Model Based Control
1
MODEL BASED CONTROL
1.
INTRODUCTION
In practice, there are two mutually exclusive, approaches to solving the process control problem:
1. The primary basis is the hardware element available for implementing the controller, with the
desired output behavior only of secondary concern;
2. The primary basis is the desired output behavior, with the hardware element for controller
implementation of secondary concern.
The first is the conventional feedback control: with the prespecified hardware element (the PID
controller), the desired output behavior must be defined somewhat more loosely, so that feasible
solutions can be found within the constraints imposed by the prespecified controller structure.
The second is the model-based control: the desired output behavior is precisely, and explicitly,
specified, and using the process model explicitly, the controller required to obtain the
prespecified behavior is derived. However, the resulting controllers may sometimes take the
familiar PID form.
There are two approaches to model-based control system design
1. Direct Synthesis: the desired output behavior is specified in the form of a trajectory, the
process model is used directly to synthesize the controller required to cause the process
output to follow this trajectory. Some of the particular techniques falling under this category
here are: Direct Synthesis Control, Internal Model Control and Generic Model Control (for
nonlinear systems).
2. Optimization: the desired output behavior is specified in the form of an objective function
(which may or may not involve an output trajectory explicitly), and the process model is used
to derive the controller required to minimize (or maximize) this objective. It is also possible
to include some known operating constraints in the optimization objective.
For optimization approachs, consider the process whose dynamics are represented by the
differential equation:
dy
= f ( y, u , d )
dt
by setting up an objective functional of the general form:
tf
Φ = G ( y (t f )) + ∫ F ( y, u, d )dt
0
where y (t f ) is the value taken by the process output at the final time t = tf; and G(.) and F(. , ., .)
are functions chosen by the control system designer to reflect the aspect of the process behavior
to be optimized. The idea is to find the control policy u(t) that minimizes (or maximizes) Φ ,
perhaps even subject to some operating constraints as:
KNU/EECS/ELEC835001
Dr. Kalyana Veluvolu
Model Based Control
2
g ( y, u ) ≤ 0
h( y , u ) = 0
In optimization approach:
1. When control action is determined according to optimal control strategy, the trajectory
followed by y is not prespecified; rather, the "best" possible trajectory, consistent with
the process model and the objective will be found and this followed by the process
output.
2. The optimal control strategy is automatically equipped to consider process operating
constraints..
3. The general optimal control problem as posed above can not be solved in closed form, or
even implicitly, and quite often intensive computation is required.
The application of these optimization techniques to the design of discrete-time multivariable
controllers for industrial processes has led to the evolution of a class of control scheme known
as Model Predictive Control; this will be discussed in somewhere else. In this topic will will
concentrate ourself to “Direct Synthesis Approaches”.
In early studies, we have already seen s controllers: the feedforward control, Smith predictor,
and the inverse-response compensator are model-based controllers. They all use the process
model explicitly and the process model is used to derive the resulting controllers; and without
making the issue of the hardware required for implementation the foremost concern, the primary
basis for each technique is a clearly defined output behavior objective. For feedforward control,
it is "perfect" disturbance rejection, while for the Smith predictor and the inverse-response
compensator it is elimination the time delay term and the right-half plane zero, respectively,
from the closed loop transfer function
2.
DIRECT SYNTHESIS CONTROLLER
2.1 Basic Concepts
The purpose of a controller is to "shape" the response of the closed loop system. The response
depends on the poles of the system.
From Fig. 2.1, it follows that,
Fig. 2.1 Feedback Control system
y ( s) =
gg c
yd ( s )
1 + gg c
(2.1)
Suppose we require that
KNU/EECS/ELEC835001
Dr. Kalyana Veluvolu
Model Based Control
3
y ( s ) = q ( s ) yd ( s )
(2.2)
where q(s) is a specific pre-determined transfer function of our choice. The choice of the
reference trajectory q(s) depends on the type of closed loop response desired. Then, the
controller synthesis problem is now posed as follows:
What is the form of the controller gc required to produce in the process the closed-loop
behavior represented by the reference trajectory q(s)?
Comparing (2.1) and (2.2),
gg c
1 + gg c
(2.3)
1 q
(
)
g 1− q
(2.4)
1
τ r s +1
(2.5)
q( s) =
Or,
gc =
Let the desired closed loop behaviour be
q( s) =
Substituting (2.5) into (2.4),
1
1 τ s +1
1
gc = ( r
)=
g 1− 1
τ r sg
τ r s +1
(2.6)
Once the process model is available, the controller can be easily obtained. It can also be shown
that even when the actual plant dynamics are not matched exactly by the model transfer function
g, provided the overall closed-loop system is stable, this direct synthesis controller does not
result in steady-state offset.
Before investigating the various types of controllers prescribed by Eq. (2.4) for various types of
process models, it is important to note that this direct synthesis controller involves the inverse of
the process model. This feature, common to all model-based controllers, will have significant
implications later on, particularly when we consider the issue of synthesis for nonminimum
phase systems.
2.2 Synthesis of Low Order Systems
Using (2.4) and (2.6), we have
Pure Gain Process: Let
g ( s) = K
gc =
KNU/EECS/ELEC835001
1
Kτ r s
Dr. Kalyana Veluvolu
Model Based Control
4
The controller is a pure Integral controller.
Pure Capacity Process: Let
g ( s) =
K
s
Then,
1
Kτ r
gc =
The controller is a pure proportional controller.
First Order Process: Let
g ( s) =
K
τ s +1
Then using (2.4) and (2.5),
gc =
τ s +1 τ ⎛
1⎞
=
⎜1 + ⎟
Kτ r s Kτ r ⎝ τ s ⎠
The controller is a PI controller with
K c = τ / Kτ r ;
τI =τ.
Example: Direct Synthesis Controller for First Order System
Design a controller for the following first-order system
g ( s) =
0.66
6.7 s + 1
using the direct synthesis approach, and given that the desirel closed loop behavior as in Eq.
(2.5), with τ r = 5 . Compare this controller with that τ r = 1 .
Solution:
The required direct synthesis controller (for τ r = 5 ) is a PI controller with Kc = 2.03 and
τ I = 6.7 .
1 ⎞
⎛
gc = 2.03 ⎜ 1 +
⎟
⎝ 6.7 s ⎠
The controller obtained with the choice τ r = 1 (the faster closed-loop trajectory) is
1 ⎞
⎛
gc = 10.15 ⎜1 +
⎟
⎝ 6.7 s ⎠
and the only difference between these controllers is that the latter has a proportional gain value
which is five times that of the former; the integral times are identical. As we would expect, a
controller with a higher gain value is required to provide the faster closed-loop response
KNU/EECS/ELEC835001
Dr. Kalyana Veluvolu
Model Based Control
5
indicated by the smaller value chosen for τ r . Further, it should be noted that a reduction in the
value of τ r by a factor of 5 resulted in a fivefold increase in the value of Kc .
3.3 Synthesis of Higher Order Systems
For a second order system,
g (s) =
K
τ s + 2ς s + 1
2 2
The direct synthesis controller , using (2.4) and (2.5), is then given by
gc ( s) =
τ 2 s 2 + 2ςτ s + 1
1
=
Kτ r s
τ r sg
(2.7a)
or,
⎛ 2ςτ
gc = ⎜
⎝ Kτ r
⎞⎛
1
τs ⎞
+ ⎟
⎟ ⎜1 +
⎠ ⎝ 2ςτ s 2ς ⎠
g=
1
(2.7b)
This is a PID controller.
Example: Suppose
( 2s + 1)( 5s + 1)
and
q=
1
5s + 1
Using (2.4)
gc =
1
(2s + 1)(5s + 1)
=
5sg
5s
or
gc = 1.4(1 +
1
+ 1.43s )
7s
PID controller with Kc=1.4, τI=7 and τD=1.43.
Now, suppose the rise time in the desired closed loop response is 5 times faster, i.e.,
q=
1
s +1
Then , using (4),
gc =
KNU/EECS/ELEC835001
1 (2 s + 1)(5s + 1)
1
⎛
⎞
=
= 7 ⎜1 + + 1.43s ⎟
sg
s
⎝ 7s
⎠
Dr. Kalyana Veluvolu
Model Based Control
6
a PID controller with Kc=7,τI=7 and τD=1.43. A reduction in the value of τ r by a factor of 5 also
resulted in a fivefold increase in the value of Kc
2.4 Synthesis of Nonminimum Phase Systems
Let the time delay process be represented by
y ( s ) = g ( s )u ( s ) =
Ke −α s
u ( s) ,
τ s +1
α >0
(2.8)
and q is as in (2.5). However, direct using (2.4),
gc =
1
(τ s + 1)eα s
=
Kτ r s
τ r sg
the controller has to predict α units of time ahead of response which is not realizable. The
desired closed response is hence modified to take into account the time delay.
e −α r s
q=
τ r s +1
(2.9)
Substituting (2.6) and (2.9) into (2.4), we get
gc =
(τ s + 1) ⎡ e − (α r −α ) s ⎤
⎢
⎥
K ⎣τ r s + 1 − e −α r s ⎦
(2.10)
gc =
(τ s + 1)
K (τ r s + 1 − e −α s )
(2.11)
Putting α=αr, we get ,
which results
u (s) =
τ s +1
K (τ r s + 1 − e −α s )
ε ( s)
(2.12)
Rearranging (2.12), we obtain,
Kτ r s
K
Ke −α s
u ( s) +
u ( s) =
u (s) + ε (s)
τ s +1
τ s +1
τ s +1
(2.13)
Let
y∗ ( s) =
K
u ( s ) = g ∗u ( s )
τ s +1
(2.14)
where
g∗ =
K
τ s +1
Substitute (2.8) and (2.14) into (2.13) to get
KNU/EECS/ELEC835001
Dr. Kalyana Veluvolu
Model Based Control
7
Kτ r s
u ( s) + y∗ (s) = y(s) + ε ( s)
τ s +1
That is,
u( s) =
τ s +1
⎡⎣ε ( s ) − ( y ∗ − y ) ⎤⎦
Kτ r s
⎛
1 ⎞
∗
= Kc ⎜1 +
⎟ ⎣⎡ε ( s ) − ( y ( s ) − y ( s ) ) ⎦⎤
s
τ
I ⎠
⎝
Furthermore,
⎛ τ ⎞⎛
1⎞
∗
u( s) = ⎜
⎟ ⎜ 1 + ⎟ ⎡⎣ε ( s ) − ( y ( s ) − y ( s ) ) ⎤⎦
⎝ Kτ r ⎠ ⎝ τ s ⎠
(2.16)
where, Kc= τ/(Kτr) and τI = τ are the parameters of a PI Controller.
Equation (2.16) corresponds to the Smith’s Predictor Controller which is represented in block
diagram form in Fig.2.2.
Fig. 2.2 Smith Predictor Controller
Approximation of Time Delay: Put
e −α s ≅ 1 − α s
Then (2.6) becomes
g=
K (1 − α s )
τ s +1
The controller given by (2.11) is expressed as
gc =
(τ s + 1)
τ
1⎞
⎛
=
⎜1 + ⎟
−α s
K (τ r s + 1 − e ) K (τ r + α ) ⎝ τ s ⎠
a PI Controller.
Pade Approximation of Time Delay
Put
KNU/EECS/ELEC835001
Dr. Kalyana Veluvolu
Model Based Control
8
e −α s ≅
1−
1+
αs
2
αs
2
Then, (10) becomes, after simplification,
⎛ αs ⎞
1+
⎛
⎞
τ
(τ s + 1)
1 ⎞⎜
⎛
2 ⎟
gc =
=
1
+
⎜
⎟
⎜
⎟
⎜
⎟
K (τ r s + 1 − e −α s ) ⎝ K (τ r + α ) ⎠ ⎝ τ s ⎠ ⎜ 1 + τ ∗ s ⎟
⎝
⎠
(2.17)
where
τ∗ =
ατ r
2(τ r + α )
(2.17) corresponds to a PID controller. The proportional gain
Kc =
τ
K (τ r + α )
and the Integral time τI= τ. The derivative part is the last product term in (2.17) which
corresponds to a gain limited derivative controller.
Example: Let
g ( s) =
0.66e −2.6 s
6.7 s + 1
and
e −2.6 s
q( s) =
5s + 1
Then,
gc =
q
6.7 s + 1
=
g (1 − q) 0.66(5s + 1 − e −2.6 s )
Put
e −2.6 s ≅ 1 − 2.6s
and simplify to get
1 ⎞
⎛
gc = 1.34 ⎜1 +
⎟
⎝ 6.7 s ⎠
a PI Controller with Kc=1.34 and τI=6.7.
Alternatively, using Pade approximation,
KNU/EECS/ELEC835001
Dr. Kalyana Veluvolu
Model Based Control
9
e −2.6 s =
1 − 1.3s
1 + 1.3s
Then, gc can be written as
1 ⎞ (1 + 1.3s )
⎛ 6.7 ⎞ ⎛
gc = ⎜
⎟ ⎜1 +
⎟
⎝ 0.66 ⎠ ⎝ 6.7 s ⎠ (7.6 + 6.5s )
That is,
1
1
⎡
⎤⎛
⎞
gc = (1.59 ) ⎢1 + + 1.09s ⎥ ⎜
⎟
⎣ 8s
⎦ ⎝ 1 + 0.86s ⎠
PID contoller with Kc=1.59, τI=8 and τD=1.09. The third product term corresponds to a low pass
filter.
Synthesis for Inverse Response Systems: Let
g (s) =
K (1 − η s )
(1 + τ 1s )(1 + τ 2 s )
The desired closed loop response will be
q( s) =
(1 − η s )
(1 + τ r1s )(1 + τ r 2 s )
Then the direct synthesis controller is given by
gc =
(1 + τ 1s )
(1 + τ 2 s )
1⎛ q ⎞
⎜
⎟=
g ⎝ 1 − q ⎠ K (η + τ r1 + τ r 2 ) s (1 + τ ∗ s )
where
τ∗ =
τ r1τ r 2
η + τ r1 + τ r 2
The controller can also be put as
⎡
⎤
1
1
+τ Ds⎥
g c = K c ⎢1 +
∗
⎣ τIs
⎦ (1 + τ s )
(2.18)
where
Kc =
τ1 + τ 2
K (η + τ r1 + τ r 2 )
τ I = τ1 + τ 2
and
τD =
KNU/EECS/ELEC835001
τ 1τ 2
τ1 + τ 2
Dr. Kalyana Veluvolu
Model Based Control
10
The third product term on the RHS of (2.18) coresponds to a first order filter term.
Example: Let
g=
1 − 3s
(2s + 1)(5s + 1)
and
q( s) =
(1 − 3s )
(1 + s )(1 + 3s )
Using (4), the controller is given, after simplification, as
1
1
⎛
⎞⎛
⎞
gc = ⎜1 + + 1.43s ⎟ ⎜
⎟
⎝ 7s
⎠ ⎝ 1 + 0.43s ⎠
a PID controller in cascade with a filter.
Synthesis for Open-Loop Unstable Systems
Consider the first-order open-loop unstable system whose transfer function model is given as:
g ( s) =
K
τ s −1
(2.19)
the direct synthesis controller required for (2.5) , resulting
gc =
τ
Kτ r
1⎞
⎛
⎜1 − ⎟
⎝ τs⎠
(2.20)
which looks like a PI controller with K c = τ Kτ r but with a very important difference that its
integral time is negative; i.e., τ I = −τ .
The problem with this controller is that it will only function properly if all the process
parameters are exactly equal to the model parameters; if this is not the case the overall closedloop system will be unstable. The controller therefore has a severe robust stability problem.
Consider the situation in which the actual open-loop unstable process has the transfer function:
g p ( s) =
Kp
τ p s −1
(2.21)
for which (2.19) is now only an approximate model. When this system is operating in the closed
loop with the controller in Eq. (2.20) derived on the basis of the approximate model of Eq.
(2.19), the closed-loop characteristic equation:
1 + g p gc = 0
becomes
Kτ rτ p s 2 + ( K pτ − Kτ r ) s − K p = 0
KNU/EECS/ELEC835001
Dr. Kalyana Veluvolu
Model Based Control
11
where K p ≠ K and τ p ≠ τ . Because of negative coefficient of K p , the system characteristic
equation will always be unstable.
The direct synthesis philosophy, which requiring the closed-loop response to be as in Eq. (2.5),
is responsible for this problem. According to Eq. (2.5), the single, closed-loop system pole is to
be located at s = −1 τ r to achieve this requires a controller that will do two things
simultaneously:
¾ cancel the RHP pole;
¾ replace it with the desired LHP pole (the one located at s = − 1 τ r ).
Such perfect pole cancellation is possible only when the location of RHP pole is exactly known.
2.6 Observations on Direct Synthesis Method
The discussion above would allow us to derive the direct synthesis tuning parameters for any
class of models and choice of reference trajectories. These formulas provide an excellent set of
tuning parameters for a wide range of problems when an adequate linear model is available for
the process. However
¾ PID parameters will be decided by a user-specified parameter: The desired closed-loop
time constant ( τ r )
o The shorter makes the action more aggressive. (larger Kc)
o The longer makes the action more conservative. (smaller Kc)
¾ For limited cases, it results PID form.
o 1st-order model without time delay: PI
o FOPDT with 1st-order Taylor series approx.: PI
o 2nd-order model without time delay: PID
o SOPDT with 1st-order Taylor series approx.: PID
o Delay modifies the Kc.
o With time delay, the Kc will not become infinite even for the perfect control
(Y/R=1).
¾ The resulting controller from direct synthesis method could be quite complex and may
not even be physically realizable.
¾ If there is RHP zero in the process, the resulting controller from direct synthesis method
will be unstable.
¾ Unmeasured disturbance and modeling error are not considered in direct synthesis
method.
These problems will be addressed in the next section by “Internal Model Control Strategy”.
KNU/EECS/ELEC835001
Dr. Kalyana Veluvolu
Model Based Control
3.
12
INTERNAL MODEL CONTROL
3.1 Motivation
Consider the process whose dynamic behavior is represented by:
y ( s ) = p ( s )u ( s ) + d
(3.1)
with the block diagram shown in Figure 3.1 Here d represents the collective effect of
unmeasured disturbances on the process output y. If it is desired to have "perfect" control in
which the output tracks the desired set-point yd perfectly, the control action required to achieve
this objective is easily obtained by substituting y = r in Eq. (3.1):
r ( s ) = p ( s )u ( s) + d
(3.2)
and then solving for u(s) to obtain:
u(s) =
1
[ r (s) − d ]
p( s)
(3.3)
Figure 3.1 Conventional Feedback Control Structure
Figure 3.2: Internal Model Control Structure
The implication is that if both d and p(s) are known, then for any given r , Eq. (3.3) provides the
controller that will achieve perfect control. Since in reality, d is unmeasured and p be only
modeled approximately by, say, p% ( s) , we may adopt the following strategy:
1. Assuming that p% is our best estimate of the plant dynamics p, then our best estimate of d
is obtained by subtracting the model prediction, p% ( s )u ( s ) , from the actual plant output y
to yield the estimate:
KNU/EECS/ELEC835001
Dr. Kalyana Veluvolu
Model Based Control
13
dˆ = y − p% ( s )u ( s )
(3.4)
2. Let us choose the notation:
u (s) =
1 ⎡
r ( s ) − dˆ ( s ) ⎤
⎦
p% ( s ) ⎣
(3.5)
where d̂ is the estimate of d given by Eq. (3.4).
A block diagrammatic representation of Eqs. (3.4) and (3.5) takes the form shown in Figure 3.2
known as the "Internal Model Control" structure (IMC) which forms the basis for the systematic
control system design methodology. Figure 3.2 is the “Internal Model Control” or “Qparametrization” structure. The IMC structure and the classical feedback structure (Figure 3.1)
are equivalent representations. Having designed q(s), its equivalent classical feedback controller
c(s) can be readily obtained via algebraic transformations, and vice-versa
c=
q
%
1 − pq
(3.6)
q=
c
%
1 + pc
(3.7)
The equivalence between the two are shown as in Fig 3.3.
Figure 3.3 Evolution of the Internal Model Control
KNU/EECS/ELEC835001
Dr. Kalyana Veluvolu
Model Based Control
14
Closed-loop transfer functions
The closed loop transfer function of Fig. 32. is given by:
%
1 − pq
pq
r +
d
1 + q( p − p% )
1 + q( p − p% )
= η ( s )r ( s ) + ε ( s )d ( s )
y =
(3.8)
where, in terms of the internal model p% and controller q(s), ε and η are defined as sensitivity
and complementary sensitivity, respectively. If no plant/model mismatch (p = p% ), they are
simplified to
% ⇒ q = p% −1η%; ε% ( s ) = 1 − η% ( s ) = 1 − pq
%
η% ( s ) = pq
(3.9)
Internal Stability
1. Assume a perfect model (p = p% ). The IMC system (Figure 3.2) is internally stable (IS) if
and only if both p and q are stable.
2. Assume that p is stable and p = p% . Then the classical feedback system (Figure 3.1) with
controller according to Equation (3.6) is IS if and only if q is stable.
Asymptotic closed-loop behavior
We need to insure that the feedback control system leads to no offset for setpoint or disturbance
changes; we thus need to define so-called Type 1 and Type 2 inputs:
Type 1 (Step Inputs): No offset to asymptotically step setpoint/disturbance changes is obtained
if
% = η% (0) = 1 ⇒ q(0) =
lim pq
s →0
1
p% (0)
(3.10)
Then regardless of the fact that p ≠ p% , (3.8) shows that y = r at steady state. Thus, so long as the
steady state gain of the controller is the same as the reciprocal of the gain of the process model.
Type 2 (Ramp Inputs): For no offset to ramp inputs, it is required that
% = η% (0) = 1
lim pq
s →0
lim
s →0
d
dη%
% )=
( pq
ds
ds
=0
(3.11)
s =0
Requirements for Physical Realizability on q
In order for q, the IMC controller, to result in physically realizable manipulated variable
responses, it must satisfy the following criteria:
1. Stability: The controller must generate bounded responses to bounded inputs; therefore
all poles of q must lie in the open Left-Half Plane.
2. Properness: In order to avoid pure differentiation of signals, we must require that q(s) be
proper, which means that the quantity
KNU/EECS/ELEC835001
Dr. Kalyana Veluvolu
Model Based Control
15
lim q ( s )
(3.12a)
| s| →∞
must be finite. We say q(s) is strictly proper if
lim | q ( s ) |= 0
(3.12b)
| s| →∞
A strictly proper transfer function has a denominator order greater than the numerator
order.
q(s) is semi-proper, that is,
lim | q ( s ) |≥ 0
(3.12c)
| s| →∞
if the denominator order is equal to the numerator order.
A system that is neither strictly proper nor semiproper is called improper.
3. Causality: q(s) must be causal, which means that the controller must not require
prediction, i.e., it must rely on current and previous plant measurements. A simple
example of a noncausal transfer function is the inverse of a time delay transfer function
q( s) =
u ( s)
= K c e +θ s
e( s )
(3.13)
The inverse transform of (3.13) relies on future inputs to generate a current output; it is
clearly not realizable:
u (t ) = K c e(t + θ )
(3.14)
Advantages:
The IMC structure offers the following benefits with respect to classical feedback:
•
No need to solve for roots of the characteristic polynomial 1+pc; one simply examines
the poles of q;
•
One can search for q instead of c without any loss of generality.
3.2 Internal Model Control Design Procedure
The IMC design procedure is a two-step approach that, although sub-optimal in a general sense,
provides a reasonable tradeoff between performance and robustness. The main benefit of the
IMC approach is the ability to directly specify the complementary sensitivity and sensitivity
functions η and ε , which as noted previously, directly specify the nature of the closed-loop
response.
IMC Design Procedure
The IMC design procedure consists of two main steps. The frist step will insure that q is stable
and causal; the second step will require q to be proper.
Step1: Factor the model p% into two parts:
p% = p% + p% −
KNU/EECS/ELEC835001
(3.15)
Dr. Kalyana Veluvolu
Model Based Control
16
p% + contains all Nonminimum Phase Elements in the plant model, that is all Right-HalfPlane (RHP) zeros and time delays. The factor p% − , meanwhile, is Minimum Phase and
invertible; an IMC controller defined as
q% = p% −−1
(3.16)
is stable and causal.
The factorization of p% + from p% is dependent upon the objective function chosen. For
example,
p% + = e −θ s ∏ ( − 遱
i + 1); Re( i ) > 0
(3.17)
i
is Integral-Absolute-Error (IAE)-optimal for step setpoint and output disturbance
changes. While, the factorization
p% + = e −θ s ∏
i
(− 遱
i + 1)
; Re( i ) > 0
(遱
i + 1)
(3.18)
is Integral-Square-Error (ISE)-optimal for step setpoint/output disturbance changes.
Using ramp, exponential, or other inputs would imply different factorizations.
% ( s ) is, in addition to
Step2: Augment q with a filter f(s) such that the final IMC controller q = qf
stable and causal, proper. With the inclusion of the filter transfer function, the final form
for the closed-loop transfer functions characterizing the system is
%%
η% = pqf
%%
ε% = 1 − pqf
(3.19)
Step3: If necessary, the IMC controller may be converted to the conventional form for
implementation by using (3.6)
The inclusion of the filter transfer function in here means that we no longer obtain “optimal
control,” as implied in Step 1. We wish to define filter forms that allow for no offset to Type 1
and Type 2 inputs; for no offset to step inputs (Type 1), we must require that η% (0) = 1 , which
requires that q (0) = p% −1 (0) and forces
f (0) = 1
(3.20)
A common filter choice that conforms to this requirement is
f ( s) =
1
(λ s + 1) n
(3.21)
The filter order n is selected large enough to make q proper, while λ is an adjustable parameter
which determines the speed-of-response. Increasing λ increases the closed-loop time constant
and slows the speed of response; decreasing λ does the opposite. λ can be be adjusted on-line to
compensate for plant/model mismatch in the design of the control system; the higher the value of
λ, the higher the robustness the control system.
KNU/EECS/ELEC835001
Dr. Kalyana Veluvolu
Model Based Control
17
For no offset to Type-2 (ramp) inputs, in addition to the requirement (3.20), the closed-loop
system must satisfy the following
d
dη%
% ) =
( pq
ds
ds
s =0
=
s =0
dp% + p% − p% −−1 f
ds
=0
(3.22)
s =0
Since p% + (0) = 1 , one such filter transfer function which meets the condition (3.22) is
f ( s) =
(2λ − p% +' (0)) s + 1
(λ s + 1)2
(3.23)
Equation (3.23) will enable us to obtain PID rules for plants with integrator.
The factorization of p% is very important, recall that for classical feedback
y = ηr + ε d
η = (1 + pc ) −1 pc
ε = (1 + pc) −1
Using the IMC structure, for no plant/model mistmatch (p = p% ), we have
%
η% = pq
%
ε% =1 − pq
(3.24)
“Perfect” control (meaning y = r for all time) is achieved when η% = 1 and ε% = 0 , which implies
that
q = p% −1
(3.25)
However, in order for u = q(r - d), the manipulated variable response, to be physically realizable,
q must be stable, proper, and causal. Nominimum phase behavior (deadtime and RHP zeros)
will cause q = p% −1 to be noncausal and unstable, respectively; if p% is strictly proper, then q will
be improper as well.
Example: Consider the plant model
p% ( s ) =
K (− 遱 + 1)e −θ s
τ 2 s 2 + 2ζτ s + 1
(3.26)
where β > 0, which implies the presence of a Right-Half Plane zero. Nonminimum phase elemets
for this plant are (− 遱 + 1)e −θ s . The “perfect” IMC controller for this system corrresponds to
τ 2 s 2 + 2ζτ s + 1 +θ s
q = p% ( s ) =
e
K (− 遱 + 1)
−1
(3.27)
While y = r using this controller, the manipulated variable response is physically unrealizable
since 1) q is unstable as a result of a Right-Half Plane pole arising from (-βs + 1); and 2) q is
noncausal because of the presence of the time lead term e+θs.
Applying an appropriate factorization to this model as described earlier results in stable, causal
control action; a correctly chosen filter order will insure properness and a physically realizable
KNU/EECS/ELEC835001
Dr. Kalyana Veluvolu
Model Based Control
18
response. One must keep in mind that the nonminimum phase elements (− 遱 + 1)e −θ s will always
form part of the closed-loop response!
3.4 Application of IMC Design to PID controller tuning
The IMC control design procedure, when applied to low-order models, will often result in PID
and PID-like controllers. Developing these is the focus of this section:
1: PI Control for the First Order Model
A PI tuning rule arises from applying IMC to the first-order model:
p% =
K
τs + 1
τ>0
(3.28)
under the condition that d and r are step input changes.
Step 1: Factor and invert p% ; since p% + = 1 , we obtain:
q% =
τ s +1
K
; no proper
Step 2: Augment with a first-order filter
f =
1
(λ s + 1)
The final form for q is
q=
τ s +1
;
K (λ s + 1)
proper
(3.29)
We can now solve for the classical feedback controller equivalent c(s) to obtain
c=
q
τ
1
=
(1 + )
τs
1 − pq K λ
(3.30)
which leads to the tuning rule for a PI controller
Kc =
τ
Kλ
τI =τ
The corresponding nominal closed-loop transfer functions for this control system are
η% =
1
;
λs + 1
q = p% −1η =
τ s +1
;
k (λ s + 1)
ε% =
λs
(λ s + 1)
Example: IMC Design for First Order Process
Design a controller for the first-order process whose transfer function is
p=
5
8s + 1
using the IMC strategy. Convert this controller to the convenentional feedback form
KNU/EECS/ELEC835001
Dr. Kalyana Veluvolu
Model Based Control
19
Solution: Observing that the transfer function is invertible, we obtain:
1 8s + 1
=
p
5
which requires only a first-order filter (n = 1) in order that c( s ) = f p be proper, Thus we have,
in this case:
q( s ) =
1
1 8s + 1
f ( s) =
p( s )
5 λs + 1
which can be implemented using a lead/lag element. Furthermore,
c( s ) =
8 ⎛
1⎞
⎜1 + ⎟
5λ ⎝ 8s ⎠
as the equivalent conventional feedback form, a PI controller whose gain depends on the filter
parameter λ .
PI Control Inverse Response Process
Consider now the first-order model with Right Half Plane (RHP) zero:
p% =
K (− β s + 1)
τs + 1
β ,τ>0
(3.31)
again under the assumption that the inputs to r and d are steps.
Step 1: Use the IAE-optimal factorization for step inputs:
p% + = (− β s + 1);
p% − =
K
;
(τ s + 1)
q% =
(τ s + 1)
K
Step 2: Use a first-order filter
f =
1
;
(λ s + 1)
q=
τ s +1
K (λ s + 1)
(3.32)
Solving for the classical feedback controller leads to another tuning rule for a PI controller:
c( s ) = K c (1 +
Kc =
τ
K (β + λ )
1
)
τs
(3.33)
τI =τ
PI with filter control
Consider now the first-order model with Left Half-Plane (LHP) zero:
p% =
K ( β s + 1)
τs + 1
β ,τ>0
(3.34)
again under the assumption that the inputs to r and d are steps.
KNU/EECS/ELEC835001
Dr. Kalyana Veluvolu
Model Based Control
20
Step 1: No nonminimum phase behavior in p% ; since p% + = 1 , we obtain:
p% − =
K ( β s + 1)
;
(τ s + 1)
q% =
(τ s + 1)
K ( β s + 1)
Step 2: Use a first-order filter (q is now strictly proper).
f =
1
;
(λ s + 1)
q=
τ s +1
K ( β s + 1)(λ s + 1)
(3.35)
Solving for the classical feedback controller,
c( s) =
Kc =
q
1
1
= K c (1 + )
τ s (τ F s + 1)
1 − pq
τ
;
K (β + λ )
τI =τ;
(3.36)
τF = β
It is interesting to note that in IMC design, the presence of a Left-Half Plane zero in the model
leads a low-pass filter element in the classical feedback controller!
PID Control
Consider now the second-order model with RHP zero:
p% =
K (− β s + 1)
(τ 1s + 1)(τ 2 s + 1)
β ,τ 1 ,τ 2 > 0
(3.37)
again under the assumption that the inputs to r and d are steps.
Step 1: Use the IAE-optimal factorization for step inputs:
p% + = (− β s + 1);
q% =
p% − =
K
(τ 1s + 1)(τ 2 s + 1)
(τ 1s + 1)(τ 2 s + 1)
K
Step 2: Use a first-order filter (even though this means that q will still be improper).
f =
1
(λ s + 1)
;
q=
(τ 1s + 1)(τ 2 s + 1)
K (λ s + 1)
(3.38)
Solving for the classical feedback controller
c( s) =
q
1
= K c (1 +
+ τ s)
%
τIs D
1 − pq
(3.39)
an ideal PID controller with
Kc =
τ1 + τ 2
;
K (β + λ )
τ I = τ1 + τ 2 ; τ D =
τ 1τ 2
τ1 + τ 2
PID with Filter Control
KNU/EECS/ELEC835001
Dr. Kalyana Veluvolu
Model Based Control
21
Consider (3.37) again and subject to step inputs to the closed-loop system. Applying the IMC
design procedure gives:
Step 1: Use the ISE-optimal factorization
p% + =
(− β s + 1)
;
( β s + 1)
p% − =
K ( β s + 1)
(τ 1s + 1)(τ 2 s + 1)
(3.40)
Step 2: A first-order filter leads to q which is semiproper:
q=
(τ 1s + 1)(τ 2 s + 1)
K ( β s + 1)(λ s + 1)
(3.41)
1
(λ s + 1)
(3.42)
f =
Solving for c(s) as before results in a filtered ideal PID controller
c( s ) = K c (1 +
1
1
+ τ D s)
τIs
(τ F s + 1)
(3.43)
with the associated tuning rule
Kc =
τ1 + τ 2
; τ I = τ1 + τ 2 ;
K (2 β + λ )
τD =
τ 1τ 2
;
τ1 + τ 2
τF =
βλ
(2 β + λ )
(3.44)
Note the insight given by IMC design procedure regarding on-line adjustment (by changing the
value for the IMC filter parameter λ).
Deadtime compensation (PI controller + Smith Predictor)
Consider the first-order with delay plant
p% =
Ke −θ s
τs + 1
τ> 0
(3.45)
and step setpoint/output disturbance changes to the closed-loop system.
Step 1: The optimal factorization (IAE, ISE, or otherwise) is p% + = e −θ s , resulting in:
q% = p% −−1 =
τ s +1
K
Step 2: A first-order filter makes q semiproper;
q=
τ s +1
K (λ s + 1)
η% =
e −θ s
;
(λ s + 1)
(3.46)
The corresponding feedback controller is
c( s) =
KNU/EECS/ELEC835001
τ s +1
K (λ s + 1 − e −θ s )
(3.47)
Dr. Kalyana Veluvolu
Model Based Control
22
which can be expressed as a PI controller using the Smith Predictor structure (same as in direct
synthesis control).
PID Tuning Rules for 1st-order with Deadtime Plants
The PID tuning rule for plants with deadtime arises from using a first-order Padè approximation
for the time delay.
Ke −θ s
p% =
τs + 1
K (−θ s 2 + 1)
≈
(τ s + 1)(θ s 2 + 1)
(3.48)
The Padè-approximated plant is a second-order plant with RHP zero; using the analysis (3.39)
leads to a PID tuning rule:
Kc =
2τ + θ
;
K (θ + 2λ )
θ
τI =τ + ;
2
τD =
τθ
2τ + θ
(3.49)
The function of λ θ independent of τ, is a measure related to robustness of the closed-loop
system. Note that at λ θ ≈ 0.8 the IMC-PID controller results in an ISE value that is only 10%
greater than optimal. The controlled variable response of the IMC-PID controller for various
settings of λ θ is shown in Figure 3.4.
Figure 3.4: IMC-PID controlled variable responses for a step setpoint change, for various
settings of λ θ solid: λ θ = 0.8 ; dotted: λ θ = 2.5 ; dashed: λ θ = 0.4 .
KNU/EECS/ELEC835001
Dr. Kalyana Veluvolu