Controllability and Observability of the DiscreteTime Structural Model
Consider now a structure in modal coordinates. Similar to the continuous-time grammians the discrete-time grammians in modal coordinates are diagonally dominant,
W0 - diag(W,1, W02,..., W0n), where Wci and W0i are 2 x 2 blocks, such that Wci = wCiI2 and W0i = w0iI2, see [98], where
4Çirni cofAt cofAt and
In the above equations Bmi is the ith block of Bm in modal coordinates, and Cmi is the ith block of Cm in modal coordinates, where Cm = [CmqQ_1 Cmv], see (2.42)
for Zs0. In the latter equation Q is the diagonal matrix of natural frequencies, Cmq is the matrix of displacement measurements, and Cmv is the matrix of rate measurements. Also wcicont and woicont denote the continuous-time controllability and observability grammians, respectively, cf. (4.45).
Note that the discrete-time controllability grammian deviates from the continuous-time controllability grammian by factor 2(1 C0^'At) , while the discrete-
time observability grammian deviates from the continuous-time observability grammian by factor 1/At. Note also that the discrete-time grammians do not converge to the continuous-time grammians, but satisfy the following conditions:
llm T7 = Wci cont and llm WoiAt = Woi
AtAt At^Q
oi cont which Is consistent with the Moore result; see [109] and Subsection 4.1.3 of this chapter.
The Hankel singular values are the square roots of the eigenvalues of the
grammian products, T = Xit2(WcWo). The approximate values of the Hankel singular values can be obtained from the approximate values of the grammians,
Note that the discrete-time Hankel singular values differ from the continuous-time values by a coefficient ki, where
mtAt
The plot of ki(«¿At) is shown in Fig. 4.5; this shows that for small sampling time, discrete- and continuous-time Hankel singular values are almost identical.
- Figure 4.5. Plot of ki versus ©¡At: For fast sampling (i.e., small At) the k value is 1.
Note that if the sampling rate is high enough (or the sampling time small enough), the discrete-time Hankel singular values are very close to the continuous-time Hankel singular values. For example, if a>tAt < 0.6 the difference is less than 3%, if a>tAt < 0.5 the difference is less than 2%, and if m^At < 0.35 the difference is less than 1%. Note also that for a given sampling time the discrete-time Hankel singular values, corresponding to the lowest natural frequencies, are closer to the continuous-time Hankel singular values than the Hankel singular values corresponding to the higher natural frequencies.
Example 4.8. Consider the discrete-time simple system as in Example 2.9. For this system k1 = k2 = k3 = 3, k4 = 0, and m1 = m2 = m3 = 1, while damping is proportional to the stiffness matrix, D = 0.01^. Determine its Hankel singular values for sampling time At = 0.7 s, and for At = 0.02 s, and compare with the continuous-time Hankel singular values.
The Hankel singular values for the continuous- and discrete-time structures with sampling times At = 0.7 s and At = 0.02 s are given in Table 4.1.
|
Continuous time |
Discrete time |
Discrete time | |
|
At = 0.7 s |
At = 0.02 s | ||
|
Mode 1 |
20.342 |
20.138 |
20.342 |
|
20.340 |
20.009 |
20.340 | |
|
Mode 2 |
4.677 |
4.324 |
4.677 |
|
4.671 |
4.225 |
4.670 | |
|
Mode 3 |
0.991 |
0.848 |
0.991 |
|
0.986 |
0.785 |
0.986 |
The table shows that for the sampling time At = 0.7 s the discrete-time Hankel singular values are smaller than the continuous-time values, especially for the third mode. In order to explain this, note that the natural frequencies are a>1 = 0.771 rad/s, m2 = 2.160 rad/s, and a>3 = 3.121 rad/s. The sampling time must satisfy condition (3.52) for each mode. For the first mode njm1 = 4.075, for the second mode njrn2 = 1.454, and for the third mode = 1.007. The sampling time satisfies the condition (3.52). However, for this sampling time, one obtains m1At = 0.540, m2At = 1.512, and m3At = 2.185. It is shown in Fig. 4.5 that the discrete-time reduction of the Hankel singular values with respect to continuous-time Hankel singular values is significant, especially for the third mode.
This is changed for the sampling time At = 0.02 s. In this case one obtains m1At = 0.015, m2At = 0.043, and m3At = 0.062. One can see from Fig. 4.5 that for these values of a>iAt the discrete-time Hankel singular values are almost equal to the continuous-time Hankel singular values.
Next, we verify the accuracy of the approximate relationship (4.83) between discrete- and continuous-time Hankel singular values. The accuracy is expressed with the coefficient ki, (4.84). The Hankel singular values were computed for different sampling times, and compared with the continuous-time Hankel singular values. Their ratio determines the coefficient ki . The plot of ki obtained for all three modes and the plot of the approximate coefficient from (4.84) are shown in Fig. 4.6. The plot shows that the approximate curve and the actual curves are close, except for ®i At very close to n.
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