ESE557 Spring 2015: Hybrid Dynamic Systems
http://www.ese.wustl.edu/~hgonzale/ese557
Lecturer: Humberto Gonzalez
Scribe: Christa Stathopoulos
Lecture 6 — January 29
6.1
Hybrid Systems
Thus far, we have studied continuous- and discrete-variable systems in isolation. Hybrid dynamical
systems are the class of systems whose variables of interest, i.e., states, inputs, and outputs, are a
combination between continuous and discrete. The biggest challenge we have ahead is to recover
as many useful properties from both continuous- and discrete-variable systems as possible (such as
uniqueness, time reversibility, continuity w.r.t. initial conditions, etc.) in our definition of hybrid
system. As we will study in the next couple of weeks, some properties follow naturally from our
definitions, while others are quite complex to even theoretically define, let alone to numerically
validate. Moreover, we add the completely new phenomena of Zeno executions, which do not exist
in either of the two original classes of systems.
Before we can define a hybrid automaton, we must first introduce several definitions that will
help us with the notation.
Definition 6.1. A hybrid time set is a sequence of intervals, τ = {Ik }N
k=0 , with N ∈ N ∪ {∞}, such
that:
1. Ik = [τk , τk0 ] for each k < N ;
2. τk ≤ τk0 = τk+1 for each k < N ; and,
0 < ∞ and I = [τ , τ 0 ], or I = [τ , ∞).
3. If N < ∞, then either τN
N
N N
N
N
We also define the number of transition of τ as hτ i = N , and the length of τ as kτ k =
Phτ i
0
k=0 τk − τk .
An illustration of three hybrid time sets is shown in Figure 6.1. As we will see later in this
lecture, the intuition behind Definition 6.1 is that each interval Ik corresponds to the time spent
by an execution in a single mode. Thus, hτ i counts the number of discrete transitions (or jumps)
of the execution, while kτ k is the total time spent in the execution.
Definition 6.2. Given a hybrid time set τ , we say that a pair (k, t) ∈ N × R is a hybrid time in τ ,
denoted (k, t) ∈ τ , iff t ∈ Ik .
Definition 6.3. We classify a hybrid time set τ as follows:
• τ is finite iff hτ i < ∞ and kτ k < ∞;
• τ is infinite iff kτ k = ∞; and,
• τ is Zeno iff hτ i = ∞ and kτ k < ∞.
Using this classification, the hybrid time sets from Figure 6.1 are either finite (τcenter ), or infinite
(τleft and τright ).
Even though the classification in Definition 6.3 will help us describe some results in the future, it
does have one problem. Execution with hτ i = kτ k = ∞ are classified as infinite under the intuition
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ESE557 Spring 2015
Lecture 6
k
3
τ3
2
1
0
τ1
τ0
k
k
3
3
2
τ2 = τ20
0
τ00
2
τ2 = τ20
1
τ10
January 29
τ1
τ0
τ10
0
τ00
t
τ2
1
τ1
τ0
τ10
τ00
t
t
Figure 6.1: Example of three hybrid time sets, denoted τleft , τcenter , and τright , respectively.
that such an execution does not have Zeno behavior. But in the case of a system which chatters for
a finite amount of time (e.g., a thermostat without hysteresis, as in the first part of Example 1.1,
whose set-point is suddenly changed), we can first “enter” and then “exit” a Zeno situation, and
this execution is hardly comparable with one where hτ i < ∞ and kτ k = ∞, which is also classified
as infinite. For the time being we will stay with the classification in Definition 6.3, but it should
be noted that a finer distinction is needed if we want to fully understand the phenomena of Zeno
executions.
Definition 6.4. Let (k1 , t1 ), (k2 , t2 ) ∈ τ be two hybrid times. We say that (k1 , t1 ) precedes (k2 , t2 ),
denoted (k1 , t1 ) ≤ (k2 , t2 ), iff either k1 = k2 and t1 ≤ t2 , or k1 < k2 .
Lemma 6.5. The binary relation ≤ defines a total ordering on each hybrid time set τ .
Proof. We only provide a sketch of the argument. Let {(kj , tj )}3j=1 . We need to show the following
properties:
• transitivity: if (k1 , t1 ) ≤ (k2 , t2 ) and (k2 , t2 ) ≤ (k3 , t3 ), then (k1 , t1 ) ≤ (k3 , t3 );
• antisymmetry: if (k1 , t1 ) ≤ (k2 , t2 ) and (k2 , t2 ) ≤ (k1 , t1 ), then (k1 , t1 ) = (k2 , t2 ); and,
• totality: either (k1 , t1 ) ≤ (k2 , t2 ) or (k2 , t2 ) ≤ (k1 , t1 ).
The proof of each of these properties is simple, and therefore left as an exercise.
Definition 6.6. Let τ, τ̂ be two hybrid time sets. We say that τ is a prefix of τ̂ , denoted τ ⊆ τ̂ ,
iff hτ i ≤ hτ̂ i, Ik = Iˆk for each k < hτ i, and Ihτ i ⊂ Iˆhτ i .
For example, using the hybrid time sets in Figure 6.1, τcenter ⊆ τleft , τcenter ⊆ τright , but neither
τleft is a prefix of τright nor vice versa.
Lemma 6.7. The binary relation ⊆ defines a partial ordering on the collection of all hybrid time
sets.
Proof. Exercise. Just need to prove transitivity and antisymmetry.
We can now formally define a hybrid dynamical system, and a hybrid dynamical execution.
First, let us consider the case of systems with no inputs.
Definition 6.8. We define an autonomous hybrid dynamical system (or an hybrid automaton) as
H = Q, D, Dinit , Dfinal , f, R, ∆ , where:
(1) Q is a finite set, denoted the discrete domain set;
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Lecture 6
January 29
(2) D ⊂ Rn is open, denoted the continuous domain set;
(3) Dinit ⊂ Q × D, denoted the initial condition set;
(4) Dfinal ⊂ Q × D, denoted the final condition set;
(5) f : Q × D → Rn , denoted the vector field ;
(6) R : Q × D → 2Q×D , denoted the reset map; and,
(7) ∆ : Q → 2D is a closed map, denoted the domain map.
Definition 6.9. Let H be a hybrid dynamical system. We say that the tuple (τ, q, x), where τ is
a hybrid time set, q : N → Q, and x : [0, ∞) → Rn , is an execution of H iff:
(1) q(t) = q(τk ) for each t ∈ Ik and each finite k ≤ hτ i;
(2) x(t) ∈ ∆ q(t) for each t ∈ τ ;
(3) q(0), x(0) ∈ Dinit ;
(4) ẋ(t) = f q(t), x(t) for each t ∈ Ik and each finite k ≤ hτ i;
(5) q(τk+1 ), x(τk+1 ) ∈ R q(τk0 ), x(τk0 ) for each k < hτ i; and,
(6) if kτ k < ∞ then q(kτ k), x(kτ k) ∈ Dfinal .
This definition is helpful in some regards. For example, it allows us to define block. But the
definition is also notationally cumbersome.
6–3