EE 264 SIST, ShanghaiTech Adaptive Pole Placement Control II YW 12-1
EE 264 SIST, ShanghaiTech
Adaptive Pole Placement Control II
YW 12-1
Contents
Adaptive Observer
APPC for general SISO plant via ss approach
Adaptive Linear Quadratic Control
Modification for Solving Stability Issue
Adaptive Pole Placement Control II 12-2
Adaptive Luenberger observer
Consider the LTI SISO plant
x = Ax+Bu, x(0) = x0
y = C>x
In the case A,B,C are known, the Luenberger observer is in the
form of
˙x = Ax+Bu+K(y − y), x(0) = x0
y = C>x
where K is chosen such that A−KC> is Hurwitz, guarantees
that x→ x exponentially fast. The existence of K is ensured by
the observability of pair (A,C>)Adaptive Pole Placement Control II 12-3
Adaptive Luenberger observer
Consider the LTI SISO plant
x = Ax+Bu, x(0) = x0
y = C>x
In the case A,B,C are known, the Luenberger observer is in the
form of
˙x = Ax+Bu+K(y − y), x(0) = x0
y = C>x
where K is chosen such that A−KC> is Hurwitz, guarantees
that x→ x exponentially fast. The existence of K is ensured by
the observability of pair (A,C>)Adaptive Pole Placement Control II 12-4
Adaptive Luenberger observer
Idea: (A,B,C)→ G(s)→ G(s)→ (A, B, C)
mapping of the 2n estimated parameters of G(s) to the n2 + 2n
parameters of A,B,C is not uniqueunless (A,B,C) is in a
observer canonical form, i.e., the plant is represented as
xo =[−ap |
In−10
]xo + bρu
y = [1, 0, . . . , 0]xo
where ap = [an−1, an−2, . . . , a0]> and bp = [bn−1, bn−2, . . . , b0]>
are the coefficients of the transfer function
G(s) = y(s)u(s) = bn−1s
n−1 + bn−2sn−2 + · · ·+ b0s
sn + an−1sn−1 + an−2sn−2 + · · ·+ a0s
Adaptive Pole Placement Control II 12-5
Adaptive Luenberger observer
Idea: (A,B,C)→ G(s)→ G(s)→ (A, B, C)
mapping of the 2n estimated parameters of G(s) to the n2 + 2n
parameters of A,B,C is not uniqueunless (A,B,C) is in a
observer canonical form, i.e., the plant is represented as
xo =[−ap |
In−10
]xo + bρu
y = [1, 0, . . . , 0]xo
where ap = [an−1, an−2, . . . , a0]> and bp = [bn−1, bn−2, . . . , b0]>
are the coefficients of the transfer function
G(s) = y(s)u(s) = bn−1s
n−1 + bn−2sn−2 + · · ·+ b0s
sn + an−1sn−1 + an−2sn−2 + · · ·+ a0s
Adaptive Pole Placement Control II 12-6
Adaptive Luenberger observer
Idea: (A,B,C)→ G(s)→ G(s)→ (A, B, C)
mapping of the 2n estimated parameters of G(s) to the n2 + 2n
parameters of A,B,C is not uniqueunless (A,B,C) is in a
observer canonical form, i.e., the plant is represented as
xo =[−ap |
In−10
]xo + bρu
y = [1, 0, . . . , 0]xo
where ap = [an−1, an−2, . . . , a0]> and bp = [bn−1, bn−2, . . . , b0]>
are the coefficients of the transfer function
G(s) = y(s)u(s) = bn−1s
n−1 + bn−2sn−2 + · · ·+ b0s
sn + an−1sn−1 + an−2sn−2 + · · ·+ a0s
Adaptive Pole Placement Control II 12-7
Adaptive Luenberger observer
Then the adaptive observer is given by
˙x = A(t)x+ bp(t)u+K(t)(y − y), x(0) = x0
y = [1, 0, . . . , 0]x
where x is the estimate of xo and
A(t) =[−ap(t) |
In−10
], K(t) = a∗ − ap(t)
ap(t) and bp(t) are the estimates of the vectors ap and bp,
respectively. a∗ ∈ Rn is chosen so that
A∗ =[−a∗|In−1
0
]is a Hurwitz matrix.Adaptive Pole Placement Control II 12-8
Adaptive Luenberger observer
Then the adaptive observer is given by
˙x = A(t)x+ bp(t)u+K(t)(y − y), x(0) = x0
y = [1, 0, . . . , 0]x
where x is the estimate of xo and
A(t) =[−ap(t) |
In−10
], K(t) = a∗ − ap(t)
ap(t) and bp(t) are the estimates of the vectors ap and bp,
respectively. a∗ ∈ Rn is chosen so that
A∗ =[−a∗|In−1
0
]is a Hurwitz matrix.Adaptive Pole Placement Control II 12-9
Adaptive Luenberger observer
Theorem: The adaptive Luenberger observer with gradient-based
algorithm guarantees the following properties:
(i) If choose u ∈ L∞ and A is a stable matrix, all signals are
bounded.
(ii) Furthermore, if choose u is sufficiently rich of order 2n, then
the state observation error |x− xo| and the parameter estimation
error θ converge to zero exponentially fast.
Brief Proof. (i) The observer equation may be written as
˙x = A∗x+ bp(t)u+(A(t)−A∗
)xo
(ii) The state observation error x = x− xo satisfies
˙x = A∗x+ bpu− apyAdaptive Pole Placement Control II 12-10
Adaptive Luenberger observer
Theorem: The adaptive Luenberger observer with gradient-based
algorithm guarantees the following properties:
(i) If choose u ∈ L∞ and A is a stable matrix, all signals are
bounded.
(ii) Furthermore, if choose u is sufficiently rich of order 2n, then
the state observation error |x− xo| and the parameter estimation
error θ converge to zero exponentially fast.
Brief Proof. (i) The observer equation may be written as
˙x = A∗x+ bp(t)u+(A(t)−A∗
)xo
(ii) The state observation error x = x− xo satisfies
˙x = A∗x+ bpu− apyAdaptive Pole Placement Control II 12-11
Contents
Adaptive Observer
APPC for general SISO plant via ss approach
Adaptive Linear Quadratic Control
Modification for Solving Stability Issue
Adaptive Pole Placement Control II 12-12
APPC via state space approach
The plant
yp = Gp(s)up, Gp(s) = Zp(s)Rp(s)
where Gp(s) is proper and Rp(s) is a monic polynomial.
The control objective is to choose up so that the closed-loop
poles are assigned to those of a monic Hurwitz polynomial A∗(s)
and yp → ym
and Assumptions are the same as before
P1. Rp(s) is a monic polynomial whose degree n is known.
P2. Zp(s), Rp(s) are coprime and degree (Zp) < n.
P3. Qm(s)ym = 0 and Qm(s) is assumed to be coprime with Zp(s).Adaptive Pole Placement Control II 12-13
APPC via state space approach
Step 1. PPC for known parameters We start by considering
the expression
e1 = Zp(s)Rp(s)
up − ym
for the tracking error. Filtering each side of with Qm(s)Q1(s) , where
Q1(s) is an arbitrary monic Hurwitz polynomial of degree q, and
using Qm(s)ym = 0, we obtain
e1 = ZpQ1RpQm
up, up = QmQ1
up
In this way, have converted the tracking problem into the
regulation problem of choosing up to regulate e1 to zero.Adaptive Pole Placement Control II 12-14
APPC via state space approach
Step 1. PPC for known parameters We start by considering
the expression
e1 = Zp(s)Rp(s)
up − ym
for the tracking error. Filtering each side of with Qm(s)Q1(s) , where
Q1(s) is an arbitrary monic Hurwitz polynomial of degree q, and
using Qm(s)ym = 0, we obtain
e1 = ZpQ1RpQm
up, up = QmQ1
up
In this way, have converted the tracking problem into the
regulation problem of choosing up to regulate e1 to zero.Adaptive Pole Placement Control II 12-15
APPC via state space approach
Step 1. PPC for known parameters We start by considering
the expression
e1 = Zp(s)Rp(s)
up − ym
for the tracking error. Filtering each side of with Qm(s)Q1(s) , where
Q1(s) is an arbitrary monic Hurwitz polynomial of degree q, and
using Qm(s)ym = 0, we obtain
e1 = ZpQ1RpQm
up, up = QmQ1
up
In this way, have converted the tracking problem into the
regulation problem of choosing up to regulate e1 to zero.Adaptive Pole Placement Control II 12-16
APPC via state space approach
Let (A,B,C) be a state-space realization of error equation in the
observer canonical form, i.e.
e = Ae+Bup, e1 = C>e
where A =[−θ∗1 |
In+q−10
], B = θ∗2, C = [1, 0, . . . , 0]> and
θ∗1, θ∗2 ∈ Rn+q are the coefficient vectors of the polynomials
Rp(s)Qm(s)− sn+q and Zp(s)Q1(s), respectively.
Note, because RpQm, Zp are coprime, any possible zero-pole
cancellation in (6.51) between Q1(s) and Rp(s)Qm(s) will occur
in R[s] < 0 due to Q1(s) being Hurwitz, which implies that
(A,B) is always stabilizable.Adaptive Pole Placement Control II 12-17
APPC via state space approach
Let (A,B,C) be a state-space realization of error equation in the
observer canonical form, i.e.
e = Ae+Bup, e1 = C>e
where A =[−θ∗1 |
In+q−10
], B = θ∗2, C = [1, 0, . . . , 0]> and
θ∗1, θ∗2 ∈ Rn+q are the coefficient vectors of the polynomials
Rp(s)Qm(s)− sn+q and Zp(s)Q1(s), respectively.
Note, because RpQm, Zp are coprime, any possible zero-pole
cancellation in (6.51) between Q1(s) and Rp(s)Qm(s) will occur
in R[s] < 0 due to Q1(s) being Hurwitz, which implies that
(A,B) is always stabilizable.Adaptive Pole Placement Control II 12-18
APPC via state space approach
Consider the feedback control law
up = −Kce, up = Q1Qm
up
where e is the state of the full-order Luenberger observer
˙e = Ae+Bup −Ko
(C>e− e1
)and Kc and Ko are solutions to the polynomial equations
det (sI −A+BKc) = A∗c(s)
det(sI −A+KoC
>)
= A∗o(s)
where A∗c and A∗o are given monic Hurwitz polynomials of degree
n+ q.Adaptive Pole Placement Control II 12-19
APPC via state space approach
The design of A∗c and A∗o polynomials
the roots of A∗c(s) = 0 represent the desired pole locations of
the transfer function of the closed-loop plant
the roots of A∗o(s) are equal to the poles of the observer
dynamics
The existence of Kc and Ko
The existence of Kc follows from the controllability of (A,B)
The existence of Ko follows from the observability of (A,C).
Adaptive Pole Placement Control II 12-20
APPC via state space approach
The design of A∗c and A∗o polynomials
the roots of A∗c(s) = 0 represent the desired pole locations of
the transfer function of the closed-loop plant
the roots of A∗o(s) are equal to the poles of the observer
dynamics
The existence of Kc and Ko
The existence of Kc follows from the controllability of (A,B)
The existence of Ko follows from the observability of (A,C).
Adaptive Pole Placement Control II 12-21
APPC via state space approach
Theorem: If Assumptions P1-P3 hold, Consider the system
yp = Gp(s)up, Gp(s) = Zp(s)Rp(s)
where Gp(s) is proper and Rp(s) is a monic polynomial. The
control law
up = Q1Qm
up, up = −Kce
˙e = Ae+Bup −Ko
(C>e− e1
)guarantees that all signals in the closed-loop plant are bounded
and e1 converges to zero exponentially fast.
Adaptive Pole Placement Control II 12-22
APPC via state space approach
Step 2. Estimation of plant parameters The adaptive law for
estimating the plant parameters is given by, for instance
θp = Γεφ
ε = z − θ>φm2s
, m2s = 1 + φ>φ
z, φ and θp =[θ>a , θ
>b
]>are as defined same as ones given in
polynomial approach.
Adaptive Pole Placement Control II 12-23
APPC via state space approach
Step 3. Adaptive control law Using the CE approach, the
adaptive control law is given by
˙e = Ae+ Bup − Ko
(c>e− e1
)up = −Kce, up = Q1(s)
Qm(s) up
where
A(t) =[−θ1(t) | In+q−1
0
], B(t) = θ2(t)
θ1(t) and θ2(t) are the coefficient vectors of the polynomials
Rp(s, t)Qm(s)− sn+q =(sn + θ>a (t)αn−1(s)
)Qm(s)− sn+q
Zp(s, t)Q1(s) = θ>b (t)αn−1(s)Q1(s)
respectively.Adaptive Pole Placement Control II 12-24
APPC via state space approach
and
Ko(t) = α∗0 − θ1(t)
α∗0 is the coefficient vector of A∗0(s); and Kc(t) is calculated at
each time t by solving the polynomial equation
det(sI − A+ BKc
)= A∗c(s)
Note, the stabilizability problem arises in adaptive control law,
where for the calculation of Kc to be possible the pair
(A(t), B(t)) has to be controllable at each time t and for
implementation purposes strongly controllable.Adaptive Pole Placement Control II 12-25
APPC via state space approach
and
Ko(t) = α∗0 − θ1(t)
α∗0 is the coefficient vector of A∗0(s); and Kc(t) is calculated at
each time t by solving the polynomial equation
det(sI − A+ BKc
)= A∗c(s)
Note, the stabilizability problem arises in adaptive control law,
where for the calculation of Kc to be possible the pair
(A(t), B(t)) has to be controllable at each time t and for
implementation purposes strongly controllable.Adaptive Pole Placement Control II 12-26
APPC via state space approach
Theorem Assume that the polynomials Zp, RpQm are strongly
coprime at each time t. Then all the signals in the closed-loop
APPC scheme via state-space approach are uniformly bounded,
and the tracking error e1 → 0 asymptotically .
Example We consider the same scalar plant
yp = b
s+ aup
where a and b are unknown constants with b 6= 0. The input up is
to be chosen so that the poles of the closed-loop plant are placed
at the roots of A∗(s) = (s+ 1)2 = 0, and yp tracks the reference
signal ym = 1.Adaptive Pole Placement Control II 12-27
APPC via state space approach
Theorem Assume that the polynomials Zp, RpQm are strongly
coprime at each time t. Then all the signals in the closed-loop
APPC scheme via state-space approach are uniformly bounded,
and the tracking error e1 → 0 asymptotically .
Example We consider the same scalar plant
yp = b
s+ aup
where a and b are unknown constants with b 6= 0. The input up is
to be chosen so that the poles of the closed-loop plant are placed
at the roots of A∗(s) = (s+ 1)2 = 0, and yp tracks the reference
signal ym = 1.Adaptive Pole Placement Control II 12-28
Contents
Adaptive Observer
APPC for general SISO plant via ss approach
Adaptive Linear Quadratic Control
Modification for Solving Stability Issue
Adaptive Pole Placement Control II 12-29
LQC
Idea: using an optimization technique to achieve our tracking or
regulation objective by minimizing a certain cost function that
reflects the performance of the closed-loop system.
Unify the regulation or tracking problem of the system as
e = Ae+Bup
e1 = C>e
where up = Q1(s)Qm(s) up, and up is to be chosen so that e ∈ L∞ and
e1 → 0 as t→∞.
Adaptive Pole Placement Control II 12-30
LQC
Idea: using an optimization technique to achieve our tracking or
regulation objective by minimizing a certain cost function that
reflects the performance of the closed-loop system.
Unify the regulation or tracking problem of the system as
e = Ae+Bup
e1 = C>e
where up = Q1(s)Qm(s) up, and up is to be chosen so that e ∈ L∞ and
e1 → 0 as t→∞.
Adaptive Pole Placement Control II 12-31
LQC
The desired up to meet this objective is chosen as the one that
minimizes the quadratic cost
J =∫ ∞
0
(e2
1(t) + λu2p(t)
)dt
where λ > 0, a weighting coefficient to be designed, penalizes the
level of the control input signal. The optimum control input upthat minimizes J is
up = −Kce, Kc = λ−1B>P
where P = P> satisfies the Riccati Equation
A>P + PA− PBλ−1B>P + CC> = 0Adaptive Pole Placement Control II 12-32
LQC
The desired up to meet this objective is chosen as the one that
minimizes the quadratic cost
J =∫ ∞
0
(e2
1(t) + λu2p(t)
)dt
where λ > 0, a weighting coefficient to be designed, penalizes the
level of the control input signal. The optimum control input upthat minimizes J is
up = −Kce, Kc = λ−1B>P
where P = P> satisfies the Riccati Equation
A>P + PA− PBλ−1B>P + CC> = 0Adaptive Pole Placement Control II 12-33
LQC
up = −Kce, Kc = λ−1B>P
where P = P> satisfies the Riccati Equation
A>P + PA− PBλ−1B>P + CC> = 0
Remark:
1. With λ > 0 and finite, the LQC guarantees A−BKc is
Hurwitz, e, e1 → 0 exponentially fast, and up ∈ L∞2. The location of the eigenvalues of A−BKc depends on the
particular choice of λ. In general, there is NO guarantee that
one can find a λ so that the closed-loop poles are equal to the
roots of the desired polynomial A∗(s)Adaptive Pole Placement Control II 12-34
LQC
Consider the state e may not be available for measurement, we
need to use
up = −Kce, Kc = λ−1B>P
where e is the state of the observer equation
˙e = Ae+Bup −Ko
(C>e− e1
)where Ko are solutions to the polynomial equations
det(sI −A+KoC
>)
= A∗o(s)
where A∗o is given monic Hurwitz polynomials of degree n+ q.
Adaptive Pole Placement Control II 12-35
LQC
Theorem The LQ control law
up = Q1(s)Qm(s) up
with
up = −Kce, Kc = λ−1B>P
guarantees that all the eigenvalues of A−BKc are in <[s] < 0,
all signals in the closed-loop plant are bounded, and e1(t)→ 0 as
t→∞ exponentially fast.
Adaptive Pole Placement Control II 12-36
ALQC
Use the CE approach to form the adaptive control law
˙e = Ae+ Bup − Ko
(c>e− e1
)up = Q1(s)
Qm(s) up, up = −Kce, Kc = λ−1B>P
where A, B, Ko are generated using the gradient-based adaptive
law (see parameter estimation schemes) and P (t) is calculated by
solving the Riccati equation
A>(t)P (t) + P (t)A(t)− P (t)B(t)λ−1B>(t)P (t) + CC> = 0
Adaptive Pole Placement Control II 12-37
ALQC
Theorem: Assume that the polynomials Rp(s, t)Qm(s) and
Zp(s, t) are strongly coprime at each time t. Then the ALQC
scheme guarantees that all signals in the closed-loop plant are
bounded and the tracking error e1 converges to zero as t→∞.Remark 1: the ALQC scheme depends on the solvability of the
algebraic Riccati equation. For the solution P (t) = P>(t) > 0 to exist,
the pair (A, B) has to be stabilizable at each time t. A sufficient
condition for (A, B) to be stabilizable is that the polynomials
Rp(s, t)Qm(s) and Zp(s, t) are coprime at each time t.
Remark 2: For P (t) to be uniformly bounded, however, we will require
Rp(s, t)Qm(s) and Zp(s, t) to be strongly coprime at each time t.Adaptive Pole Placement Control II 12-38
ALQC
Theorem: Assume that the polynomials Rp(s, t)Qm(s) and
Zp(s, t) are strongly coprime at each time t. Then the ALQC
scheme guarantees that all signals in the closed-loop plant are
bounded and the tracking error e1 converges to zero as t→∞.Remark 1: the ALQC scheme depends on the solvability of the
algebraic Riccati equation. For the solution P (t) = P>(t) > 0 to exist,
the pair (A, B) has to be stabilizable at each time t. A sufficient
condition for (A, B) to be stabilizable is that the polynomials
Rp(s, t)Qm(s) and Zp(s, t) are coprime at each time t.
Remark 2: For P (t) to be uniformly bounded, however, we will require
Rp(s, t)Qm(s) and Zp(s, t) to be strongly coprime at each time t.Adaptive Pole Placement Control II 12-39
ALQC
Theorem: Assume that the polynomials Rp(s, t)Qm(s) and
Zp(s, t) are strongly coprime at each time t. Then the ALQC
scheme guarantees that all signals in the closed-loop plant are
bounded and the tracking error e1 converges to zero as t→∞.Remark 1: the ALQC scheme depends on the solvability of the
algebraic Riccati equation. For the solution P (t) = P>(t) > 0 to exist,
the pair (A, B) has to be stabilizable at each time t. A sufficient
condition for (A, B) to be stabilizable is that the polynomials
Rp(s, t)Qm(s) and Zp(s, t) are coprime at each time t.
Remark 2: For P (t) to be uniformly bounded, however, we will require
Rp(s, t)Qm(s) and Zp(s, t) to be strongly coprime at each time t.Adaptive Pole Placement Control II 12-40
Contents
Adaptive Observer
APPC for general SISO plant via ss approach
Adaptive Linear Quadratic Control
Modification for Solving Stability Issue
Adaptive Pole Placement Control II 12-41
Loss of stabilizability
Recall scalar example
y = −ay + bu
with indirect APPC law
u = −ky, k = a+ am
b
The system loss the stabilization, when b→ 0. The solution is a
projection operation
˙b =
γ2yu if |b| > b0 or if |b| = b0 and sgn(b)yu ≥ 0
0 otherwise
Adaptive Pole Placement Control II 12-42
Loss of stabilizability
Recall scalar example
y = −ay + bu
with indirect APPC law
u = −ky, k = a+ am
b
The system loss the stabilization, when b→ 0. The solution is a
projection operation
˙b =
γ2yu if |b| > b0 or if |b| = b0 and sgn(b)yu ≥ 0
0 otherwise
Adaptive Pole Placement Control II 12-43
Modification for ensure stabilizability
a) Parameter Projection Methods: exists a convex subset C0 of the
parameter space that is assumed to have the following properties:
(i) The unknown plant parameter vector θ∗p ∈ C0.(ii) Every member θp of C0 has a corresponding level ofstabilizability greater than ε∗ for some known constant ε∗ > 0.Advantage: The projection based on gradient method is simple and does
not alter the usual properties of the adaptive law that are used in the
stability analysis.
Disadvantage: This approach relies on the rather strong assumption that
the set C0 is known. No procedure has been proposed for constructing
such a set C0 for a general class of plants.
Adaptive Pole Placement Control II 12-44
Modification for ensure stabilizability
a) Parameter Projection Methods: exists a convex subset C0 of the
parameter space that is assumed to have the following properties:
(i) The unknown plant parameter vector θ∗p ∈ C0.(ii) Every member θp of C0 has a corresponding level ofstabilizability greater than ε∗ for some known constant ε∗ > 0.Advantage: The projection based on gradient method is simple and does
not alter the usual properties of the adaptive law that are used in the
stability analysis.
Disadvantage: This approach relies on the rather strong assumption that
the set C0 is known. No procedure has been proposed for constructing
such a set C0 for a general class of plants.
Adaptive Pole Placement Control II 12-45
Modification for ensure stabilizability
a) Parameter Projection Methods: exists a convex subset C0 of the
parameter space that is assumed to have the following properties:
(i) The unknown plant parameter vector θ∗p ∈ C0.(ii) Every member θp of C0 has a corresponding level ofstabilizability greater than ε∗ for some known constant ε∗ > 0.Advantage: The projection based on gradient method is simple and does
not alter the usual properties of the adaptive law that are used in the
stability analysis.
Disadvantage: This approach relies on the rather strong assumption that
the set C0 is known. No procedure has been proposed for constructing
such a set C0 for a general class of plants.
Adaptive Pole Placement Control II 12-46
Projection operation
The gradient algorithm with projection is computed by applying
the gradient method to the following minimization problem with
constraints:
minimize J(θ)
s.t. θ ∈ S
where S is a convex subset of Rn with smooth boundary almost
everywhere. Assume that S is given by
S = {θ ∈ Rn | g(θ) ≤ 0}
where g : Rn → R is a smooth function.Adaptive Pole Placement Control II 12-47
Projection operation
The adaptive laws based on the gradient method can be modified
to guarantee that θ ∈ S by solving the constrained optimization
problem given above to obtain
θ = Pr(−Γ∇J) =
−Γ∇J if θ ∈ S0 or θ ∈ δ(S)
and − (Γ∇J)>∇g ≤ 0
−Γ∇J + Γ ∇g∇g>
∇g>Γ∇gΓ∇J otherwise
where δ(S) = {θ ∈ Rn | g(θ) = 0} and S0 = {θ ∈ Rn | g(θ) < 0}
denote the boundary and the interior, respectively, of S. E.g.
gradient algorithm based on the instantaneous cost function:
∇J = −εφAdaptive Pole Placement Control II 12-48
Modification for ensure stabilizability
Theorem The gradient adaptive laws and the LS adaptive laws
with the projection modifications retain all the properties that are
established in the absence of projection and in addition guarantee
that θ(t) ∈ S∀t ≥ 0, provided θ(0) ∈ S and θ∗ ∈ S.
Other modification methods:
b) Heuristics methods, such as re-initialization, ignore the
undesired value
c) Correction Approach
d) Persistent Excitation Approach
e) Switching Methods ...Adaptive Pole Placement Control II 12-49
Modification for ensure stabilizability
Theorem The gradient adaptive laws and the LS adaptive laws
with the projection modifications retain all the properties that are
established in the absence of projection and in addition guarantee
that θ(t) ∈ S∀t ≥ 0, provided θ(0) ∈ S and θ∗ ∈ S.
Other modification methods:
b) Heuristics methods, such as re-initialization, ignore the
undesired value
c) Correction Approach
d) Persistent Excitation Approach
e) Switching Methods ...Adaptive Pole Placement Control II 12-50