Prodromos Daoutidis

An anti-windup design for linear systems with input saturation

Abstract An anti-windup design problem is posed and it is shown that all “observer-based” anti-windup modifications solve this problem at least locally. Sufficient conditions for an observer-based anti-windup modification to solve the global version of this problem are presented. A novel observer-based anti-windup design is then proposed such that the anti-windup problem can be interpreted in terms of a reduced-order system. In particular, the proposed anti-windup design induces an invariant subspace for the dynamic behavior of the mismatch between the constrained and unconstrained closed-loops. The dynamics in this invariant subspace are identical to the behavior of the plant with input saturation starting at the origin, stabilized by linear state feedback and driven by the mismatch between the unconstrained input and this input passed through a saturation function. The second part of the paper shows how the original dynamic compensator can be modified, while retaining those dynamic features that produce a desirable closed-loop steady-state response, to ensure that the requisite invariant subspace exists and it is reasonably tuned for input saturation. Two case studies are carried out on systems that have been investigated in the literature.

Nonlinear dynamics and control of process systems with recycle

Abstract Process systems with material and energy recycle are well-known to exhibit complex dynamics and to present significant control challenges, due to the feedback interactions induced by the recycle streams. In this paper, we address the dynamic analysis and control of such process systems. Initially, we establish, through an asymptotic analysis, that (i) small recycle flowrates induce a weak coupling among individual processes, whereas (ii) large recycle flowrates induce a time scale separation, with the dynamics of individual processes evolving in a fast time scale with weak interactions, and the dynamics of the overall system evolving in a slow time scale where these interactions become significant; these slow dynamics is usually nonlinear and of low order. Motivated by this, we present (i) a model reduction methodology for deriving nonlinear low-order models of the slow dynamics induced by large recycle streams, and (ii) a controller design framework consisting of properly coordinated controllers in the fast and the slow time scales. The theoretical results are illustrated in a reaction-separation network with a large recycle compared to the throughput.

Robust control of hyperbolic PDE systems

In the previous chapter, we addressed the control of hyperbolic PDE systems without accounting explicitly for the presence of uncertainty (i.e., presence of mismatch between the model used for controller design and the actual process model) in the design of the controller. This chapter focuses on systems of quasi-linear first-order hyperbolic PDEs with uncertainty for which the manipulated variables and the controlled variables are distributed in space. The objective is to develop a framework for the synthesis of distributed robust controllers that handle explicitly time-varying uncertain variables and unmodeled dynamics. For systems with uncertain variables, the problem of complete elimination of the effect of uncertainty on the output via distributed feedback is initially considered; a necessary and sufficient condition for its solvability, as well as explicit controller synthesis formulas, is derived. Then, a distributed robust controller is derived that guarantees boundedness of the state and achieves asymptotic output tracking with arbitrary degree of asymptotic attenuation of the effect of uncertain variables on the output of the closed-loop system

Nonlinear state feedback control of second-order nonminimum-phase nonlinear systems

The present work addresses the problem of synthesizing nonlinear state feedback controllers for second-order nonminimum-phase nonlinear systems. The concept of a first-order nonlinear all-pass is first introduced. A class of static state feedback control laws is then developed that makes the closed-loop system equivalent, under an appropriate coordinate transformation, to a nonlinear first-order all-pass in series with a linear first-order lag. A particular control law from this class is calculated that results in I.%!?-optimal response. The performance of the proposed methodology in set point tracking is evaluated through numerical simulations in a CSTR example. Major contributions in the area of linear process control in the last decade have established the idea that a controller must explicitly or implicitly generate a process inverse (Garcia and Morari, 1982). When dealing with minimum-phase linear systems, such an inverse is stable and can be used for controller synthesis. When dealing with nonminimum-phase linear systems, an appropriate decomposition of the process model into a part with stable inverse and a part with unstable inverse is necessary and the con- troller must invert only the part with stable inverse. In the context of linear state feedback, the same idea arises in placing closed-loop poles at the left-half plane zeros and at the reflection of the right-half plane zeros. Such a control strategy has been shown to be ZSE-optimal for step changes. In the field of nonlinear process control, the idea of a controller that generates a process inverse is central in general synthesis methods for minimum-phase sys- tems, like the nonlinear IMC structure (Economou et al., 1986; Parrish and Brosilow, 1988), which explic- itly generates a process inverse on-line and the input/ output linearization method (Kravaris and Chung, 1987) which implicitly generates a process inverse. The latter has been shown to lead to ZSE-optimal responses for step changes (Kravaris, 1988). How- ever, the control of nonminimum-phase nonlinear systems in this vein remains a major unmet challenge. In this work, a control law for second-order non- minimum-phase nonlinear systems will be developed, that leads to an ZSE-optimal closed-loop response for changes in the set point. The proposed methodology hopes to serve as a starting point for the development of a more general framework for the control of nonminimum-phase nonlinear systems and to moti- vate further research effort in this area. In Section 2 the characterization of nonminimum- phase behavior for second-order systems will be reviewed following the approach of Bymes and Isidori (1985). In Section 3, a nonlinear analog of the linear first-order all-pass will be introduced. Section 4 will review a standard result for ZSE-optimal state feedback control of linear second-order nonmini- mum-phase systems and will motivate the develop- ment that follows. In Section 5, a class of control laws will be developed that lead to a closed-loop response of a nonlinear first-order all-pass in series with a linear first-order lag. In Section 6, a particular control law from this class will be calculated, that results in ZSE-optimal closed-loop response in the limit as the time constant of the linear lag tends to zero. Finally, Section 7 will illustrate the application of the pro- posed control methodology and evaluate its perform- ance in a CSTR example.

Continuous production of 5-hydroxymethylfurfural from fructose: a design case study

This paper focuses on the design and optimization of a process for the production of 5-hydroxymethylfurfural (HMF), a sugar derived molecule that can be used as a raw material in the production of furan based polymers. We start by reviewing the processes and kinetic mechanisms involved in its synthesis from fructose and then consider a continuous process using solvent extraction based on the semi batch concept proposed by Roman-Leshkov et al. We formulate an optimization problem in order to find the operating conditions that minimize the cost per mol of HMF produced and study the effect of different solvents, fructose prices, capital investment and kinetics on this cost.

Numerical solution of a mass structured cell population balance model in an environment of changing substrate concentration

Cell population balance models are deterministic formulations which describe the dynamics of cell growth and take into account the biological fact that cell properties are distributed among the cells of a population, due to the operation of the cell cycle. Such models, typically consist of a partial integro-differential equation, describing cell growth, and an ordinary integro-differential equation, accounting for substrate consumption. A numerical solution of the mass structured cell population balance in an environment of changing substrate concentration has been developed. The presented method is general. It can be applied for any type of single-cell growth rate expression, equal or unequal cell partitioning at cell division, and constant or changing substrate concentration. It consists of a time-explicit, one-step, finite difference scheme which is characterized by limited requirements in memory and computational time. Simulations were made and conclusions were drawn by applying this numerical method to several different single-cell growth rate expressions. A periodic behavior was observed in the case of linear growth rate, equal partitioning and constant substrate concentration. The periodicity was equal to the average doubling time of the population. In all other cases examined, a state of balanced growth was reached. Unequal partitioning resulted in broader balanced growth distributions which are reached faster. For the specific types of growth rate dependence on the substrate concentration considered, the changing substrate concentration did not affect the balanced growth-normalized distributions, except for the case of linear growth rate and equal partitioning, where the depletion of the substrate destroyed the periodic behavior observed for constant substrate concentration, and forced the system to reach a steady state.

Numerical solution of multi-variable cell population balance models. II. Spectral methods

Abstract Several Galerkin, Tau and Collocation (pseudospectral) approximations have been developed for the solution of the multi-variable cell population balance model in its most general formulation, i.e. for any set of single-cell physiological state functions. Time-explicit methods were found to be more efficient than time-implicit methods for the time integration of the system of ordinary differential equations that results after the spectral approximation in space. The Legendre and Tchebysheff polynomials that were used in Tau algorithms were shown to have significantly worse convergence and stability properties than the Galerkin and collocation algorithms that were applied with sinusoidal trial functions. The collocation method that was implemented with discrete fast Fourier transforms was found to be the most efficient from all the Galerkin and Tau algorithms that were developed. However, the method was inferior to the best finite difference algorithm that was presented in our earlier work.

Structural evaluation of control configurations for multivariable nonlinear processes

Abstract This paper addresses the problem of evaluation of alternative control configurations on the basis of structural characteristics of the process. Relative order is proposed as the main analysis tool for this purpose. Using tools from graph theory, it is shown that generic calculation of relative orders requires only structural information about the process. Relative order is interpreted as a structural measure of the initial sluggishness of the response, as well as a structural analog of dead time, which expresses fundamental structural limitations in the control quality. A matrix of relative orders of input/output pairs is introduced, which leads to a characterization of structural coupling among input and output process variables. On the basis of the above properties, general structural evaluation guidelines are proposed for alternative sets of manipulated inputs and alternative input/output pairs. The application of the theory is illustrated in the case of an evaporation unit, a chemical reactor and a network of heat exchangers.

Dynamic output feedback control of nimimum-phase nonlinear processes

Abstract This paper concerns the synthesis of dynamic output feedback controllers for minimum-phase nonlinear processes. The problem is addressed first for open-loop stable and then for general minimum-phase nonlinear processes, leading to one- and two-degree-of-freedom controllers, respectively. The synthesis of the controllers essentially involves combination of state feedback and state observers. An input/output interpretation of the resulting control structures illustrates the importance of alternative state—space realizations of the process inverse for the controller implementation. Internal stability conditions are derived for the closed-loop system. Simulation studies in a chemical reactor example illustrate the application of the control methodology developed.

Nonlinear control of diffusion-convection-reaction processes

Abstract This work addresses the nonlinear control of a non-isothermal packed-bed reactor, modeled by two quasilinear parabolic partial differential equations (PDEs). Initially, nonlinear Galerkins method and the concept of approximate inertial manifold are used to derive a minimal-order ordinary differential equation (ODE) model, which accurately describes the dynamics of the process. This model is then used for the synthesis of a nonlinear finite-dimensional controller that guarantees closed-loop stability and enforces output tracking. Computer simulations are used to evaluate the performance of the controller.