Page 311 - The Mechatronics Handbook

0066_frame_C14.fm Page 25 Wednesday, January 9, 2002 1:50 PM

with positive semi-definite constant-coefficient matrix Q and positive-definite matrix G, one finds
xt()
xt()
+1 
ut() = – tanh   G – 1 B T K et() t   ≈ – sat 1– G – 1 B T K et() t   , – 1 ≤ u ≤ 1

0
0
d
d
This controller is obtained assuming that the solution of the functional partial differential equation
can be approximated by the quadratic return function
V = 1 T
--x Σ Kx Σ
2
where K is the symmetric matrix.
Time-Optimal Control

A time-optimal controller can be designed using the functional

J = 1 ∫ t f ( x Σ Qx Σ ) td
T
--
2 t 0
Taking note of the Hamilton–Jacobi equation

∂V 1 T  ∂V  T
– ------- = min --x Σ Qx Σ +  -------- ( Ax Σ + B Σ u)

∂t – 1≤u≤1 2 ∂x Σ
the relay-type controller is found to be


u = – sgn B Σ T ∂V   – 1 ≤ u ≤ 1
-------- ,

∂x Σ
This “optimal” control algorithm cannot be implemented in practice due to the chattering phenom-
enon. Therefore, relay-type control laws with dead zone

u = – sgn   B Σ T ∂V   , – 1 ≤ u ≤ 1
--------
∂x Σ
dead zone
are commonly used.
Sliding Mode Control

Soft-switching sliding mode control laws are synthesized in [9]. Sliding mode soft-switching algorithms
provide superior performance, and the chattering effect is eliminated.
To design controllers, we model the states and errors dynamics as

x ˙ t() = Ax + Bu, – 1 ≤ u ≤ 1
e ˙ t() = Nr ˙ t() HAx HBu––

The smooth sliding manifold is

(
M = ( { t, x, e) ∈ R ≥0 × X × E υ t, x, e) = 0}
m
∩ ( { t, x, e) ∈ R ≥0 × X × E υ j t, x, e) = 0}
(
=
j=1
©2002 CRC Press LLC