achapt 9 - SYSTEM CONCEPTS 9 SYSTEM CONCEPTS Automotive...

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SYSTEM CONCEPTS 9 SYSTEM CONCEPTS Automotive vehicles, building structures and aircraft vehicles are dynamical systems but so too are actuators and sensors. Actuators and sensors have their own dynamical behavior, as well. For example, when we say that a motor prescribes a moment, what does that mean? Does it mean that it can apply a moment of any magnitude? Does the moment depend on the inertia of the motor itself? Does the applied moment depend on the physical system attached to it? The short answers to each of these questions are yes. In reality, the actuator and the physical system form a new system that has its own dynamical properties. Most dynamical systems are assemblages of subsystems that are dynamical systems in their own right. The interactions between the subsystems influence the behavior of the overall system. Ideally, it would be much simpler if the dynamical behavior of the subsystems were independent of each other; certainly the system would be much easier to analyze than if we only needed to analyze the dynamic behavior of the subsystems. In fact, this is an important goal of control system design – to put together systems MAE 461: DYNAMICS AND CONTROLS
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SYSTEM CONCEPTS composed of subsystems that act independent of one another. The principles used to accomplish this design strategy are called separation principles. In this chapter we will first introduce a notation for systems called operator notation. Next, we’ll introduce the block-diagram representation of a system. Then, we introduce the separation principle for tracking and regulation. This is arguably the most important separation principle in control system design. It serves as the basis for what is often referred to as modern controls. I. Linear Operators The previous chapters focused on single degree-of- freedom systems. The single degree-of-freedom system can be represented as (9 – 1) k dt d c dt d m L f Lx + + = = 2 2 where L is the linear operator of the physical system. In the case of dynamic systems, the linear operator is differential and not merely algebraic. The solution to Eq. (9 – 1) is represented by (9 – 2) f L x 1 = where L - 1 is the inverse of L. The operator L - 1 is symbolic of solving the differential equation. Although the notation does not explicitly deal with the system’s initial conditions, the solution depends on initial conditions so the operation L - 1 assumes that the system is acted on by certain initial conditions. MAE 461: DYNAMICS AND CONTROLS
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SYSTEM CONCEPTS The operator L is said to be linear when the following occurs: Consider the two solutions x 1 and x 2 . The operator L is linear if for any constants α 1 and 2 (9 – 3) 2 2 1 1 2 2 1 1 ) ( Lx Lx x x L + = + When the operator in Eq. (9 – 1) is linear, Eq. (9 – 1) is said to be a linear equation. The two forces that correspond to the solutions x 1 and x 2
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This note was uploaded on 09/17/2009 for the course MAE 469 taught by Professor Silverberg during the Fall '08 term at N.C. State.

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achapt 9 - SYSTEM CONCEPTS 9 SYSTEM CONCEPTS Automotive...

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