Bilinear Matrix Inequality Constraints Research Articles

Dual-Mode Robust Fuzzy Model Predictive Control of Time-Varying Delayed Uncertain Nonlinear Systems With Perturbations

For time-varying delayed nonlinear systems with parameter uncertainties and persistent disturbances, an online and an offline robust fuzzy model predictive control (RFMPC) algorithms are proposed in this paper. Both methods guarantee the input-to-state stability (ISS) of the system. Furthermore, a novel alternative optimization (AOP) approach and a dual-mode OP/AOP strategy are proposed to prevent the performance deterioration of the online optimal control (OP) approach due to the challenges in addressing the bilinear matrix inequality (BMI) constraints. With the established AOP and dual-mode OP/AOP techniques, the optimization problem constrained by BMIs is transformed into convex, and a significantly more precise approximation of the ellipsoidal minimal robust positively invariant (EmRPI) set can be calculated. Besides, the system can eventually converge into a more compact ellipsoidal set. These two alternative optimization methods can be easily extended to various nonlinear systems. A numerical example and a continuous-time stirring tank (CSTR) example are provided to validate the effectiveness and advantages of the established methodologies.

A mesh‐based partitioning algorithm for decreasing conservatism in solving bilinear matrix inequality problems

SummaryIn this paper, a mesh‐based partitioning algorithm (MBPA) to solve bilinear matrix inequality (BMI) problems is proposed, where the main idea is to divide the solution space of a non‐convex BMI problem into smaller parts in the form of Simplexes through meshing the space. In each simplex, only the concave part of BMI constraints is approximated by piecewise affine matrix (PAM), which eventually leads to a convex sub‐problem. After that, the feasible solution can be easily obtained in each simplex through the linear matrix inequality (LMI) optimization. Although the proposed MBPA exploits PAM approximation, it relaxes several restrictive conditions used in other methods, so, it can be used as an effective approach for solving BMI problems. Various numerical examples on systems from COMPleib library to design static output feedback controller are provided to validate the proposed MBPA. Simulation results reveal the satisfactory performance of MBPA in several examined problems and provide acceptable performance compared to other existing BMI solution algorithms. Comparing the percentage of superiority of MBPA with some existing prominent BMI algorithms in several control problems including spectral abscissa optimization problems, and optimization problems for the closed‐loop systems of COMPleib show that the proposed algorithm is more successful than the previous ones.

Robust edge weight synthesis for LPV multi-agent systems with integral quadratic constraints

This work implements a non-linear programming method to synthesize edge weights of an adjacency matrix for a linear parameter varying multi-agent system using bilinear matrix inequalities, which suffer uncertainties. First, convex–concave decompositions are used on the bilinear matrix inequality constraints for nominal H∞ synthesis. Then this method is improved to consider uncertainties using integral quadratic constraints with time domain representations. Agents composing the multi-agent system are depicted as longitudinal dynamics of F16 Vista aircraft at different operating conditions, which share their output information to achieve consensus. Topology of the multi-agent system is predefined and edge weights are defined as functions of the decision variable.

Constrained Reference Tracking via Structured Input–Output Finite-Time Stability

In this article, the control approach based on structured input–output finite-time stability (IO-FTS) is extended to tackle the reference tracking problem. IO-FTS was originally introduced to deal with the disturbance rejection problem. By applying the finite-time stability control to a properly augmented system, we show that it is possible to enforce a set of specific requirements on the response of the closed-loop system during the transients, taking also into account saturation constraints on the actuators. Adding IO-FTS constraints to classic-state-feedback control law leads to a feasibility problem with bilinear matrix inequality (BMI) constraints. Herein, we show how the original BMI problem can be relaxed to a linear matrix inequality (LMI) one, which comes at a price of more conservatism, but turns out to be computationally more efficient. To prove the effectiveness of the proposed approach, we consider the case of the longitudinal control of a missile.

Positivity-Preserving H∞ Model Reduction for Discrete-Time Positive Systems via a Successive Convex Optimization Algorithm

This paper considers the positivity-preserving model reduction for discrete-time positive systems. Given a stable high-order positive system, we aim to find a reduced-order model such that the approximation error is minimized within a prescribed H∞ performance and positivity is preserved. Regarding the bounded real lemma, the sufficient and necessary condition for the existence of a reduced-order model is established in terms of bilinear matrix inequality and convex semi-definite constraint, which ensures that the reduced-order system is positive and the resulted error system is stable and has an H∞ performance level. Based on the inner-approximation strategy, we approximate the bilinear constraints with convex ones, under which an iterative procedure is provided to calculate the desired reduced-order model. Finally, an example is provided to demonstrate the effectiveness and potential benefits of the presented results.

System norm regularization methods for Koopman operator approximation

Approximating the Koopman operator from data is numerically challenging when many lifting functions are considered. Even low-dimensional systems can yield unstable or ill-conditioned results in a high-dimensional lifted space. In this paper, Extended Dynamic Mode Decomposition (DMD) and DMD with control, two methods for approximating the Koopman operator, are reformulated as convex optimization problems with linear matrix inequality constraints. Asymptotic stability constraints and system norm regularizers are then incorporated as methods to improve the numerical conditioning of the Koopman operator. Specifically, the H ∞ norm is used to penalize the input–output gain of the Koopman system. Weighting functions are then applied to penalize the system gain at specific frequencies. These constraints and regularizers introduce bilinear matrix inequality constraints to the regression problem, which are handled by solving a sequence of convex optimization problems. Experimental results using data from an aircraft fatigue structural test rig and a soft robot arm highlight the advantages of the proposed regression methods.

Average Dwell-Time Minimization of Switched Systems via Sequential Convex Programming

This work finds a lower bound on the average dwell-time (ADT) of switching signals such that a continuous-time, graph-based, switched system is globally asymptotically stable, input-to-state stable, or integral input-to-state stable. We first formulate the lower bound on the ADT as a nonconvex optimization problem with bilinear matrix inequality constraints. Because this formulation is independent of the choice of Lyapunov functions, its solution gives a less conservative lower bound than previous Lyapunov-function-based approaches. We then design a numerical iterative algorithm to solve the optimization based on sequential convex programming with a convex-concave decomposition of the constraints. We analyze the convergence properties of the proposed algorithm, establishing the monotonic evolution of the estimates of the average dwell-time lower bound. Finally, we demonstrate the benefits of the proposed approach in two examples and compare it against other baseline methods.

Optimal dynamic output feedback control of Lipschitz nonlinear systems under input saturation

In this study, the problem of optimal guaranteed cost control (OGCC) for nonlinear systems under input saturation is investigated. The purpose is to design a dynamic output feedback controller such that the closed-loop system is asymptotically stable and the upper bound of the cost function is minimized. Moreover, the designed controller ensures that the control signals do not exceed their permissible values. This leads to an optimization problem with bilinear matrix inequality (BMI) constraints. The BMI conditions are converted into the linear matrix inequality conditions by using some technical lemmas for straightforward computation of the controller matrices. The simulation results show the effectiveness and advantages of the proposed theoretical results.

A successive convex optimization method for bilinear matrix inequality problems and its application to static output‐feedback control

AbstractThis article explores a successive convex optimization method for solving a class of nonconvex programming problems subject to bilinear matrix inequality (BMI) constraints. In particular, many control issues can be boiled down to BMI problems, which are typically NP‐hard. To get out of the predicament, we propose a more relaxed feasible set to approximate the original one, based on which a local optimization algorithm is developed and its convergence is analyzed. As a case of application, we consider static output‐feedback control for uncertain systems with disturbances in restricted frequency intervals. In order to strengthen the disturbance‐rejection capability over the given frequency range, we establish sufficient and necessary analysis conditions via the generalized Kalman‐Yakubovich‐Popov lemma, under which the homogeneous polynomially parameter‐dependent technique is adopted to facilitate the design. Finally, several examples are given to demonstrate the efficiency of the results.

A Novel Iterative Linear Matrix Inequality Design Procedure for Passive Inter-Substructure Vibration Control

In vibration control of compound structures, inter-substructure damper (ISSD) systems exploit the out-of-phase response of different substructures to dissipate the kinetic vibrational energy by means of inter-substructure damping links. For seismic protection of multistory buildings, distributed sets of interstory fluid viscous dampers (FVDs) are ISSD systems of particular interest. The connections between distributed FVD systems and decentralized static output-feedback control allow using advanced controller-design methodologies to obtain passive ISSD systems with high-performance characteristics. A major issue of that approach is the computational difficulties associated to the numerical solution of optimization problems with structured bilinear matrix inequality constraints. In this work, we present a novel iterative linear matrix inequality procedure that can be applied to obtain enhanced suboptimal solutions for that kind of optimization problems. To demonstrate the effectiveness of the proposed methodology, we design a system of supplementary interstory FVDs for the seismic protection of a five-story building by synthesizing a decentralized static velocity-feedback H∞ controller. In the performance assessment, we compare the frequency-domain and time-domain responses of the designed FVD system with the behavior of the optimal static state-feedback H∞ controller. The obtained results indicate that the proposed approach allows designing passive ISSD systems that are capable to match the level of performance attained by optimal state-feedback active controllers.

A Stabilization Framework for the Output Regulation of Rational Nonlinear Systems

A systematic stabilization approach is provided for systems whose regulation error dynamics is subject to rational nonlinearities given prior knowledge of the system zero-error steady-state condition and a proper internal model. The error dynamics is cast in a differential-algebraic form so as to address the synthesis of controller parameters by a numerical optimization problem subject to bilinear matrix inequality constraints. A particular case is also explored where the resulting constraints are linear matrix inequalities.

Lower bound for the cost of connecting tree with given vertex degree sequence

Abstract The optimal connecting network problem generalizes many models of structure optimization known from the literature, including communication and transport network topology design, graph cut and graph clustering, etc. For the case of connecting trees with the given sequence of vertex degrees the cost of the optimal tree is shown to be bounded from below by the solution of a semidefinite optimization program with bilinear matrix inequality constraints, which is reduced to the solution of a series of convex programs with linear matrix inequality constraints. The proposed lower-bound estimate is used to construct several heuristic algorithms and to evaluate their quality on a variety of generated and real-life datasets.

A quadratic matrix inequality based PID controller design for LPV systems

This paper develops a robust gain-scheduled proportional–integral–derivative (PID) controller design method for a linear-parameter-varying (LPV) system with parametric uncertainty. It is recognized in the literature that the robust fixed-order controller design can be formulated as a feasibility problem of a bilinear matrix inequality (BMI) constraint. Unfortunately, the search for a feasible solution of a BMI constraint is an NP hard problem in general. Previous researchers have applied a linearization method, such as a variable change technique or a congruence transformation, to transform the BMI into a LMI. The applicability of the linearization method depends on the specific structure of the problem at hand and cannot be generalized. This paper instead formulates the gain-scheduled PID controller design as a feasibility problem of a quadratic matrix inequality (QMI) constraint, which covers the BMI constraint as a special case. An augmented sequential LMI optimization method is proposed to search for a feasible solution of the QMI constraint iteratively. As an illustrative application, a vehicle lateral control problem is presented to demonstrate the applicability of the proposed algorithm to a real-world output feedback control design system.

Improving the efficiency of distribution electric networks of railways based on the multi-agent mode management method

With the increase in observability and controllability of regimes, the development of methods for managing distributed objects of the electrical network is becoming more and more important. The main research directions in smart grids are based on the theory of fuzzy sets, genetic algorithms, neural networks, stochastic control, spectral graph, bilinear matrix inequality constraints. They are aimed at solving multicriterion optimization problems of electric networks with distributed objects and are computationally-demanding and time-consuming. Meanwhile, the methods of multi-agent control of the power supply system based on the parallelization of information flows and coordination of the operation of distributed linear regulators are becoming more common. The purpose of this study is to develop methods for controlling the operating modes of smart distribution electric networks of railways using an agent-based approach for stabilizing voltages within specified limits and reducing electric power losses. This goal can be achieved by solving the problems of developing an algorithm for managing power flows based on the coordinated work of active and reactive power sources and principles of demand management of active consumers. The multi-agent power flow control was realized in the AnyLogic program, the simulation modeling of the electrical network modes was performed in Matlab Simulink with assumptions of linear characteristics of voltage loads. A method has been developed to control the operation modes of smart distribution electric networks of railways based on the presented power flow control algorithm, the hallmarks of which are the use of linearized equations for determining control actions in small increments, which allows high speed data analysis in real time without calculating steady-state modes with disturbances. The obtained simulation results prove the validity of power flow control methods for voltage stabilization based on multi-agent control and the possibility of their practical implementation on modern equipment in smart distribution networks of railways.

Enhanced LMI conditions for observer-based [formula omitted] stabilization of Lipschitz discrete-time systems

This paper deals with H∞ observer-based controller design for a class of discrete-time systems with Lipschitz nonlinearities. Usually, the observer-based control synthesis for the considered class of systems leads to the feasibility of a Bilinear Matrix Inequality (BMI). Since, solving a BMI constraint has been an NP-hard optimization problem, then linearizing this constraint to get a convex one is an interesting issue because Linear Matrix Inequalities (LMIs) are easily tractable by numerical softwares (LMI Toolboxes,..). Hence, the aim of this paper is to develop a new Linear Matrix Inequality (LMI) condition, ensuring the H∞ asymptotic convergence of the observer-based controller. Due to the introduction of a slack variable technique, the usual BMI problem is equivalently transformed to a more suitable one, which leads to less conservative and more general LMI condition compared to the existing methods in the literature. Conjointly to the slack variable technique, the Lipschitz property and the Young’s relation are used in a reformulated way to obtain additional decision variables in the LMI. In the aim to further relax the proposed LMI methodology, sliding windows of delayed states and measurements are included in the structures of the controller and the observer, respectively. The obtained LMI is more general and less conservative than the first one, which can be viewed as a particular solution. To show the effectiveness and superiority of the proposed methodology, some numerical examples and comparisons are provided.

<inline-formula> <tex-math notation="LaTeX">$\mathcal{H}_\infty$ </tex-math> </inline-formula>-Optimal Parallel Feedforward Control Using Minimum Gain

This letter presents static and dynamic parallel feedforward controller synthesis methods that render a linear time-invariant system minimum phase by augmenting its output. The system output is perturbed the least amount possible by minimizing the gain of the parallel feedforward controller while ensuring the augmented system is minimum phase. This is done by minimizing the maximum singular value of a static parallel feedforward controller or the weighted H ∞ norm of a dynamic parallel feedforward controller. Static and dynamic parallel feedforward controllers are synthesized using direct and indirect methods that involve bilinear matrix inequality constraints and are solved iteratively using linear matrix inequalities. The direct method enforces a minimum gain constraint directly on the augmented system, while the indirect method solves for an asymptotically stabilizing negative feedback controller that is inverted to obtain the parallel feedforward controller. Numerical examples are provided to demonstrate the effectiveness of the proposed controller synthesis methods.

Sequential parametric convex approximation algorithm for bilinear matrix inequality problem

The goal of this paper is to study algorithms for solving optimization problems subject to bilinear matrix inequalities (BMIs). This class of problems is known to be of great importance in engineering applications, for instance, control system designs. A main contribution is the development of a sequential convex optimization algorithm, where at each iteration step, a convex subproblem with linear matrix inequality (LMI) constraints is solved. The set of feasible points of the LMIs is a convex inner approximation of the set of feasible points of the BMI constraints around the current iteration point. Another contribution is the convergence proof of a subsequence of the iterations to a stationary point. Finally, an example of the static output-feedback controller design problem is given for comparative analysis.

Co-design of linear systems using Generalized Benders Decomposition

Design of a physical system and its controller has significant ramifications on the overall system performance. The traditional approach of first optimizing the physical design and then the controller may lead to sub-optimal solutions. This is due to the interdependence between the physical design and control parameters through the dynamic equations. Recognition of this fact paved the way for investigation into the “Co-Design” research theme wherein the overall system’s physical design and control are simultaneously optimized. In this paper, a novel approach to address the co-design problem for a class of Linear Time Invariant (LTI) dynamic systems controlled by a Linear Quadratic Regulator (LQR) feedback is presented. The considered co-design problem is formulated as a non-convex optimization problem with Algebraic Riccati Equation (ARE) constraint and convex design objective function. Using Semi-Definite Programming (SDP) duality, the ARE constraint is reduced into equivalent Bilinear Matrix Inequality (BMI) constraints. This reformulated co-design problem is solved using an iterative algorithm based on the Generalized Benders Decomposition (GBD) and Gradient Projection Method. The proposed algorithm converges to a solution which is within a specified tolerance from the nearest local minimum (in special cases global minimum) in a finite number of iterations. Necessary and sufficient conditions are developed to test minimality. Three examples are presented to show efficacy of the proposed algorithm.

Sequential LMI approach for the design of a BMI‐based robust observer state feedback controller with nonlinear uncertainties

SummaryThis paper aims at developing a robust observer–based estimated state feedback control design method for an uncertain dynamical system that can be represented as a linear time‐invariant system connected with an integral quadratic constraint–type nonlinear uncertainty. Traditionally, in existing design methodologies, a convex semidefinite constraint is obtained at the cost of conservatism and unrealistic assumptions. This paper avoids such assumptions and formulates, the design of the robust observer state feedback controller as the feasibility problem of a bilinear matrix inequality (BMI) constraint. Unfortunately, the search for a feasible solution of a BMI constraint is an NP‐hard problem in general. The applicability of a linearization method, such as the variable change method and the congruence transformation, depends on the specific structure of the problem at hand and cannot be generalized. This paper transforms the feasibility analysis of the BMI constraint into an eigenvalue problem and applies the convex‐concave–based sequential linear matrix inequality optimization method to search for a feasible solution. Furthermore, an augmentation of the sequential linear matrix inequality algorithm to improve its numerical stability is presented. In the application part, a vehicle lateral control problem is presented to demonstrate the applicability of the proposed algorithm to a real‐world estimated state feedback control design problem and the necessity of the augmentation for numerical stability.

A coordinated consistency voltage stability control method of active distribution grid

The presence of distributed generators (DGs) with high penetration poses new challenges in the management and operation of electrical grids. Due to the local character of DGs, they could in principle be used in emergency situations to prevent a voltage instability event of the grid. In this paper, a certain method is proposed to coordinate the operation of virtual power plant (VPP) and conventional voltage regulation device to improve the static voltage stability of distribution network with the multi-agent framework. The concept and the general framework of this coordinated control system is introduced, and the voltage instable nodes are determined based on the voltage instability indicator. The voltage coordinated control model of the distribution system is established according to the multi-agent consistency control theory and the coordinated controllers for agents are designed by solving a problem with bilinear matrix inequality constraints. The suggested method is implemented on an IEEE 33 nodes test system and the simulation results show its efficiency and validity.