A New Adaptive PID Controller

Model based Adaptive PID Controller with
Parallel Feedforward Compensator for Steam
Turbine Speed Control
Prof. Dr. Selahattin Ozcelik
Department of Mechanical and Industrial Engineering
Texas A&M University-Kingsville
Kingsville, TX USA
Hakan Celep
Department of Mechanical and Industrial Engineering
Texas A&M University-Kingsville
Kingsville, TX USA
Abstract— In this paper, a new design method of model based
adaptive PID controller with parallel feedforward compensator
is proposed and improvement of steam turbine control is
investigated. The method can be applied to plants, which satisfy
the Almost Strict Positive Realness (ASPR) conditions. In order
to satisfy ASPR conditions, Parallel Feedforward Compensator
(PFC) is introduced. Also, practical applications of the method
are discussed and the proposed method is applied to steam
turbine speed control system through numerical simulations.
Keywords— PID, PFC, Steam Turbine Speed Control, Adaptive
PID Control, ASPR.
I. INTRODUCTION
Apower plant comprises of boiler, gas turbine and steam
turbines as major components. Frequency stabilizing is very
important for power system security. Because of that, there
have been many signs of progress and interests in power
systems control in recent years. The new control algorithms
and techniques are applied to the power system to obtain the
best performance.
Compared to gas turbines, steam turbines have been
commonly used as for either process equipment in various
industries or electricity producer in power plants since the
industry began evolving during the 1880’s. Because it is
reliable and no matter which sector is used for, steam has an
important role for various applications.
There are different types of steam turbines, which are
dependent on operational conditions. Steam enters to steam
chest and moves to High Pressure (HP) section. Depending on
turbine type, steam passes through the Intermediate Pressure
(IP) section or leaves the turbine via Exhaust section. The
speed of turbine shaft is dependent on inlet steam
specifications and load. This process has uncertainties and
disturbs turbine speed. Large speed deviations may cause
mechanical damages and don’t ensure system security. Speed
is controlled by the governor, most of which are digital and
use conventional PID controller. PID controllers should be
tuned under changing operating conditions so that system
reliability can be satisfied.
Thus, adaptive control seems one good solution to this
problem and has a growing interest in the industrial
equipment.
Most common automatic tuning method is proposed by J.G.
Ziegler and N. B. Nichols in 1942 [1]. The method determines
Kp, Ki, and Kd gains of PID by using rules, which are
developed by Ziegler and Nichols. This method is still used in
industrial equipment and commercial sectors. However, it has
limited applications on systems, since the rules are based on
open-loop responses and cannot perform on closed loop
environment.
Since Ziegler-Nichols tuning method has restrictions on
closed-loop processes and therefore other tuning methods
have been developed. K. Aström and T. Hägglund proposed
“Phase Margin Method” [2]. However, the drawback of this
method is that it may not perform well on systems with big
time delays causing oscillations in closed loop systems.
On the other hand, Popov proposed hyperstability in [3]
and introduced the notion of strict positive realness for which
Linear Time Invariant (LTI) system is hyper stable
asymptotically if the transfer function of the system is
“strictly positive real”. M. Hakimi and H. Khaloozadeh in [4]
had studied this concept and approached the problem in the
frequency domain by using Taylor Series Expansion and
Maximum Modulus Principle and showed strict positive
realness in the frequency domain.
A robust optimal LQG controller was proposed for non-
minimum phase plant by H. Zargarzadeh and M. M. Arefi in
[14]. Robustness of the proposed controller was investigated
and for experimental purposes, the controller was later
reduced to a PI controller. As compared to loop-shaping H∞
control, proposed LQG controller showed improvement in
energy consumption and time responses.
S. Ozcelik and H. Kaufman presented Direct Model
Reference Adaptive Control (DMRAC) algorithm for Single
Input Single Output (SISO) plants in [5], where Kharitonov’s
theorem was used for feed forward compensator design to
satisfy Almost Strictly Positive Real (ASPR) conditions. This
algorithm was later extended to Multi-Input Multi-Output
(MIMO) systems in [6,12].
International Journal of Engineering Research & Technology (IJERT)
ISSN: 2278-0181http://www.ijert.org
IJERTV5IS120338
Vol. 5 Issue 12, December-2016
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Z. Iwai, I. Mizumoto and Y. Nakashima developed a
parallel feedforward compensator for (ASPR) systems and
developed tuning rules for MIMO systems in [7, 10], where
an approximate model of the system was used in order keep
the stability of the MIMO plant since ASPR conditions were
not satisfied in real applications. The drawback of the
algorithm is that it has practical and theoretical restrictions
and may not be applied to broad range systems due to its
control structure.
Hence, a new model based adaptive PID controller with
parallel feedforward compensator is proposed and applied to
steam turbine speed control system. Comparisons between
proposed algorithm and conventional PID controller are
explained and numerical simulations are pursued.
II. MODELING OF STEAM TURBINE SPEED CONTROL
SYSTEM
A. Modeling of Steam Turbine
Steam enters from main stop valve and flows to governor
valve. Governor controls the High Pressure section pressure
and then it moves to reheat. Reheated steam passes to
Intermediate Pressure section through the intermediate control
valve and piping. Finally, steam leaves turbine from exhaust
section through the crossover.
The simplified block diagram of steam turbine is shown in
Figure 1 below,
Fig.1. Block Diagram of Steam Turbine [8]
FHP, FIP and FLP being power fractions, determine power,
which comes from High Pressure, Intermediate Pressure and
Low Pressure sides. Sum of the fractions is equal to 1 (FHP +
FIP + FLP = 1). ΔE is valve power. Pg is mechanical power,
which is equal to torque. Tch, Trh, and Tco are the time
constants of steam turbine chest, re-heater, and crossover
piping, respectively. It is assumed that Tco is negligible as
compared to Trh. Thus, simplified transfer function of the
turbine system, which is the ratio of mechanical power to gate
position is [9]
)1)(1(
1
1E
gP







srhTschT
hpF
schT
hpF
(2.1)
)1)(1(
1



srhTschT
rhThpsF
(2.2)
Droop represented by R is the ratio of the steady state
frequency to load or valve gate change [15].
Percent R= (percent frequency change / percent power output
change) x100 (2.3)
B. Speed Governing Mechanism and Modeling
The Speed Governor, a type of transducer senses error in
speed and converts shaft speed to position output. Control
valves conduct input to the turbine and are activated by the
governor. Speed control mechanism has other electrical and
mechanical components such as linkages, servomotors, levers,
and amplifiers. Generic speed governing system is shown in
Figure 2 below. There are two valve nonlinearities; Rate limit
represents the saturation of the servomotor position, and
integrator limits the position of the valve. Limits are shown
by per unit (p.u). A new model based adaptive PID controller
with parallel feedforward compensator algorithm is applied to
steam turbine speed control system.
Fig.2. Generic Speed Governing System Model Presentation [5]
III. MODEL BASED ADAPTIVE PID CONTROLLER
WITH PARALLEL FEEDFORWARD COMPENSATOR
In this section, the Model Based Adaptive PID Control
algorithm is proposed. The algorithm proposed by Iwai [10]
has practical, theoretical restrictions and can’t be applied to
broad range systems due to its control structure. Hence, the
new Model Based Adaptive PID Controller with Parallel
Feedforward Compensator is proposed. If the steady-state
analysis of algorithm proposed by Iwai is studied, there are
conditions, which need to be satisfied firstly for zero steady
state response. These are,
-- GPFC = 0 (3.1)
-- G (0) ≠ 0 and ki ≠ 0 (3.2)
where GPFC is the transfer function of feedforward
compensator and ki is the integral gain of adaptive PID. GPFC
(s) is obtained from GPFC (s) = GASPR(s) – G*(s), where
GASPR(s) is an ASPR transfer function and G*(s) is the
nominal transfer function. Since GPFC (0) should be zero and
is calculated using G*(s), all physical systems cannot satisfy
the above conditions. Hence, steady state error will remain for
the controlled plant. To this effect, the new adaptive PID
controller is developed to improve the algorithm and to
eliminate the error in the system so that adaptive PID can be
applied to sophisticated systems.
1
1 + 𝑠𝑇𝑐ℎ
1
1 + 𝑠𝑇𝑟ℎ
1
1 + 𝑠𝑇𝑐𝑜
FHP FIP FLP
ΔE
ΔPg+
+
+
+
++
Turbin
e
Flow Area
Servomotor
Governor
Speed
Relay
Δ
ωr
1
1 + 𝑠𝑇𝑠𝑟
1
𝑇𝑠𝑚
KG
1
𝑠
Load
Ref.
Linearizing
Cam
Rate
Limit
Lc1
Position
Limit
1.0 CV
LC 0
- -
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A. Problem Setup
Let’s consider SISO nth
order plant:
)()(
)()()(
tpxpCtpy
tpupBtpxpAtpx


(3.3)
where xp is the state vector, up is the control vector, yp is the
output vector and Ap and Bp are matrix dimensions. The plant
can be expressed by transfer function of
p
D
p
N
p
AsI
p
Cs
p
G  )()( (3.4)
Theorem 1 [11]
Equation (3.4) is ASPR if it satisfies the conditions below,
- Np(s) is Routh-Hurwitz polynomial.
- Relative degree of the transfer function is 0 or 1.
- Leading coefficient is positive
Since most of the plants do not satisfy the ASPR conditions,
Parallel Feedforward Compensator is introduced.
B. Parallel Feedforward Compensator Design
Nominal plant parameters are known and uncertainties are
represented as additive perturbations. Plant uncertainties will
be presented by transfer functions. The aim is to develop
parallel feedforward compensator so that augmented plant
satisfies ASPR conditions and desired responses are achieved
despite changes in parameters.
Now consider a non-ASPR SISO plant in the form of Gp,
0..
1
1
0...
1
1
B
n
snB
n
snB
C
m
smC
m
smC
pG






 (3.5)
Where coefficients are Bn-j and Cm-j may take values within
the bounds below,
mjjmCjmCjmC ,...,1,0,  (3.6)
njjnBjnBjnB ,...,1,0, 
(3.7)
If the nominal plant parameters are known and Gnom (s) is
constructed, uncertainty can be shown as frequency dependent
additive perturbation. Hence, actual plant becomes Gp(s) =
Gnom (s) + Δ(s), where Δ(s) is additive perturbation.
Fig.3. Additive Perturbation and Actual Plant Representation.
Then defining,
nomGpGs  )( (3.8)
It is clear that the uncertainty is a function of plant
parameters varying within the ranges. Hence, a parallel
feedforward compensator that satisfies the worst case
uncertainty should be designed. An optimization procedure
below will be used to get the worst case uncertainty at each
frequency.
Define a vector whose elements are plant parameters in
equation (3.6),
]......[ 0101 BBBCCCV nnmm 
Maximize )j(  at each ω
V
Subject to:











n,...,1,0j,BBB
m,...,1,0j,CCC
jnjnjn
jmjmjm
(3.9)
where Δ(jω) is perturbation. Given the worst-case uncertainty
for each frequency, it is supposed that there is a rational
function which is W(s) 𝜖 RH∞ such that
  ,)(max)( jjW (3.10)
The following assumptions are made for the plant:
-- Nominal plant (Gnom) is known, minimum phase and
stable.
-- Actual plant is stable.
-- Δ(s) satisfies (3.10).
If the following augmented plant with parallel feedforward
compensator is considered as,
)()()( sPFCGspGsaG  (3.11)
The following theorem gives the design conditions of parallel
feedforward compensator. Also, augmented plant satisfies
ASPR conditions in the presence of plant uncertainty [12].
Theorem 1:
If GPFC(s) is designed as conditions below, the augmented
plant Ga(s) = Gp(s) + GPFC(s) will be ASPR.
-- GPFC(s) is stable and relative degree one.
-- The augmented nominal plant Gnom(s) + GPFC(s) is
ASPR.
-- ∆̃(s) 𝜖 RH∞ and ‖∆̃(s)‖∞ < 1
Where
)(
1
))()(()(
~
sWsPFCGsnomGs

 (3.12)
∆̃(s) is the uncertainty of the augmented plant.
(Proof of theorem)
+
+
+
+
Actual Plant
Gnom(s)
Δ(s)
GPFC(s)
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Consider the closed loop system comprised of the nominal
plant Gnom(s) and additive perturbation which is Δ(s) and
controller C(s) as shown in Figure 4, the actual plant is
)()()( ssnomGspG  (3.13)
The transfer function from w to z from can be shown as
zsCsnomGsCw ).(
1
))()(1(

 (3.14)
By using small gain theorem, the closed loop system is
stable if and only if
1P (3.15)
or it can be written as,
1)()(
1
))()(1( 



 ssCsCsnomG (3.16)
The robust stability condition by using parallel feedforward
compensator is
1
1
)1(




PFCG
nomGPFCG
(3.17)
By replacing Δ with W, one can have
1
PFCGnomG
W~



 (3.18)
C. Adaptive PID Controller Design
Suppose that augmented system is ASPR. The system can
be stabilized by using equation below,
)()()( tgtktV  (3.19)
where,
 )t(ik),t(dk),t(pk)t(k  (3.20)
 w,ay,ay)t(g  (3.21)
ayw  (3.22)
The adaptive gain vector k(t) is tuned by adaptive law below,
0
)t(y)t(g)t(k a




(3.23)
In this case 0)t(elimt  . Zero steady state error condition
is satisfied.
Error Analysis
Suppose the parallel feedforward compensator has the form
of Figure 4, then conditions below have to be satisfied,
-- ki > 0
-- Gp (0) ≠ 0
-- R(s) is step input
Then, 0)t(elimt 
Define,
)s(Guy),s(Guy),s(R)s(Gy pcpPFCcfmm  (3.24)
s/)s(CG,yyy,yyy PIDfppafmma  (3.25)
Fig.4. New Model Based Adaptive PID with Feedforward Control System
where the controller can be shown as below,
ip
2
d ksksk)s(C  (3.26)
From Figure 4, one can have the following signals,
pmfpfmpamaa yyyyyyyy)s(E  (3.27)
By taking (3.24) and (3.25) into (3.27), one can define,
)s(E
s
)s(C
)s(G)s(R)s(G
)s(Gu)s(R)s(G)s(E
apm
pcma


(3.28)
and
)s(R)s(G
s
)s(C
)s(G1)s(E mpa 





 (3.29)
)s(C)s(Gs
)s(R)s(sG
)s(E
p
m
a

 (3.30)
Using final value theorem, one can have,
)s(C)s(Gs
)s(R)s(sG
slim)s(sElime
p
m
0sa0sss

  (3.31)
0
)0(C)0(G0
s/1)0(G.0
slime
p
m
0sss 

  (3.32)
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IV. SIMULATION RESULTS
In previous chapters, steam turbine system was discussed
and design of new adaptive PID control algorithm was
introduced. In this section, an adaptive PID control system
with parallel feedforward compensator is implemented. The
objective is to control the speed of steam turbine under
changing operating conditions. Design procedure utilizes
transfer functions which are obtained from equations from
Figure 1. Steam turbine system has a governor, turbine and
load part. Under changing operating conditions of the turbine
system, the transfer function of the plant are Gplant(s)=
Ggovernor*Gturbine*Gload.. Simulations are performed using the
following nominal parameters and their respective ranges as
in Table 1.
Nominal plant transfer function Gnom(s) is









88.36106.3s83.73s32.34s53s7.6s4.0
77.1s49.42s6.1
)s(G 2345nom
(4.1)
The actual plant Gp(s) can be written as
65
2
4
3
3
4
2
5
1
32
2
1
p
cscscscscsc
bsbsb
)s(G


 (4.2)
The uncertainty is an additive perturbation in transfer
function and can be obtained from (3.9). The worst-case
uncertainty was calculated by optimization procedure by
equation (3.9) for 200 frequencies. Then ω that satisfies
equation (3.10) is given below








5s32s2.1
7.0
)s(W (4.3)
The magnitude response showing worst case uncertainty
and bounding function W(s) is shown in Figure 5,
Fig.5. Bounding Function Response, )j(w 
After getting bounding function, parallel feedforward
compensator is to be derived. By using Theorem 1 which
satisfies equation (3.12), the parallel feedforward
compensator can be found. Depending on the plant structure,
the relative degree of the compensator can be chosen either 0
or 1. Relative degree in this simulation is chosen 1 since
turbine plant structure satisfied ASPR conditions.









12
2
21)(
ss
hsh
sPFCG (4.4)
The denominator part of parallel feedforward structure is
chosen by designers regarding of the plant type. Since 2nd
degree polynomial is chosen, numerator part should be first or
second degree polynomial. If parallel feedforward structure is
formed with low degree polynomial, system response will be
faster compared to higher order polynomial structure. Parallel
feedforward compensator from optimization method using
Lagrange Multiplier is obtained as,







12
09219.01703.0
)( 2
ss
s
sGPFC
(4.5)
The augmented plant transfer function becomes as Ga(s) =
Gp(s) + GPFC(s), where











88.36s1.180s2.333306.3s155.6s46.45s553s.7s4.0
5.17724.13s38.19s24.85s7.763ss187.1s0681.0
)s(G 234567
23456
a
(4.6)
Γs , being weights for adaptive gains must be positive and can
be chosen arbitrarily depending on the system. For this
simulation, Γ1=0.04, Γ2=0.8, Γ3=0.04 are chosen. Simulations
are done for different cases to observe the changing
conditions of the steam turbine. Step input is applied for both
PID and adaptive PID controllers and results are shown.
Changes in turbine time constants are dependent on the
formula
TABLE I
DIFFERENT PLANT PARAMETERS
Parameter Nominal Range
b1 1.6 1.333 to 2
b2 4.497 3.124 to 7.028
b3 1.777 1.029 to 3.472
c1 0.4 0.4
c2 6.753 7.417 to 6.311
c3 32.54 26.68 to 41.79
c4 83.73 63.44 to 119
c5 106.3 73.19 to 168.2
c6 36.88 21.35 to 72.04
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PQo
P
VT
o
v




(4.7)
Po= Rated pressure
Qo= Rated flow out of the vessel
Time constants depend on the steam pressure, flow and
steam density. In real systems, steam coming through and
going through operation may not be stable. Changes in
pressure and flow will affect the turbine time constants and
the system. Changes in plant parameters from the nominal
values are assumed to be 25-30 % for simulations. Simulation
results for conventional PID controller were obtained for Kp=
1, Ki =0.4, Kd =0.1. Simulations are carried out for 8 different
cases
as shown in Table II below.
Simulations for the uncontrolled system responses for a
unit step input for all the 8 cases are shown in Figure 6. It is
important to note that uncontrolled system responses cannot
reach the steady-state value of unity.
Fig.6. Time vs. Unit-Step Response for Uncontrolled System
In all the simulations, turbine droop assumed as 5 %. If
there is any change in droop characteristic in the steam
turbine which usually occurs during operation, the controller
should compensate this change. During the operation of the
turbine, the load can be changed or can be disturbed.
Depending on the load, speed or valve change will be affected
and power output will be affected as well. In order to
compensate the deviation of the speed, droop needs to be
adjusted by the governor. Figure 7 shows the block diagram
of turbine governor with droop.
Fig.7. Turbine Governor Representation with Droop
As seen in Figure 8, 2% percent drop in droop causes
increasing settling time in PID controller for all the 8 cases.
However, Adaptive PID Controller can compensate this
change and performs well in all responses.
Fig.8. Time vs. Unit-Step Response under 2 % Droop Change
However, if droop drop increases further to 4%, PID cannot
control and three system responses are unacceptable and
considered unstable from a practical point of view as seen in
Figure 9. However, the new model based adaptive PID
controller can successfully control the system and maintains
desired performance for all cased.
Fig.9. Time vs. Unit-Step Response under 4 % Droop Change
Figures 10 and 11 show error history for adaptive PID and
regular PID, respectively. While errors in all the 8 cases in
adaptive PID approach zero, PID controller errors are taking
TABLE II
DIFFERENT PLANT CASES FOR SIMULATIONS
Cases Turbine Chest Turbine Re-heater Turbine Crossover
1 0.25 7. 5 0.4
2 0.3 9 0.48
3 0.2 6 0.32
4 0.2 9 0.48
5 0.3 6 0.32
6 0.25 6 0.48
7 0.3 6 0.48
8 0.3 6 0.4
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too long with three cases showing unacceptable decay.
Fig.10. Error of Model Based Adaptive PID Controller at 4 % Droop Change
Fig.11. Error of Conventional PID Controller at 4 % Droop Change
V. CONCLUSION
In this report, model based adaptive PID controller is
introduced. This new algorithm is applied to steam turbine
speed control system under droop change. From the results
obtained during simulations, the new model based adaptive
PID control algorithm can cope with changing operational
conditions and is a better alternative to conventional PID
controller.
REFERENCES
[1] J.G. Ziegler and N.B. Nichols, “ Optimum Settings for Automatic
Controllers”, Trans. of ASME, p.p. 759-768, 1942.
[2] K. Aström and T. Hägglund, “PID Controllers: Theory, Design and
Tuning” 2nd edition, 1994.
[3] B. D. O. Anderson, “A Simplified Viewpoint of Hyperstability”, IEEE
Transactions on Automatic Control, 1968.
[4] M. Hakimi and M, H. Khaloozadeh,” Revision on the Strict Positive
Realness” 5th
International Conference on Technology and Automation,
2005.
[5] S. Ozcelik, H. Kaufman, “Robust Direct Model Reference Adaptive
Controllers”, Proceedings of the 34th Conference on Decision &
Control, 1995.
[6] S. Ozcelik, H. Kaufman, “Design of MIMO Robust Direct Model
Reference Adaptive Controllers”, Proceedings of the 36th Conference
on Decision & Control, 1997.
[7] Z. Iwai, I. Mizumoto and Yuichi Nakashima,” Multivariable Stable PID
Controller Design with Parallel Feedforward Compensator”, SICE-
ICASE International Joint Conference, 3246-3251, 2006.
[8] D. Das, “Electrical Power Systems”, New Age International Publishers,
2006.
[9] P. Kundur, “Power System Stability and Control “, McGraw-Hill,
1994.
[10] Z. Iwai, I. Mizumoto, L. Liu, S.L. Shah and H. Jiang, “Adaptive Stable
PID Controller with Parallel Feedforward Compensator”, Control,
Automation, Robotics and Vision, 2006. ICARCV ’06. 9th
International
Conference on”, 2006.
[11] Kaufman, H., I. Bar-Kana and K. Sobel,” Direct Adaptive
ControlAlgorithms, Theory and Applications”. Springer-Verlag, 1994.
[12] S. Ozcelik, C. C. Palerm and H. Kaufman,” Multi-Drug Infusion
Control Using Robust Direct Adaptive Controller for Plants with Time
Delays”, Proceedings of the 7th Mediterranean Conference on Control
and Automation (MED99) Haifa, Israel, 1999.
[13] H. Celep” Model Based Adaptive PID Controller with Feedforward
Compensator for Steam Turbine Speed Control”, Proquest, 2012.
[14] H. Zargarzadeh, M. R. Jadeh Motlagn, M.M. Arefi, “Multivariable
Robust Optimal PID Controller Design For A Non-Minimum Phase
Boiler System Using Loop Transfer Recovery Technique”, 16th
Mediterranean Conf. on Control and Automation, France, p.p. 1520-
1525, 2008.
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