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Analytic results for oscillatory systems with extremal dynamic properties

Górecki H., Zaczyk M.

International Journal of Applied Mathematics and Computer Science

2014

Vol. 24, no. 4

771--784

The maximal value of the error is the most important criterion in system design. It is also the most difficult one. For that reason there exist many other criteria. The extreme value of the error represents the attainable accuracy which can be obtained and the corresponding extreme time gives information about how fast the transients are. The extreme values of the error and the corresponding time are treated here as functions of the roots of the characteristic equation. The proposed analytical formulae allow designing systems with prescribed dynamic properties.

Design of the oscillatory systems with the extremal dynamic properties

Górecki H., Zaczyk M.

Bulletin of the Polish Academy of Sciences. Technical Sciences

2014

Vol. 62, nr 2

241--253

In this article the problem of determination of such coefficients a1, a2, ..., an and eigenvalues s1, s2, ..., sn of the characteristic equation which yield required extremal values of the solution x(t) at the extremal value τ of time is solved. The extremal values of x(τ ) and τ are treated as functions of the roots s1, s2, ..., sn. The analytical formulae enable us to design the systems with prescribed dynamic properties. For solution of the problem the properties of symmetrical equations are used. The method is illustrated by an example of the equation of 4-th degree. The regions of the different kinds of the roots: real, with one pair of complex and two pairs of complex roots are illustrated. A practical problem is shown.

Design of systems with extremal dynamic properties

Górecki H., Zaczyk M.

Bulletin of the Polish Academy of Sciences. Technical Sciences

2013

Vol. 61, nr 3

563--567

In this article the problem of determination of coefficients a1, a2, . . . , an of the characteristic equation which yield required extremal values of the solution x(t) at extremal values τ of time is solved. The extremal values of x(t) and τ are treated as functions of the roots s1, s2, . . . , sn. The analytical formulae enable to design the systems with prescribed dynamic properties. The zeros and poles can be located using the known method. The extremal dynamic error x(t) is the most important property of the behaviour of the system. This extremal value of the dynamic error has fundamental role in the chemical industry where for example overrising temperature or pressure can lead to an explosion. A second very important property is the extremal time τ connected with the extremal value of the error. This property is essential in the electroenergetic system, which can be destroyed by the overvoltages waves.