Finite-time Optimal Control Research Articles

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.

Optimization-Based Approaches to Control of Probabilistic Boolean Networks

Control of gene regulatory networks is one of the fundamental topics in systems biology. In the last decade, control theory of Boolean networks (BNs), which is well known as a model of gene regulatory networks, has been widely studied. In this review paper, our previously proposed methods on optimal control of probabilistic Boolean networks (PBNs) are introduced. First, the outline of PBNs is explained. Next, an optimal control method using polynomial optimization is explained. The finite-time optimal control problem is reduced to a polynomial optimization problem. Furthermore, another finite-time optimal control problem, which can be reduced to an integer programming problem, is also explained.

Low-Rank Modifications of Riccati Factorizations for Model Predictive Control

In model predictive control ( MPC ), the control input is computed by solving a constrained finite-time optimal control ( CFTOC ) problem at each sample in the control loop. The main computational effort when solving the CFTOC problem using an active-set ( AS ) method is often spent on computing the search directions, which in MPC corresponds to solving unconstrained finite-time optimal control ( UFTOC ) problems. This is commonly performed using Riccati recursions or generic sparsity exploiting algorithms. In this paper, the focus is efficient search direction computations for AS type methods. The system of equations to be solved at each AS iteration is changed only by a low-rank modification of the previous one, and exploiting this structured change is important for the performance of AS -type solvers. In this paper, theory for how to exploit these low-rank changes by modifying the Riccati factorization between AS iterations in a structured way is presented. A numerical evaluation of the proposed algorithm shows that the computation time can be significantly reduced by modifying, instead of re-computing, the Riccati factorization. This speedup can be important for AS -type solvers used for linear, nonlinear, and hybrid MPC .

Optimal oil production under mean-reverting Lévy models with regime switching

This paper is concerned with the problem of finding optimal extraction policies for an oil field in light of various financial and economical restrictions and constraints. Taking into account the fact that the oil price in worldwide commodity markets fluctuates randomly following global and seasonal macroeconomic parameters, we model the evolution of the oil price as a mean-reverting regime-switching jump-diffusion process. We formulate this problem as a finite-time horizon optimal control problem, which we solve using the method of viscosity solutions. Moreover, we construct and prove the convergence of a numerical scheme for approximating the optimal reward function and the optimal extraction policy. A numerical example illustrating these results is presented.

Space-Time Galerkin POD with Application in Optimal Control of Semilinear Partial Differential Equations

In the context of Galerkin discretizations of a partial differential equation (PDE), the modes of the classical method of Proper Orthogonal Decomposition (POD) can be interpreted as the ansatz and trial functions of a low-dimensional Galerkin scheme. If one also considers a Galerkin method for the time integration, one can similarly define a POD reduction of the temporal component. This has been described earlier but not expanded upon -- probably because the reduced time discretization globalizes time which is computationally inefficient. However, in finite-time optimal control systems, time \textit{is} a global variable and there is no disadvantage from using a POD reduced Galerkin scheme in time. In this paper, we provide a newly developed generalized theory for space-time Galerkin POD, prove its optimality in the relevant function spaces, show its application for the optimal control of nonlinear PDEs, and, by means of a numerical example with Burgers' equation, discuss the competitiveness by comparing to standard approaches.

The effects of redundant control inputs in finite-time optimal control

This paper is concerned with the effects of redundant control inputs on the quadratic optimal performance index in finite-time optimal control. Actually, the effects of redundant control input on quadratic performance index is equal to study the effects of the redundant control input on the solution of Riccati differential equation (RDE). Hence, some sufficient conditions are presented to strictly decrease the solution of RDE with the column of input matrix increased. Especially, a necessary and sufficient condition is derived to strictly decrease the solution of the homogeneous RDE after control input extensions. Moreover, one sufficient condition is obtained, which guarantees that the minimum control energy index strictly decreases with input extension. It is shown that redundant control inputs can improve the system performance based on these results. Some examples are taken to illustrate the effectiveness of related problems.

Stabilisation of constrained uncertain systems by multi-parametric optimisation

This paper aims to construct a controlled invariant set while determining a stabilising piecewise linear control law for constrained uncertain systems. The proposed approach is based on the equivalence between the considered system and its Euler auxiliary system (EAS). The control law is obtained by the resolution of a constrained finite time optimal control (CFTOC) problem. Due to the exhaustive computing demands, a multi-parametric formulation and an offline resolution of the problem are considered to extend the set parameters variation.

Self-Triggered Model Predictive Control for Linear Systems Based on Transmission of Control Input Sequences

A networked control system (NCS) is a control system where components such as plants and controllers are connected through communication networks. Self-triggered control is well known as one of the control methods in NCSs and is a control method that for sampled-data control systems both the control input and the aperiodic sampling interval (i.e., the transmission interval) are computed simultaneously. In this paper, a self-triggered model predictive control (MPC) method for discrete-time linear systems with disturbances is proposed. In the conventional MPC method, the first one of the control input sequence obtained by solving the finite-time optimal control problem is sent and applied to the plant. In the proposed method, the first some elements of the control input sequence obtained are sent to the plant, and each element is sequentially applied to the plant. The number of elements is decided according to the effect of disturbances. In other words, transmission intervals can be controlled. Finally, the effectiveness of the proposed method is shown by numerical simulations.

Finite-time robust passive control for a class of uncertain Lipschitz nonlinear systems with time-delays

The finite-time passive control for a class of nonlinear uncertain systems with time-delays and uncertainties is studied. The nonlinear parameters are satisfied Lipschitz conditions. An optimal robust passive controller with respect to the finite-time interval is designed while the exogenous disturbances are unknown but energy bounded. Based on passive control theory, the sufficient condition for the existence of finite-time robust passive controller is given. This condition such that the resulting closed-loop system is finite-time boundedness (FTB) for all admissible uncertainties and satisfies the given passive control index. By using the constructed Lyapunov function, and applying linear matrix inequalities techniques (LMIs), the design method of the finite-time optimal passive controller is derived and can be obtained. Simulation results demonstrate the validity of the proposed approach.

Sparse stabilization and control of alignment models

Starting with the seminal papers of Reynolds (1987), Vicsek et al. (1995), Cucker–Smale (2007), there has been a lot of recent works on models of self-alignment and consensus dynamics. Self-organization has so far been the main driving concept of this research direction. However, the evidence that in practice self-organization does not necessarily occur (for instance, the achievement of unanimous consensus in government decisions) leads to the natural question of whether it is possible to externally influence the dynamics in order to promote the formation of certain desired patterns. Once this fundamental question is posed, one is also faced with the issue of defining the best way of obtaining the result, seeking for the most "economical" way to achieve a certain outcome. Our paper precisely addressed the issue of finding the sparsest control strategy in order to lead us optimally towards a given outcome, in this case the achievement of a state where the group will be able by self-organization to reach an alignment consensus. As a consequence, we provide a mathematical justification to the general principle according to which "sparse is better": in order to achieve group consensus, a policy maker not allowed to predict future developments should decide to control with stronger action the fewest possible leaders rather than trying to act on more agents with minor strength. We then establish local and global sparse controllability properties to consensus. Finally, we analyze the sparsity of solutions of the finite time optimal control problem where the minimization criterion is a combination of the distance from consensus and of the ℓ1-norm of the control. Such an optimization models the situation where the policy maker is actually allowed to observe future developments. We show that the lacunarity of sparsity is related to the codimension of certain manifolds in the space of cotangent vectors.

Experimental constrained optimal attitude control of a quadrotor subject to wind disturbances

The design and experimental verification of a Constrained Finite Time Optimal Controller (CFTOC) for attitude maneuvers of an Unmanned Quadrotor operating under severe wind conditions is the subjec ...

Optimal Selection of the Coefficient Matrix in State-Dependent Control Methods

This paper presents a state-dependent method for solving nonlinear, control-affine, finite-time optimal control problems. The problem is set in a state-dependent form and solved as a sequence of time-varying linear-quadratic regulators, for which an approach based on state transition matrices has been developed. The novelty of the method consists of the possibility of managing a number of different state-dependent coefficient factorizations of the uncontrolled dynamics, for use with various state-dependent control methods. At each iteration, a nonlinear programming problem is solved to select the state-dependent combination that maximizes the controllability, so minimizing the control effort and, likely, the objective function. Test problems are presented to show the effectiveness of the method.

Piecewise Affine Modeling and Constrained Optimal Control for a Pneumatic Artificial Muscle

In this paper, the modeling and control problem of a pneumatic artificial muscle (PAM) is being considered. The PAM is an actuator characterized by a decrease in the actuating length when pressurized. Its nonlinear nature and time-varying parameters cause difficulties in modeling their characteristics, as well as in designing controllers for high-performance positioning systems. A constrained linear and piecewise affine system model approximation is formulated, and a control scheme composed of the following is being synthesized: 1) a feedforward term regulating the control input at specific set points and 2) a constrained finite-time optimal controller handling any deviations from the system's equilibrium points. Extended experimental studies are utilized to prove the efficacy of the suggested controller.

Controlling the Movement of a Human Forearm based on Finite Time Optimal Regulator and Nonlinear Control System

human forearm trajectories structure was investigated using Optimal Regulator. First, the continuous dynamics of the human forearm were established on the basis of the relation between muscle torque and neural control signal and employed Riccati differential equation to obtain the optimal movement trajectory of the human forearm. The model and control of a human forearm is analyzed. In the advanced control of robotic manipulators, it is important for manipulators to track trajectories in a wide range of work place. If speed and accuracy is required, the control using conventional methods is difficult to realize because of the high nonlinearity of the robot system. In this work; we study the problem of human hand control carrying a mass. The equation of motion and the natural frequency of the forearm for small angular displacement are derived. We develop new methods that use vector fields in the controller construction for a set of nonlinear dynamical systems. The paper deals with compensate of non-linear system which has a similar idea as the method mentioned in linear system. We utilize an optimal regulator and suggest human arm activities under dynamic environments. Humans must pay compensation for loads arising from interaction with the physical environment. We have managed to design a control law for the non-linear arm-mass-system, in such a way that the representation of a closed loop system is affine, controllable, and a closed loop system is asymptotically stable. Throughout any motion, the forearm can be considered a one-link robot manipulator which could be exploited to benefit people with disabilities (missing extremities). An optimal controller calculates the motor command, which minimize the cost function for nonlinear arm dynamics. Therefore, the purpose of our study is to construct a computational model of arm movement. For this purpose, we use an optimal regulator as the model and compare simulated arm movements with actual nonlinear model human movements. In this paper, we present two approaches to human arm modeling: finite time optimal regulator control approach and modern nonlinear control approach. The latter one is formalized using Lie-Derivative based controllers.

An Exponential Turnpike Theorem for Dissipative Discrete Time Optimal Control Problems

We investigate the exponential turnpike property for finite horizon undiscounted discrete time optimal control problems without any terminal constraints. Considering a class of strictly dissipative systems, we derive a boundedness condition for an auxiliary optimal value function which implies the exponential turnpike property. Two theorems illustrate how this boundedness condition can be concluded from structural properties like controllability and stabilizability of the control system under consideration.

Discrete Abstraction for a Class of Stochastic Hybrid Systems Based on Bounded Bisimulation

Stochastic hybrid systems can express complex dynamical systems such as biological systems and communication networks, but computation for analysis and control is frequently difficult. In this paper, for a class of stochastic hybrid systems, a discrete abstraction method in which a given system is transformed into a finite-state system is proposed based on the notion of bounded bisimulation. In the existing discrete abstraction method based on bisimulation, a computational procedure is not in general terminated. In the proposed method, only the behavior for the finite time interval is expressed as a finite-state system, and termination is guaranteed. The obtained discrete abstract model can be used for model predictive control in which the finite-time optimal control problem is solved at each time. Furthermore, as an application, analysis of genetic toggle switches is also discussed.

Neural-network-based approach to finite-time optimal control for a class of unknown nonlinear systems

This paper proposes a novel finite-time optimal control method based on input---output data for unknown nonlinear systems using adaptive dynamic programming (ADP) algorithm. In this method, the single-hidden layer feed-forward network (SLFN) with extreme learning machine (ELM) is used to construct the data-based identifier of the unknown system dynamics. Based on the data-based identifier, the finite-time optimal control method is established by ADP algorithm. Two other SLFNs with ELM are used in ADP method to facilitate the implementation of the iterative algorithm, which aim to approximate the performance index function and the optimal control law at each iteration, respectively. A simulation example is provided to demonstrate the effectiveness of the proposed control scheme.

Finite-time optimal consensus control for second-order multi-agent systems

We propose an optimal consensus design method for solving a finite-time optimal control problem involving a second-order multi-agent system. With this method, the optimal consensus problem can be modeled as an optimal parameter selection problem with continuous state inequality constraints and free terminal time. By virtue of the constraint transcription method and a time scaling transform method, a gradient-based optimization algorithm is developed to solve this optimal parameter selection problem. Furthermore, a new consensus protocol is designed, by which the consensus value of the system velocity can be chosen to be an arbitrary value. For illustration, simulation studies are carried out to demonstrate the proposed method.

Multi-Resolution Explicit Model Predictive Control: Delta-Model Formulation and Approximation

This paper deals with the explicit solution and approximation of the constrained linear finite time optimal control problem for systems with fast sampling rates. To this aim, the recently developed explicit model predictive control (eMPC) is reformulated and characterized using the δ-operator to enjoy its promising advantages compared to the time-shift operator. Using the proposed δ-model eMPC formulation, a systematic method is proposed for first designing a low-complexity approximate eMPC solution and then improving its closed loop action without first determining an exact optimal solution that might be of prohibitive complexity. It is shown that the stability and feasibility of the proposed sub-optimal solution is guaranteed.

Finite-time optimal formation control of multi-agent systems on the Lie group SE(3)

This paper investigates the finite-time optimal formation problem of multi-agent systems on the Lie group SE(3), for the situation when the formation time and/or the cost function need to be considered. Under the condition that the formation time is given according to the task requirement, a finite-time optimal formation controller is proposed for the two-agent case to guarantee that the desired formation is achieved at the given time and the corresponding cost function is optimal. For the systems with multiple agents, the obtained finite-time optimal formation control law has second-order approximation accuracy. Finally, some numerical simulations are provided to illustrate the effectiveness of the theoretical results.