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0066_frame_C14.fm Page 34 Wednesday, January 9, 2002 1:51 PM

Using the nonquadratic Lyapunov candidate

h 2γ + j+γ +1 T j+γ +1 ς 2β + j+β+1 T j+β+1
---------------
---------------


----------------
----------------
1
1
Vt, x, e) = ∑ --------------------------- x 2γ +1   K xj t()x 2γ +1 + ∑ ---------------------------- e 2β+1   K ej t()e 2β+1
(
-
1) 
1) 
γ
(
β +
2 j ++
(
2 j +
j=0 j=0
σ 2µ + j+µ+1 T j+µ+1

----------------
----------------
1
+ ∑ ---------------------------- e 2µ+1   K ij t()e 2µ+1
1) 
µ +
2 j +(
i=0
one obtains the bounded controller as
j−γ
η

---------------
u = u as = – sin ( RTθ rm ) 0 Φ G x t()B T ∑ diag x ------------- K xj t()x j+γ +1
2γ +1
2γ +1

0 cos ( RTθ rm ) 
u bs
j=0
2j+1
ς ------------- σ 2µ+1
2j+1
-------------
1
+ G e t()B e∑ K ej t()e 2β+1 + G i t()B e∑ K ij t()--e 
T
T
s 
j=0 j=0
Here, K xj (⋅) are the unknown matrix-functions, and K ej (⋅) and K ij (⋅) are the unknown coefficients; η = 0, 1,
2,…; γ = 0, 1, 2,…; ς = 0, 1, 2,…; β = 0, 1, 2,…; σ = 0, 1, 2,…; and µ = 0, 1, 2,….
Under the assumption that X 0 , E 0 , R, D, Z, and P are admissible, the robust tracking problem is solvable
in XE. That is, the bounded real-analytic control u(⋅) guarantees the robust stability and steers the tracking
error to S e . Furthermore, stability is guaranteed, disturbance rejection is ensured, and specified input-
output tracking performance can be achieved.
Applying the controller designed, one maximizes the electromagnetic torque developed by permanent-
magnet stepper micromotors. This can be easily shown by using the expression for the electromagnetic
torque, the balanced two-phase sinusoidal voltage set (applied phase voltages u as and u bs ), as well as the
2 2
trigonometric identity sin a + cos a = 1.
The tracking controller can be designed using the tracking error. In particular, we have
2j+1
σ
ς

-------------
1
-------------
u = u as = – sin ( RTθ rm ) 0 Φ G e t()B e∑ K ei t()e 2j+1 + G i t()B e∑ K ij t()--e 2µ+1 
T
2β+1

T
0 cos ( RTθ rm )  s 
u bs
j=0 i=0
The controller design, implementation, and experimental verification are reported in [9].
14.7 Conclusions
This chapter reports the current status, documents innovative results, and researches novel paradigms
in synthesis, modeling, analysis, simulation, control, and optimization of high-performance MEMS.
These results are obtained applying reported nonlinear modeling, analysis, synthesis, control, and opti-
mization methods which allow one to attain performance assessment and predict outcomes. Novel MEMS
were devised. The application of the plate, spherical, torroidal, conical, cylindrical, and asymmetrical
motor geometry, as well as endless, open-ended, and integrated electromagnetic systems, allows one to
classify MEMS. This idea is extremely useful in the studying of existing MEMS as well as in the synthesis
of innovative high-performance MEMS. For example, asymmetrical (unconventional) geometry and
integrated electromagnetic system can be applied. Optimization can be performed, and the classifier
paradigm serves as a starting point from which advanced configurations can be synthesized and straight-
forwardly interpreted. Microscale motion devices geometry and electromagnetic systems, which play a
central role, are related. Structural synthesis and optimization of MEMS are formalized and interpreted
using innovative ideas. The MEMS classifier paradigm, in addition to being qualitative, leads one to
quantitative analysis. In fact, using the cornerstone laws of electromagnetics and mechanics (e.g., Maxwell’s,
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