ME 233 Advanced Control II Lecture 18 Stability Analysis Using The Hyperstability Theorem 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
desired 3
•  Continuous time VS discrete time •  Direct VS indirect •  MRAS VS STR
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Parallel model 7
Series-parallel model (We will use this model throughout this lecture) 8
Plant model
where
Assume  ARMA model parameters are unknown
Define:
As the unknown parameter vector 10
Collect all measurable signals in one vector
We define
as the known regressor vector 11
Plant model
where
12 Plant ARMA Model Plant estimate (series-parallel)
where
13 Plant output estimate Plant a-posteriori estimate
Plant a-priori estimate
14 Plant a-posteriori error
error:
15 A Parameter Adaptation Algorithm PAA Parameter error update law: 16 Adaptation Dynamics a-posteriori error:                       e(k) — ŷ(k)
Parameter error update law:
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δ(k) = δ(k − 1) + F φ(k − 1)υ(k) Convergence of Adaptive Systems Adaptive systems are nonlinear We need to prove that the algorithms converge: •  Output error convergence
•  Parameter error convergence I(k) → O
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Our first goal will be to prove the asymptotic convergence of the output error: e(k) → 0 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

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)
Hyperstability  Theory Applies to both continuous time and discrete time systems . Abuse of notation: We will denote the LTI block by its • transfer function
A state space description of the LTI Block: •
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. P-class nonlinearity: (passive nonlinearities)
At > 0 Yo Where is a constant which is a function of the initial conditions
At > 0 Where Yo is a constant which is a function of the initial conditions
w = f(v) υ f(v) ≥ 0
rt
Example: Dynamic P-class block — θ(t) = F φ(t)v(t) dt w(t) = φ T (t) δ(t) φ(t) E Rn θ(0) E Rn |θ(0)| < ∞ |φ(t)| < ∞ F = FT > 0 28 Example: Dynamic P-class block w(t) = φ T (t) θ(t)  
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Input is force and output is velocity
v(t) force passive mechanical system w(t) velocity Example: Passive mechanical system Input is force and output is velocity
System Energy:
31 Example: Passive mechanical system
Differentiating energy
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Differentiating energy
power input 33 Example: Passive mechanical system
integrating power, /0

Lemma:
. The parallel combination of two P-class nonlinearities is also a P-class nonlinearity.
/0

Lemma:
. The feedback combination of two P-class nonlinearities is also a P-class nonlinearity.
Hyperstability: The above feedback system is hyperstable if there exist positive bounded constants   δ1, δ2   such that, for any state space realization of G(s), At > 0 FOR ALL  P-class nonlinearities
Asymptotic Hyperstability: The above feedback system is asymptotically hyperstable if 1. lt is hyperstable 2. For all signals   |w(t)| < ∞ (I.e. bounded output  of any P-class nonlinearity), and any state space realization of G(s),
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.
ls 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 ω’s for which s = j ω is not a pole of G(s)
ls Strictly Positive Real (SPR) iff: 1. All poles of  G(s) are asymptotically stable. 2 Re{G(jω)} = G(jω) + GT(−jω) > 0 2. for all  ω,     0 ≤ ω < ∞ Nyquist Diagram Example:
For scalar rational transfer functions
1. All poles of  G(s) are asymptotically stable. Re{G(jω)} > 0   for all ω,     0 ≤ ω < ∞ 2. 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.
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
ls Strictly Positive Real (SPR) iff there exist symmetric and positive definite matrices P and Q, such that:
SPR TF implies Possitivity G(s) = C(sI − A)− 1 B + D      be SPR Let Then there exist positive definite functions V(x) > 0 and a positive  semi-definite function   λշ(x, u) ≥ 0 Such that the input u(t) output y(t) pair  satisfies
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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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by the Kalman Yakubovich, Popov lemma.
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State space description of the LTI Block: .
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• P-class nonlinearity: (passive nonlinearities) k
Where
is a bounded  constant. 50
V 1=0 51 Example: Dynamic P-class block Õ(k) = Õ(k − 1) + F ¢(k)v(k) w(k) = φ Γ(k) δ(k) F = FT > 0 φ(k) E Rn Õ(—1) E Rn ||θ(−1)|| < ∞ ||¢(k)|| < ∞ 52 Example: Dynamic P-class block w(k) = φ T (k) δ(k)
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>0 54
Example: Dynamic P-class block Õ(k) = Õ(k − 1) + F¢(k − 1)v(k) w(k) = φ T (k − 1) δ(k) φ(k) E Rn Õ(—1) E Rn F = FT > 0 ||θ(−1)|| < ∞ ||φ( k )|| < ∞ 56 Example: Dynamic P-class block
>0 58

Lemma:
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.
-75 Hyperstability: The above feedback system is hyperstable if there exist positive bounded constants   δ1, δ2   such that, for any state space realization of G(q), ||x(k)|| < δ1 [||x(0)|| + δ2] A k > 0 FOR ALL  P-class nonlinearities
Asymptotic Hyperstability: The above feedback system is asymptotically hyperstable if 1. lt is hyperstable 2. for any state space realization of G(z),
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.
ls 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.   G(ejw) + GT(e-jw) ≥ 0 for all  ω ∈ [0, π] for which z = e j ⊙ is not a pole of G(z)
ls Strictly Positive Real (SPR) iff: 1. All poles of  G(z) are asymptotically stable. G(ejw) + GT(e-jw) > 0 2. for all  0 < ω < π Example:
For scalar rational transfer functions
All poles of  G(z) are asymptotically stable. 1. Re{G(ejw)} > 0  for all   ω, 2. Note: A necessary (but not sufficient) condition for G(z) to be SPR is that its relative degree must be 0. Matrix Inequality Interpretation of SPR The transfer function
is Strictly Positive Real (SPR) if and only if there exists   P > 0   such that
SPR state-space realization fact Theorem: If G(z) = C(zl-A) 1 B + D is SPR, then D + D T > 0 Proof: Choose  P > 0 such that
Note that
Let
Such that any input u(k) output y(k) pair  satisfies
70 Proof
Define the Lyapunov function
and the function
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V(xk+1) – V(xk) = ⇒(Axk + Buk) TP(Axk + Buk) – ⇒xk Pxk
+ ⇒ [u k Cx k + u k Du k + u k D T u k + x k C T u k ] = −λ1(xk, uk) + (Cxk + Duk)Tuk = −λ1(xk,uk) + yf,uk
From the previous slide
Summing both sides of the equation yields
73 Proof of the sufficiency part of the Asymptotic Hyperstability Theorem - Discrete Time P-class . Since the nonlinearity is P-class,
•   Since LTI block is SPR, we can use the choose  P > 0 such that
74 75 0 u(k) v(k)
where
Rearranging terms,
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From the previous slide
Therefore, the feedback system is hyperstable 77 Asymptotic  Hyperstability
• monotonic nondecreasing sequence in k • bounded above
Therefore, the feedback system is asymtotically hyperstable 78 Additional Result
We have already shown that
From this we see that
Therefore,  x(k), u(k), v(k), and  w(k) converge to 0 79
Series-Parallel ID Dynamics (review) a-posteriori error:
Parameter error update law:
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PAA: θ(k) = θ(k − 1) + F φ(k − 1)υ(k)
Strictly Positive Real
δ(k) = δ(k − 1) + F φ(k − 1)υ(k) e(k) = φ T (k − 1) δ(k)    e'l'(j)v(j) ≥ i=0 ール
P-class By the sufficiency portion of Hyperstability Theorem: |v(k)| < ∞ |e(k)| < ∞
P-class By the sufficiency portion of Asymptotic Hyperstability Theorem: |v(k)| → 0 |e(k)| → 0
P-class By the sufficiency portion of Asymptotic Hyperstability Theorem: |v(k)| → 0
a-posteriori error & PAA:
Static coupling Solution: Use the a-priori error
How to we implement the PAA? a-posteriori estimate & PAA:
Static coupling Solution: Use the a-priori error
How to we implement the PAA? Õ(k) = Õ(k − 1) − F ¢(k − 1)e(k)
Therefore,
2.
3. θ(k) = θ(k − 1) + Fφ(k − 1)e(k) 91
u(k) We have  shown  that e(k) → 0 Now we will show that φ( k e°(k) → 0 Under the following assumptions:
is anti-Schur Since
Since
Thus, we know that e(k) → 0 Remember that
We have shown that
What about the parameter error  θ(k) ? 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)