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**
1.1 INTRODUCTION**

In the first section of this chapter, we consider in general the key ideas and concepts concerning
canonical representations of linear multi-input multi-output (MIMO) control systems (also
called *multivariable* control systems) with the help of the *characteristic transfer functions*
(or *characteristic gain functions*) method (MacFarlane and Belletrutti 1970; MacFarlane
*et al*. 1977; MacFarlane and Postlethwaite 1977; Postlethwaite and MacFarlane 1979). We
shall see how, using simple mathematical tools of the theory of matrices and linear algebraic
operators, one can associate a set of *N* so-called *one-dimensional characteristic systems* acting
in the complex space of input and output vector-valued signals along *N* linearly independent
directions (axes of the *canonical basis*) with an *N*-dimensional (i.e. having *N* inputs and
*N* outputs) MIMO system. This enables us to reduce the stability analysis of an interconnected
MIMO system to the stability analysis of *N independent* characteristic systems, and to
formulate the generalized Nyquist criterion. We also consider some notions concerning the
*singular value decomposition* (SVD) used in the next chapter for the performance analysis
of MIMO systems. In the subsequent sections, we focus on the structural and geometrical
features of important classes of MIMO systems - *uniform* and *normal* systems - and derive
canonical representations for their transfer function matrices. In the last section, we discuss
multivariable root loci. That topic, being immediately related to the stability analysis, is also
very significant for the MIMO system design.

**1.2 GENERAL LINEAR SQUARE MIMO SYSTEMS
**

**
1.2.1 Transfer matrices of general MIMO systems**

Consider an *N*-dimensional controllable and observable *square* (that is having the same number
of inputs and outputs) MIMO system, as shown in Figure 1.1. Here, [phi](*s*), *f*(*s*) and
[epsilon](*s*) stand for the Laplace transforms of the *N*-dimensional input, output and error vector
signals [phi](*t*), *f*(*t*) and [epsilon](*t*), respectively (we shall regard them as elements
of some *N*-dimensional complex space *[C.sup.N]*); *W*(*s*) = {*[w.sub.kr]* (*s*)}
denotes the square transfer function matrix of the open-loop system of order *N* x *N* (for simplicity, we
shall call this matrix the *open-loop transfer matrix*) with entries *[w.sub.kr]* (*s*) (*k,r*
= 1, 2, ..., *N*), which are scalar proper rational functions in complex variable *s*. The elements
*[w.sub.kk]*(*s*) on the principal diagonal of *W*(*s*) are the transfer functions of the
*separate* channels, and the nondiagonal elements *[w.sub.kr]* (*s*) (*k* [not equal to]
*r*) are the transfer functions of *cross-connections* from the *r*th channel to the *k*th.

Henceforth, we shall not impose any restrictions on the number *N* of separate channels, i.e.
on the dimension of the MIMO system, and on the *structure* (type) of the matrix *W*(*s*). At the
same time, so as not to encumber the presentation and to concentrate on the primary ideas,
later on, we shall assume that the scalar transfer functions *[w.sub.kr]* (*s*) do not have multiple poles
(we mean each individual transfer function). Also, we shall refer to the general-type MIMO
system of Figure 1.1 as simply the *general* MIMO system (so as not to introduce any ambiguity
concerning the *type* of system, which is conventionally defined in the classical control theory
as the number of pure integrators in the open-loop system transfer function).

The output *f*(*s*) and error [epsilon] (*s*) vectors, where

[epsilon](*s*) = [phi](*s*) - *f* (*s*), (1.1)

are related to the input vector [phi](*s*) by the following operator equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

are the transfer function matrices of the closed-loop MIMO system (further, for short, referred
to as the *closed-loop transfer matrices*) with respect to output and error signals, and *I* is the
unit matrix. The transfer matrices [[PHI].sub.[epsilon]](*s*) and [PHI](*s*) are usually called the
*sensitivity function matrix* and *complementary sensitivity function matrix*.

By straight forward calculation, it is easy to check that [[PHI].sub.[epsilon]](*s*) and [PHI](*s*)
satisfy the relationship:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)

From here, we come to the important conclusion that it is impossible to bring to zero the system
error if the input signal is a sum (mixture) of a reference signal and disturbances, where the
latter may be, for example, the measurement or other noises. Indeed, if the system ideally tracks
the input reference signal, that is if the matrix [[PHI].sub.[epsilon]](*s*) identically equals the zero matrix,
then, due to the superposition principle (Ogata 1970; Kuo 1995), that system also ideally reproduces
at the output the input noise [since, if [[PHI].sub.[epsilon]](*s*) = 0, then the matrix [PHI](*s*) in
Equation (1.5) is equal to the unit matrix *I*]. A certain trade-off may only be achieved provided the input
reference signal and the measurement noise have nonoverlapping (at least, partially) frequency ranges.

**1.2.2 MIMO system zeros and poles
**

**
1.2.2.1 Open-loop MIMO systems**

A single-input single-output (SISO) feedback control system with the open-loop transfer function
*W*(*s*) is depicted in Figure 1.2. That system may be regarded, if *N* = 1, as a specific case of
the MIMO system of Figure 1.1. The transfer function *W*(*s*) is a rational function in complex
variable *s* and can be expressed as a quotient of two polynomials *M*(*s*) and *D*(*s*)
with real coefficients:

*W*(*s*) = *M*(*s*)/*D*(*s*), (1.6)

where the order *m* of *M*(*s*) is equal to or less than the order *n* of *D*(*s*),
that is we consider only physically feasible systems.

From the classical control theory, we know that the *poles [p.sub.i]* of *W*(*s*) are the roots of
the denominator polynomial *D*(*s*), and *zeros [z.sub.i]* are the roots of the numerator polynomial
*M*(*s*) (Ogata 1970; Kuo 1995). In the case of usual SISO systems with *real* parameters, complex
poles and zeros always occur in complex conjugate pairs. Obviously, at the zeros *[z.sub.i]*, the transfer
function *W*(*s*) vanishes and, at the poles *[p.sub.i]*, it tends to infinity (or
1/*W*(*s*) vanishes).

In the multivariable case, the situation is not so simple, and this refers to the MIMO system zeros in particular. This indeed explains the large number of papers in which there are given different definitions and explanations of the MIMO system zeros: from the state-space positions, by means of polynomial matrices and the Smith-McMillan form, etc. (Sain and Schrader 1990; Wonham 1979; Rosenbrock 1970, 1973; Postlethwaite and MacFarlane 1979; Vardulakis 1991).

First, let us consider the *open-loop* MIMO system *poles*. We call any complex number *[p.sub.i]*
the *pole* of the open-loop transfer matrix *W*(*s*) if *[p.sub.i]* is the pole of at least one
of the entries *[w.sub.kr]* (*s*) of the matrix *W*(*s*). In fact, if at least one of the
entries *[w.sub.kr]*(*s*) of *W*(*s*) tends to infinity as *s* [right arrow]
*[p.sub.i]*, then *W*(*s*) tends (strictly speaking, by norm) to infinity. Therefore,
*[p.sub.i]* may be regarded as the pole of *W*(*s*). As a result, we count the set of the poles of
all *[w.sub.kr]* (*s*) as the poles of *W*(*s*). Such a *prima facie* formal definition
of the MIMO system pole seems evident but it leads, as we shall see later, to rather interesting results.

Let the transfer matrix *W*(*s*) be expanded, taking into account the above assumption that
*[w.sub.kr]* (*s*) have no multiple roots, into partial fractions as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)

where *n* is the total number of simple poles of *W*(*s*);

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8)

are the *residue matrices* of *W*(*s*) at the finite poles *[p.sub.i]*; and the constant matrix
*D* is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.9)

Note that the matrix *D* differs from the zero matrix if any of *[w.sub.kr]* (*s*) have the same
degree of the numerator and denominator polynomials.

The *rank [r.sub.i]* of the *i*th pole *[p.sub.i]* is defined as the rank of the residue matrix
*[K.sub.i]*, and it is called the *geometric multiplicity* of that pole. Among all poles of the open-loop
MIMO system, of special interest are those of rank *N*, which are also the poles of *all* the nonzero
elements *[w.sub.kr]*(*s*). In what follows, we shall call such poles the *absolute* poles of the
open-loop MIMO system. It is easy to see that if a complex number *[p.sub.i]* is an absolute pole of the transfer
matrix *W*(*s*), then the latter can be represented as

*W*(*s*) = 1/*s* - *[p.sub.i]* [*W*.sub.1](*s*), (1.10)

where the matrix [*W*.sub.1](*[p.sub.i]*) is nonsingular [that matrix cannot have entries with poles at the
same point *[p.sub.i]* owing to the assumption that *[w.sub.kr]* (*s*) have no multiple poles].

In a certain sense, it is more complicated to introduce the notion of *zero* of the transfer
matrix *W*(*s*), as an arbitrary complex number s that brings any of the transfer functions
*[w.sub.kr]* (*s*) to vanishing, cannot always be regarded as the zero of *W*(*s*). We
introduce the following two definitions:

1. A complex number *[z.sub.i]* is said to be an *absolute* zero of the transfer matrix
*W*(*s*) if it reduces the latter to the zero matrix.

2. A complex number *[z.sub.i]* is said to be a *local* zero of rank *k* of *W*(*s*), if
substituting it into *W*(*s*) makes the latter singular and of rank *N - k*. The local zero of
rank *N* is, evidently, the absolute zero of *W*(*s*).

Let us discuss these statements. It is clear that if a number *[z.sub.i]* is an absolute zero of
*W*(*s*), then we can express that matrix as

*W*(*s*) = (*s - [z.sub.i]*) [*W*.sub.1] (*s*), (1.11)

where [*W*.sub.1](*[z.sub.i]*) differs from the zero matrix and has rank *N*. In other words, the
absolute zero must also be the *common zero* of all the nonzero elements *[w.sub.kr]* (*s*) of
*W*(*s*).

We are not quite ready yet for detailed discussion of the notion of the open-loop MIMO
system local zero, but, as a simple example, consider the following situation. Let *[z.sub.i]* be the
common zero of all elements *[w.sub.kr]* (*s*) of the *k*th row or the *r*th column of
*W*(*s*), i.e. *[w.sub.kr]* (*[z.sub.i]*) = 0 when *k* = *const,r* = 1, 2, ...,
*N*, or when *r* = *const, k* = 1, 2, ..., *N*. Then, obviously, if
the rank of *W*(*s*) is *N* for almost all values of *s* [i.e. the *normal* rank of
*W*(*s*) is *N*], then the matrix *W*([z.sub.i]) will have at least rank *N* - 1, since,
for *s* = *[z.sub.i]*, the elements of the *k*th row or the *r*th column of
*W*(*[z.sub.i]*) are zero. Structurally, the equality to zero of all elements of the *k*th row of
*W*(*s*) means that for *s* = [z.sub.i], both the direct transfer function
*[w.sub.kk]*(*s*) of the *k*th channel and the transfer functions of all cross-connections leading to
the *k*th channel from all the remaining channels become zeros. Analogously, the equality to zero of all elements
of the *r*th column of *W*(*s*) means that for *s* = *[z.sub.i]*, both the direct
transfer function *[w.sub.rr]* (*s*) of the *r*th channel and the transfer functions of all
cross-connections leading from the *r*th channel to all the remaining channels become zeros. This situation may
readily be expanded to the case of local zero of rank *k*. Thus, if, for *s* = *[z.sub.i]*, the
elements of any *k* rows or any *k* columns of *W*(*s*) become zeros, then *[z.sub.i]* is
the local zero of rank *k*. At this point, however, a natural question arises of whether
local zeros of the matrix *W*(*s*) exist which reduce its normal rank but do not have the above
simple explanation and, if such zeros, then what is their number?

A sufficiently definite answer to that question is obtained in the following subsections, and
here we shall try to establish a link between the introduced notions of the open-loop MIMO
system poles and zeros, and the determinant of *W*(*s*). It is easy to see that both the absolute
and local zeros of *W*(*s*) make det*W*(*s*) vanishing, since the determinants of the zero
matrices as well as of the singular matrices identically equal zero. Besides, from the standard rules of
calculating the determinants of matrices (Gantmacher 1964; Bellman 1970), we have that if
some elements of *W*(*s*) tend to infinity, then the determinant det*W*(*s*) also tends to
infinity. In other words, the poles of *W*(*s*) are the poles of det*W*(*s*). Based on this, we
can represent det*W*(*s*) as a quotient of two polynomials in *s*:

det*W*(*s*) = *Z*(*s*)/[*P*(*s*) (1.12)

and call the *zeros* of *W*(*s*) the roots of the equation

*Z*(*s*) = 0 (1.13)

and the *poles* of *W*(*s*) the roots of the equation

*P*(*s*) = 0. (1.14)

Let us denote the degrees of polynomials *Z*(*s*) and *P*(*s*) as *m* and *n*,
respectively, where, in practice, *m* [less than or equal to] *n*. We shall call *Z*(*s*) the
*zeros polynomial* and *P*(*s*) the *poles polynomial*, or the
*characteristic polynomial*, of the open-loop MIMO system.

Strictly speaking, the given heuristic definition of the zeros and poles of *W*(*s*) as the roots
of Equations (1.13) and (1.14), which are obtained from Equation (1.12), is only valid if
the polynomials *Z*(*s*) and *P*(*s*) do not have coincident roots. The rigorous determination
of the zeros and poles polynomials *Z*(*s*) and *P*(*s*) can be accomplished via the
Smith-McMillan canonical form (Kailath 1980). Besides, there exists another important detail concerning the
definition of the MIMO system poles with the help of Equation (1.14), which deserves special attention and will be
discussed in Remark 1.6.

Based on the introduced notions of the open-loop MIMO system absolute poles and zeros,
we can generally write down for the matrix *W*(*s*) the following expression:

*W*(*s*) = [alpha](*s*)/ß(*s*) [W.sub.1](*s*). (1.15)

*(Continues...)*

Excerpted fromLinear and Nonlinear Multivariable Feedback ControlbyOleg GasparyanCopyright © 2007 by John Wiley & Sons, Ltd . Excerpted by permission.

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