Multivariable constrained process control via Lyapunov R-functions
Highlights
► An innovative smooth control Lyapunov function is proposed to stabilize multivariable constrained processes. ► Robustness with respect to model uncertainties is discussed. ► Constructive algorithms are given for tuning the free design parameters. ► Multivariable chemical reactions taking place in constrained Continuous Stirred Tank Reactors (CSTRs) are addressed as case studies.
Introduction
Chemical processes are continuously faced with the requirements of becoming safer, more reliable, and more economical in operation. The design of effective chemical process control systems, inherently Multi-Input Multi-Output (MIMO) and nonlinear, needs to be both rigorous and practical [12]. Moreover, the unavoidable presence of physical constraints on the process variables and in the capacity of control actuators not only limit the nominal performance of the controlled system, but can also affect the stability of the overall system. As a consequence, the stabilization of such processes is one of the most attractive research areas for the chemical and control engineering community [11].
Model Predictive Control (MPC) [22], [21], also known as Receding Horizon Control (RHC) [31], [20], can handle both state and control input constraints within an optimal control setting [34], [24]. These approaches, as well as the explicit MPC [7], [19], can be quite computationally demanding. Hence a large literature has been developed for fast computation of sub-optimal (robust) MPC solutions, see [28], [29], [37] among others in recent literature.
In [24], an interesting Lyapunov-based MPC approach has been proposed for the control of an exothermic chemical reaction, taking place in a Continuous Stirred Tank Reactor (CSTR). In particular, a quadratic Control Lyapunov Function (CLF) is used together with a horizon-1 MPC. However, an ellipsoidal set cannot accurately fit the polyhedral state constraints describing the limits on the admissible concentration of the chemical reactant and on the reactor temperature. Therefore a Quadratic CLF (QCLF) cannot guarantee stability over the whole controllable invariant set [2]. On the other hand, the estimate of the controlled invariant state-space region can be enlarged via the synthesis of Polyhedral CLFs (PCLFs) [8], composite-quadratic CLFs [17], [18], smoothed PCLFs [10], Truncated Ellipsoidal (TE) Control Lyapunov Functions [26], [36], and smoothed TE CLFs [1], [5].
This paper considers the control of constrained CSTRs and develops a technique based on the so-called Control Lyapunov R-Functions (CLRFs) for its solution. This approach has been proposed by the authors in [2], [4] and it is here made more general. The proposed CLF has inner level curves that can be made, independently from the external one, as-close-as-desired to any choice of smooth ones. The main contribution of this paper is: (a) the tuning of a free design parameter via constructive algorithms, in order to trade-off between optimality requirements and the size of the guaranteed domain of attraction; additional contributions are: (b) the discussion about robustness to model uncertainties and disturbance rejection; (c) the possibility to handle asymmetric domain of attractions. Unlike switching control strategies [14], within this novel approach both constraints and optimality arguments can be handled by a unique smooth CLF, at least for uncertain linear systems, together with a continuous control [2], [3], [4].
The paper is organized as follows. The state-feedback control of a constrained CSTR is presented as motivating example in the next section. Sections 3 On the use of R-functions, 4 Control Lyapunov R-functions for uncertain linear systems present the main theoretical tools here used, while two constructive procedures for the synthesis of suitable CLRFs are presented in Section 5. Section 6 applies the results to the CSTR case studies. In last section we conclude the paper and outline some future work. The proofs are given in Appendix A.
In denotes the n × n identity matrix. The closed k-level set of a continuous function , i.e. , is denoted by . A set is called -set if it is a convex and compact set including the origin in its interior [9]. denotes .
Section snippets
A constrained uncertain Continuous Stirred Tank Reactor as motivating example
Consider an irreversible, exothermic first-order reaction of the form , taking place in a CSTR. The inlet stream consists of pure A at flow rate F, concentration CA0 and temperature TA0. The dynamic model of the process is of the formwhere CA denotes the concentration of the species A, TR denotes the temperature of the reactor, Q is the heat input to the reactor, V is the volume of the reactor, k0, E, ΔH are, respectively,
On the use of R-functions
The framework of R-functions has been introduced in [32] for geometric applications of logic algebra. The interested reader is referred to [33] for an intensive overview of the theory on the matter. The use of R-functions for state-feedback stabilization has been firstly proposed in [1], [5]. In the following, after recalling the basic notions on R-functions, the useful composition rule introduced in [2] is reported as the basis of our developments.
Definition 1 A function is an R-function w.r.t.
Problem statement
Consider constrained uncertain linear systems of the kindwhere ,subject to the constraintsbeing full column rank, and
The objective is the state-feedback stabilization of (23) via a continuous control, static or horizon-1 optimal, guaranteeing both a large controlled set and locally-optimal performance w.r.t. to the quadratic cost J (5).
Conditions for stabilizability via control Lyapunov R-functions
In this section, the R-composed function V∧,
Constructive algorithms
In this section, two constructive algorithms are presented to build-up a suitable CLRF V∧ that is again the merging of CLFs V1 and V2 satisfying Assumption 1, Assumption 2. The two algorithms represent the trade-off between the volume of the achieved controlled invariant set versus optimality.
Roughly speaking, Algorithm 1 looks for the largest domain of attraction with an a priori-fixed “level of optimality”; Algorithm 2 looks for the best “level of optimality” after a priori-fixing a desired
Simulations
In this section, two cases of study, both modeling controlled chemical reactions in a CSTR, are simulated with both nominal conditions and randomly-taken model uncertainties and external disturbances d.
Five control algorithms are tested: the standard LQR and the linear RHC (with a 100-steps prediction and control horizons) both based on the nominal system; the horizon-1 Lyapunov-based (32) control with three different CLFs, namely the Riccati-optimal QCLF for the nominal system, the robust
Conclusion and future work
This paper considers the state-feedback stabilization of constrained uncertain linear systems, with application to the control of benchmark chemical reactions taking place in a Continuous Stirred Tank Reactor.
It has been shown that the class of control Lyapunov R-functions is particularly useful because, unlike switching control strategies, both robustness and optimality arguments can be addresses by designing a continuous control associate to a unique control Lyapunov function with a large
Acknowledgements
The authors would like to thank prof. Franco Blanchini and prof. Gabriele Pannocchia for useful discussions on Lyapunov-based and predictive control.
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