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Modeling Analysis and Design of Hybrid Control Systems Part I - - PowerPoint PPT Presentation
Modeling Analysis and Design of Hybrid Control Systems Part I - - PowerPoint PPT Presentation
Modeling Analysis and Design of Hybrid Control Systems Part I Zeno/Chatter-free Systems Joo P. Hespanha University of California at Santa Barbara Outline 1. Deterministic hybrid systems 2. Stochastic hybrid systems 3. Switched
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Example #1: Bouncing ball
x1 ú y t x1 = 0 & x2 <0 ? transition guard or jump condition state reset x2 ú – c x2
–
for any c < 1, there are infinitely many transitions in finite time (Zeno phenomena) Free fall ≡ Collision ≡
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Example #2: TCP congestion control
server client network
transmits data packets receives data packets
TCP (Reno) congestion control: packet sending rate given by
congestion 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 dropped with probability pdrop
congestion control ≡ selection of the rate r at which the server transmits packets feedback mechanism ≡ packets are dropped by the network to indicate congestion r
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Example #2: TCP congestion control
r bps rate · B bps
s( t ) ≡ queue size queue (temporary data storage)
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Example #3: Supervisory control
process controller 1 controller 2 y u 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
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Outline
1. Deterministic hybrid systems 2. Stochastic hybrid systems 3. Switched systems 4. Stability of switched systems under arbitrary switching 5. Controller realization for safe switching 6. Stability under slow switching 7. Stability of switched systems under state-dependent switching 8. Analysis tools for stochastic hybrid systems
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Stochastic hybrid systems
transition intensities (probability of transition in interval (t, t+dt])
q(t) ∈ Q = {1,2,…} ≡ discrete state x(t) ∈ Rn ≡ continuous state
continuous dynamics reset-maps
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TCP with Stochastic drops
per-packet drop prob. pckts sent per sec × pckts dropped per sec =
TCP (Reno) congestion control: packet sending rate given by
congestion 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)
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Stochastic hybrid systems with diffusion
stochastic
- diff. equation
transition intensities w ≡ Brownian motion process reset-maps
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packet-switched network
Example #4: Remote estimation
encoder decoder
white noise disturbance
x
x(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
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packet-switched network
Example #4: Remote estimation
encoder decoder
white noise disturbance
x
x(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]
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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 birth in interval (t, t+dt] probability of a death in interval (t, t+dt]
x x
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Example #6: Bio-chemical reactions
Decaying-dimerizing chemical reactions (DDR): SHS model
population of species S1 population of species S2 reaction rates
S2 S1 2 S1 S2
c1 c2 c3 c4
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Outline
1. Deterministic hybrid systems 2. Stochastic hybrid systems 3. Switched systems 4. Stability of switched systems under arbitrary switching 5. Controller realization for safe switching 6. Stability under slow switching 7. Stability of switched systems under state-dependent switching 8. Analysis tools for stochastic hybrid systems
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Switched systems with resets
parameterized family of vector fields ≡ fp : Rn → Rn p ∈
parameter set
switching 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 which 1.
- n 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
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Asymptotic stability
equilibrium point ≡ xeq ∈ Rn for which fq(xeq) = 0 ∀ q ∈ class ≡ set of functions α : [0,∞)→[0,∞) that are
- 1. continuous
- 2. strictly increasing
- 3. 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(t0) – xeq||) ||x(t0) – xeq|| x(t) t
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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≥ 0 along any solution (σ, x) ∈ to the switched system xeq β(||x(t0) – xeq||,0) ||x(t0) – xeq|| 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 stability when β(s,t) = c e-λ t s with c,λ > 0
β is independent
- f x(t0) and σ
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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)
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Outline
1. Deterministic hybrid systems 2. Stochastic hybrid systems 3. Switched systems 4. Stability of switched systems under arbitrary switching 5. Controller realization for safe switching 6. Stability under slow switching 7. Stability of switched systems under state-dependent switching 8. Analysis tools for stochastic hybrid systems
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Stability under arbitrary switching
for some admissible switching signals the trajectories grow to infinity ⇒ switched system is unstable
unstable.m
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Common Lyapunov function
Theorem: Suppose there exists a continuously differentiable, positive definite, radially unbounded function V: Rn → R such that Then 1. the equilibrium point xeq is Lyapunov stable 2. 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
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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 matrices Bq = 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
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Outline
1. Deterministic hybrid systems 2. Stochastic hybrid systems 3. Switched systems 4. Stability of switched systems under arbitrary switching 5. Controller realization for safe switching 6. Stability under slow switching 7. Stability of switched systems under state-dependent switching 8. Analysis tools for stochastic hybrid systems
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Example #11: Roll-angle control
θ
roll-angle
process u θ set-point controller θreference – + etrack set-point control ≡ drive the roll angle θ to a desired value θreference + + n
measurement noise
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Example #11: Roll-angle control
process u θ set-point controller θreference – + 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) measurement noise
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Switching controller
σ = 2 σ = 1 σ = 2 How to build the switching controller to avoid instability ? u θ θreference – + etrack + + n
measurement noise
σ switching signal taking
values in ú{1,2}
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Switched closed-loop…
u θ θreference – + etrack + + n
measurement noise
σ 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
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Outline
1. Deterministic hybrid systems 2. Stochastic hybrid systems 3. Switched systems 4. Stability of switched systems under arbitrary switching 5. Controller realization for safe switching 6. Stability under slow switching 7. Stability of switched systems under state-dependent switching 8. Analysis tools for stochastic hybrid systems
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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]
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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 systems
- 3. 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)
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Outline
1. Deterministic hybrid systems 2. Stochastic hybrid systems 3. Switched systems 4. Stability of switched systems under arbitrary switching 5. Controller realization for safe switching 6. Stability under slow switching 7. Stability of switched systems under state-dependent switching 8. Analysis tools for stochastic hybrid systems
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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 χ1 χ2
σ = 1 σ = 2 σ = 1 or 2
Thus (σ, x) ∈ [χ] if and only if x(t) ∈ χσ(t) ∀ t
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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
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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 Then 1. the equilibrium point xeq is Lyapunov stable 2. 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
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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. stability 2. Even when E is larger then {xeq } we often have M = {xeq } and can conclude asymptotic stability.
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Linear systems
Theorem (LaSalle Invariance Principle–linear system, quadratic V): Suppose there exists a positive definite matrix P A’ P + P A · – C’C · 0 Then 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)
- bservability matrix
- f the pair (C,A)
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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 z Then the switched system is stable. Moreover, if every pair (Cq,Aq), q ∈ is observable then 1. if ⊂ weak-dwell then it is asymptotically stable 2. if ⊂ p-dwell[τD,T] then it is uniformly asymptotically stable.
from general theorem
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Sets of switching signals
dwell[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
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Outline
1. Deterministic hybrid systems 2. Stochastic hybrid systems 3. Switched systems 4. Stability of switched systems under arbitrary switching 5. Controller realization for safe switching 6. Stability under slow switching 7. Stability of switched systems under state-dependent switching 8. Analysis tools for stochastic hybrid systems
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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
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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 dynamics in remote estimation
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