Robust Control of Hybrid Systems

27
temperature falls to
m
x (Fig. 1). In practical situations, exact threshold detection is
impossible due to sensor imprecision. Also, the reaction time of the on/off switch is usually
non-zero. The effect of these inaccuracies is that we cannot guarantee switching exactly at
the nominal values
m
x and
M
x . As we will see, this causes non-determinism in the discrete
evolution of the temperature.
Formally we can model the thermostat as a hybrid automaton shown in (Fig. 2). The two
operation modes of the thermostat are represented by two locations 'on' and 'off'. The on/off
switch is modeled by two discrete transitions between the locations. The continuous
variable x models the temperature, which evolves according to the following equations.

[]
εε+−∈
MM
xxx ,

[]
εε+−∈
mm
xxx ,
Off

),(

2
(,)xfxu xu
=
=− +
0
t
xM
x0
x

0xM
t
xM-e
x0
xm+e
xm
xm-e
xM+e
x

Fig. 3. Two different behaviors of the temperature starting at
0
x .
The second source of non-determinism comes from the continuous dynamics. The input

points, and the reset maps are thus the identity functions.
Finally we define the two staying conditions of the 'on' and 'off' locations as
M
xx≤+ε
and
M
xx≥−εrespectively, meaning that the system can stay at a location while the
corresponding staying conditions are satisfied.
Example 2 (Bouncing Ball). Here, the ball (thought of as a point-mass) is dropped from an
initial height and bounces off the ground, dissipating its energy with each bounce. The ball
exhibits continuous dynamics between each bounce; however, as the ball impacts the
ground, its velocity undergoes a discrete change modeled after an inelastic collision. A
mathematical description of the bouncing ball follows. Let
1
:xh
=
be the height of the ball
and
2
:xh=

(Fig. 4). A hybrid system describing the ball is as follows:

2
0
():
.
gx
x
⎡

. (2)
This model generates the sequence of hybrid arcs shown in (Fig. 5). However, it does not
generate the hybrid arc to which this sequence of solutions converges since the origin does
not belong to the jump set
D
. This situation can be remedied by including the origin in the
jump set
D . This amounts to replacing the jump set D by its closure. One can also replace
the flow set C by its closure, although this has no effect on the solutions.
It turns out that whenever the flow set and jump set are closed, the solutions of the corresponding
hybrid system enjoy a useful compactness property: every locally eventually bounded sequence of
solutions has a subsequence converging to a solution. gy −=


?0

&

0 ≺hh =
)1,0(
.
∈
−=
+
γ
γ hh


and
D be subsets of
n
ℜ
and let
f
, respectively
g
, be mappings fromC , respectively D ,
to
n
ℜ . The hybrid system :(,,,)HfgCD
=
can be written in the form

( )
()
x
f
xxC
x
g
xxD
+
=
∈
=
∈

(3)

∈

(4)
A solution to the hybrid system (4) starting at a point
0
xCD
∈
∪ is a hybrid arc x with the
following properties:
1.

0
(0,0)xx= ;
2.
given
(,) sj domx∈
, if there exists s
τ
 such that
(,) jdomxτ∈
, then, for all
[
]
,ts∈τ,
(,)xt j C∈
and, for almost all
[
]
,ts
∈

Suppose we can find a continuously differentiable function
:
n
V
ℜ
→ℜsuch that

( ): ( ), ( ) 0
( ): ( ( )) ( ) 0
c
d
ux Vx fx x C
ux Vgx Vx x D
=
∇≤∀∈
=
−≤ ∀∈
(5)
Consider (,)x ⋅⋅ a bounded solution with an unbounded hybrid time. Then there exists a value r in the
range V so that
x
tends to the largest weakly invariant set inside the set

(
)
(
)
1111
:() (0) (0)((0))
rcdd

intersection with
1
((0))
d
gu
−
. This term, however, can be very useful for zeroing in set to
which trajectories converge.
3.2 Lyapunov stability theorems
Some preliminary results on the existence of the non-smooth Lyapunov function for the hybrid
systems published in (DeCarlo, 2000). The first results on the existence of smooth Lyapunov
functions, which are closely related to the robustness, published in (Cai et al., 2005). These
results required open basins of attraction, but this requirement has since been relaxed in (Cai et
al. 2007). The simplified discussion here is borrowed from this posterior work.
Let
O be an open subset of the state space containing a given compact set A and let
0
:
≥
ω→ℜO
be a continuous function which is zero for all xA
∈
, is positive otherwise,
which grows without limit as its argument grows without limit or near the limit
O . Such a
function is called a suitable indicator for the compact set
A in the open setO . An example of
such a function is the standard function on
n
ℜ

(()) ()
xVx x
Vx fx Vx
Vgx e Vx
−
α
ω≤ ≤αω
∇≤−
≤
x
xC
xD
∀
∈
∀∈
∀∈
∩
∩
O
O
O
(7)
Suppose that such a function exists, it is easy to verify that all solutions for the hybrid
system
(,,, )fgCDfrom
(
)
CD∩∪O satisfied
Robust Control of Hybrid Systems


CD∩∪O is bounded and if
its time domain is unbounded, so it converges to
A .
According to one of the principal results in (Cai et al., 2006)
there exists a smooth Lyapunov
function for
(,,, ,,)fgCD
ω
O if and only if the set A is pre-stable and pre-attractive on O and O is
forward invariant
(i.e.,
(
)
(0,0)xCD∈ ∩∪O implies (,)xt j
∈
O for all(,) tj domx
∈
).
One of the primary interests in inverse Lyapunov theorems is that they can be employed to
establish the robustness of the asymptotic stability of various types of perturbations.
4. Hybrid control application
In system theory in the 60s researchers were discussing mathematical frameworks so to
study systems with continuous and discrete dynamics. Current approaches to hybrid
systems differ with respect to the emphasis on or the complexity of the continuous and
discrete dynamics, and on whether they emphasize analysis and synthesis results or
analysis only or simulation only. On one end of the spectrum there are approaches to hybrid
systems that represent extensions of system theoretic ideas for systems (with continuous-
valued variables and continuous time) that are described by ordinary differential equations
to include discrete time and variables that exhibit jumps, or extend results to switching
systems. Typically these approaches are able to deal with complex continuous dynamics.

⎪
⎪
=
⎨
ξ
⎡⎤
∈
ξξ∈∈
⎪
⎢⎥
⎪
⎣⎦
⎩

Η
(9)
Robust Control, Theory and Applications

32
where Q is a finite index set, for each qQ
∈
,
q
f
, :
n
qq
C
η
→ℜ are continuous functions,

{
}
qQ q
AAq
∈
=×∪
is pre-stable and globally pre-
attractive. Converse Lyapunov theorems can then be used to establish the existence of a
logic-based continuous feedback that renders the closed-loop system input-to-state stable
with respect to d . The feedback has the form

: () . () ()
T
qq q q
uk V
=
ξ−εη ξ∇ ξ
(10)
where 0ε
 and
()
q
V
ξ
is a smooth Lyapunov function that follows from the assumed
asymptotic stability when 0d
≡
. There exist class-
∞
K functions

⎫
⎛⎞
υ
⎪
⎪
⎜⎟
ξ≤α−−αξ α
⎨
⎬
⎜⎟
ε
⎜⎟
⎪
⎪
⎝⎠
⎩⎭
(11)
where
(,)dom
:sup (,)
si d
ddsi
∈
∞
= .
4.2 Control Lyapunov functions
Although the control design using a continuously differentiable control-Lyapunov function
is well established for input-affine nonlinear control systems, it is well known that not all
controllable input-affine nonlinear control system function admits a continuously
differentiable control-Lyapunov function. A well known example in the absence of this

{
}
: 1,,
q
Qm∈= … , such as
a.
for each qQ
∈
,
q
Ω
and
q
′
Ω
are open and
•

{
}
:\0
n
q
Q
qq
Q
q
∈∈
′
=ℜ = =

b.
for each qQ
∈
,
q
V is a smooth function defined on a neighborhood (relative to O )
of
q
Ω
.
c.
there exist a continuous positive definite function
α
and class-
∞
K functions
γ
and
γ
such that
•
()
(
)
()
q
xVx xγ≤ ≤γ
q
V
q

,x
q
u such that
(),(, ,) ()
qx
Vx
f
xu
q
x∇≤−α

• for each qQ∈ and
(
)
\
qrqr
x
′
∈
ΩΩ

∪∩O there exists
,x
q
u such that
(),(, ,) ()
(),(, ,) ()
qx
qx
Vx

υ , with the following data
:()
qq
ukx
=
,
(
)
\
qqrqr
C
′
=
ΩΩ

∪∩O
(12)
where
q
k is defined on
q
C , continuous and such that

()
( ), ( , ( )) 0.5 ( )
( ), ( , ( )) 0.5 ( ) \
qq q
qx qrkr
Vx fxkx x x C
nxfxkx x x

: \
rr
q
r
q
q
r
q
rQx rq x
Gx
rQx x
⎧
′′
∈∈ ∈
⎪
=
⎨
′
∈∈ ∈
⎪
⎩
Ω
ΩΩ
Ω
Ω

∩ ∪ ∩∩
∩
O, O
OO

C a compact subset of the O
that contains the origin in its interior and one takes
g
lobal
D to be a compact subset of
local
C ,
again containing the origin in its interior and such that, when using the controller
local
k ,
trajectories starting in
g
lobal
D
never reach the boundary of
local
C (Fig. 6). Finally, the hybrid
control which achieves global asymptotic stabilization while using the controller
q
k for
small signals is as follows

{
}
{}
: ( ) : :
( , ) : toggle ( ) D : :
qq
q
ukx C (x,q)xC

In this section, we review the supervisory control framework for hybrid systems. One of the
main characteristics of this approach is that the plant is approximated by a discrete-event
system and the design is carried out in the discrete domain. The hybrid control systems in
the supervisory control framework consist of a continuous (state, variable) system to be
controlled, also called the plant, and a discrete event controller connected to the plant via an
interface in a feedback configuration as shown in (Fig. 7). It is generally assumed that the
dynamic behavior of the plant is governed by a set of known nonlinear ordinary differential
equations

() ((),())xt f xt rt
=

(17)
Robust Control of Hybrid Systems

35
where
n
x ∈ℜis the continuous state of the system and
m
r
∈
ℜ is the continuous control
input. In the model shown in (Fig. 7), the plant contains all continuous components of the
hybrid control system, such as any conventional continuous controllers that may have been
developed, a clock if time and synchronous operations are to be modeled, and so on. The
controller is an event driven, asynchronous discrete event system (DES), described by a
finite state automaton. The hybrid control system also contains an interface that provides
the means for communication between the continuous plant and the DES controller.


2
xh

)(
3
xh

X

Fig. 8. Partition of the continuous state space.
The interface consists of the generator and the actuator as shown in (Fig. 7). The generator
has been chosen to be a partitioning of the state space (see Fig. 8). The piecewise continuous
command signal issued by the actuator is a staircase signal as shown in (Fig. 9), not unlike
the output of a zero-order hold in a digital control system. The interface plays a key role in
determining the dynamic behavior of the hybrid control system. Many times the partition of
the state space is determined by physical constraints and it is fixed and given.
Methodologies for the computation of the partition based on the specifications have also
been developed.
In such a hybrid control system, the plant taken together with the actuator and generator,
behaves like a discrete event system; it accepts symbolic inputs via the actuator and
produces symbolic outputs via the generator. This situation is somewhat analogous to the

Robust Control, Theory and Applications

36

time

]1[
c

can be described as a triple ;;TWBwith T ⊆ℜthe time axis, W the
signal space, and
T
BW⊂ (the set of all functions
:fT W→
) the behavior. The behavior of the
DES plant model consists of all the pairs of plant and control symbols that it can generate.
The time axis
T represents here the occurrences of events. A necessary condition for the
DES plant model to be a valid approximation of the continuous plant is that the behavior of
the continuous plant model
c
B is contained in the behavior of the DES plant model, i.e.
cd
BB⊆ .
The main objective of the controller is to restrict the behavior of the DES plant model in
order to specify the control specifications. The specifications can be described by a
behavior
s
p
ec
B . Supervisory control of hybrid systems is based on the fact that if undesirable
behaviors can be eliminated from the DES plant then these behaviors can likewise be eliminated from
the actual system. This is described formally by the relation

d s spec c s spec
BBB BBB⊆⇒ ⊆∩∩
(18)
and is depicted in (Fig. 10). The challenge is to find a discrete abstraction with behavior B
d

1
p

. (Fig.11)
shows that for a given control symbol, there are at least two possible DES plant states that
can be reached from
1
p

. Transitions within a DES plant will usually be nondeterministic
unless the boundaries of the partition sets are invariant manifolds with respect to the vector
fields that describe the continuous plant.

A

B

1
~
X

2
~
X

2
~
P

3

However, sensors/actuators that are dynamics unmodelled can substantially affect the
behavior of the system when in the loop. In this section, it is desired that the hybrid
controller provides a certain degree of robustness to such disturbances. In the following
sections, general statements are made in this regard.
5.1 Robustness via filtered measurements
In this section, the case of noise in the measurements of the state of the nonlinear system is
considered. Measurement noise in hybrid systems can lead to nonexistence of solutions.
This situation can be corrected, at least for the small measurement noise, if under global
existence of solutions,
c
C and
c
D always “overlap” while ensuring that the stability
properties still hold. The "overlap" means that for every O
ξ
∈ , either
c
eCξ+ ∈
or
c
eDξ+ ∈

all or small e . There exist generally always inflations of C and
D that preserve the
semiglobal practices asymptotic stability, but they do not guarantee the existence of
solutions for small measurement noise.
Moreover, the solutions are guaranteed to exist for any locally bounded measurement noise
if the measurement noise does not appear in the flow and jump sets. This can be carried out
by filtering measures. (Fig. 12) illustrates this scenario. The state
x is corrupted by the noise

+ is considered to be linear and defined by the
matrices
f
A ,
f
B , and
f
L , and an additional parameter 0
f
ε
> . It is designed to be
asymptotically stable. Its state is denoted by
f
x which takes value in
f
n
R . At the jumps,
f
x
is given to the current value of
y
. Then, the filter has flows given by
,
ff ff f
xAxB
y
ε
=+

(19)

1
((,))
(,)
()
(,)
()
pffc
ccffc
ff ff f
ccffc
fff
xfx Lxx
xfLxx
xAxBxe
xx
xGLxx
xABxe
+
+
+−
⎧
⎫
=+κ
⎪
⎪
⎪
=
⎪
⎬
⎪

cc
ff
cc
Lx x C
Lx x D
∈
∈
(21)
5.2 Robustness to sensor and actuator dynamics
This section reviews the robustness of the closed-loop
cl
H when additional dynamics,
coming from sensors and actuators, are incorporated. (Fig. 13) shows the closed loop
cl
H
with two additional blocks: a model for the sensor and a model for the actuator. Generally,
to simplify the controller design procedure, these dynamics are not included in the model of
the system ( , )
p
x
f
xu=

when the hybrid controller
c
H is conceived. Consequently, it is
important to know whether the stability properties of the closed-loop system are preserved,
at least semiglobally and practically, when those dynamics are incorporated in the closed
loop.
The sensor and actuator dynamics are modeled as stable filters. The state of the filter which

(, )
(,)
( , ) or
()
(,)
:
( , )

0
d
paa
ccssc
ss c c
ds ss s
da aa a ss c
cl
ccssc
ss
aa
xfxLx
xfLxx
Lx x C
xAxBxe
xAxBLxx
H
xx
xGLxx
xx
xx
ε


( , ) and
ss c c
Lx x D
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
∈τ≥τ
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎪


Controller
k

s
xFig. 13. Closed-loop system with sensor and actuator dynamics.
5.3 Robustness to sensor dynamics and smoothing
In many hybrid control applications, the state of the controller is explicitly given as a
continuous state
ξ
and a discrete state
{
}
:1, ,
q
Qn∈= , that is,
:[ ]
T
c
x
q
=ξ
. Where this is the
case and the discrete state
q
chooses a different control law to be applied to the system for
for various values of

1
k

qFig. 14. Closed-loop system with sensor dynamics and control smoothing.
(Fig. 14) shows the closed-loop system, noted that
d
cl
H
ε
, resulting from adding a block that
makes the smooth transition between control laws indexed by
q
and indicated by
q
κ . The
smoothing control block is modeled as a linear filter for the variable
q
. It is defined by the
parameter
u
ε
and the matrices
(,,)
uuu
A
BL
.

can be written as
Robust Control of Hybrid Systems

41

*
((,,))
(,)
0
(,) or
()
:
(,)
(
0
f
pcuu
ccssc
ss c c
us ss s
uu uu u
cl
cssc
ss s
uu
xfx xxLx
xfLxx
q
Lx x C
xAxBx

⎪
ε= +
⎭
⎫
=
⎪
⎪
⎡⎤
ξ
⎪
∈
⎢⎥
⎪
⎢⎥
⎣⎦
⎪
⎪
=
⎬
⎪
=
⎪
⎪
τ=
⎪
⎪
⎪
⎭



⎪
⎪
⎪
⎩
(24)
6. Conclusion
In this chapter, a dynamic systems approach to analysis and design of hybrid systems has
been continued from a robust control point of view. Stability and convergence tools for
hybrid systems presented include hybrid versions of the traditional Lyapunov stability
theorem and of LaSalle’s invariance principle.
The robustness of asymptotic stability for classes of closed-loop systems resulting from
hybrid control was presented. Results for perturbations arising from the presence of
measurement noise, unmodeled sensor and actuator dynamics, control smoothing.
It is very important to have good software tools for the simulation, analysis and design of
hybrid systems, which by their nature are complex systems. Researchers have recognized
this need and several software packages have been developed.
7. References
Rajeev, A.; Thomas, A. & Pei-Hsin, H.(1993). Automatic symbolic verification of embedded
systems, In IEEE Real-Time Systems Symposium, 2-11, DOI:
10.1109/REAL.1993.393520 .
Dang, T. (2000). Vérification et Synthèse des Systèmes Hybrides. PhD thesis, Institut National
Polytechnique de Grenoble.
Girard, A. (2006). Analyse algorithmique des systèmes hybrides. PhD thesis, Universitè Joseph
Fourier (Grenoble-I).
Ancona, F. & Bressan, A. (1999). Patchy vector fields and asymptotic stabilization, ESAIM:
Control, Optimisation and Calculus of Variations, 4:445–471, DOI:
10.1051/cocv:2004003.
Byrnes, C. I. & Martin, C. F. (1995). An integral-invariance principle for nonlinear systems, IEEE
Transactions on Automatic Control, 983–994, ISSN: 0018-9286.
Robust Control, Theory and Applications

0129.
Sanfelice, R. G.; Goebel, R. & Teel, A. R. (2005). Results on convergence in hybrid systems via
detectability and an invariance principle. Proceedings of 2005 American Control
Conference, 551–556, ISSN: 0743-1619.
Sontag, E. (1989). Smooth stabilization implies coprime factorization. IEEE Transactions on
Automatic Control, 435–443, ISSN: 0018-9286.
DeCarlo, R.A.; Branicky, M.S.; Pettersson, S. & Lennartson, B.(2000). Perspectives and results
on the stability and stabilizability of hybrid systems. Proc. of IEEE, 1069–1082, ISSN:
0018-9219.
Michel, A.N.(1999). Recent trends in the stability analysis of hybrid dynamical systems. IEEE
Trans. Circuits Syst. – I. Fund. Theory Appl., 120–134,ISSN: 1057-7122.
Halbaoui, K.; Boukhetala, D. and Boudjema, F.(2008). New robust model reference adaptive
control for induction motor drives using a hybrid controller.International Symposium on
Power Electronics, Electrical Drives, Automation and Motion, Italy, 1109 - 1113
ISBN: 978-1-4244-1663-9.
Halbaoui, K.; Boukhetala, D. and Boudjema, F.(2009a). Speed Control of Induction Motor
Drives Using a New Robust Hybrid Model Reference Adaptive Controller. Journal of
Applied Sciences, 2753-2761, ISSN:18125654.
Halbaoui, K.; Boukhetala, D. and Boudjema, F.(2009b). Hybrid adaptive control for speed
regulation of an induction motor drive,
Archives of Control Sciences,V2.
3
Robust Stability and Control of
Linear Interval Parameter Systems
Using Quantitative (State Space) and
Qualitative (Ecological) Perspectives
Rama K. Yedavalli and Nagini Devarakonda
The Ohio State University
United States of America
1. Introduction

be captured with equal ease in both frameworks. Thus it is not surprising that most of the
robustness studies of uncertain dynamical systems with real parameter variations are being
carried out in time domain state space framework and hence in this chapter, we emphasize
the aspect of robust stabilization and control of linear dynamical systems with real
parameter uncertainty.
Stability and performance are two fundamental characteristics of any feedback control
system. Accordingly, stability robustness and performance robustness are two desirable
(sometimes necessary) features of a robust control system. Since stability robustness is a
prerequisite for performance robustness, it is natural to address the issue of stability
robustness first and then the issue of performance robustness.
Since stability tests are different for time varying systems and time invariant systems, it is
important to pay special attention to the nature of perturbations, namely time varying
perturbations versus time invariant perturbations, where it is assumed that the nominal
system is a linear time invariant system. Typically, stability of linear time varying systems is
assessed using Lyapunov stability theory using the concept of quadratic stability whereas
that of a linear time invariant system is determined by the Hurwitz stability, i.e. by the
negative real part eigenvalue criterion. This distinction about the nature of perturbation
profoundly affects the methodologies used for stability robustness analysis.
Let us consider the following linear, homogeneous, time invariant asymptotically stable
system in state space form subject to a linear perturbation E:

(
)
00
(0)xAEx x x
=
+=

(1)
where A

invariant perturbation case. A methodology specifically tailored to time invariant
perturbations is discussed and included by the author in a separate publication [6].
Robust Stability and Control of Linear Interval Parameter Systems
Using Quantitative (State Space) and Qualitative (Ecological) Perspectives

45
It is also appropriate to discuss, at this point, the characterization with regard to the
boundedness of the perturbation. In the so called ‘unstructured’ perturbation, it is assumed
that one cannot clearly identify the location of the perturbation within the nominal matrix
and thus one has simply a bound on the norm of the perturbation matrix. In the ‘structured’
perturbation, one has information about the location(s) of the perturbation and thus one can
think of having bounds on the individual elements of the perturbation matrix. This
approach can be labeled as ‘Elemental Perturbation Bound Analysis (EPBA)’. Whether
‘unstructured’ norm bounded perturbation or ‘structured’ elemental perturbation is
appropriate to consider depends very much on the application at hand. However, it can be
safely argued that ‘structured’ real parameter perturbation situation has extensive
applications in many engineering disciplines as the elements of the matrices of a linear state
space description contain parameters of interest in the evolution of the state variables and it
is natural to look for bounds on these real parameters that can maintain the stability of the
state space system.
3. Robust stability and control of linear interval parameter systems under
state space framework
In this section, we first give a brief account of the robust stability analysis techniques in 3.1
and then in subsection 3.2 we discuss the robust control design aspect.
3.1 Robust stability analysis
The starting point for the problem at hand is to consider a linear state space system
described by
[
]
0

σ
σ
<
(2)
where P is the solution to the Lyapunov matrix

00
20
T
PA A P I
+
+=
(3)
See Refs [9],[10],[11] for results related to this case.
Robust Control, Theory and Applications

46
2. Time varying (real) structured variation
Case 1: Independent variations (sufficient bound) [12]-[13]

max
() ()
i
j
ti
j
i
j
Et Et
ε

ij
/ ε. For cases when ε
ij
are not known, one can take
U
eij
= |A
oij
|/|A
oij
|
max
. (·)
m
denotes the matrix with all modulus elements and (·)
s
denotes the
symmetric part of (·).
3. Time invariant, (real) structured perturbation E
ij
= Constant
Case i: Independent Variations [13]-[15]: (Sufficient Bounds). For this case, E can be
characterized as

12
ESDS
=
(6)
where S
1

−
>
<==
⎛⎞
⎡⎤
−
⎜⎟
⎢⎥
⎣⎦
⎝⎠
(7)
Notice that the characterization of E (with time invariant) in (4) is accommodated by the
characterization in [15]. ρ(·) is the spectral radius of (·).
Case ii: Linear Dependent Variation: For this case, E is characterized (as in (6) before), by

1
r
ii
i
EE
β
=
=
∑
(8)
and bounds on |β
i
| are sought. Improved bounds on |β
i
| are presented in [6].

⎤
⎢
⎥
+= −+
⎢
⎥
⎢
⎥
−+ −+ −+
⎣
⎦

Taking the nominally stable matrix to be
0
20 1
030
114
A
−
−
⎡
⎤
⎢
⎥
=−
⎢
⎥
⎢
⎥
−

000
010
010
E
⎡
⎤
⎢
⎥
=
⎢
⎥
⎢
⎥
⎣
⎦

The following are the bounds on |k
1
| and |k
2
| obtained by [15] and the proposed method.

µ
y
µ
Q
ZK [14] µ
d
[6]
0.815 0.875 1.55 1.75

uncertain systems. References [20] and [21] present methods which need the testing of
definiteness of a Lyapunov matrix obtained as a function of the uncertain parameters. In the
multimodel theory approach, [22] considers a discrete set of points in the parameter
uncertainty range to establish the stability. This paper addresses the stabilization problem
for a continuous range of parameters in the uncertain parameter set (i.e. in the context of
interval matrices). The proposed approach attacks the stability of interval matrix problem
directly in the matrix domain rather than converting the interval matrix to interval
polynomials and then testing the Kharitonov polynomials.
Robust control design using perturbation bound analysis [23],[24]
Consider a linear, time invariant system described by
xAxBu
=
+


0
(0)xx
=

Where
x is 1n × state vector, the control u is 1m
×
. The matrix pair (,)AB is assumed to
be completely controllable.
U=Gx
For this case, the nominal closed loop system matrix is given by
AABG=+ ,
1
0
T

is the design variable.
The main interest in determining
G is to keep the nominal closed loop system stable. The
reason Riccati approach is used to determine
G is that it readily renders (A+BG)
asymptotically stable with the above assumption on
Q and R
0
.
Now consider the perturbed system with linear time varying perturbations
E
A
(t) and E
B
(t)
respectively in matrices
A and B
i.e.,
[
]
[
]
() () () ()
AB
x AEtxt BEtut=+ ++


Let Δ
A and ΔB be the perturbation matrices formed by the maximum modulus deviations
expected in the individual elements of matrices

εε
=
known, we can extend the main result of equation (3) to the
linear state feedback control system of (9) and (10) and obtain the following design
observation.
Design observation 1:
The perturbed linear system is stable for all perturbations bounded by
a
ε
and
b
ε
if

()
max
1
a
mea ebm
s
PU UG
ε
μ
σε
<
≡
⎡⎤
+
⎣⎦
(9)

σ
<≡
(11)
In this context
g
μ
can be regarded as a “gain margin”.
For a given
ai
j
ε
and
bi
j
ε
, one method of designing the linear controller would be to
determine G of (3.10) by varying
c
ρ
of (3.10) such that μ is maximum. For an aircraft control
example which utilizes this method, see Reference [9].
4. Robust stability and control of linear interval parameter systems using
ecological perspective
It is well recognized that natural systems such as ecological and biological systems are
highly robust under various perturbations. On the other hand, engineered systems can be
made highly optimal for good performance but they tend to be non-robust under
perturbations. Thus, it is natural and essential for engineers to delve into the question of as
to what the underlying features of natural systems are, which make them so robust and then
try to apply these principles to make the engineered systems more robust. Towards this
objective, the interesting aspect of qualitative stability in ecological systems is considered in

seldom precisely known, but one can establish with certainty, the types of interactions that
are present. Many mathematical population models were proposed over the last few
decades to study the dynamics of eco/bio systems, which are discussed in textbooks [25]-
[26]. The most significant contributions in this area come from the works of Lotka and
Volterra. The following is a model of a predator-prey interaction where x is the prey and y is
the predator.

(,)
(,)
xx
f
x
y
yyg
x
y
=
=


(12)
where it is assumed that
(,)/ 0fxy y
∂
∂< and (,)/ 0gxy x
∂
∂>
This means that the effect of y on the rate of change of x ( x

) is negative while the effect of x

+
⎡
⎤
⎢
⎥
+
⎣
⎦

Competition

*
*
−
⎡
⎤
⎢
⎥
−
⎣
⎦

Commensalism

*
0*
+
⎡
⎤
⎢

Table 1. Types of interactions between two species in an ecosystem
In Table 1, column 2 is a visual representation of such interactions and is known as a
directed graph or ‘digraph’ [28] while column 3 is the matrix representation of the
interaction between two species. ‘*’ represents the effect of a species on itself.
In other words, in the Jacobian matrix, the ‘qualitative’ information about the species is
represented by the signs +, – or 0. Thus, the (i,j)
th
entry of the state space (Jacobian) matrix
simply consists of signs +, –, or 0, with the + sign indicating species j having a positive
influence on species i, - sign indicating negative influence and 0 indicating no influence. The
diagonal elements give information regarding the effect of a species on itself. Negative sign
means the species is ‘self-regulatory’, positive means it aids the growth of its own
population and zero means that it has no effect on itself. For example, in the Figure 1 below,
sign pattern matrices A
1
and A
2
are the Jacobian form while D
1
and D
2
are their
corresponding digraphs.
Fig. 1. Various sign patterns and their corresponding digraphs representing ecological
systems; a) three species system b) five species system