Nonconvex Bilinear Matrix Inequality Research Articles

Multi-Objective Finite-Frequency H ∞/GH 2 Static Output-Feedback Control Synthesis for Full-Vehicle Active Suspension Systems: A Metaheuristic Optimization Approach

In this paper, the problem of multi-objective control for active suspension systems with polytopic uncertainty is addressed via <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> / <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GH</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> static output feedback with a limited-frequency characteristic. For the overall analysis of the performance demanding both vehicle-ride comfort related to vertical- and transversal-directional dynamics and the time-domain constraints related to the driving maneuverability, a seven-degree-of-freedom full-vehicle model with an active suspension system is investigated. The robust static output-feedback control strategy is adopted because some state variables may not be directly measured in a realistic implementation. In designing this control, the finite-frequency H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance using the generalized Kalman—Yakubovich—Popov lemma is optimized to improve the passenger’s ride comfort, while the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GH</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> performance is optimized to guarantee the constraints concerning the suspension deflection limitation, road-holding ability, and actuator saturation problem. This control synthesis problem is formulated as nonconvex bilinear matrix inequalities and requires simultaneous consideration of different finite-frequency domain ranges for vertical and transversal motions for evaluating the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance. These design difficulties are overcome by the proposed multi-objective quantum-behaved particle swarm optimizer, which efficiently explores the relevant trade-offs between the considered multiple performance objectives and eventually provides the desired Pareto-optimal control set. Further, the numerical simulation cases of a full-vehicle active suspension system are presented to illustrate the effectiveness of the proposed control synthesis methodology in frequency and time domains.

A Convexity Approach to Dynamic Output Feedback Robust MPC for LPV Systems with Bounded Disturbances

A convexity approach to dynamic output feedback robust model predictive control (OFRMPC) is proposed for linear parameter varying (LPV) systems with bounded disturbances. At each sampling time, the model parameters and disturbances are assumed to be unknown but bounded within pre-specified convex sets. Robust stability conditions on the augmented closed-loop system are derived using the techniques of robust positively invariant (RPI) set and the S-procedure. A convexity method reformulates the non-convex bilinear matrix inequalities (BMIs) problem as a convex optimization one such that the on-line computational burden is significantly reduced. The on-line optimized dynamic output feedback controller parameters steer the augmented states to converge within RPI sets and recursive feasibility of the optimization problem is guaranteed. Furthermore, bounds of the estimation error set are refreshed by updating the shape matrix of the future ellipsoidal estimation error set. The dynamic OFRMPC approach guarantees that the disturbance-free augmented closed-loop system (without consideration of disturbances) converges to the origin. In addition, when the system is subject to bounded disturbances, the augmented closed-loop system converges to a neighborhood of the origin. Two simulation examples are given to verify the effectiveness of the approach.

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.

A proportional–integral-based robust state-feedback control method for linear parameter-varying systems and its application to aircraft

The integration of advanced linear parameter-varying and classical proportional–integral–derivative control methods has attracted great attention in control of nonlinear dynamic systems. However, linear parameter-varying proportional–integral–derivative control synthesis is a nonconvex bilinear matrix inequality problem. Although the synthesis conditions can be convexified in the case of proportional–integral control, a strong constraint on the structure of matrix variables usually leads to infeasible solutions. In this paper, a linear parameter-varying proportional–integral control design method is proposed to remove that constraint. The approach is based on the assumption of state-feedback proportional–integral control to guarantee the linear matrix inequality variables to be full matrices instead of block diagonal matrices, and extended linear matrix inequalities are proposed to synthesize the controller. This increases the likelihood of finding feasible linear matrix inequality solutions and reduces the conservatism. The proposed method is applied to control longitudinal and lateral dynamics of an F-16 aircraft and promising simulation results are obtained.

Synthesis of LPV Controllers With Low Implementation Complexity Based on a Reduced Parameter Set

A major difficulty encountered in the application of linear parameter-varying (LPV) control is the complexity of synthesis and implementation when the number of scheduling parameters is large. Often heuristic solutions involve neglecting individual scheduling parameters, such that standard LPV controller synthesis methods become applicable. However, stability and performance guarantees are rendered void, if controller designs based on an approximate model are implemented on the original plant. In this brief, a synthesis method for LPV controllers that achieves reduced implementation complexity is proposed. The method is comprised of first synthesizing an initial controller based on a reduced parameter set. Then closed-loop stability and performance guarantees are recovered with respect to the original plant, which is considered to be accurately modeled. Iteratively solving a nonconvex bilinear matrix inequality may further improve performance. A two-degrees-of-freedom (2-DOF) and three-degrees-of-freedom robotic manipulator is considered as an illustrative example, for which experimental results indicate a good performance for controllers of reduced scheduling order. Furthermore, in the 2-DOF case, controller performance has been significantly improved.

Convex Computation of the Region of Attraction of Polynomial Control Systems

We address the long-standing problem of computing the region of attraction (ROA) of a target set (e.g., a neighborhood of an equilibrium point) of a controlled nonlinear system with polynomial dynamics and semialgebraic state and input constraints. We show that the ROA can be computed by solving an infinite-dimensional convex linear programming (LP) problem over the space of measures. In turn, this problem can be solved approximately via a classical converging hierarchy of convex finite-dimensional linear matrix inequalities (LMIs). Our approach is genuinely primal in the sense that convexity of the problem of computing the ROA is an outcome of optimizing directly over system trajectories. The dual infinite-dimensional LP on nonnegative continuous functions (approximated by polynomial sum-of-squares) allows us to generate a hierarchy of semialgebraic outer approximations of the ROA at the price of solving a sequence of LMI problems with asymptotically vanishing conservatism. This sharply contrasts with the existing literature which follows an exclusively dual Lyapunov approach yielding either nonconvex bilinear matrix inequalities or conservative LMI conditions. The approach is simple and readily applicable as the outer approximations are the outcome of a single semidefinite program with no additional data required besides the problem description. The approach is demonstrated on several numerical examples.

Robust Static Output Feedback Control and Remote PID Design for Networked Motor Systems

In this paper, we investigate the problem of robust static output feedback (SOF) control for networked control systems (NCSs) subject to network-induced delays and missing data. The uncertain system matrices are assumed to lie in a convex polytope. The network-induced delays are time varying but within a given interval. The random data missing is characterized by the Bernoulli random binary distribution. Delay-dependent conditions for the exponential mean-square stability are first established in terms of matrix inequalities. Then, for the robust stabilization problem, the design of an SOF controller is presented by solving bilinear matrix inequalities (BMIs). In order to efficiently solve a nonconvex BMI, we propose an approach based on the linear matrix inequality technique. Furthermore, the developed approach is employed to design the remote proportional–integral–derivative (PID) controller for NCSs. The design of a digital PID controller is formulated as a synthesis problem of the SOF control via an augmentation method. Simulation examples illustrate the effectiveness of the proposed methods.

Robust H ∞ control for uncertain nonlinear active magnetic bearing systems via Takagi-Sugeno fuzzy models

In this paper, a systematic procedure to design the robust H ∞ fuzzy controller for a nonlinear active magnetic bearing (AMB) system affected by time-varying parametric uncertainties is presented. First, the continuous-time Takagi-Sugeno (T-S) fuzzy model is employed to represent the nonlinear AMB system. Next, based on the obtained fuzzy model, sufficient conditions are derived in terms of linear matrix inequalities (LMIs) for robust stability and H ∞ performance of the control system. The main feature of this paper is that some drawbacks existing in the previous approaches such as truncation errors and nonconvex bilinear matrix inequality (BMI) problem are eliminated by utilizing the homogeneous fuzzy model which includes no bias terms in the local state space models rather than the affine one which includes bias terms. Hence, the design method presented here will prove to be more tractable and accessible than the previous ones. Finally, numerical simulations demonstrate the effectiveness of the proposed approach.

Robust fault estimator design for uncertain networked control systems with random time delays: An ILMI approach

This paper investigates the problem of robust fault estimation for a class of uncertain networked control systems (NCSs) with random communication network-induced delays, which are described by Markov processes. Based on the Lyapunov–Razumikhin method, the existence of a delay-dependent fault estimator is given in terms of the solvability of bilinear matrix inequalities (BMIs). An iterative algorithm is proposed to change this non-convex BMI problem into quasi-convex optimization problems, which can be solved effectively by available mathematical tools. The effectiveness of the proposed design methodology is verified by an internet-based DC motor test rig.

A Barrier Certificate Approach to the Verification of the Safe Operation of a Chemical Reactor

In industrial practice, extensive simulations are performed in order to analyse the safety and the correct operation of controlled chemical processes. One aspect of verifying the safe operation is to prove that the states of the system stay within a safe region for a certain set of inputs or disturbances which is the main theme of this paper. Recently, a rigorous method for this type of verification problem has been proposed which makes use of Barrier Certificates for verifying whether an undesired set of states can be reached. If the system dynamics can be described in polynomial form, the safety of the system can be proven algorithmically. The determination of a barrier certificate is a sum-of-squares (SOS) problem which can further be transformed into a non-convex Bilinear Matrix Inequality (BMI) problem. This paper deals with proving the safety of a Continuously Stirred Tank Reactor (CSTR), a non-linear system, using barrier certificates. Uncertainties are represented by a bounded disturbance acting on the system. Safety is explicitly proven for a convex set of initial conditions and a non-convex unbounded unsafe set. Two situations are considered, the uncontrolled plant and the closed-loop system with a state-feedback controller. For the solution of the BMI problem, three different numerical approaches are compared. It turned out that solving the non-convex BMI problem directly is more efficient than solving it using the convex iterative approach.

SEMI-BLIND ROBUST IDENTIFICATION/MODEL (IN)VALIDATION WITH APPLICATIONS TO MACRO-ECONOMIC MODELLING

This paper addresses the problems of worst-case identification and model invalidation of systems subject to unknown initial conditions. While in principle these problems lead to non-convex Bilinear Matrix Inequalities (BMIs), we show that tractable convex relaxations are readily available. The potential of these techniques is illustrated by identifying and validating a subsystem of the macro-economy relating inflation and interest-rates.

Constrained quadratic stabilization of discrete-time uncertain non-linear multi-model systems using piecewise affine state-feedback

In this paper a method for non-linear robust stabilization based on solving a bilinear matrix inequality (BMI) feasibility problem is developed. Robustness against model uncertainty is handled. In different non-overlapping regions of the statespace known as clusters the plant is assumed to be an element in a polytope whose vertices (local models) are affine systems. In the clusters containing the origin in their closure, the local models are restricted to being linear systems. The clusters cover the region of interest in the state-space. A n affine state-feedback is associated with each cluster. By utilizing the affinity of the local models and the state-feedback, a set of linear matrix inequalities (LMIs) combined with a single non-convex BMI are obtained which, if feasible, guarantee quadratic stability of the origin of the closed loop. The feasibility problem is attacked by a branch-and-bound-based global approach. If the feasibility check is successful, the Lyapunov matrix and the piecewise affine state-feedback are given directly by the feasible solution. Control constraints are shown to be representable by LMIs or BMIs, and an application of the control design method to robustify constrained non-linear model predictive control is presented. In addition, the control design method is applied to a simple example.

Constrained quadratic stabilization of discrete-time uncertain nonlinear multi-model systems using piecewise affine state-feedback

In this paper a method for nonlinear robust stabilization based on solving a bilinear matrix inequality (BMI) feasibility problem is developed. Robustness against model uncertainty is handled. In different non-overlapping regions of the state-space called clusters the plant is assumed to be an element in a polytope which vertices (local models) are affine systems. In the clusters containing the origin in their closure, the local models are restricted to be linear systems. The clusters cover the region of interest in the state-space. An affine state-feedback is associated with each cluster. By utilizing the affinity of the local models and the state-feedback, a set of linear matrix inequalities (LMIs) combined with a single nonconvex BMI are obtained which, if feasible, guarantee quadratic stability of the origin of the closed-loop. The feasibility problem is attacked by a branch-and-bound based global approach. If the feasibility check is successful, the Liapunov matrix and the piecewise affine state-feedback are given directly by the feasible solution. Control constraints are shown to be representable by LMIs or BMIs, and an application of the control design method to robustify constrained nonlinear model predictive control is presented. Also, the control design method is applied to a simple example.