Terminal Cost Research Articles

Model predictive control for autonomous vehicle path tracking through optimized kinematics

Tracking performance and stability of path tracking is crucial for unmanned vehicles' navigational tasks. Researches on vehicle path tracking controllers primarily rely on dynamic models. In contrast, there are less designs and researches that focus on path tracking controller based on kinematic model. This scarcity may stem from the perceived inadequacy in the accuracy of vehicle kinematic models. This research introduces a novel method for modifying the traditional vehicle kinematic model by incorporating a front wheel steering angle modification function to enhance tracking performance of path tracking controllers based on kinematic models. In terms of control strategy selection, the study opted for nonlinear model predictive control with terminal cost. A controller which is founded on the modified model was used on a simulated vehicle on CarSim-Simulink platform to assess the tracking performance. The simulation was designed with a double-shift line reference trajectory and the initial speeds of the simulated vehicle are 5 m/s and 10 m/s, respectively. The results of the simulations indicate that employing a controller based on the modified model could reduce the peak tracking error by 76.45 % and 43.06 %, and the root mean square of the tracking error was reduced by 73.29 % and 36.06 %, respectively.

New stability theory of model predictive control: modified stage cost approach

This paper presents a promising new approach to establish the stability of finite receding horizon control with a terminal cost. Departing from the traditional approaches of using the property of the terminal cost or relaxed Lyapunov inequalities, this paper establishes its stability based on the property of a modified stage cost. First, we rotate the stage cost with the terminal cost. Then a one-step optimisation problem is defined based on this augmented stage cost. It is shown that a slightly modified Model Predictive Control (MPC) algorithm is stable if the value function of the augmented one-step cost (OSVF) is a Control Lyapunov Function (CLF). Stability for MPC algorithms with zero terminal cost or even negative terminal cost can be unified with this new approach. Combining it with the existing MPC stability theories, we are able to significantly relax the stability requirement on MPC and extend the stabilising MPC design space to the region that no existing MPC stability theories can cover. The proposed stage cost-based approach will help to further reduce the gap between stability theory and practical applications of MPC and other optimisation-based control methods.

Optimal Gliding Trajectories for Descent on Mars

In this paper, optimal gliding trajectories are analyzed, formulating an optimal control problem to be solved computationally via nonlinear programming. A multi-objective terminal cost function that minimizes the velocity, altitude, horizontal flight path angle, and the error of a desired terminal point is formulated. The results are validated via Monte Carlo simulations, analyzing the influence of the variations in the lift-to-drag ratios in the atmospheric entry. The results show feasible solutions for minimizing the distance to the desired final point, which have significant practical implications for the controlled gliding entry of a spaceplane on Mars. In this sense, the results also show that the longitudes of the landing points do not change too much for almost all trajectories, while the latitude of these points is in the interval of about 4 degrees for the majority of the trajectories. This suggests that it is possible to implement an optimal control approach with reasonable accuracy for the controlled gliding entry of a spaceplane on Mars, even in the presence of some uncertainties regarding the aerodynamic performance of the spacecraft, such as the lift-to-drag ratio.

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.

Budget impact analysis of introducing fruquintinib for metastatic colorectal cancer previously treated with fluoropyrimidine-, oxaliplatin-, and irinotecan-based chemotherapy and biologics in the United States from the payer perspective

Aims Fruquintinib is a selective small molecule tyrosine kinase inhibitor of vascular endothelial growth factor receptor (VEGFR)-1, -2, and -3 recently approved in the United States (US) for the treatment of adult patients with metastatic colorectal cancer (CRC) who have previously been treated with fluoropyrimidine-, oxaliplatin-, and irinotecan-based chemotherapy, an anti-VEGF biological therapy, and if RAS wild-type and medically appropriate, anti-epidermal growth factor receptor therapy. This study aimed to estimate the 5-year budget impact of fruquintinib from a US payer perspective (commercial and Medicare). Materials and methods A budget impact model was developed to compare two scenarios: a reference scenario in which patients received regorafenib, trifluridine/tipiracil, or trifluridine/tipiracil with bevacizumab and an alternative scenario in which patients received reference scenario treatments or fruquintinib. Market shares were evenly divided across available options. A 5-year time horizon and a hypothetical health plan of 1 million members was assumed. The model included epidemiological inputs to estimate the eligible population; clinical inputs for treatment duration, progression-free survival, overall survival, and adverse event (AE) frequency; and cost inputs for treatment, AEs, disease management, subsequent therapy, and terminal care costs. Budget impact was reported as total, per member per year (PMPY), and per member per month (PMPM). Results The model estimated an eligible population of 194 patients (39 per year) over 5 years. In the base case, the estimated 5-year budget impact of fruquintinib was $4,077,073 ($0.82 PMPY and 0.07 PMPM) for a commercial health plan. During the first year, the estimated budget impact was $627,570 ($0.63 PMPY and 0.05 PMPM). Results were robust across sensitivity analyses. PMPM costs from the Medicare perspective were greater than the base-case (commercial) ($0.17 vs. $0.07) due to higher incidence of CRC in that population. Conclusions Fruquintinib is associated with a low budget impact for payers based on proposed thresholds in the US.

Phase-Based and Lifetime Health System Costs of Care for Patients Diagnosed with Leukemia and Lymphoma: A Population-Based Descriptive Study.

Hematologic cancers, notably leukemias and lymphomas, pose significant challenges to healthcare systems globally, due to rising incidence rates and increasing costs. This study aimed to estimate the phase and lifetime health system total costs (not net costs) of care for patients diagnosed with leukemia and lymphoma in Ontario, Canada. We conducted a population-based study of patients diagnosed between 2005 and 2019, using data from the Ontario Cancer Registry linked with health administrative databases. Costs were estimated using a phase-based approach and stratified by care phase and cancer subtype. Acute lymphocytic leukemia (ALL) patients had the highest mean monthly initial (CAD 19,519) and terminal (CAD 41,901) costs among all cancer subtypes, while acute myeloid leukemia (AML) patients had the highest mean monthly cost (CAD 7185) during the continuing phase. Overall lifetime costs were highest for ALL patients (CAD 778,795), followed by AML patients (CAD 478,516). Comparatively, patients diagnosed with Hodgkin lymphoma (CAD 268,184) and non-Hodgkin lymphoma (CAD 321,834) had lower lifetime costs. Major cost drivers included inpatient care, emergency department visits, same-day surgeries, ambulatory services, and specialized cancer drugs. Since 2005, the cost structure has evolved with rising proportions of interventional drug costs. Additionally, costs were higher among males and younger age groups. Understanding these costs can help guide initiatives to control healthcare spending and improve cancer care quality.

Path Tracking Controller Design of Automated Parking Systems via NMPC with an Instructible Solution

Parking difficulties have become a social issue that people have to solve. Automated parking system is practicable for quick par operations without a driver which can also greatly reduces the probability of parking accidents. The paper proposes a Lyapunov-based nonlinear model predictive controller embedding an instructable solution which is generated by the modified rear-wheel feedback method (RF-LNMPC) in order to improve the overall path tracking accuracy in parking conditions. Firstly, A discrete-time RF-LNMPC considering the position and attitude of the parking vehicle is proposed to increase the success rate of automated parking effectively. Secondly, the RF-LNMPC problem with a multi-objective cost function is solved by the Interior-Point Optimization, of which the iterative initial values are described as the instructable solutions calculated by combining modified rear-wheel feedback to improve the performance of local optimal solution. Thirdly, the details on the computation of the terminal constraint and terminal cost for the linear time-varying case is presented. The closed-loop stability is verified via Lyapunov techniques by considering the terminal constraint and terminal cost theoretically. Finally, the proposed RF-LNMPC is implemented on a self-driving Lincoln MKZ platform and the experiment results have shown improved performance in parallel and vertical parking conditions. The Monte Carlo analysis also demonstrates good stability and repeatability of the proposed method which can be applied in practical use in the near future.

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.

Intermodal Transportation Challenges in Eastern Europe: Case Study of Romania

Abstract The purpose of this paper is to analyze data from the Romanian intermodal transportation sector and provide an overview of its main terminals (sea, river, road and rail) and the operational challenges faced by its most important logistics service providers. Intermodal transportation (by rail) is encouraged at EU level as a sustainable alternative to current predominant truck haulage. Most research articles outline issues with connectivity, terminal handling infrastructure and cost efficiency as factors hindering large scale adoption. Other authors tend to emphasize the more reliable delivery schedules, reduced road congestion and emissions as drivers of cost competitiveness on medium term. Relevant cargo volume traffic (tons, TEU and ITU) is analyzed at Romania’s main intermodal hubs in the last 10 years (2014-2023) using quantitative and qualitative data to provide a comprehensive overview of its overall competitiveness in this sector. Most Eastern European countries are faced with the same challenge in terms of intermodal transportation: the shortcomings of their supporting network due to insufficient public investment, highlighted by the results from Romania (case study method). Main results show Constanta Port had record cargo traffic in 2023, but its throughput is hindered by constant congestion issues, similar to the highly competitive road sector where intense truck traffic creates bottlenecks on the major roads leading to Western Europe. Cargo is also shipped via the Danube and freight trains, but to a limited extent due to low average speeds and inconsistent reliability along most sectors. The paper highlights the importance of a country’s strategic hubs and their connectivity (sea and/or river ports, main railway lines and road sectors) as a source of future growth and development perspectives, for terminal operators as well as neighboring regions (EU and EEA) through increased business activity.

Learning Lyapunov terminal costs from data for complexity reduction in nonlinear model predictive control

AbstractA classic way to design a nonlinear model predictive control (NMPC) scheme with guaranteed stability is to incorporate a terminal cost and a terminal constraint into the problem formulation. While a long prediction horizon is often desirable to obtain a large domain of attraction and good closed‐loop performance, the related computational burden can hinder its real‐time deployment. In this article, we propose an NMPC scheme with prediction horizon and no terminal constraint to drastically decrease the numerical complexity without significantly impacting closed‐loop stability and performance. This is attained by constructing a suitable terminal cost from data that estimates the cost‐to‐go of a given NMPC scheme with long prediction horizon. We demonstrate the advantages of the proposed control scheme in two benchmark control problems.

Remarks on a controlled degenerate diffusion process

We establish explicit estimates for a two-dimensional LQG homing problem with a zero terminal cost under the assumption that the controlled diffusion process is degenerate. The present results extend and complete those in Lefebvre (2013) which are given explicitly for a particular interesting case. Our main tool is a nonlinear integral inequality of the Gronwall type. As an application, we consider a simple example to illustrate the present results.

EV Charging Scheduling Under Demand Charge: A Block Model Predictive Control Approach

This paper studies the online scheduling of electric vehicle charging by a service provider subject to a demand charge in a distribution system. Demand charge imposes a penalty on the peak power consumption over each billing period, representing a substantial cost for the service provider with a large number of clients. Because the demand charge is calculated at the end of the billing period, it poses challenges in real-time scheduling when energy demand forecasts are inaccurate, resulting in either overly conservative power consumption or substantial demand charge. We propose a block model predictive control approach that decomposes the demand charge into a sequence of stage costs. Optimality conditions on demand patterns are also presented and analyzed. Numerical simulations demonstrate the efficacy of the proposed approach. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This paper addresses a significant practical problem of minimizing the demand charge on the real-time scheduling of deferrable demands. In particular, we consider a setting where a commercial electric vehicle (EV) charging service provider has to manage the online scheduling of a large number of arriving EVs at a charging facility subject to a maximum charging power constraint and a tariff with the demand charge. A major practical challenge is to balance the tradeoff between maximizing profit in scheduling as much EV charging as possible and the need to minimize penalty on the peak charging power. We propose a model predictive control strategy that decomposes the overall demand charge into a sequence of terminal costs. Also addressed is the practical constraint arising from the mismatched EV charging decision period and the power measurement period used to compute the demand charge. Using real data collected at the Adaptive Charging Network (ACN) testbed in simulations, the proposed approach yields 8-12% improvement in operational profit over existing benchmarks, while it has yet been tested in actual charging systems. In the future research, we will address the charging scheduling under demand charge over multiple charging stations.

Scalable Learning for Spatiotemporal Mean Field Games Using Physics-Informed Neural Operator

This paper proposes a scalable learning framework to solve a system of coupled forward–backward partial differential equations (PDEs) arising from mean field games (MFGs). The MFG system incorporates a forward PDE to model the propagation of population dynamics and a backward PDE for a representative agent’s optimal control. Existing work mainly focus on solving the mean field game equilibrium (MFE) of the MFG system when given fixed boundary conditions, including the initial population state and terminal cost. To obtain MFE efficiently, particularly when the initial population density and terminal cost vary, we utilize a physics-informed neural operator (PINO) to tackle the forward–backward PDEs. A learning algorithm is devised and its performance is evaluated on one application domain, which is the autonomous driving velocity control. Numerical experiments show that our method can obtain the MFE accurately when given different initial distributions of vehicles. The PINO exhibits both memory efficiency and generalization capabilities compared to physics-informed neural networks (PINNs).

Dynamic planning of edge sensing terminals in distribution network supporting distributed resources observable and controllable

With the advancement of low-carbon distribution networks, the heightened stochasticity introduced by a multitude of renewable energy sources in the power grid has significantly augmented the regulatory challenges faced by the power grid. Dispatching distributed resources emerges as an effective solution to this issue. However, these resources often lack observability and controllability, hindering their participation in power regulation services. To establish a reliable interaction between distributed resources and power grids, the deployment of numerous edge sensing terminals becomes essential, albeit incurring high costs. In light of this, our paper proposes a dynamic network planning method for edge sensing terminals based on node differentiation and resource observability criteria, aiming to facilitate real-time and dependable observation of distributed resources. Initially, the node weight, a metric to gauge the disparity among nodes, is computed, considering communication quality deviation, resource development synergy, and the distribution of distributed resources. Subsequently, an optimal configuration method is introduced, accounting for the terminal’s reliability under faults. Lastly, a method for dynamic terminal networking planning is presented, gradually reducing the depth of unobservable resources. An enhanced genetic algorithm is employed to address this challenge. This method was validated using an IEEE 33 node system and a 91 node actual system, demonstrating significant effectiveness in reducing terminal configuration costs.

Reinforcement Learning-Based Model Predictive Control for Discrete-Time Systems.

This article proposes a novel reinforcement learning-based model predictive control (RLMPC) scheme for discrete-time systems. The scheme integrates model predictive control (MPC) and reinforcement learning (RL) through policy iteration (PI), where MPC is a policy generator and the RL technique is employed to evaluate the policy. Then the obtained value function is taken as the terminal cost of MPC, thus improving the generated policy. The advantage of doing so is that it rules out the need for the offline design paradigm of the terminal cost, the auxiliary controller, and the terminal constraint in traditional MPC. Moreover, RLMPC proposed in this article enables a more flexible choice of prediction horizon due to the elimination of the terminal constraint, which has great potential in reducing the computational burden. We provide a rigorous analysis of the convergence, feasibility, and stability properties of RLMPC. Simulation results show that RLMPC achieves nearly the same performance as traditional MPC in the control of linear systems and exhibits superiority over traditional MPC for nonlinear ones.

Beyond the stable handling limits: nonlinear model predictive control for highly transient autonomous drifting

Autonomous vehicles that can reliably operate outside the stable handling limits would have access to a wider range of maneuvers in emergencies, improving overall safety. To that end, this paper presents a novel Nonlinear MPC approach for vehicle control with deeply saturated rear tires. Longitudinal slip management is elevated from the chassis control layer into the optimisation problem by using a coupled-slip tire model, and explicitly including wheelspeed dynamics. Terminal costs on sideslip stability help compensate for the finite horizon, while road bounds and static obstacles are encoded using slack constraints. Experiments on a racetrack with a modified Toyota GR Supra validate the controller's ability to smoothly transition from dynamic, non-equilibrium drifting to grip driving. Further experiments demonstrate robustness to significant longitudinal force and wheelspeed disturbances, and showcase the controller flexibly transitioning in and out of the sliding tire regime to balance slack constraints with tracking objectives.

Heart failure and the cost of dying: must the ferryman always be paid?

Provision of palliative care in chronic heart failure (CHF) can support complex decision-making, significantly improve quality of life and may lower healthcare costs. To examine whether healthcare costs differed in terminal admissions according to the adoption of a palliative approach. Retrospective review of medical records and costing data for all admissions resulting in death from CHF (July 2011 to December 2019),analysed as two groups (2011-2016 and 2016-2019) because of background changes in costings. Admissions with CHF resulting in death in an Australian tertiary referral centre. The cohort (n = 439) were elderly (median age 83.7 years, interquartile range (IQR) = 77.6-88.7 years) and mostly men (54.9%). Half (230, 52.4%) were referred to a specialist palliative care team, whereas over a third (172, 39.2%) received a palliative approach. Receiving a palliative approach was associated with a nonstatistically significant lower admission cost (AU$12 710 vs AU$15 978; P = 0.19) between 2011 and 2016 (n = 101, 38.8%) and a significantly lower cost (AU$11 319 vs AU$15 978; P < 0.01) between 2016 and 2019 (n = 71, 39.7%). Intensive care admission resulted in the single greatest additional cost at AU$14 624 (IQR = AU$4130-AU$44 197) (n = 48, 2011-2016). Median terminal admission cost was lower for patients with comfort goals of care (P < 0.01), without life-sustaining interventions (P < 0.01) or who received a palliative approach (P < 0.01). Referral to inpatient specialist palliative care or receiving a palliative approach resulted in comparable admission costings (AU$11 621 [IQR = AU$4705-AU$32 457] and AU$11 466 [IQR = AU$4973-AU$25 614]). A palliative approach in terminal CHF admission may improve quality at the end of life and decrease costs associated with care.

Equivalence of Optimality Criteria for Markov Decision Process and Model Predictive Control

This paper shows that the optimal policy and value functions of a Markov Decision Process (MDP), either discounted or not, can be captured by a finite-horizon undiscounted Optimal Control Problem (OCP), even if based on an inexact model. This can be achieved by selecting a proper stage cost and terminal cost for the OCP. A very useful particular case of OCP is a Model Predictive Control (MPC) scheme where a deterministic (possibly nonlinear) model is used to reduce the computational complexity. This observation leads us to parameterize an MPC scheme fully, including the cost function. In practice, Reinforcement Learning algorithms can then be used to tune the parameterized MPC scheme. We verify the developed theorems analytically in an LQR case and we investigate some other nonlinear examples in simulations.

Receding Horizon Control of Bilinear System with Hydraulic Delays

All flow systems are subject to transport delays, which are determined by the flow rates in the system. When the flow rates themselves are control inputs, the system becomes subject to input-dependent state delays, which poses significant theoretical problems. In this paper we propose a model predictive control scheme for a generic multi-variable heat transport system, where flows to the individual consumers can be manipulated by a centralized controller. The control design takes the transport delay into account by means of an explicit discretization of the transport equation, which is a partial differential equation. Different discretization methods are considered, giving rise to a high-order bilinear system model in discrete time. The control problem is formulated as a linear quadratic receding horizon problem with terminal cost. Simulation studies show that the control problem can be solved using standard software and that the Lax-Friedrichs discretization method appears to be the most suitable of the investigated methods.

Inherently Robust Economic Model Predictive Control Without Dissipativity

We establish sufficient conditions for the terminal cost and constraint such that economic model predictive control (MPC) is robustly recursively feasible and economically robust to small disturbances without any assumptions of dissipativity. Moreover, we demonstrate that these sufficient conditions can be satisfied with standard design methods. A small example is presented to illustrate the inherent robustness of economic MPC to small disturbances.