Minimal Realisation

Consider an autonomous control system

= Ax + Bu, y = Cx, (1.42)

characterized by the matrix triple (A, B, C). By (1.15) and Theorem 1.3, the output resulting from the input u and zero initial state is

y(t) = CeA(t— T1 Bu(r)dr

f

Definition The matrix

T(t) -4 C(t)1.(t, 1-)B(T) = Ce At B (1.43)

is called the weighting pattern of the system.

Two input systems which have the same weighting pattern, have the same input-output behaviour.

This may be true for systems with different state dimensions. When (1.43) holds, we say that the system (1.42) is a realisation of the weighting pattern T. The problem arises of finding alternative realisations of T, having the lowest possible state dimension. Such a realisation is called a minimal realisation. The following theorem characterizes a minimal realisation.

Theorem 1.9 The system (1.42) is a minimal realisation of the weighting pattern (1.43) if and only if the system is both completely controllable and completely observable.That is, the system

= A(t)x(t) + B(t)u(t); y(t) = C(t)x(t),

is a minimal realisation of T(t, T) = C(t).1)(t, r)B(r) iff W(0, T) and M(0, T) are both positive-definite.

Proof Sufficiency: We shall show that if the realisation is not minimal, then the matrices W(0, T) and M(0, T) cannot both be positive-definite. We assume that the given realisation is not minimal and that

i(t) = F(t)z(t) + G(t)u(t); y(t) = H(t)z(t),

is a lower dimensional realisation. For definiteness, we assume that dim x = n and dim z = v < n. It can be shown that the z-realisation of T gives rise to a decomposition

T(t, -r) = xp . (t)r . (T),

with ** having v columns and r* having v rows. Letting '(t, 0) = C(t)*(t, 0) and

r(T, o)

(t, o)r(T,

o) (or*

Pre- and post-multiplication, by V(t, 0) and ri 0) respectively, yields V(t,O)T(t, o)r(T, o)v(T, V(t, o) ,P - (t)r - (T)r/ (y, 0).

Integrating over the square 0 < t < T, 0 < T < T gives

M(0 ,

Now, from the dimension of** and r*, we know that the right side has rank less than or equal to v < n. Hence M(0, T) and W(0, T) cannot both be positive-definite.

Necessity: We let H(t) = C(t)*(t, T) and G(t) *(0, t)B(t). Since the matrices M(0, T) and W(0, T) are symmetric and positive semi-definite, it can be shown that there exist nonsingular matrices P and Q as well as so-called signature matrices S 1 and 52 such that S?=- Si, S=- S2 and

PS 1 13'

Q'S2Q We can show that for 0 <t < T

= f

=

ps l p - i-G(t) = G(t),

and

H(t) Q -1 S2 Q = H(t). We shall verify the first result by observing that

[PS1P-1 G(t) - G(t)][PS1P -1 G(t) — G(t)l i dt

PS113-1 W(0,T)P' 1 S113' — PS I P -1 W(0,T) ^— W(0,T)1 31-1 S1V --EW (0,T)

= PS 1 13-1 PS 1 P IV-1 S 1 131 —PS I P -1 PS 1 131 —PS 1 P 1 P I-1 S0+PSO

= PSIV—PS 1 V—PS1V+PS1P 1 = 0.

A similar calculation works for H.

Hence,

H(t)G(T) = H(t)Q -1 S2QPS I P -1 G(r).

It is possible to write S2QPS 1 as N1N2 with the number of columns of N1 equal to the number of rows of N2 and both equal to the rank of S2QPS 1 . The rank of S2QPS 1 equals the number of rows in G if and only if S i and S2 are nonsingular. This is the case if and only if M(0, T)and W(0, T) are both nonsingular.

This shows that a reduction of the realisation is possible for t E [0, T] and T E [0, . The minimum order of a weighting pattern restricted to a square Itj < T, IT! < T is a monotone-increasing, integer-valued function of T. Since it is bounded from above, it has a limit and there exists a finite value T , such that on the square iti < T, the order of the weighting pattern, is a maximum.

1.3.4 Stability

In this subsection, we consider the case of no control and constant A matrix such that

5c(t) = Ax(t), x(0) = xo. (1.44)

Definition A matrix A is a stability matrix if all of its eigenvalues have negative real parts.

The following theorem establishes the connection between A and the solution of (1.44).

Theorem 1.10 A necessary and sufficient condition for the solution of (1.44) to approach zero as t —> oo (in this linear, constant coefficient system, this is asymptotic stability), regardless of the value of xo, is that all the eigenvalues of A have negative real parts.

A closely related theorem establishes the connection between A and the solution of a certain linear matrix equation.

Theorem 1.11 A necessary and sufficient condition for A to be a stability matrix is that there exists a positive-definite symmetric matrix S which satisfies the Lyapunov equation

SA + AT S = —I. (1.45)

1.3.5 Feedback

A feedback system can be regarded as a coupling of input-output models. In general, we shall call a system a feedback system if it has a closed causality loop in which the output of a subsystem is coupled back to the input via other subsystems. The latter subsystems can be as simple as a single identity operator, in which case the original subsystem's input and output are one and the same. There is a great deal of arbitrariness in the whole concept, because we are required to identify at least two separate, but coupled systems. This necessarily depends on where we draw the line between the system and the environment. Usually, there is a tacit assumption that the feedback structure does indeed reflect something about the real world.

For general discussions about feedback it is most natural to use input-output system models.

Mathematically, problems stated in terms of feedback systems are fixed-point problems (A fixed point of the map T : E —> E is a value e such that Te = a.)

Many, though not all, of the approaches to the control of (1.12) yield linear expressions for the control u(t) as a function of the system state x(t). In other words, these approaches yield a matrix function of time K(t) which synthesises a control u(t) according to

u(t) = —K(t)x(t). (1.46)

K(t) is referred to as the feedback gain matrix and (1.46) as a state feedback controller for (1.12).

In certain situations (which require A and B to be constant), K turns out to be constant and then we are concerned with the question of whether or not the solutions of the closed loop system

*(t) = (A — BK)x(t)

go to zero as t oo. According to Theorem 1.9 , this occurs if and only if A — BK is a stability matrix. The problem of whether or not there exists a matrix K such that A — BK is a stability matrix, is called the stabilisability problem. A sufficient condition for (1.12) to be stabilisable is that (1.12) with constant A and B, should be completely controllable. This condition is not necessary, since if A is a stabilisability matrix and B 0, then all the solutions of (1.12) go to zero, but the system is clearly not completely controllable with this choice of A and B.