Linear Matrix Inequality Optimization Problem Research Articles

Event-Triggered Control for Linear Systems Subject to Actuator Saturation

Event-triggered control is developed to reduce the communication load in networked control systems. This means that output or actuator signals are only transmitted over the network when an event-triggering condition is violated which is designed such that a certain control performance can be guaranteed. This paper considers event-triggered control subject to actuator saturation for linear systems. Therefore we present a method to estimate the domain of attraction which represents a contractive invariant set. The contractive invariant set is estimated by an ellipsoid which is determined by solving a linear matrix inequality (LMI) optimization problem. Further a controller synthesis design considering the event-triggering condition in our criterion is given to maximize the contractive ellipsoid. Simulations are given to illustrate the results.

A multi-step robust model predictive control scheme for polytopic uncertain multi-input systems

Model predictive control (MPC) has attracted wide attention in process industries with its ability to handle constrained multivariable processes. Computational complexity can become a limiting factor when MPC is applied to large-scale systems with fast sampling times. In this paper, a control scheme known as multi-step robust MPC is presented for polytopic uncertain multi-input systems. Only one or several state feedback laws are optimized at each time interval to reduce computational complexity. A set invariance condition for polytopic uncertain systems is identified and the invariant set is determined by solving a linear matrix inequality (LMI) optimization problem. Based on the set invariance condition, a min-max multi-step robust MPC scheme is proposed. Numerical simulations show the effectiveness of the proposed scheme.

Observer-based H∞ control for vehicle handling and stability subject to parameter uncertainties

This article presents a model-based controller design approach for vehicle to improve handling and stability while the vehicle parameters are uncertain and the state of body sideslip angle is immeasurable. The variations in tire cornering stiffness, which are due to model simplification, are considered and treated as parameter uncertainties. By translating parameter uncertainties into perturbed matrices and combining with state observer, a sufficient condition for the theorem of an observer-based, optimal, and robust H∞ controller subject to parameter uncertainties is derived. In the theorem, the controller and the state observer are integrated into a linear matrix inequality optimization problem and designed together. The performances of the control system are evaluated via numerical simulation under various critical maneuvers and road conditions. The simulation results demonstrated that the vehicle dynamic control system designed by the approach proposed in this article can significantly improve vehicle handling and stability regardless of the variations on the tire cornering stiffness and the absence of the measured body sideslip angle.

A Novel Control Design on Discrete-Time Takagi–Sugeno Fuzzy Systems With Time-Varying Delays

This paper focuses on analyzing a new model transformation of discrete-time Takagi–Sugeno (T–S) fuzzy systems with time-varying delays and applying it to dynamic output feedback (DOF) controller design. A new comparison model is proposed by employing a new approximation for time-varying delay state, and then, a delay partitioning method is used to analyze the scaled small gain of this comparison model. A sufficient condition on discrete-time T–S fuzzy systems with time-varying delays, which guarantees the corresponding closed-loop system to be asymptotically stable and has an induced <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\ell_{2}$</tex></formula> disturbance attenuation performance, is derived by employing the scaled small-gain theorem. Then, the solvability condition for the induced <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\ell_{2}$</tex></formula> DOF control is also established, by which the DOF controller can be solved as linear matrix inequality optimization problems. Finally, examples are provided to illustrate the effectiveness of the proposed approaches.

Distributed model predictive control for polytopic uncertain systems subject to actuator saturation

In this paper, we present a distributed model predictive control (MPC) algorithm for polytopic uncertain systems subject to actuator saturation. The global system is decomposed into several subsystems. A set invariance condition for polytopic uncertain system with input saturation is identified and a min–max distributed MPC strategy is proposed. The distributed MPC controller is designed by solving a linear matrix inequalities (LMIs) optimization problem. An iterative algorithm is developed for making coordination among subsystems. Case studies are carried out to illustrate the effectiveness of the proposed algorithm.

Robust sensor fault estimation scheme for satellite attitude control systems

The present paper proposes two new schemes of sensor fault estimation for a class of nonlinear systems and investigates their performances by applying these to satellite control systems. Both of the schemes essentially transform the original system into two subsystems (subsystems 1 and 2), where subsystem-1 includes the effects of system uncertainties, but is free from sensor faults and subsystem-2 has sensor faults but without any uncertainties. Sensor faults in subsystem-2 are treated as actuator faults by using integral observer based approach. The effects of system uncertainties in subsystem-1 can be completely eliminated by a sliding mode observer (SMO). In the first scheme, the sensor faults present in subsystem-2 are estimated with arbitrary accuracy using a SMO. In the second scheme, the sensor faults are estimated by designing an adaptive observer (AO). The sufficient condition of stability of the proposed schemes has been derived and expressed as a linear matrix inequality (LMI) optimization problem and the design parameters of the observers are determined by using LMI techniques. The effectiveness of the schemes in estimating sensor faults is illustrated by considering an example of a satellite control system. The results of the simulation demonstrate that the proposed schemes can successfully estimate sensor faults even in the presence of system uncertainties.

LMI solutions for H-two and H-infinity decentralized controllers applied to an aerothermic process

In this paper, we present a linear matrix inequality (LMI)-based solution to implement H-two and Hinfinity decentralized robust control strategies. Appropriate parametrization of optimal H-two and H-infinity controllers is used. The general formulation of the decentralized control design leads to the optimal determination of both the state feedback gains and the observer gains of the decentralized controllers. This formulation is two folds: first, a centralized controller is obtained, and then, a simplified decentralized solution is derived by optimizing only the observer gains. The mathematical determination of these gains is formulated as an LMI optimization problem that can be easily solved using LMI solvers. As an experimental evaluation of these controllers, a real time application to an aerothermic process is carried out. A continuous-time model of the process obtained with a suitable direct continuous-time identification approach is elaborated. Results illustrating the real performance obtained from the H-two and H-infinity decentralized controllers are discussed and compared with the centralized ones.

Sampled-data filtering for linear parameter varying systems

In this paper, we address the sampled-data filter design problem for continuous-time linear parameter-varying (LPV) systems. The filtering error system obtained from augmenting a continuous-time LPV system and the sampled-data filter is a hybrid system. The sampled-data filter design objective is to ensure the error system stability and a prescribed level of the induced energy-to-energy gain (or norm) from the disturbance input to the estimation error. To this purpose, we employ a lifting method to derive an equivalent discrete-time LPV representation for the continuous-time LPV system. In the present study, the sampled-data filter synthesis conditions are formulated in terms of linear matrix inequality optimisation problems. The viability of the proposed design method to cope with variable sampling rates is illustrated through numerical examples, where reliable estimation of the LPV system outputs is achieved.

Delay-range-dependent robust 2D iterative learning control for batch processes with state delay and uncertainties

This paper proposes the design of the integrated output feedback and iterative learning control (ILC) for batch processes with uncertain perturbations and interval time-varying delays, where the main idea is to transform the design into a robust delay-range-dependent H∞ control of a 2D system described by a state-space model with varying delays. A sufficient criterion for delay-dependent H∞ noise attenuation is derived through linear matrix inequality (LMI) by introducing a comprehensive 2D difference Lyapunov–Krasovskii functional candidate and adding a differential inequality to the difference in the Lyapunov function for the 2D system. Based on the criterion obtained, the delay-range-dependent output feedback controller combined with ILC is then developed. The developed system ensures that the closed-loop system for all admissible uncertainties is asymptotically stable and has a prescribed H∞ performance level in terms of the LMI constraint. The controller is obtained by solving an LMI optimization problem with simple calculations and less constraint conditions. Moreover, the conditions can also be directly extended from delay-range-dependent to general delay-dependent stability. Applications in injection velocity control demonstrate the effectiveness and feasibility of the proposed method.

A stable fault detection observer design in finite frequency domain

This article addresses the stable fault detection observer design problem for linear time-invariant continuous-time systems in finite-frequency domain. The fault detection filter design is a synthesised optimal Luenberger observer that guarantees two requested performance indexes of fault sensitivity and stability. With the aid of generalised Kalman–Yakubovich–Popov lemma and increasing dimensions of slack variable matrix, the stability and H − performance analysis of the closed-loop system with a fault detection observer has been translated into a convex linear matrix inequality (LMI) optimisation problem to avoid the complexity of system associated with weight functions. An iterative LMI algorithm has been presented for the fault detection observer design. The effectiveness of proposed approaches is demonstrated by two numerical examples.

Static Output-Feedback Control for Vehicle Suspensions: A Single-Step Linear Matrix Inequality Approach

In this paper, a new strategy to design static output-feedback controllers for a class of vehicle suspension systems is presented. A theoretical background on recent advances in output-feedback control is first provided, which makes possible an effective synthesis of static output-feedback controllers by solving a single linear matrix inequality optimization problem. Next, a simplified model of a quarter-car suspension system is proposed, taking the ride comfort, suspension stroke, road holding ability, and control effort as the main performance criteria in the vehicle suspension design. The new approach is then used to design a static output-feedbackH∞controller that only uses the suspension deflection and the sprung mass velocity as feedback information. Numerical simulations indicate that, despite the restricted feedback information, this static output-feedbackH∞controller exhibits an excellent behavior in terms of both frequency and time responses, when compared with the corresponding state-feedbackH∞controller.

Gain-scheduling Distributed MPC for Polytopic Uncertain Systems subject to Actuator Saturation

In this paper, we present a gain-scheduling distributed model predictive control (MPC) algorithm for polytopic uncertain systems subject to actuator saturation. A large-scale system is decomposed into subsystems and sub-controllers are designed independently. An invariant set condition is provided and a min-max distributed MPC strategy is proposed based on the invariant set. The distributed MPC controller is determined by solving a linear matrix inequality (LMI) optimization problem. An iterative algorithm is provided to coordinate the sub-controllers. A numerical example is carried out to demonstrate the effectiveness of the proposed algorithm.

Control allocation for an over-actuated Satellite Launch Vehicle

Dynamic equations governing the short period dynamics of an over-actuated Satellite Launch Vehicle (SLV) with a typical eight-actuator configuration have been modeled in a separable form. Two rigid body models have been developed – one including the slosh dynamics, the other neglecting it. Effects of differential thrust, mounting misalignment and thrust misalignment are modeled as disturbances. The controller design and the allocation task have been taken up separately. The state feedback controller is designed for the virtual control inputs through H∞ disturbance rejection with pole restricted to a specified region formulated using Linear Matrix Inequality (LMI) approach. The partial state feedback control strategy is applied to the rigid body model of SLV including slosh by feeding back the states, available for measurement. The dynamic allocation is carried out by formulating a separate LMI optimization problem where the weighting matrices are selected considering the actuator priorities during allocation. The combined algorithm on controller design and allocation is tested on an attitude control problem of the SLV. The response to a step command of 2° applied to pitch, yaw and roll has been tracked. The allocation algorithm distributed the three control inputs to the eight actuators by satisfying the rate and position saturation bounds.

Delay-dependent H ∞ Control for Uncertain 2-D Discrete Systems with State Delay in the Roesser Model

This paper is concerned with the problem of H∞ control for uncertain two-dimensional (2-D) state delay systems described by the Roesser model. Based on a summation inequality, a sufficient condition to have a delay-dependent H∞ noise attenuation for this 2-D system is given in terms of linear matrix inequalities (LMIs). A delay-dependent optimal state feedback H∞ controller is obtained by solving an LMI optimization problem. Finally, a simulation example of thermal processes is given to illustrate the effectiveness of the proposed results.

Robust partially mode delay dependent ℋ∞ control of discrete-time networked control systems

This article proposes a methodology for designing a partially mode delay dependent ℋ∞ controller design for discrete-time systems with random communication delays. Communication delays between sensors and controller are modelled by a finite state Markov chain where the transition probability matrix is partially known. Stability criteria are obtained based on Lyapunov–Krasovskii functional and a novel methodology for designing a partially mode delay dependent state feedback controller has been proposed. The controller is obtained by solving linear matrix inequality optimisation problems using cone complimentarity linearisation algorithm. A numerical example is provided to illustrate the effectiveness of the proposed controller.

Wind turbine integrated structural and LPV control design for improved closed-loop performance

An iterative redesign algorithm is proposed to integrate the design of the structural parameters and a linear parameter-varying (LPV) controller for a three-bladed horizontal-axis wind turbine. The LPV controller is designed for an eighth-order lumped model of the wind turbine consisting of blades, drive-train and the tower. The lumped model response is matched with detailed open-loop numerical simulations using the Fatigue, Aerodynamics, Structures and Turbulence (FAST) code. The controller is scheduled in real-time based on the mean wind speed to account for the varying system dynamics. The objective is to track the operating trajectory meanwhile minimise the H ∞ performance index from the wind turbulence to the controlled output vector consisting of pitch angle, blade tip deflection, and the generator speed and torque. Sensitivity analysis of the closed-loop performance index with respect to the structural parameters of the system is examined. The integrated design problem is formulated as an iterative sequential controller/structure redesign to obtain the structural parameters and controller matrices corresponding to a local optimal performance index. Each step of the iterative procedure is formulated as a linear matrix inequality (LMI) optimisation problem that can be solved efficiently using available LMI solvers. The evolution of the structural parameters and performance index through the integrated design is illustrated. The FAST closed-loop simulations for two selected designs with the smallest values of the performance index demonstrate the improved performance of the overall system through the integrated structure/control redesign in both minimising the effect of the wind disturbance on the generator output power, and reducing the structural loads on the wind turbine.

Robust H ∞ static output feedback stabilization of T-S fuzzy systems subject to actuator saturation

This paper proposes a method for designing robust H∞ static output feedback stabilization of Takagi-Sugeno (T-S) fuzzy systems under actuator saturation. In this paper, the input saturation is represented by a polytopic model and the modeling error is assumed a norm-bounded uncertainty. A set invariance condition for robust H∞ static output feedback system under actuator saturation is first established. Then, the estimation of the largest domain of attraction for the system is formulated and solved as a Linear Matrix Inequality (LMI) optimization problem. Two examples are used to demonstrate the effectiveness of the proposed design method.

LMI optimization problem of delay-dependent robust stability criteria for stochastic systems with polytopic and linear fractional uncertainties

LMI optimization problem of delay-dependent robust stability criteria for stochastic systems with polytopic and linear fractional uncertaintiesThis paper studies an LMI optimization problem of delay-dependent robust stability criteria for stochastic systems with polytopic and linear fractional uncertainties. The delay is assumed to be time-varying and belong to a given interval, which means that lower and upper bounds of this interval time-varying delay are available. The uncertainty under consideration includes polytopic-type uncertainty and linear fractional norm-bounded uncertainty. Based on the new Lyapunov-Krasovskii functional, some inequality techniques and stochastic stability theory, delay-dependent stability criteria are obtained in terms of Linear Matrix Inequalities (LMIs). Moreover, the derivative of time delays is allowed to take any value. Finally, four numerical examples are given to illustrate the effectiveness of the proposed method and to show an improvement over some results found in the literature.

On estimation of attraction domain for port-controlled Hamiltonian systems subject to actuator saturation

This paper investigates the estimation of domain of attraction for nonlinear port-controlled Hamiltonian (PCH) systems with actuator saturation (AS). Several conditions are established under which an ellipsoid is contractively invariant, and thus can be employed to find the biggest ellipsoid contained in the domain of attraction. It is shown that the proposed conditions can be expressed in the form of the linear matrix inequalities (LMIs) optimization problem with constraints. Study of an illustrative example shows that the proposed method works very well in estimating the domain of attraction for some classes of nonlinear PCH systems with AS.

H∞ control for discrete-time singular Markov jump systems subject to actuator saturation

In this paper, the H∞ control problem for discrete-time singular Markov jump systems with actuator saturation is considered. A sufficient condition is obtained which guarantees that the discrete-time singular Markov jump system with actuator saturation is regular, causal, bounded state stable, and satisfies the H∞ performance. With this condition, and based on singular value decomposition approach, the design method of H∞ state feedback controller is developed by using linear matrix inequalities (LMIs) optimization problem. Numerical examples are given to illustrate the effectiveness of the proposed methods.