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ECE7850: Lecture Note 2Modeling Frameworks for Hybrid Systems
Wei Zhang
Assistant ProfessorDepartment of Electrical and Computer Engineering
Ohio State University, Columbu, Ohio, USA
Spring 2017
Lecture 2 (ECE7850 Sp17) Wei Zhang(OSU) 1 / 20
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Outline
We will first review standard discrete and continuous system models, and thenintroduce hybrid system models.
• Finite State Automaton• Differential Equation/Inclusion• Hybrid Automaton• Other Hybrid System Models
Outline Lecture 2 (ECE7850 Sp17) Wei Zhang(OSU) 2 / 20
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Automaton
An Automaton A = (Q,Σ,→, Q0, Qm), where• Q: set of states (finite or infinite)• Σ: set of input symbols (labels, or alphabet)• →⊂ Q× Σ×Q: set of transitions• Q0: set initial states• Qm: set of marked states (or final state)
Finite state automaton:
Nondeterministic automaton:
Automaton Lecture 2 (ECE7850 Sp17) Wei Zhang(OSU) 3 / 20
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Example 1 (Vending machine).
(1) Insert coin; (2) Choose tea or coffee (3) Put the cup on the tray; (4) Makedrink
Example 2 (Slot machine).
Insert coin; (2) Pull handle; (3) Win if combination is good, lose otherwise
Automaton Lecture 2 (ECE7850 Sp17) Wei Zhang(OSU) 4 / 20
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Execution of Automaton: q0σ0q1σ1 · · · qN+1 with q0 ∈ Q0, qN+1 ∈ Qm, andqi
σi−→ qi+1
The trace (string) associated with an execution q0σ0q1σ1 · · · qN+1 is:
The collection of all traces of an automaton A is called the generated language ofA, denoted by L(A).
Automaton Lecture 2 (ECE7850 Sp17) Wei Zhang(OSU) 5 / 20
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Example 3.
Q0 = Qm = {1} What is the Language?
1 2
3
Questions in formal language theory:
• Is there a finite automaton that accepts a given language?• Do two automata accept the same language?• What is the smallest automaton that accepts a given language?
Automaton Lecture 2 (ECE7850 Sp17) Wei Zhang(OSU) 6 / 20
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Model for Continuous Dynamics
ODE: ẋ = f(t, x, u), with x(0) = x0
• x ∈ Rn: state• u ∈ Rm: control input• f : R+ × Rn × Rm → Rn: (time-varying) vector field
System output y = g(x, u)
Time-invariant autonomous system:
ẋ = f(x), with x(0) = x0 (1)
Model for Continuous Dynamics Lecture 2 (ECE7850 Sp17) Wei Zhang(OSU) 7 / 20
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Solution notions to ODE (1)
Classical solution on [0, t1]: x ∈ C1 such that:
Theorem 1 (Existence).
f : Rn → Rn continuous ⇒ classical solution exists for all ICs
Example 4 (discontinuous f with no classical solution).
f(x) =
{−1 x > 01 x ≤ 0
Model for Continuous Dynamics Lecture 2 (ECE7850 Sp17) Wei Zhang(OSU) 8 / 20
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Solution notions to ODE (1)
Example 5 (Nonunique classical solution).
f(x) =√|x|
Theorem 2 (Existence& Uniqueness).
If f : Rn → Rn is locally Lipschitz1, then exists a unique classical solution for all initialconditions
1Locally Lipschitz at x̂ ∈ Rn if ∃Lx̂, � ∈ (0,∞) s.t. ‖f(x)− f(x′)‖ ≤ Lx̂‖x− x′‖Model for Continuous Dynamics Lecture 2 (ECE7850 Sp17) Wei Zhang(OSU) 9 / 20
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Caratheodory solution
Definition 1 (Absolute Continuity).
f : [a, b]→ R is absolutely continuous, if there exists a Lebesgue integrablefunction g : [a, b]→ R such that f(t) = f(a) +
∫ tag(τ)dτ,∀t ∈ [a, b]
• If f absolutely continuous, then ḟ(t) exists and ḟ(t) = g(t) almost everywhere in the senseof Lebesgue measure.
Model for Continuous Dynamics Lecture 2 (ECE7850 Sp17) Wei Zhang(OSU) 10 / 20
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Caratheodory solution
Remark: What is Lebesgue measure and “almost everywhere”?
• Lebesgue measure: µ : E → [0,∞) roughly “volume” of E ⊆ Rn
• From you intuition: µ(Rectangle) =
• Lebesgue measure for arbitrary set:µ(E) = inf{
∑∞i=1 µ(Ri) : E ⊂ ∪iRi, Rirectangle in R
n}
• A function f : Rn → Rm satisfies a property P “almost everywhere” means:
Model for Continuous Dynamics Lecture 2 (ECE7850 Sp17) Wei Zhang(OSU) 11 / 20
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Caratheodory solution
Definition 2 (Caratheodory solution to (1)).
: x(t) absolutely continuous with ẋ(t) = f(x(t)) for almost all t in the sense ofLebesgue measure.
Example 6 (Existence of Caratheodory but no classical solution).
ẋ = f(x) =
1 x > 012 x = 0
−1 x < 0
Model for Continuous Dynamics Lecture 2 (ECE7850 Sp17) Wei Zhang(OSU) 12 / 20
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Differential inclusion
Differential inclusion: ẋ ∈ F (x)• F : Rn → 2R
n
: set valued map; Often written as: F : Rn−→→Rn
• Velocity can take multiple values at any given point
• Solution of differential inclusion (in the sense of Caratheodory): x absolutelycontinuous, and ẋ(t) ∈ F (x(t)) for almost all t
Example 7.
F (x) =
−1 x > 0[−1, 1] x = 01 x < 0
, with x(0) = 2
Model for Continuous Dynamics Lecture 2 (ECE7850 Sp17) Wei Zhang(OSU) 13 / 20
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Hybrid Automaton
• Hybrid systems: coupled discrete and continuous dynamics
• One well-adopted model: Hybrid automaton:
• H = (Q,X, f, Init,Dom,E,G,R)
- Q = {q1, q2, . . .}: set of discrete states
- X = Rn: continuous state space
- f(·, ·) : Q×X → Rn: mode-dependent vector field
- Init ⊆ Q×X: set of all possible intial ”hybrid state”
- Dom(·) : Q→ 2X : mode-dependent domains for continuous state
- E ⊆ Q×Q: set of edges (defining possible mode transitions)
- G(·) : E → 2X : Guard condition
- R(·, ·) : E ×X → 2X : reset map
Hybrid Autonmaton Lecture 2 (ECE7850 Sp17) Wei Zhang(OSU) 14 / 20
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Hybrid Automaton
Example 8 (Water Tank).
pump
Hybrid Autonmaton Lecture 2 (ECE7850 Sp17) Wei Zhang(OSU) 15 / 20
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Other Hybrid System Models
• More compact representation of hybrid automaton:{ẋ = f(q, x),
(q, x) = Φ(q−, x−), q ∈ Q, x ∈ X (2)
Water tank example revisited:
• Hybrid system with continuous/discrete controls:{ẋ = f(q, x, u),
(q, x) = Φ(q−, x−, σ), q ∈ Q, x ∈ X (3)
Other HS Models Lecture 2 (ECE7850 Sp17) Wei Zhang(OSU) 16 / 20
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Other Hybrid System Models
• Switched systems:- ẋ = fq(x), q ∈ Q
- ẋ = fσ(x), σ ∈ Q
• Variable Structure Systems: ẋ = fi(x), x ∈ Pi- Piecewise affine systems:
- Piecewise linear systems:
A1 =
[0 100 0
], A2 =
[1.5 2−2 −0.5
]
ẋ =
A1x x1 < 0&x2 ∈ [0.5x1,−0.25x1]A1x x1 ≥ 0&x2 ∈ [−0.25x1, 0.5x1]A2x otherwise
−5 0 5−3
−2
−1
0
1
2
3
Other HS Models Lecture 2 (ECE7850 Sp17) Wei Zhang(OSU) 17 / 20
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Other Hybrid System Models
• Most general hybrid system model: z , (x, q){ż ∈ F (z), z ∈ Cz+ ∈ G(z), z ∈ D
(4)
- C: flow set, F : flow map, D: jump set, G: jump map- The geometries of C and D produce rich hybrid dynamical phenomena- Reduces to (3) if F and G are singletons- Example (Water Tank):
pump
Other HS Models Lecture 2 (ECE7850 Sp17) Wei Zhang(OSU) 18 / 20
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Concluding Remarks
• Understand Caratheodory solution and differential inclusion
• Familiar with three types of hybrid system models:- Hybrid Automaton- more compact representation: (3)- more general hybrid system model (4)
• For a specific system, try to use the simplest model
• Further reading [Cor08; GST09; LST12]
• Next time: Execution and solution concepts of hybrid systems
Other HS Models Lecture 2 (ECE7850 Sp17) Wei Zhang(OSU) 19 / 20
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References
[Cor08] Jorge Cortes. “Discontinuous dynamical systems”. In: IEEE control Systems 28.3(2008).
[GST09] Rafal Goebel, Ricardo G Sanfelice, and Andrew R Teel. “Hybrid dynamical systems”.In: IEEE Control Systems 29.2 (2009).
[LST12] John Lygeros, Shankar Sastry, and Claire Tomlin. “Hybrid Systems: Foundations,advanced topics and applications”. In: under copyright to be published by SpringerVerlag (2012).
Other HS Models Lecture 2 (ECE7850 Sp17) Wei Zhang(OSU) 20 / 20
OutlineAutomatonModel for Continuous DynamicsHybrid AutomatonOther Hybrid System Models