OUTLINE
• Function spaces.
• Input-output stability.
• LTI Systems.
• Small gain theorem.
• Circle criterion.
INPUT-OUTPUT
STABILITY
1
M. SAMI FADALI
EBME DEPT., UNR
2
FUNCTION SPACES
_ SPACE
• _ Space: Space of all piecewise continuous
• Vector spaces with functions as “vectors”
satisfying
functions
: Space of all
• _ Spaces
piecewise continuous functions
satisfying
_
⁄
_
_
∈
Finite _ norm of
3
•
4
EXTENDED SPACES
SPACES
_
• _2 Space: Space of all piecewise continuous
satisfying
functions
. ∈
Truncation operator
⁄
:
→
,
0,
_
Example
• _1 Space: Space of all piecewise continuous
satisfying
functions
. :
0
→
,
0,
, & ∈
_
5
6
PROPERTIES OF
INEQUALITY
Extended Space : space of all functions whose
truncations belong to the parent space
. :
Note
→
. .
∈
If
, ,
,∀ ∈
is not a normed space even though
1
1,
∈_ ,
is.
1
∞
∈_
then
∈_
• Linear operator
I.
II.
1
,∀ ,
,∀ ∈
_
∈
⁄
,∀ ∈
7
⁄
8
ASSUME
EXAMPLE
SATISFIES
_1 Space of real-valued functions
normed linear vector space of piecewise
continuous functions . :
→
with norm .
• is closed under the family of projections
, i.e.
if . ∈ ,then
∈ ,∀ ∈
,
lim
•
. :
→
. .
,
• If
0
0,
→
•
∞
_1
non-decreasing function of ∈
,i.e.
. ∈ , ∈
⇒
∈ ,then ∈ iff lim
∞ use to verify
_1
∈
lim
→
→
∈
• _ Spaces satisfy the above assumptions.
/2
,∀ ∈
∞ and
∉
9
CAUSALITY
10
CAUSALITY AND PROJECTION
• A mapping representing a physical system
→
satisfying
:
.
. ,∀ ∈ ,∀ ∈
Present outputs do not depend on future inputs
Explanation: perform two experiments
1. Apply an arbitrary input to obtain an output
then truncate it.
2. Apply the truncated input
to obtain an output
then truncate it to get
The outputs are the same for all physical systems but
not for all mathematical functions.
11
• Express causality in terms of a projection operator
• Causal operator
,
∀ ∈ ,∀ ∈
• Property expressed in terms of the input and output
and we do not need the notion of a state.
• Assume the system is relaxed: zero initial conditions
and the output is due to the input only.
12
INPUT-OUTPUT STABILITY
GAIN
• A system
is input-output -stable if
the output is in
inputs in
has a finite gain if
and
such that
= bias term which allows including
for
systems where
• If
for
the gain is
Stability depends on the function
space
If the function space is obvious we
may say that is input-output stable.
∀ ∈
∀ ∈ ,
13
EXAMPLE
•
LTI SYSTEMS
_
static nonlinearity (Figure, page 163)
for
the gain is
_
∀ ∈_
∀ ∈ ,
14
• LTI system is input output stable if its
impulse response is in the form
_
Finite gain implies input-output stability but the
converse is not true: counterexamples in Figure
_
_
15
_
• True for all the poles of the transfer
function are in the open LHP
16
CLOSED-LOOP INPUT-OUTPUT
STABILITY
_2 GAIN
• Space of square-integrable functions _
(excludes some important functions, e.g.
sinusoids)
_
∀ ∈_
∀ ∈ ,
• Feedback system: feedback
interconnection of systems
that satisfies the assumptions
for all pairs of inputs
_
⁄
_
• Using Parseval’s Theorem, we can show that
∀
17
VARIABLES
18
SMALL GAIN THEOREM
input functions including commands,
disturbances and noise.
•
output functions
error signals
•
• Assume:
no. of outputs of
= no. inputs of
•
• Consider
• If
, then the feedback
system is input-output stable.
19
20
PROOF
• Assume
ERROR
for simplicity and show that
for all pairs of inputs
Eliminate
(or
) and assume
After truncation
21
OUTPUT
22
COMMENTS
• The theorem does not address
existence and uniqueness and stability
is investigated separately assuming
existence and uniqueness.
So the output is also bounded if the error
is bounded.
23
• The theorem provides a sufficient
condition only and can be quite
conservative.
24
EXAMPLE
•
LTI system with transfer
function
•
saturation nonlinearity
with slope
APPLY SMALL GAIN THEOREM
∀
∀
System is stable even if this condition is
violated (sufficient condition only)
∀
25
EXAMPLE
•
26
SECTOR BOUND NONLINEARITY
LTI system transfer function
•
constant gain
• Small gain theorem gives the stability
condition
, i.e.
• Closed-loop transfer function
i
belongs to sector
• Closed-loop stability for
• Small gain theorem is very conservative.
27
28
ABSOLUTE STABILITY
• What are the conditions for the stability of a
strictly stable linear subsystem
and a
nonilinearity
belonging to sector
?
is replaced by
• Aizermann’s Conjecture: If
a constant gain with the stable range of a
subset of
, then the closed-loop system
with nonlinearity
is asymptotically stable:
FALSE: gives an overly optimistic estimate
TYPES OF NONLINEARITIES
1. Single-valued and time-invariant :
unique output for each input
2. With memory
(e.g. hysteresis,
backlash): output depends on the
history of the input
3. General nonlinearities : time-varying
and possibly with memory (e.g.
hysteresis)
29
30
POPOV’S METHOD
POPOV’S THEOREM
Assume:
open-loop stable
Sector bound nonlinearity
For any initial conditions, the system output is
bounded and tends to zero as
if the
polar plot of
,
∗
arbitrarily small for
with imaginary
axis poles
for poles in the open LHP.
• Use the modified frequency response
∗
31
lies entirely to the right of the Popov line
Slope
, Intercept
, and (depending on
the nonlinearity)
:
if
,
if
:
and
:
and
32
EXAMPLE: POPOV CRITERION
CIRCLE CRITERION
2
,
• Sector bound nonlinearity
2
• For any initial conditions, the system output is
if
bounded and tends to zero as
2
2
,
2
4
4
Asymptote: vertical line with real axis intercept at 0.5
1⁄0.5 2
The closed-loop system is asymptotically stable with
general nonlinearity
in the sector 0,2
∗
2
2
4
Circle Criterion
0
Imaginary Axis
-2
33
EXAMPLE: CIRCLE CRITERION
2
2
2
,
2
2
2
Circle Criterion
0
-2
Imaginary Axis
-6
-8
34
CIRCLE CRITERION
2
4
4
Asymptote: vertical line with real axis intercept at 0.5
1⁄0.5 2
The closed-loop system is asymptotically stable with general
nonlinearity
in the sector 0,2 (same as Popov)
-4
-6
-8
-4
35
• Gives conservative results.
• Can use the Hurwitz bounds to get an
optimistic stability assessment.
• The actual stable sector will be
somewhere between the conservative
estimate of the circle criterion and the
optimistic estimate of the Hurwitz
bounds.
36