- Email: [email protected]
Discussion on: “On Observability and Pseudo State Estimation of Fractional Order Systems”
274
Discussions on: “On Observability and Pseudo State Estimation of Fractional Order Systems”
validated are also important contributions of the paper under discussion.
2. Lorenzo CF, Hartley TT. Initialization of fractional-order operators and fractional differential equations, J. Comput. Nonlinear Dynamics, 2008; 23(2).
References 1. Hartley TT, Lorenzo CF. Control of initialized fractional, Nonlinear Dynamics, 2002; 29: 201–233.
Discussion on: “On Observability and Pseudo State Estimation of Fractional Order Systems” Nikolaos Kazantzis Department of Chemical Engineering, Worcester Polytechnic Institute, Worcester, Massachusetts 01609-2280, USA
A theory of derivatives of non-integer (fractional) order and the attendant ambitious goal of the rigorous development of a differential calculus capable of accommodating these new mathematical entities was first conceived by Leibnitz almost three centuries ago, followed by considerable research activity undertaken by distinguished mathematicians such as Liouville, Grunwald, Letnikov and Riemann focusing on the systematic and methodologically consistent establishment of the foundations of the above new branch of mathematical analysis, as well as the development of a theory for differential equations of an arbitrary real order [6]. During the last few decades, it has been recognized that a suitable mathematical formulation in the study of the behavior of natural and engineering systems endowed with inherent memory and hereditary properties inevitably introduces fractional derivatives to effectively overcome well-known inadequacies associated with the traditional integer-order mathematical models [4, 6]. Within such a context, significant insights and findings have been realized in a multitude of fields and application domains such as viscoelasticity, acoustics, electrical properties of certain classes of materials, model-based biological systems behavior characterization, rheology and polymeric materials, electrochemistry, electric power generation and transmission losses, socio-economic system analysis, etc as nicely demonstrated in the excellent monograph by Oldman and Spanier [4]. Specifically, the importance and relevance of fractional calculus in the analysis of linear systems as well as control system design is historically associated with the seminal pieces of research work first brought to the research scene by Oustaloup [5] and subsequently by Matignon [1–3] where both analytic E-mail: [email protected]
and algebraic frameworks offered wonderful insights and allowed the rigorous and elegant derivation of important system-theoretic results. The present paper authored by Sabatier, Farges, Merveilleaut and Feneteau [7] takes shape and is clearly influenced by the aforementioned tradition of excellence in theoretical investigations into the study of the behavior of linear fractional order control systems. In particular, the authors derive and present interesting results on a key system-theoretic property of linear fractional systems such as observability and the closely relevant problem of state estimation. Under the commensurability condition and null initial conditions (one is here reminded of the non-trivial initialization problem in fractional order system-theoretic analysis), the cornerstone of the proposed method is the so-called diffusive representation on the basis of which Luenberger-like observers are designed that allow the estimation of important state variables to be accomplished with satisfactory convergence characteristics. Given the body of valuable and very interesting work developed over the years by many groups of researchers, one would remain reasonably optimistic in anticipating equally significant advancements of knowledge to occur if future lines of research were directed towards: i) A comprehensive and practical analysis as well as unambiguous resolution of the physical realizability problem for fractional order feedback controllers and observers (and subsequently output feedback controllers of fractional orders) ii) The explicit incorporation of the inevitable delays (deadtime) into the respective frameworks of fractional order controller synthesis and observer design. It should be mentioned that the above delays naturally arise on two fronts (respectively affecting
275
Discussions on: “On Observability and Pseudo State Estimation of Fractional Order Systems”
state, input and output variables): firstly, in the modeling of natural and engineering systems with memory, and secondly due to the current state of sensor and actuator technology. Recent advances in the theory of delayed fractional order differential equations would be particularly useful in the attainment of such an objective. iii) The centrality of the robustness issue in fractional order controllers and state estimators/observers on both the theoretical as well as the practical/implementation front. Ample motivation for such investigation efforts is offered by the challenges surrounding the state space representation problem in fractional order systems as well as their occasional emergence in empirical (experimental/observational) data-driven approaches. iv) The integration of singular perturbation methods into the respective frameworks of fractional order controller synthesis and observer design in the presence of an underlying time-scale multiplicity (which is quite frequently encountered in systems and processes modeled by fractional order differential equations) v) Finally, the development of digital control analysis and design, as well as state estimation methods for linear fractional order systems. A comprehensive theory of fractional order difference equations is
currently being developed, and therefore, the associated conceptual and analytical armamentarium can be proven quite effective in the realization of such a research goal.
References 1. Matignon D, D’Andrea-Novel B. Some Results on Controllability and Observability of Finite-Dimensional Fractional Differential Systems, Computational Engineering in Systems Applications, 1996; 2: 952–956. 2. Matignon D, D’Andrea-Novel B. Observer-Based Controllers for Fractional Differential Systems, Proceedings of the 1997 Conference on Decision and Control, San Diego, CA, 1997, 4967–4972. 3. Matignon D. Stability Properties for Generalized Fractional Differential Systems, Proceedings of FDS’98: Fractional Differential Systems, Paris, France, 1998; 5: 145–158. 4. Oldham KB, Spanier J. The Fractional Calculus, (New York, NY: Academic Press) 1974. 5. Oustaloup A. Systemes asservis lineaires d’ordre fractionnaire, (Paris: Serie Automatique, Masson) 1983. 6. Podlubny I. Fractional Differential equations, (San Diego, CA: Academic Press) 1999. 7. Sabatier J, Farges C, Merveillaut M, Feneteau L. On Observability and Pseudo-State Estimation of Fractional Order Systems. European J Control, 2012; 18(3): 260–271.
Final Comment by the Authors J. Sabatier, C. Farges, M. Merveillaut, L. Feneteau
Comment to the discussion by S. Djennoune Regarding the representation suggested by S. Djennoune (relations (1) to (3)), the authors would first like to mention that a similar approach, based on fractional integration, has been proposed in a paper recently accepted at the FDA’12 workshop (see the forthcoming paper entitled “On state space description of fractional systems” by Jocelyn Sabatier, Christophe Farges, Jean-Claude Trigeassou). Moreover, the authors do not exactly agree with some of the comments in discussion paper [1] for the following reasons. In the approach mentioned in [1] that permits him to conclude to fractional systems observability and to claim that rank based criterion (7) can be extended to fractional systems, it must be noticed that the system history considered is a very restrictive special case: x (t) = x0
for − ∞ < t ≤ 0.
Note that such a condition is rarely met for a real system. Of course with such an initialization, the system history being constant, fractional integral (required to compute the system state in the future) of the system state becomes more simple to compute (see relation (5)). The same remark also holds with the representation we have proposed. Using relation (40) of our paper, a constant initialization leads to: +∞ μ(χ )e−tχ w(χ , 0)dχ yap (t) = CJ 0
with w (χ , 0) = cte = w0 (0)
∀χ ≥ 0
that thus becomes +∞ μ (χ ) e−tχ dχ . yap (t) = CJ w0 (0) 0
Discussions on: “On Observability and Pseudo State Estimation of Fractional Order Systems”
validated are also important contributions of the paper under discussion.
2. Lorenzo CF, Hartley TT. Initialization of fractional-order operators and fractional differential equations, J. Comput. Nonlinear Dynamics, 2008; 23(2).
References 1. Hartley TT, Lorenzo CF. Control of initialized fractional, Nonlinear Dynamics, 2002; 29: 201–233.
Discussion on: “On Observability and Pseudo State Estimation of Fractional Order Systems” Nikolaos Kazantzis Department of Chemical Engineering, Worcester Polytechnic Institute, Worcester, Massachusetts 01609-2280, USA
A theory of derivatives of non-integer (fractional) order and the attendant ambitious goal of the rigorous development of a differential calculus capable of accommodating these new mathematical entities was first conceived by Leibnitz almost three centuries ago, followed by considerable research activity undertaken by distinguished mathematicians such as Liouville, Grunwald, Letnikov and Riemann focusing on the systematic and methodologically consistent establishment of the foundations of the above new branch of mathematical analysis, as well as the development of a theory for differential equations of an arbitrary real order [6]. During the last few decades, it has been recognized that a suitable mathematical formulation in the study of the behavior of natural and engineering systems endowed with inherent memory and hereditary properties inevitably introduces fractional derivatives to effectively overcome well-known inadequacies associated with the traditional integer-order mathematical models [4, 6]. Within such a context, significant insights and findings have been realized in a multitude of fields and application domains such as viscoelasticity, acoustics, electrical properties of certain classes of materials, model-based biological systems behavior characterization, rheology and polymeric materials, electrochemistry, electric power generation and transmission losses, socio-economic system analysis, etc as nicely demonstrated in the excellent monograph by Oldman and Spanier [4]. Specifically, the importance and relevance of fractional calculus in the analysis of linear systems as well as control system design is historically associated with the seminal pieces of research work first brought to the research scene by Oustaloup [5] and subsequently by Matignon [1–3] where both analytic E-mail: [email protected]
and algebraic frameworks offered wonderful insights and allowed the rigorous and elegant derivation of important system-theoretic results. The present paper authored by Sabatier, Farges, Merveilleaut and Feneteau [7] takes shape and is clearly influenced by the aforementioned tradition of excellence in theoretical investigations into the study of the behavior of linear fractional order control systems. In particular, the authors derive and present interesting results on a key system-theoretic property of linear fractional systems such as observability and the closely relevant problem of state estimation. Under the commensurability condition and null initial conditions (one is here reminded of the non-trivial initialization problem in fractional order system-theoretic analysis), the cornerstone of the proposed method is the so-called diffusive representation on the basis of which Luenberger-like observers are designed that allow the estimation of important state variables to be accomplished with satisfactory convergence characteristics. Given the body of valuable and very interesting work developed over the years by many groups of researchers, one would remain reasonably optimistic in anticipating equally significant advancements of knowledge to occur if future lines of research were directed towards: i) A comprehensive and practical analysis as well as unambiguous resolution of the physical realizability problem for fractional order feedback controllers and observers (and subsequently output feedback controllers of fractional orders) ii) The explicit incorporation of the inevitable delays (deadtime) into the respective frameworks of fractional order controller synthesis and observer design. It should be mentioned that the above delays naturally arise on two fronts (respectively affecting
275
Discussions on: “On Observability and Pseudo State Estimation of Fractional Order Systems”
state, input and output variables): firstly, in the modeling of natural and engineering systems with memory, and secondly due to the current state of sensor and actuator technology. Recent advances in the theory of delayed fractional order differential equations would be particularly useful in the attainment of such an objective. iii) The centrality of the robustness issue in fractional order controllers and state estimators/observers on both the theoretical as well as the practical/implementation front. Ample motivation for such investigation efforts is offered by the challenges surrounding the state space representation problem in fractional order systems as well as their occasional emergence in empirical (experimental/observational) data-driven approaches. iv) The integration of singular perturbation methods into the respective frameworks of fractional order controller synthesis and observer design in the presence of an underlying time-scale multiplicity (which is quite frequently encountered in systems and processes modeled by fractional order differential equations) v) Finally, the development of digital control analysis and design, as well as state estimation methods for linear fractional order systems. A comprehensive theory of fractional order difference equations is
currently being developed, and therefore, the associated conceptual and analytical armamentarium can be proven quite effective in the realization of such a research goal.
References 1. Matignon D, D’Andrea-Novel B. Some Results on Controllability and Observability of Finite-Dimensional Fractional Differential Systems, Computational Engineering in Systems Applications, 1996; 2: 952–956. 2. Matignon D, D’Andrea-Novel B. Observer-Based Controllers for Fractional Differential Systems, Proceedings of the 1997 Conference on Decision and Control, San Diego, CA, 1997, 4967–4972. 3. Matignon D. Stability Properties for Generalized Fractional Differential Systems, Proceedings of FDS’98: Fractional Differential Systems, Paris, France, 1998; 5: 145–158. 4. Oldham KB, Spanier J. The Fractional Calculus, (New York, NY: Academic Press) 1974. 5. Oustaloup A. Systemes asservis lineaires d’ordre fractionnaire, (Paris: Serie Automatique, Masson) 1983. 6. Podlubny I. Fractional Differential equations, (San Diego, CA: Academic Press) 1999. 7. Sabatier J, Farges C, Merveillaut M, Feneteau L. On Observability and Pseudo-State Estimation of Fractional Order Systems. European J Control, 2012; 18(3): 260–271.
Final Comment by the Authors J. Sabatier, C. Farges, M. Merveillaut, L. Feneteau
Comment to the discussion by S. Djennoune Regarding the representation suggested by S. Djennoune (relations (1) to (3)), the authors would first like to mention that a similar approach, based on fractional integration, has been proposed in a paper recently accepted at the FDA’12 workshop (see the forthcoming paper entitled “On state space description of fractional systems” by Jocelyn Sabatier, Christophe Farges, Jean-Claude Trigeassou). Moreover, the authors do not exactly agree with some of the comments in discussion paper [1] for the following reasons. In the approach mentioned in [1] that permits him to conclude to fractional systems observability and to claim that rank based criterion (7) can be extended to fractional systems, it must be noticed that the system history considered is a very restrictive special case: x (t) = x0
for − ∞ < t ≤ 0.
Note that such a condition is rarely met for a real system. Of course with such an initialization, the system history being constant, fractional integral (required to compute the system state in the future) of the system state becomes more simple to compute (see relation (5)). The same remark also holds with the representation we have proposed. Using relation (40) of our paper, a constant initialization leads to: +∞ μ(χ )e−tχ w(χ , 0)dχ yap (t) = CJ 0
with w (χ , 0) = cte = w0 (0)
∀χ ≥ 0
that thus becomes +∞ μ (χ ) e−tχ dχ . yap (t) = CJ w0 (0) 0