Presence Of Control Constraints Research Articles

Constrained control of 1-D parabolic PDEs using sampled in space sensing and actuation

We consider distributed control of a class of 1-D parabolic PDEs under distributed in-domain point actuation and measurements in the presence of control constraints. This class includes unstable diffusion-reaction equations as well as stable Burgers’ equations, where we aim to locally improve the convergence. We suggest an observer-based control law that employs the averaged values of the observer state. This allows to regionally stabilize the system. We derive linear matrix inequalities (LMIs) conditions that provide an estimate on the set of initial conditions starting from which the state trajectories of the system are exponentially converging to zero. A numerical example validates the efficiency of the method.

Approximate-optimal control algorithm for constrained zero-sum differential games through event-triggering mechanism

This paper investigates the optimization problem of two-player zero-sum differential game with control constraints in the framework of event triggering. Relying on reinforcement learning, an adaptive dynamic programming algorithm is developed to approximate the optimal solution of zero-sum game, i.e., the saddle-point equilibrium. A single-network structure is adopted, wherein a critic neural network (NN) evaluates the action. First, the constrained Hamilton–Jacobi–Isaacs equation is mathematically derived in the presence of control constraints; the event-triggering mechanism is then incorporated to reduce calculations and actions. Then, based on the gradient-descent technique, a novel weight updating law is designed for the critic NN, which ensures the solution can converge to the optimal value online. Moreover, the stability of closed-loop system is guaranteed and the unfavorable Zeno behavior is excluded by calculating the theoretical minimum triggering interval. Finally, two numerical examples are provided to verify the reliability and effectiveness of proposed algorithm.

Enhanced trajectory linearization control based advanced guidance and control for hypersonic reentry vehicle with multiple disturbances

In this paper, the guidance and control problem for hypersonic reentry vehicle (HRV) in the presence of control constraints and multiple disturbances is handled based on unified enhanced trajectory linearization control (TLC) framework under reference-tracking methodology. First, based on the nominal trajectory and open-loop command generated by Gauss pseudo-spectral method (GPM), a time-varying feedback guidance law with integral action is synthesized to stabilize the tracking error dynamics along the nominal trajectory under the framework of TLC. Second, to improve the robustness of attitude and angular rate loop, variations of various aerodynamic coefficients and external disturbances are considered as lumped uncertainties, reduced-order linear extended state observers (LESO) with given model information are constructed to estimate the lumped uncertainties in each loop, respectively. In addition, comparisons between the estimation efficiency of LESO and reduced-order LESO are carried out. Then augmented with the disturbance estimates and TLC control law, tracking errors of the rotational dynamics can be actively rejected without sacrificing nominal performances. More importantly, fewer control consumption and smooth transient performances are achieved by using nonlinear tracking differentiator (TD) in attitude loop. The stability of the resulting closed-loop system is well established based on Lyapunov stability theory. Finally, the effectiveness of the proposed advanced guidance and control strategy is verified through extensive simulations on the six-degree-of-freedom reentry flight.

On the Regularity of the Solutions of Dirichlet Optimal Control Problems in Polygonal Domains

A linear quadratic Dirichlet control problem posed on a possibly non-convex polygonal domain is analyzed. Detailed regularity results are provided in classical Sobolev (Slobodetskii) spaces. In particular, it is proved that in the presence of control constraints, the optimal control is continuous despite the non-convexity of the domain.

A PosterioriError Estimation for Reduced Order Solutions of Parametrized Parabolic Optimal Control Problems

We consider the ecient and reliable solution of linear-quadratic optimal control problems governed by parametrized parabolic partial dierential equations. To this end, we employ the reduced basis method as a low-dimensional surrogate model to solve the optimal control problem and develop a posteriori error estimation procedures that provide rigorous bounds for the error in the optimal control and the associated cost functional. We show that our approach can be applied to problems involving control constraints and that, even in the presence of control constraints, the reduced order optimal control problem and the proposed bounds can be eciently evaluated in an oine-online computational procedure. We also propose two greedy sampling procedures to construct the reduced basis space. Numerical results are presented to conrm the validity of our approach.

Strong Stationarity for Optimal Control of the Obstacle Problem with Control Constraints

We consider the distributed optimal control of the obstacle problem with control constraints. Since Mignot proved in 1976 the necessity of a system which is equivalent to strong stationarity, it has been an open problem whether such a system is still necessary in the presence of control constraints. Using moderate regularity of the optimal control and an assumption on the control bounds (which is implied by $u_a < 0 \le u_b$ quasi-everywhere in $\Omega$ in the case of an upper obstacle $y \le \psi$), we can answer this question in the affirmative. We also present counterexamples showing that strong stationarity may not hold if $u_a < 0$ or $0 \le u_b$ are violated. (An erratum is attached.)

Sufficient conditions for strong local optimality in optimal control problems with $L_{2}$-type objectives and control constraints

We consider the optimal control problem of minimizing an objective function that is quadratic in the control over a fixed interval for a multi-input bilinear dynamical system in the presence of control constraints. Such models are motivated by and applied to mathematical models for cancer chemotherapy over an a priori specified fixed therapy horizon. The necessary conditions for optimality of the Pontryagin maximum principle are easily evaluated and give a functional description of optimal controls as continuous functions of states and multipliers. However, there is no a priori guarantee that a numerically computed extremal controlled trajectory is locally optimal. In this paper, we formulate sufficient conditions for strong local optimality that are based on the existence of a bounded solution to a matrix Riccati differential equation. The theory is applied to a $3$-compartment model for multi-drug cancer chemotherapy with cytotoxic and cytostatic agents. The numerical results are compared with those for a corresponding optimal control problem when the objective is taken linear in the controls.

Adaptive Multilevel Inexact SQP-Methods for PDE-Constrained Optimization with Control Constraints

We consider the inexact adaptive multilevel SQP-method from [J. C. Ziems and S. Ulbrich, SIAM J. Optim., 21 (2011), pp. 1--40] for the efficient solution of optimization problems governed by nonlinear partial differential equations (PDEs) and propose an extension to control constraints. For this we introduce a suitable criticality measure in the presence of control constraints as a natural generalization of the reduced gradient norm. We then obtain a fully adaptive inexact SQP-algorithm for optimal control problems with control constraints that starts on any coarse discretization of the infinite-dimensional problem and refines the meshes when and where necessary---without requiring a (suitable) a priori given hierarchy of discretizations. The implementable refinement criteria are based on local error estimators and the criticality measure. A particular refinement condition for the control due to control constraints is included. The algorithm allows inexact linear system solves and provides implementable accuracy requirements for iterative solvers of the linearized PDE and adjoint PDE on the current discretization. Appropriate decrease conditions are presented and their satisfiability under standard assumptions is shown. We prove global convergence to a stationary point of the infinite-dimensional problem. Numerical results that show the efficiency of this approach as well as the capability of defining the active set of the optimal control are presented.

A preconditioned MinRes solver for time‐periodic parabolic optimal control problems

SUMMARYThis work is devoted to the multiharmonic analysis of parabolic optimal control problems in a time‐periodic setting. In contrast to previous approaches, we include the cases of different control and observation domains, the observation in certain energy spaces, and the presence of control constraints. In all these cases, we propose a new preconditioned MinRes solver for the frequency domain equations and show that this solver is robust with respect to the space and time discretization parameters as well as the involved ‘bad’ model parameters of the state equation. Copyright © 2012 John Wiley &amp; Sons, Ltd.

Dynamical system design from a control perspective: finite frequency positive-realness approach

Nonminimum phase zeros are well known to limit the best achievable control performance when the control gain is allowed to be arbitrarily high. On the other hand, the phase crossover appears to be a limiting factor for performance when high-gain controllers are not allowed. In particular, the positive-realness in a finite frequency range seems crucial for achieving good performance in the presence of control constraints. This paper will first give multiple reasons to support this conjecture, and then develop a systematic method for designing mechanical systems to achieve the finite frequency positive-real (FFPR) property. Specifically, we present a state-space characterization of the FFPR property by generalizing the well known Kalman-Yakubovich-Popov lemma to deal with a class of frequency domain inequalities that are required to hold within a finite frequency interval. The result is further extended for uncertain systems to give a sufficient condition for satisfaction of a robust FFPR property. The (nominal) FFPR result is interpreted in the time-domain in terms of input/output signals. Finally, we show that certain sensor/actuator placement problems to achieve the FFPR property can be reduced to finite dimensional convex problems involving linear matrix inequalities. The method is applied to the shape design of a swing-arm for magnetic storage devices with the objective of maximizing the control bandwidth achievable with a limited actuator power.

Assessment of input-output controllability in the presence of control constraints

Two open-loop analysis techniques for assessing input-output controllability in the presence of control constraints are presented. The techniques used are a modified singular value analysis technique and a new optimization-based approach. The case study discussed has, depending on the choice of manipulated variables and measurable outputs, several alternative two-input two-output control schemes. The results are verified by performing closed loop simulations with PI controllers. These show that both methods are able to predict the process' closed loop performance in the presence of control constraints.

An extension of singular value analysis for assessing manipulated variable constraints

A new approach called feasible output radius analysis for linear or linearised models is introduced to address the problem of scaling dependency. This problem arises when assessing the effect of manipulated variable constraints (MVCs) on the closed-loop performance of chemical processes prior to carrying out control designs. The new indicators, R and R can be used to rank alternative control schemes on the basis that the larger R and R , the better the closed-loop performance in the presence of control constraints. These indicators are determined from extending the concept of the ‘feasible output amplitude region’ and are independent of the input scaling chosen. Theoretical analysis shows that this method is an extension of the more traditional singular value analysis approach and is more flexible in dealing with various kinds of manipulated variable constraints. A case study, i.e. a two-CSTR process, is investigated using the new method. Via the case study, some superior characteristics of the new technique are demonstrated, such as ease of calculation, and flexibility in coping with different kinds of constraints.

Assessment of input-output controllability in the presence of control constraints

Two open-loop analysis techniques for assessing input-output controllability in the presence of control constraints are presented. The techniques used are a modified singular value analysis technique and a new optimisation-based approach. The case study discussed has, depending on the choice of manipulated variables and measurable outputs, several alternative 2-input 2-output control schemes. The results are verified by performing closed loop simulations with PI controllers. These show that both methods are able to predict the process’ closed loop performance in the presence of control constraints.

A linear programming approach to on‐line constrained optimal terrain‐tracking systems

AbstractThe idea of an optimal decision strategy is applied to a class of tracking problems. The resulting pointwise optimization problem is posed as a linear programming problem, in contrast to previous work where it is posed as a quadratic programming problem. A useful result concerning the convergence of the tracking error in the presence of control constraints is derived and applied to an aircraft terrain‐following system.

An exact penalty function algorithm for optimal control problems with control and terminal equality constraints, part 1

The presence of control constraints, because they are nondifferentiable in the space of control functions, makes it difficult to cope with terminal equality constraints in optimal control problems. Gradient-projection algorithms, for example, cannot be employed easily. These difficulties are overcome in this paper by employing an exact penalty function to handle the cost and terminal equality constraints and using the control constraints to define the space of permissible search directions in the search-direction subalgorithm. The search-direction subalgorithm is, therefore, more complex than the usual linear program employed in feasible-directions algorithms. The subalgorithm approximately solves a convex optimal control problem to determine the search direction; in the implementable version of the algorithm, the accuracy of the approximation is automatically increased to ensure convergence.