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Hybrid Automaton Model and Control of Hybird Systems.

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Dev.Chem. Eng. Mineral Process., 9(lR).pp.l43-150,200I.
Hybrid Automaton Model and Control of
Hybrid Systems
Wei Zhang* and Youxian Sun
National Laboratoly of Industrial Control Technology, Institute of
Industrial Process Control, Zhejiang University, Hangzhou 310027,
PR.China
A hybrid automaton model is proposed in this paper for hybrid systems. The model,
with particular emphasis on process control applications, is based on the dynamic
features of hybrid systems. It takes into account the discrete dynamics of hybrid
systems in particular and can clearly separate the controller j i o m the closed-loop
system. The definition of controllability of hybrid systems with respect to the marked
regions is also given. An analyzing algorithm and sufficient and necessary condition
for controllability based on the hybrid automaton model are discussed. At the same
time,the propem of the loops in the trajectory of the closed-loop plant is studted.
Finally, the synthesis scheme for a hybrid controller is given. In this scheme, the
controller consists of two parts, the discrete supervisor and the continuous regulator
The closed-loopplant with the controller is state-conmilable.
Introduction
Hybrid systems, which are pervasively existing in production processes, are
complicated with both continuous and discrete dynamics. Hybrid control systems
involve both the dynarmcs and continuous and discrete controls. In recent years, with
particular emphasis on process control applications, modeling, analysis and
optimization of hybrid systems have given rise to much research [ 1-81.
Antsaklis et al. [4] studied the hybrid control of continuous systems and proposed
a DES (discrete event system) plant model. The model is a simplification of a
continuous system and interface and the simplification results in a loss of information
about the internal dynamics, leading to nondeterministic behavior. Lennartson et al.
[ 5 ] presented a general model structure for hybrid systems with particular emphasis
on process control applications. They discussed the control of hybrid systems based
on this structure. Tittus and Egardt [6] defined the controllability of hybrid systems
and proposed a synthesis scheme for integrator hybrid systems, which are simpler.
correspondingly. In references [4-61,the control for hybrid systems has been studed
through discrete control commands. The controller generates the discrete commands
first,which are then imposed on the plant after being translated into a piecewise
constant continuous control. This necessarily reduces the ability of the controller to
regulate the system in real-time and also the nondeteministic behavior is inevitable.
Branicky [7] has discussed the hybrid features in real-world systems and presented a
fiamework for hybrid systems. In the light of the terms of the hnework, the hybrid
*Authorfor correspondence (email: wzhang@i@c.zju.edu.cn).
143
W.&ng
and Y. Sun
system is named by the general hybrid dynamical system under control. In thls
structure, the hybrid controller is designed and the optimal control of the system is
studied. Deshpande and Varraiya [S] have proposed a hybrid automaton model and
have studied the viable control of hybrid systems. There are no explicit controllers
which are modeled in [7] or [El.
In this paper, as a slight generalization of the previously suggested models [4-81, a
hybrid automaton model is presented first. Then the controllability of hybrid systems
related to the marked regions is studied. Furthermore, a controller synthesis scheme is
proposed. In this scheme, the controller is composed of two parts, the discrete
supervisor and the continuous regulator. The closed-loop plant controlled by the
controller is state-controllable.
The Hybrid Automaton Model
Hybrid Features in Process Control
Hybrid systems, such as variable structure control system, Bang-Bang control system,
computer control system, flexible manufacturing system, and chemical batch process,
are existing in production processes. In chemical industries, the material flows
typically go through a number of stages of various treatments, e.g. heating, mixing,
reaction, separation, cooling, etc. The occurrence of these treatments can be viewed as
a time-discrete event. This process obviously possesses the hybrid features. Hybrid
dynamic systems contain both continuous and discrete dynamics, the evolution of the
system depends on both of the dynamics. The continuous dynamics are usually
expressed by differential equations:
where x ( t ) is the continuous state vector, u(t) is the vector of external inputs. In a
complicated hybrid dynamic system, the continuous dynamics of the system at the
different stages is often described by various differential equations. We refer to these
stages as the phases of the system. When continuous states satisfy some conditions,
the system makes a transition from one phase to the next. In general, the discrete
dynamics of hybrid systems may contain discrete input-output, discrete external
control, switching of phase, etc. We can use the automaton or Petri net to model these
discrete phenomena. In the following section we will formally give a hybrid
automaton model to describe the dynamics of hybrid systems.
The Hybrid Automaton Model
The hybrid open-loop plant is defined as a hybrid automaton. It describes the system
dynamics using discrete-event system modehg and classical continuous state-space
representation. In different discrete states, the continuous state evolves according to
different continuous dynamic laws. Now, a hybrid automaton is formally stated as a
tuple:
where Q = Q p
x Q, is the f
~ t set
e of &Crete states and C = C, u C, is the finite set
Hybrid Automaton Model and Control of Hybrid Systems
of dscrete events. Q, is the set of control modes corresponding to different settings
of the plant's control variables. These control variables, in turn,are manipulated via
signals from the controller, the control events C, . Q p is the set of physical modes of
the open-loop plant itself contains. Each physical mode corresponds to some region
p i E P , where P denotes a partition of R" , induced by the plant's discrete features.
Each region p i is associated with a unique event w , E Z p ,when the continuous state
enters p i the event wp happens. This case is finished by an event-generator
r p : P + Z p . I c Q x R " is the set of the initial states and R" is the set of
continuous states. The state of hybrid systems is a pair (q,X ) E Q x R" , the discrete
state and the continuous state.
E c Q x A(R") x Z x {R" + R")x Q
(3)
is the finite set of edges, where A(*) denotes the set of all subsets of ( 0 ) . The edges
model the discrete dynamics of the system. An e E E is a state-transition function and
denoted as
I
x,,
we .Ye
(4)
1
we say e E E is enabled when the discrete state is q, and the continuous state is in
A', . When e is enabled and the event we E C happens, the transition through e is
taken. As a result, the continuous state is reset according to the map ye and the
system enters the discrete state q; . If the number of the q; is more than one, we say
the system is nondeteministic. When the discrete state is q E Q , the continuous state
evolves according to a controlled vector field
i,(t) =
f,( X C ( O J 4 t ) )
fq :
R" x R" + R" with
(5)
F is the set of f,:
F
= { f q ( x c ( W 4 0 ) l 4 E €?I
(6)
Given an open-loop plant model and a specification for the desired closed-loop
behavior, OUT purpose is to design a controller to restrict the behavior of the
controlled-plant to agree with the specification. In this paper, we study the
controllability of hybrid systems and design a hybrid controller which generates the
discrete and continuous control laws at the same time to transfer the hybrid plant
between predefined subsets of the hybrid state space. The closed-loop system is given
in Figurel, in which the part encircled by the dashed-lines is the open-loop plant.
Controllability of Hybrid Systems
Controllability means that the state of hybrid systems can be controlled to transfer
from one state to another. In this section, we are concerned with the controllability of
hybrid systems. At first, we give some definitions.
145
W.utang and Y.Sun
.....................................................
i
..................................................................
Figure 1. Closed-loopsystem.
Some Generic Definitions
Definition 1. A marked region A, = (q,,,,x,) is a region or singular points in the
hybrid state space Q x R" with q, E Q and X, G R" .
Definition 2. A path is an edges sequence e,,...,e, if forVi = 1,...,1-1 we have
qe, = qq-, .
Definition 3. A loop is a path with qe, = q;, . A path is A simple loop if it is a loop
andsatisfiesthat V z , j = 1 , . . . , I , i # j * q , f q , .
Definition 4. An acceptable trajectory of a hybrid system H is any trajectory of
H generated by some path connecting two marked regions A,, and
A,, ,R=A,,,,ce,,.--,e, >A,,,? , with e, E E that transfers H from some initial state
(qo.x,) E &* to some f i state ( q p x , ) E ;ln2 *
Definition 5. A hybrid system is said to be state-controllable with respect to the
marked regions A,, and Am2 if there exist hybrid control inputs that give H an
acceptable trajectory along some path IC . A hybrid system is said to be controllable
if any ordered pair of marked regions is state-controllable.
According to the above definitions, the necessary condition for controllable of any
ordered pair of marked regions is that there exists at least one path of the hybrid
system connecting the two marked regions. Denote the path as
x = A,, c e, ,..-,en> A,, . It is noted that a path may contain a possible loop. Loops
are interesting from the controllability point of view. If a path contains a loop, then
146
Hybrid Automaton Model and Control of Hybrid Systems
the number of the path connecting two marked regions may be infinite. Denote a path
with a loop in a generic form as:
IC
=A,,,,< e,,...,ei(e,,,e,,,...,elm,~i~*,ei+l,...,e,
> A,,*
(7)
where the operator (-)* means that the sequence inside the parenthesis can be
repeated an arbitrary number of times. Below we first analyze the suffkient and
necessary condition for controllabilityand then study the property found in loops.
Controllability Analysis
To investigate the sufficient and necessary condition for controllability, some
notations are needed, which will be introduced below.
For f, E F , let the state evolution map be $q : R, x R" + A(R"), i.e.,4,(r,x0)
is the set of continuous states that can be reached from xo at time t under the
controlled vector field
& 0 )= fq( x ,
( 0 3
m,x , (0)= xo
(8)
for some u(t) .Where R, is the set of nonnegative real numbers. Define the backward
projection of x E R" under bq as
P,W
I&{
=
x E $q(xo)}
(9)
and for X E R" define
a,(XI = u x c xa,(4
pq(X)is the set of continuous states that can reach X
q .For e E E define the operator n, :R"
n e ( y )=
(10)
Under the dynamical law of
+ R" by
x,nu;~(P~;
(y))
(11)
n , ( Y ) is the set of continuous states fiom which a transition through e can be taken
immediately (given the discrete state is 4,) and then Y can be reached after the
passage of some time in the discrete state 4;.where y,' (.) is the reversion of the
map ye(-). For the operator ne(Y) we have the following properties.
Proposition 1. n,(Y) is monotone, i.e., Y, c Yz
= n,(Y,) c n,(Y,) .
Proposition 2. n,(Y) distributes over U, i.e., n,(Y, UY,) = n , ( Y , ) U n , ( Y , ) .
Using these notations, we have the following backward algorithm to analysis
controllabihtyfor two given marked regions Aml and Am2.
to A,,,,: R .
Algorithm Step 1. search for a path connecting A,,,,
147
W.Zhang and Y. Sun
= nen-,(2)
next i
else goto Step4.
Step4. end
Utilizing the algorithm, if the n , (2) is not null then the controllabihty of the
hybrid systems is guaranteed by the following theorem.
Theorem 1. A hybrid system is controllable with respect to the marked regions
A,,,, and Am, if and only if there exist a path a = & c e , , . * . , e >
, Amz and hybrid
(n, n, --.ne.(x,,,~)).
controls such that x,,,, E pqml
qm I
To simplify the presentation, we denote ne1(riel
(nen(.))...) as
ncln,, - . - n e m ( - )According
,
to the definition of the operator n , ( Y ) and note that
Proof.
(.a-
qL1 = qeland q;, = qm2,
we have the following equalities.
then, we have accordingly ne, --n,-,(ne,nrl.--n,m)*ne,ne,+l
- n , ( x m 2 ). Denote the
sequence I?,, as
148
Hybrid Automaton Model and Control of Hybrid Systems
fin =('JN~,
i = 1,2,-.. n
.
i=O
Then, in terms of the monotonicity of the sequence, we have
Theorem 2. If the ,'A is bounded and ye is continuous, then the limit of the
sequence
A,
exists, l k f i n = N ' .
n-m
Once N * has been generated, we can omit the loop and substitute
riel .-.ne,-l"
for the ne, --.ne,-l( n , n , ,...n,m)*ne,ne,+l
- - - n e m ( x m 2then,
) , from which we can continue
our algorithm.
Controller Synthesis
Now according to the theorem 1., we give a hybrid controller for H as a map
S(q,x ) = ( S , ,S, ) : Q x R"
+ C x R" ,
where
S , = {we[ e E IC and x
E Ze(x,,)}
S, = {u I u E R"' and 3 e E IC, x
E b q ( Z e ( x m 2 ) and
)
x 4 Ze(x,,)}
The hybrid controller consists of two parts, discrete supervisor S , and the
continuous regulator S, . The dscrete supervisor S, takes the discrete actions to
ensure that the discrete state transitions will take place as expected. In every discrete
state the continuous regulator S, dnves the continuous state to the regions where the
edges we expected will be enabled. Under the control of the hybrid controller, the
controllability of the closed-loop with respect to the marked regions will be
guaranteed by the theorem 1.
Conclusions
We have considered the hybrid automaton model of hybrid systems. Our hybrid
automaton model is a slight modification of previously suggested structures. In this
model the open-loop plant is clearly separated from the controlled closed-loop plant.
This separation is very valuable from a control theoretical point of view. The
controllability of hybrid systems related to the marked regions has also been
discussed. An algorithm and sufficient and necessary condtion for controllability
have been proposed. Based on the condition a controller synthesis scheme has been
given. The controller issues the discrete and the continuous control commands based
on the discrete and the continuous feedback information of hybrid systems.
References
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Received: 28 October 1999;Accepted afier reversion: 17 May 2000.
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