A software framework for stochastic model predictive control of nonlinear continuous-time systems (GRAMPC-S)

Stochastic linear Model Predictive Control with chance constraints – A review

A stochastic model predictive control (MPC) framework is presented in this paper for nonlinear affine systems with stability and feasibility guarantee. We first introduce the concept of stochastic control Lyapunov–Barrier function (CLBF) and provide a method to construct CLBF by combining an unconstrained control Lyapunov function (CLF) and control barrier functions. The unconstrained CLF is obtained from its corresponding semi‐linear system through dynamic feedback linearization. Based on the constructed CLBF, we utilize sampled‐data MPC framework to deal with states and inputs constraints, and to analyze stability of closed‐loop systems. Moreover, event‐triggering mechanisms are integrated into MPC framework to improve performance during sampling intervals. The proposed CLBF based stochastic MPC is validated via an obstacle avoidance example.

This paper deals with an adaptive robust tracking control using a multilayer neural network (NN) for a class of nonlinear dynamic systems with unknown time varying state delays. Typical adaptive NN backstepping controllers for uncertain nonlinear systems with time-delay give rise to computation complexity caused by the the repeated derivatives of virtual controllers and nonlinear functions. Moreover, the combined techniques usually result in only uniformly ultimately bounded (UUB) stability caused by the inherent NN approximation error. This paper presents a control scheme that uses an integral sliding mode control as a feedback term and an adaptive neural controller as a feedforward term based on the desired compensation adaptive law (DCAL) technique. First, we develop a new DCAL formulation which avoids the explosion of complexity caused by the general NN backstepping scheme to compensate for nonlinear system uncertainties, bounded system disturbances, and unknown state time delays. Then, using a Lyapunov-Krasovskii (LK) functional, it is shown that the proposed controller renders the class of uncertain nonlinear time-delay systems asymptotically stable.

High penetration of distributed renewable energy introduces significant uncertainties to active distribution networks. Optimal control methods accounting for inherent uncertainties are needed to facilitate economic and reliable operation of active distribution networks. This paper proposes a stochastic receding horizon control method based on modified stochastic model predictive control framework to integrate high penetration of distributed generation. Multiple controllable resources are jointly optimized over a finite prediction horizon while ensuring relevant security restrictions. The simplified Z-bus sensitivity for active distribution networks is developed for computationally efficient estimation of system nonlinearity with high accuracy, and is combined with the sequential linear programming to iteratively derive the linear state space model for compensation of cumulative modeling errors. Furthermore, the voltage limitations are reformulated as chance constraints to indicate the probabilistic reliability index of voltage qualification rate, and achieve tradeoffs between cost reduction and voltage regulation. The affine-disturbance feedback control policy is leveraged here to enforce close-loop control performance and analytically transform intractable chance constraints into second-order cone constraints. Comprehensive case studies based on 33-bus and 123-bus distribution systems are carried out to demonstrate the capability and effectiveness of the proposed approach in terms of modeling accuracy, control performance, cost reduction, and method scalability. The proposed approach can effectively enforce voltage regulation against uncertainties with the prescribed probability level. Control costs and constraint violation can be reduced compared with deterministic model predictive control and open-loop control strategies.

Synthesis of smoother robust control for nonlinear uncertain dynamical systems is considered. The existing saturation-type robust controller guarantees asymptotic stability or stability of uniform ultimate boundedness. It is shown that, mainly due to the part of uncertainties that is bounded by a constant, the controller may potentially have a very large rate of change in order to achieve a small ultimate bound for the state. This study, on its gain or the property of its partial derivative, reveals the need for designing smoother nonlinear robust controller along two directions. First, in a recursive design of robust control, fictitious robust control must be designed to be as smooth as possible so that the actual robust control will not have too large of a magnitude for implementation purposes. Second, if the uncertainties are known to have bounded rate of change, it is both practically and theoretically important to design a robust controller which rate of change is limited by that of the uncertainties. The objective of this paper is to design smoother robust controls by providing results on those two topics. >

This study presents a closed‐loop hierarchical control strategy for a group of connected and autonomous hybrid electric vehicles (HEVs) with the purpose of optimising fuel economy while guaranteeing traffic mobility and vehicle safety. In the hierarchical control architecture, a decentralised stochastic model predictive control (SMPC)‐based higher level controller incorporating signal phase and timing information is formulated to optimise the velocity profile of each vehicle, and an adaptive equivalent consumption minimisation strategy based lower level controller is employed for the energy management control of the HEVs. Creatively, random errors of the control variable are considered in the SMPC framework. The errors are discretised and modelled as a Markov process and the state transition matrix of the errors is generated randomly to capture the error transition dynamics. To solve the SMPC problem more efficiently, a scenario‐based SMPC is employed. Moreover, the propulsion and recuperation efficiencies of the lower level controller are calculated at each time step with measurable variables, and fed back to the higher level controller for velocity optimisation in next time step. Simulation results validate the control effectiveness and advantages of the proposed control architecture.

Decentralized neural identification and control for uncertain nonlinear systems: Application to planar robot

This study focuses on demonstrating the effectiveness and efficiency of the Stochastic Sequential Model Predictive Control (MPC) framework within the context of the Hjartdøla hydropower system. Multistage MPC, while effective in managing uncertainty, poses challenges due to its high computational demands and complex optimal control problems, particularly in applications requiring long-term forecasting, such as hydropower systems. Through a comparative simulation study with multistage MPC, this paper highlights the superior feasibility and computational speed of the Stochastic Sequential MPC framework. This work contributes to the broader understanding of MPC applications in hydropower systems.

Swarm intelligence–based distributed stochastic model predictive control for transactive operation of networked building clusters

Wind energy has been drawing considerable attention in recent years. However, due to the random nature of wind and high failure rate of wind energy conversion systems (WECSs), how to implement fault-tolerant WECS control is becoming a significant issue. This paper addresses the fault-tolerant control problem of a WECS with a probable actuator fault. A new stochastic model predictive control (SMPC) fault-tolerant controller with the Conditional Value at Risk (CVaR) objective function is proposed in this paper. First, the Markov jump linear model is used to describe the WECS dynamics, which are affected by many stochastic factors, like the wind. The Markov jump linear model can precisely model the random WECS properties. Second, the scenario-based SMPC is used as the controller to address the control problem of the WECS. With this controller, all the possible realizations of the disturbance in prediction horizon are enumerated by scenario trees so that an uncertain SMPC problem can be transformed into a deterministic model predictive control (MPC) problem. Finally, the CVaR object function is adopted to improve the fault-tolerant control performance of the SMPC controller. CVaR can provide a balance between the performance and random failure risks of the system. The Min-Max performance index is introduced to compare the fault-tolerant control performance with the proposed controller. The comparison results show that the proposed method has better fault-tolerant control performance.

Takagi-Sugeno (T-S) fuzzy model is a system described by a set of fuzzy if-then rules which give local linear representations of the underlying nonlinear system. Usually, a fuzzy control law is constructed by a set of controllers for each local linear representation to carry out the control of a T-S fuzzy system. While, simultaneous control addresses the stability and performance of multiple plants under a single feedback controller. In this paper, simultaneous linear control for nonlinear uncertain systems described by general T-S fuzzy model is considered. A procedure for simultaneous control of T-S fuzzy systems with parametric uncertainties is presented, by which robust reliability method is adopted to deal with uncertainties. Simultaneously linear controller design for an uncertain T-S fuzzy system is carried out by solving a problem of robust reliability based optimization, in which, both the robustness with respect to uncertainties and the control cost can be taken into account simultaneously. It is demonstrated by numerical simulation of the chaotic Lorenz system control that the presented method is effective and feasible.

A two-step controller synthesis method is proposed in this article for a class of uncertain non-linear systems described by piecewise affine differential inclusions. In the first step, a robust linear controller is designed for the linear differential inclusion that describes the dynamics of the non-linear system close to the equilibrium point. In the second step, a stabilising piecewise affine controller is designed that coincides with the linear controller in a region around the equilibrium point. The proposed method has two objectives: global stability and local performance. It thus enables us to use well-known techniques in linear control design for local stability and performance while delivering a global piecewise affine controller that is guaranteed to stabilise the non-linear system. To construct the required theoretical framework, a stability theorem for non-smooth Lyapunov functions is presented and proved. The new method will be applied to two examples.

In this article, a novel self-evolving data cloud-based proportional integral derivative (PID) (SEDCPID) like controller is proposed for uncertain nonlinear systems. The proposed SEDCPID controller is constructed by using fuzzy rules with nonparametric data cloud-based antecedence and PID-like consequence. The antecedent data clouds adopt the relative data density to represent the fuzzy firing strength of input variables instead of the explicit design of the membership functions in the classical sense. The proposed SEDCPID controller has the advantages of evolving structure and adapting parameter concurrently in an online manner. The density and distance information of data clouds are proposed to achieve the addition and deletion of data clouds and also a stable recursive method is proposed to update the parameters of the PID-like subcontrollers for the fast convergence performance. Based on the Lyapunov stability theory, the stability of the proposed controller is proven and the proof shows the tracking errors converge to a small neighborhood. Numerical and experimental results illustrate the effectiveness of the proposed controller in handling the uncertain nonlinear dynamic systems.

Self-triggering adaptive optimal control for nonlinear systems based on encoding mechanism

This paper deals with the problem of robust control for Uncertain Nonlinear System via fuzzy control approach. First, a fuzzy controller with robust control capability is proposed. For avoiding complex controller parameters adjustment, Multi-objective particle swarm Optimization (PSO) algorithm is adopted during the controller design. Secondly, the multi-model is employed to represent the nonlinear system with norm-bounded parameter uncertainties. Finally, a simulation example illustrates the effectiveness and the feasibility of the proposed approach.