Adaptive model predictive control for constrained nonlinear systems

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Abstract

In this paper, a method is proposed for the adaptive model predictive control of constrained nonlinear system. Rather than relying on the inherent robustness properties of standard NMPC, the developed technique explicitly account for the transient effect of parametric estimation error by combining a parameter adjustment mechanism with robust MPC algorithms. The parameter estimation routine employed guarantees non-increase of the estimation error vector. This means that the controller employs a process model which approaches that of the true system over time. These estimates are used to update the parameter uncertainty set at every time step, resulting in a gradual reduction in the conservative and/or computational effects of the incorporated robust features.

Introduction

Model predictive control (MPC) has attracted a great deal of interest from practitioners due to the relative ease constraints can be incorporated [1]. Although it has been proven that a standard implementation of MPC using a nominal model of the system dynamics exhibit nominal robustness to sufficiently small disturbances ([2], [3]), such marginal robustness guarantee may be unacceptable in practical situations. Present uncertainties must be accounted for in the computation of the control law to achieve robust stability. One way to cope with uncertainty in the system model is to employ robust MPC methods, which explicitly account for system’s uncertainties. Since robust controllers (in general) cannot learn changes in the plant, their performance is limited by the quality of the model plus the uncertainty description initially available. On the other hand, adaptive control has the potential to improve system performance as it updates the model online based on measurement data.

In general, most physical systems possess parametric uncertainties or unmeasurable parameters. Examples in chemical engineering include reaction rates, activation energies, fouling factors, and microbial growth rates. Since parametric uncertainty may degrade the performance of MPC, mechanisms to update the unknown or uncertain parameters are desirable in application. One possibility would be to use state measurements to update the model parameters off-line. A more attractive possibility is to apply adaptive extensions of MPC in which parameter estimation and control are performed online.

While some results are available for linear adaptive MPC (See [4] for example), very few adaptive MPC schemes have been developed for nonlinear systems ([5], [6]). The result in [6], implements a certainty equivalence nominal-model MPC feedback to stabilize a parametric uncertain system subject to an input constraint. The result shows that there must exist a finite time such that an excitation condition is satisfied and thus parameter convergence is achieved. There is no mechanism to decrease the identification period in any way and moreover, it is only by assumption that the true system trajectory remains bounded during the identification phase. In [5], an input-to-state stable control Lyapunov functions (iss-clf) is used to develop a MPC scheme that provides robust stabilization in the absence of parameter estimation algorithm, and ensures asymptotic stability of the closed loop with parameter adaptation. However, the work only deals with unconstrained nonlinear systems.

The design of adaptive nonlinear MPC schemes is very challenging because the “separation principle assumption” widely employed in linear control theory is not applicable to general class of nonlinear systems, in particular in the presence of constraints. A true adaptive nonlinear MPC algorithm must address the issue of robustness to model uncertainty while the estimator is evolving. Unfortunately, this may not be achieved without introducing extra degree of conservativeness and/or computational complexity in the controller calculations. The recent work [7] provided an adaptive robust MPC that deals with both state and input constraints within an adaptive framework. In the presented approach, a set valued description of the parametric uncertainty is directly adapted online to reduce the conservativeness of the solutions, especially with respect to the design of terminal penalty. The parameterization of the feedback MPC policy in terms of uncertainty set and the underlying min–max feedback MPC used in the study make the controller’s computation very challenging. The result can be viewed as a conceptual result that focus on performance improvement rather than implementation.

In this paper, it is assumed that the uncertainties in the system are due to static nonlinearities, expressible in the form of unknown (constant) model parameters. The study addresses the problem of adaptive MPC and incorporates robust features to guarantee closed loop stability and constraint satisfaction. First, a min–max feedback nonlinear MPC scheme is combined with an adaptation mechanism for the uncertainty set. The formulation accounts for the effect of future parameter estimation and automatically injects some useful excitation into the closed loop system to aid in parameter identification. Second, the developed technique is extended to a less computational algorithm based on Lipschitz bounds. In both cases, the parameter estimation routine employed guarantees non-increase of the estimation error vector. Using this estimates to update the parameter uncertainty set at every time step, results in in a gradual reduction in the conservativeness or computational demands of the algorithms.

The remainder of the paper is organized as follows. The problem description is given in Section 2 while the parameter identifier employed is outlined in Section 3. Our proposed robust adaptive MPC techniques are presented in Sections 4 Robust adaptive MPC—A min–max approach, 5 Robust adaptive MPC—A Lipschitz based approach. Simulation results are shown in Section 6 and conclusions are given in Section 7.

Nomenclature and Definitions: λ¯(M) denotes the smallest eigenvalue of matrix M. A continuous function μ:R+R+ is of class K if μ(0)=0, μ(.) is strictly increasing on R+ and is radially unbounded.

Section snippets

Problem description

The system considered is the following nonlinear parameter affine system ẋ=f(x,u)+g(x,u)θF(x,u,θ)θRnθ is the unknown parameter vector whose entries may represent physically meaningful unknown model parameters or could be associated with any finite set of universal basis functions. It is assumed that θ is uniquely identifiable and lie within an initially known compact set Θ0B(θ0,zθ0), a ball described by an initial nominal estimate θ0 and associated error bound zθ0=supsΘ0sθ0. The mapping

Parameter adaptation

In the following, we give a brief description of the identifier employed in the adaptive robust design framework. The procedure assumes the state of the system x(.) is accessible for measurement but do not require the measurement or computation of the velocity state vector ẋ(.). The algorithm is independent of the control structure employed.

Let the state predictor for (1) be denoted as xˆ and define xˆ̇=f(x,u)+g(x,u)θˆ+ke+wθˆ̇, where θˆ is the parameter estimate vector, k>0 a design constant

Robust adaptive MPC—A min–max approach

In this section, the concept of min–max robust MPC is employed to provide robustness for the MPC controller during the adaptation phase. The resulting optimization problem can either be solved in open-loop or closed-loop. In the presented approach, we choose the least conservative option by performing optimization with respect to closed loop strategies. As in typical feedback-MPC fashion, the controller chooses input u as a function of the current states. The formulation consists of maximizing

Robust adaptive MPC—A Lipschitz based approach

Due to the computational complexity associated with (feedback) min–max optimization problem for non-linear systems, it is (sometimes) more practical to use a more conservative but computationally efficient methods. Examples of such approaches include Lipschitz based methods [2], [10] and those based on the concept of reachable sets [11].

In this section, we present a Lipschitz based method whereby the nominal model rather than the unknown bounded system state is controlled, subject to conditions

Simulation example

Consider the regulation of a continuous stirred tank reactor where a first order, irreversible exothermic reaction AB is carried out. Assuming constant liquid level, the reaction is described by the following dynamic model [12]: ĊA=qV(CAinCA)k0exp(ERTr)CAṪr=qV(TinTr)ΔHρcpk0exp(ERTr)CA+UAρcpV(TcTr). The states CA and Tr are the concentrations of components A and the reactor temperature respectively. The manipulated variable Tc is temperature of the coolant stream.

It is assumed that

Conclusions

In this chapter, we presented an adaptive MPC design technique for constrained nonlinear systems with parametric uncertainties. The system’s performance is improved over time as the adaptive control updates the model online. The controller parameters is updated only when an improved parameter estimate is obtained. Robustly stabilizing MPC schemes are incorporated to ensure robustness of the algorithm to parameter estimation error during the adaptation phase. The two robust approaches, min–max

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This work was supported in part by the Natural Sciences and Engineering Council of Canada.

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