Introduction to Modern Control Theory

Several factors provided the stimulus for the development of modern control theory:

a. The necessary of dealing with more realistic models of system.

b. The shift in emphasis towards optimal control and optimal system design.

c. The continuing developments in digital computer technology.

d. The shortcoming of previous approaches.

e. Recognition of the applicability of well-known methods in other fields of knowledge.

The transition from simple approximate models, which are easy to work with, to more realistic models, produces two effects. First, a large number of variables must be included in the models. Second, a more realistic model is more likely to contain nonlinearities and time-varying parameters. Previously ignored aspects of the system, such as interactions with feedback through the environment, are more likely to be included.

With an advancing technological society, there is a trend towards more ambitious goals. This also means dealing with complex system with a large number of interacting components. The need for greater accuracy and efficiency has changer the emphasis on control system performance. The classical specifications in terms of percent overshoot, setting time, bandwidth, etc. have in many cases given way to optimal criteria such as mini mum energy, minimum cost, and minimum time  operation. Optimization of these criteria makes it even more difficult to avoid dealing with unpleasant nonlinearities. Optimal control theory often dictates that nonlinear time-varying control laws are used, even if the basic system is linear and time-invariant.

The continuing advances in computer technology have had three principal effects on the controls field. One of these relates to the gigantic supercomputers. The size and the class of the problems that can now be modeled, analyzed, and controlled are considerably large than they were when the first edition of this book was written.

The second impact of the computer technology has to so with the proliferation and wide availability of the microcomputers in homes and I the work place, classical control theory was dominated by graphical methods because at the time that was the only way to solve certain problems, Now every control designer has easy access to powerful computer packages for systems analysis and design. The old graphical methods have not yet disappeared, but have been automated. They survive because of the insight and intuition that they can provide, some different techniques are often better suited to a computer. Although a computer can be used to carry out the classical transform-inverse transform methods, it is used usually more efficient for a computer to integrate differential equations directly.

The third major impact of the computers is that they are now so commonly used as just another component in the control systems. This means that the discrete-time and digital system control now deserves much more attention than

Modern control theory is well suited to the above trends because its time-domain techniques and its mathematical language (matrices, linear vector spaces, etc.) are ideal when dealing with a computer. Computers are a major reason for the existence of state variable methods.

Most classical control techniques were developed for linear constant coefficient systems with one input and one output (perhaps a few inputs and outputs). The language of classical techniques is the Laplace or Z-transform and transfer functions. When nonlinearities ad time variations are present, the very basis for these classical techniques is removed. Some successful techniques such as phase-plane methods, describing function s, and other ad hoc methods, have been developed to alleviant this shortcoming.

However, the greatest success has been limited to low-order systems. The state variable approach of modern control theory provides a uniform and powerful method of representing systems of arbitrary order, linear or nonlinear, with time-varying or constant coefficient. It provides an ideal formulation for computer implementation and is responsible for much of the progress in optimization theory.

Modern control theory is a recent development in the field of control. Therefore, the name is justified at least as a descriptive title. However, the foundations of modern control theory are to be found in other well-established fields. Representing a system in terms of state variables is equivalent to the approach of Hamiltonian mechanics, using generalized coordinates and generalized moment. The advantages of this approach have been well-known I classical physics for many years. The advantages of using matrices when dealing with simultaneous equations of various kinds have long been appreciated in applied mathematics. The field of linear algebra also contributes heavily to modern control theory. This is due to the concise notation, the generality of the results, and the economy of thought that linear algebra provides.

Mechanism of Surface Finish Production

There are basically five mechanisms which contribute to the production of a surface which have been machined. There are:

(1) The basic geometry of the cutting process. In, for example, single point turning the tool will advance a constant distance axially per revolution of the work piece and the resultant surface will have on it, when viewed perpendicularly to the direction of tool feed motion, a series of cusps which will have a basic form which replicates the shape of the tool in cut.

(2) The efficiency of the cutting operation. It has already been mentioned that cutting with unstable built-up-edges will produce a surface which contains hard built-up-edge fragments which will result in a degradation of the surface finish. It can also be demonstrated that cutting under adverse conditions such as apply when using large feeds small rake angles and low cutting speeds, besides producing conditions whi

ch continuous shear occurring in the shear zone, tearing takes place, discontinuous chips of uneven thickness are produced, and the resultant surface is poor. This situation is particularly noticeable when machining very ductile materials such as copper and aluminum.

(3) The stability of the machine tool. Under some combinations of cutting conditions: work piece size , method of clamping, and cutting tool rigidity relative to the machine tool structure, instability can be set up in the tool which causes it to vibrate. Under some conditions the vibration will built up and unless cutting is stopped considerable damage to both the cutting tool and work piece may occur. This phenomenon is known as chatter and in axial turning is characterized by long pitch helical bands on the work piece surface and short pitch undulations on the transient machined surface.

(4) The effectiveness of removing sward. In discontinuous chip production machining, such as milling or turning of brittle materials, it is expected that the chip (sward) will leave the cutting zone either under gravity or with the assistance of a jet of cutting fluid and that they will not influence the cut surface in any way. However, when continuous chip production is evident, unless steps ate taken to control the swarf it is likely that it will impinge on the cut surface and mark it. Inevitably, this marking beside a looking unattractive, often results in a poorer surface finishing,

(5) The effective clearance angle on the cutting tool. For certain geometries of minor cutting edge relief and clearance angles it is possible to cut on the major cutting edge and burnish on the minor cutting edge. This can produce a good surface finish but, of course, it is strictly a combination of metal cutting and metal forming and is not to be recommended as a practical cutting method. However, due to cutting tool wear, these conditions occasionally arise and lead to a marked change in the surface characteristics.

Surface Finishing and Dimensional Control

Products that have been completed to their proper shape and size frequently require some type of surface finishing to enable than to satisfactorily fulfill their function. In some cases, tit is necessary to improve the physical properties of the surface material for resistance to penetration or abrasion. In many manufacturing processes, the product surface is left with dirt, chips, grease, or other harmful material upon it. Assemblies that are made of different materials, or from the same materials processed in different manners, many require some special surface treatment to provide uniformity of appearance.

Surface finishing many sometimes become an intermediate step processing. For instance, cleaning and polishing are usually essential before any kind of plating process. Some of the cleaning procedures are also used for improving surface smoothness on mating parts and for removing burrs and sharp corners, which might be harmful in later use. Another

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important need for surface finishing is for corrosion protection in a variety of environments. The type of protection procedure will depend largely upon the anticipated exposure, with due consideration to the material being protected and the economic factors involved.

Satisfying the above objectives necessitates the use of main surface-finishing methods that involve chemical change of the surface mechanical work affecting surface properties, cleaning by a variety of methods, and the application of protective coatings, organic and metallic.

In the early days of engineering, the mating of parts was achieved by machining one part as nearly as possible to the required size, machining the mating part nearly to size, and then completing its machining, continually offering the other part to it, until the desired relationship was obtained. If it was inconvenient to offer one par to the other part during machining, the final work was done at the bench by a fitter, who scraped the mating parts until the desired fit was obtained, the fitter therefore being a ‘fitter’ in the literal sense. It is obvious that the two parts would have to remain together, and in the event of one having to be replaced, the fitting would have to be done all over again. I n these days, we expect to be able to purchase a replacement for a broken part, and for it to function correctly without the need for scraping and other fitting operations.

When one part can be used ‘off the shelf’ to replace another of the same dimension and material specification, the parts are said to be interchangeable. A system of interchangeability usually lowers the production costs as there is no need for an expensive, ‘fiddling’ operation, and it benefits the customer in the event of the need to replace worn parts.

Limits and Tolerances

Machine parts are manufactured so they are interchangeable. In other words, each part of a machine or mechanism is made to a certain size and shape so it will fit into any other machine or mechanism of the same type. To make the part interchangeable, each individual part must be made to a size that will fit the mating part in the correct way. It is not only impossible, but also impractical to make many parts to an exact size. This is because machines are not perfect, and the tools become worn. A slight variation from the exact size is always allowed. The amount of this variation depends on the kind of part being manufactured. For example, a part might be made 6 in. long with a variation allowed of 0.003(three thousandths) in. above and below this size. Therefore, the part could be 5.997 to 6.003 in. and still be the correct size. These are known as the limits. The difference between upper and lower limits is called the tolerance.

A tolerance is the total permissible variation in the size of a part.

The basic size is that size from which limits of size are derived by the application of allowances and tolerances.

Sometimes the limit is allowed in only one

direction. This is known as unilateral tolerance.

Unilateral tolerancing is a system of dimensioning where the tolerance (that is variation) is shown I only one direction from the nominal size. Unilateral tolerancing allow the changing of tolerance on a hole or shaft without seriously affecting the fit.

When the tolerance is in both directions from the basic size, it is known as a bilateral tolerance (plus and minus).

Bilateral tolerancing is a system of dimensioning where the tolerance (that is variation) is split and is shown on either side of the nominal size. Limit dimensioning is a system of dimensioning where only the maximum and minimum dimensions are shown. Thus, the tolerance is the difference between these two dimensions.

Introduction of Machining of:

Machining as a shape-producing method is the most universally used and the most important of all manufacturing processes. Machining is a shape-producing process in which a power-driven device causes material to be removed in chip form. Most machining is done with equipment that supports both the work piece and cutting tool although in some cases portable equipment is used with unsupported work piece.

Low setup cost for small quantities. Machining has two applications in manufacturing. For casting, forging, and pressworking, each specific shape to be produced, even one part, nearly always has a high tooling cost. The shapes that may be produced by welding depend to a large degree on the shapes of raw material that are available. By making use of generally high cost equipment but without special tooling, it is possible, by machining, to start with nearly any form of raw material, so long as the exterior dimensions are great enough, and produce any desired shape from any material. Therefore, machining is usually the preferred method for producing one or a few parts, even when the design of the part would logically lead to casting, forging or pressworking if a high quantity were to be produced.

Close accuracies, good finishes. The second application for machining is based on the high accuracies and surface finishes possible. Many of the parts machined in low quantities would be produced with lower but acceptable tolerances if produced I high quantities by some other process. On the other hand, many parts are given their general shapes by some high quantity deformation process and machined only on selected surfaces where high accuracies are needed. Internal threads, for example, are seldom produced by any means other than machining and small holes in pressworked parts may be machined following the pressworking operations.