**Ingenieria - de - Control - Moderna - Ogata - 5ed**

Ingenieria - de - Control - Moderna - Ogata - 5ed

(Parte **3** de 6)

2.To maintain the required quality in the output,recalibration is necessary from time to time.

1–4DESIGN AND COMPENSATION OF CONTROL SYSTEMS

This book discusses basic aspects of the design and compensation of control systems. Compensation is the modification of the system dynamics to satisfy the given specifications.The approaches to control system design and compensation used in this book are the root-locus approach,frequency-response approach,and the state-space approach.Such control systems design and compensation will be presented in Chapters 6,7,9 and 10.The PID-based compensational approach to control systems design is given in Chapter 8.

In the actual design of a control system,whether to use an electronic,pneumatic,or hydraulic compensator is a matter that must be decided partially based on the nature of the controlled plant.For example,if the controlled plant involves flammable fluid,then we have to choose pneumatic components (both a compensator and an actuator) to avoid the possibility of sparks.If,however,no fire hazard exists,then electronic compensators are most commonly used.(In fact,we often transform nonelectrical signals into electrical signals because of the simplicity of transmission,increased accuracy,increased reliability,ease of compensation,and the like.)

Performance Specifications.Control systems are designed to perform specific tasks.The requirements imposed on the control system are usually spelled out as performance specifications.The specifications may be given in terms of transient response requirements (such as the maximum overshoot and settling time in step response) and of steady-state requirements (such as steady-state error in following ramp input) or may be given in frequency-response terms.The specifications of a control system must be given before the design process begins.

For routine design problems,the performance specifications (which relate to accuracy,relative stability,and speed of response) may be given in terms of precise numerical values.In other cases they may be given partially in terms of precise numerical values and

Section 1–4/Design and Compensation of Control Systems9 partially in terms of qualitative statements.In the latter case the specifications may have to be modified during the course of design,since the given specifications may never be satisfied (because of conflicting requirements) or may lead to a very expensive system.

Generally,the performance specifications should not be more stringent than necessary to perform the given task.If the accuracy at steady-state operation is of prime importance in a given control system,then we should not require unnecessarily rigid performance specifications on the transient response,since such specifications will require expensive components.Remember that the most important part of control system design is to state the performance specifications precisely so that they will yield an optimal control system for the given purpose.

System Compensation.Setting the gain is the first step in adjusting the system for satisfactory performance.In many practical cases,however,the adjustment of the gain alone may not provide sufficient alteration of the system behavior to meet the given specifications.As is frequently the case,increasing the gain value will improve the steady-state behavior but will result in poor stability or even instability.It is then necessary to redesign the system (by modifying the structure or by incorporating additional devices or components) to alter the overall behavior so that the system will behave as desired.Such a redesign or addition of a suitable device is called compensation.A device inserted into the system for the purpose of satisfying the specifications is called a compensator.The compensator compensates for deficient performance of the original system.

Design Procedures.In the process of designing a control system,we set up a mathematical model of the control system and adjust the parameters of a compensator. The most time-consuming part of the work is the checking of the system performance by analysis with each adjustment of the parameters.The designer should use MATLAB or other available computer package to avoid much of the numerical drudgery necessary for this checking.

Once a satisfactory mathematical model has been obtained,the designer must construct a prototype and test the open-loop system.If absolute stability of the closed loop is assured,the designer closes the loop and tests the performance of the resulting closedloop system.Because of the neglected loading effects among the components,nonlinearities,distributed parameters,and so on,which were not taken into consideration in the original design work,the actual performance of the prototype system will probably differ from the theoretical predictions.Thus the first design may not satisfy all the requirements on performance.The designer must adjust system parameters and make changes in the prototype until the system meets the specificications.In doing this,he or she must analyze each trial,and the results of the analysis must be incorporated into the next trial.The designer must see that the final system meets the performance apecifications and,at the same time,is reliable and economical.

1–5OUTLINE OF THE BOOK

This text is organized into 10 chapters.The outline of each chapter may be summarized as follows: Chapter 1 presents an introduction to this book.

10Chapter 1/Introduction to Control Systems

Chapter 2 deals with mathematical modeling of control systems that are described by linear differential equations.Specifically,transfer function expressions of differential equation systems are derived.Also,state-space expressions of differential equation systems are derived.MATLAB is used to transform mathematical models from transfer functions to state-space equations and vice versa.This book treats linear systems in detail.If the mathematical model of any system is nonlinear,it needs to be linearized before applying theories presented in this book.A technique to linearize nonlinear mathematical models is presented in this chapter.

Chapter 3 derives mathematical models of various mechanical and electrical systems that appear frequently in control systems.

Chapter 4 discusses various fluid systems and thermal systems,that appear in control systems.Fluid systems here include liquid-level systems,pneumatic systems,and hydraulic systems.Thermal systems such as temperature control systems are also discussed here. Control engineers must be familiar with all of these systems discussed in this chapter.

Chapter 5 presents transient and steady-state response analyses of control systems defined in terms of transfer functions.MATLAB approach to obtain transient and steady-state response analyses is presented in detail.MATLAB approach to obtain three-dimensional plots is also presented.Stability analysis based on Routh’s stability criterion is included in this chapter and the Hurwitz stability criterion is briefly discussed.

Chapter 6 treats the root-locus method of analysis and design of control systems.It is a graphical method for determining the locations of all closed-loop poles from the knowledge of the locations of the open-loop poles and zeros of a closed-loop system as a parameter (usually the gain) is varied from zero to infinity.This method was developed by W.R.Evans around 1950.These days MATLAB can produce root-locus plots easily and quickly.This chapter presents both a manual approach and a MATLAB approach to generate root-locus plots.Details of the design of control systems using lead compensators,lag compensators,are lag–lead compensators are presented in this chapter.

Chapter 7 presents the frequency-response method of analysis and design of control systems.This is the oldest method of control systems analysis and design and was developed during 1940–1950 by Nyquist,Bode,Nichols,Hazen,among others.This chapter presents details of the frequency-response approach to control systems design using lead compensation technique,lag compensation technique,and lag–lead compensation technique.The frequency-response method was the most frequently used analysis and design method until the state-space method became popular.However,since H-infinity control for designing robust control systems has become popular,frequency response is gaining popularity again.

Chapter 8 discusses PID controllers and modified ones such as multidegrees-offreedom PID controllers.The PID controller has three parameters;proportional gain, integral gain,and derivative gain.In industrial control systems more than half of the controllers used have been PID controllers.The performance of PID controllers depends on the relative magnitudes of those three parameters.Determination of the relative magnitudes of the three parameters is called tuning of PID controllers.

Ziegler and Nichols proposed so-called “Ziegler–Nichols tuning rules”as early as 1942.Since then numerous tuning rules have been proposed.These days manufacturers of PID controllers have their own tuning rules.In this chapter we present a computer optimization approach using MATLAB to determine the three parameters to satisfy

Section 1–5/Outline of the Book11 given transient response characteristics.The approach can be expanded to determine the three parameters to satisfy any specific given characteristics.

Chapter 9 presents basic analysis of state-space equations.Concepts of controllability and observability,most important concepts in modern control theory,due to Kalman are discussed in full.In this chapter,solutions of state-space equations are derived in detail.

Chapter 10 discusses state-space designs of control systems.This chapter first deals with pole placement problems and state observers.In control engineering,it is frequently desirable to set up a meaningful performance index and try to minimize it (or maximize it,as the case may be).If the performance index selected has a clear physical meaning, then this approach is quite useful to determine the optimal control variable.This chapter discusses the quadratic optimal regulator problem where we use a performance index which is an integral of a quadratic function of the state variables and the control variable.The integral is performed from t=0tot=.This chapter concludes with a brief discussion of robust control systems. q

12Chapter 1/Introduction to Control Systems

Mathematical Modeling of Control Systems

2–1 INTRODUCTION

In studying control systems the reader must be able to model dynamic systems in mathematical terms and analyze their dynamic characteristics.A mathematical model of a dynamic system is defined as a set of equations that represents the dynamics of the system accurately,or at least fairly well.Note that a mathematical model is not unique to a given system.A system may be represented in many different ways and,therefore,may have many mathematical models,depending on one’s perspective.

The dynamics of many systems,whether they are mechanical,electrical,thermal, economic,biological,and so on,may be described in terms of differential equations. Such differential equations may be obtained by using physical laws governing a particular system—for example,Newton’s laws for mechanical systems and Kirchhoff’s laws for electrical systems.We must always keep in mind that deriving reasonable mathematical models is the most important part of the entire analysis of control systems.

Throughout this book we assume that the principle of causality applies to the systems considered.This means that the current output of the system (the output at time t=0) depends on the past input (the input for t<0) but does not depend on the future input (the input for t>0).

Mathematical Models.Mathematical models may assume many different forms.

Depending on the particular system and the particular circumstances,one mathematical model may be better suited than other models.For example,in optimal control problems,it is advantageous to use state-space representations.On the other hand,for the

14Chapter 2/Mathematical Modeling of Control Systems transient-response or frequency-response analysis of single-input,single-output,linear, time-invariant systems,the transfer-function representation may be more convenient than any other.Once a mathematical model of a system is obtained,various analytical and computer tools can be used for analysis and synthesis purposes.

Simplicity Versus Accuracy.In obtaining a mathematical model,we must make a compromise between the simplicity of the model and the accuracy of the results of the analysis.In deriving a reasonably simplified mathematical model,we frequently find it necessary to ignore certain inherent physical properties of the system.In particular, if a linear lumped-parameter mathematical model (that is,one employing ordinary differential equations) is desired,it is always necessary to ignore certain nonlinearities and distributed parameters that may be present in the physical system.If the effects that these ignored properties have on the response are small,good agreement will be obtained between the results of the analysis of a mathematical model and the results of the experimental study of the physical system.

In general,in solving a new problem,it is desirable to build a simplified model so that we can get a general feeling for the solution.A more complete mathematical model may then be built and used for a more accurate analysis.

We must be well aware that a linear lumped-parameter model,which may be valid in low-frequency operations,may not be valid at sufficiently high frequencies,since the neglected property of distributed parameters may become an important factor in the dynamic behavior of the system.For example,the mass of a spring may be neglected in lowfrequency operations,but it becomes an important property of the system at high frequencies.(For the case where a mathematical model involves considerable errors,robust control theory may be applied.Robust control theory is presented in Chapter 10.)

Linear Systems.A system is called linear if the principle of superposition applies.The principle of superposition states that the response produced by the simultaneous application of two different forcing functions is the sum of the two individual responses.Hence,for the linear system,the response to several inputs can be calculated by treating one input at a time and adding the results.It is this principle that allows one to build up complicated solutions to the linear differential equation from simple solutions.

In an experimental investigation of a dynamic system,if cause and effect are proportional,thus implying that the principle of superposition holds,then the system can be considered linear.

Linear Time-Invariant Systems and Linear Time-Varying Systems.A differential equation is linear if the coefficients are constants or functions only of the independent variable.Dynamic systems that are composed of linear time-invariant lumped-parameter components may be described by linear time-invariant differential equations—that is,constant-coefficient differential equations.Such systems are calledlinear time-invariant(orlinear constant-coefficient) systems.Systems that arerepresented by differential equations whose coefficients are functions of time arecalled linear time-varyingsystems.An example of a time-varying control systemisa spacecraft control system.(The mass of a spacecraft changes due to fuel consumption.)

(Parte **3** de 6)