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

Abstract

A deterministic optimal control problem is solved for a control-affine nonlinear system with a nonquadratic cost function. We algebraically solve the HamiltonJacobi equation for the gradient of the value function. This eliminates the need to explicitly solve the solution of a Hamilton-Jacobi partial differential equation. We interpret the value function in terms of the control Lyapunov function. Then we provide the stabilizing controller and the stability margins. Furthermore, we derive an optimal controller for a control-affine nonlinear system using the state dependent Riccati equation (SDRE) method; this method gives a similar optimal controller as the controller from the algebraic method. We also find the optimal controller when the cost function is the exponential-of-integral case, which is known as risk-sensitive (RS) control. Finally, we show that SDRE and RS methods give equivalent optimal controllers for nonlinear deterministic systems. Examples demonstrate the proposed methods.1

Convex Quadratic Equation

“…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.…”

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

“…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.…”

Design of robust optimal regulator considering state and control nonlinearities

“…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.…”