitis connected in series with the armature circuit, while in the shunt motor, the twocircuits are connected in parallel. One of the central differences between the twomotors is that the shunt motor is primarily wound with a big number of turns makingthe resistance pretty high. Alternatively, a series motor with fewer number of turns,which minimizes the voltage drop across it. In some cases the two configurations arecombined to produce the compound motor. For the no load condition, this motorbehaves much like shunt motor. At higher loads, the characteristics more resemblethe series motor. Elaborate circuits are required to control compound motors[52].
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Chapter 2 PID Control of a series DC motor withinput dead-zone
2.1 Introduction
A proportional–integral–derivative controller (PID controller) is a control loop-feedback mechanism. Consequently, PID algorithm includes three essential parts:proportional, integral and derivative coefficients, which are varied to get optimalresponse and stability. The point of this algorithm is to manipulate the error. Whenthe proportional coefficient rectifies cases of error, the integral coefficient rectifiesstorage of error. Finally, the derivative coefficient rectifies present error versus errorthe last time it was checked. The main purpose of the derivative part is to cancel outthe overshoot caused by proportional and integral parts. When the error is too large,the proportional coefficient and the integral one will push the output of the controller.The response of the controller forces error to change very quickly, causing thederivative coefficient to more aggressively counteract the proportional and theintegral parts.The PID controller is perhaps the most commonly used compensator in feedbackcontrol systems. The proportional part provides the controller output which isdetermined by a function of the present state of the system. Consequently, theintegrator part gives an output that is the former state of the system. Thedifferentiation part provides a prediction of the future state of the system. One ormore of these terms, P, I, and D, are inserted into the feedback loop and their valuesadjusted to provide the best control. Each term affects the system in a slightlydifferent way.
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2.2 Application of the PID control to the problem
The proportional-integral-derivative controller obtains a desirable output in abrief time, with less overshoot and relatively minimal error. The PID controller is oneof the most commonly used controllers in the industry. The reasons behind it are theaffordable price and given benefits to the industry.Simulations were attempted for several different values of k0(integral gain), k1(proportional gain), and k2(derivative gain), and the results are presented below toprovide a demonstration of the effect of the variation of the three gains on the stabilityof the system.
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Chapter 3 Backstepping Control of series DC motor.........27
3.1 Introduction..... 27
3.2 Application of the backstepping control to the problem .... 27
3.3 Simulation ....... 33
3.3.1 Plant Model ......... 33
3.3.2 Simulations under backstepping control .......... 34
3.4 Summary ......... 38
Chapter 4 Robust Control of a series DC motor with input dead-zone ....... 39
4.1 Introduction..... 39
4.2 Application of the robust control design to the problem .... 40
4.3 Simulations ..... 54
4.4 Summary ......... 56
Chapter 4 Robust Control of a series DC motor withinput dead-zone
4.1 Introduction
In this final section in the chapter on designing control laws, we examine thesystem with uncertainties. The background information on this approach includes adiscussion of the generalized matching conditions and their importance in developinga robust control law. The theory is then applied to the system equations for the twocases of motor speed. Finally, the system will be simulated in MATLAB.To apply the robust control method to this problem, we must show that the systemmeets the so-called Generalized Matching Conditions (GMC). One importantrequirement for a system to meet the GMC is that the only type of interconnectionsbetween the subsystems may be that of feedback. The GMC include as special cases many physical systems which are seriesconnections of nonlinear subsystems. The GMC was originally introduced for linearuncertain systems but was later extended to nonlinear uncertain systems. The GMCis important in the robust control design because, as shown in the GMC proof, thosesystems which contain unmatched uncertainties satisfying the GMC can be fullycompensated by a properly designed robust control.The GMC include five major conditions:(1) Controllability condition(2) Condition to avoid singularity problem(3) Condition to reduce the effort of finding a robust control(4) Condition for simplicity of mathematical development(5) Condition requiring uncertainties to be boundedAs previously mentioned, a cascaded system is a case of the generalized matchingconditions (GMC). Our plant may be transformed into a cascaded system. We maythen apply the recursive design approach in developing a robust control law for the system with time-varying uncertainties with the assurance that, for a properlydesigned robust control, the uncertainties may be fully compensated.
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Conclusion
It has been shown that the backstepping control may be successfully applied tothe problem of designing a robust control for the nonlinear model of a series DCmotor with input dead-zone.Initially, the system was examined under the assumption that all systemparameters were perfectly known. Use of the proportional-integral technique enabledthe development of a control law. In addition, a proportional-derivative control termwas included to reduce the destabilizing effect of the PI controller. Once again theresults were quite promising. A maximum error of 12 rad/s occurred during theexpected time when load torque was changing and the motor passed through basespeed. The steady state error was once again nearly zero.In the second case, a backstepping controller is proposed. After transforming thesystem into a cascaded structure, we were able to easily apply the recursive designapproach. The resulting control law when simulated produced excellent results. Themaximum error was seen to be approximately 5 rad/s and occurred when expected.Finally, the robust control method was applied to this problem. The final speedof the motor almost exactly matched the reference speed for nearly zero steady stateerror.Although we only considered the cases when load torque and armatureinductance were unknown, the design as presented could be easily extended to handleadditional uncertainties. Further research could be conducted by including additionalnonlinear terms in the system equation. Or one might choose to consider thepossibility of the existence of uncertainties in other terms such as the moment ofinertia or flux. In any case, the superior performance of the backstepping control lawdemonstrates its value in design theory and application.The more complete the model of a system, the greater the precision that can beachieved. Such modeling usually requires that the system be represented by nonlinearequations which may contain uncertain terms. This, then, provides our motivation forcontinuing to develop and refine techniques of nonlinear control and to apply thesetechniques to physical systems. As shown, the recursive design approach may be usedto develop a nonlinear control law for the series DC motor with generally acceptableresults.
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References (abbreviated)
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