Krystyn Styczen

A modified filter SQP method as a tool for optimal control of nonlinear systems with spatio-temporal dynamics

A modified filter SQP method as a tool for optimal control of nonlinear systems with spatio-temporal dynamics Our aim is to adapt Fletchers filter approach to solve optimal control problems for systems described by nonlinear Partial Differential Equations (PDEs) with state constraints. To this end, we propose a number of modifications of the filter approach, which are well suited for our purposes. Then, we discuss possible ways of cooperation between the filter method and a PDE solver, and one of them is selected and tested.

Trigonometric approximation of smooth optimal periodic control problems

Trigonometric polynomials and penalty functions are used to approximate the constrained optimal periodic control problem for a system described by differential equations. The optimal control is assumed to be a smooth periodic function, i.e. it is continuously differentiate at least once. Sufficient conditions for the convergence of solutions of approximating problems to the optimal solution of the basic problem are given and the convergence rate is estimated. The particular usefulness of the approximation considered for singular optimal controls, which arise in chemical engineering, is emphasized.

A Review on the Direct and Indirect Methods for Solving Optimal Control Problems with Differential-Algebraic Constraints

In the article the main features of direct and indirect approaches for solving optimal control problems were presented. The mentioned methods can be effectively applied in the Model Predictive Control of the complex technological systems in electrical, chemical and aerospace engineering, often described by nonlinear differential-algebraic equations. Among the direct and indirect methods for solving optimal control problems one can mention Euler-Lagrange equations, direct optimization methods and indirect gradients methods.

Generalized trigonometric approximation of optimal periodic control problems

An optimal periodic control problem for a system described by differential equations is considered. Control units are assumed to generate control actions with the square-integrable derivative. The above problem is approximated by a sequence of discretized problems containing trigonometric polynomials, which approximate the state and control variables, and the functions in the criterion and differential equations. The conditions for a sequence of optimal solutions to discretized problems, which are to be a generalized minimizing sequence for the basic problem, are given. Extensions to more general problem formulations are presented. The possibility of application is illustrated by the example of an optimal periodic control problem for a chemical reactor.

A new optimization-based approach for aircraft landing in the presence of windshear

In the article a new approach for aircraft landing with the presence of the windshear phenomena was presented. The differential-algebraic model with variability constraints was under considerations. To transform the optimal control problem into a nonlinear optimization task, a modified direct shooting method was used. Then, to solve the obtained largescale nonlinear optimization problem, a barrier method was applied. Moreover, in the proposed optimization-based approach, the variability constraints imposed on the state trajectory were considered directly.

The matrix-based description approach for the multistage differential-algebraic processes

In the article a new insight into an optimal control problem of the multistage processes has been given. The multistage descriptor processes with differential-algebraic constraints are under considerations. The new representation of the descriptor model has been presented. Moreover, the new structures to represent the differential state variables, algebraic state variables, control function, as well the global parameters have been introduced. The generalized description enables the unified representation of a broad group of the multistage processes with differential-algebraic relations and indicates on the physical interpretation of the process variables.

Simulated annealing with constraints aggregation for control of the multistage processes

In the article the control and optimization of multistage technological processes were discussed. In the presented research, it was assumed, that in the complex technological processes, the multistage differential-algebraic constraints with unknown consistent initial conditions were considered. To rewrite the infinite-dimensional optimal control problem into the finite-dimensional optimization task, the direct shooting method was applied. Simulated annealing algorithm was proposed as the method for solving nonlinear optimization problem with constraints. Stretching function was used to allow us to locate the globally optimal solution. The complex process constraints were treated using constraints aggregation methods. The presented methodology was tested with optimal control problem of the two-reactors system. The numerical simulations were executed in MATLAB environment using Wroclaw Center for Networking and Supercomputing.

Direct Shooting Method for Optimal Control of the Highly Nonlinear Differential-Algebraic Systems

In the paper the optimal control of highly nonlinear differential-algebraic systems (DAEs) is discussed. The direct shooting method is seen as an efficient tool for control of the complex real-life technological processes, where dynamics and conservation laws are presented. To stabilize the optimization algorithm, the multiple shooting method was proposed. The multiple shooting approach introduces new decision variables and constraints to the problem, but it can preserve the stability of the process, the continuity of the differential state trajectories and enables parallel computation of the mathematical model. The conditions for the frequency of shots, to establish the well-conditioned optimization problem, are considered. The proposed method was tested on the mathematical model of the fed-batch fermentor for penicillin production process, which is a highly nonlinear multistage differential-algebraic system. The numerical simulations were executed in MATLAB environment using Wroclaw Center for Networking and Supercomputing.

The Constraints Aggregation Technique for Control of Ethanol Production

In the article a new method for control and optimization of the ethanol production process was presented. The fed batch reactor for ethanol production can be described by the system of the nonlinear differential-algebraic equations. The new control approach base on the direct shooting method, which leads to a medium- or large-scale nonlinear optimization problem. In the presented research, three main aims of the optimization with the differential-algebraic constraints are indicated. The first one is the minimization of the cost function, which represents a process performance index. The second and third aims are to provide consistent initial conditions for the system to be solved and to ensure the continuity of the differential state trajectories. For these reasons, the large number of decision variables and equality constraints, connected with both consistent initial conditions for the differential-algebraic system and continuity of the differential state trajectories, was treated using the constraints aggregation technique. The direct shooting approach together with constraints aggregation technique enables us to obtain a new form of the nonlinear optimization task.

Multiple Shooting SQP Algorithm for Optimal Control of DAE Systems with Inconsistent Initial Conditions

In the article a new approach for control of differential-algebraic systems with inconsistent initial conditions were presented. The consistent initial conditions could be difficult to obtain, for example, in the fed-batch penicillin fermentation process. Additionally difficulties were incorporated by constraints on the differential state trajectories. For this purposes a new algorithm based on a multiple shooting SQP-line search method was proposed. To ensure a stability of the solution, the multiple shooting approach were used. By division a system into smaller subsystems, as a result a large-scale problem was obtained. The proposed algorithm can be applied to a wide class of differential-algebraic systems in booth chemical and mechanical engineering. The simulations were executed in Matlab environment using Wroclaw Centre for Networking and Supercomputing.