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NCSU MAE 208 - REGULATING MULTI- DIMENSIONAL SYSTEMS

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REGULATING MULTI-DIMENSIONAL SYSTEMS MAE 461: DYNAMICS AND CONTROLS 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 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 in theory. Non-full regulation problems, althoughREGULATING MULTI-DIMENSIONAL SYSTEMS MAE 461: DYNAMICS AND CONTROLS not always practical, are quite interesting in that they can be 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) )()()()()()(222202112101tQtqtqtQtqtq=+=+ωω&&&& where)(and)(21tQtQ 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 (11 - 2) ∫−−−=∫−−−=dttqitqhtqgtQdttqitqhtqgtQ)()()()()()()()(22222221111111&& 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 gainsREGULATING MULTI-DIMENSIONAL SYSTEMS MAE 461: DYNAMICS AND CONTROLS (11 - 3) )(22222210212022221rrrrrrrrrrrrrrihgβαααααωβααα+=−+=−++= 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 in terms of the physical displacements, physical velocities, and integrals of physical displacements. This is needed in order to be able to implement the control law using actuators and sensors. 2. Physical Forces The physical forces are related to the physical sensors by transforming the modal quantities into physical quantities. From Eq. (10 - 27) and Eq. (11 - 2) (11 - 4) ∫∫∫∫∫−−−=−−−+−−−=−−−+−−−=+=ttTTTtTTTttdtdtihgdtihgdttqitqhtqgdttqitqhtqgtQtQ 0 0 2222222 0 1111111 0 2222222 0 11111112211]φφφ[φ]φφφ[φ])()()([φ])()()([φ)(φ)(xIxHGxMxxMMxMMxxMMxMMMMMφF&&&&& in whichREGULATING MULTI-DIMENSIONAL SYSTEMS MAE 461: DYNAMICS AND CONTROLS (11 - 5) ]MφφφφM[I]MφφφφM[H]MφφφφM[GTTTTTTiihhgg2221112221111222111+=+=+= Equation (11 - 4) and Eq. (11 - 5) is a PID feedback algorithm. The control gains can be set to produce any desirable performance in terms of the parameters given in Eq. (11 - 3). The algorithm can therefore produce any desirable dynamic performance, at least in theory. To implement Eq. (11 - 4) and Eq. (11 - 5), the parameters in those equations would need to be known. The parameters in those equations consist of physical parameters associated with the uncontrolled system (masses and stiffnesses) and dynamic performance parameters associated with the uncontrolled system (natural mode shapes, natural frequencies, and natural damping rates). Thus, in practice, it appears that these parameters must be known in order to produce the control forces. Controllers of this kind are also called model-based controllers. The following shows, however, that this controller is not always model-based. Equation (11 - 4) and Eq. (11 - 5) is a general form of a full regulator for a multi-dimensional system. The following considers several specialized regulators. These regulators have special characteristics that make them particularly attractive - as you’ll shortly see. Before proceeding with this, however, we’ll need two mathematical identities. First, let’s prove thatREGULATING MULTI-DIMENSIONAL SYSTEMS MAE 461: DYNAMICS AND CONTROLS (11 - 6) TT22111φφφφ +=−M Equation (11 - 6) is proven using Eq. (10 – 27). Start by writing for any x (11 - 7) MxMxMxx]φφφφ[)()(221122112211TTTTqq++=+=ϕϕϕϕϕϕ Clearly, the term in brackets in Eq. (11 - 7) is equal to the desired result, so Eq. (11 - 7) is proven. Next, let’s prove that (11 - 8) MMK ]φφφφ[2220211201TTωω+= From Eq. (10 - 27) for any x (11 - 9) MxMMxMMMMxMMxMMxKMxKKKx]φφφφ[]φφφφ[))(())(()()()(22202112012220211201222021120122112211TTTTTTTTqqωωωωϕϕωϕϕωϕϕϕϕϕϕ+=+=+=+=+= Clearly, the term pre-multiplying x on the ride side of Eq. (11 - 9) is equal to the right side of Eq. (11 - 8) which completes the proof. 3. Regulating Settling Time Consider a two degree-of-freedom system in which the interest lies in regulating settling time. Toward this end, let’s examine the response of a two degree-of-freedom system subject to PID controlREGULATING MULTI-DIMENSIONAL SYSTEMS MAE 461: DYNAMICS AND CONTROLS forces. Also, for illustrative purposes we let the response be under-damped. The r-th natural mode of vibration and the r-th modal displacement are given by (11 - 10) )sincos()(121tCtBeeAtqrrrrttrrrrrrββφϕαα++=⎟⎟⎠⎞⎜⎜⎝⎛=−−


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NCSU MAE 208 - REGULATING MULTI- DIMENSIONAL SYSTEMS

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