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Performance control of rational systems using linear-fractional representations and LMIs
Abstract
Every nonlinear system of the rational type admits a
“linear-fractional representation” (LFR), which consists of
an LTI system connected with a diagonal feedback operator linear in the
state. Using this representation, the authors can compute a quadratic
Lyapunov function that proves various properties for the system
(stability of a polytope of initial conditions, L<sub>2</sub>-induced
gain, etc.). These properties are checked by solving a convex
optimization problem over linear matrix inequalities (LMIs). The
approach can be used for state-feedback synthesis, and also for dynamic
output-feedback synthesis, provided the state equations are linear in
every state coordinate that is not measured
... Basically, the above class of systems is essentially the same one proposed by El Ghaoui and co-authors in [14], [15] namely the linear fractional representation (LFR). The main difference between our technique and the LFR one is that we consider a differential-algebraic model and Ghaoui et al interpret the system as an interconnected system (i.e., a linear system with a feedback state-dependent connection between fictitious inputs and outputs). ...
... In order to guarantee that representation (10) is well-posed, we further assume A4 The matrices Ω 2 and Φ 2 are full column rank for all x ∈ X and δ ∈ ∆. Considering Lemma 2.1 from [14] and (11), we can state the following proposition. Proposition 1: For any rational matrix function M : R n → R n×n with no singularities at origin there exist constant matrices M 1 , M 2 , and affine matrix functions Γ 1 (σ), Γ 2 (σ) with appropriate dimensions such that ...
... From the theory of nonlinear systems a level set of the Lyapunov function is normally used as an estimate of DOA, see e.g. [14]. The idea is as follows. ...
This paper proposes a convex approach to the H-infinity output feedback problem for a class of uncertain nonlinear systems in which the system matrices are allowed to be rational functions of the state and uncertain parameters. We derive sufficient linear matrix inequality (LMI) conditions for designing full-order output feedback controllers that assure the regional stability of the nonlinear system for a given energy bound on the disturbance input in the sense that the state stays inside a given region, and also minimize an upper-bound on the L2-gain of the input-output operator for the class of admissible disturbance signals. Numerical examples are used to illustrate the proposed methodology.
... However , the generalization to the nonlinear case is a difficult task that has been recently addressed by the control community using a wide diversity of approaches [4, 5, 6]. On the other hand, the stability analysis and control of nonlinear system of the type ˙ x = f (x) has been studied by several researchers using the linear matrix inequality (LMI) framework [7, 8]. In this setting, the control problem is rewritten as a set of LMIs using relaxation techniques which are then solved via interior point method algorithms [9]. ...
... The reader is referred to [11, 8] for illustrative examples of above representation applied to ordinary nonlinear systems. Remark 1 The class of system ˙ x = f (x) = Ax + Bπ, 0 = Ωx + Λπ is essentially the same one proposed in [7] if we define the vector field as ...
... From the theory of nonlinear systems a level set of the Lyapunov function is normally used as an estimate of the domain of attraction (DOA), see e.g. [7, 8]. The idea is as follows. ...
This paper proposes sufficient conditions to the regional stability analysis of switched nonlinear systems with time-varying parameters. The nonlinear sub-modes of operation are described by means of differential-algebraic equations involving the state and an auxiliary nonlinear vector. We then use piecewise polynomial Lyapunov functions and a relaxation technique that lead to a convex characterization of the problem in terms of linear matrix inequalities.
... Basically, the above class of systems is essentially the same one proposed by El Ghaoui and co-authors in [14], [15] namely the linear fractional representation (LFR). The main difference between our technique and the LFR one is that we consider a differential-algebraic model and Ghaoui et al interpret the system as an interconnected system (i.e., a linear system with a feedback state-dependent connection between fictitious inputs and outputs). ...
... In order to guarantee that representation (10) is well-posed, we further assume A4 The matrices Ω 2 and Φ 2 are full column rank for all x ∈ X and δ ∈ ∆. Considering Lemma 2.1 from [14] and (11), we can state the following proposition. Proposition 1: For any rational matrix function M : R n → R n×n with no singularities at origin there exist constant matrices M 1 , M 2 , and affine matrix functions Γ 1 (σ), Γ 2 (σ) with appropriate dimensions such that ...
... From the theory of nonlinear systems a level set of the Lyapunov function is normally used as an estimate of DOA, see e.g. [14]. The idea is as follows. ...
This paper proposes a convex approach to the H-infinity output feedback problem for a class of uncertain nonlinear systems in which the system matrices are allowed to be rational functions of the state and uncertain parameters. We derive sufficient linear matrix inequality (LMI) conditions for designing full-order output feedback controllers that assure the regional stability of the nonlinear system for a given energy bound on the disturbance input in the sense that the state stays inside a given region, and also minimize an upper-bound on the /spl Lscr//sub 2/-gain of the input-output operator for the class of admissible disturbance signals. Numerical examples are used to illustrate the proposed methodology.
... However , there are few results for the nonlinear case such as the works of Barreiro et al. (2002) which combines the bifurcation analysis and Lyapunov theory and Bean et al. (2002) which uses piecewise bilinear models and a single polynomial Lyapunov function. On the other hand, the stability and performance analysis, and control synthesis of uncertain nonlinear systems has been recently addressed by many authors via convex optimization problems, e.g. the works of (El Ghaoui and Scorletti, 1996; Dussy and El Ghaoui, 1997; Chesi et al., 2002) and (Trofino, 2000; Johansen, 2000; Coutinho, Trofino and Fu, 2002) that consider quadratic and polynomial Lyapunov functions, respectively. In general, non-quadratic Lyapunov functions are less conservative for dealing with uncertain nonlinear systems than the quadratic ones at the expense of extra computations Johansen (2000). ...
... Observe that the nonlinear decomposition (10) has an augmented space (R n ⊆ R n+m ) and the relationships between (ξ, φ) and (x, τ, λ, u) are defined by means of the constraints Ω 1 x + Ω 2 ξ = 0 and Φ 1 u + Φ 2 φ = 0. As a result, the system can only have rational nonlinearities without singularities at origin in the differential-algebraic equations (El Ghaoui and Scorletti, 1996). However, we can transform a certain differential-algebraic representation with non-rational terms into an augmented differential-algebraic form without nonrational nonlinearities. ...
This paper addresses the problem of determining robust sta- bility regions for a class of nonlinear systems with time- invariant uncertainties subject to actuator saturation. The unforced nonlinear system is represented by differential- algebraic equations where the system matrices are allowed to be rational functions of the state and uncertain parameters, and the saturation nonlinearity is modelled by a sector bound condition. For this class of systems, local stability condi- tions in terms of linear matrix inequalities are derived based on polynomial Lyapunov functions in which the Lyapunov matrix is a quadratic function of the state and uncertain pa- rameters. To estimate a robust stability region is considered the largest level surface of the Lyapunov function belonging to a given polytopic region of state. A numerical example is used to demonstrate the approach.
... For instance, the same modelling technique is used in [20] in the context of uncertain systems. In the context of nonlinear systems, the DAR is closely related to the linear fractional representation (LFR) introduced by El Ghaoui and co-authors [9, 21]. The approaches differ between each other in the way that the nonlinear systems is rewritten to apply the LMI framework. ...
... To this end, consider the following DAR form for system (61): Notice that the regularity of system (61) is guaranteed if x 1 > À1:5 which implies the regularity of the above DAR. To define the Lyapunov function candidate, we choose Y ¼ ½x 1 I 2 x 2 I 2 0 : For comparison purposes, the LFR approach is also applied to the same example considering a quadratic Lyapunov function [9] and a homogeneous one [27] with x fmg ¼ ½x 2 1 x 1 x 2 x 2 2 0 ...
This paper proposes a convex approach to regional stability and ℒ2-gain analysis and control synthesis for a class of nonlinear systems subject to bounded disturbance signals, where the system matrices are allowed to be rational functions of the state and uncertain parameters. To derive sufficient conditions for analysing input-to-output properties, we consider polynomial Lyapunov functions of the state and uncertain parameters (assumed to be bounded) and a differential-algebraic representation of the nonlinear system. The analysis conditions are written in terms of linear matrix inequalities determining a bound on the ℒ2-gain of the input-to-output operator for a class of (bounded) admissible disturbance signals. Through a suitable parametrization involving the Lyapunov and control matrices, we also propose a linear (full-order) output feedback controller with a guaranteed bound on the ℒ2-gain. Numerical examples are used to illustrate the proposed approach. Copyright © 2007 John Wiley & Sons, Ltd.
... Note that a broad class of Markovian jump nonlinear systems can be represented in the form (3), such as systems with rational nonlinearities as well as some trigonometric nonlinearities. Indeed, it can be shown that (3) includes the linear fractional representation of [12], and as such it can model the whole class of systems with rational functions of the state and uncertain parameters without singularities at the origin; for further details see [9] and [10]. ...
... Note that a broad class of Markovian jump nonlinear systems can be represented in the form (3), such as systems with rational nonlinearities as well as some trigonometric nonlinearities. Indeed, it can be shown that (3) includes the linear fractional representation of [12], and as such it can model the whole class of systems with rational functions of the state and uncertain parameters without singularities at the origin; for further details see [9] and [10]. In addition to A1 and A2, we shall adopt the following assumption to guarantee that the DAR (3) is well defined and thus, the uniqueness of the solution x is ensured. ...
This paper addresses the problem of robust stability bility analysis for a class of Markovian jump nonlinear systems subject to polytopic-type parameter uncertainty. A condition for robust local exponential mean square stability in terms of linear matrix inequalities is developed. An estimate of a robust domain of attraction of the origin is also provided. The approach is based on a stochastic Lyapunov function with polynomial dependence on the system state and uncertain parameters. A numerical example illustrates the proposed result.
... Muitos métodos têm sido propostos para obter obter tais fun¸ cões no contexto de sistemas não lineares. Por exemplo , poderiamos mencionar as técnicas baseadas nas equações de Riccati (Mracek e Sznaier et al., 1998; Langson e Alleyne, 1999; Kazakova-Frehse e Moor, 1999) e na resolução de problemas convexos em termos de LMIs (El Ghaoui e Scorletti, 1996; Pettersson e Lennartson, 1997; Dussy e El Ghaoui, 1997; Kiriakidis, 1998; Johansen, 2000; Iwasaki, 2000). Nestes métodos de determinação de funções de Lyapunov , ´ e importante que a formulação permita uma fácil extensão para problemas de síntese ou de desempenho, como por exemplo H 2 ou H ∞ . ...
... Em geral, os métodos propostos na literatura utilizam o conceito de estabilidade quadrática que aplicados a sistemas com incertezas conduzem a resultados conservadores, pois não levam em consideração a taxa de variação dos parâmetros. Por exemplo, o trabalho de (El Ghaoui e Scorletti, 1996) que utiliza funções de Lyapunov quadráticas apresenta resultados conservadores na análise de estabilidade de Sistemas Não Lineares Incertos. Em (Trofino, 2000b) foi proposto um método de análise de estabilidade local de sistemas não lineares incertos, utilizando funções de Lyapunov polinomiais nos estados ...
Neste artigo, aborda-se o problema de análise de estabilidade local do ponto de equilíbrio de sistemas não lineares incertos e de seu respectivo domínio de atração. São propostas condições em termos de inequações matriciais lineares (LMIs) para o problema de estabilidade robusta local baseadas em funções de Lyapunov que são funções polinomiais dos estados e dos parâmetros incertos. O problema de maximização da estimativa do domínio de atração consiste na maximização de uma curva de nível dentro de um dado politopo. Exemplos numéricos demonstram o desempenho desta metodologia.
... The closed-loop stability and feasibility properties of the solution can be proved via standard arguments and are here summarized. A numerical example involving a rational nonlinear plant, which admit an exact LFR description [6] is considered and comparisons with the Norm-Bounded robust approach of [3] shown. ...
... A controlled Van Der Pol nonlinear oscillator [5], [6] is taken into consideration as an illustration of the proposed MPC algorithm ...
A novel predictive control strategy for input- saturated Norm-Bounded LPV discrete-time systems is proposed. The solution is computed by minimizing an upper-bound to the "worst-case" infinite horizon quadratic cost under the constraint of steering the future state evolutions, emanating from the current state, into a feasible and positive invariant set. It will be shown that the "size" of this terminal set depends on the rate of changes of the scheduling parameter which is assumed bounded and measurable.
... On the other hand, the so-called linear matrix inequality (LMI) approach has been used widely to solve problems in linear robust control, gain-scheduling and multi-objective control (Boyd et al., 1994). Since the work (El Ghaoui and Scorletti, 1996) that showed a solution to the nonlinear problem using LMIs, re- 1 This work was partially supported by 'CAPES', Brazil, under grant BEX 0784/00-1. searchers have proposed different solutions to nonlinear robust ¦ ∞ control problems, e.g. ...
... To illustrate the potential of the proposed technique, we analyze two numerical examples. In the first one, we compute an upper bound on the ¥ 2 -gain of the system. In the second example, we design a nonlinear controller that minimizes the ¥ 2 -gain. In both examples , we assume that the sets of admissible input disturbances are given. Ghaoui and Scorletti, 1996): ...
In this paper, we consider the problem of robust H-infinity Control for a class of uncertain nonlinear systems. We derive LMI conditions for analyzing regional robust stability and performance based on Lyapunov functions which are polynomial functions of the state and uncertain parameters. More specifically, we provide an energy bound on the input disturbance which guarantees that the state of the system stays inside a given region. For the given bound on the input disturbance, we also minimize the L2-gain of the input/output operator. Through an iterative algorithm, the proposed technique is applied to control synthesis. Numerical examples are presented to illustrate our results. Keywords: H-infinity control, convex-optimization, uncertainty, Lyapunov methods. 1.
... However, the LMI framework cannot be applied in a straightforward way to deal with nonlinear dynamical systems. Nevertheless, some researchers have recently proposed sufficient LMI conditions to handle nonlinear systems ranging from quadratic [10] to more complex Lyapunov functions [11], [12], [13]. ...
This paper proposes a convex approach to the design of robust discrete-time observers to the state estimation of induction machines subject to rotor resistance variation and load disturbance. The observer gain is designed in order to minimize these effects on the state estimation in an H∞ sense, while guaranteeing the error dynamics stability. The pro- posed conditions are cast in terms of state-dependent linear matrix inequalities. Simulation results demon- strates the performance of the approach.
... In particular, considering appropriate representations of the system, the synthesis conditions can be cast as LMIs (or quasi-LMIs), which can be efficiently numerically solved and allows to consider extra requirements on performance and robustness almost straightforwardly. In this context, we can cite for instance [3], [6], [16], [17]. However, it is worth pointing out that most of these works regard continuous-time dynamics. ...
This work addresses the problem of local stabilization of rational systems considering magnitude control constraints. Based on a Recursive Algebraic Representation (RAR) of the system and on a generalized sector condition, stabilizing conditions in the form of linear matrix inequalities (LMIs) are proposed to compute a saturating state feedback control law that ensures the local asymptotic stability in a certain region of the state space. In this sense, a convex optimization problem is proposed to determine a control law aiming at maximizing an estimate of the region of attraction of the closed-loop system. An extension of the method to consider a quadratic performance criterion is also presented.
... All the computations have been carried out on a PC Pentium 4-based using the YALMIP Toolbox ( [32]). A controlled Van Der Pol nonlinear oscillator ( [34]) is taken into consideration ˙ ...
A novel moving horizon control strategy for input-saturated nonlinear polynomial systems is proposed. The control strategy
makes use of the so called sum-of-squares (SOS) decomposition, i.e. a convexification procedure able to give sufficient conditions
on the positiveness of polynomials. The complexity of SOS based numerical methods is polynomial in the problem size and, as
a consequence, computationally attractive. SOS programming is used here to derive an “off-line” model predictive control (MPC)
scheme and analyze in depth its relevant properties. The main contribution here is to show that such an approach may lead
to less conservative MPC strategies than most existing methods based on global linearization approaches. An illustrative example
is provided to show the effectiveness of the proposed SOS-based algorithm.
... Some approaches have been developed to solve the HJ inequality indirectly by reducing the problem to algebraic inequalities, but these methods are only applicable to low dimensional problems [14, 15]. Also, the linear matrix inequality approach has been used to solve linear robust control problems, and the modified LMIs and LMI framework method are proposed to solve the nonlinear robust control problem161718. Some other approaches, such as a successive approximate solution method, an inverse optimal proportional-integral-derivative control design method and a nonlinear matrix inequality algorithm method, are proposed to solve the HJ inequality more simply192021. ...
A flexible joint robot manipulator can be regarded as a cascade of two subsystems: link dynamics and the motor dynamics. Using this structural characteristic, we propose a robust nonlinear recursive control method for flexible manipulators. The recursive design is done in two steps. First, a fictitious robust control for the link dynamics is designed as if it has a direct control input. As the fictitious control, a nonlinear H
∞-control using energy dissipation is designed in the sense of L
2-gain attenuation from the disturbance caused by uncertainties to performance. In the process, Hamilton-Jacobi (HJ) inequality is solved by a more tractable nonlinear matrix inequality (NLMI) method. In the second step, the other fictitious and the actual robust control are designed recursively by using the Lyapunov’s second method. The proposed robust control is applied to a two-link robot manipulator with flexible joints in computer simulations. The simulation results show that the proposed robust control has robustness to the model uncertainty caused by changes in the link inertia and the joint stiffness.
... Na literatura esta classé e conhecida como sistema lineares com dependência paramétrica (LPV). Os trabalhos em El Ghaoui e Scorletti (1996), Feron et al. (1996, Haddad e Bernstein (1991), Kapila et al. (1997, Shamma e Xiong (1995), Trofino e de Souza (1999) e Wang e Balakrishnan (1998) tratam desta classe de sis- tema. Uma ferramenta usual para o estudo da estabilidade de sistemas LPV tem sido a função de Lyapunov, sendo que esta pode ou não apresentar uma dependência nos parâmetros do sistema. ...
Este artigo trata do problema de performance H∞ para sistemas LPV. O critério de performance H∞ é obtido empregando-se uma função de Lyapunov com dependência quadrática nos parâmetros do sistema. O critério de performance usado admite que os parâmetros do sistema e suas respectivas taxas de variação sejam confinadas numa região convexa. Sob estas considerações obtém-se LMIs que apresentam condições suficientes destinadas aos casos de análise e síntese de sistemas LPV onde um critério de performance H∞ é otimizado.
... It turns out that a large class of nonlinear systems can be decomposed into the form (12). In fact, the proposed approach is basically the same one introduced in [9] (called linear fractional representation or LFR). Moreover, taking into account [9, Lemma 2.1] and using some algebraic manipulations, one can prove that representation (12) is capable of modelling the class of rational system with no-singularities at origin. ...
This paper proposes a convex approach to the model approximation problem for a class of regionally stable uncertain nonlinear systems. More specifically, we determine an extended bilinear system which approximates in a given region of the state space the input-to-output dynamics of the nonlinear system with time-varying parameters. To this end, we use a suitable parametrization of the Lyapunov matrix in order to obtain convex model design conditions in terms of linear matrix inequalities (LMIs). The proposed approach is also extended to the model reduction case without rank constraints.
... A lot of recent research focuses on analysis and synthesis approaches in the framework of linear matrix inequalities (LMIs). Design approaches range from using quadratic Lyapunov functions ([1, 2, 3]) to those based on polynomial Lyapunov functions ([4, 5, 6, 7]). In general, non-quadratic Lyapunov functions are less conservative for dealing with uncertain and nonlinear systems than quadratic Lyapunov functions at the expense of extra computation. ...
... The last decade or so has witnessed active research work in the area of robust control of continuous-time nonlinear systems in the framework of linear matrix inequalities (LMIs). Design approaches range from using quadratic Lyapunov functions ([1, 2]) to those based on polynomial Lyapunov functions ([3, 4]). In general, non-quadratic Lyapunov functions are less conservative for dealing with uncertain and nonlinear systems than quadratic Lyapunov functions at the expense of extra computation [5]. ...
Deals with the problem of regional stability and performance analysis for a class of nonlinear discrete-time systems with uncertain parameters. We use polynomial Lyapunov functions to derive stability conditions and performance criteria in terms of linear matrix inequalities (LMIs). Although the use of polynomial Lyapunov functions is common for continuous-time systems as a way to reduce the conservatism in analysis, we point out that direct generalization of such an approach to discrete-time systems leads to intractable solutions because it results in a large number of LMIs. We introduce a novel approach to reduce the computational complexity by generalizing a result of Oliveira et al. on robust stability analysis for discrete-time systems with parameter uncertainties. We point out that the proposed method can lead to less conservative results when compared with results using quadratic Lyapunov functions.
... A lot of recent research focuses on analysis and synthesis approaches in the framework of linear matrix inequalities (LMIs). Design approaches range from using quadratic Lyapunov functions ([1, 2, 3]) to those based on polynomial Lyapunov functions ([4, 5, 6, 7]). In general, non-quadratic Lyapunov functions are less conservative for dealing with uncertain and nonlinear systems than quadratic Lyapunov functions at the expense of extra computation. ...
This paper studies the problem of regional stability and performance analysis for a class of nonlinear uncertain systems. Both continuous-time and discrete-time systems are considered. Our approach is based on the use of polynomial Lyapunov functions. We show how these functions can be used to reduce the conservatism in analysis. The conditions for analysis are given in terms of linear matrix inequalities to allow feasible computational implementation. These inequalities are derived by introducing auxilary nonlinear algebraic equations and applying Finsler's lemma. It turns out that continuous-time systems and discretetime systems require sharply di#erent techniques in order to achieve simple linear matrix inequalities. A numerical example is used to illustrate the approach and show that the proposed method can lead to less conservative results when compared with results using quadratic Lyapunov functions.
This manuscript presents my teaching and research activities from 2012 to 2021. The first part is dedicated to a brief description of these activities while the second part is focused on the synthesis of my research activities. My research subject is the development of the design methods for complex systems. These methods are based on the Robust Control and its extensions : input-output approach, dissipativity, integral quadratic constraint analysis. My main contributions in research can be split in two categories : deployment of Robust Control methods in other disciplines and development of new methods for the system design. The first category illustrates a strong potential of Robust Control for the system design and raises
ambitious scientific challenges. These challenges underline the necessity for efficient design methods that can take into account the algorithmic complexity implied by growing size and dynamic complexity of the modern technological systems. The second part of the contributions is dedicated to address these challenges. In addition to propose the design methods relevant with respect to the pragmatic problem under consideration, obtained original theoretical results are applicable to other design problems. Particularly, this includes the decentralized control synthesis in the context of Multi-Agent systems, the hierarchical analysis of the robust performance, the synthesis of LFR systems and the control and analysis of the systems with harmonically varying parameters.
This paper is concerned with design and stability analysis for a class of nonlinear systems which are exactly linearizable via nonlinear state-feedback only. First, a modeling method for linearized systems is given, which reflects the uncertainties of both the physical parameters of the original nonlinear systems and the state-dependent nonlinear terms in a non-conservative way by using linear fractional representation. Second, a sufficient condition for the stability is given for the closed-loop system, which consists of linearized system and linear compensator, in the presence of parameter variations. Third, the effectiveness of the proposed method is evaluated by experiments on tracking control using a three-link DD manipulator.
This paper addresses the design of static anti-windup compensators for a class of nonlinear systems subject to actuator saturation. Considering that a (possibly nonlinear) dynamic output feedback control law has been previously designed disregarding the saturation, a method to compute a static anti-windup gain which ensures the local stability of the closed-loop system is proposed. The results are based on a differential algebraic representation of rational systems and on a modified version of the generalized sector condition to deal with the saturation effects. From these elements, an algorithm based on the solution of LMIs is proposed to calculate an anti-windup gain to enlarge the region of attraction of the closed-loop system. Moreover, extensions of the results are presented to deal with parameter uncertainty and the presence of energy bounded disturbances. A numerical example is presented to illustrate the proposed method.
This paper aims to develop its own code of fuzzy logic control for applications in industrial systems controlled by programmable logic controllers. The algorithm is designed with flexibility for inclusion or exclusion of linguistic variables and dynamically changing the parameters of the rules of inference. The algorithm was developed on the Matlab platform and had the results compared with the Fuzzy Logic Toolbox for validation. Next was made in the implementation language in PLC ladder with different data types. The results were satisfactory, subject to the limitations regarding the type of data used in each test. Re-garding the processing, the PLC had satisfactory performance.
This paper focuses on the problem of static anti‐windup design for a class of multivariable nonlinear systems subject to actuator saturation. The considered class regards all systems that are rational on the states or that can be conveniently represented by a rational system with algebraic constraints considering some variable changes. More precisely, a method is proposed to compute a static anti‐windup gain which ensures regional stability for the closed‐loop system assuming that a dynamic output feedback controller is previously designed to stabilize the nonlinear system. The results are based on a differential algebraic representation of rational systems. The control saturation effects are taken into account by the application of a generalized sector bound condition. From these elements, LMI‐based conditions are devised to compute an anti‐windup gain with the aim of enlarging the closed‐loop region of attraction. Several numerical examples are provided to illustrate the application of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.
This paper focuses on the problem of designing stabilizing state feedback control laws for rational nonlinear systems subject to actuator saturation. The results are based on the differential-algebraic representation of rational systems and a generalized sector relation to address the saturation effects. From these elements, LMI based conditions are devised to compute a state feedback control law leading to a maximized ellipsoidal estimate of the region of attraction of the closed-loop system. Several numerical examples are provided to illustrate the potentialities of the technique.
A nonlinear missile model with time-varying uncertain parameters is controlled with a simple feedback linearisation and time-scale separation design, with synthesis based on the nominal model and full state feedback. The closed-loop system is then represented as a linear fractional transformation (LFT). A robust H∞ filter is designed for the controlled plant, to estimate unknown states. Robust stability of the closed-loop system is then verified by using a scaled linear differential inclusion (LDI) technique.
A novel predictive control strategy for input-saturated norm-bounded linear parameter varying discrete-time systems is proposed. The solution is computed by minimizing an upper bound to the ‘worst-case’ infinite horizon quadratic cost under the constraint of steering the future state evolutions, emanating from the current state, into a feasible and positive invariant set. It will be shown that the ‘size’ of this terminal set depends on the rate of change of the scheduling parameter, which is assumed to be bounded and measurable. Copyright © 2007 John Wiley & Sons, Ltd.
This paper proposes a robust H∞ filter design method for discrete-time linear parameter varying systems with the state-space model matrices depending rationally on time-varying parameters. The system is described by a difference-algebraic representation and the parameters admissible values and variations are assumed to belong to given intervals. A convex optimization approach in terms of linear matrix inequalities is devised for designing robust filters. The filter design is based on Lyapunov functions with rational dependence on the system parameters and incorporates information on available bounds on the parameters variation. The proposed method can be also applied to the design of gain-scheduled H∞ filters. Numerical examples illustrate the effectiveness of the proposed filter design.
This paper deals with the problem of robust analysis and control of a class of nonlinear discrete-time systems with (constant) uncertain parameters. For the analysis problem we use a polynomial Lyapunov function and we generalize, for nonlinear systems, the “extended stability” notion proposed by Oliveira et al. (1999) in the context of linear discrete-time uncertain systems. As a result, we propose an LMI optimization problem to maximize an estimate of the domain of attraction, and also extend this approach to the synthesis problem by considering parameter-dependent Lyapunov functions and nonlinear multipliers. Numerical examples illustrate the approach and show its potential for solving analysis and control problems of nonlinear discrete-time systems.
Two linear parameter-varying gain-scheduled controllers for active flutter suppression of the NASA Langley Research Center's Benchmark Active Control Technology (BACT) wing section are presented and compared to a previously presented gain-scheduled controller. The BACT wing section changes significantly as a function of Mach and dynamic pressure, The two linear parameter-varying (LPV) gain-scheduled controllers incorporate these changes as well as bounds on the rate of change of Mach and dynamic pressure The inclusion of rate bounds in the design process allows for improved performance over a larger range of operating conditions than previously achieved by a linear fractional transformation gain-scheduled controller. The LPV controllers differ in that one primarily reduces coupling between the trailing-edge flap and the pitch and plunge modes, whereas the second optimizes wind gust attenuation. Closed-loop stability and improved performance are demonstrated via time simulations in which both Math and dynamic pressure are allowed to vary in the presence of a Dryden wind-gust disturbance.
This paper deals with the robust filtering problem for a class of regionally stable nonlinear sys-tems subject to uncertain parameters belonging to a given polytope. The nonlinear system is represented by differential-algebraic equations where the system matrices are allowed to be rational functions of the state and uncertain parameters. Linear matrix inequality conditions based on a parameter-dependent Lyapunov function are proposed to design a robust bilinear filter which minimizes an upper-bound on the L 2 -gain of the noise-to-estimation error operator.
This chapter investigates the problem of robust ℋ∞ piecewise statefeedback control for a class of nonlinear discrete-time-delay systems via Takagi-Sugeno (T-S) fuzzy models.
The state delay is assumed to be time-varying and of an interval-like type with the lower and upper bounds. The parameter
uncertainties are assumed to have a structured linear-fractional form. Based on two novel piecewise Lyapunov-Krasovskii functionals
and some matrix inequality convexifying techniques, both delay-independent and delay-dependent controller design approaches
are developed in terms of a set of linear matrix inequalities (LMIs). Numerical examples are also provided to illustrate the
effectiveness and less conservatism of the proposed methods.
This paper focus on the problem of estimating the region of attraction of nonlinear control systems that can be put in a rational algebraic-differential form. The estimates are computed by means of invariant domains associated to rational Lyapunov functions. The saturation effects are taken into account by the use of a generalized sector condition for deadzone nonlinearities. These ingredients lead to the formulation of an LMI-optimization problem for computing regions of stability for the closed-loop system.
This paper deals with the problem of robust H∞ filtering for a class of Markov jump nonlinear systems subject to constant convex-bounded uncertain parameters. The system is described by a differential-algebraic representation, which can model the whole class of Markov jump systems with rational functions of the state and uncertain parameters, as well as some trigonometric nonlinearities. The design of mode-dependent (Markov jump) and mode-independent linear filters is considered. The proposed designs are based on the notion of exponential mean square stability together with a stochastic Lyapunov function with polynomial dependence on the system state and uncertain parameters, and are given in terms of linear matrix inequalities. A numerical example is provided to demonstrate the effectiveness of the derived results
This paper focuses on the regional stability analysis of implicit polynomial systems subject to constant uncertain parameters. A polynomial Lyapunov function is computed that assures the robust local stability of the system and also provides an estimate of the domain of attraction. The stability conditions are cast as a set of linear matrix inequalities. A numerical example illustrates the effectiveness of the proposed method.
A Luenberger-based observer is proposed to the state estimation of a class of nonlinear systems subject to parameter uncertainty and bounded disturbance signals. A nonlinear observer gain is designed in order to minimize the effects of the uncertainty, error estimation and exogenous signals in an 7-L, sense by means of a set of state- and parameterdependent linear matrix inequalities that are solved using standard software packages. A numerical example illustrates the approach.
This technical note develops a robust stability analysis method for a class of implicit polynomial systems subject to uncertain constant parameters. A polynomial (with respect to the state and uncertain parameters) Lyapunov function is computed to assess the robust local stability of the system and to provide a robust estimate of the domain of attraction. The results are given in terms of linear matrix inequalities that are affinely dependent on the system state and uncertain parameters. Numerical examples illustrate the effectiveness of the developed approach.
This study focuses on the problem of regional stability analysis of rational control systems with saturating actuators. Estimates of the region of attraction are computed by means of invariant domains associated to rational Lyapunov functions. Conditions for computing these invariant domains (regions of stability) are proposed in the form of linear matrix inequalities (LMIs). These conditions are derived considering a differential-algebraic representation of the rational system dynamics. The saturation effects are taken into account by means of a generalised sector condition for deadzone non-linearities. The obtained conditions are cast in convex optimisation schemes in order to compute a Lyapunov function which leads to a maximised estimate of the region of attraction.
In this paper, we study the asymptotic stabilization of fractional-order systems using an observer-based control law. The fractional-order systems under consideration are either linear or nonlinear and affine. A generalization of Gronwall-Bellman which is proved in the appendix is used to derive the closed-loop asymptotic stability.
The problem of controlling a liquid–gas separation process is approached by using LPV control techniques. An LPV model is derived from a nonlinear model of the process using differential inclusion techniques. Once an LPV model is available, an LPV controller can be synthesized. The authors present a predictive LPV controller based on the GPC controller [Clarke D, Mohtadi C, Tuffs P. Generalized predictive control – Part I. Automatica 1987;23(2):137–48; Clarke D, Mohtadi C, Tuffs P. Generalized predictive control – Part II. Extensions and interpretations. Automatica 1987;23(2):149–60]. The resulting controller is denoted as GPC–LPV. This one shows the same structure as a general LPV controller [El Gahoui L, Scorletti G. Control of rational systems using linear-fractional representations and linear matrix inequalities. Automatica 1996;32(9):1273–84; Scorletti G, El Ghaoui L. Improved LMI conditions for gain scheduling and related control problems. International Journal of Robust Nonlinear Control 1998;8:845–77; Apkarian P, Tuan HD. Parametrized LMIs in control theory. In: Proceedings of the 37th IEEE conference on decision and control; 1998. p. 152–7; Scherer CW. LPV control and full block multipliers. Automatica 2001;37:361–75], which presents a linear fractional dependence on the process signal measurements. Therefore, this controller has the ability of modifying its dynamics depending on measurements leading to a possibly nonlinear controller. That controller is designed in two steps. First, for a given steady state point is obtained a linear GPC using a linear local model of the nonlinear system around that operating point. And second, using bilinear and linear matrix inequalities (BMIs/LMIs) the remaining matrices of GPC–LPV are selected in order to achieve some closed loop properties: stability in some operation zone, norm bounding of some input/output channels, maximum settling time, maximum overshoot, etc., given some LPV model for the nonlinear system. As an application, a GPC–LPV is designed for the derived LPV model of the liquid–gas separation process. This methodology can be applied to any nonlinear system which can be embedded in an LPV system using differential inclusion techniques.
Every system of the form , y = g(x, u), where f and g are rational functions of the state x and linear functions of the input u, possesses a linear-fractional representation (LFR). In this LFR, the system is viewed as an LTI system, connected with a diagonal feedback element linear in the state. We devise an algorithm for computing LFRs. Based on this construction, we give sufficient conditions for various properties to hold for the open-loop system. These include checking whether a given polytope is stable, finding a lower bound on the decay rate of trajectories initiating in this polytope, computing an upper bound on the L2 gain, etc. All these conditions are obtained by analyzing the properties of a differential inclusion related to the LFR, and given as convex optimization problems over linear matrix inequalities (LMIs). We show how to use this approach for static state-feedback synthesis. We then generalize the results to dynamic output-feedback synthesis, in the case when f and g are linear in every state coordinate that is not measured. Extensions towards a class of nonrational and uncertain nonliner systems are discussed.
In this paper, we present several conditions which are both necessary and sufficient for quadratic stability of an uncertain/nonlinear system. These conditions involve multiplier matrices which characterize the uncertain/nonlinear terms in the system description. It is known that some of these conditions are sufficient for quadratic stability. One of the main contributions of this paper is to demonstrate that these conditions are also necessary conditions, hence, they are not conservative conditions for quadratic stability. By presenting multiplier matrices for many common types of uncertain/nonlinear terms, the paper also demonstrates the usefulness of the multiplier matrix approach in the analysis and control of nonlinear/time-varying/uncertain systems.
This paper presents a proposal for estimating the uncertainty and grouping quality that take place when a model is identified using a Takagi-Sugeno Fuzzy Inference System (T-S FIS). Additionally, the integration of such measures as criteria for model evaluation based on uncertainty and fuzzy partition generated during model identification is proposed. Such an index allows to identifying a model that causes a minimum uncertainty increment respecting original process data. Additionally, the index evaluates data distribution and density at obtained fuzzy sets during fuzzy modeling. The index values can be used as a complement to the final model when it is used in any model-based task (design, optimization, control, etc). Such class of tasks supposes a model with uniform uncertainty (assumed low) in all model space. Using proposed index a more realistic model uncertainty value may be calculated at any point in the model space.
This paper proposes a convex approach to design robust state observers for (extended) bilinear systems subject to parameter uncertainty, bounded disturbance and control signals, and unknown initial conditions. The estimator design problem is cast as a set of state-dependent linear matrix inequalities which can be solved very efficiently using standard software packages. The proposed methodology is applied to the problem of flux estimation in current-fed induction motors.
This paper deals with the problem of guaranteed cost analysis and control of a class of nonlinear discrete-time systems with uncertain parameters. We use polynomial Lyapunov functions to derive stability conditions with a guaranteed bound on the 2-norm of the performance output in terms of linear matrix inequalities (LMIs). We then extend this approach to the control design by considering parameter-dependent Lyapunov functions and nonlinear (stateand parameter-dependent) multipliers.
Some nonlinear control problems in industry are successfully
solved using gain-scheduling, a method primarily based on intuition from
control design for linear systems. Linear parameter varying system (LPV)
theory, introduced by Shamma et al. (1992), can be used to simplify some
of the interpolation and realization problems associated with
conventional gain-scheduling. However, many questions about the
relevance of LPV theory to nonlinear control design remain. This paper
illustrates, via examples, some of the issues
This note addresses the problem of robust stability analysis for a class of Markov jump nonlinear systems subject to polytopic-type parameter uncertainty. A condition for robust local exponential mean square stability in terms of linear matrix inequalities is developed. An estimate of a robust domain of attraction of the origin is also provided. The approach is based on a stochastic Lyapunov function with polynomial dependence on the system state and uncertain parameters. A numerical example illustrates the proposed result.
This paper deals with the problem of regional stability and performance analysis for a class of nonlinear discrete-time systems with uncertain parameters. We use polynomial Lyapunov functions to derive stability conditions and performance criteria in terms of linear matrix inequalities (LMIs). Although the use of polynomial Lyapunov functions is common for continuoustime systems as a way to reduce the conservatism in analysis, we point out that direct generalization of such an approach to discrete-time systems leads to intractable solutions because it results in a large number of LMIs. We introduce a novel approach to reduce the computational complexity by generalizing a result of Oliveira et. al. on robust stability analysis for discretetime systems with parameter uncertainties. We point out that the proposed method can lead to less conservative results when compared with results using quadratic Lyapunov functions.
.<F3.802e+05> Nonlinear control systems can be stabilized by constructing control Lyapunov functions and computing the regions of state space over which such functions decrease along trajectories of the closed-loop system under an appropriate control law. This paper analyzes the computational complexity of these procedures for two classes of control Lyapunov functions. The systems considered are those which are nonlinear in only a few state variables and which may be a#ected by control constraints and bounded disturbances. This paper extends previous work by the authors, which develops a procedure for stability analysis for these systems whose computational complexity is exponential only in the dimension of the "nonlinear" states and polynomial in the dimension of the remaining states. The main results are illustrated by a numerical example for the case of purely quadratic control Lyapunov functions.<F4.005e+05> Key words.<F3.802e+05> Lyapunov methods, nonlinear control systems, comput...
The problem of constructing Lyapunov functions for a class of
nonlinear dynamical systems is considered. The problem is reduced to the
construction of a polytope satisfying some conditions. A generalization
of the concept of sector condition that it makes possible to evaluate a
given nonlinear function by using a set of piecewise-linear functions is
proposed. This improvement greatly reduces the conservatism in the
stability analysis of nonlinear systems. Two algorithms for constructing
such polytopes are proposed, and two examples are shown to demonstrate
the usefulness of the results
This paper deals with the problem of the estimation of regions of asymptotic stability for continuous, autonomous, nonlinear systems. The first part of the work provides a comprehensive survey of the existing methods and of their applications in engineering fields. In the second part certain topological considerations are first developed and the "trajectory reversing method" is then presented together with a theorem on which it is based. In the final part, several examples of application are reported, showing the efficiency of the proposed technique for low-order (second and third) systems.
"I believe that the authors have written a first-class book which can be used for a second or third year graduate level course in the subject... Researchers working in the area will certainly use the book as a standard reference... Given how well the book is written and organized, it is sure to become one of the major texts in the subject in the years to come, and it is highly recommended to both researchers working in the field, and those who want to learn about the subject." —SIAM Review (Review of the First Edition)
"This book is devoted to one of the fastest developing fields in modern control theory---the so-called 'H-infinity optimal control theory'... In the authors' opinion 'the theory is now at a stage where it can easily be incorporated into a second-level graduate course in a control curriculum'. It seems that this book justifies this claim." —Mathematical Reviews (Review of the First Edition)
"This work is a perfect and extensive research reference covering the state-space techniques for solving linear as well as nonlinear H-infinity control problems." —IEEE Transactions on Automatic Control (Review of the Second Edition)
"The book, based mostly on recent work of the authors, is written on a good mathematical level. Many results in it are original, interesting, and inspirational...The book can be recommended to specialists and graduate students working in the development of control theory or using modern methods for controller design." —Mathematica Bohemica (Review of the Second Edition)
"This book is a second edition of this very well-known text on H-infinity theory...This topic is central to modern control and hence this definitive book is highly recommended to anyone who wishes to catch up with this important theoretical development in applied mathematics and control." —Short Book Reviews (Review of the Second Edition)
"The book can be recommended to mathematicians specializing in control theory and dynamic (differential) games. It can be also incorporated into a second-level graduate course in a control curriculum as no background in game theory is required." —Zentralblatt MATH (Review of the Second Edition)
This paper deals with the problem of automatically constructing a quadratic Lyapunov Function V = x'Ax for a high order non-linear system given by x ̇ = f(x), f(0) = 0, where f(x) is a continuous function of x which guarantees uniqueness of solutions of the system. The Lyapunov Function is found by a direct search technique so that the volume of the asymptotic stability region obtained for the system, x'Ax = 1, is maximized, thereby giving an estimate of the asymptotic stability boundary of the system. Experimental results show that such a procedure gives an excellent approximation to the exact asymptotic stability region for a system. Numerical examples are included in the paper for second, third and fourth order systems.
In this paper we present various theoretical and computational methods for estimating the domain of attraction of an autonomous nonlinear system. These methods are based on the concept of a maximal Lyapunov function, which is introduced in this paper. A partial differential equation characterizing a maximal Lyapunov function is derived, and the relationships of this equation as compared to Zubov's partial differential equation are discussed. An iterative procedure is given for solving the new partial differential equation. This procedure yields Lyapunov function candidates that are rational functions rather than polynomials. The method is applied to four two-dimensional examples and one three-dimensional example, and it is shown that the estimates obtained using this method are, in many cases, substantially better than those obtained using known methods.
In a series of recent papers, it was shown that the solution of the problem of disturbance attenuation for nonlinear system is related to the existence of solutions of a pair of Hamilton-Jacobi inequalities in n independent variables, associated with state-feedback and output-injection design. The purpose of the present paper is to discuss the necessity of the existence of solutions of the Hamilton-Jacobi inequality which determines the output-injection gain.
Previously obtained results on L 2-gain analysis of smooth
nonlinear systems are unified and extended using an approach based on
Hamilton-Jacobi equations and inequalities, and their relation to
invariant manifolds of an associated Hamiltonian vector field. On the
basis of these results a nonlinear analog is obtained of the simplest
part of a state-space approach to linear H <sub>∞</sub>
control, namely the state feedback H <sub>∞</sub> optimal
control problem. Furthermore, the relation with H <sub>∞
</sub> control of the linearized system is dealt with
Quadratic Lyapunov functions for rational systems: an LMI approach
- L Ghaoui
An LMI-based approach to enlarging a stability domain by state feedback
- L Saydy
- Y.-S Chou
- S Coraluppi
- E Abed
- A Tits