liminated using symbolic Gaussian elimination. When numerical data are available, it can be used to simplify the resulting expres- sions by eliminating irrelevant terms. 4.3 AUTOMATED SYMBOLIC LPTV SYSTEM ANALYSIS Xout=X1 Gliin(s) G1,1(s) Figure 4.23: Graphs representing a simple feedback system (top) and its open-loop HTMs (bottom). Each branch represents a (possibly time-varying) building block. The dashed lines mark the parts in common to both open-loop HTMs. dashed part is in common to both graphs and its corresponding HTM can be represented using auxiliary symbols in order to make further expressions better readable. 4.3.5.2 Eliminating the loop variables Next, the loop variables Xk(s) from (4.80) are eliminated using symbolic Gaussian elimination. For X1 (s), this implies that we solve (4.81) and substitute the result back into the equations for X2(s),X3(s), in (4.80). The result is a reduced system with one equation and one variable removed. We proceed in a likewise manner to eliminate the other loop variables until we end up with the desired input/output relation. The inversion of (I-G11(s)), implied in equation (4.81), is performed using the power series expansion (3.80) discussed in section 3.4.3.2. This expansion is truncated after the first R terms. Here, R is provided by the user, unless numerical data are available. In the latter case, its value can be determined based upon error control, as is done in symbolic approximation techniques used for analyzing LTI networks [Fern98, Gie91, Gie94, Wamb98b]. During computations, a number of simple measures can be taken to avoid wasting time on unimportant contributions. Firstly, we can limit the number of tones, i.e. the number of HTM elements, taken into account during computations. For each LPTV subsystem and interconnection of subsystems, we assume that the HTM elements Go,k(s) 22 0 for k > K with K a parameter that controls the SymbolicHTM algorithm. This assump- tion allows us to skip a detailed computation of these HTM elements. Secondly, in performing