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
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
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
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
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)
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
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
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 #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
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
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)
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
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
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
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/