An efficient method for stochastic optimal control with joint chance constraints for nonlinear systems

Stochastic model predictive control hinges on the online solution of a stochastic optimal control problem. This paper presents a computationally efficient solution method for stochastic optimal control for nonlinear systems subject to (time‐varying) stochastic disturbances and (time‐invariant) probabilistic model uncertainty in initial conditions and parameters. To this end, new methods are presented for joint propagation of time‐varying and time‐invariant probabilistic uncertainty and the nonconservative approximation of joint chance constraint (JCC) on the system state. The proposed uncertainty propagation method relies on generalized polynomial chaos and conditional probability rules to obtain tractable expressions for the state mean and covariance matrix. A moment‐based surrogate is presented for JCC approximation to circumvent construction of the full probability distribution of the state or the use of integer variables as required when using the sample average approximation. The proposed solution method for stochastic optimal control is illustrated on a nonlinear semibatch reactor case study in the presence of probabilistic model uncertainty and stochastic disturbances. It is shown that the proposed solution method is significantly superior to a standard random sampling method for stochastic optimal control in terms of computational requirements. Furthermore, the moment‐based surrogate for the JCC is shown to be substantially less conservative than the widely used distributionally robust Cantelli‐Chebyshev inequality for chance constraint approximation.     

Robust output feedback model predictive control using off-line linear matrix inequalities

A fundamental question about model predictive control (MPC) is its robustness to model uncertainty. In this paper, we present a robust constrained output feedback MPC algorithm that can stabilize plants with both polytopic uncertainty and norm-bound uncertainty. The design procedure involves off-line design of a robust constrained state feedback MPC law and a state estimator using linear matrix inequalities (LMIs). Since we employ an off-line approach for the controller design which gives a sequence of explicit control laws, we are able to analyze the robust stabilizability of the combined control laws and estimator, and by adjusting the design parameters, guarantee robust stability of the closed-loop system in the presence of constraints. The algorithm is illustrated with two examples.     

Robustness analysis of a hybrid ground coupled heat pump system with model predictive control

MPC of thermal systems usually results in robust operation with respect to uncertainties thanks to some key characteristics of the controller. However, the true limit until which these systems will actually be robust is rarely known explicitly. In this study a Hybrid Ground Coupled Heat Pump (HyGCHP) system with MPC is investigated, for which state estimation and disturbance prediction are highly uncertain, moreover, the system performance is highly sensitive to errors at these points. It has become popular to design control systems which perform explicit computations to assure robustness (e.g. min–max Robust MPC) but this framework is computationally demanding, therefore, not widely applied. An alternative is to perform robustness analysis of an MPC controlled system which is though generally avoided due to complicated theoretical formulations, implicitness and conservativeness of the approach. To tackle these issues an existing framework for robustness analysis is extended and applied to the case of a HyGCHP system with MPC to analyze robustness with respect to state estimation uncertainty. This paper presents an approach to use the original formulation, suggested for regulation/stabilization in order to analyze robustness for the case of set point tracking. The results show that the maximum allowed state estimation uncertainty found by robustness analysis of the regulation problem is confirmed by the simulated HyGCHP system with MPC, which performs set point tracking. In conclusion, the method gives a reliable guarantee for the degree of state estimation uncertainty, up to which the HyGCHP system investigated remains robust. Future research can extend the robustness analysis method towards disturbance prediction uncertainty.     

Stochastic model predictive control with active uncertainty learning: A Survey on dual control

This paper provides a review of model predictive control (MPC) methods with active uncertainty learning. System uncertainty poses a key theoretical and practical challenge in MPC, which can be aggravated when system uncertainty increases due to the time-varying nature of system dynamics. For uncertain systems with stochastic uncertainty, this paper presents the stochastic MPC (SMPC) problem in the dual control paradigm, where the control inputs to an uncertain system have a probing effect for active uncertainty learning and a directing effect for controlling the system dynamics. The complexity of the SMPC problem with dual control effect is described in connection to stochastic dynamic programming as well as Bayesian estimation for its output feedback implementation. Further, implicit and explicit dual control methods for approximating the receding-horizon control problem with dual control effect are surveyed and analyzed with the intent to discuss the key challenges and opportunities in SMPC with dual control effect.     

Adaptive model predictive control for a class of constrained linear systems based on the comparison model

This paper proposes an adaptive model predictive control (MPC) algorithm for a class of constrained linear systems, which estimates system parameters on-line and produces the control input satisfying input/state constraints for possible parameter estimation errors. The key idea is to combine the robust MPC method based on the comparison model with an adaptive parameter estimation method suitable for MPC. To this end, first, a new parameter update method based on the moving horizon estimation is proposed, which allows to predict an estimation error bound over the prediction horizon. Second, an adaptive MPC algorithm is developed by combining the on-line parameter estimation with an MPC method based on the comparison model, suitably modified to cope with the time-varying case. This method guarantees feasibility and stability of the closed-loop system in the presence of state/input constraints. A numerical example is given to demonstrate its effectiveness.     

Output‐feedback model predictive control for stochastic systems with multiplicative and additive uncertainty

This paper studies the output‐feedback model predictive control (MPC) design problem for linear systems with multiplicative and additive random uncertainty. We first present an off‐line optimization algorithm to optimize feedback gains of the observer and the dual‐mode control policy. After that, by defining a cuboid tube whose center and boundary are both time‐varying variables, we develop a set sequence with increased freedom to contain stochastic system trajectories. A quadratic performance function with analytic upper and lower bounds is minimized such that it decreases exponentially to a finite range under the expectation. The resulting MPC algorithms are proved to guarantee practically stochastic input‐to‐state stability. A numerical example of the wind turbine model illustrates the properties of the MPC algorithms.     

Computation delay compensation for real time implementation of robust model predictive control

The implementation of model predictive control (MPC) requires to solve an optimization problem online. The computation time, often not negligible especially for nonlinear MPC (NMPC), introduces a delay in the feedback loop. Moreover, it impedes fast sampling rate setting for the controller to react to uncertainties quickly. In this paper, a dual time scale control scheme is proposed for linear/nonlinear systems with external disturbances. A pre-compensator works at fast sampling rate to suppress uncertainty, while the outer MPC controller updates the open loop input sequence at a slower rate. The computation delay is explicitly considered and compensated in the MPC design. Four robust MPC algorithms for linear/nonlinear systems in the literature are adopted and tailored for the proposed control scheme. The recursive feasibility and stability are rigorously analysed. Three simulation examples are provided to validate the proposed approaches.     

Real-time optimization under parametric uncertainty: a probability constrained approach

Uncertainty is an inherent characteristic in most industrial processes, and a variety of approaches including sensitivity analysis, robust optimization and stochastic programming have been proposed to deal with such uncertainty. Uncertainty in a steady state nonlinear real-time optimization (RTO) system and particularly making robust decisions under uncertainty in real-time has received little attention. This paper discusses various sources of uncertainty within such closed loop RTO systems and a method, based on stochastic programming, that explicitly incorporates uncertainty into the RTO problem is presented. The proposed method is limited to situations where uncertain parameters enter the constraints nonlinearly and uncertain economics enter the objective function linearly. Our approach is shown to significantly improve the probability of a feasible solution in comparison to more conventional RTO techniques. A gasoline blending example is used to demonstrate the proposed robust RTO approach.     

Evacuation Transportation Planning Under Uncertainty: A Robust Optimization Approach

This paper considers evacuation via surface transportation networks in an uncertain environment. We focus on demand uncertainty which can lead to significant infeasibility cost during evacuation, where loss of life or property may appear. We develop a robust linear programming model based on a robust optimization approach where hard constraints are guaranteed within an appropriate uncertainty set. The robust counterpart solutions have been shown tractable. We show that the robustness in evacuation is important and a robust solution outperforms a nominal deterministic solution in both quality and feasibility.     

Robust constrained predictive control using comparison model

This paper proposes a quadratic programming (QP) approach to robust model predictive control (MPC) for constrained linear systems having both model uncertainties and bounded disturbances. To this end, we construct an additional comparison model for worst-case analysis based on a robust control Lyapunov function (RCLF) for the unconstrained system (not necessarily an RCLF in the presence of constraints). This comparison model enables us to transform the given robust MPC problem into a nominal one without uncertain terms. Based on a terminal constraint obtained from the comparison model, we derive a condition for initial states under which the ultimate boundedness of the closed loop is guaranteed without violating state and control constraints. Since this terminal condition is described by linear constraints, the control optimization can be reduced to a QP problem.     

Robust MPC for Linear Systems with Structured Time‐Varying Uncertainties and Saturating Actuator

In this paper, a synthesis of model predictive control (MPC) algorithm is presented for uncertain systems subject to structured time‐varying uncertainties and actuator saturation. The system matrices are not exactly known, but are affine functions of a time varying parameter vector. To deal with the nonlinear actuator saturation, a saturated linear feedback control law is expressed into a convex hull of a group of auxiliary linear feedback laws. At each time instant, a state feedback law is designed to ensure the robust stability of the closed‐loop system. The robust MPC controller design problem is formulated into solving a minimization problem of a worst‐case performance index with respect to model uncertainties. The design of controller is then cast into solving a feasibility of linear matrix inequality (LMI) optimization problem. Then, the result is further extended to saturation dependent robust MPC approach by introducing additional variables. A saturation dependent quadratic function is used to reduce the conservatism of controller design. To show the effectiveness, the proposed robust MPC algorithms are applied to a continuous‐time stirred tank reactor (CSTR) process.     

Robust investment decisions under supply disruption in petroleum markets

Energy-dependent economies and energy security strategies need to cope with oil and gas supply disruptions that are rare but persistent and can be financially catastrophic. This paper proposes a tractable approach for determining robust investment strategies in petroleum markets under the risk of supply disruption when asset prices follow geometric mean-reverting jump processes. The robust counterpart of the portfolio management problem under supply disruption is derived for several symmetric and asymmetric representations of the uncertainties in the problem. Computational experiments with real market data indicate that the robust optimization approach using uncertainty sets tailored to the characteristics of the data results in strategies with superior worst-case performance.     

Robustly asymptotically stable finite-horizon MPC

This paper presents a robustly stable finite-horizon model predictive control (MPC) scheme for linear uncertain systems, in which the uncertainty is not restricted to some specific uncertainty class (polytopic, affine, LFT, etc.). The only requirement is that the state-space matrices remain bounded over the uncertainty set. Suitable constraints are added to the MPC cost function to impose robust asymptotic stability and to deal with input/output constraints. The resulting optimization problem is solved at each time instant in a probabilistic framework using an iterative randomized ellipsoid algorithm (REA). The method is compared in simulation to the existing approach of Kothare, Balakrishnan and Morari [(1996). Robust constrained model predictive control using linear matrix inequalities. Automatica, 32(10), 1361–1379].     

A tool to analyze robust stability for constrained nonlinear MPC

A sufficient condition for robust asymptotic stability of nonlinear constrained model predictive control (MPC) is derived with respect to plant/model mismatch. This work is an extension of a previous study on the unconstrained nonlinear MPC problem, and is based on nonlinear programming sensitivity concepts. It addresses the discrete time state feedback problem with all states measured. A strategy to estimate bounds on the plant/model mismatch is proposed that can be used off-line as a tool to assess the extent of model mismatch that can be tolerated to guarantee robust stability.     

Optimization over state feedback policies for robust control with constraints

This paper is concerned with the optimal control of linear discrete-time systems subject to unknown but bounded state disturbances and mixed polytopic constraints on the state and input. It is shown that the class of admissible affine state feedback control policies with knowledge of prior states is equivalent to the class of admissible feedback policies that are affine functions of the past disturbance sequence. This implies that a broad class of constrained finite horizon robust and optimal control problems, where the optimization is over affine state feedback policies, can be solved in a computationally efficient fashion using convex optimization methods. This equivalence result is used to design a robust receding horizon control (RHC) state feedback policy such that the closed-loop system is input-to-state stable (ISS) and the constraints are satisfied for all time and all allowable disturbance sequences. The cost to be minimized in the associated finite horizon optimal control problem is quadratic in the disturbance-free state and input sequences. The value of the receding horizon control law can be calculated at each sample instant using a single, tractable and convex quadratic program (QP) if the disturbance set is polytopic, or a tractable second-order cone program (SOCP) if the disturbance set is given by a 2-norm bound.     

Methodology for robust multi-parametric control in linear continuous-time systems

This paper presents an extension of the recent multi-parametric (mp-)NCO-tracking methodology by Sun et al. [Comput. Chem. Eng. 92 (2016) 64–77] for the design of robust multi-parametric controllers for constrained continuous-time linear systems in the presence of uncertainty. We propose a robust-counterpart formulation and solution of multi-parametric dynamic optimization (mp-DO), whereby the constraints are backed-off based on a worst-case propagation of the uncertainty using either interval analysis or ellipsoidal calculus and an ancillary linear state feedback. We address the case of additive uncertainty, and we discuss approaches to dealing with multiplicative uncertainty that retain tractability of the mp-NCO-tracking design problem, subject to extra conservativeness. In order to assist with the implementation of these controllers, we also investigate the use of data classifiers based on deep learning for approximating the critical regions in continuous-time mp-DO problems, and subsequently searching for a critical region during on-line execution. We illustrate these developments with the case studies of a fluid catalytic cracking (FCC) unit and a chemical reactor cascade.     

Synthesis of Mixed Objective Output Feedback Robust Model Predictive Control

Aiming at the constrained polytopic uncertain system with energy‐bounded disturbance and unmeasurable states, a novel synthesis scheme to design the output feedback robust model predictive control(MPC)is put forward by using mixed H₂/H_∞ design approach. The proposed scheme involves an offline design of a robust state observer using linear matrix inequalities(LMIs)and an online output feedback robust MPC algorithm using the estimated states in which the desired mixed objective robust output feedback controllers are cast into efficiently tractable LMI‐based convex optimization problems. In addition, the closed‐loop stability and the recursive feasibility of the proposed robust MPC are guaranteed through an appropriate reformulation of the estimation error bound (EEB). A numerical example subject to input constraints illustrates the effectiveness of the proposed controller.     

Robust discrete-time set-based adaptive predictive control for nonlinear systems

The problem of robust adaptive predictive control for a class of discrete-time nonlinear systems is considered. First, a parameter estimation technique, based on an uncertainty set estimation, is formulated. This technique is able to provide robust performance for nonlinear systems subject to exogenous variables. Second, an adaptive MPC is developed to use the uncertainty estimation in a framework of min–max robust control. A Lipschitz-based approach, which provides a conservative approximation for the min–max problem, is used to solve the control problem, retaining the computational complexity of nominal MPC formulations and the robustness of the min–max approach. Finally, the set-based estimation algorithm and the robust predictive controller are successfully applied in two case studies. The first one is the control of anonisothermal CSTR governed by the van de Vusse reaction. Concentration and temperature regulation is considered with the simultaneous estimation of the frequency (or pre-exponential) factors of the Arrhenius equation. In the second example, a biomedical model for chemotherapy control is simulated using control actions provided by the proposed algorithm. The methods for estimation and control were tested using different disturbances scenarios.     

Conditions under which suboptimal nonlinear MPC is inherently robust

We address the inherent robustness properties of nonlinear systems controlled by suboptimal model predictive control (MPC), i.e., when a suboptimal solution of the (generally nonconvex) optimization problem, rather than an element of the optimal solution set, is used for the control. The suboptimal control law is then a set-valued map, and consequently, the closed-loop system is described by a difference inclusion. Under mild assumptions on the system and cost functions, we establish nominal exponential stability of the equilibrium, and with a continuity assumption on the feasible input set, we prove robust exponential stability with respect to small, but otherwise arbitrary, additive process disturbances and state measurement/estimation errors. These results are obtained by showing that the suboptimal cost is a continuous exponential Lyapunov function for an appropriately augmented closed-loop system, written as a difference inclusion, and that recursive feasibility is implied by such (nominal) exponential cost decay. These novel robustness properties for suboptimal MPC are inherited also by optimal nonlinear MPC. We conclude the paper by showing that, in the absence of state constraints, we can replace the terminal constraint with an appropriate terminal cost, and the robustness properties are established on a set that approaches the nominal feasibility set for small disturbances. The somewhat surprising and satisfying conclusion of this study is that suboptimal MPC has the same inherent robustness properties as optimal MPC.