Modeling Analysis and Design of Hybrid Control Systems Part I – … · 2007-05-01 · Modeling Analysis and Design of Hybrid Control Systems Part I – Zeno/Chatter-free Systems

Modeling Analysis and Design ofHybrid Control Systems

Part I – Zeno/Chatter-free Systems

João P. HespanhaUniversity of California

at Santa Barbara

Outline1. Deterministic hybrid systems2. Stochastic hybrid systems3. Switched systems

4. Stability of switched systems under arbitrary switching5. Controller realization for safe switching6. Stability under slow switching7. Stability of switched systems under state-dependent switching

8. Analysis tools for stochastic hybrid systems

Example #1: Bouncing ball

x1 ú y

t

x1 = 0 & x2 <0 ?

transition

guard or jump condition

state resetx2 ú – c x2

–

for any c < 1, there are infinitely many transitions in finite time (Zeno phenomena)

Free fall ≡

Collision ≡

Example #2: TCP congestion control

server clientnetwork

transmitsdata packets

receivesdata packets

TCP (Reno) congestion control: packet sending rate given bycongestion window (internal state of controller)

round-trip-time (from server to client and back)• initially w is set to 1• until first packet is dropped, w increases exponentially fast (slow-start)• after first packet is dropped, w increases linearly (congestion-avoidance)• each time a drop occurs, w is divided by 2 (multiplicative decrease)

packets droppedwith probability pdrop

congestion control ≡ selection of the rate r at which the server transmits packetsfeedback mechanism ≡ packets are dropped by the network to indicate congestion

r

Example #2: TCP congestion control

r bps

rate · B bps

s( t ) ≡ queue size

queue (temporary

data storage)

Example #3: Supervisory control

process

controller 1

controller 2yu

σ

2

1

σ

σ ≡ switching signal taking values in the set {1,2}

supervisor

bank of controllers

logic that selects which controller to use




Stochastic hybrid systems

transition intensities(probability of transition

in interval (t, t+dt])

q(t) ∈ Q = {1,2,…}≡ discrete statex(t) ∈ Rn ≡ continuous state

continuousdynamics

reset-maps

TCP with Stochastic drops

per-packetdrop prob.

pckts sentper sec

× pckts droppedper sec=

TCP (Reno) congestion control: packet sending rate given bycongestion window (internal state of controller)

round-trip-time (from server to client and back)• initially w is set to 1• until first packet is dropped, w increases exponentially fast (slow-start)• after first packet is dropped, w increases linearly (congestion-avoidance)• each time a drop occurs, w is divided by 2 (multiplicative decrease)

Stochastic hybrid systems with diffusion

stochasticdiff. equation

transition intensities

w≡ Brownian motion process

reset-maps

packet-switchednetwork

Example #4: Remote estimation

encoder decoder

white noisedisturbance

xx(t1) x(t2)

process state-estimator

for simplicity:• full-state available• no measurement noise• no quantization• no transmission delays

encoder ≡ determines when to send measurements to the network

decoder ≡ determines how to incorporate received measurements

packet-switchednetwork

Example #4: Remote estimation

encoder decoder

white noisedisturbance

xx(t1) x(t2)

process state-estimator

Error dynamics:

reset error to zero

prob. of sending data in [t,t+dt)depends on current error e

for simplicity:• full-state available• no measurement noise• no quantization• no transmission delays

[CDC’04, CRC Press’06]

Example #5: Ecology

For African honey bees: a1 = .3, a2 = .02, b1 = .015, b2 = .001 [Matis et al 1998]

Stochastic Logistic model for population dynamics

x(t) ≡ number of individuals of a particular species

probability of a birthin interval (t, t+dt]

probability of a deathin interval (t, t+dt]

x x

Example #6: Bio-chemical reactions

Decaying-dimerizing chemical reactions (DDR):

SHS model population of species S1

population of species S2

reaction rates

S2 0S1 0 2 S1 S2

c1 c2c3

c4




Switched systems with resets

parameterized family of vector fields ≡ fp : Rn → Rn p ∈ parameter setswitching signal ≡ piecewise constant signal σ : [0,∞) →

≡ set of admissible pairs (σ, x) with σ a switching signal and x a signal in Rn

t

σ = 1 σ = 3 σ = 2

σ = 1

switching times

A solution to the switched system is a pair (σ, x) ∈ for which1. on every open interval on which σ is constant, x is a solution to

2. at every switching time t, x(t) = ρ(σ(t), σ–(t), x–(t) )time-varying ODE

Asymptotic stability

equilibrium point ≡ xeq ∈ Rn for which fq(xeq) = 0 ∀ q ∈

class ≡ set of functions α : [0,∞)→[0,∞) that are1. continuous2. strictly increasing3. α(0)=0

s

α(s)

Definition:The equilibrium point xeq is (globally) asymptotically stable if it is Lyapunov stable and for every solution that exists on [0,∞)

x(t) → xeq as t→∞.

xeq

α(||

x(t 0)

–x e

q||)

||x(t 0

) –x e

q||

x(t)

t

Uniform asymptotic stability

Definition (class function definition):The equilibrium point xeq is uniformly asymptotically stable if ∃ β∈ :

||x(t) – xeq|| · β(||x(t0) – xeq||,t – t0) ∀ t≥ t0≥ 0along any solution (σ, x) ∈ to the switched system

xeq

β(||x

(t 0) –

x eq||

,0)

||x(t 0

) –x e

q||

x(t)

equilibrium point ≡ xeq ∈ Rn for which f(xeq) = 0

class ≡ set of functions β : [0,∞)×[0,∞)→[0,∞) s.t.1. for each fixed t, β(·,t) ∈2. for each fixed s, β(s,·) is monotone decreasing and β(s,t) → 0 as t→∞

s

β(s,t)

(for each fixed t)

t

β(s,t)(for each fixed s)

β(||x(t0) – xeq||,t)

t

We have exponential stabilitywhen

β(s,t) = c e-λ t swith c,λ > 0

β is independentof x(t0) and σ

Linear switched systems

vector fields and reset maps linear on x

Aq, Rq,q’∈ Rn× n q,q’∈

t

σ = 1 σ = 3 σ = 2

σ = 1

t0 t1 t2 t3

Theorem: For switched linear systems with state-independent switching, uniform asymptotic stability implies exponential stability (two notions are equivalent)




Stability under arbitrary switching

for some admissible switching signals the trajectories grow to infinity ⇒ switched system is unstable

unstable.m

Common Lyapunov function

Theorem:Suppose there exists a continuously differentiable, positive definite, radially unbounded function V: Rn → R such that

Then1. the equilibrium point xeq is Lyapunov stable2. if W(z) = 0 only for z = xeq then xeq is (glob) uniformly asymptotically stable.

The same V could be used to prove stability for all the unswitched systems

Algebraic conditions (linear systems)linear switched system

Theorem: If is finite all Aq, q ∈ are asymptotically stable and Ap Aq = Aq Ap ∀ p,q ∈

then the switched system is uniformly (exponentially) asymptotically stable

Theorem: If all the matrices Aq, q ∈ are asymptotically stable and upper triangular or all lower triangular then the switched system is uniformly (exponentially) asymptotically stable

(there exists a common Lyapunov function V(x) = x’ P x with P diagonal)

Theorem: I If there is a nonsingular matrix T ∈ Rn× n such that all the matricesBq = T Aq T– 1 (T–1Bq T = Aq)

are upper triangular or all lower triangular then the switched system is uniformly (exponentially) asymptotically stable

common similarity transformation

Lie Theorem actually provides the necessary and sufficient condition for the existence of such T ≡ Lie algebra generated by the matrices must be solvable




Example #11: Roll-angle control

θroll-angle

processu θset-point

controllerθreference

–

+ etrack

set-point control ≡ drive the roll angle θ to a desired value θreference

++

n

measurementnoise

Example #11: Roll-angle control

processu θset-point

controllerθreference

–

+ etrack

set-point control ≡ drive the roll angle θ to a desired value θreference

++

n

controller 1 controller 2

slow but not very sensitive to noise

(low-gain)

fast but very

sensitive to noise

(high-gain)

measurementnoise

Switching controller

σ = 2 σ = 1 σ = 2

How to build the switching controller to avoid instability ?

u θθreference

–

+ etrack

++

nmeasurement noiseσ switching signal taking

values in ú{1,2}

Switched closed-loop…

u θθreference

–

+ etrack

++

nmeasurement noiseσ

closed-loop system:

switching signal taking values in ú{1,2}

Theorem: For every family of controller transfer functions, there always exist a family a controller realizations such that the switched closed-loop systems is exponentially stable for arbitrary switching.

One can actually show that there exists a common quadratic Lyapunov function for the closed-loop.

In general the realizations are not minimal




Slow switching

switched linear systems

[τD] ≡ switching signals with “dwell-time” τD > 0, i.e., interval between consecutive discontinuities larger or equal to τD

Theorem:Assuming the sets {Aq : q ∈ } & { Rp,q : p, q∈ } are finite or compact.If all Aq, q ∈ are asymptotically stable, there exists a dwell-time τD such that the switched system is uniformly (exponentially) asymptotically stable over dwell[τD]

Slow switching on the average

switched linear systems

Theorem:Assuming the sets {Aq : q ∈ } & { Rp,q : p, q∈ } are finite or compact.If all the Aq, q ∈ are asymptotically stable, there exists an average dwell-time τD such that for every chatter-bound N0 the switched system is uniformly (exponentially) asymptotically stable over ave[τD, N0]

ave[τD, N0] ≡ switching signals with “average dwell-time” τD > 0 and “chatter-bound” N0 > 0, i.e.,

1. Same results would hold for any subset of ave[τD, N0]2. Some versions of these results also exist for nonlinear systems3. One may still have stability if some of the Aq are unstable,

provided that σ does not “dwell” on these values for a long time (switching under brief instabilities)

# of switchings in (τ,t)




Current-state dependent switching

no resets

[χ] ≡ set of all pairs (σ, x) with σ piecewise constant and x piecewise continuous such that ∀ t, σ(t) = q is allowed only if x(t) ∈ χq

Current-state dependent switching

χ ú {χq∈ Rn: q ∈ } ≡ (not necessarily disjoint) covering of Rn, i.e., ∪q∈ χq = Rn

χ1χ2

σ = 1 σ = 2

σ = 1 or 2

Thus (σ, x) ∈ [χ] if and only if x(t) ∈ χσ(t) ∀ t

Multiple Lyapunov functions

Given a solution (σ, x) and defining v(t) ú Vσ(t)( x(t) ) ∀ t ≥ 0

1. On an interval [τ, t) where σ = q (constant)

Vq : Rn → R, q ∈ ≡ family of Lyapunov functions (cont. dif., pos. def., rad. unb.)

2. But at a switching time t, where σ–(t) = p ≠ σ(t) = q,

v decreases

σ = 1 σ = 2 σ = 1 t

v=V1(x)

v=V2(x)v=V1(x)

σ = 1 σ = 2 σ = 1 t

v=V1(x)

v=V2(x)v=V1(x)

we would be okay if v would not increase at

switching times

Multiple Lyapunov functions

The Vq’s need not be positive definite and radially unbounded “everywhere”

It is enough that ∃ α1,α2∈ ∞: α1(||z||) · Vq(z) · α2(||z||) ∀ q ∈ , z ∈ χq

Theorem: ( finite)Suppose there exists a family of continuously differentiable, positive definite, radially unbounded functions Vq: Rn → R, q ∈ such that

Then1. the equilibrium point xeq is Lyapunov stable2. if W(z) = 0 only for z = xeq then xeq is (glob) uniformly asymptotically stable.

and at any z ∈ Rn where a switching signal in can jump from p to q

LaSalle’s Invariance Principle (ODE)

Theorem (LaSalle Invariance Principle):Suppose there exists a continuously differentiable, positive definite, radially unbounded function V: Rn → R such that

Then xeq is a Lyapunov stable equilibrium and the solution always exists globally.Moreover, x(t) converges to the largest invariant set M contained in

E ú { z ∈ Rn : W(z) = 0 }

M ∈ Rn is an invariant set ≡ x(t0) ∈ M ⇒ x(t) ∈ M ∀ t≥ t0

Note that:1. When W(z) = 0 only for z = xeq then E = {xeq }.

Since M ⊂ E, M = {xeq } and therefore x(t) → xeq ⇒ asympt. stability2. Even when E is larger then {xeq } we often have M = {xeq } and can

conclude asymptotic stability.

Linear systems

Theorem (LaSalle Invariance Principle–linear system, quadratic V):Suppose there exists a positive definite matrix P

A’ P + P A · – C’C · 0Then the system is stable. Moreover, x(t) converges to

M ∈ Rn is an invariant set if x(0) ∈ M ⇒ x(t) ∈ M ∀ t ≥ 0

When O is nonsingular, we have asymptotic stability(pair (C,A) is observable)

observability matrixof the pair (C,A)

Back to switched systems…

Theorem: ( finite)Suppose there exist positive definite matrices Pq∈Rn× n, q∈ such that

Aq’ Pq + Pq Aq · – Cq’Cq · 0 ∀ q∈and at any z ∈ Rn where a switching signal in [χ] can jump from p to q

z’ Pp z ≥ z’ R’q p PqRq p zThen the switched system is stable.Moreover, if every pair (Cq,Aq), q∈ is observable then1. if ⊂ weak-dwell then it is asymptotically stable2. if ⊂ p-dwell[τD,T] then it is uniformly asymptotically stable.

from general theorem

Sets of switching signalsdwell[τD] ≡ switching signals with “dwell-time” τD > 0, i.e., interval

between consecutive discontinuities larger or equal to τD

ave[τD, N0] ≡ switching signals with “average dwell-time” τD > 0 and “chatter-bound” N0 > 0, i.e.,

p-dwell[τD,T] ≡ switching signals with “persistent dwell-time” τD > 0 and “period of persistency” T > 0, i.e., ∃ infinitely many intervals of length ≥ τD on which sigma is constant & consecutive intervals with this property are separated by no more than T

weak-dwell ú ∪τD > 0 p-dwell[τD,+∞]≡ each σ has persistent dwell-time > 0

≥τD ≥τD· T · T ≥τD

dwell[τD] ⊂ ave[τD, N0] ⊂ p-dwell[γ τD,T] ⊂ weak-dwell ⊂ all




Generator of a SHS

Given scalar-valued function ψ : Q × Rn × [0,∞) → R

generator for the SHS

where

Lie derivative

reset term

Dynkin’s formula(in differential form)

diffusion term

Disclaimer: see Nonlinear Analysis’05 for technical assumptions

instantaneous variation

intensity

Lyapunov-based stability analysis

For constant rate: λ(e) = γ (exp. distributed inter-jump times)

1. E[ e ] → 0 if and only if γ > <[λ(A)]2. E[ || e ||m ] bounded if and only if γ > m <[λ(A)]

For polynomial rates: λ(e) = (e0 Q e)k Q > 0, k > 0 (reactive transmissions)

1. E[ e ] → 0 (always)2. E[ || e ||m ] bounded ∀m

getting more moments bounded requires higher comm. rates

Moreover, one can achieve the same E[ ||e||2 ]with less communication than with a constant

rate or periodic transmissions…

[CDC’04, Birkhauser’06]

error dynamicsin remote estimation

References

These slides were adapted from the courseECE229— Hybrid Control and Switched systems

taught at the University of California, Santa Barbara.

A fairly complete list of references can be found in the courses web page:http://www.ece.ucsb.edu/~hespanha/ece229/

but most of the material taught is covered by the following references:

[1] D. Liberzon. Switching Systems and Control. Birkhauser, Boston, MA, 2003.[2] J. Hespanha. Chapter Stabilization Through Hybrid Control. In Encyclopedia of Life Support Systems (EOLSS), 2004 [3] J. Hespanha. Uniform Stability of Switched Linear Systems: Extensions of LaSalle's Invariance Principle. IEEE TAC, 49(4):470-482, Apr. 2004. [4] J. Hespanha, A. S. Morse. Switching Between Stabilizing Controllers. Automatica, 38(11), Nov. 2002.

The references [2-4] can be found in the publications section ofhttp://www.ece.ucsb.edu/~hespanha/

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Modeling Analysis and Design of Hybrid Control Systems Part I – … · 2007-05-01 · Modeling Analysis and Design of Hybrid Control Systems Part I – Zeno/Chatter-free Systems