Modern Control Systems
Lecture 4
Properties of the State Space Model
Dr. Abusabah I. A. Ahmed
abusabah22@hotmail.com
Lecture Outline
❑ Introduction
❑ State Controllability
❑ Output Controllability
❑ State Observability
❑ Practice Problems
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Abusabah I. A. Ahmed
Introduction
❑ In state-space analysis we are concerned with three types of
variables that are involved in the modeling of dynamic systems:
input variables, output variables, and state variables.
❑ The dynamic system must involve elements that memorize the
values of the input for t> t1 .
❑ Since integrators in a continuous-time control system serve as
memory devices, the outputs of such integrators can be
considered as the variables that define the internal state of the
dynamic system.
❑ Thus the outputs of integrators serve as state variables.
❑ The number of state variables to completely define the
dynamics of the system is equal to the number of integrators
involved in the system.
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State Controllability
A system is completely controllable if there exists an
unconstrained control u(t) that can transfer any initial state
x(to) to any other desired location x(t) in a finite time, to ≤ t
≤ T. uncontrollable
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controllable
State Controllability
Explain Controllability Concept
Are system states are completely controllable for all u?
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State Controllability
X2 is not affected by u (it is uncontrollable)
X1 is affected by u (it is controllable)
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State Controllability
Controllability Criterion
A realization {A,B,C} is completely controllable if an only
if the controllability matrix Mc (A,B) has a full rank n.
Where:
Mc = B AB An−1 B
Example 4-1
Investigate the controllability of the three dimensional state equation
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State Controllability
Controllability Criterion
Solution
The controllability matrix Mc is independent of the transfer
function numerator coefficients b0, b1, and b2.
The determinant of the controllability matrix is |Mc| = −1 = 0, so the
state equation is controllable. The determinant is independent of the
characteristic polynomial coefficients a0, a1, and a2, so a state-space
realization in this form is always controllable.
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Output Controllability
❑ Output controllability describes the ability of an
external input to move the output from any initial
condition to any final condition in a finite time
interval.
Output controllability matrix (𝑀𝑂𝐶 ) is given as
M OC = CB CAB CA2 B CAn −1B
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State Observability
A system is said to be observable if at time t0 the system
state x(t0) can be exactly determined from observation of
the output y(t) over a finite time interval
Unobservable Observable
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State Observability
𝑦 = 𝑥2 + 2𝑢
This system is unobservable
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Abusabah I. A. Ahmed
State Observability
A realization {A, B, C } is state observable if the
observability matrix Mo has a full rank n
Where
C
CA
Observability Matrix M O = CA
2
CA
n −1
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State Observability
Example 4-2
Check State Observability Considering the system given below
0 1 0
x = x + 1 u
0 − 2
y = 0 4x
Solution
C 0 1
MO = C = 0 4 CA = 0 4 = 0 − 12
CA 0 − 2
0 4
MO =
Since 𝑟𝑎𝑛𝑘(𝑂𝑀) ≠ 𝑛therefore system
0 −12 is not completely state observable.
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Practice Problems
1. is the following five-dimensional, two-input state
controllable ?
2. Check the state controllability, state observability
and output controllability of the following system!
0 1 0
A= , B = , C = 0 1
1 0 1
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Thank you for
Attention
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