OUTLINE
INPUT-OUTPUT
STABILITY: DISCRETETIME SYSTEMS
• Gain
• Small gain theorem
• Circle criterion
M. SAMI FADALI
2
1
EBME DEPT., UNR
GAIN
NORMS
• Nonlinearity (assume zero at the origin)
• Norm of
4
3
• Define the norm
INPUT-OUTPUT STABILITY
EXAMPLE
• For any bounded input sequence
, the output sequence is also
bounded, i.e.
• Find the gain of the
saturation nonlinearity
• Input-output stable system must have
an upper bound on the ratio of the norm
of its output to that of its input.
6
5
• Upper bound = gain of nonlinear system
EXAMPLE
SOLUTION
• Find the gain of a causal linear system with
impulse response
The output is bounded by
Solution
System response: input
The norm of the output is
8
7
The gain of the system is the constant
NORM INEQUALITIES
THEOREM:
NORM
Proof
Parseval’s Identity (stable system, finite
energy for output)
9
10
Gain
PROOF
EXAMPLE
Use Parseval’s identity
Verify that
can be used to find the gain of the linear
system with transfer function
12
11
Using Parseval’s identity
SOLUTION
CLOSED-LOOP SYSTEM
Impulse response
1
Gain
Frequency response
2
13
Maximum magnitude (min denominator)
14
Block diagram of a nonlinear closedloop system.
SMALL GAIN THEOREM
PROOF
The closed-loop system is input-output
stable if the product of the gains of all the
systems in the loop is less than unity, i.e.
Output
N1 . N2 . Solve for
y
Can generalize
to
which is finite if
16
e
Norm inequalities
15
r
Error
EXAMPLE
SOLUTION
Simulate a feedback loop with linear
subsystem transfer function (T=0.01 s)
1.z+.50
0.2
z2 +0.5z+0.8
Saturation
Gain
in series with an amplifier of variable gain and
a symmetric saturation nonlinearity with
slope and saturation level unity. Investigate
the stability of the system and discuss your
results referring to the small gain theorem for
two amplifier gains (a) 1, and (b) 0.2.
Scope
Discrete
Transfer Fcn
Maximum magnitude of frequency response :
obtained using the command
>> [mag, ph]=bode(g)
18
17
Maximum magnitude  5.126.
INITIAL CONDITIONS
[1,0] AND 0.2 GAIN
INITIAL CONDITIONS
[1,0] AND UNITY GAIN.
0.6
4
0.4
0.2
2
0
0
-0.2
-0.4
0
-2
0.5
1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
1.5
Response of the closed-loop system
with initial conditions [1,0] and 0.2 gain
(stable but small gain theorem fails).
19
Response of the closed-loop system with
initial conditions [1,0] and unity gain
(Unstable)
20
-4
0
0.1
SECTOR BOUND NONLINEARITY
SECTOR BOUND NONLINEARITY
A nonlinearity
belongs to the sector
if it satisfies the condition
ABSOLUTE STABILITY
22
21
disk bounded by a circle with its
center on the negative real axis with
and
.
endpoints
THE CIRCLE CRITERION
Absolutely stable with stable
belonging to sector
if
A nonlinear feedback system is
absolutely stable with respect to the
if the origin of the state
sector
space
is globally asymptotically
stable for all nonlinearities belonging to
.
the sector
and nonlinearity
and the Nyquist plot of
enter or encircle the disc
.
does not
& Nyquist plot is the right of
&Nyquist plot of
23
& Nyquist plot of
or encircle the disc
.
inside
to the left of
does not enter
24
& Nyquist plot of
(a)
28
(d)
27
(c)
26
25
(b)
EXAMPLE
(e)
For the furnace and actuator of Example 410, determine the stability sector for the
system and compare it to the stable range
for a linear gain block.
Solution
The transfer function of the actuator and
furnace is
.
.
NYQUIST PLOT
JURY CRITERION
Nyquist Diagram
• Circle criterion gives conservative
results: the actual stable sector may be
wider than our estimates indicate.
4
Imaginary Axis
2
-2
System: l
Real: -0.935
Imag: 1.4
Frequency (rad/sec): -1.8
• Jury criterion gives the stable range
(0.099559, 3.47398). The lower bound is
approximately the same as predicted by
the circle criterion but the upper bound is
more optimistic.
-4
0
2
4
Real Axis
6
8
10
• Plot lies inside a circle of radius
which
corresponds to a sector (0.1, 0.1).
• Plot is to the right of the line z= 0.935 which
corresponds to a sector (0, 1.05).
• Cannot conclude that the system is stable in
the sector (0.1 1.05), i.e. cannot combine
stability sectors.
31
-6
-2
32
6
0
30
.
29
.
.
SIMULATION RESULTS
2.5
2
1.5
1
0.5
0
0
1
2
3
4
5
6
7
8
9
10
33
• Nonlinear system: saturation
nonlinearity with a linear gain of
unity and saturation levels (1,1).
• System is stable outside range
predicted by the circle criterion.