Online Optimization Problem Research Articles

Robust model predictive control of sampled-data Lipschitz nonlinear systems: Application to flexible joint robots

Controlling flexible joint robots has drawn the attention of many industry professionals during the past two decades. It is a difficult task because various structural features that make the control of rigid robots easier, such as passivity of the motor torque to link velocity, full actuation, and separate control of each joint, are lost when we consider joint flexibility in the control design of these robots. However, we must consider joint flexibility while designing the controller; otherwise, the system may become unstable. In this article, we devise a robust model predictive controller scheme for flexible joint robots modeled as sampled-data Lipschitz nonlinear systems with unknown bounded disturbances. It is assumed that the state of the system is accessible for feedback. Therefore, a state-feedback control law is designed using a robust stability criterion and can be computed by solving an online optimization problem. The control law optimizes the performance index by reducing its worst-case value. The proposed control design scheme is applied to the one-link flexible joint robot. Simulation results validate the effectiveness of the controller in handling nonlinearities while minimizing the effect of unknown bounded disturbances.

Efficient nonlinear model predictive and active disturbance rejection control for trajectory tracking of unmanned vehicles

This article proposes an efficient trajectory tracking control strategy of unmanned vehicles. The method is based on nonlinear model predictive control (NMPC) and active disturbance reject control (ADRC). The designed control algorithm considers three challenges including nonlinear characteristics, multiple constraints, and external disturbance. First, NMPC method is presented for the nonlinear vehicle model with multiple constraints. To relax inequality constraints and reduce the heavy calculation burden, the penalty term with the variable factor is added to the cost function, an improved continuous/generalized minimum residual method is proposed to solve NMPC online optimization problem. Then, an ADRC scheme is designed to estimate the unknown disturbance via extended state observer, and compensate them by feedback control law in real time, the corresponding parameters are obtained by a novel particle swarm optimization algorithm to further improve the control precision. It ensures the stability to a certain extent. Finally, the results indicate the designed algorithm can raise the calculation efficiency and meet the real-time requirements, and obviously increase the tracking accuracy and robustness performance.

Lyapunov-based robust model predictive tracking controller design for discrete-time linear systems with ellipsoidal terminal constraint

In this article, a robust model predictive control (MPC) method is introduced based on Lyapunov-theory to control the discrete-time uncertain linear systems due to external disturbances. The structure of the proposed controller consists of two parts designed based on the offline/online schemes. In the offline-scheme, an H ∞ -based robust tracking controller is provided to satisfy the robust and tracking performance. Based on the H ∞ performance, a new linear matrix inequality problem is developed to calculate the offline-controller gains. By considering the Lyapunov-function of the offline-controller as the terminal cost of the MPC and the designed offline-controller, the online optimisation problem (OOP) is structured. This means, the MPC is designed based on the stabilised system to satisfy the physical limitations of the overall controller and the system states. Moreover, to improve the feasibility problem of the OOP, an ellipsoidal terminal constraint is added to the proposed MPC problem. Therefore, compared with the existing min–max MPC and tube MPC, the advantages of the overall controller are feasibility improvement and reducing the computational complexity with less conservatism while the robustness against parametric uncertainties and disturbances is achieved. Two simulation examples are employed to show the superiorities of the proposed robust MPC.

Deep Learning based Model Predictive Controller on a Magnetic Levitation Ball System

The magnetic levitation (maglev) ball system is a prototypical Single-Input-Single-Output (SISO) system, characterized by its pronounced nonlinearity, rapid response, and open-loop instability. It serves as the basis for many industrial devices. For describing the dynamics of the maglev ball system precisely in the pseudo linear model, the long short-term memory (LSTM) based auto-regressive model with exogenous input variables (LSTM-ARX) is proposed. Firstly, the LSTM network is modified by incorporating the auto-regressive structure with respect to sequence input, allowing it to deduce a locally linearized model without the need for Taylor expansion. Then, the LSTM-ARX model is transformed into a linear parameter varying (LPV) state space model, and upon this foundation, a model predictive controller (MPC) is proposed. Specifically, when deducing the MPC, the deep learning-based model is linearized by fixing its state input at the current state, so that the nonlinear, non-convex optimization problem can be converted to a finite-horizon quadratic programming problem, thereby deriving the explicit form of MPC. To further enhance the efficiency of the controller in real-time control tasks, a predictive functional controller (PFC) is proposed. It employs multiple nonlinear functions to fit the control sequence, thereby reducing the number of decision variables of the on-line optimization problem in MPC. The proposed controller was successfully applied to the real-time control of the maglev ball system. Simulation and real-time control experiments have validated the improvement in transient performance and efficiency of the LSTM-ARX model-based PFC (LSTM-ARX-PFC).

Model predictive-based DNN control model for automated steering deployed on FPGA using an automatic IP generator tool

With the increase in the non-linearity and complexity of the driving system’s environment, developing and optimizing related applications is becoming more crucial and remains an open challenge for researchers and automotive companies alike. Model predictive control (MPC) is a well-known classic control strategy used to solve online optimization problems. MPC is computationally expensive and resource-consuming. Recently, machine learning has become an effective alternative to classical control systems. This paper provides a developed deep neural network (DNN)-based control strategy for automated steering deployed on FPGA. The DNN model was designed and trained based on the behavior of the traditional MPC controller. The performance of the DNN model is evaluated compared to the performance of the designed MPC which already proved its merit in automated driving task. A new automatic intellectual property generator based on the Xilinx system generator (XSG) has been developed, not only to perform the deployment but also to optimize it. The performance was evaluated based on the ability of the controllers to drive the lateral deviation and yaw angle of the vehicle to be as close as possible to zero. The DNN model was implemented on FPGA using two different data types, fixed-point and floating-point, in order to evaluate the efficiency in the terms of performance and resource consumption. The obtained results show that the suggested DNN model provided a satisfactory performance and successfully imitated the behavior of the traditional MPC with a very small root mean square error (RMSE = 0.011228 rad). Additionally, the results show that the deployments using fixed-point data greatly reduced resource consumption compared to the floating-point data type while maintaining satisfactory performance and meeting the safety conditions

Proximal-based recursive implementation for model-free data-driven fault diagnosis

We present a novel problem formulation for model-free data-driven fault diagnosis, in which possible faults are diagnosed simultaneously to identifying the linear time-invariant system. This problem is practically relevant for systems whose model cannot be identified reliably prior to diagnosing possible faults, for instance when operating conditions change over time, when a fault is already present before system identification is carried out, or when the system dynamics change due to the presence of the fault. A computationally attractive solution is proposed by solving the problem using unconstrained convex optimization, where the objective function consists of three terms of which two are non-differentiable. An additional recursive implementation based on a proximal algorithm is presented in order to solve the optimization problem online. The numerical results on a buck converter show the application of the proposed solution both offline and online.

Improved Model Predictive Speed Control of a PMSM via Laguerre Functions

This paper proposes a model predictive speed control strategy for a surface-mounted permanent magnet synchronous motor by applying Laguerre functions. The model predictive controller (MPC) incorporates an integrator. A quadratic programming procedure is applied to solve the constrained optimization problem online. The paper also provides a solution for stability. The performance efficiency of the proposed scheme is validated by comparing the results with the performance of an optimal linear quadratic regulator, conventional state-space model predictive control, and a simple MPC algorithm with integral action. Extensive simulation results confirm the efficacy of the proposed scheme, showing that it achieves good steady-state performance while maintaining a fast dynamic response.

A data‐driven Bayesian approach for optimal dynamic product transitions

AbstractIn the processing industry, dynamic product transitions are essential for achieving high product quality, minimizing the use of raw materials and energy and reducing production costs. However, optimizing dynamic product transitions is a challenging task due to the complex dynamics of the process and the uncertainty in the measurements. In this article, a data‐driven Bayesian approach for optimal dynamic product transitions is proposed. The proposed approach is based on a dynamic optimization problem that is solved using a Bayesian optimization algorithm. One of the advantages of this approach for process optimization tasks is that it does not require a first‐principles dynamic mathematical model for drawing optimal solutions. The approach is applied to three case studies, and the results are comparable in performance quality with those obtained using a traditional gradient‐based optimization approach. The results show that the proposed approach is able to find optimal transition trajectories that meet the product composition requirements using smooth control actions. The approach is also able to cope with noisy measurements, which is an important feature in real‐world applications. The proposed approach has several advantages over traditional optimization approaches, including being data driven, able to cope with noisy measurements, computationally efficient, and it requires modest computational effort. Complex online optimal control problems can benefit from adopting a data‐driven Bayesian optimization scheme.

Distributed Lyapunov-Based Model Predictive Control for AUV Formation Systems with Multiple Constraints

This paper focuses on the formation tracking issue of autonomous underwater vehicles (AUVs) subject to multiple constraints in three-dimensional space. We developed a novel distributed Lyapunov-based model predictive controller (DLMPC) with a fast finite-time extended state observer (FFTESO). Initially, the external disturbances and internal uncertainties of each AUV were precisely compensated using the designed FFTESO. Subsequently, we proposed DLMPC-based position tracking and velocity tracking controllers, which solved an online optimization problem to determine optimal velocities and control forces. This hierarchical framework effectively managed system constraints, such as state constraints and actuator saturation. Additionally, the Lyapunov-based backstepping control law was applied to construct stability constraints in the distributed optimization problem, ensuring the recursive feasibility and closed-loop system stability of the proposed scheme. Sufficient conditions and attraction regions to ensure stability were explicitly provided. Finally, the simulation results demonstrated that the proposed method improved both the convergence speed and tracking accuracy by at least 30% compared to other methods.

Distributed Online Constrained Optimization With Feedback Delays.

We investigate multiagent distributed online constrained convex optimization problems with feedback delays, where agents make sequential decisions before being aware of the cost and constraint functions. The main purpose of the distributed online constrained convex optimization problem is to cooperatively minimize the sum of time-varying local cost functions subject to time-varying coupled inequality constraints. The feedback information of the distributed online optimization problem is revealed to agents with time delays, which is common in practice. Every node in the system can interact with neighbors through a time-varying sequence of directed communication topologies, which is uniformly strongly connected. The distributed online primal-dual bandit push-sum algorithm that generates primal and dual variables with delayed feedback is used for the presented problem. Expected regret and expected constraint violation are proposed for measuring the performance of the algorithm, and both of them are shown to be sublinear with respect to the total iteration span T in this article. In the end, the optimization problem for the power grid is simulated to justify the proposed theoretical results.

Distributed online optimisation in unknown dynamic environment

In this paper, the distributed online optimisation problem is considered in an unknown dynamic environment. Compared with the existing results, an unknown dynamic environment causes the problem to be more challenging. An online optimisation algorithm is designed based on distributed mirror descent and distributed average tracking technology. The analysis of dynamic regret is presented and a bounded regret is obtained. Some simulations are given to verify the validity of the designed algorithm.

Differentially private distributed online optimization via push-sum one-point bandit dual averaging

This paper focuses on the distributed online optimization problem in multi-agent systems considering privacy preservation. Each agent exchanges local information with neighboring agents on the strongly connected time-varying directed graphs. Since the process of information transmission is prone to information leakage, a distributed push-sum dual averaging algorithm based on the differential privacy mechanism is proposed to protect the privacy of the data. In addition, to handle situations where the gradient information of the node cost function is unknown, the one-point gradient estimation is designed to calculate the true gradient information and guide the update of the decision variables. With the appropriate choice of the stepsizes and the exploration parameters, the algorithm can effectively protect the privacy of agents while achieving sublinear regret with the convergence rate O(T34). Furthermore, this paper also explores the effect of one-point estimation parameters on the regret in the online setting and investigates the relation between the convergence effect of individual regret and differential privacy levels. Finally, several federated learning experiments were conducted to verify the efficacy of the algorithm.

Approximate SDD-TMPC with Spiking Neural Networks: An Application to Wheeled Robots

Model Predictive Control (MPC) optimizes an objective function within a prediction window under constraints. In the presence of bounded disturbances, robust versions are used. Recently, a promising robust MPC was introduced that outperforms SOTA approaches. However, solving the optimization problem online is computationally expensive. An efficient approximation method, such as neural networks (NN), can be substituted to accelerate the online computation. There are discrepancies between the control inputs due to the approximation. We propose to model them as bounded state-dependent disturbances to robustly control nonlinear wheeled robots. We consider a spiking NN to ensure that small robots could use it.

PSO and NNPC-based integrative control allocation for dynamic positioning ships with thruster constraints

In order to overcome the issues of thrust saturation, uneven energy consumption and suboptimal control associated with the traditional hierarchical control strategy employed in ship dynamic positioning (DP) systems, an integrative control allocation strategy is proposed by combining the low-level thrust allocation into the high-level motion controller. The proposed scheme is implemented via the neural network-based nonlinear predictive control (NNPC) and solved via an improved particle swarm optimization (PSO) algorithm. A data-driven recurrent neural network (RNN) ship motion and thrust allocation model is off-line identified based only on the current and historical measured output and thrust input data. Utilizing the established RNN model to predict the future multi-step outputs, and combining the motion control, the thruster energy consumption and the thruster wear and tear as an integrative control objective, the integrated NNPC online optimization problem with thruster constraints is established and solved by an improved evolution algorithm of PSO. Simulations are respectively carried out for setpoint regulation and trajectory tracking. Simulation results well demonstrate the feasibility and superiority of the proposed control strategy.

Periodic Event-Triggered MPC for Continuous-Time Nonlinear Systems With Bounded Disturbances

This paper is concerned with periodic event-triggering laws in robust model predictive control (MPC) for continuous-time constrained nonlinear systems. The online optimal control problem solved at triggering times is introduced. A periodic static event-triggering condition, in which a fixed sampling time interval plays an important role in avoiding Zeno behavior, is presented to alleviate continuous checking of event detections. Then a periodic dynamic event-triggering condition is investigated to further enlarge the minimal inter-execution time. Single-mode MPC with a prediction horizon larger than the control one is considered. Sufficient conditions of recursive feasibility for the online optimal control problem are derived. In order to relax the sufficient condition of stability, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Ultimately Boundedness</i> properties are utilized in stability analysis. Finally, numerical simulation is provided to demonstrate the effectiveness of the proposed methods.

Privacy preserving distributed online projected residual feedback optimization over unbalanced directed graphs

The distributed online optimization (DOO) problem with privacy guarantees over an unbalanced directed graph is considered in this paper, where the cost function is explicitly unknown. To solve this problem, a distributed online one-point residual feedback optimization algorithm based on differential privacy is designed. For the objective function explicitly unknown, this algorithm employs the residual feedback to estimate the true gradient information. In addition, only the row stochastic adjacency matrix utilized releases the requirement of double stochastic weighting matrices, making the algorithm easier to implement on directed graphs. Theoretical results show that the algorithm not only protects the privacy information of the nodes but also achieves the same sublinear rate regret as the DOO algorithm based on two-point feedback. Moreover, our regret bound depends on weaker assumptions than the traditional DOO algorithm based on one-point feedback. Finally, the simulation results verify the effectiveness of the algorithm.

Stable Communications in Green Unmanned Aerial Relaying Systems

Green unmanned aerial relaying (UAR) systems, featuring solar-powered unmanned aerial vehicles (UAVs) to provide agile and flexible communication services, have recently gained significant attention for their wide applications. Nonetheless, due to the essential dynamics of wireless channel conditions and solar energy supply, maintaining system stability is a critical concern in practical green UAR systems, as fluctuation in these two factors can significantly affect communication performance. To address this issue, this paper proposes a scheme design for stable communications in a UAR system with dynamic solar energy supply and wireless channel conditions. Specifically, we first formulate an optimization problem, called OFL, to control the power allocation and flow data rate assignment to minimize long-term time-averaged energy consumption while ensuring the demanded data rate and system stability. Considering that solving OFL needs complete prior information about the solar power supply and wireless channel conditions, which is hardly acquired, we then explore the Lyapunov theory to transform OFL as an online optimization problem, called ONL. To find the solution to ONL, we subsequently design a distributed algorithm that enables each UAV to make its own decision locally, thus reducing the computational overhead of each UAV. Particularly, we prove that stability can be guaranteed by employing the designed algorithm. Extensive simulations are conducted to show the performance achieved by the proposed scheme.

Cooperative Double-Layer Genetic Programming Hyper-Heuristic for Online Container Terminal Truck Dispatching

In a marine container terminal, truck dispatching is a crucial problem that impacts on the operation efficiency of the whole port. Traditionally, this problem is formulated as an offline optimisation problem, whose solutions are, however, impractical for most real-world scenarios primarily because of the uncertainties of dynamic events in both yard operations and seaside loading–unloading operations. These solutions are either unattractive or infeasible to execute. Herein, for more intelligent handling of these uncertainties and dynamics, a novel cooperative double-layer genetic programming hyper-heuristic (CD-GPHH) is proposed to tackle this challenging online optimisation problem. In this new CD-GPHH, a novel scenario genetic programming (GP) approach is added on top of a traditional GP method that chooses among different GP heuristics for different scenarios to facilitate optimised truck dispatching. In contrast to traditional arithmetic GP (AGP) and GP with logic operators (LGP) which only evolve on one population, our CD-GPHH method separates the scenario and the calculation into two populations, which improved the quality of solutions in multi-scenario problems while reducing the search space. Experimental results show that our CD-GPHH dominates AGP and LGP in solving a multi-scenario function fitting problem as well as a truck dispatching problem in container terminal.

Modelling and Quadratic Dynamic Matrix Control of a DC Motor System with Non-linearities Consideration

Abstract: DC motor’s models that have often been used to test and validate control algorithms with, even the most robust ones, have usually been simplistic: limiting the generalization and applicability of the results obtained. This article presents a modeling approach for a DC motor system consisting of a Buck converter and driving a load at the end of the shaft, in which the nonlinearities of each of its sub-parts are taken into account in the model of the overall system. The model was simulated and its step responses were compared to those of the linear model and conventional nonlinear models with a significant difference (Pvalue&lt;0.001). This model was then used to test the Fast Nonlinear Quadratic Dynamic Matrix Control (FNLQDMC). The letter has demonstrated good performance in simulation in terms of setpoint tracking, constraint handling, and computational load savings during online optimization problem solving

Improved Prediction Dynamics for Robust MPC

This paper proposes a new MPC control scheme for polytopic uncertain and/or time-varying systems with state and input constraints. The MPC policies we consider employ: i) the intersection of ellipsoids to characterize the domain of attraction, ii) a time-varying Lyapunov function to bound from above the cost function, iii) a tailored ADMM algorithm to solve efficiently the online optimization problem. With respect to other well known techniques, the main advantage of the new approach is the reduced conservativeness.