Optimal Robust H ∞ Controller for an Integrating Process with Dead Time CHAPTER 4 ROBUST CONTROLLER DESIGN 4.1 Introduction The robust controller (H ∞ Controller) has been designed and its robustness is checked with the help of μ synthesis for the unstable processes with dead time. It is important to consider the various issues like disturbance rejection and the robustness of the controller due to the uncertainties present in the system. While designing the controller, the weighting functions are chosen such that the system could meet the performance requirements such as the peak value of the μ plot should be less than one [12]. The D-K iteration is also used to improve the performance of the robust controller. Once the D(s) and D -1 (s) were approximated, the plant is scaled appropriately and the H-infinity design is synthesized for the scaled plant. This procedure is repeated until the “μ” calculation for robust performance yields a value less than “1” for all frequencies. 4.2 Design of H-Infinity Controller. Dan Dai and Roy Smith [11] have discussed about the application of robust control theory for the cart-spring pendulum system with uncertainties and disturbances. The objective is to design a controller that meets the specified robust performance criteria. In [12] the authors have used the “hinfsyn” command to find the controller transfer function by defining all the input and output parameters. The partitioned matrix of the plant has to be known before using the 70
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Optimal Robust H∞ Controller for an Integrating Process with Dead Time
CHAPTER 4
ROBUST CONTROLLER DESIGN
4.1 Introduction
The robust controller (H∞
Controller) has been designed and its robustness is checked
with the help of µ synthesis for the unstable processes with dead time. It is important to consider
the various issues like disturbance rejection and the robustness of the controller due to the
uncertainties present in the system. While designing the controller, the weighting functions are
chosen such that the system could meet the performance requirements such as the peak value of
the µ plot should be less than one [12]. The D-K iteration is also used to improve the
performance of the robust controller. Once the D(s) and D-1
(s) were approximated, the plant is
scaled appropriately and the H-infinity design is synthesized for the scaled plant. This procedure
is repeated until the “µ” calculation for robust performance yields a value less than “1” for all
frequencies.
4.2 Design of H-Infinity Controller.
Dan Dai and Roy Smith [11] have discussed about the application of robust control
theory for the cart-spring pendulum system with uncertainties and disturbances. The objective is
to design a controller that meets the specified robust performance criteria. In [12] the authors
have used the “hinfsyn” command to find the controller transfer function by defining all the
input and output parameters. The partitioned matrix of the plant has to be known before using the
70
Optimal Robust H Controller for Unstable Processes with Dead Time
71
“hinfsyn” command. Through the D-K iteration method, plant has been scaled properly and the
controller is found for the synthesized plant. Controller’s robust performance has improved by
using the D-K iteration.
Roy Smith [12]has viewed into the problem of designing a controller to achieve a
performance specification for all plants, G(s), in a set of plants G. Fig. 4.1 represents the generic
synthesis interconnection structure. The lower half of this figure is the same as that for the h-
infinity design procedure. The problem is to find the C(s) such that for all ∆ ε B∆,K(s) stabilizes
Fu(G(s),∆) and ║Fu(Fl(G(s),C(s)),∆║∞ ≤ 1.Unfortunately, this problem is not yet been solved,
except in few special cases. The current approach to this problem, known as D-K iteration,
involves the iterative application of the h-infinity design technique and the upper bound µ
calculation.
Fig 4.1 The generic interconnection structure for synthesis.
Optimal Robust H Controller for Unstable Processes with Dead Time
72
The transfer function model of the distillation column is considered for the design of
robust controller for the unstable process with dead time [1]. Partitioned matrix of the plant is
obtained by considering the performance weight, disturbance weight along with the uncertainties
present in the system Fig. 4.2 which is given by
-------(4.1)
From eq. (4.1), we can define the number of control inputs and number of measurand, which
help to find the H-infinity controller for the plant using “hinfsyn” command.
Fig.4.2 Robust Controller design with weighting functions.
Optimal Robust H Controller for Unstable Processes with Dead Time
73
4.2.1 Robust Control Design Methodology
1. To have the basic idea of designing/selecting lead, lag, lead–lag compensator (It helps
to select the weighting function properly).
2. To model the expected uncertainty into the bound. (│∆│< 1) [41-43]
3. The robust performance of the controller needs to be checked in the presence of the
uncertainty in the system.
4. D-K iteration (D-scaling) method can be used to improve the performance of the
H-infinity controller design [12] for the system.
5. The weighting functions have to be selected based on the systems input requirements.
By understanding the concepts of lead-lag compensator design, the selection of
weighting functions can be made easily.
6. Robust performance can be analyzed in two ways [13]
I. ║│W1S│+│W2T│║∞< 1.
II. Peak value of the µ (D-K iteration) bound should be less than one.
7. Loop shaping method :
a) This section focuses on the tracking of a reference signal.
b) If L denotes the loop transfer function, as L=GC, then the transfer
function from reference input r to tracking error e is given by
S = L1
1 called the sensitivity function, for the specified plant
The condition for therobust performance specification is ║W1S║∞< 1.
Optimal Robust H Controller for Unstable Processes with Dead Time
74
8. The following conditions are used to check for Robust Stability, Nominal Performance,
Robust Performance
The phrases robust stability, nominal performance, and robust performance of the plant.
(i) Nominal Performance: The closed-loop system achieves nominal performance if the
performance objective is satisfied for the nominal plant model, Gnom. In this problem, it is
equivalent to: Nominal Performance ||WP(I + GnomC)–1
||∞< 1
(ii) Robust Stability:The closed-loop system achieves robust stability, if the closedloop system is
internally stable for all of the possible plant models
G .In this problem it is equivalent to a simple
norm test on a particularnominal closed-loop transfer function.
Robust Stability ||WdelCGnom(I + CGnom)–1
||∞< 1
(iii) Robust Performance:The closed-loop system achieves robust performance if theclosed-loop
system is internally stable for all
G , and in addition to that, theperformance objective,||WP(I +
GC)–1
||∞< 1,is satisfied for every
G . The property of robust performance is equivalentto a
structured singular value test (a generalization of the two H∞ norm testsin the previous
conditions) on a particular, nominal closed- loop transferfunction.
Optimal Robust H Controller for Unstable Processes with Dead Time
75
4.3 D-K Iteration for Robust Performance
1. To understand the basic concepts and design of lead-lag compensator (It helps for the
selection the weighting functions).
2. The H-infinity controller has been designed for the integrating process with dead time as
given in [1],by considering various weighting function(Wu, Wd and We).Wu is used to
reduce the error signal at low frequencies to improve tracking, while We limit the control
signal at high frequency to avoid saturation an dynamic range and bandwidth.
3. D-K iteration Algorithm :
The objective is to design a controller which minimizes the upper bound to µ for the
closed loop system ║D(ω)Fl(G(s),C(s)D(ω)-1
║∞.The major problem in doing this is that
the D-Scale that results from µ calculation is in the form frequency by frequency data
and the D-scale required above must be a dynamic system. This requires an
approximation to the upper bound D-Scale in the iteration. Looking into this issuemore
closely, it can be summarized as follows
Initialize procedure withKo(s).
Calculate the resulting closed loopFl(G(s),C(s))
Calculate D scale for µ upper bound
σmax║D(ω)Fl(G(s),C(s)D(ω)-1
║∞ . Approximate the frequency data D(ω),by
D (s) ϵRH∞ , with
D (jω) ≈ D(ω)
Design H∞ Controller for the scaled plant
D (s)G(s)
D-1
(s).
Optimal Robust H Controller for Unstable Processes with Dead Time
76
The notation D(ω) is used to emphasize that the D-Scale arises from frequency by frequency µ
analyses of G(jω)=Fl(G(jω),C(jω)) and therefore is a function of “ω”. Note that it is notthe
frequency response of some transfer function and D(jω) cannot be used as notation. The µ
analysis of the closed loop system is unaffected by the D-Scales. However the H∞ design
problem is strongly affected by scaling. The procedure aims at finding at D such that the upper
bound for the closed loop system is a closed approximation to µ for the closed loop system. At
each frequency, a scaling matrix, D(ω) , can be found such that σmax(D(ω)G(jω)D(ω)-1
) is a
closed upper bound to µ(G(jω)).
Another aspect of “µ” synthesis is to consider the iteration which approaches the optimal
“µ”value, the resulting controllers often have more and more response at high frequencies. The
above discussion used an H∞ controller to initialize the iteration. Actually any stabilizing
controller can be used. In higher order, lightly damped, interconnection structures are used, the
H-infinity design of Ko(s) may be badly conditioned. In such case the software may fail to
generate a controller, or may give controller which doesn’t stabilize the system. A different
controller can be used to get a stable closed loop system, and thereby obtain D scales.
Application of these D scales often results in a better conditioned H-infinity design problem and
the iteration can proceed.
The robust performance difference between the H∞ controller, K0(s), and C(s), can be
dramatic even after a sing D-K iteration. The H∞ problem is sensitive to the relative scaling
between v and w. The D-scale provides the significance better choice of relative scaling for
Optimal Robust H Controller for Unstable Processes with Dead Time
77
closed loop robust performance. Even the application of a constant D scale can have dramatic
benefits [12].
4.4 D-K Iteration results
The stability and robust performance measure of the plant G(s) = 60.0506 se
s
were analyzed and
those are given in the following figures Fig 4.3 to Fig 4.16.
D-K iteration attempts to minimize the quantity ║D(ω)Fl(G(s),C(s)D(ω)-1
║∞ by alternatively
minimizing this expression for the controller C(s) or the “µ” upper bound scaling D, while
holding the other constant. This process is carried out iteratively until a satisfactory controller is
constructed.
Fig 4.3 Bode plot of the plant with and without controller
The frequency roll off is increased with the presence of the controller.
10-5
10-4
10-3
10-2
10-1
100
101
102
-80
-60
-40
-20
0
20
40
60
80
Magnitude (
dB
)
Bode Diagram
Frequency (rad/sec)
Without controller
With controller
Optimal Robust H Controller for Unstable Processes with Dead Time
78
Fig. 4.4 Bode plot of the weighting functions.
Frequency response of the various weighting function is shown in the Figure 4.4.
Fig.4.5 Singular value plot for iteration1
10-3
10-2
10-1
100
101
102
-40
-30
-20
-10
0
10
20
30
40
Magnitu
de (
dB
)
Bode Diagram
Frequency (rad/sec)
w etf
w dtf
w utf
10-3
10-2
10-1
100
101
102
103
0
1
2
3
4
5
6SINGULAR VALUE PLOT: CLOSED-LOOP RESPONSE
FREQUENCY (rad/s)
MA
GN
ITU
DE
Optimal Robust H Controller for Unstable Processes with Dead Time
79
Fig.4.6. Mu bound without D Scaling
The peak value of the µ plot is about 4.4 after 1st iteration
Fig.4.7 Mu bound with D-K Scaling
10-3
10-2
10-1
100
101
102
103
2.5
3
3.5
4
4.5CLOSED-LOOP MU: CONTROLLER #1
FREQUENCY (rad/s)
MU
10-3
10-2
10-1
100
101
102
103
2.5
3
3.5
4
4.5MU bnds (solid) and ||D*M*D-1|| (dashed): ITERATION 2
Optimal Robust H Controller for Unstable Processes with Dead Time
80
Fig.4.8 Singular value plot for iteration2
Fig.4.9. Mu bound without D Scaling
The peak value of the µ plot is about 4.05 after 2nd
iteration
10-3
10-2
10-1
100
101
102
103
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5SINGULAR VALUE PLOT: CLOSED-LOOP RESPONSE
FREQUENCY (rad/s)
MA
GN
ITU
DE
10-3
10-2
10-1
100
101
102
103
2.6
2.8
3
3.2
3.4
3.6
3.8
4
4.2CLOSED-LOOP MU: CONTROLLER #2
FREQUENCY (rad/s)
MU
Optimal Robust H Controller for Unstable Processes with Dead Time
81
Fig.4.10 Mu bound with D-K Scaling
Fig.4.11 Singular value plot for iteration3
10-3
10-2
10-1
100
101
102
103
2.6
2.8
3
3.2
3.4
3.6
3.8
4
4.2MU bnds (solid) and ||D*M*D-1|| (dashed): ITERATION 3
10-3
10-2
10-1
100
101
102
103
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5SINGULAR VALUE PLOT: CLOSED-LOOP RESPONSE
FREQUENCY (rad/s)
MA
GN
ITU
DE
Optimal Robust H Controller for Unstable Processes with Dead Time
82
Fig.4.12 Mu bound without D Scaling
The peak value of the µ plot is about 4 after 3rd
iteration
Fig.4.13 Mu bound with D-K Scaling
10-3
10-2
10-1
100
101
102
103
2.5
3
3.5
4CLOSED-LOOP MU: CONTROLLER #3
FREQUENCY (rad/s)
MU
Optimal Robust H Controller for Unstable Processes with Dead Time
83
Fig.4.14 Singular value plot for iteration4
Fig.4.15 Mu bound without D Scaling
The peak value of the µ plot is about 3.9 after 4th
iteration.
Since the peak value of “µ” is not less than 1, the robust performance is not ensured for the
controller.
10-3
10-2
10-1
100
101
102
103
0
0.5
1
1.5
2
2.5
3
3.5
4SINGULAR VALUE PLOT: CLOSED-LOOP RESPONSE
FREQUENCY (rad/s)
MA
GN
ITU
DE
10-3
10-2
10-1
100
101
102
103
2.5
3
3.5
4CLOSED-LOOP MU: CONTROLLER #4
FREQUENCY (rad/s)
MU
Optimal Robust H Controller for Unstable Processes with Dead Time
84
4.5Stability Analysis
The condition used to analyze the robust stability was │K(jω)S(jω)│< │1
aW │
Fig.4.16. The robust stability analysis.
4.6 Robust PID controller satisfying the H-Infinity Principles
4.6.1 Introduction
The design of robust controller helps to satisfy the robust stability and robust
performance criteria as given in [13] under the plant parameter changes. Most of the time the
order of the robust controller (H∞) will be of higher order when compared to the process transfer
function. In this thesisan attempt has been made to design the low order controller design (PID
Controller with H-Infinity Principles for aPure Integrating Process with Dead Time (PIPDT)).
The controller design parameters such as Proportional gain, Derivative gain and Integral gain are
fixed using the principle of Hurwitz Criteria. The weighing functions of the system needs to be
selected, based on the low pass or high pass requirements of the input signal and disturbance
rejection. After finding the range of Derivative gain and the Integral gain, by sweeping the
proportional gain, we can find the various set of admissible layers of PID Controller values.
Optimal Robust H Controller for Unstable Processes with Dead Time
85
Finally, the three dimensional plots of the admissible set of PID controller values for the PIPDT
need to be plotted. Once we obtain the admissible set of PID controller settings, that satisfies the
robust performance conditions [13], it can also be easily implemented in the industries as a lower
order H-Infinity controller.
4.6.2.Robust PID Controller Design
Most of the chemical and process industries are encountered with the unstable and higher
order systems. We can model the higher order systems to FOPDT, for the purpose of controller
design. For example, in the study of the distillation column, the resultant FOPDT will have a
very large time constant, which may lead the process to settle after a very long time dead time
instants [1]. For those cases we can treat the model as a pure integrating process for the design of
controller. Once the controller has been designed, the same can be implemented for higher order
systems.
The mathematical model of the distillation column is considered for simulation studies.
Model free PID controller design is more suitable for the integrating process with various
uncertainties. The common problem with the ordinary conventional controller is that, the PID
values which are tuned will give better response only for one particular transfer function. The
tuned controller value may lead to the poor time domain response or unstable output under some
additional uncertainties added into the system. A robust controller may find solution to these
kinds of problems, the only disadvantage with the design of H-Infinity is that the order of the
controller will be very high. In order to reduce the order of the controller and to ensure the robust
performance, an H-Infinity based PID controller [10, 13, 25, 54, 55, 56, ]are proposed for the
Optimal Robust H Controller for Unstable Processes with Dead Time
86
PIPDT processes. Also the admissible set of PID controller which is found using the Hurwitz
Criterion satisfies the robust performance condition. The uncertainty is considered in the form of
dead time in PIPDT process and the PID tuning values are found for uncertain system and
plotted as a 3D plot.
4.6.3 Design Approach
Consider the mathematical model of the distillation column as 60.0506
( )se
G ss
for the
controller design [4, 59-62, 68]. The general structure of the PID controller is given by
2
( )i p dK K s K s
C ss
. Our objective is to find the values ,p iK K and dK using the Hurwitz
Criterion.
The PID values are selected by satisfying the following condition
1] ( , , , )p i ds K K K (4.2)
2] ( , , , , , )p i ds K K K (4.3)
3]1 2( ) ( ) 1W s S W s T
(4.4)
The controller has been designed for the plant with uncertainty, the uncertainty is
considered in the form of various dead time in the process. The perturbed plant with various dead
times are given below (after the pade approximation), for which the controller needs to be
designed.
Case-1: With dead time Td=6sec.
2
0.0506 0.0168( )
0.33
sG s
s s
(4.5)
Optimal Robust H Controller for Unstable Processes with Dead Time
87
Case-2: With dead time Td=4sec.
2
0.0506 0.0253( )
0.5
sG s
s s
(4.6)
Case-3: With dead time Td=2sec.
ss
ssG
0
0506.00506.0)(
2
(4.7)
The root locus plot helps to get the approximate values of Kp for all the three cases.
Optimal Robust H Controller for Unstable Processes with Dead Time
88
Fig.4.17. Root locus plot for Case 1, the maximum value of the gain is 6.63.
Root Locus
Real Axis
Imagin
ary A
xis
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.220.440.62
0.76
0.85
0.92
0.965
0.992
0.220.440.62
0.76
0.85
0.92
0.965
0.992
0.1
0.2
0.3
0.4
0.5
0.1
0.2
0.3
0.4
0.5
System: s
Gain: 6.63
Pole: 0.00231 + 0.332i
Damping: -0.00695
Overshoot (%): 102
Frequency (rad/sec): 0.332
Optimal Robust H Controller for Unstable Processes with Dead Time
89
Fig.4.18. Root locus plot for Case 2, the maximum value of the gain is 9.85.
-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0.20.40.58
0.72
0.83
0.91
0.96
0.99
0.20.40.58
0.72
0.83
0.91
0.96
0.99
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
System: s
Gain: 9.85
Pole: 0.00109 + 0.497i
Damping: -0.00218
Overshoot (%): 101
Frequency (rad/sec): 0.497
Root Locus
Real Axis
Imagin
ary A
xis
Optimal Robust H Controller for Unstable Processes with Dead Time
90
Fig.4.19.Root locus plot for Case 3, the maximum value of the gain is 19.8.
The weighting functions W1(s) and W2(s) are quite sensitive and should be chosen carefully for
all the three cases based on the frequency inputs of the plants.Let T(s) and S(s) be the
complementary sensitivity function and the sensitivity function respectively.
)()(1
)()()(
sCsG
sCsGsT
(4.8)
)()(1
1)(
sCsGsS
(4.9)
-1 -0.5 0 0.5 1 1.5 2 2.5 3-1.5
-1
-0.5
0
0.5
1
1.5
0.220.42
0.6
0.74
0.84
0.92
0.965
0.99
0.220.42
0.6
0.74
0.84
0.92
0.965
0.99
0.2
0.4
0.6
0.8
1
1.2
1.4
0.2
0.4
0.6
0.8
1
1.2
1.4
System: s
Gain: 19.8
Pole: 0.00359 + 0.996i
Damping: -0.0036
Overshoot (%): 101
Frequency (rad/sec): 0.996
Root Locus
Real Axis
Imagin
ary A
xis
Optimal Robust H Controller for Unstable Processes with Dead Time
91
Also for the case 1, the weighting functions are selected as
5
2)(1
ssW (4.10)
2
15)(2
s
ssW (4.11)
By substituting the proposed control structure C(s) and the plant transfer function P(s) eq. 4.5 in
eq. 4.8 and eq. 4.9 gives the following expressions