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Solutions to optimal control problems are usually understood to provide optimal trajectories. In this paper, we show that the optimal state-space system dynamics induce a dynamics of the active sets. More specifically, given the optimal active set at the solution obtained at the current time, its successor optimal active set (which, in turn, defines the successor solution) can be found with index set operations. These operations do not involve any optimal control (or other optimization or integration) problem, but they can be described with simple rules. These rules constitute the symbolic dynamics for active sets. The present paper treats a particular constrained nonlinear problem class, extending earlier results for the constrained linear-quadratic case.

In Model Predictive Control (MPC) closed-loop performance heavily depends on the quality of the underlying prediction model, where such a model must be accurate and yet simple. A key feature in modern MPC applications is the potential for online model adaptation to cope with time-varying changes, part-to-part variations, and complex features of the system dynamics not caught by models derived from first principles. In this paper, we propose to use a physics-informed, or gray-box, model that extends the physics-based model with a data-driven component, namely a Recurrent Neural Network (RNN). Relying on physics-informed models allows for a rather limited size of the RNN, thereby enhancing online applicability compared to pure black-box models. This work presents a method based on Moving Horizon Estimation (MHE) for simultaneous state estimation and learning of the RNN sub-model, a potentially challenging issue due to limited information available in noisy input-output data and lack of knowledge of the internal state of the RNN. We provide a case study on a quadruple tank benchmark showing how the method can cope with part-to-part variations.

We present a new simple approach to extend the admissible region of a saturated control law for discrete-time linear systems under state and input constraints. The approach is based on the interpolation of a given, performance driven, control law and a low-gain controller. The latter is designed such that its admissible set has maximum volume. The interpolation does not require on-line numerical optimization, and an analytical formula for the controller is given. Several numerical examples are presented to illustrate the control technique.

In parametric identification of Linear Parameter-Varying (LPV) systems, it is important to achieve a low variance of the model estimate by limiting the number of parameters to be identified. This is the well known “model order selection” problem, which consists of selecting the number of input and output delays and the basis functions characterizing the dependence of the LPV model parameters on the scheduling signal. Ignoring the effect of noise on the observations of the scheduling signals may lead to a bias in the final estimate and, as a consequence, also to an incorrect selection of the model order. In this paper, we introduce a “bias-corrected cost function” for the identification of LPV systems from noise-corrupted observations of the output and scheduling variable. The introduced cost function provides a bias-free parameter estimation along with model order selection. The proposed identification approach has two main advantages: (i) the problem of model order selection can be handled by adding a LASSO-like penalty term to the bias-corrected cost function; (ii) it provides a bias-free cost as a criterion to tune some hyper-parameters influencing the final parameter estimate.

The optimization of process economics within the model predictive control (MPC) formulation has given rise to a new control paradigm known as economic MPC (EMPC). Several authors have discussed the closed-loop properties of EMPC-controlled deterministic systems, however, little have uncertain systems been studied. In this paper we propose EMPC formulations for nonlinear Markovian switching systems which guarantee recursive feasibility, asymptotic performance bounds and constrained mean square (MS) stability.

In this paper we investigate the problem of optimal real-time power dispatch of an interconnection of conventional power generation plants, renewable resources and energy storage systems. The objective is to minimize imbalance costs and maximize profits whilst satisfying user demand. The managing company is able to trade energy on an electricity market. Energy prices, demand and renewable generation are considered stochastic processes. We show that under certain assumptions, the stochastic power dispatch problem over a finite horizon can be recast into a stochastic optimization formulation but with deterministic constraints. We carry out a systematic study of stochastic optimization methods to solve this problem. We also show that this problem can be approximated by a proper deterministic optimization problem using the sample average approximation method, which can then be solved by standard means.

Drinking Water Networks (DWN) are large-scale multiple-input multiple-output systems with uncertain disturbances (such as the water demand from the consumers) and involve components of linear, non-linear and switching nature. Operating, safety and quality constraints deem it important for the state and the input of such systems to be constrained into a given domain. Moreover, DWNs' operation is driven by time-varying demands and involves an considerable consumption of electric energy and the exploitation of limited water resources. Hence, the management of these networks must be carried out optimally with respect to the use of available resources and infrastructure, whilst satisfying high service levels for the drinking water supply. To accomplish this task, this paper explores various methods for demand forecasting, such as Seasonal ARIMA, BATS and Support Vector Machine, and presents a set of statistically validated time series models. These models, integrated with a Model Predictive Control (MPC) strategy addressed in this paper, allow to account for an accurate on-line forecasting and flow management of a DWN.

Extending the success of model predictive control (MPC) technologies in embedded applications heavily depends on the capability of improving quadratic programming (QP) solvers. Improvements can be done in two directions: better algorithms that reduce the number of arithmetic operations required to compute a solution, and more efficient architectures in terms of speed, power consumption, memory occupancy and cost. This paper proposes an implementation with fixed-point arithmetic of a proximal Newton method to solve optimization problems arising in input-constrained MPC. The main advantages of the algorithm are its fast asymptotic convergence rate and its relatively low computational cost per iteration since it the solution of a small linear system is required.

This paper proposes a hierarchical MPC strategy for autonomous navigation of a formation of unmanned aerial vehicles (UAVs) of quadcopter type under obstacle and collision avoidance constraints. Each vehicle is stabilized by a lower-level local linear MPC controller around a desired position, that is generated, at a slower sampling rate, by a hybrid MPC controller per vehicle. Such an upper control layer is based on a hybrid dynamical model of the UAV in closed-loop with its linear MPC controller and of its surrounding environment (i.e., the other UAVs and obstacles). The resulting decentralized scheme controls the formation based on a leader-follower approach. The performance of the hierarchical control scheme is assessed through simulations and comparisons with other path planning strategies, showing the ability of linear MPC to handle the strong couplings among the dynamical variables of each quadcopter under motor voltage and angle/position constraints, and the flexibility of the decentralized hybrid MPC scheme in planning the desired paths on-line.

In this paper we introduce a digital architecture implementing the explicit solution of a switched model predictive control problem. Given a mixed-logic dynamical system, we derive an explicit controller in the form of a possibly discontinuous piecewise-affine function. This function is then approximated by resorting to piecewise-affine simplicial functions, which can be implemented on a circuit by extending the representation capabilities of a previously proposed architecture to evaluate the control action. The architecture has been implemented on FPGA and validated on a benchmark example related to an air conditioning system.

This paper proposes a decentralized hierarchical multi-rate control scheme for large-scale dynamically-coupled linear systems subject to linear constraints on input and state variables. At the lower level, a set of decentralized and independent linear controllers stabilizes the process, without taking care of the constraints. Each controller receives reference signals from its own upper-level controller, that runs at a lower sampling frequency. By optimally constraining the magnitude and rate of variation of the reference signals to each lower-level controller, quantitative criteria are provided for selecting the ratio between the sampling rates of the upper and lower layers of control at each location, in a way that closed-loop stability is preserved and the fulfillment of the prescribed constraints is guaranteed.

For piecewise affine (PWA) systems whose dynamics are only defined in a bounded and possibly non-invariant set X,this paper proposes a numerical approach to analyze the stability of the origin and to find a region of attraction. The approach relies on introducing fake dynamics outside Xand on synthesizing a piecewise affine and possibly discontinuous Lyapunov function on a larger bounded set containing Xby solving a linear program. The existence of a solution proves that the origin is an asymptotically stable equilibrium of the original PWA system and determines a region of attraction contained in X.The procedure is particularly useful in practical applications for analyzing a posteriori the stability properties of approximate explicit model predictive control laws defined over a bounded set Xof states, and to determine whether, for a given set of initial states, the closed-loop system evolves within the domain Xwhere the control law is defined.

The cruise control problem of transferring the speed of a vehicle between two values in a fixed interval of time using a predefined sequence of gears, and minimizing a cost related to fuel consumption, is solved in this paper. This is a hybrid dynamic optimization problem since the control variables include both a continuous variable (fuel flow) and a discrete variable (the gear to apply at each instant). The solution is given in the form of a hybrid optimal control algorithm that computes the optimal switching times between gears using Dynamic Programming and the optimal fuel profile between successive gear boundaries using a gradient algorithm to approximate the optimum conditions. In order to reduce the search of the optimal switching times to a search in a finite dimension graph, a procedure based on a changing grid is used. The algorithm is illustrated by a simulation using a diesel one-dimensional car model.

We propose a switching Model Predictive Control (MPC) strategy to control vehicle steering by actuating active front steering (AFS) and electronic stability control (ESC). After describing the piecewise affine prediction model used for MPC design, where the nonlinearities arise from the relation between sideslip angles and tire forces, a switching MPC strategy is implemented, where different local MPC controllers are used depending on the current tire force conditions. The designed controller maintains most of the benefits of a previously designed hybrid model predictive controller, but it has lower complexity and allows more flexible design. The controller stability is verified and the controller behavior during challenging step steering maneuvers is tested in closed-loop simulations against a nonlinear vehicle model.

This paper addresses the problem of identification of piecewise affine (PWA) models, which involves the joint estimation of both the parameters of the affine submodels and the partition of the PWA map from data. According to ideas from set-membership identification, the key approach is to characterize the model by its maximum allowed prediction error, which is used as a tuning knob for trading off between prediction accuracy and model complexity. At initialization, the proposed procedure for PWA identification exploits a technique for partitioning an infeasible system of linear inequalities into a (possibly minimum) number of feasible subsystems. This provides both an initial clustering of the datapoints and a guess of the number of required submodels, which therefore is not fixed a priori. A refinement procedure is then applied in order to improve both data classification and parameter estimation. The partition of the PWA map is finally estimated by considering multicategory classification techniques.