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7/29/2019 Krstic _talk.pdf
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Adaptive and Nonlinear Control
Miroslav KrsticUniversity of California, San Diego
NSF-DOE Fusion Control Workshop, General Atomics, 2006
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Adaptive Control
Plantinput output
output filterinput filter
identifier
controller
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Adaptive Control
Plantinput output
output filterinput filter
identifier
controller
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Adaptive Control
Plantinput output
output filterinput filter
identifier
controller
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Adaptive Control
Plantinput output
output filterinput filter
identifier
controller
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Plant
y =B(s)
A(s)u
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Plant
y =B(s)
A(s)u
B(s) = bmsm+bm1sm1 + +b1s+b0
A(s) = ansn+an1sn1 + +a1s+a0
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Plant
y =B(s)
A(s)u
B(s) = bmsm+bm1sm1 + +b1s+b0
A(s) = ansn+an1sn1 + +a1s+a0
Unknown parameter vector
= [bm bm
1
b1 b0 an an
1
a1 a0]
T
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Controller
u =Q(s)
P(s)y
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Controller
u =Q(s)
P(s)y
P(s) and Q(s) obtained by solving a Bezout-type polynomial equation involving A(s) and
B(s) to satisfy some objectivefor example, the placement of closed-loop poles.
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Controller
u =Q(s)
P(s)y
P(s) and Q(s) obtained by solving a Bezout-type polynomial equation involving A(s) and
B(s) to satisfy some objectivefor example, the placement of closed-loop poles.
So, the coefficients of P(s) and Q(s) at each time step are determined from the estimate(t) of at each time step.
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Adaptive Control
B(s)/A(s)input output
output filterinput filter
identifier
Q(s)/P(s)
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Approaches to identifier design
Lyapunov
Estimation based/Certainty equivalence
with passive identifier (often called observer-based method)
with swapping identifier (often called the gradient method)
This talk:
Part I: State-feedback with passive identifier
Part II: Output feedback with swapping identifier
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Approaches to identifier design
Lyapunov
Estimation based/Certainty equivalence
with passive identifier (often called observer-based method)
with swapping identifier (often called the gradient method)
This talk:
Part I: State-feedback with passive identifier
Part II: Output feedback with swapping identifier
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Approaches to identifier design
Lyapunov
Estimation based/Certainty equivalence
with passive identifier (often called observer-based method)
with swapping identifier (often called the gradient method)
This talk:
Part I: State-feedback with passive identifier
Part II: Output feedback with swapping identifier
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PDE with unknown functional parameter
ut = uxx+ (x)u
Measurement: u(0)
Control: u(1)
Unstable
Infinitely many unknown parameters / infinitedimensional state
Scalar input / scalar output
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PDE with unknown functional parameter
ut = uxx+ (x)u
Measurement: u(0)
Control: u(1)
Unstable
Infinitely many unknown parameters / infinitedimensional state
Scalar input / scalar output
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PDE with unknown functional parameter
ut = uxx+ (x)u
Measurement: u(0)
Control: u(1)
Unstable
Infinitely many unknown parameters / infinitedimensional state
Scalar input /scalar output
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Plant transfer function:
u(0,s) =B(s)
A(s)u(1,s)
where
B(s) = 1
A(s) = coshs+
1
sinhs
s
Z10
sinhs(1 y)s
(y)dy
and (x), 1 are related to (x) through the solution of the PDE
pxx(x,y) = pyy(x,y) + (y)p(x,y)
p(1,y) = 0
p(x,x) = 12
Z1x
(y)dy
(x) = py(x, 0)1 = p(0, 0)
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Plant transfer function:
u(0,s) =B(s)
A(s)u(1,s)
where
B(s) = 1
A(s) = coshs+
1
sinhs
s
Z10
(y)sinh
s(1 y)s
dy
and (x), 1 are related to (x) through the solution of the PDE
pxx(x,y) = pyy(x,y) + (y)p(x,y)
p(1,y) = 0
p(x,x) = 12
Z1x
(y)dy
(x) = py(x, 0)1 = p(0, 0)
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Plant transfer function:
u(0,s) =B(s)
A(s)u(1,s)
where
B(s) = 1
A(s) = coshs+
1
sinhs
s
Z10
(y)sinh
s(1 y)s
dy
and (x), 1 are related to (x) through the solution of the PDE
pxx(x,y) = pyy(x,y) + (y)p(x,y)
p(1,y) = 0
p(x,x) = 12
Z1x
(y)dy
(x) = py(x, 0)1 = p(0, 0)
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Plant transfer function:
u(0,s) =B(s)
A(s)u(1,s)
where
B(s) = 1A(s) = cosh
s+
1
sinhs
s
Z10
(y)sinh
s(1 y)s
dy
and (x), 1 are related to (x) through the solution of the PDE
pxx(x,y) = pyy(x,y) + (y)p(x,y)
p(1,y) = 0
p(x,x) = 12
Z1x
(y)dy
(x) = py(x, 0)1 = p(0, 0)
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Compensator:
u(
1,s) =
Q(s)
P(s)u
(0
,s)
where
P(s) = coshs
1
Z0
k(1y) coshsy dy
Q(s) =
1Z
0
k(y)
1
sinh (sy)s
+sinh (
sy)
s
1yZ
0
() cosh(s)d
dy
+
1Z
0
k(y) coshs(1 y) 1Z
1y()
sinh (s(1 ))s
ddy
and
k(x) = 1 Zx
0(y)dy
Zx0
1
Zxy0
(s)ds
k(y)dy
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Compensator:
u(1,s) =Q(s)
P(s)u(0,s)
where
P(s) = coshs
1
Z0
k(1y) coshsy dy
Q(s) =
1Z
0
k(y)
1
sinh (sy)s
+sinh (
sy)
s
1yZ
0
() cosh(s)d
dy
+
1Z
0
k(y) coshs(1 y) 1Z
1y()
sinh (s(1 ))s
ddy
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Compensator:
u(1,s) =Q(s)
P(s)u(0,s)
where
P(s) = coshs
1
Z0
k(1
y) coshsy dy
Q(s) =
1Z
0
k(y)
1
sinh (sy)s
+sinh (
sy)
s
1yZ
0
() cosh(s)d
dy
+
1Z
0
k(y) coshs(1 y) 1Z
1y()
sinh (s(1 ))s
ddy
and
k(x) = 1 Zx
0(y)dy
Zx0
1
Zxy0
(s)ds
k(y)dy
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Input filter
t = xx
x(0) = 0
(1) = u(1)
Output filters
t = xx
x(0) = u(0)
(1) = 0
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Adaptive scheme
ut= uxx+ (x)u
ux(0) = 0
u(1) u(0)
(x), 1
output filterinput filter
identifier
controller
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Adaptive scheme
ut= uxx+ (x)u
ux(0) = 0
u(1) u(0)
output filterinput filter
(x), 1 identifier
controller
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Adaptive scheme
ut= uxx+ (x)u
ux(0) = 0
u(1) u(0)
(x), 1
output filterinput filter
identifier
controller
U d l (l )
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Update laws (least squares)
t(x, t) =
R10 (x,y, t)(y)dy+ 0(x, t)(0)
1 + 2 + 2(0)
v(0) (0) 1(0) +
Z10
()()d
1 =
R10 0(y, t)(y)dy+ 1(t)(0)
1 + 2 + 2(0)v(0) (0) 1(0) +
Z10
()()d
Ri ti d t ti i
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Riccati adaptation gains
t(x,y, t) = R1
0 (x,s)(s)dsR1
0 (y,s)(s)ds+ 0(x)0(y)2(0)
1 + 2 + 2(0)
(0)0(y)
R10 (x,s)(s)ds+ (0)0(x)
R10 (y,s)(s)ds
1 + 2 + 2(0)
0
(x) =
R10 (x,s)(s)ds+ 0(x)(0)
R10 0(s)(s)ds+ 1(0)1 + 2 + 2(0)
1 = R1
0 0(s)(s)ds+ 1(0)2
1 + 2 + 2(0)
Ad ti C t ll
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Adaptive Controller
u(1) =Z1
0 k(1 y)(y) + 1(y) +Z1
0 ()F(y, )d dy
with k(x) given by the integral equation in one variable
k(x) = 1 Zx
0(y)dy
Zx
0
1
Zxy
0(s)ds
k(y)dy
and
F(y, ) = 2
n=0
cos(2n+ 1)y
2cos
(2n+ 1)
2
Z10
cos(2n+ 1)s
2(s)ds
This equation is solved at each time step.
Simulation Example
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Simulation Example
ut = uxx+b(x)ux+ (x)u
ux(0) = 0
Reference signal: ur(0, t) = 3sin6t b(x) = 3 2x2 (x) = 16 + 3sin(2x)
Simulation Example
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Simulation Example
ut = uxx+b(x)ux+ (x)u
ux(0) = 0
Reference signal: ur(0, t) = 3sin6t b(x) = 3 2x2 (x) = 16 + 3sin(2x)
u(0)
tt
u(1)
Control effort Output evolution
Recommended Books on Adaptive Control
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Recommended Books on Adaptive Control
Comprehensive Ioannou and Sun
Tao
Discrete-time Mareels and Polderman
A little out-of-date but still pretty good Goodwin and Sin
Sastry and Bodson
For nonlinear systems
Krstic, Kanellakopoulos, and Kokotovic
Recommended Books on Adaptive Control
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Recommended Books on Adaptive Control
Comprehensive Ioannou and Sun
Tao
Discrete-time Mareels and Polderman
A little out-of-date but still pretty good Goodwin and Sin
Sastry and Bodson
For nonlinear systems
Krstic, Kanellakopoulos, and Kokotovic
Recommended Books on Adaptive Control
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Recommended Books on Adaptive Control
Comprehensive Ioannou and Sun
Tao
Discrete-time Mareels and Polderman
A little out-of-date but still pretty good Goodwin and Sin
Sastry and Bodson
For nonlinear systems Krstic, Kanellakopoulos, and Kokotovic
Recommended Books on Adaptive Control
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Recommended Books on Adaptive Control
Comprehensive
Ioannou and Sun
Tao
Discrete-time Mareels and Polderman
A little out-of-date but still pretty good Goodwin and Sin
Sastry and Bodson
For nonlinear systems Krstic, Kanellakopoulos, and Kokotovic
General Remarks on Adaptive Control
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General Remarks on Adaptive Control
Problems intended to solve: Trajectory tracking in the presence of large parametric uncertainties
Cancellation of unknown disturbances
Limitations:
Transient performance hard to guarantee for most adaptive schemes except for
Lyapunov-based schemes
Robustness to non-parametric uncertainties requires additional fixes
Computational effort and complexity:
Modest-to-medium (compared to optimality-driven methods)
General Remarks on Adaptive Control
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Ge e a e a s o dapt e Co t o
Problems intended to solve: Trajectory tracking in the presence of large parametric uncertainties
Cancellation of unknown disturbances
Limitations:
Transient performance hard to guarantee for most adaptive schemes except for
Lyapunov-based schemes
Robustness to non-parametric uncertainties requires additional fixes
Computational effort and complexity:
Modest-to-medium (compared to optimality-driven methods)
General Remarks on Adaptive Control
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p
Problems intended to solve: Trajectory tracking in the presence of large parametric uncertainties
Cancellation of unknown disturbances
Limitations:
Transient performance hard to guarantee for most adaptive schemes except for
Lyapunov-based schemes
Robustness to non-parametric uncertainties requires additional fixes
Computational effort and complexity:
Modest-to-medium (compared to optimality-driven methods)
General Remarks on Adaptive Control (contd)
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p ( )
Successful applications:
Many, see the book by Astrom and Wittenmark
What is needed for applications in fusion:
Further development of distributed-parameter versions of adaptive control
General Remarks on Adaptive Control (contd)
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Successful applications:
Many, see the book by Astrom and Wittenmark
What is needed for applications in fusion:
Further development of distributed-parameter versions of adaptive control
Nonlinear Control
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x = f(x) +g(x)u
y = h(x)
Most mechanical systems (vehicles, fluids) are dominated by nonlinearities.
Saturation and hysteresis are dealt with differently.
Nonlinear Control
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x = f(x) +g(x)u
y = h(x)
Most mechanical systems (vehicles, fluids) are dominated by nonlinearities.
Saturation and hysteresis are dealt with differently.
Nonlinear Control
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x = f(x) +g(x)u
y = h(x)
Most mechanical systems (vehicles, fluids) are dominated by nonlinearities.
Saturation and hysteresis are dealt with differently.
Methods
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Feedback linearization
Backstepping (robust and adaptive)
Various varieties of nonlinear optimal control (including MPC)
Books on Nonlinear Control
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Isidori (focused on structural/differential geometric properties related to controllabilitynot a design reference)
Khalil; Sastry; Vidyasagar (general textbooks that gloss over many methods)
Krstic, Kanellakopoulos, and Kokotovic (adaptive and robust design tools for uncertainnonlinear systems)
Adaptive Nonlinear ControlA Tutorial
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Backstepping Tuning Functions Design Modular Design
Output Feedback
Extensions A Stochastic Example Applications and Additional References
main source:
Nonlinear and Adaptive Control Design (Wiley, 1995)
M. Krstic, I. Kanellakopoulos and P. V. Kokotovic
Backstepping (nonadaptive)
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x1 = x2 + (x1)T , (0) = 0
x2 = u
where is known parameter vector and (x1) is smooth nonlinear function.
Goal: stabilize the equilibrium x1 = 0, x2 = (0)T = 0.
virtual control for the x1-equation:
1(x1) = c1x1 (x1)T , c1 > 0
error variables:
z1 = x1
z2 = x2 1(x1) ,
System in error coordinates:
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z1 = x1 = x2 + T = z2 + 1 +
T = c1z1 +z2z2 = x2 1 = u 1x1
x1 = u 1x1x2 +
T
.
Need to design u = 2(x1,x2) to stabilize z1 = z2 = 0.
Choose Lyapunov function
V(x1,x2) =1
2z21 +
1
2z22
we have
V = z1 (c1z1 +z2) +z2u 1
x1
x2 +
T
= c1z21 +z2u+z1 1x1 x2 +
T =c2z2
V= c1z21 c2z22
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z = 0 is globally asymptotically stable
invertible change of coordinates
x = 0 is globally asymptotically stable
The closed-loop system in z-coordinates is linear:
z1z2
= c1 1
1 c2 z1
z2
.
Tuning Functions Design
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Introductory examples:
A B C
x1 = u+ (x1)T x1 = x2 + (x1)
T x1 = x2 + (x1)T
x2 = u x2 = x3
x3 = u
where is unknown parameter vector and (0) = 0.
Degin A. Let be the estimate of and =
,
Using
u = c1x1 (x1)Tgives
x1 = c1x1 + (x1)T
To find update law for (t), choose
1 1
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V1(x, ) =1
2
x21 +1
2
T1
then
V1 = c1x21 +x1(x1)T T1 =
c
1x2
1+ T
1(x1)x1
=0
Update law:
= (x1)x1, (x1)regressor
gives
V1 = c1x21 0.By Lasalles invariance theorem, x1 = 0, = is stable and
limtx1(t) = 0
Design B. replace by in the nonadaptive design:
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z2 = x2 1(x1, ) , 1(x1, ) = c1z1 Tand strengthen the control law by 2(x1,x2, )(to be designed)
u = 2(x1,x2, ) = c2z2 z1 +1x1
x2 +
T
+2(x1,x2, )
error system
z1 = z2 + 1 + T = c1z1 +z2 + T
z2 = x2 1 = u1x1 x2 +
T1
= z1 c2z2 1x1T 1
+2(x1,x2, ) ,
or
z1z2 = c1 11 c2 z1z2 + T
1x1
T + 01
+2(x1,x2, )
=0
remaining: design adaptive law.
Choose
V ( ) V1 2 1 2 1 2 1T 1
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V2(x1,x2, ) = V1 +2z22 = 2
z21 + 2z22 + 2
T1
we have
V2 = c1z21 c2z22 + [z1, z2]
T
1x1
T
T1
= c1z21 c2z22 + T1, 1x1 z1z2 .The choice
= 2(x, ) = , 1x1 z1z2 =
1z1 1x1 z2 2
(1, 2 are called tuning functions)
makes
V2 = c1z21 c2z22,thus z = 0, = 0 is GS and x(t)
0 as t
.
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+E
+Z
T
1x1 TT T
1x1 T
c1 11 c2
'
Z'
T
E E E E E
T
'
z1
z2 2
The closed-loop adaptive system
Design C.
We have one more integrator, so we define the third error coordinate and replace in
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e a e o e o e teg ato , so e de e t e t d e o coo d ate a d ep ace
design B by potential update law,
z3 = x3 2(x1,x2, )
2(x1,x2, ) =1
2(x1,x2, ).
Now the z1,z2-system is
z1z2
=
c1 1
1
c2
z1z2
+
T
1x1
T
+
0
z3 +1
(2
)
and
V2 = c1z21 c2z22 +z2z3 +z21
(2 ) + T(2 1 ).
z3-equation is given by
2 2 2
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z3 = u
2
x1 x2 + T
2
x2
x3
2
= u 2x1
x2 +
T
2x2
x3 2
2x1
T .
Choose
V3(x, ) = V2 + 12z23 = 12z
21 + 12z22 + 12z
23 + 12T1
we have
V3 =
c1z
21
c2z
22 +z2
1
(2
)
+z3
z2 +u
2x1
x2 +
T
2x2
x3 2
+T2
2x
1
z3 1 .
Pick update law
z1
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= 3(x1,x2,x3,
) = 2 2x1 z3 = , 1x1 , 2x1 z1
z2z3
and control law
u = 3(x1,x2,x3, ) =z2
c3z3 +
2
x1 x2 + T+2
x2x3 +3,
results in
V3 = c1z21 c2z22 c3z23 +z21
(2 ) +z3
3 2
.
Notice
2 = 3 2x1
z3
we have
V3 = c1z21 c2z22 c3z23 +z3
3 2
3 +
1
2x1
z2
=0
.
Stability and regulation of x to zero follows.
Further insight:
T 0
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z1z2z3
= c1 1 01 c2 10 1 c3
z1z2z3
+T
1x1 T
2x1
T +
0
1 (2 )
3 2 3 .
2
=
3
2
x1z
3
z1z2z3
=
c1 1 01 c2 1 + 1
2x1
0 1 c3
z1z2z3
+
T
1x1 T
2x1
T
+
0
0
3 2 3
seletion of 3
z1z2z3
= c1 1 01 c2 1 + 1 2x1
0 1 1
2x1
c3 z1z2
z3
+T
1x1 T2
x1T
.
General Recursive Design Procedure
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parametric strict-feedback system:
x1 = x2 + 1(x1)T
x2 = x3 + 2(x1,x2)T
...
xn1 = xn+ n1(x1, . . . ,xn1)T
xn = (x)u+ n(x)T
y = x1
where and i are smooth.
Objective: asymptotically track reference output yr(t), with y(i)r (t), i = 1, ,n known,
bounded and piecewise continuous.
Tuning functions design for tracking (z0= 0, 0
= 0, 0
= 0)
zi = xiy(i1)r i1
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i i yr i 1
i( xi,, y
(i
1)
r ) = zi1 ciziwT
i +
i1
k=1i1xk xk+1 + i1y(k1)r y(k)r +i
i( xi, , y(i1)r ) = +
i1
i +i1k=2
k1
wizk
i( xi, , y(i1)r ) = i1 +wizi
wi( xi, , y(i
2)
r ) = ii1
k=1i
1
xk k
i = 1, . . . ,n
xi = (x1, . . . ,xi), y(i)r = (yr, yr, . . . ,y
(i)r )
Adaptive control law:
u =1
(x)
n(x, , y
(n1)r ) +y
(n)r
Parameter update law:
= n(x, , y(n1)r ) = Wz
Closed-loop system
z = A (z t)z+W(z t)T
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z = Az(z, , t)z+W(z, , t)
= W(z, , t)z ,
where
Az(z, , t) =
c1 1 0 0
1 c
2 1 + 23 2n0 1 23 . . . . . . ...... ... . . . . . . 1 + n1,n0 2n 1 n1,n cn
jk(x, ) = j1
wk
This structure ensures that the Lyapunov function
Vn =
1
2zT
z+
1
2T
1
has derivative
Vn = n
k=1
ckz2k.
Modular Design
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Motivation: Controller can be combined with different identifiers. (No flexibility for update
law in tuning function design)
Naive idea: connect a good identifier and a good controller.
Example: error system
x = x+ (x)
suppose
(t) = et
and (x) = x3
, we have
x = x+x3et
But, when |x0| >
32,
x(t) as t 13
lnx20
x20 3/2Conclusion: Need stronger controller.
Controller Design. nonlinear damping
u = x (x) (x)2x
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u x (x) (x) x
closed-loop system
x = x (x)2x+ (x) .With V= 12x
2, we have
V = x2 (x)2x2 +x(x)
= x2
(x)x 12
2+
1
42
x2 + 14
2 .
bounded (t) bounded x(t)
For higher order system
x1 = x2 + (x1)T
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x2 = u
set
1(x1, ) = c1x1 (x1)T 1|(x1)|2x1, c1, 1 > 0and define
z2 = x2 1(x1, )error system
z1 = c1z1 1||2
z1 + T
+z2
z2 = x2 1 = u1x1
x2 +
T
1
.
Consider
V2 =V1 +1
2z22 =
1
2|z|2
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2 2 2
we have
V2 c1z21 +1
41||2 +z1z2 +z2
u 1
x1
x2 +
T
1
c1z2
1+
1
41||
2
+z2u+z11x1x2+T1x1 T+1 .controller
u=z
1 c
2z
2
2 1x1 2
z2
g2 1
T
2
z2
+1
x1 x2 + T ,achieves
V2 c1z21 c2z22 +1
41+
1
42 ||2 +
1
4g2| |2
bounded , bounded (or L2) bounded x(t)
Controller design in the modular approach (z0= 0, 0
= 0)
zi = xiy(i1)r i1i 1
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i( xi,
, y
(i
1)
r ) = zi1 ciziwT
i
+
i1
k=1i1xk xk+1 + i1y(k1)r y(k)r sizi
wi( xi, , y(i2)r ) = i
i1k=1
i1xk
k
si( xi, , y(i2)r ) = i|wi|2 +gi
i1
T
2
i= 1, . . . ,n
xi = (x1, . . . ,xi), y(i)r = (yr, yr, . . . ,y
(i)r )
Adaptive control law:
u =1
(x)
n(x, , y
(n1)r ) +y
(n)r
Controller module guarantees:
If
L and
L2 or L then x
L
Requirement for identifier
error system
T T
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z = Az(z,, t)z+W(z,
, t)
T +Q(z,
, t)
T
where
Az(z, , t) = c1 s1 1 0 0
1
c2
s2 1
. . . ...
0 1 . . . . . . 0... . . . . . . . . . 1
0 0 1 cn sn
W(z, , t)T =wT
1wT2...
wTn
, Q(z, , t)T = 0
1...
n1
.
Since
1 0 01
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W(z, , t)T = 1
x1 1
. . . ...... . . . . . . 0
n1x1
n1xn1
1
F(x)T = N(z, , t)F(x)T .
Identifier properties:
(i) L and L2 or L,
(ii) if x L then F(x(t))T
(t) 0 and
(t) 0.
Identifier Design
Passive identifier
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Passive identifier
x = f+FT
x = A0 FTFP( xx) + f+F
T
c
T
'FP
Z
E
'
+
x
x
E E
T
FP
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's
= A0 F(x,u)TF(x,u)P +F(x,u)Tupdate law
= F(x,u)P , = T > 0 .
Use Lyapunov function
V= T1 + TP
its derivative satisfies
V T ()2
| |2 .
Thus, whenever x is bounded, F(x(t))T(t) 0 and (t) 0.(
(t)
0 because R0
(
)d
=
(0
)exists, Barbalats lemma...)
Swapping identifier
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x = f+FT
0 =A0 FTFP
(0 x) + f
=A0 FTFP
+F
....................................................
E
c
E
T
Z
1 +|
|2
''
T
x
0
T
++
define = x+ 0 T,
= A0 F(x,u)TF(x,u)P .
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ChooseV=
1
2T1 + P
we have
V 34
T1 +tr{T},
proves identifier properties.
Output Feedback Adaptive Designs
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x = Ax+ (y) + (y)a+
0
b
(y)u , x IRn
y = eT1x ,
A =
0...
In10
0
,
(y) =
0,1(y)...0,n(y)
, (y) = 1,1(y) q,1(y)... ...1,n(y) q,n(y)
,
unknown constant parameters:
a = [a1, . . . ,aq]T , b = [bm, . . . ,b0]
T .
State estimation filters
Filters:
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Filters:
= A0 + ky+ (y)
= A0 + (y)
= A0
+ en(y)u
vj = Aj0, j = 0, . . . ,m
T = [vm, . . . ,v1,v0, ]
Parameter-dependent state estimate
x = + T
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The vector k= [k1, . . . ,kn]T chosen so that the matrix
A0 = A keT1is Hurwitz, that is,
PA0 +AT0P = I, P = PT > 0
The state estimation error
= x xsatisfies
= A0
Parametric model for adaptation:
y = 0 + T + 2
T
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= bmvm,2 + 0 + T
+ 2 ,
where
0 = 0,1 + 2
= [vm,2,vm1,2, . . . ,v0,2, (1) + (2)]T = [0,vm1,2, . . . ,v0,2, (1) + (2)]T .
Since the states x2, . . . ,xn are not measured, the backstepping design is applied to the
system
T
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y = bmvm,2 + 0 + T
+ 2vm,i = vm,i+1 kivm,1 , i = 2, . . . , 1vm, = (y)u+ vm,+1 kvm,1 .
The order of this system is equal to the relative degree of the plant.
Extensions
Pure-feedback systems.
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xi = xi+1 + i(x1, . . . ,xi+1)T, i = 1, . . . ,n 1
xn =
0(x) + (x)Tu+ 0(x) + n(x)
T ,
where 0(
0) =
0,
1(
0) = =
n(
0) =
0,
0(
0) =
0.
Because of the dependence of i on xi+1, the regulation or tracking for pure-feedback
systems is, in general, not global, even when is known.
Unknown virtual control coefficients.
xi = bixi+1 + i(x1, . . . ,xi)T, i = 1, . . . ,n 1
T
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xn = bn(x)u+ n(x1, . . . ,xn) ,where, in addition to the unknown vector , the constant coefficients bi are also unknown.
The unknown bi-coefficients are frequent in applications ranging from electric motors to
flight dynamics. The signs of bi, i = 1, . . . ,n, are assumed to be known. In the tuningfunctions design, in addition to estimating bi, we also estimate its inverse i = 1/bi. In the
modular design we assume that in addition to sgnbi, a positive constant i is known such
that
|bi
| i. Then, instead of estimating i = 1/bi, we use the inverse of the estimate bi,
i.e., 1/bi, where bi(t) is kept away from zero by using parameter projection.
Multi-input systems.
Xi = Bi( Xi)Xi+1 + i( Xi)T, i = 1, . . . ,n 1
T
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Xn = Bn(X)u+ n(X) ,
where Xi is a i-vector, 1 2 n, Xi =XT1 , . . . ,X
Ti
T, X= Xn, and the matrices
Bi( Xi) have full rank for all Xi IRij=1j. The input u is a n-vector.
The matrices Bi can be allowed to be unknown provided they are constant and positive
definite.
Block strict-feedback systems.
xi = xi+1 + i(x1, . . . ,xi, 1, . . . ,i)T, i = 1, . . . , 1
T
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x = (x, )u+ (x, ) i = i,0( xi, i) + i( xi, i)
T, i= 1, . . . ,
with the following notation: xi = [x1, . . . ,xi]T, i =
T1 , . . . , Ti
T, x = x, and = .
Each i-subsystem is assumed to be bounded-input bounded-state (BIBS) stable with
respect to the input ( xi, i1). For this class of systems it is quite simple to modify theprocedure in the tables. Because of the dependence of i on i, the stabilizing function
i is augmented by the term + i1k=1 i1k k,0, and the regressor wi is augmented by
i1k=1 i
i1k
T.
Partial state-feedback systems. In many physical systems there are unmeasured states
as in the output-feedback form, but there are also states other than the output y = x1 that
are measured. An example of such a system is
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x1 = x2 + 1(x1)T
x2 = x3 + 2(x1,x2)T
x3 = x4 + 3(x1,x2)T
x4 = x5 + 4(x1,x2)T
x5 = u+ 5(x1,x2,x5)T .
The states x3 and x4 are assumed not to be measured. To apply the adaptive backsteppingdesigns presented in this chapter, we combine the state-feedback techniques with the
output-feedback techniques. The subsystem (x2,x3,x4) is in the output-feedback form
with x2 as a measured output, so we employ a state estimator for (x2,x3,x4) using the
filters introduced in the section on output feedback.
Example of Adaptive Stabilization in the Presence of a Stochastic Disturbance
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dx = udt+xdw
w: Wiener process with Edw2
= (t)2dt, no a priori bound for
Control laws:
Disturbance Attenuation: u = xx3Adaptive Stabilization: u = x x, = x2
Disturbance Attenuation Adaptive Stabilization
6
7
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0 0.5 1 1.5 2 2.50
0.5
1
1.5
2
2.5
3
3.5
4
4.5
t
x
dd
0 0.5 1 1.5 2 2.50
1
2
3
4
5
x
t
0 0.5 1 1.5 2 2.50
0.5
1
1.5
2
2.5
3
t
Major Applications of Adaptive Nonlinear Control
Electric Motors Actuating Robotic Loads
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Nonlinear Control of Electric Machinery, Dawson, Hu, Burg, 1998. Marine Vehicles (ships, UUVs; dynamic positioning, way point tracking, maneu-
vering)
Marine Control Systems, Fossen, 2002
Automotive Vehicles (lateral and longitudinal control, traction, overall dynamics)The groups of Tomizuka and Kanellakopoulos.
Dozens of other occasional applications, including: aircraft wing rock, compressor stall and
surge, satellite attitude control.
Other Books on Adaptive NL Control Theory Inspired by [KKK]
1. Marino and Tomei (1995),
Nonlinear Control Design: Geometric, Adaptive, and Robust
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2. Freeman and Kokotovic (1996),
Robust Nonlinear Control Design: State Space and Lyapunov Techniques
3. Qu (1998),
Robust Control of Nonlinear Uncertain Systems
4. Krstic and Deng (1998),
Stabilization of Nonlinear Uncertain Systems
5. Ge, Hang, Lee, Zhang (2001),
Stable Adaptive Neural Network Control
6. Spooner, Maggiore, Ordonez, and Passino (2002),
Stable Adaptive Control and Estimation for Nonlinear Systems: Neural and Fuzzy Approximation Tech-
niques
7. French, Szepesvari, Rogers (2003),
Performance of Nonlinear Approximate Adaptive Controllers
Adaptive NL Control/Backstepping Coverage in Major Texts
1 Khalil (1995/2002)
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1. Khalil (1995/2002),Nonlinear Systems
2. Isidori (1995),
Nonlinear Control Systems
3. Sastry (1999),
Nonlinear Systems: Analysis, Stability, and Control
4. Astrom and Wittenmark (1995),
Adaptive Control