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Control of stochastic distributed-parameter systems

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Abstract

A scheme is proposed for the feedback control of distributed-parameter systems with unknown boundary and volume disturbances and observation errors. The scheme consists of employing a nonlinear filter in the control loop such that the controller uses the optimal estimates of the state of the system. A theoretical comparison of feedback proportional control of a styrene polymerization reactor with and without filtering is presented. It is indicated how an approximate filter can be constructed, greatly reducing the amount of computing required.

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Abbreviations

a(t):

l-vector noisy dynamic input to system

A(t, a):

l-vector function

A′:

frequency factor for first-order rate law (5.68×106 sec−1)

b :

distance to the centerline between two coil banks in the reactor (4.7 cm)

B :

k-vector function defining the control action

c(χ, τ):

concentration of styrene monomer, molel −1

C p :

heat capacity (0.43 cal · g−1 · K−1)

C ij :

constants in approximate filter, Eqs. (49)–(52)

E :

activation energy (20330 cal · mole−1)

ℰ :

expectation operator

f(t,...):

n-vector function

g 0,g 1(t,...):

n-vector functions

h(t, u):

m-vector function relating observations to states

H(t):

function defined in Eq. (36)

k :

dimensionality of control vectorv(x, t)

k i :

constants in approximate filter, Eqs. (49)–(52)

K :

dimensionless proportional gain

l :

dimensionality of dynamic inputa(t)

m :

dimensionality of observation vectory(t)

n :

dimensionality of state vectoru(x, t)

P (vv)(x, s, t):

n×n matrix governed by Eq. (9)

P (va)(x, t):

n×l matrix governed by Eq. (10)

P (aa)(t):

l×l matrix governed by Eq. (11)

q i (t):

diagonal elements ofm×m matrixQ(x, s, t)

Q(x, s, t):

m×m weighting matrix

R :

universal gas constant (1.987 cal · mole−1 · K−1)

R(x, s, t):

n×n weighting matrix

R i (t):

n×n weighting matrix

s :

dimensionless spatial variable

S(x, s, t):

matrix defined in Eq. (11)

t :

dimensionless time variable

T(χ, τ):

temperature, K

u(x, t):

n-dimensional state vector

u c (t):

wall temperature

u d :

desired value ofu 1(1,t)

u c *:

reference control value ofu c

u max c :

maximum value ofu c

u min c :

minimum value of c

v(x, t):

k-dimensional control vector

W(t):

l×l weighting matrix

x :

dimensionless spatial variable

y(t):

m-dimensional observation vector

α i :

constants in approximate filter, Eqs. (49)–(52)

β :

dimensionless parameter defined in Eq. (29)

ΔH :

heat of reaction (17500 cal · mole−1)

ε :

dimensionless activation energy, defined in Eq. (29)

δ(x):

Dirac delta function

η(x, t):

m-dimensional observation noise

κ :

thermal conductivity (0.43×10−3 cal · cm−1 · sec−1 · K−1)

ρ :

density (1 g · cm−3)

τ :

time, sec

φ :

dimensionless parameter defined in Eq. (29)

χ :

spatial variable, cm

*:

reference value

^:

estimated value

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Authors and Affiliations

  1. Department of Chemical Engineering, California Institute of Technology, Pasadena, California

    T. K. Yu (Graduate Research Assistant) & J. H. Seinfeld (Associate Professor)

Authors
  1. T. K. Yu
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  2. J. H. Seinfeld
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Communicated by R. Jackson

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Yu, T.K., Seinfeld, J.H. Control of stochastic distributed-parameter systems. J Optim Theory Appl 10, 362–380 (1972). https://doi.org/10.1007/BF00935400

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  • DOI: https://doi.org/10.1007/BF00935400

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