Hierarchical Control Theory

Hierarchical control can be interpreted as an attempt to handle complex problems by decomposing them into smaller subproblems and reassembling their solutions into a "functioning" hierarchical structure. So far, heuristic approaches have been prevalent. However, they cannot guarantee that the overall solution does indeed meet the specifications. In contrast, our project aims at a formal synthesis method that can provide such a guarantee.

Description

Complexity represents a major concern in many control problems, and it is common engineering knowledge that suitable decomposition techniques form a necessary ingredient for any systematic treatment of complex control problems. Hierarchical approaches, where several control layers interact, are a particularly attractive way of problem decomposition as they provide an extremely intuitive control architecture. Complexity problems are especially pronounced for hybrid control synthesis problems, and this has motivated the particular line of research described below.

In cooperation with Thomas Moor from Erlangen University and Jen Davoren from Melbourne University, we have developed a hierarchical control synthesis framework which is general enough to encompass both continuous and discrete levels. Unlike heuristic approaches, our synthesis framework guarantees that the control layers interact "properly" and do indeed enforce the overall specifications for the considered plant model. Its elegance stems from the fact that the specifications for lower control levels can be considered suitable abstractions which may be used as a basis for the synthesis of high-level controllers. Formulating specifications for the lower control levels may rely on engineering intuition. In fact, our approach allows to encapsulate engineering intuition within a formal framework, hence exploiting positive aspects of intuition while preventing misguided aspects from causing havoc within the synthesis step.

Two-level control architecture.

To keep exposition reasonably straightforward, we focus on the two-level control architecture shown in the figure on the right. Low-level control is implemented by an intermediate layer communicating with the plant via low-level (physical) signals $u L$ and $y L$ and with the high-level supervisor via high-level (abstract) signals $u H$ and $y H$ . Apart from implementing low-level control mechanisms corresponding to high-level commands $u H$ , the intermediate layer aggregates low-level measurement information $y L$ to provide high-level information $y H$ to the high-level supervisor. Aggregation may be both in signal space and in time, i.e., the time axis for high-level signals may be "coarser" than for low-level signals. We denote the behaviours of low-level plant model, intermediate layer and high-level control by $\mathfrak{B}_{\mathrm{p}}^{\mathrm{L}}$ , $\mathfrak{B}_{\mathrm{im}}$ , and $\mathfrak{B}_{\mathrm{sup}}^{\mathrm{H}}$ , respectively. Note that in this scenario $\mathfrak{B}_{\mathrm{p}}^{\mathrm{L}}$ and $\mathfrak{B}_{\mathrm{sup}}^{\mathrm{H}}$ are behaviours on $W L$ and $W H$ , while $\mathfrak{B}_{\mathrm{im}}$ is a behaviour on $W_{\mathrm{L}} \times W_{\mathrm{H}}$ , where $W_{\mathrm{H}} := U_{\mathrm{H}} \times Y_{\mathrm{H}}$ and $W_{\mathrm{L}} := U_{\mathrm{L}} \times Y_{\mathrm{L}}$ represent the high and low-level signal sets.

The control synthesis task then is to come up with a two-level controller $(\mathfrak{B}_{\mathrm{im}},\mathfrak{B}_{\mathrm{sup}}^{\mathrm{H}})$ such that (i) the overall controller $\mathfrak{B}_{\mathrm{im}}^{\mathrm{L}}[\mathfrak{B}_{\mathrm{sup}}^{\mathrm{H}}]$ restricts the plant behaviour to a given specification behaviour $\mathfrak{B}_{\mathrm{spec}}^{\mathrm{L}}$ , (ii) the overall controller is admissible to the plant model, (iii) the high-level controller is admissible to $\mathfrak{B}_{\mathrm{im}}^{\mathrm{H}}[\mathfrak{B}_{\mathrm{p}}^{\mathrm{L}}]$ , i.e., the plant under low-level control. In this context, admissibility of a controller means the following: the controller respects the plant's input/output structure, and any trajectory that plant and controller have "agreed on" in the past can be extended into the future.

In ^[1] and ^[2], we have discussed which properties of plant model and control layers will guarantee that admissibility conditions (ii) and (iii) hold. In particular, we have shown that very weak requirements on plant model and high-level supervisor will suffice for any combination of the intermediate layers shown in the figure below.

Intermediate layers implementing switching between low-level controllers.

Intermediate layers implementing measurement aggregation in time and signal space.

In fact, these types of intermediate layers are precisely the ones that are needed in most application problems. In ^[1],^[2], we have also proposed a bottom-up design procedure for $\mathfrak{B}_{\mathrm{im}}$ and $\mathfrak{B}_{\mathrm{sup}}^{\mathrm{H}}$ : in a first step, the intended relation between high-level and low-level signals is formalised by a specification $\mathfrak{B}_{\mathrm{spec}}^{\mathrm{HL}}$ . Clearly, it is in this step where engineering intuition is "embedded" into our formal framework. In a second step, using standard control synthesis methods, we need to design the intermediate layer such that the specification $\mathfrak{B}_{\mathrm{spec}}^{\mathrm{HL}}$ holds. In a third step, synthesis of high-level control is addressed. It can be based on a suitable abstraction of $\mathfrak{B}_{\mathrm{im}}^{\mathrm{H}}[\mathfrak{B}_{\mathrm{p}}^{\mathrm{L}}]$ , i.e. the plant model under low-level control, and a suitable translation of $\mathfrak{B}_{\mathrm{spec}}^{\mathrm{L}}$ into high-level specifications. In particular, we propose to use the high-level projection of $\mathfrak{B}_{\mathrm{spec}}^{\mathrm{HL}}$ as an abstraction of the plant model under low-level control. If both synthesis steps succeed, we can guarantee that the resulting overall controller indeed enforces the original specifications $\mathfrak{B}_{\mathrm{spec}}^{\mathrm{L}}$ for the underlying plant model $\mathfrak{B}_{\mathrm{p}}^{\mathrm{L}}$ .

In ^[2], we have also discussed how the remaining degrees of freedom can be used to address performance optimisation issues. The potential of our approach has been demonstrated by applying it to a multiproduct batch control problem, where the specification is to produce the desired product volumes with minimal cost subject to quality and safety constraints ^[3] ^[2].

We are currently mostly interested in the implications of tightening/relaxing subsystem specifications, in characterising achievable performance for specific control architectures, and in demonstrating the potential of the suggested approach by applying it to specific applications. For the latter purpose, a hybrid chemical engineering benchmark for hierarchical control ^[4] has been proposed by the Fachgebiet Dynamik und Betrieb technischer Anlagen (G. Wozny, H. Arellano-Garcia) at TU Berlin and our group.

Publications

↑ ^1.0 ^1.1

Thomas Moor, Jörg Raisch, Jen M. Davoren. Admissibility Criteria for a Hierarchical Design of Hybrid Control Systems. In Proc. ADHS03 — IFAC Conference on Analysis and Design of Hybrid Systems, pages 389–394, St. Malo, France, 2003.

Bibtex
Author : Thomas Moor, Jörg Raisch, Jen M. Davoren
Title : Admissibility Criteria for a Hierarchical Design of Hybrid Control Systems
In : In Proc. ADHS03 — IFAC Conference on Analysis and Design of Hybrid Systems,
pages 389–394, St. Malo, France, 2003.
Date : 2003

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↑ ^2.0 ^2.1 ^2.2 ^2.3

Jörg Raisch, Thomas Moor. Hierarchical Hybrid Control of a Multiproduct Batch Plant, volume 322 of Lecture Notes in Control and Information Sciences, pages 99–216. Springer-Verlag, 2005.

Bibtex
Author : Jörg Raisch, Thomas Moor
In : Hierarchical Hybrid Control of a Multiproduct Batch Plant, volume 322 of Lecture Notes in Control and Information Sciences, pages 99–216. Springer-Verlag, 2005.
Date : 2005

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↑

Thomas Moor, Jörg Raisch. Hierarchical Hybrid Control of a Multiproduct Batch Plant. In Proc. 16th IFAC World Congress, Prague, Czech Republic, 2005.

Bibtex
Author : Thomas Moor, Jörg Raisch
Title : Hierarchical Hybrid Control of a Multiproduct Batch Plant
In : In Proc. 16th IFAC World Congress,
Prague, Czech Republic, 2005.
Date : 2005

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↑

H. Arellano-Garcia, S. Geist, G. Wozny, J. Raisch. Benchmark for Hierarchical Plantwide Control of Hybrid Chemical Processes. In Proc. European Symposium on Computer Aided Process Engineering – ESCAPE20, pages 541–546, 2010.

Bibtex
Author : H. Arellano-Garcia, S. Geist, G. Wozny, J. Raisch
Title : Benchmark for Hierarchical Plantwide Control of Hybrid Chemical Processes
In : In Proc. European Symposium on Computer Aided Process Engineering – ESCAPE20,
pages 541–546, 2010.
Date : 2010

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X. David-Henriet, J. Raisch,, L. Hardouin. Consistent Control Hierarchies with Top Layers Represented by Timed Event Graphs. In Proc. of the 17th International Conference on Methods and Models in Automation and Robotics, IEEE, Międzyzdroje, Poland, 2012.

Bibtex
Author : X. David-Henriet, J. Raisch,, L. Hardouin
Title : Consistent Control Hierarchies with Top Layers Represented by Timed Event Graphs
In : In Proc. of the 17th International Conference on Methods and Models in Automation and Robotics, IEEE,
Międzyzdroje, Poland, 2012.
Date : 2012

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