## Optimal Control of Switched Systems with Application to Networked Embedded Control SystemsThis thesis addresses optimal control of discrete-time switched linear systems with application to networked embedded control systems (NECSs). Part I focuses on optimal control and scheduling of discrete-time switched linear systems. The objective is to simultaneously design a control law and a switching (scheduling) law such that a cost function is minimized. This optimization problem exhibits exponential complexity. Taming the complexity is a major challenge. Two novel methods are presented to approach this optimization problem: Receding-horizon control and scheduling relies on the receding horizon principle. The optimization problem is solved based on relaxed dynamic programming, allowing to reduce complexity by relaxing optimality within predefined bounds. The solution can be expressed as a piecewise linear (PWL) state feedback control law. Stability is addressed via an a priori stability condition based on a terminal weighting matrix and several a posteriori stability criteria based on constructing piecewise quadratic Lyapunov functions and on utilizing the cost function as a candidate Lyapunov function. Moreover, a region-reachability criterion is derived. Periodic control and scheduling relies on periodic control theory. Both offline and online scheduling are studied. The optimization problem is solved based on periodic control and exhaustive search. The online scheduling solution can again be expressed as a PWL state feedback control law. Stability is guaranteed inherently. Several methods are proposed to reduce the online complexity based on relaxation and heuristics. Part II focuses on optimal control and scheduling of NECSs. The NECS is modeled as a block-diagonal discrete-time switched linear system. Various control and scheduling codesign strategies are derived based on the methods from Part I regarding the structural properties of NECSs. The methods presented in Part I and II are finally evaluated in a case study. |

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### Contents

3 | |

FiniteHorizon Control and Scheduling | 21 |

Periodic Control and Scheduling | 61 |

26 | 66 |

42 | 72 |

Conclusions and Future Work | 95 |

Optimal Control of Networked Embedded Control Systems101 6 Introduction | 103 |

Modeling | 111 |

Control and Scheduling Codesign | 121 |

Case Study | 137 |

Conclusions and Future Work | 149 |

A Supplementary Material | 155 |

Bibliography | 161 |

Zusammenfassung | 179 |

### Common terms and phrases

ˆLk admissible p-periodic switching Aj(k Bellman equation closed-loop stability closed-loop system 3.8 computation and communication considered continuous-time control and scheduling control theory control vector controller index cost function cost resulting discrete-time DPRE DRE solutions embedded control systems feedback control law fulﬁlled Furthermore inequality instant inverted pendulums Jp,adm Jsim Lyapunov function mean cost NECS NECS model NECSs networked embedded control number of DRE offline online complexity online scheduling optimal control optimization problem p-periodic switching sequence PCSoff PCSon strategy periodic control PLQR Problem 4.2 proof Pruning Algorithm PWA systems PWL closed-loop system PWQ Lyapunov function quadratic forms receding-horizon control regions relaxed dynamic programming relaxed pruning relaxed value function Remark RHCS Riccati equation RRHCS strategy sampling periods scheduling codesign Section sequence j(k stability condition stabilizable subsystems switched system 1.3 switching index switching sequence j∗(k symmetric task sequence terminal weighting matrix Theorem 3.6 time-varying system tk+1