## Optimal controllers for hybrid systems: Stability and piecewise linear explicit form (2000)

Venue: | in Proceedings of the 39th IEEE Conference on Decision and Control |

Citations: | 25 - 7 self |

### BibTeX

@INPROCEEDINGS{Bemporad00optimalcontrollers,

author = {A. Bemporad and F. Borrelli and M. Morari},

title = {Optimal controllers for hybrid systems: Stability and piecewise linear explicit form},

booktitle = {in Proceedings of the 39th IEEE Conference on Decision and Control},

year = {2000},

pages = {1810--1815},

publisher = {IEEE}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this paper we propose a procedure for synthesizing piecewise linear optimal controllers for hybrid systems and investigate conditions for closed-loop stability. Hybrid systems are modeled in discrete-time within the mixed logical dynamical (MLD) framework[8], or, equivalently [7], as piecewise affine (PWA) systems. A stabilizing controller is obtained by designing a model predictive controller (MPC), which is based on the minimization of a weighted 1/∞-norm of the tracking error and the input trajectories over a finite horizon. The control law is obtained by solving a mixed-integer linear program (MILP) which depends on the current state. Although efficient branch and bound algorithms exist to solve MILPs, these are known to be NP-hard problems, which may prevent their on-line solution if the sampling-time is too small for the available computation power. Rather than solving the MILP on line, in this paper we propose a different approach where all the computation is moved off line, by solving a multiparametric MILP (mp-MILP). As the resulting control law is piecewise affine, on-line computation is drastically reduced to a simple linear function evaluation. An example of piecewise linear optimal control of the heat exchange system [16] shows the potential of the method.

### Citations

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Citation Context ...ogramming, multi-parametric programming 1 Introduction Hybrid systems provide a unified frameworkfor describing processes evolving according to continuous dynamics, discrete dynamics, and logic rules =-=[2, 3, 20]-=-. The interest in hybrid systems is mainly motivated by the large variety of practical situations, for instance realtime systems, where physical processes interact with digital controllers. Several mo... |

226 | Control of systems integrating logic, dynamics, and constraints
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Citation Context ...ewise linear optimal controllers for hybrid systems and investigate conditions for closed-loop stability. Hybrid systems are modeled in discrete-time within the mixed logical dynamical (MLD) framework=-=[8]-=-, or, equivalently [7], as piecewise affine (PWA) systems. A stabilizing controller is obtained by designing a model predictive controller (MPC), which is based on the minimization of a weighted 1/∞-n... |

179 |
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Citation Context ...ms exist to solve MIQP/MILP [15, 17, 23], these are known to be NP-hard problems. In this paper we propose a different approach where all the computation is moved off line, by extending the result of =-=[10]-=- for linear systems to the hybrid case. By formulating the MPC problem as the minimization of a weighted 1-norm of the tracking error and command input, and by treating the current state as a vector o... |

117 | Controllers for reachability specifications for hybrid systems
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Citation Context ...ogramming, multi-parametric programming 1 Introduction Hybrid systems provide a unified frameworkfor describing processes evolving according to continuous dynamics, discrete dynamics, and logic rules =-=[2, 3, 20]-=-. The interest in hybrid systems is mainly motivated by the large variety of practical situations, for instance realtime systems, where physical processes interact with digital controllers. Several mo... |

110 | A.: Reachability analysis of dynamical systems having piecewise-constant derivatives. Theoretical Computer Science 138
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Citation Context ...ogramming, multi-parametric programming 1 Introduction Hybrid systems provide a unified frameworkfor describing processes evolving according to continuous dynamics, discrete dynamics, and logic rules =-=[2, 3, 20]-=-. The interest in hybrid systems is mainly motivated by the large variety of practical situations, for instance realtime systems, where physical processes interact with digital controllers. Several mo... |

92 | Observability and controllability of piecewise affine and hybrid systems
- Bemporad, Ferrari-Trecate, et al.
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Citation Context ...ontrollers for hybrid systems and investigate conditions for closed-loop stability. Hybrid systems are modeled in discrete-time within the mixed logical dynamical (MLD) framework[8], or, equivalently =-=[7]-=-, as piecewise affine (PWA) systems. A stabilizing controller is obtained by designing a model predictive controller (MPC), which is based on the minimization of a weighted 1/∞-norm of the tracking er... |

54 | Optimal control of hybrid systems
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Citation Context ...esulting control law is piecewise affine, on-line computation is drastically reduced to a simple linear function evaluation. An example of piecewise linear optimal control of the heat exchange system =-=[16]-=- shows the potential of the method. Keywords: Hybrid systems, model predictive control, mixed-integer programming, multi-parametric programming 1 Introduction Hybrid systems provide a unified framewor... |

51 | Interconnected automata and linear systems: A theoretical framework in discrete-time
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Citation Context ...cently, in [7] the authors proved in a constructive way that MLD systems are equivalent to piecewise affine (PWA) systems, confirming the result of equivalence between hybrid and PWA systems shown in =-=[24]-=-. PWA systems are defined by partitioning the state space into polyhedral regions, and associating with each region a different linear dynamic equation. Besides the fact that PWA systems are an import... |

45 | S.: Numerical experience with lower bounds for MIQP branchand-bound
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Citation Context ...ar Programming (MILP) problems. The main drawbackof such a control approach are its intensive on-line computation requirements. Although efficient branch and bound algorithms exist to solve MIQP/MILP =-=[15, 17, 23]-=-, these are known to be NP-hard problems. In this paper we propose a different approach where all the computation is moved off line, by extending the result of [10] for linear systems to the hybrid ca... |

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Citation Context ... treated as free variables, an MILP is solved, which provides a new integer vector (see [14] for more details). The algorithmic implementation of the mp-MILP algorithm adopted in this paper relies on =-=[12]-=- for solving mp-LP problems, and on [17] for solving MILP’s. 5 An Example Consider the following hybrid control problem for the heat exchange example proposed by Hedlund and Ranzter [16]. The temperat... |

28 |
Piecewise linear optimal controllers for hybrid systems
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Citation Context ...f an expensive mixed-integer linear program. Needless to say, this makes the approach attractive for fast processes and/or low cost control hardware. Preliminary ideas along this approach appeared in =-=[5]-=-, where the authors included a terminal state constraint to ensure stability. In this paper, we remove such a constraint and provide conditions to choose the weights in the performance index so that t... |

27 |
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Citation Context ...ted at time t − 1, is necessary for the stability of the control law, as we will detail later. Such a condition, which in the linear case can be easily avoided by using an infinite prediction horizon =-=[25]-=-, may be restrictive in certain situations, for instance in case of disturbance rejection, where one should reset the bound on σ to some nonzero value whenever the arrival of a disturbance step is det... |

23 |
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Citation Context ...13) represents an upper bound on J(v0 ,σ,x(t)), where −1k is a column vector of ones of length k, and x(k|t) = A k x(t) + � k−1 j=0 Aj (B1uk−1−j + B2δk−1−j + B3zk−1−j). Similarly to what was shown in =-=[13]-=-, it is easy to prove that the vector p � {ε u 0,...,ε u T −1 ,εx 0,...,ε x T −1 ,εx T ,ut,...,ut+T −1,δt,..., δt+T −1,zt ...,zt+T −1,σ} that satisfies equations (13) and simultaneously minimizes J(p)... |

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An algorithm for the solution of multiparametric mixed integer linear programming problems
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Citation Context ...mmand input, and by treating the current state as a vector of parameters, the optimization problem can in fact be recast as a multiparametric MILP (mp-MILP), for which efficient solvers are available =-=[1, 14]-=-. After the solution of the mp-MILP has been determined, the resulting feedbackcontrol law is piecewise affine with respect to the state. Therefore on-line computation reduces to a simple linear funct... |

18 | Verification of hybrid systems via mathematical programming - Bemporad, Morari - 1999 |

15 |
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Citation Context ...zation procedure at time t + 1. In the next section we will show how to formulate the problem (5) as a mixed integer linear program (MILP). 2.1 Stability We remarkthat an infinite horizon formulation =-=[19, 22, 6]-=- would be inappropriate in the present hybrid context for both practical and theoretical reasons. In fact, approximating the infinite horizon with a large T is computationally prohibitive, as the numb... |

14 |
Computer generated Lyapunov functions for a class of nonlinear systems
- Ohta, Gong, et al.
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Citation Context ...teed in general. An alternative is to replace the final weight �Px(T |t)�∞ by a more general piecewise linear function of x(T |t), which would require one to find a piecewise linear Lyapunov function =-=[18, 21]-=- for the hybrid system (2). Note that, although the piecewise linear weight can still be tackled by mixed-integer linear programming, the complexity of the optimization problem increases. Remark 2 Giv... |

10 |
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- Rawlings, Muske
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Citation Context ...zation procedure at time t + 1. In the next section we will show how to formulate the problem (5) as a mixed integer linear program (MILP). 2.1 Stability We remarkthat an infinite horizon formulation =-=[19, 22, 6]-=- would be inappropriate in the present hybrid context for both practical and theoretical reasons. In fact, approximating the infinite horizon with a large T is computationally prohibitive, as the numb... |

8 |
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Citation Context ...mmand input, and by treating the current state as a vector of parameters, the optimization problem can in fact be recast as a multiparametric MILP (mp-MILP), for which efficient solvers are available =-=[1, 14]-=-. After the solution of the mp-MILP has been determined, the resulting feedbackcontrol law is piecewise affine with respect to the state. Therefore on-line computation reduces to a simple linear funct... |

6 |
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Citation Context ... algorithm for synthesizing P and {un} given Q and R (which provides the solution to a sort of “equivalent” ∞-/norm-based Lyapunov equation for PWA systems), although such an algorithm is reported in =-=[4]-=- for linear systems.sExample 2.1 We slightly modify the example in [5] so that we can exploit the result of [4] for computing matrix P satisfying (7). Consider the system ⎧ ⎪⎨ x(t +1) = � cos α(t) 0.7... |

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Citation Context ...teed in general. An alternative is to replace the final weight �Px(T |t)�∞ by a more general piecewise linear function of x(T |t), which would require one to find a piecewise linear Lyapunov function =-=[18, 21]-=- for the hybrid system (2). Note that, although the piecewise linear weight can still be tackled by mixed-integer linear programming, the complexity of the optimization problem increases. Remark 2 Giv... |

5 |
Branch And Reduce Optimization Navigator
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- 2000
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Citation Context ...ar Programming (MILP) problems. The main drawbackof such a control approach are its intensive on-line computation requirements. Although efficient branch and bound algorithms exist to solve MIQP/MILP =-=[15, 17, 23]-=-, these are known to be NP-hard problems. In this paper we propose a different approach where all the computation is moved off line, by extending the result of [10] for linear systems to the hybrid ca... |

2 |
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Citation Context ...zation procedure at time t + 1. In the next section we will show how to formulate the problem (5) as a mixed integer linear program (MILP). 2.1 Stability We remarkthat an infinite horizon formulation =-=[19, 22, 6]-=- would be inappropriate in the present hybrid context for both practical and theoretical reasons. In fact, approximating the infinite horizon with a large T is computationally prohibitive, as the numb... |

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Citation Context ...y can be approximated by a piecewise linear function, some classes of discrete event systems. Examples of real-world applications that can be naturally modeled within the MLD frameworkare reported in =-=[8, 9, 11]-=-. Recently, in [7] the authors proved in a constructive way that MLD systems are equivalent to piecewise affine (PWA) systems, confirming the result of equivalence between hybrid and PWA systems shown... |