(65e) Enhanced Stability Regions For Model Predictive Control Of Nonlinear Process Systems

The operation and control of chemical processes often encounters constraints that arise out of physical limitations on the control actuators. The constraints, if not accounted for in the control design, can cause performance deterioration or even instability in the closed-loop system. Successful operation and control of a process therefore requires use of control designs that explicitly take these constraints into account. These considerations have motivated modifications in existing control approaches as well as fostered the development of controllers that explicitly account for the presence of constraints via Lyapunov-based and model-predictive control designs (see, for example, [1,2] and [3,4] for excellent reviews.

The presence of constraints limits the set of initial conditions starting from where closed-loop stability is achievable (the so-called null controllable region). Furthermore, a given controller can typically achieve stabilization only from a subset of the null controllable region. A desirable characteristic in implementing control with constraints is the ability to characterize the set of initial conditions starting from where closed-loop stability can be guaranteed in the presence of constraints, and to expand the set of stabilizable initial conditions. A comparison of the estimate of the stability region of a given controller with the null controllable region can yield a meaningful measure of how well the control effort is being utilized by the control law to achieve stabilization.

Given that process dynamics are sometimes identified or approximated by linear systems, extensive research work has focussed on designing and analyzing controllers that utilize a linear process description in computing the control action. For linear systems, characterizing the null controllable region, while being difficult, is a tractable problem and several research efforts have focussed on characterizing the null controllable region for linear systems. Furthermore, several controller designs have been proposed that allow the possibility of turning any given subset} of the null controllable region into the stability region of a proposed controller design. In model predictive approaches, which allows implementation of stability constraints demanding the state to go to some invariant neighborhood of the origin (or the origin itself), feasibility of the stability constraint can be guaranteed for infinite horizons (essentially demanding the state to go to the origin in infinite time) thereby enabling stabilization of all initial conditions in the null controllable region, albeit with the number of decision variables in the optimization problem going to infinity. When guaranteeing feasibility from a subset of the null controllable region, the results use the approach of quantifying the maximum over the minimum possible time for every point in the given set to be driven to the origin, which can be utilized in choosing the controller parameters (such as the horizon). Recently in [5] a characterization of the null controllable region has been developed where the boundary of the null controllable region can be described in terms of the constraints on the manipulated input. For some classes of linear systems (systems with real eigenvalues, low order systems with complex eigenvalues), the characterization yields explicit expressions for the boundary of the null controllable regions parameterized by the magnitude of input constraints. The work in [5] provides the characterization of the null controllable region, but does not address the problem of determining the control law that can stabilize all initial conditions in the null controllable region. We will show how this characterization can be utilized within the model predictive control framework to achieve stabilization for all initial conditions in the null controllable region, without resorting to practically infinite horizons.

For nonlinear systems, the problem of explicitly characterizing the null controllable region remains intractable. Research efforts have however, focussed on control designs that allow for explicitly characterizing the stability regions for specific controllers. Specifically in the Lyapunov-based nonlinear control designs of [6] the set of initial conditions (denoted by ) starting from where negative definiteness of the Lyapunov function derivative for the closed-loop system is guaranteed can be computed and estimates of the stability region (denoted by Ω) can be computed as the largest invariant sets within π (typically constructed using level sets of the Lyapunov function). In the model predictive control framework, several designs have been proposed that guarantee closed-loop stability contingent on the assumption of initial feasibility of the optimization problem, essentially based on the idea of driving the state to some invariant neighborhood by the end of the horizon (and therefore requiring to assume initial feasibility). More recently, in [7], Lyapunov-based and model predictive approaches were utilized within a switched controller framework to enable implementation of existing model predictive controllers with guaranteed stability region. In [8,9] (see [4] for further results and references), the stability properties of auxiliary Lyapunov-based controllers of [6] were utilized in formulating constraints in the optimization problem (specifically, requiring the Lyapunov function to decrease during the first time step). By requiring the Lyapunov function value to decrease, the predictive controllers of [8,9] at best mimic the stability region Ω of the auxiliary control designs. Specifically, for initial conditions outside Ω but inside Π, the opportunity to take control action such that successive decays in the Lyapunov function value are achievable, is lost. The auxiliary control designs of [6], do not possess a mechanism to explicitly handle such initial conditions (which may be substantial, owing to possible conservativeness in constructing invariant sets within π). The constraint handling capabilities of the predictive control approach, however, can be better utilized to expand on the set of initial conditions starting from where closed-loop stability is achieved.

Motivated by these considerations, this work considers the problem of enhancing the set of stabilizable initial conditions for nonlinear process systems subject to input constraints. The key idea is to utilize the constraints handling capabilities of the predictive control approach to fully exploit Lyapunov-based analysis tools in a way that expands the set of initial conditions that can be stabilized. To clearly illustrate the main idea behind the approach, we first consider linear systems for which characterizations of the null controllable region are available, but not control laws that can stabilize the entire null controllable region. A predictive controller is designed that utilizes this characterization to stabilize every initial condition in the null controllable region (not just subsets of the null controllable region) without resorting to (practically) infinite horizon. For nonlinear systems, where explicit characterizations of the null controllable region remain intractable, the set of initial conditions for which a decay in the Lyapunov function value is achievable (subject to constraints, and independent of the control law) are first characterized. Such characterizations are then utilized to formulate appropriate constraints in the predictive controller that enhance the set of stabilizable initial conditions compared to Lyapunov-based nonlinear control designs. The proposed method is illustrated using a chemical reactor example and the robustness with respect to parametric uncertainty and disturbances demonstrated via application to a styrene polymerization process.

References

[1] N. H. El-Farra and P. D. Christofides, ``Integrating robustness, optimality, and constraints in control of nonlinear processes,'' Chem. Eng. Sci., vol. 56, pp. 1841-1868, 2001.

[2] K. R. Muske and J. B. Rawlings, ``Model predictive control with linear-models,'' AIChE J., vol. 39, pp. 262-287, 1993.

[3] D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert, ``Constrained model predictive control: Stability and optimality,'' Automatica, vol. 36, pp. 789-814, 2000.

[4] P. D. Christofides and N. H. El-Farra, Control of Nonlinear and Hybrid Process Systems: Designs for Uncertainty, Constraints and Time-Delays. Berlin, Germany: Springer-Verlag, 2005.

[5] T. Hu, Z. Lin, and L. Qiu, ``An explicit description of null controllable regions of linear systems with saturating actuators,'' Sys. & Contr. Lett., vol. 47, pp. 65-78, 2002.

[6] N. H. El-Farra and P. D. Christofides, ``Bounded robust control of constrained multivariable nonlinear processes,'' Chem. Eng. Sci., vol. 58, pp. 3025-3047, 2003.

[7] N. H. El-Farra, P. Mhaskar, and P. D. Christofides, ``Hybrid predictive control of nonlinear systems: Method and applications to chemical processes,'' Int. J. Rob. & Non. Contr., vol. 4, pp. 199-225, 2004.

[8] P. Mhaskar, N. H. El-Farra, and P. D. Christofides, ``Predictive control of switched nonlinear systems with scheduled mode transitions,'' IEEE Trans. Automat. Contr., vol. 50, pp. 1670-1680, 2005.

[9] P. Mhaskar, N. H. El-Farra, and P. D. Christofides, ``Stabilization of nonlinear systems with state and control constraints using Lyapunov-based predictive control,'' Syst. & Contr. Lett., vol. 55, pp. 650-659, 2006.