Scalable Analysis and Control of Positive Systems

Scalable Analysis and Control ofPositive Systems

Anders Rantzer

LCCC Linnaeus Center

Lund University

Sweden

Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems

Wind Farms Need Control

Picture from http://www.hochtief.com/hochtief_en/9164.jhtml

Most wind farms today are paid to maximize power production.Future farms will have to curtial power at contracted levels.

New control objective:Minimize fatigue loads subject to fixed total production.


Minimizing Fatigue Loads

Single turbine control:Minimize tower pressure variance subject to linearizeddynamics with measurements of pitch angle and rotor speed.

Optimal controller: uloci (t)

Wind farm control:Minimize sum of all tower pressure variances subject tofixed total production of the farm:

∑mi=1 ui = 0

Optimal controller: ui(t) = uloci (t)−1m

∑mj=1 u

locj (t).

[PhD thesis by Daria Madjidian, Lund University, June 2014]


Controller Structure

C

Linear quadratic control of m identical systems and a constraint∑mi=1 ui = 0 gives an optimal feedback matrix with two parts:

One is localized (diagonal).

The other has rank one (control of the average state).


Server Farms Need Control

Picture from http://www.dawn.com/news/1017980

Single server control:Assign resources (processor speed, memory, etc.) to minimizevariance in completion time.

Server farm control:Minimize sum of all time variances with fixed total resources.


Towards a Scalable Control Theory

Process

✛Controller

✲

✛✛

✲✲

"P C

P

PC

P

P

PC

✲ ✲✲

✛

✛✛

❄❄✻ ✲✛✛

✛

✛

Linear quadratic control uses O(n3) flops, O(n2) memory

Model Predictive Control requires even more

Today: Exploiting monotone/positive systems



Process

✛Controller

✲

✛✛

✲✲

"P C

P

PC

P

P

PC

✲ ✲✲

✛

✛✛

❄❄✻ ✲✛✛

✛

✛






Process

✛Controller

✲

✛✛

✲✲

"P C

P

PC

P

P

PC

✲ ✲✲

✛

✛✛

❄❄✻ ✲✛✛

✛

✛





Outline

• Positive and Monotone Systems

○ Scalable Stability Analysis

○ Input-Output Performance

○ Trajectory Optimization

○ Combination Therapy for HIV and Cancer


Positive systems

A linear system is called positive if the state and output remainnonnegative as long as the initial state and the inputs arenonnegative:

dx

dt= Ax + Bu y= Cx

Equivalently, A, B and C have nonnegative coefficients exceptfor the diagonal of A.

Examples:

Probabilistic models.

Economic systems.

Chemical reactions.

Ecological systems.


Positive Systems and Nonnegative Matrices

Classics:

Mathematics: Perron (1907) and Frobenius (1912)

Economics: Leontief (1936)

Books:

Nonnegative matrices: Berman and Plemmons (1979)

Dynamical Systems: Luenberger (1979)

Recent control related work:

Biology inspired theory: Angeli and Sontag (2003)

Synthesis by linear programming: Rami and Tadeo (2007)

Switched systems: Liu (2009), Fornasini and Valcher (2010)

Distributed control: Tanaka and Langbort (2010)

Robust control: Briat (2013)


Example 1: Transportation Networks

Cloud computing / server farms

Heating and ventilation in buildings

Traffic flow dynamics

Production planning and logistics


A Transportation Network is a Positive System

2

1

4 3

⎡

⎢⎢⎣

x1x2x3x4

⎤

⎥⎥⎦=

⎡

⎢⎢⎣

−1− $31 $12 0 00 −$12 − $32 $23 0$31 $32 −$23 − $43 $340 0 $43 −4− $34

⎤

⎥⎥⎦

⎡

⎢⎢⎣

x1x2x3x4

⎤

⎥⎥⎦+

⎡

⎢⎢⎣

w1w2w3w4

⎤

⎥⎥⎦

How do we select $i j to minimize the gain from w to x?


Example 2: A vehicle formation


Example 2: Vehicle Formations

2

1

4 3

⎧

⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

x1 = −x1 + $13(x3 − x1) +w1x2 = $21(x1 − x2) + $23(x3 − x2) +w2x3 = $32(x2 − x3) + $34(x4 − x3) +w3x4 = −4x4 + $43(x3 − x4) +w4

How do we select $i j to minimize the gain from w to x?


Nonlinear Monotone Systems

For the nonlinear system x = f (x), let x(t) = φ(x0, t) be the solutionstarting from x0. The system is called monotone if x0 ≤ y0 impliesφ(x0, t) ≤ φ(y0, t) for all t ≥ 0.

x(0)

x(1)

x(2)

x(3)

y(0)

y(1)

y(2)

y(3)


Macroscopic Models of Traffic Flow

1 Partial differential equation by Lighthill/Whitham (1955),Richards (1956) based on mass-conservation:

0 ='ρ

't+'

'xf (ρ)

where ρ(x, t) is traffic density in position x at time t andf (ρ) expresses flow as function of density.

2 Spatial discretization by Daganzo (1994).

Both models are monotone systems!

Exploited for lines: [Gomes/Horowitz/Kurzhanskiy/Varaiya/Kwon, 2008].Exploited for networks: [Lovisari/Como/Rantzer/Savla, MTNS-14].


Outline

○ Positive and Monotone Systems

• Scalable Stability Analysis





Stability of Positive systems

Suppose the matrix A has nonnegative off-diagonal elements.Then the following conditions are equivalent:

(i) The system dxdt = Ax is exponentially stable.

(ii) There is a diagonal matrix P ≻ 0 such thatATP+ PA ≺ 0

(iii) There exists a vector ξ > 0 such that Aξ < 0.(The vector inequalities are elementwise.)

(iv) There exits a vector z > 0 such that AT z < 0.


Lyapunov Functions of Positive systems

Solving the three alternative inequalities gives three differentLyapunov functions:

ATP+ PA ≺ 0 Aξ < 0 AT z < 0

V (x) = xT Px V (x) = maxk(xk/ξk) V (x) = zT x


A Scalable Stability Test

x1 x2 x3 x4

Stability of x = Ax follows from existence of ξk > 0 such that⎡

⎢⎢⎣

a11 a12 0 a14a21 a22 a23 0

0 a32 a33 a32a41 0 a43 a44

⎤

⎥⎥⎦

︸︷︷︸

A

⎡

⎢⎢⎣

ξ1ξ2ξ3ξ4

⎤

⎥⎥⎦<

⎡

⎢⎢⎣

0

0

0

0

⎤

⎥⎥⎦

The first node verifies the inequality of the first row.

The second node verifies the inequality of the second row.

. . .

Verification is scalable!Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems

A Distributed Search for Stabilizing Gains

Suppose

⎡

⎢⎢⎣

a11 − $1 a12 0 a14a21 + $1 a22 − $2 a23 0

0 a32 + $2 a33 a32a41 0 a43 a44

⎤

⎥⎥⎦≥ 0 for $1, $2 ∈ [0, 1].

For stabilizing gains $1, $2, find 0 < µk < ξk such that

⎡

⎢⎢⎣

a11 a12 0 a14a21 a22 a23 0

0 a32 a33 a32a41 0 a43 a44

⎤

⎥⎥⎦

⎡

⎢⎢⎣

ξ1ξ2ξ3ξ4

⎤

⎥⎥⎦+

⎡

⎢⎢⎣

−1 0

1 −10 1

0 0

⎤

⎥⎥⎦

[

µ1µ2

]

<

⎡

⎢⎢⎣

0

0

0

0

⎤

⎥⎥⎦

and set $1 = µ1/ξ1 and $2 = µ2/ξ2. Every row gives a local test.

Distributed synthesis by linear programming (gradient search).


Max-separable Lyapunov Functions

Max-separable: V (x) = max{V1(x1), . . . ,Vn(xn)}

Theorem. Let x = f (x) be a monotone system such that theorigin globally asymptotically stable and the compact setX ⊂ Rn+ is invariant. Then there exist strictly increasingfunctions Vk : R+ → R+ for k = 1, . . . ,n, such thatV (x) = max{V1(x1), . . . ,Vn(xn)} satisfies

d

dtV (x(t)) = −V (x(t))

along all trajectories in X .

[Rantzer, Rüffer, Dirr, CDC-13]


Proof idea

t = 0

t = 1

t = 2

t = 3


Outline



• Input-Output Performance




Performance of Positive systems

Suppose that G(s) = C(sI − A)−1B + D where A ∈ Rn-n isMetzler, while B ∈ Rn-1+ , C ∈ R1-n+ and D ∈ R+. Define.G.∞ = supω 0G(iω )0. Then the following are equivalent:

(i) The matrix A is Hurwitz and .G.∞ < γ .

(ii) The matrix

[

A B

C D − γ

]

is Hurwitz.


Optimizing H∞/L1 Performance

LetD be the set of diagonal matrices with entries in [0, 1].Suppose B,C,D ≥ 0 and A+ ELF is Metzler for all L ∈D .

If F ≥ 0, then the following are equivalent:

(i) There exists L ∈D such that A+ ELF is Hurwitzand .C[sI − (A+ ELF)]−1B + D.∞ < γ .

(ii) There exist ξ ∈ Rn+, µ ∈ Rm+ with

Aξ + Eµ + B < 0 Cξ + D < γ µ ≤ Fξ

If ξ ,µ satisfy (ii) , then (i) holds for every L such that µ = LFξ .



2

1

4 3

⎡

⎢⎢⎣

x1x2x3x4

⎤

⎥⎥⎦=

⎡

⎢⎢⎣

−1− $31 $12 0 00 −$12 − $32 $23 0$31 $32 −$23 − $43 $340 0 $43 −4− $34

⎤

⎥⎥⎦

⎡

⎢⎢⎣

x1x2x3x4

⎤

⎥⎥⎦+

⎡

⎢⎢⎣

wwww

⎤

⎥⎥⎦

How do we select $i j ∈ [0, 1] to minimize the gain from w to∑

i xi?



A = diag{−1, 0, 0,−4} B =(

1 1 1 1)T

C =(

1 1 1 1)

K = 0

L = diag{$31, $12, $32, $23, $43, $34}

E =

⎛

⎜⎜⎝

−1 1 0 0 0 0

0 −1 −1 1 0 0

1 0 1 −1 −1 1

0 0 0 0 1 −1

⎞

⎟⎟⎠

F =

⎛

⎜⎜⎜⎜⎜⎜⎝

1 0 0 0

0 1 0 0

0 1 0 0

0 0 1 0

0 0 1 0

0 0 0 1

⎞

⎟⎟⎟⎟⎟⎟⎠

The closed loop matrix is A+ ELF.



2

1

4 3

Minimize∑

i ξ i subject to

0 ≥ −ξ1 − µ31 + µ12 + 1

0 ≥ −µ12 − µ32 + µ23 + 1

0 ≥ µ31 + µ32 − µ23 − µ43 + µ34 + 1

0 ≥ −4ξ4 + µ43 − µ34 + 1

and 0 ≤ µ i j ≤ ξ j . Then define $i j = µ i j/ξ j .

Optimal solution $12 = $32 = $43 = 1 and $31 = $23 = $34 = 0.Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems

Example 2: Vehicle Formations

⎡

⎢⎢⎣

x1x2x3x4

⎤

⎥⎥⎦=

⎡

⎢⎢⎣

−1− $13 0 $13 0

$21 −$21 − $23 $23 0

0 $32 −$32 − $34 $340 0 $43 −4− $43

⎤

⎥⎥⎦

⎡

⎢⎢⎣

x1x2x3x4

⎤

⎥⎥⎦+ Bw

2

1

4 3 2

1

4 3 2

1

4 3

B =

⎡

⎢

⎢

⎣

101011

⎤

⎥

⎥

⎦

B =

⎡

⎢

⎢

⎣

1111

⎤

⎥

⎥

⎦

B =

⎡

⎢

⎢

⎣

111010

⎤

⎥

⎥

⎦


Outline




• Trajectory Optimization



Monotone Input-output Systems

The system

x(t) = f (x(t),u(t)), x(0) = a

is called monotone if{

a0 ≤ a1u0(τ ) ≤ u1(τ ), τ ∈ [0, t]

=" φ t(a0,u0) ≤ φ t(a1,u1)

[Angeli and Sontag, 2003]


Trajectory Optimization

The monotone system x = f (x,u) is a convex monotonesystem if every row of f is also convex.

Theorem:

For a convex monotone system x = f (x,u), each component ofthe trajectory φ t(a,u) is a convex function of (a,u).

[Rantzer and Bernhardsson, 2014]


Outline





• Combination Therapy for HIV and Cancer


EvoluHon*to*resistance*in*mice*with*chronic*HIV*infecHon*

and Supplementary Table 2b). Consistent with this observation,N332K is resistant to both PGT128 and 10-1074, whereas N332Y isresistant to PGT128 but remains sensitive to high concentrations of10-1074 (Supplementary Fig. 8).

Because the selected broadly neutralizing antibodies target distinctepitopes on the HIV-1 spike, we investigated whether treatment witha combination of three (tri-mix: 3BC176, PG16 and 45-46G54W) orfive (penta-mix: 3BC176, PG16, 45-46G54W, PGT128 and 10-1074)antibodies would alter the course of infection (Fig. 2 and Supplemen-tary Table 1). These combinations neutralize all but 2 (tri-mix 98.3%)and 1 (penta-mix 99.2%) of 119 mostly tier 2 and 3 viruses from multipleclades with an IC80 (geometric mean) of 0.121mg ml21 and 0.046mgml21 for tri-mix and penta-mix, respectively (Supplementary Fig. 9).

A decline in the initial viral load was seen in 11 out of 12 tri-mix-treated mice, but 7 rebounded to pre-treatment levels (Fig. 2a,Supplementary Table 1c and Supplementary Fig. 10). In contrast tomonotherapy where mice almost never controlled viraemia beyond2 weeks of therapy, the tri-mix treatment led to prolonged and effectiveHIV-1 control in 3 of 12 animals. In 2 of these mice viral loadrebounded 20–40 days after cessation of therapy at a time when theYU2 gp120-reactive antibody concentration in serum decreased toa level below detection (Fig. 3a). In the third mouse (ID21), viraemiawas detected but remained low even in the absence of therapy for60 days.

Sequences obtained from mice that experienced viral rebound whileon tri-mix therapy showed a combination of the mutations found inthe PG16 and 45-46G54W monotherapy groups (Figs 1d and 2b, c andSupplementary Tables 2b and 3a). We verified that these mutationsrendered HIV-1YU2 resistant to PG16 and 45-46G54W by producing thecorresponding mutant pseudoviruses and testing them in neutraliza-tion assays in vitro (Supplementary Fig. 8). The pseudoviruses werenot resistant to 3BC176, confirming that this antibody did not exertselective pressure on HIV-1YU2 and therefore only 2 of the 3 antibodiesin the tri-mix were efficacious.

In contrast, sequences obtained from the mice that exhibited sus-tained viral control and rebounded after cessation of therapy eitherlacked any broadly neutralizing antibody-associated mutation, or hada mutation mapped to the 45-46G54W (K282R) or PG16 (N162P) targetsite, but not both (Fig. 3b and Supplementary Table 3a). In these mice,rebound viraemia only occurred after YU2 gp120-reactive antibodylevels decreased to below detection, indicating that the viruses thatemerged were latent and remained susceptible to the tri-mix.

All 14 mice treated with the penta-mix treatment showed a decreasein viral load 6–7 days after initiation of therapy (Fig. 2d and Sup-plementary Table 1d). However, in contrast to monotherapy and thetri-mix, all of the penta-mix-treated mice remained below baselineviral loads during the entire treatment course (Fig. 2d, Supplemen-tary Table 1d and Supplementary Fig. 10). Of 13 mice (one died), 11had viral loads below or near the limit of detection. The two mice withthe slowest reduction in viral load during treatment showed signs ofsevere graft versus host disease. We conclude that penta-mix therapyreduces the viral load to levels below detection for up to 60 days inHIV-1YU2-infected humanized mice.

Penta-mix therapy was discontinued after 31–60 days and the micewere monitored for an additional 100 days (Fig. 3c). In 7 out of 8 micethat survived, viraemia rebounded after an average of 60 days (Fig. 3c).In contrast, mice treated with antiretroviral therapy rebound after10 days following discontinuation of therapy12. Viral rebound inpenta-mix-treated mice was always correlated with decreased levels ofthe administered antibodies (Fig. 3c). Only one tri-mix (ID21) and onepenta-mix (ID129) mouse did not rebound. To determine their ability tosupport HIV-1 infection, these mice were re-infected with HIV-1YU2(57.5 ng p24) and measured for viral load 2 weeks later. One of the twomice became infected but only at very low levels compared to the initialinfection (Supplementary Fig. 11). Therefore, prolonged control wasprimarily due to the long half-life of the injected antibodies.

–20 0 20 6040102

103

104

105

106

107a

0 604020

–3–2–10123

ID16ID17ID18ID21

ID25ID22

ID43ID49

ID59ID65

ID56

Days after starting antibody treatment (tri-mix)

b160,162 276–281 458–460

1816

2243

2556

4965

gp120 residue1 100 200 300 400 500

**

Mou

se ID

D

Y

TKHA

I

NP

8888888888

N160

T162 8/88

Y

T

S

N280

A281

G458

G459

D

D

VE

K460

8/88

c

ID128ID129ID130ID132ID133ID134ID143ID144ID161ID164ID182ID187ID194

0 20 40 60

–3–2–10123

Days after starting antibody treatment (penta-mix)

d

–20 0 20 40 60

Cop

ies

ml–1

102

103

104

105

106

107

Cop

ies

ml–1

Δlog

10(c

opie

s m

l–1)

Δlog

10(c

opie

s m

l–1)

Figure 2 | HIV therapy by a combination of three (tri-mix) or five (penta-mix) broadly neutralizing antibodies in HIV-1YU2-infected humanizedmice. a, The left panel shows viral loads (RNA copies ml21, y axis) over time(days, x axis) in HIV-1YU2-infected humanized mice treated with acombination of 3BC176, PG16 and 45-46G54W (tri-mix; grey shading). Eachline represents a single mouse and symbols indicate viral load measurements(Supplementary Table 1c). Infection and viral load determination wasperformed as in Fig. 1. The right panel shows changes in log10 (RNAcopies ml21) from baseline at day 0. Red and green lines represent the averagevalues in viral load change of tri-mix and control group (Fig. 1a), respectively.b, Individual gp120 envelope sequences cloned from single mice (y axis) duringtri-mix therapy after viral rebound. gp120 sequences are represented byhorizontal grey bars with silent mutations indicated in green and amino acidreplacements in red. Black asterisks represent mutations generating a stopcodon and bold grey bars deletions. All mutations are relative to HIV-1YU2 andnumbered according to HXBc2. Selected mutations at sites highlighted in bluecan confer resistance to PG16- or 45-46G54W-mediated neutralization in vitro(Supplementary Fig. 8 and Supplementary Table 3a). c, Pie charts as in Fig. 1dillustrate distribution of amino acid changes in gp120 (b) at PG16 (left) or 45-46G54W (right) respective target sites (Supplementary Table 3a). d, As in a, forHIV-1YU2-infected humanized mice treated with a combination of 3BC176,PG16, 45-46G54W, PGT128 and 10-1074 (penta-mix; grey shading,Supplementary Table 1d). Mouse ID72 (tri-mix) and ID126 (penta-mix) diedearly and are not displayed.

RESEARCH LETTER

1 2 0 | N A T U R E | V O L 4 9 2 | 6 D E C E M B E R 2 0 1 2

Macmillan Publishers Limited. All rights reserved©2012








–20 0 20 6040102

103

104

105

106

107a

0 604020

–3–2–10123

ID16ID17ID18ID21

ID25ID22

ID43ID49

ID59ID65

ID56


b160,162 276–281 458–460

1816

2243

2556

4965

gp120 residue1 100 200 300 400 500

**

Mou

se ID

D

Y

TKHA

I

NP

8888888888

N160

T162 8/88

Y

T

S

N280

A281

G458

G459

D

D

VE

K460

8/88

c


0 20 40 60

–3–2–10123


d

–20 0 20 40 60

Cop

ies

ml–1

102

103

104

105

106

107

Cop

ies

ml–1

Δlog

10(c

opie

s m

l–1)

Δlog

10(c

opie

s m

l–1)


RESEARCH LETTER



RNA*copies/ml*

(3BC176,*PG16,*45<46G54W)** (3BC176,*PG16,*45<46G54W,*PGT128,*10<1074)*

(100,*50,*10,*100,*100)*ug/ml*(100,*50,*10)*ug/ml*

AnHbody*trimix** AnHbody*pentamix*

Days** Klein,'et'al.'Nature,'2012''

Challenge:)Design*combinaHon*of*drugs*that*gives*fastest*possible*decay*rate*of*virus*populaHon*in*spite*of*mutaHons.*

Combination Therapy is a Control Problem

Evolutionary dynamics:

x =

(

A−∑

i

uiDi

)

x

Each state xk is the concentration of a mutant. (There can behundreds!) Each input ui is a drug dosage.

A describes the mutation dynamics without drugs, whileD1, . . . ,Dm are diagonal matrices modeling drug effects.

Determine u1, . . . ,um ≥ 0 with u1 + ⋅ ⋅ ⋅+ um ≤ 1 such that xdecays as fast as possible!

[Hernandez-Vargas, Colaneri and Blanchini, JRNC 2011][Jonsson, Rantzer,Murray, ACC 2014]


Optimizing Decay Rate

Stability of the matrix A−∑

i uiDi + γ I is equivalent to

existence of ξ > 0 with

(A−∑

i

uiDi + γ I)ξ < 0

For row k, this means

Akξ −∑

i

uiDikξk + γ ξk < 0

or equivalently

Akξ

ξk−∑

i

uiDik + γ < 0

Maximizing γ is convex optimization in (logξ i,ui,γ ) !


Using Measurements of Virus Concentrations

Evolutionary dynamics:

x(t) =

(

A−∑

i

ui(t)Di

)

x(t)

Can we get faster decay using time-varying u(t) based onmeasurements of x(t) ?


Using Measurements of Virus Concentrations

The evolutionary dynamics can be written as a convexmonotone system:

d

dtlog xk(t) =

Akx(t)

xk(t)−∑

i

ui(t)Dik

Hence the decay of log xk is a convex function of the input andoptimal trajectories can be found even for large systems.


Example

A =

⎡

⎢⎢⎣

−δ µ µ 0

µ −δ 0 µµ 0 −δ µ0 µ µ −δ

⎤

⎥⎥⎦

clearance rate δ = 0.24 day−1, mutation rate µ = 10−4 day−1

and replication rates for viral variants and therapies as follows

Variant Therapy 1 Therapy 2 Therapy 3

Wild type (x1) D11 = 0.05 D21 = 0.10 D31 = 0.30Genotype 1 (x2) D12 = 0.25 D22 = 0.05 D32 = 0.30Genotype 2 (x3) D13 = 0.10 D23 = 0.30 D33 = 0.30HR type (x4) D14 = 0.30 D24 = 0.30 D34 = 0.15


Example

Optimized drug doses:

0 20 40 60 80 100 120 140 160 180 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time [days]

u1u2u3

Total virus population:

0 20 40 60 80 100 120 140 160 180 20010-2

10-1

100

101

102

103

time [days]

time-varyingconstant


Comparison*with*the*experimental*trimix*

(3BC176,*PG16,*45<46G54W)*0.5*mg*weekly/*2*per*week*injecHon*

Klein,'et'al,'Nature,'2012'

EXPERIMENT)SIMULATION)

*(0.689,*0.6712,*1.0706)*ug/ml**(3BC176,*45<46G54W,*PGT128)**

TRIMIX)








–20 0 20 6040102

103

104

105

106

107a

0 604020

–3–2–10123

ID16ID17ID18ID21

ID25ID22

ID43ID49

ID59ID65

ID56


b160,162 276–281 458–460

1816

2243

2556

4965

gp120 residue1 100 200 300 400 500

**

Mou

se ID

D

Y

TKHA

I

NP

8888888888

N160

T162 8/88

Y

T

S

N280

A281

G458

G459

D

D

VE

K460

8/88

c


0 20 40 60

–3–2–10123


d

–20 0 20 40 60

Cop

ies

ml–1

102

103

104

105

106

107

Cop

ies

ml–1

Δlog

10(c

opie

s m

l–1)

Δlog

10(c

opie

s m

l–1)


RESEARCH LETTER



0 5 10 1510−10

10−5

100

105

1010

1015

1020Nominal ,γ

∞−ind=0.80503, γ∞=0.33997

Time in days

Viru

s co

ncen

tratio

n

0 5 10 1510−50

10−40

10−30

10−20

10−10

100

1010L1,γ∞−ind=0.65329, γ

∞=0.29413

Time in days

Viru

s co

ncen

tratio

n

Fig. 3. Sum of virus populations subject to random time invariantperturbations of 5% in the dynamics for 30 different simulations for(left) a stabilizing closed loop controller comprised of antibody pentamix(0.4687,0.7815, 0.6129, 0.6279, 0.8831) µg/ml of (3BC176, PG16, 45-46G54W, PGT128, 10-1074) synthesized using the convex program (5) and(right) a robustly stabilizing closed loop controller comprised of antibodytrimix (0.6891,0.6712,1.0706) µg/ml of (3BC176, 4546-G54W, PGT128)synthesized using the L1 combination therapy algorithm.

synthesized controller with gains of (0.6891,0.6712,1.0706)µg/ml of (3BC176, 4546-G54W, PGT128) to the exper-imentally studied trimix, we chose equal concentrationsof (3BC176, PG16, 45-46G54W), namely (1, 1, 1) µg/mlfor the experimentally derived trimix. We found that eventhough total antibody concentrations were larger in ourversion of the experimental trimix, the robustly stabilizingcontroller synthesized by the L1 algorithm nonetheless per-formed overall better; the closed loop norms were kGk1 =0.2941 and kGk1�ind

= 0.6533 for the L1 controller versuskGk1=0.26433 and kGk1�ind

=0.74572 for the experimen-tal trimix.

These simulations demonstrate that although many stabi-lizing solutions to the combination therapy problem exist,the best ones are found when design parameters such as asparsity, limits on the magnitude of gains, and robustnessguarantees are simultaneously considered. Experimentallysearching for these combinations is infeasible as the numberof potential therapies and possible concentrations to consideris experimentally intractable. We propose to guide these ex-perimental activities with our ability to design and synthesizecombination therapy controllers. As such, one could generatea family of controllers based on ”design specifications”tailored not only the (viral or cellular) composition of thedisease, but to explore tradeoffs between number of therapiesused (sparsity), therapy concentrations (magnitude of thegain) and ability to support pharmacokinetic fluctuations(robustness to perturbations) and subsequently verify theseexperimentally.

V. CONCLUSION AND FUTURE WORK

Leveraging recent results in positive systems, we pro-posed a scalable SOCP based iterative algorithm for thesystematic design of sparse, small gain feedback strategiesthat stabilize the evolutionary dynamics of a generic diseasemodel. Through the addition of `1 and `2 regularizationterms to the objective function, we achieved the desiredfeedback structure. In future work, we plan to explore aprincipled integration of our methods with recent results onthe robust L1 stability of positive systems [16]. In particular,

we hope to explicitly account for model error introduced dueto model linearization, parametric uncertainty and unmodeleddynamics due to drug interactions.

VI. ACKNOWLEDGEMENTS

We would like to thank Anders Rantzer for discussionsregarding the application his results in optimal controllersynthesis for positive systems and Nikolai Matni for thereview of the technical content. We would like to thankPamela Bjorkman for discussions regarding using antibodytherapy for HIV treatment. We appreciate the help of Bjork-man laboratory staff scientist Anthony West for informationon antibody neutralization parameters.

REFERENCES

[1] D. I. S. Rosenbloom, A. L. Hill, S. A. Rabi, R. F. Siliciano, and M. A.Nowak, “Antiretroviral dynamics determines hiv evolution and predictstherapy outcome,” Nature Biotechnology, vol. 18, pp. 1378–1385, June2012.

[2] F. Michor, Y. Iwasa, and M. A. Nowak, “Dynamics of cancer progres-sion,” Nature Reviews Cancer, vol. 4, no. 3, pp. 197–205, 2004.

[3] M. Alamir and S. Chareyron, “State-constrained optimal controlapplied to cell-cycle-specific cancer chemotherapy,” Optimal ControlApplications and Methods, vol. 28, no. 3, pp. 175–190, 2007.

[4] S. Chareyron and M. Alamir, “Model-free feedback design for a mixedcancer therapy,” Biotechnology progress, vol. 25, no. 3, pp. 690–700,2009.

[5] M. Alamir, “Robust feedback design for combined therapy of cancer,”Optimal Control Applications and Methods, 2012.

[6] V. Jonsson, N. Matni, and R. M. Murray, “Reverse engineeringcombination therapy for evolutionary dynamics of disease: an H1approach,” in To appear in the IEEE Conf. on Decision and Control,2013, 2013.

[7] B. Al-Lazikani, U. Banerji, and P. Workman, “Combinatorial drugtherapy for cancer in the post-genomic era,” Nature Biotechnology,vol. 30, pp. 679–692, July 2012.

[8] F. Klein, A. Halper-Stromberg, J. A. Horwitz, H. Gruell, J. F. Scheid,S. Bournazos, H. Mouquet, L. A. Spatz, R. Diskin, A. Abadir, et al.,“Hiv therapy by a combination of broadly neutralizing antibodies inhumanized mice,” Nature, 2012.

[9] M. Eigen, “Viral quasispecies,” Scientific American, vol. 269, pp. 42–49, 1993.

[10] T. Tanaka and C. Langbort, “The bounded real lemma for internallypositive systems and h-infinity structured static state feedback,” Auto-matic Control, IEEE Transactions on, vol. 56, no. 9, pp. 2218–2223,2011.

[11] A. Rantzer, “Distributed control of positive systems,” in Decision andControl and European Control Conference (CDC-ECC), 2011 50thIEEE Conference on, Dec. 2011, pp. 6608–6611.

[12] ——, “Distributed control of positive systems,” in Automatic Control,IEEE Transactions on. In preparation, 2014.

[13] M. Li, F. Gao, J. R. Mascola, L. Stamatatos, V. R. Polonis, M. Kout-soukos, G. Voss, P. Goepfert, P. Gilbert, K. M. Greene, et al., “Humanimmunodeficiency virus type 1 env clones from acute and earlysubtype b infections for standardized assessments of vaccine-elicitedneutralizing antibodies,” Journal of virology, vol. 79, no. 16, pp.10 108–10 125, 2005.

[14] A. B. Balazs, J. Chen, C. M. Hong, D. S. Rao, L. Yang, andD. Baltimore, “Antibody-based protection against hiv infection byvectored immunoprophylaxis,” Nature, vol. 481, no. 7379, pp. 81–84,2011.

[15] R. Diskin, J. F. Scheid, P. M. Marcovecchio, A. P. West Jr, F. Klein,H. Gao, P. N. Gnanapragasam, A. Abadir, M. S. Seaman, M. C.Nussenzweig, et al., “Increasing the potency and breadth of an hivantibody by using structure-based rational design,” Science, vol. 334,no. 6060, pp. 1289–1293, 2011.

[16] C. Briat, “Robust stability and stabilization of uncertain linear positivesystems via integral linear constraints: L1-and linfinity-gains charac-terization,” arXiv preprint arXiv:1204.3554, 2012.

Robustly)stabilizing)controller,)TRIMIX)

(3BC176,*PG16,*45<46G54W)*at*(1,*1,*1)*ug/ml:**Input<output*gain*0.75**

(3BC176,*45<46G54W,*PGT128)*at*(0.689,*0.6712,*1.07)*ug/ml:*Input<output*gain*0.65*

For Scalable Control — Use Positive Systems!

Verification and synthesis scale linearly

Distributed controllers by linear programming

No need for global information

Optimal trajectiories by convex optimization


Many Research Challenges Remain

Optimal Dynamic Controllers in Positive Systems

Analyze Trade-off Between Performance and Scalability

Distributed Controllers for Nonlinear Monotone Systems


Thanks!

Enrico Vanessa Daria MartinaLovisari Jonsson Madjidian Maggio

Alessandro Bo FredrikPapadopoulos Bernhardsson Magnusson


Documents

Scalable Analysis and Control of Positive Systems