Lecture
Это окончание невероятной информации про .
...
– the range of variation of the noise from the minimum to the maximum value.
122
Fig. 2.20. Statistical characteristics of e(t)
To solve problems of system analysis and synthesis, it is important to know the answer to the question of whether the object possesses the properties of controllability and observability.
The Control System Toolbox contains the functions: ctrb, which forms the controllability matrix, and obsv, which forms the observability matrix.
In order to use these functions, one must calculate the matrices A, B, C, D using the command
>>[A,B,C,D]=ssdata(sn4s)
A =
−0.2480 −2.0709 2.2220 3.2929 −1.3184 6.3825 1.8425 −16.7833 −15.4959
B =
0.0328 −0.0287 −0.8169
C =
18.0931 0.0789 −0.1539
D = 0.
Calculation of the controllability matrix: >>My=ctrb(A,B)
123
My =
0.0328
−1.7639
40.2663
−0.0287
−5.0683
85.1294
−0.8169
13.2012
−122.7502
>>n=rank(U) n = 3
Calculation of the observability matrix:
>>Mn=obsv(A,C)
Mn =
18.0931 0.0789 −0.1539 −4.5114 −34.9894 43.0912 −34.7007 −667.7434 −901.0786
>>n=rank(N) n = 3
Thus, for the object model under study, the dimension of the state vector, determined by the size of matrices A and C, equals three, and the rank of the observability and controllability matrices is also equal to three; this allows us to conclude that the automation object is fully controllable and observable.
2.3.3.7. Main results of the identification
The identification of the spray dryer was carried out with the aim of obtaining a model of the object and studying its characteristics.
As a result of the experiment performed, a data array was obtained,
consisting of 1000 values of the input parameter of the spray dryer – gas flow rate, m3/h, and 1000 values of the output parameter – exhaust gas temperature, °C, measured at time intervals of 0.08 s.
During the identification process, the following results were obtained:
• the data were processed and converted into a single file containing the necessary information about the input and output parameters of the object, their values and measurement units; graphical dependences were obtained of the change in exhaust gas temperature versus the fuel gas flow rate at the inlet of the spray dryer;
124
•a nonparametric estimation of the initial data was performed to determine the statistical characteristics of the initial data arrays;
•as a result of parametric estimation of the experimental data, carried out with the aim of determining the parameters of a model of a given structure by minimizing a chosen model quality criterion, various structures and forms of spray dryer models were obtained:
−an autoregressive model,
−an autoregressive model with an additional input,
−an autoregressive moving-average model,
−an "input–output" model,
−a Box-Jenkins model,
−a state-variable model;
•checking the adequacy of the models showed that the state-variable model has the best degree of adequacy (89.51%). Values of the autocorrelation function of the process errors were obtained, as well as values of the cross-correlation function between the residual errors and the outputs of the object, together with the corresponding 99% confidence intervals;
•the conversion of the models made it possible to obtain the transfer functions of the spray dryer in discrete and continuous forms, needed for the further analysis and synthesis of the system:
W (z) =
−0.001236z2 +0.006901z +0.06318
,
z3 −1.773z2 +1.101z −0.2554
W (s) =
0.7165s2 + 22.12s +
264.4
;
s3 +17.06s2 +134.4s
+ 279.1
•the analysis carried out of the spray dryer model made it possible to determine the main static and dynamic characteristics of the automation object;
•analysis of the controllability and observability of the object showed that the spray dryer is fully controllable and observable. Control actions can be applied to it in order to transfer
it from an arbitrary initial state to an arbitrary predetermined final state, and for this predetermined control
125
action, it is possible to determine (measure) the initial state of the state-variable vector.
The program for constructing the system model is given in Appendix 2.
Часть 1 2.3. Software Tools for System Identification and Modeling
Часть 2 - 2.3. Software Tools for System Identification and Modeling
Comments