(654e) Robust Model Predictive Control of Nonlinear Processes with State, Input and Rate Constraints

Control systems often need to deal with uncertainty and constraints on the state and input variables. Uncertainty can arise due to a variety of sources such as modeling errors and external disturbances and, if not accounted for in the controller design, can cause loss of performance or even instability. Inherent physical limitations on the capacity of the control actuators (such as valves or pumps) or the necessity to keep state variables within certain bounds (e.g., to keep the temperature below a certain value due to safety considerations) give rise to hard constraints (that need to be satisfied at all times) on the state and input variables and on the rate of change of the input variables. Performance considerations can also be expressed as desirable bounds on the state variables (such as keeping a product concentration above a prescribed value) and on the rate of change of manipulated variables (to minimize, for instance, valve wear and tear). Such constraints in may typically be relaxed (preferably for short periods of times), and treated as soft constraints. In either cases, by limiting the control action available or enforceable, constraints impact the performance of the closed-loop system as well as stabilizability of a given initial condition. These considerations make it necessary to appropriately account for the presence of constraints and uncertainty in the controller design.

Currently, model predictive control (MPC), also known as receding horizon control (RHC), is one of the few control methods for handling state and input constraints within an optimal control setting and has been the subject of numerous research studies that have investigated the stability properties of MPC (e.g., see [1] for extensive surveys of various MPC formulations). The problem of analysis and design of predictive controllers for uncertain linear processes has been extensively investigated (see [1] for surveys of results in this area). Attention has also been focused on the problem of state constraints satisfaction and has typically been analyzed within the soft constraints framework, i.e., with the understanding that state constraints may be relaxed. In the minimum time approach, the state constraints are relaxed up-to some time, and set as hard constraints thereafter. In other approaches, they are typically relaxed for all times, and only incorporated in the objective function as appropriate penalties on state constraint violation (`softening' of state constraints). In either approach, the problem of providing explicitly the set of initial conditions starting from where stabilization can be achieved and state and input constraints are guaranteed to be feasible has not been considered.

For uncertain nonlinear processes, the problem of robust MPC design continues to be an area of ongoing research (see, for example, [2]). Several robust predictive formulations utilize, what we will refer to as the ''min-max" approach, where the manipulated input trajectory is computed by solving an optimization problem that requires minimizing the objective function (and satisfying the input and state constraints) over all possible realizations of the uncertainty. While the min-max formulations provide a natural setting within which to address this problem, computational problems with these approaches are well known, and stem in part from the nonlinearity of the model which typically makes the optimization problem non-convex and in part from performing the min-max optimization over the non-convex problem. Using min-max approaches or otherwise, the problem of determining the set of initial conditions starting from where stabilization is guaranteed in the presence of uncertainty and constraints on the state, input and rate of input variables has not been addressed.

The desire to implement control approaches that allow for an explicit characterization of their stability properties has motivated significant work on the design of stabilizing control laws, using Lyapunov techniques, that provide explicitly-defined regions of attraction for the closed-loop system. These controllers were utilized as fall-back controllers in [3] (the robust hybrid predictive control design), to allow implementation (not design) of predictive controllers with a well characterized stability region. In [4,5], a predictive controller is designed that does not assume, but provides guaranteed stabilization from an explicitly characterized set of initial conditions under input [4], and input and state constraints [5]. These results, however, were derived in the absence of rate constraints, and more importantly, in the absence of uncertainty. In summary, the problem of design of a robust predictive controller that guarantees stabilization from an explicitly characterized set of initial conditions in the presence of constraints on the state, input and rate of input change remains an open problem.

Motivated by these considerations, in this work, we propose a robust model predictive control design for the stabilization of nonlinear processes with constraints on the state, input and rate of input change. The design of the robust predictive controller uses a bounded robust controller, with its associated region of stability, as an auxiliary controller for the characterization of the stability properties. The key idea in the design of the robust predictive controller is the construction of an appropriate stability constraint (without turning it into a min-max formulation) that can be shown to be feasible from an explicitly characterized set of initial conditions. Using this approach, we address the problem for uncertain nonlinear processes where the constraints on the control input as well as rate of change of control inputs and the state variables need to be treated as hard constraints (that necessarily need to be satisfied at all times), as well as those where these constraints can be treated as soft constraints (that may be violated for some, preferably small, time). We propose a robust predictive controller formulation that guarantees stabilization and satisfaction of hard constraints from an explicitly characterized set of initial conditions. Subsequently, we employ switching between a predictive controller designed to incorporate soft constraints and the one that enforces hard constraints to handle processes with soft constraints in a way that reduces the time for which the soft constraints are violated. The implementation of the proposed method is illustrated via a chemical reactor example.

References:

[1] 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.

[2] W. Langson, I. Chryssochoos, S. V. Rakovic, and D. Q. Mayne, ``Robust model predictive control using tubes,'' Automatica, vol. 40, pp. 125-133, 2004.

[3] P. Mhaskar, N. H. El-Farra, and P. D. Christofides, ``Robust hybrid predictive control of nonlinear systems,'' Automatica, vol. 41, pp. 209-217, 2005.

[4] 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.

[5] 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., in press, 2006.