Robust State Feedback Controller Research Articles

Finite-Time Stabilization of Uncertain Switched Positive Linear Systems with Time-Varying Delays

This paper is concerned with finite-time stabilization (FTS) analysis for a class of uncertain switched positive linear systems with time-varying delays. First, a new definition of finite-time boundedness (FTB) is introduced for switched positive system. This definition can simplify FTS analysis. Taking interval and polytopic uncertainties into account, a robust state feedback controller is built such that the switched positive linear system is finite-time bounded. Finally, an example is employed to illustrate the validities of obtained results.

An Improved Memory State Feedback RMPC for Uncertain Constrained Linear Systems with Multiple Uncertain Delays

The problem of robust asymptotic stabilization is considered for a class of discrete-time uncertain linear systems with multiple uncertain time-delayed states and input constraints. Compared with other works in the literature, the proposed approach takes the information of the delayed states with the estimated time-delays indices into full consideration. Based on the predictive control principle of receding horizon optimization and Lyapunov stability theory combined with linear matrix inequalities (LMIs) techniques, a time-delayed state dependent quadratic function is considered for incorporating MPC problem formulation. The robust MPC problem is formulated to minimize the upper bound of infinite horizon cost that satisfies the sufficient conditions. The proposed approach allows for the synthesis of robust memory state feedback controllers with respect to uncertainties on the implemented delay. Since developing the improved memory state feedback controller, the novel improved method is much less conservative and more general. Finally, the numerical simulation results prove availability of the proposed method.

Robust stability analysis and H∞ control for uncertain networked flight control system with random time delays

Time delay and structured uncertainty are the main issues when faster and low–cost avionics networks are introduced to the flight control system. This paper studies the problem of robust H∞ control for uncertain networked flight control system with random time delays and external disturbances. A new sufficient delay–dependent criterion with simple mathematical forms for robust stability and stabilisation of networked flight control system is derived by a free weighting matrix approach. Attention is focused on the design of a memoryless state feedback robust controller which satisfies a prescribed H∞ performance level irrespective of the parameter uncertainties, and the maximum bound of time delay are obtained based on the solution of linear matrix inequalities (LMIs) which are established by the Lyapunov–Krasovskii stability theory. Numerical examples are given to illustrate the effectiveness of design procedure.

A class of adaptive robust controllers for uncertain dynamical systems with unknown virtual control coefficients

The problem of backstepping approach-based adaptive robust controller design is considered for a class of nonlinear systems in the presence of uncertain system parameters, external disturbances, and unknown time-varying virtual control coefficients. It is supposed that the upper bounds of external disturbances and unknown time-varying virtual control coefficients are unknown. In particular, it is also supposed that the lower bounds of time-varying virtual control coefficients are unknown. Then, the improved adaptation laws with σ-modification are employed to estimate the unknown values containing all the unknown bounds of such uncertainties. With the help of the adaptation laws, a class of backstepping approach-based continuous adaptive robust state feedback controllers is developed. Furthermore, by employing the improved Lyapunov-Krasovskii functional method, it is also shown that the proposed adaptive robust backstepping controller can guarantee the uniform asymptotic stability of such uncertain dynamical systems. Finally, a numerical example is given to demonstrate the validity of the results.

Stabilization strategies of a general nonlinear car-following model with varying reaction-time delay of the drivers

In this paper, the stabilization strategies of a general nonlinear car-following model with reaction-time delay of the drivers are investigated. The reaction-time delay of the driver is time varying and bounded. By using the Lyapunov stability theory, the sufficient condition for the existence of the state feedback control strategy for the stability of the car-following model is given in the form of linear matrix inequality, under which the traffic jam can be well suppressed with respect to the varying reaction-time delay. Moreover, by considering the external disturbance for the running cars, the robust state feedback control strategy is designed, which ensures robust stability and a smaller prescribed H∞ disturbance attenuation level for the traffic flow. Numerical examples are given to illustrate the effectiveness of the proposed methods.

Delay-range-dependent chaos synchronization approach under varying time-lags and delayed nonlinear coupling

This paper proposes a novel state feedback delay-range-dependent control approach for chaos synchronization in coupled nonlinear time-delay systems. The coupling between two systems is esteemed to be nonlinear subject to time-lags. Time-varying nature of both the intrinsic and the coupling delays is incorporated to broad scope of the present study for a better-quality synchronization controller synthesis. Lyapunov–Krasovskii (LK) functional is employed to derive delay-range-dependent conditions that can be solved by means of the conventional linear matrix inequality (LMI)-tools. The resultant control approach for chaos synchronization of the master–slave time-delay systems considers non-zero lower bound of the intrinsic as well as the coupling time-delays. Further, the delay-dependent synchronization condition has been established as a special case of the proposed LK functional treatment. Furthermore, a delay-range-dependent condition, independent of the delay-rate, has been provided to address the situation when upper bound of the delay-derivative is unknown. A robust state feedback control methodology is formulated for synchronization of the time-delay chaotic networks against the L2 norm bounded perturbations by minimizing the L2 gain from the disturbance to the synchronization error. Numerical simulation results are provided for the time-delay chaotic networks to show effectiveness of the proposed delay-range-dependent chaos synchronization methodologies.

Robust Control of a Class of Uncertain Fuzzy Systems

Robust stabilization of a class of uncertain fuzzy systems is considered in this paper. The fuzzy systems not only exit disturbances which do not satisfy the so-called matching condition and contain uncertainties in the input channel, but have uncertainties in the state matrix. By using Lyapunov function, the robust state-feedback controllers are constructed. The controllers guarantee that the states of systems are robust stabilization. A numerical example is given to show the effectiveness of our method.

Robust MPC of an unstable chemical reactor using the nominal system optimization

Abstract The continuous stirred-tank reactor with uncertain parameters was stabilized in the open-loop unstable steady state using the robust model predictive control. The gain matrices of the robust state-feedback controller were designed using the nominal system optimization and the quadratic parameter-dependent Lyapunov functions. The controller was verified by simulations using the non-linear model of the reactor and compared with the robust model predictive controller designed using the worst-case system optimization. The values of the quadratic cost function and the consumption of coolant were observed. Both robust model predictive controllers stabilized the reactor despite constrained control inputs and states. The robust model predictive control based on the nominal system optimization improved control responses and decreased the consumption of coolant.

SoC-based solution for multi-axis control systems using high-level synthesis

Recent advances in multi-axis control systems highly rely on new control algorithms and system architectures. The system-on-chip technology is able to realize the novel multi-axis control systems efficiently. But the system-on-chip design and implementation are still challenging and time-consuming for most control system practitioners. This article presents a system-on-chip-based solution for multi-axis control systems. With the high-level synthesis technology, the proposed system can be used as a rapid prototyping for multi-axis control or a test platform for modern control algorithms. A study case of the proposed solution is given to prove its feasibility. The advantage of this implementation is efficient system integration with rich functionality due to the high flexibility of system-on-chip. In order to verify the proposed solution of using system-on-chip to implement modern control algorithms, a mixed H2/ H∞ state-feedback robust controller via linear matrix inequalities is designed for multi-axis synchronization. Experimental results are shown to prove the efficiency of the proposed system-on-chip-based solution.

Adaptive control of a class of non-linear systems preceded by backlash-like hysteresis

We consider a class of uncertain non-linear systems preceded by unknown backlash-like hysteresis, which is modelled by a differential equation. We propose a new state feedback robust adaptive control scheme using a backstepping technique and properties of the differential equation. In this control scheme, we construct a new continuous function to design an estimator to estimate the unknown constant parameters and the unknown bound of a ‘disturbance-like’ term. The transient performance of the output tracking error can be guaranteed by the introduction of pre-estimates of the unknown parameters in our controller together with update laws. We do not require bounds on the ‘disturbance-like’ term or unknown system parameters in this scheme. The global stability of the closed-loop system can be proved.

Finite-Horizon Approximate Optimal Guaranteed Cost Control of Uncertain Nonlinear Systems With Application to Mars Entry Guidance.

This paper studies the finite-horizon optimal guaranteed cost control (GCC) problem for a class of time-varying uncertain nonlinear systems. The aim of this problem is to find a robust state feedback controller such that the closed-loop system has not only a bounded response in a finite duration of time for all admissible uncertainties but also a minimal guaranteed cost. A neural network (NN) based approximate optimal GCC design is developed. Initially, by modifying the cost function to account for the nonlinear perturbation of system, the optimal GCC problem is transformed into a finite-horizon optimal control problem of the nominal system. Subsequently, with the help of the modified cost function together with a parametrized bounding function for all admissible uncertainties, the solution to the optimal GCC problem is given in terms of a parametrized Hamilton-Jacobi-Bellman (PHJB) equation. Then, a NN method is developed to solve offline the PHJB equation approximately and thus obtain the nearly optimal GCC policy. Furthermore, the convergence of approximate PHJB equation and the robust admissibility of nearly optimal GCC policy are also analyzed. Finally, by applying the proposed design method to the entry guidance problem of the Mars lander, the achieved simulation results show the effectiveness of the proposed controller.

Guaranteed cost control of linear uncertain discrete-time impulsive systems

The guaranteed cost control problem for a class of linear uncertain discrete-time impulsive systems is considered. The parametric uncertainties are assumed to be time-varying and norm-bounded. The problem is to design a robust state feedback controller such that the resulting closed-loop system is robustly exponentially stable, and the closed-loop value of a specified quadratic cost function is not more than a certain upper bound for all admissible uncertainties and for all admissible impulse time sequences. A sufficient condition for the existence of guaranteed cost state feedback controllers is derived via a time-varying Lyapunov function approach. This condition is expressed in terms of linear matrix inequalities. Furthermore, the problem of selecting a suboptimal guaranteed cost controller is formulated as a convex optimization problem. An example is provided to demonstrate the effectiveness of the proposed results.

Discretisation and control of polytopic systems with uncertain sampling rates and network-induced delays

This paper investigates the problems of uncertain sampling rate discretisation and the networked control of uncertain time-invariant continuous-time linear systems in polytopic domains. The sampling period is assumed to be unknown but belonging to a given interval. To avoid the difficulty of dealing with the exponential of uncertain matrices, a discrete-time model is obtained by applying a Taylor series expansion of degree ℓ to the original system. The resulting discrete-time model is composed of homogeneous polynomial matrices with parameters lying in the Cartesian product of simplexes, called a multi-simplex, plus an additive norm-bounded term representing the discretisation residual error. The original continuous-time system is controlled through a communication network that introduces a time delay in the process. Linear matrix inequality relaxations that include a scalar parameter search are proposed for the design of a digital robust state feedback controller that guarantees the closed-loop stability of the networked control system. Numerical experiments are presented to illustrate the versatility of the proposed method, which can be applied to a more general class of networked control problems than the existing approaches in the literature.

Robust exponential [formula omitted] control for uncertain time-varying delay systems with input saturation: A Markov jump model approach

This paper is concerned with the state feedback stabilization problem for uncertain time-varying delay systems with input saturation. Based on the stochastic property of time-varying delay, the considered system can be transformed into a continuous Markov jump system with stochastic delays, where the growth trend and the maximum value of the time-varying delay are limited. The problem we address is to design a robust state feedback controller which can switch with time-varying delay in terms of Markov process, such that the system is exponentially mean-square stable with disturbance attenuation γ. Conditions for the existence of solutions to this problem are obtained in terms of linear matrix equalities (LMIs). When these LMIs are feasible, the desired robust controller is given. Finally, two examples are provided to demonstrate the effectiveness of the proposed approach.

Robust adaptive fault‐tolerant control for linear systems with actuator failures and mismatched parameter uncertainties

In this study, a direct adaptive control scheme is developed to solve the robust fault-tolerant control problem for linear systems with mismatched parameter uncertainties, disturbances and actuator faults including loss of effectiveness, outage and stuck. Although the unparameterisable time-varying stuck faults and the upper bounds of disturbances are unknown, the adaptive laws are proposed to estimate the unknown controller parameters online. Then, by using the information from the adaptive mechanism, a class of robust adaptive state feedback controllers are constructed for automatically compensating the actuator faults, external disturbances and mismatched parameter uncertainties. Moreover, on the basis of Lyapunov stability theory, it is proved that the resulting adaptive closed-loop system can be guaranteed to be asymptotically stable in the presence of actuator faults, disturbances and mismatched parameter uncertainties. Finally, simulation results are presented to illustrate the efficiency of the proposed fault-tolerant design approach.

Robust flatness-based tracking control for brushless direct current motor drives

The trajectory tracking problem for nonlinear brushless direct current drive is solved by combined robust and flatness state feedback control. The drive's nonlinear model is shown to have the flatness property. The proposed controller consists of two parts, linear and nonlinear. Linear matrix inequalities (LMI) optimization is used to design the linear part which achieves robust stability against system uncertainties, desired swiftness, and guaranteed cost performance. System uncertainty due to changes in the drive's parameters is represented with a norm-bounded structure. The nonlinear control part solves the motion planning problem through flatness which avoids integrating the differential equations of the dynamics. The main advantages of this technique are that the LMI algorithm includes an optimal part to preclude high control efforts, and the control burden is heavily placed on the linear part to achieve flatness properties. In some systems, in which flatness cannot be achieved, adding robust linear control can overcome or alleviate this problem.

Robust stability and stabilization of linear stochastic systems with Markovian switching and uncertain transition rates

This paper is concerned with the robust stabilization problem for a class of linear uncertain stochastic systems with Markovian switching. The uncertain stochastic system with Markovian switching under consideration involves parameter uncertainties both in the system matrices and in the mode transition rates matrix. New criteria for testing the robust stability of such systems are established in terms of bi-linear matrix inequalities (BLMIs), and sufficient conditions are proposed for the design of robust state-feedback controllers. A numerical example is given to illustrate the effectiveness of our results.

Robust state-feedback control of uncertain LPV systems: An LMI-based approach

In this paper, the problem of designing an LPV state-feedback controller for uncertain LPV systems that can guarantee some desired bounds on the H∞ and the H2 performances and that satisfies some desired constraints on the closed-loop poles location is considered. In the proposed approach, the vector of varying parameters is used to schedule between uncertain LTI systems. The resulting idea consists in using a double-layer polytopic description so as to take into account both the variability due to the parameter vector and the uncertainty. The first polytopic layer manages the varying parameter and is used to obtain the vertex uncertain systems, where the vertex controllers are designed. The second polytopic layer is built at each vertex system so as to take into account the model uncertainties and add robustness into the design step. Under some assumptions, the problem reduces to finding a solution to a finite number of LMIs, a problem for which efficient solvers are available nowadays. The solution to the multiobjective design problem is found both in the case when a single fixed Lyapunov function is used and when multiple parameter-varying Lyapunov functions are used. The validity and performance of the theoretical results are demonstrated through a numerical example.

Robust Stability and Stabilization of Positive Interval Systems Subject to Actuator Saturation

AbstractThis paper addresses the stabilization problem for a class of uncertain positive linear systems (PLSs) in the presence of saturating actuators. The objective is to obtain sufficient conditions for the robust stability of PLSs and to design robust state feedback control laws such that the closed‐loop uncertain system is asymptotically stable and positive at the origin with a large domain of attraction. Several sufficient conditions for robust stabilization and positivity are derived via the Lyapunov function approach and convex analysis method for both the discrete‐time and the continuous‐time cases, respectively. The state feedback controller design and the estimation of the domain of attraction are presented by solving a convex optimization problem with linear matrix inequalities (LMIs) constraints. A numerical example is given to show the effectiveness of the proposed methods.

H∞ State Feedback Control for Switched Linear Systems: Application to an Engine Air Path System

This paper presents the results of an engine air-path H∞ state feedback switched linear controller. The main advantage of this approach is that controller gains are calculated offline, with the robust specifications allowing fast calculation and is cost-effective in calculation and resources.The air-path system dynamics are governed by the Saint-Venant equations, difficult to handle in control synthesis due to its nonlinear properties (states - control inputs coupling, nonlinear functions). To employ advanced control techniques, a switched linear state space representation is proposed. The model is simplified considering that slow dynamics are quite constant. The proposed approach provides a robust state feedback control toward perturbations (model error, unmodeled dynamics…) by considering these disturbances as unknown inputs.