Design of parameter-scheduled state-feedback controllers using shifting specifications

Abstract

In this paper, the problem of designing a parameter-scheduled state-feedback controller is investigated. The paper presents an extension of the classical regional pole placement, H 2 control and H ∞ control problems, so as to satisfy new specifications, that will be referred to as shifting pole placement control, shifting H 2 control and shifting H ∞ control, respectively. By introducing some parameters, or using the existing ones, the controller can be designed in such a way that different values of these parameters imply different regions where the closed-loop poles are situated, or different performances in the H 2 or H ∞ sense. The proposed approach is derived within the so-called Lyapunov shaping paradigm, where a single quadratic Lyapunov function is used for ensuring stability and desired performances in spite of arbitrary parameter time variation. The problem is analyzed in the continuous-time LPV case, even though the developed theory could be applied to LTI systems in cases when it is desired to vary the control system performances online. Results obtained in simulation demonstrate the effectiveness and the relevant features of the proposed approach.

Design of State-Feedback Controllers for Linear Parameter Varying Systems Subject to Time-Varying Input Saturation

“…The term 2αP can be added to the inequalities (16) to ensure a guaranteed decay rate of the derivative of the Lyapunov function, which can be used to tune the closed-loop transient properties [24], thus obtaining (13).…”

“…It makes sense that a change in the saturation function should be tied to a change in the performance achieved by the closed-loop control system (e.g., if the maximum possible input decreases, the system's response should become slower). For this reason, the time-varying saturation limits are addressed using shifting specifications, following some ideas found, for example, in References [13,14]. This means that some parameters are introduced which, on the one hand, they are scheduled by the time-varying saturations and, on the other hand, they schedule the performance criteria in such a way that different values of these parameters imply different performances (in this paper, we will consider the guaranteed decay rate but the developed results can be extended straightforwardly to other criteria, for example, pole clustering or H ∞ /H 2 guaranteed bounds).…”

“…As in Reference [13], the proposed approach is obtained using quadratic Lyapunov functions with constant matrices and, therefore, the results might be somehow conservative when compared to other types of Lyapunov functions, for example, parameter-dependent [22] or piecewise [23], which lead to more complex mathematical calculations and are beyond the scope of this paper. The final design procedure is developed using the theory of ellipsoidal invariant sets [24].…”

Análisis y diseño de sistemas lineales con parámetros variamtes utilizando LMIs

“…Sin embargo, en el caso de sistemas con varios parámetros inciertos, una caracterización elipsoidal de la incertidumbre podría ser preferible para evitar problemas computacionales relacionados con el crecimiento exponencial del número total de vértices politópicos (Tanaka et al, 2000). En el segundo caso, detallado en (Rotondo et al, 2015a), el parámetro θ es utilizado para inducir variaciones en indicadores de prestación como la región LMI D o el desempeño H ∞ γ, obteniendo respectivamente D(θ) o γ(θ). Este paradigma, denominado mudable (del inglés shifting), es apropiado para todas las situaciones en las cuales una degradación de las prestaciones, en unas condiciones concretas de funcionamiento, sea deseable.…”

“…Teniendo en cuenta que las matrices de vértices A 1 y A 2 no son controlables a través de (b 1 , 0, 0) T debido a su diagonalidad de bloques (aunque sí son cuadráticamente estables), se intenta encontrar una ganancia de controlador K(θ(t)) que aumente la tasa de decaimiento exponencial de la función de Lyapunov para las matrices de vértices A 3 , A 4 , A 5 y A 6 . De esta forma, se obtendrá un sistema en lazo cerrado que se vuelva más rápido para valores crecientes de |ω h |, siguiendo así el mismo espíritu del paradigma mudable detallado en (Rotondo et al, 2015a) en el semiplano a la izquierda Re(s) < λ L (ver Sección 3.1) y la estabilidad cuadrática de las matrices A 1 , A 2 que, a través del cambio de variable Γ(θ) = K(θ)P, conduce al siguiente conjunto de LMIs:…”