Cuadro 2 Esquema de entrevista sobre creatividad
6.3. CATEGORÍA 3: LA CREATIVIDAD EN LA ESCUELA En el siguiente apartado se recogen los resultados de la categoría 3 titulada la
6.4.4. CREATIVIDAD Y EVALUACIÓN.
M U LTIVARIABLE SMITH PREDICTOR WITH ADDITIVE
MODELLING ERRORS
The Smith predictor determines that the control law should be of the form:
u(y) = (I+ KH)- 1 K (8.6. 1 )
where
H( y) = (D( y)-E)G( y) (8.6.2)
When this control law is applied, the closed-loop can be described by Figure 8.6. 1
r + u * y
K G
h
H
Figure 8.6.1
This block diagram can be rearranged as Figure 8.6.2
r u * y K . G A A * G G + Figure 8.6.2
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In this representation i t i s clear that i f the model i s exact, the outer-loop,
which is the difference between the true output and the expected (modelled) output, 6y, is zero and the controller is effectively feed-forward. If there is any noise in
the process and the process is open-loop unstable then the Smith predictor scheme will fail to control the process as was shown in Section 8.2.
However, in general the model will not always exactly equal the real process and mismatch will occur. It is of interest to know how much mismatch can exist before the controller fails to stabilise the process. If a controller will continue to stabilise a process even when the process is not modelled exactly, then the system is said to have a degree of Robust Stability.
A wide range of literature exists that investigates the robustness of delay free processes. However, much less is known about the robustness of time-delayed processes. It is known, however, that the traditional Smith predictor can be sensitive to modelling errors. In this section, the robust stability result of Owens and Raya ( 1 982) is extended to include the multi variable Smith predictor. Owens and Raya ( 1 982) investigated the case when the errors were assumed to be additive and the matrix delay-operator, D(y), acted only on the outputs and was a square matrix with non-zero elements only on the main diagonal. The extension to the non-square case is straightforward and relies only on the inclusion of the E matrix that is used in the construction of the multi variable Smith predictor.
Definition
8.2:
A vector norm is a measure of the size of a vector. It has the following properties:llxll � 0 Jlaxll = lalllxll
There are several commonly used vector norms. These include the Euclidean norm
and the absolute norm.
Definition
8.3:
The induced matrix operator norm has the propertiesIIXII � 0
llaXII = laiiiXII
IIX + Yll � IIXII + IIYII
For example 11 ':11 = max I,l(.)ijl is the matrix norm induced by the vector norm IHI i j
= max l(.)il 1
Definition
8.4: A
process is Bounded-Input Bounded-Output (BIBO) stable if for any finite initial conditions and for any u(t) such that llu(t)ll exists and is finite, the outputs y(t) exist and have finite norm.Theorem 8.3: Contraction mapping theorem.
If the mapping T: U --7 U and IIT(u)- T(v)ll � llu - vll then the mapping has a
convergence point.
Proof: See Ortega and Rheinboldt ( 1970)
By defining U to be the range of possible actuations, u, let Y be the
space of possible outputs, y, and Z to be the space of possible vectors, z. Suppose
that U 0 is a linear subs pace of U,Y 0 is a linear subspace of Y and Zo is a linear
subspace of Z; these are regarded as spaces of stable inputs, outputs and process-
outputs, z, respectively. Assuming that these vector spaces have a norm topology
with respect to which they are Banach Spaces, it is possible to state the following theorem.
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Theorem 8.4: A multivariable Smith predictor is Bounded-Input Bounded-Output (BIBO) stable if
1\
(a) The delay-free plant G(y) and its model G (y) maps Uo into Zo
(b) The delay component D, its model f> and E map Z0 into Z0•
(c) The restriction to Yo of the delay free mapping
(d) (e)
1\
r � Ua= (I+ KEG t l Kr
has range in U and has finite induced nonn.
1\ 1\
llt = 11 (I + KEG t l KmG 11 < 1
1 1\
ll2 = l -Ilt 11 (I + KEG tl KDL1GII<l
Comment: Conditions (a) and (b) are a requirement that the plant and the model
are both open-loop stable. Condition (c) requires that the closed-loop model is stable and bounded and finally conditions (d) and (e) provide upper bounds on the additive mismatches of the delays and the delay-free dynamics.
These requirements of stability echo those of the Internal Model Control
structure discussed in Chapter 6.
Proof: (Adapted from Owens and Raya, 1982).
By assumption G(y) and D(y) are stable and bounded so it is sufficient to prove
that ue Uo whenever re Yo.
From Figure 8.6.2 it can be seen that
1\ 1\ 1\
u = K(r - EG u -(DG-DG ) u)
which can be written as
1\ 1\ 1\ u = (I + KEG tiK[r-(DG - DG ) u]
This is an equation in U of the form
u = Wr(u)
Conditions (a) to (c) ensure that Wr maps Uo into Uo for all r in Yo. Now 1\ 1\ 1\ Ao = 11(1 + KEG t I K(DG -DG )11 1\ 1\ =11(1 + KEG t 1 K(D�G + WG )11 1\ 1\ A
< 11(1 + KEG tiKD�G 11 + 11(1 + KEG tlKmG 11
(8.6.4)
(8.6.5)
The two absolute values on the right hand side of this inequality can be replaced using assumptions (d) and (e) giving:
Ao < A20-At) + AI < (I-AI) + AI = I
Thus it can be seen that
IIW,(u)ll � 11 u 11
(8.6.6)
(8.6.7)
Hence W r(u) is a contraction mapping of Uo into itself. By the contractive mapping theorem W,(u) has only one fixed point. Since the process output Gu is
finite (and since Gu E Zo from Theorem 8.4, assumption (a)), the process is BIBO as required.
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8.7 R EVIEW
In this chapter it has been proved that the poles of a Smith predictor controlled time-delayed process can be divided into two sets: the poles of the equivalent delay-free closed-loop process and the poles of the open-loop process. The proof utilises the results of Chapter 3 in order to decompose the transfer function in an appropriate manner. Hence it can be seen that a time-delayed open-loop unstable process cannot be stabilised using a Smith predictor.
Two ways in which modelling errors be accounted for in process descriptions were examined. It was seen that some robustness results relating to Smith predictors could be derived for inexactly modelled process descriptions when either additive or multiplicative model errors were assumed.