7. CRITERIOS DE ESTILO VISUAL Y MARCA
7.16 Aplicaciones Audiovisuales
7.16.8 Textos destacados sobre video
In general, chattering must be eliminated for the controller to perform properly. This can be achieved by smoothing out the control discontinuity in a thin boundary layer neighboring the switching surface
0 boundary layer width. Fig. 7.6.a illustrates boundary layer for the casen=2.
φ
ε
ε boundary l
ayer x&
x
Fig. 7.6.a The boundary layer Fig. 7.6.b illustrates this concept:
- Out side ofB(t), choose the control law u as before (7.5)
Fig. 7.6.b Control interpolation in the boundary layer Given the results of section 7.1.1, this leads to tracking to within a guaranteed precision ε , and more generally guarantees that for all trajectories starting inside B(t=0)
1
Example 7.2________________________________________
Consider again the system (7.10): x&&=−a(t)x&2cos3x+u ,
Switching control law:
~)
The tracking performance with switching control law is given in Fig. 7.7 and with smooth control law is given in Fig. 7.8.
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4
Fig. 7.7 Switched control input and tracking performance
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4 Tracking Error (x10-3)
Fig. 7.8 Smooth control input and tracking performance
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⊗ Note that:
- The smoothing of control discontinuity inside B(t) essentially assigns a low-pass filter structure to the local
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Chapter 7 Sliding Control 34
dynamics of the variable s , thus eliminating chattering.
Recognizing this filter-like structure then allows us, in essence, to tune up the control law so as to achieve a trade-off between tracking precision and robustness to un-modeled dynamics.
- Boundary layer thickness φ can be made time-varying, and can be monitored so as to well exploit the control
“bandwidth” available.
Consider again the system (7.1): x(n) =f(x)+b(x)u, with ˆ =1
= b
b . In order to maintain attractiveness of the boundary layer now that φ is allowed to vary with time, we must actually modify condition (7.5). Indeed, we now need to guarantee that the distance to the boundary layer always decreases.
≥φ
s ⇒ (s− )φ ≤−η dt
d
−φ
≤
s ⇒ (s− )φ ≥η dt
d
Thus, instead of simply required that (7.5) be satisfy outside the boundary layer, we now required that
≥φ
s ⇒ s s
dt
d ( )
2
1 2≤ φ&-η (7.26)
The additional term φ& s in (7.26) reflects the fact that the boundary layer attraction condition is more stringent during boundary layer contraction (φ&<0) and less stringent during boundary layer expansion (φ&>0).In order to satisfy (7.26), the quantity φ− is added to control discontinuity gain& k(x), i.e., in our smooth implementation the term
) sgn(
)
( s
k x obtained from switched control law u is actually replaced by k(x)sat(s/φ), where
φ x x)= ( )−&
( k
k (7.27)
and sat is the saturation function
=
≤
=
otherwise y
y
y if y y
) sgn(
) sat(
1 )
sat(
and can be seen graphically as in the following figure
−1 ) sat(y
1 y
Accordingly, control law becomesu=uˆ−k(x)sat(s/φ). Now, we consider the system trajectories inside the boundary layer.
They can be expressed directly in terms of the variable s as
) ( )
( x
x φs f k
s&=− −∆ (7.28)
where ∆f =fˆ−f . Since k and ∆ are continuous in x , f using (7.4) to rewrite (7.28) in the form
(
( ) ( ))
)
( s f Oε
k
s=− + −∆ x +
x φ
& (7.29)
We can see from (7.29) that the variable s (which is a measure of the algebraic distance to the surfaceS(t)) can be view as the output of the first order filter, whose dynamics only depend on the desired state x , and whose input are, to d the first order, “perturbations”, i.e., uncertainty ∆f x( d). Thus chattering can be eliminated, as long as high-frequency un-modeled dynamics are not excited.
Conceptually, the perturbations are filtered according to (7.29) to give s , which in turn provides tracking error x~ by further low-pass filtering, according to definition (7.3)
s x~
1storder filter
(7.29) ( ) 1
1 + n−
p λ )
( )
( Oε
f d +
∆
− x
φ of
choice definitionofs Fig. 7.9 Structure of the closed-loop error dynamics Control action is a function of x andx . Since d λis break-frequency of filter (7.3), it must be chosen to be “small” with respect to high-frequency modeled dynamics (such as un-modeled structural modes or neglected time-delays).
Furthermore, we can now turn the boundary layer thickness φ so that (7.29) also presents a first-order filter of bandwidthλ. It suffices to let
=λ φ x ) ( d
k (7.30)
which can be written from (7.27) as )
( d k x φ
φ&+λ = (7.31)
(7.27) can be rewritten as φ x x
x)= ( )− ( )+λ
( k k d
k (7.32)
⊗ Note that:
- The s-trajectory is a compact descriptor of the closed-loop behavior: control activity directly depends on s , while tracking error x~ is merely a filtered version of s - The s-trajectory represents a time-varying measure of the
validity of the assumptions on model uncertainty.
- The boundary layer thickness φ describes the evolution of dynamics model uncertainty with time. It is thus particularly informative to plot s(t),φ(t), and −φ(t)on a single diagram as illustrated in Fig. 7.11b.
Example 7.3________________________________________
Consider again the system described by (7.10):
u x x t a
x&&=− ()&2cos3 + . Assume thatφ(0)=η/λ withη=0.1,
=20
λ . From (7.31) and (7.32)
( ) ( )
φ
φ
&
&
&
&
− +
=
+ +
− +
=
η
λ η η
x x
x x x
x x
k d d
3 cos 5 . 0
3 cos 5 . 0 3
cos 5 . 0 ) (
2
2 2
where, φ&=−λφ+(0.5x&d2 cos3x +η). The control law is now
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Chapter 7 Sliding Control 35
]
- The arbitrary constant η(which formally, reflects the time to reach the boundary layer starting from the outside) is chosen to be small as compared to the average value ofk x( d), so as to fully exploit our knowledge of the structure of parametric uncertainty.
- The value of λis selected based on the frequency range of un-modeled dynamics.
The control input, tracking error, and s -trajectories are plotted in Fig. 7.11.
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4 Tracking Error (x10-3)
Fig. 7.11a Control input and resulting tracking performance
-8
Fig. 7.11b s-trajectories with time-varying boundary layer We see that while the maximum value of the time-varying boundary layer thickness φ is the same as that originally chosen (purposefully) as the constant value of φ in Example 7.2, the tracking error is consistently better (up to 4 times better) than that in Example 7.2, because varying the thickness of the boundary layer allow us to make better use of the available bandwidth.
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Example 7.4________________________________________
A simplified model of the motion of an under water vehicle can be written (7.21): mx&&+cx& x& =u. The a priori bounds on m and c are: 51≤ m≤ and0.5≤ c≤1.5. Their estimate values are mˆ= 5andcˆ=1. λ=20, η=0.1. The smooth control input using time-varying boundary layer, as describe above is designed as follows:
x And the controller is
The results are given in Fig. 7.12
0 0.5 1.0 1.5 2.0 2.5 4 acceleration (m/s2 ) velocity (m/s)
a. References b. Control input
0 1 2 3 4 5 6 Tracking Error (x10-2)
0 1 2 3 4 5 6
- The desired trajectory x must itself be chosen smooth d enough not to excite the high frequency un-modeled dynamics.
- An argument similar to that of the above discussion shows that the choice of dynamics (7.3) used to define sliding surfaces is the “best-conditioned” among linear dynamics, in the sense that it guarantees the best tracking performance given the desired control bandwidth and the extent of parameter uncertainty.
- If the model or its bounds are so imprecise that F can only be chosen as a large constant, then φ from (7.31) is
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Chapter 7 Sliding Control 36
constant and large, so that the term ksat( φs/ )simply equals λs/βin the boundary layer.
- A well-designed controller should be capable of gracefully handling exceptional disturbances, i.e., disturbances of intensity higher than the predicted bounds which are used in the derivation of the control law.
- In the case that λ is time-varying, the term x
u′=−λ&~should be added to the corresponding uˆ , while the augmenting gain k(x) according by the quantity
) 1 ( −
′ β
u . It will be discussed in next section.