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.
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
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]
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
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).
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
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.
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
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
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
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
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
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
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
Outline
• Positive and Monotone Systems
○ Scalable Stability Analysis
○ Input-Output Performance
○ Trajectory Optimization
○ Combination Therapy for HIV and Cancer
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
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.
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive 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)
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
Example 1: Transportation Networks
Cloud computing / server farms
Heating and ventilation in buildings
Traffic flow dynamics
Production planning and logistics
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
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?
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
Example 2: A vehicle formation
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
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?
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
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)
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
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].
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
Outline
○ Positive and Monotone Systems
• Scalable Stability Analysis
○ Input-Output Performance
○ Trajectory Optimization
○ Combination Therapy for HIV and Cancer
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
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.
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
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
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
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).
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
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]
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
Proof idea
t = 0
t = 1
t = 2
t = 3
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
Outline
○ Positive and Monotone Systems
○ Scalable Stability Analysis
• Input-Output Performance
○ Trajectory Optimization
○ Combination Therapy for HIV and Cancer
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
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.
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
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 .
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
Example 1: Transportation Networks
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?
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
Example 1: Transportation Networks
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.
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
Example 1: Transportation Networks
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
⎤
⎥
⎥
⎦
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
Outline
○ Positive and Monotone Systems
○ Scalable Stability Analysis
○ Input-Output Performance
• Trajectory Optimization
○ Combination Therapy for HIV and Cancer
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
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]
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
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]
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
Outline
○ Positive and Monotone Systems
○ Scalable Stability Analysis
○ Input-Output Performance
○ Trajectory Optimization
• Combination Therapy for HIV and Cancer
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
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
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
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]
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
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 ) !
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
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) ?
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
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.
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive 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
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
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
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
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)
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
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
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
Many Research Challenges Remain
Optimal Dynamic Controllers in Positive Systems
Analyze Trade-off Between Performance and Scalability
Distributed Controllers for Nonlinear Monotone Systems
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems
Thanks!
Enrico Vanessa Daria MartinaLovisari Jonsson Madjidian Maggio
Alessandro Bo FredrikPapadopoulos Bernhardsson Magnusson
Anders Rantzer, LCCC Linnaeus center Scalable Analysis and Control of Positive Systems