Page 316 - The Mechatronics Handbook

0066_frame_C14.fm Page 30 Friday, January 18, 2002 4:36 PM

Suppose that a set of admissible control U consists of the Lebesgue measurable function u(·). It was
demonstrated that the Hamilton–Jacobi theory can be used to find control laws, and the minimization
of nonquadratic performance functionals leads one to the bounded controllers.
Letting u = Φ(t, e, x), one obtains a set of admissible controllers. Applying the error and state feedback
we define a family of tracking controllers as

)
T1∂Vt, e, x(
)
T ∂Vt, e, x(


d
u = Ω x()Φ t, e, x) = – Ω x()Φ G E t()B E t, x) --------------------------- + G X t()Bt, x) ------------------------- , s = -----
(
(
(


∂e
s
dt
∂x
where Ω(·) is the nonlinear function; G E (·) and G X (·) are the diagonal matrix-functions defined on [t 0 ,∞);
B E (·) is the matrix-function; and V(·) is the continuous, differentiable, and real-analytic function.
Let us design the Lyapunov function. This problem is a critical one and involves well-known difficulties.
The quadratic Lyapunov candidates can be used. However, for uncertain nonlinear systems, nonquadratic
functions V(t, e, x) allow one to realize the full potential of the Lyapunov-based theory and lead us to
the nonlinear feedback maps which are needed to achieve conflicting design objectives. We introduce
the following family of Lyapunov candidates:
ς i+β+1 T i+β+1 η i+γ +1 T i+γ+1
2β +
2γ +

----------------
1
---------------
---------------

1
----------------
(
Vt, e, x) = ∑ ---------------------------- e 2β+1   K Ei t()e 2β+1 + ∑ ---------------------------- x 2γ +1   K Xi t()x 2γ +1
1) 
1) 
2 i ++
2 i +
β +
γ
(
(
i=0 i=0
where K Ei (·) and K Xi (·) are the symmetric matrices; ζ, β, η, and γ are the nonnegative integers; ζ = 0, 1,
2,…; β = 0, 1, 2,…; η = 0, 1, 2,…; and γ = 0, 1, 2,…
The well-known quadratic form of V(t, e, x) is found by letting ζ = β = η = γ = 0, and we have
(
Vt, e, x) = --e K E0 t()e + 1 T
1 T
--x K X0 t()x
2 2
By using ζ = 1, β = 0, η = 1, and γ = 0, one obtains a nonquadratic candidate:
T T
Vt, e, x) = --e K E0 t()e + --e K E1 t()e + 1 T 1 2 2
1 T
(
1 2
--x K X0 t()x +
2
--x K X1 t()x
2 4 2 4
One obtains the following tracking control law:
i−β
 ς -------------- i+β+1
----------------
1
u = – Ω x()Φ G E t()B E t, x( ) T ∑ diag et() 2β+1 K Ei t()--et() 2β+1

 s
i=0
η ------------- i+γ+1
i−γ
--------------- 
(
+ G X t()Bt, x) T ∑ diag xt() 2γ +1 K Xi t()xt() 2γ +1 
i=0 
i−β
-------------
2β+1
e 1 0 L 0 0
i−β
-------------
2β+1
i−β 0 L 0 0
------------- e 2
diag et() 2β+1 = M M O M M
i−β
-------------
2β+1
0 0 L e b−1 0
i−β
-------------
2β+1
0 0 M 0 e b
©2002 CRC Press LLC