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4.3.6  Lightly damped flexible plants; collocated and non-collocated control

Some of the poles and zeros of flexible plants' transfer functions are only lightly damped, with the damping coefficients as small as 1% and even 0.1%. The loop gain responses of the loops with such plants exhibit sharp peaks and notches as in the examples in Figs. 4.32 and 4.34.

For the closed-loop system to be stable, the modes should be gain- or phase-stabilized. The modes which need attention are the modes that are not already gain-stabilized, i.e., those resulting in the loop gain falling within the interval from - x to x1 , as the modes 2 and 3 shown in Fig. 4.34(a). Increasing the modal damping can reduce the value of the modal peak and notch and gain-stabilize the mode. Otherwise, the mode needs to be phase-stabilized as shown in Fig. 4.34(b).

 
(a) (b)

Fig. 4.34  Modes (a) on the loop gain response and
(b) on the L-plane Nyquist diagram

To phase-stabilize mode 3, a phase lag might be added to the loop to center the mode at the phase lag of -360°, so the resonance loop will be kept away from the critical points -180° and -540° as illustrated in Fig. 4.34. The required phase lag can be obtained by introducing a low-pass filter in the loop (this is a better solution than adding n.p. lag since the filter will provide the additional benefit of attenuating modes of higher frequencies).

If the plant's phase uncertainty at the frequency of the flexible mode is large, phase-stabilization is not feasible and the mode must be gain-stabilized. A typical case of the gain stabilization of a structural mode is shown in Fig. 4.35. The Bode step allows a steep roll-off at frequencies beyond the step. Gain-stabilization of the mode reduces the feedback in the functional band. The average loop gain at the frequency of the mode must be no higher than (20 log Q + x) dB. Damping of the high-frequency mode would allow increased feedback bandwidth.

Fig. 4.35  Gain-stabilization
of a high-frequency mode

Fig. 4.36  Mechanical plant with flexible
appendages and the sensor collocated with
the actuator

Next, consider flexible plant model for translational motion shown in Fig. 4.36, consisting of rigid bodies with masses M1, M2, ... connected with springs. The actuator applies a force to the first body. The motion sensor S1 is collocated with the actuator and senses the motion of the first body, so the control is called collocated.

If the sensors are velocity sensors, the transfer function from the actuator force to the sensor S1 is, in fact, the plant driving point impedance (or mobility). The driving point impedance of a passive system is positive real (see Appendix 3), and its phase belongs to the interval [-90°, 90°]. The impedance function of a lossless plant has purely imaginary poles and zeros which alternate along the frequency axis, and the phase of the plant transfer function alternates between 90° and -90°. Flexible plants are discussed in more detail in Chapter 7.

   

Example 1. The plant of the control system having the loop response shown in Fig. 4.35 is non-collocated since the plant has a pole at zero frequency, and then the mode's pole and zero follow, i.e., a pole follows a pole. If the mode pole-zero order were reversed, the control would be collocated.

     
   

Example 2. A collocated force-to-velocity translational transfer function of a body with two flexible appendages resonating respectively at 3.32 and 7.35 rad/sec and negligible damping is

.

The masses of the appendages in this example are approximately 10 times smaller than the mass of the main body which explains why the poles are rather close to the zeros. The gain and phase responses are plotted in Fig. 4.36 with

    n = conv([1 0 10],[1 0 50]);
    d = conv([1 0],[1 0 11]); d = conv(d,[1 0 54]);
    w = logspace(0, 1, 1000); bode(n,d,w)

When the sensor is a position sensor or an accelerometer, an extra integrator or differentiator should be added to this transfer function which changes the slope respectively by - 6 dB/oct or 6 dB/oct.

When the control is collocated, a flexible appendage adds a zero-pole pair to the loop response as shown in Figs. 4.34 (two lower-frequency modes) and 4.37. The modes do not destabilize the system since the phase lag only decreases by 180° between the added zero-pole pair. (However, the mode reduces the integral of phase, and therefore the average gain slope and the available feedback somewhat decrease.)

Fig. 4.37  Bode diagram for a collocated mechanical plant
with two flexible appendages

Placing the sensor on any other body makes the control non-collocated. In this case the spring connecting the bodies introduces an extra unwelcome phase lag into the loop.

Thus, the sensor location defines whether the control is collocated or non-collocated. The trade-off associated with where to place the sensor, on M1, on M2 , or on M3 in Fig. 4.36, is very often encountered in practice. The sensor must be placed within the power train someplace from the actuator to the tip of the tool or other object of control. When the sensor is placed closer to the actuator, i.e., on M1, the feedback bandwidth can be widened but it is the position of the first body that is controlled. The flexibility between the bodies will introduce the error in the position of the tip. However, when the sensor is placed on the tool, i.e., on M2 , or on the tip of the tool, on M3, then we are controlling exactly the variable that needs to be controlled, but the feedback bandwidth must be reduced as shown in Fig. 4.35. The best results can be obtained by combining these sensors.

The collocated and non-collocated control will be further discussed in Sections 7.8.3 and 7.8.4.

     
   

Example 3. A Nyquist diagram for a flexible plant (Saturn V controller) is given in Appendix 13, Fig. A13.26. Many flexible modes are seen on this diagram. While the controller was being designed, serious discussions were going on about where to place the gyros: closer to the location that needed to be better controlled (in this case the control would be non-collocated), or closer to the engines where the control would be collocated and would be easier to implement. It was eventually decided to play it safe and place the gyros closer to the engines.

Most of the modes are gain-stabilized, only the large mode seen in the upper right sector of the diagram is phase-stabilized. The loop phase lag at the mode's central frequency is 315°. The plant parameter variation must not reduce this phase lag by more than 45° or else the stability margins will be violated. The control at these frequencies is analog. A digital controller with insufficiently high sampling frequency would cause large phase uncertainty (as will be discussed in Section 5.10.7) and would make the system unstable.

     
   

Example 4. In pneumatic systems, compressibility of the air in the cylinder of an actuator creates a series "spring" between the actuator and the plant. This makes the control non-collocated and reduces the available feedback bandwidth.

     
   

Example 5. In Example 1 of Section 4.2.5, the slosh modes are non-collocated. Because of this, gain-stabilization of the modes is required and is implemented as shown in Fig. 4.14.

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