Optimal control using an algebraic method for control-affine non-linear systems

Won, Chang-Hee; Biswas, Saroj

doi:10.1080/00207170701411375

Cited by 21 publications

(26 citation statements)

References 10 publications

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“…Notably, subject to equality and/or inequality constraints, the QP constitutes the basis for an extension of the renowned Newton's method [4]. Associated with the CQF, the convex quadratic equation (CQE) serves as a fundamental element and thus demands a comprehensive understanding (before investigating into CQF), which has been attracting attention among the control and optimization communities [5,6]. In particular, the field of nonlinear control design has devoted efforts to further uncover its importance.…”

Section: Introductionmentioning

confidence: 99%

Convex Quadratic Equation

Liang

Hsieh

2020

J Optim Theory Appl

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Two main results (A) and (B) are presented in algebraic closed forms. (A) Regarding the convex quadratic equation, an analytical equivalent solvability condition and parameterization of all solutions are formulated, for the first time in the literature and in a unified framework. The philosophy is based on the matrix algebra, while facilitated by a novel equivalence/coordinate transformation (with respect to the much more challenging case of rank-deficient Hessian matrix). In addition, the parameter-solution bijection is verified. From the perspective via (A), a major application is reexamined that accounts for the other main result (B), which deals with both the infinite and finite-time horizon nonlinear optimal control. By virtue of (A), the underlying convex quadratic equations associated with the Hamilton-Jacobi equation, Hamilton-Jacobi inequality, and Hamilton-Jacobi-Bellman equation are explicitly solved, respectively. Therefore, the long quest for the constituent of the optimal controller, gradient of the associated value function, can be captured in each solution set. Moving forward, a preliminary to exactly locate the optimality using the state-dependent (resp., differential) Riccati equation scheme is prepared for the remaining symmetry condition.

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Section: Introductionmentioning

confidence: 99%

Convex Quadratic Equation

Liang

Hsieh

2020

J Optim Theory Appl

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“…Although these techniques are verified by numerical simulations, neither approximation errors are considered, nor proofs of convergence demonstrated [21][22][23]. In [24][25][26][27], the authors tried to solve the continuous optimal regulation and tracking problem via analytical optimization. Only alternatives to HJB equations are investigated.…”

Section: Introductionmentioning

confidence: 99%

Tensor Product‐Based Model Transformation and Optimal Controller Design for High Order Nonlinear Singularly Perturbed Systems

Bouzaouache

2019

Asian Journal of Control

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This paper investigates the optimization of a general class of nonlinear singularly perturbed systems. Tensor productbased modeling, algebraic properties, and singular perturbation theory played a significant role in the formulation of the derived reduced model. The obtained reduced output is a nonlinear state and input dependent vector. In this case, solving the optimal control problem when minimizing a quadratic performance index becomes more difficult because the weighting matrices vary as functions of states. So the reformulation of the initial problem is needed and the resolution is discussed from an analytical point of view. A two-step controller design procedure is suggested and an algorithm is proposed for the calculus of the gain matrices of the nonlinear control law. The Simulation results are presented to demonstrate the effectiveness of the approach and the power of the implemented algorithm.

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“…Then the optimal control of the SDRE has been investigated by Cloutier, D'Souza, and Mracek (1996), Khaloozadeh and Abdollahi (2002). Over the past two decades, SDRE has been used more extensively in optimizing the control of nonlinear systems (Abbaszadeh & Marquez, 2012;Aliyu, 2000;Amato, 2006;Amato, Colacino, Cosentino, & Merola, 2014;Amato, Cosentino, & Merola, 2010;Banks, Kwon, Toivanen, & Tran, 2006;Bernábe-Loranca, Coello-Coello, & Osorio-Lama, 2012;Bouzaouache & Braiek, 2008;Elloumi & Braiek, 2012;Mohan & Miller, 2008;Shamma & Cloutier, 2003;Strano & Terzo, 2015;Won & Biswas, 2007). The SDRE method can be used to control and stabilize a wide range of systems, such as satellites (Cyril, Jaar, & Misra, 1995;Flores-Abad & Ma, 2012;Inaba, Oda, & Asano, 2006;Tarabini et al, 2007), spacecraft (Kaiser, Sjöberg, Delcura, & Eilertsen, 2008;Xin & Pan, 2011), helicopter (Bogdanov CONTACT Mohammad Ali Nekoui manekoui@eetd.kntu.ac.ir et al, 2003;Dimitrov & Yoshida, 2004;Liu, Wu, & Lu, 2007;Yoshida, Dimitrov, & Nakanishi, 2006), robots (Abiko & Hirzinger, 2007;Huang, Wang, Meng, & Liu, 2015;Huang, Wang, Meng, Zhang, & Liu, 2016), chaotic system (Choi, 2012;Sinha, Henrichs, & Ravindra, 2000;Yuan, Liu, Lin, Hu, & Gong, 2017) and more.…”

Section: Introductionmentioning

confidence: 99%

Design of robust optimal regulator considering state and control nonlinearities

Rudposhti

Nekoui

Teshnehlab

2018

Systems Science & Control Engineering

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Most real systems can be considered as a nonlinear and parameters of the systems may have uncertainties and external disturbances. Hence, in this paper, using the power series algorithm (PSA) based on State Dependent Riccati Equations (SDRE) and considering the proposed conditions that guarantee the asymptotic stability, the optimal regulator problem for a particular class of nonlinear systems is solved. Also, according to the specified pattern in the modified PSA (MPSA) method, weighting matrices are used as a function of state variables to achieve the better regulatory responses. Simulations are carried out on the Lorenz's chaotic system with uncertainties and external disturbances. The efficiency of optimal regulators the PSA and the MPSA methods in eliminating the external disturbances and being robust to uncertainties are compared together. The results show that the values of the performance index and the control cost in the MPSA method are smaller than the PSA method. Then the regulatory response in the MPSA method is more efficient.

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Optimal control using an algebraic method for control-affine non-linear systems

Cited by 21 publications

References 10 publications

Convex Quadratic Equation

Convex Quadratic Equation

Tensor Product‐Based Model Transformation and Optimal Controller Design for High Order Nonlinear Singularly Perturbed Systems

Design of robust optimal regulator considering state and control nonlinearities

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