Continuous-time Model Predictive Control Research Articles

Continuous-time nonlinear model predictive control based on Pontryagin Minimum Principle and penalty functions

Pontryagin's Minimum Principle (PMP) is a powerful tool for solving Nonlinear Model Predictive Control problems (NMPC), enabling the handling of time-varying input constraints and cost functions. However, applying PMP encounters challenges when state constraints must be satisfied. This arises because the optimal trajectory often requires a blend of unconstrained and constrained arcs with unknown junction points. To address this issue, relaxation methods are frequently explored, where state constraints are replaced with penalty functions. The contributions of this paper are as follows. First, a method of penalty functions allowing for coping with soft state constraints is examined. We prove the recursive feasibility of this method and demonstrate its efficiency in a numerical example. Second, the finite-time practical stability for the optimal reference tracking NMPC problem is addressed. By appropriately choosing the terminal cost, one can guarantee the convergence of the state vector to a predefined neighbourhood of the target state.

Stability of Progressively Tightening Model Predictive Control in Continuous Time

We consider a continuous-time nonlinear model predictive control formulation that is progressively tightening in path costs and constraints. Under standard assumptions, we prove asymptotic stability of the origin for the corresponding closed-loop system and extend this result to formulations employing an auxiliary dynamic system. The theoretical results are illustrated on a numerical example.

Computationally Efficient Continuous-Time Model Predictive Control of a 2-DOF Helicopter via B-Spline Parameterization.

This paper investigates one way to reduce the computational burden of continuous-time model predictive control (MPC) laws by representing the input/output signals and related models using B-spline functions. Such an approximation allows to implement the resulting feedback control law more efficiently, requiring less online computational effort. As a result, the proposed controller formulates the control signals as continuous polynomial spline functions. All constraints assumed over the prediction horizon are then expressed as constraints acting on the B-splines control polygon vertices. The performance of the proposed theoretical framework has been demonstrated with several real-time experiments using the well-known 2-DOF laboratory helicopter setup. The aim of the presented experiments was to track given step-like reference trajectories for pitch and yaw angles under notable parameter uncertainties. In order to suppress the influence of uncertainties, the control algorithm is implemented in an adaptive mode, equipped with the recursive least squares (RLS) estimation of model parameters and with the adaptation of stabilizing terminal set and terminal cost calculations. Thanks to the presented framework, it is possible to significantly reduce the computational burden, measured by the number of decision variables and input constrains, indicating the potential of the proposed concept for real-time applications, even when using embedded control hardware.

Influence of Discretization in Dynamically Embedded Model Predictive Control

The paper analyzes the closed-loop stability of Dynamically Embedded Model Predictive Control for input-constrained continuous-time nonlinear systems. Given a stabilizing continuous-time optimal control problem, the proposed method performs a discrete approximation to obtain a finite number of optimization variables. The resulting optimization problem is then embedded into a dynamic feedback law that evolves in parallel to the system. Using Input-to-State Stability, it is shown that the dynamic interconnection between the ideal continuous-time model predictive controller and the dynamically embedded solver is asymptotically stable for a sufficiently small discretization step and sufficiently fast solver dynamics. Numerical results, however, highlight a counter-intuitive behavior: as the discretization step decreases, the stability of the closed-loop system tends to deteriorate. This suggests that, although the discretization should be sufficiently accurate to correctly capture the behavior of the system, oversampling the system dynamics may be just as harmful as undersampling.

Continuous-Time Nonlinear Model Predictive Tracking Control with Input Constraints Using Feedback Linearization

This paper presents a tracking control scheme for nonlinear systems with input constraints by combining the continuous-time model predictive control and the feedback linearization. Although there are some similar combinations for nonlinear systems presented in literature, their formulations are complex and massive computations are unavoidable. This study aims to simplify the formulations and reduce the computational loads by imposing the Laguerre functions to approximate the control signals. Since the Laguerre functions have the property of orthogonality, the tracking control problem, by applying the combination, leads to a constrained quadratic optimization problem in terms of the coefficients associated with the Laguerre functions, where the input constraints are converted so as to be state-dependent, based on feedback linearization. The Hildreth’s quadratic programming algorithm is used to solve the optimization problem, so as to determine the coefficients. Moreover, this study also summarizes some common linearization schemes and shows their pros and cons. Furthermore, the proposed approach is applied to two illustrative examples, and the control performances are compared with those by linear control strategies combined with those linearization schemes.

Constant Power Load Stabilization in DC Microgrids Using Continuous-Time Model Predictive Control

Despite its advantages over its AC counterparts, DC microgrids present a lot of challenges. One of these challenges is the instability issues caused by constant power loads (CPLs). CPLs deteriorate the system’s performance due to their incremental negative impedance characteristics. In this paper, a DC microgrid composed of a PV/battery system feeding a pure CPL was considered. A continuous-time model predictive control combined with a disturbance observer was applied to the DC–DC bidirectional converter. The purpose of the composite controller is to address the nonlinearity of the CPL and to maintain the stability of the system in a large operating region under load and PV generation variations. To show the performance of the system, several tests were performed under PV power and CPL power variations. Simulation results show good performance in terms of transient response, optimal tracking, and stability in a large operating region.

Revisiting SIR in the Age of COVID-19: Explicit Solutions and Control Problems

The non-population conserving SIR (SIR-NC) model to describe the spread of infections in a community is proposed and studied. Unlike the standard SIR model, SIR-NC does not assume population conservation. Although similar in form to the standard SIR, SIR-NC admits a closed form solution while allowing us to model mortality, and also provides different, and arguably a more realistic, interpretation of the model parameters. Numerical comparisons of this SIR-NC model with the standard, population conserving, SIR model are provided. Extensions to include imported infections, interacting communities, and models that include births and deaths are presented and analyzed. Several numerical examples are also presented to illustrate these models. Two control problems for the SIR-NC epidemic model are presented. First we consider the continuous time model predictive control in which the cost function variables correspond to the levels of lockdown, the level of testing and quarantine, and the number of infections. We also include a switching cost for moving between lockdown levels. A discrete time version that is more amenable to computation is then presented along with numerical illustrations. We then consider a multi-objective and multi-community control where we can define multiple cost functions on the different communities and obtain the minimum cost control to keep the value function corresponding to these control objectives below a prescribed threshold.

An Improved Model Predictive Fast Frequency Control for Power System Stability Against Unknown Time-Delay Switch Attack

A modern power grid is a cyber-physical system, which are vulnerable to cyber attacks. A recently found attack, the time-delay switch attack (TDSA), is made by inserting time delays into communication channels. A TDSA can be highly destructive to a power system as it can lead to instability. This paper presents a novel model predictive control (MPC) for fast frequency controller in a power system which can effectively mitigate the unknown TDSA. The MPC recently has received great attentions to be applied as FFC in a power system. Most of the MPC design are based on discrete-time model, whose future plant behaviour is calculated through iteration, rather than convolution. Nevertheless, one crucial step in the derivation of discrete-time MPC (DTMPC) is to capture the control trajectory over a finite prediction horizon. This imposes a challenge in designing a DTMPC to counteract the time-delay with unknown time length. Thus, a continuous-time MPC (CTMPC) is proposed to deal with TDSA. To overcome the unknown time delay, we synthesize an accurate time-delay estimator and sequential state predictor (SSP), are designed to accurately estimate and effectively counteract the unknown and random TDSA. All presented case studies are based on a real Taipower system and justification of the effectiveness of the proposed method was verified.

Continuous‐time receding‐horizon reinforcement learning and its application to path‐tracking control of autonomous ground vehicles

AbstractReinforcement learning (RL) and approximate dynamic programming (ADP) have been recently studied to solve nonlinear optimal control problems (OCPs) of continuous‐time (CT) systems. However, online learning efficiency and reliability are two major concerns to be further improved. Motivated by the above issues, in this paper we propose a receding‐horizon reinforcement learning (RHRL) algorithm for near‐optimal control of CT systems under control constraints. Different from classic RL and ADP, in the proposed approach, the infinite‐horizon OCP is decomposed as a series of finite‐horizon ones solved with an actor‐critic structure according to the receding horizon strategy, which can improve the online learning efficiency and reliability. The unknown dynamics of the system are identified offline using a sparse kernel‐based neural network structure whose weights are also updated online in the RHRL framework to improve the control performance. Moreover, the convergence of the modeling error is proven. To verify the effectiveness of our approach, we apply the RHRL algorithm to the autonomous ground vehicle for realizing near‐optimal path‐tracking control. Compared with CT model predictive control using a nominal model and other model‐free tracking controllers such as pure pursuit, heuristic dual programming, and the soft actor‐critic algorithm, RHRL performs better in terms of control performance.

Constrained reference learning for continuous-time model predictive tracking control of autonomous systems

Abstract Often, systems need to adapt their behavior to other systems in their surroundings while obeying constraints to achieve good performance or due to safety reasons. We consider repetitive applications, where the reference for the controller stems from noisy sensor data. Including preview information of the reference, e.g. extrapolating from previous cycles or similar instances, can significantly improve the overall tracking performance and ensure constraint satisfaction. We propose a learning-supported predictive controller that uses Gaussian processes as reference generators for its control task. A Gaussian process is used to learn, filter, and predict the references. It learns references tailored to model predictive control, taking into account continuous-time system dynamics and constraints via constrained hyperparameter optimization. We illustrate the benefits concerning approximation and control performance of the informed Gaussian-process training by a cooperative, mobile robot example.

Recursive Feasibility of Continuous-Time Model Predictive Control Without Stabilising Constraints

We consider sampled-data Model Predictive Control (MPC) of nonlinear continuous-time control systems. We derive sufficient conditions to guarantee recursive feasibility and asymptotic stability without stabilising costs and/or constraints. Moreover, we present formulas to explicitly estimate the required length of the prediction horizon based on the concept of (local) cost controllability. For the linear-quadratic case, cost controllability can be inferred from standard assumptions. In addition, we extend results on the relationship between the horizon length and the distance of the initial state to the boundary of the viability kernel from the discrete-time to the continuous-time setting.

Steering control based on model predictive control for obstacle avoidance of unmanned ground vehicle

In this paper, a strategy based on model predictive control consisting of path planning and path tracking is designed for obstacle avoidance steering control problem of the unmanned ground vehicle. The path planning controller can reconfigure a new obstacle avoidance reference path, where the constraint of the front-wheel-steering angle is transformed to formulate lateral acceleration constraint. The path tracking controller is designed to realize the accurate and fast following of the reconfigured path, and the control variable of tracking controller is steering angle. In this work, obstacles are divided into two categories: static and dynamic. When the decision-making system of the unmanned ground vehicle determines the existence of static obstacles, the obstacle avoidance path will be generated online by an optimal path reconfiguration based on direct collocation method. In the case of dynamic obstacles, receding horizon control is used for real-time path optimization. To decrease online computation burden and realize fast path tracking, the tracking controller is developed using the continuous-time model predictive control algorithm, where the extended state observer is combined to estimate the lumped disturbances for strengthening the robustness of the controller. Finally, simulations show the effectiveness of the proposed approach in comparison with nonlinear model predictive control, and the CarSim simulation is presented to further prove the feasibility of the proposed method.

Non‐linear‐disturbance‐observer‐enhanced MPC for motion control systems with multiple disturbances

This study addresses the optimised tracking problem for motion control systems with multiple disturbances by a non-linear-disturbance-observer-enhanced continuous-time model predictive control (MPC) method. The core is to predict the future tracking error and desired control input (including the lumped effects of disturbances/uncertainties) in the receding-horizon by a higher-order sliding mode disturbance observer, which is designed based upon a rough nominal system. Different from the direct compensation approach in most existing composite MPC methods, disturbance estimates are taken full advantage in the optimisation. The explicit relationship between asymptotic stability and weights in the performance index is provided. Simulations on position control of the robot arm system and experiments on speed regulation of the permanent magnet synchronous motor servo system are both presented to demonstrate the workability.

Explicit spline-based continuous-time MPC: a study on design and performance

This paper presents a continuous-time model predictive control scheme based on B-spline functions used for signals and model approximation. The proposed controller offers two interesting advantages. First, it formulates the control signal as a continuous polynomial spline function, the nature of which is determined by its control polygon that is subject of optimization. Second, all continuous constraints assumed over prediction horizon are consistently transformed into constraints imposed on a finite number of elements of this control polygon. Using parametric quadratic programming we further show how to obtain an explicit representation of the proposed controller, which is known for its efficient online implementation. The featured simulation study demonstrates that by a suitable choice of number and position of knots of the spline function over the prediction horizon it is possible to substantially reduce the number of critical regions of the explicit controller while preserving control performance, and to mitigate the direct correlation between number of regions and chosen length of prediction horizon.

Control of Variable-Speed Variable-Pitch Wind Turbines using Mpc and Qft

Tracing of operating point where maximum power is extracted, regulation of power and speed are the major control objectives for the main component of WECS i.e., wind turbine. A controller with objective designed for one operating region may not sustain better performance in any other operating region. To overcome such issues and achieve improved nature of performance in any operating region using a new advancement in controller with future prediction capability and thus attain the required control objectives. In this work, such objectives are achieved by using continuous-time multi model predictive control and Quantitative Feedback Theory control (QFT). Both the controllers’ performance is compared using simulink studies for a wind turbine model. Simulation results will show the performance of the above mentioned control scheme. The entire simulation work will be carried out on MATLAB/Simulink environment.

Kautz Function Based Continuous-Time Model Predictive Controller for Load Frequency Control in a Multi-Area Power System

A continuous-time Model Predictive Controller was proposed using Kautz function in order to improve the performance of Load Frequency Control (LFC). A dynamic model of an interconnected power system was used for Model Predict... | Find, read and cite all the research you need on Tech Science Press

Continuous-time Model Predictive Control for Real-time Flexibility Scheduling of Plugin Electric Vehicles

Abstract This paper proposes a continuous-time model predictive control (MPC) for co-optimizing the charging flexibility of plugin electric vehicles (PEVs) and generation schedule of generating units in real-time power systems operation. A continuous-time queuing model is developed to aggregate and cluster a large population of PEVs, which represents their aggregate flexibility to the power system operator. The proposed model integrates the most recent information about the system load and PEV arrival, and utilizes the flexibility of PEVs and generating units to supply the ramping requirements of load, while ensuring the delay-based and deadline-based service quality constraints of PEVs. The proposed model is implemented on the IEEE Reliability Test System, using PEV and load data of California. The simulation results demonstrate effectiveness of the proposed model to utilize the flexibility of PEVs in real-time power systems operation, which considerably reduces ramping requirements from conventional generating units.

Feedback Linearizing Model Predictive Excitation Controller Design for Multimachine Power Systems

In this paper, a nonlinear excitation controller is designed for multimachine power systems in order to enhance the transient stability under different operating conditions. The two-axis models of synchronous generators in multimachine power systems along with the dynamics of the IEEE Type-II excitation systems are considered to design the proposed controller. The partial feedback linearization scheme is used to simplify the multimachine power system as it allows decoupling a multimachine power system based on the excitation control inputs of synchronous generators. A receding horizon-based continuous-time model predictive control scheme is used for partially linearized power systems to obtain linear control inputs. Finally, the nonlinear control laws, which also include receding horizon-based control inputs, are implemented on the IEEE 10-machine, 39-bus New England power system. The superiority of the proposed scheme is evaluated by providing comparisons with a similar existing nonlinear excitation controller, where the control input for the feedback linearized model is obtained using the linear quadratic regulator (LQR) approach. The simulation results demonstrate that the proposed scheme performs better as compared to the LQR-based partial feedback linearizing excitation controller in terms of enhancing the stability margin.

An Improved Continuous-Time Model Predictive Control of Permanent Magnetic Synchronous Motors for a Wide-Speed Range

This paper proposes an improved continuous-time model predictive control (CTMPC) of permanent magnetic synchronous motors (PMSMs) for a wide-speed range, including the constant torque region and the flux-weakening (FW) region. In the constant torque region, the mathematic models of PMSMs in dq-axes are decoupled without the limitation of DC-link voltage. However, in the FW region, the mathematic models of PMSMs in dq-axes are cross-coupled together with the limitation of DC-link voltage. A nonlinear PMSMs mathematic model in the FW region is presented based on the voltage angle. The solving of the nonlinear mathematic model of PMSMs in FW region will lead to heavy computation load for digital signal processing (DSP). To overcome such a problem, a linearization method of the voltage angle is also proposed to reduce the computation load. The selection of transiting points between the constant torque region and FW regions is researched to improve the performance of the driven system. Compared with the proportional integral (PI) controller, the proposed CTMPC has obvious advantages in dealing with systems’ nonlinear constraints and improving system performance by restraining overshoot current under step torque changing. Both simulation and experimental results confirm the effectiveness of the proposed method in achieving good steady-state performance and smooth switching between the constant torque and FW regions.

Critically damped stabilization of inverted-pendulum systems using continuous-time cascade linear model predictive control

In this paper, a cascade continuous-time linear MPC controller with 7 well defined parameters is proposed to stabilize the inverted pendulum system around the desired equilibrium points. The proposed controller scheme is composed of two MPC controllers in cascade. These controllers are tuned so as to obtain a doubly critically damped behavior for the inner and outer loops. Explicit generalized expressions of the MPC controller parameters in relation to the inverted pendulum characteristics and the chosen horizon time parameter with the robustness analysis of the controlled system are given to facilitate the controller design for a wide class of inverted pendulum systems. Simulation results show that the MPC controller is compared favorably to the well known parallel two-PID control scheme.