Robust astatic control system of rotation frequency of the gas turbine engine, страница 2

Finally, after adding integrator (3) to regulator (5) we’ll get equations of the search astatic regulator:

The said requirements to closed system may be carried out for robust astatic regulator synthesis based dual observer [3]. In this case the model of the plant (1) extends by adding integrator to its input

and in stead of (1) is used the following model of extending plant

where input signal of integrator u(t) is the control signal now.

Dynamic regulator for extended plant (11) bases minimal order dual observer. The equations of such regulator write as [3]:

where qÎR^g=n=5 – observer states vector; JÎR^n+m=6 – vector signal, feedbacks by observer states and the plant output;  – (n+m)´m feedback transfer matrix; other matrixes considering structure of  from (11) define as:

where F – some matrix of m´n size. Arcs above mean, that synthesis carries out for extended plant (11).

For matrixes  and F finding are used procedures, analogous (6), (7):

where cost matrixes  and  for implementation of robustness property (2) are chose as

 – some m´m constant matrix.

After removing the integrator (10) from the model of extended control plant (11) to the regulator (12), we’ll get equations of the search astatic regulator:

For comparative estimation of offered methods we’ll show few variants of regulator parameter (9) and (14) setting: for optional chose of the cost matrixes and for choosing these matrixes according to references (8), (13).

For optional chose matrixes Q=diag(10⁵; 1; 1; 1; 1; 10^-5) and Y=I₅ astatic regulator (9) was find (based Luenberger’s observer). Its transfer function model is

Analysis of the plot of returnable difference function in logarithmical scale for closed system (1), (15) shows (fig. 2), that 20lg(g_min)= –4.2db, g_min=0.61, so stability margins of the system (4.1db for amplitude and 35.6^о for phase) are small.

Using algorithm (8) we set , Y=BB^T. For values b=1 and b=200 transfer functions of regulators are:

and the plots of returnable difference function (fig. 2) show that with increasing b (b={1, 200}) the value g_min (g_min={0.71, 0.93}) increases too.

The result of the synthesis astatic regulator (14) based dual observer for optional chose matrixes =diag(10⁵; 1; 1; 1; 10^-5) and =diag(10³; 1; 1; 1; 1; 1) is the transfer function

Closed system (1), (17) such as system (1), (15) has small stability margins (g_min=0.62), but they may be increased using method (13) for cost matrixes choosing. So for , =СС^Т and b=1, b=200 regulators models identical (16) were got:

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Fig. 2. Plots of function 20lg|g(jw)|	Fig. 3. Plots of transitional processes in the closed systems

Plots of transitional processes (fig. 3) in the closed systems show that quality of regulation is acceptable, static accuracy requirements are implemented.

REFERENCES

[1]  O.V. Avdeev, V.U. Chelmadeev, A.I. Golodnyi, U.V. Sadomtsev, “Approximation model of the gas turbine engine at the behaviour speeding with starter”, Problems fine mechanics and control, Saratov: SSTU, 2004, pp. 142-145.

[2]  I.V. Lutsenko, U.V. Sadomtsev, “Robust actatic regulator synthesis based Luenberger's minimal order observer”, Analytical theory automatic control and her addendum, Saratov: SSTU, 2005. pp.113-115.

[3]  I.V. Lutsenko, “Robust actatic regulator synthesis based dual minimal order observer”, Information technology in the science, development and social country, Saratov: Scientisic book, 2005.