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0066_Frame_C24 Page 26 Thursday, January 10, 2002 3:45 PM

that the system S 1 (the plant) is strictly proper, i.e., D 1 = 0. We then obtain

x ˙ 1 t() x 1 t()
= A 1 – B 1 D 2 C 1 B 1 C 2 + B 1 D 2 u t() (24.150)
x ˙ 2 t() – B 2 C 1 A 2 x 2 t() B 2

x 1 t()
y t() = [ C 1 0] (24.151)
x 2 t()

The same results apply, mutatis mutandis, to discrete-time interconnected systems. More details can be
found elsewhere, e.g., in [15].

24.6 System Properties

Controllability, Reachability, and Stabilizability
A very important question that we must be interested in regarding control systems using state space models
is whether or not we can steer the state via the control input to certain locations in the state space. We
must remember that the states of a system frequently are internal variables like temperature, pressure,
level of tanks, or others. These are sometimes critical variables that we want to keep between specific
values.

Controllability
The issue of controllability is concerned with whether or not a given initial state x 0 can be steered to the
origin in finite time using the input u(t).

Example 24.11
If we examine the model defined in (24.152), we note that the input u(t) has no effect over the state x 2 (t).

x ˙ 1 t() = 0 1 x 1 t() + 1 ut()
x ˙ 2 t() 0 0 x 2 t() 0 (24.152)

T
Given an initial state [x 1 (0), x 2 (0)] , the input u(t) can be chosen to steer x 1 (t) to zero, while x 2 (t)
remains unchanged.
Formally, we have the following definition:
Definition 24.1 A state x o is said to controllable if there exists a finite interval [0, T] and an input
{u(t), t ∈ [0, T]} such that x(T) = 0. If all states are controllable, then the system is said to be completely
controllable.
Reachability
A related concept is that of reachability, used sometimes in discrete-time systems. It is formally defined
as follows:
Definition 24.2 A state x ≠ 0 is said to be reachable, from the origin, if given x(0) = 0, there exists a
x
finite time interval [0, T] and an input {u(t), t ∈ [0, T]} such that x(T) = . If all states are reachable
the system is said to be completely reachable.
For continuous, time-invariant, linear systems, there is no distinction between complete controlla-
bility and reachability. However, the following example illustrates that there is a subtle difference in the

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