Robust optimal stabilization of balance systems with parametric variations

This paper proposes a method of robust optimal stabilization for balance systems such as rocket and missile systems, Segway human transportation systems, and inverted pendulum systems. Constant-gain controllers such as linear-quadratic regulators may not guarantee stability let alone optimality for balance systems affected by parametric variations. The robust stability and robust performance achieved through the proposed variable-gain controller are better than those of the linear-quadratic regulator. The proposed controller consists of two components, one of which is designed offline for nominal values of parametric variations and one of which is updated online for off-nominal values of parametric variations. A salient feature of the proposed method is a linear transformation that converts the vector control input of balance systems into scalar control input for application of the proposed method. A fourth-order linearized model of an inverted pendulum system is simulated to show the efficacy of the proposed method.

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[1] Liu Z, Yuan C, Zhang Y. Linear parameter varying control of unmanned quadrotor helicopter with mass variation and battery drainage. In: IEEE 2016 Chinese Guidance, Navigation and Control Conference; 2016.
[2] Trilla L, Bianchi FD, Bellmunt O. Gomis. Linear parameter-varying control of permanent magnet synchronous generators for wind power systems. IET Power Electron 2014; 7: 692-704.
[3] Zaffar S, Memon A. Robust and optimal stabilization of uncertain linear systems using LQR methods. In: UKACC 2014 International Conference on Control; 2014.
[4] Bemporad A, Borrelli F, Morari M. Min-max control of constrained uncertain discrete-time linear systems. IEEE T Automat Contr 2003; 48: 1600-1606.
[5] Jungers M, Oliveira RC, Peres PL. MPC for LPV systems with bounded parameter variations. Int J Control 2011; 84: 24-36.
[6] Chang H, Krieger A, Astol A, Pistikopoulos E. Robust multi-parametric model predictive control for LPV systems with application to anaesthesia. J Process Contr 2014; 24: 1538-1547.
[7] Rotondo D, Puig V, Nejjari F. Linear quadratic control of LPV systems using static and shifting speci cations. In: 2015 European Control Conference; 2015.
[8] Luspay T, Kulcsar B, Grigoriadis K. Mean-square optimal control of linear parameter varying systems with noisy scheduling parameter measurements. In: 2013 American Control Conference; 2013.
[9] Shamma JS, Mohammadpour J, Scherer CW. Control of Linear Parameter Varying Systems with Applications. Berlin, Germany: Springer, 2012.
[10] Willems J. Least squares stationary optimal control and the algebraic Riccati equation. IEEE T Automat Contr 1971; 16: 621-634.
[11] Strejc V. State space synthesis of discrete linear systems. Kybernetika 1972; 8: 83-113.
[12] Houska B, Diehl M. Robust nonlinear optimal control of dynamic systems with affine uncertainties. In: Proceedings of the 48th IEEE Conference on Decision and Control, 2009 Held Jointly with the 2009 28th Chinese Control Conference; 2009.
[13] Balakrishnan V, Vandenberghe L. Connections between duality in control theory and convex optimization. In: Proceedings of the 1995 American Control Conference; 1995.
[14] Higham NJ, Kim HM. Numerical analysis of a quadratic matrix equation. IMA J Numer Anal 2000; 20; 499-519.
[15] Ogata K. Modern Control Engineering. 4th ed. Upper Saddle River, NJ, USA: Prentice Hall, 2001.