Theme 6

Pole Placement and Observer Design

6.1. Controllability
6.2. Observability
6.3. Design via Pole Placement
6.4. State Observer

The concepts of controllability and observability were introduced by R.

E. Kalman.
A system is said to be controllable if it is possible by means of an
unbounded control vector to transfer the system from any initial state to any
other state in a finite number of sampling periods.
A system is said to be observable if, with the system in state , it is
possible to determine this state from the observation of the output and control
vectors over a finite number of sampling periods.
The design method based on pole placement coupled with state observers
is one of the fundamental design methods available to control engineers. If the
system is completely controllable, then the desired closed-loop poles in the
plane can be selected and the system that will give such closed-loop poles can
be designed. The design approach of placing the closed-loop poles in the
desired locations in the plane is called the pole placement design technique;
that is, in the pole placement design technique we feed back all state variables
so that all pole of the closed-loop system are placed at desired locations. In
practical control system, however, measurement of all state variables may not
be possible; in that case, not all state variables will be available for feedback.
To implement a design based on state feedback, it becomes necessary to
estimate the unmeasurable state variables. Such estimation can be done by use
of state observers.
The pole placement design process of control system may be separated
into two phases. In the first phase, we design the system assuming that all state
variables are available for feedback. In the second phase, we design the state

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observer that estimates all state variables (or only those of that are not directly
measurable) that are required for feedback to complete the design.

6.1. Controllability
A control system is said to be completely controllable if it is possible to
transfer the system from any arbitrary initial state to any desired state (also
arbitrary state) in a finite time period. That is, a control system is controllable
if every state variable can be controlled in a finite time period by some
unconstrained control signal. If any state variable is independent of the control
signal, then it is impossible to control this state variable and therefore the
system is uncontrollable.
Complete State Controllability for a Linear Time-Invariant Discerete-
Time Control System. Consider the discrete-time control system defined by

, (6.1)
where
- state vector (n-vector) at -th sampling instant
- control signal at -th sampling instant
- matrix
- matrix
- sampling period.
We assume that is constant for .
The discrete-time control system given by Equation (6.1) is said to be
completely controllable or simply state controllable if there exist a piecewise-
constant control signal defined over a finite number of sampling periods
such that, starting from any initial state, the state can be transferred to
desired state in at most sampling periods. The necessary and sufficient
condition for complete state controllability is the rank of the following matrix
to be , or

. (6.2)
The matrix is called the controllability matrix.

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 Complete State Controllability in the Case Where is a Vector. If
the system is defined by
,
where is an -vector, - is an -vector, is an matrix, and is
an matrix, then it can be proved that the condition for complete state
controllability is that the matrix

be of rank , or that
.
Determination of Control Sequence to Bring the Initial State to a
Desired State. If the matrix

is of rank and is a scalar, then it is possible to find linearly

independent scalar equations from which a sequence of unbounded control
signals , , can be uniquely determined such that any
initial state is transferred to the desired state in sampling periods.
Note also that, if the control signal is not a scalar, but a vector, then the
sequence of is not unique. Then there exists more than one sequence of
control vector to bring the initial state to a desired state in not more
than sampling periods.
Condition for Complete State Controllability in the Plane. The
condition for complete state controllability can be stated in terms of pulse
transfer function: A necessary and sufficient condition for complete state
controllability is that no cancellation occurs in the pulse transfer function. If
cancellation occurs, the system cannot be controlled in the direction of the
canceled mode.
Complete 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 necessary to define separately
complete output controllability. Consider the discrete-time control system
defined by

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 , (6.3)

, (6.4)
where
- state vector (n-vector) at -th sampling instant
- control signal (scalar) at -th sampling instant
- output vector ( -vector) at -th sampling instant
- matrix
- matrix
- matrix.
The system defined by equation (6.3) and (6.4) is said to be completely
output controllable, or simply output controllable, if it is possible to construct
an unconstrained control signal defined over a finite number of sampling
periods such that, starting from any initial state , the output
can be transferred to the desired point (an arbitrary point) in the
output space in at most sampling periods. A necessary and sufficient
condition for the system to be completely output controllable is that

. (6.5)
It must be mentioned, that in the system defined by equation (6.3) and
(6.4), complete state controllability implies complete output controllability if
and only if the rows of are linearly independent.
Consider the discrete-time control system defined by

(6.6)

, (6.7)
where
- state vector ( -vector) at -th sampling instant
- control signal ( -vector) at -th sampling instant
- output vector ( -vector) at -th sampling instant

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 - matrix
- matrix
- matrix
- matrix.
A necessary and sufficient condition for the system defined by equation
(6.6) and (6.7) to be completely output controllable is that the matrix

be of rank ,
or
.
It is noted that the presence of matrix in the system output equation
always helps to establish complete output controllability.
Example 6.1: Consider the following pulse transfer function

Clearly, cancellation of factors in the numerator and denominator

occurs. Thus, one degree of freedom is lost. Because of this cancellation, this
system is not completely controllable.
The same conclusion can be obtained, of course, by writing this pulse
transfer function in the form of state equation. A possible state-space
representation for this system is

Since

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The rank of is 1. Therefore, we arrive at the same conclusion: that the
system is not completely state controllable.
6.2. Observability
Consider the discrete-time control system defined by

(6.8)

, (6.9)
then

and
.

Since the matrices and are known and is also known, the
second and third terms on the right-hand of the last equation are known
quantities. Therefore, they may be subtracted from the observed value of .
Hence, for determining a necessary and sufficient condition for complete
observability, it is suffices to consider the system described by the equations
(6.10)

, (6.11)
where

- state vector ( -vector) at -th sampling instant

- output vector ( -vector) at -th sampling instant
- matrix
- matrix.
The system is said to be completely observable if every initial state can
be determined from the observation of over a finite number of sampling
periods.
The concept of obserbability is useful for solving the problem of
reconstructing unmeasurable state variables. In practice, however, the difficulty
encountered with state feedback control systems is that some of the state
variables are not accessible for direct measurement. Then it becomes necessary

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to estimate the unmeasurable state variables in order to construct the feedback
control signals.
Complete Observability of Discrete-Time Systems. A necessary and
sufficient condition for the system defined by equations (6.10) and (6.11) to be
completely observable is that the rank of the matrix

(6.12)
be . The matrix given by (6.12) is called the observability matrix. Note that in
(6.12) asterisks indicate conjugate transposes. If matrices and are real, then
the conjugate transpose notation such as may be changed to the transpose
notation such as .
Condition for Complete State Observability in the Plane. The condition
for complete state observability can also be stated in terms of pulse transfer
function: A necessary and sufficient condition for complete state observability
is that no pole-zero cancellation occur in the pulse transfer function. If
cancellation occurs, the canceled mode cannot be observed in the output.
The pulse transfer function has no cancellation if and only if the system is
completely state controllable and completely observable.
Principle of Duality. Consider the system defined by the equations

(6.13)

, (6.14)
where

- state vector ( -vector) at -th sampling instant

- control signal ( -vector) at -th sampling instant
- output vector ( -vector) at -th sampling instant
- matrix
- matrix
- matrix.
Consider the system defined by the equations

(6.15)

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 , (6.16)
where

- state vector ( -vector) at -th sampling instant

- control signal ( -vector) at -th sampling instant
- output vector ( -vector) at -th sampling instant
- conjugate transpose of
- conjugate transpose of
- conjugate transpose of .
The principle of duality states that system defined by equations (6.13)
and (6.14) is completely state controllable (observable) if and only if system
defined by equations (6.15) and (6.16) is completely observable (state
controllable). By use of this principle, the observability of a given system can
be checked by testing the state controllability of its dual.
Example 6.2: Show that the following system is not completely
observable

,
where

Note that the control signal does not affect the complete
observability of the system. To examine complete observability, we may set
. For this system, we have

Notice that

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 .

Hence, the rank of the matrix is less than 3. Therefore, the

system is not completely observable.
In fact, in this system a pole-zero cancellation occurs in the pulse transfer
function of the system. The pulse transfer function between and is

And the pulse transfer function between and is

Then, the pulse transfer function between and is

Clearly, the factors in the numerator and denominator cancel each

other. This means that there are nonzero initial states that cannot be
determined from the measurement of .
6.3. Design via Pole Placement
In this lecture we shall present a design method called pole placement.
We assume that all variables are measurable and are available for feedback. If
the system considered is completely controllable, then poles of the closed-loop
system may be placed at any desired locations by means of state feedback
through an appropriate state feedback gain matrix.
The present design technique begins with a determination of the desired
closed-loop poles based on transient-response and/or frequency response
requirements such as speed, damping ratio, or bandwidth. Given such
considerations, let us assume that we decide that the desired closed-loop poles
are to be at , , …, . Then, by choosing an appropriate gain matrix
for state feedback, it is possible to force the system to have closed-loop poles at
the desired locations, provided that the original system is completely state
controllable.

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 Necessary and Sufficient Condition for Arbitrary Pole Placement.
Consider the open-loop control system shown in Figure 6.1a). the state
equation is
(6.17)
where
- state vector ( -vector) at -th sampling instant
- control signal (scalar) at -th sampling instant
- output vector ( -vector) at -th sampling instant
- matrix
- matrix.
We assume that the magnitude of the control signal is unbounded. If
the control signal is chosen as
,
where is the state feedback gain matrix (a matrix), then the system
becomes a closed-loop control system as shown in Figure 6.1b), and its state
equation becomes
. (6.18)
Note that we choose matrix such that the eigenvalues of are the
desired closed-loop poles .

a)

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 b)
Figure 6.1: a) Open-loop control system; b) Closed loop control system
It can be proven that a necessary and sufficient condition for arbitrary
pole placement is that the system be completely state controllable.
The state feedback gain matrix is determined in such a way that the
error will reduce to zero with sufficient speed. The matrix is not unique for a
given system. It depends on the desired closed-loop pole locations selected.
The selection of the desired closed-loop poles or the desired characteristic
equation is a compromise between the rapidly of the response of the error
vector and the sensitivity to disturbances and measurement noises. That is, if
we increase the speed of error response, then the adverse effect of disturbances
and measurement noises generally increase. In determining the state feedback
gain matrix for a given system, it is desirable to examine several matrices
based on several different desired characteristic equations and to choose the
one that gives the best overall system performance.
Once the desired characteristic equation is selected, the corresponding
state feedback gain matrix for the system can be determine as follow:
Substitute into characteristic equation

and then match the coefficients of powers in of this characteristic equation

with equal powers in of the desired characteristic equation

.
Such a direct calculation of matrix may be simpler for low-order
system.
Deadbeat Control. The concept of deadbeat response is unique to
discrete-time control system. There is no such thing as deadbeat response in
continuous-time control system. In deadbeat control, any nonzero error vector

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will be driven to zero in at most sampling periods if the magnitude of the
scalar control is unbounded. The settling time depends on the sampling
periods, since the response settles down in at most sampling periods. If the
sampling period is chosen very small, the settling time will also be very
small, which implies that the control signal must have an extremely large
magnitude. Otherwise, it will not be possible to bring the error response to zero
in a short time period.
In deadbeat control, the sampling period is the only design parameter.
Thus, if the deadbeat response is desired, the designer must choose the sampling
period carefully so that an extremely large control magnitude is not required.
Note that it is not physically possible to increase the magnitude of the control
signal without bound. If the magnitude is increased sufficiently, the saturation
phenomenon always takes place. If saturation occurs in the magnitude of the
control signal, then the response can no longer be deadbeat. The settling time
will be more than sampling periods. In the actual design of deadbeat control
systems, the designer must be aware of the trade-off that must be made between
the magnitude of the control signal and the response speed.
Pole Placement When the Control Signal is a Vector. We have
considered the pole placement design problem when the control signal is a
scalar. If the control signal is a vector ( -vector), the response can be speeded
up, because we have more freedom to choose control signals
to speed up the response. For example, in the case of the th-order system with
a scalar control, the deadbeat response can be achieved in at most sampling
periods. In the case of the vector control , the deadbeat response can be
achieved in less than sampling periods.
Control System With Reference Input. Thus far, we have considered
regulator systems. In the regulator system, the reference input is fixed for a long
period, and external disturbances create nonzero states. The characteristic
equation for the system determines the speed by which the nonzero states
approach the origin. Now, we shall consider the case where the system has a
reference input. Consider the system shown in Figure 6.2. The plant is
described by the state and output equations

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 Figure 6. 2. State feedback control system
The control signal is
.
By eliminating from the state equation, we have
.
The characteristic equation for the system is
.
As stated earlier, if the system is completely controllable, then the feedback
matrix can be determined to yield the desired closed-loop poles.
It is important to point out that state feedback can change the characteristic
equation for the system, but in doing so the steady-state gain of the entire
system is changed. Therefore, it is necessary to have an adjustable gain in the
system. This gain should be adjusted such that the unit-step response of the
system at steady state is unity, or .
Example 6.3: Consider the system given by

where , , .

Determine the state feedback gain matrix such that when the control
signal is given by the closed-loop system (regulator system)
exhibits the deadbeat response to an initial state .
The characteristic equation of the closed-loop system is

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 .

The desired characteristic equation is .

By comparing the coefficients of equal powers of , we obtain

, .
This gives the desired state feedback gain matrix.
Let us verify that the response of this system to an arbitrary initial state
is indeed the deadbeat response. Since the closed-loop state equation becomes
,

= .

For and initial states we obtain

For

Therefore, the state for becomes zero and the response is

indeed deadbeat.
For the control signal we can write

It is obviously, that with decreasing of the sampling period the value

of the control signal is increasing.

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 Example 6.4: Consider the system defined by

where , , .

Design a control system such that the desired closed-loop poles of the
characteristic equation are at , .

Thus, the desired characteristic polynomial is given by

.
The characteristic equation of closed-loop system is

By comparing the coefficients of equal powers of , we obtain

, .
Using this matrix, the state equation becomes
.
For we obtain

The pulse transfer function for this system is

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 To determine gain constant , we use the condition that the steady-state
output for the unit-step input is unity, or

Therefore .
The unit-step response of this system is shown in Figure 6. 3.

1.6

1.4

1.2

1
y(k)

0.8

0.6

0.4

0.2

0
0 5 10 15 20 25 30 35 40
k

Figure 6. 3. Unit-step response for the system of Example 6. 4

6.4. State Observers

In practice, however, not all state variables are available for direct
measurement. In many practical cases, only a few state variables are
measurable. For instance, it may be that only the output variables are
measurable. Hence, it is necessary to estimate the state variables that are not
directly measurable. Such estimation is called observation. In a practical
system it is necessary to observe the unmeasurable state variables from the
output and control variables.
A state observer is a subsystem in the control system that performs an
estimation of the state variables based on the measurements of the output and
control variables.
In the following discussions of state observers, we shall use the notation
to designate the observed state vector. Figure 6.4. shows a schematic
diagram of a state observer. The state observer will have and as inputs
and as output.

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 Figure 6.4. Schematic diagram of the state observer
In our lecture we shall discuss the design of the full-order state observer.
Full order state observation means that we observe all state variables
regardless of whether some state variables are available for direct measurement.
Observation of only the unmeasurable state variables is referred to as minimum-
order state observation. Observation of all unmeasurable state variables plus
some (but not all) of the measurable state variables is referred to as reduced-
order state observation.
Necessary and Sufficient Condition for State Observation. Figure 6.5.
shows a regulator system with a state observer. The state and output equations
of the system are
(6.19)
, (6.20)
where
- state vector ( -vector)
- control signal ( -vector)
- output vector ( -vector)
- matrix
- matrix
- matrix.
It can be proved that the necessary and sufficient condition for state observation
is that the system be completely observable. In other words, if equation

is satisfied then can be determined from and

.

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 Figure 6.5. Regulator system with a state observer
Full-Order State Observer. The order of the state observer that will be
discussed is the same as that of the system. We assume that the actual state
cannot be measured directly and the observed state cannot be compared
with the actual state . Since the output can be measured, it is
possible to compare with .
Consider the state feedback control system shown in Figure 6.6. The
system equations are
(6.21)
, (6.22)

where
- state vector ( -vector)
- control signal ( -vector)
- output vector ( -vector)
- matrix
- matrix
- matrix
- state feedback gain matrix ( matrix).

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 Figure 6.6. State feedback control system

Figure 6.7. Observed-state feedback control system

We assume that the system is completely state controllable and completely
observable, but is not available for direct measurement. Figure 6.7 shows a
state observer incorporated into the system of Figure 6.6. The observed state
is used to form the control vector , or
.
From Figure 6.7 we have
,
where is the observer feedback gain matrix. This last equation can be
modified as
. (6.23)

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The state observer given by equation (6.23) is called a prediction observer,
since the estimate is one sampling period ahead of the measurement .
The eigenvalues of are called the observer poles.
Error Dynamics of the Full-Order State Observer. Notice that if
then equation (6.23) becomes
,
which is identical to the state equation of the system. To obtain the observer
error equation, let us subtract equation (6.23) from equation (6.21)

(6.24)
Now let us define the error . Then equation (6.24) becomes

. (6.25)
From equation (6.25) we see that the dynamic behavior of the error
signal is determined by the eigenvalues of . If matrix is a stable
matrix, the error vector will converge to zero for any initial error . That is,
will converge to regardless of the values of and . If the
eigenvalues of are located in such a way that the dynamic behavior of
the error vector is adequately fast, then any error will tend to zero with
adequate speed. One way to obtain fast response is to use deadbeat response.
This can be achieved if all eigenvalues of are chosen to be zero.
Example 6. 5. Consider the system

where , , .

Design a full-order state observer, assuming that the system

configuration is identical to that shown in Figure 6.7. The desired eigenvalues
of the observer matrix are , .
Since the configuration of the state observer is specified as shown in
Figure 6.7, the design of the state observer reduces to the determination of an
appropriate observer feedback gain matrix . First, let us examine the
observability matrix. The rank of

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is 2. Hence, the system is completely observable and determination of the
desired observer feedback gain matrix is possible.
The characteristic equation of the observer is

Since the desired characteristic equation is

,
by comparing the coefficients of equal powers of , we obtain

.
Text:
Ogata, K. Discrete-Time Control Systems, 2nd ed., Prentice-Hall, Upper Saddle
River, N.J.

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