European Journal of Control (2004)10:338–340 # 2004 EUCA
Discussion on: ‘‘Generalized LFT-Based Representation of Parametric Uncertain Models’’ Juan C. Cockburn Department of Computer Engineering, Rochester Institute of Technology, Rochester, NY 14623-5603, USA
1. Introduction
(a)
In order to use robust control techniques such as structured singular value () analysis and synthesis of feedback systems it is necessary to model the plant as a linear fractional transformation of a system matrix and a perturbation matrix as shown in Fig. 1. The input–output mapping G corresponding to the block diagrams (a) and (b) is, respectively,
¼ M11 þ M12 ðI M22 Þ1 M21 , e , Þ GðÞ ¼ F l ðM e 12 , e 22 þ M e 21 ðI M e 11 Þ1 M ¼M
M11
M12
M21
M22
u
~ M11 ~ M21
~ M12 ~ M22
u
(b)
Δ Fig 1. LFT-representations of systems with uncertainty: (a) upper and (b) lower LFT-representation.
where k M
i Iri
i¼1
is a perturbation that belongs to a bounded set that describes size and type of uncertainty and ~ ) which represents M ¼ ½M11 , M12 , M21 , M22 (resp. M the information about where the uncertainty enters the system. F u ð, Þ is called an upper LFTrepresentation and F l ð, Þ a lower LFT-representation. It is important to note that the use of upper or lower LFT-representations for modeling uncertain systems is a matter of convenience since one can be obtained E-mail:
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y
y
GðÞ ¼ F u ðM, Þ
¼ diagð1 Ir1 , . . . , k Irk Þ ¼
Δ
from the other by reordering the inputs and outputs. Without loss of generality only upper LFT-representations will be discussed. Therefore, a fundamental problem in robust design and analysis is to find a coefficient matrix M and a structured perturbation such that GðÞ ¼ F u ðM, Þ. This is the problem addressed by the paper under discussion. It is known that finding a LFT-representation of a system with parametric uncertainty is equivalent to a multidimensional realization problem and as such inherits all the difficulties of the latter [4]. The main difficulty is that joint controllability and observability of multidimensional
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Discussion on: ‘‘Generalized LFT-Representation’’
systems is not equivalent to minimality and thus, obtaining minimal realizations from non-minimal ones is extremely difficult and still an open problem. Therefore, practical algorithms for LFTrepresentations aim at obtaining lower order representations by exploiting the structure of the problem. An excellent summary of existing algorithms for standard LFT-representations can be found in [6]. Inspired by the state-space theory of descriptor systems Hecker and Varga introduced a generalized LFT-representation that has the potential of leading to lower order representations. The main objective of this discussion is to describe a general approach to normalization of LFT-representations and to highlight the need to develop model reduction techniques for descriptor systems.
2.1. Standard LFT-Normalization This transformation can be represented by a standard Þ as LFT-representation F u ðN, 2
3 2 3 Lk N N I I r i¼1 i ri 4 11 12 5 ¼ 4 5: Lk Lk N21 N22 i¼1 i ði i ÞIri i¼1 i Iri Note that for the common normalization presented by the authors i ¼ i;nom , i ¼ 0 and i i ¼ ðiþ i Þ=2. When the nominal parameters, i;nom are not centered, that is, i;nom 6¼ ði þ iþ Þ=2, a suitable normalization can be defined by choosing i , i and i such that i ¼ 0 ) i ¼ i;nom , i ¼ 1 ) i ¼ i and i ¼ þ1 ) i ¼ iþ leading to i ¼ i;nom , ði;nom i Þ ðiþ i;nom Þ , ðiþ i Þ þ ði;nom i Þ i ðiþ i;nom Þ : i ¼ i i;nom ðiþ i Þ
2. Normalization
i ¼
A fundamental step that is necessary to set up the robust control problem properly is normalization of the uncertain parameters. In systems with parametric uncertainty the perturbation matrix has the form ¼ diagð1 Ir1 , . . . , k Irk Þ ¼
k M
i Iri ,
i¼1
where i 2 ½i , iþ , i;nom 2 ½i , iþ corresponds to the P nominal model and r ¼ ki¼1 ri denotes the order of the LFT-representation. For analysis and design it is necessary to work ¼ k i Ir such that with normalized parameters i i¼1 ji j 1 and i;nom ¼ 0. This is convenient because the nominal system now corresponds to a normalized system without perturbations, for example, ¼ 0, and the set of allowable perturbations is characterized by all structured matrices such that k < 1. k In Hecker and Varga the most common normalization where i;nom ¼ ði þ iþ Þ=2 was presented. The main limitation of this normalization is that it requires that the nominal values of the uncertain parameters be at the center of the uncertainty interval which may not be true for many models. Note that this restriction is unnecessary since any linear fractional transformation between i and i , such as i ¼ i
1 þ i I , 1 þ i i
i ¼ 1, . . . , rk
can be used for normalization.
The normalized LFT-representation is then obtained ÞÞ ¼ by the composition, GðÞ ¼ F u ðM, F u ðN, Þ where L is given by the star product of N F u ðð, LÞ, and M by 2 4
F l ðN,M11 Þ M21 ðIN22 M11 Þ1 N21
3 N12 ðIM11 N22 Þ1 M12 5 : F u ðM,N22 Þ
2.2. Generalized LFT-Normalization The normalization discussed above is extended to generalized LFT-representations as follows. Let GðÞ ¼ F u ðM, Þ denote a generalized upper LFTrepresentation with coefficient matrix 2 M10 M¼4
M11 M21
3 M12 5 M22
and uncertainty block ¼ 0 Ir0 diagð1 Ir1 , . . . , k Irk Þ: Note that a new block matrix, 0 Ir0 , must be augmented to the perturbation structure of a standard Þ represents the stanLFT-representation. If F u ðN, Þ is the , 0 Ir0 dard LFT-normalization, then F u ðN
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Discussion on: ‘‘Generalized LFT-Representation’’
generalized LFT-normalization, where 2 3 I I 0 N 0 N r r r 11 r 12 0 0 5: ¼4 o N 0r0 N21 Ir0 N22 This clearly reveals the structure of Eqs (13) and (14) 22 ¼ nom , of the paper by Hecker and Varga where N N21 ¼ sl , N11 ¼ 0ro 0r and N21 ¼ Iro Ir . The normalized LFT-representation can be ob ÞÞ ¼ tained by the composition, GðÞ ¼ F u ðM, F u ðN, F u ðL, Þ where 10 , L10 ¼ N 22 Þ1 M11 N 21 , 11 þ N 12 ðM10 M11 N L11 ¼ N 22 Þ1 M12 , 12 ðM10 M11 N L12 ¼ N 22 ðM10 M11 N 21 , 22 Þ1 M11 þ IÞN L21 ¼ M21 ðN 22 Þ1 M12 : L22 ¼ M22 þ M21 N22 ðM10 M11 N These formulas extend Lemma 3.1 to the general normalization presented above. The star product does not appear to generalize naturally to descriptor LFTrepresentations.
realizations [3]. Furthermore, it is also possible to obtain reduced order models of stable multidimensional systems [2]. In the paper under discussion only one-dimensional model reduction techniques for generalized LFTrepresentations based on realization theory of descriptor systems have been used. It is expected that an extension of the multidimensional model reduction techniques of [1] to descriptor systems would lead to better results.
4. Concluding Remarks The paper by Hecker and Varga presents a generalized LFT-representation that often leads to lower order models than those obtained using standard LFTrepresentations. In this note it was shown that any linear fractional transformation relating the uncertain parameters to the normalized parameters can be used for normalization. It is important to note that normalization must be performed as late as possible in the realization procedure and must be followed by model reduction. The extension of multidimensional model reduction techniques to descriptor systems would be a valuable addition to this approach.
3. Model Reduction When the modeling system is combined with parametric uncertainty in addition to the time dimension, each uncertain parameter becomes an additional dimension. Model reduction is an important step in obtaining minimal state-space realizations of multiinput, multi-output (MIMO) systems and also in obtaining reduced order models. Minimality of onedimensional systems is equivalent to joint controllability and observability and thus can be obtained by similarity and truncation. On the other hand this is not true for multi-dimensional systems where the characterization of minimality is more complex. Sequential one-dimensional model reduction was proposed in [5] to reduce the order of LFTrepresentations with respect to individual parameters. Later it was shown that a notion of minimality can be defined by imposing a non-commutativity condition on parameters [1] leading to a generalized Kalman decomposition where ‘‘uncontrollable’’ and ‘‘unobservable’’ can be removed to obtain ‘‘minimal’’
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