Terminal State Constraints Research Articles

Time optimal Zermelo’s navigation problem with moving and fixed obstacles

In this paper, we consider a time optimal Zermelo’s navigation problem (ZNP) with moving and fixed obstacles. This problem can be formulated as an optimal control problem with continuous inequality constraints and terminal state constraints. By using the control parametrization technique together with the time scaling transform, the problem is transformed into a sequence of optimal parameters selection problems with continuous inequality constraints and terminal state constraints. For each problem, an exact penalty function method is used to append all the constraints to the objective function yielding a new unconstrained optimal parameters selection problem. It is solved as a nonlinear optimization problem. Different scenarios are considered in the simulation, and the results obtained show that the proposed method is effective.

Generalized terminal state constraint for model predictive control

A terminal state equality constraint for Model Predictive Control (MPC) laws is investigated, where the terminal state/input pair is not fixed a priori but it is a free variable in the optimization. The approach, named “generalized” terminal state constraint, can be used for both tracking MPC (i.e. when the objective is to track a given steady state) and economic MPC (i.e. when the objective is to minimize a cost function which does not necessarily attains its minimum at a steady state). It is shown that the proposed technique provides, in general, a larger feasibility set with respect to the existing approaches, given the same prediction horizon. Moreover, a new receding horizon strategy is introduced, exploiting the generalized terminal state constraint. Under mild assumptions, the new strategy is guaranteed to converge in finite time, with arbitrarily good accuracy, to an MPC law with an optimally-chosen terminal state constraint, while still enjoying a larger feasibility set. The features of the new technique are illustrated by an inverted pendulum example in both the tracking and the economic contexts.

Optimal feedback control for dynamic systems with state constraints: An exact penalty approach

In this paper, we consider a class of nonlinear dynamic systems with terminal state and continuous inequality constraints. Our aim is to design an optimal feedback controller that minimizes total system cost and ensures satisfaction of all constraints. We first formulate this problem as a semi-infinite optimization problem. We then show that by using a new exact penalty approach, this semi-infinite optimization problem can be converted into a sequence of nonlinear programming problems, each of which can be solved using standard gradient-based optimization methods. We conclude the paper by discussing applications of our work to glider control.

Economic model predictive control with self-tuning terminal cost

In this paper, we propose an economic model predictive control (MPC) framework with a self-tuning terminal weight, which builds on a recently proposed MPC algorithm with a generalized terminal state constraint. First, given a general time-varying terminal weight, we derive an upper bound on the closed-loop average performance which depends on the limit value of the predicted terminal state. After that, we derive conditions for a self-tuning terminal weight such that bounds for this limit value can be obtained. Finally, we propose several update rules for the self-tuning terminal weight and analyze their respective properties. We illustrate our findings with several examples.

A maximum principle for fully coupled forward–backward stochastic control systems with terminal state constraints

We study a stochastic optimal control problem where the controlled system is described by a fully coupled forward–backward stochastic differential equation (FBSDE), while the forward state is constrained in a convex set at the terminal time. By introducing an equivalent backward control problem, we use terminal variation approach to obtain a stochastic maximum principle. Applications to the utility optimization problem in the financial market and state constrained stochastic linear quadratic control models are investigated.

Min–max optimal control of linear systems with uncertainty and terminal state constraints

In this paper, a class of min–max optimal control problems with continuous dynamical systems and quadratic terminal constraints is studied. The main contribution is that the original terminal state constraint in which the disturbance is involved is transformed into an equivalent linear matrix inequality without disturbance under certain conditions. Then, the original min–max optimal control problem is solved via solving a sequence of semi-definite programming problems. An example is presented to illustrate the proposed method.

Study on Stochastic Linear Quadratic Optimal Control with Quadratic and Mixed Terminal State Constraints

This paper studies the indefinite stochastic LQ control problem with quadratic and mixed terminal state equality constraints, which can be transformed into a mathematical programming problem. By means of the Lagrangian multiplier theorem and Riesz representation theorem, the main result given in this paper is the necessary condition for indefinite stochastic LQ control with quadratic and mixed terminal equality constraints. The result shows that the different terminal state constraints will cause the endpoint condition of the differential Riccati equation to be changed. It coincides with the indefinite stochastic LQ problem with linear terminal state constraint, so the result given in this paper can be viewed as the extension of the indefinite stochastic LQ problem with the linear terminal state equality constraint. In order to guarantee the existence and the uniqueness of the linear feedback control, a sufficient condition is also presented in the paper. A numerical example is presented at the end of the paper.

Robust Control of Constrained Linear Systems With Bounded Disturbances

This technical note develops a novel robust control algorithm for linear systems subject to additive and bounded disturbances. The approach is based on the constraint tightening method. While this problem can be tackled using existing robust model predictive control techniques, the proposed method has an advantage in that it is computationally efficient and avoids the need to solve repeatedly an online optimization problem, while the optimization problem solved at initialization is a simple linear programming problem. The algorithm elaborated in this technical note guarantees convergence to a minimal disturbance invariant set, and the terminal predicted state constraint set is allowed to be larger than the minimal disturbance invariant set. As an illustration, the developed algorithm is applied to constrained roll control of a ship operating in a wave field. Simulation results show that the proposed approach reduces the ship roll motion while the input and dynamic stall constraints are satisfied.

Model Predictive Control with generalized terminal state constraint

The properties of a particular terminal state constraint for the design of Model Predictive Control (MPC) laws are reviewed, and the technique is applied to an inverted pendulum example. The new approach, named “generalized” terminal state constraint, induces the recursive feasibility of the underlying optimization problem and recursive satisfaction of state and input constraints; moreover, it may provide, in general, a larger feasibility set with respect to existing approaches, given the same computational complexity.

Integration of RTO with MPC through the gradient of a convex function

Predictive controllers are usually implemented as part of a hierarchical structure of the process operation where the Real Time Optimization (RTO) and the Model Predictive Control (MPC) are executed in separated layers. Here, it is proposed to use the gradient of a convex function that links the optimizing targets to the MPC controller. The convex function is a nu+1 dimensional paraboloid of the process inputs, with the minimum corresponding to the input targets resulting from the RTO routine. For this MPC controller, the infinite prediction horizon and terminal state constraint are considered, and it operates according to the zone control strategy. Simulation results of a linear system are provided.

Explicit MPC for LPV Systems: Stability and Optimality

This paper considers high-speed control of constrained linear parameter-varying systems using model predictive control. Existing model predictive control schemes for control of constrained linear parameter-varying systems typically require the solution of a semi-definite program at each sampling instance. Recently, variants of explicit model predictive control were proposed for linear parameter-varying systems with polytopic representation, decreasing the online computational effort by orders of magnitude. Depending on the mathematical structure of the underlying system, the constrained finite-time optimal control problem can be solved optimally, or close-to-optimal solutions can be computed. Constraint satisfaction, recursive feasibility and asymptotic stability can be guaranteed a priori by an appropriate selection of the terminal state constraints and terminal cost. The paper at hand gathers previous developments and provides new material such as a proof for the optimality of the solution, or, in the case of close-to-optimal solutions, a procedure to determine a bound on the suboptimality of the solution.

An Exact Penalty Function Method for Continuous Inequality Constrained Optimal Control Problem

In this paper, we consider a class of optimal control problems subject to equality terminal state constraints and continuous state and control inequality constraints. By using the control parametrization technique and a time scaling transformation, the constrained optimal control problem is approximated by a sequence of optimal parameter selection problems with equality terminal state constraints and continuous state inequality constraints. Each of these constrained optimal parameter selection problems can be regarded as an optimization problem subject to equality constraints and continuous inequality constraints. On this basis, an exact penalty function method is used to devise a computational method to solve these optimization problems with equality constraints and continuous inequality constraints. The main idea is to augment the exact penalty function constructed from the equality constraints and continuous inequality constraints to the objective function, forming a new one. This gives rise to a sequence of unconstrained optimization problems. It is shown that, for sufficiently large penalty parameter value, any local minimizer of the unconstrained optimization problem is a local minimizer of the optimization problem with equality constraints and continuous inequality constraints. The convergent properties of the optimal parameter selection problems with equality constraints and continuous inequality constraints to the original optimal control problem are also discussed. For illustration, three examples are solved showing the effectiveness and applicability of the approach proposed.

A New Approach for Solving Optimal Nonlinear Control Problems Using Decriminalization and Rationalized Haar Functions

This paper presents a numerical method based on quasilinearization and rationalized Haar functions for solving nonlinear optimal control problems including terminal state constraints, state and control inequality constraints. The optimal control problem is converted into a sequence of quadratic programming problems. The rationalized Haar functions with unknown coefficients are used to approximate the control variables and the derivative of the state variables. By adding artificial controls, the number of state and control variables is equal. Then the quasilinearization method is used to change the nonlinear optimal control problems with a sequence of constrained linear-quadratic optimal control problems. To show the effectiveness of the proposed method, the simulation results of two constrained nonlinear optimal control problems are presented.

A Pontryagin Maximum Principle for Infinite-Dimensional Problems

A basic idea of the classical approach for obtaining necessary optimality conditions in optimal control is to construct suitable “needle-like control variations.” We use this idea to prove the main result of the present paper—a Pontryagin maximum principle for infinite-dimensional optimal control problems with pointwise terminal constraints in arbitrary Banach state space. By refining the classical variational technique we are able to replace the differentiability of the norm of the state space (guaranteed by the strict convexity of its dual norm, which is assumed in the known results) by a separation argument. We also drop another key assumption which is common in the existing literature on infinite-dimensional control problems—that the set of variations (in the state space) of the state trajectory's endpoint (resulting from the control variations) be finite-codimensional. Instead, we require only that it has nonempty interior in its closed affine hull. As an application of the abstract result we present an illustrative example—an optimal control problem for an age-structured system with pointwise terminal state constraints.

The Maximum Principles for Stochastic Recursive Optimal Control Problems Under Partial Information

A maximum principle for partially observed stochastic recursive optimal control problems is obtained under the assumption that control domains are not necessarily convex and forward diffusion coefficients do not contain control variables. Such kind of recursive optimal control problems have wide applications in finance and economics such as recursive utility optimization and principal-agent problems. By virtue of a classical spike variational approach and a filtering technique, the maximum principle is obtained, and the related adjoint processes are characterized by the solutions of forward-backward stochastic differential equations in finite-dimensional spaces. Then our theoretical results are applied to study a partially observed linear-quadratic recursive optimal control problem. In addition, for the case with initial and terminal state constraints, the corresponding maximum principle is also obtained by using Ekeland's variational principle.

Multiple Method 2-D Trajectory Optimization Satisfying Waypoints and No-Fly Zone Constraints

Minimum time to target is one of the primary goals of a global strike mission. The Hypersonic Cruise Vehicle and the Common Aero Vehicle are currently being investigated for mission effectiveness. Additional mission requirements include passage through intermediate waypoints and avoidance of no-fly zones. Thus, a real-time or near real-time autonomous trajectory generation technique is desired to minimize the flight time, satisfy terminal and multiple intermediate state constraints, and remain within specified control limitations. The research herein presents a baseline technique, an analytical geometric trajectory optimization technique, and a dynamic optimization technique. Numerical examples for constant speed trajectories as well as decelerating flight are used to demonstrate and compare the presented techniques. These results show the significant time savings achievable through optimization, the accuracy and computation efficiency of the geometric solution, and the robustness and application of the dynamic optimization technique.

Constrained Model Predictive Control for Nonholonomic Vehicle Regulation Problem

A model predictive control architecture based on discrete time nonlinear car model is derived to solve regulation(“parking”) problem. Parameters of the proposed controller are chosen by considering terminal state constraints. This setup combined with terminal state penalty in the cost function could assure control stability. The generated trajectory satisfies minimum curvature requirements and actuator saturations of the vehicle are considered in controller design. Obstacle avoidance is realized by considering distance constraints in the open-loop optimization process. Simulation results are given to illustrate the feasibility of the proposed control architecture.

Optimal control of the delay linear systems with allowance for the terminal state constraints

Consideration was given to the linear problem of optimal control of one type of the delay systems where the delay appears in one equation of the system mathematical model. The terminal states of the system are bounded, the optimal control is realized by the discrete control actions obeying the geometrical constraints. Consideration was given to two types of solutions--program and positional. A dual method of calculation of the optimal programs was presented. Described was an algorithm of the optimal controller generating in real time the current values of the positional solution (optimal feedback). The results obtained were illustrated by the example of control of a system with the fourth-order delay.

MODEL PREDICTIVE CONTROL FOR NONLINEAR TIME-DELAY SYSTEMS

This paper presents a model predictive controller with guaranteed closed-loop stability for nonlinear time-delay systems. The basic idea of the proposed nonlinear model predictive control (NMPC) scheme is based on previous results in NMPC for systems without delayed states. In particular, an expanded zero terminal state constraint, that takes the delayed states of the system into account, is used in order to guarantee closed-loop stability. The applicability of the proposed NMPC controller is demonstrated considering the control of a continuous stirred tank reactor with recycle stream.

On the stability of constrained MPC without terminal constraint

The usual way to guarantee stability of model predictive control (MPC) strategies is based on a terminal cost function and a terminal constraint region. This note analyzes the stability of MPC when the terminal constraint is removed. This is particularly interesting when the system is unconstrained on the state. In this case, the computational burden of the optimization problem does not have to be increased by introducing terminal state constraints due to stabilizing reasons. A region in which the terminal constraint can be removed from the optimization problem is characterized depending on some of the design parameters of MPC. This region is a domain of attraction of the MPC without terminal constraint. Based on this result, it is proved that weighting the terminal cost, this domain of attraction of the MPC controller without terminal constraint is enlarged reaching (practically) the same domain of attraction of the MPC with terminal constraint; moreover, a practical procedure to calculate the stabilizing weighting factor for a given initial state is shown. Finally, these results are extended to the case of suboptimal solutions and an asymptotically stabilizing suboptimal controller without terminal constraint is presented.