Finding initial costates in finite‐horizon nonlinear‐quadratic optimal control problems

A procedure for obtaining the initial value of the costate in a regular, finite‐horizon, nonlinear‐quadratic problem is devised in dimension one. The optimal control can then be constructed from the solution to the Hamiltonian equations, integrated on‐line. The initial costate is found by successively solving two first‐order, quasi‐linear, partial differential equations (PDEs), whose independent variables are the time‐horizon duration T and the final‐penalty coefficient S. These PDEs need to be integrated off‐line, the solution rendering not only the initial condition for the costate sought in the particular (T, S)‐situation but also additional information on the boundary values of the whole two‐parameter family of control problems, that can be used for design purposes. Results are tested against exact solutions of the PDEs for linear systems and also compared with numerical solutions of the bilinear‐quadratic problem obtained through a power‐series' expansion approach. Bilinear systems are specially treated in their character of universal approximations of nonlinear systems with bounded controls during finite time‐periods. Copyright © 2007 John Wiley & Sons, Ltd.     

Hierarchical optimal control of a binary distillation column

This paper presents a two‐level optimal control for the binary distillation column. From the control point of view, the binary distillation column is a high‐order system consisting of several interconnected subsystems (trays) that aim to maximize output purity. Controlling the system in a hierarchical manner not only decreases complexity and solution time but also can improve the control structure's reliability. In this study, the distillation column is decomposed into N_T second‐order subsystems, each representing one stage (one tray). Accordingly, the cost function is decomposed, and thus, the overall problem is converted into N_T lower‐order subproblems where each tray has its own controller. A two‐level interaction prediction approach provides optimal control for the overall system. The task of the first level is to solve the subproblems using the predicted values of the coordination parameters, whereas the second level acts as a coordinator to update the coordination parameters. Simulation results show the capability and efficacy of the proposed two‐level control method in finding the optimal solution with less complexity and lower solution time than those of the centralized method.     

An improved smoothing technique‐based control vector parameterization method for optimal control problems with inequality path constraints

An improved control vector parameterization (CVP) method is proposed to solve optimal control problems with inequality path constraints by introducing the l₁ exact penalty function and a novel smoothing technique. Both the state and control variables are allowed to appear explicitly in the inequality path constraints simultaneously. By applying the penalty function and smoothing technique, all the inequality path constraints are firstly reformulated as non‐differentiable penalty terms and incorporated into the objective function. Then, the penalty terms are smoothed by using a novel smooth function, leading to a smooth optimal control problem with no inequality path constraints. With discretizing the control space, a corresponding nonlinear programming (NLP) problem is derived, and error between the NLP problem and the original problem is discussed. Results reveal that if the smoothing parameter is sufficiently small, the solution of the NLP problem is approximately equal to the original problem, which shows the convergence of the proposed method. After clarifying some theories of the proposed approach, a concomitant numerical algorithm is put forward with furnishing the updating rules of both the penalty parameter and smoothing parameter. Simulation examples verify the advantages of the proposed method for tackling nonlinear optimal control problems with different inequality path constraints. Copyright © 2016 John Wiley & Sons, Ltd.     

Polynomial matrix solution of H2 optimal control problem for state‐space systems

A polynomial matrix solution to the H₂ output feedback optimal control problems is obtained for systems represented in state‐equation form. The proof does not invoke the separation principle but is obtained in the z‐domain. The cost function includes weighted states, which allows the so‐called standard system model problem to be solved. This encompasses the class of inferential control problems. The results also enable the two‐degree‐of‐freedom optimal control solution properties to be explored. Copyright © 2002 John Wiley & Sons, Ltd.     

An extended model for pairwise conflict resolution in air traffic management

A conflict resolution model for aircraft traversing planar intersecting trajectories is developed and solved. The model incorporates the flight dynamics of each aircraft. Three types of optimal conflict avoidance manoeuvres are examined. The first is the minimum‐time safe deviation to a designated objective and the second and third are the minimum‐time safe deviation to and from a designated ground track. These objectives lead to singular optimal control problems where the control variable constraint is defined in terms of the maximum allowable turn rate ∣u(t)∣⩽u_M,0⩽t⩽t_f, for the aircraft executing the avoidance manoeuvre and the non‐autonomous state variable constraint is defined in terms of the radius r_p of the protected zone about the potentially conflicting aircraft. A fast, accurate procedure is derived for computing the optimal steering program which requires the solution of a coupled differential‐algebraic system of equations to determine the control law on any boundary arc of an extremal trajectory. An example pairwise trajectory conflict is analysed in detail. The incorporation of variable airspeed v(t) and an acceleration/deceleration control u₂(t) is also discussed. Copyright © 1999 John Wiley & Sons Ltd.     

Hamiltonian two‐degrees‐of‐freedom control of chemical reactors

A novel unified approach to two‐degrees‐of‐freedom control is devised and applied to a classical chemical reactor model. The scheme is constructed from the optimal control point of view and along the lines of the Hamiltonian formalism for nonlinear processes. The proposed scheme optimizes both the feedforward and the feedback components of the control variable with respect to the same cost objective. The original Hamiltonian function governs the feedforward dynamics, and its derivatives are part of the gain for the feedback component. The optimal state trajectory is generated online, and is tracked by a combination of deterministic and stochastic optimal tools. The relevant numerical data to manipulate all stages come from a unique off‐line calculation, which provides design information for a whole family of related control problems. This is possible because a new set of PDEs (the variational equations) allow to recover the initial value of the costate variable, and the Hamilton equations can then be solved as an initial‐value problem. Perturbations from the optimal trajectory are abated through an optimal state estimator and a deterministic regulator with a generalized Riccati gain. Both gains are updated online, starting with initial values extracted from the solution to the variational equations. The control strategy is particularly useful in driving nonlinear processes from an equilibrium point to an arbitrary target in a finite‐horizon optimization context. Copyright © 2010 John Wiley & Sons, Ltd.     

A priori and a posteriori error estimates of H1‐Galerkin mixed finite element method for parabolic optimal control problems

In this exposition, we study both a priori and a posteriori error analysis for the H¹‐Galerkin mixed finite element method for optimal control problems governed by linear parabolic equations. The state and costate variables are approximated by the lowest order Raviart‐Thomas finite element spaces, whereas the control variable is approximated by piecewise constant functions. Compared to the standard mixed finite element procedure, the present method is not subject to the Ladyzhenskaya‐Babuska‐Brezzi condition and the approximating finite element spaces are allowed to be of different degree polynomials. A priori error analysis for both the semidiscrete and the backward Euler fully discrete schemes are analyzed, and $L^{\infty }(L^2)$ convergence properties for the state variables and the control variable are obtained. In addition, L²(L²)‐norm a posteriori error estimates for the state and control variables and $L^{\infty }(L^2)$‐norm for the flux variable are also derived.     

Optimal control problems with Atangana‐Baleanu fractional derivative

In this paper, we study fractional‐order optimal control problems (FOCPs) involving the Atangana‐Baleanu fractional derivative. A computational method based on B‐spline polynomials and their operational matrix of Atangana‐Baleanu fractional integration is proposed for the numerical solution of this class of problems. With this numerical technique, the FOCPs are reduced to a system of equations which are solved for the unknown parameters with the help of Mathematica® software. Our results show the applicability and usefulness of the numerical technique.     

Exact penalty functions for optimal control problems I: Main theorem and free‐endpoint problems

In this two‐part study, we develop a general approach to the design and analysis of exact penalty functions for various optimal control problems, including problems with terminal and state constraints, problems involving differential inclusions, and optimal control problems for linear evolution equations. This approach allows one to simplify an optimal control problem by removing some (or all) constraints of this problem with the use of an exact penalty function, thus allowing one to reduce optimal control problems to equivalent variational problems and apply numerical methods for solving, eg, problems without state constraints, to problems including such constraints, etc. In the first part of our study, we strengthen some existing results on exact penalty functions for optimisation problems in infinite dimensional spaces and utilise them to study exact penalty functions for free‐endpoint optimal control problems, which reduce these problems to equivalent variational ones. We also prove several auxiliary results on integral functionals and Nemytskii operators that are helpful for verifying the assumptions under which the proposed penalty functions are exact.     

Exact penalty functions for optimal control problems II: Exact penalization of terminal and pointwise state constraints

The second part of our study is devoted to an analysis of the exactness of penalty functions for optimal control problems with terminal and pointwise state constraints. We demonstrate that with the use of the exact penalty function method one can reduce fixed-endpoint problems for linear time-varying systems and linear evolution equations with convex constraints on the control inputs to completely equivalent free-endpoint optimal control problems, if the terminal state belongs to the relative interior of the reachable set. In the nonlinear case, we prove that a local reduction of fixed-endpoint and variable-endpoint problems to equivalent free-endpoint ones is possible under the assumption that the linearized system is completely controllable, and point out some general properties of nonlinear systems under which a global reduction to equivalent free-endpoint problems can be achieved. In the case of problems with pointwise state inequality constraints, we prove that such problems for linear time-varying systems and linear evolution equations with convex state constraints can be reduced to equivalent problems without state constraints, provided one uses the L^∞ penalty term, and Slater's condition holds true, while for nonlinear systems a local reduction is possible, if a natural constraint qualification is satisfied. Finally, we show that the exact L^p-penalization of state constraints with finite p is possible for convex problems, if Lagrange multipliers corresponding to the state constraints belong to L^p′, where p^′ is the conjugate exponent of p, and for general nonlinear problems, if the cost functional does not depend on the control inputs explicitly.     

A priori and a posteriori error estimates of finite‐element approximations for elliptic optimal control problem with measure data

We analyze both a priori and a posteriori error analysis of finite‐element method for elliptic optimal control problems with measure data in a bounded convex domain in (d = 2or3). The solution of the state equation of such type of problems exhibits low regularity due to the presence of measure data, which introduces some difficulties for both theory and numerics of the finite‐element method. We first prove the existence, uniqueness, and regularity of the solution to the optimal control problem. To discretize the control problem, we use continuous piecewise linear elements for the approximations of the state and co‐state variables, whereas piecewise constant functions are used for the control variable. We derive a priori error estimates of order for the state, co‐state, and control variables in the L²‐norm. Further, global a posteriori upper bounds for the state, co‐state, and control variables in the L²‐norm are established. Moreover, local lower bounds for the errors in the state and co‐state variables and a global lower bound for the error in the control variable are obtained in the case of two space dimensions (d = 2). Numerical experiments are provided, which support our theoretical results.     

On L1‐minimization in optimal control and applications to robotics

In this paper, we analyze optimal control problems with control variables appearing linearly in the dynamics. We discuss different cost functionals involving the L^p‐norm of the control. The case p = 0 represents the time‐optimal control, the case p > 1 yields a standard smooth optimal control problem, whereas the case p = 1 leads to a nonsmooth cost functional. Several techniques are developed to deal with the nonsmooth case p = 1. We present a thorough theoretical discussion of the necessary conditions. Two types of numerical methods are developed: either a regularization technique is used or an augmentation approach is applied in which the number of control variables is doubled. We show the precise relations between the L¹‐minimal control and the bang–bang or singular controls in the augmented problem. Using second‐order sufficient conditions (SSC) for bang–bang controls, we obtain SSC for L¹‐minimal controls. The different techniques and results are illustrated with an example of the optimal control for a free‐flying robot which is taken from Sakawa. Copyright © 2006 John Wiley & Sons, Ltd.     

A novel approach for the numerical investigation of optimal control problems containing multiple delays

This paper deals with the numerical solution of optimal control problems with multiple delays in both state and control variables. A direct approach based on a hybrid of block‐pulse functions and Lagrange interpolating polynomials is used to convert the original problem into a mathematical programming one. The resulting optimization problem is then solved numerically by the Lagrange multipliers method. The operational matrix of delay for the presented framework is derived. This matrix plays an imperative role to transfer information between 2 consecutive switching points. Furthermore, 2 upper bounds on the error with respect to the L²‐norm and infinity norm are established. Several optimal control problems containing multiple delays are carried out to illustrate the various aspects of the proposed approach. The simulation results are compared with either analytical or numerical solutions available in the literature.     

Space-time a posteriori error analysis of finite element approximation for parabolic optimal control problems: A reconstruction approach

We derive space-time a posteriori error estimates of finite element method for linear parabolic optimal control problems in a bounded convex polygonal domain. To discretize the control problem, we use piecewise linear and continuous finite elements for the approximations of the state and costate variables, whereas piecewise constant functions are employed for the control variable. The temporal discretization is based on the backward Euler implicit scheme. An elliptic reconstruction technique in conjunction with energy argument is used to derive a posteriori error estimates for the state, costate, and control variables in the L^∞(0,T;L²(Ω))-norm. Moreover, numerical experiments are performed to illustrate the performance of the derived estimators.     

A necessary and sufficient condition of optimality for a class of multidimensional control problems

In this paper, we introduce a necessary and sufficient condition of optimality for a new class of multidimensional optimal control problems governed by path-independent curvilinear integral functionals and mixed constraints involving first-order partial differential equations (PDEs) of m-flow type. Furthermore, as a consequence, we establish the equivalence between the class of (strongly) b-invex functionals and the class of (strongly) b-pseudoinvex functionals. The mathematical development in the paper is supported by illustrative examples, as well.     

Short Communication: Bilinear optimal control of a Kirchoff plate to a desired profile

We consider the problem of driving a Kirchhoff plate towards a desired state or profile at a final time T while taking into account a quadratic cost of implementing the control. The control enters the equation as a bilinear term in the state equation. Our purpose is to prove that an optimal control of this type exists, may be characterized as a solution to an optimality system and is unique for time T sufficiently small.     

Direct optimization of dynamic systems described by differential‐algebraic equations

This paper presents a method for the optimization of dynamic systems described by index‐1 differential‐algebraic equations (DAE). The class of problems addressed include optimal control problems and parameter identification problems. Here, the controls are parameterized using piecewise constant inputs on a grid in the time interval of interest. In addition, the DAE are approximated using a Rosenbrock–Wanner (ROW) method. In this way the infinite‐dimensional optimal control problem is transformed into a finite‐dimensional nonlinear programming problem (NLP). The NLP is solved using a sequential quadratic programming (QP) technique that minimizes the L_∞ exact penalty function, using only strictly convex QP subproblems. This paper shows that the ROW method discretization of the DAE leads to (i) a relatively small NLP problem and (ii) an efficient technique for evaluating the function, constraints and gradients associated with the NLP problem. This paper also investigates a state mesh refinement technique that ensures a sufficiently accurate representation of the optimal state trajectory. Two nontrivial examples are used to illustrate the effectiveness of the proposed method. Copyright © 2008 John Wiley & Sons, Ltd.     

A minimal energy control problem for second‐order linear hyperbolic systems with two independent variables

In this paper, we investigate controllability and minimal energy optimal control for Goursat–Darboux problem for the second‐order linear hyperbolic systems with two independent variables. The equation describes a relation between functions u: D→?^m and z: D→?ⁿ. We get an integral representation of the Goursat–Darboux problem by means of Riemann's matrix. The first half of the paper considers conditions under which there exists a control u for which the solution z of dynamics satisfies z(x₁, y₁) = p for any given p. The studied problem is reduced to the moments problem. The optimal control was found in a closed analytic form. Further, degeneracy of the matrix constructed by means of Riemann's matrix is shown to be a necessary and sufficient condition of controllability. Copyright © 2011 John Wiley & Sons, Ltd.     

Subsystem‐level optimal control of weakly coupled linear stochastic systems composed of N subsystems

In this paper we introduce a transformation for the exact closed‐loop decomposition of the optimal control and Kalman filtering tasks of linear weakly coupled stochastic systems composed of N subsystems. In addition to having obtained N completely independent reduced‐order subsystem Kalman filters working in parallel, we have obtained the exact solution of the algebraic regulator and filter Riccati equations in terms of the solutions of the corresponding reduced‐order subsystem algebraic Riccati equations. The introduced transformation produces a lot of savings especially for on‐line computations since it allows parallel processing of information with lower‐order‐dimensional Kalman filters. The methodology presented is applied to a 17th‐order cold‐rolling mill. Copyright © 1999 John Wiley & Sons, Ltd.     

2,α‐norm optimal model reduction for optimal control problems subject to parabolic and hyperbolic evolution equations

This paper deals with a numerical solution method for optimal control problems subject to parabolic and hyperbolic evolution equations. Firstly, the problem is semi‐discretized in space with the boundary or distributed controls as input and those parts of the discretized state appearing in the cost functional as output variables. The corresponding transfer function is then approximated optimally with respect to the _2,α‐norm providing an optimally reduced optimal control problem, which is finally solved by a first‐discretize‐then‐optimize approach. To enable the application of this reduction method, a new constrained optimal model reduction problem subject to reduced systems with real system matrices is considered. Necessary optimality conditions and a transformation procedure for the reduced system to a canonical form of real matrices are presented. The method is illustrated with numerical examples where also complicated controls with many bang‐bang arcs are investigated. The approximation quality of the optimal control and its correlation to the decay rate of the Hankel singular values of the system are numerically studied. A comparison to the approach of using Balanced Truncation for model reduction is applied. Copyright © 2013 John Wiley & Sons, Ltd.