SC 617 Adaptive Control Theory Lecture 12 - 03/03/2015

Back Stepping and Non-CE methodLecturer: Srikant Sukumar Scribe: Srinath Dama

1 Double integral systems (Continued)

Consider the Double integrator system dynamics given as follows:

x1 = x2 + f(x1)

x2 = u (1)

1. Assuming is knownIn order to design a controller for the above system we can use the classical backstepping designapproach. The control objective given as follows:

limt

x1 0

limt

x2 x2d

Where,

x2d = K1x1 f(x1) (2)

Consider V1() and V2() as follows:

V1 =1

2x1

2

V2 =1

2(x2 x2d)2

V2 = (x2 x2d)(u x2d) (3)

Consider the update law, u = x2d K2(x2 x2d). Hence, the overall Lyapunov function is given asfollows:

V = V1 + V2

V = x1x1 K2(x2 x2d)2

= x1x2 + x1f(x1)K2(x2 x2d)2

= x1x2 + x1(x2d K1x1)K2(x2 x2d)2

(K1 1

2)x1

2 (K2 1

2)(x2 x2d)2

V 0 (4)

2. Assuming is unknown

First control x2d = K1x1 f(x1)if x2 = x2d

x1 = K1x1 + f(x1)

1

Let us take V1 =1

2x1

2 +1

22

V1 = x1x1 1

= x1(K1x1 + f(x1))1

= K1x12 + (x1f(x1)

)

First Update = x1f(x1)

V1 K1x12

Let us take V2 =1

2(x2 x2d)2

V2 = (x2 x2d)(x2 x2d)u = x2d K2(x2 x2d)

x2d = K1x1 x1f2(x1) f(x1)

= K1(x2 + f(x1)) x1f2(x1) f(x1)

= (K1 + f

x1)(x2 + f(x1)) x1f2(x1)

x2d = (K1 + f

x1)(x2 + f(x1)) x1f2(x1)

= x2d + f(x1)(K1 + f

x1)

Where =

Let the overall Lyapunov function V = V1 + V2, where, V1() and V2() are given as follows:

V1 =1

2x1

2 +1

22

V2 =1

2(x2 x2d)2 +

1

22

Therefore, V evaluated as follows:

V = x1x2 + x1f(x1)1

x1f(x1) + (x2 x2d)(u x2d)

1

second control u = x2d K2(x2 x2d)

= x2d + f(x)(K1 + f

x1)

= x1(x2 x2d) + x1x2d + x1f(x1) x1f(x1)K2(x2 x2d)2

+ (x2 x2d)f(x1)(K1 + f

x1)) 1

= K1x12 K2(x2 x2d)2 + x1(x2 x2d) 0

Second update = (x2 x2d)f(x1)(K1 + f

x1)

2

Hence,by signal chasing we can prove x1 0, x2 0 and x2 x2d, x2 x2d

2 Creating an attractive set for parameter estimation error

Non-certainty Equivalence

= (5)

Consider the scalar first order system

x = ax+ bu (6)

where, a, b are the unknown constants.

Control objective:

limt

x(t) r(t) = 0

The error dynamics given as follows:

e = ax+ bu r(t)

= b(u+a

bx 1

br(t))

Let, 1 =ab and 2 =

1b . Reconsider the control law, u = 1x 2(Ke r)

e = Ke+ b(u+ abx+

1

b(Ke r))

e = Ke+ b([x, (Ke r)] + u)

Where, = [1, 2 ]T and W = [x, (Ke r)]. The filters are given as follows:

ef = ef + eWf = Wf +Wuf = uf + uef = Kef + b(Wf + uf )

If non CE uf = Wf ( + )

Where - Dynamics and - static

ef = Kef + b(Wf Wf ( + ))

= Kef bWf ( + )= Kef bWfZ

where Z = + , Z = + , = WfT efSgn(b)

Z = + Wf

TefSgn(b) +Wf

T (Kef bWfZ)Sgn(b)

let = Wf

TefSgn(b) +KWf

T efSgn(b)

Z = | b |WfTWfZSgn(b)

3

Consider the Lyapunov function V = Z2

2 +e2f2

V = ZZ + ef ef

= Z( | b |WfTWfZ) + ef (Kef bWfZ)= | b | WfZ 2 Ke2f befWfZ

( | b | b2

2)WfZ2 (K

1

2)e2f

0

Likewise, signal chasing we can proove the following,

ef 0WfZ 0

e 0

4