### Table 5. Full state and output feedback control laws with sensor numbers.

1998

"... In PAGE 20: ... The ideal case is to eliminate the acoustic sensors entirely and use the model with coupling along with structural data to reconstruct the acoustic state. For the simulations presented here, three sensor con gurations were considered as sum- marized in Table5 . In all cases, the number of sensors measuring the potential was taken to be N = 0 in (2.... In PAGE 22: ... In comparing the rms values and time plots of the three compensators, it is noted that the performance of Compensator I with measurements of pressure, displacement and velocity is only 1-2 dB better than that of Compensators II and III. Recall from Table5 that Compensator III employs only 5 velocity sensors for the actual state reconstruction. The pre- computed gains and coupled model provide the remaining information required for accurate state estimation and control computation.... In PAGE 25: ... Hence while signi cant attenuation is achieved throughout most of the cavity, optimization issues concerning patch number and orientation should be investigated to attain global attenuation. Similar results obtained with Compensators I and III described in Table5 are plotted in Figure 9. The small patch having radius R=12 was employed as an actuator and rms sound pressure levels along axis 2 are reported in the gure.... In PAGE 27: ... The example we consider in this section reinforces the tenet held by many acousticians that this strategy is not e ective in general and should be used only for certain exogenous frequencies (see, for example, [13, 20]). It also illustrates the bene ts of utilizing a compensator for the coupled system which employs only structural sensors (see Compensator III of Table5 ) rather than a purely structural controller. For the structural acoustic system in this work, a purely structural controller would be designed for the discretized plate model quot; KP 0 0 MP # quot; _ #(t) #(t) # = quot; 0 KP ?KP ?CP # quot; #(t) _ #(t) # + quot; 0^ B # u(t) + quot; 0 ^ g(t) # + ^ D (t) where again, #(t) contains the generalized Fourier coe cients for displacement and MP ; KP and CP are the mass, sti ness and damping matrices for the plate (see Section 2).... ..."

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### Table 2 Static Gain controlled systems with output feedback

"... In PAGE 5: ... These are shown in Table 3. The static gain EA design that was obtained is shown in the left column of Table2 , and the time response of the system to an unit disturbance in sideslip angle is shown in Figure 2. This is a fairly acceptable time response in terms of performance.... In PAGE 5: ... Now consider the situation where sideslip angle is no longer available as a measurement. The design requirements are considerably more difficult to attain, and the right-hand column of Table2 shows that this closed-loop system has bad decoupling (a higher condition number) and low robustness. Figure 3 is an indication of the deterioration in performance, where the same regulation task as with the 4-output system is attempted.... ..."

### Table 4.1: Static output feedback control examples

### Table 3b: Example 1: Uncertainties in amplitude only: { Comparison of H1 controller with simulated full state feedback and output feedback with compensator rms voltage for the time interval [ 3 75; 10 75].

"... In PAGE 21: ...1 66.8 Table3 : Example 1: Uncertainties in amplitude only: { Comparison of H1 controller with simulated full state feedback, output feedback with compensator, and open loop pressure levels at cavity point (0:3; 0:1) for the time interval [ 3 75; 10 75]. Considering rst the open loop case, it is noted that increasing values of lead to monotone increasing rms sound pressure levels in accordance with the monotone increasing levels in the driving force.... In PAGE 21: ... The former case exhibits optimal reduction when no noise is present in the force whereas the latter exhibits monotone decreasing sound pressure levels as is increased. Finally, we found that the rms levels of the controlling voltages in both cases decrease monotonically as is increased as depicted in Table3 b below. (rms) Input Voltage - (volts), = 0:00... In PAGE 22: ... The previously discussed H1 full state feedback control results depend upon y and r being balanced in the sense that they are generated by the same forcing function (in the form F and F, respectively). Deviation from this balance by adding either positive or negative error to the amplitude leads to degradation of control authority as noted in the full state feedback column of Table3 . In this example, the inconsistency between the measured force driving r and the actual force driving y leads to higher sound pressure levels when either positive and negative amplitude errors are present.... ..."

### Table 2: Real zeros ki and inertias i for the reactor. H(k) is equal to nN = 6 is I1 =] ? 22; ?20[, therefore the three plants are simultaneously stabilizable by a static output feedback u = ky for any value of k such that ?22 lt; k lt; ?20:

"... In PAGE 5: ... During the life of the reactor, some representative values of E are 20, 25 and 30. Assuming that only y = x1 is available for feedback, the N = 3 linearized systems of order n = 2 to be simultaneously stabilized are given by p1(s)=q1(s) = (0:5 ? 0:25s)=(11 ? 5s + s2) p2(s)=q2(s) = (?0:5 ? 0:25s)=(?2:25 ? 2:25s + s2) p3(s)=q3(s) = (?0:5 ? 0:25s)=(?3:5 ? 3:5s + s2): Applying the algorithm described in the previous section, we obtain the values of ki and i given in Table2 . The only interval for which the number of positive eigenvalues of... ..."

### Table 1: Real zeros ki and inertias i for the aircraft. H(k) is equal to nN = 12 is the interval I0 therefore the four plants are simultaneously stabilizable by a static output feedback u = ky for any nite value of k such that k lt; ?0:5764

"... In PAGE 4: ...or feedback is the second state component, i.e. y = x2: The values of the parameters a11; a12; a13; a21; a22; a23 and b1 at each operation point are given in [Howitt and Luus, 1991]. The N = 4 corresponding scalar transfer functions of order n = 3 are given by p1(s)=q1(s) = (?351:1 ? 367:6s)=(?113:0 + 51:46s + 31:84s2 + s3) p2(s)=q2(s) = (?676:5 ? 346:6s)=(?31:50 + 38:53s + 31:32s2 + s3) p3(s)=q3(s) = (?455:4 ? 978:4s)=(?262:5 + 84:85s + 33:12s2 + s3) p4(s)=q4(s) = (?538:7 ? 790:3s)=(576:7 + 71:46s + 31:74s2 + s3): Applying the algorithm described in the previous section, we obtain the values of ki and i given in Table1 . The only interval for which the number of positive eigenvalues of... ..."

### Table 7: Parameter Matrix (Output Feedback with No Probing Noise) 2

"... In PAGE 13: ...irst, assume w = 0, i.e., no probing noise in the feedback channel. The identi ed parameter matrix and loss function matrix are shown in Table7 and Table 8. The loss function of the second order feedback (backward) model is zero.... ..."

### Table 8: Loss Function Matrix (Output Feedback With no Probing Noise) 2

### Table 2: Experiments with augmented matrices ~ A; ~ B; ~ C, where ~ A is 2n 2n, formed with random A; B; C, n = 5, nu = 2, ny = 2. Algorithm 1 always nds a static controller, as predicted by the generic stabilizability (by a full-order controller) of (A; B; C).

1997

"... In PAGE 6: ... We then know that the augmented triple is (generically) stabilizable by a static output-feedback controller. Table2 shows that the algorithm was again... ..."

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### Table 6.2: Time interval [3/75,10/75]. The improvement seen in Table 6.2 when compared to Table 6.1 can be attributed to the fact that the compensator has an initial learning period before converging to the state of the plant. The time trajectory of pressure at c2 = (0:30; 0:10) for 0% and 5% disturbance level is shown in Figure 3 for the uncontrolled case, and the state and output feedback cases. The controlled cases use full state feedback controller and a controller based on the state estimator (the state observer uses 11 sensors/microphones: 3 displacement and 3 velocity sensors placed at = 0:1; 0:3; 0:5 and 5 microphones placed at points p1; : : :; p5, as shown in Figure 2). By comparing the results of Figure 3 (and Figure 6 of [6]) we can see that even with a 5% disturbance the compensator does perform satisfactorily. One should keep in mind that even though the full state feedback performs somewhat better than the output feedback (but both perform much better than the open loop case), the compensator does not require full information on the plant(which is not implementable) but rather 11 measurements of the beam velocity and displacement and the cavity pressure.

1996

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