Now suppose the temperature of the inflowing liquid changes to a constant lower value ( Figure 5.12). There is then no error signal and consequently no current to the heating element. Initially, take the temperature of the liquid in the bath to be at the set value. Consider the above example in Figure 5.8 of the amplifier as the proportional controller. Proportional controllers have limitations. Thus, when the controller is designed by pole assignment, the sampling interval must not be excessively short to avoid an ill-conditioned matrix in Eq. ![]() From the discussion of pole-zero matching of Section 6.3.2, it can be deduced that the poles of the discretized plant approach the zeros as the sampling interval is reduced (see also Section 12.2.2). We therefore require that the roots of the polynomials P( z) and Q( z) be sufficiently different to avoid an ill-conditioned matrix. The structure of the matrix shows that it will be almost singular if the coefficients of the numerator polynomial P( z) and denominator polynomial Q( z) are almost identical. ![]() The condition number becomes larger as the matrix becomes almost singular. As discussed in Section 9.2.3, the matrix must have a small condition number for the system to be robust with respect to errors in the known parameters. It can be shown that the matrix on the LHS is nonsingular if and only if the polynomials P( z) and Q( z) are coprime, which we assume.
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