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EE 264 SIST, ShanghaiTech
Model Reference Adaptive Control
YW 9-1
Contents
Introduction
Motivating Examples
MRC for SISO plant
Model Reference Adaptive Control 9-2
General Structure
Figure: General Structure of MRAC
Categories:
i) Direct and Indirect MRAC
ii) Adaptive law with and without normalizationModel Reference Adaptive Control 9-3
General Structure
Figure: General Structure of MRAC
Categories:
i) Direct and Indirect MRAC
ii) Adaptive law with and without normalizationModel Reference Adaptive Control 9-4
Contents
Introduction
Motivating Examples
MRC for SISO plant
Model Reference Adaptive Control 9-5
Adaptive regulation
Example: one parameter case adaptive regulation
Consider
x = ax+ u, x(0) = x0
where a is a constant but unknown. The control objective is find
proper input signal u such that x→ 0 as t→∞.
Reference model
x = −amx, am > 0
Control law
u = −k(t)x
with k∗ = am + a.Model Reference Adaptive Control 9-6
Adaptive regulation
Example: one parameter case adaptive regulation
Consider
x = ax+ u, x(0) = x0
where a is a constant but unknown. The control objective is find
proper input signal u such that x→ 0 as t→∞.
Reference model
x = −amx, am > 0
Control law
u = −k(t)x
with k∗ = am + a.Model Reference Adaptive Control 9-7
Adaptive regulation
Example: one parameter case adaptive regulation
Consider
x = ax+ u, x(0) = x0
where a is a constant but unknown. The control objective is find
proper input signal u such that x→ 0 as t→∞.
Reference model
x = −amx, am > 0
Control law
u = −k(t)x
with k∗ = am + a.Model Reference Adaptive Control 9-8
Adaptive Regulation
Indirect approach:
k(t) = am + θ, θ = Φ(x, u)
Direct approach: rewrite the plant as
x = −amx+ k∗x+ u = −amx− kx˙k = k := ϕ(x)
with k = k − k∗. Consider a Lyapunov candidate function
V (x, k) = x2
2 + k2
2γ
Model Reference Adaptive Control 9-9
Adaptive Regulation
Indirect approach:
k(t) = am + θ, θ = Φ(x, u)
Direct approach: rewrite the plant as
x = −amx+ k∗x+ u = −amx− kx˙k = k := ϕ(x)
with k = k − k∗. Consider a Lyapunov candidate function
V (x, k) = x2
2 + k2
2γ
Model Reference Adaptive Control 9-10
Adaptive Regulation
Indirect approach:
k(t) = am + θ, θ = Φ(x, u)
Direct approach: rewrite the plant as
x = −amx+ k∗x+ u = −amx− kx˙k = k := ϕ(x)
with k = k − k∗. Consider a Lyapunov candidate function
V (x, k) = x2
2 + k2
2γ
Model Reference Adaptive Control 9-11
Adaptive Regulation
To generate the adaptive law for k. Taking the time derivative of
V as
V = −amx2 − kx2 + kϕ
γ
Take ϕ(x) = γx2, we have
V = −amx2
indicates
x, k, k ∈ L∞
x ∈ L2, x ∈ L∞, therefore x→ 0 as t→∞.
Note, we cannot establish that k → k∗.Model Reference Adaptive Control 9-12
Adaptive Regulation
To generate the adaptive law for k. Taking the time derivative of
V as
V = −amx2 − kx2 + kϕ
γ
Take ϕ(x) = γx2, we have
V = −amx2
indicates
x, k, k ∈ L∞
x ∈ L2, x ∈ L∞, therefore x→ 0 as t→∞.
Note, we cannot establish that k → k∗.Model Reference Adaptive Control 9-13
Adaptive Regulation
To generate the adaptive law for k. Taking the time derivative of
V as
V = −amx2 − kx2 + kϕ
γ
Take ϕ(x) = γx2, we have
V = −amx2
indicates
x, k, k ∈ L∞
x ∈ L2, x ∈ L∞, therefore x→ 0 as t→∞.
Note, we cannot establish that k → k∗.Model Reference Adaptive Control 9-14
Adaptive Tracking
Example 2: Two parameter case adaptive tracking. Consider
x = −ax+ bu
a, b unknown, but sgn(b) is assume to be known. Reference model
xm = bm
s+ amr
It is assumed that am, bm, and r are chosen so that xm represents
the desired state response of the plant. The control goal is
x(t)→ xm(t)
Model Reference Adaptive Control 9-15
Adaptive Tracking
Example 2: Two parameter case adaptive tracking. Consider
x = −ax+ bu
a, b unknown, but sgn(b) is assume to be known. Reference model
xm = bm
s+ amr
It is assumed that am, bm, and r are chosen so that xm represents
the desired state response of the plant. The control goal is
x(t)→ xm(t)
Model Reference Adaptive Control 9-16
Adaptive Tracking
The optimal control law
u = −k∗x+ l∗r
where k∗ and l∗ verifyx(s)r(s) = bl∗
s+ a+ bk∗= bm
s+ am= xm(s)
r(s)results in
l∗ = bm
b, k∗ = am − a
b
Since a, b unknown, we propose the control law
u = −kx+ lr
where k, l are estimates for k∗, l∗ respectively.Model Reference Adaptive Control 9-17
Adaptive Tracking
The optimal control law
u = −k∗x+ l∗r
where k∗ and l∗ verifyx(s)r(s) = bl∗
s+ a+ bk∗= bm
s+ am= xm(s)
r(s)results in
l∗ = bm
b, k∗ = am − a
b
Since a, b unknown, we propose the control law
u = −kx+ lr
where k, l are estimates for k∗, l∗ respectively.Model Reference Adaptive Control 9-18
Adaptive Tracking
The optimal control law
u = −k∗x+ l∗r
where k∗ and l∗ verifyx(s)r(s) = bl∗
s+ a+ bk∗= bm
s+ am= xm(s)
r(s)results in
l∗ = bm
b, k∗ = am − a
b
Since a, b unknown, we propose the control law
u = −kx+ lr
where k, l are estimates for k∗, l∗ respectively.Model Reference Adaptive Control 9-19
Adaptive Tracking
Express the plant equation in terms of k∗ and l∗
x = −amx+ bmr + b(k∗x− l∗r + u)
Define the error e := x− xm, we have a B-SPM
e = b
s+ am(k∗x− l∗r + u)
Alternatively, we can express the error equation as
e = −ame+ b(k∗x− l∗r + u)
= −ame+ b(−kx+ lr)
with k := k − k∗, l := l − l∗.Model Reference Adaptive Control 9-20
Adaptive Tracking
Express the plant equation in terms of k∗ and l∗
x = −amx+ bmr + b(k∗x− l∗r + u)
Define the error e := x− xm, we have a B-SPM
e = b
s+ am(k∗x− l∗r + u)
Alternatively, we can express the error equation as
e = −ame+ b(k∗x− l∗r + u)
= −ame+ b(−kx+ lr)
with k := k − k∗, l := l − l∗.Model Reference Adaptive Control 9-21
Adaptive Tracking
Express the plant equation in terms of k∗ and l∗
x = −amx+ bmr + b(k∗x− l∗r + u)
Define the error e := x− xm, we have a B-SPM
e = b
s+ am(k∗x− l∗r + u)
Alternatively, we can express the error equation as
e = −ame+ b(k∗x− l∗r + u)
= −ame+ b(−kx+ lr)
with k := k − k∗, l := l − l∗.Model Reference Adaptive Control 9-22
Adaptive Tracking
Express the plant equation in terms of k∗ and l∗
x = −amx+ bmr + b(k∗x− l∗r + u)
Define the error e := x− xm, we have a B-SPM
e = b
s+ am(k∗x− l∗r + u)
Alternatively, we can express the error equation as
e = −ame+ b(k∗x− l∗r + u)
= −ame+ b(−kx+ lr)
with k := k − k∗, l := l − l∗.Model Reference Adaptive Control 9-23
Adaptive Tracking
This motivates the following Lyapunov candidate function for
designing the adaptive laws
V (e, k, l) = e2
2 + k2
2γ1|b|+ l2
2γ2|b|
where γ1 > 0, γ2 > 0. The time derivative is given by
V = −ame2 − bkex+ bler + |b|k
γ1
˙k + |b|lγ2
˙l = −ame2
Because |b| = b sgn(b), the indefinite terms disappear if we choose
k = γ1ex sgn(b), l = −γ2er sgn(b)
Model Reference Adaptive Control 9-24
Adaptive Tracking
This motivates the following Lyapunov candidate function for
designing the adaptive laws
V (e, k, l) = e2
2 + k2
2γ1|b|+ l2
2γ2|b|
where γ1 > 0, γ2 > 0. The time derivative is given by
V = −ame2 − bkex+ bler + |b|k
γ1
˙k + |b|lγ2
˙l = −ame2
Because |b| = b sgn(b), the indefinite terms disappear if we choose
k = γ1ex sgn(b), l = −γ2er sgn(b)
Model Reference Adaptive Control 9-25
Adaptive Tracking
Properties:
e, k, l ∈ L∞, x, u ∈ L∞
e ∈ L2 and e, k, l ∈ L∞ if r ∈ L∞
if r is s.r. of order 2, e→ 0, k → k∗ , l→ l∗ exponentially fast.
Alternatively, we can generate a, b first, then combined with
u = kx+ lr where
k(t) = am + a(t)b(t)
, l(t) = bm
b(t)
This is a typical indirect MRAC method.
Note, for now, methods presented do not have normalization.Model Reference Adaptive Control 9-26
Adaptive Tracking
Properties:
e, k, l ∈ L∞, x, u ∈ L∞
e ∈ L2 and e, k, l ∈ L∞ if r ∈ L∞
if r is s.r. of order 2, e→ 0, k → k∗ , l→ l∗ exponentially fast.
Alternatively, we can generate a, b first, then combined with
u = kx+ lr where
k(t) = am + a(t)b(t)
, l(t) = bm
b(t)
This is a typical indirect MRAC method.
Note, for now, methods presented do not have normalization.Model Reference Adaptive Control 9-27
Adaptive Tracking
Properties:
e, k, l ∈ L∞, x, u ∈ L∞
e ∈ L2 and e, k, l ∈ L∞ if r ∈ L∞
if r is s.r. of order 2, e→ 0, k → k∗ , l→ l∗ exponentially fast.
Alternatively, we can generate a, b first, then combined with
u = kx+ lr where
k(t) = am + a(t)b(t)
, l(t) = bm
b(t)
This is a typical indirect MRAC method.
Note, for now, methods presented do not have normalization.Model Reference Adaptive Control 9-28
Adaptive Tracking
Properties:
e, k, l ∈ L∞, x, u ∈ L∞
e ∈ L2 and e, k, l ∈ L∞ if r ∈ L∞
if r is s.r. of order 2, e→ 0, k → k∗ , l→ l∗ exponentially fast.
Alternatively, we can generate a, b first, then combined with
u = kx+ lr where
k(t) = am + a(t)b(t)
, l(t) = bm
b(t)
This is a typical indirect MRAC method.
Note, for now, methods presented do not have normalization.Model Reference Adaptive Control 9-29
Adaptive Tracking
Properties:
e, k, l ∈ L∞, x, u ∈ L∞
e ∈ L2 and e, k, l ∈ L∞ if r ∈ L∞
if r is s.r. of order 2, e→ 0, k → k∗ , l→ l∗ exponentially fast.
Alternatively, we can generate a, b first, then combined with
u = kx+ lr where
k(t) = am + a(t)b(t)
, l(t) = bm
b(t)
This is a typical indirect MRAC method.
Note, for now, methods presented do not have normalization.Model Reference Adaptive Control 9-30
Contents
Introduction
Motivating Examples
MRC for SISO plant
Model Reference Adaptive Control 9-31
Problem Statement
Consider the SISO LTI plant
xp = Apxp +Bpup, xp(0) = x0 ∈ Rnp
yp = C>p xp
where yp, up ∈ R and Ap, Bp, Cp have the appropriate dimensions.
The transfer function of the plant :
yp = Gp(s)up = kpZp(s)Rp(s)up
where Zp(s), Rp(s) are monic known polynomials, constant kp is
a.k.a. high-frequency gain.Model Reference Adaptive Control 9-32
Problem Statement
Reference model
xm = Amxm +Bmr, xm(0) = xm0 ∈ Rnm
ym = C>mxm
where ym ∈ R and r ∈ R is the reference input.
The transfer function
ym = Wm(s)r
is expressed as
Wm(s) = kmZm(s)Rm(s)
where Zm(s), Rm(s) are monic polynomials and km is a constant.Model Reference Adaptive Control 9-33
Problem Statement
Reference model
xm = Amxm +Bmr, xm(0) = xm0 ∈ Rnm
ym = C>mxm
where ym ∈ R and r ∈ R is the reference input.
The transfer function
ym = Wm(s)r
is expressed as
Wm(s) = kmZm(s)Rm(s)
where Zm(s), Rm(s) are monic polynomials and km is a constant.Model Reference Adaptive Control 9-34
Problem Statement
MRC problem: The MRC objective is to determine the plant
input up so that all signals(xp, yp, up) are bounded and the plant
output yp tracks the reference model output ym as close as
possible for any given reference input r(t) which is a uniformly
bounded piecewise continuous function of time.
Model Reference Adaptive Control 9-35
Problem assumptions
Plant :
P1. Zp(s) is a monic Hurwitz polynomial with degree of mp.
P2. An upper bound n of the degree np of Rp(s) is known.
P3. The relative degree n∗ = np −mp of Gp(s) is known.
P4. The sign of the high-frequency gain kp is known.
Reference model:
M1. Zm(s), Rm(s) are monic Hurwitz polynomials of degree
qm, pm, respectively, where pm ≤ n
M2. The relative degree n∗m = pm − qm of Wm(s) is the same as
that of Gp(s), i.e., n∗m = n∗
Model Reference Adaptive Control 9-36
Problem assumptions
Plant :
P1. Zp(s) is a monic Hurwitz polynomial with degree of mp.
P2. An upper bound n of the degree np of Rp(s) is known.
P3. The relative degree n∗ = np −mp of Gp(s) is known.
P4. The sign of the high-frequency gain kp is known.
Reference model:
M1. Zm(s), Rm(s) are monic Hurwitz polynomials of degree
qm, pm, respectively, where pm ≤ n
M2. The relative degree n∗m = pm − qm of Wm(s) is the same as
that of Gp(s), i.e., n∗m = n∗
Model Reference Adaptive Control 9-37
MRC scheme
In addition, for MRC problem, we assume Gp(s) is known exactly.
A trivial choice for up is
up = km
kp
Zm(s)Rm(s)
Rp(s)Zp(s) r
which leads to the closed-loop transfer function
yp
r= km
kp
Zm
Rm
Rp
Zp
kpZp
Rp= Wm(s)
Drawbacks: This control law may involve zero-pole cancellations
outside C− when Rp(s) is not Hurwitz.
Core design logic : transfer function matchingModel Reference Adaptive Control 9-38
MRC scheme
In addition, for MRC problem, we assume Gp(s) is known exactly.
A trivial choice for up is
up = km
kp
Zm(s)Rm(s)
Rp(s)Zp(s) r
which leads to the closed-loop transfer function
yp
r= km
kp
Zm
Rm
Rp
Zp
kpZp
Rp= Wm(s)
Drawbacks: This control law may involve zero-pole cancellations
outside C− when Rp(s) is not Hurwitz.
Core design logic : transfer function matchingModel Reference Adaptive Control 9-39
MRC scheme
In addition, for MRC problem, we assume Gp(s) is known exactly.
A trivial choice for up is
up = km
kp
Zm(s)Rm(s)
Rp(s)Zp(s) r
which leads to the closed-loop transfer function
yp
r= km
kp
Zm
Rm
Rp
Zp
kpZp
Rp= Wm(s)
Drawbacks: This control law may involve zero-pole cancellations
outside C− when Rp(s) is not Hurwitz.
Core design logic : transfer function matchingModel Reference Adaptive Control 9-40
MRC scheme
Let us consider the feedback control law
up = θ∗>1α(s)Λ(s)up + θ∗>2
α(s)Λ(s)yp + θ∗3yp + c∗0r
where Λ(s) = Λ0(s)Zm(s) and Λ0(s) is monic, Hurwitz, and of
degree n0 = n− 1− qm.
α(s) , αn−2(s) =[sn−2, sn−3, . . . , s, 1
]> for n ≥ 2
α(s) , 0 for n = 1
c∗0, θ∗3 ∈ R; θ∗1, θ∗2 ∈ Rn−1 are constant parameters.
Hence, the controller parameter vector to be designed is
θ∗ =[θ∗>1 , θ∗>2 , θ∗3, c
∗0
]>∈ R2n
Model Reference Adaptive Control 9-41
MRC scheme
Let us consider the feedback control law
up = θ∗>1α(s)Λ(s)up + θ∗>2
α(s)Λ(s)yp + θ∗3yp + c∗0r
where Λ(s) = Λ0(s)Zm(s) and Λ0(s) is monic, Hurwitz, and of
degree n0 = n− 1− qm.
α(s) , αn−2(s) =[sn−2, sn−3, . . . , s, 1
]> for n ≥ 2
α(s) , 0 for n = 1
c∗0, θ∗3 ∈ R; θ∗1, θ∗2 ∈ Rn−1 are constant parameters.
Hence, the controller parameter vector to be designed is
θ∗ =[θ∗>1 , θ∗>2 , θ∗3, c
∗0
]>∈ R2n
Model Reference Adaptive Control 9-42
MRC scheme
We can now meet the control objective if we select θ∗ so that the
closed-loop poles are stable and the closed-loop transfer functiony(s)r(s) = Gc(s) = Wm(s), i.e., the matching equation
c∗0kpZpΛ(Λ− θ∗>1 α
)Rp − kpZp
(θ∗>2 α+ θ∗3Λ
) = kmZm
Rm
is satisfied for all s ∈ C. Choosing
c∗0 = km
kp
and using Λ(s) = Λ0(s)Zm(s), the matching equation becomes(Λ− θ∗>1 α
)Rp − kpZp
(θ∗>2 α+ θ∗3Λ
)= ZpΛ0Rm
Model Reference Adaptive Control 9-43
MRC scheme
We can now meet the control objective if we select θ∗ so that the
closed-loop poles are stable and the closed-loop transfer functiony(s)r(s) = Gc(s) = Wm(s), i.e., the matching equation
c∗0kpZpΛ(Λ− θ∗>1 α
)Rp − kpZp
(θ∗>2 α+ θ∗3Λ
) = kmZm
Rm
is satisfied for all s ∈ C. Choosing
c∗0 = km
kp
and using Λ(s) = Λ0(s)Zm(s), the matching equation becomes(Λ− θ∗>1 α
)Rp − kpZp
(θ∗>2 α+ θ∗3Λ
)= ZpΛ0Rm
Model Reference Adaptive Control 9-44
MRC scheme
Equating the coefficients of the powers of s on both sides, we can
express the matching equation in terms of the algebraic equation
Sθ∗ = p
where θ∗ =[θ∗>1 , θ∗>2 , θ∗3
]>; S is an (n+ np − 1)× (2n− 1)
matrix that depends on the coefficients of Rp, kp, Zp, and Λ; and
p is an n+ np − 1 vector with the coefficients of ΛRp − ZpΛ0Rm.
Lemma Let the degrees of Rp, Zp,Λ,Λ0, and Rm be as specified
in assumptions. Then the solution θ∗ of Sθ∗ = p always exists.In
addition, if Rp, Zp are coprime and n = np, then the solution θ∗ is
unique.Model Reference Adaptive Control 9-45
MRC scheme
Equating the coefficients of the powers of s on both sides, we can
express the matching equation in terms of the algebraic equation
Sθ∗ = p
where θ∗ =[θ∗>1 , θ∗>2 , θ∗3
]>; S is an (n+ np − 1)× (2n− 1)
matrix that depends on the coefficients of Rp, kp, Zp, and Λ; and
p is an n+ np − 1 vector with the coefficients of ΛRp − ZpΛ0Rm.
Lemma Let the degrees of Rp, Zp,Λ,Λ0, and Rm be as specified
in assumptions. Then the solution θ∗ of Sθ∗ = p always exists.In
addition, if Rp, Zp are coprime and n = np, then the solution θ∗ is
unique.Model Reference Adaptive Control 9-46
MRC scheme
Equating the coefficients of the powers of s on both sides, we can
express the matching equation in terms of the algebraic equation
Sθ∗ = p
where θ∗ =[θ∗>1 , θ∗>2 , θ∗3
]>; S is an (n+ np − 1)× (2n− 1)
matrix that depends on the coefficients of Rp, kp, Zp, and Λ; and
p is an n+ np − 1 vector with the coefficients of ΛRp − ZpΛ0Rm.
Lemma Let the degrees of Rp, Zp,Λ,Λ0, and Rm be as specified
in assumptions. Then the solution θ∗ of Sθ∗ = p always exists.In
addition, if Rp, Zp are coprime and n = np, then the solution θ∗ is
unique.Model Reference Adaptive Control 9-47
MRC scheme
A state-space realization of the control law
ω1 = Fω1 + gup, ω1(0) = 0 ∈ Rn−1
ω2 = Fω2 + gyp, ω2(0) = 0 ∈ Rn−1
up = θ∗>ω
where θ∗ =[θ∗>1 , θ∗>2 , θ∗3, c
∗0
]T, ω = [ω>1 , ω>2 , yp, r]> and
F =
−λn−2 −λn−3 −λn−4 · · · −λ0
1 0 0 · · · 0
0 1 0 · · · 0...
.... . .
. . ....
0 0 · · · 1 0
, g =
1
0...
0
λi are the coefficients of Λ(s)
Λ(s) = sn−1 + λn−2sn−2 + · · ·+ λ1s+ λ0 = det(sI − F )
Model Reference Adaptive Control 9-48
MRC scheme
Example: Let us consider the second-order plant
yp = −3(s+ 4)s2 − 3s+ 2up
and the reference model
ym = 1s+ 1r
1) check assumptions of plant and reference: np = 2., n∗ = 1 is
equal to that of the reference model.
2) choose the polynomial Λ(s) = s+ 2 = Λ0(s) and the control
input
up = θ∗11
s+ 2up + θ∗21
s+ 2yp + θ∗3yp + c∗0r
Model Reference Adaptive Control 9-49
MRC scheme
Example: Let us consider the second-order plant
yp = −3(s+ 4)s2 − 3s+ 2up
and the reference model
ym = 1s+ 1r
1) check assumptions of plant and reference: np = 2., n∗ = 1 is
equal to that of the reference model.
2) choose the polynomial Λ(s) = s+ 2 = Λ0(s) and the control
input
up = θ∗11
s+ 2up + θ∗21
s+ 2yp + θ∗3yp + c∗0r
Model Reference Adaptive Control 9-50
MRC scheme
Example: Let us consider the second-order plant
yp = −3(s+ 4)s2 − 3s+ 2up
and the reference model
ym = 1s+ 1r
1) check assumptions of plant and reference: np = 2., n∗ = 1 is
equal to that of the reference model.
2) choose the polynomial Λ(s) = s+ 2 = Λ0(s) and the control
input
up = θ∗11
s+ 2up + θ∗21
s+ 2yp + θ∗3yp + c∗0r
Model Reference Adaptive Control 9-51
MRC scheme
3) solve θ∗ by matching equationyp(s)r(s) = −3c∗0(s+ 4)(s+ 2)
(s+ 2− θ∗1) (s− 1)(s− 2) + 3(s+ 4) (θ∗2 + θ∗3(s+ 2)) = Gc(s)
Forcing Gc(s) = 1s+1 , we have c∗0 = −1
3 , and the matching
equation becomes
(θ∗1 − 3θ∗3) s2+(−3θ∗1 − 3θ∗2 − 18θ∗3) s+2θ∗1−12θ∗2−24θ∗3 = −8s2−18s−4
4) Implement control law as
ω1 = −2ω1 + up
ω2 = −2ω2 + yp
up = −2ω1 − 4ω2 + 2yp −(1
3
)r.
Model Reference Adaptive Control 9-52
MRC scheme
3) solve θ∗ by matching equationyp(s)r(s) = −3c∗0(s+ 4)(s+ 2)
(s+ 2− θ∗1) (s− 1)(s− 2) + 3(s+ 4) (θ∗2 + θ∗3(s+ 2)) = Gc(s)
Forcing Gc(s) = 1s+1 , we have c∗0 = −1
3 , and the matching
equation becomes
(θ∗1 − 3θ∗3) s2+(−3θ∗1 − 3θ∗2 − 18θ∗3) s+2θ∗1−12θ∗2−24θ∗3 = −8s2−18s−4
4) Implement control law as
ω1 = −2ω1 + up
ω2 = −2ω2 + yp
up = −2ω1 − 4ω2 + 2yp −(1
3
)r.
Model Reference Adaptive Control 9-53