- Email: [email protected]

Multi-stage and Two-stage Robust Nonlinear Model Predictive Control S. Lucia ∗ S. Engell ∗ ∗

Process Dynamics and Operations Group, Department of Biochemical and Chemical Engineering, Technische Universit¨ at Dortmund, 44221 Dortmund, Germany (e-mail: {sergio.lucia, sebastian.engell} @bci.tu-dortmund.de).

Abstract: This papers presents a non-conservative robust Nonlinear Model Predictive Control (NMPC) scheme based on a multi-stage and a two-stage formulation. Both control schemes can satisfy the constraints for all the possible values of the uncertainty and, as illustrated by a simulation example, the resulting average cost is smaller than using standard NMPC or min-max (feedback) NMPC approaches. Since Multi-stage NMPC is based on a discrete-time formulation and most problems in the process industry are modeled by a set of ODE’s, different techniques for the discretization of the optimal control problem are considered for the multi-stage NMPC case, and compared for a highly nonlinear bioreactor example. Keywords: Robust control; Model predictive control; Multistage optimization; Collocation; Multiple Shooting. 1. INTRODUCTION One of the main problems that Nonlinear Model Predictive Control (NMPC) has to face is the fact that stability and performance rely strongly on the accuracy of the model and therefore it is sensitive to plant-model mismatch and model disturbances. As real-world systems are subject to uncertainties, robust nonlinear model predictive control has become a very active area of research in the last years. Most robust NMPC controllers are based on a min-max approach (see Campo and Morari [1987]), that tries to minimize the cost of the worst-case realization of the uncertainty. They are usually classified into open-loop and closed-loop (or feedback) approaches. The open loop approaches optimize over a sequence of control inputs, leading to very conservative solutions (see Scokaert and Mayne [1998]), whereas the second ones optimize over a sequence of control policies, what results in an infinite dimension problem which is very difficult to solve in general for nonlinear systems. As a solution to this problem, the assumption of a fixed structure in the control policy can be adopted, leading to a suboptimal solution of the problem. Tube-based approaches (Rawlings and Mayne [2009], Mayne et al. [2011], Yu et al. [2011], Rakovic et al. [2011]) solve the nominal problem using standard NMPC and then apply an ancillary controller that introduces feedback information and takes care of maintaining the system evolution within a certain neighborhood of the nominal ⋆ The research leading to these results has received funding from the European Union Seventh Framework Programme FP7/20072013 under grant agreement number FP7-ICT-2009-4 248940 (EMBOCON) and from the Deutsche Forschungsgemeinschaft (DFG, German Research Council) in the context of the research cluster: Optimization-based control of uncertain systems, under grand agreement number EN 152/39-1.

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trajectory for all the possible values of the uncertainty. In order to guarantee robust constraint satisfaction, the constraint sets for the solution of the nominal problem have to be tightened. In our approach we investigate the multi-stage and twostage robust NMPC formulation proposed in Dadhe and Engell [2008], Engell [2009] and Lucia et al. [2012]. Making use of ideas already applied successfully to scheduling problems in Cui and Engell [2010], the uncertainty is modeled as a moving scenario tree, in a similar manner as explained in Scokaert and Mayne [1998]. In this way, the feedback structure is represented by a finite-dimensional open-loop optimization problem, in contrast to feedback min-max approaches. Some work using similar ideas for linear systems has been reported in Mu˜ noz de la Pe˜ na et al. [2005] and Bernardini and Bemporad [2009]. Assuming that the scenario tree describes the uncertainty perfectly, our approach represents the online decision problem exactly and therefore provides the best possible solution. The main drawback of the approach is that it results in large optimization problems, the size of which grows exponentially with the prediction horizon and with the number of uncertainties and uncertainty levels. After this introductory section, the paper is organized as follows. In Section 2 the multi-stage NMPC problem is introduced together with the two-stage formulation. Section 3 gives an overview about the different discretization methods for the transformation of the multi-stage control problem into a Nonlinear Programming Problem (NLP). Section 4 presents a comparison of multi-stage, two-stage, min-max and standard NMPC for a discrete nonlinear double integrator, as well as a comparison of the different discretization methods for a nonlinear bioreactor. Finally, some conclusions and directions for future work are stated in Section 5.

10.3182/20120823-5-NL-3013.00015

IFAC NMPC'12 Noordwijkerhout, NL. August 23-27, 2012

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Fig. 2. Scenario tree representation of the uncertainty evolution with robust horizon for multi-stage NMPC Then the optimization problem is formulated as: N X min ωi J i u ˜

In the multi-stage robust NMPC approach the uncertainty is modeled by a tree of discrete scenarios (see Fig. 1) where the uncertainty is resolved subsequently at each node and the control inputs are separated into stages. Each path from the root node x0 to the leaf nodes is called a scenario. At each node of the tree, a decision is computed based on the information up to that time, taking into account explicitly the uncertainty about future evolutions as well as the future decisions on these branches. It is necessary to impose that decisions based on the same information have to be equal in order to model the real-time decision problem correctly. This is imposed by the so-called nonanticipativity constraints. They force all control inputs that branch at one node to be the same (i.e., in Fig. 1 u10 = u20 = u30 , u11 = u21 = u31 , ...). We consider a discrete-time formulation of an uncertain nonlinear system described by: xjk+1 = f (xk , uk , dk ), (1) where xjk ∈ X ⊆ Rn , ujk ∈ U ⊆ Rm are the j-th state and control vectors at stage k, and dlk describes the uncertainty at stage k. It can include parametric uncertainties or unknown disturbances within known bounds. The state at stage k, xjk , depends on the values of the previous stage, denoted as (xk , uk , dk ), following the structure of the tree depicted in Fig. 1. It is assumed that the evolution of the uncertainty can be represented by a tree of discrete realizations. For simplicity, the uncertainty and therefore the scenario tree are assumed to be uniform, i.e., it has the same number of branches at all nodes, given by dlk ∈ {d1k , d2k ...dsk } at stage k for s different possible values of the uncertainty. For ease of notation, the vector of nodes is defined as x ˜ = [x0 , x11 , x21 , ..., xN Np ], and the vector of control 1 2 N inputs as u ˜ = [u1 , u1 , ..., uNp ], where N is the number of scenarios (or leaf nodes) and Np is the prediction horizon. These vectors contain all states and controls in the scenario tree. Finally, we define the scenario vector Si , that contains all states and controls belonging to the scenario i. 182

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The multi-stage NMPC is a general framework that also includes the possibility of the formulation of a closed-loop min-max approach by simply substituting the summation in (2) by the max operator. In addition to this, it is possible to extend the formulation in order to guarantee robust constraint satisfaction and stability, as well as recursive feasibility. For this purpose it is necessary to include and extend the usual ingredients used for nominal and min-max NMPC stability. An average decrease of the terminal cost is required for stability, while the use of a robust invariant set as terminal region together with a terminal control policy facilitates the recursive feasibility. Robust constraint satisfaction is ensured directly if the optimization problem is feasible (cf. Lucia and Engell [2012]). 2.1 Multi-stage NMPC with robust horizon The main drawback of the multi-stage approach is that the size of the resulting optimization problem grows exponentially with the prediction horizon, and with the number and levels of the uncertainties. One possibility to avoid the exponential growth with the prediction horizon is to

IFAC NMPC'12 Noordwijkerhout, NL. August 23-27, 2012

3.1 Euler implicit discretization Using implicit Euler’s discretization, the discretized system can be written as: xk+1 = xk + ∆T (Φ(xk+1 , uk , dk )). (5) After the discretization the resulting NLP is: min f (xopt ) (6a) opt x

xl ≤xopt ≤ xu , (6b) opt bl ≤Ax ≤ bu , (6c) opt cl ≤c(x ) ≤ cu , (6d) opt where x is the augmented optimization vector. It includes all the states (nodes) and control inputs as indicated N 2 1 in Fig. 1. That is, xopt = [x0 , x11 , x21 , ..., xN Np , u1 , u1 , ..., uNp ]. The objective function f in (6a) includes all the necessary terms to create an objective function in the form of (2). The constraints in (6b) denote the state and control constraints imposed on the real system. The linear constraints (6c) include the non-anticipativity constraints and the nonlinear constraints (6d) include the discretized model for all the nodes of the tree. The sparsity and structure of the problem can be exploited explicitly leading to a faster solution despite of the size of the problem (see Steinbach [2000]), but this issue is part of our future work. s.t.

Fig. 3. Scenario tree representation of the uncertainty evolution for two-stage NMPC consider that the scenario tree branches only up to a certain stage (robust horizon) and after this point the uncertainty is assumed to be constant. Similar ideas have been applied in scheduling Cui and Engell [2010] or in linear min-max MPC Mu˜ noz de la Pe˜ na et al. [2006]. An example of the resulting scenario tree can be seen in Fig. 2. 2.2 Two-stage NMPC Another possibility to simplify the multi-stage optimization problem is to consider a two-stage formulation. In this case, it is assumed that the uncertainty fully resolves in stage two (see Fig. 3). Therefore, the tree branches only in the first stage and thereafter all scenarios are independent. This approach (successfully applied to online scheduling problems in Sand and Engell [2004]) may be appropriate together with the application of scenario-based decomposition algorithms that take advantage of the fact that the non-anticipativity constraints exist only in the first stage. Possible algorithms are Augmented Lagrangian Decomposition or Progressive Hedging Algorithm. See Ruszczynski [1997] for a review. 3. DISCRETIZATION METHODS AND IMPLEMENTATION ISSUES The formulation presented in the previous section assumes a discrete nonlinear model. However, most chemical processes are modeled using Ordinary Differential Equations (ODE’s), that can be written as follows: x˙ = Φ(x, u, d).

(4)

Here we describe suitable discretization approaches to transform the ODE’s into a Nonlinear Programming Problem (NLP) such that is solvable by standard optimization methods, and their adaptation for the solution of multistage and two-stage optimization problems in the context of model predictive control. 183

3.2 Euler collocation For highly nonlinear problems simple Euler discretization will not provide a sufficient accuracy for a satisfactory performance of the NMPC controller. A simple option consists of using more points for the model discretization on each branch between nodes of the original scenario tree. However, all the control inputs belonging to the same branch are required to be equal, since a piecewise control input will be implemented to the plant after each NMPC iteration. Better direct collocation approaches are based on Gauss pseudospectral collocation. See Benson [2005] for a comparison and a detailed explanation of such methods. 3.3 Multiple shooting Well-known multiple shooting methods, first presented in Bock and Plitt [1984], for the solution of optimal control problems fit perfectly to the structure of the multi-stage optimization problem. In this case each node of the tree is the initial condition of an Initial Value Problem (IVP). A parallelization of the integration is possible not only on the time scale as in standard multiple shooting, but on each of the branchings at each node. For more details on the implementation of the multiple shooting method the reader is referred to the extensive literature. 4. SIMULATION RESULTS In this section we present an illustrative comparison of the performance of nominal, multi-stage, two-stage and min-max MPC for a nonlinear integrator example with an additive disturbance. Furthermore, the different discretization methods presented previously together with the use of different robust horizons are compared for a highly nonlinear bioreactor, including the case when there are active state constraints.

IFAC NMPC'12 Noordwijkerhout, NL. August 23-27, 2012

4.1 Comparison of Nominal, Multi-stage, Two-stage and Min-max NMPC

States mean 0.6 multi−stage two−stage min−max nominal

where x1 (k) and x2 (k) are the states at time k, u(k) is the control input and d(k) is an additive disturbance that takes at each time k one of the following discrete values d = {−0.05, 0, 0.05} with probabilities {0.2, 0.4, 0.4}. A prediction horizon Np = 3 and a sampling time Ts = 1 s are used. The stage cost is a standard quadratic cost of the form L(x, u) = x′ Qx + u′ Ru, where 10 and R = 0.15. Q= 01 The optimization problem is formulated as presented in (2). The problem is solved using the sparse nonlinear solver SNOPT via MATLAB/TOMLAB in a computer with a 4core (2.66 GHz) Intel i5 processor for 50 different runs, with the additive disturbance varying at each sampling time between the previously specified values with its corresponding probability. The average computation times for one iteration of the NMPC for the multi-stage, twostage, min-max and nominal case are 0.32 s., 0.61 s., 0.42 s., and 0.023 s. respectively. The mean and standard deviation after 50 runs for the first state x1 together with the accumulated cost are represented in Fig. 4. The accumulated cost is calculated as J(k + 1) = J(k) + x′p Qxp + u′p Rup , where xp is the state of the real plant after introducing the optimizing input up . It can be seen that -as expected- the multi-stage solution has the lowest accumulated cost and the min-max solution has the biggest cost. Since the additive disturbance is persistent, it is not possible to converge to the origin, but the average nominal NMPC performance has a bigger steady state error than the average multi-stage or twostage NMPC controllers. This is due to the fact that information about the disturbance (probabilities) is taken into account in the multi-stage and two-stage approaches.

4.2 Comparison of Different Discretization Methods for Multi-stage NMPC of a Nonlinear Bioreactor In this section the different discretization methods presented in section 3 are compared for a highly nonlinear bioreactor. The model is presented in Ungar [1990] as a benchmark for difficult nonlinear control problems. The system is described by the following ordinary differential equations: x˙ 1 = −x1 u + x1 (1 − x2 )ex2 /Γ (8) 1+β 1 + β − x2 where x1 and x2 are the dimensionless state variables, bounded between zero and one. They represent the cell x˙ 2 = −x2 u + x1 (1 − x2 )ex2 /Γ

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Fig. 4. Comparison of multi-stage, two-stage, min-max and nominal NMPC for 50 different samples. mass and the amount of nutrients in a constant volume tank. The control input u represents the flow rate of nutrients into the tank and also the rate at which contents are removed from the tank, in order to maintain a constant volume. It is restricted to be between bounds, u ∈ [0, 2]. In this case, we consider that Γ is the only uncertain constant parameter and it can take the following values Γ = {0.8 Γnom , Γnom , 1.2 Γnom } where Γnom = 0.48 and we consider no uncertainty of the parameter β, i.e., β = βnom = 0.02. For the NMPC controller, a prediction horizon Np = 4, and sampling time ∆T = 0.5 s is used. The control task consists of the tracking of the cell mass (x1 ) for possibly changing set points. The cost function includes a quadratic tracking term for x1 and a quadratic penalty term for the control movements, i.e., the stage ′ opt ′ cost has the form L(·) = (x − xopt ss ) Q(x − xss ) + ∆u R∆u, where 200 0 and R = 75. Q= 0 0 In order to avoid steady state error, a bias term is introduced. At each sampling time, the set point used in the optimizer (xopt ss ) is updated using a proportional rule, opt ref i.e., xopt ss ← xss + K(xss − xp ), with K = 0.05 where ref xss is the real setpoint and xp is the state of the real plant. The resulting NLP is solved again using the sparse nonlinear solver SNOPT via MATLAB/TOMLAB. In Fig. 5, the responses for a step in the setpoint of x1 of both standard NMPC with no uncertainty (a) and multi-stage NMPC (b) are shown for different discretization methods. The results of the tracking problem where there is a plant model mismatch, i.e., the real parameter in the plant is Γ = 1.2 Γnom can be seen in Fig. 5 (c) for standard NMPC and in Fig. 5 (d) for multi-stage NMPC. As expected, for all the cases the solution of the collocation strategies converges to the one obtained using multiple shooting when increasing the number of collocation points (”col1” means implicit Euler discretization, ”col2” denotes the use of 2 Euler collocation points). The best cost is obtained for

IFAC NMPC'12 Noordwijkerhout, NL. August 23-27, 2012

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If there are state constraints in the process the use of a robust approach such as Multi-stage NMPC is even more important. In this case, a state constraint for the cell mass x1 is added such that x1 ≥ 0.065. A step in the uncertainty, i.e., a change of the value of the parameter Γ from Γnom to 1.2 Γnom takes place at time t = 20 s, while the setpoint is maintained constant. In Fig. 6 it can be seen that the standard NMPC cannot fulfill the state constraint regardless of the solution method used. Fig. 7 shows the multi-stage NMPC performance for different robust horizons using an Euler discretization as the solution method (solid lines) and for the case with robust horizon 3 and multiple shooting (dotted line). In this case, the 185

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the nominal parameter using standard NMPC, because in this case the model is perfect. However, under the presence of uncertainty, the accumulated cost for standard NMPC (Fig. 5 (c)) is significantly bigger than for multi-stage NMPC (Fig. 5 (d)) regardless of the discretization method, showing the advantages of robust multi-stage NMPC. In addition, it can be seen that a better discretization method does not guarantee a lower cost (see for instance Fig. 5 (c)) because the approximated discretization may be more similar to the uncertain system than the real one. Furthermore, for the multi-stage case with uncertainty (Fig. 5 (d)) the cost is even lower than for the multistage case with the nominal parameter (Fig. 5 (b)). This is due to the fact that in this case the uncertainty is helping to achieve the control task and the multi-stage approach takes advantage of this, in sharp contrast to other robust approaches (like tube-based methods) that track the nominal solution.

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Fig. 7. Performance of multi-stage NMPC for different robust horizons using Euler implicit discretization for a step in the uncertain parameter Γ at time t = 20 s. 5. CONCLUSIONS In this paper, a robust optimization algorithm based on a multi-stage or a two-stage formulation for nonlinear model predictive control has been presented. The main idea is that the uncertainty is taken into account explicitly by considering discrete disturbances that are represented as a scenario tree. In order to deal with the rapidly increasing problem size, it is considered that the scenario tree branches only up to a certain stage (robust horizon) and afterwards the uncertainty is considered to be constant. The paper illustrates the influence that the implementation issues (discretization method, robust horizon) have on the performance of multi-stage NMPC as well as the differences between standard, two-stage, multi-stage and minmax NMPC by simulation examples of nonlinear systems with additive disturbances and parametric uncertainties. It was shown that the multi-stage formulation provides a promising framework for robust NMPC having better performance than standard or min-max approaches. The main challenge of this scheme is the efficient solution of the very big numerical optimization resulting problem. Future work will be focused on the explicit exploitation of the structure in the solution of the resulting NLP, on the use of decomposition approaches in order to achieve a realtime robust NMPC scheme even for systems with several uncertainties as well as on an optimal way to choose the discrete values of the uncertainty to build the scenario tree. REFERENCES D. Benson. A Gauss Pseudospectral Transcription for Optimal Control. PhD thesis, Massachusetts Institute of Technology, 2005. D. Bernardini and A. Bemporad. Scenario-based model predictive control of stochastic constrained linear systems. In 48th IEEE Conference on Decision and Control, 2009., pages 6333–6338, 2009. H. Bock and K. Plitt. A multiple shooting algorithm for direct solution of optimal control problems. In 9th IFAC World Congress Budapest, 1984. P.J. Campo and M. Morari. Robust model predictive control. In American Control Conference, pages 1021– 1026, 1987. 186

J. Cui and S. Engell. Medium-term planning of a multiproduct batch plant under evolving multi-period multiuncertainty by means of a moving horizon strategy. Computers & Chemical Engineering, 34:598–619, 2010. K. Dadhe and S. Engell. Robust nonlinear model predictive control: A multi-model nonconservative approach. In Book of Abstracts, Int. Workshop on NMPC, Pavia, page 24, 2008. S. Engell. Online optimizing control: The link between plant economics and process control. In 10th International Symposium on Process Systems Engineering, volume 27, pages 79 – 86. Elsevier, 2009. M. Lazar, D. Mu˜ noz de la Pe˜ na, W.P.M.H. Heemels, and T. Alamo. On input-to-state stability of min-max nonlinear model predictive control. Systems & Control Letters, 57(1):39 – 48, 2008. S. Lucia and S. Engell. Stability properties of multi-stage robust nonlinear model predictive control. In 51st IEEE Conference on Decision and Control (submitted), 2012. S. Lucia, T. Finkler, D. Basak, and S. Engell. A new robust NMPC scheme and its application to a semibatch reactor example. In International Symposium on Advanced Control of Chemical Processes, 2012. D. Mayne, E. C. Kerrigan, and P. Falugi. Robust model predictive control: advantages and disadvantages of tube-based methods. In 18th IFAC World Congress Milano, 2011. D. Mu˜ noz de la Pe˜ na, A. Bemporad, and T. Alamo. Stochastic programming applied to model predictive control. In 44th IEEE Conference on Decision and Control, pages 1361–1366, 2005. D. Mu˜ noz de la Pe˜ na, T. Alamo, A. Bemporad, and E.F. Camacho. Feedback min-max model predictive control based on a quadratic cost function. In American Control Conference, pages 1575–1580, 2006. S. Rakovic, B. Kouvaritakis, M. Cannon, C. Panos, and R Findeisen. Fully parameterized tube MPC. In 18th IFAC World Congress Milano, 2011. J.B. Rawlings and D.Q. Mayne. Model Predictive Control Theory and Design. Nob Hill Pub, 2009. Andrzej Ruszczynski. Decomposition methods in stochastic programming. Mathematical Programming, 79:333– 353, 1997. G. Sand and S. Engell. Modeling and solving real-time scheduling problems by stochastic integer programming. Computers & Chemical Eng., 28:1087 – 1103, 2004. P.O.M. Scokaert and D.Q. Mayne. Min-max feedback model predictive control for constrained linear systems. IEEE Trans. on Aut. Control, 43(8):1136–1142, 1998. M.C. Steinbach. Hierarchical sparsity in multistage convex stochastic programs. In Stochastic Optimization: Algorithms and Applications, pages 363–388, 2000. L. H. Ungar. Neural networks for control. chapter A bioreactor benchmark for adaptive network-based process control, pages 387–402. MIT Press, 1990. S. Yu, H. Chen, and F. Allg¨ower. Tube MPC scheme based on robust control invariant set with application to lipschitz nonlinear systems. In 20th IEEE Conference on Decision and Control, 2011.

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