Novel developments in process optimisation using predictive control

In industrial practice, constrained steady state optimisation and predictive control are separate, albeit closely related functions within the control hierarchy. This paper presents a method which integrates predictive control with on-line optimisation with economic objectives. A receding horizon optimal control problem is formulated using linear state space models. This optimal control problem is very similar to the one presented in many predictive control formulations, but the main difference is that it includes in its formulation a general steady state objective depending on the magnitudes of manipulated and measured output variables. This steady state objective may include the standard quadratic regulatory objective, together with economic objectives which are often linear. Assuming that the system settles to a steady state operating point under receding horizon control, conditions are given for the satisfaction of the necessary optimality conditions of the steady-state optimisation problem. The method is based on adaptive linear state space models, which are obtained by using on-line identification techniques. The use of model adaptation is justified from a theoretical standpoint and its beneficial effects are shown in simulations. The method is tested with simulations of an industrial distillation column and a system of chemical reactors.

Dynamic integrated system optimization and parameter estimation for discrete time optimal control of nonlinear systems

An algorithm for solving nonlinear discrete time optimal control problems with model-reality differences is presented. The technique uses Dynamic Integrated System Optimization and Parameter Estimation (DISOPE), which achieves the correct optimal solution in spite of deficiencies in the mathematical model employed in the optimization procedure. A version of the algorithm with a linear-quadratic model-based problem, implemented in the C+ + programming language, is developed and applied to illustrative simulation examples. An analysis of the optimality and convergence properties of the algorithm is also presented.

A new approach for the solution of the optimal set-point tracking problem for nonlinear systems

In this paper, a discrete time dynamic integrated system optimisation and parameter estimation algorithm is applied to the solution of the nonlinear tracking optimal control problem. A version of the algorithm with a linear-quadratic model-based problem is developed and implemented in software. The algorithm implemented is tested with simulation examples.

Hierarchical control and optimisation of the Tennessee Eastman process

Based on integrated system optimisation and parameter estimation a method is described for on-line steady state optimisation which compensates for model-plant mismatch and solves a non-linear optimisation problem by iterating on a linear - quadratic representation. The method requires real process derivatives which are estimated using a dynamic identification technique. The utility of the method is demonstrated using a simulation of the Tennessee Eastman benchmark chemical process.

Optimal control of nonlinear differential algebraic equation systems

A novel iterative procedure is described for solving nonlinear optimal control problems subject to differential algebraic equations. The procedure iterates on an integrated modified linear quadratic model based problem with parameter updating in such a manner that the correct solution of the original non-linear problem is achieved. The resulting algorithm has a particular advantage in that the solution is achieved without the need to solve the differential algebraic equations . Convergence aspects are discussed and a simulation example is described which illustrates the performance of the technique. 1. Introduction When modelling industrial processes often the resulting equations consist of coupled differential and algebraic equations (DAEs). In many situations these equations are nonlinear and cannot readily be directly reduced to ordinary differential equations.

Using zero-norm constraint for sparse probability density function estimation

A new sparse kernel probability density function (pdf) estimator based on zero-norm constraint is constructed using the classical Parzen window (PW) estimate as the target function. The so-called zero-norm of the parameters is used in order to achieve enhanced model sparsity, and it is suggested to minimize an approximate function of the zero-norm. It is shown that under certain condition, the kernel weights of the proposed pdf estimator based on the zero-norm approximation can be updated using the multiplicative nonnegative quadratic programming algorithm. Numerical examples are employed to demonstrate the efficacy of the proposed approach.

Optimal hedging with higher moments

This study proposes a utility-based framework for the determination of optimal hedge ratios (OHRs) that can allow for the impact of higher moments on hedging decisions. We examine the entire hyperbolic absolute risk aversion family of utilities which include quadratic, logarithmic, power, and exponential utility functions. We find that for both moderate and large spot (commodity) exposures, the performance of out-of-sample hedges constructed allowing for nonzero higher moments is better than the performance of the simpler OLS hedge ratio. The picture is, however, not uniform throughout our seven spot commodities as there is one instance (cotton) for which the modeling of higher moments decreases welfare out-of-sample relative to the simpler OLS. We support our empirical findings by a theoretical analysis of optimal hedging decisions and we uncover a novel link between OHRs and the minimax hedge ratio, that is the ratio which minimizes the largest loss of the hedged position. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark

Predictive computed-torque control of a PUMA 560 manipulator robot

This paper describes the integration of constrained predictive control and computed-torque control, and its application on a six degree-of-freedom PUMA 560 manipulator arm. The real-time implementation was based on SIMULINK, with the predictive controller and the computed-torque control law implemented in the C programming language. The constrained predictive controller solved a quadratic programming problem at every sampling interval, which was as short as 10 ms, using a prediction horizon of 150 steps and an 18th order state space model.

Optimal linearization trajectories for tangent linear models

We examine differential equations where nonlinearity is a result of the advection part of the total derivative or the use of quadratic algebraic constraints between state variables (such as the ideal gas law). We show that these types of nonlinearity can be accounted for in the tangent linear model by a suitable choice of the linearization trajectory. Using this optimal linearization trajectory, we show that the tangent linear model can be used to reproduce the exact nonlinear error growth of perturbations for more than 200 days in a quasi-geostrophic model and more than (the equivalent of) 150 days in the Lorenz 96 model. We introduce an iterative method, purely based on tangent linear integrations, that converges to this optimal linearization trajectory. The main conclusion from this article is that this iterative method can be used to account for nonlinearity in estimation problems without using the nonlinear model. We demonstrate this by performing forecast sensitivity experiments in the Lorenz 96 model and show that we are able to estimate analysis increments that improve the two-day forecast using only four backward integrations with the tangent linear model. Copyright © 2011 Royal Meteorological Society

On the universal TRO of a JC*-triple, ideals and tensor products

In a development from material introduced in recent work, we discuss the interconnections between ternary rings of operators (TROs) and right C*-algebras generated by JC*-triples, deducing that every JC*-triple possesses a largest universally reversible ideal, that the universal TRO commutes with appropriate tensor products and establishing a reversibility criterion for type I JW*-triples.

Synergistic effects of mixing cocksfoot and sainfoin on in vitro rumen fermentation: role of condensed tannins

Some bioactive secondary metabolites in forage legumes can cause digestive interactions, so that the rumen fermentation pattern of a mixture of forages can differ from the average values of its components. The objective of this study was to investigate the potential role of condensed tannins (CT) on the synergistic effects between one grass species, cocksfoot, and one CT-containing legume species, sainfoin, on in vitro rumen fermentation characteristics. Cocksfoot and sainfoin in different proportions (in g/kg, 1000:0, 750:250, 500:500, 250:750 and 0:1000) were incubated under anaerobic conditions in culture bottles containing buffered rumen fluid from sheep. Incubations were carried out using artificial saliva with and without polyethylene glycol (PEG), which binds and thus inactivates CT. Rumen fermentation parameters describing the degradation and the fate of the energetic and nitrogenous substrates were measured at 3.5 and 24 h. At the early fermentation stage, when the sainfoin level increased from 0 to 1000 g/kg, the ammonia concentration in the medium quadratically decreased from 3.20 to 0.53 mmol/l in absence of PEG (P<0.01) but not in its presence. This result demonstrates that sainfoin CT decreased the rumen degradation of the proteins in the whole mixture, including the proteins in cocksfoot, rather than just the proteins in sainfoin. Interestingly, the total gas and methane productions were lower in mixtures incubated in absence of PEG than in presence of PEG (P<0.001) while no significant PEG effect was observed on digestibility. At the late fermentation stage, a positive quadratic effect on dry matter digestibility was detected without PEG (P<0.05), indicating a synergistic action of cocksfoot plus sainfoin on plant substrate degradation due to CT. The presence of PEG increased gas production (P<0.001) and NH3-N concentration in the medium (P<0.001). Our results suggest that CT could allow a better utilization of plant substrates in mixtures by the rumen ecosystem by improving the partitioning of degraded substrates toward lower gas losses, and decreasing the protein degradation.

On the approximation of local and linear radiative damping in the middle atmosphere

The validity of approximating radiative heating rates in the middle atmosphere by a local linear relaxation to a reference temperature state (i.e., ‘‘Newtonian cooling’’) is investigated. Using radiative heating rate and temperature output from a chemistry–climate model with realistic spatiotemporal variability and realistic chemical and radiative parameterizations, it is found that a linear regressionmodel can capture more than 80% of the variance in longwave heating rates throughout most of the stratosphere and mesosphere, provided that the damping rate is allowed to vary with height, latitude, and season. The linear model describes departures from the climatological mean, not from radiative equilibrium. Photochemical damping rates in the upper stratosphere are similarly diagnosed. Threeimportant exceptions, however, are found.The approximation of linearity breaks down near the edges of the polar vortices in both hemispheres. This nonlinearity can be well captured by including a quadratic term. The use of a scale-independentdamping rate is not well justified in the lower tropical stratosphere because of the presence of a broad spectrum of vertical scales. The local assumption fails entirely during the breakup of the Antarctic vortex, where large fluctuations in temperature near the top of the vortex influence longwave heating rates within the quiescent region below. These results are relevant for mechanistic modeling studies of the middle atmosphere, particularly those investigating the final Antarctic warming.

Symmetric stability of compressible zonal flows on a generalized equatorial β plane

Sufficient conditions are derived for the linear stability with respect to zonally symmetric perturbations of a steady zonal solution to the nonhydrostatic compressible Euler equations on an equatorial � plane, including a leading order representation of the Coriolis force terms due to the poleward component of the planetary rotation vector. A version of the energy–Casimir method of stability proof is applied: an invariant functional of the Euler equations linearized about the equilibrium zonal flow is found, and positive definiteness of the functional is shown to imply linear stability of the equilibrium. It is shown that an equilibrium is stable if the potential vorticity has the same sign as latitude and the Rayleigh centrifugal stability condition that absolute angular momentum increase toward the equator on surfaces of constant pressure is satisfied. The result generalizes earlier results for hydrostatic and incompressible systems and for systems that do not account for the nontraditional Coriolis force terms. The stability of particular equilibrium zonal velocity, entropy, and density fields is assessed. A notable case in which the effect of the nontraditional Coriolis force is decisive is the instability of an angular momentum profile that decreases away from the equator but is flatter than quadratic in latitude, despite its satisfying both the centrifugal and convective stability conditions.

Nonlinear stability of Euler flows in two-dimensional periodic domains

The non-quadratic conservation laws of the two-dimensional Euler equations are used to show that the gravest modes in a doubly-periodic domain with aspect ratio L = 1 are stable up to translations (or structurally stable) for finite-amplitude disturbances. This extends a previous result based on conservation of energy and enstrophy alone. When L 1, a saturation bound is established for the mode with wavenumber |k| = L −1 (the next-gravest mode), which is linearly unstable. The method is applied to prove nonlinear structural stability of planetary wave two on a rotating sphere.

A Hamiltonian weak-wave model for shallow-water flow

A reduced dynamical model is derived which describes the interaction of weak inertia–gravity waves with nonlinear vortical motion in the context of rotating shallow–water flow. The formal scaling assumptions are (i) that there is a separation in timescales between the vortical motion and the inertia–gravity waves, and (ii) that the divergence is weak compared to the vorticity. The model is Hamiltonian, and possesses conservation laws analogous to those in the shallow–water equations. Unlike the shallow–water equations, the energy invariant is quadratic. Nonlinear stability theorems are derived for this system, and its linear eigenvalue properties are investigated in the context of some simple basic flows.