Fractional Order Differentiation and Robust Control Design

Sabatier, Jocelyn; Lanusse, Patrick; Melchior, Pierre; Oustaloup, Alain

doi:10.1007/978-94-017-9807-5

Cited by 119 publications

(97 citation statements)

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“…Remark This corollary is less conservative than Theorem 4.6 in the work of Sabatier et al, and it is easier to apply due to no complex entries in the matrix. Applying Theorem 4.6 in the aforementioned work to the Caputo fractional‐order linear control system above, we can derive a sufficient condition for the system to be externally stable as

\begin{matrix} alignleft \\ align-1 [\begin{array}{ccc} \overset{r}{true} A^{T} P + r P A & P B & \overset{r}{true} C^{T} \\ B^{T} P & - γ^{2} & 0 \\ r C & 0 & - I \end{array}] < 0, & align-2 \end{matrix}

where r := e (1− α ) j ( π /2) . According to the Schur complement, it is equivalent to

\begin{matrix} alignleft \\ align-1 [\begin{array}{cc} \overset{r}{true} A^{T} P + r P A + C^{T} C & P B \\ B^{T} P & - γ^{2} \end{array}] < 0 . & align-2 \end{matrix}

As we can see, the sufficient condition in Corollary is the same as the inequality above as α =1.…”

Section: Lyapunov‐like Function Methodsmentioning

confidence: 88%

“…Corollary requires A T P + PA <0, which verifies that the eigenvalues of A are located in the left complex half plane, while the inequality just above verifies that the eigenvalues of rA are located in the left complex half plane. Due to the fact that the complex eigenvalues of a real matrix appear in pair of conjugates, the eigenvalues of A in the inequality just above are those such that

| \arg (eig (A)) | > π - \frac{α π}{2} .

Since π − απ /2> π /2 for all α ∈(0,1), Corollary is less conservative than Theorem 4.6 in the work of Sabatier et al…”

Section: Lyapunov‐like Function Methodsmentioning

confidence: 96%

“…For fractional‐order linear control systems, there are already some results such as H ∞ control, L p norm finiteness property, and H 2 norm computation . However, except the works of Rakhshan et al and Gallegos and Duarte‐Mermoud, no other related results for Caputo fractional‐order nonlinear control systems have been reported.…”

Section: Introductionmentioning

confidence: 99%

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External stability of Caputo fractional‐order nonlinear control systems

Ren

2019

Intl J Robust & Nonlinear

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Summary This paper investigates external stability of Caputo fractional‐order nonlinear control systems. Following the idea of a traditional Lyapunov function method, we point out the problems that would appear when applying it for fractional external stability. These problems are shown to be solvable by employing results on smoothness of solutions, but this method generalized for Caputo fractional‐order nonlinear control systems requires strong conditions to be imposed on vector field functions and inputs. To further explore the fractional external stability, diffusive realizations and Lyapunov‐like functions are taken into consideration. Specifically, a Caputo fractional‐order nonlinear control system with certain assumptions proves to be equivalent to its diffusive realization; a Lyapunov‐like function based on the realization exhibits properties useful to prove the external stability. As expected, this Lyapunov‐like method has weaker requirements. Finally, it is applied to the external stabilization of a Caputo fractional‐order Chua's circuits with inputs.

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\begin{matrix} alignleft \\ align-1 [\begin{array}{ccc} \overset{r}{true} A^{T} P + r P A & P B & \overset{r}{true} C^{T} \\ B^{T} P & - γ^{2} & 0 \\ r C & 0 & - I \end{array}] < 0, & align-2 \end{matrix}

where r := e (1− α ) j ( π /2) . According to the Schur complement, it is equivalent to

\begin{matrix} alignleft \\ align-1 [\begin{array}{cc} \overset{r}{true} A^{T} P + r P A + C^{T} C & P B \\ B^{T} P & - γ^{2} \end{array}] < 0 . & align-2 \end{matrix}

As we can see, the sufficient condition in Corollary is the same as the inequality above as α =1.…”

Section: Lyapunov‐like Function Methodsmentioning

confidence: 88%

| \arg (eig (A)) | > π - \frac{α π}{2} .

Since π − απ /2> π /2 for all α ∈(0,1), Corollary is less conservative than Theorem 4.6 in the work of Sabatier et al…”

Section: Lyapunov‐like Function Methodsmentioning

confidence: 96%

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External stability of Caputo fractional‐order nonlinear control systems

Ren

2019

Intl J Robust & Nonlinear

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“…It deals with the concept of

\frac{d^{n} y true(t true)}{d t^{n}}

and

\frac{d^{α} y true(t true)}{d t^{α}}

with n and α which are integer and non‐integer (even complex possibility) numbers, respectively. NIO operator a

{normalD}_{normalt}^{α}

creates the Fractional Order (FO) operator of derivative and integral, and can be expressed as follows :

{}_{a}D_{t}^{α} = true{\begin{matrix} \frac{d^{α}}{d t^{α}} \\ 1 \\ \int_{a}^{t} {(d τ)}^{α} \end{matrix} \begin{matrix} ℜ true(α true) > 0 \\ ℜ true(α true) = 0 \\ ℜ true(α true) < 0 \end{matrix}

…”

Section: Design Of Niopid Based Dpc Strategymentioning

confidence: 99%

“…The FO of s‐domain can be approximated using a NIO transfer function in the predefined frequency range [ω l, ω h ]. Among the relevant approximations, Crone approximation is usually opted owing to the common trends . This approximation provides zeros and poles, i.e.,

s^{α} \approx K \prod_{n = 1}^{N} \frac{1 + (\frac{s}{ω_{z, n}})}{1 + (\frac{s}{ω_{p, n}})}

…”

Section: Design Of Niopid Based Dpc Strategymentioning

confidence: 99%

Augment dynamic and transient capability of DFIG using optimal design of NIOPID based DPC strategy

Falehi

2017

Env Prog and Sustain Energy

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To exactly control the active and reactive power of Doubly Fed Induction Generator (DFIG), a robust controller along with an effective strategy should be applied. Since the generated power by DFIG depends on the wind speed, both active and reactive power has been essentially followed out according to the entrance wind in the steady state and turbulent conditions. Hence, the Non‐Integer Order Proportional Integral Derivative (NIOPID) controller based Direct Power Control (DPC) strategy is here introduced to simultaneously minimize the deviation of both the active and reactive power so that precise tracking of these powers to be attained. Toward this subject, both the dynamic and transient stability of DFIG have been scheduled as main targets of this paper. Due to multi‐objective nature of the design problem, a multi‐objective technique is assigned to figure out the optimal result. Considering the high performance of Multi Objective Particle Swarm Optimization (MOPSO) for unraveling the nonlinear objectives, it has been employed to reveal the optimal parameters NIOPID controller. To well verify and validate the performance of NIOPID controller, it has been evaluated under the affected power system caused by short circuit and flicker incidents. The simultaneous coordination scheme based on MOPSO has been withal scheduled with respect to three aforementioned operating conditions. To sum up, the simulation results have transparently validated the dynamic and transient performance of NIOPID based DPC. © 2017 American Institute of Chemical Engineers Environ Prog, 37: 1491–1502, 2018

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Performance of redox flow battery in combined frequency and voltage control of multi‐area multi‐source system using CFOPDN‐FOPIDN controller

Dekaraja

Saikia

2021

Int Trans Electr Energ Syst

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Summary This paper presents the combined voltage and frequency control of three‐area multi‐source interconnected power systems, having thermal‐solar thermal power plant, thermal‐wind and hydrothermal‐geothermal power plants. A maiden attempt has been made to incorporate one of the energy storage devices called redox flow battery (RFB) in all areas. Appropriate system non‐linearity constraints are considered for both thermal and hydro plants. A new cascade controller called cascade fractional‐order PD with filter and fractional‐order proportional‐integral‐derivative (PID) with filter (CFOPDN‐FOPIDN) is proposed as a secondary controller in the unified automatic load frequency control (ALFC) and automatic voltage regulator (AVR) system. A novel optimization algorithm called artificial flora algorithm is used to optimize the controller parameters. From the observation of the system dynamic responses, it reveals that the CFOPDN‐FOPIDN controller is superior to the FOPID, PIDN and PI controllers. The unified ALFC‐AVR study exhibits that the AVR loop enhances the system dynamics considerably. It is observed that the system oscillations are substantially reduced along with the peak deviation and settling time in the presence of RFB. Moreover, the combined frequency deviation and area control error as an input signal to the RFB outperforms the individual one. The system performance improves with an increase in the RFB gain value. Furthermore, it is divulged that a larger frequency deviation leads to a faster rate of change of frequency. Eventually, the sensitivity analysis reveals the sturdiness of the proposed controller.

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Fractional Order Differentiation and Robust Control Design

Cited by 119 publications

References 0 publications

External stability of Caputo fractional‐order nonlinear control systems

External stability of Caputo fractional‐order nonlinear control systems

Augment dynamic and transient capability of DFIG using optimal design of NIOPID based DPC strategy

Performance of redox flow battery in combined frequency and voltage control of multi‐area multi‐source system using CFOPDN‐FOPIDN controller

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