This paper presents a motion planning approach for rigid-soft hybrid robots that incorporates continuous expression and input mapping, aiming to achieve both tracking and collision avoidance. The conventional piecewise constant curvature model utilized in soft robots suffers from issues of discontinuity and singularity. These inherent drawbacks pose significant challenges when attempting to address the nonlinear problems associated with motion planning. To overcome these limitations, we establish a relationship between the length of drive cable and the coordination of robot terminal. Through the application of Taylor transformation, we effectively eliminate the control variation present in the denominator of this expression. Then, the nonlinear model predictive control (NMPC) with this model expression and input mapping is proposed for motion planning. By directly constructing the current input and output of the motion planning based on the combination of previous data and substituting the linearized part of the model with this data, we mitigate the impact of model inaccuracies on the solution process. Both the obstacles and robot are modeled as polyhedra, and the collision problem is reformulated in the form of continuous nonlinear inequalities. These expressions are then incorporated into the NMPC optimization function as collision avoidance constraints. This approach enables the attainment of a larger feasible movement area and more precise tracking results during the motion planning process. Finally, experimental and simulation results conducted on a rigid-soft hybrid robot demonstrate the feasibility and superiority of the proposed method, validating its effectiveness in practical applications.

The data-driven sliding mode control (SMC) method proves to be highly effective in addressing uncertainties and enhancing system performance. In our previous work, we implemented a co-design approach based on an input-mapping data-driven technique, which effectively improves the convergence rate through historical data compensation. However, this approach increases computational complexity in multi-input and multi-output (MIMO) systems due to the dependency of the number of online optimization variables on system dimensions. To improve applicability, this paper introduces a novel input-mapping-based online learning SMC strategy with low computational complexity. First, a new sliding mode surface is established through online convex combination of pre-designed offline surfaces. Then, an input-mapping-based online learning sliding mode control (IML-SMC) strategy is designed, utilizing a reaching law with adaptively adjusted convergence and switching coefficients to minimize chattering. The input-mapping technique employs the mapping relationship between historical input and output data for predicting future system dynamics. Accordingly, an optimization problem is formulated to learn from the past dynamics of the uncertain system online, thereby enhancing system performance. The optimization problem in this paper features fewer variables and is independent of system dimension. Additionally, the stability of the proposed method is theoretically validated, and the advantages are demonstrated through a MIMO system. Note to Practitioners—The design of control strategies that reduce the impact of mismatches between practical systems and models on system performance, while also ensuring applicability, is crucial. To address this issue, this paper proposes a low-complexity IML-SMC strategy. This strategy uses historical real input-output information and the mapping relationship with future dynamics to compensate for the impact of unknown dynamics and improve the system’s convergence rate. Notably, the control strategy introduced in this paper significantly reduces online computational complexity, ensuring applicability, and stability is proven. When there is a deviation between the model and the actual system, practitioners can implement the low-complexity IMC-SMC strategy proposed in this paper to more quickly achieve the control objectives in the actual system.

This study proposes a mixed traffic flow control structure, utilizing multi-lane fleets composed of a small portion of connected and autonomous vehicles (CAVs) in mixed flow as mobile actuators. Multiple fleets formed within the mixed traffic flow feature a novel structure and a valve-like function that regulates both traffic volume and speed, referred to as “valve fleet”. Specifically, the travel speed and structural spacing of valve fleets are controllable parameters, which can regulate the surrounding traffic speed and the flow through the fleet based on the downstream traffic state to decongest bottlenecks. Then, a control-oriented mobile cell transmission model (MCTM) is developed to characterize the macroscopic traffic dynamics with the presence and influence of valve fleets and bottleneck areas on freeways. Moreover, a hierarchical framework for mixed traffic flow regulation is designed, where the upper-level traffic optimization model dynamically determines all fleets’ parameters in a rolling-horizon fashion to minimize total travel time and suppress local congestion. The decentralized lower-level fleet controller adopts model predictive control (MPC) to coordinate CAVs’ motions and handles interactions between CAVs and human-driven vehicles (HDVs). To evaluate the proposed method, we conduct microscopic experiments in the SUMO simulator to implement the proposed traffic control method in realistic traffic environments. The comprehensive comparison results demonstrate the proposed method’s superiority in enhancing traffic efficiency and alleviating congestion at freeway bottleneck segments.

Research on the longitudinal control of mixed platoons has garnered significant attention, particularly addressing the uncertainties of human-driven vehicles (HDVs) and the computational inefficiency of centralized control for large platoons. This paper presents a novel distributed control strategy for multiple mixed platoons, utilizing a flexible model structure to account for dynamic uncertainties and lateral disturbances. A distributed control framework is introduced, where each platoon’s states and constraints are coupled with neighboring platoons, improving scalability and resilience in large-scale mixed traffic environments. Each platoon’s dynamic uncertainties are characterized by convex hulls in the subsystem model. A distributed data-driven input mapping method is proposed, where subsystems dynamically map the process data linearly from both local and neighboring platoons to local control law, eliminating the need for pre-collected datasets or offline training. To reduce iterations and communication requirements, a distributed implementation algorithm is proposed by designing subsystem basic laws with robust model predict control, then the control problem is solved online using convex quadratic programming, enabling fast computation of control actions and enhancing adaptability to real-time traffic variations. Additionally, a theoretical analysis guarantees the stability and optimality of the system, providing conditions that assist in subsystem parameter design for practical implementation. Simulation results validate the effectiveness and efficiency of the proposed approach, particularly in comparison to traditional methods that rely on pre-collected data.

This paper examines matrix-scaled consensus problems on switching networks where each agent holds time-varying matrix-valued scaling matrices that are either positive definite or negative definite. Matrix-scaled consensus amongst all agents can be achieved if the switching networks have frequent spanning trees and the scaling matrices remain unchanged within a time interval with a spanning tree. On time-varying networks, agents with the same time-varying scaling matrix will converge to the same point, differing from the virtual consensus value by the inverse of the scaling matrix. Both discrete-time and continuous-time cases are discussed. Simulation results are provided to demonstrate the theory.

Path tracking is a key problem in autonomous vehicles, but uncertainties in vehicle models and environmental parameters as well as the couple of lateral and longitudinal behaviors pose a major challenge to path following accuracy and efficiency. To solve this problem, a layered vehicle lateral and longitudinal controller based on model predictive control is proposed in this paper. Firstly, a lateral controller based on a synthesis of robust model predictive control is used to reduce the influence of model uncertainty. Then a longitudinal nonlinear model predictive controller is proposed to achieve the coordination between the lateral following error and the longitudinal forward speed of the vehicle. Finally, the bottom PID controller is used to control the throttle and brake of the vehicle. The effectiveness of the proposed method is verified by experiments, which show that it is robust to the uncertain parameters and the computation is relatively small.

This article examines for the first time an integrated structure of smart injection molding systems (IMS) based on Industry 4.0 technologies and provides a system-level solution for manufacturing smart products. The fully automated smart IMS structure allows manufacturers to produce thermoplastic products directly from raw materials without requiring any human labor. Following this, we focus on the control problem associated with the auxiliary robot manipulators that support the smart IMS. A cooperative distributed predictive control (DPC) algorithm with one-tap memory is proposed to achieve optimal closed-loop performance for multiple robot manipulators simultaneously performing their respective tasks. Using one-tap memory in smart IMS, we optimize the local performance index, which includes penalized terms diverging from it, and the global cooperation effort with limited memory acceleration to reach manifold consensus, without requiring manipulators to exchange information iteratively at each step. The cooperative DPC algorithm is also applied to five robot manipulators in the smart IMS, which require them to work cooperatively to achieve rhythmic and synchronized movements. Industrial experiment results demonstrate the feasibility of smart IMS combined with the cooperative DPC. Moreover, the cooperative DPC method outperforms the other two predictive control methods based on three metrics of the smart IMS.

The coordination of traffic flow among regions is necessary for a large-scale road traffic network to avoid local congestions and improve the overall traffic efficiency. In this paper, by incorporating the random characteristic of traffic flow, we formulate the problem of perimeter traffic flow control for a multi-region traffic network as a Markov decision process with adaptive state definition. Based on stochastic macroscopic fundamental diagrams (MFD) of regions, a state transition probability model is proposed to describe the state changes of the multi-region traffic network under different perimeter control policies. With the stochastic MFD-based state transition probabilities rather than counting from the historical data, a policy iteration algorithm with perturbation analysis is introduced to get the optimal perimeter control policy in real-time without the requirement of online or offline learning. The proposed method is compared with the classic perimeter control methods by simulation, which indicates its effectiveness in mitigating the congestion and improving the network throughput, as well as the promising implementation prospect.

This study proposes a cooperative platoon formation scheme in a multi-lane traffic environment with connected and autonomous vehicles (CAVs). It coordinates the lane-changing decisions and longitudinal trajectories of CAVs to form platoons based on each vehicle’s target lane, aiming to reduce the negative impact of lane-changing maneuvers on traffic flow and improve platoon formation efficiency. Mathematically, a vehicle model for platoon formation is developed which couples the lateral lane-changing and longitudinal car-following behaviors of vehicles. A model predictive control-based mixed integer linear programming (MILP) problem is formulated, which optimizes all vehicles’ lateral and longitudinal trajectories and improves maneuverability and efficiency. In contrast to the majority of existing studies, the configurations of platoons to be formed are not predefined and can be jointly optimized to further improve flexibility. Moreover, the framework that integrates configuration decisions, trajectory planning and control is executed dynamically with a rolling horizon based on real-time traffic states to enhance reliability. To validate the proposed scheme, we conduct simulation experiments in SUMO to implement the cooperative platoon formation in a typical three-lane traffic flow with different traffic demand levels. The extensive comparison results indicate the superiority of the proposed method in improving traffic flow speed and platoon formation efficiency.

Batch processes are typically nonlinear systems with constraints. Model predictive control (MPC) and iterative learning control (ILC) are effective methods for controlling batch processes. By combining batch-wise ILC and time-wise MPC, this article proposes a multirate control scheme for constrained nonlinear systems. Two-dimensional (2-D) framework is used to combine historical batch data with current measurements. The ILC part uses run-to-run control with previous iteration data, and the MPC part uses real-time control with current sampled measurements. Real-time feedback-based MPC in the time axis and run-to-run ILC in the batch axis are combined to optimize the current inputs based on previous batch input–output data and real-time system measurements. Rather than achieving control objectives in a single batch, our design allows multiple batches to be executed successively. To establish the stability of the combined scheme, rigorous theoretical analysis is presented next. The combined scheme with improved performance is then validated through two illustrative numerical examples.

In this article, we study the optimal iterative learning control (ILC) for constrained systems with bounded uncertainties via a novel conic input mapping (CIM) design methodology. Due to the limited understanding of the process of interest, modeling uncertainties are generally inevitable, significantly reducing the convergence rate of the control systems. However, huge amounts of measured process data interacting with model uncertainties can easily be collected. Incorporating these data into the optimal controller design could unlock new opportunities to reduce the error of the current trail optimization. Based on several existing optimal ILC methods, we incorporate the online process data into the optimal and robust optimal ILC design, respectively. Our methodology, called CIM, utilizes the process data for the first time by applying the convex cone theory and maps the data into the design of control inputs. CIM-based optimal ILC and robust optimal ILC methods are developed for uncertain systems to achieve better control performance and a faster convergence rate. Next, rigorous theoretical analyses for the two methods have been presented, respectively. Finally, two illustrative numerical examples are provided to validate our methods with improved performance.

Lane change for automated vehicles (AVs) is an important but challenging task in complex dynamic traffic environments. Due to difficulties in guaranteeing safety as well as a high efficiency, AVs are inclined to choose relatively conservative strategies for lane change. To avoid the conservatism, this paper presents a cooperation-aware lane change method utilizing interactions between vehicles. We first propose an interactive trajectory prediction method to explore possible cooperations between an AV and the others. Further, an evaluation on safety, efficiency and comfort is designed to make a decision on lane change. Thereafter, we propose a motion planning algorithm based on model predictive control (MPC), which incorporates AV’s decision and surrounding vehicles’ interactive behaviors into constraints so as to avoid collisions during lane change. Quantitative testing results show that compared with the methods without an interactive prediction, our method enhances driving efficiencies of the AV and other vehicles by 14.8% and 2.6%, respectively, which indicates that a proper utilization of vehicle interactions can effectively reduce the conservatism of the AV and promote the cooperation between the AV and others.

In this article, we develop data-driven optimal synchronization control architectures for leader-follower multiagent systems with additive disturbances and unknown system matrices. To minimize output synchronization error, algebraic Riccati equations (AREs) are derived, and unique feedback gains are determined by policy iteration. On that basis, two data-driven optimal synchronization control algorithms are developed without relying on the dynamics of the system, which guarantee output synchronization while minimizing synchronization errors and rejecting disturbances. The first algorithm uses the output synchronization error data to perform online data-driven learning (DDL), while the second algorithm uses the input data to perform DDL, where both data sample requirements are transformed into rank conditions. We have presented rigorous theoretical analyses of our proposed algorithms, which demonstrate that if an initial control protocol can make the system achieve output synchronization under mild conditions, our proposed two algorithms can take advantage of the data from reaching synchronization to optimize the closed-loop performance. Finally, a numerical example is provided to emphasize the effectiveness of our methods.

The robust iterative learning control (RILC) can deal with the systems with unknown time-varying uncertainty to track a repeated reference signal. However, the existing robust designs consider all the possibilities of uncertainty, which makes the design conservative and causes the controlled process converging to the reference trajectory slowly. To eliminate this weakness, a data-driven method is proposed. The new design intends to employ more information from the past input-output data to compensate for the robust control law and then to improve performance. The proposed control law is proved to guarantee convergence and accelerate the convergence rate. Ultimately, the experiments on a robot manipulator have been conducted to verify the good convergence of the trajectory errors under the control of the proposed method.

In batch processes, the ability to learn from previous process data results in high-value and batch-improved products. For batch processes with constraints and state-dependent uncertainty, this article presents a conic iterative learning control (ILC) approach, which uses cone theory to incorporate historical process data into optimization-based ILC design. The proposed conic ILC approach uses rank conditioning to select distinct data samples and conic mapping to map the data to the to-be-optimized control input variables, since adding all historical process data would be computationally intensive. Our method yields a tradeoff between learning ability from historical experience and computational efficiency from solving the optimization problem. Provable constraint satisfaction and robust stability are considered separately. To demonstrate the proven properties and effectiveness of the approach, we present a case study of the injection molding process.

Forecasting the trajectories of neighbor vehicles is a crucial step for decision making and motion planning of autonomous vehicles. This paper proposes a graph-based spatial-temporal convolutional network (GSTCN) to predict future trajectory distributions of all neighbor vehicles using past trajectories. This network tackles spatial interactions using a graph convolutional network (GCN), and captures temporal features with a convolutional neural network (CNN). The spatial-temporal features are encoded and decoded by a gated recurrent unit (GRU) network to generate future trajectory distributions. Besides, we propose a weighted adjacency matrix to describe the intensities of mutual influence between vehicles, and the ablation study demonstrates the effectiveness of our scheme. Our network is evaluated on two real-world freeway trajectory datasets: I-80 and US-101 in the Next Generation Simulation (NGSIM). Comparisons in three aspects, including prediction errors, model sizes, and inference speeds, show that our network can achieve state-of-the-art performance.

Bin-packing problem (BPP) is a typical combinatorial optimization problem whose decision-making process is NP-hard. This article examines BPPs in varying environments, where random number and shape of items are to be packed in different instances. The objective is to find a unified model to derive optimal decision process that maximizes the utilization of bins. To this end, by mimicking the experience-based reasoning process of humans, this article proposes a novel brain-inspired experience reinforcement model, which takes advantage of both biological and engineering systems. By learning experience from similar situations, the model is adaptive, such as the human brain for sophisticated scenarios and varying environments. The proposed model mimics the functional coordination among brain regions by knowledge representation and knowledge extraction modules. The former one corresponds to the part of information processing and experience storage. The latter one includes two parts that can train reasoning strategies and improve the decision performance. The proposed model is applied to instances of random number and shape of items of BPP. The obtained results outperform the state-of-the-art methods for BPPs in varying environments.

Skeleton-based human action recognition has drawn considerable research interest since it can robustly accommodate dynamic circumstances and complex backgrounds. By modeling the human body skeletons as graph structure, graph convolution network (GCN) has achieved great success in this field. However, these methods based on GCN are difficult to capture global features and relations only through a single-layer network. To obtain the relations between distant skeleton joints of human body, it is necessary to stack multiple graph convolution layers. In addition, the topology of the graph needs to be set manually in graph convolution operation and it is shared in all layers and time dimensions. In this paper, a novel mix-hops graph convolutional network (MHGCN) is proposed to recognize human action from skeleton data. The proposed module can fuse local features with global features through a layer of graph convolutional network. Besides, the topological structure of graph in our model changes with the time dimension and it can be individually learned in an end-to-end way through the BP algorithm. The experiments on several benchmark datasets show remarkable performance for human action recognition, demonstrating the effectiveness of our method.

Effective traffic signal control strategies are critical for traffic management in urban traffic networks. Most existing optimization-based urban traffic control approaches update the traffic signal at regular time instants, where the length of the fixed update time interval is determined based on a trade-off between the computational efficiency and the control performance. Since event-triggered control (ETC) allows for more flexible and more efficient control than conventional time-triggered control by triggering the control action by events, and since it can refrain from redundant optimization while retaining a satisfactory behavior, we use an ETC scheme for traffic light control. In addition, based on the geographically distributed feature of traffic networks, a distributed paradigm is adopted to reduce the computational complexity for the optimization. We propose a distributed threshold-based event-triggered control strategy, where the independent triggering of agents leads to an asynchronous update of traffic signals in the system. The triggered agent then solves a mixed-integer linear programming problem and updates its traffic signals. The proposed approach is evaluated under various traffic demands by simulation, and is shown to yield the best trade-off between control performance and computational complexity compared to other control strategies.

This paper presents a new method to estimate real-time queue lengths at a signalized intersection by utilizing limited data extracted from surveillance videos. This method focuses on reconstructing vehicles' trajectories on an entire road segment. The real-time queue length can be derived from these reconstructed trajectories. In order to improve the accuracy of the trajectory reconstruction, a built-up car-following model is proposed to reconstruct the trajectories of vehicles joining and leaving the queue, respectively, which are further corrected by a fusion algorithm. The proposed method was validated in the Next Generation Simulation dataset, and the queue length for each signal cycle can be estimated with a high accuracy. The results show that the proposed method has a higher precision compared to three baseline models.

This paper investigates consensus problem of second-order multi-agent system on matrix-weighted networks. It is shown that when the null space of the Gauge transformed graph Laplacian is spanned by the Kronecker product of an all-one vector and a set of orthogonal vectors, the algebraic multiplicity of eigenvalue zero cannot exceed the nullity of the graph Laplacian, thus admitting a proper blocking of the system matrix's Jordan normal form. Second-order bipartite consensus is thereby achieved independent of the structural balance of the network. Simulation examples are provided to demonstrate the theoretical results.

We present an assume-guarantee contract framework for cyber-physical system design under probabilistic requirements. Given a stochastic linear system and a set of requirements captured by bounded Stochastic Signal Temporal Logic (StSTL) contracts, we propose algorithms to check contract compatibility, consistency, and refinement, and generate a sequence of control inputs that satisfies a contract. We leverage encodings of the verification and control synthesis tasks into mixed integer optimization problems, and conservative approximations of probabilistic constraints that produce sound and tractable problem formulations. We illustrate the effectiveness of our approach on three case studies, including the design of controllers for aircraft power distribution networks.

Since urban traffic congestion has become a major problem for big cities in recent years, we propose a distributed control scheme for traffic lights in the network. First, a new criterion called traffic process ability which implies the balance between the traffic demand and traffic capacity of each road is introduced. Moreover, the congestion of a road is mitigated by utilizing the traffic process ability of the neighbors more effectively, where different weights are assigned to roads in each agent according to their importance or the real-time traffic conditions. As only local information is needed, a distributed control scheme in which the road network is divided among several agents is proposed. Furthermore, in order to accelerate the congestion dissipation process, the aggregated state of each agent is introduced into the performance index and balanced with its neighboring agents. The control signals are calculated by agents in a parallel way and the optimization problem is solved iteratively to reach a convergence. Finally, the effectiveness of the proposed control scheme is evaluated by simulation under different scenarios and the performance is compared with the traffic responsive control method SCOOT.

Traffic state prediction has been a popular topic, since traffic congestion occurs in most cities and creates inconvenience to human daily life. In this paper, we propose a predicting method for a city's overall traffic state, in order to help people avoid possible future congestion. Based on the variable-order Markov model theory and probability suffix tree, the proposed method makes use of the association rules to improve forecasting performance. Since the association rules are extracted from the historical traffic data and describe the traffic state relations among different regions, the proposed method can improve the predictive accuracy. The traffic system in Shanghai is considered as our experimental case because of its complicated and gigantic coupling transport network. The experimental results indicate more accuracy compared with other methods in long-term traffic status prediction.

This paper examines the event-triggered consensus problem of the multi-agent system on signed networks. An event-based distributed control protocol is proposed for multiagent networks where the dynamics of each agent is characterized by a controllable linear time-invariant system ( A, B). By exploring the Gauge transformation between Laplacian matrix and signed Laplacian matrix, we show that the bipartite consensus and trivial consensus can be achieved, respectively, in terms of the structural balance of the underlying signed network. Furthermore, the Zeno phenomenon of the closed-loop system is shown to be non-existed when employing the proposed event-based control protocol. Simulation results are finally provided to demonstrate our results.

In this paper, we present a linear least squares method to identify a damping rate function for a non-Markovian single qubit system. The dynamics of the system is described by a time-convolutionless master equation where the unknown damping rate function contains all the information of the environment. By expressing the function as a polynomial of time with unknown coefficients, we convert the identification problem into a parameter estimation problem. We transform the master equation into a Bloch differential equation, and obtain a reduced system whose output is the time trace of an observable. Then we use a linear least squares method to estimate unknown coefficients in the polynomial by utilizing the measured outputs. Finally, the effectiveness of our method is shown in an example of a two-level atom non-Markovian system.

We develop an assume-guarantee contract framework for the design of cyber-physical systems, modeled as closed-loop control systems, under probabilistic requirements. We use a variant of signal temporal logic, namely, Stochastic Signal Temporal Logic (StSTL) to specify system behaviors as well as contract assumptions and guarantees, thus enabling automatic reasoning about requirements of stochastic systems. Given a stochastic linear system representation and a set of requirements captured by bounded StSTL contracts, we propose algorithms that can check contract compatibility, consistency, and refinement, and generate a controller to guarantee that a contract is satisfied, following a stochastic model predictive control approach. Our algorithms leverage encodings of the verification and control synthesis tasks into mixed integer optimization problems, and conservative approximations of probabilistic constraints that produce both sound and tractable problem formulations. We illustrate the effectiveness of our approach on a few examples, including the design of embedded controllers for aircraft power distribution networks.

In this paper, we present a gradient algorithm to identify a damping rate function for a non-Markovian single qubit system. The dynamics of the single qubit system in a non-Markovian environment are assumed to obey a time convolutionless master equation, where all the non-Markovian effects of the environment are combined in the unknown damping rate function. To identify the damping rate function, we measure time trace observables of the qubit such that we can formulate the identification procedure as an optimization problem. Thus, we design a gradient algorithm to optimally reveal the damping rate function.

In the consideration of constraints, repetitive model predictive control is an effective method to track a periodic signal as well as reject a periodic disturbance. However, in practical systems, it is difficult to determine the period of the disturbance; meanwhile, too much information is needed for the design of repetitive controller, thus will make it difficult for the design of controller. In this paper, a system subject to periodic disturbance with Markov jump uncertainty is considered and a new method of repetitive model predictive control with Markov jump model has been introduced to reject the disturbance. The simulation demonstrates that the proposed method is effective.

For automatic train operation (ATO) system, the main object of ATO controller is to control the moving of the train as close as possible to the target curve, so that the demands for time, safety, travelling comfort and energy saving can be met. In this paper, a model predictive control (MPC) strategy is introduced to control the train system due to its good performance and ability to handle system with time-delay and constraints. Model parameters can be identified by measured data. The proposed MPC algorithm with corresponding switching strategy is designed. The simulation results illustrate the feasibility and effectiveness of the proposed control scheme.

In this paper, constrained model predictive control (MPC) designs for a class of structured uncertain time-delay systems have been developed, where state delays as well as a random input delay have been taken into account. By replacing the control strategy with a freer one, an improved MPC design with larger initial feasible region has been developed. Furthermore, by analyzing the obtained algorithm, some useful properties of the solutions to the MPC optimization problem have been established. With these properties, a variant of the obtained algorithm with dramatically reduced number of online optimizing variables and hence the online computational burden has also been developed. However, the control performance degrades slightly. The two algorithms haven been proved to stabilize the closed loop system in the mean square sense and to guarantee the satisfaction of constraints. Finally a numeric example is given to illustrate the proposed results.