Predictive Control Of Nonlinear Systems Research Articles

Iterative distributed model predictive control for nonlinear systems with coupled non‐convex constraints and costs

AbstractThis paper proposes a distributed model predictive control (DMPC) algorithm for dynamic decoupled discrete‐time nonlinear systems subject to nonlinear (maybe non‐convex) coupled constraints and costs. Solving the resulting nonlinear optimal control problem (OCP) using a DMPC algorithm that is fully distributed, termination‐flexible, and recursively feasible for nonlinear systems with coupled constraints and costs remains an open problem. To address this, we propose a fully distributed and globally convergence‐guaranteed framework called inexact distributed sequential quadratic programming (IDSQP) for solving the OCP at each time step. Specifically, the proposed IDSQP framework has the following advantages: (i) it uses a distributed dual fast gradient approach for solving inner quadratic programming problems, enabling fully distributed execution; (ii) it can handle the adverse effects of inexact (insufficient) calculation of each internal quadratic programming problem caused by early termination of iterations, thereby saving computational time; and (iii) it employs distributed globalization techniques to eliminate the need for an initial guess of the solution. Under reasonable assumptions, the proposed DMPC algorithm ensures the recursive feasibility and stability of the entire closed‐loop system. We conduct simulation experiments on multi‐agent formation control with non‐convex collision avoidance constraints and compare the results against several benchmarks to verify the performance of the proposed DMPC method.

Delay-dependent recursive feasibility based switched model predictive control for nonlinear systems

This paper presents a coordinated design of a model predictive controller (MPC) and switching law for delayed nonlinear switched systems with Lipschitz property. The article derives delay-dependent recursive feasibility constraints to handle disturbances in a polytopic form and incorporates them into the optimization problem. The control gain at each step is determined to ensure the feasibility of the optimization problem during the execution time of each sub-system. Additionally, constraints related to the cost function and H∞ performance conditions are introduced in the developed linear matrix inequality problem to minimize the cost function with an infinite predictive horizon and guarantee controller robustness against unknown constant delays. The coordinated design of the MPC and switching law is achieved using the multiple Lyapunov–Krasovskii functional, which reduces the strictness of the controller constraints compared to the switched Lyapunov–Krasovskii functional. However, it imposes a dwell-time limitation on the switching law. By adopting the PDT structure, the dwell-time limitation is reduced compared to common structures. To evaluate the proposed design, it is applied to a water pollution system, and its performance is assessed. The results demonstrate superior performance compared to previous works.

Contraction-Based Stochastic Model Predictive Control for Nonlinear Systems With Input Delay Using Multidimensional Taylor Network

For nonlinear systems with stochastic uncertainties and input delay, the existing control approaches based on the robust mechanism are generally conservative in most practical scenarios. Within this context, a stochastic model predictive control (MPC) scheme based on uncertainty contraction is proposed here to reduce conservativeness and improve real-time performance. In terms of the Lyapunov-Razumikhin theorem, a robust variational controller (RVC) is deduced as the primary controller for the variational dynamic with input delay. A variational vector field in Wasserstein metric space is constructed to detect the minimum geodesic between the uncertainty distribution and the desired one. Integrating RVC along the minimum geodesic, the modified auxiliary controller is developed for tracking the nominal trajectory with less conservatism. Subsequently, the chance constraints of stochastic states are transformed into the deterministic quantile constraints of nominal states. The sparse multi-dimensional Taylor network based on Bayesian compressive sensing is designed to parameterize the ambiguity set in Wasserstein metric space for higher computational efficiency.A tractable MPC scheme is then formulated to achieve the reference index with an improved trade-off between robustness and real-time performance.The stochastic input-to-state stability of the considered system is verified theoretically by the Lyapunov-Krasovskii theorem. The effectiveness of the proposed scheme is confirmed by a numerical simulation derived from a practical industrial process.

Linear parameter-varying model predictive control for nonlinear systems using general polytopic tubes

This paper deals with model predictive control (MPC) for nonlinear systems using linear-parameter varying (LPV) embedding of the nonlinear dynamics (LPVMPC). The proposed LPVMPC can incorporate information of the future evolution of the scheduling parameter over the MPC prediction horizon with uncertainty bounds, which are used to construct anticipated scheduling tubes for robustification. Therefore, this approach is less conservative than the methods that only consider knowledge of the bounds on the scheduling parameter’s rate of variation. The scheduling tubes are employed to synthesize online general polytopic invariant state tubes. The optimization problem of the proposed LPVMPC is a single quadratic program. Recursive feasibility is proven. A numerical example is presented for demonstrating the effectiveness of the proposed LPVMPC algorithm compared to nonlinear MPC and other standard approaches.

Dual‐mode event‐triggered predictive control for nonlinear systems with bounded disturbances

AbstractIn this article, we propose a dual‐mode event‐triggered predictive control method for nonlinear systems with bounded disturbances. The proposed method contains two triggering mechanisms, namely, the hybrid threshold‐based event‐triggered model predictive control (HETMPC) mechanism and the event‐triggered linear quadratic regulator mechanism. The former triggering mechanism is designed based on the error between the real state and the optimal state and also the disturbance information acted on the investigated system. Compared with the traditional fixed triggering threshold, the designed HETMPC has a fewer triggering numbers and reduces the computational burden of online real‐time optimization. This event‐triggered mechanism will be adopted before the states go into the terminal invariant set. The latter event‐triggered mechanism is designed based on the derivation of the system state and it will be adopted after the states enter the terminal invariant set. The feasibility and the input‐to‐state practical stability analysis of the designed strategy is presented. Some simulations, including the application to a mass‐spring‐damper system, are provided to show the correctness and feasibility of the designed algorithms.

Homothetic Tube Model Predictive Control for Nonlinear Systems

Homothetic tube model predictive control is generalized to the setting of nonlinear discrete time systems. The developed generalization is natural in the sense that the utilized assumptions in the nonlinear setting are suitably generalized variants of the related assumptions deployed in the linear case. The main analysis is carried out for Lipschitz systems, and it is subsequently extended to set–valued systems and other classes of nonlinear systems as well as particularized for parameterically uncertain linear systems. All desirable computational and structural properties of homothetic tube MPC are preserved.

Generalised Predictive Control of Nonlinear Systems Using the Echo State Network and the Nelder-Mead Method

Generalised Predictive Control of Nonlinear Systems Using the Echo State Network and the Nelder-Mead Method

Security-based dynamic output-feedback model predictive control for nonlinear systems in T-S fuzzy form subject to deception attacks

This paper is concerned with the security-based fuzzy model predictive control (FMPC) problem for a class of discrete-time Takagi-Sugeno (T-S) fuzzy systems with deception attacks on the measured outputs. With respect to the unmeasurable system states, the nonlinearity of the system and the destructiveness of deception attacks, the dynamic output-feedback control in the framework of FMPC is adopted, meanwhile, the worst-case optimization problem over the infinite moving horizon is formulated for the performance analysis and control synthesis. By means of the quadratic function approach and the singular value decomposition technique, the non-convexity caused by couplings between decisive variables is coped with and conditions are derived to suffice the terminal constraint set. In addition, to mitigate the destruction of attacks to the recursive feasibility, the inequality analysis technique is applied with the aid of the introduction of special scalars. Based on the establishments, a certain auxiliary optimization problem with solvability is put forward to find the desired controllers, and sufficient conditions are obtained to guarantee that the underlying system subject to deception attacks under the proposed FMPC-based controllers is mean square secure in H2-sense. Finally, an illustrative example is used to demonstrate the validity of the proposed methods.

Economic optimization and predictive control for nonlinear systems using Lyapunov based nonlinear cost function design

Optimal control of physical processes depends on the formulation of objective functions and contemplation of appropriate predictive control. In this article, we propose a new framework, to blend the concepts of the Lyapunov-based economic model predictive control (Ly-EMPC) scheme with the novel cost-efficient profit function design. Furthermore, Lyapunov’s stability criteria is employed to verify the existence of duality and optimality for the designed economic stage costs. This method differs from the existing EMPC techniques by implementing Lagrange multipliers in the rotated cost function as a measure of stability. The closed-loop performance using the regularized Ly-EMPC is validated against a standard tracking objective-based nonlinear MPC for three benchmark examples through an econometrics gauge factor. Interestingly, in contrast to tracking NMPC, the proposed Ly-EMPC scheme excels in economic performance, while the formation of nonlinear economic stage cost is conceptually simpler, novel, and practically profitable.

Modeling and predictive control of nonlinear processes using transfer learning method

AbstractThis work develops a transfer learning (TL) framework for modeling and predictive control of nonlinear systems using recurrent neural networks (RNNs) with the knowledge obtained in modeling one process transferred to another. Specifically, transfer learning uses a pretrained model developed based on a source domain as the starting point, and adapts the model to a target process with similar configurations. The generalization error for TL‐based RNN (TL‐RNN) is first derived to demonstrate the generalization capability on the target process. The theoretical error bound that depends on model capacity and the discrepancy between source and target domains is then utilized to guide the development of pretrained models for improved model transferability. Subsequently, the TL‐RNN model is utilized as the prediction model in model predictive controller (MPC) for the target process. Finally, the simulation study of chemical reactors via Aspen Plus Dynamics is used to demonstrate the benefits of transfer learning.

Machine learning-based predictive control of nonlinear time-delay systems: Closed-loop stability and input delay compensation

The purpose of this work is to study machine-learning-based model predictive control of nonlinear systems with time-delays. The proposed approach involves initially building a machine learning model (i.e., Long Short Term Memory (LSTM)) to capture the process dynamics in the absence of time delays. Then, an LSTM-based model predictive controller (MPC) is designed to stabilize the nonlinear system without time delays. Closed-loop stability results are then presented, establishing robustness of this LSTM-based MPC towards small time-delays in the states. To handle input delays, we design an LSTM-based MPC with an LSTM-based predictor that compensates for the effect of input delays. The predictor is used to predict future states using the process measurement, and then the predicted states are used to initialize the LSTM-based MPC. Stabilization of the time-delay system with both state and input delays around the steady state is achieved through the featured design. The approach is applied to a chemical process example, and its performance and robustness properties are evaluated via simulations.

Funnel Model Predictive Control for Nonlinear Systems with Relative Degree One

Funnel Model Predictive Control for Nonlinear Systems with Relative Degree One

On Designing the Event-Triggered Multistep Model Predictive Control for Nonlinear System Over Networks With Packet Dropouts and Cyber Attacks

In this article, the event-triggered multistep model predictive control for the discrete-time nonlinear system over communication networks under the influence of packet dropouts and cyber attacks is studied. First, the interval type-2 Takagi-Sugeno fuzzy model is applied to express the discrete-time nonlinear system and an event-triggered mode, which is capable of determining whether the sampled signal ought to be delivered into the unreliable network, is designed to economize communication resources. Second, two Bernoulli processes are introduced to represent the randomly happening packet dropouts in the unreliable network and the randomly occurring deception attacks on the actuator side from the adversaries. Third, under the assumption that the system states are unmeasurable, a multistep parameter-dependent model predictive controller is synthesized via optimizing one series of feedback laws for a given period of time, which leads to improved control performance than that of the one-step approach. Moreover, the results on the recursive feasibility and closed-loop stability related to the networked system are achieved, which explicitly consider the external disturbance and input constraint. Finally, simulation experiments on the mass-spring-damping system are carried out to illustrate the rationality and effectiveness of the provided control strategy.

Online machine learning modeling and predictive control of nonlinear systems with scheduled mode transitions

AbstractThis work develops a model predictive control (MPC) scheme using online learning of recurrent neural network (RNN) models for nonlinear systems switched between multiple operating regions following a prescribed switching schedule. Specifically, an RNN model is initially developed offline to model process dynamics using the historical operational data collected in a small region around a certain steady‐state. After the system is switched to another operating region under a Lyapunov‐based MPC with suitable constraints to ensure satisfaction of the prescribed switching schedule policy, RNN models are updated using real‐time process data to improve closed‐loop performance. A generalization error bound is derived for the updated RNN models using the notion of regret, and closed‐loop stability results are established for the switched nonlinear system under RNN‐based MPC. Finally, a chemical process example with the operation schedule that requires switching between two steady‐states is used to demonstrate the effectiveness of the proposed RNN‐MPC scheme.

Event-based model predictive control for nonlinear systems with dynamic disturbance

In this paper, we investigate the event-based model predictive control (MPC) for constrained nonlinear systems with dynamic disturbance. An event-triggered disturbance prediction MPC (DPMPC) scheme and a self-triggered counterpart, which explicitly consider the disturbance dynamics, are proposed. For the event-triggered DPMPC scheme, the triggering condition relying on the state prediction error and the predicted disturbance sequence, updates at each time step based on the system states. For the self-triggered DPMPC scheme, the next triggering instant is determined by using the optimal state sequence and predicted disturbance sequence. In both event-based schemes, the optimal control problems are solved only at triggering instants, thus reducing the consumption of computational resource. The effectiveness of the two schemes is demonstrated by a simulation example.

Self‐triggered predictive control of nonlinear systems using approximation model

AbstractThis article studies discrete‐time model predictive control of continuous‐time nonlinear systems with measurement noises and exogenous disturbances. We consider the co‐design of discrete‐time finite horizon optimal control problem (FHOCP) and the associated self‐triggering schemes that schedule the time instants for sampling the state and computing the FHOCP. The state‐dependent nature of the self‐triggered scheduling can not only dynamically adjust the inter‐sampling period according to the system status, but dynamically discretize the model used in the FHOCP as well, in order to reduce the complexity of the FHOCP if designed appropriately. It is shown that the system can be stabilized with uniform ultimate boundedness, as long as the scheduling scheme matches the approximation model such that the one‐step approximation error between the predicted state and the actual state is below a threshold related to the cost function. These results can be applied to most existing model approximation methods with either fixed or time‐varying sampling rates.

Model-free predictive control of nonlinear systems under False Data Injection attacks

The control systems attacked by False Data Injection (FDI) force the equipment out of action. This paper presents a model-free predictive control framework based on polynomial regressors that attenuates adverse effects of FDI attacks on control systems modeled by the nonlinear systems. An FDI attacker targets at tampering the state estimation results, thereby destroy the security of control systems. In order to guarantee its stability, the polynomial regression vectors are considered. The novel point of this paper is the improvement of existing attack datasets by the polynomial regression which combines previous recorded datasets and attack datasets. The polynomial regression vectors can ensure the stable operation of the nonlinear systems guaranteed under FDI attacks. Finally, the simulation is employed to verify our points.

Multiple Model Predictive Control for Nonlinear Systems Based on Self-Balanced Multimodel Decomposition

Multiple Model Predictive Control for Nonlinear Systems Based on Self-Balanced Multimodel Decomposition

Hybrid Koopman model predictive control of nonlinear systems using multiple EDMD models: An application to a batch pulp digester with feed fluctuation

In the pulping process, feed fluctuations often occur due to the supply of raw materials from various and unconventional sources, such as recycle to meet the increasing market demand for paper, and strict environmental regulations. However, such feed fluctuation can significantly extend the operating range of the process, which may cause differing local dynamics that degrade the performance of a single model-based controller. Motivated by these concerns, in this work, a hybrid Koopman model predictive control (KMPC) framework for a batch pulping process is developed to regulate the Kappa number and the cell wall thickness (CWT) of fibers to produce pulp with desired properties in the presence of feed fluctuations. Specifically, multiple local models are constructed by clustering the time-series operation data from the pulping process and identifying lifted state–space models for each cluster using extended dynamic mode decomposition (EDMD). Subsequently, a local EDMD model-based controller for each cluster is developed. In the closed-loop system, one local controller is selected based on the current system state at every time step. Consequently, in the numerical experiments, the derived multiple EDMD models successfully predicted the local behavior of the batch pulping process, and the developed hybrid KMPC system was able to obtain the desired Kappa number and CWT values in the presence of feed fluctuations.

Membership-function-dependent model predictive control for nonlinear systems in a piecewise-fuzzy framework

This paper is concerned with the membership-function-dependent model predictive control (MPC) problem for a class of Takagi-Sugeno (T-S) fuzzy systems with hard constraints. In order to design a set of membership-function-dependent controllers in the framework of MPC, the original T-S fuzzy systems are equivalently converted into a piecewise-fuzzy model by means of the partition method of the premise variable space. Then, instead of calculating vertices to construct the fuzzy MPC controller as adopted in the traditional T-S fuzzy control strategies, some staircase membership functions are employed to approximate the continuous membership functions, such that the information of membership functions can be adequately considered in the control synthesis to reduce the conservatism of the controller design. Furthermore, associated with the piecewise-fuzzy model, an online optimization problem based on the membership-function-dependent terminal constraint set is formulated in terms of the piecewise-Lyapunov-function-based MPC to calculate the feedback gains. Besides, by employing polynomial constraints based on the shape information of the membership functions, errors between the staircase membership functions and the original continuous membership functions are taken into account for facilitating the feasibility investigation and the stability analysis. Finally, two illustrative examples are used to illustrate the validity of the proposed membership-function-dependent MPC strategy.