Feedback passivation plus tracking-error-based multivariable control for a class of free-radical polymerisation reactors

ABSTRACT

This paper proposes a tracking-error-based multivariable control to stabilise a nonlinear system at the desired trajectory (including the open-loop unstable equilibrium manifold). The control approach is developed on the basis of feedback passivation and then applied to stabilise globally exponentially a class of free-radical polymerisation reactors. More precisely, under certain conditions the system dynamics can be rendered strictly input/output passive through the use of an appropriate input coordinate transformation. A canonical form related to the so-called port-Hamiltonian representation of passive system is consequently derived and provides physical interpretations such as dissipative/non-dissipative term and supply rate. A feedback law based on tracking-error is then designed for the global exponential stabilisation at a reference trajectory passing through the desired set-point. The theoretical developments are illustrated for polystyrene production in a continuous stirred tank reactor. Numerical simulations show that the system trajectory converges globally exponentially to the reference trajectory despite effects of disturbance.

KEYWORDS:

• Feedback passivation
• passivity-based control
• tracking-error
• canonical form
• multivariable system
• polymerisation

Acknowledgments

The authors are grateful to the University of Malaya and the Ministry of Higher Education in Malaysia for supporting this collaborative work under Fundamental Research Grant Scheme (FRGS) under grant number FP064-2015A.

The authors also acknowledge the Viet Nam National Foundation for Science and Technology Development (NAFOSTED) for all the support under grant number 104.99-2018.40.

The authors wish to thank Professor Romeo Ortega (Laboratoire des Signaux et Systèmes, CNRS UMR 8506, Supélec, France) and the reviewers for helpful discussions and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. A passive input–output system is defined on the basis of dissipativity property with respect to an appropriate supply rate. In other words, a passive system is necessarily dissipative, but not in the opposite direction (Bao & Lee, 2007; Brogliato, Lozano, Maschke, & Egeland, 2007; Van der Schaft, 2000).

2. We refer the readers to Hoang, Dochain, Couenne, and Le Gorrec (2017), Ortega, Van der Schaft, Castanos, and Astolfi, (2008), and references therein, for further information on passive and cyclo-passive concepts.

3. For the sake of simplicity, the notation x can be used instead of x(t). Discussions on the (strict) separability condition can be found in Hoang et al. (2017) and Favache, Dochain, and Winkin (2011).

4. Such decomposition is made arbitrarily as long as the requirements are met. Note also that in Hudon, Hoang, García-Sandoval, and Dochain (2015) and Guay and Hudon (2016), the authors proposed a geometric decomposition technique, based on the Hodge decomposition theorem, to re-express a given vector field f(x) into a potential-driven form. More precisely, this allows to encode the divergence of the given vector field into its exact and anti-exact components, and into its co-exact and anti-coexact components.

5. Even if it is difficult to identify sufficiently m components of f_nd (x), we can arbitrarily choose k elements with 1 ⩽ k < m such that $f_{nd}\left(x\right)={\sum }_{i=1}^k{f}_{nd,i}\left(x\right)$ while other components (i.e. n components) are equal to zero.

6. We refer the readers to Bao and Lee (2007) and Byrnes, Isidori, and Willems (1991) for more details on the necessary and sufficient condition (including the relative degree and weakly minimum phase behaviour of the system) of output feedback passivation.

7. Note also that H(e) = H(x − x_d ) and thus, $\frac{\partial H\left(e\right)}{\partial e}=\frac{1}{2}\left[\frac{\partial H\left(x\right)}{\partial x}-\frac{\partial H\left(x_d\right)}{\partial x_d}\right]$.

8. The dynamics of the error state vector e are naturally formatted in a port-Hamiltonian structure.

Additional information

Funding

Fundamental Research Grant Scheme (FRGS) [grant number FP064-2015A]; Viet Nam National Foundation for Science and Technology Development (NAFOSTED) [grant number 104.99-2018.40].