Page 312 - The Mechatronics Handbook

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The time-varying nonlinear switching surface is υ(t, x, e) = K υxe (t, x, e) = 0. The soft-switching control
law is given as
ut, x, e) = – Gφυ(), – 1 ≤ u ≤ 1, G > 0
(

where φ(⋅) is the continuous real-analytic function of class C ( ≥ 1), for example, tanh and erf.
Constrained Control of Nonlinear MEMS: Hamilton–Jacobi Method
Constrained optimization of MEMS is a topic of great practical interest. Using the Hamilton–Jacobi theory,
the bounded controllers can be synthesized for continuous-time systems modeled as

MEMS MEMS MEMS 2w+1 MEMS
(
(
x ˙ t () = F s x ) + B s x )u , y = Hx
u min ≤ u ≤ u max , x MEMS () x 0 MEMS
t 0 =
Here, x MEMS ∈ X s is the state vector; u ∈ U is the vector of control inputs: y ∈ Y is the measured output;
F s (⋅), B s (⋅) and H(⋅) are the smooth mappings; F s (0) = 0, B s (0) = 0, and H(0) = 0; and w is the nonnegative
integer.
To design the tracking controller, we augment the MEMS dynamics
MEMS MEMS MEMS 2w+1 MEMS
(
(
(
x ˙ t () = F s x ) + B s x )u y = H x )

u min ≤ u ≤ u max , x MEMS () x 0 MEMS
t 0 =
ref
with the exogenous dynamics x ˙ () = Nr y = Nr H x MEMS ).
(
–
–
t
Using the augmented state vector
MEMS
x
x = ∈ X
x ref
one obtains
x MEMS
x ˙ t() = Fx, r) + Bx()u 2w+1 , u min ≤ u ≤ u max , xt 0 = x 0 , x =
(
()
x ref
F s x( MEMS ) B s x MEMS )
(
(
Fx, r) = + 0 r, Bx() =
(
– H x MEMS ) N 0
The set of admissible control U consists of the Lebesgue measurable function u(⋅), and a bounded
controller should be designed within the constrained control set
U = { u ∈ u imin ≤ u i ≤ u imax , i = 1,…,m}.
m
We map the control bounds imposed by a bounded, integrable, one-to-one, globally Lipschitz, vector-

valued continuous function Φ ∈C ( ≥ 1). Our goal is to analytically design the bounded admissible state-
feedback controller in the closed form as u = Φ(x). The most common Φ are the algebraic and transcen-
dental (exponential, hyperbolic, logarithmic, trigonometric) continuously differentiable, integrable, one-
to-one functions. For example, the odd one-to-one integrable function tanh with domain (−∞, +∞) maps
−1
the control bounds. This function has the corresponding inverse function tanh with range (−∞, +∞).
The performance cost to be minimized is given as

∫
T
–
1
2w
J = ∫ ∞ [ W x x() + W u u()] t = ∫ ∞ W x x() + ( 2w + 1) ( Φ ()) G diag u ) u d t
(
1
–
d
d
u
t 0 t 0
−1
where G ∈ m×m is the positive-definite diagonal matrix.
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