Granular Computing Application to Hybrid Systems Control

Francesco Rago Ph.D., Member IEEE,

M3 Comp. LLC Newark, DE USAfrancesco.rago@megatris.com

Summary

The purpose of this speech is to illustrate a possible approach to control complex systems.

The report presents a new approach to integrate event calculus with the control theory of hybrid systems using the glue of granular computing.

A successful management of a system with a large net of sensors requires a conceptual framework powerful enough to make correct predictions.

We have used granular computing to define granule of information starting from sensors net events. The information granules were able to have a sufficient level of abstraction to extract a meaning from events to put under control.

The control model is modelled by Piece Wise Hybrid System that consists of an automaton and, for each discrete mode, of an affine systems on a polyhedral set.

Complex cum + plexus = interweaved

“The field of complex systems focuses on certain questions about parts, wholes and relationships.” (http://necsi.edu/guide/study.html

) Complex system can be monitored using

sensors networks.

Complex Systems Definition

Sensor Networks: definition

A "sensor" is a transducer (or actuator) that provides functions to generate a correct representation of a sensed or controlled quantity (e.g., temperature, pressure, strain, flow, pH, etc.).

A sensors network is a web of sensor nodes that collect data on sensor field, process these data and send them to other sensor nodes or to a special node called gateway, which communicates with a base station through the Internet or an Intranet.

Monitoring generates a large amount of sensor data and efficiently dealing with this large amount of data in a resource-constrained wireless sensor network is a challenge

Hayek & Modeling

One of Hayek's main contributions to early systems complexity theory is his distinction between the human capacity to predict the behavior of simple systems and its capacity to predict the behavior of complex systems through modeling. He believed that complex phenomena could not be modeled after the sciences that deal with essentially simple phenomena like physics .Complex phenomena can only allow pattern predictions, through modeling.

Modelling

The structure of any system is often just as important in determining its behavior as the individual components themselves.

It is also claimed that in some cases the behavior of the whole cannot be explained in terms of the behavior of the parts.

In real world modeling is one of the powerful tool to manage complex systems, but requires to capture a large set of data using distribute sensors.

Distribute sensors environments are growing in complexity and the governance of complex and hybrid systems is difficult.

Granular Computing (GC) In a complex environment like a sensor network the

complexity level of events is huge. It is not possible to define an easy-to-manage relationships

matrix because the number of events and their complex structure.

A natural approach to fix the problem is granular computing that can help to aggregate events in meaningful clusters or granules.

Information granules are collections of entities that are arranged together due to their similarity or functional adjacency.

Information granules as all abstraction of our reality are aimed at building efficient views of the world and supporting perception of the physical world.

A primitive notion of granular computing is a granule representing a part of a whole. Like system theory, GC explores the composition of parts, their interrelationship, and connections to the whole. A real-world problem consists of a web of interacting and interrelated parts. In order to have a practical understanding and solution , it necessary to extract approximate structures that are tractable and easy to analyze.

GC exploits structures in terms of granules, levels, and hierarchies based on multi-level and multi-view representations.

Structured Thinking

“Granular computing attempts to unify reductionist thinking and systems thinking into structured thinking”( Y. Yao)

Model

A Model is a structure T=<MAT,EXP,TRA>MAT: mathematical or formal

modelEXP:experimental dataTRA: translation

Models Interaction

Theory 1 translation able in Theory 2Theorem: if A is a theorem of T1

the translation A* is a Theorem of T2

Example: A is related to events A* is related to Control model

Translation of primitive concepts

The Truth respect to the Model

A is a mathematical or logical relationship among variables g1,…gn

A(g1,…gn) is true respect to a physical system a measure results of g1,…,gn admit values that satisfy in MAT the relationship ASemantic consequences:

A=False not A(A&B)=True (A=True & B=True )

not vice versa

Affine System n

GWs

Events

d<,>

Q (Automata)

Affine System 1

G1

Thinking with Interacting Models GC + PL

…Gn

u/y

System

Granule Definition

G=<X,G,A,C>X a spaceG framework Ai:XG(X)

A=(A1,…,An) information granuleC family for communication procedures

C permits communication Between Granular Words

Decision systems are information systems of the form: U={U,At,d)U finite non empty set of objects At is a finite set of attributesd is a decision rules

Decision System

Decision Rules

V={Va;aAt} Vd

Atomic formulas over B At V descriptor on B

V are expression of the form a=v descriptor on V where a B and v Va

)()( vdva aBa

Events calculus

The Event Calculus was originally introduced by Kowalski and Sergot as a logic-programming framework for representing and reasoning about events.

The basic concepts of the model of time and change of Event Calculus are those of event and property.

Event Definition

O observable SO Space of values of O

X SO

(O,X) is an event

Event Calculus Formalization(R. Miller and M. Shanahan)

The Event Calculus can be described as a sorted predicate calculus with equality, with sorts: A for actions (variables a1, a2,…),

F for fluents (variables f1, f2,...),

T for time-points (integer variables t, t1, t2,…)

X for variables (x, x1, x2,…),

Properties are fluents, which hold during time periods initiated and terminated by events occurrences.

An event occurrence is represented by associating the event with time point at which it occurred by means of a clause.

The relation between events and properties can be defined by means of initiate and terminate clauses.

The concepts of property and event can be viewed as the Event Calculus counterparts of fluent and action of Situation Calculus of McCarthy.

Five predicates (others than equality):Happens AxTHoldsAt FxTInitiates AxFxTTerminates AxFxT< TxT

Auxiliary predicates

Clipped T x F x T (Fluent F terminates between time T1 and T2)

DeClipped T x F x T (Fluent F initiates between time T1 and T2)

GC EC

u F u DomU D D={P D : set of propositions in Event Calculus}

(description of DomU) A dependency is a set of valuations or decisions Dependency proposition in : P in D d(P)=True A state of a domain description D is a pair

<,v>, where is maximal among dependencies and relative to D and v is a decision.

A structure is a pair < , > where D is a dependency and is a partial function from strings of actions A into < ,v>.

}){,( dATU

Affine System n

GWs

Events

d<,>

Q (Automata)

Affine System 1

G1

Thinking with Interacting Models GC + PL

…Gn

u/y

System

How control a complex hybrid system There is not a unique solution Reductionist approach A complex and non linear systems can be controlled

with a sub-problems definition: Each sub-control problem can be solved using

affine linear system that approximate a larger non linear control algorithm

Piecewise Affine Hybrid System (PAHS) consists of an automaton and, for each discrete mode, of an affine system on a polyhedral set.

A GW is a problem or an aspect to control with a time scale and it permits to model part of control problem because it is less complex and limited to a specific plan or tool.

Piecewise-affine hybrid system (PAHS) A continuous piecewise-affine hybrid system consists at the discrete

level of a finite-state automaton and at the continuous level, at each state of the automaton, of an affine system on a polytope.

Every affine system is assumed to be controlled by an input function that in our case is the fuzzy or crisp result of valuation calculus on information related to granule sets.

When the continuous state reaches the boundary of the corresponding polyhedral set, a discrete event occurs, transferring the system to a new discrete mode. Also the continuous state is restarted and will continue to evolve according to the laws of the affine system corresponding to the new discrete mode.

The affine system and polytope is different for every discrete mode. In every discrete mode the discrete event that occurs upon leaving the corresponding polytope, depends on the facet through which the polytope is left.

Definition PAHS

PAHS =(Q,E,f,U, {(Xq,Aq)|qQ}, {(Gq(e),Rq(e)) |(q,e)

dom(f)} (Q,E,f) Automaton

f:QxE Q in combination with a |Q|-tuple of affine systems Aq=(Aq,Bq,aq)

Polytopes Xq(q Q) with input in the polytope U. The automaton and the affine system interacts via guard sets , Gq(e), each of

which is a union of finitely many facets, and affine reset maps Rq(e).

The evolution of the hybrid system satisfies the following affine finite difference equation on the full-dimensional polytope Xq with an affine control with xq Xq and u U

xq=Aqx(t)+Bqu(t)+aq,yq (t)=Cqxq(t)+Dq)u(t)+cq

A detailed description. A PAHS consists of A closed convex polyhedral input set U An observation set Y Rp

A finite discrete state set Q A set of discrete events, comprising a set

Ein of input events and a set of Ect of dynamivally generated events

A discrete state transiction function , which is a partial function QxE Q

For each discrete state qQ:A convex polyhedral continuous state

space XqRqn

A convex polyhedral initial state set Xq

initRqn

An affine systema Aq given by:

x=Aqx(t)+Bqu(t)+aq,An affine output map Cq:XqxUY given

by:

y (t)=Cqx(t)+Dq)u(t)+cq

For each event e E, and each discrete state q Q such that (e,q) is defined ,A closed convex polyhedral guard

set X(q,e)guard Xq

An affine continuous state transition function F(q,e): X(q,e)

guardX (e,q)

The state space X of PAHS is the set: qQ{q}x Xq

Whenever the continuous status leaves the polyhedral set Xq, a discrete event e occur, corresponding to the guard set Gq(e) that is crossed. The discrete transition according to the transition map f is :

).,,(

)(lim

exqfq

thenEeiforGsxxif

q

qqq

In the new discrete mode q+, the evolution of the new continuous state xq

+ is described by an other difference equation, with q replaced by q+, and with initial value x+.

Problem of Control of PAHS (Habets,van Shuppen)

If the Algorithm halts with QsQc Q the problem is solvable and a solution is given by the control law K={kq|q Qc}

Observability

A system is observable in time T if the initial state can be determined from the output function.

Reachability

Solutions to reachability and conditions for solvability of control problem:Habets, Van Shuppen: A control

problem for affine dynamical systems on a full dimensional polytope

An easy example

Fluent f0: man working in a room Rn Fluent f1: room Rn heated Fluent f3: room Rn controlled

action rtn : RFID whitelisted room Rn TAG=mmm entrance action ron : RFID whitelisted room Rn TAG=mmm exit action pck: power on computer Rn.Cm action phl : power on heating action pof : power off heating human choice action tim: temperature increases in Roomm action ow: window open action dc: room temperature decreases by man

Happens(rt1,t); Initiates(rt1,f0,t); Happens(ph,t); Initiates(rt1,f1,t);

HoldsAt(f0,t2)[Happens(rt1,t1) & Initiates(rt1,f0,t1) & (t1<t2 & Clipped(t1,f0,t2)]

HoldsAt(f0,t2)[Happens(rt1,t1) & Terminates(rt2,f0,t2) & (t1<t2 & Declipped(t1,f0,t2)]

HoldsAt(f4,t2)[Happens(rt2,t1) & Initiates(rt2,f0,t1) & (t1<t2 & Clipped(t1,f4,t2) & HoldsAt(f0,t2)]

Events

(Happen

s)

Fluent 0 Fluent 1 Fluent 3

Happens(rt,tx) Initiates(rt,f0,tx); NA Initiates(rt,f3,tx)

Happens(ph,tx) HoldsAt(ph,f0,tx) Initiates(ph,f1,tx) Opt(ph, f3,tx)

Happens(po,tx’); HoldsAt(po,f0,tx) Terminate(po,f1,t

x)Warning(po, f3,tx

)

Happens(dc,tx’); HoldsAt(dc,f0,tx) Chilled(dc,f1,tx) Warning(dc, f3,tx)

Happens(ti,tx’); HoldsAt(ti,f0,tx) Heated(ti,f1,tx) Opt(ti, f3,tx)

Happens(ow,tx’); HoldsAt(ph,f0,tx) Chilled(ow,f1,tx) Warning(ow, f3,tx)

Happens(ro,tx’); Terminated(ro,f0,tx);

NA Terminate(rt,f3,tx

)

1=(HoldsAt(f1,tn) HoldsAt(f0,tn+1) T>22C)

m: ActionsSet 1(d=v)

<1,v> q1Q x=Aq1x(t)+Bq1u(t)+aq1,

)()( vdva aBa

Experience

It was very hard to perform optimal abstractions to cover all aspects of a complex system because it is very difficult to define granule that are meaningful for a total system control. For this reason specific views of systems were chosen to have models related to control objectives.

The information granule were used to create aggregations of fluents and actions. The type of description and interaction dictates the level of granularity. In this sense, we regard information granules as a usefully vehicle of carry out efficient input to control computing.

The actions and fluents were defined together to predicates and decision rules. Linguistic variables and membership functions appropriately chosen and associated.

The discrete problem naturally suggests the Piecewise-Affine Hybrid Control Systems approach. The granules collection suggests a criteria to divide the system in pieces. For each piece was created the affine control(s) that works fine in almost stable situation with limited fluctuations.

Of course the holistic view can be lost if the guard set is not correct.

Conclusion

The report presents a new approach to integrate event calculus with the control theory of hybrid systems using the glue of granular computing. A successful management of a system with a large net of sensors requires a conceptual framework powerful enough to make correct predictions.

We have defined granule of information starting from sensors net events. The information granules were able to have a sufficient level of abstraction to extract a meaning from uncorrelated events.

Information Granules are conceptual building blocks usefull to describe the problem .

An other key result was the simplification of the problem, even if a general rule to define information granule to control a system is not definable in my knowledge at the actual status of art.

The non linearity and complexity of environments suggested a reductionist approach with Piecewise-Affine Hybrid Control System.

The granules collection suggested how to divide the system in pieces takings into account the relative significance of the available information (discrete and continuous).

Any questions?