Linear Matrix Inequality Problem Research Articles

A Robust Fault-Tolerant Predictive Control for Discrete-Time Linear Systems Subject to Sensor and Actuator Faults.

In this paper, a robust fault-tolerant model predictive control (RFTPC) approach is proposed for discrete-time linear systems subject to sensor and actuator faults, disturbances, and input constraints. In this approach, a virtual observer is first considered to improve the observation accuracy as well as reduce fault effects on the system. Then, a real observer is established based on the proposed virtual observer, since the performance of virtual observers is limited due to the presence of unmeasurable information in the system. Based on the estimated information obtained by the observers, a robust fault-tolerant model predictive control is synthesized and used to control discrete-time systems subject to sensor and actuator faults, disturbances, and input constraints. Additionally, an optimized cost function is employed in the RFTPC design to guarantee robust stability as well as the rejection of bounded disturbances for the discrete-time system with sensor and actuator faults. Furthermore, a linear matrix inequality (LMI) approach is used to propose sufficient stability conditions that ensure and guarantee the robust stability of the whole closed-loop system composed of the states and the estimation error of the system dynamics. As a result, the entire control problem is formulated as an LMI problem, and the gains of both observer and robust fault-tolerant model predictive controller are obtained by solving the linear matrix inequalities (LMIs). Finally, the efficiency of the proposed RFTPC controller is tested by simulating a numerical example where the simulation results demonstrate the applicability of the proposed method in dealing with linear systems subject to faults in both actuators and sensors.

H ∞ bounded real lemma for singular fractional-order systems

ABSTRACT This paper derives the bounded real lemma corresponding to norm of the singular fractional-order system (SFOS). The bounded real lemma established for both cases of and has no conservatism and it turns the original complex computation of norm into linear matrix inequality (LMI) problem, which can be performed in a computationally efficient fashion. In addition, it can test the system stability automatically, which is critical for system analysis. From this advantage, the synthesis problem of control for the SFOS is addressed in the form of LMI. The simulation study is performed to validate the effectiveness and efficiency of the proposed methods.

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.

A unifying framework for form-finding and topology-finding of tensegrity structures

This paper presents a unifying framework for the form-finding and topology-finding of tensegrity structures. The novel computational framework is based on rank-constrained linear matrix inequalities. For form-finding, given the topology (i.e., member connectivities), the determination of the member force densities is formulated into a linear matrix inequality (LMI) problem with a constraint on the rank of the force density matrix. The positive semi-definiteness and rank deficiency condition of the force density matrix are well managed by the rank-constrained LMI-based formulation. A Newton-like algorithm is employed to solve the rank-constrained LMI problem. Two methods, named direct method and indirect method, are proposed to determine the nodal coordinates once the force densities have been obtained. For topology-finding, given the geometry (i.e., nodal coordinates), the determination of the topology is also formulated into an LMI problem with a constraint on the rank of the tangent stiffness matrix. Numerical examples demonstrate that different types of form-finding problems (such as tensegrity structures with single and with multiple self-stress states, symmetric and irregular tensegrity structures) can be uniformly and efficiently solved by the proposed approach. Furthermore, three well-known tensegrity structures are reproduced to verify the effectiveness of the proposed formulation on the topology-finding of tensegrity structures.

Parameter-dependent actuator fault estimation for vehicle active suspension systems based on RBFNN

The actuator fault estimation (FE) problem is addressed in this study for the quarter-car active suspension system (ASS) with consideration of the sprung mass variation. Firstly, the ASS is modeled as a parameter-dependent system with actuator fault and external disturbance input. Then, a parameter-dependent FE observer is designed by using the radial basis function neural network (RBFNN) to approximate the actuator fault. In addition, the design conditions are turned into a linear matrix inequality (LMI) problem which can be easily solved with the aid of LMI toolbox. Finally, simulation and comparison results are given to show the accuracy and rapidity of the proposed FE method, as well as good adaptability against the sprung mass variation. Moreover, a simple FE-based active fault-tolerant control (AFTC) strategy is provided to further demonstrate the effectiveness and applicability of the proposed FE method.

A RobustH∞ Fault Tolerant Controller for Uncertain Systems Described by Linear Fractional Transformation Model

In this article, a robust H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> fault tolerant control law is addressed for a class of the uncertain dynamical systems represented via linear fractional transformation. To this objective, a state-feedback controller law is utilized for achieving the control objective. Thus, a linear matrix inequality based performance condition would be derived to guarantee that the disturbance suppression is accomplished in the uncertain system. Hence, the gains of the robust H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> controller would be suitably determined by checking the feasibility of such a linear matrix inequality problem. The proposed control technique is numerically simulated in two dynamical uncertain systems (i.e., a typical control system and a mechanical robotic arm). Considering the disturbance rejections and transient responses, the results illustrate the efficiency of the recommended robust technique compared with the existing control methods.

Robust Stochastic Observer-Based Attack-Tolerant Missile Guidance Control Design Under Malicious Actuator and Sensor Attacks

In this study, the robust H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> stochastic observer-based attack-tolerant guidance control strategy is designed for the nonlinear stochastic missile guidance control system under the external disturbance and measurement noise as well as actuator attack signal and sensor attack signal. To simplify the attack signal estimation, a novel nonsingular smoothed dynamic model is introduced to efficiently describe the actuator and sensor attack signals. Consequently, the state/attack signal estimation can be easily achieved by using conventional Luenberger observer. Next, to attenuate the effect of external disturbance, measurement noise and approximation errors of attack signal on the missile guidance control system, the robust H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> attack-tolerant guidance control performance is considered and the design condition is derived in terms of nonlinear Hamilton-Jacobi inequality (HJI) constrained problem. Since HJI cannot be easily solved analytically or numerically, the Takagi-Sugeno (T-S) fuzzy modelling method is utilized to facilitate the robust H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> attack-tolerant guidance control strategy design. Thereafter, the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> observer-based attack-tolerant control design problem is transformed into linear matrix inequalities problem (LMIP) which can be solved very efficiently by using the convex optimization techniques. Simulation example, with the comparison between the proposed method and conventional robust missile guidance strategy, is given to illustrate the design procedures and validate the performance of the proposed method.

Computation of robust control invariant sets with predefined complexity for uncertain systems

AbstractThis paper presents an algorithm that computes polytopic robust control‐invariant (RCI) sets for rationally parameter‐dependent systems with additive disturbances. By means of novel linear matrix inequalities (LMI) feasibility conditions for invariance along with a newly developed method for volume maximization, an iterative algorithm is proposed for the computation of RCI sets with maximized volumes. The obtained RCI sets are symmetric around the origin by construction and have a user‐defined level of complexity. Unlike many similar approaches, the proposed algorithm directly computes the RCI sets without requiring control inputs to be in a specific feedback form. In fact, a specific control input is obtained from the LMI problem for each extreme point of the RCI set. The outcomes of the proposed algorithm can be used to construct a piecewise‐affine controller based on offline computations.

Tensor product model transformation‐based control for fractional‐order biological pest control systems

AbstractAs environmental pollution and safety of human and creatures have been an issue of concern, pest management based on nonchemical use is preferable. Biological pest control can satisfy the regulation of pest population without causing environmental problems. Moreover, representation of dynamical systems in the form of fractional‐order dynamic models can explain several phenomena better than the integer‐order dynamic models can. The aim of this study is to determine the biological pest control policy for the fractional‐order pest control systems, which is a complex nonlinear multiple input and multiple output control problem. Here, the system is represented in the form of a fractional‐order Lotka–Volterra model. In order to avoid complexity in the nonlinear feedback controller design process, the tensor product (TP) model transformation was applied to automatically transform the biological pest control system to the TP polytopic model which allows the designer to synthesize the controller under linear framework. Moreover, the design process can be easily carried out by solving an linear matrix inequality (LMI) problem, which is very well applicable to high‐dimensional multiple input and multiple output control problems. Simulation studies were conducted to demonstrate the ease of the TP‐based design process. The simulation results showed that the designed control policy effectively manipulates the pest populations of the fractional‐order Lotka–Volterra model to the desired level which is economically viable.

Supervised-distributed control with joint performance and communication optimisation

This paper is concerned with the control of systems composed of multiple coupled subsystems. In such architectures, communication between different local controllers is desired in order to achieve a better overall control performance. Any resultant improvement in control performance needs, however, to be significant enough to warrant the additional design complexity and higher energy consumption and costs associated with introducing communication channels between controllers. A practical distributed control design aims, therefore, to achieve an acceptable balance between minimising the use of communication between controllers and maximising the system-wide performance. In this article, a new approach to the problem of synthesising stabilising distributed control laws for discrete-time linear systems that balances performance and communication is presented. The approach employs a supervisory agent that, periodically albeit not necessarily at every sampling instant, solves an optimisation problem in order to synthesise a stabilising state feedback control law for the system. The online optimisation problem, which maximises sparsity of the control law while minimising an infinite-horizon performance cost, is formulated as a bilinear matrix inequality (BMI) problem; subsequently, it is then relaxed to a linear matrix inequality (LMI) problem, and (i) convergence to a solution as well as (ii) that early termination guarantees a feasible (but suboptimal) control law are proved. Stability of the closed-loop system under what is a switched control law is guaranteed by the inclusion of dwell-time constraints in the LMI problem. Finally, the efficacy of the approach is demonstrated through numerical simulation examples.

Fault estimation for continuous‐time non‐linear switched systems with time‐varying delay based on intermediate estimator

In this work, the fault estimation (FE) problem is studied for continuous-time non-linear switched systems with time-varying delay over fading channel under average dwell time switching. A novel FE method based on the intermediate estimator is used to eliminate the observer matching constraint condition, where an intermediate variable is introduced to estimate the state and fault simultaneously. Meanwhile, the fading channel of data transmission is modelled as Bernoulli distributed white sequences. Furthermore, because the observer matching conditions are eliminated by the intermediate estimator, both the wirtinger integral inequality and an advanced reciprocal convex combination method are utilised to provide a tighter estimation on continuous derivative of certain Lyapunov–Krasovskii function over fading channel, leading to a less conservative result. Moreover, the solution of the parameters for intermediate estimator is characterised in the feasibility of linear matrix inequality problem. Finally, two numerical examples indicate the proposed technique is valid and superior.

A Framework for Solving Chance-Constrained Linear Matrix Inequality Programs

We propose a novel partial sample average approximation (PSAA) framework to solve the two main types of chance-constrained linear matrix inequality (CCLMI) problems: CCLMI with random technology matrix and CCLMI with random right-hand side. We propose a series of computationally tractable PSAA-based approximations for CCLMI problems, analyze their properties, and derive sufficient conditions that ensure convexity for the two most popular—normal and uniform—continuous distributions. We derive several semidefinite programming PSAA reformulations efficiently solved by off-the-shelf solvers and design a sequential convex approximation method for the PSAA formulations containing bilinear matrix inequalities. The proposed methods can be generalized to other continuous random variables whose cumulative distribution function can be easily computed. We carry out a comprehensive numerical study on three practical CCLMI problems: robust truss topology design, calibration, and robust control. The tests attest to the superiority of the PSAA reformulation and algorithmic framework over the scenario and sample average approximation methods. Summary of Contribution: In line with the mission and scope of IJOC, we study an important type of optimization problems, chance-constrained linear matrix inequality (CCLMI) problems, which require stochastic linear matrix inequality (LMI) constraints to be satisfied with high probability. To solve CCLMI problems, we propose a novel partial sample average approximation (PSAA) framework: (i) develop a series of computationally tractable PSAA-based approximations for CCLMI problems, (ii) analyze their properties, (iii) derive sufficient conditions ensuring convexity, and (iv) design a sequential convex approximation method. We evaluate our proposed method via a comprehensive numerical study on three practical CCLMI problems. The tests attest the superiority of the PSAA reformulation and algorithmic framework over standard benchmarks.

Reduction of SISO H-infinity output feedback control problem

We consider the linear matrix inequality (LMI) problem of H∞ output feedback control problem for a generalized plant whose control input, measured output, disturbance input, and controlled output are scalar. We provide an explicit form of the optimal value. This form is the unification of some results in the literature of H∞ performance limitation analysis. To obtain the form of the optimal value, we focus on the non-uniqueness of perpendicular matrices, which appear in the LMI problem. We use the null vectors of invariant zeros associated with the dynamical system for the expression of the perpendicular matrices. This expression enables us to reduce and simplify the LMI problem. Our approach uses some well-known fundamental tools, e.g., the Schur complement, Lyapunov equation, Sylvester equation, and matrix completion. We use these techniques for the simplification of the LMI problem. Also, we investigate the structure of dual feasible solutions and reduce the size of the dual. This reduction is called a facial reduction in the literature of convex optimization.

Robust model predictive control for nonlinear parameter varying systems without computational delay

SummaryThis article proposes a one‐step ahead robust model predictive control (MPC) for discrete‐time Lipschitz nonlinear parameter varying (NLPV) systems subject to disturbances. Within the proposed design framework, the optimization that generates the MPC policy to be implemented at next time instant is executed in advance during the current sampling period based on future state prediction. This new feature allows avoidance of the online computational delay existing in the traditional MPC settings and improves the control performance. The proposed MPC is proved to be recursively feasible with the guarantee for robust closed‐loop system stability and satisfaction of input and output constraints. A tractable linear matrix inequality (LMI) optimization problem is formulated to compute the control gains at each time instant. The computational complexity of the obtained LMI problem is also analyzed. The one‐step ahead robust MPC is further developed to cover discrete‐time Lipschitz NLPV systems with disturbance compensation. Efficacy and performance improvement of the design are demonstrated through a numerical example and an application to adaptive cooperative cruise control for automated vehicles under variable road geometry.

Dynamic output feedback control for continuous-time Markov jump linear systems with hidden Markov models

In this work, we study the design of dynamic output feedback controllers for continuous-time Markov jump linear systems in the worst-case scenario of partial observation of both, the Markov chain and the state, through an iterative method. We assume that the controller has only access to the output of a fault-detection and isolation device and present design conditions for the , , and mixed controller that switches according only to the detector. As a by-product of our formulation, we are also able to design mode-dependent and independent controllers, and briefly discuss the design of filters depending on the detector. The results are given in terms of bilinear matrix inequalities that allow for an iterative separation procedure method in which in each step, we solve a linear matrix inequality problem. We also present an illustrative example in the context of systems subject to faults.

Low-conservative robust composite nonlinear feedback control for singular time-delay systems

A low-conservative composite nonlinear feedback controller is proposed for singular time-delay systems with time-varying delay. The proposed composite nonlinear feedback controller not only improves the transient responses of the closed-loop system but it also has less conservatism than other composite nonlinear feedback controllers. The gain of the linear part of the composite nonlinear feedback controller is obtained by precise mathematical calculation to depend not only on the upper bound of the delay but also on the delay range and rate of its changes. More advantages of the proposed composite nonlinear feedback controller are its accurate operation in the presence of actuator saturation, model uncertainties, and system singularities. The linear and nonlinear parts of the proposed controller are designed by solving a linear matrix inequality problem confirmed through a theorem using Lyapunov stability analysis. The theoretical achievements are endorsed by computer simulation through numerical and practical examples.

Characterization of the dual problem of linear matrix inequality for H-infinity output feedback control problem via facial reduction

This paper deals with the minimization of H_infty output feedback control. This minimization can be formulated as a linear matrix inequality (LMI) problem via a result of Iwasaki and Skelton 1994. The strict feasibility of the dual problem of such an LMI problem is a valuable property to guarantee the existence of an optimal solution of the LMI problem. If this property fails, then the LMI problem may not have any optimal solutions. Even if one can compute parameters of controllers from a computed solution of the LMI problem, then the computed H_infty norm may be very sensitive to a small change of parameters in the controller. In other words, the non-strict feasibility of the dual tells us that the considered design problem may be poorly formulated. We reveal that the strict feasibility of the dual is closely related to invariant zeros of the given generalized plant. The facial reduction is useful in analyzing the relationship. The facial reduction is an iterative algorithm to convert a non-strictly feasible problem into a strictly feasible one. We also show that facial reduction spends only one iteration for so-called regular H_infty output feedback control. In particular, we can obtain a strictly feasible problem by using null vectors associated with some invariant zeros. This reduction is more straightforward than the direct application of facial reduction.

Nonlinear control design using Takagi-Sugeno fuzzy applied to under-actuated visual servo system

This paper presents the Takagi-Sugeno (TS) fuzzy control design for nonlinear stabilization and tracking control of a ball on plate system. To deal with the plant nonlinearity and the fuzzy convergence issue, we formulate the parallel distributed compensator (PDC) TS fuzzy model to characterize the global behaviour of the nonlinear system and synthesize a feasible control framework using a velocity compensation scheme. The nonlinear dynamics of the ball on plate system is obtained using the Euler-Lagrangian energy based approach. To identify the moving objects in the video stream, a background subtraction algorithm using thresholding technique is formulated. Moreover, the stability analysis of the TS fuzzy control is reduced to linear matrix inequality (LMI) problem and solved using the Lyapunov direct method. The potential benefits of the proposed control structure for real time test cases are experimentally assessed using hardware in loop (HIL) testing on a ball on plate system. Experimental results substantiate that the TS fuzzy scheme can significantly improve not only the tracking performance but also the robustness of the closed loop system.

Induced [formula omitted]-gain computation for rational LPV systems using Finsler’s lemma and minimal generators

This paper proposes a novel method to compute an upper bound on the induced L2-gain for a linear parameter varying (LPV) system with rational parameter dependence. The proposed method relies on a standard dissipation inequality condition. The storage function is a quadratic function of the state and a rational function of the parameters. The specific parameter dependence is restricted to involve (fixed) rational functions and an affine term with free decision variables. Finsler’s lemma and affine annihilators are used to formulate sufficient linear matrix inequality (LMI) conditions for the dissipativity relation. The dimension and conservatism of the resulting LMI problem are reduced by the joint application of minimal generators and maximal annihilators. An LPV model of a pendulum–cart system is used to demonstrate the proposed method and compare it to existing techniques in the literature.

Gain‐scheduled controller design for linear parameter varying systems subject to pole assignment

SummaryThis paper addresses a pole‐assignment control problem for discrete‐time linear parameter varying (LPV) systems. Based on LPV modeling approach, time‐varying systems can be mathematically described via combining several linear systems and a specific weighting function. For the LPV systems, gain‐scheduled (GS) control scheme is applied to deal with stabilization problem subject to pole‐assignment constraint. Moreover, two cases of Lyapunov function are respectively employed to derive some sufficient conditions which belong to linear matrix inequality (LMI) problems. Solving those derived conditions, the corresponding GS scheduled controller can be designed such that the asymptotical stability and pole assignment of LPV systems are guaranteed. Finally, a controller problem of truck‐trailer system is used to demonstrate the applicability and effectiveness of the proposed design methods.