(504c) Nonlinear Observer Design for Process Monitoring in the Presence of Fast and Slow Modes | AIChE

(504c) Nonlinear Observer Design for Process Monitoring in the Presence of Fast and Slow Modes

Authors 

Duan, Z. - Presenter, Texas A&M University
Kravaris, C., Texas A&M University
Sensors are widely used in chemical processes for safe operation and product quality monitoring purposes. However, in many practical applications, only some of the variables critical for quality control are available for measurement. This is particularly significant in bioreactor systems, where some important species concentrations are either unavailable or only available with significant delay and high cost, due to the lack of appropriate online estimating devices or the high price of sensors. In these circumstances, state observers play an important role in process monitoring.

Processes with multi-time-scale dynamics are common in chemical engineering. In fact, many bioreactor systems involve both fast and slow dynamics owing to the presence of multiple microbial cultures.

These systems could pose challenges to process simulation due to stiffness issues, and are thus computationally costly for optimization. To overcome this, one option is to do model reduction and properly eliminate the fast modes.[1]

It is also a challenging problem to design observers for systems showing spectral gap. On one hand, there may be ill conditioning problems, and the selection and placement of observer poles could have a critical effect. On the other hand, high gain observers may perform well, but could result in severe peaking problems and are likely to fail in the presence of measurement noises. Many researchers have studied observer design problems for linear multi-time-scale systems. In recent work by Yoo at al.[2], the authors decomposed a singular perturbed system into fast and slow subsystems through a series of linear transformations, and placed the observer poles separately for the two subsystems. The overall observer shows less peaking problem and converges faster than the traditional observer in their numerical example. For nonlinear systems, nonlinear observers generally show better stability and performance properties, as the linear observer is designed only for local validity. But in the presence of spectral gap, the design problem is even harder than in the linear case. Kazantzis et al.[3] has proposed an approach to design observers for slow states in singular perturbed systems, by first designing observers for the corresponding reduced model and then applying it back to the detailed model. Inspired by these works, we will consider more general nonlinear multi-time-scale systems and propose two approaches to design nonlinear observers for them.

In the presentation, we will explore two alternative design approaches. The first approach is to introduce a systematic way to properly select the design parameters for the observer PDEs [4] for the detailed model. The first step in this approach will be to design a linear observer for the linearized model and use the matrices of the resulting error dynamics as the design parameters for the nonlinear observer PDE. The second approach is to combine observer design with model reduction using the invariant manifold method [1]. Similarly to the work of Kazantzis et al. [3], a nonlinear observer designed for reduced model will be applied on the detailed model. We will show that this observer derived from the reduced system will also converge for the original detailed model.

The theoretical methods will be applied to two bioreactor case studies. In the first case study, the aerobic digestion system involves three states: the concentrations of biomass and substrate, and the dissolved oxygen tension (DOT). With the measurement of biomass concentration and DOT, we will estimate the substrate concentration. In this system, DOT dynamics is generally much faster than the other two. The second approach would then be based on a reduced model without DOT.

The second case study is for an anaerobic digestion system, with the concentrations of substrate, VFA, acidogens and methanogens to be the four states. By measuring the total biomass concentration and the VFA concentration, the unmeasured three states are to be estimated. Stamatelatou et al.[5] has studied the properties of this system and shown that the substrate concentration is associated with much faster dynamics than the other three. In this case, the reduced model derived by Stamatelatou et al.[5] will be used for the second approach. We will also compare the two methods and evaluate their performance on the case studies.

References

[1] Kazantzis, N., Kravaris, C. and Syrou, L., 2010. A new model reduction method for nonlinear dynamical systems. Nonlinear Dynamics, 59(1-2), p.183.

[2] Yoo, H. and Gajic, Z., 2018. New designs of linear observers and observer-based controllers for singularly perturbed linear systems. IEEE Transactions on Automatic Control, 63(11), pp.3904-3911.

[3] Kazantzis, N., Huynh, N. and Wright, R.A., 2005. Nonlinear observer design for the slow states of a singularly perturbed system. Computers & Chemical Engineering, 29(4), pp.797-806.

[4] Kazantzis, N. and Kravaris, C., 1998. Nonlinear observer design using Lyapunov’s auxiliary theorem. Systems & Control Letters, 34(5), pp.241-247.

[5] Stamatelatou, K., Syrou, L., Kravaris, C. and Lyberatos, G., 2009. An invariant manifold approach for CSTR model reduction in the presence of multi-step biochemical reaction schemes. Application to anaerobic digestion. Chemical Engineering Journal, 150(2-3), pp.462-475.