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Optimal Stabilization of Uncertain System via Invariant Ellipsoids Approach

Наименование конференции: 

  • Lyapunov Memorial Conference LMC, Kharkiv, Ukraine


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Description and control of dynamic systems suggests an account for exogenous disturbances and uncertainties in the system coefficients. Here we adopt the unknown-but-bounded model for both, which involves no statistical properties, rates of variation, etc. Instead, the uncertainties are assumed to be arbitrary, and only bounds for their admissible values are known. In many cases, of the most adequate models of exogenous disturbances are the so-called persistent disturbances, which are the subject of study in l_1-optimization theory. However, application of l_1-optimization techniques to control system design often leads to high-dimensional controllers and is very hard to implement in the continuous-time case. Also, precise description of reachable sets for systems subjected to persistent disturbances is extremely cumbersome.A natural way to overcome these difficulties is to appeal to the invariant sets ideology in order to reduce complexity and attain the control objectives. Among various possible shapes of invariant sets, ellipsoids should be distinguished because of their simple structure and direct connection to the quadratic Lyapunov functions.On top of that, in the framework of the ellipsoidal description, a powerful apparatus of linear matrix inequalities (LMIs) and semidefinite programming can be used as a technical solution tool.In this note we propose a simple yet universal approach to rejection of persistent exogenous disturbances robustly against norm-bounded matrix uncertainties by means of linear static state feedback. Our approach is based on the method of invariant ellipsoids, by which means the optimal control design reduces to finding the minimal invariant ellipsoid for the closed-loop system. With such an ideology, the original problem can be reformulated in terms of LMIs, and the control design problem directly reduces to a semidefinite program and one-dimensional minimization. These are straightforward to implement numerically using any of the appropriate toolboxes that are presently available, e.g., Matlab-based toolboxes SeDuMi and Yalmip.The technique and results presented here generalize the previous toward the presence of model uncertainty. Another attractive property of the approach is that it is equally applicable to discrete-time systems. The efficacy of the technique is illustrated through application to the double pendulum problem.

Библиографическая ссылка: 

Щербаков П.С., Хлебников М.В. Optimal Stabilization of Uncertain System via Invariant Ellipsoids Approach / . -: -, 2007. С. -.