Stochastic model predictive control of nonlinear and uncertain systems

This thesis attempts to shed some additional light on pressingquestions regarding the control of uncertain systems. Specialfocus is given to systems with uncertain uncertainty (inexactlyknown distribution), numerical optimization methodsto enable the use of proposed advanced optimization methodsin practice and systems controlled by an economic controllerwhere stability is not always the primary objective.Current state-of-the-art methods often neglect that underlyinguncertainty in a stochastic model is, in fact, uncertainas well in the sense that its probability distribution functionis unknown and can only be (inaccurately) estimated fromdata. Hence, theoretical guarantees obtained by such methods,e.g., mean-square stability, may not be satisfied in practice.Moreover, many control methods use convex costs asa performance index to be optimized, which may not be themost descriptive choice for real-world problems. Here, weendeavour to remedy the shortcomings of such methods. Wefocus on three important extension in particular, i) theoreticaldevelopments to deal with non-convex performance indicesin stochastic optimal control problems and ii) novel methodsto deal with “uncertainty in the uncertainty” in a rigorousand theoretically sound way and iii) numerical optimizationmethods to solve these problems efficiently.Model predictive control (MPC) is an advanced control methodthat has found its way into many practical applications. Sinceits introduction and popularization in the 80’s in the processindustry, it has now taken a long way to automotive applications,large scale networks and robotics. MPC itself uses amathematical model of a system to predict its possible futuretrajectories. A sequence of control actions is then calculatedby solving an optimization problem by minimizing a performanceindex of the state and input cost along predicted trajectories. When the system moves to a new state, the new stateis sampled and the whole procedure is applied again. Part ofits popularity stems from the fact that the MPC frameworkcan also incorporate state and input constraints and handlemultiple input output systems naturally.Stochastic economic model predictive control is concernedwith problems with non-convex costs which are readily foundin real-world applications. Rather than minimizing a deviationfrom a prescribed (optimal/best) set-point or a trackingreference, the main objective is to optimize a given economiccost functional. The control paradigm that optimizes the processeconomics within the MPC formulation is usually knownas economic MPC (EMPC). Several research directions havediscussed the closed-loop properties of EMPC-controlled deterministicsystems, however, little have uncertain systemsbeen studied. In this thesis we propose EMPC formulationsfor nonlinear Markovian switching systems which guaranteerecursive feasibility, asymptotic performance bounds andconstrained mean square (MS) stability. For nonlinear systemswe provide design guidelines based on the system linearizationusing only mild assumptions on the system dynamicsand stage cost function.Risk-averse model predictive control is an approach to bridgethe gap between two popular control strategies dubbed stochasticand robust MPC. In robust MPC, modeling errors and disturbancesare assumed to be unknown-but-bounded quantitiesand the performance index is minimized with respectto the worst-case realization of the uncertainty (min-max) approach).However, such worst-case events which are unlikelyto occur in practice and render robust MPC severely conservativesince all statistical information, typically availablefrom past measurements, is completely ignored. On the otherhand, in stochastic MPC it is assumed that the underlying uncertaintyis a random vector following some probability distribution.In reality, not always can the probability distributionbe accurately estimated from available data, nor does itremain constant in time. Nonetheless, theoretical guaranteesof such algorithms hinge on this unrealistic assumption. Usingthe theory of risk measures, which originated in the fieldof stochastic finance, we devise a novel algorithmic and theoreticalsolutions to combine advantages of robust and stochasticoptimal control by proposing a unifying framework thatextends and contains both as special cases. In this thesis,we propose risk-averse formulations where the total cost ofthe MPC problem is expressed as a nested composition ofconditional risk mappings. We focus on constrained nonlinearMarkovian switching systems and derive Lyapunov-typerisk-averse stability conditions. Moreover, for the nonlinearsystem we prescribe a linearization based controller designprocedure and we show that linearized system locally inheritsstability properties of its linear counterpart.Finally, we propose a splitting for risk-averse problems whichmakes the problem a candidate for proximal algorithms. Usually,risk-averse problems are solved using stochastic dualdynamics programming approaches or generic interior pointmethod solvers. Both of these approaches are are not adept todeal with problems of large dimension. However, we showthat risk-averse problems posses a rich structure that we canexploit to devise very efficient and massively parallelisablemethods to solve them.