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Journal ArticleDOI

LQR based improved discrete PID controller design via optimum selection of weighting matrices using fractional order integral performance index

TL;DR: The impact of fractional order (as any arbitrary real order) cost function on the LQR tuned PID control loops is highlighted in the present work, along with the achievable cost of control.
About: This article is published in Applied Mathematical Modelling.The article was published on 2013-03-15 and is currently open access. It has received 112 citations till now. The article focuses on the topics: PID controller & Linear-quadratic regulator.
Citations
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Journal ArticleDOI
TL;DR: In this paper, the stabilization and trajectory tracking of magnetic levitation system using PID controller whose controller gains are determined via Linear Quadratic Regulator (LQR) approach is considered.

75 citations


Cites background or methods from "LQR based improved discrete PID con..."

  • ...Here, the points which are important for determining the controller gain alone are explained and the further detail can be referred in [10]....

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  • ...Classical optimal control theory has evolved over decades to formulate the well known Linear Quadratic Regulators which minimizes the excursion in state trajectories of a system while requiring minimum controller effort [10]....

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  • ...[10] have given a formulation for tuning the PID controller gains via LQR approach with guaranteed pole placement....

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Journal ArticleDOI
TL;DR: It is observed that effort required in fractional order control is smaller as compared with its integer counterpart for obtaining the same system performance.
Abstract: The aim of this paper is to employ fractional order proportional integral derivative ( FO-PID ) controller and integer order PID controller to control the position of the levitated object in a magnetic levitation system ( MLS ), which is inherently nonlinear and unstable system. The proposal is to deploy discrete optimal pole-zero approximation method for realization of digital fractional order controller. An approach of phase shaping by slope cancellation of asymptotic phase plots for zeros and poles within given bandwidth is explored. The controller parameters are tuned using dynamic particle swarm optimization ( dPSO ) technique. Effectiveness of the proposed control scheme is verified by simulation and experimental results. The performance of realized digital FO-PID controller has been compared with that of the integer order PID controllers. It is observed that effort required in fractional order control is smaller as compared with its integer counterpart for obtaining the same system performance.

70 citations

Journal ArticleDOI
TL;DR: The developed Integrated IoT architecture is experimentally validated in real-time lab-scale fluid transportation pipeline system and the performance of Linear Quadratic Regulator-PID controller to regulate pressure and flow rate of the fluid being tansported is analyzed by comparing with convnetional controllers like Internal-Mode controller and Zigler–Nichols controller.

49 citations

Journal ArticleDOI
TL;DR: An improved tuning methodology of PID controller for standard second order plus time delay systems (SOPTD) is developed using the approach of Linear Quadratic Regulator (LQR) and pole placement technique to obtain the desired performance measures.
Abstract: An improved tuning methodology of PID controller for standard second order plus time delay systems (SOPTD) is developed using the approach of Linear Quadratic Regulator (LQR) and pole placement technique to obtain the desired performance measures. The pole placement method together with LQR is ingeniously used for SOPTD systems where the time delay part is handled in the controller output equation instead of characteristic equation. The effectiveness of the proposed methodology has been demonstrated via simulation of stable open loop oscillatory, over damped, critical damped and unstable open loop systems. Results show improved closed loop time response over the existing LQR based PI/PID tuning methods with less control effort. The effect of non-dominant pole on the stability and robustness of the controller has also been discussed.

47 citations


Cites methods from "LQR based improved discrete PID con..."

  • ...Many techniques have been developed and still research is going on for better tuning of the PID controller using complex numerical optimization procedures [8,9]....

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Journal ArticleDOI
TL;DR: The performance of the proposed collaborative scheme is compared with existing control schemes and basic proportional-integral-derivative control method for obtaining the controlled tracking performance and has a potential application in robotic manipulator control for different applications.

46 citations

References
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Book
01 Jun 1979
TL;DR: In this article, an augmented edition of a respected text teaches the reader how to use linear quadratic Gaussian methods effectively for the design of control systems, with step-by-step explanations that show clearly how to make practical use of the material.
Abstract: This augmented edition of a respected text teaches the reader how to use linear quadratic Gaussian methods effectively for the design of control systems. It explores linear optimal control theory from an engineering viewpoint, with step-by-step explanations that show clearly how to make practical use of the material. The three-part treatment begins with the basic theory of the linear regulator/tracker for time-invariant and time-varying systems. The Hamilton-Jacobi equation is introduced using the Principle of Optimality, and the infinite-time problem is considered. The second part outlines the engineering properties of the regulator. Topics include degree of stability, phase and gain margin, tolerance of time delay, effect of nonlinearities, asymptotic properties, and various sensitivity problems. The third section explores state estimation and robust controller design using state-estimate feedback. Numerous examples emphasize the issues related to consistent and accurate system design. Key topics include loop-recovery techniques, frequency shaping, and controller reduction, for both scalar and multivariable systems. Self-contained appendixes cover matrix theory, linear systems, the Pontryagin minimum principle, Lyapunov stability, and the Riccati equation. Newly added to this Dover edition is a complete solutions manual for the problems appearing at the conclusion of each section.

3,254 citations


"LQR based improved discrete PID con..." refers background in this paper

  • ...Introduction Classical optimal control theory has evolved over d cades to formulate the well known Linear Quadratic Regulators which minimizes t he excursion in state trajectories of a system while requiring minimum controller effort [1]....

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Book
01 Jan 1987
TL;DR: Discrete-time control systems, Discrete- time control systems , مرکز فناوری اطلاعات و ا�ل squares رسانی, کسورزی.
Abstract: Discrete-time control systems , Discrete-time control systems , مرکز فناوری اطلاعات و اطلاع رسانی کشاورزی

2,098 citations


"LQR based improved discrete PID con..." refers background in this paper

  • ...The basics of discrete time optimal quadratic regul ator is introduced here [27]....

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  • ...Discrete Time Quadratic Regulator Theory Applied to Optimal Digital PID Controller Design It is well known that discrete time realization of PID controllers are now more preferred than their continuous time counterpart [1 3], [27] since the gains of a digital PID controller can be changed, switched or scheduled on line so as to control complicated time...

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Journal ArticleDOI
TL;DR: In this article, the state-of-the-art on generalized (or any order) derivatives in physics and engineering sciences is outlined for justifying the interest of the noninteger differentiation.
Abstract: The state-of-the-art on generalized (or any order) derivatives in physics and engineering sciences, is outlined for justifying the interest of the noninteger differentiation. The problems subsequent to its use in real-time operations are then set out so as to motivate the idea of synthesizing it by a recursive distribution of zeros and poles. An analysis of the existing work is also proposed to support this idea. A comprehensive study is given of the synthesis of differentiators with integer, noninteger, real or complex orders, and whose action is limited to any given frequency bandwidth. First, a definition, in the operational and frequency domains, of a frequency-band complex noninteger order differentiator, is given in a mathematical space with four dimensions which is a Banach algebra. Then, the determination of its synthesized form, by a recursive distribution of complex zeros and poles characterized by complex recursive factors, is presented. The complex noninteger differentiation order is expressed as a function of these recursive factors. The number of zeros and poles is calculated to be as low as possible while still ensuring the stability of the synthesized differentiator to be synthesized. A time validation is presented. Finally, guidelines are proposed for the conception of the synthesized differentiator.

1,361 citations

Posted Content
TL;DR: In this paper, a solution to the more than 300-year old problem of geometric and physical interpretation of fractional integration and dieren tiation is suggested for the Riemann-Liouville fractional Integration and Dieren Tiation, the Caputo fractional dierentiation, and the Riesz potential.
Abstract: A solution to the more than 300-years old problem of geometric and physical interpretation of fractional integration and dieren tiation (i.e., integration and dieren tiation of an arbitrary real order) is suggested for the Riemann-Liouville fractional integration and dieren tiation, the Caputo fractional dieren tiation, the Riesz potential, and the Feller potential. It is also generalized for giving a new geometric and physical interpretation of more general convolution integrals of the Volterra type. Besides this, a new physical interpretation is suggested for the Stieltjes integral.

905 citations


"LQR based improved discrete PID con..." refers background in this paper

  • ...For 1 Λ > , the values would be higher than that given by 1 Λ = due to the inherent nature of the memory effect pre sent in the fractional integral [29]-[32]....

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  • ...8 also shows that the ITSE which is ba sed on a first order integration approaches towards a steady value monotonically as time increases whereas with a low order of fractional integration ( Λ ) the integration no longer remains monotonic funct ion which justifies the fact that the integrand in (31) is changing its shape over time [29], [32]....

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  • ...[30] and Podlubny [32] have given the geometric il lustration of fractional order differentiation and integration in a lucid manner....

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  • ...Thus if ( )v τ is the measured velocity of a body, then the wrong value of distance covered is given by the integral ( ) ( )10 0 t I v t v dτ τ= ∫ (29) whereas the real or actual distance passed is givenby ( ) ( ) ( )0 0 t I v t v dgα τ τ= ∫ (30) Gutierrez et al. [30] and Podlubny [32] have given the geometric illustration of fractional order differentiation and integration in a lucid manner....

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Book
01 Jun 2011
TL;DR: In this article, a modern approach to solve the solvable system of fractional and other differential equations, linear, non-linear; without perturbation or transformations, but by applying physical principle of action-and-opposite-reaction, giving approximately exact series solutions.
Abstract: When a new extraordinary and outstanding theory is stated, it has to face criticism and skeptism, because it is beyond the usual concept. The fractional calculus though not new, was not discussed or developed for a long time, particularly for lack of its application to real life problems. It is extraordinary because it does not deal with ordinary differential calculus. It is outstanding because it can now be applied to situations where existing theories fail to give satisfactory results. In this book not only mathematical abstractions are discussed in a lucid manner, with physical mathematical and geometrical explanations, but also several practical applications are given particularly for system identification, description and then efficient controls. The normal physical laws like, transport theory, electrodynamics, equation of motions, elasticity, viscosity, and several others of are based on ordinary calculus. In this book these physical laws are generalized in fractional calculus contexts; taking, heterogeneity effect in transport background, the space having traps or islands, irregular distribution of charges, non-ideal spring with mass connected to a pointless-mass ball, material behaving with viscous as well as elastic properties, system relaxation with and without memory, physics of random delay in computer network; and several others; mapping the reality of nature closely. The concept of fractional and complex order differentiation and integration are elaborated mathematically, physically and geometrically with examples. The practical utility of local fractional differentiation for enhancing the character of singularity at phase transition or characterizing the irregularity measure of response function is deliberated. Practical results of viscoelastic experiments, fractional order controls experiments, design of fractional controller and practical circuit synthesis for fractional order elements are elaborated in this book. The book also maps theory of classical integer order differential equations to fractional calculus contexts, and deals in details with conflicting and demanding initialization issues, required in classical techniques. The book presents a modern approach to solve the solvable system of fractional and other differential equations, linear, non-linear; without perturbation or transformations, but by applying physical principle of action-and-opposite-reaction, giving approximately exact series solutions.Historically, Sir Isaac Newton and Gottfried Wihelm Leibniz independently discovered calculus in the middle of the 17th century. In recognition to this remarkable discovery, J.von Neumann remarked, the calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more equivocally than anything else the inception of modern mathematical analysis which is logical development, still constitute the greatest technical advance in exact thinking.This XXI century has thus started to think-exactly for advancement in science & technology by growing application of fractional calculus, and this century has started speaking the language which nature understands the best.

695 citations


"LQR based improved discrete PID con..." refers background in this paper

  • ...The FO integral represents an area under a transformed function which is dependent on he time step and fractional order with changing limits [29]....

    [...]

  • ...8 also shows that the ITSE which is ba sed on a first order integration approaches towards a steady value monotonically as time increases whereas with a low order of fractional integration ( Λ ) the integration no longer remains monotonic funct ion which justifies the fact that the integrand in (31) is changing its shape over time [29], [32]....

    [...]

  • ...For 1 Λ > , the values would be higher than that given by 1 Λ = due to the inherent nature of the memory effect pre sent in the fractional integral [29]-[32]....

    [...]

  • ...In Chapter 5 of [29], it has been illustrated in a detailed ma nner that for the FO integrals like that...

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  • ...Fractional Order Integral Performance Indices and Their Impact on the LQR Based PID Design Fractional calculus is a 300 year’s old subject and has found wide application in many branches of engineering and science [29]-[31]....

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