Iterative Linear Matrix Inequality Research Articles

Iterative LMI-Tuned PID Controller for Robust Decentralized Multi-Area Automatic Generation Control in Deregulated Electricity Markets

This study proposes an Iterative Linear Matrix Inequality (LMI)-Tuned Proportional-Integral-Derivative (PID) controller for robust decentralized AGC in the context of deregulated electricity markets. The study addresses the challenges posed by market uncertainties and diverse generation sources, aiming to enhance the stability and performance of power systems. This study is driven by the urgent need to develop robust AGC strategies that are adapted to the complex dynamics of decentralized power systems operating in deregulated markets. The conventional centralized control approaches are often inadequate to address the diverse and distributed nature of modern generation units. Furthermore, the intermittent and variable nature of renewable energy sources adds a layer of complexity that demands innovative control solutions. The impact of bilateral contracts on the dynamics is taken into consideration while modifying the conventional LFC of interconnected power networks. To model these bilateral contracts, the distribution company (DISCO) participation matrix (DPM) is presented, and the multi-area block diagram reflects these concepts. Simulation studies are conducted to evaluate and compare the performance of the proposed Iterative LMI-Tuned PID Controller and Conventional PID Controller. Results are analyzed against key performance metrics, considering scenarios reflective of diverse market conditions. Comparative studies with the conventional PID controller provide insights into the efficacy and robustness of the proposed approach.

H∞ observer-based controller synthesis for fractional order systems over finite frequency range

This paper focuses on the H∞ observer-based control problem for fractional order systems (FOSs) over finite frequency range. Firstly, as an extension to the KYP lemma, the structure singular value–μ and its upper bound are analyzed and determined. The necessary and sufficient conditions of the H∞ performance for FOSs over finite frequency range are given through the generalized μ analysis method. This method is different from the traditional μ-analysis method, which is only suitable for integer order systems. Secondly, according to these obtained LMI-based criteria for H∞ performance index and a key projection lemma, we design an H∞ observer-based controller to stabilize the estimation error and decrease the H∞ norm for the FOS simultaneously. With the help of the matrix congruence transformation, the feedback gain matrix is parameterized by a scalar matrix. Thirdly, to deal with the nonlinear terms in matrix inequalities, an iterative linear matrix inequality (ILMI) algorithm is developed. Finally, two numerical examples are provided to explain the correctness of the established results.

An iterative output feedback controller-design strategy for energy-to-peak structural vibration control

ABSTRACT To address the issue of inadequate control effect resulting from the challenge of measuring state feedback information comprehensively, this paper proposes an iterative output feedback energy-to-peak control method that solely depends on partial external observation data. This paper presents the computational steps for achieving a suboptimal solution to the output feedback bilinear matrix inequality constrained problem using an iterative linear matrix inequality approach. The proposed method overcomes the challenge of numerically solving the optimization problem. The unknown variables in the output feedback linear matrix inequality optimization problem are transformed equivalently to obtain a new linear matrix inequality optimization problem. The numerical results of the two optimization problems are then used to iterate with each other and obtain a suboptimal solution. A comprehensive analysis was conducted on an 8-story building structure, utilizing state feedback, output feedback, and iterative output feedback energy-to-peak control techniques. The results of the study indicate that the proposed iterative output feedback energy-to-peak control approach efficiently reduces the displacement and acceleration response of the structure, exhibiting superior control effectiveness than both the state feedback and output feedback energy-to-peak control methods. The findings demonstrate the practicality and versatility of the suggested method.

Finite‐horizon H∞ output fault‐tolerant consensus for multi‐agent systems with switching topology and deception attacks

AbstractThis article investigates the output fault‐tolerant consensus problem for multi‐agent systems with switching topology and deception attacks. In order to reduce the effects of switching topology, the differences between topologies can be modeled as topology uncertainties. In practice, the deception attack may occur in a random way and assumed to satisfy Bernoulli distributions. The purpose of this research is to design an output fault‐tolerant controller such that, in the presence of switching topology, actuator faults, and deception attacks, the finite‐horizon output fault‐tolerant consensus can be achieved under a prespecified performance. An initial‐state‐based performance requirement is constructed, and this bypasses the need for the zero initial condition in the conventional theory. After this, the controller gains are computed by employing an iterative linear matrix inequality scheme. Finally, a numerical example is provided to show the effectiveness of the proposed methods.

Linear Quadratic Differential Games based MIMO-PID design for Load Frequency Control of a 2-area interconnected power system: An Iterative LMI approach

In this paper a Multi-input-Multi-output (MIMO)-Proportional-Integral-Derivative (PID) control design method via sub-optimal solution of Linear Quadratic Differential Games (LQDG) for Load frequency Control (LFC) of a 2-area interconnected power system is proposed. The main motivation behind the proposed centralised LFC design scheme is to solve multi-objective optimization problem that can accomplish improvement in transient response & disturbance rejection simultaneously. The proposed iterative optimization algorithm in Linear matrix Inequality (LMI) framework yields appropriate sub-optimal PID gains. A 2-area LFC problem is solved using the proposed design and comparative results suggests that the proposed design can significantly attenuate the load disturbances as well as provide improved transient response specifications.

Design of a Robust Decentralized Power System Stabilizer for Sudan National Grid

Power system stabilizers (PSS) are used to generate supplementary control signals for the excitation system in order to damp the low frequency, generator angle, speed variation, generator power, voltage magnitude and power flow oscillations. This paper presents robust decentralized power system stabilizer (PSS) design approaches for a power system consisting of 41 machines with 167 buses. This paper mainly focuses on developing robust decentralized control techniques for power systems, with special emphasis on problems that can be expressed as minimizing a linear objective function under linear matrix inequalities [LMI] in tandem with bilinear matrix inequalities [BMI] constraints. The design problem is considered the natural extension of the reduced order for decentralized dynamic output H2/H∞ -norm controller’s synthesis for power systems. The resulting optimization problem has a general bilinear matrix inequalities (BMIs) form which can be solved using an iterative linear matrix inequalities [LMIs] programming method. Simulations were carried out using loss of line without fault tests at transmission line on Sudan grid.

Takagi–Sugeno Fuzzy Realization of Stability Performance-Based Fault-Tolerant Control for Nonlinear Systems

This article is dedicated to studying realization issues of the stability performance-based nonlinear fault-tolerant control framework via Takagi–Sugeno (T–S)fuzzy models. To this end, the nonlinear fault-tolerant control strategy with an online fault detection system monitoring the system stability performance degradation induced by faults is first introduced by means of the stable image and kernel representations. On this basis, the T–S fuzzy models are applied to approximate the nonlinear system, and a design approach of the fuzzy observer-based controller is proposed for the system stabilization via the iterative linear matrix inequality method. With the controller gains, the fuzzy-model-based nominal stable image representation of the system is formulated, which leads to the generation of the input and output error signals. Then, with the reference signal and system input and output error signal data, a data-driven algorithm is given to online estimate the evaluation function defined in terms of system uncertainties and faults. By virtue of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L_{2}$</tex-math></inline-formula> input–output stability of the controller stable kernel representation, a threshold calculation method is presented and, thus, the stability performance-based fault detection system based on fuzzy models is realized. Furthermore, for fault-tolerant purpose, the fault-tolerant controller design is discussed, which aims to retain the system stability. Two examples are provided in the end to illustrate the proposed results.

Stability analysis for a class of switched systems with state constraints via time-varying Lyapunov functions

This article investigates the stability analysis for a class of continuous-time switched systems with state constraints under pre-specified dwell time switchings. The state variables of the studied system are constrained to a unit closed hypercube. Firstly, based on the definition of set coverage, the system state under saturation is confined to a convex polyhedron and the saturation problem is converted into convex hull. Then, sufficient conditions are derived by introducing a class of multiple time-varying Lyapunov functions in the framework of pre-specified dwell time switchings. Such a dwell time is an arbitrary pre-specified constant which is independent of any other parameters. In addition, the proposed Lyapunov functions can efficiently eliminate the “jump” phenomena of adjacent Lyapunov functions at switching instants. The feature of this paper is that the definition of set coverage is utilized to replace the restriction on the row diagonally dominant matrices with negative diagonal elements to analyze stability. The other feature of the constructed time-varying Lyapunov functions is that there are two time-varying functions. One of the two time-varying functions contains the jump rate, which will present a certain degree of freedom in designing the dwell time switching signal. An iterative linear matrix inequality (LMI) algorithm is presented to verify the sufficient conditions. Finally, two examples are presented to show the validity of the method.

Stability analysis and stabilisation of continuous-discrete fractional-order 2D Fornasini–Marchesini first model

This paper focuses on the structural stability and stabilisation of the continuous-discrete fractional-order two-dimensional Fornasini-Marchesini first model. The sufficient and necessary stability conditions are given in polynomial form at first. Secondly, to make the problem solvable, based on the property of Kronecker product and the Generalized–Kalman–Yakubovich–Popov Lemma, the new sufficient and necessary stability conditions are established in the form of linear matrix inequalities. Thirdly, with the help of the Projection Lemma, the stabilisation conditions are obtained after introducing a state feedback controller, which can be solved with an iterative linear matrix inequality algorithm. In the end, the effectiveness of the proposed results is verified by two numerical examples.

A New Power Flow Model With a Single Nonconvex Quadratic Constraint: The LMI Approach

In this paper, we propose a new mathematical model for power flow problem based on the linear and nonlinear matrix inequality theory. We start with rectangular model of power flow (PF) problem and then reformulate it as a Bilinear Matrix Inequality (BMI) model. A <b>Theorem</b> is proved which is able to convert this BMI model to a Linear Matrix Inequality (LMI) model along with One Nonconvex Quadratic Constraint (ONQC). Our proposed LMI-ONQC model for PF problem has only one single nonconvex quadratic constraint irrespective of the network size, while in the rectangular and BMI models the number of nonconvex constraints grows as the network size grows. This interesting property leads to reduced complexity level in our LMI-ONQC model which in turn makes it easier to solve for finding a PF solution. The non-conservativeness, iterative LMI solvability, well-defined and easy-to-understand geometry and pathwise connectivity of feasibility region are other important properties of proposed LMI-ONQC model which are discussed in this paper. An illustrative two-bus example is carefully studied to show different properties of our LMI-ONQC model. We have also tested our LMI-ONQC model on 30 different power-system cases including four ill-conditioned systems and compared it with a group of existing approaches. The numerical results show the promising performance of our LMI-ONQC model and its solution algorithm to find a PF solution.

Robust Feedback Control for Discrete-Time Systems Based on Iterative LMIs with Polytopic Uncertainty Representations Subject to Stochastic Noise

This paper deals with the design of linear observer-based state feedback controllers with constant gains for a class of nonlinear discrete-time systems in the form of a quasi-linear representation in presence of stochastic noise. For taking into account nonlinearities in the design of linear observer-based state feedback controllers, a polytopic modeling approach is investigated. An optimization problem is formulated to reduce the sensitivity of the controlled system towards stochastic input, state, and output noise with a predefined covariance. Due to the nonlinearities, the separation principle does not hold, thus, the controller and the observer have to be designed simultaneously. For this purpose, a Lyapunov-based method is used, which provides, in addition to the controller and observer gains, a stability proof for the nonlinear closed loop in a predefined polytopic domain. In general, this leads to nonlinear matrix inequalities. To solve these nonlinear matrix inequalities efficiently, we propose an approach based on linear matrix inequalities (LMIs) with a superposed iteration rule. When using this iterative LMI approach, a minimization task can be solved additionally, which desensitizes the closed loop to stochastic noise. The proposed method additionally enables the consideration of different linear closed loop structures by a unified Lyapunov-based framework. The efficiency of the proposed approach is demonstrated and compared with a classical LQG approach for a nonlinear overhead traveling crane.

Proportional Tracking Control of Positive Linear Systems

This letter considers the reference tracking of positive linear systems via a proportional controller, which remains an open problem. Through using a virtual reference method in state-space models, optimal tracking performance is guaranteed for single-input multi-output (SIMO) systems via a bisection method. The proportional tracking for multi-input multi-output (MIMO) systems is reformulated as a bi-convex programming problem and a convergence-guaranteed algorithm is developed through using iterative linear matrix inequalities. Numerical simulations are provided to verify the proposed results.

Tuning of passive TMD for floating offshore wind turbine using linear matrix inequalities

Over the last several years, the number of offshore wind turbine installations has increased exponentially. A change from shallow to deep marine waters will be seen in future installations. As we move further away from the shore, the loading on the floating offshore wind turbine (FOWT) system increases. This higher load may have a negative impact on the wind turbine's efficiency. It may also lead to power fluctuations, large tower displacements and increased fatigue loads. In order to circumvent these effects, passive secondary devices are installed. A tuned mass damper (TMD) is one such device that may be utilized for load mitigation. In this paper, an efficient method is presented for the design of TMD for FOWT. The design problem is posed as feedback response control problem for the plant represented by the FOWT-TMD system. The design objective is to evaluate the static controller such that the response of the FOWT is minimized. The controller gains correspond to the TMD properties. The TMD parameters are obtained using iterative LMI, which uses a combination of convex and metaheuristic optimization. The proposed TMD is found to be useful in lowering the FOWT response. The methods described in this paper can be utilized to determine the best TMD configuration.

Weighted sensitivity design of multivariable PID controllers via a new iterative LMI approach

This paper addresses the weighted sensitivity design of multivariable proportional–integral–derivative (PID) controllers for multi-input multi-output (MIMO) processes. First, the PID reference tracking problem is formulated into an H∞ PID control problem by using the weighted sensitivity synthesis, and the problem is further casted into an H∞ SOF control problem. Second, based on the idea of T–K iteration, a new iterative LMI algorithm has been developed to locally solve the optimal H∞ SOF control problem. Then, a design procedure is presented for weighted sensitivity design of multivariable PID Controllers for MIMO processes. Finally, three application examples are provided to show the effectiveness and good performance of our method for practical industrial processes.

Optimal TMD design for floating offshore wind turbines considering model uncertainties and physical constraints

The wind resources can be efficiently tapped using floating offshore wind turbines (FOWTs). However, the increased loading due to coupled action of wind and waves can affect the efficiency of the wind turbines. Structural control utilizing tuned mass damper (TMD) is an effective solution for response control of FOWT. An efficient framework for optimal design of TMD is presented in this paper. The main features of the proposed framework are: (1) The design of TMD is cast in the form of feedback control problem. The parameters of the TMD like stiffness and damping acts like a static-controller, whose gains are evaluated such that the performance specifications (described by a certain norm of the FOWT and/or TMD responses) are minimized, (2) the performance specifications are reduced to a convex program using linear matrix inequalities (LMIs), (3) the optimization problem is solved using an iterative LMI procedure, (4) the design procedure incorporates model uncertainties and, (5) the physical constraints (like TMD relative displacement) can also be incorporated by formulating design problem as multi-objective optimization. The optimal TMD design is illustrated for a case where TMD is installed in nacelle. Finally, the nonlinear dynamic response of the FOWT with the optimized TMD is evaluated using aero-hydro-servo-elastic analysis. Five different load cases with varying wind and wave loading are considered for simulations. The performance assessment is carried out in terms of structural response, damage equivalent fatigue loads and fluctuations in power generation. The designed TMDs are found to be effective for all the load cases. The proposed methodology is found to be effective for optimal tuning of the TMDs for response control of FOWT.

반복적인 선형행렬부등식을 이용한 연속시간 T-S 퍼지 시스템에 대한 정적 출력궤환 제어

This investigation deals with the design method of a static output feedback controller for continuous-time T-S fuzzy systems via iterative linear matrix inequalities. First, a matrix inequality that guarantees the stability of the nonlinear system is introduced. Based on this, an algorithm for finding the stabilization control gains, using the cone complementary linearization method, for continuous-time T-S fuzzy systems is presented. The validity of the proposed algorithm is verified through a simple example.

Filter Design for Positive T–S Fuzzy Continuous-Time Systems With Time Delay Using Piecewise-Linear Membership Functions

This article focuses on the filtering problem and stability analysis for positive Takagi-Sugeno (T-S) fuzzy systems with time delay under L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -induced performance. Due to the importance of estimation of system states but the few filter design results on positive nonlinear systems, it is an attractive and meaningful topic well worth studying. In order to fully exploit and take advantage of the positivity of positive T-S fuzzy systems, many commonly used methods, for instance free-weighting matrix approach and similarity transformation are probably not suitable for positive systems. To address the hard-nut-to-crack problem, an auxiliary variable is introduced so that the augmentation approach can be employed to carry out the positivity and stability analysis of filtering error systems. In addition, another obstacle that cannot be ignored is the existence of nonconvex terms in the stability and positivity conditions. For getting around this barrier, some iterative linear matrix inequality algorithms have been proposed in the literature. However, considering the weakness that these methods cannot guarantee the convergence to a numerical solution and the iterative process is exhaustive, we present an effective matrix decoupling method to convert the nonconvex conditions into convex ones in this article. Furthermore, a linear copositive Lyapunov function, which incorporates the positivity of system states and time delay at the same time is chosen so that the positivity characteristic of filtering error systems can be captured further. However, because of plenty of valuable information of membership functions being ignored, hence, the obtained results are conservative. For the sake of relaxing the conservativeness, the advanced piecewise-linear membership functions approximate method is utilized to facilitate the stability and positivity analysis. Therefore, the relaxed stability and positivity conditions, which are cast as sum of squares (SOS) are obtained and can be solved numerically. Finally, the effectiveness of the designed fuzzy filtering strategy with satisfying L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -induced performance are demonstrated by a simulation example.

Annulus-event-based fault detection for state-saturated nonlinear systems with time-varying delays

In this paper, the annulus-event-based fault detection method is proposed for a class of nonlinear systems subject to state saturation and time-varying delays. In particular, in order to save the network resources, a novel communication strategy called annulus-event-based triggering mechanism is considered. That is to say, the measurement output is sent to the filter only when the error between the measured value at the current moment and the data at the last transmission moment is within the prescribed triggering-annulus. Moreover, the system states at each instant are bounded by the convex hull via introducing a free matrix whose infinite norm is less than or equal to 1. Subsequently, sufficient conditions are obtained to ensure the global asymptotic stability and the H∞ performance of the filtering error dynamics system, which are expressed in line with some nonlinear matrix inequalities that can be handled via the given iterative linear matrix inequality method. Finally, two numerical simulations are employed to illustrate the validity of the developed fault detection filtering algorithm.

Iterative Solution of Linear Matrix Inequalities for the Combined Control and Observer Design of Systems with Polytopic Parameter Uncertainty and Stochastic Noise

Most research activities that utilize linear matrix inequality (LMI) techniques are based on the assumption that the separation principle of control and observer synthesis holds. This principle states that the combination of separately designed linear state feedback controllers and linear state observers, which are independently proven to be stable, results in overall stable system dynamics. However, even for linear systems, this property does not necessarily hold if polytopic parameter uncertainty and stochastic noise influence the system’s state and output equations. In this case, the control and observer design needs to be performed simultaneously to guarantee stabilization. However, the loss of the validity of the separation principle leads to nonlinear matrix inequalities instead of LMIs. For those nonlinear inequalities, the current paper proposes an iterative LMI solution procedure. If this algorithm produces a feasible solution, the resulting controller and observer gains ensure robust stability of the closed-loop control system for all possible parameter values. In addition, the proposed optimization criterion leads to a minimization of the sensitivity to stochastic noise so that the actual state trajectories converge as closely as possible to the desired operating point. The efficiency of the proposed solution approach is demonstrated by stabilizing the Zeeman catastrophe machine along the unstable branch of its bifurcation diagram. Additionally, an observer-based tracking control task is embedded into an iterative learning-type control framework.

Global Consensus of Positive Edge System With Sector Input Nonlinearities

This paper investigates the positive edge consensus problems for undirected networks, where each edge is modeled by a continuous-time positive system with sector input nonlinearities. Based on a given nodal network, the corresponding edge network is presented, and some relationships between the given nodal network and its edge network are given. The observer-based protocol for non-negative consensus is proposed. Rigorous theoretical analysis which considers both the input nonlinearities and the positivity constraint is introduced to achieve a global consensus result. Simultaneous stabilization problems are constructed for edge-based non-negative consensus using the maximum and the minimum nonzero eigenvalues of the edge Laplacian matrix. Furthermore, the robustness of the positive edge consensus to parameter perturbation is illustrated. Sufficient non-negative consensus conditions related to the bounds of system matrices are given, and the non-negative edge consensus conditions for star nodal networks are also given only using the vertex number. Iterative linear matrix inequalities (LMIs)/LMI algorithm is presented to obtain the feedback matrix and the observer matrix. The given theoretical results are finally illustrated by some simulations.