Active Set Prediction using Machine Learning Methods for Complexity Reduction in Nonlinear MPC

Active Set Prediction using Machine Learning Methods for Complexity Reduction in Nonlinear MPC Raphael Dyrska∗a, Karol Kišb, Martin Klaučob, and Martin Mönnigmanna aAutomatic Control and Systems Theory, Ruhr-Universität Bochum, Germany bInstitute of Information Engineering, Automation, and Mathematics, Slovak University of Technology in Bratislava, Slovakia November 2025 Preprint Abstract We use Machine Learning (ML) methods to simplify the Nonlinear Program (NLP) arising in Nonlinear Model Predictive Control (nonlinear MPC, NMPC). Since every solution to an NMPC problem defines a set of active and a set of inactive constraints, we propose to use predictions about these sets to reduce the complexity of the NLP to be solved. Specifically, we use ML methods to predict active sets for NMPC problems. The required classification networks are simple enough to be evaluated online, i.e., during the runtime of the controller. They can be trained to a high accuracy, qualifying as suitable candidates for an application to NMPC. The results are evaluated using numerical simulations for a model of a continuous stirred-tank reactor. Keywords: nonlinear model predictive control, machine learning, neural networks, classification, active sets ∗Corresponding author. Email address: [email protected] 1 1 Introduction In automatic control, the concept of Nonlinear Model Predictive Control (Nonlinear MPC, NMPC) was one of the main research topics addressed during the last decades, due to the ability of NMPC to handle both constraints and systems with multiple inputs and outputs (see, e.g., [1, 2, 3, 4, 5]). In NMPC, the control signal applied to the system is determined by solving a constrained nonlinear program (NLP) in every time step. Solving the NLP in every time step can be computationally demanding and might be restrictive for certain applications. Thus, intense research focused on approaches reducing the computational effort for solving the underlying NLP. In the classic NMPC literature, these approaches can be divided into contributions addressing the algorithmic performance to speed-up the solution process itself (see, e.g., [6, 7]) and approaches exploiting information about the structure of the solution to either simplify the underlying NLP before solving it (see, e.g., [8, 9]) or to even avoid its solution (see, e.g., [10, 11, 12]). The combination of linear and nonlinear MPC with Machine Learning (ML) methods is another well-studied field. Many contributions use neural networks to replace the controller completely (see, e.g., [13, 14, 15, 16, 17]). However, ML methods also have the potential to assist NMPC and viceversa. ML methods are, for example, often used to form the prediction model (see, e.g., [18]) or to tune the cost function of the optimization problem (see, e.g., [19]). Conversely, MPC methods can be used to provide safety guarantees to learning-based controllers (see, e.g., [20]). More recent approaches use ML methods to cope with uncertainties, by predicting plant-model mismatch, for example, but also other sources of uncertainty (see, e.g., [21, 22, 23]). Another branch of approaches uses ML methods to accelerate the solution of the MPC problem (see, e.g., [24, 25]). In MPC, a solution to the underlying optimization problem determines the active set, i.e., the set of inequality constraints that actually hold with equality at the optimal solution. The authors in [26] propose to use classification algorithms to predict the set of constraints that are expected to be active for the optimal solution of a linear MPC problem and to use this prediction for warm-starting an active-set algorithm used for solving the underlying optimization problem. In this article, we extend the idea of predicting active sets using ML 2 methods as presented by [26] from linear to nonlinear MPC problems. However, instead of using the predicted active sets on an algorithmic level as proposed by [26], we use a straightforward approach for simplifying the NLP before solving it. In previous works (see [8, 27], see also [28, 29]) we used a priori information about active and thus inactive constraints to simplify the underlying NLP by rearranging and removing constraints. While the prediction of the active set was based on the principle of optimality for NMPC on a shrinking horizon in [27], we used predictions based on the solution of the previous time step to simplify the NLP also for the classic, receding horizon case in [8]. Following these works, we now propose to use ML methods to predict active constraints and to apply the same simplification approach to the underlying NLP. The non-convexity of the underlying NLP poses challenges for the generation of training data since, for example, several local minima with the same active set and several local minima with different active sets may exist for the same initial state (see [30] for properties of NMPC solutions resulting from non-convexity). In Sections 2.1 and 2.2, we state the class of NMPC problems investigated here and introduce a model of a continuous stirred-tank reactor (CSTR) that serves as an example throughout the paper, respectively. Section 3 states the methods applied in this article, specifically the process of generating training data (Sec. 3.1), the type of classification networks and their training for active constraint prediction (Sec. 3.2), and the simplification approach for a known set of active constraints (Sec. 3.3). In Section 4, we provide closedloop simulation results for the CSTR introduced in Section 2.2. Conclusions and an outlook to future work are given in Section 5. 2 Problem Statement 2.1 NMPC Problem Class We consider the class of nonlinear dynamical systems in discrete time given by difference equations x(k+ 1) = f(x(k), u(k)), k = 0,1,..., (1) with system states x(k)∈Rnand input signals u(k)∈Rm. We assume the nonlinear mapping f:Rn×Rm→Rnto be twice continuously differentiable 3 and we assume f(0,0) = 0. To regulate system (1) to the origin we solve, in every time step, the constrained nonlinear optimization problem min X,U ∥x(N)∥2 P+ N−1 X k=0 ∥x(k)∥2 Q+∥u(k)∥2 R(2a) s.t. x(k+ 1) = f(x(k), u(k)), k ∈NN−1 0,(2b) x(k)∈ X, k ∈NN−1 0,(2c) u(k)∈ U, k ∈NN−1 0,(2d) x(N)∈ T ,(2e) x(0) = x(k),(2f) for horizon length N, with optimization variables X= (x(1)⊺, ..., x(N)⊺)⊺ and U= (u(0)⊺, ..., u(N−1)⊺)⊺describing the predicted states and inputs, respectively. The matrices P,Qand Rhave obvious dimensions and are assumed to be positive definite. The sets Xand Uare assumed to contain the origin in their interiors and to be defined by finite numbers of linear inequalities, which implies they are convex polyhedra. The terminal set T in (2e) is an ellipsoid here (see, e.g., [31] for this problem class) T={x|x⊺Px ≤γ},(3) which implies it is represented by one additional inequality. We assume there exists a stabilizing control law κ(x)in Tsuch that the terminal cost function ∥x(N)∥2 P, the terminal set T, and κ(x)fulfill the stability conditions summarized as in [1], A1–A4. To simplify the notation, we introduce z= (X⊺, U⊺)⊺and rewrite the optimization problem (2) in the compact form min z∥z∥2 H(4a) s.t. F(x(0), z)=0,(4b) G(x(0), z)≤0,(4c) with H∈RN(n+m)×N(n+m),H≻0. In (4), the equality constraints are 4 abbreviated as F(x(0), z) :=      x(1) −f(x(0), u(0)) . . . x(N)−f(x(N−1), u(N−1))      , and the inequality constraints described by (2c)–(2e) are summarized in G(x(0), z). Let the optimal solution to (4) for an initial state x(0) be denoted by z⋆(x(0)), which we often abbreviate by z⋆. If not stated otherwise, the optimal solution to (4) is assumed to be a local minimum. Let Q={1, ..., q}denote the set of indices of all qconstraints. A constraint i∈ Q is called active for the optimizer z⋆if row iof Gis zero, i.e. Gi(x(0), z⋆)=0. We recall that inactive constraints are fulfilled with strict inequality, i.e., Gi(x(0), z⋆)<0. We introduce the active and inactive set Aand Icollecting the indices of active and inactive constraints, respectively. We write A(x)for an active set that results for the initial state x. 2.2 Benchmark Process: Continuous Stirred-Tank Reactor We use the following CSTR model [32] as an illustrating example throughout the paper: dcA dt=q V(cA,f−cA)−k0e−E RT cA(5a) dT dt=q V(TA,f−T)−∆Hk0 ρcp e−E RT cA+UA V ρcp (Tc−T).(5b) State variables cAand Tdescribe the concentration of substance Aand the reactor temperature, respectively, and the manipulated variable Tcis the temperature of the coolant stream. Parameter values are collected in Table 1. The control task is to regulate the system to an unstable steady state. Following [32], we define the state and input vectors x1=cA−0.5 0.5, x2=T−350 20 , u =Tc−300 20 by shifting the variables to the unstable equilibrium. The system is discretized with a sampling time of Ts= 0.03 min, and the weighting matrices 5 Table 1: Parameter values of the CSTR. parameter value parameter value ρ1000 g L−1cp0.239 J g−1K−1 ∆H−5×104J mol−1E R8750 K k07.2×1010 min−1UA5×104J min−1K−1 q100 L min−1TA,f350 K V100 L cA,f1.0 mol L−1 in the NMPC cost function (2a) and the prediction horizon are chosen to be P= 265.88 49.39 49.39 125.99!, Q = 10 0 0 10!, R = 1, N = 10.(6) Furthermore, we introduce input and terminal constraints −1≤u(k)≤3.5,(7) x(N)∈ T ,T={x∈ X | x⊺P x ≤γ}, γ = 9.3353,(8) and the terminal controller as κ(x) = KTx, with KT= (−1.6154,−3.6644), such that the terminal weighting matrix, the terminal set, and the terminal controller fulfill the stability conditions referred to in Section 2.1. 3 Active Set Prediction using Classification We first introduce the generation of suitable training data, followed by a description of the training process of a classification network Φ(x)for predicting A(x). In the last subsection, we state a simplification approach for the NLP (4) as one possibility to benefit from the prediction of active sets. 3.1 Generation of Training Data For a reliable prediction of an optimal active set, the training data should be representative for the expected initial states that occur during closedloop control and consider features of the system at hand. As shown in [30], the non-convexity of the NLP can lead to the existence of several local minima with different active sets for the same initial state. This may lead to inconclusive training data for parts of the state space such as, e.g., areas 6 of states with mixed active sets, which will in general impede the training process in terms of convergence and resulting accuracy. We showed in [30] that the resulting minimum strongly depends on the initial guess forwarded to the optimizer. During control, we often apply a warm-start that results from extending the solution of the previous time step using the terminal controller κ(x)together with the prediction model. To generate the training data, we follow the strategy described in [26] for the linear case and regulate a well-distributed set of initial states closedloop into the terminal set. The use of the same warm-start here is essential to minimize the influence of varying local minima corrupting the training data. Further, by generating the training data in a closed-loop fashion, the highest concentration of data will result for the area around the terminal set, which is also expected to be the area containing most of the states for which problem (4) needs to be solved when controlling the corresponding system. Formally, we summarize pairs of a specific xiand Aias a data tuple Difor the training. Initial conditions resulting in the same active set will be summarized, and we call the resulting set a class for brevity. All optimization problems solved for generating training and testing data and during the computational study in Section 4 were implemented in Matlab using the nonlinear optimizer fmincon and the interior-point algorithm. For the CSTR, we generated a grid of feasible initial states with a step size of ∆x1= ∆x2= 0.01. Regulating all states of the grid into the terminal set, in total 134 736 data tuples of states and corresponding optimal active set were determined. Tuples Diwith equal active sets Aiwere summarized into one class, resulting in 185 different classes. Figure 1(a) shows the resulting state space, where initial states that yield the same optimal active set and thus belong to the same class are shown with the same color. Since the training data was generated by controlling all states of the initial grid into the terminal set T, the highest concentration of training data results for the area around T. The training process can be expected to benefit from a balanced training set. Classes with only few members can be a sign for outliers or numerically unstable results. We opt for a heuristic reduction of the number of classes while simultaneously maintaining a high resolution of the training data. Since the data tuples of each class will be split into a training and a verification set for the internal training process in the ratio 3:1(see Sec. 3.2 7 (a) Original training data. (b) Processed training data, with removed states marked in red. Figure 1: Training data for the classification network. The color of the states indicates the corresponding class, omitted states are shown in red. below), we propose to remove classes with 3or less pairs. The histogram in Figure 2 visualizes the distribution of the 134 736 states into the 185 different classes. Obviously, the majority of states belongs to the class on the very left. At the same time, classes on the very right form only a fraction of the overall simulation data. Removing classes with 3or less data tuples, the number of classes could be reduced by around 15.68%, from 185 to 156, while only 0.04% of all data tuples were removed. The states of the training set that have been removed are marked in red in Figure 1(b). The Figure 2: Number of data tuple per class. With 37 478 tuples, the class labeled as ’1’ is the highest represented class. Note that classes are sorted for cardinality and do not represent the enumeration depicted in Fig. 1. 8 position near the boundary of the feasible set indicates that (i) numerical issues might be a reason for outlier results and (ii) removing these states will not severely influence the results since in general most initial states will be located in the interior of the feasible set rather than close to its boundary. 3.2 Active Set Classification Network A feed-forward neural network (NN) consisting of nLfully connected layers is used as classification network Φ(x), mapping states xto a corresponding active set Aby classifying them into a class representing one unique combination of active constraints. The transformation for each hidden layer l with 1≤l < nLfollows v(l)=ω(l)a(l−1) +ξ(l),(9) with vthe vectorized input to each node, weights ω, biases ξ, and activation function atargeting the previous layer. For the activation function of the hidden layers, the Rectified Linear Unit (ReLU) function a(l)=ReLU(v(l)),with ReLU(v) = max(0, v), is chosen to capture nonlinear behavior in the relation x→ A. For the multiclass classification between nadifferent classes, the activation function for each input iof the output layer nLis chosen as the softmax function a(nL) i=σ(v(nL) i) = exp(v(nL) i) Pna c= 1 exp(v(nL) c),(10) to generate a probability for each of the nadifferent classes. An illustration of the overall prediction process is shown in Figure 3. Weights Ω=(ω(1), . . . , ω(nL))and biases Ξ=(ξ(1), . . . , ξ(nL))for all nL layers are the training parameters of the classification network Φ(x). They are optimized by solving the cross-entropy loss function min Ω,Ξ− na X c= 1 yclog σ(v(nL) c),(11) which is a standard choice for multi-class classification problems. In (11), ycindicates the presence of class cas for the input xi∈ Di, and the softmax 9 problem to train classification networks with a precision of more than 94%. The predicted active sets were used to simplify the nonlinear programs solved in every time step. With a CSTR model as example, we could present a reduction of the average number of iterations necessary to find a control input of almost 10%. Future work will investigate different forms of classification networks, and investigate techniques such as transfer learning to deal with, e.g., changing parameters of the optimization problem. Acknowledgment This paper is funded by the Alexander von Humboldt Foundation research group linkage cooperation program and by the European Commission under the grant no. 101079342 (Fostering Opportunities Towards Slovak Excellence in Advanced Control for Smart Industries). 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