Simple Robust Normalized PI Control for Controlled Objects with One-order Modelling Error

267

1
'( ) '( )
()
() ()
ss
s
ss
uvor
v
uv
ωωλ
ω
ωω
==−
=
(21)
Solutions of control parameters:
Solving these simultaneous equations, the following functions can be obtained:

(, )
(, , ) ( 1,2, , )
j
isj
p
Kgp j
s
ω fa

2
22
()
2
on
nn
K
Gs
ss
ω
ς
ωω
=
++

(23)()
1
s
K
Hs
s
ε
=
+

(24)


=

(26)
432
(1 )( 1)
()
(2 1) ( 2 ) ( 1)
i
ii
Kpss
Ws
s
ssKpsK
ε
εςε ες
++
=
++++ +++

(27)

Stable conditions by Hurwits approach with four parameters:
a. In the case of a certain time constant
IPL&IPS Common Region:
Advances in Reinforcement Learning

268

εςε ςε ςε
+
+−− +

(30)22 22
22
3
2
[{4 2 2 } (2 1)]
8(21)
2
p
p
k
p
ς ε ςε ς ε ςε
ες ε ςε
ε
++−−+
+
+++


(31)

22
0422where p for

++−>
ςε
ςε ςε ς ε
ςε
ςε ςε ς ε
ςε ςε ς ε

(32)

It can be proven that
3
k
>0 in the IPS region, and

23
,0kk whenp→∞ →∞ →

(33)

IP0 Region: 2
2
2( 2 1)
00
(2 1)
i
Kwherep
ςε ςε

p
where k p
ςε
=22
422for
ς
ε
ς
ε
ς
ε
+
+<

(36)
2
4( 1)
min 1kwhen
p
ε
ς
ς
ε
+


2
( 0,0 0.707), 0
(1 2 )
p
when p
ς
ες
ς
=><<=
−

c. Robust loop gain margin
The following loop gain margin is obtained from eqs. (28) through (38) in the cases of certain
and uncertain parameters:

iUL
i
K
gm
K


(39)

where
iUL
K
is the upper limit of the stable loop gain
i

the bilinear transform.
Robust loop gain margin:

(_ )
g
mPLregion
=
∞

(42)

It is risky to increase the loop gain in the
IPL region too much, even if the system does not
become unstable because a model order error may cause instability in the IPL region. In the
IPL region, the sensitivity of the disturbance from the output rises and the flat property of
the gain curve is sacrificed, even if the disturbance from the input can be isolated to the
output upon increasing the control gain.
Frequency transfer function:

222
0.5
22 2 2 2
{1 }
()[ ]
(2) {1 }
__ /
()1
+
−+−+
→=

ω
will be obtained without
considering the evaluation function on
i
K
alone.
Stationary points and the integral gain:
Using the
Stationary Points Investing for Fraction Equation approach based on Lagrange’s
undecided multiplier approach with equality restriction, the following two loop gain
equations on
x are obtained. Both identities can be used to check for miscalculation.

22
1
0.5{ 2(2 1) 1}/{2 ( 1) }
i
K
xxxp
ςς
=+−++−

(44)22
2
2
0.5{3 4(2 1) 1}/{2 (2 1) }
0

4. Numerical results
In this section, the solutions of double same integral gain for a tuning region at the
stationary point of the gain curve of the closed system are shown and checked in some
parameter tables on normalized proportional gains and normalized damping coefficients.
Moreover, loop gain margins are shown in some parameter tables on uncertain time
constants of one-order modeling error and damping coefficients of original controlled
objects for some tuning regions contained with safest only I region. 1.08120.34800.7598-99-991.2
1.41161.30681.18921.04960.86120.8
-99
-99
0.6999
0.9
1.16471.04460.89320.64301.1
1.24571.13351.00000.81861.0
1.32711.21971.09630.94240.9
1.101.051.000.95
1.08120.34800.7598-99-991.2
1.41161.30681.18921.04960.86120.8
-99
-99
0.6999
0.9
1.16471.04460.89320.64301.1
1.24571.13351.00000.81861.0
1.32711.21971.09630.94240.9
1.101.051.000.95


1.101.051.000.95

Table 2.
12ii
KK=
values for
ς
and p in IPL tuning by the first tuning method
Simple Robust Normalized PI Control for Controlled Objects with One-order Modelling Error

271
Table 1 lists the stationary points for the first tuning method. Table 2 lists the integration
gains (
12ii
K
K=
) obtained by substituting Eq. (46) into Eqs. (44) and (45) for various
damping coefficients.
Table 3 lists the integration gains (
12ii
K
K=
) for the second tuning method.

1.01.2501.6672.505.001.7
0.55560.62500.71430.83331.01.3
2.50
1.667
1.250
0.9

ε
for
each controlled
ς
by IPL (
p
=1.5) control is very useful for analysis of robustness. Then, the
unstable region, the unstable region, which does not become unstable even if the loop gain
becomes larger, and robust stable region in which uncertainty of the time constant, are
permitted in the region of
ε
.
Figure 3 shows a reference step up-down response with unknown input disturbance in the
continuous region. The gain for the disturbance step of the IPL tuning is controlled to be
approximately 0.38 and the settling time is approximately 6 sec.
The robustness on indicial response for the damping coefficient change of ±0.1 is an
advantageous property. Considering
Zero Order Hold. with an imperfect dead-time
compensator using 1
st
-order Pade approximation, the overshoot in the reference step
response is larger than that in the original region or that in the continuous region.

󲪔 󲪕󲪔 󲪖󲪔 󲪗󲪔 󲪘󲪔 󲪙󲪔 󲪚󲪔
󲪑󲪔󲪒󲪙
󲪔
󲪔󲪒󲪙
󲪕
󲪕󲪒󲪙
󲪖

󲪖
󲪕󲪔
󲪘
󲪝󲪔
󲪕󲪜󲪔
󲪖󲪛󲪔
󲪗󲪚󲪔
󲪘󲪙󲪔
󲪸
󲫓
󲪞
󲪄
󲪳
󲫙
󲫘
󲪌
󲪕
󲪍
󲪪󲫖󲫓󲫑󲪞󲪄󲪭󲫒󲪌󲪖󲪍
󲪕󲪔
󲪑󲪖
󲪕󲪔
󲪔
󲪕󲪔
󲪖
󲪕󲪔
󲪘
󲪦󲫓󲫈󲫉󲪄󲪨󲫍󲫅󲫋󲫖󲫅󲫑
󲪪󲫖󲫉󲫕󲫙󲫉󲫒󲫇󲫝󲪄󲪄󲪌󲫖󲫅󲫈󲪓󲫗󲫉󲫇󲪍
󲪱
00
( 1 0.1, 1.0, 1.5, 1.005, 1, 199.3, 0.0050674)
in
Kp s k
ςως
=± = = = = = =−

Fig. 3. Robustness of IPL tuning for damping coefficient change.
Then, Table 4 lists robust loop gain margins (
1gm >
) using the stability limit by Eq.(37) and
the loop gain by the second tuning method on uncertain
ε
in the region of
(0.1 10)
ε
≤≤
for
each controlled
ς
(>0.7) by IPL(
p
=1.5) control. The first gray row shows the area that is also
unstable
.
Advances in Reinforcement Learning

272

e
p
s/zita 0.4 0.5󲅸 0.6 0.7 0.8
0.1 1.189 1.832󲅸 2.599 3.484 4.483
0.6 1.066 1.524󲅸 2.021 2.548 3.098
1 1.097 1.492󲅸 1.899 2.312 2.729
2.1 1.254 1.556󲅸 1.839 2.106 2.362
10 1.717 1.832󲅸 1.924 2.003 2.073
Table 5. Robust loop gain margins on uncertain
ε
in each region for each controlled
ς
at IPS
(
p
=0.01)

eps/zita 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.1 0.6857 1.196 1.835 2.594 3.469 4.452 5.538 6.722
0.4 0.6556 1.087 1.592 2.156 2.771 3.427 4.118 4.84
0.5 0.6604 1.078 1.556 2.081 2.645 3.24 3.859 4.5
0.6 0.6696 1.075 1.531 2.025 2.547 3.092 3.655 4.231
1 0.7313 1.106 1.5 1.904 2.314 2.727 3.141 3.556
2.1 0.9402 1.264 1.563 1.843 2.109 2.362 2.606 2.843
10 1.5722 1.722 1.835 1.926 2.004 2.073 2.136 2.195
9999 1.9995 2 2 2 2 2 2 2
Table 6. Robust loop gain margins on uncertain
ε
in each region for each controlled
ς

󲪑󲪕󲪔
󲪑󲪙
󲪔
󲪙
󲪕󲪔
󲫞󲫍󲫘󲫅 󲪎󲪖󲪔
󲪰󲫓󲫓󲫔󲪄󲫋󲫅󲫍󲫒󲪄󲫑󲫅󲫖󲫋󲫍󲫒󲪄󲫓󲫊󲪄󲪭󲪴󲪄󲫇󲫓󲫒󲫘󲫖󲫓󲫐󲪄󲪌󲫔󲪡󲪕󲪒󲪙󲪍
󲫉󲫔󲫗󲪎󲪕󲪔
󲫋󲫑
󲪔
󲪕󲪔
󲪖󲪔
󲪗󲪔
󲪘󲪔
󲪔
󲪙󲪔
󲪕󲪔󲪔
󲪕󲪙󲪔
󲪖󲪔󲪔
󲪑󲪕󲪔
󲪑󲪙
󲪔
󲪙
󲪕󲪔
󲫞󲫍󲫘󲫅󲪎󲪖󲪔
󲫐󲫓󲫓󲫔󲪄󲫋󲫅󲫍󲫒󲪄󲫑󲫅 󲫖󲫋󲫍󲫒󲪄󲫓󲫊󲪄󲪭󲪴󲪄󲫇󲫓󲫒󲫘󲫖󲫓󲫐󲪌󲫔󲪡󲪕󲪒󲪔󲪍
󲫉󲫔󲫗󲪎󲪕󲪔
󲫋󲫑
󲪔
󲪕󲪔

󲪙
󲪕󲪔
󲫞󲫍󲫘󲫅 󲪎󲪖󲪔
󲪰󲫓󲫓󲫔󲪄󲫋󲫅󲫍󲫒󲪄󲫑󲫅󲫖󲫋󲫍󲫒󲪄󲫓󲫊󲪄󲪭󲪴󲪄󲫇󲫓󲫒󲫘󲫖󲫓󲫐󲪄󲪌󲫔󲪡󲪔󲪒󲪔󲪕󲪍
󲫉󲫔󲫗󲪎󲪕󲪔
󲫋󲫑

(a)
p
=1.5 (b)
p
=1.0 (c)
p
=0.5 (d)
p
=0.01or 0
Fig. 4. Mesh plot of closed loop gain margin
Next, we call the larger attenuation region with more than 1>
γ
and less than 2 loop gain
margin to the weak robust segment region in which region uncertainty time constant of
one-order modeling error is only allowed in some region over some larger loop gain margin
and some larger change of attenuation is not allowed. The third and the forth segment is
almost unstable. Especially, notice that the joint of each segment is large bending so that the
sensitivity of uncertainty for loop gain margin is larger more than the imagination. 󲪔
󲪔󲪒󲪙
󲪕

󲪖󲪔
󲪔
󲪔󲪒󲪙
󲪕
󲪕󲪒󲪙
󲪖
󲪖󲪒󲪙
󲫞󲫍󲫘󲫅
󲪸󲫌󲫉󲪄󲫛󲫓󲫖󲫗󲫘󲪄󲫐󲫍 󲫒󲫉󲪄󲫓󲫊󲪄 󲫐󲫓󲫓󲫔󲪄󲫋 󲫅󲫍 󲫒󲪄󲫑󲫅󲫖󲫋 󲫍󲫒󲫉󲪄 󲫊󲫓󲫖󲪄󲪭󲪴󲪷󲪄󲫘󲫙󲫒󲫍󲫒󲫋󲪄 󲫖󲫉󲫋󲫍󲫓󲫒
󲫉󲫔󲫗
󲫋󲫑
󲪔
󲪔󲪒󲪖
󲪔󲪒󲪘
󲪔󲪒󲪚
󲪔󲪒󲪜
󲪕
󲪔
󲪙
󲪕󲪔
󲪕󲪙
󲪖󲪔
󲪔
󲪔󲪒󲪙
󲪕
󲪕󲪒󲪙
󲪖
󲪖󲪒󲪙
󲫞󲫍󲫘 󲫅
󲪸󲫌󲫉󲪄󲫛󲫓󲫖󲫗󲫘󲪄󲫐󲫍󲫒󲫉󲪄󲫓󲫊󲪄󲫇󲫐󲫓󲫗󲫉󲫈󲪄󲫐󲫓󲫓󲫔󲪄󲫋󲫅󲫍󲫒󲪄󲫑󲫅󲫖󲫋󲫍󲫒󲪄󲫅󲫗󲪄󲫘󲫌󲫉󲪄󲫔󲫅󲫖󲫅󲫑󲫉󲫘󲫉󲫖󲪄󲫞󲫍󲫘󲫅󲪄󲫓󲫊󲪄󲫇󲫓󲫒󲫘󲫖󲫓󲫐󲫉󲫈󲪄󲫓󲫆󲫎󲫉󲫇󲫘

p
=1.5 (b)
p
=0.01 (c)
p
=0 (d)
p
=1.5, 1.0, 0.5, 0.01
Fig. 5. The various worst lines of loop gain margin in a parameter plane (certain&uncertain)
Moreover, the readers had to notice that the strong robust region and weak robust region
of IPL is shift to larger damping coefficient region than ones of IPS and IP0. Then, this is
also one of risk on IPL tuning region and change of tuning region from IP0 or IPS to IPL
region.
5. Conclusion
In this section, the way to convert this IP control tuning parameters to independent type PI
control is presented. Then, parameter tuning policy and the reason adopted the policy on the
controller are presented. The good and no good results, limitations and meanings in this
chapter are summarized. The closed loop gain curve obtained from the second order example
with one-order feedback modeling error implies the butter-worth filter model matching
method in higher order systems may be useful. The Hardy space norm with bounded
window was defined for I, and robust stability was discussed for MIMO system by an
expanssion of small gain theorem under a bounded condition of closed loop systems.
Advances in Reinforcement Learning

274
- We have obtained first an integral gain leading type of normalized IP controller to
facilitate the adjustment results of tuning parameters explaining in the later. The
controller is similar that conventional analog controllers are proportional gain type of PI
controller. It can be converted easily to independent type of PI controller as used in recent
computer controls by adding some converted gains. The policy of the parameter tuning is

It was not good results that the analytical solution and the stable region were
complicated to obtain for higher order systems with higher order modeling error
though they were easy and primary. Then, it was unpractical.
6. Appendix
A. Example of a second-order system with lag time and one-order modelling error
In this section, for applying the robust PI control concept of this chapter to systems with
lag time, the systems with one-order model error are approximated using Pade
approximation and only the simple stability region of the integral gain is shown in the
special proportional tuning case for simplicity because to obtain the solution of integral
gain is difficult.
Here, a digital IP control system for a second-order controlled object with lag time L without
sensor dynamics is assumed. For simplicity, only special proportional gain case is shown.
Transfer functions:

2
22
(1 0.5 )
(1 0.5 )
()
1 ( 1)(0.5 1) 2
Ls
n
nn
K
Ls
Ke K Ls
Gs
Ts Ts Ls s s
ω
ς

s
ε
=
+

(A3)

Normalized operation:
The normalize operations as same as above mentioned are done as follows. ,
n
n
s
sLLω
ω


(A4)2
(1 0.5 )
()
21
Ls
Gs
ssς
−

ωω
=

(A8)111
() ( ) () ( )
iin
n
Cs K p Cs K p
ss
ω
ω
=+ = +

(A9)

Closed loop transfer function:
The closed loop transfer function is obtained using above normalization as follows;

2
2
43 2
1 (1 0.5 )
()
(21)
()
1 (1 0.5 )
1( )


(A.10)43 222
0.5
(1 0.5 )( 1)
()
(2 1) ( 2 0.5 )
i
ii
if p L then
KLss
Ws
ss LKssK
ε
εςε ες
=
−+
=
++++− ++
(A11)

432
0
(1 0.5 )( 1)
()
(2 1) ( 2 ) (1 0.5 )
i
ii

ςε
+
<<< >>
−22 2
{(2 1)(2 0.5 ) } (2 1) 0.5
ii
LK K when p L
ςε ς ε ε ςε
++− −> + =

(A13)2
22
2( 2 1)
0.5
(2 1){(2 1) 0.5 }
i
K
when p L
L
ς
εςε
ςε ςε
+
+

2
00.5
(1 0.5 )
i
K
when p L
L
ς
<< =
+

(A16)

Analytical solution of Ki for flat gain curve using Stationary Points Investing for Fraction
Equation approach is complicated to obtain, then it is remained for reader’s theme.
In the future, another approach will be developed for safe and simple robust control.
B. Simple soft M/A station
In this section, a configuration of simple soft M/A station and the feedback control system
with the station is shown for a simple safe interlock avoiding dangerous large overshoot.
B.1 Function and configuration of simple soft M/A station
This appendix describes a simple interlock plan for an simple soft M/A station that has a
parameter-identification mode (manual mode) and a control mode (automatic mode).
The simple soft M/A station is switched from automatic operation mode to manual
operation mode for safety when it is used to switch the identification mode and the control
mode and when the value of Pv exceeds the prescribed range. This serves to protect the
plant; for example, in the former case, it operates when the integrator of the PID controller
varies erratically and the control system malfunctions. In the latter case, it operates when
switching from P control with a large steady-state deviation with a high load to PI or PID
control, so that the liquid in the tank spillovers. Other dangerous situations are not
considered here because they do not fall under general basic control.

Mv at auto mode
Switch
Conditions
On Pv
Conditions
On M/A
Switch
M
A
Integrated
Switching
Logic
Self-holding
Logic
S
W
I
T
C
H
T Pv’
Pv
Mv at manual mode
Cv
Mv
Mv at auto mode
Switch
Conditions
On Pv
Conditions

Gs e
Ts
−
=
+

2.
Sensor & Signal Conditioner:
()
1
s
s
s
K
Gs
Ts
=
+

3.
Controller:
2
1
( ) 0.5 ( 0.5 )
i
Cs K L
s
=+

4.

Auto Out of Service
Mv Pv
Cv
Av

(a) Switching example from auto mode to
out of service by Pv High
(b) Switching example from auto mode to
manual mode by Pv High
Fig. B3 Simulation results for 2 kinds of switching mode
C. New norm and expansion of small gain theorem
In this section, a new range restricted norm of Hardy space with window(Kohonen T., 1995)
w
H
∞
is defined for I, of which window is described to notation of norm with superscript w,
and a new expansion of small gain theorem based on closed loop system like general
w
H
∞

control problems and robust sensitivity analysis is shown for applying the robust PI
control concept of this chapter to MIMO systems.
The robust control was aims soft servo and requested internal stability for a closed loop
control system. Then, it was difficult to apply process control systems or hard servo systems
which was needed strong robust stability without deviation from the reference value in the
steady state like integral terms.
The method which sets the maximum value of closed loop gain curve to 1 and the results of
this numerical experiments indicated the above sections will imply the following new
expansion of small gain theorem which indicates the upper limit of Hardy space norm of a

+
(C-2)
Simple Robust Normalized PI Control for Controlled Objects with One-order Modelling Error

279
then the following inequality on the open loop transfer function is hold in a region of
frequency.

min max
1
()() , 1 [ , ]
1
w
Gj Hj for
∞
ωω≤ γ≥ ω∈ωω
γ−
(C-3)
In the same feedback system, G(s) holds the following inequality in a region of frequency.

min max
() , 1 [ , ]
1
w
Gj for
∞
γ
ω≤ γ≥ ω∈ω ω
γ−
(C-4)

(C-5)

() 1 ()() 1 () ()
1
1()
()
wwww
w
w
Gj Gj Hj Gj Hj
Hj
Gj
∞
∞∞∞
∞
∞
ω≤+ ω ω≤+ ω ω
−ω≤
ω
(C-6)

1
() 1, 1
1
()
1
1()
w
w
w

280
D. Parametric robust topics
In this section, the following three topics (Bhattacharyya S. P., Chapellat H., and Keel L. H., 1994.)
are introduced at first for parametric robust property in static one, dynamic one and stable one as
assumptions after linearizing a class of non-linear system to a quasi linear parametric variable
(QLPV) model by Taylar expansion using first order reminder term. (M.Katoh, 2010)
1.
Continuity for change of parameter
Boundary Crossing Theorem
1) fixed order polynomials P(λ,s)
2) continuous polynomials with respect to one parameter λ on a fixed interval I=[a,b].
If P(a,s) has all its roots in S, P(b,s) has at least one root in U, then there exists at least
one ρ in (a,b] such that:
a) P(ρ,s) has all roots in S U∂S
b) P(ρ,s) has at least one root in ∂S

P(a,s)
P(b,s)
P(ρ,s)

Fig. D-1 Image of boundary crossing theorem
2.
Convex for change of parameter
Segment Stable Lemma
Let define a segment using two stable polynomials as follows.
12
() () (1 ) ()
s
ss
λ

λ
δλ
≠∈∂∈

3.
Worst stability margin for change of parameter
Parametric stability margin (PSM) is defined as the worst case stability margin within
the parameter variation. It can be applied to a QLPV system of a class of non-linear
system. There are non-linear systems such as becoming worse stability margin than
linearized system although there are ones with better stability margin than it. There is a
case which is characterized by the one parameter m which describes the injection rate of
I/O, the interpolation rate of segment or degree of non-linearity.
E. Risk and Merit Analysis
Let show a summary and enhancing of the risk discussed before sections for safety in the following
table.
Simple Robust Normalized PI Control for Controlled Objects with One-order Modelling Error

281
Kinds Evaluation of influence Countermeasure
1) Disconnection of
feedback line
2) Overshoot over limit
value
1) Spill-over threshold
2) Attack to weak material
Auto change to manual
mode by M/A station
Auto shut down
Change of tuning region
from IPS to IPL by making

in process control and hard
servo area

It is dislike property in soft
servo and robot control
because of hardness for
disturbance
There is a strong robust
stability damping region in
which the closed loop gain
margin for any uncertainty
is over 2 and almost not
changing.
It is uniform safety for
some proportional gain
tuning region and changing
of damping coefficient.
For integral loop gain
tuning, it recommends the
simple limiting sensitivity
approach.
1) Because the region is
different by proportional
gain, there is a risk of grade
down by the gain tuning.
There is a weak robust
stability damping region in
which the worst closed loop
gain margin for any
uncertainty is over given

Katoh M., (2009). Loop Gain Margin in Simple Robust Normalized IP Control for Uncertain
Parameter of One-Order Model Error, International Journal of Advanced Computer
Engineering, Vol.2, No.1, January-June, pp.25-31, ISSN:0974-5785, Serials
Publications, New Delhi (India)
Katoh M and Imura N., (2009). Double-agent Convoying Scenario Changeable by an
Emergent Trigger, Proceedings of the 4
th
International Conference on Autonomous
Robots and Agents, Feb 10-12, Wellington, New Zealand, pp.442-446
Katoh M. and Fujiwara A., (2010). Simple Robust Stability for PID Control System of an
Adjusted System with One-Changeable Parameter and Auto Tuning, International
Journal of Advanced Computer Engineering, Vol.3, No.1, ISSN:0974-5785, Serials
Publications, New Delhi (India)
Katoh M.,(2010). Static and Dynamic Robust Parameters and PI Control Tuning of TV-MITE
Model for Controlling the Liquid Level in a Single Tank”, TC01-2, SICE Annual
Conference 2010, 18/August TC01-3
Krajewski W., Lepschy A., and Viaro U.,(2004). Designing PI Controllers for Robust Stability
and Performance, Institute of Electric and Electronic Engineers Transactions on Control
System Technology, Vol. 12, No. 6, pp. 973- 983.
Kohonen T.,(1995, 1997). Self-Organizing Maps, Springer
Kojori H. A., Lavers J. D., and Dewan S. B.,(1993). A Critical Assessment of the Continuous-
System Approximate Methods for the Stability Analysis of a Sampled Data System,
Institute of Electric and Electronic Engineers Transactions on Power Electronics, Vol. 8,
No. 1, pp. 76-84.
Miyamoto S.,(1998). Design of PID Controllers Based on H
∞
-Loop Shaping Method and LMI
Optimization, Transactions of the Society of Instrument and Control Engineers, Vol. 34,
No. 7, pp. 653-659. (in Japanese)
Namba R., Yamamoto T., and Kaneda M., (1998). A Design Scheme of Discrete Robust PID

linear results to some nonlinear models and use passivity theory to write nonlinear fault
tolerant controllers. In this chapter several controllers are proposed for different problem
settings: a) Linear time invariant (LTI) certain plants, b) uncertain LTI plants, c) LTI models
with input saturations, d) nonlinear plants affine in the control with single input, e) general
nonlinear models with constant as well as time-varying faults and with input saturation. We
underline here that we focus in this chapter on the theoretical developments of the controllers,
readers interested in numerical applications should refer to (6; 8).
2. Preliminaries
Throughout this chapter we will use the L
2
norm denoted ||.||,i.e.forx ∈ R
n
we define
||x|| =
√
x
T
x. The notation L
f
h denotes the standard Lie derivative of a scalar function h(.)
along a vector function f(.). Let us introduce now some definitions from (40), that will be
frequently used in the sequel.
Definition 1 ((40), p.45): The solution x
(t, x
0
) of the system
˙
x = f (x), x ∈ R
n
, f locally

)|| ⇒ 0, ∀| |
˜
x
0
− x
0
|| <
r(x
0
) and
˜
x
0
∈ Z, the solution is asymptotically stable conditionally to Z.Ifr(x
0
) → ∞,
the stability is global.
Definition 2 ((40), p.48): Consider the system H :
˙
x
= f (x, u), y = h(x, u), x ∈ R
n
, u, y ∈ R
m
,
with zero inputs, i.e.
˙
x
= f (x,0), y = h(x,0) and let Z ⊂ R
n


t
1
t
0
|ω(u(t), y(t))|dt < ∞, ∀ t
0
≤ t
1
. S is called the storage function. If the storage function is
differentiable the previous conditions writes as
˙
S
(x(t)) ≤ ω(u(t), y(t)).
The system H is said to be passive if it is dissipative with the supply rate w
(u, y)=u
T
y.
Definition 4 ((40), p.36): We say that H is output feedback passive (OFP(ρ)) if it is dissipative
with respect to ω
(u, y)=u
T
y − ρy
T
y for some ρ ∈ R.
We will also need the following definition to study the case of time-varying faults in Section
8.
Definition 5 (24): Afunction
x : [0, ∞) → R
n

of the function f : R
n
→ R. We also mean by semiglobal stability of the equilibrium point
x
0
for the autonomous system
˙
x = f(x), x ∈ R
n
with f a smooth function, that for each
compact set K
⊂ R
n
containing x
0
, there exist a locally Lipschitz state feedback, such that x
0
is asymptotically stable, with a basin of attraction containing K ((44), Definition 3, p. 1445).
3. FTC for known LTI plants
First, let us consider linear systems of the form
˙
x
= Ax + Bαu,(1)
where, x
∈ R
n
, u ∈ R
m
are the state and input vector, respectively, and α ∈ R
m×m

m
→ R
m
, s.t. [ψ(t, y) −K
1
y]
T
[ψ(t, y) −
K
2
y] ≤ 0, ∀(t, y),withK = K
2
− K
1
= K
T
> 0, where K
1
= diag(k1
1
, ,k1
m
), K
2
=
diag(k2
1
, ,k2
m
),issaidtobelongtothesector[K

domain if the origin is uniformly asymptotically (UA) stable within a finite domain.
We can now introduce the idea used here, which is as follows:
Let us associate with the faulty system (1) a virtual output vector y
∈ R
m
˙
x
= Ax + Bαu
y
= Kx,
(3)
and let us write the controller as an output feedback
u
= −y.(4)
From (3) and (4), we can write the closed-loop system as
˙
x
= Ax + Bv
y
= Kx
v
= −α(t)y.
(5)
We have thus transformed the problem of stabilizing (1), for all bounded matrices α
(t),tothe
problem of stabilizing the system (5) for all α
(t). It is clear that the problem of GUA stabilizing
(5) is a Lure’s problem in (2), with the linear time varying stationarity ψ
(t, y)=α(t)y,and
where the ‘nonlinearities’ admit the sector bounds K

2
ij
)

P
ˆ
A
(K)+
ˆ
A
T
(K)P (
ˆ
C
T
− P
ˆ
B)W
−1
((
ˆ
C
T
− P
ˆ
B)W
−1
)
T
−I

D
+
ˆ
D
T
)
0.5
and {
ˆ
A(K),
ˆ
B(K),
ˆ
C(K),
ˆ
D(K)} is a minimal realization
of the transfer matrix
ˆ
G
=[I + K(sI − A)
−1
B][ I + 
1
× I
m×m
K(sI − A)
−1
B]
−1
,(8)

,
where G
(s)=K(sI − A)
−1
B, is strictly positive real (SPR). Now, using the KYP lemma as
presented in (Lemma 6.3, (22), p. 240), we can write that a sufficient condition for the GUA
stability of x
= 0 along the solution of (1) with u = −Kx is the existence of P = P
T
> 0, L and
W, s.t.
P
ˆ
A
(K)+
ˆ
A
T
(K)P = −L
T
L − P,  > 0
P
ˆ
B
(K)=
ˆ
C
T
(K) − L
T

ˆ
A
T
(K)P = −L
T
L − P,  > 0
P
ˆ
B
(K)=
ˆ
C
T
(K) − L
T
W
W
T
W =
ˆ
D
(K)+
ˆ
D
T
(K)
rank
⎡
⎢
⎢

T
=(
ˆ
C
T
− P
ˆ
B)W
−1
. Finally, from the first equation in (10), we arrive at the following
condition on P
P
ˆ
A
(K)+
ˆ
A
T
(K)P +(
ˆ
C
T
− P
ˆ
B)W
−1
((
ˆ
C
T

B)W
−1
)
T
−I

< 0. (11)
Thus, to solve equation (10) we can solve the constrained optimal problem
287
Passive Fault Tolerant Control
min
k
ij
(
∑
i=m
i
=1
∑
j=n
j
=1
k
2
ij
)

P
ˆ
A

⎢
⎢
⎣
K
KA
.
.
.
KA
n−1
⎤
⎥
⎥
⎥
⎦
= n. 
(12)
Note that the inequality constraints in (7) can be easily solved by available LMI algorithms, e.g.
feasp under Matlab. Furthermore, to solve equation (10), we can propose two other different
formulations:
1. Through nonlinear algebraic equations: Choose W
= W
T
which implies by the third
equation in (10) that W
=(
ˆ
D
(K)+
ˆ

⎢
⎢
⎣
K
KA
.
.
.
KA
n−1
⎤
⎥
⎥
⎥
⎦
= n.
(13)
To solve (13) we can choose 
=
˜

2
and P =
˜
P
T
˜
P, which leads to the nonlinear algebraic
equation
F

n and l
ij
, i =
1, ,m, j = 1,
˜
n are the elements of K,
˜
P and L, respectively. Equation (14) can then be
resolved by any nonlinear algebraic equations solver, e.g. fsolve under Matlab.
2. Through Algebraic Riccati Equations (ARE): It is well known that the positive real lemma
equations, i.e. the first three equations in (10) can be transformed to the following ARE ((3),
pp. 270-271):
P
(
ˆ
ˆ
A
−
ˆ
BR
−1
ˆ
C)+(
ˆ
ˆ
A
T
−
ˆ
C

n
×
˜
n
, R =
ˆ
D
(K)+
ˆ
D
T
(K) > 0. Then, if a solution P = P
T
> 0is
found for (15) it is also a solution for the first three equation in (10), together with
W
= −VR
1/2
, L =(P
ˆ
B −
ˆ
C
T
)R
−1/2
V
T
, VV
T

.
.
KA
n−1
⎤
⎥
⎥
⎥
⎦
= n,
(16)
where P is the symmetric solution of the ARE (15), that can be directly computed by
available solvers, e.g. care under Matlab.
There are other linear controllers for LPV system, that might solve the problem stated in
Section 3.1, e.g. (1). However, the solution proposed here benefits from the simplicity of the
formulation based on the absolute stability theory, and allows us to design FTCs for uncertain
and saturated LTI plants, as well as nonlinear affine models, as we will see in the sequel.
Furthermore, reformulating the FTC problem in the absolute stability theory framework may
be applied to solve the FTC problem for several other systems, like infinite dimensional
systems, i.e. PDEs models, stochastic systems and systems with delays (see (26) and the
references therein). Furthermore, compared to optimal controllers, e.g. LQR, the proposed
solution offers greater robustness, since it compensates for the loss of effectiveness over
[
1
,1]. Indeed, it is well known that in the time invariant case, optimal controllers like LQR
compensates for a loss of effectiveness over
[1/2, 1] ((40), pp. 99-102). A larger loss of
effectiveness can be covered but at the expense of higher control amplitude ((40), Proposition
3.32, p.100), which is not desirable in practical situations.
Let us consider now the more practical case of LTI plants with parameter uncertainties.

∈ R
n×m
},
α
= diag(α
11
, ,α
mm
),0< 
1
≤ α
ii
≤ 1 ∀i ∈{1, ,m},andA, B, x, u as defined before.
4.1 Problem statement
Find a state feedback controller u(x) such that the closed-loop controlled system (17) admits x = 0 as a
globally asymptotically (GA) stable equilibrium point
∀α(s.t.0< 
1
≤ α
ii
≤ 1), ∀ΔA ∈◦A, ΔB ∈
◦B
.
4.2 Problem solution
We first re-write the model (17) as follows:
˙
x
=(A + ΔA)x +(B + ΔB) v
y
= Kx

L
T
B
T
+ A
˜
H −BL
˜
K ≤ 0 ∀L ∈ L
v
,
˜
Q =
˜
Q
T
> 0,
˜
H > 0
−
˜
Q
+
˜
HΔA
T
−
˜
K
T

◦A, ◦B respectively.
Proof: Under Assumption 1, and using Theorem 5 in ((15), p. 330), we can write the stabilizing
static state feedback u
= −Kx,whereK is such that, for a given H > 0, Q = Q
T
> 0wehave

Q
+(A −BLK)
T
H + H(A −BLK) ≤ 0 ∀L ∈ L
v
−Q +((ΔA −ΔBLK)
T
H + H(ΔA − ΔBLK)) < 0 ∀(ΔA, ΔB,Ł) ∈◦A
v
×◦B
v
× L
v
,
(20)
where, L
v
is the set containing the vertices of {
1
I
m×m
, I
m×m

HA
T
−
˜
K
T
L
T
B
T
+ A
˜
H − BL
˜
K ≤ 0 ∀L ∈ L
v
,
˜
Q =
˜
Q
T
> 0,
˜
H > 0
−
˜
Q +
˜
HΔA

the available actuators have limited maximum amplitudes. For this reason, it is more realistic
to consider bounded control amplitudes in the design of the fault tolerant controller.
5. FTC for LTI plants with control saturation
We consider here the system (1) with input constraints |u
i
|≤u
max
i
, i = 1, ,m, and study the
following FTC problem.
5.1 Problem statement
Find a bounded feedback controller, i.e. |u
i
|≤u
max
i
, i = 1, ,m, such that the closed-loop controlled
system (1) admits x
= 0 as a uniformly asymptotically (UA) stable equilibrium point ∀α(t)(s.t.0<

1
≤ α
ii
(t) ≤ 1), i = 1, , m, within an estimated domain of attraction.
5.2 Problem solution
Under the actuator constraint |u
i
|≤u
max
i

)min{1, |y
i
|}.
Thus we have rewritten the system (1) as a MIMO Lure’s problem with a generalized sector
condition, which is a generalization of the SISO case presented in (16).
Next, we define the two functions ψ
1
: R
n
→ R
m
, ψ
1
(x)=−
1
I
m×m
sat(Kx) and
290
Robust Control, Theory and Applications
ψ
2
: R
n
→ R
m
, ψ
2
(x)=−sat(Kx).
We can then write that v is spanned by the two functions ψ

i=1
γ
i
(t)ψ
i
(x),
i=2
∑
i=1
γ
i
(t)=1, γ
i
(t) ≥ 0 ∀ t}.
Note that in the SISO case, the problem of analyzing the stability of x
= 0 for the system (22)
under the constraint (23) is a Lure’s problem with a generalized sector condition as defined in
(16).
Let us recall now some material from (16; 17), that we will use to prove Proposition 4.
Definition 8 ((16), p.538): The ellipsoid level set ε
(P, ρ) := {x ∈ R
n
: V(x)=x
T
Px ≤ ρ}, ρ >
0, P = P
T
> 0 is said to be contractive invariant for (22) if
˙
V

(ρ)={x ∈ R
n
/ V(x) ≤ ρ} and a set of functions
ψ
i
(u), i ∈{1, , N}. Suppose that for each i ∈{1, , N}, L
V
(ρ) is contractively invariant
for
˙
x
= Ax + Bψ
i
(u).Letψ(u, t) ∈ co{ψ
i
(u), i ∈{1, , N}} for all u, t ∈ R,thenL
V
(ρ) is
contractively invariant for
˙
x
= Ax + Bψ(u, t).
Theorem 1((17), p. 353): Given an ellipsoid level set ε
(P, ρ), if there exists a matrix H ∈ R
m×n
such that
(A + BM(v, K, H))
T
P + P(A + BM(v, K, H)) < 0,
for all

m
+(1 −v
m
)h
m
⎤
⎥
⎦
, (24)
then ε
(P, ρ) is a contractive domain for
˙
x = Ax + Bsat(Kx).
We can now write the following result:
Proposition 4: Under Assumption 1, the system (1) admits x
= 0 as a UA stable equilibrium
1
Hereafter, h
i
, k
i
denote the ith line of H, K, respectively.
291
Passive Fault Tolerant Control