# (585d) Fast Estimation of Plant Steady State, with Application to Static RTO

#### AIChE Annual Meeting

#### 2016

#### 2016 AIChE Annual Meeting

#### Computing and Systems Technology Division

#### Estimation and Control of Uncertain Systems

#### Wednesday, November 16, 2016 - 4:09pm to 4:27pm

**Fast Estimation of Plant Steady State, with Application to Static RTO**

**D. Rodrigues, M. Amrhein, J. Billeter and D. Bonvin**

Laboratoire dâ??Automatique, Ecole Polytechnique FÃ©dÃ©rale

de Lausanne

EPFL â?? Station 9, 1015 Lausanne, Switzerland

In the operation of continuous processes, many tasks

require the knowledge of plant steady state at various operating points. This

is for example the case in the context of kinetic modeling, response surface

modeling and real-time optimization. If the computational techniques are in

principle straightforward, the time needed to reach steady state represents the

main limiting factor. This work proposes a novel way of speeding up the *estimation *of plant steady state through

the use of feedback control and rate estimation. It must be emphasized here

that rate estimation requires only some structural information of the plant and

no rate model. Generally speaking, the context of the present investigation

corresponds to industrial practice, where there is significant plant-model

mismatch, which typically calls for the use of measurements to feed data-driven

techniques.

We will illustrate the fast estimation of plant steady state in

the context of static optimization of continuous reactors. Real-time optimization (RTO) is typically implemented via

some iterative scheme that uses steady-state plant measurements. The cost and

constraints of the optimization problem are functions of the input and output steady-state

values. At the k^{th} iteration, the constant inputs **u**_{k}

are usually applied to the plant in open loop and,

once steady state is reached, the outputs **yÌ?*** _{k}* are measured and the cost and constraint values are evaluated. However, depending

on the dominant time constant of the plant, the time necessary to reach steady

state may be rather long. Hence, it would be useful to be able to speed up

convergence to steady state, or at least speed up the

*estimation*of plant steady state. This will be done in this work through

combination of feedback control (to speed up a specific part of the plant) and

rate estimation (to estimate the steady-state values of the remaining part of

the plant so that the cost and constraint functions can be evaluated).

For this,

let us consider a time-invariant dynamic system with the inputs **u** and the

state variables **x** and **z** that can be described by the differential

equations:

**xÌ?**(*t*) = **f**(**x**(*t*),**u**(*t*)) **x**(0)

= **x**_{0}

**zÌ?**(*t*) = **h**(**x**(*t*),**u**(*t*)) â?? **Î©**(**x**(*t*),**u**(*t*)) **z**(*t*) **z**(0) = **z**_{0}

The particular

structure of this dynamic system can be exploited when the *slow* and *fast* states **z**

and **x** are associated with slow and fast dynamics, respectively, and the slow states do not affect the fast states**.**

The idea is to use feedback control to speed up the convergence of the fast states

**x** to **xÌ?** and the inputs **u** to **uÌ?**, provided that the

states **x** are accessible, and then compute the steady-state values **zÌ?** of the slow

states as:

**zÌ?**

= **Î©**^{-1}(**xÌ?**,**uÌ?**) **h**(**xÌ?**,**uÌ?**)

This

computation relies on nonparametric estimation of the rates **h**(**xÌ?**,**uÌ?**)

and **Î©**(**xÌ?**,**uÌ?**) using measurements and structural information

of the plant. Note that **zÌ?** can be estimated long before the state variables **z** converge to their steady-state

values.

RTO is implemented via a two-layer

approach. In the inner layer, feedback control is used to rapidly drive **x **to the setpoints **x*** _{sp}* by manipulating

the inputs

**u**and rate estimation is

used to compute

**zÌ?**from

**xÌ?**

and

**uÌ?**. In the outer (optimization) layer, the RTO algorithm

computes optimal values for the setpoints

**x**

*. The inner layer is*

_{sp}described in more detail next for chemical reactors:

â?¢

The objective of the

control scheme is to drive the fast states** x**, typically some reactant concentrations,

to their constant setpoints in the shortest possible time after a step change

in the setpoints **x*** _{sp}*. This time is shorter than

the time needed by the open-loop plant to reach steady state after a step change

in the inputs

**u**, typically inlet flowrates. Multivariable

control is implemented via input-output feedback linearization [1], pole

placement or optimal control and typically involves the measurement or

estimation of

**x**. Note that the gains

to control the fast states (reactant concentrations)

**x**are typically lower than those necessary to control both the fast

(reactant) concentrations

**x**and the slow (product) concentrations

**z**, thereby making the control

scheme with

**x**less sensitive to

measurement noise. Also note that the

separation into slow and fast states assumes that the rates

**f**(

**x**,

**u**)

are independent of the slow (product) concentrations

**z**.

â?¢

The rate estimation

relies on the knowledge of the stoichiometry and the inlet concentrations, and

on measurements of flowrates, reactor temperature, volume, and some of the

concentrations *at steady state*. The

rates are computed through numerical differentiation of the reaction variants

using a Savitzky-Golay filter [1]. These reaction variants are easily computed

from measured concentrations [2]. It will also be shown that the proposed

Savitzky-Golay filter is the optimal convolution filter for rate estimation.

A simulated homogeneous CSTR [3] is used to

illustrate the implementation of â??fast static RTOâ? and address the following questions:

(i) how to best combine feedback control and rate estimation, (ii) which controlled

and manipulated variables to choose, (iii) how to eliminate the slow states in the

objective and constraint functions, and (iv) how to deal with measurement noise

in this measurement-based RTO algorithm.

[1]

D. Rodrigues, J. Billeter, and D. Bonvin. Control of reaction systems via rate

estimation and feedback linearization. In *PSE-12/ESCAPE-25*, Copenhagen,

2015.

[2] D. Rodrigues, S. Srinivasan, J. Billeter, and D.

Bonvin. Variant and invariant states for chemical reaction systems, *Comput. Chem. Engng*, 73:23-33, 2015.

[3] B. Srinivasan, L. T. Biegler, and D. Bonvin. Tracking

the necessary conditions of optimality with changing set of active constraints

using a barrier-penalty function. *Comput. Chem. Engng*, 32(3):572â??579,

2008.