On the importance of tailored modeling data for model-based control Xavier Bombois CNRS Laboratoire Amp` ere (Ecole Centrale de Lyon) IUAP study day - 28 November 2017 Xavier Bombois (CNRS) Identification for control IUAP study day 1 / 34
Introduction Without models, no advanced engineering Without models, no satisfactory control systems Engineering systems become more and more complex A perfect model is an illusion Good news: for a given objective (e.g. control), only a few of the system properties are really relevant to be modeled Specification for an appropriate model for control: a model with little uncertainty for these relevant properties; large uncertainty allowed for all other properties. Xavier Bombois (CNRS) Identification for control IUAP study day 2 / 34
Introduction Without models, no advanced engineering Without models, no satisfactory control systems Engineering systems become more and more complex A perfect model is an illusion Good news: for a given objective (e.g. control), only a few of the system properties are really relevant to be modeled Specification for an appropriate model for control: a model with little uncertainty for these relevant properties; large uncertainty allowed for all other properties. Xavier Bombois (CNRS) Identification for control IUAP study day 2 / 34
Introduction Without models, no advanced engineering Without models, no satisfactory control systems Engineering systems become more and more complex A perfect model is an illusion Good news: for a given objective (e.g. control), only a few of the system properties are really relevant to be modeled Specification for an appropriate model for control: a model with little uncertainty for these relevant properties; large uncertainty allowed for all other properties. Xavier Bombois (CNRS) Identification for control IUAP study day 2 / 34
Without data, no model Data are always required to determine/identify a model of a system Data obtained by applying an excitation signal at the input of the system and by measuring the effect of this excitation at the output The excitation signal perturbs the systems and leads to an economical cost - particularly important when the system is a production unit Objective: to obtain an appropriate model for control at a reasonable cost Xavier Bombois (CNRS) Identification for control IUAP study day 3 / 34
Without data, no model Data are always required to determine/identify a model of a system Data obtained by applying an excitation signal at the input of the system and by measuring the effect of this excitation at the output The excitation signal perturbs the systems and leads to an economical cost - particularly important when the system is a production unit Objective: to obtain an appropriate model for control at a reasonable cost Xavier Bombois (CNRS) Identification for control IUAP study day 3 / 34
Objective: to obtain an appropriate model for control at a reasonable cost Our solution: to design the excitation signal in an optimal way This signal must be optimized in order to uniquely capture those relevant system properties for control = ⇒ minimal economical cost leading nevertheless to an appropriate model for control Xavier Bombois (CNRS) Identification for control IUAP study day 4 / 34
Objective: to obtain an appropriate model for control at a reasonable cost Our solution: to design the excitation signal in an optimal way This signal must be optimized in order to uniquely capture those relevant system properties for control = ⇒ minimal economical cost leading nevertheless to an appropriate model for control Xavier Bombois (CNRS) Identification for control IUAP study day 4 / 34
Objective: to obtain an appropriate model for control at a reasonable cost Our solution: to design the excitation signal in an optimal way This signal must be optimized in order to uniquely capture those relevant system properties for control = ⇒ minimal economical cost leading nevertheless to an appropriate model for control Xavier Bombois (CNRS) Identification for control IUAP study day 4 / 34
Combination Identification and Control e.g. high level controller design for an industrial process disturbances PROCESS input= output excitation IDENTIFICATION EXPERIMENT The excitation signal perturbs the system and lead to an economical cost - that can e.g. be measured by the power of the excitation signal Xavier Bombois (CNRS) Identification for control IUAP study day 5 / 34
Combination Identification and Control e.g. high level controller design for an industrial process disturbances PROCESS input= output excitation IDENTIFICATION EXPERIMENT The excitation signal perturbs the system and lead to an economical cost - that can be e.g. measured by the power of the excitation signal Xavier Bombois (CNRS) Identification for control IUAP study day 5 / 34
Combination Identification and Control Identification disturbances Data → Uncertain model PROCESS input= output excitation IDENTIFICATION EXPERIMENT Performance evaluated a-priori by Control robustness analysis wrt. Model → Controller the model uncertainty Xavier Bombois (CNRS) Identification for control IUAP study day 6 / 34
Combination Identification and Control Identification disturbances Data → Uncertain model PROCESS input= output excitation IDENTIFICATION EXPERIMENT Performance evaluated a-priori by Control robustness analysis wrt. Model → Controller the model uncertainty Xavier Bombois (CNRS) Identification for control IUAP study day 6 / 34
Combination Identification and Control Identification disturbances Data → Uncertain model PROCESS input= output excitation IDENTIFICATION EXPERIMENT Performance evaluated a-priori by Control robustness analysis wrt. Model → Controller the model uncertainty Xavier Bombois (CNRS) Identification for control IUAP study day 6 / 34
Combination Identification and Control Identification disturbances Data → Uncertain model PROCESS input= output excitation IDENTIFICATION EXPERIMENT Performance MODEL-BASED Control CONTROL SYSTEM evaluated a-priori by disturbances robustness analysis Model → Controller wrt. output Controller PROCESS input the model uncertainty Xavier Bombois (CNRS) Identification for control IUAP study day 6 / 34
Identification of an appropriate model for control: a complex problem Identification and Robust Control: well established scientific fields (prediction error identification, H ∞ control, µ -analyzis), but in a rather independent way An optimal combination of these two fields is a complex problem In other words, obtaining appropriate models for control via identification is a difficult task Gevers, “Towards a joint design of identification and control?”, Essays on Control: Perspectives in the Theory and its Applications , 1993 Xavier Bombois (CNRS) Identification for control IUAP study day 7 / 34
Identification of an appropriate model for control the uncertainty of the identified model plays a central role in the interaction identification-control Xavier Bombois (CNRS) Identification for control IUAP study day 8 / 34
A bit of theory: uncertainty of an identified model True system: y ( t ) = G ( q , θ 0 ) u ( t ) + H ( q , θ 0 ) e ( t ) ⇒ Data: Z N = { u ( t ) , y ( t ) t = 1 ... N } Excitation signal u ( t ) = Estimation of θ 0 ∈ R k via prediction error identification: N H ( q , θ ) − 1 ( y ( t ) − G ( q , θ ) u ( t )) � 2 ˆ � � θ N = arg min θ t =1 ˆ θ N ∼ N ( θ 0 , P θ ) = ⇒ θ 0 ∈ U = { θ | ( θ − ˆ θ ( θ − ˆ θ N ) T P − 1 θ N ) < α } with a probability Pr ( χ 2 ( k ) < α ) Xavier Bombois (CNRS) Identification for control IUAP study day 9 / 34
A bit of theory: uncertainty of an identified model True system: y ( t ) = G ( q , θ 0 ) u ( t ) + H ( q , θ 0 ) e ( t ) ⇒ Data: Z N = { u ( t ) , y ( t ) t = 1 ... N } Excitation signal u ( t ) = Estimation of θ 0 ∈ R k via prediction error identification: N H ( q , θ ) − 1 ( y ( t ) − G ( q , θ ) u ( t )) � 2 ˆ � � θ N = arg min θ t =1 ˆ θ N ∼ N ( θ 0 , P θ ) = ⇒ θ 0 ∈ U = { θ | ( θ − ˆ θ ( θ − ˆ θ N ) T P − 1 θ N ) < α } with a probability Pr ( χ 2 ( k ) < α ) Xavier Bombois (CNRS) Identification for control IUAP study day 9 / 34
A bit of theory: uncertainty of an identified model True system: y ( t ) = G ( q , θ 0 ) u ( t ) + H ( q , θ 0 ) e ( t ) ⇒ Data: Z N = { u ( t ) , y ( t ) t = 1 ... N } Excitation signal u ( t ) = Estimation of θ 0 ∈ R k via prediction error identification: N H ( q , θ ) − 1 ( y ( t ) − G ( q , θ ) u ( t )) � 2 ˆ � � θ N = arg min θ t =1 ˆ θ N ∼ N ( θ 0 , P θ ) = ⇒ θ 0 ∈ U = { θ | ( θ − ˆ θ ( θ − ˆ θ N ) T P − 1 θ N ) < α } with a probability Pr ( χ 2 ( k ) < α ) Xavier Bombois (CNRS) Identification for control IUAP study day 9 / 34
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