Terminal Set Research Articles

A linear-time algorithm for finding a one-to-many 3-disjoint path cover in the cube of a connected graph

Given two disjoint vertex sets S={s} and T={t1,t2,t3} of a connected graph, a one-to-many 3-disjoint path cover joining S and T is a vertex-disjoint path cover {P1,P2,P3} such that each path Pi joins s and ti. In this paper, we present an efficient algorithm that builds, if exists, a one-to-many 3-disjoint path cover in the cube of a connected graph G joining the two terminal sets. Interestingly enough, we show that a carefully selected spanning tree or spanning unicyclic subgraph of G is all we need to find the desired disjoint path cover in time linear to the size of G.

Ellipsoidal Lyapunov-based hybrid model predictive control for mixed logical dynamical systems with a recursive feasibility guarantee

This paper proposes an ellipsoidal hybrid model predictive control approach to solve the robust stability problem of uncertain hybrid dynamical systems modelled by the mixed logical dynamical framework. In this approach, the traditional terminal equality constraint is replaced by an ellipsoid that results in a maximal positive invariant set for the closed-loop system. Then, a Lyapunov decreasing condition along with the robustness criterion is introduced to the optimization problem to achieve the robust stability of the closed-loop system. As the main advantages, the ellipsoidal terminal set proposed in this paper attains a larger domain of attraction along with the recursive feasibility guarantee. Moreover, the stability and robustness constraints are achieved by a lower prediction horizon, which leads to a smaller dimension optimization problem. In addition, to reduce the computational complexity of the corresponding optimization problem, a suboptimal version of the proposed algorithm is introduced. Finally, numerical and car suspension system examples show the capabilities of the proposed method.

Experimental validation of model predictive control stability for autonomous driving

This paper addresses the design of time-varying model predictive control of an autonomous vehicle in the presence of input rate constraints such that closed-loop stability is guaranteed. Stability is proved via Lyapunov techniques by adding a terminal state constraint and a terminal cost to the controller formulation. The terminal set is the maximum positive invariant set of a multi-plant description of the vehicle linear time-varying model. The terminal cost is an upper-bound on the infinite cost-to-go incurred by applying a linear–quadratic regulator control law. The proposed control design is experimentally tested and successfully stabilizes an autonomous Scania construction truck in an obstacle avoidance scenario.

Dynamic path planning and trajectory tracking using MPC for satellite with collision avoidance

This paper proposes a dynamic path planning and trajectory tracking algorithm for an autonomous satellite, released from the space station, to get to the desired position for performing space tasks. The complex construction of the space station results in the presence of a geometric channel constraint for the obstacles avoidance. In addition, a three dimension B-spline template with minimizing the curvature of the path is designed, which could guarantee the continuity of the curvature to make the trajectory smooth and avoid the satellite from stopping at discontinuities waypoints. Then, the reference states and inputs are solved by a new projection method, which provides a foundation for the subsequent trajectory tracking. Subsequently, a finite horizon model predictive control method is constructed for the path tracking. The benefits of this approach are to take constraints into consideration, and to get optimal performance by minimizing the fuel consumption compared with other tracking controllers. The closed-loop stability is guaranteed by the feedback controller, terminal penalty, and a newly terminal constraint set. In simulation experiments, results illustrate the effectiveness and practicality of the algorithm.

Network Sparsification for Steiner Problems on Planar and Bounded-Genus Graphs

We propose polynomial-time algorithms that sparsify planar and bounded-genus graphs while preserving optimal or near-optimal solutions to Steiner problems. Our main contribution is a polynomial-time algorithm that, given an unweighted undirected graph G embedded on a surface of genus g and a designated face f bounded by a simple cycle of length k , uncovers a set F ⊆ E ( G ) of size polynomial in g and k that contains an optimal Steiner tree for any set of terminals that is a subset of the vertices of f . We apply this general theorem to prove that: — Given an unweighted graph G embedded on a surface of genus g and a terminal set S ⊆ V ( G ), one can in polynomial time find a set F ⊆ E ( G ) that contains an optimal Steiner tree T for S and that has size polynomial in g and | E ( T )|. — An analogous result holds for an optimal Steiner forest for a set S of terminal pairs. — Given an unweighted planar graph G and a terminal set S ⊆ V ( G ), one can in polynomial time find a set F ⊆ E ( G ) that contains an optimal (edge) multiway cut C separating S (i.e., a cutset that intersects any path with endpoints in different terminals from S ) and that has size polynomial in | C |. In the language of parameterized complexity, these results imply the first polynomial kernels for S teiner T ree and S teiner F orest on planar and bounded-genus graphs (parameterized by the size of the tree and forest, respectively) and for (E dge ) M ultiway C ut on planar graphs (parameterized by the size of the cutset). Additionally, we obtain a weighted variant of our main contribution: a polynomial-time algorithm that, given an undirected plane graph G with positive edge weights, a designated face f bounded by a simple cycle of weight w ( f ), and an accuracy parameter ε > 0, uncovers a set F ⊆ E ( G ) of total weight at most poly(ε -1 ) w ( f ) that, for any set of terminal pairs that lie on f , contains a Steiner forest within additive error ε w ( f ) from the optimal Steiner forest.

The Greening of Terminal Concessions in Seaports

Port authorities around the world are pursuing a greening of port management in view of safeguarding their license to operate, and increasing their economic and environmental competitiveness. This paper analyzes how port authorities, via the design and implementation of concession agreements, can contribute to a further greening of port management. The paper presents a typology of green instruments applicable to a terminal concession setting. The instruments are evaluated on overall feasibility and suitability in a concession context, but also on more specific criteria related to implementation issues, contribution to green strategies, and targets of port authorities and regulatory/enforcement aspects. The evaluation matrix is based on the output of a structured workshop with port managers and concession experts in a sample of European ports. We demonstrate that a variety of regulatory, investment, and pricing measures are available to port authorities to include green targets in terminal concession agreements. Not all instruments have the same likeliness of being embraced or implemented by port authorities, in part because of a low perceived contribution, high associated regulation costs, or simply because the port authority is unlikely to have jurisdiction in that specific area. Measures related to information reporting and some types of harm-based standards, design standards, and technology specifications are relatively easy to implement in a concession setting. The results also show that many of the measures with a higher expected contribution to innovation and environmental objectives are typically also the ones with higher regulation costs, which might post a higher complexity in terms of their implementation in a concession setting. We further argue that initiatives toward the greening of concession procedures can only reap full benefits if these actions are embedded in a chain approach toward the environment (ship, port, terminal, warehouse, and inland transport).

Efficiency of night interruption treatments with red and far-red light-emitting diodes (LEDs) in preventing bud set in Norway spruce seedlings

Terminal bud set can be prevented by interrupting night with short pulses of light when the natural photoperiod is too short to maintain growth. Norway spruce (Picea abies Karst.) seedlings originating from 61°N and 64°N were grown in growth chambers under conditions that mimic growth conditions in a heated greenhouse in early spring in Finland (experiment 1) or under constant growth conditions (experiment 2). The seedlings were exposed to the following night interruption (NI) treatments using light-emitting diodes (LEDs) that generated red (R, peak at 660 nm) and far-red (FR, peak at 735 nm) wavelengths in 20 s pulses at 15 min intervals: (i) red light alone (R); (ii) R combined with FR (R + FR); and (iii) control (no NI treatment). The R + FR treatment was more effective in preventing terminal bud set than the R treatment. Seedling responses depended on the provenance and growth conditions. The R treatment reduced the proportion of seedlings with terminal buds in the 61°N seedlings and delayed bud set in the 64°N seedlings. The fluctuating growth conditions or longer dark period between the photoperiod and NI treatments reduced the efficiency of the R + FR treatment. A combination of R and FR LEDs with adequate light intensity and duration is suitable for intermittent NI treatment in Norway spruce seedlings.

An Exponential Lower Bound for Cut Sparsifiers in Planar Graphs

Given an edge-weighted graph G with a set Q of k terminals, a mimicking network is a graph with the same set of terminals that exactly preserves the size of minimum cut between any partition of the terminals. A natural question in the area of graph compression is to provide as small mimicking networks as possible for input graph G being either an arbitrary graph or coming from a specific graph class. We show an exponential lower bound for cut mimicking networks in planar graphs: there are edge-weighted planar graphs with k terminals that require 2^{k-2} edges in any mimicking network. This nearly matches an upper bound of mathcal {O}(k 2^{2k}) of Krauthgamer and Rika (in: Khanna (ed) Proceedings of the twenty-fourth annual ACM-SIAM symposium on discrete algorithms, SODA 2013, New Orleans, 2013) and is in sharp contrast with the upper bounds of mathcal {O}(k^2) and mathcal {O}(k^4) under the assumption that all terminals lie on a single face (Goranci et al., in: Pruhs and Sohler (eds) 25th Annual European symposium on algorithms (ESA 2017), 2017, arXiv:1702.01136; Krauthgamer and Rika in Refined vertex sparsifiers of planar graphs, 2017, arXiv:1702.05951). As a side result we show a tight example for double-exponential upper bounds given by Hagerup et al. (J Comput Syst Sci 57(3):366–375, 1998), Khan and Raghavendra (Inf Process Lett 114(7):365–371, 2014), and Chambers and Eppstein (J Gr Algorithms Appl 17(3):201–220, 2013).

Beam‐domain SWIPT in massive MIMO system with energy‐constrained terminals

In this study, the authors consider the simultaneous wireless information and power transfer (SWIPT) protocol design for the massive multiple-input multiple-output (MIMO) system in the beam-domain. In this system, the base station (BS) simultaneously serves a set of half-duplex energy-constrained terminals that are uniformly distributed within its coverage area. Based on the beam-domain distributions of channels, the BS can intelligently schedule terminals to mitigate the interference between terminals and improve the transmission spectral efficiency. The entire protocol can be divided into two phases. The first phase is designed for terminals energy harvesting as well as downlink training. During this phase, the BS transmits energy signals to the terminals. The terminals utilise the received energy signals for energy harvesting and downlink channel estimation. In the second phase, the BS forms the receive beamformers to receive signals transmitted by terminals. The transmit powers at the BS and the time switching ratio are optimised under the constraints of the current available energy and minimum transmission rate of terminals, so that the system can achieve the maximum sum-rate performance. Simulation results show that compared with traditional massive MIMO SWIPT protocols, the proposed SWIPT protocol can achieve better spectral efficiency performance.

Information geometry on the curved [formula omitted]-exponential family with application to survival data analysis

Information geometry and statistical manifold have attracted wide attention in the past few decades. The Amari–Chentsov structure plays a central role both in optimization and statistical inference because of the invariance under sufficient statistics. This paper discusses the Amari–Chentsov structure on statistical manifolds induced by the curved q-exponential family with Type-I censored data. The reliability function and cumulant generating function of the family are obtained, where the cumulant generating function contains a random variable depends on the experimental design. After the construction of the statistical manifold, the geometric quantities, i.e., Fisher metric, Amari tensor and connection coefficients are investigated. We find that the affine connection is not equal to 0, which is different from the existing results. The relationship among the geometric quantities based on the q-exponential family and curved q-exponential family is discussed. The relationship between the curvature on the manifold in the normal and tangent directions is obtained. Employing the maximum likelihood and Bayesian methods, the parameter vector and its dual are estimated. The main results are illustrated with the two-parameter q-exponential distribution and a real data analysis based on a terminal human cancer data set.

Approximating Steiner trees and forests with minimum number of Steiner points

Let R be a finite set of terminals in a convex metric space (M,d). We give approximation algorithms for problems of finding a minimum size set S⊆M of additional points such that the unit-disc graph G[R∪S] of R∪S satisfies some connectivity properties. Let Δ be the maximum number of independent points in a unit ball. For the Steiner Treewith Minimum Number of Steiner Points problem we obtain approximation ratio 1+ln⁡(Δ−1)+ϵ, which in R2 reduces to 1+ln⁡4+ϵ<2.3863+ϵ; this improves the ratios ⌊(Δ+1)/2⌋+1+ϵ of [19] and 2.5+ϵ of [7], respectively. For the Steiner Forestwith Minimum Number of Steiner Points problem we give a simple Δ-approximation algorithm, improving the ratio 2(Δ−1) of [21]. We also simplify the Δ-approximation of [4], when G[R∪S] should be 2-connected.

Mixed stochastic-deterministic tube MPC for offset-free tracking in the presence of plant-model mismatch

This paper presents a stochastic-tube model predictive control (MPC) strategy that systematically handles plant-model mismatch, guarantees stability in the presence of changing operating conditions, and ensures offset-free tracking for all reachable operating conditions. The key notion of this work is to separate the system uncertainty into two distinct sources of additive bounded state error. The first source is a non-random uncertainty that represents mismatch between the linear model of the controller and the true plant (as well as other types of persistent disturbances). The second source represents random fluctuations due to either the intrinsic stochastic variability of the system, or exogenous disturbances. Recursive feasibility and stability of the stochastic-tube MPC strategy is guaranteed by defining the terminal invariant set to include the effect of changes in the steady-state operating conditions. Offset-free tracking is achieved with a disturbance estimator that can be tuned independently of the controller. To reduce the conservatism inherent to robust control performance, a new filter model for deterministic disturbances is proposed that predicts uncertainty in the future evolution of the disturbances based on a bound on how quickly the plant-model mismatch can vary. The stochastic-tube MPC strategy is demonstrated on two simulation case studies—a benchmark DC-DC converter and an industrially-motivated fluidized-bed catalytic cracking unit.

Sequential distributed predictive control of a 3D tower crane

An asymptotically stabilizing sequential distributed model predictive control (MPC) of a 3D tower crane is proposed. Stability is ensured by employing three locally stabilizing MPC control laws. In the case of Lipschitz continuous local MPC control laws, a terminal cost and a terminal set constraint are used as stabilizing ingredients while robust control invariant feasible set is used as an additional constraint to guarantee recursive feasibility. On the other hand, in the case of an arbitrary cost function, switching to a robust dual-mode local control law is used inside of the terminal set to guarantee asymptotic stability.

Construction of the Solvability Set in Differential Games with Simple Motion and Nonconvex Terminal Set

We consider planar zero-sum differential games with simple motion, fixed terminal time, and polygonal terminal set. The geometric constraint on the control of each player is a convex polygonal set or a line segment. In the case of a convex terminal set, an explicit formula is known for the solvability set (a level set of the value function, maximal u-stable bridge, viability set). The algorithm corresponding to this formula is based on the set operations of algebraic sum and geometric difference (the Minkowski difference). We propose an algorithm for the exact construction of the solvability set in the case of a nonconvex polygonal terminal set. The algorithm does not involve the additional partition of the time interval and the recovery of intermediate solvability sets at additional instants. A list of half-spaces in the three-dimensional space of time and state coordinates is formed and processed by a finite recursion. The list is based on the polygonal terminal set with the use of normals to the polygonal constraints on the controls of the players.

Learning Model Predictive Control for Iterative Tasks. A Data-Driven Control Framework

A learning model predictive controller for iterative tasks is presented. The controller is reference-free and is able to improve its performance by learning from previous iterations. A safe set and a terminal cost function are used in order to guarantee recursive feasibility and nondecreasing performance at each iteration. This paper presents the control design approach, and shows how to recursively construct terminal set and terminal cost from state and input trajectories of previous iterations. Simulation results show the effectiveness of the proposed control logic.

The Steiner tree in [formula omitted]-free split graphs—A Dichotomy

Given a connected graph G and a terminal set R⊆V(G), the Steiner tree problem (STREE) asks for a tree that includes all of R with at most r vertices from V(G)∖R, for some integer r≥0. It is known from (Garey et al., 1977) that STREE is NP-complete in general graphs. A Split graph is a graph which can be partitioned into a clique and an independent set. White et al. (1985) have established that STREE in split graphs is NP-complete. In this paper, we present an interesting dichotomy: we show that STREE on K1,4-free split graphs is polynomial-time solvable, whereas STREE on K1,5-free split graphs is NP-complete. We investigate K1,4-free and K1,3-free (also known as claw-free) split graphs from a structural perspective. Further, using our structural study, we present polynomial-time algorithms for STREE in K1,4-free and K1,3-free split graphs. Although, polynomial-time solvability of K1,3-free split graphs is implied from K1,4-free split graphs, we wish to highlight our structural observations on K1,3-free split graphs which may be of use in solving other combinatorial problems.

Optimization-free robust MPC around the terminal region

We present a novel dual-mode MPC scheme that significantly reduces the computational effort of robust MPC (RMPC). Specifically, we propose a method for the computation of a large set C on which no optimal control problem (OCP) needs to be solved online. The method is motivated by the trivial observation that, for classical MPC, no optimization is required for the states in the terminal set T, because the unconstrained linear–quadratic regulator is optimal there. While this observation cannot be directly transferred to RMPC, we show that suitable sets C exist in the neighborhood of T and state an algorithm for their computation. We stress that the resulting sets C are significantly larger than robust positively invariant sets that are typically exploited in RMPC and on which it is well-known that no OCP needs to be solved online. The approach is illustrated with three examples for which we observe a reduction of the numerical effort between 22.36% and 95.60%.

The Initial and Terminal Cluster Sets of an Analytic Curve

AbstractFor an analytic curve γ (a, b) → ℂ, the set of values approached by γ(t), as t ↘ a and as t ↗ b can be any two continua of ℂ ⋃ {∞}.

Fuzzy Model Predictive Control of Discrete-Time Systems with Time-Varying Delay and Disturbances

In this paper, model predictive control (MPC) of discrete T–S fuzzy systems subjected to bounded time-varying delay and persistent disturbances is investigated. The Razumikhin approach is adopted for time-delay systems because it involves a Lyapunov function associated with the original nonaugmented state space of system dynamics when compared to the Krasovskii approach. As such, the Razumikhin approach has a good potential to avoid the inherent complexity of the Krasovskii approach especially in the presence of large delays and disturbances. Based on which, both online and offline MPC approaches for systems with time-varying delay are provided. In addition, persistent disturbances are considered that robust positive invariance and input-to-state stability under such circumstances are realized. In the offline approach, a sequence of explicit control laws that correspond to a sequence of robust constraints sets are computed offline. And it is proved that system states including all possibly delayed states can be steered to the terminal constraint set in finite time. Moreover, it allows the exact time delay to be unknown in the proposed two approaches. In particular, for systems with time-varying delay, the special positively invariant set theory and finite-time control theory based on the Razumikhin approach are directly revealed via the proposed offline approach.

Linear discrete systems controllability from a set of initial states to a terminal set

Linear discrete systems controllability from a set of initial states to a terminal set