Modern Control Systems (MCS)
Lecture-35-36
Design of Control Systems in Sate Space
Reduced Order State Observer
Dr. Imtiaz Hussain
email: imtiaz.hussain@faculty.muet.edu.pk
URL :http://imtiazhussainkalwar.weebly.com/
Introduction
• The state observers discussed in previous lecture was
designed to estimate all the state variables.
• In practice however, some of the state variables may be
accurately measured.
• Therefore, such accurately measurable state variables need
not be estimated.
• In that case a reduced order state observer may be
designed to estimate only those state variables which are
not directly measurable.
Introduction
• Suppose that the state vector x is an n-vector and the
output vector y is an m-vector that can be measured.
• Since m output variables are linear combinations of the
state variables, m state variables need not be estimated.
• We need to estimate only n-m state variables.
• Then the reduced-order observer becomes an (n-m)th
order observer.
• Such an (n-m)th order observer is the minimum-order
observer.
Introduction
• Following figure shows the block diagram of a system with
a minimum-order observer.
Minimum Order State Observer
• To present the basic idea of the minimum-order observer we
will consider the case where the output is a scalar (that is,
m=1).
• Consider the system
𝒙 = 𝑨𝒙 + 𝑩𝑢
𝑦 = 𝑪𝒙
• where the state vector x can be partitioned into two parts
xa (a scalar) and xb [an (n-1)-vector].
• Here the state variable xa is equal to the output y and thus
can be directly measured, and xb is the unmeasurable
portion of the state vector.
Minimum Order State Observer
• Then the partitioned state and output equations become
Minimum Order State Observer
• Then the partitioned state and output equations become
• The equation of measured portion of the state is given as
• Or
• The terms on the left hand side of above equation can be
measure, therefore this equation serves as an output
equation.
Minimum Order State Observer
• Then the partitioned state and output equations become
• The equation of unmeasurable portion of the state is given
as
• Noting that terms Abaxa and Bbu are known quantities.
• Above equation describes the
unmeasured portion of the state.
dynamics
of
the
Minimum Order State Observer
• The design procedure can be simplified if we utilize the design
technique developed for the full-order state observer.
• Let us compare the state equation for the full-order observer
with that for the minimum-order observer.
• The state equation for the full-order state observer is
• The state equation for the minimum order state observer is
• The output equations for the full order and minimum order
observers are
Minimum Order State Observer
• List of Necessary Substitutions for Writing the Observer Equation for
the Minimum-Order State Observer
Table–1
Minimum Order State Observer
• The observer equation for the full-order observer is given by :
• Then, making the substitutions of Table–1 into above
equation, we obtain
• Error dynamics are given as
Minimum Order State Observer
• The characteristic equation for minimum order state observer
is
Minimum Order State Observer
• Design methods
• 1. Using Transformation Matrix
Minimum Order State Observer
• Design methods
• 1. Using Ackerman’s Formula
Example-1
• Consider a system
• Assume that the output y can be measured accurately so
that state variable x1 (which is equal to y) need not be
estimated. Let us design a minimum-order observer.
Assume that we choose the desired observer poles to be
at
Example-1
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END OF LECTURES-35-36