Root Locus Design Example #3


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1 Root Locus Design Example #3 A. Introduction The system represents a linear model for vertical motion of an underwater vehicle at zero forward speed. The vehicle is assumed to have zero pitch and roll angles. The only force acting on the vehicle is the error between weight and buoyancy. The weight of the vehicle must be reduced in order to rise; this is accomplished by expelling water from depth control tanks. In order to dive, additional water is taken into the depth control tanks to make the vehicle heavier. The transfer function for the system is G p (s) = C(s) U(s) = M e s 3 s ( ) = M e s 2 ( ) = 8 7 s 2 (2s ) = 4 7 s 2 (s.5) The output signal is the estimated vertical velocity, and the control signal is water flow rate into and out of the depth control tanks. Performance specifications on the system are: Overshoot in the closedloop step response: P O specified %; Settling time in the closedloop step response: specified 5 seconds; Steadystate error in the closedloop step response: e ss specified =. B. Choosing s as the Dominant ClosedLoop Pole The openloop system is Type 2, so the steadystate error specification is already satisfied. The steadystate error will be zero for both step and ramp inputs from both reference and output disturbance signals. No disturbances acting at the system input are being considered. Assuming that the equations for the standard secondorder system are valid, the overshoot specification imposes the following conditions: () ζ = ln (P O specified /) =.592 (2) π 2 ln 2 (P O specified /) θ = cos (ζ) = 53.76, slope = tan (θ) =.3644 = Im [s ] Re [s ] The settling time specification imposes the following condition: Re [s ] = (3) 4 specified =.267 (4) Combining the constraints in (3) and (4), the following point is chosen to be a dominant closedloop pole: s = Re [s ] ( j slope) =.267 j.364 (5) Figure shows the root locus plot for the uncompensated system, and a zoomed view of the plot. It is clear in the zoomed view that the branches beginning at s = go immediately into the righthalf plane, so compensation is needed just to stabilize the system in addition to making the root locus pass through s. The plant makes an angle at s of
2 2.8 Uncompensated System.4 Zoomed View.6.3 s Fig.. Root locus plots for the uncompensated system. G p (s ) = 2 tan [(.364 ) / (.267 )] tan [(.364 ) / (.267 (.5))] = 2 ( ) = 2 (26.24 ) = = 3.3 so the compensator must provide a phase shift of (6) G c (s ) = 8 ( ) = = (7) at s in order to place that point on the root locus. C. Compensator Designs The following table shows the results of the designs of four different compensators for this system. Each of the designs places a closedloop pole at s = s. The first three designs are single stage; the fourth design is a twostage compensator (s z c ) d zc z c (s p c ) d pc p c K c, 634 4, , 67, 34, 8 P O 38.4% 37.3% 3.8% 8.2% 7.6 s 52.6 s 62.4 s.2 s poles.267 ± j ± j ± j ± j ± j The design procedure is described in detail in Compensator Design to Improve Transient Performance Using Root Locus.
3 3 Figure 2 shows the root locus plots for the four compensated systems. In each case, the branches of the root locus beginning at the origin have been pulled into the lefthalf plane, allowing the closedloop system to be stable for certain gain values. Figure 3 shows zoomed views of the four compensated root locus plots. It is clear in this figure that the point s = s is on the root locus, so that design goal has been met in each case. Figure 4 shows the closedloop step responses for each of the four compensated systems. None of these designs satisfy the overshoot specification, even though s is a closedloop pole in each case. The additional poles and zeros in the compensated system have reduced the validity of the secondorder equations, particularly in terms of overshoot. The values for overshoot and settling time are shown in the table above. More iterations in choosing locations for the compensator pole and zero could be attempted, but it is not clear that significant improvements in performance can be made. Different locations for s can also be selected. In general it is wise to choose a point corresponding to less overshoot than the specified value. D. Derivative on Output Only Configuration Assuming that the compensator pole and zero are in the lefthalf plane, the singlestage phase lead compensator is identical in structure to the real implementation of a ProportionalDerivative (PD) controller, with the following relationships among the parameters: G c (s) = K c (s z c ) (s p c ) = K P K Ds τs K c >, z c <, p c <, z c / p c = α < K P >, K D >, τ > (8) K c = K P τ K D, z c = τ K P K P τ K D, p c = τ K P = K cz c p c = αk c, K D = (K c K P ) τ, τ = p c () The phase lead and PD compensators have the block diagram structures shown in the top two plots of Figure 5. The parallel structure of the PD compensator can be modified so that the derivative action acts only on the output signal, rather than on the error signal. This prevents taking the derivative of a step input. The bottom diagram in Figure 5 illustrates this structure. Figure 6 shows the closedloop step response for this derivative on output only configuration of the PD compensator, using compensator shown in the table. The overshoot for this system is 9.7%, and the settling time is 42.4 seconds, so both specifications have been satisfied. The choice of s allowed the settling time specification to be satisfied, and that coupled with the modified structure of the compensator allowed the overshoot specification to be satisfied. The values of the compensator parameters (for compensator ) in PD format are: K P = , K D = τ = () Not only does the derivative on output only configuration provide much better performance than the original configuration, at least in terms of overshoot, the maximum magnitude of the control signal u(t) is also smaller than with the normal configuration. Figure 7 shows the control signal for each of the four compensator designs. These control signals are the inputs to G p (s) in response to a unit step reference signal, and they are the signals producing the output signals shown in Figure 4. Also shown in the graphs is the control signal using the derivative on output only configuration with (9)
4 4.8 Using Compensator.8 Using Compensator Using Compensator 3 4 Using Compensator Fig. 2. Root locus for 4 leadcompensated systems.
5 5.4 Using Compensator.4 Using Compensator 2.3 s.3 s Using Compensator 3.4 Using Compensator 4.3 s.3 s Fig. 3. Zoomed views of the compensated root locus plots.
6 6.4 Using Compensator.4 Using Compensator PO = 38.4% = 7.6 sec.8.6 PO = 37.3% = 52.6 sec Using Compensator 3.4 Using Compensator PO = 3.8% = 62.4 sec.8.6 PO = 8.2% =.2 sec Fig. 4. Closedloop step responses for the four compensated systems.
7 7 R( s) c ( c ) ( s p ) K s z c U ( s) G ( ) p s C( s) R( s) K P KDs τ s U ( s) G ( ) p s C( s) R( s) K P U( s) G ( ) p s C( s) KDs τ s Fig. 5. Phase lead, PD, and PD with derivative on output only block diagrams.
8 8.4 Using Compensator PO = 9.7% = 42.4 sec.4.2 = Normal Configuration 2 = Derivative on Output Only Fig. 6. Closedloop step response for the modified structure of derivative on output only, using compensator. compensator. It seen to be considerably smaller in maximum magnitude than the others, being almost invisible in the fourth plot. Assuming that G p (s) is strictly proper (a reasonable assumption for most real systems), moving the derivative action out of the forward path prevents an impulse from appearing as part of the control signal u(t) when the reference input is a step function. The value of u(t) at t = is equal to K c for the phase lead or PD compensator in the normal configuration, while it is K p = αk c in the derivative on output only configuration, with α <.
9 9 2 Using Compensator 5 Using Compensator Control Signal 6 4 Control Signal 2 PD DOO 2 PD DOO x 4 Using Compensator 3 4 x 5 Using Compensator Control Signal Control Signal PD DOO Fig. 7. Control signals for the 4 phase lead compensators and the derivative on output only compensator.
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