Studies of robust-model controller design

Application to Benchmark structural model

4.9 Studies of robust-model controller design

4.9.1 Overview

The robust-controller design developed in Chapter 3 explicitly considers the un- certainty in the models of the benchmark system in the selection of the "optimal"

controller. The probability models for the parametric uncertainty of the input filter are used to compute the probable robust performance that serves as the controller objective function. Often, the calculated performance is not very sensitive to the particular form of the probability distributions for the uncertain parameters, pro- vided the distributions exhibit the same general behavior, i.e., provided the most probable values and the local shape of the distribution about that peak are similar among the distributions under consideration. However, the performance does de- pend on the most probable value of the parameter and its variance. The sensitivity of the controller design to the probability models and their parameters is the subject

of Section 4.9.2.

4.9.2 Robust controller design for acceleration feedback

Sensitivity to peak values for uncertain-variable probability density func- tions.

The variations in the optimal uncertain-model control design with the value for the most-probable parameters are explored in this section. The probability models for the uncertain parameters are chosen to be log-normal distributions for the Kanai- Tajimi filter frequency, damping ratio, and input magnitude, as discussed previously in Section 4.4. A standard log-normal distribution (Benjamin and Cornell1970) is scaled by the most-probable values of each parameter, and this standard distribution is described by

( 4.23) ((}) 1 1 log (} - ^/-Llog o

[ ( )2]

p - exp - -

- (}(Jlogov'2if 2 ()logO '

where ^/-Llog o represents the mean of the natural logarithm of(} and ^()logo is the log of the variance of(}, termed the "log-variance." Throughout this section, each of the parameter probability distributions has a log-variance of 0.2.

The variations in the optimal controller gains with respect to the most probable values for the uncertain parameters are summarized in Table 4.14 (using wb

=

307r rad/sec). In this table, the most-probable parameter values are indicated with a "0" superscript, while the parameter values that serve as the design points for the asymptotic approximation to (2.21) (see Section 2.5.4) are indicated with an

"*·" The set of values for

wg

is, in radjsec, {37.3, 50.0, 75.0, 100.0}, (₉ is chosen from {0.30, 0.50, 0.60}, and (Jx₉ is kept constant at 0.12 g. For each pair of rows in the table, the first row displays the most-probable parameter values, while the second row contains the parameter values found as the "design points," that is, the parameters that maximize the integrand of (2.21).

The robust performance is also calculated for the uncontrolled system for the various most-probable parameter values described above. The results of this study

are shown in Table 4.15. For the robust performance calculations in Table 4.15, recall that failure probability performance measure is an approximation to the failure probability of the system, and is obtained by summing the failure probabilities for each failure possibility.

One comment on the robust controller optimization is that the failure probability performance for the robust example is often quite flat with respect to the controller parameters. This sometimes leads to difficulties in the optimization, requiring a large number of iterations for convergence. This behavior is also seen in the figures in Chapter 3, which plot the failure probability as a function of a single controller gain. The flatness in the objective function with respect to the controller gains is a consequence of the nonlinear failure-probability-based objective function. Since most of the controllers in this range achieve nearly equal performance, any one that provides satisfactory performance could be used.

Sensitivity to form of probability models

Alternative probability models could be considered for the parameter uncertainty besides log-normally distributed uncertainty. Other popular models would include the

x

²-distributionor a normal (or truncated normal) one. For each PDF considered, the probability distribution parameters can be selected so the different distributions exhibit very similar behavior around their most probable values. For example, in Figure 4.24 the PDFs and CDFs of a normal (solid line), log-normal (dashed line), and

x

² (dotted line) distribution appear quite similar when the distribution parameters are chosen appropriately. Hence, since the asymptotic approximation depends largely on the shape of the probability density function at its maximum point (as this peak typically determines the maximum of the integrand from the total probable performance), the effects at the tails from the different distributions are assumed to be negligible, at least compared within the accuracy of the asymptotic approximation to evaluate the performance.

Table 4.14 Performance, "design" point for asymptotic integration, and controller parameters for uncertain-model controller, ^wb

=

207r radfsec and ^Wdf = 601r radfsec.

Model parameters Controller parameters

CPU time ^,u * _{a~ a~}

robust Pf w₉, w₉

(g, (;

^{Xgl Xg} kt k2 k3

(sec) (radjsec) (g)

1.68x10³ 1.17x1o-¹ 37.3 0.300 0.120

0.110 0.225 0.103 36.7 0.261 0.145

1.67x 10³ 2.56x1o-² 37.3 0.500 0.120

0.0914 0.144 0.134 36.5 0.409 0.166

1.97x 10³ 1.30x 10-² 37.3 0.600 0.120

II

o.o924 0.144 0.134 36.2 0.480 0.174

1.10x10³ 5.47x1o-² 50.0 0.300 0.120

0.103 0.210 0.1031 39.7 0.260 0.149

2.07x10³ 1.29x1o-² 50.0 0.500 0.120

0.0826 0.155 0.127 40.8 0.408 0.168

1.95x10³ 6.54x1o-³ 50.0 0.600 0.120

0.0809 0.149 0.130 41.1 0.479 0.176

1.83x10³ 1.79x1o-³ 75.0 0.300 0.120

0.107 0.156 0.110 45.0 0.271 0.163

1.80x 10³ 8.76x1o-⁴ 75.0 0.500 0.120

0.0857 0.145 0.125 49.4 0.429 0.183

2.43x10³ 5.21x10-⁴ 75.0 0.600 0.120

0.0774 0.147 0.120 50.8 0.500 0.191

2.65x10³ 6.28x1o-⁵ 100.0 0.300 0.120

0.0669 0.157 0.115 53.4 0.292 0.186

2.37x10³ 7.77x1o-⁵ 100.0 0.500 0.120

0.0849 0.134 0.125 63.7 0.462 0.209

1.74x10³ 5.77x1o-⁵ 100.0 0.600 0.120

0.0895 0.130 0.127 66.9 0.534 0.215

Table 4.15 Robust performance and "design" point for asymptotic in- tegration for uncontrolled system.

Model parameters

I ^"PJ'' I ^{w~, w;} I

^{(o (*}

I

^O"¥u'

O"~g

(rad/sec) ⁹' 9 (g) 8.98x10-¹ 37.3 0.300

I

^0.120

36.5 0.278 0.124

1.0 37.3 0.500

I

^0.120

35.9 0.477 0.117 7.95x10-¹

I

37.3 0.600

I

^0.120

35.8 0.572 0.117 7.60x1o-¹

II

50.0 0.300 0.120 42.8 0.280 0.127 6.96x1o-¹

II

50.0 0.500 0.120 47.3 0.477 0.118 5.58x 10-¹

II

50.0 0.600 0.120 46.8 0.565 0.120 2.15x10-¹

II

75.0 0.300 0.120 61.6 0.293 0.131 2.24x1o-¹

II

75.0 0.120

I

63.1 0.133

75.0 0.120

I

63.0 0.136

5.53x10-² 100.0 75.0 6.67x10-² 100.0 77.8 5.76x10-² 100.0 77.8

0.045.---,---.,.---, 0.04

0.035 _i

!A

!

! 0.03 _I il

0.025 0.02 0.015 0.01 0.005

50 1QO

ro₉ (rad/sec) 150

0.9 0.8

f~

l

j

0.7 0.6 0.5 0.4 0.3 0.2

0.1 o~~~~--~--~

J

0 50 100

ro₉ (rad/sec) 150

Figure 4.24 Normal, log-normal, and

x

² probability density functions and cumulative distribution functions for w₉•

4.9.3 Robust control design including roll-off frequency as a con- troller parameter

Similarly to the nominal-model controller design approach, the roll-off frequency of the frequency-weighting filter is included as a controller parameter. The results of the controller optimization including this parameter are shown in Table 4.16.