- Email: [email protected]
Model Structure Evolution Continuous Fraction Descriptions
4th IFAC Symposium on System, Structure and Control Università Politecnica delle Marche Ancona, Italy, Sept 15-17, 2010
Model Structure Evolution Continuous Fraction Descriptions Nicos Karcanias ∗ and Giorgos Grigoriou ∗∗ ∗ Systems and Control Centre, School of Engineering and Mathematical Sciences, City University London, Northampton Square EC1V 0HB, UK (e-mail: [email protected]). ∗∗ Systems and Control Centre, School of Engineering and Mathematical Sciences, City University London, Northampton Square EC1V 0HB, UK (e-mail: [email protected]).
Abstract: The paper deals with a form of complexity inherent in the new paradigm of Structure Evolving Systems (SES), [N.Karcanias (2008)]. This new family emerges in many challenging applications where there is variability of the system structure, its components and environment, in a way that implies evolution of system structure and properties. Of special interest are problems of integrated system design where there is variability and evolution of the models used from early to late stages of design. We are interested in working out a representation of the evolution of this complexity and then having a representation of the chains of such models, study the evolution of structural properties within the derived chains. The central question driving our study is: ”How do system structural properties evolve, when model complexity increases from the low to high complexity?”. Given that progressing from simple to more complex requires deep knowledge of the particular system, we reverse the process and we consider chains of models generated by model reduction; in this process we use the additional requirement that model simplification may be expressed in a structural sense that may allow study of evolution of system properties. We consider linear systems and we focus on the SISO case and use the McMillan degree as an indicator of complexity. The need for parameterizing the chain of models motivates the use of methods for approximation based on Continuous Fractions. Different forms of Continuous Fractions are considered and their role in parameterizing Chains of Models are considered, as well as their link to control related representations and Control problems. Central to deriving such chain representations are different forms of the Euclidian division for polynomials. The paper provides the basis for studying structural properties evolution in the context of Structure Evolving Systems. Keywords: Algebraic System Theory, Euclidian Division, Long Division, Continued Fraction Expansion, Structure Evolution 1. INTRODUCTION
criteria; this is a consequence of overspecialization and lack of a holistic co-ordination of design methodology. Similar nature problems arise in the re-engineering of existing systems/networks to upgrade them, such that they meet new requirements and performance demands. This may involve physical addition (growth), or removal (death) of parts of the system and represents evolution of a given system shell along a number of possible paths; in such a process we may intervene on the subsystem components, process synthesis/topology of interconnections, and overall instrumentation. Key questions that arise relate to the representation and modelling of such forms of evolution, and then express model evolution in terms of the structural features and properties of the respective models. The major challenge is managing complexity; in our case this corresponds to controlling, or directing such an evolutionary process along ”paths” with desirable properties. The research centers around the question: ”How do system structural properties evolve, when model complexity increases from the low to high complexity?”. Having a parametric description of the models for the transition from low to
Existing methods in Systems and Control deal predominantly with ”fixed systems”, that is those where the components, interconnection topology, measurementactuation schemes, systems environment and control structures are fixed. The process of overall design of a system (process synthesis - global instrumentation - control) has a cascade nature [N.Karcanias (1994, 2008, 1995)], and this introduces a notion of evolution that has a model ”shaping” role and thus affects the properties of the resulting system. In fact, as we go through the different design stages we have an evolution of structure, topology, properties and behaviour of the overall system. Design is an iterative process and thus it is characterized by ”early” and ”late” stages. ”Early design” requires evaluation of many alternatives using simple models and methodology [J.M.Douglas (1988)], [J.E.Rijnsdorp (1991)] whereas ”late design” uses models of greater complexity and accuracy and requires more detailed evaluation of performance. Decision making at each stage is largely based on local 978-3-902661-83-8/10/$20.00 © 2010 IFAC
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10.3182/20100915-3-IT-2017.00028
SSSC 2010 Ancona, Italy, Sept 15-17, 2010
high complexity representations is crucial for our work. The overall objective is the study of evolution of system properties as a function of the model complexity. Given that progressing from simple to more complex requires deep knowledge of the particular system, we reverse the process and we consider chains of models generated by model reduction; in this process we use the additional requirement that model simplification may be expressed in a structural sense that may allow study of evolution of system properties. We consider linear systems and we focus on the SISO case where the McMillan degree is used as an indicator of complexity. Amongst the alternative model reduction techniques we concentrate on methods based on Continuous Fractions Representations (CFR) because of their ability to generate parametrisations of the approximants. Many different forms of Continuous Fractions may be defined [14] and we focus our attention to four which see to have the potential in parameterizing the Chains of resulted Models. CFRs have emerged in the work of Kalman [R.E.Kalman (1963)] as well in Network Synthesis [Valkenburg (1960)] and are instrumental in deriving the parametric expression of the nested family of models. The link of such representations to feedback and structural recursiveness and to Control problems is also considered. Central to deriving such chain representations are different forms of the Euclidian division for polynomials. The paper provides the basis for studying structural properties evolution in the context of Structure Evolving Systems.
The exact nature of the graph depends on the stage of the design (early, late) and this is affected by the nature of models for local processes and the description of the physical interconnection streams. We may define the following notion of a graph associated with the system: Definition 1. Let us denote every subsystem Σi by a pair of vertices (ei , wi ) , denoting inputs and outputs, and an edge gi providing an input-output description of Σi . If we denote fik by the physical (information) streams connecting the wi output and the ek input, the set {(ei , wi ), gi , fi k∀i, k = 1, 2, . . . , µ} will be called the kernel graph of the system. This graph model is the simplest early system representation, it is denoted as Mc and it is referred to as the kernel model. Mc contains the basic information linked to subsystems and physical streams, defines a primitive form of structure that stems from the conceptual model of the system and provides the minimal information on the physical interconnection topology. At later stages the dimensionality of physical interconnection streams may change, if more than one variable is associated with the physical streams, as we increase our requirements for modelling. This variability from 1-dimensional vertices, edges to many dimension vertices, edges respectively describes a form of evolution defined as Dimensional Variability of Graphs. Fundamental issues related to the dimensional variability of the graph relate to the classification of the properties of the directed graph, which are independent from the dimensionality of nodes and those which depend on their dimensionality.
2. EARLY MODELS AND THE DESIGN TIME EVOLUTION NESTING
Having defined the kernel model, we proceed to developing models of increasing complexity, which however are generated from the same Mc model. This is done by preserving the generic structure of the interconnection rule, referred to here as kernel graph, and successively using models with increasing complexity for the sub-process. This leads to the following nested set of models:
A special form of system evolution is linked to the need for variable complexity modelling (VCM) in the design process as we move from early to late design. In fact, by assuming that we have a fixed interconnection structure throughout the design, then at the early stages we require simple modelling for sub-processes and physical interconnections, whereas at the late stages of design more detailed, full dynamic models are required for both sub-processes and physical interconnection structures. The study of such problems requires the development of a framework that permits the transition from simple graphs to full dynamic models and allows study of Systems and Control properties in a unifying way. We abstract the general problem of system design and start from the first and fundamental stage of system conceptualization. This process is linked to the problem of Conceptual Modelling where we transform Design Requirements and Objectives to sets of preliminary forms of system synthesis that may be referred to here as conceptual process flow-sheets Mci . In the case of Chemical Engineering this procedure is described in [J.M.Douglas (1988)]. The general outcome of this early design process (performed by e and the experienced engineers) is a family models which is denoted by: M = {Mci , i = 1, 2, . . . , k} . In this family the basic elements in modelling are:
Fig. 1. Nested set of models of variable complexity Clearly, the process of model building continues beyond the construction of Mo, which is the simplest nonlinear model. This nesting described expresses an evolution of the overall system model, parameterized by complexity (McMillan degree for linear systems) and it is referred to
(i) The general interconnection rule defining the associated graph. (ii) The early description of sub-processes in terms of simple models. 39
SSSC 2010 Ancona, Italy, Sept 15-17, 2010
as Design Time Evolution and it is denoted as in Figure 2.
(1) Standard Euclidean Division: Qeuc = {qj (s) ∈ R[s], j = 1, 2, ..., k, k + 1} (2) Long Division: Q`d = {˜ qj (s) ∈ R(s), j = 1, 2, ..., m, m + 1} (3) Reverse Long Division: Qr`d = {ˆ qj (s) ∈ R(s), j = 1, 2, ..., ρ, ρ + 1} (4) Reverse Inverse Long Division: Qri`d = {˘ qj (s) ∈ R(s), j = 1, 2, ..., τ, τ + 1} and thus continuous fraction representations which for the case (1) has the form: 1 1 1 b t = = q0 + + + . . . + (2) a q1 q2 qk+1 The four representations considered above are equivalent and since they are based on the polynomial Euclidean division are uniquely defined. Each one of the three provides a representation of a coprime set {b(s), a(s)} in terms of set of polynomials, or rational functions. In the paper, we carry out an investigation on the links between the different representations and their impact in developing links between them and the system structural properties. The implications of the different forms of CFEs for deriving chains of models with variable complexity are then investigated. We focus on the Euclidean representation where a number of results related to invariance under output feedback are established.
Fig. 2. Model Embedding Process Representation of Nesting We may approach this problem by a splitting into two stages. First, we adopt a procedure for simplification of description of subsystem models by using Model Reduction, while we preserve the Graph Structure. The latter implies a fixed input, output structure for the subsystems. This can generate the sequences of models: Ma0 ⊂ Ma1 ⊂ Ma2 ⊂ . . . Mak ⊂ Mak+1 ⊂ Mak+2
(1)
3.1 The Standard Euclidean Division and the Continuous Fraction Expansion (SFE)
where Ma0 is the kernel model, Ma1 is the linear steady state model, corresponds to first order dynamics, Mak+1 may be a nonlinear model with simple Voltera description [S.Sastry (1999)], etc. It is very important to note that there is a reversibility between Model Complexity Evolution and Model Simplification Approach. Model Evolution and Model Reduction may become completely reverse processes, if we use fixed input, output subsystem structures and interconnection graphs. This may be referred to as duality between model reduction and model complexity evolution. An important research task is the generation of such chains of models and the development of representations that may permit the evaluation of evolution of properties.
Of special interest for the development of the system evolution is the study of a set of invariants introduced by CFEs. In the case of the CFE based on the standard Euclidean Division we have: Theorem 2. Continuous Fraction Representation. Let a(s), b(s) be two polynomials and let d(s) = gcd{a(s), b(s)}, deg{a(s)} > deg{b(s)} and let qj (s), j = 1, 2, ..., k, k + 1 be the partial quotients of the successive divisions of the Euclidean Algorithm based on {a(s), b(s)} pair. The rational function t(s) = b(s)/a(s) ∈ R(s) is then represented by the set: P = {d(s) : qj (s), j = 1, 2, ..., k, k + 1} (3) or by the continuous fraction representation:
3. CONTINUOUS FRACTION EXPANSIONS (CFE) AND EUCLIDEAN DIVISIONS
t(s) =
In the following we consider the derivation of convenient model reduction techniques which have a potentially recursive nature and thus provide useful tools for evaluation of evolution of system properties. We use as a natural tool the model reduction methods based on four different types of Continued Fraction Expansion (CFE), which in a way are based on different forms of the Euclidean division. The four types are the Standard Euclidean Division, Long Division, Reverse Long Division and Reverse Inverse Long Division. In the paper we develop these forms of CFEs for a rational function t(s) = b(s)/a(s) ∈ R(s) where b(s), a(s) ∈ R[s] and investigate their properties for the case of strictly proper rational functions. In particular, we consider the forms of continuous fractions which lead to a representation of given rational function in terms of sets of polynomials, or rational functions leading to corresponding representations:
b(s) a(s)
1
=
1
q1 (s) +
1
q2 (s) +
1
q3 (s) +
.. .
.... + .... +
1 qk (s) +
1 qk+1 (s) (4)
Corollary 3. If P = {d(s) : qj (s), j = 1, 2, ..., k, k + 1} is CFR set of t(s) ∈ R(s) which is strictly proper then for all elements qj (s), deg{q1 (s)} ≥ 0 and for j = 2, , k + 1 : deg{qj (s)} > 0. The reverse problem is then considered ie whether any set P with the above properties can be the continuous fractional representation (CFR) set of some t(s) ∈ R(s). 40
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Theorem 4. Inverse Problem. Given the set P = {d(s); qj (s), j = 1, ..., k, k + 1, d(s), qj (s) ∈ R[s], deg{q1 (s)} ≥ 0, deg{qj (s)} > 0}∀j}, j 6= 1, there always exists a uniquely defined rational function t(s) = b(s)/a(s) ∈ R(s) which is proper and has the set P as its CFR set, d(s) = gcd{a(s), b(s)}.
Step (5): Fifth division of r˜3 (s) by r˜4 (s): (−3/5s) = (3) (−1/5s) + 0
The process terminates, since we have found the last remainder to be 0. Then, this form of Continuous Fraction Expansion is expressed as: b(s) s+2 1 1 1 1 t(s) = = 2 = q˜1 + = a(s) s + 4s + 3 q˜2 + q˜3 + q˜4 + q˜5 1 1 1 1 1 = + + + + s −s2 /2 −4/5s −25s2 /6 −1/5s (15) or b(s) 1 1 t(s) = = + 1 a(s) s −s2 /2 + 1 −4/5s + 1 −25s2 /6 + −1/5s (16)
This establishes the conditions under which a rational function may be associated with the set P and a procedure for constructing the rational function from its set P is also given. 3.2 Examples In the following we demonstrate four types of Continuous Fraction description based on different types of division and we introduce them in terms of examples. We consider the case of strictly proper rational function t(s) = b(s)/a(s) ∈ R(s) where we assume that deg{a(s)} > deg{b(s)}. Example 5. Consider the rational function: b(s) s+2 t(s) = = 2 (5) a(s) s + 4s + 3
Reverse Long Division: We perform the reverse long divisions by dividing b(s) by a(s) etc where first we write the polynomials in ascending order. We follow the steps described below:
Standard Euclidean Division: The standard continuous Euclidean division of a(s), b(s) polynomials may be expressed for the case a(s) and b(s) of (5) as:
Step (1): First division of b(s) by a(s): (2 + s) = (3 + 4s + s2 ) (2/3) + (−5s/3 − 2s2 /3)
step (1): first division: (s2 + 4s + 3) = (s + 2) (s + 2) + (−1) , a(s) , b(s) , q1 (s) , r1 (s) step (2): second division: (s + 2) = (−1) (−s − 2) + (0) , b(s) , r1 (s) , q2 (s) , r2 (s) From the above two steps it follows that: s+2 b(s) = 2 = t(s) = a(s) s + 4s + 3 1 1 = = r1 (s) 1 q1 (s) + (s + 2) + b(s) −(s + 2)
, b(s)
(6)
Step (1): First division of b(s) by a(s): (s + 2) = (s2 + 4s + 3) (1/s) + (−2 − 3/s)
, q˜2 (s)
, a(s)
(7)
, r˜2 (s)
, q˜3 (s)
, r˜2 (s)
, r˜3 (s) , q˜4 (s)
, rˆ1 (s)
(9)
, rˆ1 (s)
, qˆ2 (s)
, rˆ2 (s) (18)
, rˆ2 (s)
, qˆ3 (s)
, rˆ3 (s) (19)
Step (4): Fourth division of rˆ2 (s) by rˆ3 (s): (14s/5 + s2 ) = (−3s2 /42) (−196/5s) + (s2 ) , rˆ2 (s)
, rˆ3 (s)
, qˆ4 (s)
, rˆ4 (s) (20)
Step (5): fifth division of rˆ3 (s) by rˆ4 (s): (−3s2 /42) = (s2 ) (−1/14) + 0 (10)
, rˆ3 (s)
, rˆ4 (s) , qˆ5 (s) , rˆ5 (s)
(21)
Reverse Inverse Long Division: We perform the reverse inverse long divisions by writing the polynomials in ascending order and in the inverse form dividing a(s) by b(s) etc. We follow the steps described below: Step (1): First division of a(s) by b(s): (3 + 4s + s2 ) = (2 + s) (3/2) + (5s/2 + s2 ) , a(s)
, r˜3 (s) (12)
, r˜4 (s)
(17)
(−5s/3 − 2s2 /3) = (14s/5 + s2 )(−25/42) + (−3s2 /42)
(8)
, r˜2 (s) (11)
Step (4): Fourth division of r˜2 (s) by r˜3 (s): (5s/2 + 3) = (−3/5s)(−25s2 /6) + 3
, rˆ1 (s)
Step (3): Third division of rˆ1 (s) by rˆ2 (s):
Step (3): Third division of r˜1 (s) by r˜2 (s): (−2 − 3/s) = (5s/2 + 3) (−4/5s) + (−3/5s) , r˜1 (s)
, qˆ1 (s)
(3 + 4s + s2 ) = (−5s/3 − 2s2 /3)(−9/5s) + (14s/5 + s2 )
, b(s) , a(s) , q˜1 (s) , r˜1 (s) Step (2): Second division of a(s) by r˜1 (s): (s2 + 4s + 3) = (−2 − 3/s) (−s2 /2) + (5s/2 + 3) , r˜1 (s)
, a(s)
Step (2): Second division of a(s) by rˆ1 (s):
Long Division: We perform long divisions by dividing b(s) by a(s) etc following the steps:
, a(s)
(14)
, r˜3 (s) , r˜4 (s) , q˜5 (s) , r˜5 (s)
, b(s) , q˘1 (s)
, r˘1 (s)
(22)
Step (2): Second division of b(s) by r˘1 (s): (2 + s) = (5s/2 + s2 ) (4/5s) + s/5 , b(s)
(13)
, r˘1 (s)
, q˘2 (s) , r˘2 (s)
Step (3): Third division of r˘1 (s) by r˘2 (s): 41
(23)
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(5/2 + s) = (1/5) (25/2) + s , r˘1 (s) , r˘2 (s) , q˘3 1(s) , r˘3 (s) Step (4): Fourth division of r˘2 (s) by r˘3 (s): (1/5) = (s) (1/5s) + 0 , r˘2 (s)
Remark 7. The four representations considered above are equivalent since they provide alternative representations for the same rational function and since they are based on the polynomial Euclidean division are uniquely defined. However, the mapping between these alternatives is not explicitly known. Each one of them provides a representation of a coprime set {b(s), a(s)} in terms of set of polynomials, or rational functions ie: Qeuc = {qj (s) ∈ R[s], j = 1, 2, ..., k, k + 1} Q`d = {˜ qj (s) ∈ R(s), j = 1, 2, ..., m, m + 1} (31) Qr`d = {ˆ qj (s) ∈ R(s), j = 1, 2, ..., ρ, ρ + 1} Qri`d = {˘ qj (s) ∈ R(s), j = 1, 2, ..., τ, τ + 1}
(24)
(25)
, r˘3 (s) , q˘4 (s) , r˘4 (s)
The process terminates, since we have found the last remainder to be 0. Then, this form of Continuous Fraction Expansion is expressed as:
t(s) =
b(s) s+2 = 2 = a(s) s + 4s + 3 1 1 1 1 = = + + + q˘1 (s) q˘2 (s) q˘3 (s) q˘4 (s) 1 1 1 1 = + + + 3/2 4/5s 25/2 1/5s
Such sets provide alternative representations for a rational function and their properties is the subject of subsequent investigation.
(26)
In the case of non-proper rational functions the standard Euclidean division:
or
b(s) = q0 (s)a(s) + r0 (s), deg{r0 (s)} < deg{a(s)} (32) t(s) =
b(s) = a(s)
1 3/2 + 4/5s +
introduces the decomposition:
(27)
1
t(s) =
1 25/2 +
1 1/5s
Example 6. Consider the non-proper rational function: s3 + 5s2 + 8s + 5 b(s) = (28) t(s) = a(s) s2 + 4s + 3 where now deg{b(s)} > deg{a(s)}. We can always perform the division of b(s) by a(s) as indicated below ie: s3 + 5s2 + 8s + 5 = (s2 + 4s + 3) (s + 1) + (s + 2) , a(s)
, q0 (s)
(33)
and the sets in 31 take the extended forms: Q∗ euc = {qj (s) ∈ R[s], j = 0, 1, 2, ..., k, k + 1} Q∗ `d = {q0 (s) ∈ R[s], q˜j (s) ∈ R(s), j = 1, 2, ..., m, m + 1} Q∗ r`d = {q0 (s) ∈ R[s], qˆj (s) ∈ R(s), j = 1, 2, ..., ρ, ρ + 1} Q∗ ri`d = {q0 (s) ∈ R[s], q˘j (s) ∈ R(s), j = 1, 2, ..., τ, τ + 1} (34)
3.3 Observations and Remarks
, b(s)
r0 (s) b(s) = q0 (s) + a(s) a(s)
4. GENERATION OF A FAMILY OF NESTED MODELS USING THE REVERSE INVERSE LONG DIVISION
, r0 (s) (29)
from which we have that: b(s) s3 + 5s2 + 8s + 5 t(s) = = = a(s) s2 + 4s + 3 (30) s+2 r0 (s) = (s + 1) + 2 = q0 (s) + s + 4s + 3 a(s) where r0 (s)/a(s) is strictly proper. Clearly, the above indicate that in all previous divisions non-properness of the function introduces a q0 (s) ∈ R[s], whereas all the rest hold true.
In this section we use Reverse Inverse Long Division in order to generate a family of nested models based on the special structure of the continued fraction expansion indicating a chain of models of increasing complexity using a method of model reduction introduced by [C.F.Chen (1971)]. Furthermore we proceed into demonstrating the nestedness property through the corresponding block diagrams and the state space realizations.
Reverse Inverse Long Division Expansion The above example clearly suggest that there are many methods that can be used to work out a Continuous Fraction Expansion. All of them are based on some form of division of the type described in the example. We shall refer to them as Standard Euclidean Fractional Representation (SEFR), Long Division Fractional Representation (LDFR), Reverse Long Division Fractional Representation (RLDFR) and Reverse Inverse Long Division Fractional Representation (RILDFR) correspondingly, if the vehicle for their computation are the Standard Euclidean Division, the Long Division ,the Reverse Long Division or the Reverse Inverse Long Division of polynomials.
Let us consider the following strictly proper transfer function:
G(s) =
b1 sn−1 + b2 sn−2 + . . . + bn−1 s + bn sn + a1 sn−1 + a2 sn−2 + . . . + an−1 s + an
(35)
the continued fraction expansion derived from the Reverse Inverse Long Division will give:
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1
G(s) = q1 +
s q2 + q3 + s
1 ..
. qk 1 + qk+1 + · · · s .. . q2n + s
(36)
Remark 8. The number of the quotients (q1 , q2 , . . . , q2n ) is the double of the order of the transfer function G(s) which is n. The quotients (q1 , q2 , . . . , q2n ) can be any real number except from zero.
Fig. 3. Block Diagram Representation Figure 4 represents the second order truncated model. We can observe that all the inner loops that correspond to the discarded quotients have been left out as well and only the blocks that correspond to the first four quotients have remained.
Model Reduction Method for SISO Considering the continued fraction expansion of (36). According to [C.F.Chen (1971)] we truncate the fraction at the q2k quotient where 2k is an even number less than 2n and we discard the last 2n − 2k quotients. 1
G(s) = q1 +
s q2 1 + q3 + s ..
(37) .
q2k s In this way obtain a reduced model of order k which means that the numerator of the reduced model will be of (k − 1) − th order and the denominator of k − th order.
Gk (s) =
sk
Fig. 4. Block Diagram Representation of the Reduced Model By comparison of the two figures we are noticing the nesting characteristics implied making the model more dense as we move from a simple to a more complex model. Moreover, we are going to develop a state space formulation. Considering the diagram of the Fig.3, if we assign z as it’s shown in the diagram, the output of every integrator (1/s) the state variable and as u and y the input and output respectively of the system we obtain:
b1 sk−1 + b2 sk−2 + . . . + bk−1 s + bk (38) + a1 sk−1 + a2 sk−2 + . . . + ak−1 s + ak
As we can see by examining (36) and (37) the first 2k quotients are the same in both representations such that the continued fraction expansion of the reduced model is included in the corresponding expansion of the original model indicating a nestedness relation between the two of them.
q1 q2 x˙ 1 q1 q2 x˙ 2 q1 q2 x˙ 3 x˙ 4 = − q1 q2 .. .. . . x˙ n q1 q2
Block Diagrams and State Space Realizations The continued fraction expansion can be graphically illustrated as the diagram of Fig.3 with constant feedback loops and integral forward loops. As the quotients in the expansion form descend lower and lower in position, or equivalently the blocks develop to more and more inner loops, they influence less and less the performance of the system.
q1 q4 ··· (q1 + q3 )q4 · · · (q1 + q3 )q4 · · · (q1 + q3 )q4 . . . (q1 + q3 )q4
q1 q2n x1 (q1 + q3 )q2n x2 (q1 + q3 + q5 )q2n x3 x . . 4 ··· . . .. . . ··· . xn · · · (q1 + q3 + · · · + q2n−1 )q2n
1
1 1 + 1 u .. . 1
(39)
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x1 x2 x 3 y = [q2 q4 q6 · · · q2n ] . ..
(40)
xn Remark 9. By inspection the above equations we note that: • The entries in the state matrix are easily obtained from the quotients of the continued fraction expansion. • The entries under the main diagonal have the value of the main diagonal element of corresponding column. • This kind of representation gives us a very clear description of the nesting property that allows us to obtain the state space realizations of all the nested models simply by partition. In other words, if we want a second order model, we simply take the upper lefthand corner of the original matrix as the reduced one:
Fig. 5. Standard Feedback Configuration with w(s) closed loop transfer Function The above allows the derivation of a recursive feedback interpretation of the CFR descriptions and their link to the system invariants under different forms of compensation. The set up provided by the CFR enables the investigation of the evolution of system structure and properties within chains models with variable complexity. Using the above description we may rewrite the expression (42)as:
x˙ 1 q1 q2 q1 q4 x1 1 =− + u x˙ 2 q1 q2 (q1 + q3 )q4 x2 1 t(s) =
(41) y = [q2 q4 ]
x1 x2
5. FEEDBACK INTERPRETATION OF THE CFR
w ˆ1,2 (s) =
The different froms of CFR have been the subject of interpretation in different areas and this underlines the principle of recursiveness that haw emerged in many areas. In fact, early descriptions of this recursiveness in [J.Rissanen (1971)] on the problem of partial realization [A.J.Tether (1970)], [R.E.Kalman (1963)], [A.C.Antoulas (1986)], [W.B.Gragg and A.Lindquist (1983)] and in Pad´e Approximation by [C.F.Chen (1971)] and in the areas of passive network synthesis In this section we introduce an alternative interpretation of nestedness which is linked to the standard Euclidean Division. The advantage of this new interpretation is that it is linked to feedback design and there are indications of being linked to the passive realization problem. A feedback interpretation of the CFR is readily derived and this demonstrates the nested structure of the representation that has close links to modelling and state space realizations and in particular the problem of partial realization and the development of chains of reduced models. The principle of the recursiveness implied by CFR is based on the fact that the closed loop transfer function of the standard feedback configuration of Figure 5 is expressed as in the equation below: 1 1 h(s) + g(s)
1 q1 (s) + q2 (s) + w ˆ1,2 (s)
(43)
where t(s) is expressed in terms of q1 (s), q2 (s) and w ˆ1,2 (s). Note that w ˆ1,2 (s) has the nested structure that corresponds to all but the first two q1 (s), q2 (s), ie:
For a model of higher complexity we only have to use the nesting property and take a bigger k×k minor from the upper left-hand side of the original matrix.
t(s) =
1
1 1 q3 (s) + q4 (s) + w ˆ1,2,3,4 (s)
(44)
Furthermore, w ˆ1,2,3,4 (s) defining the rest of the continuous fraction decomposition, that is: 1
w ˆ1,2,3,4 (s) =
(45)
1
q5 (s) + q6 (s) +
1 .. . 1 qk (s) +
1 qk+1 (s)
which again may be expressed in the previously defined nested way. Clearly, the above process continues and the overall transfer function t(s) may be defined in a nested way in terms of the qi (s), i = 1, 2, . . . , k, k + 1 and the nested transfer functions w ˆ1,2 (s) ,w ˆ1,2,3,4 (s) etc. It is clear that the nature of the central nesting (last element)w ˆ1,2,...,k (s), depends on whether k, the number of elements in P = {d(s) : qj (s), j = 1, 2, . . . , k, k + 1} is even, or odd. Thus, we distinguish the cases:
(42)
(i) k+1 odd, then: 44
SSSC 2010 Ancona, Italy, Sept 15-17, 2010
w ˆ1,2,...,k (s) =
1 qk+1 (s)
6. CONCLUSIONS
(46)
The problem of evolution of models in the early to late design has been discussed and the process of generating models of variable complexity has been considered using the notion of Continuous Fraction Description (CFR). Out of four CFRs which have been considered the Reverse Inverse Long Division Expansion [C.F.Chen (1971)], provides a direct link to the chains of variable complexity models. It seems that the other methods also relate to deriving models of variable complexity and in particular the Standard Euclidean Division provides links to the Partial Realization Problem [R.E.Kalman (1963)], [A.C.Antoulas (1986)], [A.J.Tether (1970)], [W.B.Gragg and A.Lindquist (1983)] etc. The detailed study of the other CFRs to model chain generation is under investigation. The parametric description of the variable complexity family is linked to notions of recursiveness [A.C.Antoulas (1986)]. Alternative forms of recursiveness have been introduced and the one linked to the Euclidean division has the advantage that it provides links to output feedback invariants and design. The current investigation was restricted to the SISO case. The parametrizations based on the other CFRs and the generalizations to the MIMO case are issues under investigation. The study of evolution of system properties follows the parametrization of the families of nested models.
(ii) k+1 even, then: w ˆ1,2,...,k−1 (s) =
1 qk (s) +
1
(47)
qk+1 (s)
The above process is represented by the following block diagrams below:
Fig. 6. Nested Structure of t(s) based on w ˆ1,2 (s) partial remainder
REFERENCES A.C.Antoulas (1986). On recursiveness and related topics in linear systems. IEEE Transactions on Automatic Control, AC-31, 1121–1135. A.J.Tether (1970). Construction of minimal linear statevariable models from finite input-output data. IEEE Transactions on Automatic Control, AC-15, 427–436. C.F.Chen (1971). Model reduction of multlivariable control systems by means of matrix continued fractions. Ph.D. thesis, University of Cambridge. J.E.Rijnsdorp (1991). Integrated process control and automation. Elsevier. J.M.Douglas (1988). Conceptual design of chemical processes. McGraw-Hill Int. Ed. Chem. Eng. Series, New York. J.Rissanen (1971). Recursive identification of linear systems. SIAM J. Control, 9(3), 420–430. N.Karcanias (1994). Global process instrumentation: issues and problems of a systems and control theory framework. Measurement, 14, 103–113. N.Karcanias (1995). Integrated process design: A generic control theory/design based framework. Computers in Industry, 26, 291–301. N.Karcanias (2008). Structure evolving systems and control in integrated design. IFAC Annual Reviews in Control, 32, 161–182. R.E.Kalman (1963). Mathematical description of linear dynamical systems. SIAM J. Control, 1, 152–192. S.Sastry (1999). Nonlinear systems: analysis, stability and control, volume 10. Springer, New York. Valkenburg, M.V. (1960). Introduction to Modern Network Synthesis. John Wiley. W.B.Gragg and A.Lindquist (1983). On the partial realization problem. Linear Algebra and its Applications, 50, 277–319.
Fig. 7. Nested Structure of on w ˆ1,2 (s) based the partial remainder w ˆ1,2,3,4 (s) Note that in these diagrams the kernel is defined by for the k + 1 odd case by w ˆ1,2,...,k (s) which is a transfer function based simply on qk+1 (s) , whereas for the k + 1 even case the kernel is defined by the feedback configuration 47 based on qk (s) and qk−1 (s). Remark 10. The nested structure is defined by the feedback (odd q) parallel, (even q) and kernel and may be symbolically represented by: t(s) I q1 (s), q2 (s), w ˆ1,2 (s); w ˆ1,2 (s) I q3 (s), q4 (s), w ˆ1,2,3,4 (s); . . . (48)
This nested form involves feedback and has an impact on the uncertainty, especially, when we define approximants of t(s) (Pad´e). Examining the impact of the feedback on the approximations when we neglect terms is an issue that requires attention. Natural questions that arise have to do with how many terms we require from the nested representation to achieve certain accuracy. 45
Model Structure Evolution Continuous Fraction Descriptions Nicos Karcanias ∗ and Giorgos Grigoriou ∗∗ ∗ Systems and Control Centre, School of Engineering and Mathematical Sciences, City University London, Northampton Square EC1V 0HB, UK (e-mail: [email protected]). ∗∗ Systems and Control Centre, School of Engineering and Mathematical Sciences, City University London, Northampton Square EC1V 0HB, UK (e-mail: [email protected]).
Abstract: The paper deals with a form of complexity inherent in the new paradigm of Structure Evolving Systems (SES), [N.Karcanias (2008)]. This new family emerges in many challenging applications where there is variability of the system structure, its components and environment, in a way that implies evolution of system structure and properties. Of special interest are problems of integrated system design where there is variability and evolution of the models used from early to late stages of design. We are interested in working out a representation of the evolution of this complexity and then having a representation of the chains of such models, study the evolution of structural properties within the derived chains. The central question driving our study is: ”How do system structural properties evolve, when model complexity increases from the low to high complexity?”. Given that progressing from simple to more complex requires deep knowledge of the particular system, we reverse the process and we consider chains of models generated by model reduction; in this process we use the additional requirement that model simplification may be expressed in a structural sense that may allow study of evolution of system properties. We consider linear systems and we focus on the SISO case and use the McMillan degree as an indicator of complexity. The need for parameterizing the chain of models motivates the use of methods for approximation based on Continuous Fractions. Different forms of Continuous Fractions are considered and their role in parameterizing Chains of Models are considered, as well as their link to control related representations and Control problems. Central to deriving such chain representations are different forms of the Euclidian division for polynomials. The paper provides the basis for studying structural properties evolution in the context of Structure Evolving Systems. Keywords: Algebraic System Theory, Euclidian Division, Long Division, Continued Fraction Expansion, Structure Evolution 1. INTRODUCTION
criteria; this is a consequence of overspecialization and lack of a holistic co-ordination of design methodology. Similar nature problems arise in the re-engineering of existing systems/networks to upgrade them, such that they meet new requirements and performance demands. This may involve physical addition (growth), or removal (death) of parts of the system and represents evolution of a given system shell along a number of possible paths; in such a process we may intervene on the subsystem components, process synthesis/topology of interconnections, and overall instrumentation. Key questions that arise relate to the representation and modelling of such forms of evolution, and then express model evolution in terms of the structural features and properties of the respective models. The major challenge is managing complexity; in our case this corresponds to controlling, or directing such an evolutionary process along ”paths” with desirable properties. The research centers around the question: ”How do system structural properties evolve, when model complexity increases from the low to high complexity?”. Having a parametric description of the models for the transition from low to
Existing methods in Systems and Control deal predominantly with ”fixed systems”, that is those where the components, interconnection topology, measurementactuation schemes, systems environment and control structures are fixed. The process of overall design of a system (process synthesis - global instrumentation - control) has a cascade nature [N.Karcanias (1994, 2008, 1995)], and this introduces a notion of evolution that has a model ”shaping” role and thus affects the properties of the resulting system. In fact, as we go through the different design stages we have an evolution of structure, topology, properties and behaviour of the overall system. Design is an iterative process and thus it is characterized by ”early” and ”late” stages. ”Early design” requires evaluation of many alternatives using simple models and methodology [J.M.Douglas (1988)], [J.E.Rijnsdorp (1991)] whereas ”late design” uses models of greater complexity and accuracy and requires more detailed evaluation of performance. Decision making at each stage is largely based on local 978-3-902661-83-8/10/$20.00 © 2010 IFAC
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high complexity representations is crucial for our work. The overall objective is the study of evolution of system properties as a function of the model complexity. Given that progressing from simple to more complex requires deep knowledge of the particular system, we reverse the process and we consider chains of models generated by model reduction; in this process we use the additional requirement that model simplification may be expressed in a structural sense that may allow study of evolution of system properties. We consider linear systems and we focus on the SISO case where the McMillan degree is used as an indicator of complexity. Amongst the alternative model reduction techniques we concentrate on methods based on Continuous Fractions Representations (CFR) because of their ability to generate parametrisations of the approximants. Many different forms of Continuous Fractions may be defined [14] and we focus our attention to four which see to have the potential in parameterizing the Chains of resulted Models. CFRs have emerged in the work of Kalman [R.E.Kalman (1963)] as well in Network Synthesis [Valkenburg (1960)] and are instrumental in deriving the parametric expression of the nested family of models. The link of such representations to feedback and structural recursiveness and to Control problems is also considered. Central to deriving such chain representations are different forms of the Euclidian division for polynomials. The paper provides the basis for studying structural properties evolution in the context of Structure Evolving Systems.
The exact nature of the graph depends on the stage of the design (early, late) and this is affected by the nature of models for local processes and the description of the physical interconnection streams. We may define the following notion of a graph associated with the system: Definition 1. Let us denote every subsystem Σi by a pair of vertices (ei , wi ) , denoting inputs and outputs, and an edge gi providing an input-output description of Σi . If we denote fik by the physical (information) streams connecting the wi output and the ek input, the set {(ei , wi ), gi , fi k∀i, k = 1, 2, . . . , µ} will be called the kernel graph of the system. This graph model is the simplest early system representation, it is denoted as Mc and it is referred to as the kernel model. Mc contains the basic information linked to subsystems and physical streams, defines a primitive form of structure that stems from the conceptual model of the system and provides the minimal information on the physical interconnection topology. At later stages the dimensionality of physical interconnection streams may change, if more than one variable is associated with the physical streams, as we increase our requirements for modelling. This variability from 1-dimensional vertices, edges to many dimension vertices, edges respectively describes a form of evolution defined as Dimensional Variability of Graphs. Fundamental issues related to the dimensional variability of the graph relate to the classification of the properties of the directed graph, which are independent from the dimensionality of nodes and those which depend on their dimensionality.
2. EARLY MODELS AND THE DESIGN TIME EVOLUTION NESTING
Having defined the kernel model, we proceed to developing models of increasing complexity, which however are generated from the same Mc model. This is done by preserving the generic structure of the interconnection rule, referred to here as kernel graph, and successively using models with increasing complexity for the sub-process. This leads to the following nested set of models:
A special form of system evolution is linked to the need for variable complexity modelling (VCM) in the design process as we move from early to late design. In fact, by assuming that we have a fixed interconnection structure throughout the design, then at the early stages we require simple modelling for sub-processes and physical interconnections, whereas at the late stages of design more detailed, full dynamic models are required for both sub-processes and physical interconnection structures. The study of such problems requires the development of a framework that permits the transition from simple graphs to full dynamic models and allows study of Systems and Control properties in a unifying way. We abstract the general problem of system design and start from the first and fundamental stage of system conceptualization. This process is linked to the problem of Conceptual Modelling where we transform Design Requirements and Objectives to sets of preliminary forms of system synthesis that may be referred to here as conceptual process flow-sheets Mci . In the case of Chemical Engineering this procedure is described in [J.M.Douglas (1988)]. The general outcome of this early design process (performed by e and the experienced engineers) is a family models which is denoted by: M = {Mci , i = 1, 2, . . . , k} . In this family the basic elements in modelling are:
Fig. 1. Nested set of models of variable complexity Clearly, the process of model building continues beyond the construction of Mo, which is the simplest nonlinear model. This nesting described expresses an evolution of the overall system model, parameterized by complexity (McMillan degree for linear systems) and it is referred to
(i) The general interconnection rule defining the associated graph. (ii) The early description of sub-processes in terms of simple models. 39
SSSC 2010 Ancona, Italy, Sept 15-17, 2010
as Design Time Evolution and it is denoted as in Figure 2.
(1) Standard Euclidean Division: Qeuc = {qj (s) ∈ R[s], j = 1, 2, ..., k, k + 1} (2) Long Division: Q`d = {˜ qj (s) ∈ R(s), j = 1, 2, ..., m, m + 1} (3) Reverse Long Division: Qr`d = {ˆ qj (s) ∈ R(s), j = 1, 2, ..., ρ, ρ + 1} (4) Reverse Inverse Long Division: Qri`d = {˘ qj (s) ∈ R(s), j = 1, 2, ..., τ, τ + 1} and thus continuous fraction representations which for the case (1) has the form: 1 1 1 b t = = q0 + + + . . . + (2) a q1 q2 qk+1 The four representations considered above are equivalent and since they are based on the polynomial Euclidean division are uniquely defined. Each one of the three provides a representation of a coprime set {b(s), a(s)} in terms of set of polynomials, or rational functions. In the paper, we carry out an investigation on the links between the different representations and their impact in developing links between them and the system structural properties. The implications of the different forms of CFEs for deriving chains of models with variable complexity are then investigated. We focus on the Euclidean representation where a number of results related to invariance under output feedback are established.
Fig. 2. Model Embedding Process Representation of Nesting We may approach this problem by a splitting into two stages. First, we adopt a procedure for simplification of description of subsystem models by using Model Reduction, while we preserve the Graph Structure. The latter implies a fixed input, output structure for the subsystems. This can generate the sequences of models: Ma0 ⊂ Ma1 ⊂ Ma2 ⊂ . . . Mak ⊂ Mak+1 ⊂ Mak+2
(1)
3.1 The Standard Euclidean Division and the Continuous Fraction Expansion (SFE)
where Ma0 is the kernel model, Ma1 is the linear steady state model, corresponds to first order dynamics, Mak+1 may be a nonlinear model with simple Voltera description [S.Sastry (1999)], etc. It is very important to note that there is a reversibility between Model Complexity Evolution and Model Simplification Approach. Model Evolution and Model Reduction may become completely reverse processes, if we use fixed input, output subsystem structures and interconnection graphs. This may be referred to as duality between model reduction and model complexity evolution. An important research task is the generation of such chains of models and the development of representations that may permit the evaluation of evolution of properties.
Of special interest for the development of the system evolution is the study of a set of invariants introduced by CFEs. In the case of the CFE based on the standard Euclidean Division we have: Theorem 2. Continuous Fraction Representation. Let a(s), b(s) be two polynomials and let d(s) = gcd{a(s), b(s)}, deg{a(s)} > deg{b(s)} and let qj (s), j = 1, 2, ..., k, k + 1 be the partial quotients of the successive divisions of the Euclidean Algorithm based on {a(s), b(s)} pair. The rational function t(s) = b(s)/a(s) ∈ R(s) is then represented by the set: P = {d(s) : qj (s), j = 1, 2, ..., k, k + 1} (3) or by the continuous fraction representation:
3. CONTINUOUS FRACTION EXPANSIONS (CFE) AND EUCLIDEAN DIVISIONS
t(s) =
In the following we consider the derivation of convenient model reduction techniques which have a potentially recursive nature and thus provide useful tools for evaluation of evolution of system properties. We use as a natural tool the model reduction methods based on four different types of Continued Fraction Expansion (CFE), which in a way are based on different forms of the Euclidean division. The four types are the Standard Euclidean Division, Long Division, Reverse Long Division and Reverse Inverse Long Division. In the paper we develop these forms of CFEs for a rational function t(s) = b(s)/a(s) ∈ R(s) where b(s), a(s) ∈ R[s] and investigate their properties for the case of strictly proper rational functions. In particular, we consider the forms of continuous fractions which lead to a representation of given rational function in terms of sets of polynomials, or rational functions leading to corresponding representations:
b(s) a(s)
1
=
1
q1 (s) +
1
q2 (s) +
1
q3 (s) +
.. .
.... + .... +
1 qk (s) +
1 qk+1 (s) (4)
Corollary 3. If P = {d(s) : qj (s), j = 1, 2, ..., k, k + 1} is CFR set of t(s) ∈ R(s) which is strictly proper then for all elements qj (s), deg{q1 (s)} ≥ 0 and for j = 2, , k + 1 : deg{qj (s)} > 0. The reverse problem is then considered ie whether any set P with the above properties can be the continuous fractional representation (CFR) set of some t(s) ∈ R(s). 40
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Theorem 4. Inverse Problem. Given the set P = {d(s); qj (s), j = 1, ..., k, k + 1, d(s), qj (s) ∈ R[s], deg{q1 (s)} ≥ 0, deg{qj (s)} > 0}∀j}, j 6= 1, there always exists a uniquely defined rational function t(s) = b(s)/a(s) ∈ R(s) which is proper and has the set P as its CFR set, d(s) = gcd{a(s), b(s)}.
Step (5): Fifth division of r˜3 (s) by r˜4 (s): (−3/5s) = (3) (−1/5s) + 0
The process terminates, since we have found the last remainder to be 0. Then, this form of Continuous Fraction Expansion is expressed as: b(s) s+2 1 1 1 1 t(s) = = 2 = q˜1 + = a(s) s + 4s + 3 q˜2 + q˜3 + q˜4 + q˜5 1 1 1 1 1 = + + + + s −s2 /2 −4/5s −25s2 /6 −1/5s (15) or b(s) 1 1 t(s) = = + 1 a(s) s −s2 /2 + 1 −4/5s + 1 −25s2 /6 + −1/5s (16)
This establishes the conditions under which a rational function may be associated with the set P and a procedure for constructing the rational function from its set P is also given. 3.2 Examples In the following we demonstrate four types of Continuous Fraction description based on different types of division and we introduce them in terms of examples. We consider the case of strictly proper rational function t(s) = b(s)/a(s) ∈ R(s) where we assume that deg{a(s)} > deg{b(s)}. Example 5. Consider the rational function: b(s) s+2 t(s) = = 2 (5) a(s) s + 4s + 3
Reverse Long Division: We perform the reverse long divisions by dividing b(s) by a(s) etc where first we write the polynomials in ascending order. We follow the steps described below:
Standard Euclidean Division: The standard continuous Euclidean division of a(s), b(s) polynomials may be expressed for the case a(s) and b(s) of (5) as:
Step (1): First division of b(s) by a(s): (2 + s) = (3 + 4s + s2 ) (2/3) + (−5s/3 − 2s2 /3)
step (1): first division: (s2 + 4s + 3) = (s + 2) (s + 2) + (−1) , a(s) , b(s) , q1 (s) , r1 (s) step (2): second division: (s + 2) = (−1) (−s − 2) + (0) , b(s) , r1 (s) , q2 (s) , r2 (s) From the above two steps it follows that: s+2 b(s) = 2 = t(s) = a(s) s + 4s + 3 1 1 = = r1 (s) 1 q1 (s) + (s + 2) + b(s) −(s + 2)
, b(s)
(6)
Step (1): First division of b(s) by a(s): (s + 2) = (s2 + 4s + 3) (1/s) + (−2 − 3/s)
, q˜2 (s)
, a(s)
(7)
, r˜2 (s)
, q˜3 (s)
, r˜2 (s)
, r˜3 (s) , q˜4 (s)
, rˆ1 (s)
(9)
, rˆ1 (s)
, qˆ2 (s)
, rˆ2 (s) (18)
, rˆ2 (s)
, qˆ3 (s)
, rˆ3 (s) (19)
Step (4): Fourth division of rˆ2 (s) by rˆ3 (s): (14s/5 + s2 ) = (−3s2 /42) (−196/5s) + (s2 ) , rˆ2 (s)
, rˆ3 (s)
, qˆ4 (s)
, rˆ4 (s) (20)
Step (5): fifth division of rˆ3 (s) by rˆ4 (s): (−3s2 /42) = (s2 ) (−1/14) + 0 (10)
, rˆ3 (s)
, rˆ4 (s) , qˆ5 (s) , rˆ5 (s)
(21)
Reverse Inverse Long Division: We perform the reverse inverse long divisions by writing the polynomials in ascending order and in the inverse form dividing a(s) by b(s) etc. We follow the steps described below: Step (1): First division of a(s) by b(s): (3 + 4s + s2 ) = (2 + s) (3/2) + (5s/2 + s2 ) , a(s)
, r˜3 (s) (12)
, r˜4 (s)
(17)
(−5s/3 − 2s2 /3) = (14s/5 + s2 )(−25/42) + (−3s2 /42)
(8)
, r˜2 (s) (11)
Step (4): Fourth division of r˜2 (s) by r˜3 (s): (5s/2 + 3) = (−3/5s)(−25s2 /6) + 3
, rˆ1 (s)
Step (3): Third division of rˆ1 (s) by rˆ2 (s):
Step (3): Third division of r˜1 (s) by r˜2 (s): (−2 − 3/s) = (5s/2 + 3) (−4/5s) + (−3/5s) , r˜1 (s)
, qˆ1 (s)
(3 + 4s + s2 ) = (−5s/3 − 2s2 /3)(−9/5s) + (14s/5 + s2 )
, b(s) , a(s) , q˜1 (s) , r˜1 (s) Step (2): Second division of a(s) by r˜1 (s): (s2 + 4s + 3) = (−2 − 3/s) (−s2 /2) + (5s/2 + 3) , r˜1 (s)
, a(s)
Step (2): Second division of a(s) by rˆ1 (s):
Long Division: We perform long divisions by dividing b(s) by a(s) etc following the steps:
, a(s)
(14)
, r˜3 (s) , r˜4 (s) , q˜5 (s) , r˜5 (s)
, b(s) , q˘1 (s)
, r˘1 (s)
(22)
Step (2): Second division of b(s) by r˘1 (s): (2 + s) = (5s/2 + s2 ) (4/5s) + s/5 , b(s)
(13)
, r˘1 (s)
, q˘2 (s) , r˘2 (s)
Step (3): Third division of r˘1 (s) by r˘2 (s): 41
(23)
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(5/2 + s) = (1/5) (25/2) + s , r˘1 (s) , r˘2 (s) , q˘3 1(s) , r˘3 (s) Step (4): Fourth division of r˘2 (s) by r˘3 (s): (1/5) = (s) (1/5s) + 0 , r˘2 (s)
Remark 7. The four representations considered above are equivalent since they provide alternative representations for the same rational function and since they are based on the polynomial Euclidean division are uniquely defined. However, the mapping between these alternatives is not explicitly known. Each one of them provides a representation of a coprime set {b(s), a(s)} in terms of set of polynomials, or rational functions ie: Qeuc = {qj (s) ∈ R[s], j = 1, 2, ..., k, k + 1} Q`d = {˜ qj (s) ∈ R(s), j = 1, 2, ..., m, m + 1} (31) Qr`d = {ˆ qj (s) ∈ R(s), j = 1, 2, ..., ρ, ρ + 1} Qri`d = {˘ qj (s) ∈ R(s), j = 1, 2, ..., τ, τ + 1}
(24)
(25)
, r˘3 (s) , q˘4 (s) , r˘4 (s)
The process terminates, since we have found the last remainder to be 0. Then, this form of Continuous Fraction Expansion is expressed as:
t(s) =
b(s) s+2 = 2 = a(s) s + 4s + 3 1 1 1 1 = = + + + q˘1 (s) q˘2 (s) q˘3 (s) q˘4 (s) 1 1 1 1 = + + + 3/2 4/5s 25/2 1/5s
Such sets provide alternative representations for a rational function and their properties is the subject of subsequent investigation.
(26)
In the case of non-proper rational functions the standard Euclidean division:
or
b(s) = q0 (s)a(s) + r0 (s), deg{r0 (s)} < deg{a(s)} (32) t(s) =
b(s) = a(s)
1 3/2 + 4/5s +
introduces the decomposition:
(27)
1
t(s) =
1 25/2 +
1 1/5s
Example 6. Consider the non-proper rational function: s3 + 5s2 + 8s + 5 b(s) = (28) t(s) = a(s) s2 + 4s + 3 where now deg{b(s)} > deg{a(s)}. We can always perform the division of b(s) by a(s) as indicated below ie: s3 + 5s2 + 8s + 5 = (s2 + 4s + 3) (s + 1) + (s + 2) , a(s)
, q0 (s)
(33)
and the sets in 31 take the extended forms: Q∗ euc = {qj (s) ∈ R[s], j = 0, 1, 2, ..., k, k + 1} Q∗ `d = {q0 (s) ∈ R[s], q˜j (s) ∈ R(s), j = 1, 2, ..., m, m + 1} Q∗ r`d = {q0 (s) ∈ R[s], qˆj (s) ∈ R(s), j = 1, 2, ..., ρ, ρ + 1} Q∗ ri`d = {q0 (s) ∈ R[s], q˘j (s) ∈ R(s), j = 1, 2, ..., τ, τ + 1} (34)
3.3 Observations and Remarks
, b(s)
r0 (s) b(s) = q0 (s) + a(s) a(s)
4. GENERATION OF A FAMILY OF NESTED MODELS USING THE REVERSE INVERSE LONG DIVISION
, r0 (s) (29)
from which we have that: b(s) s3 + 5s2 + 8s + 5 t(s) = = = a(s) s2 + 4s + 3 (30) s+2 r0 (s) = (s + 1) + 2 = q0 (s) + s + 4s + 3 a(s) where r0 (s)/a(s) is strictly proper. Clearly, the above indicate that in all previous divisions non-properness of the function introduces a q0 (s) ∈ R[s], whereas all the rest hold true.
In this section we use Reverse Inverse Long Division in order to generate a family of nested models based on the special structure of the continued fraction expansion indicating a chain of models of increasing complexity using a method of model reduction introduced by [C.F.Chen (1971)]. Furthermore we proceed into demonstrating the nestedness property through the corresponding block diagrams and the state space realizations.
Reverse Inverse Long Division Expansion The above example clearly suggest that there are many methods that can be used to work out a Continuous Fraction Expansion. All of them are based on some form of division of the type described in the example. We shall refer to them as Standard Euclidean Fractional Representation (SEFR), Long Division Fractional Representation (LDFR), Reverse Long Division Fractional Representation (RLDFR) and Reverse Inverse Long Division Fractional Representation (RILDFR) correspondingly, if the vehicle for their computation are the Standard Euclidean Division, the Long Division ,the Reverse Long Division or the Reverse Inverse Long Division of polynomials.
Let us consider the following strictly proper transfer function:
G(s) =
b1 sn−1 + b2 sn−2 + . . . + bn−1 s + bn sn + a1 sn−1 + a2 sn−2 + . . . + an−1 s + an
(35)
the continued fraction expansion derived from the Reverse Inverse Long Division will give:
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1
G(s) = q1 +
s q2 + q3 + s
1 ..
. qk 1 + qk+1 + · · · s .. . q2n + s
(36)
Remark 8. The number of the quotients (q1 , q2 , . . . , q2n ) is the double of the order of the transfer function G(s) which is n. The quotients (q1 , q2 , . . . , q2n ) can be any real number except from zero.
Fig. 3. Block Diagram Representation Figure 4 represents the second order truncated model. We can observe that all the inner loops that correspond to the discarded quotients have been left out as well and only the blocks that correspond to the first four quotients have remained.
Model Reduction Method for SISO Considering the continued fraction expansion of (36). According to [C.F.Chen (1971)] we truncate the fraction at the q2k quotient where 2k is an even number less than 2n and we discard the last 2n − 2k quotients. 1
G(s) = q1 +
s q2 1 + q3 + s ..
(37) .
q2k s In this way obtain a reduced model of order k which means that the numerator of the reduced model will be of (k − 1) − th order and the denominator of k − th order.
Gk (s) =
sk
Fig. 4. Block Diagram Representation of the Reduced Model By comparison of the two figures we are noticing the nesting characteristics implied making the model more dense as we move from a simple to a more complex model. Moreover, we are going to develop a state space formulation. Considering the diagram of the Fig.3, if we assign z as it’s shown in the diagram, the output of every integrator (1/s) the state variable and as u and y the input and output respectively of the system we obtain:
b1 sk−1 + b2 sk−2 + . . . + bk−1 s + bk (38) + a1 sk−1 + a2 sk−2 + . . . + ak−1 s + ak
As we can see by examining (36) and (37) the first 2k quotients are the same in both representations such that the continued fraction expansion of the reduced model is included in the corresponding expansion of the original model indicating a nestedness relation between the two of them.
q1 q2 x˙ 1 q1 q2 x˙ 2 q1 q2 x˙ 3 x˙ 4 = − q1 q2 .. .. . . x˙ n q1 q2
Block Diagrams and State Space Realizations The continued fraction expansion can be graphically illustrated as the diagram of Fig.3 with constant feedback loops and integral forward loops. As the quotients in the expansion form descend lower and lower in position, or equivalently the blocks develop to more and more inner loops, they influence less and less the performance of the system.
q1 q4 ··· (q1 + q3 )q4 · · · (q1 + q3 )q4 · · · (q1 + q3 )q4 . . . (q1 + q3 )q4
q1 q2n x1 (q1 + q3 )q2n x2 (q1 + q3 + q5 )q2n x3 x . . 4 ··· . . .. . . ··· . xn · · · (q1 + q3 + · · · + q2n−1 )q2n
1
1 1 + 1 u .. . 1
(39)
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x1 x2 x 3 y = [q2 q4 q6 · · · q2n ] . ..
(40)
xn Remark 9. By inspection the above equations we note that: • The entries in the state matrix are easily obtained from the quotients of the continued fraction expansion. • The entries under the main diagonal have the value of the main diagonal element of corresponding column. • This kind of representation gives us a very clear description of the nesting property that allows us to obtain the state space realizations of all the nested models simply by partition. In other words, if we want a second order model, we simply take the upper lefthand corner of the original matrix as the reduced one:
Fig. 5. Standard Feedback Configuration with w(s) closed loop transfer Function The above allows the derivation of a recursive feedback interpretation of the CFR descriptions and their link to the system invariants under different forms of compensation. The set up provided by the CFR enables the investigation of the evolution of system structure and properties within chains models with variable complexity. Using the above description we may rewrite the expression (42)as:
x˙ 1 q1 q2 q1 q4 x1 1 =− + u x˙ 2 q1 q2 (q1 + q3 )q4 x2 1 t(s) =
(41) y = [q2 q4 ]
x1 x2
5. FEEDBACK INTERPRETATION OF THE CFR
w ˆ1,2 (s) =
The different froms of CFR have been the subject of interpretation in different areas and this underlines the principle of recursiveness that haw emerged in many areas. In fact, early descriptions of this recursiveness in [J.Rissanen (1971)] on the problem of partial realization [A.J.Tether (1970)], [R.E.Kalman (1963)], [A.C.Antoulas (1986)], [W.B.Gragg and A.Lindquist (1983)] and in Pad´e Approximation by [C.F.Chen (1971)] and in the areas of passive network synthesis In this section we introduce an alternative interpretation of nestedness which is linked to the standard Euclidean Division. The advantage of this new interpretation is that it is linked to feedback design and there are indications of being linked to the passive realization problem. A feedback interpretation of the CFR is readily derived and this demonstrates the nested structure of the representation that has close links to modelling and state space realizations and in particular the problem of partial realization and the development of chains of reduced models. The principle of the recursiveness implied by CFR is based on the fact that the closed loop transfer function of the standard feedback configuration of Figure 5 is expressed as in the equation below: 1 1 h(s) + g(s)
1 q1 (s) + q2 (s) + w ˆ1,2 (s)
(43)
where t(s) is expressed in terms of q1 (s), q2 (s) and w ˆ1,2 (s). Note that w ˆ1,2 (s) has the nested structure that corresponds to all but the first two q1 (s), q2 (s), ie:
For a model of higher complexity we only have to use the nesting property and take a bigger k×k minor from the upper left-hand side of the original matrix.
t(s) =
1
1 1 q3 (s) + q4 (s) + w ˆ1,2,3,4 (s)
(44)
Furthermore, w ˆ1,2,3,4 (s) defining the rest of the continuous fraction decomposition, that is: 1
w ˆ1,2,3,4 (s) =
(45)
1
q5 (s) + q6 (s) +
1 .. . 1 qk (s) +
1 qk+1 (s)
which again may be expressed in the previously defined nested way. Clearly, the above process continues and the overall transfer function t(s) may be defined in a nested way in terms of the qi (s), i = 1, 2, . . . , k, k + 1 and the nested transfer functions w ˆ1,2 (s) ,w ˆ1,2,3,4 (s) etc. It is clear that the nature of the central nesting (last element)w ˆ1,2,...,k (s), depends on whether k, the number of elements in P = {d(s) : qj (s), j = 1, 2, . . . , k, k + 1} is even, or odd. Thus, we distinguish the cases:
(42)
(i) k+1 odd, then: 44
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w ˆ1,2,...,k (s) =
1 qk+1 (s)
6. CONCLUSIONS
(46)
The problem of evolution of models in the early to late design has been discussed and the process of generating models of variable complexity has been considered using the notion of Continuous Fraction Description (CFR). Out of four CFRs which have been considered the Reverse Inverse Long Division Expansion [C.F.Chen (1971)], provides a direct link to the chains of variable complexity models. It seems that the other methods also relate to deriving models of variable complexity and in particular the Standard Euclidean Division provides links to the Partial Realization Problem [R.E.Kalman (1963)], [A.C.Antoulas (1986)], [A.J.Tether (1970)], [W.B.Gragg and A.Lindquist (1983)] etc. The detailed study of the other CFRs to model chain generation is under investigation. The parametric description of the variable complexity family is linked to notions of recursiveness [A.C.Antoulas (1986)]. Alternative forms of recursiveness have been introduced and the one linked to the Euclidean division has the advantage that it provides links to output feedback invariants and design. The current investigation was restricted to the SISO case. The parametrizations based on the other CFRs and the generalizations to the MIMO case are issues under investigation. The study of evolution of system properties follows the parametrization of the families of nested models.
(ii) k+1 even, then: w ˆ1,2,...,k−1 (s) =
1 qk (s) +
1
(47)
qk+1 (s)
The above process is represented by the following block diagrams below:
Fig. 6. Nested Structure of t(s) based on w ˆ1,2 (s) partial remainder
REFERENCES A.C.Antoulas (1986). On recursiveness and related topics in linear systems. IEEE Transactions on Automatic Control, AC-31, 1121–1135. A.J.Tether (1970). Construction of minimal linear statevariable models from finite input-output data. IEEE Transactions on Automatic Control, AC-15, 427–436. C.F.Chen (1971). Model reduction of multlivariable control systems by means of matrix continued fractions. Ph.D. thesis, University of Cambridge. J.E.Rijnsdorp (1991). Integrated process control and automation. Elsevier. J.M.Douglas (1988). Conceptual design of chemical processes. McGraw-Hill Int. Ed. Chem. Eng. Series, New York. J.Rissanen (1971). Recursive identification of linear systems. SIAM J. Control, 9(3), 420–430. N.Karcanias (1994). Global process instrumentation: issues and problems of a systems and control theory framework. Measurement, 14, 103–113. N.Karcanias (1995). Integrated process design: A generic control theory/design based framework. Computers in Industry, 26, 291–301. N.Karcanias (2008). Structure evolving systems and control in integrated design. IFAC Annual Reviews in Control, 32, 161–182. R.E.Kalman (1963). Mathematical description of linear dynamical systems. SIAM J. Control, 1, 152–192. S.Sastry (1999). Nonlinear systems: analysis, stability and control, volume 10. Springer, New York. Valkenburg, M.V. (1960). Introduction to Modern Network Synthesis. John Wiley. W.B.Gragg and A.Lindquist (1983). On the partial realization problem. Linear Algebra and its Applications, 50, 277–319.
Fig. 7. Nested Structure of on w ˆ1,2 (s) based the partial remainder w ˆ1,2,3,4 (s) Note that in these diagrams the kernel is defined by for the k + 1 odd case by w ˆ1,2,...,k (s) which is a transfer function based simply on qk+1 (s) , whereas for the k + 1 even case the kernel is defined by the feedback configuration 47 based on qk (s) and qk−1 (s). Remark 10. The nested structure is defined by the feedback (odd q) parallel, (even q) and kernel and may be symbolically represented by: t(s) I q1 (s), q2 (s), w ˆ1,2 (s); w ˆ1,2 (s) I q3 (s), q4 (s), w ˆ1,2,3,4 (s); . . . (48)
This nested form involves feedback and has an impact on the uncertainty, especially, when we define approximants of t(s) (Pad´e). Examining the impact of the feedback on the approximations when we neglect terms is an issue that requires attention. Natural questions that arise have to do with how many terms we require from the nested representation to achieve certain accuracy. 45