A First Course in FUZZY and NEURAL CONTROL

6.2. INVERSE DYNAMICS 203

x(k+3) = Ax(k+2)+Bu(k+2)

= A

°

A^2 x(k)+ABu(k)+Bu(k+1)

¢

+Bu(k+2)

= A^3 x(k)+A^2 Bu(k)+ABu(k+1)+Bu(k+2)

x(k+4) = Ax(k+3)+Bu(k+3)

= A

°

A^3 x(k)+A^2 Bu(k)+ABu(k+1)+Bu(k+2)

¢

+Bu(k+3)

= A^4 x(k)+A^3 Bu(k)+A^2 Bu(k+1)+ABu(k+2)+Bu(k+3)

andfinally

x(k+n)=Anx(k)+An−^1 Bu(k)+∑∑∑+ABu(k+n−2) +Bu(k+n−1)

= Anx(k)+WU

where
W=

£

An−^1 BAn−^2 B ∑∑∑ A^2 BABB

§

is then◊ncontrollability matrix discussed in Section 2.3, and

U=

£

u(k) u(k+1) ∑∑∑ u(k+n−2) u(k+n−1)

§T

If the matrixWis nonsingular, we can computeUdirectly fromWandx:

U = W−^1 (x(k+n)−Anx(k))

= φ(x(k),x(k+n))

This is a control law for the system that is derived directly from the plant
dynamics by computing the inverse dynamics.

Example 6.1TakeA=





010

001

− 5 − 2 − 3



andB=





0

0

9



. Then

W=

£

A^2 BABB

§

=





900

−27 9 0

63 −27 9





and

U =







1

9 00

1

3

1

9 0

2

9

1

3

1

9











x(k+3)−







− 5 − 2 − 3

15 1 7

−35 1 − 20





x(k)







=







1

9 00

1

3

1

9 0

2

9

1

3

1

9





x(k+3)−







−^59 −^29 −^13

0 −^59 −^29

00 −^59





x(k)

=







1

9 x^1 (k+3)+

5

9 x^1 (k)+

2

9 x^2 (k)+

1

3 x^3 (k)

1

3 x^1 (k+3)+

1

9 x^2 (k+3)+

5

9 x^2 (k)+

2

9 x^3 (k)

2

9 x^1 (k+3)+

1

3 x^2 (k+3n)+

1

9 x^3 (k+3)+

5

9 x^3 (k)







= φ(x(k),x(k+n))