Title: DESIGN OF ROBUST CONTROL SYSTEM FOR A PARAMETRIC UNCERTAIN JET ENGINE POWER PLANT
1DESIGN OF ROBUST CONTROL SYSTEM FOR A PARAMETRIC
UNCERTAIN JET ENGINE POWER PLANT
Prof. P.S.V. Nataraj Systems
Control Engineering Indian Institute of
Technology Bombay Mumbai, India
2Overview
 Introduction to Jet Engine
 Various Gas Turbine Configurations
 Performance and Control of Turbojet Gas Turbine
Engine  Control Requirements of aero Gas Turbine
 Design Steps for the Controller Design
 Robust Control Design Techniques
3FUNDAMENTALS
 For an aircraft turbine engine, it is necessary
to achieve maximum thrust with minimum engine
weight.  Control system should ensure all components
operate at mechanical or thermal limits for at
least one of the engines critical operating
conditions.
4FUNDAMENTALS
Cont.
 Single input single output controls are used on
commercial engines where emphasis is on economic
operation.  Multivariable controls provide enhanced
performance for military engines especially
those with variable flow path geometry.  In both case the maximum use of the available
performance within the structural and
aerothermal limitations is required.
5 Reheat burners
Compressor Variable Geometry ( VG)
Nozzle actuators
Main Burners
Afterburner flow Distributor
Reheat
Nozzle
Hydromechanical Systems
MainFuel
VG
Manual Fuel Control Linkage
Fuel in
PLA
Gearbox
Digital Electronic Control Unit
Engine System Feedback back
JET ENGINE WITH INTEGRATED CONTROLS
7
6Thrust Augmentation By Afterburning
 Afterburner accomplishes short period increase in
the rated thrust of the gas turbine engine  Objective is to improve the take off, climb and
maximum speed characteristics of jet propelled
aircraft
7Various Gas Turbine Configurations
 A Single Spool Turbojet Power Plant consists of
an intake, a compressor, a combustion chamber, a
turbine and a propelling nozzle.  A Twin Spool turbofan power plant consists of an
intake, a low pressure compressor, high pressure
compressor, a combustion chamber, a high pressure
turbine, a low pressure turbine, a mixer, and a
propelling nozzle.
8Dynamic Performance of Turbojet Engine
 Dynamic equations are obtained using the power
balance, continuity and energy equations.  The steady state power balance equation simply
implies that the power output of the turbine must
equal the power absorbed by the compressor. 
9Control System Design
 The increased gas turbine engine complexity has
resulted in a corresponding increase in the
complexity of the control system.  Control requirements applied to gas turbine
engines consist of ensuring safe, stable engine
operation.  An accurate and reliable control system is
required to ensure needed engine performance and
operational stability throughout the flight
envelope. Cont.
10Control System Design Cont.
 The control system must sense Pilots commands,
air frame requirements as well as critical engine
parameters.  It must then compute the necessary schedules and
actuate system variable for total engine control
over the full range of operation.  Full authority digital electronic control system
consists of two controllers, one of which is the
digital electronic computer section and the other
the backup hydro mechanical section.
11(No Transcript)
12Control Requirements of aero Gas Turbine Engine
 Relationship between the fuel flow control
variable and the engine rotational speed over the
flight envelope.  Relationship between the fuel flow control
variable and the engine compression ratio over
the flight envelope.  Relationship between the Power lever angle input
into the control system and the rotational speed
of the engine so as to incorporate a governing
action and control engine thrust.
.Cont.
13Control Requirements of Aero Gas Turbine Engine
Cont.
 Set limits on fuel flows so as to ensure safe
operation of engine without excess rotational
speeds, temperatures and pressures at specific
stations of the engine.  Position control of exhaust nozzle as per nozzle
control schedule depending on Power Lever Angle
(PLA) position.  Reheat fuel flow selection by Pilot through PLA.
14Control Structure of Military Jet Engine
15Design Steps for the Controller of Jet Engine
 Selection of the basic operating requirements for
each element of control.  Evaluation of engine requirements and control
variables to select the mode of control that will
provide the best operation.  Selection of types of control regulators and
computing system .  Evaluation of stability requirements and basic
performance requirements of the system
16Design Steps for the Controller of Jet Engine
Cont.
 Establishing the ability of the control
components to meet the physical requirements of
endurance, environment and vibration.  Evaluating the final system by analysis or
testing to establish its ability to perform as
required under actual operating conditions
17Digital Control of Jet Engine
 The primary control objectives of gas turbine
i.e., thrust, is a complex and nonlinear
function of flight conditions like altitude, mach
number and control variables comprising of main
engine fuel flow, reheat fuel flow and nozzle
area.  These parameters are to be controlled with
complex schedules to a high accuracy.
Cont.
18Digital Control of Jet Engine Cont.
 Full authority digital engine control system
executes these schedules to provide the most
optimum performance.  The control system components are the hydro
mechanical fuel management units with built in
pumps, metering values and position feedback
transducers and nozzle control system and their
corresponding digital control units.
19MAJOR CONTROL LOOPS OF JET ENGINE POWER PLANT
 HIGH PRESSURE COMPRESSOR ACCELERATION CONTROL
LOOP  LOW PRESSURE COMRESSOR SPEED CONTROL LOOP
 VARIABLE GEOMETRY CONTROL LOOP
 NOZZLE CONTROL LOOP
 AFTERBURNER FUEL FLOW CONTROL LOOP
20 MAIN FUEL FLOW CONTROLS NL
 COMPRESSOR VARIABLE GEOMETRY CONTROLS NH
 NOZZLE AREA SCHEDULED AS FUNCTION OF ALTITUDE AND
PILOT THROTTLE POSITION  AFTERBURNER FUEL FLOW SCHEDULED AS A FUNCTION OF
NOZZLE AREA AND ALTITUDE
21BLOCK DIAGRAM OF HIGH PRESSURE COMPRESSOR SPEED
ACCELERATION LOOP OF JET ENGINE
22 3 Parametric uncertainties in Plant.
 Uncertainty 5
 Performance requirements for the inner
 Acceleration loop
 1. Stability Margins Gain Margin 6 dB
 Phase Margin 450
 2. Tracking Specifications Rise time of
closed loop transfer 
function between 0.4 to 0.8 sec  1. No overshoot

23BLOCK DIAGRAM OF THE LOW PRESSURE COMPRESSOR
SPEED CONTROL OF JET ENGINE
24 5 Parametric uncertainties in Plant.
Uncertainty 5 Performance
requirements for the outer Speed loop 1.
Stability Margins Gain Margin 6 dB
Phase Margin 450 2. Tracking
Specifications Rise time of closed loop
transfer
function between 0.7 to 0.95 sec 1.
No overshoot
25BLOCK DIAGRAM OF EXHAUST NOZZLE POSITION CONTROL
SYSTEM
Actuator position
LVDT Linear variable differential transducer
(to measure stroke)
26 2 Parametric uncertainties in Plant.
Uncertainty 4
Performance requirements for the Variable C
Nozzle control loop 1. Stability Margins
Gain Margin 6 dB Phase Margin
450 2. Tracking Specifications Rise time
of closed loop transfer
function between 0.12 to
0.18 sec 1. No overshoot
27BLOCK DIAGRAM OF AFTERBURNER FUEL FLOW CONTROL
LOOP
Demanded Afterburner Fuel flow
Servo current
Throttle
Actual Afterburner fuel flow
Operating schedule
Afterburner H/M System
Controller
Temperature
Operating Schedule Afterburner fuel flow as a
function of nozzle area
28 2 Parametric uncertainties in Plant.
Uncertainty 4 Performance
requirements for the Afterburner fuel flow
control loop 1. Stability Margins Gain
Margin 6 dB Phase Margin
450 2. Tracking Specifications Rise time
of closed loop transfer
function between 0.55 to
0.78 sec 1. No overshoot
29BLOCK DIAGRAM OF THE VARIABLE GEOMETRY CONTROL
SYSTEM OF JET ENGINE POWER PLANT
VG Variable geometry IGV Inlet guide vane H/M
Hydromechanical
Pilot Throttle
Desired Compressor IGV position
Operating schedule
VG H/M System
Compensator

Inlet Temperature
Actual IGV position
Input to VG H/M system Servo valve
current Output of VG H/M system IGV Actuator
Position
30 3 Parametric uncertainties in Plant.
Uncertainty 4 Performance
requirements for the variable Geometry
control loop 1. Stability Margins Gain
Margin 6 dB Phase Margin
450 2. Tracking Specifications Rise time
of closed loop transfer
function between 0.15 to
0.24 sec 1. No overshoot
31 An Interval Analysis Approach for Design of
Robust First Order Compensator for Jet Engine
32Basic Concepts
 Consider a strictly proper interval plant family
P comprising of  plants of the form.


 where interval bounds are a priori given for each
uncertain coefficient qi and ri . Let C(s) be a
compensator in a feedback structure for this
interval plant. If C(s) is such that it
stabilizes the entire P, then C(s) is said to
robustly stabilize P.  If the compensator for an interval plant is
first order, the stability of only sixteen
extreme plants is necessary and sufficient
to stabilize the entire interval family.
33 Block diagram of Compressor Speed control Loop
of Jet Engine
Desired Compressor Speed
Throttle
Operating Schedule
Compressor
Engine
Inlet Temperature
Compressor Speed
34 Block diagram of Ndot controller of Jet Engine
Desired Ndot
Ndot Schedule
Inlet Temperature
Compressor
Engine
Compressor Speed
Differentiator
35Algorithm for Compensator Synthesis
Let 4 denote the set 1,2,3,4. Let Ni1 (s) , i1
4 , denote the Kharitonov polynomials
associated with NP(s,q).
.Cont.
36Algorithm for Compensator Synthesis
..Cont.
Consider that a robustly stabilizing PI
controller
is to be
synthesized for an interval plant family P with
Kharitonovs polynomials N1 (s), N2 (s), N3 (s)
and N4 (s) and D1 (s), D2 (s), D3 (s) and D4 (s)
for the numerator and denominator respectively.
37The sixteen extreme plants are defined by
with i1, i2 1,2,3,4.
38The compensator synthesis Algorithm given by
Barmish et al.
Algorithm Synthesis of first order compensator
(due to Barmish et. al.) Begin Algorithm
(i) Set up sixteen Routh tables for closed loop
polynomials associated with each extreme plant.
(ii) Enforce positivity for each of the first
column entries which are functions of K1 and K2.
This leads to set of inequalities involving K1
and K2. (iii) To obtain the final controller,
(K1, K2) should stabilize all sixteen extreme
plants simultaneously. Let Ki1,i2, i1,i2
1,2,3,4
39 Denote the set of stabilizing gains
corresponding to the i1th and i2th
Kharitonov polynomial for the numerator and
denominator, respectively.  Solve the inequalities for each of the sixteen
extreme plants independently by gridding method
and obtain the set of stabilizing gains for each
of them.  (iv) Obtain the desired set of stabilising
gains for the interval system as intersection of
these results evaluated for the sixteen extreme
plants.  END Algorithm.
40Remarks
 Here, the bounds for the stabilising gains are
obtained in the selected range of K1 and
K2. It is not possible to obtain all the
solutions. Further, the results are guaranteed
only at the gridding points. It may happen that a
point between the selected grid points may not be
a feasible solution.  The proposed algorithm using interval analysis
evaluates the bounds for the stabilising gains,
in which all feasible solutions are obtained as a
set of interval boxes of specified accuracy. Any
point within these interval box is a guaranteed
stabilising gain for the interval system.
41Algorithm Synthesis of first order Compensator
using interval analysis
Begin Algorithm i) Obtained by replacing the
gridding technique in step (iii) of Barmish et
als algorithm given above with the subdefinite
computations technique (to solve the set of
inequalities involving the sixteen extreme
plants). ii) Using subdefinite computation
technique, obtain initial bounds on K1 and K2 in
which all feasible solutions must lie. iii) Find
all feasible solutions as interval boxes of
specified accuracy in the initial bounds obtained
at (ii) above. End Algorithm
42Remark 1 The solution set obtained as a set of
interval boxes contains all the feasible
solutions. Any point within this box is a
guaranteed stabilizing gain for the interval
system. Remark 2 A necessary and sufficient
condition for the existence of a robust
stabilizing controller is nonemptiness of the
set of gains in (iii) above. Remark 3 The
interval plant P can be stabilized by selecting
any (K1,K2) K.
43 A computer program has been developed for the
above controller synthesis technique. 
 Consider the SISO Jet engine interval plant
with input as fuel flow and output as
acceleration of compressor speed, Ndot. 
 P(s,)
 The uncertainty bounds are
 qo 940, 980, r1 97, 107, r2
215,230
44Let us synthesise a compensator of the
form
to stabilize the interval plant
P(s,) The Kharitonov polynomials for the
numerator and denominator of the effective plant
of acceleration loop are N1(s) 940N2(s)
980 D1(s) s2 97s 215 D2(s) s2
107s 215 D3(s) s2107s230 D4(s) s2
97s 230
45Thus, there are 8 different extreme plants. Using
these extreme plants, and PI compensator C(s),
the associated closed loop polynomials are
derived as follows. p1,1(s) s3 97s2
(215940K1)s 940K2 p1,2(s) s3 107s2
(215940K1)s 940K2 p1,3(s) s3 107s2
(230 940K1)s 940K2 p1,4,(s) s3 97s2
(230 940K1)s 940K2 p3,1(s) s3 97s2
(215 980K1)s 980K2 p3,2(s) s3 107s2
(215 980K1)s 980K2 p3,3(s) s3 107s2
(230 940K1)s 980K2 p3,4(s) s3 97s2
(230980K1)s 980K2
46 Routh table is set up for all the closed loop
polynomials.  Inequalities associated with each closed loop
polynomial are obtained.  These inequalities are solved using proposed
interval analysis algorithm to obtain bounds on
stabilizable controller parameters.  Any value within these bounds will stabilize
the uncertain plant
47Results obtained
 3 Inequalities associated with each closed loop
polynomial  Thus 24 inequalities are solved using proposed
interval analysis algorithm  Initial bounds on K1 and K2 are obtained as
 0, 2.19131e6
 Thus the given plant is stabilizable even with a
very large value of K1 K2  However, large values of gain are not practical.
 Practical gains can be obtained by implementing
additional performance constraints. 
48(No Transcript)
49(No Transcript)
50(No Transcript)
51CONCLUSION
 We have developed an interval analysis based
algorithm for evaluation of set of robustly
stabilizing first order compensator for an
interval plant of jet engine.  The algorithm guarantees that all feasible robust
stabilizers lie within the bounds of computed
interval boxes.  The technique guarantees stability for the entire
interval plant set.  It also finds if a robust first order compensator
for an individual loop is feasible or not. If
feasible, the algorithm gives the entire solution
set.  Although synthesis procedure is for robustly
stabilizing compensator, appropriate controller
meeting desired performance specification can
also be obtained using constraints on compensator
coefficients and verifying the performance
through simulation in simulink.
52DESIGN OF A ROBUST 2 DOF CONTROLLER FOR VARIABLE
EXHAUST NOZZLE CONTROL OF MILITARY AERO GAS
TURBINE
53 Military aero gas turbines are fitted with
variable exhaust nozzle system as it ensures that
engine operating line is not affected by the
increased volume of gas stream due to reheat
combustion  We present a methodology, based on Quantitative
Feedback Theory (QFT), for design of a robust 2
Degree of Freedom (DOF) controller for the
control of variable exhaust nozzle area.  The advantage of this methodology is that a
single robust controller can be used for the full
flight envelope control of the variable exhaust
nozzle, which is contrary to the conventional
controller incorporating gain scheduling for
different locations in the flight envelope
54 The design is done in frequency domain and the
evolved robust controller stabilizes the system
that does not have a distinct set of poles and
zeros but a range over which each of the poles
and zeros might lie.  The methodology is successfully applied to design
a robust 2 DOF compensator for variable exhaust
nozzle system of an experimental aero gas turbine
engine.
55Nozzle Electro Hydromechanical System
Demanded Nozzle Area
Desired Actuator Stroke
Throttle
Area to stroke converter
Operating schedule
Conventional controller
Servo valve
Actuator
Pump
Temperature
Actual Actuator stroke
LVDT
Block diagram of exhaust nozzle position control
system
56(No Transcript)
57Block diagram of the two degrees of freedom
control system
58 The nozzle control system is basically a position
control system functioning as closed servo loop.  The desired nozzle position is a function of
altitude and throttle position.  The hydro mechanical system of a variable exhaust
nozzle being a closed system is only a function
of the engine speed.  The nonlinear plant dynamics are approximated
using piecewise linearisation between idle and
maximum operating speed of the gas turbine
engine.
59 The model of this system at various operating
speeds is integrated together to form a
parametric uncertain system  Model uncertainties are also taken into account.
 A robust 2 DOF controller is then designed which
stabilizes the system that does not have a
distinct set of poles and zeros but a range over
which each of the poles and zeros might lie.
60 The designed controller must fulfill two
objectives Nozzle position keeping and nozzle
position changing .  In the first case, the control objective is to
maintain the nozzle actuator position. In the
second case, the aim is to implement the change
of position without oscillations and in the
shortest time possible  In both situations, the operability of the system
must be independent of the uncertainties in the
model of the nozzle electro hydro mechanical
system .  The mathematical model of the nozzle electro
hydro mechanical system between the hydraulic
actuator stroke Y(s) and the drive current to the
servo valve I (s) at various operating speeds
between idle and maximum engine speed is
approximated by the transfer function
61
 where k and a are different constant
parameters at various engine speeds.  Considering the model uncertainties at
various speeds and integrating it with variation
of k and a parameters owing to nonlinearity,
interval bounds on these parameters are
specified.  Thus despite the fact that the model is
nonlinear, the QFT model for linear SISO systems
with parametric uncertainty is used,
incorporating the two degrees of freedom control
system
Y(s) k I (s)
s (as 1)
62  The design of the controller includes a cascade
compensator, G (s) and a prefilter, F (s) (both
LTI ) in order to reduce the variations in the
output of the system caused by the uncertainties
in plant parameters.  The system must fulfill robust stability and
robust tracking specifications. For the robust
stability margins, the phase margin angle should
be at least 450 and the  gain margin 6 dB. Thus the robust stability
specification is defined by 
P (j) G (j)
2.3
1 P (j) G (j)
63 The robust tracking, must be defined within an
acceptable range of variation. This is generally
defined in the time domain but is normally
transferred to the frequency domain, being
expressed by TRL (j) TR (j) TRU
(j) where TR(s) represents the closed loop
transfer function and TRL(s) TRU(s) the
equivalent transfer functions of the lower and
upper tracking bounds
64 Design Illustration
 The 2 DOF controller design has been carried out
for following uncertainty in the k and a
parameters which are pertaining to a variable
exhaust nozzle system of an aero gas turbine
under development.  k 0.4, 0.9
 a 0.2, 0.4
 Acceptable range of variation of rise time for
change in nozzle position is specified as 0.12 to
0.18 sec. Following set of frequencies for the
design has been used.  0.1, 1, 4, 7, 9, 10, 12, 13, 20, 30, 50,
100 
65  On the basis of the performance specifications
and the plant templates, the robust stability and
robust tracking bounds are computed  For the design of the G(s) controller, the
Nichols chart is used, adjusting the nominal open
loop transfer function Lo PoG (Po is the
nominal plant) in such a way that no bounds are
violated  The controller obtained is
 G (s) 0.965 ( s 87.21) / ( 4.34 x 106 s2
2.08 x 103 s 1)
66  With this controller, the robust stability
specification is fulfilled but not the robust
tracking specification. This is obtained by
design of the prefilter  F(s) 16.66 x 103 ( s 60) / ( 1.54 x 103
s2 9.2 x 102 s 1)
67 Plant templates
68Robust tracking bounds.
Robust stability bounds.
69Intersection of bounds.
Shaping of Lo (jw) on the Nichols chart for the
nominal plant
70Frequency response of nozzle control system
Closed loop response of the nozzle control system
for unit step command of nozzle actuator position
71ROBUST QFT CONTROLLER DESIGN FOR TWIN SPOOL GAS
TURBINE SPPED CONTROL
 Acceleration of highpressure compressor spool
forms the inner loop. Direct control of
acceleration, rather than speed, allows tighter
control of engine acceleration thereby improving
transient response and reducing mechanical
stress.  The controller must fulfill two objectives Speed
keeping and speed changing. In the first case,
the control objective is to maintain the gas
turbine low pressure compressor spool speed
following the desired speed as per control
schedule  In the second case, the aim is to implement the
change of speed without oscillations and in the
shortest time possible.
72(No Transcript)
73 k1, k2, k3 ,a ,b ,c 5 Parametric
uncertainties.  k1 285 , 330 k2 7.5 , 8.0 k3
10, 15  a a 0.01, 0.03 b 0.4 , 0.5 c 0.27 ,
0.33 
 Performance requirements for the inner
loop  1. Stability Margins Gain Margin 6 dB
Phase Margin 450  2. Tracking Specifications Rise time of
closed loop transfer 
function between 0.4 to 0.8 sec  1. No overshoot
 Performance requirements for the outer
loop  1. Stability Margins Gain Margin 6 dB
Phase Margin 450  2. Tracking Specifications Rise time of
closed loop tr. fun. 
between 0.7 to 0.95 sec  3. No overshoot
74(No Transcript)
75(No Transcript)
76An Algorithm for Controller Synthesis of Jet
Engine Power Plant with Parametric Uncertainties
77Proposed Synthesis Procedure
 A key concept for our work is the KTR
 Given a GKS with vertex polynomial P1 and P2
 Construct two polynomials PA and PB as follows

 PA P1odd P2even
 PB P1even P2odd

 The polytope defined by the polynomials P1PAP2PB
is defined as the KTR  KTR is a two dimensional polytope in the
coefficient space, with four exposed edges
78Proposed Synthesis Procedure
Cont.
 Consider any GKS associated with the C(s) and
given P.  Construct a KTR for each GKS.
 Set up Routh tables for all the vertex
polynomials of all KTR.  Finally, solve the total set of inequalities
using an appropriate nonlinear solver to get a
robustly stabilizing compensator C(s) for P.
79Jet Engine Application
 Consider the following single input single output
jet engine interval plant with manipulated
variable as fuel flow and controlled variable as
compressor speed. 
 P(s,)
 The uncertainty bounds are
 r3 0.004, 0.005, r2 0.4, 0.5
80Jet Engine Application
Cont.
 Let us synthesize a lead lag compensator of the
form  C(s) Nc(s)/Dc(s) a(1bs)/(1cs)
 Constraint on c is selected in order to have a
dominating effect of zero as compared to pole of
compensator.
81Jet Engine Application
Cont.
 From the GKT, the robust stability of the closed
loop system is equivalent to that of the set of
GKS. There are four GKS in our example  1. 4.3a(1bs) (1cs) (0.001 0.004) s3
0.5s2 s  2. 4.3a(1bs) (1cs) (0.001 0.004) s3
(0.1 0.4) s2 s  3. 4.3a(1bs) (1cs) (0.001 0.005) s3
(0.1 0.4) s2 s  4. 4.3a(1bs) (1cs) (0.001 0.005) s3
0.4s2 s
82Jet Engine Application
Cont.
 For each of these 4 GKS, KTR is constructed as
per definition.  4 vertex polynomials of KTR corresponding to the
each GKS are obtained.  Routh tables constructed for all the 16 vertex
polynomials.  Enforcement of the positivity requirement for
the first column of the Routh tables leads to a
set comprising of 80 inequality constraints.
83Jet Engine Application
Cont.
 This set of inequalities is solved using the
UniCalc solver based on interval analysis
computation technique.  One of the feasible coefficients of the
compensator in the solution set are a 0.7025,  b 0.5074 and c 0.1062.
 All the roots of the 16 vertex polynomials were
found to lie in the left half of the s plane
thereby confirming stability.
84CONCLUSION
 In this work, we have developed a new interval
analysis based methodology for synthesis of a
robust general order compensator for a parametric
uncertain jet engine power plant.  The technique guarantees stability for the entire
interval plant set.  It also finds if a compensator of any desired
structure is feasible or not. If feasible, the
algorithm gives the entire solution set.
85CONCLUSION Cont.
 Although synthesis procedure is for robustly
stabilizing compensator, appropriate controller
meeting desired performance specification could
also be obtained using appropriate constraints on
compensator coefficients and verifying the
performance through closed loop simulation.  Alternately, desired performance constraints can
be derived as additional constraints and this
synthesis technique can be extended to design jet
engine robust compensators meeting both stability
and performance requirements.
86DESIGN OF ROBUSTLY STABILIZING CONTROLLERS FOR
JET ENGINE USING SET INVERSION
87Introduction
 The performance requirements of modern, high
technology aircraft has placed severe demands to
engine control capability.  All existing systems are subject to various
disturbances and uncertainties. Mathematically,
we can only approximate an existing system with a
transfer function depending upon the information
available about a system and the observations
over a certain period of time.  In this work, an algorithm using interval
analysis approach is proposed for the synthesis
of a robustly stabilizing first order compensator
of a jet engine interval plant having parametric
uncertainties.
88Introduction (contd)
 Our aim is to develop algorithmic results which
will enable us to determine if a robust first
order stabilizer exists for a given interval
plant.  If it exists, then we obtain the set of
stabilizing compensator parameters as a union of
interval vectors (or boxes).  Any value within this union represents a
guaranteed robustly stabilizing compensator for
the given interval plant.
89Set Inversion Problem
 Let be a nonlinear
function from Then the problem of
set inversion can be posed as a problem of
characterization of  In this work we address the problem of
characterizing S defined by a set of nonlinear
inequalities
90Compensator Synthesis Algorithm
 Consider a interval plant family P comprising of
plants of the form  Where interval bounds are priory given for each
uncertain coefficient  A compensator C(S), if exists, which stabilizes
the entire plant family P is then said to
robustly stabilize P.  If the compensator for an interval plant is of
first order, the stability of only sixteen
extreme plants is necessary and sufficient to
stabilize the entire interval family
(Barmish1994).
91Synthesis of first order compensator by set
inversion
 A compensator of first order of structure
is designed as follows.  Set up sixteen Routh tables (Barmish etal, 1992)
for closed loop polynomials associated with each
extreme plant.  Enforce positivity for each of the first column
entries which are functions of K1 and K2. This
leads to set of inequalities involving K1 and K2.  Solve the inequalities for the sixteen extreme
plants by the proposed set inversion algorithm
and obtain the set of stabilizing gains in the
given initial bounds.
92Set Inversion Algorithm
 Let be the initial box . The set
inversion algorithm encloses the portion S
contained in X0, between two partitions Kin and
Kout in the sense that
 where and Ke is the
list of indeterminate boxes.  The monotonicity test form is given as



 where is the set of integers i
such that  properly contains zero
93Set Inversion Algorithm
 BEGIN ALGORITHM
 Initialize XX0, Kin, Kout, LX
 Remove all boxes from list L and evaluate FMT
over all the boxes.  Deposit all boxes for which F(X)gt0 for all
i1,,m in the lists Kin and Kout.  Discard all those remaining boxes for which sup
Fi (X)lt0 for any i1,,m.  Deposit all those remaining boxes for which w(X)lt
in list Kout
94Set Inversion Algorithm (Contd)
 Find all those boxes for which Fi is
monotonically increasing or decreasing in every
direction for, i1,. ,m. Apply Algorithm TPB to
discard infeasible parts of these boxes.  Bisect all remaining boxes in the maximum width
coordinate direction k, getting subboxes V1, V2
such that Deposit all these
subboxes in L  If the list L is empty, EXIT algorithm. Else, go
to step 2.  END ALGORITHM

95Throwing part boxes Algo. (TPB)
 If f is monotonically increasing/decreasing in
every direction on X, then algorithm TPB locates
the subbox on which the inequality
fgt0 is certainly feasible, and outputs a list LX
of boxes whose union is the complement of C in X  Algorithm TPB parameterizes the line joining
points  in terms of single parameter .
 Use NewtonRaphson to find , such that f()0
96TPB (Contd)
 Construct a subbox on which fgt0
is certainly infeasible.  Using the box complementation algorithm in
(Kearfott 1996), find the complement of box C in
X to get a list LX such that
97 TPB (Contd)
X\C
X
C
98Jet Engine Problem
 A SISO Jet engine interval plant with input as
fuel flow and output as acceleration of
compressor speed, Ndot
Desired Ndot
99Jet Engine Problem (Contd)
 The transfer function is
 Where
 Compensator of the form is
required to be designed. 
100Values of K1 and K2
101Conclusions
 The set inversion algorithm gives values of K1
and K2 for the system to be robustly stable.  The algorithm guarantees that all feasible robust
stabilizers lie within the bounds of computed
interval boxes.
102Design of Robust 2 x 2 MIMO Control System for
Gas Turbine with Variable Geometry
103Design Method
 The MIMO design is carried out using Equivalent
Disturbance Attenuation method based on QFT.  Plant uncertainties are transferred into its
equivalent disturbance sets.  Thus, the design problem becomes the disturbance
attenuation problem.  The fundamental point is that if G is chosen so
that the output T satisfies the system
tolerances over the entire VT set, then the
original MIMO specifications are satisfied
104Basic Concepts
R PG ( FR C)
T (I L0)1 VT V I  P0 P1
105A 2 x 2 MIMO Design Problem
Controller Structure for 2 x 2 MIMO System
106Controller Synthesis Procedure
 Translation of time domain specifications into
frequency specifications.  Translation of tracking specifications into its
equivalent disturbance specifications.  Bound generation for stability and disturbance.
 Loopshaping and filter design.
107Gas Turbine Application
k1
P11 
as2 (0.0169s1) s
k2 (s
0.2125) P12

bs2 (0.0029.s2 0.1622s1)
s
0.848 k3 P22

cs2 (0.0191s1) s
k4 P21

ds2 (0.0169s1) s where,
k1 8.70,9.70 k2 5.70,6.30 k3
1.15,1.25 k4 6.20,6.50 a 0.45,0.55
b 0.40,0.46 c 0.63,0.64 d
0.45,0.55
Inputs Fuel flow VG Outputs LP HP
Compressor Speed
108 Performance Specifications
 The rise time of closed loop transfer function
for speed control loop should lie between 0.3 to
0.5 sec.  The rise time of closed loop transfer function
for variable geometry loop should lie between
0.26 to 0.4 sec.  The stability margins for both the loops are 6
dB gain margin and 45 degrees phase margin.  The cross coupling effects are t12/t22 0.05,
 t21/t11 0.1
109Tolerances on tij
110Bounds on l110 and Loopshaping on l110
111 Bounds on l220 and Loopshaping on l220
112 Compensator and the filter obtained are as
follows. 3.836
x105 g11 
 ( s7.902
)(s2400s160000)
54 f11 
( s5.4 )(s10)
1.514 x
105 (s 3.5) g22 
 ( s27.75
)(s280s6400) 1.78 ( s
45) f22 
( s8 )(s10)
113 Closed Loop Step Responses
114Conclusion
 No plant templates need to be calculated and the
design is obtained much faster.  It is a very viable technique to design robust
control system of modern gas turbine with
variable geometry which are widely used for power
generation, naval and air propulsion.
115Thank You