Constraint Satisfaction Research Articles

Approximate Dynamic Programming for Control of Wave Energy Converters With Implementation and Validation on a Point Absorber Prototype

It is recognized that the application of constrained optimal control for wave energy converters (WECs), represented by model predictive control (MPC), is hindered by its computation burden resulting from online optimization, especially when the model contains nonlinearities. In this article, a novel control solution based on approximate dynamic programming (ADP) is proposed, in which an explicit controller is solved by model sampling and neural network training. In this way, the major computation load is moved offline to enable fast online execution, and the need for wave prediction is eliminated. The effectiveness of ADP is verified by a simulation case study based on a nonlinear WEC system. Using nonlinear MPC as the performance benchmark, over 90% energy efficiency is achieved by ADP under various sea states with constraint satisfaction and dramatically faster computation. The ADP controller is further experimentally validated on a newly constructed WEC prototype through wave tank testing, where successful implementation and satisfactory performance are demonstrated.

The chemostat reactor: A stability analysis and model predictive control

This contribution develops the model predictive control for an unstable chemostat reactor. Initially, we analyze the system’s model — a nonlinear first-order hyperbolic partial integro-differential equation (PIDE) — and carry the model linearization around an unstable operating condition. Employing the structure-preserving Cayley–Tustin transformation, we obtain a discrete-time model representation of the continuous model. Subsequently, we solve the operator Ricatti equations in the discrete-time setting to derive a full state feedback controller that stabilizes the closed-loop and design a Luenberger observer for state reconstruction given the system output measures. Finally, we formulate a dual-mode MPC ensuring constraint satisfaction and optimality, integrating the gain-based unconstrained full-state feedback optimal control obtained from the Ricatti equation. This dual-mode strategy describes an optimization problem where the predictive controller acts only if constraints become active within the control horizon. Simulation studies validate the controller performance, where the MPC only takes action if the constraints are predicted to be active within the control horizon while also guaranteeing closed-loop stabilization under only output feedback. This type of controller can be easily implemented with other control strategies and significantly decreases the computational costs of solving the optimal control problems when compared to other MPC approaches.

Four versions of transtheoretical stances, and the Bernese view

A brief characterization of transtheoretical stances to which existing approaches can be allocated is followed by a description of the "Bernese view", that is, what Klaus Grawe and his colleagues, including the authors of this article have developed: the origins, a model of the multiple constraint satisfaction construction of therapist action, a discussion of psychotherapy integration, the crucial role of supervisors in an integrative multiple constraint satisfaction approach, and a discussion of when and how trainees should be introduced to a transtheoretical stance.

Convex distributed robust model predictive control for collision and obstacle avoidance

AbstractThe article proposes a convex distributed robust model predictive control algorithm for collision avoidance in multi‐agent systems with additive disturbances, which utilizes the concept of tube model predictive control to address the disturbances. To tackle the coupled collision avoidance constraints, each agent integrates assumed nominal position and input trajectories from its neighbors, rather than relying on actual ones. Compatibility constraints are then designed using the normal vector of the constructed separating hyperplanes for collision avoidance and the residual collision avoidance margin of the optimal solutions at the last time step to restrict deviations between assumed and actual trajectories and ensure consistency among the agents. The nonconvex collision and obstacle avoidance constraints are convexified using the concept of safe sets, transforming them into time‐varying closed polyhedral constraints while accounting for the impact of disturbances. Particularly, the residual collision avoidance margin is incorporated into the construction of safe sets for the satisfaction of coupled collision avoidance constraints. Consequently, the original collision avoidance optimal control problems can be efficiently and simultaneously solved for all agents using standard quadratic programming techniques. Under the design of local terminal constraint sets for each agent, the proposed algorithm ensures a rigorous analysis of robust constraint satisfaction for collision avoidance constraints, as well as an assessment of recursive feasibility and closed‐loop stability. The effectiveness of the algorithm is demonstrated through an illustrative example, and simulation results validate its ability to successfully achieve collision and obstacle avoidance even in the presence of disturbances.

Autonomous spacecraft collision avoidance with a variable number of space debris based on safe reinforcement learning

The avoidance of multiple space debris collisions by autonomous spacecraft has garnered significant interests worldwide. Applying deep reinforcement learning (DRL) to autonomous spacecraft collision avoidance problems is still difficult because of limitations on constraint satisfaction and environment state perception, even if DRL is a suitable model-free and data-driven framework. In this research, a state-of-the-art penalized proximal policy optimization (P3O) method is applied to address the spacecraft's autonomous collision avoidance problem, which is formalized as a constrained Markov decision process (CMDP). In contrast with traditional DRL methods, P3O promises to satisfy multiple constraints in actual spaceship operations while also facilitating efficient learning in multi-dimensional, continuous state and action spaces. The scalability of the P3O algorithm is enhanced by combining the feature extraction capabilities and variable-length input capabilities of the long short-term memory (LSTM) networks, enabling the P3O to adapt to a variable number of space debris without the need for network retraining. The performance of the proposed method is compared with other five methods through simulation cases, which verifies the superior performance of the proposed method in terms of scalability, energy consumption, collision probability and constraint satisfaction.

On Classifying Continuous Constraint Satisfaction Problems

A continuous constraint satisfaction problem (CCSP) is a constraint satisfaction problem (CSP) with an interval domain $U \subset \mathbb{R}$. We engage in a systematic study to classify CCSPs that are complete of the Existential Theory of the Reals, i.e., ER-complete. To define this class, we first consider the problem ETR, which also stands for Existential Theory of the Reals. In an instance of this problem we are given some sentence of the form $\exists x_1, \ldots, x_n \in \mathbb{R} : \Phi(x_1, \ldots, x_n)$, where $\Phi$ is a well-formed quantifier-free formula consisting of the symbols $\{0, 1, +, \cdot, \geq, >, \wedge, \vee, \neg\}$, the goal is to check whether this sentence is true. Now the class ER is the family of all problems that admit a polynomial-time many-one reduction to ETR. It is known that NP $\subseteq$ ER $\subseteq$ PSPACE. We restrict our attention on CCSPs with addition constraints ($x + y = z$) and some other mild technical conditions. Previously, it was shown that multiplication constraints ($x \cdot y = z$), squaring constraints ($x^2 = y$), or inversion constraints ($x\cdot y = 1$) are sufficient to establish ER-completeness. We extend this in the strongest possible sense for equality constraints as follows. We show that CCSPs (with addition constraints and some other mild technical conditions) that have any one well-behaved curved equality constraint ($f(x,y) = 0$) are ER-complete. We further extend our results to inequality constraints. We show that any well-behaved convexly curved and any well-behaved concavely curved inequality constraint ($f(x,y) \geq 0$ and $g(x,y) \geq 0$) imply ER-completeness on the class of such CCSPs.

Predefined-time stabilization of stochastic nonlinear systems with application to UAVs

The paper presents a new Lyapunov-type predefined-time stabilization control algorithm for stochastic high-order nonlinear systems with asymmetric output constraints. In contrast to stochastic finite-time and fixed-time stabilization, the average value of the settling-time function for stochastic predefined-time stabilization control is independent of both the initial value and the control factors. To mitigate the significant uncertainties arising from the asymmetric output constraint, a tan-type barrier Lyapunov function is formulated. Furthermore, by harnessing the previously mentioned barrier Lyapunov function and integrating the power integrator technique, a controller design strategy is formulated based on the backstepping method. The rigorous analysis in this study proves that the designed controller ensures both the attainment of predefined-time convergence of the system states to the origin in probability and the satisfaction of the output constraint. Finally, an example of a roll angle subsystem for quadrotor UAVs and a numerical illustration are presented to corroborate the theoretical analysis.

Optimal Coordination Of Multiple Setting Group Type Overcurrent Relays

Relay coordination optimization issues are multi-objective, multi-constrained problems that aim to reduce each relay's operation time while satisfying all relevant constraints. Fulfillment of the coordination time interval (CTI) and time limitations for relay operations are the coordination constraints. To solve coordination optimization issues, decision-making variables are defined such that the objective function is minimized with due satisfaction of necessary constraints. The majority of the most recent research that is currently available defines relay coordination for fixed network topologies, but in practice, it varies for a variety of reasons, including maintenance and element failures. Over current relays now have the capability of storing multiple relay settings but the number of relay settings that can be stored in relays is far less than the possible number of network configurations, which is why a suitable clustering technique such as K-Means can be used to cluster out various network topologies using a proper clustering index. Because both the objective function and the constraints are time dependent, time dependent attributes such as average relay operating time are used as the clustering index instead of the conventional index fault current deviation.

Sketching Approximability of All Finite CSPs

A constraint satisfaction problem (CSP), \(\textsf {Max-CSP}(\mathcal {F})\) , is specified by a finite set of constraints \(\mathcal {F}\subseteq \lbrace [q]^k \rightarrow \lbrace 0,1\rbrace \rbrace\) for positive integers q and k . An instance of the problem on n variables is given by m applications of constraints from \(\mathcal {F}\) to subsequences of the n variables, and the goal is to find an assignment to the variables that satisfies the maximum number of constraints. In the (γ ,β)-approximation version of the problem for parameters 0 ≤ β ≤ γ ≤ 1, the goal is to distinguish instances where at least γ fraction of the constraints can be satisfied from instances where at most β fraction of the constraints can be satisfied. In this work, we consider the approximability of this problem in the context of sketching algorithms and give a dichotomy result. Specifically, for every family \(\mathcal {F}\) and every β < γ, we show that either a linear sketching algorithm solves the problem in polylogarithmic space or the problem is not solvable by any sketching algorithm in \(o(\sqrt {n})\) space. In particular, we give non-trivial approximation algorithms using polylogarithmic space for infinitely many constraint satisfaction problems. We also extend previously known lower bounds for general streaming algorithms to a wide variety of problems, and in particular the case of q = k =2, where we get a dichotomy, and the case when the satisfying assignments of the constraints of \(\mathcal {F}\) support a distribution on \([q]^k\) with uniform marginals. Prior to this work, other than sporadic examples, the only systematic classes of CSPs that were analyzed considered the setting of Boolean variables q = 2, binary constraints k =2, and singleton families \(|\mathcal {F}|=1\) and only considered the setting where constraints are placed on literals rather than variables. Our positive results show wide applicability of bias-based algorithms used previously by [ 47 ] and [ 41 ], which we extend to include richer norm estimation algorithms, by giving a systematic way to discover biases. Our negative results combine the Fourier analytic methods of [ 56 ], which we extend to a wider class of CSPs, with a rich collection of reductions among communication complexity problems that lie at the heart of the negative results. In particular, previous works used Fourier analysis over the Boolean cube to initiate their results and the results seemed particularly tailored to functions on Boolean literals (i.e., with negations). Our techniques surprisingly allow us to get to general q -ary CSPs without negations by appealing to the same Fourier analytic starting point over Boolean hypercubes.

Learning to Branch: Generalization Guarantees and Limits of Data-Independent Discretization

Tree search algorithms, such as branch-and-bound, are the most widely used tools for solving combinatorial and non-convex problems. For example, they are the foremost method for solving (mixed) integer programs and constraint satisfaction problems. Tree search algorithms come with a variety of tunable parameters that are notoriously challenging to tune by hand. A growing body of research has demonstrated the power of using a data-driven approach to automatically optimize the parameters of tree search algorithms. These techniques use a training set of integer programs sampled from an application-specific instance distribution to find a parameter setting that has strong average performance over the training set. However, with too few samples, a parameter setting may have strong average performance on the training set but poor expected performance on future integer programs from the same application. Our main contribution is to provide the first sample complexity guarantees for tree search parameter tuning. These guarantees bound the number of samples sufficient to ensure that the average performance of tree search over the samples nearly matches its future expected performance on the unknown instance distribution. In particular, the parameters we analyze weight scoring rules used for variable selection. Proving these guarantees is challenging because tree size is a volatile function of these parameters: we prove that, for any discretization (uniform or not) of the parameter space, there exists a distribution over integer programs such that every parameter setting in the discretization results in a tree with exponential expected size, yet there exist parameter settings between the discretized points that result in trees of constant size. In addition, we provide data-dependent guarantees that depend on the volatility of these tree-size functions: our guarantees improve if the tree-size functions can be well approximated by simpler functions. Finally, via experiments, we illustrate that learning an optimal weighting of scoring rules reduces tree size.

Architectural design space exploration of complex engineered systems with management constraints and preferences

During the architectural design phase, the decisions made have a crucial impact on developing complex systems. These decisions often limit the performance bounds and steer subsequent research directions. However, the multitude of architectural components and intricate interrelationships cause component combination explosion. Additionally, the diverse requirements of different stakeholders add complexity to determining the weight of indicators, posing challenges to complex system architecture generation and trade-off. This paper proposes a method for exploring the architectural design space in complex systems realisation, which aims to resolve issues related to the formal representation of architectures, selection within the architectural design space, and trade-offs. Firstly, the architectural design space is represented by formulating and mathematically describing a morphological matrix. Then, the construction of the Constraint Satisfaction Problem model aims to achieve a reduction in the scale of the design space. Moreover, through the evolutionary algorithm, the Pareto front is identified. Weight sensitivity analysis and the improved Technique for Order Preference by Similarity to the Ideal Solution method are combined to assist designers in finding a satisfactory architecture from the design space with many combinations. Finally, the proposed method is verified by the design of the launch vehicle’s primary and secondary separation system.

Safety enhancement for nonlinear systems via learning-based model predictive control with Gaussian process regression

This paper proposes a novel safe Gaussian process model predictive control scheme for discrete-time nonlinear systems subject to additive uncertainties. The scheme is implemented using an online learning framework that provides safety guarantees based on a nominal model, and employs Gaussian process regression (GPR) to learn the uncertainties to improve the control performance. The advantage of this framework lies in the ability to decouple safety and performance in the robust controller design. Furthermore, the inaccuracy measure given by GPR can be used to adaptively tighten the constraints, leading to the less conservative behaviors. A rigorous analysis of the constraint satisfaction, recursive feasibility and closed-loop stability is also presented. Two simulation examples, including a regulation problem of a nominally linear system and a tracking problem of a car whose model is nonlinear, are performed to verify the effectiveness of the proposed scheme.

A Distributed Model Predictive Control Strategy for Constrained Multi-Agent Systems: The Uncertain Target Capturing Scenario

In this paper, a distributed control architecture is presented for addressing the target capturing problem of a multi-agent system whose dynamics is described by double integrator models subject to bounded disturbance effects. Starting from novel kinematic models used as reference trajectories, the aim consists in driving the multi-agent system within the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">containment</i> region without entering the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">distancing</i> one and, whenever necessary, there remaining confined. The underlying control problem has been tackled by means of model predictive control arguments. In particular, two different distributed strategies have been developed, and adequately switched to each other, in order to guarantee constraint satisfaction within the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">capturing</i> region despite any disturbance realization. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This paper proposes a methodological solution for dealing with the capturing problem for multi-agent systems operating in uncertain environments where the target is defined by the ring resulting from the distancing and containment ellipsoids. Differently from the existing literature, the proposed approach combines into a unique framework properties coming from potential fields theory and distributed model predictive control philosophy. It is interesting to put in light that the underlying control strategy allows the users to face surveillance and rescue operations, to cite a few, in computational affordable way due to the required memory resources because most of computations can be straightforwardly moved in the off-line phase.

Cascade NMPC-PID control strategy of active heave compensation system for ship-mounted offshore crane

In this paper, the problem associated with active heave motion compensation during heavy-lift operations performed by ship-mounted offshore cranes beneath rough marine environment is investigated. To effectively decouple the correlated motion of the suspended payload from the support vessel via the secondary regulated active heave compensation (SRAHC) system, a novel hierarchical control strategy is proposed by integrating a cascade control structure with an adaptive robust control scheme that incorporates a vertical motion forecast algorithm based on autoregressive integrated moving average (ARIMA) model, a conventional proportional-integral-derivative (PID) framework, and a nonlinear model predictive control (NMPC) method. The introduction of the cascade structure is motivated by its prompt error response against control lag and efficient system order reduction for computational burden alleviation, which forms a crucial foundation for the NMPC-based real-time regulation of PID gains, ensuring optimal evolution of the SRAHC system while improving its practical feasibility. Furthermore, the enhancements in heave compensation and trajectory tracking performance, noise resistance, constraint satisfaction, and engineering application potential of the proposal are demonstrated through a thorough comparative analysis conducted in a practical co-simulation research system.

A novel tri-stage with reward-switching mechanism for constrained multiobjective optimization problems

AbstractThe effective exploitation of infeasible solutions plays a crucial role in addressing constrained multiobjective optimization problems (CMOPs). However, existing constrained multiobjective optimization evolutionary algorithms (CMOEAs) encounter challenges in effectively balancing objective optimization and constraint satisfaction, particularly when tackling problems with complex infeasible regions. Subsequent to the prior exploration, this paper proposes a novel tri-stage with reward-switching mechanism framework (TSRSM), including the push, pull, and repush stages. Each stage consists of two coevolutionary populations, namely $${\text {Pop}}_1$$ Pop 1 and $${\text {Pop}}_2$$ Pop 2 . Throughout the three stages, $${\text {Pop}}_1$$ Pop 1 is tasked with converging to the constrained Pareto front (CPF). However, $${\text {Pop}}_2$$ Pop 2 is assigned with distinct tasks: (i) converging to the unconstrained Pareto front (UPF) in the push stage; (ii) utilizing constraint relaxation technique to discover the CPF in the pull stage; and (iii) revisiting the search for the UPF through knowledge transfer in the repush stage. Additionally, a novel reward-switching mechanism (RSM) is employed to transition between different stages, considering the extent of changes in the convergence and diversity of populations. Finally, the experimental results on three benchmark test sets and 30 real-world CMOPs demonstrate that TSRSM achieves competitive performance when compared with nine state-of-the-art CMOEAs. The source code is available at https://github.com/Qu-jq/TSRSM.

High‐level decision‐making for autonomous overtaking: An MPC‐based switching control approach

AbstractThe key motivation of this paper lies in the development of a high‐level decision‐making framework for autonomous overtaking maneuvers on two‐lane country roads with dynamic oncoming traffic. To generate an optimal and safe decision sequence for such scenario, an innovative high‐level decision‐making framework that combines model predictive control (MPC) and switching control methodologies is introduced. Specifically, the autonomous vehicle is abstracted and modelled as a switched system. This abstraction allows vehicle to operate in different modes corresponding to different high‐level decisions. It establishes a crucial connection between high‐level decision‐making and low‐level behaviour of the autonomous vehicle. Furthermore, barrier functions and predictive models that account for the relationship between the autonomous vehicle and oncoming traffic are incorporated. This technique enables us to guarantee the satisfaction of constraints, while also assessing performance within a prediction horizon. By repeatedly solving the online constrained optimization problems, we not only generate an optimal decision sequence for overtaking safely and efficiently but also enhance the adaptability and robustness. This adaptability allows the system to respond effectively to potential changes and unexpected events. Finally, the performance of the proposed MPC framework is demonstrated via simulations of four driving scenarios, which shows that it can handle multiple behaviours.

Enhancing constraint programming via supervised learning for job shop scheduling

Constraint programming (CP) is a powerful technique for solving constraint satisfaction and optimization problems. In CP solvers, the variable ordering strategy used to select which variable to explore first in the solving process has a significant impact on solver effectiveness. To address this issue, we propose a novel variable ordering strategy based on supervised learning, which we evaluate in the context of job shop scheduling problems. Our learning-based methods predict the optimal solution of a problem instance and use the predicted solution to order variables for CP solvers. Unlike traditional variable ordering methods, our methods can learn from the characteristics of each problem instance and customize the variable ordering strategy accordingly, leading to improved solver performance. Our experiments demonstrate that training machine learning models is highly efficient and can achieve high accuracy. Furthermore, our learned variable ordering methods perform competitively compared to four existing methods. Finally, we showcase the benefits of integrating machine learning-based variable ordering methods with conventional domain-based approaches through tie-breaking.

ConSequence: Synthesizing Logically Constrained Sequences for Electronic Health Record Generation

Generative models can produce synthetic patient records for analytical tasks when real data is unavailable or limited. However, current methods struggle with adhering to domain-specific knowledge and removing invalid data. We present ConSequence, an effective approach to integrating domain knowledge into sequential generative neural network outputs. Our rule-based formulation includes temporal aggregation and antecedent evaluation modules, ensured by an efficient matrix multiplication formulation, to satisfy hard and soft logical constraints across time steps. Existing constraint methods often fail to guarantee constraint satisfaction, lack the ability to handle temporal constraints, and hinder the learning and computational efficiency of the model. In contrast, our approach efficiently handles all types of constraints with guaranteed logical coherence. We demonstrate ConSequence's effectiveness in generating electronic health records, outperforming competitors in achieving complete temporal and spatial constraint satisfaction without compromising runtime performance or generative quality. Specifically, ConSequence successfully prevents all rule violations while improving the model quality in reducing its test perplexity by 5% and incurring less than a 13% slowdown in generation speed compared to an unconstrained model.

Conversational Modeling for Constraint Satisfaction

Many problems, from Sudoku to factory scheduling, can be regarded as constraint satisfaction problems. A key component of real world problem solving is a conversation between a constraint programming expert and a problem domain expert to specify the problem to be solved. This presentation argues that the time is ripe for progress in automating the constraint programmer side of this conversation and suggests promising avenues for this pursuit.

Pure-Past Action Masking

We present Pure-Past Action Masking (PPAM), a lightweight approach to action masking for safe reinforcement learning. In PPAM, actions are disallowed (“masked”) according to specifications expressed in Pure-Past Linear Temporal Logic (PPLTL). PPAM can enforce non-Markovian constraints, i.e., constraints based on the history of the system, rather than just the current state of the (possibly hidden) MDP. The features used in the safety constraint need not be the same as those used by the learning agent, allowing a clear separation of concerns between the safety constraints and reward specifications of the (learning) agent. We prove formally that an agent trained with PPAM can learn any optimal policy that satisfies the safety constraints, and that they are as expressive as shields, another approach to enforce non-Markovian constraints in RL. Finally, we provide empirical results showing how PPAM can guarantee constraint satisfaction in practice.