Time-minimal set point transition for nonlinear SISO systems under different constraints

Abstract

Set point transition of nonlinear plants plays an important role in many applications where dynamic process management has to be considered. This transition should be rapid – as operation in the new set point increases the economical benefit – but in compliance with all safety regulations. We present a feedforward approach for a time-minimal set point transition of single-input, single-output nonlinear plants with respect to input, state and output constraints. The conceptual idea is based on the design of an optimization problem utilizing a coordinate change of the plant and a special setup function for the output trajectory. This allows the simultaneous planning of the trajectory and determination of the corresponding control signal. The formulation of the set point transition as an optimization problem provides a flexible design with respect to the integration of inequality constraints. Additional flexibility is provided by the type of setup functions, which permit any adjustment of the output trajectory between the set points. In contrast to other works, we focus on the stationarity of the system output, which allows a faster transition compared to the requirement to reach a steady state of all states. Finally, we present an example from the field of process systems engineering to demonstrate the applicability of the proposed methodology.

Introduction

This contribution considers the classical control engineering task of time-minimal set point transition. This entails the question of how to design a controller that brings a plant from one stationary set point ${\hat{y}}_{\mathrm{0}}$ to a final value ${\hat{y}}_T$ as fast as possible. This type of problem occurs in many applications, such as robotics, aerospace or process systems engineering. An overview about different time-optimal state transition tasks can be found in the introduction of Devasia (2011). Frequently, the integration of different types of constraints during the transition process, to satisfy safety regulations or proper operation conditions, is of particular importance. Especially, if time-minimal set point transitions are required, bang–bang solutions should be avoided, as this can lead to increased wear of equipment.

To illustrate our approach, we focus on a process systems engineering perspective of the set point transition scenario. We assume a hierarchical structure of the process operation management, where an upper control layer (e.g. a Real-Time Optimization Layer) specifies the new set points to a controller in a layer below, see Engell, 2007, Seborg et al., 2016 and Skogestad (2004). Fig. 1 illustrates this hierarchical control structure.

Due to changing process conditions it might be efficient to adapt these set points to ensure an overall objective, e.g. profit maximization. The exact values for such set points depend on the specification of the operational objective, which is not discussed in detail in this contribution.

A suitable control structure has to generate a manipulating signal in a way, that the system moves from an initial output level to a final one. In this context, different constraints have to be satisfied to ensure a feasible and safe transition.

Basically, one can distinguish between optimal output and state transition, see Devasia, 2011, Devasia, 2012 and Perez and Devasia (2003). While the former refers to the plant output $y$ the later refers to the state coordinates which should move from an initial value ${\hat{x}}_{\mathrm{0}}$ to a final value. Both problems are related to each other such that state transition techniques can be used to achieve a controlled output transition (Devasia, 2011).

An additional way of classifying the transition approaches depends on whether optimization techniques are used to determine the solution or not. For instance, the transition problem in Graichen and Zeitz (2008) is explained by a two-point boundary value problem. However, representatives of the first group are Model Predictive Controllers (MPC), where there exist many subtypes. The main idea of any MPC is that for a given reference signal, the MPC will compute a control signal by repeatedly solving an optimal control problem. In this context, the future plant behavior is predicted in every control step to ensure different kinds of constraints. For further information regarding MPC see Grüne and Pannek, 2013, Matschek et al., 2018 and Mayne, Rawlings, Rao, and Scokaert (2000).

An important aspect is the reference signal, which is classically determined offline in advance. For the definition of the reference signal, it is useful to distinguish between trajectory tracking and path following, see Faulwasser (2013).

The previously described controllers are usually applied online, i.e. in active operation. An alternative option is the use of feedforward control strategies where the manipulation signal is determined offline in advance. Here, the concept of differential flatness is an important aspect which is widely used in the literature for system transformation and determination of the input signal based on certain ansatz functions, see Guay, 2005, Oldenburg and Marquardt, 2002 and Varigonda, Georgiou, and Daoutidis (2004). For a general introduction to this, the reader is referred to Fliess, Levine, Martin, and Rouchon (1995) and Lévine (2009). A wide variety of theoretical concepts exists for feedforward design to achieve a set point tracking, see e.g. Devasia et al., 1996, Graichen, 2006 and Piazzi and Visioli (2001a).

One of the results of a feedforward control is the reference trajectory defining the set point transition, see Springer, Gattringer, and Staufer (2013). It can be represented by a time-dependent setup function. In the literature, various types of trajectory references can be distinguished. While in Graichen and Zeitz (2008) or Piazzi and Visioli (2001b) polynomial or cosine-series are used as reference, in Fliess, Mounier, Rouchon, and Rudolph (1998) Gevrey functions and in Treuer, Weissbach, and Hagenmeyer (2011) splines are applied to avoid oscillations during set point changes.

As mentioned above, the integration of different constraints during the transition process is of particular importance. In Faiz, Agrawal, and Murray (2001) path and actuator constraints are considered for a differentially flat system during the tracking. For the case of non-flat systems, in Graichen and Zeitz (2008) a method is proposed to satisfy input and output constraints. The conceptual idea lies in the implementation of saturation functions on a plant in input–output normal form. This strategy was also successfully implemented on a discretized tubular reactor model with input constraints, see Wieland, Meurer, Graichen, and Zeitz (2006). Furthermore, in Käpernick and Graichen (2013) this technique has been extended for the application to optimal control problems for a class of nonlinear systems. The result is an unconstrained optimization problem based on a transformed system dynamic.

The method described in the present article can be classified as an optimization-based offline method for output transition, where the output trajectory is planned during optimization.

The main contribution of this work is the design of an optimization problem to achieve a time-minimal set point transition using a coordinate change and a parameterized setup function similar to Graichen and Zeitz (2008). However, there are two main differences to that work. First, a novel type of setup function is used that allows to build up an optimization problem to guarantee a time-minimal set point transition of the set point. Second, we do not need saturation functions in our formulation to include inequality constraints. This provides more flexibility when integrating additional constraints.

The remainder of this paper is structured as follows. In Section 2 some general information about the system are presented as well as the transition problem is defined. At the beginning of Section 3 we present some theoretical aspects, followed by a brief description of a classical solution strategy. In Section 4, we propose an optimization-based approach that addresses some of the challenges that the classical approach entails. A numerical example to demonstrate the time-minimal set point transition is presented in Section 5, which is followed in Section 6 by the conclusion.