Modeling Analysis and Design of Hybrid Control Systems Part I – Zeno/Chatter-free Systems João P. Hespanha University of California at Santa Barbara
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
Example #1: Bouncing ball Free fall ≡ x 1 ú y Collision ≡ for any c < 1, there are infinitely many transitions in finite time (Zeno phenomena) t guard or jump condition x 1 = 0 & x 2 <0 ? transition x 2 ú – c x 2 – state reset
Example #2: TCP congestion control transmits receives data packets data packets server client network r packets dropped with probability p drop congestion control ≡ selection of the rate r at which the server transmits packets feedback mechanism ≡ packets are dropped by the network to indicate congestion 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)
Example #2: TCP congestion control queue (temporary data storage) r bps rate · B bps s ( t ) ≡ queue size
Example #3: Supervisory control logic that selects supervisor which controller to use bank of controllers σ controller 1 process u y controller 2 σ ≡ switching signal taking values in the set {1,2} σ 2 1
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
Stochastic hybrid systems continuous dynamics transition intensities (probability of transition in interval ( t , t + dt ] ) reset-maps q ( t ) ∈ Q = {1,2,…} ≡ discrete state x ( t ) ∈ R n ≡ continuous state
TCP with Stochastic drops per-packet pckts dropped pckts sent × = drop prob. per sec 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)
Stochastic hybrid systems with diffusion stochastic diff. equation transition intensities reset-maps w ≡ Brownian motion process
Example #4: Remote estimation process state-estimator white noise disturbance x x ( t 1 ) x ( t 2 ) decoder encoder packet-switched network for simplicity: • full-state available • no measurement noise • no quantization • no transmission delays decoder ≡ determines how to incorporate received measurements encoder ≡ determines when to send measurements to the network
Example #4: Remote estimation process state-estimator white noise disturbance x x ( t 1 ) x ( t 2 ) decoder encoder packet-switched network for simplicity: • full-state available • no measurement noise • no quantization • no transmission delays Error dynamics: prob. of sending data in [ t , t + dt ) depends on current error e reset error to zero [CDC’04, CRC Press’06]
Example #5: Ecology Stochastic Logistic model for population dynamics x ( t ) ≡ number of individuals of a particular species probability of a birth probability of a death in interval ( t , t + dt ] in interval ( t , t + dt ] x x For African honey bees: a 1 = .3, a 2 = .02, b 1 = .015, b 2 = .001 [Matis et al 1998]
Example #6: Bio-chemical reactions Decaying-dimerizing chemical reactions (DDR): c 3 c 1 c 2 S 1 0 S 2 0 2 S 1 S 2 c 4 population of SHS model species S 1 population of species S 2 reaction rates
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
Switched systems with resets f p : R n → R n p ∈ � parameterized family of vector fields ≡ 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 R n switching times σ = 2 σ = 1 σ = 1 σ = 3 t A solution to the switched system is a pair ( σ , x ) ∈ � for which on every open interval on which σ is constant, x is a solution to 1. time-varying ODE at every switching time t , x ( t ) = ρ ( σ ( t ) , σ – ( t ) , x – ( t ) ) 2.
Asymptotic stability equilibrium point ≡ x eq ∈ R n for which f q ( x eq ) = 0 ∀ q ∈ � class � ≡ set of functions α : [0, ∞ ) → [0, ∞ ) that are α ( s ) 1. continuous 2. strictly increasing 3. α (0)=0 s Definition : The equilibrium point x eq is (globally) asymptotically stable if it is Lyapunov stable and for every solution that exists on [0, ∞ ) x ( t ) → x eq as t →∞ . α (|| x ( t 0 ) – x eq ||) || x ( t 0 ) – x eq || x ( t ) x eq t
Uniform asymptotic stability β ( s , t ) (for each fixed t ) equilibrium point ≡ x eq ∈ R n for which f ( x eq ) = 0 class �� ≡ set of functions β : [0, ∞ ) × [0, ∞ ) → [0, ∞ ) s.t. s 1. for each fixed t , β ( · , t ) ∈ � β ( s , t ) 2. for each fixed s , β ( s , · ) is monotone (for each fixed s ) decreasing and β ( s , t ) → 0 as t →∞ t Definition (class �� function definition): The equilibrium point x eq is uniformly asymptotically stable if ∃ β ∈ �� : || x ( t ) – x eq || · β (|| x ( t 0 ) – x eq ||,t – t 0 ) ∀ t ≥ t 0 ≥ 0 along any solution ( σ , x ) ∈ � to the switched system β is independent β (|| x ( t 0 ) – x eq ||,0) of x ( t 0 ) and σ β (|| x ( t 0 ) – x eq ||, t ) || x ( t 0 ) – x eq || We have exponential stability x ( t ) when x eq β (s,t) = c e - λ t s t with c, λ > 0
Linear switched systems A q , R q,q’ ∈ R n × n q,q’ ∈ � vector fields and reset maps linear on x σ = 2 σ = 1 σ = 1 σ = 3 t 0 t 2 t 3 t 1 t Theorem: For switched linear systems with state-independent switching, uniform asymptotic stability implies exponential stability (two notions are equivalent)
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
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 : R n → R such that Then 1. the equilibrium point x eq is Lyapunov stable 2. if W ( z ) = 0 only for z = x eq then x eq 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 A q , q ∈ � are asymptotically stable and ∀ p,q ∈ � A p A q = A q A p then the switched system is uniformly (exponentially) asymptotically stable Theorem: If all the matrices A q , 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 ∈ R n × n such that all the matrices common similarity B q = T A q T – 1 ( T –1 B q T = A q ) transformation are upper triangular or all lower triangular then the switched system is uniformly (exponentially) asymptotically stable Lie Theorem actually provides the necessary and sufficient condition for the existence of such T ≡ Lie algebra generated by the matrices must be solvable
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
Example #11: Roll-angle control roll-angle θ measurement noise set-point control ≡ drive the roll angle θ to a desired value θ reference n e track θ θ reference + + u set-point process + controller –
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