Robust State Feedback Control Law Research Articles

Trajectory Tracking for Autonomous Vehicles Using Robust Model Predictive Control

This paper proposes a novel trajectory tracking control method using a robust model predictive control (RMPC) design technique based on matrix inequality and the theoretical approach of the hybrid & switched system. Firstly, considering bounded model uncertainties and norm-bounded external disturbances, the system states and control matrices are modified. In addition, vehicle time-varying longitudinal speed is considered, and a vehicle lateral dynamic model based on Linear Parameter Varying (LPV) is established by utilizing a polytope with finite vertices. Then the Min-Max robust MPC state feedback control law is obtained at every timestamp by solving a set of matrix inequalities which are derived from Lyapunov stability and the minimization of the worst-case in infinite-horizon quadratic objective function. The numerical simulation results demonstrated that the proposed robust LPV MPC can effectively track vehicle lateral dynamics on both straight and curved roads with varying longitudinal speeds.

Robust efficient cruise control for high-speed train movement based on the self-triggered mechanism

This paper investigates the robust efficient self-triggered cruise control problem for high-speed trains to reduce the consumptions of the computational resources and wireless communication bandwidth. By capturing the dynamic evolution of high-speed trains in the actual environment, an error dynamic state-space model is established considering uncertain parameters, input constraints and disturbances to the train motion. According to the event-triggered control framework, a co-design method is introduced to design the cruise controller and the event-triggered condition jointly. Based on the Lyapunov stability method, a sufficient condition for the existence of the robust state-feedback control law and the event-triggered condition parameter are presented in terms of linear matrix inequalities (LMIs), which ensures that the high-speed train tracks the desired speed, and the coupler deviations are stable at the equilibrium point under external disturbances. With the fact that the event-triggered controller often requires a dedicated detector to continuously monitor the system state, a self-triggered implementation is further developed to dynamically calculate the next triggered instant at each update time based on recently collected information. In the end, numerical experiments are provided to illustrate the effectiveness of the proposed approach in tracking the desired movement profiles, and thereby demonstrates the advantages of the proposed self-triggered controller over the traditional periodic triggered control method.

Matrix Inequalities Based Robust Model Predictive Control for Vehicle Considering Model Uncertainties, External Disturbances, and Time-Varying Delay.

In this paper, we design a robust model predictive control (MPC) controller for vehicle subjected to bounded model uncertainties, norm-bounded external disturbances and bounded time-varying delay. A Lyapunov-Razumikhin function (LRF) is adopted to ensure that the vehicle system state enters in a robust positively invariant (RPI) set under the control law. A quadratic cost function is selected as the stage cost function, which yields the upper bound of the infinite horizon cost function. A Lyapunov-Krasovskii function (LKF) candidate related to time-varying delay is designed to obtain the upper bound of the infinite horizon cost function and minimize it at each step by using matrix inequalities technology. Then the robust MPC state feedback control law is obtained at each step. Simulation results show that the proposed vehicle dynamic controller can steer vehicle states into a very small region near the reference tracking signal even in the presence of external disturbances, model uncertainties and time-varying delay. The source code can be downloaded on https://github.com/wenjunliu999.

Design of nonlinear active noise control earmuffs for excessively high noise level.

When exposed to high levels of noise, earmuffs are often used to avoid hearing loss. However, active noise control earmuffs may exhibit nonlinearities under excessive levels of noise, due to their low-power characteristics of the loudspeakers, and thus nonlinear control algorithms are required to improve the control performance. In this paper, an analytical model of a nonlinear active noise control earmuff is investigated. Based on this model, a robust state feedback control law is designed in the framework of linear matrix inequalities with respect to the parametric uncertainties of the loudspeaker and the limitation of control input. Then the backstepping approach is adopted to force the nonlinear part of the loudspeaker to track the derived state feedback signal and estimate the unknown parameters. Both recorded vehicle noise and multi-frequency noise are used to test the effectiveness of the proposed controller and the control performance is compared with that of a widely accepted nonlinear generalized functional link artificial neural network algorithm. Simulation results demonstrate that the proposed controller is capable of attenuating the interior noise and reducing harmonic and intermodulation distortions significantly.

Robust partial sampled-data state feedback control of Markov jump linear systems

This paper proposes new conditions for the design of a robust partial sampled-data state feedback control law for Markov jump linear systems (MJLS). Although, as usual, the control structure depends on the Markov mode, only the state variable is sampled in order to cope with a specific network control structure. For analysis, an equivalent hybrid system is proposed and a two-point boundary value problem (TPBVP) that ensures minimum or cost is defined. For control synthesis, it is rewritten as a convex set of sufficient conditions leading to minimum guaranteed cost of the mentioned performance classes. The optimality conditions are expressed through differential linear matrix inequalities (DLMIs), a useful mathematical device that can be handled by means of any available LMI solver. Examples are included for illustration.

On finite time stability with guaranteed cost control of uncertain linear systems

This paper deals with the design of a robust state feedback control law for a class of uncertain linear time varying systems. Uncertainties are assumed to be time varying, in one-block norm bounded form. The proposed state feedback control law guarantees finite time stability and satisfies a given bound for an integral quadratic cost function. The contribution of this paper is to provide a sufficient condition in terms of differential linear matrix inequalities for the existence and the construction of the proposed robust control law. In particular, the construction of the feedback control law is brought back to a feasibility problem which can be solved inside the convex optimization framework. The effectiveness of the proposed approach is shown by means of the results obtained on a numerical and a physical example.

Robust Local Stabilization of Discrete-Time Systems with Time-Varying State Delay and Saturating Actuators

The robust local stabilization of uncertain discrete-time systems with time-varying state delayed and subject to saturating actuators is investigated in this work. A convex optimization method is proposed to compute robust state feedback control law such that the uncertain closed-loop is locally asymptotically stable if the initial condition belongs to an estimate of the region of attraction for the origin. The proposed procedure allows computing estimates of the region of attraction through the intersection of ellipsoidal sets in an augmented space, reducing the conservatism of the estimates found in the literature. Also, the conditions can handle the amount of delay variation between two consecutive samples, which is new in the literature for the discrete-time case. Although the given synthesis conditions are delay dependent, the proposed control law is delay independent, yielding to easier real time implementations. A convex procedure is proposed to maximize the size of the set of safe initial conditions. Numerical examples are provided to illustrate the effectiveness of our approach and also to compare it with other conditions in the literature.

Robust decentralised state feedback control design for large-scale networked control system

ABSTRACTThis paper deals with robust decentralised feedback control for large-scale networked control systems (NCSs). A NCS model is extended to a class of large-scale system. Less conservative sufficient conditions are formulated in the form of a nonconvex matrix inequality, of which a feasible solution can be obtained by solving an optimisation problem in terms of linear matrix inequalities. The sufficient conditions are constructed by a set of new Lyapunov–Krasovskii functionals, which are based mainly on the lower and upper bounds of delay and packet loss. An explicit expression for the desired robust decentralised state feedback control law is also given. In order to verify the proposed method, it is applied to a numerical example and the load frequency problem in a multi-area power system. The simulation results show good performance and more robust stability against digital network communications delays, packet loss and external disturbances.

Robust approach for attitude tracking and nonlinear disturbance rejection of rigid body spacecraft

This paper studies the attitude tracking and nonlinear external disturbances rejection problem of rigid body spacecraft by a robust state feedback control law motivated by nonlinear internal model design. This control law, independent of inertia matrix, can not only achieve asymptotic tracking of the desired attitude, but also reject nonlinear external disturbances in the control torque, generated by a so-called nonlinear exosystem. It is also worth mentioning that our control law is a much simpler and smooth one, which can beat the inherent drawback of a discontinuous one that might result in undesired chattering effect in the real applications. With the introduction of nonlinear internal model, we convert the attitude tracking and disturbance rejection problem of spacecraft to the global robust stabilisation problem of a complicated nonlinear multi-input, multi-output augmented system, and then perform a rigorous and sophisticated stability analysis of such nonlinear augmented system. Simulation results will also be provided to demonstrate the effectiveness of our design.

Bộ điều khiển tối ưu bền vững cho hệ thống điều khiển số có trễ

This paper presents a method of designing robust optimal controllers for digital control systems with control delay. A predictor similar to the Smith one is used. The robust optimal state feedback control law is obtained by dynamic programming method. The uncertainty in system parameters is taken care of by an additional term in the Riccati equation. The case of output feedback controller is also dicussed.

Linear quadratic networked control of uncertain polytopic systems

SummaryThis paper investigates the problem of designing robust linear quadratic regulators for uncertain polytopic continuous‐time systems over networks subject to delays. The main contribution is to provide a procedure to determine a discrete‐time representation of the weighting matrices associated to the quadratic criterion and an accurate discretized model, in such a way that a robust state feedback gain computed in the discrete‐time domain assures a guaranteed quadratic cost to the closed‐loop continuous‐time system. The obtained discretized model has matrices with polynomial dependence on the uncertain parameters and an additive norm‐bounded term representing the approximation residual error. A strategy based on linear matrix inequality relaxations is proposed to synthesize, in the discrete‐time domain, a digital robust state feedback control law that stabilizes the original continuous‐time system assuring an upper bound to the quadratic cost of the closed‐loop system. The applicability of the proposed design method is illustrated through a numerical experiment. Copyright © 2015 John Wiley & Sons, Ltd.

Robust Observer Based Control for Stability of Linear Systems: Application to Polytopic Systems

This paper investigates the problem of global stabilization by output feedback for linear time-invariant systems. We give first a procedure to design a robust observer for the linear system. Then using this robust observer with the robust state feedback control law developed by Molander and Willems (1980, “Synthesis of State Feedback Control Laws With a Specified Gain and Phase Margin,” IEEE Trans. Autom. Control, 25(5), pp. 928–931), we construct an output feedback which yields a closed loop system with robustness characteristics. That is, we establish a separation principle. Finally, we give sufficient conditions to establish a robust output feedback for linear polytopic systems.

State Feedback Guaranteed Cost Control of Parameter Uncertain Nonlinear System

In an actual system, the effects of nonlinear factors are inevitable. So in real practice, when a model for complex system is being built, all the features in it will be linearized. Though simplifying the designing and analyzing process, the model being built in this way is thought to be incapable of revealing the true characteristics of the system. In order to solve this problem, the paper analyzes a model combining both linear and nonlinear features while taking the parameter perturbation of the linear part into consideration, which enables the model to retain as many characteristics of the actual system as possible. Provided that the nonlinear function satisfies the Lipschitz constraint conditions, the robust guaranteed cost state feedback control law of nonlinear system is deduced using the Lyapunov function and then converted into the feasible solutions of linear matrix inequality (LMI). The proposed method optimizes the design of controller by modifying the previous oversimplified models that fail to reveal the real characteristics of the actual system, and the effectiveness of the proposed method is being verified through an algorithm simulation example.

Robust ℋ2 model matching from frequency domain specifications

This study addresses the problem of designing compensators for robust model matching. Firstly, the open-loop reference model (obtained, for instance, from classical frequency domain specifications), Gm(s), is recast as the product between two transfer functions GR(s) and Gr(s). A robust state feedback control law is synthesised such that an upper-bound on the ℋ2-norm of the error system between the uncertain system and Gr(s) is minimised, where the state feedback gain is determined through a convex optimisation problem over linear matrix inequality constraints. In addition, block GR(s) behaves as a series compensator in a unity feedback strategy aiming a robust steady state performance. A robust observer scheme is also proposed whenever the states are not available online to the state feedback controller. Numerical examples are presented to illustrate the approach.

Robust control of delayed singular systems with linear fractional parametric uncertainties

Abstract This paper deals with the problem of robust H ∞ control for delayed singular systems with parametric uncertainties. The parametric uncertainties are assumed to be of a linear fractional form, which includes the norm bounded uncertainty as a special case and can describe a class of rational nonlinearities. A strict linear matrix inequality (LMI) design approach is developed such that, when the LMI is feasible, a desired robust state feedback control law can be constructed, which guarantees that, for all admissible uncertainties, the resulting closed-loop system is not only regular, impulse free and stable, but also meets an H ∞ -norm bound constraint on disturbance attenuation. A numerical example is provided to demonstrate the application of the proposed method.

Delay-dependent robust control for uncertain discrete singular systems with time-varying delay

The design problem of delay-dependent robust control for uncertain discrete singular systems with time-varying delay is addressed in this paper. The uncertainty is assumed to be norm-bounded. By establishing a finite sum inequality based on quadratic terms, a new delay-dependent robust stability condition is derived and expressed in terms of linear matrix inequalities (LMIs). A suitable robust state feedback control law is presented, which guarantees that the resultant closed-loop system is regular, causal and stable for all admissible uncertainties. Numerical examples are given to demonstrate the applicability of the proposed method.

A Robust nonlinear state feedback control law for a stirred tank chemical reactor with recycling

A nonlinear state feedback control law for a stirred tank chemical reactor with recycling is studied in this paper. The system is described by nonlinear delay differential equations and the projected control law depends on the system variables at present and past times. Such feedback control law drives the output of the system, the reactor temperature, at the desired value, with exponential error decay rate. Moreover, the closed loop system is locally input-to-state stable with respect to a disturbance forcing, as the control law, the jacket temperature. Such disturbance may well describe sensor and actuator errors, as well as errors due to parameters uncertainty such as the time-delay. Copyright @ IFAC, 7th Workshop on Time-Delay Systems, Nantes, September 2007.

Robust Control Design for Retarded Discrete-Time Systems

This paper focuses on the analysis and synthesis of a robust stabilizing controller for linear discrete time systems with norm-bounded time varying uncertainties. Delay independent robust stability conditions are derived and two synthesis methods are presented. One method is to construct a robust memoryless state feedback control law from the solutions of linear matrix inequalities. The other method consists of designing robust observer-based output feedback controller. The results are expressed in termes of linear matrix inequalities. A comparison with μ∕LDI tests is presented. Furthermore, numerical examples are given for illustration.

Robust Stabilization of Delayed Singular Systems with Linear Fractional Parametric Uncertainties

This paper deals with the problem of robust stabilization for delayed singular systems with parametric uncertainties. The parametric uncertainties are assumed to be of a linear fractional form involving all system matrices. Necessary and sufficient conditions for quadratic stability and quadratic stabilization are obtained. Moreover, the results generalize and improve previous works on delayed singular systems with norm-bounded parametric uncertainties. A strict linear matrix inequality (LMI) design approach is developed such that, when the LMI is satisfied, a desired robust state feedback control law can be constructed. A numerical example is provided to demonstrate the application of the proposed method.

Non-fragile H∞ vibration control for uncertain structural systems

The paper deals with the robust non-fragile H ∞ control problem for uncertain structural systems with additive controller gain variations. The parameter uncertainties for the mass, damping and stiffness of the structural systems are unknown but norm bounded. Based on the H ∞ control theory and a linear matrix inequality formulation, a new method for designing a robust state-feedback control law is presented. The objective is to reduce the disturbance on the controlled output to a prescribed level for all admissible parametric uncertainties and controller gain variations. A four-degree-of-freedom building model subject to seismic excitation is used to illustrate the effectiveness of the approach through simulation.