1. Manual Control of an Unstable System
with a Saturating Actuator
Karl J. Åström and Sabina Brufani‡
Department of Automatic Control
Lund Institute of Technology
Box 118, S-221 00 LUND, Sweden
Abstract.
This paper describes an interesting problem that has
arisen in connection with control of high performance
aircrafts. A simple prototype problem that captures
some of the key features is formulated and solved.
1. Introduction
Automatic control is increasingly being used in mis-
sion critical applications. Control of high perfor-
mance aircrafts is a typical example. Considerable
benefits can be obtained by having an aircraft that
is unstable in certain flight conditions and providing
stability with a control system. Unfortunately the
unstable flight conditions include landing and take
off. An aircraft has hydraulic actuators that satu-
rate. During saturation the feedback loop is broken
and the unstable states may then diverge to situa-
tions which cannot be recovered. The presence of a
pilot is yet another complication because the pilot
may also drive the system unstable through man-
ual control actions. Design of control systems for
such situations is a significant challenge as has been
pointed out in [6] and [5]. Aircraft manufacturers
have also encountered severe difficulties with prob-
lems of this type, see [4]. One solution adopted is to
try to divide the available control authority between
control and stabilization. In this paper we will dis-
cuss a simple problem that captures some of the dif-
ficulties.
2. An Example
Consider the system
dx1
dt
x1 − sat u
dx2
dt
sat u,
(1)
∗ This work has been partially supported by the Swedish Research
Council for Engineering Science, contract 95-759 and the EU
ERASMUS Program.
‡ Universita Degli Studi di Roma, La Sapienza, Roma, Italy
where sat is the saturation function. The purpose
of the control is to stabilize the state x1 at the
equilibrium point x1 0, and to permit the manual
control of the variable x2. The control strategy should
be such that if the system starts in a stabilizable
state it should remain in this start irrespective of
the control actions.
Stabilizable States
If the state x1 becomes larger than one it diverge
towards infinity irrespective of the control actions
taken. The set of stabilizable states is thus Ss
{x, x1 < 1}.
If ls > 1 the system is stabilized with the strategy
us ls x1 (2)
Let r be the command signal if there were no
saturation in (1) the controller
um l1x1 + l2(x2 − r) (3)
would give good command signal following. Reason-
able values of the gain are l1 ω2
0 + 2ω0ζ + 1 and
l2 ω2
0 which give the characteristic polynomial
s2
+ 2ζω0 + ω2
0.
3. A Hybrid Strategy
The control strategy (2) stabilizes the system, pro-
vided that the initial conditions are such that x1 <
1. The control strategy (3) gives manual control with
good response but the system can easily be driven
unstable by inappropriate manual control actions.
We are therefore faced with the problem of combining
the strategies. It is intuitively clear that the stabi-
lizing strategy should be given the highest priority
and that we should allow manual control only when
this is safe. One possibility to achieve this is to use
the hybrid system shown in Figure 3, where the con-
troller Cm is defined by um (l1 − ls)x1 + l2(x2 − r).
and the output of Cm is limited by the saturation
block with limits the output to ±vm, where um0 < 1.
WM12-6 3:10 Proc of the 36th IEEE CDC San Diego, CA
0-7803-4187-2 964-965

2. If there were no saturations the combined effect of
the controllers Cs and Cm are thus identical to (3).
When the limiter in the controller saturates the sig-
nal um becomes either um0 or −um0. The outer loop is
broken and only the stabilizing controller operates.
um
us
x1
x2
Cm Σ
Process
Cs
r
u
Figure 1 An hybrid strategy
4. Properties
The closed loop system obtained with the hybrid
strategy is a piece wise linear system. Since the
phase plane is R2
the behavior of the system can
be understood from a phase plane analysis. There
are five different regions:
Ω0 us + um ≤ 1 um ≤ um0
Ω+
1 us + um0 ≤ 1 um ≥ um0
Ω−
1 us − um0 ≤ 1 um ≤ −um0
Ω+
2 us + um ≥ 1
Ω−
2 us + um ≤ −1
Both controllers are active in region Ω0 and there
are no saturations.The manual control strategy satu-
rates in regions Ω1+ and Ω−
1 . The output of the man-
ual controller is constant, either um0 or −um0, where
um0 < 0 defines the authority given to the manual
control. The process saturates in regions Ω2+ and
Ω−
2 . If x1 < 1 control signal will however drive the
system in the right direction and the state will re-
main in the stability region provided that there are
no disturbances.
Since the system is of second order its behavior can
be analyzed from a phase portrait. This is shown in
Figure 4 which is calculated for r 0 and um0 0.5.
The figure shows that the strategy has the desired
behavior.
5. Generalizations
The idea described in this paper can be applied to a
wide range of systems. Determination of stabilizable
regions and and stabilizing controllers is well known,
see [3]. [2], [3]. [1] It is of interest to note that the
structure with cascaded saturations were used in
[7] to stabilize systems with cascaded integrators.
Simultaneous stabilization and control of the pivot
of a pendulum is an interesting application which is
easy to verify experimentally.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−15
−10
−5
0
5
10
15
x1
x2
Ω+
1
Ω−
1
Ω0
Ω+
2
Ω−
2
Figure 2 Phase-plots of the system controlled by the
hybrid strategy
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