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Journal of the Taiwan Institute of Chemical Engineers 0 0 0 (2017) 1–15
Contents lists available at ScienceDirect
Journal of the Taiwan Institute of Chemical Engineers
journal homepage: www.elsevier.com/locate/jtice
Optimization approach for the analytical design of an industrial PI
controller for the optimal regulatory control of first order processes
under operational constraints
Rodrigue Tchamna, Moonyong Lee∗
School of Chemical Engineering, Yeungnam University, Gyeongsan 712-749, Republic of Korea
a r t i c l e
i n f o
Article history:
Received 10 November 2016
Revised 21 March 2017
Accepted 3 August 2017
Available online xxx
Keywords:
Constrained optimal regulatory control
Proportional-integral (PI) controller design
Operational constraints
Optimization based approach
Stable first order process
a b s t r a c t
In this paper, an optimization-based approach for the closed-form design of an industrial proportionalintegral (PI) controller was proposed for the optimal regulatory control of first order process under three
typical operational constraints. An ingenious parameterization with Lagrangian multiplier method was
used to convert the constrained optimal control problem in the time domain to an unconstrained optimization problem to derive an analytical solution for the optimal regulatory control. Three typical operational constraints could be taken into account in the controller design stage, explicitly. The proposed analytical design method required no complicated optimization steps and guaranteed global optimal closedloop performance and stability. The proposed analytical approach also provides useful insights into the
optimal controller design and analysis. A practical and facile procedure for designing optimal PI parameters and a feasible constraint set was also proposed.
© 2017 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
1. Introduction
For economic and/or safety reasons, the control and operation
of many modern chemical processes are restricted by the stringent
operational constraints associated with the process variable, manipulated variable, and its allowable rate of change. For example, if
a control system cannot consider the actuator limitation explicitly,
the control algorithm may supply a control signal that the actuator
cannot achieve. This will lead to actuator saturation, which in turn
results in severe degradation of the control performance and even
instability of the control loop, even in the case of an open-loop
stable process. This tough environment in process control highlights the need to design more precise controllers that can cope
with both the control performance and operational constraints in
an efficient manner.
In order to manage the performance and robustness together
in optimal control, the control objective should take into account
not only the variations in the controlled variable, but also in the
controller output. Optimal control of a process may become even
more challenging when constraints need to be considered in the
design stage. PID controllers used for industrial processes rarely
tackle the constraints in an explicit manner [1]. Optimal control
is a remarkable challenge when multiple constraints need to be
∗
Corresponding author.
E-mail address: mynlee@yu.ac.kr (M. Lee).
considered, i.e., constrained optimal control, in its design. In a popular approach using Pontryagin’s principle or the Hamilton–Jacobi–
Bellman equation for a classical optimal control framework [2–6],
the optimal controller parameters are obtained by solving the nonlinear constrained optimization numerically. On the other hand,
despite the use of complicated optimization packages for solving
non-linear optimization, the existing numerical methods neither
guarantee a global optimal solution nor provide truthful insights
and physical interpretations regarding the complicated relationships and effects among the process parameters and optimal solutions.
The limitations of the conventional black-box approach by solving non-linear optimization has prompted research into novel analytical approaches for finding the optimal controller parameters,
but mainly focusing on integral processes [7–12]. Recently, Thu and
Lee [13] extended the analytical approach to constrained optimal
servo control of first order processes and showed the resulting PI
controller guarantees the global optimality without the necessity
of sophisticated optimization packages. In addition to servo control, regulatory control against disturbances is major consideration
in most process control loop.
This study extends the analytical design approach for the servo
optimal control to the regulatory control. The proposed PI controller provides the globally optimal control performance, whilst
satisfying the given operational limits of the process variable, controller output, and its rate of change. A formulation of the optimal
regulatory control problem under constraints was first provided
http://dx.doi.org/10.1016/j.jtice.2017.08.012
1876-1070/© 2017 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
Please cite this article as: R. Tchamna, M. Lee, Optimization approach for the analytical design of an industrial PI controller for the
optimal regulatory control of first order processes under operational constraints, Journal of the Taiwan Institute of Chemical Engineers
(2017), http://dx.doi.org/10.1016/j.jtice.2017.08.012
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R. Tchamna, M. Lee / Journal of the Taiwan Institute of Chemical Engineers 000 (2017) 1–15
Fig. 1. Block diagram of the feedback control of a stable first order process.
for a first order process that has a clear practical significance as
a representative basic and starting process dynamics in all model
reductions for process control purpose. By applying the Lagrange
multiplier method, re-express it in the form of a corresponding
unconstrained optimization problem in terms of two new parameters that provide a clear physical meaning and useful graphical
interpretation for controller design. The shape and trends of the
constraints and objective function mainly depend on the parameterization of the control parameters. Since the graphical analysis
essentially depends on this parameterization, the choice of these
new parameters should be done such as the graphical analysis
can be easily performed. After the parameterization of the system,
the analytical design for an optimal PI controller and feasible constraint set was obtained from meticulous graphical examination of
the possible location of the global optimum. The proposed optimal
controller enhanced the closed-loop performance significantly by
ensuring a global optimal solution whilst taking care of the critical functioning requirements in an explicit way. The other good
point of the proposed design technique is that, it relates the control parameters to the plant parameters in a closed-form/explicit
way which is very appealing to control practitioners. To the authors’ best knowledge, this study is the first work to develop the
analytical PI controller design of optimal regulatory control for the
first order process under operational constraints. It also offers valuable insights into the optimal performance and its physical understanding, which is not available in black-box based approaches applying the numerical non-linear optimization directly.
2.2. Formulation of the constrained optimal regulatory control
problem
The goal of the constrained optimal regulatory problem is to
minimize the weighted sum of the process variable error, e(t), and
the rate of change in the manipulated variable, u (t), for a given
step disturbance change, D, subjected to the following three typical operational constraints: maximum allowable limit in (1) the
controlled variable, ymax , (2) the rate of change in the manipulated
variable, u max , and (3) the manipulated variable, umax . Consequently, the optimal controller parameters is obtained by minimizing the performance index in Eq. (4.1) for a step disturbanceD/s,
and subjected to the constraints given in Eqs. (4.2), (4.3), and (4.4).
min =
∞
0
(4.2)
u (t ) ≤ u max ,
(4.3)
|u(t )| ≤ umax .
(4.4)
After some mathematical operations, the above performance index can be transformed to the form expressed in Eqs. (5.1)–(5.4),
with ζ and τ c the two new design parameters. (See the Appendix
for the derivation details).
β 1
1
+ 2
2
τc 4 ζ
ε
β
1
(τc − τ )2
3 2
= ατc ζ +
+
,
τc 4 ζ 2
τ2
Fig. 1 presents a schematic diagram of the system. The Laplace
transfer function of the system in Fig. 1 is expressed as follows:
τI s + 1
D(s ),
ε τc τI s 2 + ε τI s + 1
(1)
τ
τ 1
=
(1 + K Kc ) ε K Kc
(3.1)
1
.
1 − ττc
(3.2)
and
ε= The closed-loop damping ratio of the above system is
ζ=
1
2
ε τI 1
=
τc
2
τ τI
.
τ − τc τc
(5.1)
τc g(ζ ) ≤ γg ,
(5.2)
h(ζ , τc ) ≤ γh
(5.3)
f (ζ , τc ) ≤ γf ,
(5.4)
(2)
where
τc =
ατc3 ζ 2 +
subject to
and
U (s ) = −
(4.1)
|y(t )| ≤ ymax ,
min (ζ , τc ) =
2.1. Control system description
K τε τc τI s
Y (s ) =
D (s )
ε τc τI s 2 + ε τI s + 1
ωy (y(t ) − ysp (t ))2 + ωu (u (t ))2 dt
subject to
2. Optimal PI controller design
∗
(3.3)
where
K 2
ωu
τ ymax α = 2 ω y D
; β=
(D )2 ; γg = ;
2
K D
τ
u u γh = max ; γ f = max .
D
D
(6)
g(ζ ), h(ζ , τ c ), and f(ζ , τ c ) are given in Eqs. (A13), (A38), and
(A28.1), (A28.2) in the Appendix, respectively. In order to obtain
the coefficients in Eq. (6), the plant parameters, the weighting
factors, the constraints and the disturbance size must be known.
Please cite this article as: R. Tchamna, M. Lee, Optimization approach for the analytical design of an industrial PI controller for the
optimal regulatory control of first order processes under operational constraints, Journal of the Taiwan Institute of Chemical Engineers
(2017), http://dx.doi.org/10.1016/j.jtice.2017.08.012
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In practice, the size of the disturbance is usually unknown and
changed. Choosing a maximum value expected would be a practical way for specifying the disturbance size for controller design
purpose. Disturbance can also be estimated by using a disturbance
observer. For disturbance estimation, interested readers may refer
to [14].
As shown in Eqs. (5.2)–(5.4), the three constraints are only
function of ζ and τ c . Therefore, a simple graphical examination
of the contour of the objective function and the constraints in
(ζ , τ c )space allows to find the location of global optimal solution
without complex numerical optimization process.
2.3. Optimal PI controller design
The Lagrangian multiplier [15] can be applied in order to convert the constrained optimization problem in Eqs. (5.1)–(5.4) to an
equivalent unconstrained problem as follows:
min L(ζ , τc , 1 , 2 , 3 , σ1 , σ2 , σ3 )
= (ζ , τc ) + 1 (γh − h(ζ , τc ) − σ12 ) + 2
γ f − f (ζ , τc ) − σ32
β
1
(τc − τ )2
3 2
= ατc ζ +
+
τc 4 ζ 2
τ2
+1 (γh − h(ζ , τc ) − σ12 ) + 2 γg − g(ζ )τc − σ22
+3 γ f − f (ζ , τc ) − σ32
+3
γg − g(ζ )τc − σ22
3
the corresponding conditions of the Lagrange multipliers and slack
variables as follows:
†
†
Case A (ϖ1 = ϖ2 = ϖ3 = 0): The extreme point, (ζ , τc ), which is
located inside the feasible region, is therefore the global optimum.
Solving Eqs. (8) and (9) simultaneously, the global optimum can be
determined in explicit form as:
ζ =
†
τc† =
1
1
+ 2
2
τ
1
ζ†
β
4α
β
4α
1/2 1/2
(14.1)
1/4
(14.2)
Case B (σ 1 = ϖ2 = ϖ3 = 0): The global optimum, symbolized as
(ζ ∗h , τc∗h ), is positioned on the constraint, γ h = h(ζ , τ c ). ζ ∗h and
τc∗h can be obtained by substituting σ 1 = ϖ2 = ϖ3 = 0 into Eqs. (8)–
(10), thus solving the following system of equations:
γh − h(ζ , τc ) = 0
(15.1)
−1
∂ ∂ ∂h
∂h
−
= 0.
∂ζ ∂ τc ∂ τc
∂ζ
(15.2)
Case C (ϖ1 = σ 2 = ϖ3 = 0): For this case, the global optimum,
(ζ ∗g , τc∗g ), is located on the constraint, γ g = τ c g(ζ ). ζ ∗g and τc∗g are
obtained by substituting ϖ1 = σ 2 = ϖ3 = 0 into Eqs. (8), (9) and (11),
(7)
where ϖi and σ i are the Lagrange multiplier and the slack variable,
respectively.
The necessary conditions for an optimal solution are
∂L
∂
∂ h ( ζ , τc )
=
+ 1 −
+ 2 [−g(ζ )]
∂ τc
∂ τc
∂ τc
∂ f ( ζ , τc )
+3 −
= 0,
∂ τc
(8)
∂L ∂
∂ h ( ζ , τc )
∂ g( ζ )
∂ f ( ζ , τc )
=
− 1
− 2 τc
− 3
= 0,
∂ζ
∂ζ
∂ζ
∂ζ
∂ζ
and solving the following system of equations:
γg − g(ζ )τc = 0
(16.1)
∂
∂ ∂g
g − τc
=0
∂ζ
∂ τc ∂ζ
(16.2)
Case D (σ 1 = σ 2 = ϖ3 = 0): Using Eqs. (10) and (11), the global
gh
optimum represented by (ζ gh , τc ) is located on the intersection
gh
point by γ h = h(ζ , τ c ) andγ g = τ c g(ζ ). ζ gh and τc can be calculated by solving
γg − τc g(ζ ) = 0
(9)
(17.1)
γh − h(ζ , τc ) = 0
(17.2)
(ζ ∗ f , τc∗ f ),
∂L
= γh − h(ζ , τc ) − σ12 = 0,
∂ 1
(10)
∂L
= γg − τc g(ζ ) − σ22 = 0,
∂ 2
Case E (ϖ1 = ϖ2 = σ 3 = 0): The global optimum,
is lo∗f
cated on the constraint, γ f = f (ζ , τc ). ζ ∗f and τc can be found by
solving the following system of equations:
(11)
γf − f (ζ , τc ) = 0
(18.1)
∂L
= γ f − f (ζ , τc ) − σ32 = 0,
∂ 3
(12)
∂ f ∂ ∂ f ∂
−
=0
∂ τc ∂ζ ∂ζ ∂ τc
(18.2)
∂L
∂L
∂L
= −21 σ1 = 0;
= −22 σ2 = 0;
= −23 σ3 = 0.
∂ σ1
∂ σ2
∂ σ3
(13)
The simultaneous solutions of Eqs. (8)–(13) for possible combinations of σ i = 0, σ i = 0, ϖi = 0, and ϖi = 0 are associated with the
corresponding optimal cases. Note that instead of introducing the
slack variables, Karush–Kuhn–Tucker conditions can be utilized for
solving the optimization problem by Eqs. (5.1)–(5.4).
Fig. 2 presents seven possible cases for the location of global
optimum: the global optimum can be found inside the feasible region (case A), or on the boundary of one constraint (cases B, C, and
E), or on the intersection point of two constraints (cases D, F, and
G).
The global optima of the seven cases can be evaluated by inspecting their geometrical characteristics in (ζ , τc )space as well as
Case F (ϖ1 = σ 2 = σ 3 = 0): Using Eqs. (11) and (12), the global
gf
optimum, (ζ g f , τc ), is on the intersection point created by γ f =
f (ζ , τc ) and γ g = τ c g(ζ ). ζ gf and τc are calculated by solving the
following equations:
gf
γf − f (ζ , τc ) = 0
(19.1)
γg − τc g(ζ ) = 0
(19.2)
Case G (σ 1 = ϖ2 = σ 3 = 0): Using Eqs. (10) and (12), the global
fh
optimum, (ζ fh , τc ), is located on the intersection point of the
fh
curved γ f = f (ζ , τc ) and γ h = h(ζ , τ c ). ζ fh and τc are calculated
by solving the following set of equations:
γ f − f ( ζ , τc ) = 0
(20.1)
γh − h(ζ , τc ) = 0
(20.2)
Please cite this article as: R. Tchamna, M. Lee, Optimization approach for the analytical design of an industrial PI controller for the
optimal regulatory control of first order processes under operational constraints, Journal of the Taiwan Institute of Chemical Engineers
(2017), http://dx.doi.org/10.1016/j.jtice.2017.08.012
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Fig. 2. Typical contours and constraints for the seven possible cases with respect to the location of the global optimum.
After the global optimum is determined in (ζ , τc ) space, the optimal PI parameters corresponding to each case can then be calculated using Eqs. (3.1)–(3.3):
Kcopt =
τ
1
K
ε τ
opt
opt
c
; τIopt =
4
ε opt
(ζ opt )2 τcopt ,
(21.1)
every condition for discriminating the seven possible global optimum cases was derived.
Table 1 lists the results for the conditions and characteristics
of the global optima. Fig. 4 also presents the overall procedure for
determining the global optimum quickly.
where
ε opt = 1
opt
1 − τcτ
.
(21.2)
The conditions for the seven possible cases were also evaluated based on a meticulous analysis of the graphical shape of the
constraints and contours in (ζ , τ c ) space. The concept of the relative locations between the extreme point and its projections to the
constraints were used mainly to develop the conditions to discriminate each case associated with the global optimum.
Fig. 3 shows an example of the projection of the extreme point
and its notation rule used in this study. The notation, ζ †f , represents the abscissa of the projection of the extreme point on the
†h
†g
constraint curve by f(ζ , τ c ). Similarly, τc and τc indicate the ordinate of the projection of the extreme point on the constraint
†
†h
curves by h(ζ , τ c ) and g(ζ ), respectively. If τc is such that τc ≤
†g
†
τc ≤ τc , then the global optimum is above the constraint, h(ζ ,
τ c ), and below the constraint, g(ζ ). Referring to Fig 2, this corresponds to cases A, E or possibly G. In this case, it is apparent from
Fig. 2 that if ζ †f < ζ †, it belongs to case A (i.e., the extreme point
is the global optimum), otherwise it belongs to either case E or G.
∗f
Case E and G can be distinguished simply by comparing τc and
fh
fh
τc , where τc is the ordinate of the intersection of the constraints
∗f
fh
by f and h. As shown in Fig. 2, if τc > τc , the global optimum
case belongs to case E, otherwise case G. Using similar reasoning,
3. Design of a feasible constraint set
3.1. Effect of the constraints on the feasible region
Fig. 5 shows the effects of the constraint specifications on
the feasible region in (ζ , τ c ) space. The constraint imposed by
Eq. (5.4) lays a vertical line that shifts rightward but bends leftward as umax decreases. The constraint given in Eq. (5.3) shifts up
as u max decreases, whereas the constraint in Eq. (5.2) shifts down
as ymax decreases. Note that the feasible region exists only when
the constraint curve of u max is below that ofymax . As indicated in
Fig. 5, the feasible region shrinks as u max and ymax decrease, and
will eventually cease to exist if the constraints are lower than certain minimum allowable values.
Fig. 6 presents the tangent (ζ t , τct ), where the two constraint
curves by u max and ymax touch each other in a single point in
(ζ , τ c )space, for the different specifications of u max and ymax . The
tangent consists of the smallest feasible u max for a given ymax (or
the smallest feasible ymax for a given u max ). As ymax decreases, the
smallest feasible u max that can still yield a feasible region must increase, as expected from the inherent tradeoff between the process
variable and the manipulated variable in the control response. As
ymax decreases, the controller requires more control actions, which
indicates increases in Kc to cope with the tight constraint; thus, τct
decreases in (ζ , τ c )space, as shown in Fig. 6.
Please cite this article as: R. Tchamna, M. Lee, Optimization approach for the analytical design of an industrial PI controller for the
optimal regulatory control of first order processes under operational constraints, Journal of the Taiwan Institute of Chemical Engineers
(2017), http://dx.doi.org/10.1016/j.jtice.2017.08.012
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Fig. 3. Projection of the extreme point on the constraints in (ζ , τ c ) space.
Table 1
Global optimums of the constrained optimization problem in Eqs. (5.1)–(5.4).
Case
Constraint
specification
Condition
Global optimum
Location of global optimum
Calculation of global optimum
A
mild u max mild ymax
mild umax
ζ †f ≤ ζ † and τc†h ≤ τc† ≤
(ζ † , τc† )
In the interior of the
constraint set.
ζ † = [ 12 +
B
tight u max mild ymax
mild umax
ζ †f ≤ ζ ∗ h ≤ ζ gh and τc† < τc†h
(ζ ∗h , τc∗h )
on γ h = h(ζ , τ c )
{ ∂
C
mild u max tight ymax
mild umax
ζ †f ≤ ζ ∗ g ≤ ζ gh and τc∗ f ≤ τcg f
γ
and τc† ≥ g(ζg† )
(ζ ∗g , τc∗g )
on τc = g(ζg )
D
tight u max tight ymax
mild
[ζ ∗ h ≥ ζ gh and τc† < τc† ] or
γ
[ζ ∗ g > ζ gh and τc† ≥ g(ζg† ) ]
(ζ gh , τcgh )
E
mild u max mild ymax
tight umax
ζ † < ζ †f and τcf h ≤ τc∗ f or ζ gh >
ζ ∗ g and τcg f ≥ τc∗ f ≥ τcf h
(ζ ∗ f , τc∗ f )
F
mild u max tight ymax
ζ† > ζ
γg
g( ζ † )
h
f
∗g
>ζ
gh
∗f
c
and τ
>τ
gf
c
on the vertex by
γg
and γh = h (ζ , τc )
g( ζ )
G
tight u max mild
ymax tight umax
[ζ † < ζ †f and τc∗ f ≤ τcf h ] or
[ζ gh ≥ ζ ∗ h ≥ ζ †f and τc† < τc†h ]
(ζ f h , τcf h )
3.2. Discrimination and design of the feasible constraint set
In a constrained optimization problem, the existence of the
feasible solution region should be checked first for a given constraint set. Moreover, it is often required to control a system on
the tightest allowable constraint set. On the other hand, the three
constraint specifications cannot be selected randomly or independently due to their interrelation. For this reason, the constraint set
must be determined not only by considering the process requirements, but also by satisfying a feasibility condition.
3.2.1. Necessary condition for feasible ymax , u max , and umax
ymax and umax should be set to ymax ≥ 0 and umax
≥ |D|because lim y(t ) = 0 and | lim u(t )| = |D|. Furthermore,
t→∞
t→∞
γ h − h ( ζ , τc ) = 0
{
γg − τc g(ζ ) = 0
γ h − h ( ζ , τc ) = 0
on γ f = f(ζ , τ c )
γ − f (ζ , τc ) = 0
{ ∂ f f ∂ ∂ f ∂
on the vertex by
γ − f (ζ , τc ) = 0
{ f
γg − τc g(ζ ) = 0
γ f = f (ζ , τc ) and τc =
tight umax
]1/2 ; τc† = ζ1† ( 4βα )1/2
γg − g(ζ )τc = 0
{ ∂
∂ ∂g
∂ζ g − τc ∂ τc ∂ζ = 0
τc =
gf
c
1/2
( βα )
∂ ∂ h −1 ∂ h
∂ζ − ∂ τc [ ∂ τc ] ∂ζ = 0
γ
(ζ , τ )
gf
1
2τ 2
∂ τc ∂ζ − ∂ζ ∂ τc = 0
γg
g( ζ )
on the vertex by γ f = f(ζ ,
τ c ) and γ h = h(ζ , τ c )
{
γf − f (ζ , τc ) = 0
γ h − h ( ζ , τc ) = 0
when ζ approaches ∞, Eq. (A38) becomes
lim h(ζ , τc ) =
ζ →∞
τ − τc
= γh .
τ τc
(22)
Let τch (∞ ) be the solution of Eq. (22). τch (∞ ) can be found as
τch (∞ ) = γ ττ+1 . Hence, τch (∞ ) is always positive as long as u max
h
≥ 0. Therefore, ymax ,u max , and umax should be set at least to satisfy the following:
ymax ≥ 0, umax ≥ |D|, umax ≥ 0
(23)
3.2.2. Condition for a feasible (ymax , u max )
For a given ymax , there is a minimum available u max value below which the optimal control problem is not feasible. Let u t max
be the tightest/smallest possible value of u max . As stated in the
Please cite this article as: R. Tchamna, M. Lee, Optimization approach for the analytical design of an industrial PI controller for the
optimal regulatory control of first order processes under operational constraints, Journal of the Taiwan Institute of Chemical Engineers
(2017), http://dx.doi.org/10.1016/j.jtice.2017.08.012
JID: JTICE
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Fig. 4. Flow chart for finding the global optimum and PI parameters.
Fig. 5. Effect of the constraint specifications, ymax , umax , and u max , on the feasible region.
Please cite this article as: R. Tchamna, M. Lee, Optimization approach for the analytical design of an industrial PI controller for the
optimal regulatory control of first order processes under operational constraints, Journal of the Taiwan Institute of Chemical Engineers
(2017), http://dx.doi.org/10.1016/j.jtice.2017.08.012
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Fig. 6. Tangent points by the constraint curves of u max and ymax .
previous section, u t max can be obtained when ζ = ζ t . ζ t can be calculated by solving the following equation:
dh
dh(ζ , τc )
=
dζ
ζ , γg g−1
= 0.
dζ
(24)
umax = h ζ t |D|.
h
ζ , γg g
−1
and
dh
= γh
ζ , γg g−1
=0
dζ
(25)
(26)
(27)
Finally, if ymax ≥ ytmax , the constraint set (ymax , u max ) is feasible. Otherwise, it is not feasible.
3.2.3. Feasible umax for a given feasible set (ymax , u max )
For a feasible (ymax ,u max ), it can be inferred intuitively from
geometrical analysis of the constraint curves that any positive
umax will lead to a feasible region if there is no intersection between the constraint curves by τ c g(ζ ) = γ g and h(ζ , τ c ) = γ h . Furthermore, in the case where the intersection, ζ gh , exists, the constraint, umax , is feasible if ζ gh ≥ ζ gf . To evaluate the existence of an
intersection between the two constraint curves, i.e., by τ c g(ζ ) = γ g
and h(ζ , τ c ) = γ h , it is important to calculate τ c (∞), which is
the value of τ c where the two constraint curves are secant when
ζ → ∞. τ c (∞) of each constraint curve can be obtained by solving
the following equations:
[τc (∞ )g(ζ ) = γg ]ζ →∞
Because
(28.2)
lim g(ζ ) = 1 and
ζ →∞
lim h(ζ , τc ) = (τ − τc )/τ τc = γh ,
ζ →∞
τcg (∞ ) = γg
t , then the constraint set (y
Finally, if u max ≥ umax
max , u max ) is
feasible. Otherwise, it is infeasible.
Similarly, for a given u max , there is a minimum available ymax
value below which the optimal control problem is not feasible. Let
ytmax be the tightest/smallest possible ymax . The values of ytmax and
ζ t can be obtained by solving the following system of equations
simultaneously:
{h[ζ , τc (∞ )] = γh }ζ →∞
τ c (∞) of the two constraint curves are
t
Once ζ t is obtained, ymax
can be derived as follows:
t
and
(28.1)
and
τch (∞ ) =
(29.1)
τ
.
γh τ + 1
(29.2)
A vertex ζ gh exists when τc (∞ ) ≥ τch (∞ ). Therefore,
g
τ
γg ≥
.
γh τ + 1
(30)
Substitution of Eq. (6) for γ g and γ h into Eq. (31) yields
ymax
|K D|
u
τ max + 1 ≥ 1.
|D|
(31)
Overall, for a given feasible (ymax ,u max ), umax is feasible if
either 1) Eq. (31) is not satisfied or 2) Eq. (31) is satisfied and
ζ gh ≥ ζ gf . Otherwise, umax is not feasible and should be increased until one of the conditions is satisfied. Note that if
ymax
u max
|K D| (τ |D| + 1 ) < 1, no vertex point is formed by γ g = τ c g(ζ )
and γ h = h(ζ , τ c ), i.e., case D does not exist. This situation is likely
to happen if the specification, u max and/or ymax , are set mildly
with a large value. In this situation, ζ gh can be considered as a sufficiently large value, for an evaluation of the conditions presented
in Table 1 and Fig. 4.
Fig. 7 presents the overall procedure to test the feasibility of
a constraint set (ymax ,u max ,umax ) and design a feasible constraint
set.
4. Simulation of the closed-loop performance
Consider the following stable first order process as
G p (s ) =
10
10s + 1
(32)
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The proposed PI controllers provide the optimal responses satisfying the ymax , umax , and u max constraints strictly.
5. Discussion
5.1. Effect of the weighting coefficient
Fig. 7. Overall procedure for designing and checking a feasible constraint set.
Table 2
Constraint specifications and resulting optimal PI parameters.
Example
1
2
3
4
5
6
7
case
A
B
C
D
E
F
G
Constraint specification
PI parameter
ymax
umax
u max
KC
τI
0.70
0.70
0.36
0.285
0.70
0.30
0.70
2.70
2.70
2.70
2.70
1.105
1.20
1.20
2.70
1.11
2.70
2.10
2.70
2.70
1.37
1.32
1.11
1.83
2.10
1.76
2.07
1.37
1.32
1.11
1.43
0.76
2.18
0.92
1.36
The values of the weighting coefficients were selected arbitrarily as ωy = ωu = 0.5. Examples of cases A–G were simulated for
various constraint specifications as listed in Table 2. Figs. 8 and 9
present the responses of y(t), u(t), and u (t) for the seven examples.
The weighting coefficient, ω, is a main parameter to regulate
the tradeoff between the robustness and performance in optimal
control. The weighting factor ωy is used to emphasize on the control performance. As ωy increases, the optimal control results in a
tighter control response. While on the other, if the main control
objective is to have a smooth control action rather than fast response, then larger values of ωu should be chosen. Fig. 10 presents
the effects of the weighting coefficient on the trajectories of the
global optimum and the extreme point as the weighting coefficient
increases. The process in Eq. (32) was considered with ymax = 0.36,
umax = 2.7, u max = 2.7, and ωu = 0.5.
When ωy is small, e.g. ωy = 0.05 (or conversely ωu is large
enough), the variation of u (t) mainly determines the performance
measure of the optimal control. Accordingly, the controller yields
a tight response of u (t) and the response is possibly constrained
by ymax . In this case, the extreme point is likely to be above the
boundary of γ g = g(ζ ) in (ζ , τc ) space, and the global optimum
will be on γ g = g(ζ ), i.e., case C. For a larger ωy , the variation
of y(t) governs the performance measure more, and the optimal
controller yields a tight response of y(t). Accordingly, the extreme
point shifts down in (ζ , τc ) space, as predicted from Eq. (14.2), and
becomes the global optimal point, i.e., case A. This is physically
logical in that as a largerωy is applied, a larger Kc is required to
obtain a tight response of the process variable, which corresponds
to the lower position of the extreme point in (ζ , τ c ) space. As ωy
is increased further to a sufficiently large value, the optimum is
then likely to be on the constraint, τc2 = γh h(ζ ), i.e., case B, where
the response is constrained by the u max specification, as seen in
Fig. 10.
Fig. 11 presents the corresponding closed-loop responses of
the proposed optimal PI controller for the process examined in
Fig. 8. Responses of y(t), u (t), and u(t) for cases A to D.
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Fig. 9. Responses of y(t), u (t), and u(t) for cases E to G.
Fig. 10. Effects of the weighting factor on the trajectories of the extreme point and global optimum in (ζ , τ c ) space.
Fig. 10 for different weighting factors of ωy = 0.05, 0.85, and 0.98.
The inherent trade-off between the process variable and the manipulated variable can be seen from this figure. An increase in
ωy leads to a faster and tighter control response as increases,
and conversely, a decrease in ωy (or an increase in ωu ) makes
the manipulated variable and its rate smoother and smaller. For
ωy = 0.05, only ymax constrains the optimal control response, i.e.,
case C. For ωy = 0.85, the process variable departs from the constraint, ymax . None of the three constraints constrains the optimal responses, i.e., case A. Thereafter, the constraint on the
controlled variable is released completely as ωy increases to
ωy = 0.98. Instead, u max constrains the optimal response, i.e., case
B. These features of the optimal responses as the weighting factor varies match perfectly with those indicated in Fig. 10. The proposed optimal controller strictly satisfies all the constraints specifications on ymax , umax , and u max , regardless of the value of the
weighting factor.
opt
Remark 1. For case A or an unconstrained case, the ratio of Kc to
τIopt is related only to the weighting factors ωy and ωu as follows:
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Fig. 11. Closed-loop responses by the proposed PI controller for various weighting coefficients.
Fig. 12. Effect of the process gain on the extreme point and global optimum in (ζ , τ c ) space.
Kcopt
opt
I
τ
=
ωy
.
ωu
the process variable and that of the rate of change of the manipulated variable.
(33)
Eq. (33) shows and interesting result: the ratio of the optimal
PI control parameters for the regulatory control of first order systems is totally independent on process parameters. Whatever the
values of the plant parameters, the ratio of the optimal PI control
parameters only depends on the ratio of the weighing factors of
5.2. Effect of the process parameters
The process gain and time constant affect the trend and location of the constraints as well as the location of the global optimum in (ζ , τ c ) space, which results in different behaviors of
the optimal controller. Figs. 12 and 13 show how the extreme
point, constraint curves, and global optimum vary according to
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Fig. 13. Effect of the time constant on the extreme point and global optimum in (ζ , τ c ) space.
Fig. 14. Closed-loop responses by the proposed PI controller for various process gains.
the time constant and process gain. The process in Eq. (32) was
considered again with ymax = 0.36, umax = 2.7, u max = 2.7 ,
and ωy = ωu = 0.5 but with varying process parameters.
As shown in Fig. 12, as the process gain increases, the constraint
curve byymax and the extreme point move down in (ζ , τ c )space
while both the constraint curves by umax and u max are independent of the process gain. This tendency of the extreme point and
each constraint for varying the process gain is matched well with
that predicted from each corresponding equation, i.e., Eqs. (14.2),
(A13), (A38) , (A28.1) and (A28.2). As a result, in this particular
example, as the process gain increases from K = 3–15, the optimal response shifts from the unconstraint case, i.e., case A, through
case C constrained by ymax , and finally to case D constrained by
both ymax and u max .
On the other hand, as shown in Fig. 13, the extreme point
and the constraint curve by ymax shift down in (ζ , τ c ) space, as
Please cite this article as: R. Tchamna, M. Lee, Optimization approach for the analytical design of an industrial PI controller for the
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Fig. 15. Closed-loop responses by the proposed PI controller for various process time constants.
the time constant decreases. The constraint curve by umax rotates
slightly leftwards as the time constant decreases, whereas that by
u max is insensitive to changes in the time constant. These observations match well with those predicted from their corresponding
equations. As a result, as the process time constant is decreased
from τ = 20 to 7, the optimal response of this process shows unconstraint case A, then case C, and finally case D.
Figs. 14 and 15 show the corresponding closed loop responses
for various process gains and time constants examined in Figs.
12 and 13, respectively. As observed in Fig. 14, the optimal response shifts from the unconstraint case A to case C and finally
to case D as the process gain increases from K = 3–15. The reverse
phenomenon can be observed in Fig. 15; an increase in the process
time constant makes the system shift from case D to C and then to
A. These observations validate what are observed in (ζ , τ c ) space
in Figs. 13 and 14. Note that the proposed PI controller always provides the optimal responses with no constraint violations in every
case examined.
The proposed optimization-based analytical approach offers the
following useful insights, which are difficult to obtain using blackbox or numerical approaches, into the optimal design of the stable
process with operational constraints: (1) this approach can explain
the effects of the process and control parameters on the optimal
responses; (2) it can determine if a constraint set is feasible in the
constrained optimal problem, and if not, it suggests how to design
the operational constraints to make the problem feasible; and (3)
it provides the optimal PI parameters of the constrained optimal
control problem without the need for complex optimization packages, and guarantees its stability and global optimality. The same
analysis presented in this paper can be extended to more general
cases such as three term PID controllers, higher order and dead
time processes by taking a hybrid method combining analytical
and numerical approaches and/or a dead time compensator such
as Smith Predictor and IMC.
Remark 2. The validity of the results in this paper is under the assumption that the system model is a perfect model. When a process model is not perfect, the responses of the system will possibly
violate the constraints. One practical way to cope with the constraint violation problem by the plan-model mismatch would be
to use a conservative or tighter constraint value considering these
uncertainties.
This work was supported by the 2016 Yeungnam University Research Grant.
6. Conclusions
y(t ) =
A novel analytical design method was developed for the optimal
regulatory control of stable first order process under operational
constraints. The original constrained optimal control problem was
formulated in (ζ , τ c ) space using clever parameterization, and then
converted to an equivalent unconstrained problem using the classical Lagrangian multiplier method to derive the analytical solutions
for the optimal PI parameters. Seven possible cases for the constrained optimal control problem were then analyzed in terms of
the location of the global optimum in (ζ , τ c ) space.
Acknowledgments
Appendix A. Proof of the Performance Index in Eq. (5.1)
When a step change in the disturbance, D(s) = D/s is applied
to the stable first order process presented in Fig. 1, y(t) and u (t)
given in Eqs. (1) and (2) are derived as
D r 1 r 2 τ I e r 1 t − e r 2 t
Kc
u (t ) = −D
r1 − r2
for r1 = r2
(A1)
r r 1 2
τI r1 er1 t − r2 er2 t + er1 t − er2 t for r1 = r2 ,
r1 − r2
(A2)
where r1 and r2 are the roots of the characteristic equation,
ε τc τI s2 + ε τI s + 1 = 0.
r1 =
−1 + xi
−1 − xi
, r2 =
2 τc
2 τc
(A3)
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and
x=
∀ζ , g(ζ ) > 0; consequently, Eq. (4.2) can be expressed as
1 − ζ2
for 0 < ζ ≤ 1
.
ζ
ζ2 − 1
=
for ζ > 1
ζ
(A4)
= ωy
ωy
∞
∞
(y(t ) )2 dt + ωu
0
2 2
D τ I
r1 r2
1
0
Kc
r1 − r2
−
+
2
u (t ) dt
2r1
r1 + r2
2
1
−
+
2r1
r1 + r2
2r2
2r2
s +
1
τc
s+
1
ε τc τI
2
n
τc
; r1 r2 = ωn2 =
1
ε τc τI
− cos
x τ − 2τ
x c
t +
sin
t
2 τc
xτ
2 τc
(A15)
(A5)
(A6)
; r1 − r2 =
t
τ − 2 τc
−1 +
t
for ζ = 1
2τ
2τ τ
t c x c τ −2τ
x c
= −D 1 + exp −
− cosh
t +
sinh
t
2 τc
2 τc
xτ
2 τc
for ζ > 1
From the characteristic equation of the system, along with Eq.
(A3), the roots of the characteristic equation satisfy the following
set of equations:
−1
t
2 τc
The steady-state value of u(t) is
uss (t → ∞ ) = −D.
= s + 2ζ ωn s + ω = 0.
r1 + r2 = − 2ωn ζ =
= −D 1 + exp −
.
2
u(t ) = −D 1 + exp −
The characteristic equation of the system is given as
2
(A14)
u(t) for a step change in the disturbance can be derived using
the inverse Laplace transform of Eq. (2) as follows:
for 0 < ζ < 1
r r 2 r
r
2r1 r2
1 2
+ ωu (−D )2
τI2 − 1 − 2 +
r1 − r2
2
2
( r1 + r2 )
2
1
1
+
ypeak < ymax ⇒ |K D| τc g(ζ ) ≤ ymax .
τ
Appendix C. Proof of Eq. (5.4)
Note that the above notation for xare used all along this Appendix. The optimal control performance index can be derived as
follows:
=
13
xi
τc
(A7)
(A16)
Furthermore, the peak time for the largest peak of u(t) can be
obtained from du(t)/dt = 0 as follows:
⎧
2
⎪
2 τc
⎪
−1 2ζ (τc − τ ) ζ − 1
⎪
tanh
for ζ > 1
⎪
⎨ x
2 ζ 2 ( τc − τ ) + τ
tupeak =
.
⎪
2 ζ ( τc − τ ) 1 − ζ 2
2 τc
⎪
−1
⎪
tan
+ kπ , k ∈ Z for ζ < 1
⎪
⎩
x
2 ζ 2 ( τc − τ ) + τ
By substituting Eq. (A7) into Eq. (A5), can be found as follows:
β 1
1
β
1
(τc − τ )2
3 2
= ατc ζ +
+
= ατc ζ +
+
.
τc 4 ζ 2 ε 2
τc 4 ζ 2
τ2
3
2
(A8)
Appendix B. Proof of Eq. (5.2)
y(t) for a step change in the disturbance can be obtained using
the inverse Laplace transform of Eq. (1) as follows:
x t
2DK τc
exp −
sin
t
y(t ) =
τ x 2 τc
2 τc
t
DK
for 0 < ζ < 1
(A9)
Therefore, the steady-state value of y(t) is
yss (t → ∞ ) = 0.
(A10)
Taking the derivative of y(t), the peak time for the largest peak
of y(t) can be obtained as
2τc atan(x )
x
= 2 τc
2τc atanh(x )
=
x
tpeak =
for
0<ζ <1
for
ζ =1
for
ζ >1
.
where
K
τ
τc g ( ζ ),
(A11)
(A12)
tupeak <0 ⇔ 2ζ (τc −τ )+τ > 0 ⇒ ζ <
τ
2 ( τ − τc )
=
ε
2
= ζ0 f
(A18)
To avoid a negative peak time, and also to ensure that
the peak time will be the smallest peak time among all
the peak time sequences, the peak time must be chosen as
follows:
if
ζ < ζ0 f
tupeak =
else
Therefore, the peak of y(t) is as follows:
ypeak = D
However, the theoretical values of the peak times do not always
correspond to the actual peak time.
Since τc = 1+τK Kc ⇒ ττc = 1 + K Kc > 1. This means that τ > τ c ,
hence τ c − τ < 0.
for ζ < 1, the peak time of u(t) must satisfy two conditions:
tupeak must be the smallest time among the time series in (A17),
and must be a positive value. For k = 0, e.g., the peak time will be
negative if the following relation is satisfied
2
exp −
t
for ζ = 1
τ
2τc
x t
2DK τc
=
exp −
sinh
t
for ζ > 1.
τ x
2 τc
2 τc
=
(A17)
tupeak =
2 τc
2ζ 2 (τc − τ )x
tan−1
x
2 ζ 2 ( τc − τ ) + τ
2 τc
2ζ 2 (τc − τ )x
tan−1
x
2 ζ 2 ( τc − τ ) + τ
+π
.
(A19)
Thus, the peak time for ζ < 1 can be
⎧
2 τc
2ζ 2 (τc − τ )x
⎪
tan−1
+π
⎪
2
⎪
2 ζ ( τc − τ ) + τ
⎪
⎪ x
⎪
⎨
2ζ 2 (τc − τ )x
tupeak = 2τc tan−1
2
⎪
x
2 ζ ( τc − τ ) + τ
⎪
⎪
⎪
⎪
⎪ 2 ( τc − τ ) ⎩
2 τc
2 τc − τ
expressed as
for 0 < ζ < ζ0 f and
for
ζ0 f ≤ ζ < 1
ζ <1
.
tan −1 x
2
g( ζ ) = √
exp −
for 0 < ζ < 1
for ζ = 1
2
x
1+x
= 2 exp(−1 ) ,
(A13)
(A20)
for ζ = 1
−1
tanh x
2
= cite this exp
−
for ζ > 1
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For ζ > 1, it should be noted that tupeak > 0⇒ζ > ζ 0f .
On the other hand, since lim [tanh(x )] = 1, the following relation
x→∞
tanh (x) = λ physically holds only if λ ≤ 1. Hence,
x
2ζ (τ − τ )ζ 2 − 1
c
tanh
t
=
2τc upeak
2 ζ 2 ( τc − τ ) + τ
2 ζ ( τc − τ ) ζ 2 − 1
⇒
≤1
2 ζ 2 ( τc − τ ) + τ
2 ζ ( τc − τ ) ζ 2 − 1
1
τ
=1⇒ζ = = ζ1 f
2 τc ( τ − τc )
2 ζ 2 ( τc − τ ) + τ
τ
ζ >1
τ
τc ≥
2
f ( ζ , τc ) = 1
else
(A21)
(A22)
(A23)
2 τc
for
if
ζ 0f and ζ 1f need to be compared. ζ 1f /ζ 0f can be expressed as
ζ1 f
=
ζ0 f
and
⎧
4 −τ τc + τc2 ζ 2 + τ 2
⎪
⎪
⎪
1+
⎪
⎪
τ 2
⎪
⎪
⎪
2
⎪
⎨ exp − 1 tanh−1 2ζ (τc − τ )x
x
2 ζ 2 ( τc − τ ) + τ
f ( ζ , τc ) =
⎪
for
ζ
<
ζ
<
ζ
⎪
0f
1f
⎪
⎪
⎪
⎪
⎪
1 for ζ = ζ1 f
⎪
⎪
⎩
undefined f or ζ > ζ1 f
(A28.2)
Hence,
ζ1 f > ζ0 f
for
ζ1 f < ζ0 f
for
τc <
τc >
ζ1 f = ζ0 f = 1 for τc =
τ
with
2
τ
(A24)
2
f or
ζ ≥ ζ1 f , tanh
x
tupeak = 1 ⇒ tupeak = ∞.
2 τc
(A25)
τc >
τ
upeak = D f (ζ , τc )
where
f ( ζ , τc ) = 1 +
4 −τ τc + τ
2
c
2
τ
ζ +τ
(A29)
(A30)
u (t) can be derived by taking the derivative of equation (A15)
t τ − τ x D
c
exp −
cos
t
τc
2 τc
τ
2 τc
1 − x 2 τ − 2 τc
x
−
sin(
t)
for 0 < ζ < 1
2 xτ
2 τc
t τ − 2τ t
D
τ −τ u (t ) = −
=−
exp −
c
−
+
τc
2 τc
τ
4 τc
τ
t τ − τ x D
c
=−
exp −
cosh
t
τc
2 τc
τ
2 τc
1 + x 2 τ − 2 τc
x
−
sinh(
t)
for ζ > 1
2 xτ
2 τc
c
for ζ =1
(A31)
The steady-state value of u (t) is
(A26)
uss (t → ∞ ) = 0.
(A27)
f or
(A32)
The peak time of u (t) can be obtained by solving du (t)/dt = 0
as follows:
ζ < 1, 2 τc
tu peak =
2
1
τ
2 τc ( τ − τc )
Appendix D. Proof of Eq. (5.3)
elsei f
Therefore,
ζ1 f =
;
|D| · f (ζ , τc ) ≤ umax
2
tupeak = ∞
τ
τc <
⎧ 2
2
⎪
2 τc
−1 2ζ (τc − τ ) ζ − 1
⎪
⎨ tanh
for ζ0 f < ζ < ζ1 f
x
2 ζ 2 ( τc − τ ) + τ
tupeak =
⎪
for ζ = ζ1 f
⎪∞
⎩
unde f ined
for ζ > ζ1 f
else
2 τc
ζ
−1
tupeak =
tanh
x
ζ2 − 1
τ
2 ( τ − τc )
Note that ∀(ζ , τ c ),f(ζ , τ c ) > 0.
Therefore, Eq. (4.4) can be converted to
2
Using (A24) and (A25), the peak time for ζ > 1 can be expressed as
if
ζ0 f =
τ
Note that ζ 1f can be seen like a cutoff damping ratio. This
means that, if ζ > ζ 1f , set ζ = ζ 1f . This implies that
x
tan
−1
4 ζ 2 ( τc − τ ) + τ
4 ζ 2 ( τc − τ ) − 2 τ c + 3 τ
1 − ζ2
ζ
+ kπ , k ∈ Z
2
1
2ζ 2 (τc − τ )x
× exp − tan−1
x
2 ζ 2 ( τc − τ ) + τ
(A33)
−
π
x
ζ <1
4 −τ τc + τc2 ζ 2 + τ 2
= 1+
τ2
1
2ζ 2 (τc − τ )x
× exp − tan−1
for ζ0 f ≤ ζ <1
x
2ζ 2 (τc −τ )+τ
2 τ −τ |τ − 2τc |
( c
)
= 1+
exp −
for ζ = 1 (A28.1)
τ
2 τc − τ
For k = 0, for example, the sign of the peak time is given as
tu peak > 0 for
ζ < ζ01h or ζ > ζ02h
< 0 for
for 0 < ζ < ζ0 f and
with
ζ01h =
τ
ζ01h < ζ < ζ02h
4 ( τ − τc )
;
ζ02h =
Considering that (1) tu
peak
3 τ − 2 τc
.
4 ( τ − τc )
(A34)
(A35)
must be the smallest positive time
value among the above series of times, and (2) for ζ > ζ 01h , the
Please cite this article as: R. Tchamna, M. Lee, Optimization approach for the analytical design of an industrial PI controller for the
optimal regulatory control of first order processes under operational constraints, Journal of the Taiwan Institute of Chemical Engineers
(2017), http://dx.doi.org/10.1016/j.jtice.2017.08.012
ARTICLE IN PRESS
JID: JTICE
R. Tchamna, M. Lee / Journal of the Taiwan Institute of Chemical Engineers 000 (2017) 1–15
theoretical maximum of |u (t)| obtained from du (t)/dt = 0 is always
lower than |u (0)|, and tu
can be expressed as
tu peak =
peak
2 τc
4 ζ ( τc + τ ) − τ
tan −1
x
4 ζ 2 ( τc + τ ) − 2 τc − 3 τ
2
1−ζ
2
ζ
ζ < ζ01h
= 0 for ζ ≥ ζ01h
for
(A36)
In fact, it can be shown graphically that for ζ ≥ ζ 01h , the curves
of |u (t)| start from the value 1 at t = 0, then decrease and start oscillating with the maximum peaks lower than 1. This means that
the mathematical value of the peak of |u (t)|is not the actual peak.
The actual peak when ζ ≥ ζ 01h is obtained at the initial time.
Therefore, the peak of u (t)can be expressed as
upeak = Dh(ζ , τc )
(A37)
with
h ( ζ , τc ) =
1
2 τ τc ζ
4 τc ( τc − τ ) ζ 2 + τ 2
1
× exp − tan−1
x
u ( 0 ) = τ − τc
= D τ τc
4ζ 2 (τc −τ )+τ
x
2
4ζ (τc −τ )+3τ − 2τc
for
ζ ≥ ζ01h
for
ζ < ζ01h
(A38)
Therefore, Eq. (4.3) can be expressed as
|D|h(ζ , τc ) ≤ umax
(A39)
[m5G;August 24, 2017;12:55]
15
References
[1] Chiu T, Christofides PD. Nonlinear control of particulate processes. AIChE J
1999;45:1279–97.
[2] Liou C-T, Chien YS. The effect of nonideal mixing on input multiplicities in a
CSTR. Chem Eng Sci 1991;46:2113–16.
[3] Rao AS, Chidambaram M. Control of unstable processes with two RHP poles, a
zero and time delay. Asia Pacific J Chem Eng 2006;1:63–9.
[4] Ali E, Al-humaizi K. Temperature control of ethylene to butene-1 dimerization
reactor. Ind Eng Chem Res 20 0 0;39:1320–9.
[5] Uma S, Chidambaram M, Rao AS. Enhanced control of unstable cascade processes with time delays using a modified smith predictor. Ind Eng Chem Res
2009;48:3098–111.
[6] Agrawal P, Lim HC. Analysis of various control schemes for continuous bioreactors. Adv Biochem BioTechnol 1984;30:61–90.
[7] Seki H, Ogawa M, Ooyama S, Akamatsu K, Ohshima M, Yang W. Industrial application of a nonlinear model predictive control to polymerization reactors.
Control Eng Pract 2001;9:819–28.
[8] Shin J, Lee J, Park S, Koo K-K, Lee M. Analytical design of a proportionalintegral controller for constrained optimal regulatory control of inventory loop.
Control Eng Pract 2008;16:1391–7. http://dx.doi.org/10.1016/j.conengprac.2008.
04.006.
[9] Lee M, Shin J. Constrained optimal control of liquid level loop using a conventional proportional-integral controller. Chem Eng Commun 2009;196:729–45.
doi:10.1080/00986440802557393.
[10] Lee M, Shin J, Lee J. Implement a constrained optimal control in a conventional
level controller. Hydrocarb Process 2010;89:71–6.
[11] Lee M, Shin J, Lee J. Implement a constrained optimal control in a conventional
level controller. Hydrocarb Process 2010;89:81–5.
[12] Nguyen VH, Yoshiyuki Y, Lee M. Optimization based approach for industrial
PI controller design for optimal servo control of integrating process with constraints. J Chem Eng Jpn 2011;44:345–54.
[13] Thu HCT, Lee M. Analytical design of proportional-integral controllers for the
optimal control of first-order processes with operational constraints. Korean J
Chem Eng 2013;30:2151–62. doi:10.1007/s11814- 013- 0153- 1.
[14] Xiong Y, Saif M. Unknown disturbance inputs estimation based on a state functional observer design. Automatica 2003;39:1389–98.
[15] Vapnyarskii IB. Lagrange multipliers. Heidelberg: encyclopedia of mathematics.
Springer; 2001.
Please cite this article as: R. Tchamna, M. Lee, Optimization approach for the analytical design of an industrial PI controller for the
optimal regulatory control of first order processes under operational constraints, Journal of the Taiwan Institute of Chemical Engineers
(2017), http://dx.doi.org/10.1016/j.jtice.2017.08.012
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