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

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

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

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

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

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

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

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

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

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

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

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

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

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

Hybrid Automaton

Example 8 (Water Tank).

pump

Hybrid Autonmaton Lecture 2 (ECE7850 Sp17) Wei Zhang(OSU) 15 / 20

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

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

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

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

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