12.1: The Transfer Function Matrix

It is evident from (10.20) that the transfer function matrix for the system, which relates the input transform to the output transform when the initial condition is zero, is given by

\[H(z)=C(z I-A)^{-1} B+D \ \tag{12.1}\]

For a multi-input, multi-output (MIMO) system with \(m\) inputs and \(p\) outputs, this results in a \(p \times m\) matrix of rational functions of \(z\). In order to get an idea of the nature of these rational functions, we express the matrix inverse as the adjoint matrix divided by the determinant, as follows:

\[H(z)=\frac{1}{\operatorname{det}(z I-A)} C[\operatorname{adj}(z I-A)] B+D\nonumber\]

The \(n\) determinant \(det(zI - A)\) in the denominator is an \(n\)th-order monic (i.e. coefficient of \(z\) is 1) polynomial in \(z\), known as the characteristic polynomial of \(A\) and denoted by \(a(z)\). The entries of the adjoint matrix (the cofactors) are computed from minors of \((zI - A)\), which are polynomials of degree less than \(n\). Hence the entries of the matrices

\[(z I-A)^{-1}=\frac{1}{\operatorname{det}(z I-A)} \operatorname{adj}(z I-A)\nonumber\]

and

\[H(z)-D=\frac{1}{\operatorname{det}(z I-A)} C \operatorname{adj}(z I-A) B\nonumber\]

are strictly proper, i.e. have numerator degree strictly less than their denominator degree. With the \(D\) term added in, \(H(z)\) becomes proper that is all entries have numerator degree less than or equal to the degree of the denominator. For \(|z| \nearrow \infty, H(z) \rightarrow D\).

The polynomial \(a(z)\) forms the denominators of all the entries of \((zI - A)^{-1}\) and \(H(z)\), except that in some, or even all, of the entries there may be cancellations of common factors that occur between \(a(z)\) and the respective numerators. We shall have a lot more to say later about these cancellations and their relation to the concepts of reachability (or controllability) and observability. To compute the inverse transform of \((zI - A)^{-1}\) (which is the sequence \(A^{k-1}\)) and the inverse transform of \(H(z)\) (which is a matrix sequence whose components are the zero-state unit sample responses from each input to each output), we need to find the inverse transform of rationals whose denominator is \(a(z)\) (apart from any cancellations). The roots of \(a(z)\) - also termed the characteristic roots or natural frequencies of the system, thus play a critical role in determining the nature of the solution. A fuller picture will emerge as we proceed.

Multivariable Poles and Zeros

You are familiar with the definitions of poles, zeros, and their multiplicities for the scalar transfer functions associated with single-input, single-output (SISO) LTI systems. For the case of the \(p \times m\) transfer function matrix \(H(z)\) that describes the zero-state input/output behavior of an \(m\)-input, \(p\)-output LTI system, the definitions of poles and zeros are more subtle. We include some preliminary discussion here, but will leave further elaboration for later in the course.

It is clear what we would want our eventual definitions of MIMO poles and zeros to specialize to in the case where \(H(z)\) is nonzero only in its diagonal positions, because this corresponds to completely decoupled scalar transfer functions. For this diagonal case, we would evidently like to say that the poles of \(H(z)\) are the poles of the individual diagonal entries of \(H(z)\), and similarly for the zeros. For example, given

\[H(z)=\operatorname{diagonal}\left(\frac{z+2}{(z+0.5)^{2}}, \frac{z}{(z+2)(z+0.5)}\right)\nonumber\]

we would say that \(H(z)\) has poles of multiplicity 2 and 1 at \(z = -0.5\), and a pole of multiplicity 1 at \(z = -2\); and that it has zeros of multiplicity 1 at \(-2\), at \(z = 0\), and at \(z = \infty\). Note that in the MIMO case we can have poles and zeros at the same frequency (e.g. those at \(-2\) in the above example), without any cancellation! Also note that a pole or zero is not necessarily characterized by a single multiplicity; we may instead have a set of multiplicity indices (e.g. as needed to describe the pole at \(-0.5\) in the above example). The diagonal case makes clear that we do not want to define a pole or zero location of \(H(z)\) in the general case to be a frequency where all entries of \(H(z)\) respectively have poles or zeros.

For a variety of reasons, the appropriate definition of a pole location is as follows:

• Pole Location: \(H(z)\) has a pole at a frequency \(p_{0}\) if some entry of \(H(z)\) has a pole at \(z=p_{0}\).

The full definition (which we will present later in the course) also shows us how to determine the set of multiplicities associated with each pole frequency. Similarly, it turns out that the appropriate definition of a zero location is as follows:

• Zero Location: \(H(z)\) has a zero at a frequency \(\eta_{0}\) if the rank of \(H(z)\) drops at \(z=\eta_{0}\).

Again, the full definition also permits us to determine the set of multiplicities associated with each zero frequency. The determination of whether or not the rank of \(H(z)\) drops at some value of \(z\) is complicated by the fact that \(H(z)\) may also have a pole at that value of \(z\); however, all of this can be sorted out very nicely.