Analytical formulation of the integral square error for linear stable feedback control system

Shahemabadi, Ali Rafiei; Noor, Samsul Bahari Mohd; Taip, Farah Saleena

doi:10.1109/iccsce.2013.6719951

Cited by 4 publications

(2 citation statements)

References 5 publications

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“…It is defined as

ISE = \int_{0}^{\infty} e^{2} false(t false) normald t

If the convergence of integral in (16) is assumed, then from Parseval's theorem, the integral can be rewritten as

ISE = \frac{1}{2 π j} \int_{+ j normal∞}^{- j normal∞} E false(s false) E false(- s false) normald s

where j is the imaginary value in the complex plane and

E false(s false)

is the Laplace transform of error signal. For the unity feedback control system,

E false(s false)

can be written in terms of parameters of I_RTF in (15) as follows:

E false(s false) = ()(, 1 - T_{normalI_RTF} false(s false)) = \frac{s + 2 ζ ω_{n}}{s^{2} + 2 ζ ω_{n} s + ω_{n}^{2}}

Through the integral tables [40], the value of ISE depending on the damping ratio and natural frequency is found as the following:

ISE = \frac{1 + {(2 ζ)}^{2}}{4 ζ ω_{n}}

By assuming a natural frequency constant, the optimal damping ratio can be found as

\frac{\partial ISE}{\partial ζ} = \frac{{(2 ζ)}^{2} - 1}{4 ζ^{2} ω_{n}} false\Rightarrow ζ = \frac{1}{2}

Therefore, optimal I_RTF in (15) is then obtained as

T_{...}

…”

Section: Optimal Integer and Fractional‐order Reference Transfer Fumentioning

confidence: 99%

Optimal fractional‐order controller design using direct synthesis method

Yumuk

Güzelkaya

Eksin

2020

IET Control Theory & Appl

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“…It is defined as

ISE = \int_{0}^{\infty} e^{2} false(t false) normald t

If the convergence of integral in (16) is assumed, then from Parseval's theorem, the integral can be rewritten as

ISE = \frac{1}{2 π j} \int_{+ j normal∞}^{- j normal∞} E false(s false) E false(- s false) normald s

where j is the imaginary value in the complex plane and

E false(s false)

is the Laplace transform of error signal. For the unity feedback control system,

E false(s false)

can be written in terms of parameters of I_RTF in (15) as follows:

E false(s false) = ()(, 1 - T_{normalI_RTF} false(s false)) = \frac{s + 2 ζ ω_{n}}{s^{2} + 2 ζ ω_{n} s + ω_{n}^{2}}

Through the integral tables [40], the value of ISE depending on the damping ratio and natural frequency is found as the following:

ISE = \frac{1 + {(2 ζ)}^{2}}{4 ζ ω_{n}}

By assuming a natural frequency constant, the optimal damping ratio can be found as

\frac{\partial ISE}{\partial ζ} = \frac{{(2 ζ)}^{2} - 1}{4 ζ^{2} ω_{n}} false\Rightarrow ζ = \frac{1}{2}

Therefore, optimal I_RTF in (15) is then obtained as

T_{...}

…”

Section: Optimal Integer and Fractional‐order Reference Transfer Fumentioning

confidence: 99%

Optimal fractional‐order controller design using direct synthesis method

Yumuk

Güzelkaya

Eksin

2020

IET Control Theory & Appl

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“…To illustrate the procedure, Figure 3 shows the SITL block diagram. This index was chosen because a system based on it would have reasonable damp-227 ing and a satisfactory transient response, punishing errors in the quadratic weights 228 during the experiment, regardless of the time they occur [24,25].…”

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confidence: 99%

Study of the Influence of a Nonlinear Control Allocation Frequency Variation on a Quadrotor Tilt-Rotor Aircraft Stability

Santos¹,

Mercorelli²,

Honório³

et al. 2021

Preprint

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This paper presents a study on the influence of the frequency variation of a nonlinear1 control allocation technique execution, developed by the author [1], named by Fast Control2 Allocation (FCA) for the Quadrotor Tilt-Rotor (QTR) aircraft. Then, through Software In The3 Loop (SITL) simulation, the proposed work considers the use of Gazebo, QGroundControl, and4 Matlab applications, where different frequencies of the FCA can be implemented separated in5 Matlab, always analyzing the QTR stability conditions from the virtual environment performed in6 Gazebo. TheresultsshowedthattheFCAneedsatleast200HzoffrequencyfortheQTRsafeflight7 conditions, i. e., 2 times smaller than the main control loop frequency, 400 Hz. Lower frequencies8 than this one would case instability or crashes during the QTR Operation.

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