Finite-time Optimal Control Problem Research Articles

Adaptive Finite-Time Optimal Formation Control for Second-Order Nonlinear Multiagent Systems

This article addresses an adaptive backstepping finite-time optimal formation control problem for second-order multiagent systems (MASs) with unknown nonlinear dynamics. Neural networks (NNs) are used to identify the unknown uncertain terms in the controlled system. Then, the finite-time optimal formation control is designed by constructing a novel optimal performance index function containing exponential power terms based on the framework of identifier–actor–critic. It is proved that all signals in the system are bounded in finite time, and the formation control is simultaneously achieved at minimum cost. The effectiveness and superiority of the proposed control algorithm are verified by simulation comparisons and data analyses.

Scenario trajectory optimization and control on STEP

In this paper we present optimized actuator trajectories, evolving in time and space, of non-inductive ramp-up scenarios for the Spherical Tokamak for Energy Production (STEP). These trajectories are computed by solving a non-linear, multi-objective, constrained, finite-time optimal control problem. A method unique to STEP ramp-up studies that provides an alternative to existing trajectory search strategies which rely on manually adjusting trajectories to reach a desired state. To navigate a non-linear parameter space which is densely populated with local minima, we demonstrate an iterative objective function construction process whereby costs and constraints are included successively and re-optimized after each inclusion to improve convergence and feasibility. This method is particularity useful when the initial trajectory is far from the desired operating space. We use the RApid Plasma Transport simulatOR (RAPTOR) code to self-consistently solve four coupled, 1-D state equations; poloidal flux, electron temperature, ion temperature and electron density. Our STEP actuator trajectories lasting 1500s, consist of 8 Gaussian electron cyclotron heating and current drive beams distributed across the minor radius, a Deuterium/Tritium particle source and plasma current. We also introduce a modified transport Bohm–gyroBohm model and a new actuator module to RAPTOR which were required to adequately simulate the STEP operating scenarios.

Reduced order model predictive control for parametrized parabolic partial differential equations

Model Predictive Control (MPC) is a well-established approach to solve infinite horizon optimal control problems. Since optimization over an infinite time horizon is generally infeasible, MPC determines a suboptimal feedback control by repeatedly solving finite time optimal control problems. Although MPC has been successfully used in many applications, applying MPC to large-scale systems – arising, e.g., through discretization of partial differential equations – requires the solution of high-dimensional optimization problems and thus poses immense computational effort. We consider systems governed by parametrized parabolic partial differential equations and employ the reduced basis method as a low-dimensional surrogate model for the finite time optimal control problem. The reduced order optimal control serves as feedback control for the original large-scale system. We analyze the proposed RB-MPC approach by first developing a posteriori error bounds for the errors in the optimal control and associated cost functional. These bounds can be evaluated efficiently in an offline-online computational procedure and allow us to guarantee asymptotic stability of the closed-loop system using the RB-MPC approach in several practical scenarios. We also propose an adaptive strategy to choose the prediction horizon of the finite time optimal control problem. Numerical results are presented to illustrate the theoretical properties of our approach.

Finite-time optimal control for Markov jump systems with singular perturbation and hard constraints

This study addresses the finite-time optimal control problem for Markov jump systems subject to singular perturbations and hard constraints. The transition probabilities are considered to vary randomly within a finite set, which can be governed by a higher-level nonhomogeneous Markov process. The input signal is quantized dynamically before being passed to the controller, and a generalized framework is formulated based on the quantized control input and actuator faults. The goal is to ensure the finite-time boundedness of Markov jump singularly perturbed systems by designing a controller that satisfies the hard constraints. Then, by adopting a novel Lyapunov functional, the singularly perturbed parameter-independent sufficient conditions are obtained. For the preset finite-time interval, the optimization problem of minimizing the upper bound of the state trajectory is skillfully solved. Finally, a simulation example is presented to verify the effectiveness of the proposed design method.

Data-Driven Optimization for Deposition with Degradable Tools

We present a data-driven optimization approach for robotic controlled deposition with a degradable tool. Existing methods make the assumption that the tool tip is not changing or is replaced frequently. Errors can accumulate over time as the tool wears away and this leads to poor performance in the case where the tool degradation is unaccounted for during deposition. In the proposed approach, we utilize visual and force feedback to update the unknown model parameters of our tool-tip. Subsequently, we solve a constrained finite time optimal control problem for tracking a reference deposition profile, where our robot plans with the learned tool degradation dynamics. We focus on a robotic drawing problem as an illustrative example. Using real-world experiments, we show that the error in target vs actual deposition decreases when learned degradation models are used in the control design.

Finite-time optimal feedback control mechanism for knowledge transmission in complex networks via model predictive control

In this paper, the finite-time (FT) optimal feedback control problems of knowledge transmission processes in complex networks via model predictive control (MPC) have been studied. Firstly, we build a knowledge transmission Susceptible–Infected–Hesitation (SIH) model in complex networks. Secondly, interventional control strategies are designed to regulate the system parameters to improve the performance of knowledge dissemination, including improving self-learning ability, acquaintance influence, and review rate. With the help of the Lyapunov-based HJB optimal control method, the existence of the optimal solution to the economic optimal problem of the knowledge transmission control model is guaranteed. Then, the optimal control solution is derived by using Pontryagin’s maximum principle. To focus on the performance indicators and state trajectories of the system and enable the controller to modify in real-time according to the state in a fixed time interval as soon as possible, an MPC based on FT feedback is proposed for the first time. Furthermore, under feedback control and initial conditions, the control knowledge dissemination model is FT stable. Numerical simulations are provided to verify the proposed method.

Log–log convexity of an optimal control problem for positive linear systems

This paper investigates the finite-time optimal control problems for positive linear systems with a time-varying control input. Cost functional of the system state is constructed. Under some assumptions on the designed parameters and cost functionals, the optimization problem with piecewise-constant matrix functions is first proved to be log–log convex and can be solved via geometric programming. Then, the log–log convex result is further extended to the optimization problem with continuous functions. Finally, an optimal control problem is investigated to verify the effectiveness of the proposed optimization framework.

Traction Adaptive Motion Planning and Control at the Limits of Handling

In this article, we address the problem of motion planning and control at the limits of handling, under locally varying traction conditions. We propose a novel solution method where traction variations over the prediction horizon are represented by time-varying tire force constraints, derived from a predictive friction estimate. A constrained finite time optimal control problem (CFTOC) is solved in a receding horizon fashion, imposing these time-varying constraints. Furthermore, our method features an integrated sampling augmentation procedure that addresses the problems of infeasibility and sensitivity to local minima that arise at abrupt constraint alterations, for example, due to sudden friction changes. We validate the proposed algorithm on a Volvo FH16 heavy-duty vehicle, in a range of critical scenarios. Experimental results indicate that traction adaptive motion planning and control improves the vehicle’s capacity to avoid accidents, both when adapting to low local traction, by ensuring dynamic feasibility of the planned motion, and when adapting to high local traction, by realizing high traction utilization.

Cooperative signal-free intersection control using virtual platooning and traffic flow regulation

The emerging technologies of connectivity and automation enable the potential for signal-free intersection control. In this context, virtual platooning is posited to be an innovative, decentralized control strategy that maps two-dimensional vehicle movements onto a one-dimension virtual platoon to enable intersection operations. However, the effectiveness of virtual platooning-based control can be limited or degraded by parametric inaccuracies and unparameterized disturbances in vehicle dynamics, heavy traffic congestion, and/or uncoordinated platoons in multi-lane intersections. To explicitly address these limitations, this study proposes a hybrid cooperative intersection control framework consisting of microscopic-level virtual platooning control and macroscopic-level traffic flow regulation for traffic environments with connected autonomous vehicles. In virtual platooning control, vehicles approaching an intersection are organized into coordinated independent virtual platoons to avoid potential conflicts triggered by platoon formation changes. Through coordination, vehicles in a platoon are grouped into compatible passing sets to maintain desired safe spacing when proceeding through the intersection. We propose a distributed adaptive sliding mode controller (DASMC) which uses the backstepping control method and model reference adaptive control method to address parametric inaccuracies, and the sliding mode control method to consistently suppress the negative effects of the unparameterized disturbances. Each vehicle approaching the intersection utilizes the kinematic information from neighboring vehicles to implement the DASMC in a distributed manner such that vehicles within the same virtual platoon can achieve consensus safely. However, virtual platooning control cannot preclude excessive traffic from approaching the intersection, which can cause undesired spillbacks and degrade intersection control performance. To address this issue, traffic flow regulation is integrated with the virtual platooning control using an iterative feedback loop mechanism. In each iteration of the iterative feedback loop, a constrained finite-time optimal control (CFTOC) problem is solved to determine the optimal input flow permitted to proceed through the intersection, and the virtual platooning control provides feedback on the queue status to the CFTOC to initiate the next iteration. The effectiveness of the proposed intersection control framework is evaluated through numerical experiments. The results indicate that the proposed virtual platooning DASMC controller can mitigate the effects of parametric inaccuracies and unparameterized disturbances to achieve consensus for approaching vehicles, as well as guarantee string stability. Further, the proposed framework can alleviate traffic spillbacks and travel delays effectively through traffic flow regulation.

Finite-time optimal tracking control using augmented error system method

This paper proposes a design method of finite-time optimal tracking controller for quadratic performance index. A novel nonlinear finite-time optimal controller is designed to solve the finite-time optimal tracking control problem for linear system. By applying augmented error system method and finite-time optimal state feedback control, sufficient conditions involving V-function are obtained, and eventually, a nonlinear controller based on the V-function is derived. In addition, the construction of a specific V-function satisfying those conditions is discussed in detail. The nonlinear controller which we introduced makes closed-loop system finite-time stable, so that the tracking error of original system converges in finite time. The simulation example illustrates the efficiency of our results.

Adaptive Neural Networks Finite-Time Optimal Control for a Class of Nonlinear Systems

This article addresses the finite-time optimal control problem for a class of nonlinear systems whose powers are positive odd rational numbers. First of all, a finite-time controller, which is capable of ensuring the semiglobal practical finite-time stability for the closed-loop systems, is developed using the adaptive neural networks (NNs) control method, adding one power integrator technique and backstepping scheme. Second, the corresponding design parameters are optimized, and the finite-time optimal control property is obtained by means of minimizing the well-defined and designed cost function. Finally, a numerical simulation example is given to further validate the feasibility and effectiveness of the proposed optimal control strategy.

Suboptimal closed‐form feedback control of input‐affine non‐linear systems

A closed-form feedback solution to finite-time optimal control problems for input-affine non-linear systems is presented here. Based on the state-dependent Riccati equation (SDRE) framework, this suboptimal solution, called balanced-cost SDRE (BCSDRE), is proposed to minimise the overall control effort required for driving the system from an arbitrary initial condition to an acceptable neighbourhood around origin, within a pre-specified final time. The state error is dynamically penalised to balance the overall state error and the control cost, resulting in a modified SDRE. Unlike the existing finite-horizon SDRE methods, the proposed technique does not require to solve algebraic Riccati and Lyapunov equations at every time step, which makes this approach computationally efficient. Based on the Lyapunov theory, the origin of the BCSDRE controlled system is shown to be locally exponentially stable. Furthermore, the state profile is also shown to be locally predefined time ultimately bounded, under appropriate tuning of the proposed controller. The proposed technique is implemented to design a controller for the Van der Pol oscillator and impact angle guidance design problem, and its performance is compared with the existing finite-horizon SDRE-based techniques.

Approximate Solutions to the Hamilton-Jacobi Equations for Generating Functions

For a nonlinear finite time optimal control problem, a systematic numerical algorithm to solve the Hamilton-Jacobi equation for a generating function is proposed in this paper. This algorithm allows one to obtain the Taylor series expansion of the generating function up to any prescribed order by solving a sequence of first order ordinary differential equations recursively. Furthermore, the coefficients of the Taylor series expansion of the generating function can be computed exactly under a certain technical condition. Once a generating function is found, it can be used to generate a family of optimal control for different boundary conditions. Since the generating function is computed off-line, the on-demand computational effort for different boundary conditions decreases a lot compared with the conventional method. It is useful to online optimal trajectory generation problems. Numerical examples illustrate the effectiveness of the proposed algorithm.

Direct Parallel Computations of Second-Order Search Directions for Model Predictive Control

Model predictive control ( MPC ) is one of the most widely spread advanced control schemes in industry today. In MPC , a constrained finite-time optimal control ( CFTOC ) problem is solved at each iteration in the control loop. The CFTOC problem can be solved using, for example, second-order methods, such as interior-point or active-set methods, where the computationally most demanding part often consists of computing the sequence of second-order search directions. Each search direction can be computed by solving a set of linear equations that corresponds to solving an unconstrained finite-time optimal control ( UFTOC ) problem. In this paper, different direct (noniterative) parallel algorithms for solving UFTOC problems are presented. The parallel algorithms are all based on a recursive variable elimination and solution propagation technique. Numerical evaluations of one of the parallel algorithms indicate that a significant boost in performance can be obtained, which can facilitate high-performance second-order MPC solvers.

Reconfigurable Model Predictive Control for Multievaporator Vapor Compression Systems

This paper considers the control of a multievaporator vapor compression system (ME-VCS) where individual evaporators are permitted to turn ON or OFF. We present a model predictive controller (MPC) that can be easily reconfigured for different ON/OFF configurations of the system. In this approach, only the cost function of the constrained finite-time optimal control problem is updated depending on the system configuration. Exploiting the structure of the system dynamics, the cost function is modified by zeroing elements of the state, input, and terminal cost matrices. The advantage of this approach is that cost matrices for each configuration of the ME-VCS do not need to be stored or computed online. This reduces the effort required to tune and calibrate the controller and the amount of memory required to store the controller parameters in a microprocessor. The reconfigurable MPC is compared with a conventional approach in which individual model predictive controllers are independently designed for each ON/OFF configuration. The simulations show that the reconfigurable MPC method provides a similar closed-loop performance in terms of reference tracking and constraint satisfaction to the set of individual model predictive controllers. Further, we show that our controller requires substantially less memory than the alternative approaches. Experiments on a residential two-zone vapor compression system further validate the reconfigurable MPC method.

MPC-based Co-design of Control and Routing for Wireless Sensor and Actuator Networks

A wireless sensor and actuator network (WSAN) is a class of networked control systems. In WSANs, sensors and actuators are located in a distributed way, and communicate to controllers through a wireless communication network such as a multi-hop network. In this paper, we propose a model predictive control (MPC) method for co-design of control and routing of WSANs. MPC is an optimal control strategy based on numerical optimization. The control input is calculated by solving the finite-time optimal control problem at each discrete time. In the proposed method, a WSAN is modeled by a switched linear system. In the finite-time optimal control problem, a control input and a mode corresponding to a communication path are optimized simultaneously. The proposed method is demonstrated by a numerical example.

Strong Stationarity Conditions for Optimal Control of Hybrid Systems

We present necessary and sufficient optimality conditions for finite-time optimal control problems for a class of hybrid systems described by linear complementarity models. Although these optimal control problems are difficult in general due to the presence of complementarity constraints, we provide a set of structural assumptions ensuring that the tangent cone of the constraints possesses geometric regularity properties. These imply that the classical Karush–Kuhn–Tucker conditions of nonlinear programming theory are both necessary and sufficient for local optimality, which is not the case for general mathematical programs with complementarity constraints. We also present sufficient conditions for global optimality. We proceed to show that the dynamics of every continuous piecewise affine (PWA) system can be written as the optimizer of a mathematical program, which results in a linear complementarity model satisfying our structural assumptions. Hence, our stationarity results apply to a large class of hybrid systems with PWA dynamics. We present simulation results showing the substantial benefits possible from using a nonlinear programming approach to the optimal control problem with complementarity constraints instead of a more traditional mixed-integer formulation.

Approximate MLD System Model of Switched Linear Systems for Model Predictive Control

In this paper, to reduce the computation time for solving the finite-time optimal control problem (i.e., the model predictive control (MPC) problem), an approximate model of switched linear systems is proposed. A switched linear system is one of the typical subclasses of hybrid systems, and has several applications such as networked control systems. First, an approximation of the state variable in switched linear systems is derived. Next, using auxiliary binary and continuous variables, an approximate model given by a mixed logical dynamical system model is derived. Using this approximate model, the dimension of the auxiliary continuous variable can be reduced. As a result, the online computation time in MPC can decrease. Finally, the effectiveness of the proposed method is presented by a numerical example.

Cone separation, quadratic control systems and control of spin dynamics in the presence of decoherence.

In this paper, we study some control problems related to the control of coupled spin dynamics in the presence of relaxation and decoherence in nuclear magnetic resonance spectroscopy. The decoherence is modelled through a master equation. We study some model problems, whereby, through an appropriate choice of state variables, the system is reduced to a control system, where the state enters linearly and controls quadratically. We study this quadratic control system. Study of this system gives us explicit bounds on how close a coupled spin system can be driven to its target state and how much coherence and polarization can be transferred between coupled spins. Optimal control for the quadratic control system can be understood as the separation of closed cones, and we show how the derived results on optimal efficiency can be interpreted in this formulation. Finally, we study some finite-time optimal control problems for the quadratic control system.This article is part of the themed issue 'Horizons of cybernetical physics'.

Boundary Control for Optimal Mixing by Stokes Flows

We discuss the optimal boundary control problem for mixing an inhomogeneous distribution of a passive scalar field in an unsteady Stokes flow. The problem is motivated by mixing the fluids within a cavity or vessel at low Reynolds numbers by moving the walls or stirring at the boundary. It is natural to consider the velocity field which is induced by a control input tangentially acting on the boundary of the domain through the Navier slip boundary conditions. Our main objective is to design an optimal Navier slip boundary control that optimizes mixing at a given final time. This essentially leads to a finite time optimal control problem of a bilinear system. In the current work, we consider a general open bounded and connected domain \(\Omega \subset \mathbb {R}^{d}, d=2,3\). We employ the Sobolev norm for the dual space \((H^{1}(\Omega ))'\) of \(H^{1}( \Omega )\) to quantify mixing of the scalar field in terms of the property of weak convergence. A rigorous proof of the existence of an optimal control is presented and the first-order necessary conditions for optimality are derived.