Theory, Design, and Applications of Interval Estimation for Nabla Fractional Order Systems
Nabla分數階系統的區間估計理論、設計及其應用研究
Student thesis: Doctoral Thesis
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Award date  10 Oct 2023 
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Permanent Link  https://scholars.cityu.edu.hk/en/theses/theses(464ad54f7f344cce88c39044ecb75d6a).html 

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Abstract
State estimation has been regarded as a critical problem in the control community and extensively investigated for application fields. Unfortunately, state estimation cannot preserve convergence to the actual state due to the ubiquitous model uncertainties. Interval estimation becomes a powerful tool in solving the problem, which is also considered as a promising approach given its simplicity and straightforward concepts. This deterministic approach merely assumes that the uncertainties are unknown but roughly bounded, thereby avoiding some very restrictive assumptions. Namely, one can usually limit the estimation state fluctuate in a certain range and the estimation error dynamics are both positive and stable. Fractional calculus is an important issue and has been widely applied in the field of systems and control. Due to various models exhibit fractional characteristics in practice, fractional order system analysis and observer synthesis become critical topics to be explored. Thus, achieving interval estimation for fractional order systems becomes a burning problem. However, as for fractional order systems, existing work on interval estimation is still weak and faces great challenge. On the one hand, fractional order positive system theory and stability theory under different system classes are still incomplete thus far. On the other hand, the historydependent characteristics caused by the memory of fractional operators lead to complication in computation which brings great challenge to the development of interval estimation for such systems. Notice that nabla fractional order systems described by fractional difference and sum operators have attracted much attention owing to the findings of special phenomenon and characteristics over modeling. It is thus inevitable and crucial to achieve interval estimation for nabla fractional order systems.
In this thesis, the framework of interval estimation for nabla fractional order systems is systematically constructed from both theoretical and applied perspectives. The main results of this thesis are outlined in the following.
Firstly, the problem of positivity analysis for nabla fractional order systems is investigated and the corresponding positivity conditions are given, which is a prerequisite for interval estimation. To determine positivity of the considered systems, timedomain responses are derived and described as the MittagLeffler form and the shift form, respectively. The positivity criteria can thus be provided sufficiently and necessarily utilizing the timedomain responses. Based on the positivity conditions, a significant property regarding matrix discrete MittagLeffler function is proved rigorously. The results are also extended to the nabla fractional order systems involving timevarying delays.
Secondly, stability analysis of nabla fractional order systems is comprehensively addressed, which is also an essential condition of interval estimation. To be specific, discrete MittagLeffler stability is defined and several significant criteria for discrete MittagLeffler stability and asymptotic stability are developed. Stability of nabla fractional order nonlinear systems can thus be determined applying the fractional direct Lyapunov method. Due to the fact that fractional difference of a Lyapunov function is usually hard to obtain, a useful and general inequality is provided to improve the practicality of the fractional direct Lyapunov method. Moreover, stability analysis of nabla fractional order LTI systems and delayed positive systems are also studied which further enrich the stability theory and facilitate the development of interval estimation.
Then, interval estimation for nabla fractional order systems is implemented by directly applying the developed positivity and stability conditions. The standard scheme is to find an appropriate gain matrix to ensure the positive and stable command of estimation error dynamics. Specifically, a classic Luenbergertype interval observer design procedure for the considered systems is given. Besides, due to the possible absence of an appropriate gain matrix in practice, a more general interval estimation method is developed by using the coordinate transformation technique. Interval estimation method is also extended to the nabla fractional order delayed systems and nonlinear systems, which further completes the content of interval estimation for fractional order systems.
Finally, some applications of the developed interval estimation scheme are provided for nabla fractional order systems. Interval estimation is powerful to deal with the state estimation involving bounded uncertainties. Thus, many applied problems can be solved using the estimated upper and lower states from interval estimation. Stabilization of nabla fractional order nonlinear systems is achieved. The considered systems can be stabilized using the state of the interval estimation feedback. The feedback gain of the controller can be obtained by solving a linear matrix inequality. Also, state estimation of nabla fractional order RLC type electrical circuit is addressed. In addition, fault detection of nabla fractional order systems is realized by utilizing the residuals and robustness of the interval estimation against bounded uncertainties.
The framework of interval estimation for nabla fractional order systems is uniquely proposed and completed, which further enriches the theory and application of fractional order systems. Some illustrative examples are provided to validate the usefulness and flexibility of the elaborated interval estimation scheme. The proposed framework is remarkably useful in solving the state estimation problem of different nabla fractional order systems with bounded model uncertainties.
In this thesis, the framework of interval estimation for nabla fractional order systems is systematically constructed from both theoretical and applied perspectives. The main results of this thesis are outlined in the following.
Firstly, the problem of positivity analysis for nabla fractional order systems is investigated and the corresponding positivity conditions are given, which is a prerequisite for interval estimation. To determine positivity of the considered systems, timedomain responses are derived and described as the MittagLeffler form and the shift form, respectively. The positivity criteria can thus be provided sufficiently and necessarily utilizing the timedomain responses. Based on the positivity conditions, a significant property regarding matrix discrete MittagLeffler function is proved rigorously. The results are also extended to the nabla fractional order systems involving timevarying delays.
Secondly, stability analysis of nabla fractional order systems is comprehensively addressed, which is also an essential condition of interval estimation. To be specific, discrete MittagLeffler stability is defined and several significant criteria for discrete MittagLeffler stability and asymptotic stability are developed. Stability of nabla fractional order nonlinear systems can thus be determined applying the fractional direct Lyapunov method. Due to the fact that fractional difference of a Lyapunov function is usually hard to obtain, a useful and general inequality is provided to improve the practicality of the fractional direct Lyapunov method. Moreover, stability analysis of nabla fractional order LTI systems and delayed positive systems are also studied which further enrich the stability theory and facilitate the development of interval estimation.
Then, interval estimation for nabla fractional order systems is implemented by directly applying the developed positivity and stability conditions. The standard scheme is to find an appropriate gain matrix to ensure the positive and stable command of estimation error dynamics. Specifically, a classic Luenbergertype interval observer design procedure for the considered systems is given. Besides, due to the possible absence of an appropriate gain matrix in practice, a more general interval estimation method is developed by using the coordinate transformation technique. Interval estimation method is also extended to the nabla fractional order delayed systems and nonlinear systems, which further completes the content of interval estimation for fractional order systems.
Finally, some applications of the developed interval estimation scheme are provided for nabla fractional order systems. Interval estimation is powerful to deal with the state estimation involving bounded uncertainties. Thus, many applied problems can be solved using the estimated upper and lower states from interval estimation. Stabilization of nabla fractional order nonlinear systems is achieved. The considered systems can be stabilized using the state of the interval estimation feedback. The feedback gain of the controller can be obtained by solving a linear matrix inequality. Also, state estimation of nabla fractional order RLC type electrical circuit is addressed. In addition, fault detection of nabla fractional order systems is realized by utilizing the residuals and robustness of the interval estimation against bounded uncertainties.
The framework of interval estimation for nabla fractional order systems is uniquely proposed and completed, which further enriches the theory and application of fractional order systems. Some illustrative examples are provided to validate the usefulness and flexibility of the elaborated interval estimation scheme. The proposed framework is remarkably useful in solving the state estimation problem of different nabla fractional order systems with bounded model uncertainties.
 Nabla fractional order systems, Interval estimation, Positive system theory, Stability analysis, Coordinate transformation, Stabilization, Fault detection