Advances in Linear Matrix Inequality Methods in Control

Control of the bubble radius is essential in many medical and industrial applications. In this paper, the nonlinear governing dynamic of single spherical bubble in a perfect incompressible liquid is introduced. In continuation, the state space form of the nonlinear dynamic is presented and then observability and controllability are checked by Lie derivative concepts and finally the linearized state space is obtained around its equilibrium point. Second part of this paper is devoted to designing control strategies for the nonlinear governing dynamics of bubble, where the ultrasonic pressure plays the role of control input and the output is radius of bubble which is intended to follow desired radius in order to prevent from bubble collapse phenomenon. A novel robust control is proposed to ensure the desired level of performance in the presence of parametric uncertainty. In this method, the stability of the closed loop system is guaranteed by Lyapunov theorem and the stabilizing gain is achieved by Linear Matrix Inequality (LMI) optimization approach. For comparison, Linear Quadratic Regulator method is selected which is a very classical approach. The optimal feedback gain is also achieved through LMI approach. Simulation results indicate the effectiveness and efficiency of the robust over optimal controller.

This paper investigates the design of a robust linear parameter-varying (LPV) controller for speed regulation of surface permanent magnet synchronous motors (SPMSMs). Motor dynamics is defined in the α–β stationary frame and a parameter-varying model formulation is provided to describe its nonlinear dynamics. A robust gain-scheduled LPV dynamic output-feedback controller is designed to guarantee the asymptotic stability and performance requirements, as well as, guarantee robustness against parameter perturbations and torque load disturbances. As a novel aspect in the SPMSM control, the real-time impact of temperature variation on the winding resistance and magnet flux during motor operation is explicitly considered in the LPV modeling as a scheduling parameter and the subsequent LPV control design compensating for demagnetization effects in the motor response. The controller synthesis conditions guaranteeing stability and robust performance to norm-bounded uncertainty in the system matrices are formulated in a convex linear matrix inequality (LMI) optimization framework. Finally, the validity of the proposed method is assessed in simulation studies. The closed-loop simulation studies demonstrate that the proposed LPV controller provides improved transient response with respect to settling time, overshoot, and disturbance rejection in tracking the velocity profile under the influence of parameter perturbation, temperature variations, and load disturbances.

International audience

Linear matrix inequality (LMI) methods, novel techniques in solving optimisation problems, were introduced as a unified approach for vehicle's active suspension system controller design. LMI methods were used to provide improved and computationally efficient controller design techniques. The active suspension problem was formulated as a standard convex optimisation problem involving LMI constraints that can be solved efficiently using recently developed interior point optimisation methods. An LMI based controller for a vehicle system was developed. The controller design process involved setting up an optimisation problem with matrix inequality constraints. These LMI constraints were derived for a vehicle suspension system. The resulting LMI controller was then tested on a quarter-car model using computer simulations. The LMI controller results were compared with an optimal PID controller design solution. The LMI controller was further tested by incorporating a nonlinear term in the vehicle's suspension model; the LMI's controller degraded response was enhanced by using gain-scheduling techniques. The LMI controller with gain-scheduling gave good results in spite of the unmodelled dynamics in the suspension system, which was triggered by large deflections due to off-road driving.

The PID controller is most commonly used for industrial applications. After studying the standard nested two loop controller structure in a permanent magnet brushless DC motor speed servo system applications, this paper proposes a state space parametrization model which allows to convert robust PID controller design into an linear matrix inequality(LMI) optimization problem. An analysis of experiment results demonstrates that the proposed robust controller with LMI optimization, offers noticeable improvements in performance especially in cases of(relatively large) model uncertainties. And the proposed technique guarantees uniform exponential stability of the closed loop system, a desired rate of convergence, and Hinfin norm bound.

This paper reviews robust feedback control applications in power system. Robust control concept in power system control area evolves by the way of uncertainty modeling. Uncertainties can be caused by variation of loading, load parameters, network topology and machine parameters. One direct way to take into consideration of uncertainties is to enumerate various system operating conditions and use a set of linear time invariant systems to model the power systems. Enumeration method is used in robust control signal selection and robust controller design. Another way is to model uncertainties as structured or unstructured uncertainties. Controllers are designed based on nominal system condition to tolerate uncertainty as much as possible. The controller synthesis problem is solved by either Algebraic Ricatti Equation (ARE) or Linear Matrix Inequality (LMI). LMI is shown to be more flexible in controller design since it provides a way to specify system dynamic performance. Most recently, linear parameter varying (LPV) concept expands the use of enumeration method to express a set of linear time invariant (LTI) systems with state space dependent on linear parameters. LPV based control methods extend gain scheduling to design linear control for each operating conditions and overall a nonlinear controller for the varying parameter system.

We propose a congestion control algorithm and design a state feedback controller for regulating the congestion of available bit rate (ABR) traffic in asynchronous transfer mode (ATM) networks. ABR service is specified as the rate-based feedback mechanism for congestion control and provides a control signal to throttled sources from a bottleneck node in order to adjust the throttled source rates based on available capacity. Our main purposes are as follows: (1) to design a state feedback controller; (2) to place the closed-loop poles in any desired regions based on a pole placement constraint; (3) to use the LMI (linear matrix inequalities) approach for finding the optimal state feedback gain K. The simulation results show that, under pole placement and LMI approaches, our state feedback congestion controller for ABR traffic in ATM networks can possibly guarantee the system stability, provide no buffer overflow, determine the optimal source traffic rates and place the closed-loop poles in any desired regions as well.

A robust delay-dependent guaranteed cost PID multivariable output feedback controller design for time-varying delayed systems: An LMI optimization approach

Decentralized robust guaranteed cost control for a class of interconnected singular large-scale systems with time-delay and parameter uncertainty

The robust control for a class of distributed parameter switched systems with time-delay is studied. The sufficient condition for the existence of the hybrid state feedback robust controller for the closed-loop switched systems is obtained based on Green formula and Lyapunov function method. Using the linear matrix inequality (LMI) approach, the design of the robust controller is converted into finding feasible solutions of a group of LMIs, which can be efficiently carried out using Matlab LMI toolbox. Simultaneously, the method to reduce the conservativeness of the design for the robust control system is also investigated. Finally, a numerical example is presented to show the validity of the proposed design method.

In this paper, a linear parameter-varying (LPV) control design method is evaluated experimentally on an active magnetic bearing (AMB) system. Linear matrix inequality (LMI) conditions for control design of affine parameter-dependent systems using parameter-dependent Lyapunov functions are proposed. A speed-dependent LPV model of the AMB system is derived. Speed-dependent model uncertainties are identified using artificial neural networks (ANNs), and a parameter-dependent uncertainty weighting function is approximated for LPV control synthesis. Experiments are conducted to verify the robustness of LPV controllers for a wide range of rotor speeds. This LPV control approach eliminates the need for gain-scheduling, and provides better performance and less conservativeness over a wide range of rotational speeds than controllers designed with constant uncertainty weighting functions.

Aiming at the problem of transverse vibration of high-speed elevator caused by surface roughness excitation of guide rail and relative motion between car and car frame, firstly, this paper establishes an eight-degree-of-freedom active control model for the transverse vibration of the high-speed elevator separated of car from the car frame. Secondly, the transverse vibration state equation of the high-speed elevator is derived. The H2 norm is used to minimize the vibration acceleration of the car system. Based on Linear matrix inequality (LMI) optimization technology, a robust controller for the high-speed elevator is designed. Finally, the robust controller is solved by MATLAB and its simulation analysis is carried out. The results show that the vibration of the car and the car frame is inconsistent in actual operation. At the same time, the simulation results also show that the root mean square value, mean value and maximum value of the acceleration of the transverse vibration of the car and the car frame are reduced by more than 19% after using the robust controller. Therefore, the robust controller designed in this paper can effectively suppress the transverse vibration of the vibration car system.

For the speed control system of autonomous electric vehicle (AEV), challenge happens with how to determine an appropriate driving speed to satisfy the dynamic environment while resisting uncertainty and disturbance. Therefore, this paper proposes a robust optimal speed control approach based on hierarchical architecture for AEV through combining deep reinforcement learning (DRL) and robust control. In decision‐making layer, a deep maximum entropy proximal policy optimization (DMEPPO) algorithm is presented to obtain an optimal speed via dynamic environment information, heuristic target entropy and adaptive entropy constraint. In motion control layer, to track the learned optimal speed while resisting uncertainty and disturbance, a robust speed controller is designed by the linear matrix inequality (LMI). Finally, simulation experiment results show that the proposed robust optimal speed control scheme based on hierarchical architecture for AEV is feasible and effective.

This paper proposes a new robust FIR equalizer for discrete time systems subject to parameter uncertainties in the matrices of the system state-space model from an H ∞ perspective. The uncertain parameters are supposed to belong to a given convex bounded polyhedral domain. The discrete algebraic Riccati inequality (DARI) and linear matrix inequality (LMI) provide an extremely superior methodology in the H ∞ control fields. By the H ∞ and robust methods in conjunction with the bounded real lemma, the robust FIR equalizer design is transformed into a standard LMI optimization problem. An example is given to illustrate the proposed methodology.

A Linear Matrix Inequality (LMI) approach for designing the H∞ Proportional-Integral (PI) controller for a nonlinear dynamic system is studied. A local linear model containing time-varying norm-bounded uncertain parameters is identified at first. The robust PI control design problem based on the norm-bounded uncertain linear model is then transformed into a standard H∞ control problem, where the latter is further formulated as LMIs. By adopting the LMI expression, a symmetric positive definite matrix P with guaranteed overall system’s quadratic stability can be easily determined. The matrix P can finally be used to infer the robust PI controller parameters. One chemical process, a non-isothermal continuous-stirred-tank reactor (CSTR) with a first-order exothermic reaction, is illustrated to demonstrate the effectiveness of the proposed LMI-based H∞ PI controller design method for nonlinear dynamic processes.