Parameter-dependent Lyapunov Function Research Articles

Worst-Case Mahler Measure in Polytopic Uncertain Systems

The Mahler measure provides a way to quantify the unstable and plays a key role in stabilization problems. This technical brief addresses the computation of the worst-case Mahler measure in systems depending polynomially on uncertain parameters constrained in a polytope. A sufficient condition for establishing an upper bound of the worst-case Mahler measure is provided in terms of linear matrix inequality (LMI) feasibility tests, where a homogeneous parameter-dependent quadratic Lyapunov function (HPD-QLF) is searched for. Moreover, it is shown that the best upper bound guaranteed by this condition can be obtained by solving generalized eigenvalue problems. Then, the conservatism of this methodology is investigated, showing that the upper bound is monotonically nonincreasing with the degree of the HPD-QLF, and that there exists a degree for which the upper bound is guaranteed to be tight. Some numerical examples illustrate the proposed results.

Gain-scheduled control of two-dimensional discrete-time linear parameter-varying systems in the Roesser model

This paper is concerned with gain-scheduled control of two-dimensional discrete-time linear parameter-varying systems described by a Roesser state-space model with matrices depending affinely on time-varying scheduling parameters. The parameter admissible values and variations are assumed to belong to given intervals. Linear matrix inequality based methods are devised for designing static state feedback gain-scheduled controllers with either an H∞ or quadratic regulator-type performance. The control designs build on quadratically parameter-dependent Lyapunov functions and allow for incorporating information on available bounds on the parameters variation. The proposed controller gain can be independent, affine, quadratic, or a matrix fraction of quadratic polynomial matrices in the scheduling parameters.

Compensadores dinâmicos para sistemas discretos no tempo com parametros variantes e aplicação a um sistema fuzzy Takagi-Sugeno

Neste artigo, considera-se o problema de síntese de compensadores dinâmicos por realimentação de saídas para sistemas lineares, discretos no tempo, com parâmetros variantes. Sob a hipótese que os parâmetros variantes podem ser medidos ou estimados em tempo-real, são consideradas duas estruturas de compensadores dinâmicos, com ordem igual a da planta: uma parcialmente dependente e outra completamente dependente dos parâametros. A estabilidade e o desempenho do sistema em malha-fechada são garantidos via a utilização de uma funçao de Lyapunov dependente de parâmetros e condiçõs LMIs são estabelecidas e utilizadas para a síntese dos controladores. Um aspecto importante considerado, e que objetiva a aplicação prática dos resultados, particularmente no controle de sistemas não lineares, é a validade do modelo LPV utilizado para representar localmente o sistema a controlar. Os resultados propostos são aplicados sobre um sistema não linear representado localmente por um modelo fuzzy de Takagi-Sugeno.

Stability analysis for a special class of dynamical networks

In this article, stability analysis and decentralised control problems are studied for a special class of linear dynamical networks. Necessary and sufficient conditions for stability and stabilisability under a decentralised control strategy are given for this type of linear networks. Especially, two types of linear regular networks, star-shaped networks and globally coupled networks, are studied in detail, respectively. A dynamical network can be viewed as a large-scale system composed of some subsystems with some coupling structures, based on this, the relationship between the stability of a network and the stability of its corresponding subsystems is studied. Different from the discussions that the subsystems in networks vary with different coupling structures (Duan, Z.S., Wang, J.Z., Chen, G.R., and Huang, L. (2008), ‘Stability Analysis and Decentralised Control of a Class of Complex Dynamical Networks’, Automatica, 44, 1028–1035), the subsystems in network discussed in this article remain unchanged with different interconnections which is the same as in general large-scale system. It is also pointed out that some subsystems must be made unstable for the whole network to be stable in some special cases. Moreover, the controller design method based on parameter-dependent Lyapunov function is provided.

Leader-following consensus problem with an accelerated motion leader

In this paper, we consider a multi-agent consensus problem with an accelerated motion leader and variable interconnection topology. To track such a leader, a neighbor-based local controller together with a neighbor-based state-estimation rule is proposed for each second-order follower-agent. The neighbor-based estimation rule is used to estimate the acceleration of the leader, which is assumed not to be measured by follower agents directly. By constructing a parameter-dependent common Lyapunov function, a sufficient condition is established to guarantee that each agent can follow the leader although the leader moves with an unknown time-variant acceleration. Moreover, the tracking error is estimated for the case that the unknown acceleration part of the leader has bounded derivative. Finally, a numerical example is given to illustrate the obtained result.

Performance analysis for uncertain multivariable systems obtained by system identification

This article provides a framework for robust performance analysis of linear time-invariant uncertain multivariable systems obtained by classical system identification methods. The performance measures are in terms of H 2 or H ∞ norm of a closed-loop transfer matrix. An upper bound for the performance analysis criterion is computed via an LMI-based optimisation problem. The LFT description is used as a tool for uncertainty modelling. The proposed performance conditions are derived based on using the parameter-dependent Lyapunov functions and are deduced via a parametrisation for the set of multipliers corresponding to the ellipsoidal uncertainty set delivered by system identification procedure. The effectiveness of the proposed analysis approach is demonstrated by a numerical example.

Multi-objective and reliable control for trajectory-tracking of rendezvous via parameter-dependent Lyapunov functions

A reliable, multi-objective and state-feedback controller design algorithm is presented for trajectory-tracking of circular-orbit rendezvous, in which H∞ performance, H2 performance and regional pole placement are taken into account. Given actuator failures and exogenous disturbances, the dynamical model is described as a polytopic system. By applying parameter-dependent Lyapunov functions and introducing slack matrices, the condition for the existence of multi-objective controller is described as a non-convex optimization subject to nonlinear matrix inequality constraints. Then, a hybrid genetic algorithm is proposed to obtain the desired controller. Numerical simulations are given to show that the proposed method can (a) handle actuator failures and exogenous disturbances effectively, and (b) provide better disturbance-attenuation performances than conventional methods.

Output tracking control with enhanced damping of internal dynamics and its output boundedness for static synchronous compensator system

This study investigates an important and challenging problem in non-linear systems: achieving output tracking control with a tolerable error bound as well as providing enhanced damping control of the internal dynamics. This problem is solved by using a damping controller in an input-output linearised static synchronous compensator system. The controllability and observability analyses direct the authors how to determine the damping controller gain. The proposed controller provides improved damping for lightly damped internal dynamics with degraded but tolerable output tracking performance. Finding a parameter-dependent Lyapunov function for the zero dynamics of error dynamics proves the stability of the damped internal dynamics in the time-invariant and time-varying systems. Stability of the closed-loop system is established by designing a composite Lyapunov function and it ensures that the tracking error of the output, in the approximated input-output linearised system, remains within a specified bound. Through stability analysis, the damping controller and the desired reference trajectory providing uniform boundedness of tracking error in output can be designed.

Generalized guaranteed cost control with D-stability and multiple output constraints

This paper is concerned with the problem of robust generalized guaranteed cost control with D-stability and multiple output constraints for a class of linear uncertain systems. Being a combination of output performance indices, a generalized cost function is considered to the linear polytopic uncertain systems described in a unified framework. The aim is to design a state feedback controller, such that the closed-loop system is robust D-stable, and the upper bound of the generalized cost function is as small as possible subject to multiple output constraints. Based on parameter-dependent Lyapunov functions, convex conditions for the existence of such controllers are presented in terms of linear matrix inequality. The proposed approach shows a unified treatment of the linear systems in the differential, shift, and delta domains. Numerical examples are provided to illustrate the effectiveness of the design method.

A kind of synergic control for a collection of vehicles

In many fields and even in our daily life, besides the usual feedback controls, it is also very useful for us to control our business by regulating the connections among the subsystems directly. In this paper, the problem of a kind of synergic control among a collection of vehicles performing a shared task using intervehicle communication and internal output feedback are considered, in which the weights of the corresponding edges are considered as the control variables and called connection coefficient gain in this paper. Firstly, when the weights of the corresponding edges are fixed, necessary and sufficient conditions for stability and stabilizability under a special decentralized control strategy are given for the corresponding closed loop network system. Corresponding controllers design methods are proposed via the parameter-dependent Lyapunov function. Especially, two types of networks, star-shaped networks and globally coupled networks, are studied in detail. Secondly, the Nyquist criterion is presented, which uses the eigenvalues of the weights matrix to determine the effects of the connection coefficients regulation on system stability. Thirdly, a general description of this kind of synergic control is proposed via the block diagram of the information flow. Finally, the synergic control among a collection of vehicles with different linear dynamical model is investigated simply.

Robust Gain-Scheduled Control of Switched-Mode DC–DC Converters

This paper presents a robust control synthesis framework for switched DC-DC converters. The framework is based on an LMI formulation which can be solved automatically by efficient convex optimization algorithms. The method considers parameter-dependent Lyapunov functions such that it can take into account the uncertainty of converter parameters, nonlinear dynamics (such as state-dependence), as well as transient and steady-state performances that can be imposed beforehand. The result of the proposed synthesis method is a gain-scheduled controller that guarantees stability despite the accounted nonlinear dynamics and can provide excellent performances. Two different synthesis examples are shown for a DC-DC boost converter and their performance and robustness are compared with a standard control approach as current-mode control, both in nominal and non-nominal conditions. Finally, the proposed approach is verified with experimental results.

Output chattering attenuation between two tracking controllers

In this paper, an output chattering attenuation scheme is proposed for two tracking controllers. The controllers have integral actions and are used to eliminate the steady-state tracking errors. By designing a compensator and reducing the discontinuous switching, the output chattering at the switching points can be attenuated significantly. Moreover, combining with the generalized Kalman-Yakubovich-Popov (KYP) lemma and the parameter-dependent Lyapunov function method, this problem can be characterized in terms of linear matrix inequality (LMI) formulation. Finally, the simulation results demonstrate the effectiveness of the proposed method.

LMI Relaxations for Reduced-Order Robust ${\cal H}_{\infty}$ Control of Continuous-Time Uncertain Linear Systems

This technical note is concerned with the problem of reduced order robust <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> dynamic output feedback control design for uncertain continuous-time linear systems. The uncertain time-invariant parameters belong to a polytopic domain and affect all the system matrices. The search for a reduced-order controller is converted in a problem of static output feedback control design for an augmented system. To solve the problem, a two-stage linear matrix inequality (LMI) procedure is proposed. At the first step, a stabilizing state feedback scheduled controller with polynomial or rational dependence on the parameters is determined. This parameter-dependent state feedback controller is used at the second stage, which synthesizes the robust (parameter-independent) output feedback <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> dynamic controller. A homogeneous polynomially parameter-dependent Lyapunov function of arbitrary degree is used to assess closed-loop stability with a prescribed <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> attenuation level. As illustrated by numerical examples, the proposed method provides better results than other LMI based conditions from the literature.

On parameter-dependent Lyapunov functions for robust fault detection filter design with application in power systems

This study investigates the fault detection problem for uncertain linear time invariant (LTI) systems subject to polytopic uncertainties, exploiting some properties provided by the observer-based robust fault detection filter (RFDF), which has possible applications in practical power systems. By means of parameter-dependent Lyapunov functions, the existence condition of RFDF is assessed through solving a group of linear matrix inequalities (LMIs). In order to further reduce the conservativeness, an efficient algorithm in terms of LMIs by generating homogeneous polynomial parameter-dependent Lyapunov functions of arbitrary degree on the uncertain parameters is presented, which includes as special case existing conditions for RFDF design. It can be established that as the degree of the polynomial increases, the number of LMIs and free variables increases and the test becomes less conservative. Moreover, the fault sensitivity H− index can be optimized via a convex optimization algorithm, leading to the optimal RFDF. The methodology proposed can also be applied to other relevant aspects such as determining the threshold. An uncertain LTI power system model is adopted as an illustrated example to demonstrate the efficacy of the proposed methods when compared to other methods from the literature.

MPC for LPV Systems Based on Parameter-Dependent Lyapunov Function with Perturbation on Control Input Strategy

In this paper, the model predictive control (MPC) algorithm for linear parameter varying (LPV) systems is proposed. The proposed algorithm consists of two steps. The first step is derived by using parameter-dependent Lyapunov function and the second step is derived by using the perturbation on control input strategy. In order to achieve good control performance, the bounds on the rate of variation of the parameters are taken into account in the controller synthesis. An overall algorithm is proved to guarantee robust stability. The controller design is illustrated with two case studies of continuous stirred-tank reactors. Comparisons with other MPC algorithms for LPV systems have been undertaken. The results show that the proposed algorithm can achieve better control performance.

Exponential stability of a class of networked control systems with time delays and packet dropouts

This paper studies the exponential stability problem for a class of networked control systems (NCSs) with time delays and packet dropouts. The state-feedback closed-loop NCS is modeled as a new discrete-time switched system. Through using the parameter-dependent Lyapunov function, a sufficient condition is obtained for the exponential stability of the NCS under a packet-dropout rate. A linear matrix inequality approach is employed to design the state-feedback controller. An illustrative example is presented to demonstrate the effectiveness of the proposed method.

Stabilisation of polytopic singularly perturbed linear systems

This article deals with two time-scale polytopic linear systems in the singular perturbation form and in discrete-time. The aim is to propose well-conditioned convex conditions for stability analysis and control design. As the classical LTI time-scale separation-based approach is not successful in the case of polytopic singularly perturbed systems, our method leads to linear matrix inequality conditions independent of the singular perturbation parameter and offers an interesting solution from a numerical point of view. In addition, the conservatism of the proposed approach is limited as it is based on parameter-dependent Lyapunov functions. A numerical example taken from the literature illustrates the effectiveness of the given stabilisation conditions.

An Observer Based State Feedback LFT LPV Controller for an Industrial Manipulator

In this paper, an observer based state feedback control law for linear parameter varying (LPV) systems represented via linear fractional transformations (LFTs) is synthesized for the control of the first axis of an industrial manipulator. Using parameter dependent Lyapunov functions, the induced ℒ2 performance of the observer and the state feedback can be guaranteed over the whole parameter space while taking into account a limited parameter variation rate. Parameter dependent weights allow for an adjustable performance over the workspace of the manipulator. The pole regions of the closed loop, frozen at certain parameter values, are assigned by introducing pole region constraints in the synthesis.

Improving the Performance of Robust MPC Using the Perturbation on Control Input Strategy Based on Nominal Performance Cost

Abstract This paper proposes a strategy to improve the control performance of robust MPC by using a sequence of free control inputs. For a given feedback gain which asymptotically stabilizes the closed-loop system, a sequence of free control inputs is calculated by minimizing the nominal performance cost over the perturbation horizon N . The nominal model of the process is included in the controller design in order to improve control performance. The optimization problem at each time step is formulated as the convex optimization problem involving linear matrix inequalities (LMI). The proposed strategy can be applied to the feedback gains derived from both a single Lyapunov function and parameter dependent Lyapunov function. For both cases, robust stability is proved to be guaranteed. The controller design is illustrated with an example in chemical process. The results show that the proposed strategy can significantly improve control performance of the conventional robust MPC.

Robust Constrained MPC based on Nominal Performance Cost with Applications in Chemical Processes

Abstract In this paper, a synthesis approach to robust constrained model predictive control (MPC) for uncertain polytopic discrete-time systems is presented. An overall algorithm is derived by using parameter-dependent Lyapunov function. The nominal model of the plant is included in the controller design in order to improve control performance. The optimization problem at each time step is formulated as the convex optimization problem involving linear matrix inequalities (LMI). Thus, the algorithm is computationally tractable. The algorithm is proved to guarantee robust stability. The controller design is illustrated with an example of continuous stirred-tank reactor.