Linear Matrix Inequality Problem Research Articles

Lyapunov-based robust model predictive tracking controller design for discrete-time linear systems with ellipsoidal terminal constraint

In this article, a robust model predictive control (MPC) method is introduced based on Lyapunov-theory to control the discrete-time uncertain linear systems due to external disturbances. The structure of the proposed controller consists of two parts designed based on the offline/online schemes. In the offline-scheme, an H ∞ -based robust tracking controller is provided to satisfy the robust and tracking performance. Based on the H ∞ performance, a new linear matrix inequality problem is developed to calculate the offline-controller gains. By considering the Lyapunov-function of the offline-controller as the terminal cost of the MPC and the designed offline-controller, the online optimisation problem (OOP) is structured. This means, the MPC is designed based on the stabilised system to satisfy the physical limitations of the overall controller and the system states. Moreover, to improve the feasibility problem of the OOP, an ellipsoidal terminal constraint is added to the proposed MPC problem. Therefore, compared with the existing min–max MPC and tube MPC, the advantages of the overall controller are feasibility improvement and reducing the computational complexity with less conservatism while the robustness against parametric uncertainties and disturbances is achieved. Two simulation examples are employed to show the superiorities of the proposed robust MPC.

Event-triggered [formula omitted]-fault-tolerant consistency control for multi-agent systems

This paper investigates the issue of active fault-tolerant control for multi-agent systems using an event-triggered mechanism, aiming to achieve state-consistency control under a leader–follower framework. The proposed control protocol consists of two main components: leader input estimation and fault tolerant controller design under an event-triggered mechanism. Firstly, a leader–follower error system is constructed, where the leader’s input is considered as a supplementary vector for the system state, resulting in an augmented error system. Then, a distributed state observer is designed for this augmented error system to transform the leader’s input estimation problem into an augmented state estimation issue. Furthermore, an event-triggered mechanism is designed to optimize network communication resources by updating the control law only when specific triggered conditions are met. Next, taking into account the errors introduced by the distributed observer and event-triggered mechanism, the problem of determining the state error feedback gain matrix is transformed into a linear matrix inequality problem using H∞ robustness design and Lyapunov stability analysis methods. Finally, the effectiveness of this theoretical framework is validated through a practical example involving several VTOLs.

Power Regulation of a Three-Phase L-Filtered Grid-Connected Inverter Considering Uncertain Grid Impedance Using Robust Control

Uncertain grid impedance is often common in power distribution networks; therefore, it is crucial to design an efficient controller in this situation. An issue that frequently occurs is the problem of unpredictable grid impedance, which can cause voltage fluctuations, power quality problems, and potential damage to equipment. This work provides a systematic control strategy to tackle these issues by supplying well-regulated power from a DC source to an AC power grid. A linear matrix inequality (LMI)-based robust optimal control is proposed in this paper to provide stability to the inverter system without offset error at the output side. The convergence time to steady state is minimized by solving the LMI problem to maximize the eigen value of the closed-loop system with the inclusion of the uncertainty of the filter parameter and grid impedance. Furthermore, the uncertainties in this study include the potential variation of values for the filters and the grid's impedance. These uncertainties occur because the grid impedance can fluctuate fast in the event of a fault or termination of a transmission line, while the filter's impedance can also be affected by changes in operating temperature. The simulation study of this proposed control includes a comparison between wide and narrow uncertainty ranges, as well as a performance comparison under uncertain parameters. Furthermore, this approach exhibits a lower power ripple in comparison to existing PI control method.

Fuzzy PD-sliding mode control design for networked system with time delays

This research specifically addresses the problem of sliding mode control (SMC) in a particular group of T–S fuzzy systems that experience output disturbances and time delays using a networked control system. First, a model for a proportional–derivative sliding mode observer (SMO) is established, subsequently followed by the design of a controller based on SMO to maintain closed-loop system stability. The authors establish a novel mechanism to optimize the bandwidth utilization of the communication network by introducing a newly adjusted event-triggering parameter. After that, addressed a linear matrix inequality (LMI) problem to determine the controller gain and observer coefficients. This was done to guarantee the asymptotic stability of the closed-loop plant. Furthermore, the authors were able to directly identify the disturbances for these plants using the proposed descriptor SMO. Moreover, a parameter-based novel event-triggered scheme, along with a new resultant closed-loop system, will be employed to modify trigger frequency parameters within a unified framework. The controller gains will be determined by solving a linear matrix inequality, subject to certain conditions that ensure sufficiency. These conditions will be deduced to ensure asymptotic stability. In the final section, the authors presented examples from the Wind Energy System (WES) that illustrate the feasibility and effectiveness of our proposed methodology.

Robust optimal power flow considering uncertainty in wind power probability distribution

This paper proposes an optimal power flow model that takes into account the uncertainty in the probability distribution of wind power. The model can schedule controllable generators under any possible distribution of wind power to ensure the safe and economic operation of the system. Firstly, considering the incompleteness of historical wind power data, the paper models the uncertainty of wind power using second-order moments of probability distribution and their fluctuation intervals. Subsequently, a robust optimal power flow model based on probability distribution model and joint chance constraints is established. The Lagrangian duality theorem is then employed to eliminate random variables from the optimization model, transforming the uncertainty model into a deterministic linear matrix inequality problem. Finally, a convex optimization algorithm is used to solve the deterministic problem, and the results are compared with traditional chance-constrained optimal power flow model. The feasibility and effectiveness of the proposed method are validated through case study simulations.

Delay-dependent recursive feasibility based switched model predictive control for nonlinear systems

This paper presents a coordinated design of a model predictive controller (MPC) and switching law for delayed nonlinear switched systems with Lipschitz property. The article derives delay-dependent recursive feasibility constraints to handle disturbances in a polytopic form and incorporates them into the optimization problem. The control gain at each step is determined to ensure the feasibility of the optimization problem during the execution time of each sub-system. Additionally, constraints related to the cost function and H∞ performance conditions are introduced in the developed linear matrix inequality problem to minimize the cost function with an infinite predictive horizon and guarantee controller robustness against unknown constant delays. The coordinated design of the MPC and switching law is achieved using the multiple Lyapunov–Krasovskii functional, which reduces the strictness of the controller constraints compared to the switched Lyapunov–Krasovskii functional. However, it imposes a dwell-time limitation on the switching law. By adopting the PDT structure, the dwell-time limitation is reduced compared to common structures. To evaluate the proposed design, it is applied to a water pollution system, and its performance is assessed. The results demonstrate superior performance compared to previous works.

Event‐triggered security control for fuzzy‐model‐based cyber‐physical systems under Denial‐of‐Service attacks and actuator faults

AbstractThis article addresses the issue of event‐triggered security control for nonlinear cyber‐physical systems, characterized by Denial‐of‐Service (DoS) attacks, time‐varying delays, and actuator faults. These complexities are captured using the Takagi–Sugeno fuzzy model. Significantly, this study focuses on DoS attacks targeting the controller‐to‐actuator channel, and subsequently introduces a model for actuator faults that considers the prolonged utilization or degradation of component parts. To optimize communication resource allocation, a model‐independent event‐triggered mechanism is proposed. Utilizing the Lyapunov–Krasovskii function approach, the objective of achieving stochastic stability is transformed into a linear matrix inequalities problem, thus deriving a set of sufficient conditions. The efficiency and validity of the proposed algorithms is demonstrated through simulation results.

Linear Quadratic Optimal Robust Multivariable Proportional-Integral Control Design for a Boiler-Turbine Unit

Abstract Boiler-turbine unit is a highly nonlinear, coupled, and ill-conditioned system. Moreover, the presence of physical constraints, such as actuator magnitude and rate limits, makes the system difficult to control. This paper employs a fixed multivariable proportional-integral (PI) controller of the I–P structure for robust linear quadratic (LQ) compensation of a nonlinear boiler-turbine benchmark. In order to ensure that a single PI controller works for the whole operating region of this nonlinear system, the linearized model of the system is represented as a norm-bounded, time-varying uncertain system. The ranges of the uncertain parameters of this linearized model are determined from different operating points of the nonlinear system. To design the PI controller for the uncertain system, first, it is transformed into a state feedback design for an augmented uncertain system and then the state feedback gains satisfying some LQ performance limit are computed by solving a linear matrix inequality (LMI) problem. As the uncertainty in the feedforward matrix of the linearized model cannot be considered in the above design process, an LMI-based method is developed to check if the designed PI controller performance in H∞ sense is close to the one if the neglected uncertainty is included. The performance of the controller is tested on the nonlinear boiler-turbine unit under several operating conditions and physical constraints. Comparisons are also made with some existing PI controllers, to show the superiority of the proposed robust PI controller.

On the Functional Observer Design With Sampled Measurement

In this paper, a functional observer for continuous systems with sampled measurement is designed for a class of nonlinear systems, operating under unknown inputs, and the existence conditions are derived. The nonlinear function, as a function of full states, is decomposed to Lipschitz term and an unknown one. The stability and boundedness of the estimation dynamics are analyzed and accordingly, the design procedure is formulated as a Linear Matrix Inequality (LMI) problem. Moreover, the design algorithm for some special cases is obtained. The numerical simulations are performed to evaluate the effectiveness of the proposed approach.

Relaxed nonquadratic H∞ robust constrained event-triggered fuzzy controller design for discrete-time nonlinear systems

This paper introduces an [Formula: see text] robust constrained event-triggered fuzzy controller for discrete-time nonlinear systems in the presence of system parameter uncertainties, the parameter uncertainties of the output matrices and the disturbances, simultaneously. To reduce the communication resources and improve the robust performance, the event-triggered mechanism is incorporated with the [Formula: see text] robust fuzzy controller design. Moreover, by employing the fuzzy modeling of the actuator saturation and a nonquadratic Lyapunov function (NQLF), the ultimate bounded stability (UBS) of the closed-loop system is guaranteed subject to the amplitude limitations of the control signal. This means, a new linear matrix inequality (LMI) problem is derived to ensure the robustness and constraints in an offline scheme based on the event-triggered strategy. Besides, to meet the practical problem to further improve the controlled system performance, the control input rate constraint is also considered and a new LMI condition is provided with no online computational burden. Therefore, the computational burden of the proposed controller is greatly reduced with less conservatism. Two simulation examples including a numerical dynamic system and a truck–trailer system are employed to illustrate the less conservatism and computational complexity of the proposed method compared with the existing approaches.

Incipient Fault Detection and Reconstruction Using an Adaptive Sliding-Mode Observer for the Actuators of Fixed-Wing Aircraft

This paper proposes a new method of detection and reconstruction of the incipient fault of fixed-wing actuators based on an adaptive sliding-mode observer. First, a mathematical model of a fixed-wing aircraft is derived under certain assumptions, considering nonlinear terms and system disturbances. Second, by introducing a nonsingular coordinate transformation, the incipient faults are separated from the disturbances. For a subsystem with no disturbances, the Luenberger observer can estimate the incipient fault. For a subsystem with disturbances, the sliding-mode observer is robust against these disturbances. The Lyapunov stability theory guarantees dynamic error convergence and system stability. The evaluation function was designed to realize residual evaluation and threshold judgment. Third, based on the concept of equivalent output injection, an adaptive sliding-mode observer method is proposed to reconstruct actuator faults precisely under the condition of the uncertain system structure. The design steps of the proposed reconstruction method are introduced in the form of a linear matrix inequality problem, which provides an effective method for calculating the design parameters. Finally, the simulation results of the De Havilland DHC-2 “Beaver” aircraft demonstrate the correctness and effectiveness of the proposed method.

Nonlinear feedback switching rule design for sector‐bounded switched systems

AbstractThis article presents a robust switching rule design technique for a class of nonlinear and uncertain switched systems. The approach is based on a state transformation that allows for nonlinear state feedback. The results are given in terms of linear matrix inequality problems and they guarantee global asymptotic stability of the closed‐loop system even if sliding modes occur on any switching surface. The class of systems considered is general enough to include multiple time varying sector‐bounded nonlinearities associated with real‐time changes in the operation point. The potential of the method is illustrated through photovoltaic systems, in which the nonlinear I–V characteristic of the modules is treated as a sector‐bounded function. As the slope of the sectors depend on the environmental conditions, robust sectors containing the nonlinearities for any uncertain uniform operation condition are derived.

Geometric approach to robust stability analysis of Linear Parameter-Varying systems: Computational trade-offs between the exact and the simplex convex hulls

Robust stability is investigated for continuous-time, affine, Linear Parameter-Varying (LPV) models, with bounded variation rates of uncertainty. Towards this end, affine Parameter Dependent Lyapunov Functions (PDLF) are considered to certificate stability in the Lyapunov sense. In the literature, the search for PDLFs amounts to a non conservative Linear Matrix Inequality (LMI) problem, at expense of a factorial growth in its complexity. Remedies to overcome this complexity have been proposed recently, exploiting geometrical aspects of the problem, however, they can be conservative. This paper presents new stability tests that are a trade-off between the exact and the fastest solutions. We offer an analytical procedure to indicate when the proposed tests are prone to reduce conservativeness. Also, a simple procedure is introduced to possibly improve existing and new tests, without impacts on the computation effort. Numerical simulations illustrate the improvements of the proposed strategies.

H2model order reduction of bilinear systems via linear matrix inequality approach

AbstractThis paper proposes an H2‐optimal model order reduction (MOR) method for bilinear systems based on the linear matrix inequality (LMI) approach. In this method, to reduce the computational complexity, at first, a reduced middle‐order approximation of the system is derived based on common bilinear MOR methods. Next, the H2norm of the error system is minimized to obtain the reduced‐order bilinear model. Generalized Lyapunov equations are added to the optimization problem as LMI constraints to guarantee the specification of type II Gramians of the bilinear system to improve accuracy. Besides, two stability conditions are included to the optimization problem as its constraints to preserve stability of reduced‐order bilinear model. One of advantages of the proposed method is the need for only one of the Gramians of controllability or observability. Since the proposed H2‐optimal MOR problem is a polynomial matrix inequality (PMI) problem, an iterative method is used to convert the PMI to the LMI problems and solve the optimization problem. Three bilinear test systems are considered to show the proposed method's efficiency, while its performance is compared with some classical methods. Results show that the proposed methods lead to a more accurate reduced‐order model than other MOR methods.

Overlapping internal boundary control of lane-free automated vehicle traffic

Lane-free vehicle driving has been recently proposed for connected automated vehicles on highways or arterials. Lane-free traffic implies that incremental changes of the road width lead to corresponding incremental changes of traffic flows and capacity. In these conditions, internal boundary control (IBC) was proposed for flexible sharing of the total road width and capacity among the two traffic directions of the highway in real-time in order to maximize the cross-road infrastructure utilization and minimize congestion and delays. Centralized solutions , e.g. LQ (linear quadratic) regulators, requiring information from the whole highway stretch under consideration, have already been proposed, which, however, may be cumbersome for long highways with respect to the offline controller design effort; the extent of real-time communications; and the physical system architecture. This paper investigates two different overlapping decentralized control schemes for IBC in lane-free automated vehicle traffic. The first approach is based on a contractible controller, which is designed and employed in a decomposed way, for each subsystem of an extended highway system. In the second approach, the overall discrete-time LQ control problem is transformed into a linear matrix inequalities (LMI) problem. After selecting the overlapping feedback structure and solving the overall LMI problem, the correspondingly structured gain matrix is obtained, enabling a decentralized overlapping control deployment in real-time. Simulation investigations, involving a realistic highway stretch and demand scenarios, demonstrate that the proposed decentralized regulators are almost as efficient as the centralized control solutions for the problem at hand.

Spatiotemporal adaptive state feedback control of a linear parabolic partial differential equation

AbstractThis article deals with the issue of asymptotic stabilization for a linear parabolic partial differential equation (PDE) with an unknown space‐varying reaction coefficient and multiple local piecewise uniform control. Clearly, the unknown reaction coefficient belongs to a function space. Hence, the fundamental difficulty for such issue lies in the lack of a conceptually simple but effective parameter identification technique in a function space. By the Lyapunov technique combined with a variant of Poincaré‐Wirtinger inequality, an update law is derived for estimate of the unknown reaction coefficient in a function space. Then a spatiotemporal adaptive state feedback control law is constructed such that the estimate of the unknown coefficient is bounded and the closed‐loop PDE is asymptotically stable in the sense of spatial norm if a sufficient condition given in terms of space‐time varying linear matrix inequalities (LMIs) is fulfilled for the estimated coefficient and the control gains. Both analytical and numerical approaches are proposed to construct a feasible solution to the space‐time varying LMI problem. With the aid of the semigroup theory, the well‐posedness and regularity of the closed‐loop PDE is also analyzed. Moreover, two extensions of the proposed adaptive control scheme are discussed: the PDE in ‐D space and the PDE with unknown diffusion and reaction coefficients. Finally, numerical simulation results are presented to support the proposed spatiotemporal adaptive control design.

Optimal constrained integral sliding mode control design for fuzzy-based nonlinear systems

This study introduces a novel H∞ optimal constrained integral sliding mode control (OCISMC) for nonlinear systems due to the matched/unmatched external disturbances based on Takagi–Sugeno (TS) fuzzy models. Based on the Lyapunov function, the appropriate conditions are provided to reach the sliding mode from any initial condition and robustness against the matched disturbances is guaranteed. The optimal nominal part of the proposed OCISMC is designed based on an online optimization problem. To reach robustness against the unmatched disturbances and constrained controller, the H∞ performance and the limitations of the control signal are included in the design procedure of the suggested OCISMC. Furthermore, by considering a non-quadratic Lyapunov function (NQLF), a new linear matrix inequality problem with less conservatism is derived. Finally, two examples are simulated and compared with the existing works to illustrate the capability of the proposed OCISMC.

Observer-Based Robust Fault Predictive Control for Wind Turbine Time-Delay Systems with Sensor and Actuator Faults

This paper presents a novel observer-based robust fault predictive control (OBRFPC) approach for a wind turbine time-delay system subject to constraints, actuator/sensor faults, and external disturbances. The proposed approach is based on an augmented state-space representation that contains state-space variables and estimation errors. The proposed augmented representation is then used to synthesize a robust predictive controller. In addition, an observer is developed and used to estimate both state variables and actuator/sensor faults. To ensure that the proposed approach has disturbance rejection capabilities, the disturbance estimates were merged with the prediction model. In addition, the disturbance rejection capabilities and fault tolerance were insured by formulating the control process as an optimization problem subject to constraints in terms of linear matrix inequalities (LMIs). As a result, the controller gains are acquired by solving an LMI problem to guarantee input-to-state stability in the presence of sensor and actuator faults. A simulation example is conducted on a nonlinear wind turbine (1 MW) model with 3 blades, a horizontal axis, and upwind variable speed subject to actuator/sensor faults in the pitch system. The results demonstrate the ability of the proposed method in dealing with nonlinear systems subject to external disturbances and keeping the control performance acceptable in the presence of actuator/sensor faults.

A Novel Operation Optimization Method Based on Mechanism Analytics for the Quality of Molten Steel in the BOF Steelmaking Process

This paper investigates a dynamic analytics and operation optimization method for the basic oxygen furnace (BOF) steelmaking process, which is a multi-stage process. First, based on reaction kinetics, fluid dynamics and conservation of mass, that a novel discrete-time nonlinear system with process disturbance, time delay and the constrained operation input is established characterizes the dynamics of the quality (the carbon content and the temperature) of molten steel in the BOF steelmaking process. It is difficult and complicated to realize analytics and operation optimization for the BOF steelmaking process by using the established nonlinear system directly, so a surrogate-model-based discrete-time switched system with time delay and actuator saturation is given to describe the multi-stage BOF steelmaking process. Then, sufficient conditions are derived under which the stability is guaranteed and the tracking performance is achieved. The parameters of analytics-based operation optimization algorithm can be obtained by solving linear matrix inequality problems (LMIPs). Finally, the effectiveness of the proposed method is verified by a BOF steelmaking numerical experiment example. Note to Practitioners—This paper deals with the problem of dynamic analytics and operation optimization arising from the BOF steelmaking process. Based on the real-time estimation of the carbon content and the temperature of molten steel, the plan for the operation of oxygen lance and the addition of auxiliary materials is given to produce the qualified molten steel. This paper establishes a system model based on the mechanism, and the model parameters are derived from papers verified by a large amount of data to ensure the correctness of the model. In addition, actuator saturation is applied to describe the phenomenon of restricted operation variables. Through a surrogate model approximation and theoretical analysis, the parameters of dynamic analytics and operation optimization algorithm can be obtained by solving LMIPs, which is a convex optimization problem that is easy to solve. The numerical experiment results show that the proposed method can give operators some desired references.

Functional Observer Design for Parallel Connected Li-Ion Battery: A Descriptor Systems Theory Approach

This letter investigates cell-level state of charge (SOC) estimation in a parallel connected lithium-ion (Li-ion) battery pack under a reduced sensing scenario. The internal states of individual cells within a battery pack are likely to differ due to cell inconsistencies caused by manufacturing tolerances and usage conditions. This creates a need for cell-level estimation in a parallel connected battery pack. The dynamics of parallel connected battery packs require solving differential algebraic equations (DAEs). This letter considers the reduced sensing scenario, measuring only the total current and voltage for estimating individual cell SOCs and local currents in a battery pack. A novel functional observer, designed under milder conditions, is proposed and the observer design is formulated in terms of a feasible linear matrix inequality (LMI) problem. In the simulation, the SOC of two cylindrical Li-ion battery cells are estimated, and the results illustrate a fast convergence and hence the effectiveness of our approach.