Parameter-dependent Lyapunov Function Research Articles

Gain-Scheduled Observers Using Inexact Scheduling Parameters

This note addresses the design problem of Gain-Scheduled (GS) observers for discrete-time Linear Parameter-Varying (LPV) systems via Parameter-Dependent Lyapunov Functions (PDLFs). We give a sufficient condition for our problem in terms of Parameter-Dependent Linear Matrix Inequalities (PDLMIs). Our method not only guarantees the stability of estimation error dynamics but also optimizes the estimation errors under exogenous disturbances to LPV systems. A numerical example is included to illustrate our results.

Sufficient Dilated LMI Conditions for H∞ Static Output Feedback Robust Stabilization of Linear Continuous‐Time Systems

New sufficient dilated linear matrix inequality (LMI) conditions for the H∞ static output feedback control problem of linear continuous‐time systems with no uncertainty are proposed. The used technique easily and successfully extends to systems with polytopic uncertainties, by means of parameter‐dependent Lyapunov functions (PDLFs). In order to reduce the conservatism existing in early standard LMI methods, auxiliary slack variables with even more relaxed structure are employed. It is shown that these slack variables provide additional flexibility to the solution. It is also shown, in this paper, that the proposed dilated LMI‐based conditions always encompass the standard LMI‐based ones. Numerical examples are given to illustrate the merits of the proposed method.

Less Conservative Control Design for Linear Systems with Polytopic Uncertainties via State‐Derivative Feedback

The motivation for the use of state‐derivative feedback instead of conventional state feedback is due to its easy implementation in some mechanical applications, for example, in vibration control of mechanical systems, where accelerometers have been used to measure the system state. Using linear matrix inequalities (LMIs) and a parameter‐dependent Lyapunov functions (PDLF) allowed by Finsler’s lemma, a less conservative approach to the controller design via state‐derivative feedback, is proposed in this work, with and without decay rate restriction, for continuous‐time linear systems subject to polytopic uncertainties. Finally, numerical examples illustrate the efficiency of the proposed method.

Fuzzy control of non-linear systems using parameter-dependent polynomial fuzzy model

This study investigates the stability of parameter-dependent polynomial-fuzzy-model-based (PDPFMB) control systems. A parameter-dependent polynomial fuzzy model is proposed to represent a non-linear plant. A parameter-dependent polynomial fuzzy controller is then employed to stabilise the non-linear plant. The stability of PDPFMB control systems is investigated using the sum-of-squares (SOS) technique based on a parameter-dependent Lyapunov function candidate. With the consideration of the information of system parameters, parameter-dependent SOS-based stability conditions are obtained to determine the system stability and polynomial feedback gains. A feasible solution to the stability conditions can be obtained numerically using the third-party Matlab toolbox SOSTOOLS. Unlike the membership functions, as the system parameters are not single signed functions, the traditional analysis approach obtaining the stability conditions cannot be applied. Instead, the membership functions and system parameters are considered as symbolic variables for the construction of the parameter-dependent SOS-based stability conditions. A simulation example is given to demonstrate the merits of the proposed approach.

Parameter-dependent H∞ filter design for LPV systems and an autopilot application

This paper considers the design problem of parameter dependent H ∞ filters for linear parameter varying (LPV) systems whose parameters are measurable. Conditions for existence of parameter-dependent Lyapunov function are proposed via parametrical linear matrix inequality (LMI) constraints. Based on the solutions to the LMIs, an algorithm for the gain matrices of LPV filter is presented. The design method is applied to a missile system to demonstrate the effectiveness.

Robust State-Feedback Controller Design for Uncertainty Singular Systems

In this paper the problem of robust stability and stabilization of a class of uncertain singular Systems with uncertainties in both the derivative and state matrices is studied. By using a parameter dependent Lyapunov function, we derive the linear matrix inequalities (LMIs) based sufficient conditions for the stability and stabilization of the system. By solving these LMIs, the robust controller is derived. Finally, the numerical example is given to show the effectiveness of the proposed theorems.

Robust PD State Feedback Controller Design for Uncertain Singular Systems

This paper is concerned with the problem of robust stability and stabilization of singular systems with uncertainties in both the derivative and state matrices. By using a parameter dependent Lyapunov function, we derive the LMI-based sufficient conditions for the stabilization of the singular systems. Secondly, by solving these LMIs, a proportional plus derivative (PD) state feedback controller is designed for the closed-loop systems to be quadratically normal and quadratically stable (QNQS). Finally, the numerical example is given to show the effectiveness of the proposed theorems.

Gain-scheduled L-one control for linear parameter-varying systems with parameter-dependent delays

This paper deals with the problem of gain-scheduled L-one control for linear parameter-varying (LPV) systems with parameter-dependent delays. The attention is focused on the design of a gain-scheduled L-one controller that guarantees being an asymptotically stable closed-loop system and satisfying peak-to-peak performance constraints for LPV systems with respect to all amplitude-bounded input signals. In particular, concentrating on the delay-dependent case, we utilize parameter-dependent Lyapunov functions (PDLF) to establish peak-to-peak performance criteria for the first time where there exists a coupling between a Lyapunov function matrix and system matrices. By introducing a slack matrix, the decoupling for the parameter-dependent time-delay LPV system is realized. In this way, the sufficient conditions for the existence of a gain-scheduled L-one controller are proposed in terms of the Lyapunov stability theory and the linear matrix inequality (LMI) method. Based on approximate basis function and the gridding technique, the corresponding controller design is cast into a feasible solution problem of the finite parameter linear matrix inequalities. A numerical example is given to show the effectiveness of the proposed approach.

H2 analysis for LPV systems by parameter-dependent Lyapunov functions

H2 analysis for LPV systems by parameter-dependent Lyapunov functions

Gain-scheduled output-feedback controllers depending solely on scheduling parameters via parameter-dependent Lyapunov functions

In this paper, we propose a new design method for Gain-Scheduled Output Feedback (GSOF) controllers for continuous-time Linear Parameter-Varying (LPV) systems via Parameter-Dependent Lyapunov Functions (PDLFs). The GSOF controllers depend solely on scheduling parameters. Although our method requires a line search to obtain suboptimal controllers, it produces practical GSOF controllers, being independent of the derivatives of scheduling parameters. Our method is proved to be no more conservative than conventional design methods via constant Lyapunov functions as well as particularly structured PDLFs.

Distributed H<sub>∞</sub> Control for Leader-Following Coordination with Switching Topology

In this paper, we consider multi-agent H∞consensus control problems with external disturbance under the undirected switching topologies. The agent dynamics is expressed in the form of a second-order model and the control laws are neighbor-based feedback laws. By using the model transformation, the multi-agent H∞consensus control problem is converted into H∞ control problem for a switching linear system with special structure. A sufficient condition is established under which all agents can reach consensus with the desired H∞performance in switching topology case by constructing a parameter-dependent common Lyapunov function. Moreover, we give an explicit estimation expression to estimate H∞ performance index. Finally, a numerical example is provided to illustrate the effectiveness of our results.

Linear Parameter Varying Analysis of Switched Controllers for Attitude Control Systems

T HE on-orbit life of a satellite is decomposed into phases, allowing the convergence from the separation state to the mission state, the orbit control, and the safe state in case of fault. For each phase ormode, a specific set of attitude pointing requirements is to be met by using a specific set of actuators and sensors. Inside a mode, several attitude control and estimation algorithms can be designed, for example, to meet increasing performance levels requirements. Over the past 30 years, the number of attitude and orbit control system (AOCS) modes has been reduced so that generic architectures now typically consist of three modes (one safe-hold mode, one mission mode, and one orbit control mode). In the same time, the software complexity has increased to meet ever more demanding requirements. Thus, the switching between different AOCS modes and software requires permanently enhanced robustness levels to cope with more stringent kinematics conditions (satellite angular rate and position)while also improving the pointing accuracy. In this Note, the problem of the reaction wheel control software switching inside the mission mode of the Centre National d’Etudes Spatiales (CNES) microsatellite DEMETER (detection of electromagnetic emissions transmitted from earthquake regions) [1–4] is considered. Because of the limited speed and torque capacities of the reaction wheels, the linear proportional-derivative controller specifically designed for small pointing errors can no longer be used when the latter get too large, which is typically the case when the mission mode is activated. Then, to avoid torque and speed saturations during this phase,whichwill be referred to as the rough pointing phase of the mission mode, a dedicated controller is used to maintain the angular rate around a constant target until the pointing error becomes small enough. More interesting, as soon as the speed target is reached, the commanded torque is null. The parameters of the control system are tuned in such a way that the delivered control input signal remains continuous when switching occurs for prespecified pointing and rate errors. However, because of external perturbations and AOCS sampling constraints, the nominal conditions are rarely met when the control system switches from rough to fine pointing phases, which locally induces some undesirable (and what it is hoped to be transient) behaviors. As a result, it becomes rather difficult to obtain stability and robustness proofs, which have to be addressed through extensive and time-consuming nonlinear simulations campaigns. In this context, the central idea of this Note consists of rewriting the switching-based control law in a quasi-linear-parameter-varying form thanks to which the following properties hold: 1) The continuity of the control signals is guaranteed at anytime, since the new system is based on a smooth transition from rough to fine pointing phases. 2) The stability and robustness proofs for the nonlinear closedloop plant can be obtained via enhanced linear-parameter-varying (LPV) analysis techniques, which are developed in the Note. While many methods are now available to evaluate the stability of LPVplants,most of thewell-established results are numerically quite demanding as soon as the size of the plant increases, togetherwith the complexity of the selected Lyapunov function. In this Note, a new algorithm based on parameter-dependent Lyapunov (PDL) functions is then developed, thanks to which the stability analysis can be performed at a reasonable cost. The key idea of the proposed iterative technique consists of a first step of optimizing the Lyapunov functions on a rough grid of the parameter space and, in a second step, to check the validity on the continuous space. The grid is then updated if necessary until the whole space is cleared. This approach may be viewed as an original adaptation of a similar idea [5], which was exploited 10 years ago in a -based robustness analysis. The Note is organized as follows. After a brief description of the AOCS control loops and switching-based controllers in Sec. II, the quasi-LPV formulation is detailed in Sec. III. Stability analysis of LPV systems via PDL functions is then described in Sec. IV, where the aforementioned algorithm is detailed. Section V is devoted to the presentation of stability analysis and time-domain simulation results. Finally, some concluding remarks end the Note.

Parameter-dependent Lyapunov functions for state-derivative feedback control in polytopic linear systems

In some practical problems, for instance, the suppression of vibration in mechanical systems, the state-derivative signals are easier to obtain than the state signals. Using Linear Matrix Inequalities (LMIs), and applying the reciprocal projection lemma in a Parameter-dependent Lyapunov Function (PDLF), this article proposes a method for the design of state-derivative feedback applied to uncertain linear systems. The control design aims the system stabilisation without and with decay rate restriction. When considering only the system stability, the proposed methodology becomes practically equivalent to the Common Quadratic Lyapunov Function (CQLF) technique. Otherwise, when the decay rate is taken in account, the proposed methodology is shown to be less conservative. Numerical examples illustrate its efficiency.

Dilated LMI conditions for time-varying polytopic descriptor systems: the discrete-time case

This article addresses the admissibility analysis and state-feedback control synthesis problems of discrete-time polytopic descriptor systems with possibly time-varying parameters. First, we present a new necessary and sufficient strict linear matrix inequality (LMI) condition for the admissibility analysis of linear time-invariant (LTI) descriptor systems. Then, based on the concept of poly-quadratic admissibility, introduced in this article, we extend this result to the admissibility analysis of possibly time-varying parameter-dependent descriptor systems. This extension, which applies to both uncertain and measurable parameters, uses parameter-dependent Lyapunov functions and employs two slack variables. The extended conditions are also expressed as strict LMI conditions which are easily tractable numerically compared to the non-strict ones often encountered when dealing with descriptor systems. Note that we have proposed two separate admissibility analysis conditions: one directly exploitable for state estimation and the other for state-feedback control. The need for different admissibility analysis conditions for each synthesis problem is motivated by the fact that the duality between state estimation and state-feedback control, which hold in the case of measurable parameters, does not hold when dealing with uncertain ones. In our approach, we overcome the absence of such duality by considering a dilation only on the dynamical part of the descriptor system. The application of our analysis result to both robust state-feedback control and polytopic state-feedback control are also presented in this article. Our analysis results extend to descriptor systems, some existing results developed for regular systems.

Delay-Dependent L1 Filtering for LPV Systems With Parameter-Varying Delays

The paper investigates the problems of delay-dependent L1 filtering for linear parameter-varying (LPV) systems with parameter-varying delays, in which the state-space data and the time delays are dependent on parameters that are measurable in real-time and vary in a compact set with bounded variation rate. The attention is focused on the design of L1 filter that guarantees the filtering error system to be asymptotically stable and satisfies the worst-case peak-to-peak gain of the filtering error system. In particular, we concentrate on the delay-dependent case, using parameter-dependent Lyapunov function, the decoupled peak-to-peak performance criterion is first established for a class of LPV systems. Under this condition, the admissible filter can be found in terms of linear matrix inequality (LMI) technology. According to approximate basis function and the gridding technique, the filter design problem is transformed into feasible solution problem of the finite parameter LMIs. Finally, a numerical example is provided to illustrate the feasibility of the developed approach.

Robust analysis and control of parameter-dependent uncertain descriptor systems

In this paper, we address the robust admissibility analysis and control of descriptor systems with parameter-dependent, possibly time-varying, uncertainties. First, we propose a new necessary and sufficient condition for the admissibility of linear time-invariant (LTI) descriptor systems. This condition uses a strict linear matrix inequality (LMI) formulation and involves two additional matrix variables usually called slack variables. Then, we present an extension of this condition to robust admissibility of time-varying parameter-dependent descriptor systems. Using parameter-dependent Lyapunov functions and employing two slack variables, this extension states robust admissibility LMI conditions that are less restrictive, as supported by our numerical examples, than the existing ones. Our robust admissibility analysis result, which can be directly used for state estimation, cannot be so for state-feedback control design. This is due to the absence of duality between state estimation and state-feedback control in the case of parameter-dependent systems with uncertain parameters as recently reported in the literature. In order to overcome this problem, we also present in this paper a new robust admissibility analysis result, involving only one slack variable, which we apply to state-feedback control design.

Synthesis of a linearly interpolated gain scheduling controller for large flexible spacecraft ETS-VIII

This paper illustrates a design procedure for a linearly interpolated gain scheduling controller for Engineering Test Satellite VIII (ETS-VIII) using its linear parameter-varying (LPV) model. The LPV model here consists of piecewise-linear functions of the paddle rotation angle and a norm-bounded perturbation. The main purpose of this research is to derive a simple structured scheduling law that can be easily implemented in a satellite onboard computer. The derived gain has only two grid points and is scheduled simply by linear interpolation, which is desirable from the standpoint of implementability. Moreover, since the synthesis condition is based on parameter-dependent Lyapunov functions, it gives less conservative results than existing methods. Simulation results are presented to show the effectiveness of the proposed synthesis.

H ∞ controller design for linear systems with time-invariant uncertainties

This paper considers the problem of designing robust H ∞ state feedback controller for linear continuous-time systems with time-invariant uncertainties. The main result given here concerns H ∞ controller design using the parameter dependent Lyapunov function approach and a new control law. A new condition ensuring the linear systems to be is asymptotically stable with a prescribed H ∞ performance is proposed in terms of a set of linear matrix inequalities (LMIs). Theoretic proof is given to show that the proposed condition is less conservative than existing results in the literature. An example is provided to demonstrate the efficiency of the proposed method.

Multivariable RBF-ARX model-based robust MPC approach and application to thermal power plant

For a class of smooth nonlinear multivariable systems whose working-points vary with time and the future working-points knowledge are unknown, a combination of a local linearization and a polytopic uncertain linear parameter-varying (LPV) state-space model is built to approximate the present and the future system’s nonlinear behavior, respectively. The combination models are constructed on the basis of a matrix polynomial multi-input multi-output (MIMO) RBF-ARX model identified offline for representing the underlying nonlinear system. A min–max robust MPC strategy is designed to achieve the systems’ output-tracking control based on the approximate models proposed. The closed loop stability of the MPC algorithm is guaranteed by the use of time-varying parameter-dependent Lyapunov function and the feasibility of the linear matrix inequalities (LMIs). The effectiveness of the modeling and control methods proposed in this paper is illustrated by a case study of a thermal power plant simulator.

Modified non-linear damping of internal dynamics via feedback linearisation for static synchronous compensator

Static synchronous compensator (STATCOM) is a non-linear system so that the conventional linear output feedback controller with synthesised feedback control cannot guarantee uniform tracking performance, although it improves the stability margin in an inductive operating range. Recently, the input-output linearisation via feedback has been applied to STATCOM and it shows uniform transient performance but owing to lightly damped internal dynamics it results in current ripples in DC-side capacitors. To reduce the ripples, a damping term, time derivative of active current multiplied by a constant gain, was added to the input-output linearisation. This damping controller brings improved internal stability by moving the poles of internal dynamics from the imaginary axis. However, there exists one important limitation that it cannot guarantee stability of internal dynamics at every operating point for some system parameters. A modified non-linear damping controller is proposed to improve the limitation. The proposed method has the damping term with a variable gain, thus provides improved transient performance over the whole operating range. Stability of the closed-loop system is verified by the parameter-dependent Lyapunov function. Owing to the damping term, the exact input-output linearisation is not performed; thus the output is not decoupled from internal dynamics and oscillatory transient response appears. Using the parameter-dependent Lyapunov function, it is also shown that the oscillatory output response is bounded. Improved transient performance in time-domain is validated by simulations with the topological model.