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NCSU MAE 208 - REGULATING THE REFERENCE PATH

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REGULATING THE REFERENCE PATH (DISCRETELY-ACTING ACTUATORS) 8 REGULATING THE REFERENCE PATH (DISCRETELY-ACTING ACTUATORS) This chapter considers the regulation problem restricted to the case in which the control force is produced by discretely-acting actuators. The feedback is therefore nonlinear. The class of nonlinear forcing functions is very broad – really any force that is not a linear function of the state variables. This class of forcing functions is too broad to treat in a comprehensive manner. Instead, this section treats a few popular types of nonlinear feedback methods – the ones commonly implemented by discretely-acting actuators. Examples of actuators that, by design, are discretely-acting are solenoids, pneumatic and hydraulic devices, and braking systems. The most common type of discretely-acting actuator operates in a bang-off-bang manner. The bangs refer to constant levels of force in one direction or another. One can distinguish between bang-off-bang actuators by the proportion of bang time. At one MAE 461: DYNAMICS AND CONTROLSREGULATING THE REFERENCE PATH (DISCRETELY-ACTING ACTUATORS) extreme, the actuator is always on – banging in one direction or another. This is called bang-bang control. At the other extreme, the actuator on-time is very small – banging over short bursts of time. This is called impulse control. It turns out that time-optimal control is of the bang-bang type, and that fuel-optimal control is of the impulse type. Therefore, in practice, bang-off-bang control can be thought of as a compromise between fuel-optimality and time-optimality. This chapter first considers bang-bang control, and then it treats impulse control. In each case, we’ll see how the dynamic performance is affected by the control. 1. Bang-bang Feedback Consider an undamped single degree-of-freedom system subject to bang-bang feedback. Assume that the bang-bang feedback is produced by dry friction with a friction coefficient denoted by.μThe equation governing the motion of the system and the bang-bang force are (8 – 1) ⎩⎨⎧>−<==+00xmgxmgffkxxm&&&&μμ Notice that the bang-bang force is the same as a dry friction force of constant value in a direction that opposes the velocity. Dividing Eq. (8 – 1) by m, (8 – 2) )sgn(20xgxx&&&μω−=+ where sgn(x&) is the sign function, valued at 1 or –1 depending on the sign of x&. The response of the MAE 461: DYNAMICS AND CONTROLSREGULATING THE REFERENCE PATH (DISCRETELY-ACTING ACTUATORS) system is found by solving Eq. (8 – 2) over consecutive time intervals over which the control force is constant. Let tLi denote the lower limit of the i-th time interval, and let tUi denote the upper limit of the i-th time interval. The particular solution and the homogeneous solution over the i-th time interval are (8 – 3) 0)()()()())(sin())(cos()sgn()1(0020===<<−+−=−=− UiLiiULiUiLiLiLihptxtxtxtxtttttCttBxxgx&&&ωωωμ From Eq. (8 – 3) the complete response over the i-th time interval is (8 – 4) UiLiLiLiLitttttxgtxtxgx<<++−= )cos(]))(sgn()([))(sgn(02020ωωμωμ&& As shown in Fig. 8 - 1, the bang-bang response decays linearly at the linear rate (8 – 5) 02πωμγg= MAE 461: DYNAMICS AND CONTROLSREGULATING THE REFERENCE PATH (DISCRETELY-ACTING ACTUATORS) (a) (b) Figure 8 – 1 (a) Bang-bang and (b) Impulse control From Eq. (8 – 5), the response decays at a rate that is inversely proportional to the system’s natural frequency of oscillation. Thus, the larger the natural frequency, the slower the system is brought to rest. Also, note among the possible control forces bounded by μg, that the response given in Eq. (8 – 4) brings the system to rest in minimum-time. 2. Impulse Feedback Consider again an undamped single degree-of-freedom system subject to impulses. Before proceeding with impulse control, let’s examine the effect of a unit impulse on the response of the system. First, notice when the unit impulse is applied that the associated fuel cost is ∫∫=== .1)()( dttdttfCFδ Indeed, a unit amount of fuel is consumed. Secondly, notice that the reduction of energy caused by the unit impulse depends on the velocity of the system at the time of the impulse. Indeed, MAE 461: DYNAMICS AND CONTROLSREGULATING THE REFERENCE PATH (DISCRETELY-ACTING ACTUATORS) 2)(222vvmmvEΔ−−=Δ in which mv1=Δ is the change in velocity caused by the unit impulse. Whereas the change in velocity caused by the unit impulse depends only on the magnitude of the impulse, the change in energy depends on the velocity v of the system at the instant of the impulse. Indeed, from above, mvE21−=Δ As v increases, the energy removed from the system by the impulse increases, too. Thus, an impulse applied at an instant of peak velocity, in the opposite direction of the velocity, is fuel-optimal. Now, consider the undamped mass-spring system subject to impulses applied at “local” values of the maximum velocity (every time x = 0). The equation governing the motion of the system and the impulse force are (8 – 6) ),,2,1()sgn()(1nrxIIttIffkxxmrnrrrL&&&=−=−==+∑= where is the magnitude of the r-th impulse. The magnitudes of the impulses are free to vary without loosing the fuel-optimality of the solution. However, the times at which the impulses are applied must be set equal to the instances rIMAE 461: DYNAMICS AND CONTROLSREGULATING THE REFERENCE PATH (DISCRETELY-ACTING ACTUATORS) MAE 461: DYNAMICS AND CONTROLS ),,2,1(, nrtrK= when x(tr) = 0. When the impulses are set to be uniform, the impulse response decays linearly, as shown. The linear decay rate of the displacement is (8 – 7) mIπγ= Notice that the linear decay rate associated with uniform impulses is independent of the system’s natural frequency. Thus, the time required to bring the system to rest is independent of the system’s natural frequency (See Fig. 8 – 1). Dead-band Discretely acting actuators, unlike continuously acting actuators, do not have the ability to apply arbitrarily small forces. This causes a problem during the “end time” of the control action. During the end time of the control action, during which relatively small forces are needed to stop the motion, the discretely-acting actuator is not capable of bringing the system to rest. If the actuator is not turned


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