Stabilization of an unstable tubular reactor by nonlinear passive output feedback control

Highlights

• The stabilization of an unstable multi–jacket tubular reactor is addressed by the construction of a PDE model-based MIMO decentralized output feedback controller.

• The jacket partition and per jacket sensor locations are design degrees of freedom.

• The MIMO output controller is a pointwise observer-based passive state feedback controller derived from notions and tools from chemical reactor engineering and passivity-based control.

• Closed-loop stability criteria in terms of sensor locations and control gains are derived together with tuning guidelines and efficient late lumping on-line implementation that precludes unduly on-line computational load.

• The control is tested with an example through simulation and favorably compared with adaptive controllers.

Abstract

The multiple–input multiple–output (MIMO) output feedback (OF) control problem of an exothermic multi-jacket tubular open-loop unstable reactor is addressed. Over its axial length, the reactor has several equally sized cooling jackets. The controller must adjust the jacket temperatures on the basis of per jacket temperature measurements so that the closed-loop system is robustly stable. The problem is solved within a constructive framework, by combining notions and tools from chemical reactor engineering and partial differential equations (PDEs) control systems theory. The result is a MIMO nonlinear OF dynamic control design with (i) a decentralized MIMO passive state feedback (SF) controller implemented with a pointwise observer (PWO), (ii) closed-loop stability conditions in terms of sensor set and control gains, and (iii) efficient late lumping-based on-line implementation. The design is put in perspective with industrial PI and inventory control, and applied to a representative example through numerical simulation with favorable comparison against adaptive controllers.

Introduction

Exothermic jacketed tubular reactors are important spatially distributed units in chemical industry with complex nonlinear dynamics, underlain by the interplay of stabilizing mass and heat convection–dispersion transport with destabilizing heat generation by chemical reaction [1], including multi-stability, limit cycling, and bifurcation phenomena [2], [3].

Industrial tubular reactors are, by far, controlled with PI control [4], [5], [6], [7] driven by a temperature sensor at the sensitive (with largest temperature slope change) location [8], [9], [10]. PI control has low cost, reliable functioning, and acceptance by practitioners, but its implementation requires experience, insight, and testing dosage [11], [12]. Recently, model predictive control for tubular reactors has been proposed [13], but its advantages against PI control in terms of cost, reliability, on-line computational load, and acceptance by practitioners are not clear [12]. These considerations motivate the development of application-oriented advanced partial differential equation (PDE) model-based output feedback (OF) control designs for tubular reactors in the light of systematicity and scalability against development-maintenance costs.

Advanced output feedback controllers for open-loop stable and unstable tubular reactors have been designed with two rather disconnected approaches [14], [15]: (i) early lumping, with spatial discretization techniques followed by modal geometric, Lyapunov redesign, model predictive, LQR, economic model predictive [16], [17], [18], [19], [20], [21] as well as backstepping [22], and passivity based [9] ordinary differential equations (ODEs)-based control, with direct on-line implementation, and (ii) late lumping, with PDE model-based flatness, backstepping [15], [23], [24], [25], [26], discontinuous feedback [27], and adaptive [28], [29], [30] control, followed by spatial discretization for on-line implementation. In the context of thermodynamics-based passive control design, with exploitation of the reactor physics, it has been established that: (i) finite dimensional controllers can stabilize an infinite dimensional system [31], [32], and (ii) distributed stabilizing controllers can be designed with the Lyapunov’s direct method [33]. Motivated by analogous earlylumping-based control approaches [34], [35], passive state feedback (SF) controllers for single-state transport-reaction PDEs have been designed. The number of sensor and their locations for state estimation and control of tubular reactors have been drawn with early or late lumping-based Kalman and Gramian local observability measures [36], [37], nonlinear optimal control [17], [21], [38] and data-driven optimization [10].

The early lumping approach yields a ready to implement control according to the mature control theory for ODE systems, but most designs lack reliability assurance via PDE model-based closed-loop stability [39]. At the cost of more mathematical complexity, the late lumping approach retains throughout control design the reactor physics inherent to the PDE model, and formal closed-loop stability conditions. However, the early and late lumping approaches overlook a key applicability subject: the extent of spatial lumping in the light of on-line computational load, which reflects number of ODEs and their ill-conditioning [40].

Eventhough the preceding early and late lumping approach-based studies contain valuable and complementary results and insight, there are still open problems in both approaches, among them are: (i) the multiple-input multiple-output (MIMO) OF control of multi-jacket reactors, (ii) efficient lumping for preclusion of unduly on-line computational load, and (iii) the formal connection between early and late lumping approaches. These considerations motivate the scope of the present study: the PDE model-based MIMO OF control design for tubular reactors with exploitation of the system characteristics and supplementation with an efficient late lumping.

In this study, the problem of regulating with MIMO OF control a multi–jacket exothermic tubular (possibly open-loop unstable) reactor is addressed, with the jacket partition and sensor set as key process-control design degrees of freedom. The problem is solved within a constructive framework [41], by combining notions and tools from chemical reactor engineering [1], [42], modal analysis [43], and SF passive control [34], [35] in the light of industrial PI [12] and inventory control [11]. First (in Section 2), the control problem is technically stated, and interpreted (in Section 3) as a favorable transport stabilizing versus reaction destabilizing compromise. Then (in Section 4), the SF controller is designed in two steps: (i) firstly, a stabilizing MIMO output temperature regulatory SF control, with structure that resembles the one of an industrial PI control, is constructed, and (ii) secondly, such MIMO SF controller is supplemented with a passive SF component for behavior improvement, with structure that resembles the one of an industrial calorimetric controller [44], [45]. The resulting SF controller is implemented (in Section 5) with a pointwise observer (PWO), leading to a MIMO OF control design with: (i) a closed-loop stability condition in terms of the sensor set and control gains, (ii) an application-oriented control gain-measurement tuning procedure, and (iii) reduced on-line computational load via efficient late lumping. The proposed control design is (in Section 6) applied to a case example through numerical simulation, with favorable comparison against adaptive control. The proposed approach extends the previous results on passive control for transport-reaction processes presented in in [34], [35], [46] for the MIMO case.