Aerospace Science and Technology, 1999. no. 3. 127-140
Analysis and tuning of a 'Total Energy Control System'
control law using eigenstructure assignment
L.P. Faleiro a*, A.A. Lambregts
a
b
Institute of Robotics and System Dynamics. DLR Deutsches Zentrum fUr Luft- und Raumfahrt e. V,
Oberpfaffenhofen, 82234 WeBling, Germany
b Advanced Controls, Federal Aviation Authority, Seattle, USA
(Received 14 October 1998. revised 21 December 1998. accepted 5 February 1999)
Faleiro L.E. Lambregts A.A., Aerospace Science and Technology, 1999, no. 3, 127-140
Abstract
The Total Energy Control System (TECS) is a complete airplane longitudinal dynamics flight control
concept for autopilot operational control modes and Fly-By-Wire command augmentation for civil airplanes.
Unlike conventional strategies. it facilitates fully integrated control of the airplane elevator and engines. Th is
system, which is based on simple proportional and integral control of the energy states of the airplane, is
much easier to design and understand than most conventional airplane controllers.
This paper describes why the high visibility of the two input, two output command augmentation structure of TECS is an improvement over current flight control system architectures. Deriving a control law for
TECS is currently a heuristic process. However. there is potential for tuning to be done more systematically,
and this is examined by using eigenstructure analysis and assignment . To illustrate the concepts, a linear
model of the longitudinal dynamics of the Aerospace Technologies Demonstrator (AID) airplane is used. A
heuristically designed TECS controller for this model is first described. The controlled airplane is then analysed using eigenstructure analysis and the results are utilised to produce an improved TECS controller for
the ATD model using eigenstructure assignment . © Elsevier, Paris
Total Energy Control System I eigenstructure assignment I integrated Dight control I control decoupiing
Zusammenfassung
Analyse und Anpassung elnes 'Total Energy Control System' Flugregelgesetzes mit dem Verrahren
der Eigenstrukturvorgabe. Das 'Total Energy Control System' (TECS) ist ein intergriertes
Steuerungssystem fiir die manuelle und automatische Steuerung der Flugzeuglangsbewegung von
Zivilflugzeugen. Anders als herkomrnliche Strategien ermoglicht es eine vollintegrierte Steuerung von
Hohenruder und Triebwerken. Dieses System, das auf proportionaler und integraler Riickkopplung der
Energiezustande des F1ugzeuges basiert, ist einfacher zu entwerfen und systemtechnisch zu verstehen als die
meisten traditionellen Flugzeugsteuerungsansatze. Dieser Beitrag beschreibt, warum die Reglerstruktur von
TECS. die auf zwei Eingangs- und zwei Ausgangsgrolsen basiert, eine Verbesserung der herkommlichen
Architektur von F1ugsteuerungssystemen darstellt. Die parametrische Auslegung eines Steuergesetzes fUr
TECS ist derzeit ein heuristischer ProzeB. Jedoch kann dieses 'tuning' systematisien werden. Dies geschieht
hier durch Anwendung der Eigenstrukturanalyse und -vorgabe. Um die Konzepte zu veranschaulichen , wird
ein Modell der Langsbewegung des DaimlerChry sler Aerospace Airbus Technologie-Demonstrators (ATD)
verwendet. Ausgehend von einer heuristisch entworfenen TECS-Steuerung fUr dieses Modell wird das geregelte F1ugzeug durch Eigenstrukturanalyse untersucht. Mit diesen Resultaten wird gezeigt, wie ein verbesserter TECS-Regler fiir den ATD durch Eigenstrukturvorgabe ausgelegt werden kann. © Elsevier. Paris
TECS I Eigenstrukturvorgabe I integrierte Flugsteuerung I entkoppelnde Steuerung
• Correspondence and reprints
Aerospace Science and Technology. 121~9638. 99103f Q Elsevier, Pari.
128
L.F. Faleiro, A.A. Lambregts
Notation and operations
A,B,C,D Airplane dynamics in state-space form
x,y,u,r Airplane states, outputs, inputs
and references
Airplane drag force (N)
D
Elevator movement (rad)
8e
Throttle movement (rad)
8t h
Airplane total energy (J)
ET
E
Airplane specific total energy rate
g
Acceleration due to gravity (9.81 ms- 2)
y
Airplane vertical flight path angle (rad)
h
Airplane altitude (m)
nth order identity matrix
In
K
Feedback gain matrix
L
Airplane specific energy distribution rate
A,A
Eigenvalue, matrix of eigenvalues
Mi
ith mode
N
Engine core speed (%max)
n
Number of airplane states
Airplane pitch rate (rad-s -1)
Eigenvalue, eigenvector sensitivity matrices
s
Laplace operator
T
Airplane thrust force (N)
Time (s)
Time period (s)
8
Airplane angle of attitude (rad)
u
Airplane body axis forward speed (ms -1)
v
Airplane measured airspeed (ms -I)
V,W
Matrices of right and left eigenvectors
W
Airplane weight (N)
w
Airplane body axis downward speed (ms -1)
Airplane earth axis downward disp. (m)
z
Matrix of zeros
o
X,X,
XT
XCL
Xl
xP
Xd,Xd
X-I
XQ,XO
Augmented matrix, vector of X, x
Matrix transpose
Closed-loop system matrix
Integral operations matrix
Proportional operations matrix
Desired values of X, x
Inverse of X
Initial value of x, x
Commanded variable
Error variable
1. Introduction
State of the art avionics/flight control systems for civil
airplanes have evolved into very capable, albeit very
complex and costly systems. Nearly every function on
the flight deck, formerly executed manually by the pilot,
can now be performed automatically. As demands on the
flight crew increase, they have been able to delegate
tasks to automatic systems, which give them relief from
routine work and high workloads. This is especially vital
in terminal area operations, where errors can quickly
lead to disaster. Here, automation has produced its largest payoff in safety and airline revenue. The most automated operation, Category III automatic landing - which
is designed to require minimal routine pilot interaction has an admirable safety record. So, what are today's
avionics/flight controls problems?
The simple answer is the sometimes unnecessary
costs due to complexity during design development, in
operations, maintenance, spares and in training. The
great majority of automation deficiencies and unnecessary complexities are the result of a bad formulation of
requirements and adhering too long to outdated technologies, engineering design concepts and processes.
Perhaps in a misguided attempt to minimise cost and
risk, flawed design concepts have been carried forward
into new generation airplanes and for 50 years deficiencies have been addressed by an incremental 'Band Aid'
approach, driven in part by considerations of maintaining
design commonality and training.
This paper describes a flight control systems strategy
which has the potential to solve most of these problems.
The paper is in three parts. The first (section 2) describes
the historical development of traditional (upto the
1980's) and current (1980's onwards) flight control system architectures, and how they have led to the need for
a new control strategy with a more transparent architecture. The second part (sections 3-6) describes the structure of one such control strategy, the Total Energy
Control System (TECS), and why it can solve current
problems. The third part (sections 7-11) describes how
the detailed design of a TECS controller can be improved beyond its current state. This is demonstrated
through the examination of the performance of a TECS
control law using eigenstructure analysis, and the subsequent tuning of the TECS controller to give a better solution to a particular design requirement, using eigenstructure assignment.
2. Deficiencies of the traditional flight control
design approach
"In the traditional design approach, the design of each
operational control mode is approached as a separate
problem at each flight condition and for each combination with other control modes."
Aerospace Science and Tuhnology
Analysis and tuning of a 'Total Energy Control System' control law using eigenstructure assignment
Analyse undAnpassung eines 'Total Energy Control System' Flugregelgesetzes mitdem Veifahren der Eigenstrukturvorgabe
As a consequence, any specific combination of
control mode (for instance, ALTITUDE SELECT and
SPEED HOLD) must be designed to work with one another for each desired flight condition, else they tend to
be incompatible. This is one of the basic reasons for the
proliferation of control modes, as the desired functionality must be extended to the full flight regime. Another
reason for this proliferation is that the historic development process has always assigned control of the flight
path (used to control altitude) to the elevator, then later,
assigned control of speed to the throttle. This produced
the single input single output (SISO) control laws for the
'autopilot' and the 'autothrottle', as depicted injigure I.
This SISO control strategy has serious limitations and
deficiencies which then in themselves became the justification for the design of new specialised control modes.
2.1. Problems with the SISO design approach
The fundamental problem with the SISO control design approach is that the airplane responses to command
in elevator and thrust are highly coupled. This means that
while one variable is being controlled by one controller,
the other variable is being driven away from its equilibrium state. unless both controllers are commanded in a
properly co-ordinated fashion. A direct result of this
situation is that autothrottle speed control at constant elevator tends to cause an unstable flight path response (for
example on all Boeing B747's) and at constant thrust, the
autopilot effort to control flight path using the elevator
will induce a speed deviation.
The basic safety of a mixed control mode operation
with the pilot manually controlling the flight path and the
speed controlled automatically or vice versa, is also
questionable. Because the actions of the autothrottle are
not tactically co-ordinated with autopilot flight path
contro\, the autothrottle constantly upsets autopilot path
control and vice versa, resulting in a notorious control
coupling problem, familiar to every airline pilot. It manifests itself especially when excited by turbulence or
r····--····· ..· · · · · ,
:Autothrottle
Engine
.
r
.
,
Airplane
: Autopilot
Figure 1. Traditional pitchautopilot and autothrottle.
1999. no. 3
129
windshear, to the point where tracking performance and
ride quality become unacceptable. The old remedy to
break this coupling was to change the autopilot control
mode to an ATTITUDE HOLD mode (for example in the
older B747-200/300). This problem has now been reduced to an acceptable level for cruise operation after a
very difficult and costly development process, implementing provisions such as separation of the control frequency between autopilot and autothrottle by going to
very low autothrottle feedback gain, application of 'energy compensation', turbulence compensation and nonlinear windshear detection and compensation.
Due to the lack of proper control co-ordination, the
autopilot ALTITUDE SELECT and VERTICAL SPEED
control modes never functioned satisfactorily, because a
climb or descent command without regard to the thrust
required caused speed to diverge if not managed by the
pilot or autothrottle. Even with an autothrottle on, speed
will still run away, if the autopilot manoeuvre is large
enough to result in the thrust being commanded to its
limit in an attempt to hold speed. It should be noted that
traditional autopilots had no knowledge of the airplane's
flight path steady state performance limit and no speed
envelope protection for any of its control modes. Proper
use and careful monitoring of the automatic systems has
always been considered a pilot responsibility. But
humans are error prone and are not particularly well
adapted to do monitoring jobs.
These problems resulted in the development of the
FLIGHT LEVEL CHANGE (FLC) control mode that
was first implemented on the Airbus A310 and later on
the B757n67. However, there have been a number of
incidents where the FLC control mode did not properly
execute the pilot's commands. A lack of co-ordination of
throttle and elevator control in one case resulted in
serious speed bleed down (lATA report on Automation
[ I D, arrested only by the pilot taking over control.
2.2. Airplane energy state
Current autopilot designs using a SISO control strategy cannot properly manage the airplane's energy state,
because that requires functionally integrated flight path
and speed control, taking airplane performance and
envelope limits into account. An energy management
deficiency was a contributing factor in an Airbus A330
crash [2]. In that accident scenario the ALTITUDE CAPTURE control mode continued to command a normal
two engine capture trajectory after one engine was
brought back to idle to simulate an engine failure. The
result was an excessive pitch up and speed bleed off to
stall, because the autopilot did not properly limit control
authority to take the available thrust level into account
and the speed envelope protection function inadvertently
did not provide coverage for this control submode.
Testing on Boeing airplanes after the A330 crash showed
similar deficiencies.
130
"The basic deficiency of SISO control strategy of
autopilot and autothrottle systems is that they do not properly co-ordinate their actions to manage airplane energy."
The speed control autothrottle has a basic conceptual
flaw of leaving out the most important state variable flight path angle - in determining required thrust. The
path control autopilot has the basic deficiency of not
knowing the airplane's steady state climb and descent
limits. Basic safety regulations state that the elevator
must have sufficient authority to maintain pitch control
at any flight condition. In the past, designers have used
this higher control authority of the elevator to tighten
autopilot flight path control, thereby converting path
deviation into speed deviation. This has put the speed
control autothrottle in a 'no win' situation and sets up a
basic energy management conflict.
Lift is affected by both speed and angle of attack. On
landing approach the sensitivity of airspeed to a change
in flight path is ~ 4 knots/deg (~ 2 ms -l/deg). Flight
path control is also very sensitive to speed deviations,
2 knots (l ms- l) being enough result in ~ 20 ft (6 m)
deviation in 5 seconds if left uncorrected.
As a result there have been countless autothrottle and to a lesser extent autopilot - improvement programs,
attempting to overcome poor system damping, command
overshoot, unacceptably high control activity, poor speed
and flight path control in turbulence and windshear.
Despite this, even the latest generation designs have not
satisfactorily solved the energy management and control
coupling problems. The fundamental physics are summed up nicely in an article by Soule [24].
L.F. Faleiro, A.A. Lambregts
concepts. The outcome of this work, conducted at
Boeing between 1979 and 1982, was a completely new
approach to the design and up-front integration of all the
required automatic flight guidance and control modes, as
well as the manual Fly-By-Wire control mode. This new
architecture featured only two subsystems, the Flight
Management Computer (FMC) and the Flight Control
Computer (FCC), shown infigure 2. It was based on the
concept of rational partitioning of airline operations
oriented functions, hosted in the FMC as before, and
pilot workload relief and safety oriented control functions, hosted solely in the restructured FCC.
FCC
Velocity vector control
Speed hold
Attitude hold
Vertical speed
Vertical path
Horizontal path
Envelope protection
Thrust limiting
Figure 2. Improved control mode architecture.
2.3. Control mode overlap
4. The Total Energy Control System (TECS)
The result of all this is that typical current system
architectures have functionally overlapping SISO flight
path and speed control modes in the autopilot, the autothrottle and the Flight Management Computer (FMC).
Inconsistencies have crept into the design of the many
different control modes. The excessive amount of control
modes and submodes, the functional overlap and the
inconsistency of operation and performance between the
FMC, the autopilot and the autothrottle has become a
real challenge for any pilot to understand, remember and
effectively manage. These issues, and their history, have
been explored in much greater detail in [17]. It is evident
that there is a legitimate need for better and simpler
conceptual designs.
With this improved concept, airplane control modes
can be designed in the FCC to be completely generic and
independent of specific airplane characteristics. The airplane's flight trajectory is entirely defined by specifying
the trajectory acceleration as a function of time, which
for commercial airplanes must be kept within the same
limits, regardless of airplane type. Trajectory control is a
point mass kinematic problem and should be treated
independently of airplane specific characteristics.
Stability augmentation, to correct for undesirable dynamic characteristics ofthe basic airplane, should only take
place in inner-loop design.
3. Improved guidance and control architecture
4.1. Flight path and speed control function
integration through energy management
Not satisfied with the operation and performance of
traditional automatic guidance and control laws, NASA
Langley sponsored research to develop improved
The development of an integrated, generalised multi
input, multi output (MIMO) control strategy can now be
based on first principles:
A,rospace Science and Technology
Analysis and tuning of a 'Total Energy Control System' control law using eigenstructure assignment
Analyse und Anpassungeines 'TotalEnergy ControlSystem' Flugregelgesetzes mit dem vetfahren der Eigenstrukturvorgabe
From the point mass airplane energy states it follows
that
Total energy
= Kinetic energy + Potential energy
Er =
ET
~ W V 2 + Wh
2g
V
h
(1)
-=-+WV
g
V
E=(V/g)+y.
(4)
From the point mass airplane force equations it follows that for small flight path angles
W«V/g) + y).
(5)
In level flight. initial thrust is trimmed against the
drag of the airplane. So the correct thrust control strategy is to develop the incremental thrust command as follows:
(6)
where the subscript e refers to the error in that variable.
From (4) and (6), it follows that
(7)
Thus, altering the thrust of the airplane will proportionally alter the specific rate of energy into the airplane,
increasing the sum of the airplane flight path angle and
acceleration along the flight path. Therefore thrust
should be used for total energy control. From this, we
can formulate a simple control law to relate desired
changes in energy rate of the airplane to commands in
thrust:
KTI) .
tc = ( KTP + -se..
(8)
Integral control is necessary to allow the control law
to trim the thrust at any desired condition. The thrust will
now alter to drive the error in specific energy rate, i;
to zero.
It is also well known that elevator control is energy
conservative with good approximation. This allows the
pilot to exchange potential energy for kinetic energy and
vice versa using the elevator, without significant short
term energy loss. Thus. the elevator may be considered
for energy distribution control. A similar proportional
and integral control law can now be used to transform
errors in the desired energy distribution rate t.; into elevator angle command:
8ec = ( KEP
where
1999. no. 3
KEI) .
+ -st;
L=(v/g)-y.
As this control concept is based on energy considerations, it is called the Total Energy Control System
(TECS). Equations (8) and (9) form the TECS core algorithm, and are based on natural control behaviour of the
airplane and its control surfaces.
(2)
(3)
T - D=
131
(9)
(10)
4.2. Total Energy Control System (TECS) core
algorithm
The structure of the TECS core algorithm is shown in
figure 3. To form the control error signals
and t,
requires Ye, Y, (V /g)e and (V/g) signals. The selected
structure of the control algorithm to develop coordinated thrust and elevator commands uses proportional and integral control feedbacks. However, for the proportional terms only Eand L feedbacks are used instead
of the error signals, to avoid the creation of unwanted
zeros in the system transfer functions, which would
result in an undesirable overshoot in the command response.
Usually, controlling i; with the elevator requires an
inner-loop control law to stabilise the airplane's short
period mode, for example a conventional pitch rate/pitch
attitude feedback inner-loop. This is designed to yield
pitch attitude dynamics that match the engine dynamics
as much as is possible, providing co-ordinated control.
This inner-loop, where Be will be transformed into Sec,
has been excluded from figure 3.
The core feedback gains K T I, K T P and K E I,
K E P used in the development of the thrust command
and the elevator command are designed to yield identical dynamics for energy rate error Ee and energy distribution rate error L, for either a flight path angle command or a longitudinal acceleration command. The
MIMO control law allows precise thrust and elevator
control co-ordination to achieve decoupled command
responses. As a result, a flight path angle command will
not cause a significant speed deviation and vice versa.
Another way to look at command response decoupling is
to realise that for a flight path angle or an acceleration
and of i; need
command the dynamic response of
to be identical, otherwise energy is added to or subtrac-
s,
e,
Yc
-..-
Y6
+
11
+
t,
Y
+
v
s
8c
ve
vt:
-s
L
g
+
Figure 3. TECScore algorithm.
LI!
132
L.F. Faleiro, A.A. Lambregts
ted from the variable that is commanded to be held
constant.
4.3. TECS architecture and control mode
hierarchy
Figure 4 shows a typical TECS architecture and
control mode hierarchy. As TECS is derived from the
energy states of the airplane, its operational control
modes can be linked to energy based principles. This
idea was developed in [13-16], and has been further
explored for optimum trajectory guidance in [25].
For the four autopilot control modes in figure 4 using
a defined path in space, the altitude error is first normalised into a vertical speed command by multiplication
with the factor K h and next converted into a flight path
angle command by dividing vertical speed command by
airspeed. Similarly, errors for the three speed control
modes are first converted into a true airspeed error
signal, which is then multiplied by the factor K v to form
the longitudinal acceleration command and then normalised by dividing the acceleration command by the gravity constant g.
Because altitude and speed errors are normalised in
terms of their relative energy, the normalisation constants
K hand K v should theoretically be selected as equal,
as altitude and speed control must operate at the
same bandwidth to achieve perfect co-ordinated decoupled control and efficient energy management during
simultaneous flight path and speed manceuvres, This has
never been recognised in traditional autopilot and autothrottle designs.
4.4. Fly-By-Wire (FBW) control mode
The TECS core flight path angle and speed control
algorithm inherently provides about 80% of what is needed for an advanced FBW manual control mode.
Correctly designed, a FBW manual control mode with
TECS can satisfy all Level I handling qualities criteria. It
has been shown that the pilot cannot induce airplane
Flight path angle
I
I <>~•
Altitude
i
lope
Vertical navigation
I
h
r--i
~
KhI~
~
T,
Yc
PBW
TECS
core
Airspeed
Mach number
Time navigation
J--,.,
f---
Ve
g
ee
pilot coupling and can control the flight path angle
directly. This pilot-established flight path angle will be
maintained automatically, regardless of airplane envelope and configuration changes, or external disturbances
due to turbulence and windshear. Details on these aspects
of TECS can be found in [17].
5. TECS in practice
All the TECS principles described above, including
the FBW concept, were developed and evaluated in
flight test on the NASA Transport Systems Research
Vehicle (TSRV) B737-100 with great success [3, 4]. To
facilitate common type rating, it was also demonstrated
in piloted simulations (conducted on the B737 and B747
simulators) that standardised flying characteristics between different airplanes could easily be obtained. No
special control mode engage computations were needed
and the control mode transition point was fully adapted
to alterations in speed.
Likewise, transitioning from any initial path control
mode to a glide slope control mode worked, and allowed
transient-free captures from any initial flight path angle,
either from below or above, while holding speed very
close to the command target. This design concept eliminated the need for command processors and capture
control submodes. Compared to conventional systems,
the TECS design approach eliminated control function
overlap, reduced hardware by up to 50% and control law
software by up to 70%. The reusable TEes design simplified and shortened the overall design development
cycle and reduced overall cost and risk. In addition to the
NASA TSRV-B737, TECS was applied and flown on the
'Condor' high altitude long endurance unpiloted airplane technology demonstration program (no unclassified
references currently available).
Although these programs have been highly successful, the TECS designs have not yet been introduced in
production airplanes. The reasons for this are complex
and include the difficulty of making needed design process and organisational changes, the lack of a single corporate organisation that can claim ownership of the
diverse subsystems that will be affected by this dramatic
consolidation of functions and issues of commonality,
pilot training and perceived risk of new technology.
For the sake of brevity, discussion of most of the design implementation details has been omitted here. More
details about TECS. its architectural development, and
full implementation of all required operational modes
and envelope protection, can be found in references [3, 4,
13-17].
KWIl I - - - - - - t
~
Figure 4. TEeS control mode architecture.
6. Example of a TEeS flight control law
To demonstrate the basic application of the TECS
principles in flight control law design, a simple TECS
Aerospace Science and Technology
Analysis and tuning of a 'Total Energy Control System' control law using eigenstructure assignment
Analyse undAnpassung eines 'Total Energy Control System' Flugregelgesetzes mit dem Veifahren der Eigenstrukturvorgabe
controller was designed at DLR Oberpfaffenhofen for a
non-linear model of the longitudinal dynamics of the
Aerospace Technologies Demonstrator (AID) airplane,
as modelled in [18]. This non-linear model was linearised at a nominal flight condition, and the resulting statespace model (A, B, C), which will be used in this paper,
is given in the appendix.
The simple control problem that will be addressed in
this paper was defined as a set of performance requirements. A controller was required to provide tracking
capability for demands in airspeed V and altitude h. For
operational reasons, the responses to step demands in
these two variables had to conform to limits in rise times
and decoupling. For brevity, this paper concentrates only
on simple decoupling requirements. For the AID, these
were:
• For a step demand of 10 ms -I in airspeed, altitude
should not exceed 8 m from its trimmed value.
• For a step demand of 10m in altitude, airspeed
should not exceed 0.2 ms -I from its trimmed value.
Although these requirements are based on small commands for a linear problem, linear design methodologies
can usually be expanded to a non-linear controller with
scheduling, without undue problem. The TECS control ler architecture itself provides "automatic" gain scheduling to give consistent performance. Hence, the control
design problem addressed in this paper is physically
meaningful.
The TECS controller structure shown in figures 3 and
4 was used to construct a controller for the non-linear
AID model. Nominal TECS gains - K E P, K E I, K T P
and KT 1- were based on a heuristic design philosophy,
where the designer manually tuned them to produce suitable performance responses to command signals. They
were determined as
KEP
= KTP = 1,
KEI
= KTI = 0.4.
(11)
The guidance error normalisation gains K v and K h
were also designed heuristically and, based on the principles described earlier, were given by
Kv
= 0.1,
Kh = 0.1.
(12)
The time taken to design these was relatively short, as
the controller structure is visible. These gains provided
identical dynamics in both the thrust and pitch attitude
paths of the TECS, as required by the TECS principles.
This paper uses a linear subset of the non-linear
controller model, with only the ALTITUDE SELECT
and SPEED HOLD modes implemented. The nominal
control gains given in (11) and (12) were implemented in
this linear structure.
Figure 5 shows the responses of the linear model to a
10 ms -I step demand in airspeed . Airspeed is attained
with an oscillatory altitude displacement of up to 4.3 m.
From equations (4) and (10), it can be seen that for a res1999, no. 3
133
E
~f1or----.----"-"""""-"""'-::=~-""----"''----'
i
5
5
10
15
20
25
30
35
40
25
30
35
40
25
30
35
40
1irre(s)
5
10
15
20
T1rre(s)
5
10
15
20
l1rre(s)
Figure 5. Responses to a step demand in airspeed.
ponse in airspeed with no change in y, the plots of Eand
L should be exactly the same. Figure 5 shows that this
did not occur for the airspeed demand, as Elagged L in
phase, which is what resulted in the adversely coupled
behaviour that was observed. This lag is caused by the
slower dynamics of engine response than elevator actuation response in the aircraft.
Figure 6 shows the response of the airplane to a step
demand in altitude. Altitude is attained with virtually no
coupling of airspeed (0.03 ms- I deviation). From (4)
and (10), it can be seen that TECS effects this behaviour
through an increase in E and an exactly opposing response in L. As shown in figure 6, the energy rates were
in anti-phase, but were not of the same magnitude, resulting in very slight coupling.
Although this TECS controlled AID satisfies the
decoupling specifications, it is desirable to explore the
possibility of reducing coupling (certainly to improve
performance during an airspeed demand) even further.
Although in theory, the TEeS gains should be identical in the thrust and pitch attitude paths, it is evident that
these will have to be altered from their nominal values to
effect this result. However, it is time consuming to do
134
L.F. Faleiro, A.A. Lambregts
tions will be examined using eigenstructure analysis. The
results of this analysis will be used to re-design the
TEes controller for the ATD using eigenstructure assignment.
E
l10
::J
....
~
h
In 5
i;
:0Q)
0
8.
Ul
....
<c
8. Producing a state-space model of TECS
V
10
0
20
30
40
30
40
30
40
Time(s)
-3
10X 10
Ul
....
Q)
~
...
2;;
Q)
c:
w
10
20
Time(s)
10
20
Time(s)
Before beginning any analysis, the airplane model has
to be available in a suitable format. It was assumed that
eigenstructure analysis will be pursued in order to examine aspects of the TECS core algorithm control law
only, and so the pitch and thrust inner loops of the system are kept fixed. These are a simple combination of
feedforward and feedback gains, and they are used in
combination with the linear open-loop model (A, B) to
give the ACL and BCL matrices of the closed-loop
dynamics model given in the appendix, corresponding to
the n airplane states x and the two control inputs u.
The TEeS core algorithm requires the feedback of E
and t: These are a linear combination of the airplane
states x and control inputs u, and define the CCL and
DCL matrices of the closed-loop model, also given in the
appendix.
In order to facilitate analysis, the TECS core algorithm structure (figure 3) itself is mapped into the matrix
based controller structure shown in figure 7.
Based on the TECS structure shown in figure 3, the
control gains infigure 7 are given by:
Figure 6. Responses to a step demand in altitude.
o ],
KEP
this by heuristically tuning the TECS control law gains
due to the interactions within the airplane dynamics.
7. An alternative to the heuristic process
Although using manual tuning has resulted in TEeS
control laws that have been implemented in the NASA
TSRV-B737, there is little doubt that it would be advantageous to have systematic and analytical tools to provide the benefits described below. Therefore, it is proposed
that the principles of the TECS philosophy be used to
obtain analytical measures of the performance of the
TECS-controlled airplane. These measures should then
be used as a base from which to:
• Analyse a TECS with more (mathematical) rigour.
• Provide a quicker solution to a control problem.
• Thereby reduce the time and money taken to produce a TECS controller.
One way in which the closed-loop TEeS can be analysed is by an examination of its dynamic modes and the
way in which they affect the energy states of the airplane. In the remainder of this paper, these model interac-
K J = [KTI
0
The controlled system in matrix form can be obtained
by augmenting the closed-loop system matrices
(ACL, BCL, CCL, DcL> with matrices corresponding
to the sta~s 19r !lte i9tegrals of e. and
This resulting
system (A, B, C, D) is given by
t:
A = [ACL
CCL
Ii = [BCL]
(14)
c=[C~L ~lD=[DO~L]
(15)
0],
0
DCL
u
. j(
:
r
'
Figure 7. Controlled airplane and TECS core algorithm in
blockdiagram form.
Aerospace Science and Technology
Analysis and tuning of a 'Total Energy Control System' control law using eigenstructure assignment
Analyse und Anpassung eines 'Total Energy Control System' Flugregelgesetzes mitdem Veifaltren derEigenstrukturvorgabe
KP and KI were used to form the feedback control law
-KG
u=
(16)
(17)
where
The TECS controlled airplane is now described by:
x = <A -
BK(l2
+ Di}-IC)X' + [~J r, y = G
fEe f LelT
Eigenstructure analysis can now be applied on the
controlled airplane system given in equation (18).
10.1. What is eigenstructure?
(20)
u = [Thruste (Jel T
(21)
.
10. Eigenstructure analysis of TECS
(19)
y = [E L fEe f Lef
.
Almost all the dynamics possess satisfactory damping
and natural or comer frequencies for airplane operation.
However, A4 appears to be in a nearly neutral position.
Very little can be deduced about this pole without a more
thorough knowledge of how it affects the airplane. This
analysis can be furthered by an examination of the eigenstructure of the TECS controlled airplane.
(18)
where the augmented state and output vectors and the
airplane input and reference vectors are given respectively by:
x= [u ui e q N
T
(22)
r = [Ee Lel .
The augmented model given in equation (18) is therefore a representation of the airplane with a TECS, and
can now be used in eigenstructure analysis. To avoid
c,9J!lplicated matrix algebra, it is now assumed that
DK « 1. This is valid for most TECS systems, where
the measured accelerations that would result in a D
matrix are extremely small.
In general terms, the seven (n + 2) eigenvalues and
eigenvectors of the TECS augmented airplane system are
given by:
A
= [AI' .. A; ... An+2],
Now that the closed-loop system is in a state-space
form, its stability can be examined. The poles (A) of the
nominal TECS controlled airplane, along with their damping ratio (0 and natural frequency (wnad can be determined from the system matrix <A - BKc>, where the
nominal controller, from (11), is:
0
I
0.4
0
0 ]
0.4 .
(23)
These poles are shown in table I. As all these poles
have a negative real component, the airplane is stable.
Table I. Poles of the nominal TEeS controlled ATD.
Index
Eigenvalue
,\\
-6.631
'\2
-3.947
'\3'\3.
-0.596 ± 0.722i
'\4
-0.002
'\5
-0.588
'\6
-0.227
1999.no. 3
S
Wnol
0.636
0.936
V
= [v I ... v; ... Vn+2]
- --(A-BKC)VA.
where
(24)
(25)
The left, or dual basis eigenvectors of the same system
are in tum defined by W:
WT
= [WI"
'W;"
,w n+2]
- --W(A -BKC) = AW.
where
9. Stability of the nominal TECS control law
K = [Io
135
(26)
(27)
Solving the equations given in (25) and (27), together
with the analytical expression for time response, gives a
direct link between the eigenvalues and eigenvectors of
the controlled airplane and its energy state time behaviour:
n+2
y(t)
= L Cv;wT eAltX'o
;=1
+
ECv;wT Jot
;=1
eA;(t-r) [: ] r(r)dr.
(28)
2
Given a set of inputs and initial conditions, the output
y(t) becomes a linear function of the right eigenvectors
v; of the system. The index i in (28) refers to each of the
(n + 2) dynamic modes of the airplane. In conventional
longitudinal airplane dynamics, for instance, these are
the short period pitching oscillation (SPPO) mode and
the phugoid mode. The dynamic behaviour of airplane
systems is composed of a collection of first-order and
second-order modes.
Each mode is composed of two elements. The first is
one that describes the transient (time decay and frequency) behaviour of the mode. In classical control theory.
this is the pole of the mode. In eigenstructural terms, this
is the eigenvalue of the mode. The second component of
each mode is a magnitude; this is the residue of the
136
L.F. Faleiro, A.A. Lambregts
mode. More specifically, what matters are the relative
residues of the mode. These directly influence the prevalence of the mode in each of the airplane states in the
and each of the measured outputs in the
state vector
output vector y. In classical control theory, the residues
of a mode are dependent on two things, the poles and the
zeros of the system. In eigenstructure analysis, the residues of each mode are related to the eigenvector of that
mode. Together, the eigenvalues and eigenvectors form
the eigenstructure of the airplane dynamics.
Interactions between modes and airplane outputs can
b~ inspected by using the mode-output coupling vectors
CV in exactly the same way as the eigenvectors V can
be examined to determine interactions between airplane
modes and airplane states.
x
Table II. Scaledmode-output coupling vectors of ATD.
Mode Mi and its eigenvalue Ai
MI
M2
M3
M4
M5
M6
-6.63 I -3.947 -0.596 ± 0.722i -0.002 -0.588 -0.227
Table III. Reference input-mode coupling of the ATD.
Inputs
Mi
Be
10.2. Eigenstructure analysis of the nominal TEeS
control law on the ATD
The above concepts were applied to determine the
mode-output coupling eigenstructure of the nominal
TECS control law for the AID, shown in table II. The
magnitudes and directions of the output-coupling vectors, CCLVi, relate to the coupling of the closed-loop system modes with Eand L. The phases of the modes are
measured clockwise in relation to the negative real axis
of the s-plane. Each of the modes in the table can now be
attributed to airplane dynamic characteristics.
Taking the mode MI as an example, its eigenvalue
shows that it contributes a first order behaviour with a
corner frequency of 6.631 rads -1 to the time response.
Its column eigenvector shows that if this mode is excited,
it will cause a large change in E. It will also cause a
change in i; but this effect in i. will only be about 67%
of the value of the peak influence in E.
Information can also be gleaned from the magnitudes
of the rows along the eigenvectors. For instance, table II
shows that Ehas much larger components of modes M2
and M6 than of any other mode. This means that a similar perturbation in ail of the dynamic modes will cause
the total energy rate of the airplane to describe the contribution from these two modes more than any other. A
more complete analysis of the table shows that:
1. All modes, other than M4, are coupled to
E and L.
2. MI and M2 describe distinctly more in-phase
motion between E and L.
3. M3 describes distinctly more anti-phase motion
between Eand L.
4. M3 and M5 are mainly coupled to
L.
5. M6 is mainly coupled to B.
SO far, the discussion has centred on the homogeneous
motion in the airplane dynamics. When the relationship
between system reference inputs r and system modes
MI
M2
M3
M4
M5
M6
Lc
4.1
-29
19L74"
5
-14
29L - 80'
-33
-I
7
-9
5
38
needs to be analysed, the reference-mode coupling vectors, w[O 12]\ can be examined. For the nominal TECS
controlled AID, these are shown in table III.
Reference-mode coupling is interpreted in a similar
way to mode-output coupling. The values in the table
provide the factors by which each dynamic mode is excited by a particular reference input. From table III, we can
discern three things:
I. M I and M4 are least perturbed by the references.
2. M2 and M6 are more perturbed by Be than by t.;
3. M3 and M5 are more perturbed by i; than by Be.
Overall analysis of tables I and II now demonstrate
the following about this TECS controlled ATD.
• M4 is uninvolved in the behaviour of this controlled
airplane. Hence, we now know that its near-instability is not critical with these control gains.
• M I and M2 ill"e associated with airplane motion
where E and L are nearly in phase.
• M3 is associated with airplane motion where E and
i. are more in anti-phase.
• M3 and M5 are associated with L.
• M2 and M6 are associated with E.
Eigenstructure analysis has provided a way of understanding how output coupling comes about through the
coupling of dynamic modes of the TECS controlled airplane to its performance output. This information cannot
be obtained easily by other means, and now, it can be
used to tune the TECS control law.
Aerospace Science and Technology
Analysis and tuning of a 'Total Energy Control System' control law using eigenstructure assignment
Analyse und Anpassung eines 'Total Energy Control System' Flugregelgesetzes mitdem Veifahren der Eigenstrukturvorgabe
11.
Eigenstructure assignment of TECS
137
11.2. TECS control law tuning
Equation (28) shows that. given a set of initial conditions and inputs to an airplane system. the time response
of the outputs y can be manipulated by altering the
eigenstructure of the system. What is required is a
method of specifying the eigenvalues and eigenvectors
of the closed-loop system directly.
The use of eigenstructure assignment (EA) to do this
in more traditionally structured modem flight control
systems is well documented (see [9] and [19] for real airplane implementation and [8, 20-22] for theoretical studies). and the method of EA and its abilities and restrictions are best summarised in [6. 7. 11 . 12.22], and will
not be repeated here.
There is sufficient design freedom to alter the four
gains K T P, K E P, K T I, K E Ito allow the assignment
of a set of four desired eigenvalues to the closed-loop
system. In a MIMO full output feedback system. EA
additionally offers the ability to choose the four closedloop system eigenvectors that correspond to these eigenvalues. Let these four desired eigenvalues and eigenvectors of the airplane closed-loop system be defined by:
To demonstrate the ability of EA to be used solely for
improving couplingldecoupling, the desired eigenvalues
were left very near the values of the nominal TECS
controller shown in table I. The desired eigenvectors
were based on the eigenstructure analysis described in
section 10.2, which led to the following requirements:
• M I should be equally coupled to Eand i:
• M2 should be equally coupled to E and i; and possibly more strongly coupled to E.
• M3 should be oppositely coupled to E and L. and
possibly more strongly coupled to L.
• M5 should be decoupled from E.
• M6 should be decoupled from i:
The resulting desired eigenstructure is shown in
table IV. The entries for M4 show that we are not particularly worried about assignment of this mode. the reason being that eigenstructure analysis showed that it had
no significant effect on the rest of the controlled airplane.
Table IV. Desired eigenstructure for full freedom design
M1
M2
M3
M4
M5
M6
-6.632 -3.948 - 0.597 ± 0.723; -0.002 -0.589 -0.228
where
(29)
Equation (29) is an implicit expresgjon that can now
be used to determine the gain matrix K , given a desired
eigenstructure (Ad. Vd). Limitations exist on how Ad
and Vd can be assigned. and these are detailed in refe-
rences [6. 11. 12]. The EA algorithm used to produce a
controller in this paper is the same as that used in [6]. and
will not be repeated here.
The task for the control law designer is to decide on
how to specify a desired eigenstructure for TECS.
Desired eigenvalues can be chosen in the same way as
closed-loop system poles are chosen in conventional
synthesis methods. They will depend on design requirements such as providing the controlled airplane with
minimum rise times and overshoots to step commands.
Desired eigenvectors can be specified to reflect differing
requirements. coupling and decoupling being potentially
the most useful for TECS.
11.1. Couplingldecoupling improvement
It has been shown that the eigenvectors can be related
to the contribution that each mode makes to the state vector x. Hence. if we want to quantify the influence of a
mode in a state. we can set the corresponding desired
eigenvector element to the required value. If we wish to
decouple this influence. we set it to zero.
1999. no. 3
There are two things to consider for the final design .
The first is that we only have the freedom to assign four
eigenvalues and eigenvectors to the system. These were
chosen as M2. M3 (a complex conjugate mode) and MS.
as they were found to be the most sensitive of the dynamic modes (mode sensitivity is described in [II D.
The second thing to consider is that with full feedback. EA normally has the freedom of assignment to
effect a good solution. With the TECS. there are
constraints on controller structure. Out of the possible 8
feedback gains that could be assigned from each of the
outputs in (20) to each of the inputs in (21). only four are
used to form TEeS. Thus. we have to somehow
constrain the EA process so that the remaining four gains
become zero.
For the AID. this was done by a process of assigning
the nominal control law eigenvectors (shown in table Tf)
as the initial desired eigenvectors. and then iteratively
altering these towards the desired eigenvectors shown in
table IV to provide a compromise between full freedom
EA and limited freedom design. The resulting desired
eigenstructure is shown in table V.
This desired eigenstructure was used to produce the
controller matrix given by
K=
[
1.057
-0.242
0.027
1.038
0.772
0.197
0.050] .
0.446
(30)
L.F. Faleiro, A.A. Lambregts
138
Table V. Desired eigenstructure for limited freedom design.
M3
M2
-3.948
-0.597 ± 0.732;
M5
-0.589
E
L
E
r~
~ -50
This controller is not yet in the correct structure for a
TECS strategy. Calvo-Ramon [5] describes measures of
the rate of change of an eigenvalue or eigenvector with a
change in a feedback control gain. As previously
demonstrated on an airplane control system [23] these
measures can be used to form gain sensitivity matrices
for the EA control law in (30), given by
dA
dK
SA __ [0.39 0.01
0.03 0.33
dV
SV _ [0.26
- 0.08
----..r ;:}
dK ;:}
0.13 0.02]
0.01 0.19
K _ [1.027
-
0
°
0.783
a
0]
0.451'
The time responses to step inputs in airspeed and altitude, given by the bold lines in figures 8 and 9, show that
the final control gains are an improvement over the heuristically tuned TECS gains (whose time responses are
shown by the dotted lines). The total energy rate and
energy distribution rates are more in-phase. This has
been effected by the large increase in KEP, which is now
95% larger. The result is a higher throttle rate to compensate for the slowness of engine response. However, as
can be seen in figures 8 and 9, the actual control effort
has not altered adversely. Thus,
• For a 10 ms -I step demand in airspeed, altitude
deviated between only 2.3 m from its trim value.
• For a 10 m ste~ demand in altitude, airspeed deviated 0.08 ms - from its trim value.
EA can therefore be used to tune the TEes control
law systematically and effectively when addressing output decoupling.
20
lIme(s)
25
cO.02
0
o
5
10
15
20
lIme(s)
25
30
o
5
ro
~
w
lIme(s)
~
35
~~
30
fj~::
~
35
40
I
40
:1
~
40
Nominal cortrollaw
EA timed control law
Figure 8. Responses to a step demandin airspeed.
s
i'lZ
~
0
10
<I:
(33)
15
w
0.04 0.38 0.13]
(32)
0.58 0.12 0.40 .
1.045
10
'~~
~:~
:
(31)
These matrices are of the same structure as the
controller, and each of their elements describes the sensitivity of the eigenvalues and eigenvectors of.!,he system
to changes in the corresponding elements of K.
These show that the four gains that contribute the least
to eigenstructure perturbation were the same as the gains
that could not be implemented in the TECS structure,
and they were suppressed using a method demonstrated
in [23], which distributes the effect of the loss of these
gains onto the remaining four gains, which are KT P,
KEP, KTf and KEf.
The resulting control law is given by the matrix
5
20
Time (5)
30
40
-3
X10_~_
r~-"""I:lI::l:=-=~ __I
o
10
30
20
40
Time (5)
: I
30
40
Nominal comol1_
EA lunltd co"lrollQW
Figure 9. Responses to a step demand in altitude.
Aerospace Science and Technology
Analysis and tuning of a 'Total Energy Control System' control law using eigenstructure assignment
Analyse undAnpassung eines 'Total Energy Control System' Flugregelgesetzes mitdem Veifahren derEigenstrukturvorgabe
12.
Conclusions
This paper discussed some aspects of the evolution of
SISO automatic flight control systems, where incremental development of operational flight control modes over
time has resulted in inefficient system architectures.
These have considerable functional overlap, unnecessarily high systems complexity, too much hardware/software and less than optimum system performance.
A new control strategy was described - the Total
Energy Control System (TECS) for longitudinal flight
path and speed control. TECS uses a MIMO control
algorithm to provide complete operational and performance consistency for all operational modes and flight
conditions. It is based on the consideration of energy
management, and the result is a visible structure with
pilot-like, energy efficient operation that solves most of
the problems of traditional autopilots and autothrottles.
It was proposed that improvements can be made to
TECS control law tuning. The classical structure of
TECS was first mapped onto a state-space based structure and notation. The resulting matrices were used to
demonstrate that eigenstructure analysis is a tool that can
be used to
Appendix: The ATD model
Two linear models are used in this paper; the openloop airplane at a nominal condition and the airplane
with inner loop stability augmentation added. The
models shown here are used for simulation. However, as
the z state (altitude) is not required for analysis and synthesis, reduced order models that did not have z as a state
(obtained by removing the relevant rows and columns of
the airplane matrices) were used in these cases.
• The basic airplane (A, B, q
The nominal flight condition at which the AID nonlinear longitudinal dynamics model is trimmed using the
DYMOLA modelling environment is: total weight of
14 tonnes, an airspeed of 173 knots (88 ms -1) and an
altitude of 1000 ft (304.8 m). The resulting linear model
is expressed in the form of a linear, time-invariant statespace system.
A
q
-0.1359
-0.1119
0.0078
0
0
'"
iJ
til
0.0001
-0.0009
0
0
0
0
-9.9396
88.2923
0
-1.9932
\
0
-9.74$1
-1.097\
-88.8500
0
0
0
._]["]
-0.0037
w
o
z
0.00\9
q
9
N
o
-1.7500
B
0
• provide information on the amount of each dynamic
mode present in an airplane state or output.
+
o
0,3786]
-8.3777
o
-5.~140 [&,,]
o
0
[
• provide information on how a reference command
might affect the dynamic modes of the airplane.
1999. no. 3
0.1482
-0.8949
0.9937
-0.0440
0
0
['J r"m
Ii>
Z
• examine the dynamic modes of a TECS controlled
airplane.
This information was assessed, and it was shown that
eigenstructure assignment (EA) can be used to force total
energy rate and energy distribution rate of the airplane to
become more in-phase, consequently providing a TEeS
that is better tuned, despite restrictions on the structure of
the controller.
The improvement in airplane performance that this
provides through the ability to decouple airplane modes
from airplane outputs was explored on a linear model of
the Aerospace Technologies Demonstrator successfully.
Now that it has been demonstrated that the TECS core
algorithm can be supported by methods of systematic
and quantitative analysis and design - in this case eigenstructure assignment - to shape airplane behaviour, further work should involve the concurrent tuning of the
core control law and the inner loops, and the concurrent
tuning of the core control law and the operational modes.
Interactive multi-objective synthesis tuning. as elaborated in [10], is well suited to support this process.
This should lead to an overall analysis and synthesis
method that contains the visibility of the TEeS structure
combined with the practicality and rigour of using a systematic analysis and design method, thus reducing the
time taken to produce a well tuned TECS control law.
139
263.~778
a,it,
Eigenvalue
Dynamicmode
-1.4414 ± 1.9109i
-0.0091
SPPO
± 0.1506i
Phugoid
Heave
Engine
-0.0007
-1.7500
• The airplane model with pitch and thrust command
functions added (ACL,BcL. eeL,DCL)
This is the basic airplane (A, B) augmented with
inner-loops. These are a linear thrust to throttle feedforward gain and pitch attitude/pitch rate feedback to the
elevator.
Act
Ii>
i
q
9
-01359
-0.1119
0.0078
0
0
[' J= [_"m
N
01482
-0.8949
0.9937
-0.0440
0
0
0.0001
-0.0009
0
0
0
0
-9.5611
79.9146
0
-7.9072
\
0
'-]["]
-9.5558
-5.2859
-0.0037
-88.8500
o
-2.9570
0
0
0.0019
o
-1.7500
w
l
q
9
N
140
L.F. Faleiro, A.A. Lambregts
3786
-8.3777
0.
0
l
o
+
o
1
-3'~140 [~~]
0
14~83
[LE]= [-0.0017
-0.0042
-0.0060
0.0163
0
0
:a~~563
-0.0563
-00516
-2.0516
0.0069]
0.0069
r~1
q
6
N
D CL
+ [~
][ ]
=~:~~:: ~~
Eigenvaiue
Dynamic mode
-0.1146 ±0.0795i
-6.8728
-1.7134
-0.0004
-1.7500
Pseudo-Phugoid
fast mode
fast mode
Heave
Engine
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