krstic _talk.pdf

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


2/90

Adaptive Control

Plantinput output

output filterinput filter

identifier

controller


3/90

Adaptive Control

Plantinput output


identifier

controller


4/90

Adaptive Control

Plantinput output


identifier

controller


5/90

Adaptive Control

Plantinput output


identifier

controller


6/90

Plant

y =B(s)

A(s)u


7/90

Plant

y =B(s)

A(s)u

B(s) = bmsm+bm1sm1 + +b1s+b0

A(s) = ansn+an1sn1 + +a1s+a0


8/90

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


9/90

Controller

u =Q(s)

P(s)y


10/90

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.


11/90

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.


12/90

Adaptive Control

B(s)/A(s)input output


identifier

Q(s)/P(s)


13/90

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


14/90


Lyapunov




This talk:




15/90


Lyapunov




This talk:




16/90

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


17/90


ut = uxx+ (x)u

Measurement: u(0)

Control: u(1)

Unstable


Scalar input / scalar output


18/90


ut = uxx+ (x)u

Measurement: u(0)

Control: u(1)

Unstable


Scalar input /scalar output


19/90

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)


20/90


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



p(1,y) = 0

p(x,x) = 12

Z1x

(y)dy

(x) = py(x, 0)1 = p(0, 0)


21/90


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



p(1,y) = 0

p(x,x) = 12

Z1x

(y)dy

(x) = py(x, 0)1 = p(0, 0)


22/90


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



p(1,y) = 0

p(x,x) = 12

Z1x

(y)dy

(x) = py(x, 0)1 = p(0, 0)


23/90

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


24/90

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


25/90

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


26/90

Input filter

t = xx

x(0) = 0

(1) = u(1)

Output filters

t = xx

x(0) = u(0)

(1) = 0


27/90

Adaptive scheme

ut= uxx+ (x)u

ux(0) = 0

u(1) u(0)

(x), 1


identifier

controller


28/90

Adaptive scheme

ut= uxx+ (x)u

ux(0) = 0

u(1) u(0)


(x), 1 identifier

controller


29/90

Adaptive scheme

ut= uxx+ (x)u

ux(0) = 0

u(1) u(0)

(x), 1


identifier

controller

U d l (l )


30/90

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


31/90

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


32/90

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


33/90

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


34/90

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


35/90


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



36/90



Tao



Sastry and Bodson

For nonlinear systems

Krstic, Kanellakopoulos, and Kokotovic



37/90



Tao



Sastry and Bodson

For nonlinear systems Krstic, Kanellakopoulos, and Kokotovic



38/90


Comprehensive

Ioannou and Sun

Tao



Sastry and Bodson

For nonlinear systems Krstic, Kanellakopoulos, and Kokotovic

General Remarks on Adaptive Control


39/90


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)



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Ge e a e a s o dapt e Co t o



Limitations:








41/90

p



Limitations:






General Remarks on Adaptive Control (contd)


42/90

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)


43/90

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


44/90

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


45/90

x = f(x) +g(x)u

y = h(x)



Nonlinear Control


46/90

x = f(x) +g(x)u

y = h(x)



Methods


47/90

Feedback linearization

Backstepping (robust and adaptive)

Various varieties of nonlinear optimal control (including MPC)

Books on Nonlinear Control


48/90

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


49/90

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)


50/90

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:


51/90

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


52/90

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


53/90

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


54/90

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:


55/90

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


56/90

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

.


57/90

+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


58/90

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


59/90

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


60/90

= 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


61/90

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


62/90

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


63/90

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


64/90

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


65/90

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


66/90

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


67/90

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


68/90

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


69/90

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


70/90

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


71/90

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


72/90

Passive identifier

x = f+FT

x = A0 FTFP( xx) + f+F

T

c

T

'FP

Z

E

'

+

x

x

E E

T

FP


73/90

'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


74/90

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 .


75/90

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

krstic _talk.pdf

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