Page 313 - The Mechatronics Handbook

0066_frame_C14.fm Page 27 Wednesday, January 9, 2002 1:50 PM

Performance integrands W x (⋅) and W u (⋅) are real-valued, positive-definite, and continuously differen-
−1
tiable integrand functions. Using the properties of Φ one concludes that inverse function Φ is integrable.
Hence, integral

∫ ( Φ ()) G diag u ) u
T
1
(
1
–
2w
–
u
d
exists.
Example
Consider a nonlinear dynamic system
dx 3
------ = ax + bu , u min ≤ u ≤ u max
dt
Taking note of
∫
T
(
1
2w
1
W u u() = ( 2w + 1) ( Φ ()) G diag u ) u
–
–
d
u
one has the positive-definite integrand
∫
--u +
--u tanh u +
W u u() = 3 tanh uG u u = 1 3 – 1 1 2 1 ( 1 u ), G – 1 = 1
2
–
1
1
2
–
--ln
–
--
d
6
3
3
6
In general, if the hyperbolic tangent is used to map the saturation effect, for the single-input case, one has
2w+1
1u
1 u
u
2 ∫
∫
–
W u u() = ( 2w + 1) u tanh --- u = u 2w+1 tanh --- – k --------------- u
–
2w
d
d
k
2
k
u
k –
Necessary conditions that the control function u(·) guarantees a minimum to the Hamiltonian
∫
T
H = W x x() + ( 2w + 1) ( Φ ()) G diag u ) u + ∂Vx() T [ ) + Bx()u 2w+1 ]
1
----------------- Fx, r(
–
–
(
2w
1
d
u
∂x
are: first-order necessary condition n1,
∂H
------- = 0
∂u
and second-order necessary condition n2,
2
∂ H
--------------------- > 0
∂u × ∂u T
κ
The positive-definite return function V(·), V ∈ C , κ ≥ 1, is
()
Vx 0 = inf Jx 0 , u( ) = inf Jx 0 , Φ ·()) ≥ 0
(
u∈U
The Hamilton–Jacobi–Bellman equation is given as
∂V  – 1 T – 1 2w ∂Vx() T 2w+1 
∫
(
[

u
d
– ------- = min W x x() + ( 2w + 1) ( Φ ()) G diag u ) u + ----------------- Fx, r( ) + Bx()u ] 
∂t u∈U  ∂x 
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