CONTROLLABILITY

1
CONTROLLABILITY OBSERVABILITY
When I complete this chapter, I want to be able
to do the following.

• Determine degrees of freedom for control
• Select state variables that are observable
• select input/output designs that are controllable
• understand various meanings of controllability
  apply appropriately

2
Multiloop Performance
We are here, and making progress all the time!

• Defining control objectives
• Controllability Observability
• Interaction Operating window
• The Relative Gain
• Multiloop Tuning
• Performance and the RDG
• SVD and Process directionality

• Robustness
• Integrity
• Control for profit
• Optimization-based design methods
• Process design
• - Series and self-regulation
• - Zeros (good/bad/ugly)
• - Recycle systems
• - Staged systems

3
CONTROLLABILITY OBSERVABILITY
Outline of the lesson.

• Placing key questions in perspective
• Degrees of Freedom
• Controllability - Definitions Meanings
• Controllability class exercises
• Observability
• Tie together Observability and Controllability
• Workshops
• Self-Study Guides

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CONTROLLABILITY OBSERVABILITY
Control Design Form 1. Safety 2. Environmental
protection 3. Equipment protection 4. Smooth
operation 5. Product quality 6. Profit 7.
Monitoring and diagnosis
Process design 1. Valves 2. By passes 3.
Equipment capacity 4. Precision of the MVs
Manipulated variables
Controlled variables
Can we control the CVs with the MVs?
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CONTROLLABILITY OBSERVABILITY
For a single-loop system, what is the minimum
requirement for control to be possible?
The process gain must not be zero Kp ? 0.
OK, lets apply this test to every control
loop in a multiloop design. Would this be a
valid test for controllability?
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CONTROLLABILITY OBSERVABILITY
Can we control this process?
How will we go about building a base for
analysis? 1. Weve always heard that degrees of
freedom are important 2. Is control possible with
the MVs and CVS available well call this
controllability. 3. Can we determine key
variables from the measurements well call this
observability.
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CONTROLLABILITY
Degrees of Freedom - Three words with many
meanings!!
For the CSTR with an exothermic reaction, how
many degrees of freedom exist?
vc
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CONTROLLABILITY
DOF Lets figure this out! Complete the
following table to determine the number of
degrees of freedom.

• Number of
• equations
• dependent variables
• external manipulated
• external disturbance
• constants
• degrees of freedom

How many variables can be controlled?
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CONTROLLABILITY
The preceding provides lots of process insight.
Is there an easier method for determining the
maximum number of controlled variables?
What is the maximum number variables that can
be controlled?
DOF Method 1 The maximum number controlled
variables is equal to ... (relate to equipment)
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CONTROLLABILITY
DOF Method 2 The degrees of freedom for control
is equal to the number of independent . (relate
to the streams)
Weigh the meanings of the two definitions and
select the best for the control design.
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CONTROLLABILITY
DOF Method 1 The maximum number controlled
variables is equal to the number of adjustable
final elements.
DOF Method 2 The degrees of freedom for control
is equal to the number of independent material
and energy streams that can be manipulated.
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CONTROLLABILITY
Apply both DOF methods to the following problem.
There is one stream and two valves. What is the
maximum number variables that can be
controlled? Quick test Does this make sense?
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CONTROLLABILITY
Apply both DOF methods to the following problem.
There is one stream and two valves. What is the
maximum number variables that can be controlled?
Number of valves Quick test Does this make
sense? Yes, we reduce the pressure to protect
downstream equipment and control the flow. This
design also improves the accuracy of head flow
sensors is the source pressure changes.
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CONTROLLABILITY
DEGREES OF FREEDOM is perhaps less helpful than
we first anticipated. Lets move
on. CONTROLLABILITY - A characteristic of the
process that determines whether a specified
dynamic behavior can be achieved with a defined
set of controlled and manipulated variables.
We seek a fundamental property of the process
independent of a specific control design or
structure.
Various specifications for dynamic behavior are
possible we will review the most commonly used.
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CONTROLLABILITY
The most common definition Pointwise State
Controllable - A system is controllable if it is
possible to adjust the manipulated variables u(t)
so that the system will be forced from an
arbitrary state x(t0) to x(t1) in a finite time.
x(t1)
x(t0)
Originally due to Kalman this is sometimes
termed reachability, which assures that the final
state is not simply x(t1) 0
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CONTROLLABILITY
Pointwise State Controllable A simplified
derivation of the criterion for p.s.
controllability.
Linear, time-invariant system
The ?s are the (distinct) eigenvalues of A and
the columns of P are the associated
eigenvectors. Through these transformations, the
system is diagonalized, as given in the following.
The system is uncontrollable if any row of ? has
only zeros. If this is the case, no input
affects an output.
0
0
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CONTROLLABILITY
Pointwise State Controllable A process example
for p.s. controllability.
The process is a series of four tanks. The input
is the temperature of the inlet stream, or the
heat to the inlet stream. The states are the
temperatures in the four tanks. Is the system
p.s. controllable?
at t 0, Ti 0.0 for i 1,4 at t 400, T1
1, T2 -1, T3 1 and T4 -1
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CONTROLLABILITY
Pointwise State Controllable A process example
for p.s. controllability.
From Skogestad Postlethwaite, 1996
The control performance can be achieved! Look at
the manipulated inlet temperature.
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CONTROLLABILITY
(Pointwise) Output Controllable - A system is
controllable if it is possible to adjust the
manipulated variables u(t) so that the system
will be forced from an arbitrary state y(t0) to
y(t1) in a finite time.
The system is output controllable if and only if
all rows of G(s) are linearly independent.
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CONTROLLABILITY
An alternative definition Functional
Controllable - A system is controllable if it is
possible to adjust the manipulated variables u(t)
so that the system will follow a (smooth) defined
path from y(t0) to y(t1) in a finite time.
x(t1)
x(t0)
This definition was originally proposed by
Rosenbrock.
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CONTROLLABILITY
Functional Controllable Statements of the
criteria for functional controllability.
I. A system G(s) is (output) functionally
controllable if and only if 1. The dimensions of
y and u are the same (say n) and 2. The rank of
G(s) n Stated differently, G-1(s) exists for
all s Stated again, ?min(G(j?)) gt 0 minimum
singular value II. If the system has more
inputs than outputs, then, the rank of G(s) must
equal n, the output dimension. III. The system
cannot be output functional controllable if the
number of inputs is less than the number of
outputs.
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CONTROLLABILITY
Functional Controllability A process example for
functional controllability.
The process is a series of four tanks. The input
is the temperature of the inlet stream, or the
heat to the inlet stream. The states are the
temperatures in the four tanks. Is the system
functionally controllable?
T2 x2
T3 x3
T4 x4
T1 x1
Consider two cases 1. y T1 T2 T3 T4T 2. y
T4
u T0
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CONTROLLABILITY
Compare Pointwise State Controllability and
Functional Controllability -for the airplane
example below.
Then, recommend when to use each of the
definition/tests in the process industries.
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CONTROLLABILITY
Limitations to classical controllability
analysis 1. Does not give a criterion for the
types of behavior that we desire for many of our
closed-loop systems. 2. No limit is placed on the
adjustments to the manipulated variables (u could
reach infinity). 3. Some (many) states can be of
no importance (within rather large
bounds). 4. The result is either yes/no, while we
seek information on how well control is likely to
perform. 5. The results are for a point and can
change with operations.

• Processes that are (either p.s., p. output, or
  functional) controllable might perform poorly
  under feedback,
• Processes that are not p.s. or functional
  controllable might perform well under feedback.

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CONTROLLABILITY
We have seen the math now, lets build some
physical insights.We will investigate a few
simple processes to determine their
controllability. For the purpose of these
exercises, we will use a less restrictive version
of function controllability.
The system will be deemed controllable if the
steady-state I/O gain matrix can be inverted,
i.e., Det G(0) ? 0 G(0) -1 exists This
is only applicable to open-loop stable plants.
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CONTROLLABILITY

• For process Example 1 the blending process
• Are the CVs independently controllable?
• Does interaction exist?

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CONTROLLABILITY

• For process Example 1 the blending process
• Are the CVs independently controllable?
• Does interaction exist?

Yes, this system is controllable!
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CONTROLLABILITY

• For process Example 2 the distillation tower
• Are the CVs independently controllable?
• Does interaction exist?

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CONTROLLABILITY
For process Example 2 the distillation tower
Det (K) 1.54 x 10-3 ? 0 Small but not zero
(each gain is small) The system is controllable!
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CONTROLLABILITY

• For process Example 3 the non-isothermal CSTR
• Are the CVs independently controllable?
• Does interaction exist?

A ? B -rA k0 e -E/RT CA
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CONTROLLABILITY

• For process Example 3 the non-isothermal CSTR
• Are the CVs independently controllable?
• Does interaction exist?

A ? B -rA k0 e -E/RT CA
The interaction can be strong In general, the
temperature and conversion (extent of reaction)
can be influenced. The system is
controllable. (See Appendix C for examples)
v1
v2
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CONTROLLABILITY

• For process Example 4 the mixing tank
• Are the CVs independently controllable?
• Does interaction exist?

v1
v2
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CONTROLLABILITY

• For process Example 4 the mixing tank
• Are the CVs independently controllable?
• Does interaction exist?

v1
0
0
v2
Nothing affects composition at S-S the system is
NOT controllable.
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CONTROLLABILITY

• For process Example 5 the non-isothermal CSTR
• Are the CVs independently controllable?
• Does interaction exist?

A ? B -rA k0 e -E/RT CA
v1
v2
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CONTROLLABILITY

• For process Example 5 the non-isothermal CSTR
• Are the CVs independently controllable?
• Does interaction exist?

Solution continued on next slide
Both valves have the same effects on both
variables the only difference is the magnitude
of the flow change (? constant).
A ? B -rA k0 e -E/RT CA
v1
v2
Det (K) 0 not controllable!
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For process Example 5 the non-isothermal
CSTR In this case, both MVs affect ONE common
variable, and this common variable affects both
CVs. We can change both CVs, but we cannot move
the CVs to independent values!
Solution continued on next slide
37
For process Example 5 the non-isothermal
CSTR For input contraction, multivariable
feedback control is not possible the system is
not controllable! We can change both CVs, but we
cannot move the CVs to independent values!
Solution complete
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CONTROLLABILITY

• For process Example 6 the non-isothermal CSTR
• Are the CVs independently controllable?
• Does interaction exist?

A ? B 2C -rA k0 e -E/RT CA
v1
v2
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CONTROLLABILITY

• For process Example 6 the non-isothermal CSTR
• Are the CVs independently controllable?
• Does interaction exist?

Solution continued on next slide
A ? B 2C -rA k0 e -E/RT CA
Using the symbol Ni for the number of moles of
component i that reacts, we have the following.
v1
Because of the stoichiometry, NC 2 NB and the
system is not controllable!
v2
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CONTROLLABILITY

• For process Example 6 the non-isothermal CSTR
• Are the CVs independently controllable?
• Does interaction exist?

Solution continued on next slide
A ? B 2C -rA k0 e -E/RT CA
v1
Det (K) 0 not controllable!
v2
41
For output contraction, both MVs affect both CVs,
but the CVs are related through the physics and
chemistry. We can change both CVS, but we cannot
move the CVs to independent values!
Solution continued on next slide
42
In this case, multivariable feedback control is
not possible the system is uncontrollable!
Solution complete
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CONTROLLABILITY
CONTROLLABILITY Conclusions about determining
controllability
These are generally easy to determine.
Lack of controllability when
1. One CV cannot be effected by any valve 2. One
MV has no effect on CVs 3. Lack of independent
effects. Look for contractions
This requires care and process insight to
determine.
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CONTROLLABILITY
Class exercise Summarize the underlying
principles that can lead to a contraction and
to a loss of controllability. These principles
will be applicable to many process examples,
although the specific variables could be
different. Hint Review the examples in the
lecture and identify the root cause for each.
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CONTROLLABILITY
Class exercise Summarize the underlying
principles that can lead to a contraction and
to a loss of controllability.
1. Material Balance 2. Energy balance 3.
Equilibrium 4. Stoichiometry (elementary
balance) 5. Kirchhoffs first law (conservative
of electrical charge) 6. Mechanic linkage
between elements of system (valves) 7.
Regulatory control system (if considered part of
the process). For example, one variable is
controlled by two regulatory controllers with the
same values for their set points.
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OBSERVABILITY
The system is state observable if for any time t1
gt 0 the initial state x(0) can be determined
from a time history of the input u(t) and output
y(t) over the interval 0, t1.
A state is not observable if it is not measured
and does not affect a measurement.
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OBSERVABILITY
Observability criterion
For the system to be observable, the canonical
variables must affect the measurements.
Therefore, the system is observable if and only
if no column of CP has all zero elements.
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OBSERVABILITY
Observability A process example
The process is a series of four tanks. The input
is the temperature of the inlet stream, or the
heat to the inlet stream. The states are the
temperatures in the four tanks. Are the states
observable from the measurement of only T4?
T2 x2
T3 x3
T4 x4
T1 x1
T4
u T0
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OBSERVABILITY
Limitations to classical observability
analysis 1. The analysis does not consider
practical issues such as high order derivatives
needed to determine a variable from measured
outputs. 2. Observability requires an exact model
and parameters. 3. Noise will limit the practical
estimation from measurements. 4. Full state
observability is not typically required for good
process process control.
As a result, some observable variables might not
be practically determined.
However, observability is required for practical
estimation of unmeasured variables from measured
variables.
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CONTROLLABILITY OBSERVABILITY
A system could have up to four types of variables.
Controllable Observable
inputs
outputs
Controllable , Not Observable
Not Controllable, but Observable
Neither Controllable or Observable
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CONTROLLABILITY OBSERVABILITY
Summarize the essential minimum conditions for
process variables

• All unstable modes to be controllable
• All key state variables to be observable and
  controllable. Here key includes safety,
  equipment protection, product quality and
  production rate.
• All unobservable and uncontrollable states to
  remain within an acceptable range

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CONTROLLABILITY OBSERVABILITY
What can we use from the preceding material? 1.
Degrees of Freedom - The number of inputs ?
controlled variables. 2. Controllability Given
(1) above, - The system must also be p.s.
controllable. - Since this is rigorous at one
point, we must investigate whether this will
change within the operating region. - Acceptable
dynamic performance is not ensured. 3.
Observability - All key variables must be
observable. - This does not ensure useful
variables for control given likely sensor errors,
noise and process changes.
More to come!
More to come!
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CONTROLLABILITY OBSERVABILITY
Class exercise Define the criteria that you
think are required for a system to be
controllable and observable.
1. Well restrict this exercise to continuously
operating processes. 2. Your answer will be
(much) more than a summary of the topics in the
lecture. 3. Define the characteristics or
features of the system. Report in order from the
least restrictive to the most restrictive. 4. Dete
rmine tests for the system at each definition, as
you include more restrictions.
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CONTROLLABILITY OBSERVABILITY Workshop 1
Discuss what happens to the degrees of freedom
when control is added to a process. Be sure to
address the external variables
Without control
With control
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CONTROLLABILITY OBSERVABILITY Workshop 2
Functional output controllability seems to
provide a useful statement and criterion.
Discuss reasons why you might not use this
criterion. Be sure to address limits to feedback
control performance.
I. A system G(s) is (output) functionally
controllable if and only if 1. The dimensions of
y and u are the same (say n) and 2. The rank of
G(s) n Stated differently, G(s) -1 exists
for all s Stated again, ?min(G(j?)) gt 0
minimum singular value
56
CONTROLLABILITY OBSERVABILITY Workshop 3
Evaluate the controllability of the CSTR. The
concentration of component B is to be controlled
by manipulating the feed flow rate.
A ? B ? C
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CONTROLLABILITY OBSERVABILITY Workshop 4
We need to control the mixing tank effluent
temperature and concentration. You have been
asked to evaluate the design in the
figure. Discuss good and poor aspects and decide
whether you would recommend the design.
58
CONTROLLABILITY OBSERVABILITY Workshop 5
The sketch describes a simplified boiler for the
production of steam. The boiler has two fuels
that can be manipulated independently. Analyze
the controllability of this system and determine
the loop pairing.
59
CONTROLLABILITY OBSERVABILITY Workshop 6
The sketch describes a simplified flash drum. A
design is proposed to control the temperature and
pressure of the vapor section. Analyze the
controllability of this system and determine if
the loop pairing is correct.
60
CONTROLLABILITY
When I complete this chapter, I want to be able
to do the following.

• Determine degrees of freedom for control
• Select state variables that are observable
• select input/output designs that are controllable
• understand various meanings of controllability
  apply appropriately

• Lots of improvement, but we need some more
  study!
• Read the references
• Review the notes, especially learning goals and
  workshop
• Try out the self-study suggestions
• We will develop lots more technology on these
  issues!

61
CONTROLLABILITY LEARNING RESOURCES

• SITE PC-EDUCATION WEB
• - Instrumentation Notes
• - Interactive Learning Module (Chapter 20)
• - Tutorials (Chapter 20)
• BOOKS
• - Skogestad and Postlethwaite, Multivariable
  Feedback Control, Wiley, New York, 1996 (Sec 4.2,
  4.3, 6.3)
• - Friedland, Control System Design, McGraw-Hill,
  New York, 1986
• (Ch 5.)
• - Chen, Linear System Theory and Design, Holt,
  Rhinehart and Winston, New York, 1984 (Sec. 5.6
  and 5.7)
• - Kailath, Linear Systems, Prentice-Hall,
  Englewood Cliffs, 1980. (Sections 2.3 and 2.4)

62
CONTROLLABILITY SUGGESTIONS FOR SELF-STUDY
1. Lack of controllability can be due to the
process structure or to specific parameters. To
determine structural controllability, which is
independent of parameter values, see - Barton,
et al, Comp Chem Engr, 9, 547-555, 1985 -
Hopkins et.al., J. Proc. Cont., 8, 57-68,
(1998) 2. Some attractive control designs could
be missed if we limited ourselves to open-loop
stable processes. See, for example, -
Skogestad, et al., IEC Res. 29, 2339-46 (1990)
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CONTROLLABILITY SUGGESTIONS FOR SELF-STUDY
3. See additional examples of uncontrollable
mechanical and electrical systems in the
following citation. - Friedland, Control System
Design, McGraw-Hill, New York, 1986 (Ch 5.)