economic model predictive control of transport-reaction processes

Economic Model Predictive Control of Transport-ReactionProcesses

Liangfeng Lao, Matthew Ellis and Panagiotis D. Christofides

Department of Chemical & Biomolecular Engineering

Department of Electrical Engineering

University of California, Los Angeles

AIChE Meeting

San Francisco, CA

November 7, 2013

INTRODUCTION

Transport-Reaction Processes

• Examples:

⋄ Fixed bed and tubular reactors

⋄ Fluidized reactors

⋄ Chemical vapor deposition processes

• Process state variables are characterized by strong spatial variations and

nonlinear behavior

⋄ Diffusion and convection phenomena

⋄ Complex reaction mechanisms / Arrhenius dependence of reaction rates on

temperature

• First-principles modeling leads to nonlinear parabolic partial differential equations

(PDEs)

TUBULAR REACTOR EXAMPLE

Concentration Trajectory

• Diffusion-convection-reaction described by a quasi-linear parabolic PDE system:

∂T

∂t= −v

∂T

∂z+

k

ρCp

∂2T

∂z2+

(−∆H

ρCp

)e−E/RTC2

A −hAs

ρCp(T − TC)

∂CA

∂t= −v

∂CA

∂z+DA

∂2CA

∂z2− k0e

−E/RTC2A

BACKGROUND ON CONTROL OF PARABOLIC PDES

• Standard approach: (Ballas, IJC, 1979; Ray, McGraw-Hill, 1981; Curtain, Springer-Verlag, 1978)

⋄ Derivation of ODE models using eigenfunction expansions

⋄ Controller design using methods for ODEs

⋄ High-dimensionality of the controller

• Synthesis of nonlinear low-order controllers: (Christofides, Birkhauser, 2001)

⋄ Derivation of low-order ODE models using Galerkin’s method and

approximate inertial manifolds

⋄ Nonlinear and robust controller synthesis

• Other approaches:

⋄ Passivity-based control approach (Ydstie et al., Syst. & Contr. Lett., 1997)

⋄ Backstepping boundary control (Krstic and Smyshlyaev, SIAM, 2008)

• Model predictive control with quadratic cost functions (Dubljevic et al., Inter. J. Rob. & Non.

Contr., 2006, Comp. & Chem. Eng., 2005)

• Economic model predictive control of transport-reaction processes is an open issue

CONVENTIONAL MPC VS. ECONOMIC MPC

• Conventional MPC

⋄ Quadratic cost function for MPC (Q,R: positive definite matrices)

J =

∫ N∆

0

[(x(τ)− xset)TQ(x(τ)− xset) + (u(τ)− uset)

TR(u(τ)− uset)]dτ

⋄ Penalize the deviation of states and inputs from the steady-state

⋄ Force the system to the economically optimal steady-state

• Lyapunov-based Economic MPC (LEMPC)

⋄ General cost function which accounts for the process economics

J =

∫ N∆

0

L(x(τ), u(τ))dτ

⋄ Use MPC for economic optimization and process control

⋄ Need constraints for closed-loop stability

⋄ Key ideas of our approach:

Utilize Lyapunov-based techniques in the economic MPC formulation

Take advantage of the stability region of an explicit Lyapunov-based controller to

ensure closed-loop stability under LEMPC

LYAPUNOV-BASED ECONOMIC MPC

Two Mode Control Strategy

(M. Heidarinejad, et al., AIChE J. 2012, JPC 2012; S&CL, 2012; 2013)

minimizeu∈S(∆)

∫ tk+N

tk

L(x(τ), u(τ))dτ

subject to ˙x(t) = f(x(t), u(t), 0)

x(tk) = x(tk)

u(t) ∈ U, ∀ t ∈ [tk, tk+N )

V (x(t)) ≤ ρ ∀ t ∈ [tk, tk+N ),

if V (x(tk)) ≤ ρ and tk < t′

∂V

∂xf(x(tk), u(tk), 0)

≤ ∂V

∂xf(x(tk), h(x(tk)), 0)

if V (x(tk)) > ρ and tk ≥ t′

• Economic cost function

• Dynamic model

• Initial condition

• Input constraints

• Mode 1: Boundedness

of closed-loop state

• Mode 2: Convergence

to the origin

LYAPUNOV-BASED ECONOMIC MPC

Two Mode Control Strategy

Ωρ

Ωρ

xs

x(t0)

x(t′)

• Mode 1: Boundedness

of closed-loop state

• Mode 2: Convergence

to the origin

MOTIVATION OF EMPC FOR PDE SYSTEMS

Tubular reactor

• Control objective: maximize the production rate of the product B subject to a

constraint on the available reactant material

⋄ Formulate a model predictive controller that works to directly optimize

objective:

J(x, u) =

∫ tf

0

∫ L

0

k exp

(−E

RT (z, t)

)(CA(z, t))

2 dz dt

⋄ Account for constraint on available reactant material:

1

tf

∫ tf

0

u(τ)dτ = uavailable

• Challanges of applying EMPC to a PDE system

⋄ Construct a finite dimensional model using finite difference method can result

in a high-order model (can be computational inefficient)

⋄ Formulate EMPC with a finite dimensional model of the system

PRESENT WORK

(Lao et al., Ind. & Eng. Chem. Res., 2013; J. Process Contr., submitted)

• Scope:

⋄ Nonlinear parabolic PDEs

⋄ State-feedback Economic Model Predictive Control for PDE systems using

low-order/high-order finite-dimensional models of the PDE system

⋄ Output-feedback Economic Model Predictive Control for PDE systems using

low-order/high-order finite-dimensional models of the PDE system

• Objective:

⋄ Optimally operate a nonlinear parabolic PDE system with EMPC

• Approach:

⋄ Formulate the transport-reaction process as an infinite-dimensional system

⋄ Use Galerkin’s method for modal decomposition of the system

⋄ Take advantage of the stability region of the Lyapunov-based controller

⋄ Perform state-feedback/output-feedback low-order/high-order EMPC design

⋄ Application to a tubular reactor

CLASS OF NONLINEAR PARABOLIC PDE SYSTEMS

• System description

∂x

∂t= A

∂x

∂z+B

∂2x

∂z2+Wu(t) + f(x(z, t))

yj(t) =

∫ 1

0

cj(z)x(z, t)dz, j = 1, · · · , p

⋄ Boundary conditions and initial condition:

∂xj

∂z= g0xj , z = 0;

∂xj

∂z= g1xj , z = 1; x(z, 0) = η(z, 0)

⋄ x ∈ Rn: state vector of the system

⋄ u(t) ∈ Rm: manipulated input vector

⋄ f(x(z, t)) ∈ Rn: nonlinear vector function

⋄ yj(t): the jth measured output

⋄ cj(z): jth sensor distribution function

⋄ z ∈ [0, 1]: the spatial coordinate

⋄ u(t) ∈ U = u ∈ Rm | |ui(t)| ≤ ui,max, i = 1, . . . ,m⋄ A, B, W , g0 and g1: constant matrices and vectors

APPLICATION TO A TUBULAR REACTOR

• A non-isothermal tubular reactor where an irreversible second-order reaction of

the form A → B (exothermic) takes place

• Dynamic model of the process:

∂T

∂t= −v

∂T

∂z+

k

ρCp

∂2T

∂z2+

(−∆H)

ρCpexp

(−E

RT

)CA

2 − hAs

ρCp(T − TC)

∂CA

∂t= −v

∂CA

∂z+DA

∂2CA

∂z2− k0 exp

(−E

RT

)CA

2

• Boundary conditions:

z = 0 :∂T

∂z=

ρCpv

K(T − Tf ),

∂CA

∂z=

v

DA(CA − CAf )

z = L :∂T

∂z= 0,

∂CA

∂z= 0

• Manipulated input: the inlet concentration of species A, CAf

DIMENSIONLESS PDE DYNAMIC MODEL

• Dimensionless form (combining the non-homogeneous part of the boundary

conditions with the differential equation):

∂x1

∂t= −∂x1

∂z+

1

Pe1

∂2x1

∂z2+ βT (Ts − x1) +BTBC exp

(γx1

1 + x1

)(1 + x2)

2

+ δ(z − 0)Ti

∂x2

∂t= −∂x2

∂z+

1

Pe2

∂2x2

∂z2−BC exp

(γx1

1 + x1

)(1 + x2)

2 + δ(z − 0)u

where δ is the standard Dirac function

• Boundary conditions:

z = 0 :∂x1

∂z= Pe1x1,

∂x2

∂z= Pe2x2

z = 1 :∂x1

∂z= 0,

∂x2

∂z= 0

• Process Parameters: Pe1 = 7, P e2 = 7, BT = 2.5, BC = 0.1, βT = 2, Ts = 0, Tf = 0

and γ = 10

GALERKIN’S METHOD

• Solve the eigenvalue problem of the spatial differential operator, Ai subject to:

Aiϕij = −dϕij

dz+

1

Pei

d2ϕij

dz2= λjϕij

dϕij

dz= Peiϕij , z = 0;

dϕij

dz= 0, z = 1; i = 1, 2, j = 1, . . . ,∞

• Apply Ps, Pf to get the infinite-dimensional nonlinear system

x = Ax+ Bu+ f(x), x(0) = x0

⋄ Ps, Pf : orthogonal projection operators in the space of ϕij

• Derive out set of infinite ODEs

dxs

dt= Asxs +Bsu+ fs(xs, xf ) (m)

dxf

dt= Afxf +Bfu+ ff (xs, xf ) (l)

y(t) = Csxs(t) + Cfxf (t)

⋄ xs(t) = Psx(t): state vector corresponding to slow eigenmodes

⋄ xf (t) = Pfx(t): state vector corresponding to fast eigenmodes

⋄ Cs: measurement operator corresponding to slow eigenmodes

EMPC FORMULATION

Economic Cost & Reactant Material Constraint

• Control Objective: Maximize the total reaction rate along the length of the

reactor over one operation period of tf = 1

• The economic cost function:

J(x, u) =

∫ tf

0

L(CA, T, u) dt

L(CA, T, u) =

∫ L

0

k exp

(−E

RT (z, t)

)(CA(z, t))

2 dz

• Since the reaction is second-order, can maximize the rate by feeding in the most

amount of material

• Consider the total amount of reactant material during one period to be fixed

(practical consideration)

• The reactant material constraint:

1

tf

∫ tf

0

u(τ)dτ = 0.5

EMPC FORMULATION 1

Low-order State-feedback EMPC Formulation

• EMPC formulation:

maxu∈S(∆)

∫ tk+N

tk

L(as(τ), u(τ)) dτ

s.t. as(t) = Asas(t) + Fs(as(t), 0) +Bsu(t)

umin ≤ u(t) ≤ umax, ∀t ∈ [tk, tk+N )

as(tk) =

∫ 1

0

x(tk)ϕsdz

u ∈ gk

[as(t)− ass]TP [as(t)− ass] < ρ, ∀ t ∈ [tk, tk+N )

⋄ Model based on first two modes (m = 2) computed from 200 measured

state-feedback points

⋄ ass: steady states value of the coefficients as

⋄ gk: the reactant material constraint

⋄ Prediction horizon and sampling time: N = 3, ∆ = 0.02 hr = 72 sec,

umin = −1 and umax = 1

CASE 1: LOW-ORDER STATE-FEEDBACK EMPC


Temperature Trajectory

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

u(t)

t

Low−order EMPC with Input ConstraintUniform Input Distribution

Manipulated Input Trajectories

• Initially, feed in the maximum allow-

able reactant material

• At t = 0.75, feed minimum allowable

reactant material to satisfy material

constraint

• Able to obtain the optimal trajectory

with a reduced-order model (2 modes)


Comparison with Uniform Material Distribution


Temperature Trajectory

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

t

Rea

ctio

n R

ate,

L(x

,u)

Low−order EMPC with Input ConstraintUniform Input Distribution

Reaction Rate Trajectory

• Average reaction rate from the sys-

tem under EMPC is 9.45% greater

than that from the system under

uniform in time reactant material

distribution


Comparison with Finite-difference EMPC

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

u(t)

t

Low−order EMPCFinite−Difference EMPCUniform Input Distribution

Manipulated Input Trajectory

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Simulation Time

Eva

luat

ion

Tim

e (s

)

Finite Difference EMPCLow−Order EMPC

EMPC Computational Time Trajectory

• Finite-difference EMPC uses 400th− order ODE model

• Input, state and reaction rate trajectories are overlapping

• Low-order state-feedback EMPC shows significant computational efficiency

improvement

• Finite-difference EMPC based on a 4th− order ODE model does not compute the

optimal trajectory

EMPC FORMULATION 2

High-order State-feedback EMPC Formulation with State and Input Constraints

• Consider a state constraint on the maximum allowable temperature in the reactor


maxu∈S(∆)

∫ tk+N

tk

L(as(τ), af (τ), u(τ)) dτ

s.t. as(t) = Asas(t) + Fs(as(t), af (t)) +Bsu(t)

af (t) = Afaf (t) +Bfu(t)

umin ≤ u(t) ≤ umax, ∀ t ∈ [tk, tk+N )

amin ≤ a(t) ≤ amax, ∀ t ∈ [tk, tk+N )

as(tk) =

∫ 1

0

x(tk)ϕsdz, af (tk) =

∫ 1

0

x(tk)ϕfdz

a(t) = as(t) + af (t)

u ∈ gk

[a(t)− ass]TP [a(t)− ass] < ρ, ∀ t ∈ [tk, tk+N )

⋄ Slow subsystem based on the first 10 modes

⋄ Fast subsystem based on the next 190 modes

CASE 2: HIGH-ORDER STATE-FEEDBACK EMPC

State Constraint

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

u(t)

t

High−order EMPC with State & Input ConstraintsUniform Input Distribution


0 0.2 0.4 0.6 0.8 11.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

x 1,m

ax

t

High−order EMPC with State & Input Constraints

Maximum Allowable Temperature

Maximum Temperature Trajectory

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

t

Rea

ctio

n R

ate,

L(x

,u)

High−Order EMPC with State & Input ConstraintsUniform Input Distribution


• The reactor temperature must not ex-

ceed the maximum allowable operat-

ing temperature (state constraint)

• EMPC determines the optimal time-

varying operating strategy

• Average reaction rate from the system

under EMPC is 6.91% greater

EMPC FORMULATION 3

State Estimation Using Output Feedback Methodology

• State Estimation: Use a finite number, p, of measured outputs yj(t) (here, we

choose state-feedback measured points) (j = 1, · · · , p) to compute estimates of asand af with the assumption:

p = m

C−1s exists

• The estimates of the slow modes, as(t) is obtained from a direct inversion of the

measured output map:

as(t) = C−1s y(t)

• The fast dynamics af (t) can be ignored (given that Af includes eigenvalues with

large negative real part) compared with the slow dynamics as(t)

• An explicit form for the estimated fast modes, af :

af (t) = −A−1f [Bfu(t) + ff (as(t), 0)]

EMPC FORMULATION 3

Low-order Output-feedback EMPC


maxu∈S(∆)

∫ ttk+N

tk

L(as(τ), u(τ))dτ

s.t. ˙as(t) = Asas(t) + Fs(as(t), 0) + Bsu(t)

as(tk) = C−1s y(tk)

umin ≤ u(t) ≤ umax

u ∈ gk

as,min ≤ as(t) ≤ as,max

[as(t)− ass]TP [as(t)− ass] < ρ, ∀ t ∈ [tk, tk+N )

⋄ Model based on first 11/21 modes (m = 11/21) computed from 11 measured

state-feedback points (p = 11/21)

⋄ as(t): predicted slow mode function of as(t)

⋄ Prediction horizon and sampling time: N = 3, ∆ = 0.01 hr = 36 sec,

umin = −1 and umax = 1

CASE 3: LOW-ORDER OUTPUT-FEEDBACK EMPC

Maximum Temperature and Manipulated Input Trajectories

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1

−0.5

0

0.5

1

1.5

t

u(t)

11 Slow Modes21 Slow ModesUniform Input, u(t) = 0.5


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

t

x 1,m

ax

21 Slow Modes11 Slow ModesMaximum Allowable Temperature


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

t

Rea

ctio

n R

ate,

L(x

,u)

11 Slow Modes21 Slow ModesUniform Input, u(t) = 0.5


• EMPC based on 21 slow modes computes

a smoother manipulated input profile

• Average reaction rate from the EMPC

based on 21 slow modes is 5.13% greater

than that of EMPC based on 11 slow

modes and 8.45% greater than that of the

system with uniform in time reactant ma-

terial distribution

EMPC FORMULATION 4High-order Output-feedback EMPC Formulation


maxu∈S(∆)

∫ ttk+N

tk

L(as(τ), af (τ), u(τ))dτ

s.t. ˙as(t) = Asas(t) + Fs(as(t), af (t)) + Bsu(t)

as(tk) = C−1s y(tk)

af (t) = −A−1f [Bfu(t) + ff (as(t), 0)]

a(t) = as(t) + af (t)

umin ≤ u(t) ≤ umax

u ∈ gk

amin ≤ a(t) ≤ amax

[a(t)− as]TP [a(t)− as] < ρ, ∀ t ∈ [tk, tk+N )

⋄ Model based on first 30 modes which include 11 slow modes and 19 fast modes

(m = 11, l = 19) computed from 11 measured state-feedback points (p = 11)

⋄ af (t): predicted fast mode function of af (t)

⋄ a(t): predicted finite mode function of a(t)

CASE 4: HIGH-ORDER OUTPUT-FEEDBACK EMPC

Maximum Dimensionless Temperature and Manipulated Input Trajectories

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.2

−0.8

−0.4

0

0.4

0.8

1.2

t

u(t)

30 Modes (11 Slow Modes, Output Feedback)


30 Modes (11 Slow Modes, State Feedback)

Uniform Input, u(t) = 0.5


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

t

x 1,m

ax

30 Modes (11 Slow Modes, Output Feedback)11 Modes (11 Slow Modes, Output Feedback)30 Modes (11 Slow Modes, State Feedback)Maximum Allowable Temperature


• Additional 19 fast modes reduces the input fluctuation and increase the maximum

dimensionless temperature

• State-feedback EMPC with same number of slow and fast modes uses 100

measured points to construct the reduced-order model used in EMPC

• State-feedback EMPC shows its advantage of model accuracy due to more

available measurement points

CASE 4: HIGH-ORDER OUTPUT-FEEDBACK EMPC

Reaction Rate and Computational Time Trajectories

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

t

Rea

ctio

n R

ate,

L(x

,u)



30 Modes (11 Slow Modes, State Feedback)

Uniform Input, u(t) = 0.5


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

t

Com

puta

tiona

l Tim

e Pr

ofile

/ se

c

30 Modes (11 Slow Modes, Output Feedback)11 Modes (11 Slow Modes, Output Feedback)30 Modes (11 Slow Modes, State Feedback)

EMPC Computational Time Trajectory

• Average reaction rate from the high-order output feedback EMPC is 0.89%

greater than that from the low-order output feedback EMPC system

• Average reaction rate from the high-order output feedback EMPC system is

0.73% less than that from the EMPC system with state feedback

• The average computational efficiency of the high-order output feedback EMPC

system is comparable to that of the state feedback EMPC system

CONCLUSIONS

• Proposed EMPC formulation for nonlinear parabolic PDE systems

⋄ Use Galerkin’s method to realize order reduction of nonlinear parabolic PDE

model

⋄ Perform low/high-order state/output-feedback EMPC formulations

• Application to a tubular reactor

⋄ Demonstrated the ability to maximize the economic cost by following a

time-varying input trajectory

⋄ Demonstrated the ability to satisfy a state constraint on the maximum

allowable temperature

⋄ Compared the output-feedback and state-feedback EMPC performance on

model accuracy, objective optimization and computational efficiency

ACKNOWLEDGEMENT

Financial support from the National Science Foundation and Department of Energy is

gratefully acknowledged

economic model predictive control of transport-reaction processes

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