Stabilization in a Chemostat with Sampled and Delayed Measurements
Michael Malisoff
Joint with Jerome Harmand and Frederic Mazenc
0/11 Models: Represent cell or microorganism growth, wastewater treatment, or natural environments like lakes..
States: Microorganism and substrate concentrations, prone to incomplete measurements and model uncertainties..
Our goals: Input-to-state stabilization of equilibria with uncertain uptake functions using only delayed sampled substrate values
O. Bernard, D. Dochain, J. Gouze, J. Harmand, J. Monod, ....
Background and Motivation
Chemostat: Laboratory apparatus for continuous culture of microorganisms, many biotechnological applications..
1/11 States: Microorganism and substrate concentrations, prone to incomplete measurements and model uncertainties..
Our goals: Input-to-state stabilization of equilibria with uncertain uptake functions using only delayed sampled substrate values
O. Bernard, D. Dochain, J. Gouze, J. Harmand, J. Monod, ....
Background and Motivation
Chemostat: Laboratory apparatus for continuous culture of microorganisms, many biotechnological applications..
Models: Represent cell or microorganism growth, wastewater treatment, or natural environments like lakes..
1/11 Our goals: Input-to-state stabilization of equilibria with uncertain uptake functions using only delayed sampled substrate values
O. Bernard, D. Dochain, J. Gouze, J. Harmand, J. Monod, ....
Background and Motivation
Chemostat: Laboratory apparatus for continuous culture of microorganisms, many biotechnological applications..
Models: Represent cell or microorganism growth, wastewater treatment, or natural environments like lakes..
States: Microorganism and substrate concentrations, prone to incomplete measurements and model uncertainties..
1/11 O. Bernard, D. Dochain, J. Gouze, J. Harmand, J. Monod, ....
Background and Motivation
Chemostat: Laboratory apparatus for continuous culture of microorganisms, many biotechnological applications..
Models: Represent cell or microorganism growth, wastewater treatment, or natural environments like lakes..
States: Microorganism and substrate concentrations, prone to incomplete measurements and model uncertainties..
Our goals: Input-to-state stabilization of equilibria with uncertain uptake functions using only delayed sampled substrate values
1/11 Background and Motivation
Chemostat: Laboratory apparatus for continuous culture of microorganisms, many biotechnological applications..
Models: Represent cell or microorganism growth, wastewater treatment, or natural environments like lakes..
States: Microorganism and substrate concentrations, prone to incomplete measurements and model uncertainties..
Our goals: Input-to-state stabilization of equilibria with uncertain uptake functions using only delayed sampled substrate values
O. Bernard, D. Dochain, J. Gouze, J. Harmand, J. Monod, ....
1/11 Background and Motivation
Chemostat: Laboratory apparatus for continuous culture of microorganisms, many biotechnological applications..
Models: Represent cell or microorganism growth, wastewater treatment, or natural environments like lakes..
States: Microorganism and substrate concentrations, prone to incomplete measurements and model uncertainties..
Our goals: Input-to-state stabilization of equilibria with uncertain uptake functions using only delayed sampled substrate values
S. Pilyugin, G. Robledo, H. Smith, P. Waltman, G. Wolkowicz, ...
1/11 Background and Motivation
Chemostat: Laboratory apparatus for continuous culture of microorganisms, many biotechnological applications..
Models: Represent cell or microorganism growth, wastewater treatment, or natural environments like lakes..
States: Microorganism and substrate concentrations, prone to incomplete measurements and model uncertainties..
Our goals: Input-to-state stabilization of equilibria with uncertain uptake functions using only delayed sampled substrate values
S. Pilyugin, G. Robledo, H. Smith, P. Waltman, G. Wolkowicz, ...
1/11 Background and Motivation
Chemostat: Laboratory apparatus for continuous culture of microorganisms, many biotechnological applications..
Models: Represent cell or microorganism growth, wastewater treatment, or natural environments like lakes..
States: Microorganism and substrate concentrations, prone to incomplete measurements and model uncertainties..
Our goals: Input-to-state stabilization of equilibria with uncertain uptake functions using only delayed sampled substrate values
S. Pilyugin, G. Robledo, H. Smith, P. Waltman, G. Wolkowicz, ...
1/11 Background and Motivation
Chemostat: Laboratory apparatus for continuous culture of microorganisms, many biotechnological applications..
Models: Represent cell or microorganism growth, wastewater treatment, or natural environments like lakes..
States: Microorganism and substrate concentrations, prone to incomplete measurements and model uncertainties..
Our goals: Input-to-state stabilization of equilibria with uncertain uptake functions using only delayed sampled substrate values
S. Pilyugin, G. Robledo, H. Smith, P. Waltman, G. Wolkowicz, ...
1/11 Contents go in and out of culture vessel at constant rate F (l3/t). Constant culture volume V (l3). 3 Constant input nutrient concentration sin (mass/l ). rate of change of nutrient = input - washout - consumption. rate of change of organism = growth - washout.
Review of Simple Chemostat
Three vessels: feed bottle, culture vessel, collection vessel.
2/11 Constant culture volume V (l3). 3 Constant input nutrient concentration sin (mass/l ). rate of change of nutrient = input - washout - consumption. rate of change of organism = growth - washout.
Review of Simple Chemostat
Three vessels: feed bottle, culture vessel, collection vessel. Contents go in and out of culture vessel at constant rate F (l3/t).
2/11 3 Constant input nutrient concentration sin (mass/l ). rate of change of nutrient = input - washout - consumption. rate of change of organism = growth - washout.
Review of Simple Chemostat
Three vessels: feed bottle, culture vessel, collection vessel. Contents go in and out of culture vessel at constant rate F (l3/t). Constant culture volume V (l3).
2/11 rate of change of nutrient = input - washout - consumption. rate of change of organism = growth - washout.
Review of Simple Chemostat
Three vessels: feed bottle, culture vessel, collection vessel. Contents go in and out of culture vessel at constant rate F (l3/t). Constant culture volume V (l3). 3 Constant input nutrient concentration sin (mass/l ).
2/11 rate of change of organism = growth - washout.
Review of Simple Chemostat
Three vessels: feed bottle, culture vessel, collection vessel. Contents go in and out of culture vessel at constant rate F (l3/t). Constant culture volume V (l3). 3 Constant input nutrient concentration sin (mass/l ). rate of change of nutrient = input - washout - consumption.
2/11 Review of Simple Chemostat
Three vessels: feed bottle, culture vessel, collection vessel. Contents go in and out of culture vessel at constant rate F (l3/t). Constant culture volume V (l3). 3 Constant input nutrient concentration sin (mass/l ). rate of change of nutrient = input - washout - consumption. rate of change of organism = growth - washout.
2/11 s = concentration of nutrient in culture vessel. msx 3 Consumption: a+s , x = concentration of organism (mass/l ). m = maximum growth rate (1/t). a = half-saturation constant. ( 0 ms x s = (sin − s)D − a+s γ 0 ms (SC) x = x a+s − D
Review of Simple Chemostat
0 Without organisms or consumption, (Vs) (t) = sinF − s(t)F.
3/11 msx 3 Consumption: a+s , x = concentration of organism (mass/l ). m = maximum growth rate (1/t). a = half-saturation constant. ( 0 ms x s = (sin − s)D − a+s γ 0 ms (SC) x = x a+s − D
Review of Simple Chemostat
0 Without organisms or consumption, (Vs) (t) = sinF − s(t)F. s = concentration of nutrient in culture vessel.
3/11 m = maximum growth rate (1/t). a = half-saturation constant. ( 0 ms x s = (sin − s)D − a+s γ 0 ms (SC) x = x a+s − D
Review of Simple Chemostat
0 Without organisms or consumption, (Vs) (t) = sinF − s(t)F. s = concentration of nutrient in culture vessel. msx 3 Consumption: a+s , x = concentration of organism (mass/l ).
3/11 ( 0 ms x s = (sin − s)D − a+s γ 0 ms (SC) x = x a+s − D
Review of Simple Chemostat
0 Without organisms or consumption, (Vs) (t) = sinF − s(t)F. s = concentration of nutrient in culture vessel. msx 3 Consumption: a+s , x = concentration of organism (mass/l ). m = maximum growth rate (1/t). a = half-saturation constant.
3/11 Review of Simple Chemostat
0 Without organisms or consumption, (Vs) (t) = sinF − s(t)F. s = concentration of nutrient in culture vessel. msx 3 Consumption: a+s , x = concentration of organism (mass/l ). m = maximum growth rate (1/t). a = half-saturation constant. ( 0 ms x s = (sin − s)D − a+s γ 0 ms (SC) x = x a+s − D
3/11 Review of Simple Chemostat
0 Without organisms or consumption, (Vs) (t) = sinF − s(t)F. s = concentration of nutrient in culture vessel. msx 3 Consumption: a+s , x = concentration of organism (mass/l ). m = maximum growth rate (1/t). a = half-saturation constant. ( 0 ms x s = (sin − s)D − a+s γ 0 ms (SC) x = x a+s − D
3/11 X 0(t) = G(t, X(t), X(t − τ(t))), X(t) ∈ X (Σ)
t0−t |X(t)| ≤ γ1 e γ2(|X|[t0−τ¯,t0]) (UGAS)
γi ∈ K∞: 0 at 0, strictly increasing, unbounded. supt≥0 τ(t) ≤ τ¯.
0 X (t) = G t, X(t), X(t − τ(t)),δ (t) , X(t) ∈ X (Σpert)
t0−t |X(t)| ≤ γ1 e γ2(|X|[t0−τ¯,t0]) + γ3(|δ|[t0,t]) (ISS)
Find γi ’s by building Lyapunov-Krasovskii functionals (LKFs). Equivalent to KL formulation; see Sontag 1998 SCL paper.
Input-to-State Stable (ISS)
ISS (Sontag, ’89) generalizes uniform global asymptotic stability.
4/11 t0−t |X(t)| ≤ γ1 e γ2(|X|[t0−τ¯,t0]) (UGAS)
γi ∈ K∞: 0 at 0, strictly increasing, unbounded. supt≥0 τ(t) ≤ τ¯.
0 X (t) = G t, X(t), X(t − τ(t)),δ (t) , X(t) ∈ X (Σpert)
t0−t |X(t)| ≤ γ1 e γ2(|X|[t0−τ¯,t0]) + γ3(|δ|[t0,t]) (ISS)
Find γi ’s by building Lyapunov-Krasovskii functionals (LKFs). Equivalent to KL formulation; see Sontag 1998 SCL paper.
Input-to-State Stable (ISS)
ISS (Sontag, ’89) generalizes uniform global asymptotic stability.
X 0(t) = G(t, X(t), X(t − τ(t))), X(t) ∈ X (Σ)
4/11 0 X (t) = G t, X(t), X(t − τ(t)),δ (t) , X(t) ∈ X (Σpert)
t0−t |X(t)| ≤ γ1 e γ2(|X|[t0−τ¯,t0]) + γ3(|δ|[t0,t]) (ISS)
Find γi ’s by building Lyapunov-Krasovskii functionals (LKFs). Equivalent to KL formulation; see Sontag 1998 SCL paper.
Input-to-State Stable (ISS)
ISS (Sontag, ’89) generalizes uniform global asymptotic stability.
X 0(t) = G(t, X(t), X(t − τ(t))), X(t) ∈ X (Σ)
t0−t |X(t)| ≤ γ1 e γ2(|X|[t0−τ¯,t0]) (UGAS)
γi ∈ K∞: 0 at 0, strictly increasing, unbounded. supt≥0 τ(t) ≤ τ¯.
4/11 t0−t |X(t)| ≤ γ1 e γ2(|X|[t0−τ¯,t0]) + γ3(|δ|[t0,t]) (ISS)
Find γi ’s by building Lyapunov-Krasovskii functionals (LKFs). Equivalent to KL formulation; see Sontag 1998 SCL paper.
Input-to-State Stable (ISS)
ISS (Sontag, ’89) generalizes uniform global asymptotic stability.
X 0(t) = G(t, X(t), X(t − τ(t))), X(t) ∈ X (Σ)
t0−t |X(t)| ≤ γ1 e γ2(|X|[t0−τ¯,t0]) (UGAS)
γi ∈ K∞: 0 at 0, strictly increasing, unbounded. supt≥0 τ(t) ≤ τ¯.
0 X (t) = G t, X(t), X(t − τ(t)),δ (t) , X(t) ∈ X (Σpert)
4/11 Find γi ’s by building Lyapunov-Krasovskii functionals (LKFs). Equivalent to KL formulation; see Sontag 1998 SCL paper.
Input-to-State Stable (ISS)
ISS (Sontag, ’89) generalizes uniform global asymptotic stability.
X 0(t) = G(t, X(t), X(t − τ(t))), X(t) ∈ X (Σ)
t0−t |X(t)| ≤ γ1 e γ2(|X|[t0−τ¯,t0]) (UGAS)
γi ∈ K∞: 0 at 0, strictly increasing, unbounded. supt≥0 τ(t) ≤ τ¯.
0 X (t) = G t, X(t), X(t − τ(t)),δ (t) , X(t) ∈ X (Σpert)
t0−t |X(t)| ≤ γ1 e γ2(|X|[t0−τ¯,t0]) + γ3(|δ|[t0,t]) (ISS)
4/11 Equivalent to KL formulation; see Sontag 1998 SCL paper.
Input-to-State Stable (ISS)
ISS (Sontag, ’89) generalizes uniform global asymptotic stability.
X 0(t) = G(t, X(t), X(t − τ(t))), X(t) ∈ X (Σ)
t0−t |X(t)| ≤ γ1 e γ2(|X|[t0−τ¯,t0]) (UGAS)
γi ∈ K∞: 0 at 0, strictly increasing, unbounded. supt≥0 τ(t) ≤ τ¯.
0 X (t) = G t, X(t), X(t − τ(t)),δ (t) , X(t) ∈ X (Σpert)
t0−t |X(t)| ≤ γ1 e γ2(|X|[t0−τ¯,t0]) + γ3(|δ|[t0,t]) (ISS)
Find γi ’s by building Lyapunov-Krasovskii functionals (LKFs).
4/11 Input-to-State Stable (ISS)
ISS (Sontag, ’89) generalizes uniform global asymptotic stability.
X 0(t) = G(t, X(t), X(t − τ(t))), X(t) ∈ X (Σ)
t0−t |X(t)| ≤ γ1 e γ2(|X|[t0−τ¯,t0]) (UGAS)
γi ∈ K∞: 0 at 0, strictly increasing, unbounded. supt≥0 τ(t) ≤ τ¯.
0 X (t) = G t, X(t), X(t − τ(t)),δ (t) , X(t) ∈ X (Σpert)
t0−t |X(t)| ≤ γ1 e γ2(|X|[t0−τ¯,t0]) + γ3(|δ|[t0,t]) (ISS)
Find γi ’s by building Lyapunov-Krasovskii functionals (LKFs). Equivalent to KL formulation; see Sontag 1998 SCL paper.
4/11 µ1(s) µ(s) = 1+γ(s) , with a unique maximizer sM ∈ (0, sin]
Assumption 1: The function µ is C1 and µ(0) = 0. Also, there is 0 a constant sM ∈ (0, sin] such that µ (s) > 0 for all s ∈ [0, sM ) and 0 µ (s) ≤ 0 for all s ∈ [sM , ∞). Finally, µ(s) > 0 for all s > 0.
Uncertain Controlled Chemostat with Sampling
( s˙(t) = D(s(t − τ(t))[sin − s(t)] − (1 + δ(t))µ(s(t))x(t) (C) x˙ (t) = [(1 + δ(t))µ(s(t)) − D(s(t − τ(t))]x(t)
τ , t ∈ [0, τ ) τ(t) = f f τf + t − tj , t ∈ [tj + τf , tj+1 + τf ) and j ≥ 0
0 < 1 ≤ ti+1 − ti ≤ 2. δ :[0, ∞) → [d, ∞), with d ∈ (−1, 0].
5/11 Assumption 1: The function µ is C1 and µ(0) = 0. Also, there is 0 a constant sM ∈ (0, sin] such that µ (s) > 0 for all s ∈ [0, sM ) and 0 µ (s) ≤ 0 for all s ∈ [sM , ∞). Finally, µ(s) > 0 for all s > 0.
Uncertain Controlled Chemostat with Sampling
( s˙(t) = D(s(t − τ(t))[sin − s(t)] − (1 + δ(t))µ(s(t))x(t) (C) x˙ (t) = [(1 + δ(t))µ(s(t)) − D(s(t − τ(t))]x(t)
τ , t ∈ [0, τ ) τ(t) = f f τf + t − tj , t ∈ [tj + τf , tj+1 + τf ) and j ≥ 0
0 < 1 ≤ ti+1 − ti ≤ 2. δ :[0, ∞) → [d, ∞), with d ∈ (−1, 0].
µ1(s) µ(s) = 1+γ(s) , with a unique maximizer sM ∈ (0, sin]
5/11 Uncertain Controlled Chemostat with Sampling
( s˙(t) = D(s(t − τ(t))[sin − s(t)] − (1 + δ(t))µ(s(t))x(t) (C) x˙ (t) = [(1 + δ(t))µ(s(t)) − D(s(t − τ(t))]x(t)
τ , t ∈ [0, τ ) τ(t) = f f τf + t − tj , t ∈ [tj + τf , tj+1 + τf ) and j ≥ 0
0 < 1 ≤ ti+1 − ti ≤ 2. δ :[0, ∞) → [d, ∞), with d ∈ (−1, 0].
µ1(s) µ(s) = 1+γ(s) , with a unique maximizer sM ∈ (0, sin]
Assumption 1: The function µ is C1 and µ(0) = 0. Also, there is 0 a constant sM ∈ (0, sin] such that µ (s) > 0 for all s ∈ [0, sM ) and 0 µ (s) ≤ 0 for all s ∈ [sM , ∞). Finally, µ(s) > 0 for all s > 0.
5/11 Uncertain Controlled Chemostat with Sampling
( s˙(t) = D(s(t − τ(t))[sin − s(t)] − (1 + δ(t))µ(s(t))x(t) (C) x˙ (t) = [(1 + δ(t))µ(s(t)) − D(s(t − τ(t))]x(t)
τ , t ∈ [0, τ ) τ(t) = f f τf + t − tj , t ∈ [tj + τf , tj+1 + τf ) and j ≥ 0
0 < 1 ≤ ti+1 − ti ≤ 2. δ :[0, ∞) → [d, ∞), with d ∈ (−1, 0].
(?) µ1(s) µ(s) = 1+γ(s) , with a unique maximizer sM ∈ (0, sin]
1 Lemma: Under Assumption 1, there are µ1 ∈ C ∩ K∞ and a nondecreasing C1 function γ : R → [0, ∞) such that (?) holds for 0 0 all s ≥ 0, µ1(s) > 0 on [0, ∞), and γ (s) > 0 on [sM , ∞).
5/11 Uncertain Controlled Chemostat with Sampling
( s˙(t) = D(s(t − τ(t))[sin − s(t)] − (1 + δ(t))µ(s(t))x(t) (C) x˙ (t) = [(1 + δ(t))µ(s(t)) − D(s(t − τ(t))]x(t)
τ , t ∈ [0, τ ) τ(t) = f f τf + t − tj , t ∈ [tj + τf , tj+1 + τf ) and j ≥ 0
0.8
0.6
0.4
0.2
0.5 1.0 1.5 2.0 2.5 (1 + δ)µ(s) for Different√ Constant δ Choices, sM = 1/ 2 and sin = 1
5/11 Uncertain Controlled Chemostat with Sampling
( s˙(t) = D(s(t − τ(t))[sin − s(t)] − (1 + δ(t))µ(s(t))x(t) (C) x˙ (t) = [(1 + δ(t))µ(s(t)) − D(s(t − τ(t))]x(t)
τ , t ∈ [0, τ ) τ(t) = f f τf + t − tj , t ∈ [tj + τf , tj+1 + τf ) and j ≥ 0
0 < 1 ≤ ti+1 − ti ≤ 2. δ :[0, ∞) → [d, ∞), with d ∈ (−1, 0].
(?) µ1(s) µ(s) = 1+γ(s) , with a unique maximizer sM ∈ (0, sin]
1 Lemma: Under Assumption 1, there are µ1 ∈ C ∩ K∞ and a nondecreasing C1 function γ : R → [0, ∞) such that (?) holds for 0 0 all s ≥ 0, µ1(s) > 0 on [0, ∞), and γ (s) > 0 on [sM , ∞).
5/11 Uncertain Controlled Chemostat with Sampling
( s˙(t) = D(s(t − τ(t))[sin − s(t)] − (1 + δ(t))µ(s(t))x(t) (C) x˙ (t) = [(1 + δ(t))µ(s(t)) − D(s(t − τ(t))]x(t)
τ , t ∈ [0, τ ) τ(t) = f f τf + t − tj , t ∈ [tj + τf , tj+1 + τf ) and j ≥ 0
0 < 1 ≤ ti+1 − ti ≤ 2. δ :[0, ∞) → [d, ∞), with d ∈ (−1, 0].
µ1(s) µ(s) = 1+γ(s) , with a unique maximizer sM ∈ (0, sin]
Goal: Under suitable conditions on an upper bound τM for the delay τ(t), and for constants s∗ ∈ (0, sin), design the control D to render the dynamics for X(t) = (s(t), x(t)) − (s∗, sin − s∗) ISS.
5/11 (a) Assume that µ1(sin) − µ1(s∗) > 0 1+γ(sin) 1+γ(sin−µ1(s∗)sinτM ) (b) √1 1 and τM < max , 2ρ s µ (s ) , with s∗ < sin. 2sin 2ρmωl l in 1 in
Theorem 1: For all componentwise positive initial conditions, all solutions of the chemostat system (C) with δ(t) = 0 and
µ1(s∗) D(s(t − τ(t))) = 1+γ(s(t−τ(t))) (1) 2 remain in (0, ∞) and converge to (s∗, sin − s∗).
Main Result for Unperturbed Case
0 0 0 ωs = inf µ1(s) , ωl = sup µ1(s) , ρl = sup γ (s), s∈[0,s ] in s∈[0,sin] s∈[0,sin] ρ2 µ2(l+1.1µ (s )s τ ) ρ = l max 1 1 ∗ in M , where µ(s) = µ1(s) m 2ωs 1+γ(l) 1+γ(s) l∈[0,sin]
6/11 Theorem 1: For all componentwise positive initial conditions, all solutions of the chemostat system (C) with δ(t) = 0 and
µ1(s∗) D(s(t − τ(t))) = 1+γ(s(t−τ(t))) (1) 2 remain in (0, ∞) and converge to (s∗, sin − s∗).
Main Result for Unperturbed Case
0 0 0 ωs = inf µ1(s) , ωl = sup µ1(s) , ρl = sup γ (s), s∈[0,s ] in s∈[0,sin] s∈[0,sin] ρ2 µ2(l+1.1µ (s )s τ ) ρ = l max 1 1 ∗ in M , where µ(s) = µ1(s) m 2ωs 1+γ(l) 1+γ(s) l∈[0,sin] (a) Assume that µ1(sin) − µ1(s∗) > 0 1+γ(sin) 1+γ(sin−µ1(s∗)sinτM ) (b) √1 1 and τM < max , 2ρ s µ (s ) , with s∗ < sin. 2sin 2ρmωl l in 1 in
6/11 Main Result for Unperturbed Case
0 0 0 ωs = inf µ1(s) , ωl = sup µ1(s) , ρl = sup γ (s), s∈[0,s ] in s∈[0,sin] s∈[0,sin] ρ2 µ2(l+1.1µ (s )s τ ) ρ = l max 1 1 ∗ in M , where µ(s) = µ1(s) m 2ωs 1+γ(l) 1+γ(s) l∈[0,sin] (a) Assume that µ1(sin) − µ1(s∗) > 0 1+γ(sin) 1+γ(sin−µ1(s∗)sinτM ) (b) √1 1 and τM < max , 2ρ s µ (s ) , with s∗ < sin. 2sin 2ρmωl l in 1 in
Theorem 1: For all componentwise positive initial conditions, all solutions of the chemostat system (C) with δ(t) = 0 and
µ1(s∗) D(s(t − τ(t))) = 1+γ(s(t−τ(t))) (1) 2 remain in (0, ∞) and converge to (s∗, sin − s∗).
6/11 z = −(x − sin + s∗) − (s − s∗) = −X2 − X1. X= error variable.
Step 2: Build T ∈ K∞ and a constant c¯ > 0 such that U (s) = R s−s∗ m dm, (2) 1 0 sin−s∗−m satisfies
˙ (s(t)−s∗)(µ1(s∗)−µ1(s(t))) U1(t) ≤ 2[1+γ(s(t−τ(t)))] (3) R t ˙ 2 ¯ + ρmτM t−τ(t)(s(m)) dm + c|s(t) − s∗||z(t)|
for all t ≥ T (|X(0)|) where X(t) = (s(t), x(t)) − (s∗, sin − s∗).
Proof Outline for δ = 0 Unperturbed Case
Step 1: For any fixed s¯ ≥ sin, show that z = sin − s − x satisfies −tµ1(s∗) |z(t)| ≤ |z(0)|e 1+γ(¯s) for all t ≥ 0 . (ES)
7/11 Step 2: Build T ∈ K∞ and a constant c¯ > 0 such that U (s) = R s−s∗ m dm, (2) 1 0 sin−s∗−m satisfies
˙ (s(t)−s∗)(µ1(s∗)−µ1(s(t))) U1(t) ≤ 2[1+γ(s(t−τ(t)))] (3) R t ˙ 2 ¯ + ρmτM t−τ(t)(s(m)) dm + c|s(t) − s∗||z(t)|
for all t ≥ T (|X(0)|) where X(t) = (s(t), x(t)) − (s∗, sin − s∗).
Proof Outline for δ = 0 Unperturbed Case
Step 1: For any fixed s¯ ≥ sin, show that z = sin − s − x satisfies −tµ1(s∗) |z(t)| ≤ |z(0)|e 1+γ(¯s) for all t ≥ 0 . (ES) z = −(x − sin + s∗) − (s − s∗) = −X2 − X1. X= error variable.
7/11 Proof Outline for δ = 0 Unperturbed Case
Step 1: For any fixed s¯ ≥ sin, show that z = sin − s − x satisfies −tµ1(s∗) |z(t)| ≤ |z(0)|e 1+γ(¯s) for all t ≥ 0 . (ES) z = −(x − sin + s∗) − (s − s∗) = −X2 − X1. X= error variable.
Step 2: Build T ∈ K∞ and a constant c¯ > 0 such that U (s) = R s−s∗ m dm, (2) 1 0 sin−s∗−m satisfies
˙ (s(t)−s∗)(µ1(s∗)−µ1(s(t))) U1(t) ≤ 2[1+γ(s(t−τ(t)))] (3) R t ˙ 2 ¯ + ρmτM t−τ(t)(s(m)) dm + c|s(t) − s∗||z(t)| for all t ≥ T (|X(0)|) where X(t) = (s(t), x(t)) − (s∗, sin − s∗).
7/11 Step 4: The sum U3 of a quadratic Lyapunov function for the z variable and U2 admits a constant c3 > 0 such that ˙ U3(t) ≤ −c3U3(st , z(t)) (6) for all t ≥ T (|X(0)|).
Proof Outline for δ = 0 Unperturbed Case
Step 3: Find constants ci > 0 such that
R s(t)−s∗ m R t R t 2 U2(st ) = dm + 2ρmτM (s˙(m)) dm d`. (4) 0 sin−s∗−m t−τM ` satisfies ˙ 2 U2(t) ≤ −c1U2(st ) + c2z (t) + c¯|s(t) − s∗||z(t)| (5) for all t ≥ T (|X(0)|).
8/11 Proof Outline for δ = 0 Unperturbed Case
Step 3: Find constants ci > 0 such that
R s(t)−s∗ m R t R t 2 U2(st ) = dm + 2ρmτM (s˙(m)) dm d`. (4) 0 sin−s∗−m t−τM ` satisfies ˙ 2 U2(t) ≤ −c1U2(st ) + c2z (t) + c¯|s(t) − s∗||z(t)| (5) for all t ≥ T (|X(0)|).
Step 4: The sum U3 of a quadratic Lyapunov function for the z variable and U2 admits a constant c3 > 0 such that ˙ U3(t) ≤ −c3U3(st , z(t)) (6) for all t ≥ T (|X(0)|).
8/11 ISS with respect to δ(t) without upper bounds on |δ|∞... (a) (1+d)µ1(sin) − µ1(s∗) > 0 ...(1 + δ(t))µ(s(t))... 1+γ(sin) 1+γ(sin−µ1(s∗)sinτM )
U2(st ) = Z s(t)−s∗ Z t Z t m 2 dm + 2ρmτ (s˙(m)) dm d`. sin−s∗−m M 0 t−τM `
Functions γi from ISS condition measure distance from equilibria at all times, providing both transient and asymptotic information
Mazenc, F., J. Harmand, and M. Malisoff, “Stabilization in a chemostat with sampled and delayed measurements and uncertain growth functions,” Automatica, 2017.
Extensions and Applications
9/11 (a) (1+d)µ1(sin) − µ1(s∗) > 0 ...(1 + δ(t))µ(s(t))... 1+γ(sin) 1+γ(sin−µ1(s∗)sinτM )
U2(st ) = Z s(t)−s∗ Z t Z t m 2 dm + 2ρmτ (s˙(m)) dm d`. sin−s∗−m M 0 t−τM `
Functions γi from ISS condition measure distance from equilibria at all times, providing both transient and asymptotic information
Mazenc, F., J. Harmand, and M. Malisoff, “Stabilization in a chemostat with sampled and delayed measurements and uncertain growth functions,” Automatica, 2017.
Extensions and Applications
ISS with respect to δ(t) without upper bounds on |δ|∞...
9/11 U2(st ) = Z s(t)−s∗ Z t Z t m 2 dm + 2ρmτ (s˙(m)) dm d`. sin−s∗−m M 0 t−τM `
Functions γi from ISS condition measure distance from equilibria at all times, providing both transient and asymptotic information
Mazenc, F., J. Harmand, and M. Malisoff, “Stabilization in a chemostat with sampled and delayed measurements and uncertain growth functions,” Automatica, 2017.
Extensions and Applications
ISS with respect to δ(t) without upper bounds on |δ|∞... (a) (1+d)µ1(sin) − µ1(s∗) > 0 ...(1 + δ(t))µ(s(t))... 1+γ(sin) 1+γ(sin−µ1(s∗)sinτM )
9/11 Functions γi from ISS condition measure distance from equilibria at all times, providing both transient and asymptotic information
Mazenc, F., J. Harmand, and M. Malisoff, “Stabilization in a chemostat with sampled and delayed measurements and uncertain growth functions,” Automatica, 2017.
Extensions and Applications
ISS with respect to δ(t) without upper bounds on |δ|∞... (a) (1+d)µ1(sin) − µ1(s∗) > 0 ...(1 + δ(t))µ(s(t))... 1+γ(sin) 1+γ(sin−µ1(s∗)sinτM )
U2(st ) = Z s(t)−s∗ Z t Z t m 2 dm + 2ρmτ (s˙(m)) dm d`. sin−s∗−m M 0 t−τM `
9/11 Mazenc, F., J. Harmand, and M. Malisoff, “Stabilization in a chemostat with sampled and delayed measurements and uncertain growth functions,” Automatica, 2017.
Extensions and Applications
ISS with respect to δ(t) without upper bounds on |δ|∞... (a) (1+d)µ1(sin) − µ1(s∗) > 0 ...(1 + δ(t))µ(s(t))... 1+γ(sin) 1+γ(sin−µ1(s∗)sinτM )
U2(st ) = Z s(t)−s∗ Z t Z t m 2 dm + 2ρmτ (s˙(m)) dm d`. sin−s∗−m M 0 t−τM `
Functions γi from ISS condition measure distance from equilibria at all times, providing both transient and asymptotic information
9/11 Extensions and Applications
ISS with respect to δ(t) without upper bounds on |δ|∞... (a) (1+d)µ1(sin) − µ1(s∗) > 0 ...(1 + δ(t))µ(s(t))... 1+γ(sin) 1+γ(sin−µ1(s∗)sinτM )
U2(st ) = Z s(t)−s∗ Z t Z t m 2 dm + 2ρmτ (s˙(m)) dm d`. sin−s∗−m M 0 t−τM `
Functions γi from ISS condition measure distance from equilibria at all times, providing both transient and asymptotic information
Mazenc, F., J. Harmand, and M. Malisoff, “Stabilization in a chemostat with sampled and delayed measurements and uncertain growth functions,” Automatica, 2017.
9/11 Mathematica Simulations of (C)
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0 50 100 150 200 = , µ( ) = 0.5s , = . ,δ ( ) = . sin 1 s 1+0.25s+2s2 tj 0 24j t 0 s(t) in Red, x(t) in Blue, D(t) in Green.
10/11 Mathematica Simulations of (C)
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0 20 40 60 80 100 0.26 0.24 0.22 0.20 0.18 0.16 0.14
0 20 40 60 80 100 = , µ( ) = 0.5s , = . ,δ ( ) = . ( + ( )). sin 1 s 1+0.25s+2s2 tj 0 24j t 0 15 1 sin t s(t) in Red, x(t) in Blue, D(t) in Green.
10/11 Mathematica Simulations of (C)
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0 10 20 30 40 50 60 70 = , µ( ) = 0.5s , = . ,δ ( ) = . ( + ( )). sin 1 s 1+0.25s+2s2 tj 0 24j t 0 15 1 sin t s(t) in Red, x(t) in Blue, D(t) in Green.
10/11 Chemostats model substrate and species interactions. They have uncertainties, delays, and discrete measurements. Discretization of continuous time controls can produce errors. Our control only needs discrete delayed substrate values. We can allow general nonmonotone growth functions. Our barrier functions gave ISS with uncertain growth functions. We can also cover delayed perturbed multispecies chemostats. We plan to study PDE models of age-structured chemostats.
Thank you for your attention!
Conclusions and Future Work
11/11 They have uncertainties, delays, and discrete measurements. Discretization of continuous time controls can produce errors. Our control only needs discrete delayed substrate values. We can allow general nonmonotone growth functions. Our barrier functions gave ISS with uncertain growth functions. We can also cover delayed perturbed multispecies chemostats. We plan to study PDE models of age-structured chemostats.
Thank you for your attention!
Conclusions and Future Work
Chemostats model substrate and species interactions.
11/11 Discretization of continuous time controls can produce errors. Our control only needs discrete delayed substrate values. We can allow general nonmonotone growth functions. Our barrier functions gave ISS with uncertain growth functions. We can also cover delayed perturbed multispecies chemostats. We plan to study PDE models of age-structured chemostats.
Thank you for your attention!
Conclusions and Future Work
Chemostats model substrate and species interactions. They have uncertainties, delays, and discrete measurements.
11/11 Our control only needs discrete delayed substrate values. We can allow general nonmonotone growth functions. Our barrier functions gave ISS with uncertain growth functions. We can also cover delayed perturbed multispecies chemostats. We plan to study PDE models of age-structured chemostats.
Thank you for your attention!
Conclusions and Future Work
Chemostats model substrate and species interactions. They have uncertainties, delays, and discrete measurements. Discretization of continuous time controls can produce errors.
11/11 We can allow general nonmonotone growth functions. Our barrier functions gave ISS with uncertain growth functions. We can also cover delayed perturbed multispecies chemostats. We plan to study PDE models of age-structured chemostats.
Thank you for your attention!
Conclusions and Future Work
Chemostats model substrate and species interactions. They have uncertainties, delays, and discrete measurements. Discretization of continuous time controls can produce errors. Our control only needs discrete delayed substrate values.
11/11 Our barrier functions gave ISS with uncertain growth functions. We can also cover delayed perturbed multispecies chemostats. We plan to study PDE models of age-structured chemostats.
Thank you for your attention!
Conclusions and Future Work
Chemostats model substrate and species interactions. They have uncertainties, delays, and discrete measurements. Discretization of continuous time controls can produce errors. Our control only needs discrete delayed substrate values. We can allow general nonmonotone growth functions.
11/11 We can also cover delayed perturbed multispecies chemostats. We plan to study PDE models of age-structured chemostats.
Thank you for your attention!
Conclusions and Future Work
Chemostats model substrate and species interactions. They have uncertainties, delays, and discrete measurements. Discretization of continuous time controls can produce errors. Our control only needs discrete delayed substrate values. We can allow general nonmonotone growth functions. Our barrier functions gave ISS with uncertain growth functions.
11/11 We plan to study PDE models of age-structured chemostats.
Thank you for your attention!
Conclusions and Future Work
Chemostats model substrate and species interactions. They have uncertainties, delays, and discrete measurements. Discretization of continuous time controls can produce errors. Our control only needs discrete delayed substrate values. We can allow general nonmonotone growth functions. Our barrier functions gave ISS with uncertain growth functions. We can also cover delayed perturbed multispecies chemostats.
11/11 Thank you for your attention!
Conclusions and Future Work
Chemostats model substrate and species interactions. They have uncertainties, delays, and discrete measurements. Discretization of continuous time controls can produce errors. Our control only needs discrete delayed substrate values. We can allow general nonmonotone growth functions. Our barrier functions gave ISS with uncertain growth functions. We can also cover delayed perturbed multispecies chemostats. We plan to study PDE models of age-structured chemostats.
11/11 Conclusions and Future Work
Chemostats model substrate and species interactions. They have uncertainties, delays, and discrete measurements. Discretization of continuous time controls can produce errors. Our control only needs discrete delayed substrate values. We can allow general nonmonotone growth functions. Our barrier functions gave ISS with uncertain growth functions. We can also cover delayed perturbed multispecies chemostats. We plan to study PDE models of age-structured chemostats.
Thank you for your attention!
11/11 References with Hyperlinked Paper Titles
Mazenc, F., G. Robledo, and M. Malisoff, “Stability and robustness analysis for a multispecies chemostat model with delays in the growth rates and uncertainties," Discrete and Continuous Dynamical Systems Series B, to appear. Mazenc, F., J. Harmand, and M. Malisoff, “Stabilization in a chemostat with sampled and delayed measurements and uncertain growth functions," Automatica, Volume 78, April 2017, pp. 241-249. Karafyllis, I., M. Malisoff, and M. Krstic, “Sampled-data feedback stabilization of age-structured chemostat models," in Proceedings of the 2015 American Control Conference (Chicago, IL, 1-3 July 2015), pp. 4549-4554.
11/11 References with Hyperlinked Paper Titles
Mazenc, F., and M. Malisoff, “Stability and stabilization for models of chemostats with multiple limiting substrates," Journal of Biological Dynamics, Volume 6, Issue 2, 2012, pp. 612-627. Mazenc, F., and M. Malisoff, "Strict Lyapunov function constructions under LaSalle conditions with an application to Lotka-Volterra systems," IEEE Transactions on Automatic Control, Volume 55, Issue 4, April 2010, pp. 841-854. Mazenc, F., M. Malisoff, and J. Harmand, “Further results on stabilization of periodic trajectories for a chemostat with two species," IEEE Transactions on Automatic Control, Volume 53, Special Issue on Systems Biology, January 2008, pp. 66-74.
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