ABSTRACT
The book deals with the IO and ISO mathematical models of the time-invariant linear continuous-time systems. The study presented for them can be developed to their Input-Internal dynamics-Output (IIO) mathematical models introduced in [148]. In order to be clear and precise in using di¤erent notions (e.g., system regime,
system desired regime, nominal control), we have presented their de…nitions and the procedures on how to determine them. These issues are very simple, but constitute the basis of the control systems theory, which is often ignored in the literature on the …rst control systems course. Every dynamic physical system transfers and transmits simultaneously ac-
tions and in‡uences of both the input vector and all initial conditions. The system transfer function matrix G(s) does not and cannot express and/or describe how the linear time-invariant (continuous-time, discrete-time or hybrid) system transforms the actions and in‡uence of nonzero initial conditions on the system into its internal and output behavior. This lack of G(s) is the consequence of its de…nition and validity only for zero initial conditions. Mathematically considered, the obstacle to treating the transmission of the
in‡uence of nonzero initial conditions is the double sum in initial conditions of Laplace transform of the n-th order Input-Output (IO) linear system di¤erential equation. This obstacle has been removed by solving this mathematical problem of how to put the double sum in the equivalent form to G(s)I(s) that characterizes the product of the system transfer function matrix G(s) and Laplace transform I(s) of the input vector I(t). Once this has been solved, we have become able to determine the complex domain description independent of the input vector and of all initial conditions, which completely expresses and describes how the system transfers, transmits and transforms in‡uences of both the input vector and all initial conditions on the system internal and output behavior. Such description is the system full transfer function matrix F (s). It has the same characteristics as the system transfer function matrix G(s): - the independence of the input vector, - the independence of all initial conditions,
- the invariance relative to the input vector and to all initial conditions, - the system order, dimension, structure and parameters completely de-
termine F (s). After presenting the de…nitions of the system full transfer function matrix
F (s) and of its submatrices for every type of the system mathematical models, we presented and proved how we can easily determine them by using the same mathematical knowledge that we apply to determine the system transfer function matrix G(s). In this context we discovered the existence of row (non)degenerate, column (non)degenerate and (non)degenerate matrix functions. We should use the full transfer function matrix FIOCS"(s) of the control
system rather than only its transfer function matrix GIOCSd(s) or GIOCS"yd(s) (Theorem 306, Section 10.2, and Theorem 310, Section 10.3). The latter are insu¢ cient for the analysis or synthesis of the control system from the tracking point of view, while the former is both necessary and su¢ cient. Moreover, the system full transfer function matrix F (s) expresses channeling
information and system structure through its submatrices. We explained the physical meaning of F (s). This book discovers how the use of the system full transfer function matrix
F (s) permits new results on the pole-zero cancellation, which is permissible if, and only if, it is possible in the system full transfer function matrix F (s), or at least in its appropriate submatrix that links the system reaction with its cause.
