Optimal control of robots under stochastic uncertainty: robust feedback control

The most important aspect in the optimal control and design of manipulators is the determination of the basic movement, i.e. the calculation of the optimal trajectory on which the robot has to move. Having an optimal reference trajectory and an optimal open-loop control, there is the need of control corrections by applying a certain feedback control. Different attempts exist for this. In this article a method will be shown which is based on classical control theory, that works with cost functions being minimized. The aim is to take into account stochastic parameter variations in order to obtain robust optimal feedback controls. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal cash management under uncertainty

We solve an agent’s optimization problem of meeting demands for cash over time with cash deposited in bank or invested in stock. The stock pays dividends and uncertain capital gains, and a commission is incurred in buying and selling of stock. We use a stochastic maximum principle to obtain explicitly the optimal transaction policy.     

Optimal budgetary and monetary policies under uncertainty: A stochastic control approach

In this paper, we determine optimal budgetary and monetary policies for Austria using a small macroeconometric model. We use a Keynesian model of the Austrian economy, called FINPOL1, estimated by ordinary least squares, which relates the main objective variables of Austrian economic policies, such as the growth rate of real gross domestic product, the rate of unemployment, the rate of inflation, the balance of payments, and the ratio of the federal budget deficit to GDP, to fiscal and monetary policy instruments, namely expenditures and revenues of the federal budget and money supply. Optimal fiscal and monetary policies are calculated for the model under a quadratic objective function using the algorithm OPTCON for the optimum control of nonlinear stochastic dynamic systems. Several control experiments are performed in order to assess the influence of different kinds of uncertainty on optimal budgetary and monetary policies. Apart from deterministic optimization runs, different assumptions about parameter uncertainties are introduced; the results of these different stochastic optimum control experiments are compared and interpreted.     

Optimal reinsurance under risk and uncertainty

This paper deals with the optimal reinsurance problem if both insurer and reinsurer are facing risk and uncertainty, though the classical uncertainty free case is also included. The insurer and reinsurer degrees of uncertainty do not have to be identical. The decision variable is not the retained (or ceded) risk, but its sensitivity with respect to the total claims. Thus, if one imposes strictly positive lower bounds for this variable, the reinsurer moral hazard is totally eliminated.Three main contributions seem to be reached. Firstly, necessary and sufficient optimality conditions are given in a very general setting. Secondly, the optimal contract is often a bang–bang solution, i.e., the sensitivity between the retained risk and the total claims saturates the imposed constraints. Thirdly, the optimal reinsurance problem is equivalent to other linear programming problem, despite the fact that risk, uncertainty, and many premium principles are not linear. This may be important because linear problems may be easily solved in practice, since there are very efficient algorithms.     

Optimal hedging under output price uncertainty

This paper provides a theoretical and empirical analysis of optimal hedging under output price uncertainty. The theoretical analysis is facilitated by exploiting the duality between production and cost while the empirical implementation uses the envelope theorem and the indirect expected utility function. Empirically estimable equations are derived by approximating the indirect expected utility function by a Taylor series approximation. The model is tested by using live cattle data as output while using prices of corn, soybeans, and the feeder cattle as inputs. The results support the theoretical predictions and the evidence shows that live cattle farmers exhibit decreasing absolute risk aversion.     

Optimal staffing of specialized programme trainees under uncertainty

We propose a stochastic model and provide an easy-to-implement optimization tool for admission decisions to specialized training programmes designed for service industries. The model can be applied for staffing of trainees in medical residency programmes, vocational schools, management trainee programmes, and similar. Especially towards graduation, trainees in these programmes substantially contribute to workforce of their affiliated institutions, thus having a targeted number of advanced level students become a potential performance metric for administration. For uncertain attrition rates and study duration, we model and provide an iterative solution algorithm to find the optimal annual admission number for these programmes. Our numeric analysis results show that the solution is robust to changes in attrition and study duration probabilities; hence, our model is robust against specification errors for these parameters, which could be hard to estimate due to data unavailability and fluctuations in educational and economic conditions.     

Optimal Open-Loop Feedback Control of Robots under uncertainty

The aim of this presentation is to construct an optimal open-loop feedback controller for robots, which takes into account stochastic uncertainties. This way, optimal regulators being insensitive with respect to random parameter variations can be obtained. Usually, a precomputed feedback control is based on exactly known or estimated model parameters. However, in practice, often exact informations about model parameters, e.g. the payload mass, are not given. Supposing now that the probability distribution of the random parameter variation is known, in the following, stochastic optimisation methods will be applied in order to obtain robust open-loop feedback control. Taking into account stochastic parameter variations, the method works with expected cost functions evaluating the primary control expenses and the tracking error. The expectation of the total costs has then to be minimized. Corresponding to Model Predictive Control (MPC), here a sliding horizon is considered. This means that, instead of minimizing an integral from a starting time point t₀ to the final time t_f, the future time range [t; t+T], with a small enough positive time unit T, will be taken into account. The resulting optimal regulator problem under stochastic uncertainty will be solved by using the Hamiltonian of the problem. After the computation of a H-minimal control, the related stochastic two-point boundary value problem is then solved in order to find a robust optimal open-loop feedback control. The performance of the method will be demonstrated by a numerical example, which will be the control of robot under random variations of the payload mass. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal PID controller of robots under stochastic uncertainty

In practice often it is not possible to specify exact model parameters. Hence, precomputed controller based on some parameter estimates can produce bad results. In this presentation the aim is to combine classical PID control theory and stochastic optimisation methods in order to obtain robust optimal feedback control. The method works with cost functions being minimized and takes into account stochastic parameter varations. After Taylor expansion to calculate expected cost functions and a few transformations an approximate deterministic substitute PID control problem follows. Here, retaining only linear terms, approximation of expectations and variances of the expected cost functions can be calculated explicitly. By means of splines, numerical approximations of the objective function and the differential equations are obtained then. Using stochastic optimization methods, random parameter variations are incorporated into the optimal control process. Hence, robust optimal feedback controls are obtained. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal production in a competitive market under yield uncertainty

Motivated by the U.S. influenza vaccine market, we study the impact of random yield and production capacity on the number of firms, total supply, consumer surplus and social welfare in a market with identical suppliers. We formulate a two-stage game with endogenous entry, where each entering firm aims to maximize its profit under yield uncertainty subject to a production capacity constraint. We show that if firms produce full capacity in the equilibrium, then there are fewer firms in the equilibrium compared to the social optimum even for small levels of yield uncertainty. Furthermore, we prove that if firms do not produce full capacity in the equilibrium, they will not produce full capacity in the social optimum.     

Optimal control of the obstacle in a quasilinear elliptic variational inequality

In this paper we consider an obstacle control problem where the state satisfies a quasilinear elliptic variational inequality and the control function is the obstacle. The state is chosen to be close to the desire profile while the H² norms of the obstacle is not too large. Existence and necessary conditions for the optimal control are established.     

Optimal feedback control for a linear-quadratic control problem with a state inequality constraint

In this paper, we consider the linear-quadratic control problem (LQCP) for systems defined by evolution operators with an inequality constraint on the state. It is shown that, under suitable assumptions, the optimal control exists, is unique, and has a closedloop structure. The synthesis of the feedback control requires one to solve two Riccati integral equations.     

Optimal control under persistent disturbances

Let f be an orientation-preserving Morse-Smale diffeomorphism of an n-dimensional (n ≥ 3) closed orientable manifold M ⁿ . We show the possibility of representing the dynamics of f in a “source-sink” form. The roles of the “source” and “sink” are played by invariant closed sets one of which, A _f , is an attractor, and the other, R _f , is a repeller. Such a representation reveals new topological invariants that describe the embedding (possibly, wild) of stable and unstable manifolds of saddle periodic points in the ambient manifold. These invariants have allowed us to obtain a classification of substantial classes of Morse-Smale diffeomorphisms on 3-manifolds. In this paper, for any n ≥ 3, we describe the topological structure of the sets A _f and R _f and of the space of orbits that belong to the set M ⁿ \ (A _f ∪ R _f ).     

An uncertainty inequality for Fourier-Dunkl series

An uncertainty inequality for the Fourier-Dunkl series, introduced by the authors in [Ó. Ciaurri, J.L. Varona, A Whittaker-Shannon-Kotel’nikov sampling theorem related to the Dunkl transform, Proc. Amer. Math. Soc. 135 (2007) 2939-2947], is proved. This result is an extension of the classical uncertainty inequality for the Fourier series.     

On one class of control problems under uncertainty

An approach to solving a control problem for a nonlinear dynamic system containing controlled and uncertain parameters with terminal performance functional is described. The parameters of the controlled process are such that the uncertainty vector dominates over the control vector. A game control in the class of counterstrategies is suggested such that the value of the functional depends on the realized uncertainty. The quality of a counterstrategy is estimated by comparing the values of the functional under the game control and under the optimal control. The results of calculating the control and values of the functional for test parameters of the model and of the class of uncertainties are presented.     

Decentralized optimal control of dynamical systems under uncertainty

The problem of optimal control of a group of interconnected dynamical objects under uncertainty is considered. The cases are examined in which the centralized control of the group of objects is impossible due to delay in the channel for information exchange between the group members. Optimal self-control algorithms in real time for each dynamical object are proposed. Various types of a priori and current information about the behavior of the group members and about uncertainties in the system are examined. The proposed methods supplement the earlier developed optimal control methods for an individual dynamical system and the methods of decentralized optimal control of deterministic objects. The results are illustrated with examples.