Lyapunov Redesign – Min-Max Approach Lecture Notes by B.Yao

Lyapunov Redesign – Min-Max Approach

System ( , ) ( , ) [ ( , , )]x f t x G t x u t x u (M1)

where , , andn px R u R f G are known, and pR represents matched model uncertainties.

Assumptions: A1. The origin of the nominal closed-loop system with ( , )u t x ( , ) ( , ) ( , )x f t x G t x t x (M2) is uniformly asymptotically stable. A2. We know a Lyapunov function for (M2), i.e., we have a continuously differentiable function ( , )V t x that satisfies the inequalities:

1 2( , )x V t x x

3V V f G xt x

(M3)

where 1 2 3, , and are class K functions. A3. With ( , ) ,u t x v the matched uncertain term is bounded by

( , , ) ( , ) , 0 1o ot x v t x v (M4)

where is a known non-negative continuous function.

Lyapunov Redesign – Min-Max Approach Lecture Notes by B.Yao

Objective Design an additional feedback control v such that the overall control u v stabilizes the actual system (M1) in the presence of uncertainties.

Solution The derivative of V in A2 along the trajectories of the actual system (M1) is:

V VV f G Gv Gt x

(M5)

Noting (M3) and letting T Vw Gx

where :pw R

3( ) ( )TV x w v (M6)

Thus 0V if we can pick up v such that

( ) 0Tw v (M7)

which can be done as follows if discontinuous control laws are allowed.

Lyapunov Redesign – Min-Max Approach Lecture Notes by B.Yao

Case 1: (M4) is satisfied for 2

In this case, let

22

( , ) wv t x vw

(M8)

where ( , )t x is a scalar bounding function to be determined. Then,

2 2 2 2( ) ( , )T Tw v t x w w w w

2 2 2( , ) ow w t x v

2 2(1 ) 0o w w (M9)

if

1 o

(M10)

Lyapunov Redesign – Min-Max Approach Lecture Notes by B.Yao

Case 2: (M4) is satisfied for (i.e., component wise).

In this case, let

1sgn( )

( , )sgn( ), sgn( )sgn( )p

wv t x w w

w

(M11)

where is a bounding function to be determined. Then, v and

1 1 11

( )p

Ti i i

iw v w w w w w

1o v w

11 0o w (M12)

if

1( , ) ( , )

1 ot x t x

(M13)

With the above discontinuous control law, 3V x and the origin is uniformly asymptotically stable. However, the same as in the SMC, control input chattering will occur in implementation. To avoid this practical problem,

Lyapunov Redesign – Min-Max Approach Lecture Notes by B.Yao

as well as the theoretical question of the existence of the solution due to the use of discontinuous controls, let us consider the following continuous feedback law for (M8) instead:

2

2

22

( , ) if ( , )

( , ) ( , )

wt x t x ww

vwt x t x w

(M14)

With (M14), when 2w , (M9) is satisfied and thus from (M6):

3V x (M15)

When 2w , (M6) becomes:

2

2 23

TwV x w

23 2 2 2

1ox w w v

23 2 211 ox w w

Lyapunov Redesign – Min-Max Approach Lecture Notes by B.Yao

2

3 21 1 11

2 4ox w

31 14 ox (M16)

Therefore, for any (t, x), (M15) and (M16) imply that

31

4oV x

(M17)

Thus,

13 3

1 12 2

o ox x

and

312

V x (M18)

By Theorem L.17, the solutions are uniformly ultimately bounded (UUB), i.e., there exists a finite time to such that the solution is bounded above by:

Lyapunov Redesign – Min-Max Approach Lecture Notes by B.Yao

1, ,o o ox x t t t t t

1 1 11 2 1 2 3 1

1( ) ,2

ox t t (M19)

where is a class KL function. Note that the bound of the steady-state error in (M19) is proportional to and, theoretically, can be made arbitrarily small by showing a small enough . However, too small will essentially lead to the same control input chattering problem as the original discontinuous control law due to the negated high frequency dynamics.