Linear Matrix Inequality Optimization Problem Research Articles

Observer‐based control with enlarged domain of attraction for one‐sided Lipschitz systems subject to input saturation

AbstractThis article deals with the problem of observer‐based controller (OBC) design for a class of nonlinear systems in the presence of input saturation. The nonlinearities are assumed to satisfy the one‐sided Lipschitz and the quadratically inner‐bounded conditions. Sufficient conditions for the existence of OBC are derived by Lyapunov stability theory. The designed OBC not only ensures the asymptotic stability of the overall closed‐loop system with a domain of attraction as large as possible, but also guarantees the actuator saturation avoidance. This problem is converted into a linear matrix inequality optimization problem by judicious use of Young's relation for straightforward computation of the OBC gains. Finally, the effectiveness of the proposed design method is demonstrated by simulation results.

Robust [formula omitted] model predictive control for constrained Lipschitz non-linear systems

This paper is concerned with the robust H∞ model predictive control problem for a class of non-linear systems subject to state and input constraints and with norm bounded disturbances. It is well known that the disturbance degrades the control performance remarkably and can cause system instability. As well as, design the H∞ model predictive control based on the non-linear model is much more challenging, because the control problem changes to a non-convex non-linear problem. The main contribution is the introduction of the novel robust controller to stabilize the perturbed Lipschitz non-linear systems. The objective is to minimize the L2 gain between the disturbance input and the controlled output. The non-convex control problem is formulated as a linear matrix inequality optimization problem. The designed controller guarantees the closed-loop asymptotic stability with a prescribed H∞ disturbance attenuation level. In order to reduce conservatism, a sum of squares optimization problem is proposed to obtain the optimal value of Lipschitz coefficient. The proposed algorithm is applied to two non-linear systems, a laboratory tank and a DC/AC converter to evaluate its applicability and effectiveness.

Dynamic Output Feedback MPC of Polytopic Uncertain Systems: Efficient LMI Conditions

We focus on the design of an efficient dynamic output feedback model predictive control (DOFMPC) for linear systems with both polytopic and additive uncertainties. We tackle the problem of bilinear matrix inequality formulation which is often encountered in this type of work. In order to formulate a numerically traceable optimization problem, expressed in terms of linear matrix inequalities (LMIs), we propose a new convex stability condition by the utilization of a relaxation formula. Moreover, other inequality conditions in the optimization problem are tailored so that an efficient LMI optimization problem is formulated. Unlike in previous works, the proposed approach does not require any special structure of system matrices, which results in more flexibility. The effectiveness of the proposed approach is verified through a numerical example.

Average Dwell Time Based Smooth Switching Linear Parameter-Varying Proportional-Integral-Derivative Control for an F-16 Aircraft

Switching control methods have great potential applications in flight control system design. However, nonsmooth control commands at switching instants could lead to performance degradations or bring in safety risks. This paper presents a smooth switching proportional-integral-derivative (PID) control method for linear parameter-varying (LPV) systems to solve this issue. By introducing a transition parameter subspace between any two neighboring LPV subsystems, the switched controller gains are designed to be smoothly varied when the parameter trajectory passes through these subspaces, and the Lyapunov function in the transition subspace is designed independently to make the algorithm less conservatism. The controller synthesis condition is formulated as a linear matrix inequality (LMI) optimization problem using Finsler’s lemma. The concept of average dwell-time (ADT) is employed to analyze the performance of the resulting closed-loop system. The effectiveness of the ADT-based smooth switching control strategy is demonstrated by applying it to the nonlinear longitudinal model of the F-16 aircraft, and the simulation results are compared with the traditional ADT switching control method. The results show that the proposed method can achieve a better transition performance with a simpler control structure.

Pareto-Optimal Pilot Design for Cellular Massive MIMO Systems

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

Robust 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.

Optimizing the performance of the feedback controller for state‐based switching bilinear systems

SummaryThis article is concerned with the design and performance optimization of feedback controllers for state‐based switching bilinear systems (SBLSs), where subsystems take the form of bilinear systems in different state space polyhedra. First, by further dividing the subregions into smaller regions and designing region‐dependent feedback controllers in the resulting regions, the SBLSs can be transformed into corresponding switching linear systems (SLSs). Then, for these SLSs, by imposing contractility conditions on the Lyapunov functions, an upper bound on the infinite horizon quadratic cost can be obtained. Optimizing this upper bound yields the controller design. The optimization problem is formulated as a linear matrix inequalities optimization problem, which can be solved efficiently. Finally, the stability of the close‐loop system under the proposed controller is established step by step through a decreasing overall Lyapunov function.

Low-Latency Robust MPC for CACC under Variable Road Geometry

Abstract Cooperative adaptive cruise control (CACC) has attracted much research attention, due to its great potential in improving traffic throughput, safety and energy efficiency. This paper aims to address the following problems that are rarely investigated in the literature: (i) the time delay caused by online computation of the optimal control action, and (ii) robustly stable CACC under variable road geometry. To this end, a one-step ahead robust model predictive control (MPC) is developed for achieving CACC and lane keeping (LK) of the followers in the platoon, by leveraging vehicle-to-vehicle (V2V) and vehicle-to-infrastructure (V2I) communications. In the proposed design, the current MPC policy is generated one-step ahead during the previous sampling period to avoids the optimization-induced time delay existing in the traditional MPC. LK control is incorporated with CACC to ensure vehicle lateral stability and vehicle following under variable road geometry. The MPC design is formulated as an easily solved linear matrix inequality (LMI) optimization problem with consideration of the control input limits and constraints on platooning errors and lateral displacement. Effectiveness of the proposed MPC and its advantages over the traditional MPC are verified by simulating a vehicle platoon on roads with time-varying bank, curvature and grade.

An iterative approach for the discrete‐time dynamic control of Markov jump linear systems with partial information

SummaryThe , and mixed dynamic output feedback control of Markov jump linear systems in a partial observation context is studied through an iterative approach. By partial information, we mean that neither the state variable x(k) nor the Markov chain θ(k) are available to the controller. Instead, we assume that the controller relies only on an output y(k) and a measured variable coming from a detector that provides the only information of the Markov chain θ(k). To solve the problem, we resort to an iterative method that starts with a state‐feedback controller and solves at each iteration a linear matrix inequality optimization problem. It is shown that this iterative algorithm yields to a nonincreasing sequence of upper bound costs so that it converges to a minimum value. The effectiveness of the iterative procedure is illustrated by means of two examples in which the conservatism between the upper bounds and actual costs is significantly reduced.

A Detector-Based Approach for the Constrained Quadratic Control of Discrete-Time Markovian Jump Linear Systems

This paper considers the quadratic control problem of discrete-time Markov jump linear systems with constraints on the norm of the state and control variables. We assume that the Markov chain parameter is not available, and instead, there is a detector, which emits signals providing information on this parameter. It is desired to derive a feedback linear control using the information provided by this detector in order to stochastically stabilize the closed-loop system, satisfy the constraints whenever the initial conditions belong to an invariant set, and minimize an upper bound for the quadratic cost. We show that a linear matrix inequality (LMI) optimization problem can be formulated in order to obtain a solution for this problem. Two other related problems, one for minimizing the guaranteed quadratic cost considering fixed initial conditions and the other for maximizing an estimate of the domain of an invariant set, can also be formulated using our LMI approach. This paper is concluded with some numerical simulations.

Robust consensus for linear multi-agent systems with structured uncertainties

ABSTRACT This paper addresses a robust consensus problem for linear multi-agent systems subject to structured uncertainty and external disturbances under the leaderless framework. A distributed dynamic output-feedback protocol is proposed, which utilises not only relative output information of neighbouring agents but also relative state information of neighbouring controllers. Through model transformations, the robust consensus control problem of multi-agents network is reduced to a set of independent stabilisation problems for single linear subsystems. For robust consensus, it is shown that the analysis and full state-feedback synthesis conditions for such subsystems can be formulated as linear matrix inequality (LMI) optimisation problems. On the other hand, the synthesis condition for dynamic output-feedback protocol is formulated as non-convex bilinear matrix inequality (BMI) optimisation problem. An iterative LMI algorithm is then presented to solve the resulting BMI optimisation problem. An example of multiple mass-spring-damper systems has been used to demonstrate theoretical results.

Fault Detection of continuous time linear switched systems using combination of Bond Graph method and switching observer

In this paper, a new Fault Detection (FD) scheme based on combination of switching observer and Bond Graph (BG) method for linear continuous time switched systems with Average Dwell Time (ADT) approach is proposed. The proposed scheme is a BG-based two-stage FD system in which the compact state space representation and Global Analytical Redundancy Relations (GARRs) are derived based on the BG model. In the first stage, a switched observer is designed considering disturbance attenuation level, fault sensitivity and rapid fault detection criteria by solving a weighted Linear Matrix Inequality (LMI) optimization problem. Next, a new form of GARRs which is based on output estimation error of the observer and is called Error-based Global Analytical Redundancy Relations (EGARRs) is developed to combine the observer and BG method. The output estimation errors from the observer, which are adequately sensitive to faults and simultaneously the effects of disturbances are attenuated therein, are given to the EGARRs to generate the residuals of the FD system. The proposed method may be used for fault detection of switched linear systems with ADT based on the BG model of the system. Finally, two case studies including a two-tank system and a buck converter-driven DC motor are considered to show the efficiency and real-time implementation of the proposed method.

Robust Guaranteed Cost Sampled-Data Fuzzy Control for Uncertain Nonlinear Time-Delay Systems

This paper presents a robust guaranteed cost sampled-data fuzzy control (GCSDFC) design for Takagi–Sugeno fuzzy systems with parametric uncertainties and time-delay. Initially, a robust guaranteed cost sampled-data fuzzy controller is developed to stabilize exponentially the closed-loop fuzzy system while providing an upper bound for the quadratic cost function. In order to make full information of the actual sampling pattern, a novel time-dependent Lyapunov functional is subsequently constructed to derive the condition for the existence of the proposed controller which is given in terms of linear matrix inequalities (LMIs). Then, to minimize the upper bound of the cost function, a suboptimal robust GCSDFC problem can be formed as an LMI optimization problem. Finally, two examples are given to illustrate the effectiveness of the proposed method.

Observer-based H∞ control for second-order neutral systems with actuator saturation and the application to mechanical system

The observer-based H∞ control problem for second-order neutral systems with actuator saturation is considered. First, a sufficient delay-dependent condition is proposed using Wirtinger-based integral inequality, under which such type neutral system with time-delay and actuator saturation is stable and satisfies H∞ performance. Then, based on the derived condition, the design method of observer-based controller is developed using linear matrix inequalities (LMIs) optimization problem. Finally, a numerical example is provided to illustrate the application of the second-order mechanical systems and the effectiveness of the theoretical results.

Non-fragile sampled-data control of network systems subject to time-delay

ABSTRACTThis paper studies the consensus problem of non-linear network systems subject to time-delay through aperiodic sampled-data control, which is more desirable and flexible than periodic sampling control. By receiving the relative information of the neighboring nodes, each node can obtain control input signals at the discrete sampling instants. A new stability criterion is established for network systems via using a time-dependent Lyapunov functional and the free-weighting matrix method. The maximum upper bound of sampling interval can be achieved by solving Linear matrix inequality optimization problem. Under the basic work of a non-fragile sampled-data controller, network systems with time-delay are asymptotically stable. In the end, two numerical simulation examples can be provided to illustrate the validity of the application of the proposed algorithm.

Stability analysis and design of model predictive reset control for nonlinear time-delay systems with application to a two-stage chemical reactor system

This paper presents a systematic Lyapunov-based approach for reset control of nonlinear time-delay system. The reset controller consists of a base controller and a reset law that enforces resets to the controller states. The reset actions are triggered if predefined reset conditions are satisfied and reset law is used to determine reset values of the controller states. A model predictive strategy is proposed to design the reset law by solving a linear matrix inequality optimization problem. Based on Lyapunov–Krasovskii approach, the reset controller with guaranteed asymptotical stability is synthesized. Finally, we apply the developed method to the control design of a two-stage chemical reactor with delayed recycle streams, and the simulation results verify the effectiveness of the main results.

Exponential Hidden Markov Models for <inline-formula> <tex-math notation="LaTeX">${H} _\infty$ </tex-math> </inline-formula> Control of Jumping Systems

This letter addresses the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> -control problem for continuous-time Markov jump linear systems where the current mode of operation is estimated by an imperfect detector. Different from existing approaches, the joint jump process is defined in this letter by a continuous-time hidden Markov model, with the hidden process representing the dynamics of the real system while the emitted signal is represented by the information emitted from the detector. We consider static output feedback controllers which depend only on the information emitted from the detector and that stabilize the closed loop system in the mean-square sense. We develop a linear matrix inequality optimization problem which minimizes an upper bound for the H∞ norm of the closed-loop system in order to design our static output feedback controller. This letter is concluded with a numerical example.

Improved receding horizon H∞ temperature spectrum tracking control for Debye media in microwave heating process

This paper considers the problem of tracking control for the coupled thermal-electromagnetic model with temperature-dependent permittivity in microwave heating process, where the incident electrical field is subject to the asymmetrical saturation. Due to the nonhomogeneous Neumann boundary condition and inherent infinite-dimensional characteristics for microwave heating process, it is difficult for us to readily design and implement tracking controller. With the help of improved spectral Galerkin method, the traditional partial differential equation (PDE) model is transformed into the linear finite-dimensional ordinary differential equation (ODE) model which can be subsequently used for controller design. Then, the receding horizon H∞ control strategy is improved to stabilize the derived uncertain ODE error model with asymmetrical input saturation and bounded external disturbance. The outcome of improved algorithm is formulated as a linear matrix inequalities (LMIs) optimization problem, which can be on-line updated by applying the finite-different time-domain (FDTD) method. Finally, the improved control strategy is successfully applied in one-dimensional cavity model for tracking temperature spectrum.

Interval Fuzzy Model for Robust Aircraft IMU Sensors Fault Detection.

This paper proposes a data-based approach for a robust fault detection (FD) of the inertial measurement unit (IMU) sensors of an aircraft. Fuzzy interval models (FIMs) have been introduced for coping with the significant modeling uncertainties caused by poorly modeled aerodynamics. The proposed FIMs are used to compute robust prediction intervals for the measurements provided by the IMU sensors. Specifically, a nonlinear neural network (NN) model is used as central prediction of the sensor response while the uncertainty around the central estimation is captured by the FIM model. The uncertainty has been also modelled using a conventional linear Interval Model (IM) approach; this allows a quantitative evaluation of the benefits provided by the FIM approach. The identification of the IMs and of the FIMs was formalized as a linear matrix inequality (LMI) optimization problem using as cost function the (mean) amplitude of the prediction interval and as optimization variables the parameters defining the amplitudes of the intervals of the IMs and FIMs. Based on the identified models, FD validation tests have been successfully conducted using actual flight data of a P92 Tecnam aircraft by artificially injecting additive fault signals on the fault free IMU readings.

$$H_{\infty }$$ H ∞ memory feedback control for uncertain singular Markov jump systems with time-varying delay and input saturation

This paper concerns with the $$H_{\infty }$$ robust exponential stability and memory state feedback stabilization of singular Markov jump systems with time-delay and input saturation. Linear matrix inequality conditions are given to guarantee regularity, impulse-freeness and exponential stability for the singular Markov jump system. Moreover, sufficient conditions are presented to ensure the $$H_{\infty }$$ disturbance attenuation level, and the design method of memory feedback controller is developed by solving linear matrix inequalities optimization problem without any decompositions of system matrices and equivalent transformation. Furthermore, the function in the proof procedure belongs to multiple Lyapunov-like functions whose advantage lies in their flexibility. Finally, numerical examples are employed to verify the effectiveness of the proposed methods and to illustrate the significant improvement on the conservativeness of some reported results in the literature.