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ME 233 Advanced Control II
Lecture 18
Stability Analysis Using
The Hyperstability Theorem
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Adaptive Control
Basic Adaptive Control Principle
Controller parameters are not constant, rather,
they are adjusted in an online fashion by a
Parameter Adaptation Algorithm (PAA)
When is adaptive control used?
• Plant parameters are unknown
• Plant parameters are time varying
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Example of a system with varying parameters
• Temperature control system
flowmeter
flowcontroller
desiredflowrate
hot
cold
in
agitator
temperature
controller
measured temperature
dq
q
q
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Adaptive Control Classification
• Continuous time VS discrete time
• Direct VS indirect
• MRAS VS STR
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Model Reference Adaptive Systems
(MRAS)
u(t)u (t)c
y(t)
y (t)dreference
model
plantadjustablecontroller
parameteradaptationalgorithm
+-
e(t)
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Self-Tuning Regulators (STR)
u(t) y(t)plant
adjustablecontroller
modelidentification
controllerdesign
reference
performancespecification
plantparametersestimates
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y(k)-dq B( )-1q
A( )-1q
+
-dq B( ,t)-1q
A( ,t)-1q
-
y(k)^^
^
e(k)u(k)
Identification of a LTI system7
PAA
Parallel model
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y(k)-dq B( )-1q
A( )-1q
+
- y(k)^
e(k)
u(k)
Identification of a LTI system8
Parameter
Adaptation
Algorithm
Series-parallel model
regressor
(We will use this model throughout this lecture)
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Plant ARMA Model
Plant model
where
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Unknown plant parameters
Assume ARMA model parameters are unknown
As the unknown parameter vector
Define:
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Regressor vector
Collect all measurable signals in one vector
as the known regressor vector
We define
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Plant ARMA Model
Plant model
where
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Plant ARMA Model
Plant estimate (series-parallel)
where
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Plant output estimate
Plant a-posteriori estimate
Plant a-priori estimate
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Plant a-posteriori error
error:
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A Parameter Adaptation Algorithm
PAA
Parameter error update law:
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Adaptation Dynamics
a-posteriori error:
Parameter error update law:
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0
-
+
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Adaptation Dynamics
PAA
PAA:
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Convergence of Adaptive Systems
Adaptive systems are nonlinear
We need to prove that the algorithms converge:
• Output error convergence
• Parameter error convergence
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Output error Convergence
Our first goal will be to prove the asymptotic convergence of the
output error:
Two frequently used methods of stability analysis are:
• Stability analysis using Lyapunov’s direct method
– State space approach
• Stability analysis using the Passivity or Hyperstability
theorems
– Input/output approach
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Hyperstability
Hyperstability Theory
• Developed by V.M. Popov to analyze the stability of a class
of feedback systems (monograph published in 1973)
• Popularized by I.D. Landau for the analysis of adaptive
systems (first book published in 1979)
LTI
P-Classnonlinearity
vu
w
0
-
+
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Hyperstability Theory
Hyperstability Theory
• Applies to both continuous time and discrete time systems
• Abuse of notation: We will denote the LTI block by its
transfer function
LTI
P-Classnonlinearity
vu
w
0
-
+
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CT Hyperstability Theory
• A state space description of the LTI Block:
G(s)
P-ClassNL
v(t)u(t)
w(t)
0
-
+
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CT Hyperstability Theory
• P-class nonlinearity: (passive nonlinearities)
G(s)
P-ClassNL
v(t)u(t)
w(t)
0
-
+
Where is a constant which is a function of the
initial conditions
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CT Hyperstability Theory
Where is a constant which is a function of the
initial conditions
P-class
nonlinearity
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Example: Static P-class NL
v
(v)f
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Example: Static P-class NL
v
(v)f
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Example: Dynamic P-class block
P-NL
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Example: Dynamic P-class block
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Example: Passive mechanical system
K
M
B
v(t)
w(t) = x(t)
Input is force and output is velocity
passive
mechanical
systemforce velocity
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Example: Passive mechanical system
Input is force and output is velocity
K
M
B
v(t)
w(t) = x(t)
System Energy:
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Example: Passive mechanical system
Input is force and output is velocity
K
M
B
v(t)
w(t) = x(t)
Differentiating energy
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Example: Passive mechanical system
Input is force and output is velocity
K
M
B
v(t)
w(t) = x(t)
Differentiating energy
power input
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Example: Passive mechanical system
Input is force and output is velocity
K
M
B
v(t)
w(t) = x(t)
integrating power,
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Examples of P-class NLLemma:
• The parallel combination of two P-class
nonlinearities is also a P-class nonlinearity.
NL
+
NL1
2
w v+
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NL
-
+NL
1
2
w v
Examples of P-class NLLemma:
• The feedback combination of two P-class
nonlinearities is also a P-class nonlinearity.
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CT Hyperstability
Hyperstability: The above feedback system is hyperstable if
there exist positive bounded constants such that,
for any state space realization of G(s),
G(s)
P-ClassNL
v(t)u(t)
w(t)
0
-
+
FOR ALL P-class nonlinearities
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CT Asymptotic Hyperstability
Asymptotic Hyperstability: The above feedback system is
asymptotically hyperstable if
1. It is hyperstable
2. For all signals (I.e. bounded output of
any P-class nonlinearity), and any state space realization
of G(s),
G(s)
P-ClassNL
v(t)u(t)
w(t)
0
-
+
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CT Hyperstability Theorems
Hyperstability Theorem: The above feedback system is
hyperstable iff the transfer function G(s) of the LTI block
is Positive Real.
Asymptotical Hyperstability Theorem: The above feedback
system is asymptotically hyperstable iff the transfer
function G(s) of the LTI block is Strictly Positive Real.
G(s)
P-ClassNL
v(t)u(t)
w(t)
0
-
+
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CT Positive Real TF
Is Positive Real iff:
1. G(s) does not have any unstable poles (I.e. no Re{s} > 0).
2. Any pole of G(s) that is in the imaginary axis does not
repeat and its associated residue (I.e. the coefficient
appearing in the partial fraction expansion) is non-
negative.
3.
for all real w’s for which s = j w is not a pole of G(s)
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Strictly Positive Real (SPR) TF
Is Strictly Positive Real (SPR) iff:
1. All poles of G(s) are asymptotically stable.
2.
for all Nyquist Diagram
Real Axis
Imag
inary
Axis
0 0.2 0.4 0.6 0.8 1
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
Example:
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Strictly Positive Real (SPR) TF
For scalar rational transfer functions
1. All poles of G(s) are asymptotically stable.
2. for all
Nyquist Diagram
Real Axis
Imag
inary
Axis
0 0.2 0.4 0.6 0.8 1
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
Note:
A necessary (but not sufficient)
condition for G(s) to be SPR is that
its relative degree must be less than or
equal to 1.
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Kalman Yakubovich Popov Lemma
Is Strictly Positive Real (SPR) if and only if
• there exist a symmetric and positive definite matrix P,• matrices L and K,• and a constant ε > 0 such that
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Kalman Yakubovich Popov Lemma
Is Strictly Positive Real (SPR) iff there exist symmetric and
positive definite matrices P and Q, such that:
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SPR TF implies Possitivity
Let be SPR
Then there exist positive definite functions
and a positive semi-definite function
Such that the input u(t) output y(t) pair satisfies
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SPR TF implies Passivity
Proof: We consider a strictly causal transfer function
which is SPR, with state space realization
By the Kalman Yakubovich, Popov lemma, there exist
symmetric and positive definite matrices P and Q, such that
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SPR TF implies Passivity
Proof: Define the PD function
and compute:
by the Kalman Yakubovich, Popov lemma.
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SPR TF implies Passivity
Proof: Thus, since
Define the PD function and integrate
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DT Hyperstability Theory
• State space description of the LTI Block:
G(q)
NL
v(k)u(k)
w(k)
0
-
+
P-Class
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DT Hyperstability Theory
• P-class nonlinearity: (passive nonlinearities)
Where is a bounded constant.
G(q)
NL
v(k)u(k)
w(k)
0
-
+
P-Class
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Example: Static nonlinearity:
v
(v)f
P-block
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Example: Dynamic P-class block
P-block
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53
Example: Dynamic P-class block
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54
Example: Dynamic P-class block
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55
Example: Dynamic P-class block
P-block
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56
Example: Dynamic P-class block
P-block
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57
Example: Dynamic P-class block
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58
Example: Dynamic P-class block
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59
Example: Dynamic P-class block
P-block
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60
Examples of P-class NLLemma:
• The parallel combination of two P-class
nonlinearities is also a P-class nonlinearity.
• The feedback combination of two P-class
nonlinearities is also a P-class nonlinearity.
NL
-
+NL
1
2
w v
NL
+
NL1
2
w v+
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61
DT Hyperstability
Hyperstability: The above feedback system is hyperstable if
there exist positive bounded constants such that,
for any state space realization of G(q),
FOR ALL P-class nonlinearities
G(q)
NL
v(k)u(k)
w(k)
0
-
+
P-Class
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62
DT Asymptotic Hyperstability
Asymptotic Hyperstability: The above feedback system is
asymptotically hyperstable if
1. It is hyperstable
2. for any state space realization of G(z),
G(q)
NL
v(k)u(k)
w(k)
0
-
+
P-Class
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63
DT Hyperstability Theorems
Hyperstability Theorem: The above feedback system is
hyperstable iff the transfer function G(z) of the LTI block
is Positive Real.
Asymptotical Hyperstability Theorem: The above feedback
system is asymptotically hyperstable iff the transfer
function G(z) of the LTI block is Strictly Positive Real.
G(q)
NL
v(k)u(k)
w(k)
0
-
+
P-Class
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64
Positive Real TF
Is Positive Real iff:
1. G(z) does not have any unstable poles (I.e. no |z| > 1).
2. Any pole of G(z) that is in the unit circle does not repeat
and its associated residue (i.e. the coefficient appearing in
the partial fraction expansion) is non-negative.
3.
for all for which z = e j w is not a pole of G(z)
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65
Strictly Positive Real (SPR) TF
Is Strictly Positive Real (SPR) iff:
1. All poles of G(z) are asymptotically stable.
2.
for all
Example:
Nyquist Diagram
Real Axis
Ima
gin
ary
Ax
is
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
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66
Strictly Positive Real (SPR) TF
For scalar rational transfer functions
1. All poles of G(z) are asymptotically stable.
2. for all
Note:
A necessary (but not sufficient)
condition for G(z) to be SPR is that
its relative degree must be 0.
Nyquist Diagram
Real Axis
Ima
gin
ary
Ax
is
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
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67
Matrix Inequality Interpretation of SPR
The transfer function
is Strictly Positive Real (SPR) if and only if
there exists such that
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SPR state-space realization fact
Theorem: If G(z) = C(zI-A)-1B + D is SPR, then
Proof: Choose such that
Note that
68
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SPR TF is P-classLet be SPR
Then there exist positive definite functions
Such that any input u(k) output y(k) pair satisfies
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Shorthand notation70
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ProofLet be SPR
Choose such that
Define the Lyapunov function
and the function
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Proof72
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Proof73
From the previous slide
Summing both sides of the equation yields
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Proof of the sufficiency part of the Asymptotic
Hyperstability Theorem - Discrete Time
• Since the nonlinearity is P-class,
74
G(q)
NL
v(k)u(k)
w(k)
0
-
+
P-Class
• Since LTI block is SPR, we can use the choose
such that
SPR
P-class
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Hyperstability
75
G(q)
NL
v(k)u(k)
w(k)
0
-
+
P-Class
SPR
From the previous proof (SPR TF is P-class), we have
where
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76
G(q)
NL
v(k)u(k)
w(k)
0
-
+
P-Class
From the P-class nonlinearity:
SPR
Rearranging terms,
Hyperstability
Therefore,
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77
G(q)
NL
v(k)u(k)
w(k)
0
-
+
P-Class
SPR
From the previous slide
Hyperstability
Therefore, the feedback system is hyperstable
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78
G(q)
NL
v(k)u(k)
w(k)
0
-
+
P-Class
SPR
Asymptotic Hyperstability
• monotonic nondecreasing sequence in k
• bounded above
Therefore, the feedback system is asymtotically hyperstable
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79
G(q)
NL
v(k)u(k)
w(k)
0
-
+
P-Class
SPR
Additional Result
Therefore, x(k), u(k), v(k), and w(k) converge to 0
We have already shown that
From this we see that
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Stability analysis of Series-parallel ID80
y(k)-dq B( )-1q
A( )-1q
+
- y(k)^
e(k)
u(k)
Parameter
Adaptation
Algorithm
regressor
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Series-Parallel ID Dynamics
(review)a-posteriori error:
Parameter error update law:
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0
-
+
82
Series-Parallel ID Dynamics
(review)
PAA
PAA:
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Stability analysis of Series-parallel ID
0
-
+
PAA
Strictly Positive Real
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Stability analysis of Series-parallel ID
0
-
+
PAA
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Stability analysis of Series-parallel ID
0
-
+
PAA
SPR
P-class
By the sufficiency portion of Hyperstability Theorem:
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Stability analysis of Series-parallel ID
0
-
+
PAA
SPR
P-class
By the sufficiency portion of Asymptotic Hyperstability
Theorem:
Q.E.D.
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Stability analysis of Series-parallel ID
0
-
+
PAA
SPR
P-class
By the sufficiency portion of Asymptotic Hyperstability
Theorem:
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How to we implement the PAA?
a-posteriori error & PAA:
Solution: Use the a-priori error
Static
coupling
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89
How to we implement the PAA?
a-posteriori estimate & PAA:
Solution: Use the a-priori error
Static
coupling
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90
How to we implement the PAA?
Multiply by
Therefore,
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How we implement the PAA
1.
2.
3.
91
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92
Stability analysis of Series-parallel ID
Under the following assumptions:
is anti-Schur
We have shown that
Now we will show that
Since
Since
y(k)-dq B( )-1q
A( )-1q
+
- y(k)^
e(k)
u(k)
Parameter
Adaptation
Algorithm
regressor
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93
Stability analysis of Series-parallel ID
Thus, we know that
Remember that
y(k)-dq B( )-1q
A( )-1q
+
- y(k)^
e(k)
u(k)
Parameter
Adaptation
Algorithm
regressor
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94
Stability analysis of Series-parallel ID
What about the parameter error ?
We have shown that
since
However, this does not imply that the parameter error goes to zero
We need to impose another condition on u(k) to guarantee that the parameter error goes to zero. (persistence of excitation)
y(k)-dq B( )-1q
A( )-1q
+
- y(k)^
e(k)
u(k)
Parameter
Adaptation
Algorithm
regressor