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)     

A primal‐dual method for the Meyer model of cartoon and texture decomposition

In this paper, we study the original Meyer model of cartoon and texture decomposition in image processing. The model, which is a minimization problem, contains an l₁‐based TV‐norm and an l_∞‐based G‐norm. The main idea of this paper is to use the dual formulation to represent both TV‐norm and G‐norm. The resulting minimization problem of the Meyer model can be given as a minimax problem. A first‐order primal‐dual algorithm can be developed to compute the saddle point of the minimax problem. The convergence of the proposed algorithm is theoretically shown. Numerical results are presented to show that the original Meyer model can decompose better cartoon and texture components than the other testing methods.     

On the numerical solution of certain boundary control problems for vibrating media in one space-dimension

This paper is concerned with the numerical solution of control problems which consist of minimizing certain quadratic functionals depending on control functions in L ₂[0,1] for some given time T > 0 and bounded with respect to the maximum norm. These control functions act upon the boundary conditions of a vibrating system in one space-dimension which is governed by a wave equation of spatial order 2n They are to be chosen in such a way that a given initial state of vibration at time zero is transferred into the state of rest. This requirement can be expressed by an infinite system of moment equations to be satisfied by the control functions

The control problem is approximated by replacing this infinite system by finitely many, say N, equations (truncation) and by choosing piecewise constant functions as controls (discretization). The resulting problem is a quadratic optimization problem which is solved very efficiently by a multiplier method

Convergence of the solutions of the approximating problems to the solution of the control problem, as N tends to infinity and the discretization is infinitely refined, is shown under mild assumptions. Numerical results are presented for a vibrating beam     

Behavioral portfolio selection with loss control

In this paper we formulate a continuous-time behavioral (à la cumulative prospect theory) portfolio selection model where the losses are constrained by a pre-specified upper bound. Economically the model is motivated by the previously proved fact that the losses occurring in a bad state of the world can be catastrophic for an unconstrained model. Mathematically solving the model boils down to solving a concave Choquet minimization problem with an additional upper bound. We derive the optimal solution explicitly for such a loss control model. The optimal terminal wealth profile is in general characterized by three pieces: the agent has gains in the good states of the world, gets a moderate, endogenously constant loss in the intermediate states, and suffers the maximal loss (which is the given bound for losses) in the bad states. Examples are given to illustrate the general results.     

Min-max control of systems approximated by simple models:L 1-Type cost functionals

A control problem for actual processes, modeled by simple dynamic linear systems, is studied. A min-max approach is taken, since the problem is reduced to the control of a model, whose output is affected by an input-dependent signal constrained in norm.Particular attention is given to the features of the penalty functional, since it is desired to synthesize the min-max control by means of a feedback on the states of the model; for example, a quadratic functional does not have this property. AnL ₁-type functional is then proposed, which is not differentiable; by means of duality techniques, however, the minimization can be carried out. The min-max is obtained both by iterative algorithms, both as a function of the actual value of the model state: this function is the solution of a set of partial differential equations.This work was supported by the CENS and the Gruppo Nazionale per l'Analisi Funzionale e le sue Applicazioni, Consiglio Nazionale delle Ricerche (CNR), Roma, Italy.     

Minimizing the natural fuel requirement for nuclear reactor power systems: A nonstandard optimal control problem

A class of model problems in nuclear reactor economics is defined and shown to be equivalent to a linear optimal control problem to which present versions of the maximum principle apparently cannot be applied. It is shown that the search for an optimal control can be restricted tocontrols of maximum fuel utilization (Comfu), and that theComfu's are in a one-to-one correspondence with the functions which satisfy certain inequalities and are solutions of a nonlinear Volterra integral equation containing the value of the cost functional as a parameter. In the general case, one can establish an iterative procedure, involving solution of the integral equation at each iteration, for approximating the optimalComfu. For some important special cases, a point on the solution corresponding to the optimalComfu is knowna priori, and thus the optimalComfu can be obtained by solving the integral equation only once. Some possible generalizations of the original economic model are also discussed.This research was sponsored by the US Atomic Energy Commission under contract with the Union Carbide Corporation.     

Strong controllability and optimal control of the heat equation with a thermal source

In this paper we consider an optimal control system described byn-dimensional heat equation with a thermal source. Thus problem is to find an optimal control which puts the system in a finite time T, into a stationary regime and to minimize a general objective function. Here we assume there is no constraints on control. This problem is reduced to a moment problem.We modify the moment problem into one consisting of the minimization of a positive linear functional over a set of Radon measures and we show that there is an optimal measure corresponding to the optimal control. The above optimal measure approximated by a finite combination of atomic measures. This construction gives rise to a finite dimensional linear programming problem, where its solution can be used to determine the optimal combination of atomic measures. Then by using the solution of the above linear programming problem we find a piecewise-constant optimal control function which is an approximate control for the original optimal control problem. Finally we obtain piecewise-constant optimal control for two examples of heat equations with a thermal source in one-dimensional.     

Diffusion approximation forGI/G/1 controlled queues

A queueing model is considered in which a controller can increase the service rate. There is a holding cost represented by functionh and the service cost proportional to the increased rate with coefficientl. The objective is to minimize the total expected discounted cost.Whenh andl are small and the system operates in heavy traffic, the control problem can be approximated by a singular stochastic control problem for the Brownian motion, namely, the so-called reflected follower problem. The optimal policy in this problem is characterized by a single numberz ^* so that the optimal process is a reflected diffusion in [0,z ^*]. To obtainz ^* one needs to solve a free boundary problem for the second order ordinary differential equation. For the original problem the policy which increases to the maximum the service rate when the normalized queue-length exceedsz ^* is approximately optimal.     

Quickest search over Brownian channels

In this paper we resolve an open problem proposed by Lai, Vincent Poor, Xin, and Georgiadis [Quickest search over multiple sequences. IEEE Trans. Inf. Theory 57(8) (2011), pp. 5375–5386]. Consider a sequence of Brownian motions with unknown drift equal to one or zero, which may be observed one at a time. We give a procedure for finding, as quickly as possible, a process which is a Brownian motion with non-zero drift. This original quickest search problem, in which the filtration itself is dependent on the observation strategy, is reduced to a single filtration impulse control and optimal stopping problem, which is in turn reduced to an optimal stopping problem for a reflected diffusion, which can be explicitly solved.     

Minimal data rates and entropy in digitally networked systems

In digitally networked systems the assumption of classical control theory that information can be transmitted instantaneously, lossless and with arbitrary precision is violated. This raises the question about the smallest data rate above which a control task can be solved. For a single control loop and the problem to make a set Q of states invariant, the minimal data rate can be described by an entropy-like quantity, the so-called invariance entropy. Under some controllability and hyperbolicity assumptions, the invariance entropy can be expressed in terms of Lyapunov exponents. Furthermore, one can show that for making Q invariant with a data rate close to the smallest, no strategies more complicated than stabilization at periodic trajectories are necessary. For a network with n subsystems, which can all communicate with each other, there are different ways to formulate the question about the smallest data rate for the invariance problem, but also in this setting entropy-like quantities yield important information. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Heavy traffic resource pooling in parallel‐server systems

We consider a queueing system with r non‐identical servers working in parallel, exogenous arrivals into m different job classes, and linear holding costs for each class. Each arrival requires a single service, which may be provided by any of several different servers in our general formulation; the service time distribution depends on both the job class being processed and the server selected. The system manager seeks to minimize holding costs by dynamically scheduling waiting jobs onto available servers. A linear program involving only first‐moment data (average arrival rates and mean service times) is used to define heavy traffic for a system of this form, and also to articulate a condition of overlapping server capabilities which leads to resource pooling in the heavy traffic limit. Assuming that the latter condition holds, we rescale time and state space in standard fashion, then identify a Brownian control problem that is the formal heavy traffic limit of our rescaled scheduling problem. Because of the assumed overlap in server capabilities, the limiting Brownian control problem is effectively one‐dimensional, and it admits a pathwise optimal solution. That is, in the limiting Brownian control problem the multiple servers of our original model merge to form a single pool of service capacity, and there exists a dynamic control policy which minimizes cumulative cost incurred up to any time t with probability one. Interpreted in our original problem context, the Brownian solution suggests the following: virtually all backlogged work should be held in one particular job class, and all servers can and should be productively employed except when the total backlog is small. It is conjectured that such ideal system behavior can be approached using a family of relatively simple scheduling policies related to the cμ rule. This revised version was published online in June 2006 with corrections to the Cover Date.     

Input-output dynamic model for optimal environmental pollution control

This paper describes the allocation of a wastewater treatment fund within a region based on a dynamic input-output model. Considering the complexity of the input-output process, many indeterminate factors must be included in the model. For example, with the aging of machines, an unexpected loss will be caused by the retention of raw materials during an operation; this can be realistically considered as a random variable, because of the sufficiently large amount of historical data. By contrast, actions such as a temporary transfer or inexperienced operators can only be regard as uncertain variables, because of a lack of historical data. First, the pollution control model is formulated in an uncertain environment by including both human uncertainty and objective randomness. Second, an optimal control model subject to an uncertain random singular system is established; this model can be transformed into an equivalent optimization problem. To solve such a problem, recurrence equations are presented based on Bellman’s principle, and these were successfully applied to address the optimal control problem in two special cases. Moreover, two algorithms are formulated for solving the pollution control problem. Finally, the optimal distribution strategies of the pollution control fund used to control the emissions of COD and NH₃-H, which are two indicators of wastewater in China, were obtained through the proposed algorithms.     

Least squares data fitting with implicit functions

This paper discusses the computational problem of fitting data by an implicitly defined function depending on several parameters. The emphasis is on the technique of algebraic fitting off(x, y; p) = 0 which can be treated as a linear problem when the parameters appear linearly. Various constraints completing the problem are examined for their effectiveness and in particular for two applications: fitting ellipses and functions defined by the Lotka-Volterra model equations. Finally, we discuss geometric fitting as an alternative, and give examples comparing results.     

Optimal switching control of a fed-batch fermentation process

Considering the hybrid nature in fed-batch culture of glycerol biconversion to 1,3-propanediol (1,3-PD) by Klebsiella pneumoniae, we propose a state-based switching dynamical system to describe the fermentation process. To maximize the concentration of 1,3-PD at the terminal time, an optimal switching control model subject to our proposed switching system and constraints of continuous state inequality and control function is presented. Because the number of the switchings is not known a priori, we reformulate the above optimal control problem as a two-level optimization problem. An optimization algorithm is developed to seek the optimal solution on the basis of a heuristic approach and control parametrization technique. Numerical results show that, by employing the obtained optimal control strategy, 1,3-PD concentration at the terminal time can be increased considerably.     

An algebraic approach to assignment problems

For assignment problems a class of objective functions is studied by algebraic methods and characterized in terms of an axiomatic system. It says essentially that the coefficients of the objective function can be chosen from a totally ordered commutative semigroup, which obeys a divisibility axiom. Special cases of the general model are the linear assignment problem, the linear bottleneck problem, lexicographic multicriteria problems,p-norm assignment problems and others. Further a polynomial bounded algorithm for solving this generalized assignment problem is stated. The algebraic approach can be extended to a broader class of combinatorial optimization problems.     

Comparative studies on dynamic programming and integer programming approaches for concave cost production/inventory control problems

This paper is concerned with classical concave cost multi-echelon production/inventory control problems studied by W. Zangwill and others. It is well known that the problem with m production steps and n time periods can be solved by a dynamic programming algorithm in O(n ⁴ m) steps, which is considered as the fastest algorithm for solving this class of problems. In this paper, we will show that an alternative 0–1 integer programming approach can solve the same problem much faster particularly when n is large and the number of 0–1 integer variables is relatively few. This class of problems include, among others problem with set-up cost function and piecewise linear cost function with fewer linear pieces. The new approach can solve problems with mixed concave/convex cost functions, which cannot be solved by dynamic programming algorithms.     

Min-max optimal control of systems approximated by finite-dimensional models: Nonquadratic cost functional

An open-loop control problem with nonquadratic performance criterion for a dynamical system, described by an abstract linear equation of evolution and approximated by a finite-dimensional model, is solved. A min-max approach is taken: the value of the cost functional in theworst-case output error between the system and the model, under the assumption that the norm of this output error is estimated by the norm of the input, is minimized.The form of the cost functional reproduces itself under maximization, so that the min-max control problem, when only thedistance between the model and the system is given, has the same features and proprieties of the control problem when the system is thorougly known.Existence and uniqueness theorems for the optimal control are proven, using the spectral proprieties of the model transfer function; and, in the case of a time-invariant model, the min-max control computation is reduced to the solution of a constant-coefficients Sturm-Liouville problem followed by the search for the zeros of a very simple numerical function.This work was made within the Gruppo Nazionale per l'Analisi Funzionale e le sue Applicazioni, Consiglio Nazionale delle Ricerche.     

Fast Iterative Solution of Saddle Point Problems in Optimal Control Based on Wavelets

In this paper, wavelet techniques are employed for the fast numerical solution of a control problem governed by an elliptic boundary value problem with boundary control. A quadratic cost functional involving natural norms of the state and the control is to be minimized. Firstly the constraint, the elliptic boundary value problem, is formulated in an appropriate weak form that allows to handle varying boundary conditions explicitly: the boundary conditions are treated by Lagrange multipliers, leading to a saddle point problem. This is combined with a fictitious domain approach in order to cover also more complicated boundaries.Deviating from standard approaches, we then use (biorthogonal) wavelets to derive an equivalent infinite discretized control problem which involves only ₂-norms and -operators. Classical methods from optimization yield the corresponding optimality conditions in terms of two weakly coupled (still infinite) saddle point problems for which a unique solution exists. For deriving finite-dimensional systems which are uniformly invertible, stability of the discretizations has to be ensured. This together with the ₂-setting circumvents the problem of preconditioning: all operators have uniformly bounded condition numbers independent of the discretization.In order to numerically solve the resulting (finite-dimensional) linear system of the weakly coupled saddle point problems, a fully iterative method is proposed which can be viewed as an inexact gradient scheme. It consists of a gradient algorithm as an outer iteration which alternatingly picks the two saddle point problems, and an inner iteration to solve each of the saddle point problems, exemplified in terms of the Uzawa algorithm. It is proved here that this strategy converges, provided that the inner systems are solved sufficiently well. Moreover, since the system matrix is well-conditioned, it is shown that in combination with a nested iteration strategy this iteration is asymptotically optimal in the sense that it provides the solution on discretization level J with an overall amount of arithmetic operations that is proportional to the number of unknows N _J on that level.Finally, numerical results are provided.     

Dynamics and control of underactuated mechanical systems: analysis and simple experimental verification

Underactuated mechanical systems are systems with fewer control inputs than the degrees of freedom, m < n, the relevant technical examples being e.g. cranes, aircrafts and flexible manipulators. The determination of an input control strategy that forces an underactuated system to complete a set of m specified motion tasks (servo-constraints) is a demanding problem. The solution is conditioned to differential flatness of the problem, denoted that all 2n state variables and m control inputs can algebraically be expressed, at least theoretically, in terms of the desired m outputs and their time derivatives up to a certain order. A more practical formulation, motivated hereafter, is to pose the problem as a set of differential-algebraic equations, and then obtain the solution numerically. The theoretical considerations are illustrated by a simple two-degree-of-freedom underactuated system composed of two rotating discs connected by a flexible rod (torsional spring), in which the pre-specified motion of the first disc is actuated by the torque applied to the second disc, n = 2 and m = 1. The determined control strategy is then verified experimentally on a laboratory stand representing the two-disc system. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Minimization of a functional over the set of causal operators of causal Hilbert space

The minimization problem for a quadratic functional defined on the set of nonwarning (causal) operators acting in a causal Hilbert space can be regarded as an abstrat analog of the Wiener problem on constructing the optimal nonwarning filter. A similar problem also arises in the linear control problem with the quadratic performance criterion (in this case the transfer operators of a closed control system serve as causal ones). The introduction of causal operators in filtering theory and control theory is a mathematical expression of the causality principle, which must be taken into account for a number of problems. In the present paper we attempt to systematize the mathematical foundations of the abstract linear filtering theory, for which its basic results are expressed in terms of operators describing the filtering problem. We introduce and study a class of finite operators, a natural generalization of the class of causal operators, and give a solution of the minimization problem for a quadratic positive functional defined on the set of causal operators acting in a “discrete” causal space. Bibliography: 54 titles. Translated fromProblemy Matematicheskogo Analiza, No. 14, 1995. pp. 143–187.