Robust Positive Invariant Set Research Articles

Predictive fault-tolerant control for nonlinear batch processes with small time delays via iterative learning control: A Lyapunov–Razumikhin approach

For batch processes with small time delays and actuator partial fault, the existing methods based on iterative learning control still have some limitations, including the conservative and computationally burdensome of stability conditions and the limited fault-tolerant control capabilities. For this background, an iterative learning robust predictive fault-tolerant control method is developed, which integrates the Lyapunov–Razumikhin function method and derives stability conditions based on robust positive definite invariant set and terminal constraint set. With small time delays, the stability conditions of the system deduced using the Lyapunov–Razumikhin function are solved at a lower computational cost, which is due to the fact that the dimensionality of the stabilization condition is directly related to the size of the time delay, and thus the small time delay implies a lower dimensionality. Especially, the computational effort for solving the stability conditions online is reduced, allowing real-time control law gains to be obtained and combined with historical batches of control inputs, reducing the learning cycles of the system, and realizing stable tracking of the setpoints within shorter operating batches. Moreover, the robust positive invariant set and the set of terminal constraints are able to constrain the state of the system within a safe range for all possible uncertainties, bounded disturbances, and faults. This makes the proposed methods based on them more robust and fault tolerant. Finally, a nonlinear batch reactor is used as an example to demonstrate the effectiveness and feasibility of the developed method.

Dual‐mode resilient model predictive control for cyber‐physical systems against DoS attacks

AbstractIn this article, a resilient model predictive control algorithm incorporating a dual‐mode control strategy is proposed for a class of cyber‐physical systems subject to state and input constraints, additive disturbances and denial‐of‐service (DoS) attacks. An adversary aims to disrupt the controller‐to‐actuator channel by deploying adversarial jamming signals. By exploring the prediction uncertainty between a nominal model and an actual model and utilizing a constraint tightening technique, a novel set of constraints is developed, which guarantees robust constraint satisfaction in open‐loop mode caused by DoS attacks. A robust positive invariant set is designed, which allows the control mode to switch from rolling optimization to a state‐feedback control law when the state enters its interior, thus saving computational resources. We explicitly determine a critical duration for DoS attacks and, through theoretical analysis, establish that when the duration of DoS attacks falls below this critical value, the proposed algorithm ensures guaranteed recursive feasibility and robust asymptotic stability. Finally, some comparisons are provided to illustrate the effectiveness of the proposed approach.

Safe Motion Planning and Control Framework for Automated Vehicles with Zonotopic TRMPC

Model mismatches can cause multi-dimensional uncertainties for the receding horizon control strategies of automated vehicles (AVs). The uncertainties may lead to potentially hazardous behaviors when the AV tracks ideal trajectories that are individually optimized by the AV's planning layer. To address this issue, this study proposes a safe motion planning and control (SMPAC) framework for AVs. For the control layer, a dynamic model including multi-dimensional uncertainties is established. A zonotopic tube-based robust model predictive control scheme is proposed to constrain the uncertain system in a bounded minimum robust positive invariant set. A flexible tube with varying cross-sections is constructed to reduce the controller conservatism. For the planning layer, a concept of safety sets, representing the geometric boundaries of the ego vehicle and obstacles under uncertainties, is proposed. The safety sets provide the basis for the subsequent evaluation and ranking of the generated trajectories. An efficient collision avoidance algorithm decides the desired trajectory through the intersection detection of the safety sets between the ego vehicle and obstacles. A numerical simulation and hardware-in-the-loop experiment validate the effectiveness and real-time performance of the SMPAC. The result of two driving scenarios indicates that the SMPAC can guarantee the safety of automated driving under multi-dimensional uncertainties.

Robust Shared Control for Four-Wheel Steering Considering Driving Comfort and Vehicle Stability

Although the four-wheel steering system expands the flexibility of vehicle control, it also brings the problem of difficult coordination between driver comfort and vehicle stability. To this end, this paper proposes robust coordinated control for a four-wheel steering (4WS) vehicle considering driving comfort and vehicle stability. First, the vehicle dynamics model is constructed to reflect the lateral motion characteristics of a 4WS vehicle. Then, the driver model is coupled into the 4WS vehicle model to describe the driver’s handling characteristics. To suppress the system perturbation caused by the uncertainties of driver behavior and vehicle states, the Takagi-Sugeno fuzzy robust control method is developed to design the human-machine co-driving system. Moreover, the robust positive invariant set theory is used to guarantee the stability and safety constraints of the vehicle. Finally, the proposed human-machine shared robust control for 4WS vehicle is verified through the driving simulator platform. The results indicate that the fuzzy robust shared control approach comprehensively improves the driving comfort, vehicle stability, and path tracking.

Iterative learning hybrid robust predictive fault-tolerant control for nonlinear batch processes with partial actuator faults

For nonlinear batch processes with uncertainties, disturbances and partial actuator faults, an iterative learning robust predictive fault-tolerant control approach is developed. Based on the conclusion of robustness analysis, the stability conditions combining the robust positive invariant set with the terminal constraint set can guarantee that the system is stable under faults. Furthermore, the real-time optimal gains of control law solved online can significantly reduce the learning cycles of the controller, which in turn can meet the demand for rapid production of batch processes. The injection process is employed as an example to ultimately confirm the viability and efficacy of the developed approach.

A learning- and scenario-based MPC design for nonlinear systems in LPV framework with safety and stability guarantees

ABSTRACT This paper presents a learning- and scenario-based model predictive control (MPC) design approach for systems modelled in the linear parameter-varying (LPV) framework. Using input-output data collected from the system, a state-space LPV model with uncertainty quantification is first learned through the variational Bayesian inference Neural Network (BNN) approach. The learned probabilistic model is assumed to contain the true dynamics of the system with a high probability and is used to generate scenarios that ensure safety for a scenario-based MPC. Moreover, to guarantee stability and enhance the performance of the closed-loop system, a parameter-dependent terminal cost and controller, as well as a terminal robust positive invariant set are designed. Numerical examples will be used to demonstrate that the proposed control design approach can ensure safety and achieve desired control performance.

Rigid-Tube Nonlinear Model Predictive Control for Path Following

While rigid-tube model predictive control introduces robustness with respect to bounded uncertainty, its application to reference tracking may result in poor control performance due to improper reference-trajectory design or disturbances. To circumvent this, the combination of path-following and rigid-tube model predictive control is proposed for systems that require merely a geometric transition without an additional time specification. Therefore, we present results on how to combine rigid-tube model predictive control and path-following. Furthermore, we show for a nonlinear system that the proposed control method steers the system, subject to disturbances, to a robust positive invariant set around the path, while repeated feasibility, stability, and convergence can be guaranteed under mild assumptions. Moreover, the effectiveness and gained flexibility of the predictive controller in handling disturbances is demonstrated in a simulation example considering an autonomous mobile robot.

A Zonotopic-Based Watermarking Design to Detect Replay Attacks

This paper suggests the use of zonotopes for the design of watermark signals. The proposed approach exploits the recent analogy found between stochastic and zonotopic-based estimators to propose a deterministic counterpart to current approaches that study the replay attack in the context of stationary Gaussian processes. In this regard, the zonotopic analogous case where the control loop is closed based on the estimates of a zonotopic Kalman filter (ZKF) is analyzed. This formulation allows to propose a new performance metric that is related to the Frobenius norm of the prediction zonotope. Hence, the steady-state operation of the system can be related with the size of the minimal Robust Positive Invariant set of the estimation error. Furthermore, analogous expressions concerning the impact that a zonotopic/Gaussian watermark signal has on the system operation are derived. Finally, a novel zonotopically bounded water-mark signal that ensures the attack detection by causing the residual vector to exit the healthy residual set during the replay phase of the attack is introduced. The proposed approach is illustrated in simulation using a quadruple-tank process.

Robust iterative learning model predictive control for repetitive motion of maglev planar motor

This paper presents a robust iterative learning model predictive control (RILMPC) scheme for repetitive trajectory tracking of a magnetically levitated (maglev) planar motor. The motivation lies in the improvement of tracking performance and disturbance rejection ability for the maglev system. Relying on the past error in the repetitive motion, the model discrepancy is persistently compensated with iterations to improve the prediction accuracy. The cost function that serves as the upper bound of the tracking error is then derived from the past control trajectory based on the Lipschitz property of disturbances. In the proposed RILMPC scheme, a minimal robust positive invariant set is introduced into the MPC optimisation to cope with unmodeled dynamics and disturbances. Furthermore, it is theoretically shown that the perturbed closed-loop system is stable, and the proposed RILMPC scheme is recursively feasible with perfect tracking performance under some conditions. Finally, comparative experiments carried out on a maglev planar motor demonstrate that the proposed control strategy possesses satisfactory transient/steady-state tracking performance and robustness against disturbances.

Hybrid variables-dependent event-triggered model predictive control subject to polytopic uncertainties

This paper focuses on the model predictive control (MPC) issue where the measurable state is released under an event-triggered mechanism (ETM) to implement MPC over an infinite horizon. A new dynamic ETM (DETM) is devised to conserve network resources, which contains an additive internal dynamic variable (IDV), a multiplicative adaptively adjusting variable, a time-varying weighting matrix and several flexible scalars. The MPC problem is formulated as a ‘min–max’ optimisation problem (OP), where a hard constraint and robust positive invariant set on the predictive state/IDV are considered simultaneously. By resorting to a Lyapunov-like function that depends on the IDV, we put forward an auxiliary OP with matrix-inequality-based constraints. By the feasible solutions of such an auxiliary OP, the feedback gain matrix is designed which ensures the asymptotic stability of the closed-loop system. Two examples are presented to demonstrate the validity of the devised DETM and the DETM-based MPC algorithm. The study verifies that the devised DETM has advantages over an existing counterpart in conserving network resources while achieving the desired performance.

Robust Path Tracking and Obstacle Avoidance using Tube-based Model Predictive Control for Surface Vehicles

This paper describes a tube-based model predictive control (TMPC) approach to robust path tracking and obstacle avoidance for surface vehicles. The TMPC algorithm consists of a nominal model predictive control and a state feedback control, which ensure that the state and input constraints are always satisfied under uncertain environmental disturbances. For obstacle avoidance, a robust positively invariant (RPI) set that contains the vehicle's position must be effectively calculated. To this end, the vehicle's dynamics are decoupled into surge and sway-yaw subsystems based on the vehicle's error dynamics with regard to the desired path, and zonotopes are used for the RPI set calculation. A TMPC is then designed using the obtained RPI set for each subsystem. Simulation results are presented to verify the effectiveness of the proposed control algorithm.

A Robust Impedance Controller for Improved Safety in Human–Robot Interaction

Abstract This paper presents a novel impedance controller modified with a switching strategy for the purpose of improving safety in human–robot interactions. Under normal operating conditions, an impedance controller is enabled when adequate tracking performance is maintained in the presence of bounded disturbances. However, if disturbances are greater than anticipated such that tracking performance is degraded, the proposed controller temporarily switches modes to a control strategy better apt to limit control inputs. With disturbances returning to the prescribed bounds, tracking performance will be restored and the impedance controller will resume for nominal operation. The control parameters are constrained by a few conditions necessary for smooth operation. First, a pair of equality constraints is required for the control signal to be continuous when switching control modes. Second, a Lyapunov analysis is performed to formulate an equality constraint on the control parameters to ensure only a single switch occurs when changing control modes to avert control chatter. Third, a matrix inequality constraint is necessary to ensure a robust positive invariant set is formed for when impedance control is active. Numerical simulations are provided to illustrate the controller and conditions. The simulation results successfully validate the presented theory, demonstrating how the constraints yield a continuous control signal, eliminate switching chatter, and permit robustness to disturbances.

Time-Varying Tube-Based Output Feedback Robust MPC for T–S Fuzzy Systems

For Takagi–Sugeno fuzzy systems subject to inexact membership functions, bounded disturbances, and noises, an output feedback robust model predictive control (RMPC) approach with time-varying robust tubes is investigated. The membership functions errors are bounded within convex sets via the properties of zonotopes and interval matrices. An offline table stores a series of structures that include nested robust positive invariant sets with the corresponding nominal feedback controller gains, ancillary controller gains, and observer gains. According to bounds of real-time estimation error sets, the time-varying structures in the offlined table are searched. Then, the output feedback RMPC problem with time-varying tightened constraints on inputs and states is optimized to stabilize the nominal system. The output feedback RMPC approach can not only update bounds of the estimation errors and uncertain terms resulting from inexact membership functions, but also reduce the computational burden. The proposed RMPC algorithm with recursive feasibility guarantees the robust stability of the controlled systems.

Terminal‐cost design for model predictive control with linear stage‐costs: A set‐theoretic method

AbstractThis article presents a method for designing stabilizing terminal‐costs for linear model predictive controllers whose stage‐costs are a mixture of weighted 1‐norms and ∞‐norms of the system states and inputs. This problem is abstracted by replacing the 1/∞‐norm stage‐costs with convex Minkowski functions. We show that a convex Minkowski terminal‐cost is a control Lyapunov function if and only if the dual of its defining set satisfies a set‐inclusion condition. Furthermore, we show that this set‐inclusion is satisfied by all robust positive invariant sets of a certain dual system. In particular, we show that the minimal convex set that satisfies the set‐inclusion produces the optimal terminal‐cost which is the cost‐to‐go of the corresponding unconstrained infinite‐horizon optimal control problem with convex Minkowski stage‐costs. To reduce the number of constraints in the MPC, we show that the defining set of the Minkowski function can be used as an invariant terminal‐constraint. Finally, we demonstrate our terminal‐cost design procedure for three case studies; a double‐integrator, a galvanometer laser scanner, and a nanosatellite.

Robust Motion Planning for Uncertain Systems With Disturbances Using the Invariant-Set Motion Planner

The invariant-set motion planner uses a collection of safe sets to find a collision-free path through an obstacle-filled environment [1] – [4] . This article extends the invariant-set motion planner to systems with persistently varying disturbances and parametric model uncertainty. This is accomplished by replacing the previously used positive invariant sets with robust positive invariant sets. Since the model uncertainty obfuscates the relationship between the invariant sets in the state space, and the references and obstacles in the output space, we reformulate the dynamics in velocity form so that the system output appears directly in the modified system state. Since the persistently varying disturbances will prevent the closed-loop system from converging to the desired reference, we introduce a new robust connection rule where references are connected when the invariant set of one reference contains the minimal volume robust invariant-set of another. In addition, we bound the time required to transition between invariant sets to ensure safety when the obstacles are moving. By parameterizing the invariant sets using a precomputed input-to-state Lyapunov function, we reduce the real-time computational complexity of our motion planner. The robust invariant-set motion planner is demonstrated for an automated highway driving case study.

Observer-based output feedback robust MPC via zonotopic set-membership state estimation for LPV systems with bounded disturbances and noises

For linear parameter varying (LPV) systems with bounded disturbances and noises, this paper presents a synthesis approach to observer-based output feedback robust model predictive control (RMPC) via the zonotopic set-membership state estimation. In the off-line optimization problem, the state observer sub-gains with a minimal ellipsoidal robust positively invariant (RPI) set for the zonotopic estimation error set are designed. In the on-line control optimization problem, bounds of the current and future zontopic estimation error sets are outer approximated by scaling the minimal ellipsoidal RPI set. Then, an on-line constrained optimization problem that considers bounds of the predicted zonotopic estimation error sets is solved to guarantee robust stability of the closed-loop observer system. In the on-line receding horizon optimization, the zonotopic estimation error set is refreshed based on zonotopic properties, and its outer approximated ellipsoidal set is optimized via the technique of S-procedure. By properly considering the relationship between the zonotopic estimation error set and its outer approximated ellipsoidal set, recursive feasibility of the optimization problem can be ensured. The on-line optimization problem guarantees that the nominal estimation error system and nominal closed-loop observer system converge to the origin. When bounded disturbances and noises are considered, the estimation error system and closed-loop observer system respectively bounded within time-varying RPI sets and robust control invariant (RCI) sets are steered to a neighborhood of the origin such that robust stability of the controlled system is guaranteed. A simulation example is given to illustrate the proposed method.

Energy efficient convex-lifting-based robust control of a heat exchanger

This paper analyses control performance and energy savings reached by application of four robust control approaches implemented on the laboratory plate heat exchanger. The output temperature of the cold medium is the controlled variable and the flow rate of the hot medium is the manipulated variable. Closed-loop control of the heat exchanger aims to ensure offset-free setpoint temperature tracking. First, the computationally demanding robust model predictive control (RMPC) strategy is investigated. Improvements ensured by implementing three modifications of the convex-lifting-based robust control methods are analysed. These modifications are based on the (i) non-tunable, (ii) tunable, and (iii) multiple tunable robust positive invariant (RPI) sets. Designed robust control approaches significantly improved control performance. For considered control setup, the settling time was reduced up to 70%. Energy savings were increased by 82%, when compared to RMPC. Moreover, the considered methods are proper for real-time implementation as they significantly reduce computational demands. The designed robust convex-lifting-based methods are suitable for industrial hardware with limited computational requirements.

Robust dynamic positioning of autonomous surface vessels with tube-based model predictive control

This paper proposes two robust dynamic positioning (DP) approaches for autonomous surface vessels when full states are measurable and when only partial states are available with measurement errors. High fidelity nonlinear hydrodynamics are considered which are approximated as linear models in the local area of DP setpoint. The linearization errors are seen as bounded unmodeled dynamics and are accommodated in the tube-based model predictive control (MPC) together with environmental disturbances. The tube-based MPC controller contains all the possible uncertain trajectories in a tube that is based on a precomputed robust positive invariant set and a nominal trajectory solved online. The total controller consists of a feedforward part compensating the predicted environmental forces, a nominal part guiding the vessel towards and stabilizing it at the origin, and an affine feedback part bounding the uncertain vessel trajectory within the tube. Furthermore, an output feedback robust DP controller is proposed by utilizing a simple Luenberger observer to estimate the system states when full states are not available. The resultant estimation and measurement errors are also incorporated in the tube-based MPC. Simulation results show that the proposed full state and output feedback robust DP controllers can achieve the DP goals within system constraints.

Tracking Control for Directional Drilling Systems Using Robust Feedback Model Predictive Control

A rotary steerable system (RSS) is a drilling technology which has been extensively studied and used for over the last 20 years in hydrocarbon exploration and it is expected to drill complex curved borehole trajectories. RSSs are commonly treated as dynamic robotic actuator systems, driven by a reference signal and typically controlled by using a feedback loop control law. However, due to spatial delays, parametric uncertainties and the presence of disturbances in such an unpredictable working environment, designing such control laws is not a straightforward process. Furthermore, due to their inherent delayed feedback, described by delay differential equations (DDE), directional drilling systems have the potential to become unstable given the requisite conditions. This paper proposes a Robust Model Predictive Control (RMPC) scheme for industrial directional drilling, which incorporates a simplified model described by ordinary differential equations (ODE), taking into account disturbances and system uncertainties which arise from design approximations within the formulation of RMPC. The stability and computational efficiency of the scheme are improved by a state feedback strategy computed offline using Robust Positive Invariant (RPI) sets control approach and model reduction techniques. A crucial advantage of the proposed control scheme is that it computes an optimal control input considering physical and designer constraints. The control strategy is applied in an industrial directional drilling configuration represented by a DDE model and its performance is illustrated by simulations.

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.