Linear Matrix Inequality Algorithm Research Articles

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.

Distributed Robust Fault Estimation for Multiagent Systems Based on Transition Variable Estimator.

This article investigates the distributed robust fault estimation for a class of multiagent systems with actuator faults and nonlinear uncertainties. To estimate the actuator faults and system states simultaneously, a novel transition variable estimator is constructed. Compared with existing similar results, the fault estimator existing condition is not necessary for designing the transition variable estimator. Furthermore, the bounds of the faults and their derivatives can be unknown in designing the estimator for each agent in the system. The parameters of the estimator are calculated by using Schur decomposition and linear matrix inequality algorithm. Finally, the performance of the proposed method is demonstrated through experiments of wheeled mobile robots.

Observer-based robust preview tracking control for a class of continuous-time Lipschitz nonlinear systems

<p>In this paper, a novel observer-based robust preview tracking controller design method is proposed for a class of continuous-time Lipschitz nonlinear systems with external disturbances and unknown states. First, a state observer is designed to reconstruct unknown system states. Second, using differentiation, the state lifting technique, the differential mean value theorem, and several ingenious mathematical manipulations, an augmented error system (AES) containing the previewable information of a reference signal is constructed, thereby transforming the tracking control problem into a robust $ H_{\infty} $ control problem. Based on linear parameter-varying (LPV) system theory, a sufficient condition for asymptotic stability of a closed-loop system with a robust $ H_{\infty} $ performance level is established in terms of the linear matrix inequality (LMI). Furthermore, a tracking controller, which includes observer-based feedback control, integral control, and preview feedforward compensation, is established for the original system. In particular, the tracking controller design is simplified by computing the observer and tracking controller gains simultaneously via only a one-step LMI algorithm. Finally, numerical simulation results demonstrate that the proposed controller leads to superior improvement in the output tracking performance compared with the existing methods.</p>

A Takagi–Sugeno fuzzy controller for minimizing cancer cells with application to androgen deprivation therapy

Androgen deprivation therapy (ADT) is frequently used to treat prostate cancer which is a widespread disease having a very low survival rate. A prolonged course of ADT can increase toxicity and drug resistance. This study proposes an adaptive therapy combining chemotherapy or immunotherapy with the discontinuation of hormone therapy to overcome these obstacles. The super-twisting sliding mode control (STSMC) algorithm is found to be one of the effective approach as an ADT model for obtaining suitable dosage adaptively. The primary objective is to rapidly reduce the number of cancer cells and the duration of drug exposure. The Takagi–Sugeno fuzzy controller-based active control algorithm is introduced, and it’s performance is compared with the STSMC algorithm. While maintaining global asymptotic stability, the Takagi–Sugeno fuzzy controller reduces the duration of therapy to six months. The controllers are implemented utilizing the linear matrix inequality (LMI) algorithm and the yet another LMI (YALMIP) toolset for MATLAB, and their efficacy is validated utilizing MATLAB and Simulink simulations. This study presents a novel approach to improve prostate cancer treatment outcomes by integrating nonlinear control algorithms and adaptive dosage strategies to reduce treatment duration and minimize drug exposure, thereby improving patient outcomes in prostate cancer management.

Robust output regulation of a class of smart actuators described by a minimum phase LPV dynamics

This paper presents a new control strategy for a class of minimum phase linear parameter-varying (LPV) systems representative of smart material actuators. In position control applications of such devices, one is typically interested in achieving output regulation with an arbitrary exponential decay rate. In principle, this problem can be tackled by assigning an exponential decay rate to the full system state, by means of well-established linear matrix inequalities (LMI) methods. For the class of systems under investigation, however, this approach leads to excessively large controller gains in case the desired decay rate is faster than the open-loop zero dynamics. In addition, the existence of unmeasurable state variables related to the material microstructure makes it not possible to directly implement full state feedback laws which are commonly adopted in LPV theory. To overcome these issues, a new design strategy is proposed. The key idea relies on arbitrarily shaping the output exponential decay rate without requiring fast convergence of the full state vector. This goal is achieved by means of a partial state feedback control law which solely depends on the measurable system states. A LMI algorithm is also developed to systematically address the design of the partial state feedback controller. The method is validated by means of simulations on generic examples, as well as via experiments on a mechatronic positioning system based on a dielectric elastomer membrane. In case the desired output convergence speed is faster than the open-loop zero dynamics, it is shown that the new approach leads to better transient behaviors and significantly smaller controller gains than standard LPV design techniques.

Robust H[formula omitted]-PID control Stability of fractional-order linear systems with Polytopic and two-norm bounded uncertainties subject to input saturation

This paper deals a type of H∞ proportional–integral–derivative (PID) control mechanism for a type of structural uncertain fractional order linear systems by convex Polytopic and two-norm bounded uncertainties subject to input saturation which mainly focuses on the case of a fractional order α such that 0<α<1. The Gronwall–Bellman lemma and the sector condition of the saturation function are investigated for system stability analysis and stabilization. The main strategy of the presented strategy is to restore fractional order PID controller design under input saturation problem from static output feedback controller design. Unlike existing strategies, non-iterative strategy is used to get optimal output feedback based on the LMI. On the premise of a linear matrix inequality algorithm, the SOF control laws can be obtained. After that, the fractional-order PID controller is recovered from the SOF controller. A numerical example is provided in order to show the validity and superiority of the proposed method.

Distributed event-triggered consensus control for multiple Euler–Lagrange systems with transmission delay under directed networks

This article addresses the distributed consensus control for multiple Euler–Lagrange systems under directed networks with time-varying transmission delays. Only some followers can access the information of the leader; communication is directed between agents. A distributed adaptive event-triggered control protocol is designed for the two cases where the generalized velocity of the leader is constant/time varying. When the generalized velocity of the leader is time varying, a distributed adaptive observer is designed to make followers obtain the time-varying information of the leader. Then, Lyapunov stability theory and the Linear matrix inequality algorithm prove the tracking errors can converge to zero in a completely distributed chattering-free manner. With the event-triggered mechanism, the number of communications can be decreased, and the system update frequency and the energy consumption can be reduced effectively. Meanwhile, the elimination of the Zeno behavior is also proved. Finally, the simulation results verify the effectiveness of the two controllers.

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.

Robust Speed Tracking Control for Future Electric Vehicles under Network-Induced Delay and Road Slope Variation.

Integrated motor-transmission (IMT) powertrain systems are widely used in future electric vehicles due to the advantages of their simple structure configuration and high controllability. In electric vehicles, precise speed tracking control is critical to ensure good gear shifting quality of an IMT powertrain system. However, the speed tracking control design becomes challenging due to the inevitable time delay of signal transmission introduced by the in-vehicle network and unknown road slope variation. Moreover, the system parameter uncertainties and signal measurement noise also increase the difficulty for the control algorithm. To address these issues, in this paper a robust speed tracking control strategy for electric vehicles with an IMT powertrain system is proposed. A disturbance observer and low-pass filter are developed to decrease the side effect from the unknown road slope variation and measurement noise and reduce the estimation error of the external load torque. Then, the network-induced delay speed tracking model is developed and is upgraded considering the damping coefficient uncertainties of the IMT powertrain system, which can be described through the norm-bounded uncertainty reduction method. To handle the network-induced delay and parameter uncertainties, a novel and less-conservative Lyapunov function is proposed to design the robust speed tracking controller by the linear matrix inequality (LMI) algorithm. Meanwhile, the estimation error and measurement noise are considered as the external disturbances in the controller design to promote robustness. Finally, the results demonstrate that the proposed controller has the advantages of strong robustness, excellent speed tracking performance, and ride comfort over the current existing controllers.

Output Feedback Model Predictive Control for NCSs with Input Quantization

This paper addresses the robust output feedback model predictive control (MPC) schemes for networked control systems (NCSs) with input quantization. The logarithmic quantizer is considered in this paper, and the sector bound approach is applied, which appropriately treats the quantization error as a sector‐bounded uncertainty. The presented method involves an offline designed state observer using linear matrix inequality (LMI) and online robust output feedback MPC algorithms which optimize one free control move followed by the output feedback using the estimated state. Moreover, due to the uncertainty of estimation error, a technique of refreshing the bound of estimation error which involves the quantization error is provided so as to guarantee the recursive feasibility of the optimization problem. The proposed MPC schemes inherit the characteristics of the synthesis approach of MPC, guaranteeing the recursive feasibility of the optimization problem and the stability of a closed‐loop system, and explicitly account for quantization error. Two simulation examples are given to illustrate the effectiveness of the proposed methods.

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.

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.

Event-triggered controller design for linear systems with saturated input and bounded disturbance

This paper deals with the stabilisation problem of linear systems subject to saturated input and bounded disturbance via event-triggered control strategy. In a hybrid systems framework, both a state feedback law and a static output feedback law are proposed to design the event-triggered controllers, which provide solutions to the event-triggered control collaborative design problem. For the case of using static output feedback, an iterative linear matrix inequality algorithm is proposed to obtain the feedback gain. In addition, the local uniform ultimate boundedness of the closed-loop system is ensured. Finally, two examples are given to illustrate the effectiveness of the proposed controllers.

Towards the Autonomy: Control Systems for the Ship in Confined and Open Waters.

The concept of the Marine Autonomous Surface Ship (MASS) requires new solutions in many areas: from law, through economics, social sciences, environmental issues to the technology and even ethics. It also plays a central role in the work of numerous research teams dealing with the ship motion control systems. This article presents the results of the experiments with application of the selected control methods in automatic steering of the movement of an autonomous ship in the two regimes: during the maneuvers at low speed (in a harbor confined waters) and during the lake trials in open water conditions. In the first case, multidimensional state controller synthesized with Linear Matrix Inequalities (LMI) algorithms was used, while, in the second case, Model Predictive Control (MPC) control was adopted. The object for which the experiments were carried out was 1:24 scale model of the Liquefied Natural Gas (LNG) carrier. The paper presents also the design of the measurement and control system and the user interface. The experiments were conducted in the natural conditions on the lake. The results of the experiments indicate the fundamental role of the measurement system in the process of controlling an autonomous ship.

Spatially Local Piecewise Fuzzy Control for Nonlinear Delayed DPSs With Random Packet Losses

This article introduces a spatially local piecewise fuzzy control (SLPFC) for nonlinear delayed distributed parameter systems (DPSs) with random packet losses. Due to the limited bandwidth of the channels, the random packet losses could occur simultaneously in the communication channels from the sensor to the controller and from the controller to the actuator, which are assumed to obey the Bernoulli random binary distribution. An SLPFC design is given such that, for all possible random packet losses, the closed-loop delayed DPS is mean square exponentially stable. Delay-dependent conditions are obtained and then two procedures are given for designing the spatially local piecewise fuzzy controller: one casts the controller design into an iterative linear matrix inequality (LMI) algorithm and the other casts the controller design into a parameter-dependent LMI problem. Two examples are provided to illustrate the usefulness of the presented design approach.

Finite frequency fault estimation observer design for T-S Fuzzy discrete-time descriptor systems

This paper deals with the problem of fault estimation observer design for T-S fuzzy nonlinear discrete-time descriptor systems in the finite frequency domain. First, a novel fault estimation observer for the augmented descriptor system is given. Based on Parseval's theorem and Lyapunov theory, the performance in the finite frequency domain is derived via time-domain method. Then, new sufficient conditions of admissibility and robust performance index for the dynamic error system are obtained as bilinear matrix inequalities. Meanwhile, an iterative linear matrix inequality algorithm is given to obtain the optimal solution. Finally, an example is given to demonstrate the effectiveness of our method.

Delay‐dependent robust control of stochastic systems with convex polynomial uncertainty

SummaryThis article addresses a robust control problem of time‐varying stochastic systems with time‐delays. Through Linear Parameter Varying (LPV) modeling approach, the time‐varying parameters can be described via the convex combination. Therefore, the LPV stochastic system is interpreted by a weighting function and multiplicative noised linear systems. Furthermore, stabilization problem for the systems is investigated via Gain‐Scheduled (GS) technique to increase robustness. For the problem, some sufficient conditions are derived by Integral Lyapunov‐Krasovskii Function (ILKF) such that the conservatism caused by derivative of weighting function is ignored. Besides, an Iterative Linear Matrix Inequality algorithm is used to determine the auxiliary variable such that the problem can be solved via the convex calculation. Finally, some simulations are provided to demonstrate effectiveness and applicability of this article.

Optimized state feedback control of quarter car active suspension system based on LMI algorithm

Suspension systems are applied in vehicles so as to improve the passengers ride comfort, vehicle stability and better road handling when the vehicle moves on roads with bumps and terrains. In this paper, Linear Matrix Inequality (LMI) and linear quadratic regulator (LQR) controllers were proposed and this was also compared with passive suspension system. Using nonlinear quarter car active suspension system model with hydraulic actuator to reduce the effect of bumpy road surfaces on vehicle and hence improve passenger ride comfort. Vehicle body suspension travel and wheel deflection were measured to determine the performance of the active suspension system. Mean absolute error (MAE) and integral square error (ISE) were used as the performance indexes, thus a MAE and ISE of passive 75.01%, LQR 82.65% and LMI 89.11% and passive 4.11x10−6, LQR 3.24x10−6 and LMI 2.19x10−6 respectively were recorded. Based on the simulation results and analysis, the LMI based control performs better in minimizing vibration effects and also guarantee better ride quality and vehicle stability when compared to the conventional LQR and the passive suspension system.

Event-triggered communication scheme for stochastic systems in wireless sensor networks

With the development of sensors and network technology, wireless sensor networks are widely used in various scenes. However, in most applications, the energy of the sensor nodes cannot be supplemented. In addition, since the lifetime of the electronic switching element is measured by the number of trigger times, which makes the effective use of energy, extending the working life of the network becomes one of the key factors to be considered. This paper discusses an event-triggered control scheme for stochastic systems. The mechanism determines when the controller sends control data in order to balance system performance and drive frequency. In this paper, the control scheme is deduced by a class of quadratic performance index function. Then, the formula of performance index is designed based on state feedback and output feedback, and the upper bound of performance index is deduced and the conditions in the theorem can be constructed by solving a set of linear matrix inequalities, which can be easily tested with linear matrix inequality algorithms. Finally, a numerical example is given to illustrate the results.