Antiwindup for stable linear systems with input saturation: An LMI-based synthesis

Abstract

Abstract-This paper considers closed-loop quadratic stability and 2 performance properties of linear control systems subject to input saturation. More specifically, these properties are examined within the context of the popular linear antiwindup augmentation paradigm. Linear antiwindup augmentation refers to designing a linear filter to augment a linear control system subject to a local specification, called the "unconstrained closed-loop behavior." Building on known results on and LPV synthesis, the fixed order linear antiwindup synthesis feasibility problem is cast as a nonconvex matrix optimization problem, which has an attractive system theoretic interpretation: the lower bound on the achievable 2 performance is the maximum of the open and unconstrained closed-loop 2 gains. In the special cases of zero-order (static) and plant-order antiwindup compensation, the feasibility conditions become (convex) linear matrix inequalities. It is shown that, if (and only if) the plant is asymptotically stable, plant-order linear antiwindup compensation is always feasible for large enough 2 gain and that static antiwindup compensation is feasible provided a quasi-common Lyapunov function, between the open-loop and unconstrained closed-loop, exists. Using the solutions to the matrix feasibility problems, the synthesis of the antiwindup augmentation achieving the desired level of 2 performance is then accomplished by solving an additional LMI.Index Terms-Antiwindup analysis, antiwindup synthesis, control systems, cost optimal control, finite 2 gain, linear matrix inequalities (LMIs), linear parameter varying (LPV).

A non-square sector condition and its application in deferred-action anti-windup compensator design

“…This non-square sector condition is applied with the S-procedure [2] to the deferred-action AW problem proposed by [22]. The arising deferred-action AW synthesis conditions are much closer to those found in the standard ("immediate") case of [9]: the LMI's are of similar complexity, and contain the inequalities of [9] as special cases, unlike the LPV-based results in [22,26]; the increase in complexity as the number of control inputs increased is linear rather than polynomial (as for [22]). It will also be shown that the results derived here are less conservative than the sector-based results found in [26].…”

“…Many modern approaches to AW design are based on tools from convex optimisation ( [18,13,9,31,8,21,19,21,5,1,17]) and although these provide an attractive framework for AW synthesis, the compensators produced can be conservative: while stability is guaranteed, observed time-domain performance may be disappointing. This has two main sources: (i) the quadratic Lyapunov functions and sector bounds used in the synthesis algorithms; and (ii) the linearity of the AW compensator, which means its behaviour is identical for both small and large excursions beyond the control limits.…”

“…The original deferredaction scheme of [22] involved writing the system equations in pseudo-LPV form and then using a scaled small gain approach to guarantee stability. Although examples undoubtedly demonstrated the appeal of the "delayed" AW approach, there are a number of issues with the scheme: the pseudo-LPV modelling of the system meant that 2 m sets of LMI's needed to be solved, obviously making the approach unattractive when the number of control inputs, m, is large; and the order of compensator produced is of the combined order of the plant (n p ) and controller (n c ), whereas standard full-order approaches ( [9]), only require n p states for optimal performance. In later work [24][25][26], some of these problems were addressed by using a sector-approach.…”

Estratégia de controle para o seguimento de referências em sistemas de tempo discreto com atuadores saturantes

“…O objetivo da compensação de anti-windup é o estabelecimento de uma correção automática dos estados do controlador, quando da ocorrência da saturação, a fim de minimizar a perda de performance ou maximizar a região de atração do sistema em malha fechada (ver (Kothare et al, 1994), (Grimm et al, 2003), (Gomes da Silva Jr. and Tarbouriech, 2005) e referências). Considerando o caso de estabilidade local, em (Hu et al, 2002), é proposta uma formulação baseada em inequações matriciais bilineares (BMIs) relaxadas utilizando um modelo politópico do termo de saturação.…”

“…Neste tipo de modelagem, a estrutura da condição de setor clássica é relaxada, permitindo a obtenção de condições diretamente na forma de LMIs. Resultados relativos a compensadores dinâmicos de anti-windup são apresentados em (Grimm et al, 2003), e . Do ponto de vista prático, o problema de seguimento de referências em sistemas com restrição de controle é de grande interesse.…”