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EE 264 SIST, ShanghaiTech
Adaptive Pole Placement Control
YW 11-1
Contents
Introduction
Motivating Examples
APPC scheme for general SISO plant via Polynomial approach
Adaptive Pole Placement Control 11-2
APPC = a pole placement control law + parameter adaptive law
The desired properties of the plant are expressed in terms of
desired pole locations to be placed by the controller
applicable to both minimum- and nonminimum-phase systems
includes direct and indirect two categories.
the use of adaptive laws with normalization and without
normalization does not extend to APPC.
Adaptive Pole Placement Control 11-3
APPC = a pole placement control law + parameter adaptive law
The desired properties of the plant are expressed in terms of
desired pole locations to be placed by the controller
applicable to both minimum- and nonminimum-phase systems
includes direct and indirect two categories.
the use of adaptive laws with normalization and without
normalization does not extend to APPC.
Adaptive Pole Placement Control 11-4
APPC = a pole placement control law + parameter adaptive law
The desired properties of the plant are expressed in terms of
desired pole locations to be placed by the controller
applicable to both minimum- and nonminimum-phase systems
includes direct and indirect two categories.
the use of adaptive laws with normalization and without
normalization does not extend to APPC.
Adaptive Pole Placement Control 11-5
APPC = a pole placement control law + parameter adaptive law
The desired properties of the plant are expressed in terms of
desired pole locations to be placed by the controller
applicable to both minimum- and nonminimum-phase systems
includes direct and indirect two categories.
the use of adaptive laws with normalization and without
normalization does not extend to APPC.
Adaptive Pole Placement Control 11-6
Contents
Introduction
Motivating Examples
APPC scheme for general SISO plant via Polynomial approach
Adaptive Pole Placement Control 11-7
Scalar Example: Adaptive Regulation
Consider the scalar plant
y = ay + bu
where a and b 6= 0 are unknown constants, and the sign of b is
known.
The control objective: choose u so that the closed-loop pole is
placed at −am < 0, and y → 0 as t→∞.
The idea control law
u = −k∗y, k∗ = a+ amb
Adaptive Pole Placement Control 11-8
Scalar Example: Adaptive Regulation
Consider the scalar plant
y = ay + bu
where a and b 6= 0 are unknown constants, and the sign of b is
known.
The control objective: choose u so that the closed-loop pole is
placed at −am < 0, and y → 0 as t→∞.
The idea control law
u = −k∗y, k∗ = a+ amb
Adaptive Pole Placement Control 11-9
Scalar Example: Adaptive Regulation
Consider the scalar plant
y = ay + bu
where a and b 6= 0 are unknown constants, and the sign of b is
known.
The control objective: choose u so that the closed-loop pole is
placed at −am < 0, and y → 0 as t→∞.
The idea control law
u = −k∗y, k∗ = a+ amb
Adaptive Pole Placement Control 11-10
Scalar Example: Adaptive Regulation
The closed-loop plant
y = −amy
achieves the control objective. Consider a, b unknown, use the
certainty equivalence approach to form the APPC as
u = −ky
with the estimate k is generated in two different ways:
Direct approach : ˙k = φ(y)
Indirect approach: k = a+am
b, b 6= 0
Adaptive Pole Placement Control 11-11
Scalar Example: Adaptive Regulation
The closed-loop plant
y = −amy
achieves the control objective. Consider a, b unknown, use the
certainty equivalence approach to form the APPC as
u = −ky
with the estimate k is generated in two different ways:
Direct approach : ˙k = φ(y)
Indirect approach: k = a+am
b, b 6= 0
Adaptive Pole Placement Control 11-12
Scalar Example: Adaptive Regulation
The closed-loop plant
y = −amy
achieves the control objective. Consider a, b unknown, use the
certainty equivalence approach to form the APPC as
u = −ky
with the estimate k is generated in two different ways:
Direct approach : ˙k = φ(y)
Indirect approach: k = a+am
b, b 6= 0
Adaptive Pole Placement Control 11-13
Direct Adaptive Regulation
Add and subtract the term −amy in the plant equation to obtain
y = −amy + (a+ am) y + bu
= −amy + b (k∗y + u)
= −amy − bky
where k , k − k∗ is the parameter error. Equation relates the
parameter error term bky with the regulation error y and
motivates the Lyapunov function
V = y2
2 + k2|b|2γ
Adaptive Pole Placement Control 11-14
Direct Adaptive Regulation
Add and subtract the term −amy in the plant equation to obtain
y = −amy + (a+ am) y + bu
= −amy + b (k∗y + u)
= −amy − bky
where k , k − k∗ is the parameter error. Equation relates the
parameter error term bky with the regulation error y and
motivates the Lyapunov function
V = y2
2 + k2|b|2γ
Adaptive Pole Placement Control 11-15
Direct Adaptive Regulation
time derivative of V is
V = −amy2 − bky2 + |b|γk ˙k = −amy2 − bky2 + bk
˙kγ
sgn(b)
The adaptive law
k = γy2 sgn(b)
leads to V = −amy2 ≤ 0 which implies
y, k, k ∈ L∞ and y ∈ L2.
y, u ∈ L∞; (Barbalat’s lemma) and y(t)→ 0 as t→∞
However, this does not guarantee that the closed-loop pole of the
plant is placed at −am even asymptotically with time.Adaptive Pole Placement Control 11-16
Direct Adaptive Regulation
time derivative of V is
V = −amy2 − bky2 + |b|γk ˙k = −amy2 − bky2 + bk
˙kγ
sgn(b)
The adaptive law
k = γy2 sgn(b)
leads to V = −amy2 ≤ 0 which implies
y, k, k ∈ L∞ and y ∈ L2.
y, u ∈ L∞; (Barbalat’s lemma) and y(t)→ 0 as t→∞
However, this does not guarantee that the closed-loop pole of the
plant is placed at −am even asymptotically with time.Adaptive Pole Placement Control 11-17
Direct Adaptive Regulation
time derivative of V is
V = −amy2 − bky2 + |b|γk ˙k = −amy2 − bky2 + bk
˙kγ
sgn(b)
The adaptive law
k = γy2 sgn(b)
leads to V = −amy2 ≤ 0 which implies
y, k, k ∈ L∞ and y ∈ L2.
y, u ∈ L∞; (Barbalat’s lemma) and y(t)→ 0 as t→∞
However, this does not guarantee that the closed-loop pole of the
plant is placed at −am even asymptotically with time.Adaptive Pole Placement Control 11-18
Indirect Adaptive Regulation
Add and subtract the term −amy in the plant equation to obtain
y = −amy + (a+ am) y + bu
= −amy + (a− a+ bk)y + bu
= −amy − ay − bu+ bu = −amy − ay − bu
where we have using the algebraic equation
k = a+ am
b, u = ky
and a = a− a, b = b− b. This equation relates the regulation or
estimation error y with the parameter errors a, b.Adaptive Pole Placement Control 11-19
Indirect Adaptive Regulation
Add and subtract the term −amy in the plant equation to obtain
y = −amy + (a+ am) y + bu
= −amy + (a− a+ bk)y + bu
= −amy − ay − bu+ bu = −amy − ay − bu
where we have using the algebraic equation
k = a+ am
b, u = ky
and a = a− a, b = b− b. This equation relates the regulation or
estimation error y with the parameter errors a, b.Adaptive Pole Placement Control 11-20
Indirect Adaptive Regulation
Motivates the Lyapunov candidate function
V = y2
2 + a2
2γ1+ b2
2γ2
for some γ1, γ2 > 0. The time derivative V is given by
V = −amy2 − ay2 − buy + a ˙aγ1
+ b ˙bγ2
Choosing˙a = ˙a = γ1y
2, ˙b = ˙b = γ2yu
we have
V = −amy2 ≤ 0Adaptive Pole Placement Control 11-21
Indirect Adaptive Regulation
Motivates the Lyapunov candidate function
V = y2
2 + a2
2γ1+ b2
2γ2
for some γ1, γ2 > 0. The time derivative V is given by
V = −amy2 − ay2 − buy + a ˙aγ1
+ b ˙bγ2
Choosing˙a = ˙a = γ1y
2, ˙b = ˙b = γ2yu
we have
V = −amy2 ≤ 0Adaptive Pole Placement Control 11-22
Indirect Adaptive Regulation
Motivates the Lyapunov candidate function
V = y2
2 + a2
2γ1+ b2
2γ2
for some γ1, γ2 > 0. The time derivative V is given by
V = −amy2 − ay2 − buy + a ˙aγ1
+ b ˙bγ2
Choosing˙a = ˙a = γ1y
2, ˙b = ˙b = γ2yu
we have
V = −amy2 ≤ 0Adaptive Pole Placement Control 11-23
Indirect Adaptive Regulation
To guarantee b 6= 0, introduce a projection
˙a = γ1y2,
˙b =
γ2yu if |b| > b0 or if |b| = b0 and sgn(b)yu ≥ 0
0 otherwise
where b(0) is chosen so that b(0) sgn(b) ≥ b0. Furthermore, the
time derivative V satisfies
V =
−amy2 if |b| > b0 or if |b| = b0 and sgn(b)yu ≥ 0
−amy2 − byu if |b| = b0 and sgn(b)yu < 0
Adaptive Pole Placement Control 11-24
Indirect Adaptive Regulation
To guarantee b 6= 0, introduce a projection
˙a = γ1y2,
˙b =
γ2yu if |b| > b0 or if |b| = b0 and sgn(b)yu ≥ 0
0 otherwise
where b(0) is chosen so that b(0) sgn(b) ≥ b0. Furthermore, the
time derivative V satisfies
V =
−amy2 if |b| > b0 or if |b| = b0 and sgn(b)yu ≥ 0
−amy2 − byu if |b| = b0 and sgn(b)yu < 0
Adaptive Pole Placement Control 11-25
Indirect Adaptive Regulation
The projection introduces the extra term −byu
byu = byu− byu = (|b| − |b|) sgn(b)yu = (b0 − |b|) sgn(b)yu ≥ 0
Hence the projection can only make V more negative and
V ≤ −amy2 ∀t ≥ 0
always hold.
y, a, b ∈ L∞, y ∈ L2, and |b(t)| ≥ b0∀t ≥ 0, which implies that
k ∈ L∞.
y ∈ L∞, and y(t)→ 0 as t→∞Adaptive Pole Placement Control 11-26
Indirect Adaptive Regulation
The projection introduces the extra term −byu
byu = byu− byu = (|b| − |b|) sgn(b)yu = (b0 − |b|) sgn(b)yu ≥ 0
Hence the projection can only make V more negative and
V ≤ −amy2 ∀t ≥ 0
always hold.
y, a, b ∈ L∞, y ∈ L2, and |b(t)| ≥ b0∀t ≥ 0, which implies that
k ∈ L∞.
y ∈ L∞, and y(t)→ 0 as t→∞Adaptive Pole Placement Control 11-27
Indirect Adaptive Regulation
The projection introduces the extra term −byu
byu = byu− byu = (|b| − |b|) sgn(b)yu = (b0 − |b|) sgn(b)yu ≥ 0
Hence the projection can only make V more negative and
V ≤ −amy2 ∀t ≥ 0
always hold.
y, a, b ∈ L∞, y ∈ L2, and |b(t)| ≥ b0∀t ≥ 0, which implies that
k ∈ L∞.
y ∈ L∞, and y(t)→ 0 as t→∞Adaptive Pole Placement Control 11-28
Scalar example : Adaptive Tracking
Consider the plant
y = −ay + bu
with a, b are unknown constants, but the sign of b 6= 0 is known.
Control objective:
i) closed-loop pole is at −am and u, y ∈ L∞ii) y(t) tracks the reference signal ym(t) = c, ∀t ≥ 0, where c 6= 0
is a known constant.
Adaptive Pole Placement Control 11-29
Scalar example : Adaptive Tracking
Consider the plant
y = −ay + bu
with a, b are unknown constants, but the sign of b 6= 0 is known.
Control objective:
i) closed-loop pole is at −am and u, y ∈ L∞ii) y(t) tracks the reference signal ym(t) = c, ∀t ≥ 0, where c 6= 0
is a known constant.
Adaptive Pole Placement Control 11-30
Scalar example : Adaptive Tracking
Define the tracking error e = y − c satisfies
e = ae+ ac+ bu
Then, the tracking objective becomes e(t) = y(t)− ym → 0 as
t→∞ exponentially fast. If a, b, c are known, we can choose an
idea control law
u = −k∗1e− k∗2, k∗1 = a+ amb
, k∗2 = ac
b
leading to
e = −ame
Adaptive Pole Placement Control 11-31
Scalar example : Adaptive Tracking
Define the tracking error e = y − c satisfies
e = ae+ ac+ bu
Then, the tracking objective becomes e(t) = y(t)− ym → 0 as
t→∞ exponentially fast. If a, b, c are known, we can choose an
idea control law
u = −k∗1e− k∗2, k∗1 = a+ amb
, k∗2 = ac
b
leading to
e = −ame
Adaptive Pole Placement Control 11-32
Direct Adaptive Tracking
The CE control law
u = −k1e− k2
As before, we rewrite the error equation as
e = −ame+ b (u+ k∗1e+ k∗2)
= −ame− b(k1e+ k2)
where k1 = k1 − k∗1, k2 = k2 − k∗2,which relates the tracking error
e with the parameter errors k1, k2 and motivates the Lyapunov
function
V = e2
2 + k21|b|2γ1
+ k22|b|2γ2
Adaptive Pole Placement Control 11-33
Direct Adaptive Tracking
The CE control law
u = −k1e− k2
As before, we rewrite the error equation as
e = −ame+ b (u+ k∗1e+ k∗2)
= −ame− b(k1e+ k2)
where k1 = k1 − k∗1, k2 = k2 − k∗2,which relates the tracking error
e with the parameter errors k1, k2 and motivates the Lyapunov
function
V = e2
2 + k21|b|2γ1
+ k22|b|2γ2
Adaptive Pole Placement Control 11-34
Direct Adaptive Tracking
The time derivative V is forced to satisfy
V = −ame2
by choosing
k1 = γ1e2 sgn(b), k2 = γ2e sgn(b)
The APPC scheme may be written as
u = −k1e− γ2 sgn(b)∫ t
0e(τ)dτ
k1 = γ1e2 sgn(b)
which is referred to as the direct adaptive proportional plus
integral (API) controller.Adaptive Pole Placement Control 11-35
Direct Adaptive Tracking
The time derivative V is forced to satisfy
V = −ame2
by choosing
k1 = γ1e2 sgn(b), k2 = γ2e sgn(b)
The APPC scheme may be written as
u = −k1e− γ2 sgn(b)∫ t
0e(τ)dτ
k1 = γ1e2 sgn(b)
which is referred to as the direct adaptive proportional plus
integral (API) controller.Adaptive Pole Placement Control 11-36
Indirect Adaptive Tracking
The CE control law
u = −k1e− k2
with
k1 = a+ am
b, k2 = ac
b
provided b 6= 0. Again we rewrite the error equation as
e = −ame+ (a+ am) e+ ac+ bu
= −ame+ ame+ ay + bu
= −ame− ay − bu+ ay + bu
= −ame− ay − bu
Adaptive Pole Placement Control 11-37
Indirect Adaptive Tracking
The CE control law
u = −k1e− k2
with
k1 = a+ am
b, k2 = ac
b
provided b 6= 0. Again we rewrite the error equation as
e = −ame+ (a+ am) e+ ac+ bu
= −ame+ ame+ ay + bu
= −ame− ay − bu+ ay + bu
= −ame− ay − bu
Adaptive Pole Placement Control 11-38
Indirect Adaptive Tracking
The CE control law
u = −k1e− k2
with
k1 = a+ am
b, k2 = ac
b
provided b 6= 0. Again we rewrite the error equation as
e = −ame+ (a+ am) e+ ac+ bu
= −ame+ ame+ ay + bu
= −ame− ay − bu+ ay + bu
= −ame− ay − bu
Adaptive Pole Placement Control 11-39
Indirect Adaptive Tracking
Leading to an adaptive law
˙a = γ1ey,
˙b =
γ2eu if |b| > b0 or if |b| = b0 and sgn(b)eu ≥ 0
0 otherwise
where b(0) satisfies b(0) sgn(b) ≥ b0. One can verify the
boundedness of and convergence of the tracking error by
considering the Lyapunov function
V = e2
2 + a2
2γ1+ b2
2γ2
Adaptive Pole Placement Control 11-40
Indirect Adaptive Tracking
Leading to an adaptive law
˙a = γ1ey,
˙b =
γ2eu if |b| > b0 or if |b| = b0 and sgn(b)eu ≥ 0
0 otherwise
where b(0) satisfies b(0) sgn(b) ≥ b0. One can verify the
boundedness of and convergence of the tracking error by
considering the Lyapunov function
V = e2
2 + a2
2γ1+ b2
2γ2
Adaptive Pole Placement Control 11-41
Contents
Introduction
Motivating Examples
APPC scheme for general SISO plant via Polynomial approach
Adaptive Pole Placement Control 11-42
PPC
Consider the SISO LTI 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)
yp is required to follow a certain class of reference signals ym
Remark: In general, by assigning the closed-loop poles to those of
A∗(s), we can guarantee closed-loop stability and convergence of
the plant output yp to zero, provided that there is no external
input.Adaptive Pole Placement Control 11-43
PPC
Consider the SISO LTI 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)
yp is required to follow a certain class of reference signals ym
Remark: In general, by assigning the closed-loop poles to those of
A∗(s), we can guarantee closed-loop stability and convergence of
the plant output yp to zero, provided that there is no external
input.Adaptive Pole Placement Control 11-44
PPC
Consider the SISO LTI 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)
yp is required to follow a certain class of reference signals ym
Remark: In general, by assigning the closed-loop poles to those of
A∗(s), we can guarantee closed-loop stability and convergence of
the plant output yp to zero, provided that there is no external
input.Adaptive Pole Placement Control 11-45
PPC
Consider the SISO LTI 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)
yp is required to follow a certain class of reference signals ym
Remark: In general, by assigning the closed-loop poles to those of
A∗(s), we can guarantee closed-loop stability and convergence of
the plant output yp to zero, provided that there is no external
input.Adaptive Pole Placement Control 11-46
PPC
Assumptions
P1. Rp(s) is a monic polynomial whose degree n is known.
P2. Zp(s), Rp(s) are coprime and degree (Zp) < n.
P3. The reference signal ym ∈ L∞ is assumed to satisfy
Qm(s)ym = 0
where Qm(s), known as the internal model of ym, is a known
monic polynomial of degree q with all roots in <[s] ≤ 0 and
with no repeated roots on the jω -axis. The internal model
Qm(s) is assumed to be coprime with Zp(s).Adaptive Pole Placement Control 11-47
PPC
Assumptions
P1. Rp(s) is a monic polynomial whose degree n is known.
P2. Zp(s), Rp(s) are coprime and degree (Zp) < n.
P3. The reference signal ym ∈ L∞ is assumed to satisfy
Qm(s)ym = 0
where Qm(s), known as the internal model of ym, is a known
monic polynomial of degree q with all roots in <[s] ≤ 0 and
with no repeated roots on the jω -axis. The internal model
Qm(s) is assumed to be coprime with Zp(s).Adaptive Pole Placement Control 11-48
PPC
Assumptions
P3. The reference signal ym ∈ L∞ is assumed to satisfy
Qm(s)ym = 0
and Qm(s) is assumed to be coprime with Zp(s).
Example
if ym(t) = c+ sin(3t), where c is any constant, then
Qm(s) = s(s2 + 9
)and, according to P3, Zp(s) should not have s
or s2 + 9 as a factor.
Adaptive Pole Placement Control 11-49
Indirect APPC via Polynomial Approach
The design of the APPC scheme is based on three steps:
1. a control law is developed that can meet the control objective
in the known parameter case. i.e. PPC
2. an adaptive law is designed to estimate the plant parameters
online. The estimated plant parameters are then used to
calculate the controller parameters at each time t
3. the APPC is formed by replacing the controller parameters in
Step 1 with their online estimates.
Adaptive Pole Placement Control 11-50
APPC via polynomial approach
Step 1. PPC for known parameters. We consider the control law
up = − P (s)Qm(s)L(s) (yp − ym)
where P (s), L(s) are polynomials (with L(s) monic) of degree
q + n− 1, n− 1, respectively, chosen to satisfy the polynomial
equation
L(s)Qm(s)Rp(s) + P (s)Zp(s) = A∗(s)
Assumptions P2 and P3 guarantee that Qm(s)Rp(s) and Zp(s)
are coprime, which guarantees hat L(s), P (s) exist and are unique.
Adaptive Pole Placement Control 11-51
APPC via polynomial approach
Step 1. PPC for known parameters. We consider the control law
up = − P (s)Qm(s)L(s) (yp − ym)
where P (s), L(s) are polynomials (with L(s) monic) of degree
q + n− 1, n− 1, respectively, chosen to satisfy the polynomial
equation
L(s)Qm(s)Rp(s) + P (s)Zp(s) = A∗(s)
Assumptions P2 and P3 guarantee that Qm(s)Rp(s) and Zp(s)
are coprime, which guarantees hat L(s), P (s) exist and are unique.
Adaptive Pole Placement Control 11-52
APPC via polynomial approach
The closed-loop plant
yp = ZpP
A∗ym
and the control law
up = RpP
A∗ym
if ym ∈ L∞ and ZpPA∗ ,
RpPA∗ are proper with stable poles, it follows
that yp, up ∈ L∞.
The tracking error e1 = yp − ym can be shown to satisfy
e1 = −LRpA∗
Qmym = −LRpA∗
[0]
it is easily to see e1 → 0 as t→∞.Adaptive Pole Placement Control 11-53
APPC via polynomial approach
The closed-loop plant
yp = ZpP
A∗ym
and the control law
up = RpP
A∗ym
if ym ∈ L∞ and ZpPA∗ ,
RpPA∗ are proper with stable poles, it follows
that yp, up ∈ L∞.
The tracking error e1 = yp − ym can be shown to satisfy
e1 = −LRpA∗
Qmym = −LRpA∗
[0]
it is easily to see e1 → 0 as t→∞.Adaptive Pole Placement Control 11-54
APPC via polynomial approach
The closed-loop plant
yp = ZpP
A∗ym
and the control law
up = RpP
A∗ym
if ym ∈ L∞ and ZpPA∗ ,
RpPA∗ are proper with stable poles, it follows
that yp, up ∈ L∞.
The tracking error e1 = yp − ym can be shown to satisfy
e1 = −LRpA∗
Qmym = −LRpA∗
[0]
it is easily to see e1 → 0 as t→∞.Adaptive Pole Placement Control 11-55
APPC via polynomial approach
Since L(s) is not necessarily Hurwitz, the control law
up = − P (s)Qm(s)L(s) (yp − ym)
may have unstable poles. An alternative realization is
Λ(s)up = Λ(s)up −Qm(s)L(s)up − P (s) (yp − ym)
where Λ(s) is any monic Hurwitz polynomial of degree n+ q − 1.
Filtering each side with 1Λ(s) , we obtain
up = Λ− LQmΛ up −
P
Λ (yp − ym)
The control law is implemented using 2(n+ q − 1) integrators to
realize the proper stable transfer functions Λ−LQm
Λ , PΛ .Adaptive Pole Placement Control 11-56
APPC via polynomial approach
Since L(s) is not necessarily Hurwitz, the control law
up = − P (s)Qm(s)L(s) (yp − ym)
may have unstable poles. An alternative realization is
Λ(s)up = Λ(s)up −Qm(s)L(s)up − P (s) (yp − ym)
where Λ(s) is any monic Hurwitz polynomial of degree n+ q − 1.
Filtering each side with 1Λ(s) , we obtain
up = Λ− LQmΛ up −
P
Λ (yp − ym)
The control law is implemented using 2(n+ q − 1) integrators to
realize the proper stable transfer functions Λ−LQm
Λ , PΛ .Adaptive Pole Placement Control 11-57
APPC via polynomial approach
Since L(s) is not necessarily Hurwitz, the control law
up = − P (s)Qm(s)L(s) (yp − ym)
may have unstable poles. An alternative realization is
Λ(s)up = Λ(s)up −Qm(s)L(s)up − P (s) (yp − ym)
where Λ(s) is any monic Hurwitz polynomial of degree n+ q − 1.
Filtering each side with 1Λ(s) , we obtain
up = Λ− LQmΛ up −
P
Λ (yp − ym)
The control law is implemented using 2(n+ q − 1) integrators to
realize the proper stable transfer functions Λ−LQm
Λ , PΛ .Adaptive Pole Placement Control 11-58
APPC via polynomial approach
Step 2. Estimation of plant polynomials The following PM can be
derived:
z = θ∗>p φ
where
z = sn
Λp(s)yp, θ∗p =[θ∗>b , θ∗>a
]>, φ =
[α>n−1(s)
Λp(s) up,−α>n−1(s)
Λp(s) yp
]>αn−1(s) =
[sn−1, . . . , s, 1
]>, θ∗a = [an−1, . . . , a0]> , θ∗b = [bn−1, . . . , b0]> ,
and Λp(s) is a monic Hurwitz polynomial.
Note that, if the degree of Zp(s) is less than n− 1, then some of
the first elements of θ∗b will be equal to zero.Adaptive Pole Placement Control 11-59
APPC via polynomial approach
Step 2. Estimation of plant polynomials The following PM can be
derived:
z = θ∗>p φ
where
z = sn
Λp(s)yp, θ∗p =[θ∗>b , θ∗>a
]>, φ =
[α>n−1(s)
Λp(s) up,−α>n−1(s)
Λp(s) yp
]>αn−1(s) =
[sn−1, . . . , s, 1
]>, θ∗a = [an−1, . . . , a0]> , θ∗b = [bn−1, . . . , b0]> ,
and Λp(s) is a monic Hurwitz polynomial.
Note that, if the degree of Zp(s) is less than n− 1, then some of
the first elements of θ∗b will be equal to zero.Adaptive Pole Placement Control 11-60
APPC via polynomial approach
A wide class adaptive laws can be designed to estimate θ∗p. For
example, the gradient algorithm
θp = Γεφ, ε =z − θTp φm2s
, m2s = 1 + φTφ
where Γ = Γ> > 0 is the adaptive gain and
θp =[bn−1, . . . , b0, an−1, . . . , a0
]Tare the estimated plant parameters form the plant polynomial
Rp(s, t) = sn + an−1(t)sn−1 + · · ·+ a1(t)s+ a0(t),
Zp(s, t) = bn−1(t)sn−1 + · · ·+ b1(t)s+ b0(t)
Adaptive Pole Placement Control 11-61
APPC via polynomial approach
A wide class adaptive laws can be designed to estimate θ∗p. For
example, the gradient algorithm
θp = Γεφ, ε =z − θTp φm2s
, m2s = 1 + φTφ
where Γ = Γ> > 0 is the adaptive gain and
θp =[bn−1, . . . , b0, an−1, . . . , a0
]Tare the estimated plant parameters form the plant polynomial
Rp(s, t) = sn + an−1(t)sn−1 + · · ·+ a1(t)s+ a0(t),
Zp(s, t) = bn−1(t)sn−1 + · · ·+ b1(t)s+ b0(t)
Adaptive Pole Placement Control 11-62
APPC via polynomial approach
Step 3. Adaptive control law
up =(Λ(s)− L(s, t)Qm(s)
) 1Λ(s)up − P (s, t) 1
Λ(s) (yp − ym)
with online estimates L(s, t), P (s, t) calculated at each frozen
time t using the polynomial equation
L(s, t) ·Qm(s) · Rp(s, t) + P (s, t) · Zp(s, t) = A∗(s)
Remark: stabilizability problem in indirect APPCThe existence and uniqueness of L(s, t), P (s, t) is guaranteed, provided
that Rp(s, t)Qm(s), Zp(s, t) are coprime at each frozen time tm which
means that at certain points in time the solution L(s, t) P (s, t) may not
exist.Adaptive Pole Placement Control 11-63
APPC via polynomial approach
Step 3. Adaptive control law
up =(Λ(s)− L(s, t)Qm(s)
) 1Λ(s)up − P (s, t) 1
Λ(s) (yp − ym)
with online estimates L(s, t), P (s, t) calculated at each frozen
time t using the polynomial equation
L(s, t) ·Qm(s) · Rp(s, t) + P (s, t) · Zp(s, t) = A∗(s)
Remark: stabilizability problem in indirect APPCThe existence and uniqueness of L(s, t), P (s, t) is guaranteed, provided
that Rp(s, t)Qm(s), Zp(s, t) are coprime at each frozen time tm which
means that at certain points in time the solution L(s, t) P (s, t) may not
exist.Adaptive Pole Placement Control 11-64
APPC via polynomial approach
Theorem: Assume that the estimated plant polynomials
RpQm, Zp are strongly coprime at each time t. Then all the
signals in the closed loop are uniformly bounded, and the tracking
error converges to zero asymptotically with time.
Remark: By strongly coprime we mean that the polynomials do
not have roots that are close to becoming common. For example,
the polynomials s+ 2 + ε, s+ 2, where s is arbitrarily small, are
not strongly coprime.
Adaptive Pole Placement Control 11-65
APPC via polynomial approach
Theorem: Assume that the estimated plant polynomials
RpQm, Zp are strongly coprime at each time t. Then all the
signals in the closed loop are uniformly bounded, and the tracking
error converges to zero asymptotically with time.
Remark: By strongly coprime we mean that the polynomials do
not have roots that are close to becoming common. For example,
the polynomials s+ 2 + ε, s+ 2, where s is arbitrarily small, are
not strongly coprime.
Adaptive Pole Placement Control 11-66
APPC via polynomial approach
Example: Consider the plant
yp = b
s+ aup
where a and b are constants. The desired roots
A∗(s) = (s+ 1)2
and the reference signal ym = 1.
Adaptive Pole Placement Control 11-67