Showing papers in "Lecture Notes in Control and Information Sciences in 1990"

Time-discretization of the Zakai equation for diffusion processes observed in correlated noise

Abstract: A time discretization scheme is provided for the Zakai equation, a stochastic PDE which gives the conditional law of a diffusion process observed in white-noise. The case where the observation noise and the state noise are correlated, is considered. The numerical scheme is based on a Trotter-like product formula, which exhibits prediction and correction steps, and for which an error estimate of order δ is proved, where δ is the time discretization step. The correction step is associated with a degenerate second-order stochastic PDE, for which a representation result is available in terms of stochastic characteristics [8,9,11]. A discretization scheme is then provided to approximate these stochastic characteristics. Under an additional assumption on the correlation coefficient, an error estimate of order √β is proved for the overall numerical scheme. This has been proved to be the best possible error estimate by Elliott-Glowinski [5].

Genetic breeding of non-linear optimal control strategies for broom balancing

Abstract: This paper describes a search for the time-optimal “bang bang” control strategy for the three dimensional broom balancing (inverted pendulum) problem by genetically breeding populations of control strategies using a recently developed new “genetic computing” paradigm. The new paradigm produces results in the form of a control strategy consisting of a composition of functions, including arithmetic operations, conditional logical operations, and mathematical functions. This control strategy takes the problem’s state variables as its input and generates the direction from which to apply the “bang bang” force as its output.

Discrete approximation of the minimal time function for systems with regular optimal trajectories

Abstract: In this paper we prove an estimate of the rate of convergence of the approximation scheme for the nonlinear minimum time problem presented in [2]. The estimate holds provided the system have time-optimal controls with bounded variation. This estimate is of order v with respect to the discretization step in time, if the minimal time function is Holder continuous of exponent v. The proof combines the convergence result obtained in [2] by PDE methods, with direct control-theoretic arguments.

A dynamic programming based Gas Pipeline Optimizer

Abstract: A dynamic programming based Gas Pipeline Optimizer (GPO) has been developed at Scientific Software-Intercomp for the HBJ gas transmission pipeline system in India. Used as an operating and planning tool, the GPO will determine the discharge pressures at the compressor stations and the number of compressor trains to operate at each compressor station so that fuel consumption and start-up/shut-down costs for the entire HBJ system are minimized under steady state conditions. Further, the optimization will satisfy specified delivery flow rate and minimum delivery pressure requirements at the receiving terminals and ensure that minimum line pack (inventory) requirements are met for each section of the pipeline. Excessive starting and stopping of compressor trains will also be avoided.

An algebraic method for optimizing resources in timed event graphs

Abstract: A min-linear system theory has been developed for timed event graphs using dioid algebraic structures. This in particular allows modeling and evaluating performance of flexible workshops and distributed processing systems. In this paper, we use this theory to study the problem of resource optimization, i.e., minimizing the cost of some resources (machines, pallets, processors) in order to achieve a given rate of production, or optimizing the rate of production itself. This problem reduces to an integer linear programming problem. We use rational symbolic computation in the dioid algebra to simplify and generate the corresponding simplex.

Moduloïds and pseudomodules

Abstract: We determine here sufficient conditions for finite dimensional moduloids and pseudomodules to be lattices. In particular, we show that completeness of the dioid D of scalars is such a condition. The simplicity conditions for pseudomodules, which make the classification problem tractable, are also shown to be sufficient for the lattice structure. Since these conditions are clearly unrelated, both results show that neither one is necessary. A concluding example illustrates this remark.

Discrete normalized coprime factorization

Abstract: In this paper a reliable algorithm is developed to perform a normalized coprime factorization of proper discrete time finite dimensional linear time invariant systems. Instead of using the bilinear transform the factorization is calculated directly. The system is allowed to have a singular state-space matrix. It is shown that a modified discrete time Riccati equation plays a crucial role to obtain a state-space realization for the factorization.

Output regulation of a flexible robot arm

Abstract: The problem of controlling the motion of a one-link flexible robot arm is considered. A nonlinear dynamic model is derived assuming that the flexibility is represented by elastic springs along the link. Taking as output the end-effector angular position, the resulting zero-dynamics is unstable. Therefore, inversion-based controllers cannot be used for tracking of output trajectories. Instead, it is shown here that a nonlinear regulator approach solves the problem, allowing asymptotic trajectory tracking with internal stability. Simulation results show the good performance of the overall controller and in particular the benefits achieved by a nonlinear design of the regulator.

Controllability of delay-differential systems

Abstract: The concept of controllability is introduced and investigated for the class of AR delay-differential systems with separable AR descriptions. For this class of systems, it is shown that a system Σ described by the AR equation R(σ 1,σ 2)w=o (with σ 1 the differentiation- and σ 2 the delay-operator) is controllable if and only if rank R(λ,e −λ) is constant for all λ∈C.

An interior-point algorithm for large-scale quadratic problems with box constraints

Abstract: We present computational experience with an interior-point algorithm for large-scale quadratic programming problems with box constraints. The algorithm requires a total of O(√nL) number of iterations, where L is the size of the input data of the problem, and O(n 3) arithmetic operations per iteration. The algorithm has been implemented using vectorization and tested on an IBM 3090-600S computer with vector facilities. The computational results suggest that the efficiency of the algorithm depends on an appropriate choice of parameters. Computational results with various large-scale problems, including examples of obstacle problems, are presented.

Minimal rational interpolation and Prony’s method

Abstract: A new method is proposed for dealing with the rational interpolation problem. It is based on the reachability of an appropriately defined pair of matrices. This method permits a complete clarification of several issues raised, but not answered, by the so-called Prony method of fitting a linear model to given data.

Discontinuous predictive control, inversion and singularities. Application to a heat exchanger

Abstract: Predictive control by inversion is applied to nonlinear systems. Problems related to singularities and non minimum-phase, which often arise in this context, are most simply overcome by introducing a discontinuous control approach. Such a solution is illustrated on a heat exchanger example.

A new sufficient condition for the well-posedness of non-linear least square problems arising in identification and control

Abstract: We show how simple 1-D geometrical calculations (but along all maximal segments of the parameter or control setl) can be used to establish the weliposedness of a non-linear leastsquare (NLLS) problem and the absence of local minima in the corresponding error function. These sufficient conditions, which are shown to be sharp by elementary examples, are based on the use of the recently developed “size x curvature”, conditions for proving that the output set is strictly quasiconvex. The use of this geometrical theory as a numerical or theoretical tool is discussed. Finally, application to regularized NLLS problem is shown to give new information on the choice of the regularizing parameter.

On observers for smooth nonlinear digital systems

Abstract: The problem of constructing an observer for a nonlinear discrete-time system is analyzed as one of left-inverting a certain map from the states to the measurements. Practical and theoretical consequences are derived.

A combinatorial approach of the nonlinear sampling problem

Abstract: In this paper, results of combinatorial interest taken from the well-known Baker-Campbell-Hausdorff formula are proposed. They are used to analyze algebraic relations between nonlinear continuous time systems and their discretized analogues.

Nonlinear control of missiles through a geometric approach

Abstract: Many theorical studies have been conducted about decoupling and linearization of nonlinear systems. This approach is a means to synthesize the control laws of single-input single-output systems. It is used here for the piloting of nonrolling missiles, of which transverse acceleration is to be controlled. In this paper, it is proved that for this kind of problem, it is possible to associate such a method to simplifications of the model on the one hand and separation of the system into multiple time scale cascade subsystems on the other. Thus, the dimension of the unobservable part can be minimized and the control law is simpler.

Optimality conditions for utility maximization in an incomplete market

Abstract: The problem of maximizing expected utility of final wealth in an incomplete market is investigated The incomplete market is modelled by a bond and a finite number of stocks, the latter being driven by a d-dimensional Brownian motion The coefficients of the bond and stock price processes are adapted to this Brownian motion, and the number of stocks is less than or equal to the dimension of the driving Brownian motion It is shown that there is a way to “fictitiously” complete this market so that the optimal protfolio for the resulting completed market coincides with the optimal portfolio for the original incomplete market A number of equivalent characterizations of the fictitious completion are given, and examples are provided

Aggregation and multi-level control in discrete event dynamic systems

Abstract: In this paper we consider higher-level aggregate modelling and control of discrete-event dynamic systems (DEDS). The higher-level models considered correspond to associating specified sequences of events in the original system to single macroscopic events in the higher-level model. We also consider the problem of designing a compensator that can be used to restrict microscopic behavior so that the system will only produce strings of these primitive sequences or tasks. With this lower level control in place we can construct higher-level models which typically have many fewer states and events than the original system. A complete treatment of the topics presented here can be found in [5].

Analyse de l’algorithme multigrille FMGH de résolution d’équations d’Hamilton-Jacobi-Bellman

Abstract: In this paper, the resolution of Hamilton-Jacobi-Bellman equations by multigrid methods is studied The Howard-multigrid algorithm FMGH is presented and, under some regularity assumptions, a convergence result is established In addition, it is shown that the complexity of this algolithm is in the order of the nmnber of discretisation points Some numerical examples are reported I n t r o d u c t i o n L'algorithme FMGH (introduit dans [1]) est un Mgorithme de r~solution numr on pent, ~ partir d 'un hombre fixe d'itr la complexit~ de rdsolution des probl~mes tests soit minimiser une fonction coflt J sur l'ensemble des contrSles admissibles Uad C )T/k, l'6tat &rant un processus stochastique )/-'t r6gi par une 6quation diff6rentielle stochastique dans ~ Donnons l'exemple d 'un problbme horizon infini dont l'6tat est arrSt& stir la fronti~re de l'ouvert Yl C JT~ d, il s'~crit dXt = g(Un X,) dt + a(X,) dWt I'Vt processus de Wiener x ( o ) = zo J(U(),Xo) = E (fo*e-~tc(Ut, X,)dt + e-~*f(XT))

Approximation of Lyapunov exponents of stochastic differential systems on compact manifolds

Abstract: We consider a stochastic differential system on a compact manifold; under our hypotheses, its Lyapunov exponents are deterministic. We propose an algorithm of numerical computation of these exponents, and we give a theoretical estimate for the approximation error. The method is based upon the discretization of the linearized stochastic flow of diffeomorphisms generated by the differential system.

An analysis of regularity conditions in nonlinear synthesis

Abstract: Many results are now available for output nulling, system inversion and dynamic input-output decoupling, each with its own set of regularity conditions. In this paper, we present a framework, based upon the canonical map from initial conditions and inputs to outputs, in which these regularity conditions can be reformulated and compared. As a byproduct, it is seen how to enlarge the class of systems for which an inverse system can be constructed.

A forward accessibility algorithm for nonlinear discrete time systems

Abstract: In this paper we present a SAMDI algorithm (Symbolic Algebraic Manipulation for DIscrete time system), which permits us to know whether a system is accessible for nonlinear discrete time. This algorithm uses the symbolic language REDUCE. In our algorithm we use a set of Gij vector field, which allows us to work with the same geometrical tools as for the continuous time system. Therefore, the SAMDI algorithm uses the same procedures of the SAM (Symbolic Algebraic Manipulation) programme, which was realized for continuous time systems.

Asymptotic properties in rational l2-approximation

Abstract: This paper is concerned with the problem of best rational approximation of given order n in the Hardy space H 2. We show that, generically, all critical points converge to the function in H 2 as n increases to infinity. This property shows in turn that local maxima can appear only for a finite range of orders. This has consequences on an algorithm to find local minima previously described by some of the authors [3].

CRONE Control: Principle, synthesis, performances with non-linearities and robustness-input immunity dilemma

Abstract: The area of this paper concerns the robustness of stability degree, and more particularly the robustness of the damping of the control versus the parameters of the plant. The approach of the CRONE Control is presented as resulting from the non integer order differential equation which represents the dynamic model directing a natural robust relaxation. A frequency illustration of robustness is given in the Nickol-Black plane through an open loop frequency template. Such a template is synthesized by means of a CRONE variable phase regulator from the phase diagram of an agricultural mobile robot using a highly non linear error detector. The corresponding robustness performances are presented. At last, a dilemma is established by considering two templates of different lenghts. The imput of the plant is given for step responses to the reference input. It shows that the greater the robustness, the lower the input immunity.

Fuel savings by optimal aircraft cruise with singular and chattering control

Abstract: Fuel minimization in aircraft cruise is considered as an optimal periodic control problem. Optimality conditions for flight trajectories with singular arcs and state veriable constraints are derived. For the numerical investigation, an optimization program based on the method of multiple shooting is applied.

A new canonical form for descriptor systems with outputs

Abstract: We construct a new canonical form for descriptor systems (E,A,B,C) under proportional feedback, proportional output injection and change of bases of input space, output space and state space. This form is compared with that of Van Der Weiden and Bosgra (1980).

Composite optimization: Second order conditions, value functions and sensityvity

Abstract: By composite optimization we mean a class of optimization problems without constraints involving composite functions of the form f(X)=g(F(X)), where F is a smooth map and g is a convex (in general non-smooth) function. Such problems typically appear as reduction forms for many (if not practically all) important classes of optimization problems with constraints. In the paper we give a formula for the second order lower epi-derivative of composite functions and further apply it to obtain new second order conditions and sensitivity estimates in composite optimization.