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11 CONTROL FUNDAMENTALS 83 11 CONTROL FUNDAMENTALS 11.1 Introduction 11.1.1 Plants, Inputs, and Outputs Controller design is about creating dynamic systems that behave in useful ways. Many target systems are physical; we employ controllers to steer ships, fly jets, position electric motors and hydraulic actuators, and distill alcohol. Controllers are also applied in macro-economics and many other important, non-physical systems. It is the fundamental concept of controller design that a set of input variables acts through a given “plant” to create an output. Feedback control then uses sensed plant outputs to apply corrective plant inputs: Plant Inputs Outputs Sensors Jet aircraft elevator, rudder, etc. altitude, hdg altimeter, GPS Marine vessel rudder angle heading gyrocompass Hydraulic robot valve position tip position joint angle U.S. economy fed interest rate, etc. prosperity, inflation inflation, M1 Nuclear reactor cooling, neutron flux heat, power level temp., pressure 11.1.2 The Need for Modeling Effective control system design usually benefits from an accurate model of the plant, although it must be noted that many industrial controllers can be tuned up satisfactorily with no knowledge of the plant. Ziegler and Nichols, for example, developed a general heuristic recipe which we detail later. In any event, plant models simply do not match real-world systems exactly; we can only hope to capture the basic components in the form of differential or other equations. Beyond prediction of plant behavior based on physics, system identification generates a plant model from actual data. The process is often problematic, however, since the measured response could be corrupted by sensor noise or physical disturbances in the system which cause it to behave in unpredictable ways. At some frequency high enough, most systems exhibit effects that are difficult to model or reproduce, and this is a limit to controller performance. 11.1.3 Nonlinear Control The bulk of this subject is taught using the tools of linear systems analysis. The main reason for this restriction is that nonlinear systems are difficult to model, difficult to design controllers for, and difficult overall! Within the paradigm of linear systems, there are many
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11 CONTROL FUNDAMENTALS 84 sets of powerful tools available. The reader interested in nonlinear control is referred to the book by Slotine and Li (1991). 11.2 Partial Fractions Partial fractions are presented here, in the context of control systems, as the fundamental link between pole locations and stability. Solving linear time-invariant systems by the Laplace Transform method will generally create a signal containing the (factored) form K ( s + z 1 )( s + z 2 ) ··· ( s + z m ) Y ( s )= . (1) ( s + p 1 )( s + p 2 ) ( s + p n ) Although for the moment we are discussing the signal Y ( s ), later we will see that dynamic systems are described in the same format: in that case we call the impulse response G ( s ) a transfer function. A system transfer function is identical to its impulse response, since L ( δ ( t )) = 1.
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