Convex Linear Matrix Inequality Research Articles

A Robust Control for a Controllable Suspension System with an Input Time Delay

In this study, a state feedback robust control strategy with time delay dependence for the uncertainty, fault, and input time delay of a vehicle’s controllable suspension system is designed. Firstly, based on the working principle of the adjustable damping suspension system, the total damping force of the suspension is divided into “foundation damping force” and “controllable damping force”, taking the adjustable damping interval as the constraint boundary, which varies with the actuating velocity. Furthermore, based on the sensitivity analysis of the input time delay’s relationship with damping force disturbance, it exerts influence on the controllable damping force part. The associated uncertainty and fault are defined through linear fraction transformation and proportional gain, respectively, and the two are decoupled from the control mechanism. Then, based on the Lyapunov stability theory, a robust control law for the time-delay-dependent H∞ state feedback is designed and transformed into a linear matrix inequality convex optimization problem to solve. Finally, the feasibility and effectiveness of the proposed method are verified by designing different combinations of uncertainty and fault tests and by comparing and analyzing these with the time-delay-independent H2/H∞ robust control strategy under random and pulse road excitation.

Common errors on using convex modelling and linear matrix inequalities for nonlinear control

This note discusses common errors on using convex modelling and linear matrix inequalities for nonlinear control, a methodology that has become increasingly popular due to its systematicness and numerical implementability. Illustrations on common problems are made from existing literature: they are classified and discussed; advices are given to prevent them. Convex modelling is employed in linear parameter varying, Takagi-Sugeno models, and other convex structures in order to subsume or rewrite a nonlinear system for analysis or design via the direct Lyapunov method. Convexity plays a central role in allowing a finite set of vertex conditions in the form of linear matrix inequalities to be sufficient for the corresponding task. In contrast with other nonlinear methodologies, this one produces expressions resembling linear results, which makes it easier to grasp while often inducing subtle mistakes.

Nonlinear Modeling and Feedback Control of Boom Barrier Automation

We address modeling and control of a gate access automation system. A model of the mechatronic system is derived and identified. Then, an approximate explicit feedback linearization scheme is proposed, which ensures almost linear response between the electronic driver duty cycle input and the delivered torque. A nonlinear optimization problem is solved offline to generate a feasible trajectory associated with a feedforward action, and a low-level feedback controller is designed to track it. The feedback gains can be conveniently tuned by solving a set of convex linear matrix inequalities, performing a multiobjective tradeoff between disturbance attenuation and transient response. The proposed control strategy is tested on an industrial device. The experiments show that it can effectively meet the requirements in terms of robustness, load disturbance rejection, and tracking performance.

A sufficient condition to design unknown input observers for nonlinear systems with arbitrary relative degree

AbstractThis article addresses the problem of state and unknown inputs (UIs) estimation for nonlinear systems with arbitrary relative degree with respect to the UIs. For this purpose, a novel nonlinear unknown input observer (UIO) is proposed, which is able to decouple the UIs by using the derivatives of the output signal. The error dynamics is attained by an exact handling and a factorization of its gradient to obtain a local polytopic representation suitable for input‐affine nonlinear systems. For that representation, a novel design condition based on convex optimization and linear matrix inequalities is proposed to exponentially stabilize the estimation error and to guarantee the validity of the proposed nonlinear UIO. Numerical simulations indicate the effectiveness of the proposed approach for different classes of nonlinear systems, for which the UIs could be totally decoupled from the state estimation.

Robust decentralized plug and play voltage tracker of islanded microgrids under loads and lines uncertainties by the invariant ellipsoids

Microgrids (MGs) comprising of distributed generation units (DGs) are subjected to plug-and-play operation (PnP), lines (connecting the DGs) parameters uncertainty, and load changes. Robust stability and an authentic operation for islanded microgrids can be guaranteed through a robust decentralized voltage tracker developed in this paper. The proposed controller design has the following merits: 1) fully decentralized, 2) scalable, and 3) maintains robust stability against PnP of DGs, load changes, and lines parameters’ uncertainties. A sufficient condition is developed by linear matrix inequality convex optimization is exploited to solve the problem. The MGs’ changes in load and the line's parameters are modelled as norm-bounded uncertainties. The suggested controller uses local measurements from DGs, i.e., decentralized by the decomposition of the global system into subsystems. For each subsystem, the rest of the system's impact is considered disturbances, whose influence must be minimized. The proposed disturbance rejection control algorithm is based on the method of invariant-ellipsoids. Several time-domain scenarios such as connecting and disconnecting a number of DGs, local load changes, and variation of transmission line parameters are executed to assess the suggested controller's effectiveness, using MATLAB /Sim PowerSystems Toolbox.

An Improved Integral Inequality for Delay-Dependent Gain-Scheduled LPV Control

The present work examines the delay-dependent gain-scheduling feedback control with guaranteed closed-loop stability and induced L2 norm performance for continuous-time linear parameter-varying (LPV) systems with arbitrary time-varying delay. An extension of Lyapunov stability utilizing Krasovskii functionals is considered to derive stability analysis and synthesis conditions for delay-dependent dynamic output feedback LPV control design. The main challenges associated with this approach are selecting appropriate Lyapunov-Krasovskii functionals (LKFs) and finding efficient integral inequalities to bound the derivative of the LKF. Accordingly, a novel modified parameter-dependent LKF candidate along with an affine version of Jensen’s inequality bounding technique are employed leading to the derivation of less conservative sufficient conditions expressed in terms of convex linear matrix inequalities (LMIs). The proposed methodology is compared with past work in the literature in terms of conservatism reduction and performance improvement through a numerical example. Finally, the application of the proposed output-feedback LPV control design is evaluated on the automated mean arterial blood pressure (MAP) regulation in critical patient resuscitation via vasoactive drug infusion. Closed-loop simulation results are presented to illustrate the potential of the introduced LPV gain-scheduling design to provide MAP set-point tracking in the presence of disturbances and varying input delays.

Invariant set design of decentralized control for islanded microgrids under plug-and-play operation

This article presents a new invariant-set design for voltage regulation of islanded microgrids (MGs) composed of several distributed generators (DGs). The advantages of the proposed control are: (1) the control structure is decentralized, and (2) it tracks the reference inputs quickly and maintains robust stability against the MG’s load changes and plug- and-play operation. To achieve these objectives, the global system is decomposed into several subsystems. Each DG subsystem is modeled as a linear time-invariant system with norm-bounded uncertainty representing load changes. For a specific subsystem, the effect of the rest of subsystems is represented as a disturbance. The effect of the disturbance on the subsystem is reduced by minimizing the volume of its invariant ellipsoid (approximating and bounding its invariant set). The changes in the disturbance is modeled by the norm-bounded form. The controller design is based on a derived sufficient condition in the form of nonlinear matrix inequality (solved iteratively by fixing a positive scalar as a convex linear matrix inequality, LMI, optimization problem). To evaluate the effectiveness of the proposed control strategy, different time-domain simulations are conducted using MATLAB/SimPowerSystems Toolbox. Simulations demonstrate that the suggested controller provides excellent performance in terms of voltage tracking under load changes and MG’s plug-and-play operation.

LPV Delay-Dependent Sampled-Data Output-Feedback Control of Fueling in Spark Ignition Engines

We propose a delay-dependent sampled-data output-feedback LPV control technique to address the air-fuel ratio (AFR) regulation problem in spark ignition (SI) engines. AFR control and advanced fueling strategies are essential for maximizing fuel economy while minimizing harmful exhaust emissions. The fuel path of the SI engine, as well as the three-way catalyst (TWC) simplified dynamics, have been captured by a continuous-time linear parameter-varying (LPV) system with varying time delay, where the system dynamics rely on the engine speed, defined as the system's scheduling parameter. The interconnection of the continuous-time plant and a digital controller through analog-to-digital and digital-to-analog converter devices forms a hybrid closed-loop configuration. Therefore, in order to benefit from continuous-time control synthesis tools, the input-delay method has been employed to transform the hybrid closed-loop system into the continuous-time domain with system inherent time delay and an additional delay imposed by the mapping approach. The designed sampled-data gain scheduled output-feedback controller seeks to establish the closed-loop asymptotic stability and a prescribed level of performance for the LPV system with an arbitrarily varying time delay and varying sampling time, where the synthesis results are provided in a convex linear matrix inequality (LMI) constraint setting. Finally, several closed-loop simulation scenarios are conducted, and comparisons are provided to demonstrate the proposed methodology's performance in achieving precise reference AFR tracking and disturbance attenuation.

Robust linear parameter-varying output-feedback control of permanent magnet synchronous motors

This paper investigates the design of a robust linear parameter-varying (LPV) controller for speed regulation of surface permanent magnet synchronous motors (SPMSMs). Motor dynamics is defined in the α–β stationary frame and a parameter-varying model formulation is provided to describe its nonlinear dynamics. A robust gain-scheduled LPV dynamic output-feedback controller is designed to guarantee the asymptotic stability and performance requirements, as well as, guarantee robustness against parameter perturbations and torque load disturbances. As a novel aspect in the SPMSM control, the real-time impact of temperature variation on the winding resistance and magnet flux during motor operation is explicitly considered in the LPV modeling as a scheduling parameter and the subsequent LPV control design compensating for demagnetization effects in the motor response. The controller synthesis conditions guaranteeing stability and robust performance to norm-bounded uncertainty in the system matrices are formulated in a convex linear matrix inequality (LMI) optimization framework. Finally, the validity of the proposed method is assessed in simulation studies. The closed-loop simulation studies demonstrate that the proposed LPV controller provides improved transient response with respect to settling time, overshoot, and disturbance rejection in tracking the velocity profile under the influence of parameter perturbation, temperature variations, and load disturbances.

New Robust Stability Criteria for Lur’e Systems with Time-varying Delays and Sector-bounded Nonlinearities

This paper deals with a robust stability problem for uncertain Lur’e systems with time-varying delays and sector-bounded nonlinearities. An improved delay-dependent robust stability criterion is proposed via a modified Lyapunov-Krasovskii functional (LKF) approach. Firstly, a modified LKF consisting of delay-dependent matrices and double-integral items under two delay subintervals is constructed, thereby making full use of the delay and its derivative information. Secondly, the stability criteria can be expressed as convex linear matrix inequality (LMI) via the properties of quadratic function application. Thirdly, to further reduce the conservatism of stability criteria, the quadratic generalized free-weighting matrix inequality (QGFMI) is used. Finally, some numerical examples, including the Lur’e system and the general linear time-delayed system, are presented to show the improvement of the proposed approach.

Hankel Norm Model Reduction of Discrete-Time Interval Type-2 T–S Fuzzy Systems With State Delay

This article focuses on the model reduction problem of discrete-time time-delay interval type-2 Takagi-Sugeno (T–S) fuzzy systems. Compared with the type-1 T–S fuzzy system, the interval type-2 T–S fuzzy system has more advantages in expressing nonlinearity and capturing uncertainties. In addition, in order to simplify the analysis process, complex high-order systems can be approximated as low-order systems, which is called model reduction. In previous studies, there are few researches on model reduction of the interval type-2 T–S fuzzy system with time delay. Hankel norm is adopted to limit the error after model reduction. Based on Jensen's inequality, a linear matrix inequality (LMI) condition for the Hankel norm performance of the error system is obtained. A membership-function-dependent method based on piecewise linear membership functions is utilized to deal with mismatched membership functions where information of membership functions will be used for relaxing analysis results. Next, by a convex linearization design, the model reduction problem is formulated as a convex LMI feasibility/optimization condition. Numerical examples are given to verify the validity of the analysis.

Pareto-Optimal Pilot Design for Cellular Massive MIMO Systems

We introduce a non-orthogonal pilot design scheme that simultaneously minimizes two contradicting targets of channel estimation errors of all base stations (BSs) and the total pilot power consumption of all users in a multi-cell massive MIMO system, subject to the transmit power constraints of the users in the network. We formulate a multi-objective optimization problem (MOP) with two objective functions capturing the contradicting targets and find the Pareto optimal solutions for the pilot signals. Using weighted-sum-scalarization technique, we first convert the MOP to an equivalent single-objective optimization problem (SOP), which is not convex. Assuming that each BS is provided with the most recent knowledge of the pilot signals of the other BSs, we then decompose the SOP into a set of distributed non-convex optimization problems to be solved at individual BSs. Finally, we introduce an alternating optimization approach to cast each one of the resulting distributed optimization problems into a convex linear matrix inequality (LMI) form. We provide a mathematical proof for the convergence of the proposed alternating approach and a complexity analysis for the LMI optimization problem. Simulation results confirm that the proposed approach can significantly reduce pilot power, whilst maintaining the same level of channel estimation error as in a baseline.

Robust delay‐dependent LPV synthesis for blood pressure control with real‐time Bayesian parameter estimation

Mean arterial blood pressure (MAP) dynamics estimation and its automated regulation could benefit the clinical and emergency resuscitation of critical patients. In order to address the variability and complexity of the MAP response of a patient to vasoactive drug infusion, a parameter-varying model with a varying time delay is considered to describe the MAP dynamics in response to drugs. The estimation of the varying parameters and the delay is performed via a Bayesian-based multiple-model square root cubature Kalman filtering approach. The estimation results validate the effectiveness of the proposed random-walk dynamics identification method using collected animal experiment data. Following the estimation algorithm, an automated drug delivery scheme to regulate the MAP response of the patient is carried out via time-delay linear parameter-varying (LPV) control techniques. In this regard, an LPV gain-scheduled output-feedback controller is designed to meet the MAP response requirements of tracking a desired reference MAP target and guarantee robustness against norm-bounded uncertainties and disturbances. In this context, parameter-dependent Lyapunov-Krasovskii functionals are used to derive sufficient conditions for the robust stabilization of a general LPV system with an arbitrarily varying time delay and the results are provided in a convex linear matrix inequality (LMI) constraint framework. Finally, to evaluate the performance of the proposed MAP regulation approach, closed-loop simulations are conducted and the results confirm the effectiveness of the proposed control method against various simulated clinical scenarios.

H-infinity adaptive observer enhancements for vehicle chassis dynamics-based navigation sensor fault construction

The issues of chassis dynamics-based navigation sensor fault and state estimation in land vehicles are specialized in this study. Owing to the essential attributes of robust theory-based observers, an H-infinity adaptive observer is proposed to implement the fault reconstructions of faulty sensors, offering a reference to vehicles for further favorable control decision-making. This observer fuses a linear matrix inequality convex optimization strategy, with the dynamics of land vehicles established mathematically, the consequent problems associated with augmented descriptor system state-space model, Lyapunov stability and linear matrix inequality convex optimization are discussed in detail. The numerical simulations on vehicular systems that suffered with single-existing deadlocking, gain scheduling, and constant deflection sensor fault are conducted. The results indicate, the fault channel outputs fairly reflect the variations of real faults under severe step-type fault input circumstances, so that the applicability of the fault observer against sensor failures is guaranteed. The proposed sensor fault construction idea is further extended to a loosely coupled inertial measurement units/global positioning system (GPS) illustration with GPS unavailable in its north velocity channel. After reconstructing the priori system state for “State one-step prediction” of Kalman filter, the compensated navigation parameters by state estimator exhibit consistent with the references as expected, the vehicle chassis dynamics-based sensor fault construction method, therefore, may be recognized as an effective measure to a class of integrated navigation systems experiencing some unknown sensor failures.

A high precision image registration method for measurement based on the stereo camera system

Image registration is the major problem to many applications including measurement in the area of computer vision. In this paper, a scheme of high-precision measurement based on the stereo camera system is proposed. To take advantages of the speed up robust features and the optical flow features, they are chosen to be the registration basis for estimating and refining the initial geometrical transformation, This paper reviews the existing methods in the optimization procedure of image registration. By analyzing the weakness and strength recent progress in global polynomial optimization, a hybrid method which combines the merits of the SVD method, interleaving parameterization algorithm and the convex linear matrix inequalities relaxation method together is presented to achieve the global solution of the fundamental matrix and the rotation matrix problems. The experimental results confirms the high precision and stability of the proposed method.

Certifiably Optimal and Robust Camera Pose Estimation From Points and Lines

The Perspective-n-Point-and-Line (PnPL) problem, as a cornerstone in geometric computer vision, seeks to estimate the absolute pose of a calibrated camera from 3D-to-2D point and line correspondences. In this paper, we present a certifiably globally optimal and robust solution to the PnPL problem with a large number of outliers among the correspondences. Our first contribution is to reformulate the general PnPL problem as a novel constrained global optimization problem of Sum-of-Squares (SOS) polynomials using generalized geometric distances for both points and lines. Our second contribution is to efficiently solve this non-convex optimization problem by reducing it to an equivalent convex Linear Matrix Inequality (LMI) problem via a series of SOS relaxations. With these two contributions, we can develop a non-minimal solver, named SPnPL, for the outlier-free cases. The third contribution is to further add a Graduated Non-Convexity (GNC) cost function to SPnPL so as to remove outliers through closed-form iterations, which leads to a robust solver, named GNC-SPnPL. Both synthetic and real-data experiments confirm that SPnPL can mostly outperform the existing state-of-the-art PnPL algorithms and GNC-SPnPL can render accurate results even when 85% of the correspondences are outliers.

A Robust Guiding Torque Control Method for Automatic Steering Using LMI Algorithm

The existing path tracking methods usually neglect the effect of the drivers on the steering control. This paper proposes a robust steering control method of human-machine steering torque superposition based on linear matrix inequality (LMI) algorithm. First, the model for solving steering superposition torque introduces the steering system and steering resistance torque model in addition to the vehicle model, which increases the nonlinearity and uncertainty of system, and the human in torque superposition control also increases the external interferences. Therefore, this paper proposes a LMI robust control algorithm to reduce the external interference and the influence of uncertain factors on the system and improve the tracking performance of system, by use of Lyapunov stability theory and Schur complement property to convert the region pole assignment and robust control constraint conditions into LMI convex optimization problem. The next, the nonlinear vehicle dynamics solving model including Fiala tire model, steering column model is established; the nonlinear tire model is linearized by use of affine function, and the steering superposition control law is solved by use of LMI. Then, the union CarSim and Simulink simulation is conducted under different situations to verify the robustness and control performance of control system. Finally, through establishing the hardware-in-the-loop experiment table based on LabVIEW-RT, the effectiveness of control strategy is verified. The test results show that the method solves the model uncertainty and the robustness decrement problem resulting from human intervention, ensuring a good tracking performance, and a stable system at the same time.

Improved robust stability criteria for uncertain linear neutral‐type systems via novel Lyapunov‐Krasovskii functional

AbstractThe stability problem for the uncertain time‐varying delayed neutral‐type system is concerned in this paper. By introducing a novel Lyapunov‐Krasovskii functional (LKF) related to a delay‐product‐type function and two delay‐dependent matrices, some new delay‐dependent robust stability sufficient conditions are derived, which are based on convex linear matrix inequality (LMI) framework. The sufficient conditions in this paper can reduce the conservativeness of some recent previous ones. In the end, some numerical examples, including a linear neutral‐type system, the partial element equivalent circuit and a general linear system, show the effectiveness of the proposed method.

New Results on Stability Analysis of Uncertain Neutral-Type Lur’e Systems Derived from a Modified Lyapunov-Krasovskii Functional

This paper is concerned with the problem of the absolute and robustly absolute stability for the uncertain neutral-type Lur’e system with time-varying delays. By introducing a modified Lyapunov-Krasovskii functional (LKF) related to a delay-product-type function and two delay-dependent matrices, some new delay-dependent robustly absolute stability criteria are proposed, which can be expressed as convex linear matrix inequality (LMI) framework. The criteria proposed in this paper are less conservative than some recent previous ones. Finally, some numerical examples are presented to show the effectiveness of the proposed approach.

Decentralised robust T‐S fuzzy controller for a parallel islanded AC microgrid

This study proposes a decentralised robust fuzzy control strategy for islanded operation of an AC microgrid with voltage source inverters. The objective is to design a robust controller for regulating the load voltage and sharing power among distributed generators (DGs) in the presence of uncertainties in the system and non-linear loads. The AC microgrid consists of parallel DGs connected to a main AC bus. A Takagi-Sugeno fuzzy approach is developed in this article to achieve stability and desired performance in dealing with non-linearities in the islanded microgrid and H ∞ criterion is used to obtain a robust control strategy in presence of uncertainties raised by unmodelled and high frequency dynamics. Therefore, a non-convex condition in H ∞ optimisation problem is converted to a convex linear matrix inequality (LMI) condition and is solved by MATLAB LMI toolbox. In order to develop a decentralized system, P–f and Q–V droop controllers are used to specify set-points for local inverter controllers in each DG. The effectiveness of the proposed control strategy in presence of constant power load and DG accidental outage is validated by the simulation of a microgrid test system in MATLAB SimPowerSystems toolbox. Comparison with cascaded proportional integral controller shows advantages of robust T-S controller.