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Abstract
This thesis focuses on the theoretical underpinnings of model predictive control (MPC) for linear stochastic systems. The plant model comprises a deterministic and stochastic part. Each part is modeled by a linear timeinvariant (LTI) system and parametrized through its respective transfer function (TRF). Only singleinputsingleoutput (SISO) systems are considered. We show how this setup gives rise to a linear stochastic state space model in continuous time comprising a linear stochastic diﬀerential equation (SDE). This is done by rigorous application of the distribution theory of Laurent Schwartz. We use the convention that continuoustime white noise should be what results by diﬀerentiating the sample paths of Brownian Motion in the sense of distributions. The derivation leads directly to the notion of Wiener integrals. It shows why the linear SDE framework does not conﬂict with deterministic system theory allowing distributionvalued input. The external behaviour of an LTI SISO system L is characterized by its impulse response h. Using distribution theory we show how the output of L is described by a Wiener integral in terms of h in the case of continuoustime white noise input. The derivations require that h be of locally ﬁnite variation. For deterministic linear systems the formula Ypsq“ HpsqUpsq relates the Laplace transforms Upsq and Ypsq of input and output respectively for a causal system at rest prior to the onset of excitation. It is assumed that the Laplace transform H of h exists and that h is locally of ﬁnite variation. We prove that the formula retains its validity also in case the input is the distributional derivative of a sample path of Brownian Motion. Consider a SISO LTI system L with square integrable h with continuoustime white noise input. The ﬁnitedimensional probability distributions of the output will converge with time to a family of distributions which in turn deﬁne a stationary process. We show that there exists a version of this process which has almost surely (a.s.) continuous sample paths, as long as h is globally of bounded variation. This is a special case of a general theorem on Gaussian processes, the proof of which is quite involved. Here we oﬀer a simpler proof for the relevant special case by exploiting a result from Fourier Analysis. We apply MPC to a linear stochastic system L in continuous time. Assuming equidistant sampling and zeroorder hold (ZOH) input, an equivalent discretetime linear stochastic model is established. We derive suﬃcient conditions on the TRF ensuring that the resulting Kalman ﬁlter be stabilizing, in particular that the relevant discretetime algebraic Riccati equation (DARE) has a unique positive deﬁnite stabilizing solution. Implementations of MPC in discretetime often feature an optimal control problem (OCP) with a cost function comprising a term which is quadratic in the input rate of change. We propose a continuoustime analogue of this OCP and for an LTI system in state space form. Suﬃcient conditions are derived for the minimizer of the continuoustime OCP to provide feedback that ensures nominal stability. Using MPC we treat the problem of minimizing the mean square tracking error for the output with respect to some reference trajectory. Assuming that the referecence trajectory is constant throughout each sampling interval we provide a transcription of this problem to an OCP in discrete time. This discretetime OCP is in a form which permits the use of Riccati iteration solver the complexity of which is approximately linear in the prediction horizon. Using work by C. Van Loan formulae are derived allowing for the eﬃcient calculation of the parameters of the discretetime OCP. We consider discretizations for the continuoustime OCP proposed. No constraints on the state vector are imposed but both input and input rate of change are subject to constraints. For ZOH discretization we establish the convergence of the minimizers punq of the discretized problems to the continuoustime solution u˚ P H1 as the (uniform) discretization step tends to zero. The convergence takes place in L2norm.
Original language  English 

Place of Publication  Kgs. Lyngby 

Publisher  Technical University of Denmark 
Number of pages  248 
Publication status  Published  2019 
Series  DTU Compute PHD2018 

Volume  491 
ISSN  09093192 
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Projects
 1 Finished

Model Predictive Control based on Stochastic Differential Equations  An Artificial Pancreas with Fast Insulin, Glucagon and Multiple Sensors
Hagdrup, M., Jørgensen, J. B., Madsen, H., Pedersen, M., Scherer, C. W., Cannon, M., Poulsen, B. & Poulsen, N. K.
Technical University of Denmark
01/09/2014 → 06/02/2019
Project: PhD