achapt 11 - REGULATING MULTI-DIMENSIONAL SYSTEMS 11...

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REGULATING MULTI-DIMENSIONAL SYSTEMS 11 REGULATING MULTI- DIMENSIONAL SYSTEMS This chapter considers the regulation problem restricted to the case in which the actuators are continuously-acting. We’ll consider two types of regulation problems; full regulation problems and non-full regulation problems. Full regulation problems arise when the number of actuators is equal to or greater than the number of degrees-of- freedom that are being controlled. Non-full regulation problems arise when fewer actuators than controlled degrees-of-freedom are used. Full regulation is prevalent in engineering systems. When a system is fully regulated the multi-degree- of-freedom system can be treated as single degree- of-freedom systems, in which each mode is regulated one independent of the other and the regulation problem can be separated from the tracking problem. We’ll also see that full regulation tends to be more cost efficient than non- full regulation, and less sensitive to unknown parameters. In non-full regulation problems, just a few actuators can regulate a system’s response, at least MAE 461: DYNAMICS AND CONTROLS
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REGULATING MULTI-DIMENSIONAL SYSTEMS in theory. Non-full regulation problems, although not always practical, are quite interesting in that they resemble a “juggling act” that appears on the surface to be impossible. This chapter concludes with the problem of non-full regulation. 1. PID Regulation of Modes Recall that the modal equations of an undamped two degree-of-freedom system are (11 - 1) ) ( ) ( ) ( ) ( ) ( ) ( 2 2 2 20 2 1 1 2 10 1 t Q t q t q t Q t q t q = + = + ω ω & & & & where now represent modal control forces. Since Eq. (11 - 1) is in the form of single degree-of-freedom systems, we can employ PID feedback to regulate each mode. Consider PID feedback modal forces ) ( and ) ( 2 1 t Q t Q (11 - 2) = = dt t q i t q h t q g t Q dt t q i t q h t q g t Q ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 2 2 2 2 2 2 1 1 1 1 1 1 1 & & Notice in Eq. (11 - 2) that the feedback forces are independent of each other, i.e., that the first modal force is a function of the first modal coordinate and that the second modal force is a function of the second modal coordinate. This independence is consistent with the observation made earlier that the first modal force affects only the first modal coordinate and that the second modal force affects only the second modal coordinate. Comparing Eq. (11 - 1) and Eq. (11 - 2) with Eq. (7 - 30) and Eq. (7 - 35), we get the PID modal control gains MAE 461: DYNAMICS AND CONTROLS
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REGULATING MULTI-DIMENSIONAL SYSTEMS (11 - 3) ) ( 2 2 2 2 2 2 1 0 2 1 2 0 2 2 2 2 1 r r r r r r r r r r r r r r i h g β α α α α α ω β α α α + = + = + + = Equation (11 - 2) is a modal control law . It expresses modal forces in terms of modal displacements, modal velocities and integrals of modal displacements. But actuators produce physical forces not modal forces and sensors perform measurements of physical quantities not measurements of modal quantities. The question arises how to obtain a physical control law , that is, a control law that expresses the physical forces
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