Page 309 - The Mechatronics Handbook

0066_frame_C14.fm Page 23 Wednesday, January 9, 2002 1:50 PM

The matrices of coefficients are A, B, and H. The smooth mapping fields of the nonlinear model are
denoted as F z (⋅), B p (⋅), and H(⋅).
It should be emphasized that the control is bounded. For example, using the IC duty ratio d D as the
control signal, we have 0 ≤ d D ≤ 1 or −1 ≤ d D ≤ +1. Four-quadrant ICs are used due to superior
performance, and −1 ≤ d D ≤ +1. Hence, we have −1 ≤ u ≤ +1. However, in general, u min ≤ u ≤ u max .

Proportional-Integral-Derivative Control
Many MEMS can be controlled by the proportional-integral-derivative (PID) controllers, which, taking
note of control bounds, are given as [9]

de
min ∫
ut() = sat u u max   e, et, ----- 
d
dt
 ς ------------- σ ------------- α 2j+1
2j+1
2j+1
u  ∑ 2β+1 + ∑ ∫ 2µ+1 dt + ∑ 2γ+1
-------------
= sat u  k pj e k ij e k dj e ˙  , u min ≤ ≤
max
min  j=0 j=0 j=0  uu max
 proportional integral derivative 
where k pj , k ij , and k dj are the matrices of the proportional, integral, and derivative feedback gains; ς, β, σ,
µ, α, and γ are the nonnegative integers.
In the nonlinear PID controllers, the tracking error is used. In particular,
et() = rt() – yt()
reference/command output
Linear bounded controllers can be straightforwardly designed. For example, letting ς = β = σ = µ = 0,
we have the following linear PI control law:

∫
ut() = sat u u max  k p0 et() + k i0 et t  
d
min
The PID controllers with the state feedback extension can be synthesized as

u
ut() = sat u ( e, x)
max
min
 
 ς ------------- σ ------------- α 2j+1 
2j+1
2j+1
(
∂Ve, x)
u  ∑ k pj e 2β+1 + ∑ ∫ 2µ+1 ∑ ------------- + Gt()B--------------------- 
2γ +1
= sat u  k ij e dt + k dj e ˙  , u min ≤ uu max
≤
max
min  j=0 j=0 j=0 ∂ e 
 x 
 proportional integral derivative 
where V(e, x) is the function that satisfies the general requirements imposed on the Lyapunov pair [9],
e.g., the sufficient conditions for stability are used.
It is evident that nonlinear feedback mappings result, and the nonquadratic function V(e, x) can be
synthesized and used to obtain the control algorithm and feedback gains.
Tracking Control
Tracking control is designed for the augmented systems, which are modeled using the state variables and
the reference dynamics. In particular, from
ref
x ˙ t() = Ax + Bu, x ˙ () = rt() yt() = rt() Hx t()
–
–
t
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