Supervisory Control of Hybrid Systems

Supervisory Control of Hybrid Systems

Written by X. D. Koutsoukos et al.

Presented by Wu, Jian

04/16/2002

Outline

• Introduction

• Hybrid system modeling

• Supervisory control design

• Summary

Introduction

• Hybrid dynamical systems are characterized by interacting continuous and discrete dynamics.

• Hybrid control systems typically arise from computer-aided control of continuous processes. Its study is essential in designing the supervisory controller.

Introduction

Introduction

• The ‘plant’ is composed of the continuous process to be controlled and any continuous controllers.

• The ‘controller’ includes a discrete decision process that is typically a discrete-event system described by a finite automaton.

• The ‘interface’ makes it possible for these different processes to communicate with each other.

Introduction

• This supervisory control framework is quite flexible and can describe from the modern engineering systems to a switching control system.

• One of the main characteristics of the supervisory control has been the emphasis and explicit identification of the interface between the continuous and discrete dynamics.

Hybrid system modeling

• Continuous plant: The plant is in general a nonlinear, time-invariant system represented by a set of ordinary differential equations:

.

).(,)(.interval

),[,,

,)()(

))(),(()(

0 Xxstatesinitialforsolutionsofuniquenessand

existenceforconditionsthemeetsandXincontinuous

isffunctiontheRtrfixedeachFor

timesomebatwithRRRXandlyrespective

vectorsinputandstatetheareRtrandXtxwhere

trtxftx

mn


• This representation of the plant is quite general, e.g. a linear switching system consisting of m subsystems can be described by

.

1)(

,,1)(},1,0{:,

)()()(

1

1

1

1

timeat

actuatedisAsystemthethatimpliesrwhere

RtandtrRrRxwhere

txAtrtx

ii

m

i iin

m

iii


• Controller: the controller (supervisor) is a discrete event system that is modeled as a deterministic finite automaton. This automaton is specified by

.:,

:,

,,

),,,,(

~~

~~~~

~~

~~~

functionoutputtheisRSandfunctiontransition

statetheisSXSsymbolscontrollerofsettheisR

symbolsplantofsettheisXstatesofsettheisSWhere

RXSS


The action of the controller is described by the equations

where . The index n is analogous to a time index in that it specifies the order of the symbols in the sequence. The input and output signals associated with the controller are sequences of symbols.

])[(][

])[],1[(][~~

~~~

nsnr

nxnsns

~~~~~~

][,][,][ RnrandXnxSns


• Interface– The controller and plant cannot communicate directly

in a hybrid control system because each utilizes different type of signals. Thus, an interface is required that can convert continuous time signals to the sequences of the symbols and vice versa.

– To a great extent, the way that this conversion is accomplished determines the nature of the overall hybrid control system.

– The interface consists of the two simple subsystems: the generator and the actuator.


– The generator is the subsystem of the interface that converts the continuous-time output (state) of the plant to an symbolic input for the controller. To perform this task, two processes must be in place:

• A triggering mechanism is required, which will determine when a plant symbol should be generated. In the generator, the triggering mechanism is based on the idea of the plant events.

Let the sequence of plant events be denoted by e, where e[n]=i means the nth plant event was triggered by crossing the hypersurface defined by hi . Let the sequence of plant event instants be given by , where is the time of the nth plant event and .

e ][ne.0]0[ e


A simple way of expressing the conditions for the generation of plant events is by

Note: When the crossing occurs exactly at a point where

, one must use the following conditions:

0))(()/(

0))((

txhdtd

txh

i

i

0))(()/( txhdtd i


and

The first group contains three conditions: 1) at the time

of the plant event the plant state lies on the triggering hypersurface; 2) immediately after the event the plant state lies on the negative (positive) side of the triggering hypersurface; and 3) prior to reaching the triggering hypersurface, the plant lies on the positive (negative) side. The fourth condition concerns the ordering of the sequences. It requires that plant events be ordered chronologically and simultaneous plant events be ordered to their number.



• A process to determine which particular plant symbol should be generated is required. At each time in the sequence , a plant symbol is generated according to the function . The sequence of plant symbols can defined as

where i identifies the hypersurface that was crossed.

eventsilent

eventnonsilentnxx ei

,

])),[((~

][nei


– The actuator converts the sequence of controller symbols to a plant input signal, using the function as follows:

where is a characteristic function taking on the value of unity over the time interval and zero elsewhere is the time of the nth control symbol.

,:~

mRR

])1[],[,(])[()(0

~

nntInrtr ccn

),,( 21 tI

),[ 21 ][nc


• Example -- Thermostat/Furnace System:

The plant can be modeled as

The thermostat partitions the state space of the plant with two hypersurfaces as follows:


The associated functions are very simple in this case: . So there are two plant symbols: and

.

The output function of the controller is defined as

And the actuator operates as

21 and

ii xx~

)( 1

~

x2

~

x


Supervisory control design

• The objective is to develop methodologies that, given the system description and performance specifications, extract discrete-event controllers that supervises the plant to guarantee these specifications are satisfied.

• In the other words, in the supervisory control paradigm, the objective of the controller is to restrict the behavior of a given uncontrolled DES in order to satisfy prescribed specifications on the languages generated by the system.


• Given the DES plant and desired language K, the objective in the supervisory control problem for hybrid systems is to build a supervisor S such that L(S, G) K. In addition, it is required that the supervisor is maximally permissive, meaning that the language L(S,G) is as large as possible.

),,,,(~~~

RXPG


• Controllability:

when the above condition is true, the desired language K is said to be controllable, and provided that K is prefix-closed, a controller can be designed that will restrict the system to the language K.

_~_

KLXK u


• If the desired language K is not attainable for a given DES, it may be possible to find a more restricted language. Since we want the least restricted behavior, it is desirable the find the supremal element of the family of controllable sublanguages of K.

• The supremal controllable language is the largest subset of K that can be attained by a controller and can be found via following iterative procedure:


The basis of the algorithm is a fixed point iteration of a certain operator on languages. The largest fixed point of the iteration is computed by iterative applications of the operator.

Furthermore, the is a regular language, and therefore, it can be realized by a supervisor described by a finite automaton.

K


• Example – more complex DES plant model


• The language generated by this DES is

, where

• The control task for this DES is that let it never enter state . By simply removing the transitions to and then compute the resulting language, this desired language is therefore

mLL_

4

~

p

4

~

p


• In this example, the language K is not controllable. This can be seen by considering the string for which may generate and enter state . By using the iterative scheme, the supremal controllable sublanguage of K can be found,

Kxxx 6

~

3

~

2

~

5

~

x

4

~

p


Summary

• The general scheme of the hybrid control system is presented. Three main components ‘plant’, ‘controller’ and ‘interface’ are introduced.

• The supervisory control problem for hybrid system is formulated and algorithms for supervisory control design based on the controllability of the specification language have been discussed.

Documents

Supervisory Control of Hybrid Systems