Robustness analysis of nonlinear systems subject to state feedback linearization

Rohr, Eduardo; Pereira, Luís Fernando Alves; Coutinho, Daniel

doi:10.1590/s0103-17592009000400003

Cited by 11 publications

(12 citation statements)

References 19 publications

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“…Now, from Lemmas 3 and 4 and the ideas presented in (Rohr et al, 2009), it is possible to derive some sufficient conditions to ensure that D with V (x) = x ′ P x, P = P ′ ≻ 0, is an estimate of the region of attraction of the nonlinear system (1) within the set X. It is important to emphasize that the calculation of this region D is done for the representation of (1) in the form of the NLDI (5).…”

Section: Estimation Of Regions Of Attractions For Nonlinear Systems Umentioning

confidence: 99%

“…is a constant matrix of proper dimension and f : X → R n is a nonlinear function of class C 1 . Here, the subset X of R n represents a state-space region of interest of (1) given by (Rohr et al, 2009)…”

Section: Problem Formulationmentioning

confidence: 99%

“…Next Lemma provides sufficient conditions to ensure that the region of attraction D as defined by D := {x : x ′ P x ≤ 1, P = P ′ ≻ 0} is bounded by the state-space region X (Rohr et al, 2009).…”

Section: Estimation Of Regions Of Attractions For Nonlinear Systems Umentioning

confidence: 99%

“…Lemma 5 provides an estimate D ⊂ X of the region of attraction of the nonlinear system (1), using the representation of this system in the form of the NLDI (5). In order to find the largest estimate D inside X, we may solve the LMIs in Lemma 5 as an optimization problem by minimizing the trace of matrix P , as suggested in (Rohr et al, 2009).…”

Section: Estimation Of Regions Of Attractions For Nonlinear Systems Umentioning

confidence: 99%

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A new approach for modeling and control of nonlinear systems via norm-bounded linear differential inclusions

Kuiava

Ramos

Pota

2012

Sba Controle & Automação

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A systematic approach to model nonlinear systems using norm-bounded linear differential inclusions (NLDIs) is proposed in this paper. The resulting NLDI model is suitable for the application of linear control design techniques and, therefore, it is possible to fulfill certain specifications for the underlying nonlinear system, within an operating region of interest in the state-space, using a linear controller designed for this NLDI model. Hence, a procedure to design a dynamic output feedback controller for the NLDI model is also proposed in this paper. One of the main contributions of the proposed modeling and control approach is the use of the mean-value theorem to represent the nonlinear system by a linear parameter-varying model, which is then mapped into a polytopic linear differential inclusion (PLDI) within the region of interest. To avoid the combinatorial problem that is inherent of polytopic models for medium- and large-sized systems, the PLDI is transformed into an NLDI, and the whole process is carried out ensuring that all trajectories of the underlying nonlinear system are also trajectories of the resulting NLDI within the operating region of interest. Furthermore, it is also possible to choose a particular structure for the NLDI parameters to reduce the conservatism in the representation of the nonlinear system by the NLDI model, and this feature is also one important contribution of this paper. Once the NLDI representation of the nonlinear system is obtained, the paper proposes the application of a linear control design method to this representation. The design is based on quadratic Lyapunov functions and formulated as search problem over a set of bilinear matrix inequalities (BMIs), which is solved using a two-step separation procedure that maps the BMIs into a set of corresponding linear matrix inequalities. Two numerical examples are given to demonstrate the effectiveness of the proposed approach.

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Section: Estimation Of Regions Of Attractions For Nonlinear Systems Umentioning

confidence: 99%

Section: Problem Formulationmentioning

confidence: 99%

Section: Estimation Of Regions Of Attractions For Nonlinear Systems Umentioning

confidence: 99%

Section: Estimation Of Regions Of Attractions For Nonlinear Systems Umentioning

confidence: 99%

See 2 more Smart Citations

A new approach for modeling and control of nonlinear systems via norm-bounded linear differential inclusions

Kuiava

Ramos

Pota

2012

Sba Controle & Automação

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“…Recentemente, em Rohr et al (2009) foi proposto uma téc-nica para assegurar uma certa robustez à lei de controle linearizante para uma classe de sistemas não lineares através da escolha adequada da dinâmica livre que resulta da realimentação linearizante. Em resumo, esse trabalho utiliza uma função quadrática e uma descrição das condições de estabilidade em termos de desigualdades matriciais lineares (ou LMIs) dependentes dos estados para garantir a estabilidade robusta do sistema de controle frente a incertezas paramé-tricas.…”

Section: Introductionunclassified

Controle robusto por realimentação linearizante parcial de bioreatores em modo de operação descontínua com alimentação

Pimentel¹,

Coutinho

2012

Sba Controle & Automação

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Robust Partial Feedback Linearizing Control of Bioreactors in Fed-Batch ModeA methodology for the robust control design of a class of bioreactors operating in fed-batch mode is proposed. Firstly, a general model is presented to describe the growing dynamics of the bacteria Escherichia Coli and the yeast Saccharomyces Cerevisiae cultures, which are the most employed microorganisms applied in biotechnological industry. From this model, the paper proposes a control law that regulates the by-product concentration in small levels (i.e., close to zero) aiming to maximize the biomass concentration. To this end, the nonlinear dynamics of the bioreactor is rewritten in terms of time-varying parameters (quasi-LPV approach) in order to apply the linear matrix inequality framework for designing the free dynamics resulting from a partial feedback linearizing strategy. The proposed design conditions provide the robust stability of the closed-loop system while ensuring some performance w.r.t. disturbance signals. To verify the behavior of the closed-loop system, several simulation based tests are carried out to evaluate the controller performance against some recent strategies proposed in the specialized literature. RESUMOUma metodologia para o projeto de controladores robustos foi desenvolvida para bioreatores operando em modo descontínuo com alimentação. Inicialmente, apresenta-se um modelo geral que descreve a dinâmica do crescimento da bacté-ria Escherichia Coli e da levedura Saccharomyces Cerevisiae que são atualmente os dois microorganismos mais utilizados na indústria biotecnológica. A partir do modelo dinâmico da cultura de microorganismos, busca-se projetar uma lei de controle que mantenha a concentração do produto secundá-rio em níveis próximos a zero visando, desta forma, maximizar a produção de biomassa. Com este objetivo, a dinâmica não linear é descrita em termos de parâmetros variantes no tempo (abordagem quasi-LPV) possibilitando a utilização da formulação por desigualdades matriciais lineares para o projeto da dinâmica livre resultante da aplicação de uma lei de controle do tipo linearizante parcial. As condições propostas permitem garantir a estabilidade robusta do sistema em malha fechada frente a incertezas paramétricas, além de assegurar um certo desempenho com relação a perturbações. Para verificar o comportamento da metodologia proposta, vários testes por simulação são realizados para avaliar o comportamento da estratégia proposta em relação a trabalhos disponí-veis na literatura especializada.

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Gain‐scheduled control design for discrete‐time nonlinear systems using difference‐algebraic representations

Reis

Araujo

Tôrres

et al. 2020

Intl J Robust & Nonlinear

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Summary This paper addresses the local stabilization problem of nonlinear systems described by Difference‐Algebraic Representations (DAR). A novel set of sufficient Linear Matrix Inequalities (LMI) conditions are developed to design gain‐scheduled state feedback controllers. The proposed approach uses parameter‐dependent Lyapunov functions and new auxiliary decision variables aiming to obtain less conservative results. Two optimization problems are proposed to either obtain the largest estimated Domain‐of‐Attraction (DOA) or minimize the ℓ2‐gain from the energy‐bounded disturbance input to the performance output. Furthermore, this investigation considers control input saturation and system states constrained in a polyhedral region. Numerical examples are provided to illustrate the effectiveness of the proposed methodology, showing favorable comparisons with recently published similar approaches.

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Robustness analysis of nonlinear systems subject to state feedback linearization

Cited by 11 publications

References 19 publications

A new approach for modeling and control of nonlinear systems via norm-bounded linear differential inclusions

A new approach for modeling and control of nonlinear systems via norm-bounded linear differential inclusions

Controle robusto por realimentação linearizante parcial de bioreatores em modo de operação descontínua com alimentação

Gain‐scheduled control design for discrete‐time nonlinear systems using difference‐algebraic representations

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