Maciej Szymkat

Optimal control of a fedbatch fermentation process: Numerical methods, sufficient conditions and sensitivity analysis

Bang-bang and singular optimal controls in a fedbatch fermentation process are computed for a range of time horizons. Numerical algorithms for determining the optimal control structure and computing the switching times are presented. Second order sufficient optimality conditions and sensitivity of optimal solutions are discussed.

The method of Monotone Structural Evolution for dynamic optimization of switched systems

The paper presents application of the Monotone Structural Evolution (MSE) to dynamic optimization of switched systems. The MSE is a direct computational method of optimal control which automatically identifies the optimal structure. Its distinctive feature is that the decision space undergoes gradual evolution, driven by the discrepancy from the Maximum Principle conditions. The approach is illustrated with two examples: the benchmark fishing problem, and a three-mode problem with right-hand sides linear in the state.

Extended variable parameterization method for optimal control

An extension of the variable parameterization method is presented. Conditions for switching generation and reduction are analysed. The previously used quasi-Newton algorithm is completed with the Newton method. The transversality conditions are modified. The ideas are illustrated by a numerical example showing the performance of the algorithm also for singular cases. A comparison with direct and indirect collocation approaches is given.

Time optimal control for the pendulum-cart system in real-time

An experiment in practical implementation of simulated time-optimal control in a laboratory cart-pendulum system is described. The simulation results are compared with two types of measured trajectories of the real system: generated by an implementation of the time-optimal control, and generated by a rule-based algorithm. The technologies of rapid prototyping, real-time simulation and hardware-in-the-loop simulation are used for direct synthesis of control algorithms. The feasibility of construction of time-optimal controller in the real system has been pointed out.

Consistent Control Procedures in the Monotone Structural Evolution. Part 2: Examples and Computational Aspects

The consistent control procedures for state-constrained and interior arcs are implemented in the MSE, and their performance demonstrated on numerical examples. For state constrained problems with index 1, a two-phase technique is proposed which ensures the exact fulfillment of the state constraint. To enhance efficiency of the method of prototype adjoints applied to consistent representation of interior control arcs, a new ‘freezing’ technique is used.

Computation of time optimal controls by gradient matching

Time optimal control of nonlinear systems is considered. To find the optimal horizon, a sequence of fixed-horizon problems of minimizing an auxiliary cost functional is solved by gradient methods, with the use of adjoint trajectories. Basically, the control switching times are decision variables. The proposed computational methods of search for optimal horizon are based on an estimation-continuation strategy. The approach proved effective in the case of a cart-pendulum system.

Consistent Control Procedures in the Monotone Structural Evolution. Part 1: Theory

The concept of consistent control procedures is introduced in optimal control computations. The stock of such procedures of the MSE, a direct method of dynamic optimization, is extended to handle state-constrained and interior arcs. Thus equipped, the MSE can automatically identify optimal control structures and yield arbitrarily exact approximations of optimal solutions by adjusting a bounded number of parameters.

A Generalization of the Orlando Formula — Symbolic Manipulation Approach

In this paper some new results on polynomials whose roots are linear combinations (with fixed integer coefficients) of the roots of other polynomial are presented. This research is motivated by a new method for calculation of a generalized integral criterion with the integrand function being an arbitrary polynomial of the transient error.

Robust near time-optimal trajectory planning by intermediate targets assignment

Robustified time-optimal trajectories are constructed by concatenation of arcs connecting appropriately selected intermediate targets. Stochastic methods are used for the examination of the vicinity of the time-optimal trajectory to find a set of displacement points such that: (i) time T necessary to reach the final target along the concatenated trajectory is close to the original optimal control time, and (ii) global sensitivity of T with respect to state disturbances with given stochastic properties is sufficiently small. This is obtained by backward tracking admissible and inadmissible terminal states of randomly generated trajectories. Geometric properties are discussed. The method of robustification is illustrated by an example of a nonlinear fourth-order cart-pendulum system.

Neighboring optimization for constrained control problems in real time

A neighboring trajectory closed-loop control is proposed for fixed-horizon constrained optimal control problems. The lower level of the algorithm adjusts the switching times based on the linearization of the optimal controller and performs reduced optimization with a change of control structure, whereas the upper level finds the optimal control and recalculates the linearization each time the deviation from the optimal solution becomes too large. The linearized switching controller is analytically derived. A comparison with full repetitive optimization shows that the new method gives good disturbance rejection at low computational cost.