Department of Electromechanical Engineering

Modern Control System

1 / 23
Chapter-3: Stability, Controllability, Observability of
Control Systems

1 Stability
BIBO stability
Internal stability

2 Controllability
State Controllability
Output Controllability

3 Observability

2 / 23
 Stability

Stability

The stability of a system can be thought as a continuity in its dy-

namic behavior.

If a small perturbation arises in the system inputs or initial condi-

tions, a stable system will present small modifcations in its perturbed
response.

In an unstable system, any perturbation, no matter how small, will

make states or outputs grow unbounded or until the system disinte-
grates or saturates.

Stability is the first objective in control design and a basic require-

ment of dynamic systems that perform operations or process signals.

3 / 23
 Stability
BIBO stability

BIBO stability
A system is BIBO (bounded-input bounded-output) stable if every
bounded input produces a bounded output.

Recall that the response of a LTI system is composed of

Response to initial conditions + Response to inputs

The concept of Input-Output Stability refers to stability of the re-

sponse to inputs only, assuming zero initial conditions.

When the system is represented by state equations

ẋ = Ax + Bu
y = Cx + Du
4 / 23
Stability
BIBO stability

The BIBO stability will depend on the eigenvalues of the matrix A,

since every pole of G(s) is an eigenvalue of A.

Cadj(sI − A)B
G(s) = C(sI − A)−1 B + D = +D
det(sI − A)

thus if all eigenvalues of A have negative real part, all the poles
of G(s) will have negative real part, and the system will be BIBO
stable.

Note that not every eigenvalue of A is a pole of G(s), since there

may be pole-zero cancellations while computing G(s).

Thus, a state equation may be BIBO stable even when some eigen-
values of A do not have negative real part.

5 / 23
Stability
BIBO stability

Example. Although the system

   
−1 10 −2
ẋ = x+ u
0 1 0
 
y = −2 3 x − 2u

has one eigenvalue with positive real part λ = 1, it is BIBO stable,

since its transfer function
−2(s + 2)
G(s) = C(sI − A)−1 B + D =
s +1
has a single pole at s = −1.

6 / 23
 Stability
Internal stability

Internal stability

The concept of Internal Stability refers to stability of the system

response to initial conditions only, assuming zero inputs.

In other words, we now study the stability of the response of the

state equation
ẋ = Ax, with x(0) = x0
Because the solution of this equation is given by

x(t) = e At x(0)

stability is determined by the eigenvalues of A. If the eigenvalues of

A have all negative real part, the system response will decay to 0 as
t goes to ∞.

7 / 23
Stability
Internal stability

Lyapunov Stability: The system ẋ(t) = Ax(t) is Lyapunov stable,

or marginally stable, or simply stable, if every finite initial state x0
excites a bounded response x(t).

Asymptotic Stability: The system ẋ(t) = Ax(t) is asymptotically

stable if every finite initial state x0 excites a bounded response x(t)
that approaches 0 as t goes to ∞.

Theorem (Internal Stability): The equation ẋ(t) = Ax(t) is

Lyapunov stable if and only if all the eigenvalues of A have
zero or negative real parts, and those with zero real part are
associated with a Jordan block of order 1.
Asymptotically stable if and only if all eigenvalues of A have
negative real parts.

8 / 23
Stability
Internal stability

Example 1. Consider
 
0 0 0
ẋ = 0 0 0  x
0 0 −1
The matrix A has eigenvalues λ1 = 0 with multiplicity 2, and λ2 =
−1 with multiplicity 1. The eigenvalue λ1 = 0 is associated to
Jordan blocks of order 1, so the equation is Lyapunov stable.

Example. Now consider the equation

 
0 1 0
ẋ = 0 0 0  x
0 0 −1
The matrix A has the same eigenvalues and same multiplicities of the
previous example. Now, however, the repeated eigenvalue λ1 = 0 is
associated to a Jordan block of order 2, so the equation is unstable.
9 / 23
Stability
Internal stability

Indeed, we know that the solution of this equation is given by

  
1 t 0 x1 (0)
x = 0 1 0  x2 (0)
0 0 e −t x3 (0)
 
x1 (0) + tx2 (0)
= x2 (0) 
−t
e x3 (0)
from which we see that x1 (t) grows unbounded if x2 (0) 6= 0.

Example 3: Consider the system

   
−2 0 1
ẋ = x+ u
0 3 2
 
y= 1 0 x
Eigenvalues of A are -2(stable) and 3(unstable). Output is equal to
the first state, which is decoupled from the second state: y (t) =
x1(t). 10 / 23
Stability
Internal stability

The transfer function of this system:

s −3 1
G (s) = =
(s − 3)(s + 2) s +2

After pole-zero cancellation the transfer function has only a stable

pole (-2)

Now lets look at the states

Z t
−2t
x1 (t) = e x1 (0) + e −2(t−τ ) u(τ )dτ = y (t)
Z0 t
x2 (t) = e 3t x1 (0) + 2 e 3(t−τ ) u(τ )dτ
0

So the first state and output are fine, however, x2 (t) will grow un-
bounded.
11 / 23
Stability
Internal stability

1
The transfer function G (s) = s+2 is input/output stable.

Its state space realization given above is unstable.(internally unstable

realization of a stable transfer function).

12 / 23
 Controllability

Controllability

The concepts of controllability and observability were introduced

by Kalman. They play an important role in the design of control
systems in state space.

In fact, the conditions of controllability and observability may govern

the existence of a complete solution to the control system design
problem. The solution to this problem may not exist if the system
considered is not controllable.

A system is said to be controllable at time t0 , if it is possible by

means of an unconstrained control u(t) that can transfer the system
from any initial state x(t0 ) to any other state in a finite interval of
time.

13 / 23
 Controllability
State Controllability

State Controllability
Consider the continuous-time system.
ẋ = Ax + Bu
The system described above is said to be state controllable at t = t0
if it is possible to construct an unconstrained control signal that will
transfer an initial state to any final state in a finite time interval
t0 ≤ t ≤ t1 . If every state is controllable, then the system is said to
be completely state controllable.

we can assume that the final state is the origin of the state space
and the initial time is zero, t0 = 0.
Z t1
At1
x(t1 ) = 0 = e x(0) + e A(t1 −τ ) Bu(τ )dτ
0
Z t1
x(0) = − e −Aτ Bu(τ )dτ
0 14 / 23
Controllability
State Controllability

e −Aτ can be written

n−1
X
e −Aτ = ak (τ )Ak
k=0

combining the above two equations gives

n−1
X Z t1
k
x(0) = − A B ak (τ )u(τ )dτ
k=0 0

Let us put Z t1
bk = ak (τ )u(τ )dτ
0
Then the initial state equation becomes
n−1
X
x(0) = − Ak Bbk
k=0

15 / 23
Controllability
State Controllability

 
b0

 ...


 b1 

x(0) = − B ... AB ... . . . ..
h i
n−1
. A B 
 ...

 ..

 .

 ... 
bn−1
If the system is completely state controllable, then, given any initial
state x(0), the above equation
h must be satisfied. This requires
i that
. . .
the rank of the nxn matrix B .. AB .. . . . .. An−1 B be n.

Therefore, The system given by the equation given above is com-

pletely state controllable if and only if the hvectors B, AB, ..., An−1 B
are linearly independent, or the nxn matrix B ... AB ... . . . ... An−1 B
i

has a rank of n.
16 / 23
Controllability
State Controllability

The matrix B ... AB ... . . . ..

h i
. A B is commonly called the
n−1

controllability matrix.

Example: Let us consider a system represented by the two state

equation

ẋ1 = −2x2 + u
ẋ2 = −3x2 + dx1

and determine the condition for state controllability

Exercise: Consider the following transfer function:

y (s) s + 2.5
=
u(s) (s + 2.5)(s − 1)

cheek the sate controllability of the system?

17 / 23
 Controllability
Output Controllability

Output Controllability
In the practical design of a control system, we may want to control
the output rather than the state of the system.

Complete state controllability is neither necessary nor sufficient for

controlling the output of the system. For this reason, it is desirable
to define separately complete output controllability.

ẋ = Ax + Bu
y = Cx + Du

The system described above is said to be completely output con-

trollable if it is possible to construct an unconstrained control vec-
tor u(t) that will transfer any given initial output y (t0 ) to any final
output y (t1 ) in a finite time interval t0 ≤ t ≤ t1 .
18 / 23
Controllability
Output Controllability

The system described by the above equations is completely output

controllable if and only if the px(n + 1)m matrix

.. .. .. .. ..
h i
CB 2
. CAB . CA B . . . . . CA B .D n−1

is of rank p. Note that the presence of the Du term always helps to

establish output controllability.

Example: Consider the system described by

      
ẋ1 1 1 x1 0
= + u
ẋ2 −2 −1 x2 1
 
  x1
y= 1 0
x2

Is the system sate and output controllable

19 / 23
 Observability

Observability
A system is said to be observable at time t0 , with a system state
x(t0 ), it is possible to determine this state from the observation of
the output over a finite time interval.

Consider the unforced system described by the following equations:

ẋ = Ax
y = Cx

The system is said to be completely observable if every state x(t0 )

can be determined from the observation of y (t) over a finite time
interval, t0 ≤ t ≤ t1 .

The system is, therefore, completely observable if every transition

of the state eventually affects every element of the output vector
20 / 23
Observability

Consider the unforced system described above. The output vector

y(t) is
y (t) = Ce At x(0)
where
n−1
X
−At
e = ak (t)Ak
k=0
Hence, we obtain
n−1
X
y (t) = ak (t)CAk x(0)
k=0
or
y (t) = a0 (t)Cx(0) + a1 (t)CAx(0) + ... + an−1 (t)CAn−1 x(0)
If the system is completely observable, then, given the output y(t)
over a time interval t0 ≤ t ≤ t1 , x(0) is uniquely determined from
the above equation.
21 / 23
Observability

It can be shown that this requires the rank of the npxn matrix
 
C
 ... 
 
 CA 
 
 ... 
 
 .. 
 . 
 
 ... 
CAn−1
to be n.
From this analysis, we can state the condition for complete observ-
ability as follows: The unforced system given above is completely
observable if and only if the nxnp matrix
.. .. ..
h i
? ? ?
C . A C . . . . . (A ) C ? n−1 ?

has rank of n or has n linearly independent column vectors. This

matrix is called the observability matrix.
22 / 23
Observability

All the poles of the closed loop system can be placed arbitrarily
in the complex plane if and only if the system is observable
and controllable. Observability refers to the ability to estimate
a state variable.

Example: Consider the system described by

      
ẋ1 1 1 x1 0
= + u
ẋ2 −2 −1 x2 1
 
  x1
y= 1 0
x2

Is the system observable?

23 / 23