Control Of Uncertain Linear Systems Research Articles

Reliable State Feedback Control for Uncertain Linear Systems With Terminal Reliability Constraint

Reliable State Feedback Control for Uncertain Linear Systems With Terminal Reliability Constraint

Guaranteed cost event-triggered H∞ control of uncertain linear system via output disturbance observer

This paper is concerned with anti-disturbance output feedback control of uncertain systems via dynamic guaranteed cost triggering mechanism. An output disturbance observer is constructed to attenuate matched disturbances and the mismatched one is attenuated by the H∞ approach. A new dynamic event-triggered mechanism is established in terms of guaranteed cost triggering condition and triggering threshold. A composite controller is built on an observer based state feedback controller and an output disturbance compensator. By employing the Lyapunov stability method, a sufficient condition is presented in the framework of linear matrix inequality to ensure the asymptotic stability of the closed-loop system with the prescribed cost. Comparative simulation studies are supplied to verify the effectiveness of the proposed composite control scheme.

Formal Synthesis of Controllers for Uncertain Linear Systems Against -Regular Properties: A Set-Based Approach

In this paper, we present how to synthesize controllers to enforce $\omega$-regular properties over linear control systems affected by bounded disturbances. In particular, these controllers are synthesized based on so-called hybrid controlled invariant (HCI) sets. To compute these sets, we first construct a product system between the linear control system and the deterministic Streett automata (DSA) modeling the desired property. Then, we compute the maximal HCI set over the state set of the product system by leveraging a set-based approach. To ensure termination of the computation of the HCI sets within a finite number of iterations, we also propose two iterative schemes to compute approximations of the maximal HCI set. Finally, we show the effectiveness of our results via two case studies.

Data-Driven Safe Control of Uncertain Linear Systems Under Aleatory Uncertainty

Safe control of constrained uncertain linear systems under aleatory uncertainty is considered. Aleatory uncertainty characterizes random noises and is modeled by a probability distribution function (PDF). Data-based probabilistic safe controllers are designed for the cases where the noise PDF is 1) zero-mean Gaussian with a known covariance, 2) zero-mean Gaussian with an uncertain covariance, and 3) zero-mean non-Gaussian with an unknown distribution. Easy-to-check model-based conditions for guaranteeing probabilistic safety are provided for the first case by introducing probabilistic <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\lambda$</tex-math></inline-formula> -contractive sets. These results are then extended to the second and third cases by leveraging distributionally-robust probabilistic safe control and conditional value-at-risk ( <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">CVaR</b> ) based probabilistic safe control, respectively. Data-based implementations of these probabilistic safe controllers are then considered. Moreover, an upper bound on the minimal risk level, under which the existence of a safe controller is guaranteed, is learned using collected data. A simulation example is provided to show the effectiveness of the proposed approach.

Discretization and state feedback control for uncertain linear systems—A new approach considering linear multistep method theory

This paper proposes a novel procedure for discretizing uncertain time-invariant continuous-time linear systems in polytopic domains. The approach is based on multistep method theory and involves mixing models obtained with different multiples of a fixed step-size (sampling time), thereby increasing data availability. An optimization procedure is employed to determine the coefficients that minimize the residual discretization error, and then these coefficients are combined to construct an augmented structure for the discrete-time system. The discretization error is computed using a grid search and incorporated into the discrete-time system formulation. Using the resulting discrete-time system, a state feedback methodology is applied to synthesize digital controllers capable of stabilizing the original continuous-time system. The paper presents two sufficient Linear Matrix Inequality (LMI)-based conditions for designing robust controllers specific to the proposed structure. The first condition utilizes a constant Lyapunov function, while the second condition employs a parameter-dependent Lyapunov function. In both cases, the proposed conditions guarantee asymptotic stability for the continuous-time system in closed-loop. A numerical experiment is conducted to illustrate the validity and effectiveness of the proposed method.

Low-order sliding mode repetitive control for uncertain linear systems perturbed by band-limited periodic disturbances

This article presents a systematic method for designing discrete-time low-order sliding mode repetitive controller for a class of uncertain linear systems. A linear system considered in this class is perturbed by band-limited periodic disturbances and parametric uncertainties. The digital controller design method we propose combines two control strategies, namely the repetitive control and the sliding mode control techniques. The proposed controller is required to be low order and is intended to reject the periodic disturbances with specific dominant frequencies and to establish robustness against plant parametric uncertainties. Moreover, the proposed method yields a repetitive controller with only a small number of delay terms and can perform fast transient responses. We show through mathematical analyses that the linear system applying the low-order sliding mode repetitive controller satisfies sliding mode stability and robustness criteria. In addition, we demonstrate via a numerical example that a servomotor controlled by the low-order sliding mode repetitive controller can track a frequency-shifted triangular signal with a short transient period and minor errors. Other examples are also presented and show that the low-order sliding mode repetitive controller exhibits better performance than other relevant control methods.

Design and analysis of higher-order repetitive sliding mode controller for uncertain linear systems with time-varying periodic disturbances

In this article, a discrete-time design of higher-order repetitive sliding mode controller (HO-RSMC) for uncertain linear systems with time-varying periodic disturbances is presented. In the design, a higher-order repetitive control (HORC) is employed to significantly suppress the periodic exogenous disturbance with period variation. The HORC-based control system offers a slow transient response due to the multiple control delays. Moreover, since the system stability is easily affected by model uncertainties, an output-feedback sliding mode controller (SMC) is added to overcome the shortcomings of HORC. Stability analysis of the resulting closed-loop system is rigorously provided. The simulation results and comparison studies confirm the effectiveness and good performance of the proposed design.

Model Predictive Control for Linear Uncertain Systems Using Integral Quadratic Constraints

In this work, we propose a tube-based MPC scheme for state- and input-constrained linear systems subject to dynamic uncertainties characterized by dynamic integral quadratic constraints (IQCs). In particular, we extend the framework of $\rho$-hard IQCs for exponential stability analysis to external inputs. This result yields that the error between the true uncertain system and the nominal prediction model is bounded by an exponentially stable scalar system. In the proposed tube-based MPC scheme, the state of this error bounding system is predicted along with the nominal model and used as a scaling parameter for the tube size. We prove that this method achieves robust constraint satisfaction and input-to-state stability despite dynamic uncertainties and additive bounded disturbances. A numerical example demonstrates the reduced conservatism of this IQC approach compared to state-of-the-art robust MPC approaches for dynamic uncertainties.

Scalable tube model predictive control of uncertain linear systems using ellipsoidal sets

Scalable tube model predictive control of uncertain linear systems using ellipsoidal sets

Online learning stochastic model predictive control of linear uncertain systems

AbstractThis article presents an online learning stochastic model predictive control method for linear uncertain systems with state‐dependent additive uncertainties, where the uncertainty is modeled as Gaussian process. The proposed scheme utilizes the probabilistic reachable sets as time‐varying tubes, which are formulated by forecasting the variance propagation of uncertainty via Gaussian process regression, to embody the chance constraints. The proposed learning based stochastic model predictive control algorithm is designed by refining the active data dictionary to train the Gaussian process model of uncertainty online. In particular, the data points in the active data dictionary are selected from the raw data around the predicted optimal nominal trajectories, which reduces the computational load as well as preserves the control performance. Then the algorithm feasibility and closed‐loop stability of the developed algorithm are analyzed. Finally, the efficacy and superiority over existing methods are verified by simulation studies.

Observer-Based Quadratic Guaranteed Cost Control for Linear Uncertain Systems with Control Gain Variation

This study proposes a method for designing observer-based quadratic guaranteed cost controllers for linear uncertain systems with control gain variations. In the proposed approach, an observer is designed, and then a feedback controller that ensures the upper bound on the given quadratic cost function is derived. This study shows that sufficient conditions for the existence of the observer-based quadratic guaranteed cost controller are given in terms of linear matrix inequalities. A sub-optimal quadratic guaranteed cost control strategy is also discussed. Finally, the effectiveness of the proposed controller is illustrated by a numerical example. The result shows that the proposed controller is more effective than conventional methods even if system uncertainties and control gain variations exist.

Performance-improved finite-time fault-tolerant control for linear uncertain systems with intermittent faults: an overshoot suppression strategy

This paper presents a novel prespecified finite-time and overshoot-restrained fault-tolerant tracking control scheme to compensate the intermittent faults in linear uncertain systems. Starting from the system tracking error dynamics model, we design a speed excitation function to improve the response speed of the control. A finite-time performance index function is established in the zero-sum game framework for the fault-free optimal control policy. Based on Lyapunov stability theory, a fault compensation oriented fault estimation and diagnosis method is proposed to ensure the uniformly ultimately bounded stability of tracking errors. It is noteworthy that the proposed speeding fault-tolerant control can accelerate the trajectory tracking, narrow the upper bound of tracking errors and restrain the overshoot. Numerical and practical experiments fully illustrate the effectiveness of the proposed strategy.

Synthesis of Computationally Efficient Recursively Feasible Multi-Stage Predictive Controllers for Uncertain Linear Systems

Synthesis of Computationally Efficient Recursively Feasible Multi-Stage Predictive Controllers for Uncertain Linear Systems

Delay-dependent Robust Control for Uncertain Linear Systems with Distributed and Multiple Delays

This paper considers robust control of uncertain linear neutral systems with multiple state and state derivative delays. With equivalent descriptor representation, the stabilization problem is extended to more general class of neutral-type uncertain linear systems with discrete and distributed delays. The parametric uncertainties are time varying and unknown but norm bounded. Two delay-dependent/independent approaches are proposed to design robust controllers for a class of uncertain linear neutral systems with parametric uncertainty, discrete and distributed multiple delays. Using a presented descriptor model and an appropriate Lyapunov functional, sufficient conditions for closed loop stability are given in terms of linear matrix inequalities (LMIs). Solving the LMI problems, a robust memoryless state feedback is designed for all admissible uncertainties. The results depend on the size and varying rate of the delays. Two examples are provided to show the effectiveness of the proposed strategy.

Adaptive event-triggered control for uncertain linear systems with process time-varying delays

This paper concerns an event-triggered control for uncertain linear systems with process time-varying delays. An adaptive event trigger (AET) is employed for reducing the data transmission amount from the sensor to the controller. Because a positive constant and a positive variable are simultaneously involved in the AET, the AET condition may be difficult to be violated and the data transmission amount may be reduced. A discrete-time feedback control law is employed to guarantee the uniformly ultimately bounded (UUB) of the closed-loop systems. If the gain matrix of the time-delayed state is compromised with the sufficient stability condition, the desired stability can be always achieved regardless of the time-varying delays. In addition, the bound region can be specified by regulating the threshold of the AET. It is proved that the unwanted Zeno phenomena is naturally excluded since the minimal event-triggered time-interval of the AET is a positive real number. Furthermore, the maximal event-triggered time interval can also be quantitatively approximated. A numerical example is provided to demonstrate the effectiveness of the proposed method, and the influences of the AET threshold and time delay are discussed as well.

Fault-Tolerant Funnel Control for Uncertain Linear Systems

We study adaptive fault-tolerant tracking control for uncertain linear systems. Based on recent results in funnel control, and the time-varying Byrnes-Isidori form, we develop a low-complexity model-free controller that achieves prescribed performance of the tracking error for any given sufficiently smooth reference signal. Within the considered system class, we allow for more inputs than outputs as long as a certain redundancy of the actuators is satisfied. An important role in the controller design is played by the controller weight matrix. This is a rectangular input transformation chosen such that in the resulting system, the zero dynamics, which are assumed to be uniformly exponentially stable, are independent of the new input. We illustrate the fault-tolerant funnel controller by an example of a linearized model for the lateral motion of a Boeing 737 aircraft.

Discrete‐time modified repetitive sliding mode control for uncertain linear systems

SummaryThis article presents a design strategy and stability analysis of modified repetitive sliding mode controller for uncertain linear systems. A modified repetitive controller is adopted to simultaneously track and reject periodic signals. A discrete‐time sliding mode controller is combined to compensate the slow response of repetitive control and to provide robustness against plant parameters uncertainties. Stability analysis is provided to prove boundedness of the proposed control law and the convergence of sliding function and the tracking error. Comparative simulation results demonstrate that the proposed method is able to accurately track reference signal and to reject disturbance with fast transient response. The results also indicate that the closed‐loop system remains stable in the presence of plant parameter changes.

Optimal Selection of Basis Functions for Robust Tracking Control of Uncertain Linear Systems—With Application to Three-Dimensional Printing

Abstract There is growing interest in the use of the filtered basis functions (FBF) approach to track linear systems, especially nonminimum phase (NMP) plants, because of its distinct advantages compared to other tracking control methods in the literature. The FBF approach expresses the control input to the plant as a linear combination of basis functions with unknown coefficients. The basis functions are forward filtered through the plant dynamics, and the coefficients are selected such that tracking error is minimized. Similar to other feedforward control methods, the tracking accuracy of the FBF approach deteriorates in the presence of uncertainties. However, unlike other methods, the FBF approach presents flexibility in terms of the choice of the basis functions, which can be used to improve its accuracy. This paper analyzes the effect of the choice of the basis functions on the tracking accuracy of FBF, in the presence of uncertainties, using the Frobenius norm of the lifted system representation (LSR) of FBF's error dynamics. Based on the analysis, a methodology for optimal selection of basis functions to maximize robustness is proposed, together with an approach to avoid large control effort when it is applied to NMP systems. The basis functions resulting from this process are called robust basis functions. Applied experimentally to a desktop three-dimensional (3D) printer with uncertain NMP dynamics, up to 48% improvement in tracking accuracy is achieved using the proposed robust basis functions compared to B-splines, while utilizing much less control effort.

Plug-and-Play Distributed Safety Verification for Linear Control Systems With Bounded Uncertainties

Control and, in particular, learning-based control, is challenging in large-scale and safety-critical system networks due to interactions between subsystems, which can potentially be time varying. This article presents a plug-and-play safety framework that can be applied together with high-performance control algorithms, e.g., emerging from learning techniques. The framework ensures constraint satisfaction for a network of uncertain linear control systems during control, learning, and during topology changes in the form of agents joining or leaving the network with prior plug-in or plug-out requests. The presented approach is based on safe sets and tube-based tracking controllers, which can be designed in a distributed fashion, i.e., using only local information. The presented plug-and-play procedure requires only the update of the safe sets of the agents directly involved in the plug-and-play operation making the operation computationally efficient compared to completely redesigning all safe sets. The capabilities of the safety framework are illustrated using numerical examples.

Predictive extended state observer-based robust control for uncertain linear systems with experimental validation

This paper describes a predictive extended state observer-based robust control for uncertain process control applications. The technique discussed in the article uses the extended state observer (ESO) that can estimate the dynamics of the system as well as total disturbance encountered in the system. The disturbances, parametric uncertainties associated with the processes are treated as an extended state variable to be estimated in real-time using ESO. With the implementation of a predictive algorithm with an ESO, the proposed control structure extends its applicability to time-delayed higher-order processes. The proposed control technique utilizes the simple first-order modified predictive ESO even in the case of higher-order processes. The novel predictive ESO is able to obtain a delay less estimation of total disturbance as compared with existing normal ESO. Also, novel predictive ESO maintains its stability margin in presence of time delay as well provides better response as compared with normal ESO. Numerical simulations show that the proposed scheme provides a significant improvement in transient response as compared with internal model control-based proportional-integral-derivative (IMC-PID) control. The proposed scheme requires less knowledge of the process as compared with the IMC-PID structure. The implementation of the proposed control is tested on a real-life single tank level control system. Because of its merit, the suggested technique can be used as automatic for online tuning, as it is less reliant on the process model.