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Process systems, passivity and the second law of thermodynamics
Computers chem. Engng Vol. 20, Suppl., pp. S I 119-S1124, 1996
Pergamon
S0098-1354(96)00194.9
Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0098-1354196 $15.00+0.00
Process Systems, Passivity and the Second Law of Thermodynamics A N T O N I O A. A L O N S O and B. ERIK Y D S T I E Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh PA 15213 A b s t r a c t - " In this p a p e r We use the first and second laws of t h e r m o d y n a m i c s to motivate a theory for nonlinear process control. O u r m a i n tenets are: positive entropy production, b o u n d e d n e s s of entropy in t e r m s of energy, concavity of t h e entropy density and Helmholtz free energy as a storage function. These give t h e process s y s t e m a causal i n p u t - o u t p u t description, zero s t a t e detectability and stabilizability. To make the theory apply to practical s y s t e m s we follow ideas from classical irreversible t h e r m o d y n a m i c s and extend the concept of entropy of the non-equilibrium by a s s u m i n g local equilibrium.
1
Introduction
In 1867 R u d o l p h Clausius 1 proposed t h e following inequality for the entropy S of a body undergoing a t r a n s f o r m a t i o n from an equilibrium s t a t e to a n o t h e r via heating: s 2 - s l >_
~
2 dq 7
(1)
dq is t h e i n c r e m e n t a l h e a t i n g and T t h e absolute t e m p e r a t u r e . Inequality (1) is called the second law of t h e r m o d y namics. A r o u n d t h e s a m e time as Clansius developed inequality (1) it also became apparent t h a t h e a t and work are equivalent. T h i s idea is expressed as an equality called the first law of t h e r m o d y n a m i c s : dq = du + dw
(2)
were du s t a n d s for t h e change in t h e internal energy and dw t h e incremental work performed by deformation. By combining t h e first a n d second laws we relate entropy to energy t h r o u g h t the definition of a new t h e r m o d y n a m i c q u a n t i t y referred to as Helmholtz free energy, A = U - TS, where U is the internal energy. For an equlibrium t r a n s f o r m a t i o n we have U2 - T~S2 > V l - T l S ~
(3)
By defining t h e reference s y s t e m at absolute zero so t h a t T1 -- 0"1 = 0 we establish a lower b o u n d for t h e Helmholtz energy and it follows t h a t A can be used a storage function candidate for control s y s t e m design. Internal disspation gives stabilization a n d a connection is m a d e between nonlinear control theory as developed in [9] and [1] and nonequilibrium t h e r m o d y n a m i c theory [4] [2] [8] t h r o u g h the notion of stability of t h e zero dynamics.
2
Process systeins~ potentials and actions
Let Y and T be s u b s e t s of R~. and R repectively. A thermodynamic system is a pair (V, II). T h e elements v • ]2 have t h e form v = ( v l , v 2 , , . . . , ~ , ) where the v~'s are m e a s u r e s on Y and t h e elements r • II have t h e form x = ( r l , *r2,, ..., t n ) where the ,ri's are signed m e a s u r e s on V x T. We will refer to r , as a process on the s y s t e m a n d to v, as t h e potential corresponding to t h e process t i . T h e subscript i m a y be replaced by letters so t h a t M, E, S, A refers to m a s s , energy, entropy a n d Helmholtz free energy respectively. T h e s u b s e t s "V are referred to as volumes and their closures B are referred to as the boundaries. T h e subsets 7- = It, t + r) are intervals of the real line where t is referred to as t i m e a n d r takes values in ( - ~ o , oc). T h e m e a s u r e v is a potential for the process x a n d t h e process performed by a s y s t e m is therefore represented by the transition
• (t, ~) = ~(t + ~) - ~(t)
(4)
A change in potential is initiated by non-zero process. T h e process itself satisfies a unique decomposition
=p+~
(5)
T h e set-function l0 : V x T ~-* R defines production of v in ]) during t h e interval of time 7" and ~ is t h e supply of v to t h e s y s t e m from t h e exterior t h r o u g h t h e b o u n d a r y B. In this way ~ represents an action on t h e s y s t e m (13, II). From t h e theory of integration on p r o d u c t spaces it follows t h a t for every point in R~. x t t we define a unique derivative f of v so t h a t v = f v fd]]" T h e vector function ] = f (x, y, ~, t) is t h e density of v at the point (x, y, z, t) • R 4. An n vector of densities ] = (fa, ..., ]~) defines the s t a t e of a s y s t e m and is referred to as t h e field. From t h e definitions above it t h e n follows t h a t the process v induces a t r a n s f o r m a t i o n on the state space so t h a t ( r , f ) ~-* M ~ f T h e set of processes II0 = {x(t, r ) : ~ = M r f } is referred to cycle is a process which takes t h e s t a t e of t h e s y s t e m back to as t h e s t a t e space. We now enter into t h e main s u b j e c t process control. Existence of such processes are properties
(6)
as t h e set of thermodynamic cycles at f . A t h e r m o d y n a m i c to its s t a r t i n g point. T h e set F = {f : II0 # ~} is referred and define a process pair of e x t r a o r d i n a r y i m p o r t a n c e for of what we call a process system.
1 "Uber ver~chiedene ffir die Anwendung bequeme Forraen der Hauptgleichnungen der mechanischen Warmetheorie," Abhandlungen fiber die mechanische Wgrmetheorie, Vol. II, Viewegm Braumchweig , 1867, pp 1-44. S1119
S1120
European Symposium on Computer Aided Process Engineering--6. Part B
1. The process r is said to be a Clausius process if ~b _< r. 2. A process r is said to be a conservation process if $ = r . On this background we define the dissipation function d for a Clausius process r so that d=r-$ From property 1 it follows that dissipation is non-negative and that the Clausius process r is irreversible in the following sense: d>0and$<0 for a l l r E I I 0 In other words, it is not possible to maintain or return to any state I E F without dissipation. The state set
e'={M,l:d=0} is called the set of passive states. A passive state has minimum dissipation in the sense that d = 0 with equality if and only if I E F*. A description of the thermodynamic system and the topological structure of F is proposed by Coleman and Owen [2]. The notion of a passive state in the context of classical Hamiltonian systems is described by Dani~ls [3] and dissipativity goes at least back to Rayleigh who introduced the dissipation function [4]. Similar notions play i m p o r t a n t rhles in nonlinear control theory [1]. The intuitive basis for these definitions are similar and we will be able to make strong connections. P r o p o s i t i o n 2.1 A Clausius process converges so that limt_oo I = 1", a passive state if l i m s u p ~ _ o o r ( t , r ) < c~ and liminf~_~¢ $(t, r ) > - c ~ . / f l i m i n f ~ _ ~ $(t, r) > 0 then l i m t _ ~ ¢(t, r ) = 0
Proofi • = d + ~b < ~ . It follows that limsup~_~¢ d(t, r ) < ~ . Let r > 0 be real number and n and integer. We have r,-1
~--~d(t +,r, r)=J(t,
nr) <
/=0
Since d(t, r ) is non-negative we must conclude that l i m t _ ~ d(t, r ) = 0 for all r > 0. We get convergence of I since d is positive definite on F*. We have limt_oo(r(t, r ) - $(t, r ) ) = 0. The second part of the result then follows since r is a Clausius process. [] The thermodynamic system P = (V, II) is said to be a process system if there exist processes {rE, r s } so that 1. r E is a conservation process with a potential E called the energy. 2. r s is a Clausius process with potential S called the entropy. 3. The entropy density s = s(1 ) is a concave, homogeneous function (order 1) in f. The first and second statements are referred to as the first and second laws of thermodynamics respectively. The third statement gives structural properties needed to establish passivity and causality. We define an algebra for process systems. The addition of process systems, P = P1 • P2, is defined by adding potentials and processes so that v = vl + v~,r = rl + r2 A multiplication, P = a • P1, is defined so that v = a v l , x = at1. P r o p o s i t i o n 2.2 The algebra of process systems is closed with respect to addition and multiplication with positive a.
Proof: If v is a conservation process then we have r = r l + r2 = ~kl + ~2 = ~b so that ¢ is a conservation process as well. Similarly, if r is a Clausius process r = xl + r2 _> ~bl + ~2 = ~b so that r is a Clausius process. Finally S = f v " ( 1 1 ) + s(12)dl; so that s ( f ) = s(11) + s(12) with f =
I2
and s is concave with respect to t since
O2 s
OfOr T < O. The result for multiplication follows in the same manner.
[]
T h e r m o d y n a m i c stability theory is concerned with deviations from states which often are passive. Of particular importance are the equilibrium, or so called Gibbs states and the stationary non-equilibrium steady states. Recently there has also been interest in studying deviations from non-stationary states. The deviation of a process system PI with respect to a process system P2, P = P1 O P2, is defined so that so that v = vl - v 2 , r = ~rl - r2 The deviation is positive if dl > d2.
Proposition 2.3 The algebra of process systems is closed with respect to positive deviations. Proof: If x is a conservation process then we have r = ,bl - ~2 = ~b It follows that x is a conservation process. If ~r is Clausius process we have x = dl - d2 + ~1 - ~b2 > ~b if and only if d~ - d2 > O. Finally s ( f ) - s ( f l ) - s(f2) so that s ( t ) is concave with respect to 11 since a/t-'~o5 °2~ < 0 . []
European Symposium on Computer Aided Process Engineering--6. Part B
3
A
Clausius
process
with
a passive
S 1121
action
A Clausius process r s is said to have a passive action if the the mapping J ~ A(]) is causal,
.(,,.>= f <
a
with
B=/A(f)TJ.dB
and S(0) = 0. dB is the unit vector normal to B which by convention is pointing outward from 12. This definition induces a Hilbert space structure on the input output space. The definition of a system with a passive action corresponds to what Byrnes, Isidori and Willems refers to as a passive system [1] and we therefore retain their terminology in all but two respects. They emphasize the passivity of the system and require dissipativity of 7rs. We emphasize the passivity of the action and require that r s is Clausius. This is a subtle, albeit important distinction. The other distinction is that the thermodynamic system is infinite dimensional whereas the paper [1] deals with finite dimensional systems. By combining equations (4) and (5) we get the "macroscopic balance":
~(t + ~) - ~(t) = p(t, ~) + ¢(t, r)
(7)
We define the density of production t, and the density of flow J so that
Substituting these definitions into equation (7) and using the Gauss-Reynolds transport theorems the following system of partial differential equations is obtained: a t / + 0x~ Jk = ~ (8) These equations, which give the local form of the conservation laws and are called field equations and are the infinitessimal generators for the group action (6). We find, in the same way as equation (8) was derived, that the density of dissipation satisfies the equation
Ots+O=kJsk =~rs>O
ds(t,r)=/7./v~SdVdT-
(9)
This equation is not independent of the field equations since s = s(f) is a function of state. Friedrichs and Lax refers to (9) as an extension of the field. The following result is adapted from [7] and [8]. L e m m a 3.1 The density of entropy can be represented as the scalar product s = A ( f ) T f where f satisfies equation (8) and A (t) is concave. P r o o f i We adjoin the field equations (8) to inequality (9) using Lagrange multipliers A so that
Ors + O.~kJsk -- A r (Ot] + 8xk Jk -- ~r) > 0 This we re-arrange so that
(0~, -
:)
o , / + (oj Js~ - : o : J ~ ) ax~/+ A~ > 0
From this we find that the Olausius condition imposes the following restrictions on the field 0fs O.tJs~ Art,
= = >
a AOIJI~ 0
(10) (11) (12)
Differentiating (10), we obtain 0]A = Oi.]js for all i,j and it follows from concavity of s(f) that the vector of Lagrange multipliers A ( / ) defines a concave function. Now, s(f) is a homogeneous function of order 1 so we get s(f) = (Oys)] using Euler' s formula. By multiplying through with IT in equation (10) we get ]TO/s = f T A and the result follows by defining A (1) = A (]). [] The system of partial differential equations (8) are conservation laws whereas the balance equation for entropy (9) is a derived quantity in the sense that s, ~s and Jsk are functions of ] and J. Since s is a concave function in ] we conclude that the initial value problem is well posed and that a unique solution exists in the neighborhood of the initial surface [5]. Hyperbolicity guarantees finite wave speed propagation and causality. C o r o l l a r y 3.1 o's and Jsk are related to the field 1 by the following relations:
O's = Jsk
=
(OqakA) T J~ "4-ATer AT jk
Proof: By using the relation derived for s in proposition 3.1 and the fact that entropy density is a homogeneous function of first order the following relation is obtained after differentiating s over time: 0ts - A T ( l ) 0 t t . Substituting (8) and reordering terms: Ors + Ü~k ( JkA) = (Ox~A) T Jk + ATo " (13)
S1122
European Symposium on Computer Aided Process Engineering--6. Part B
The result follows by comparing equations (9) and (13) and equating terms.
[2
The Corollary above establishes that the entropy production can be written as a bilinear form [4] or = y T x
where X is a vector of thermodynamic forces and Y is a vector of thermodynamic fluxes. The forces satisfy the relation ea[f I < IX[ _< e21Yl (14) where ] is the state of a deviation system chosen using a passive state as a reference and ex and e2 are constants. It follows that the thermodynamic force X vanishes in the passive state. Furthermore, in classical non-equlibrium theory, the thermodynamic flux is a function of the thermodynamic force so that Y = LX
with
L = L~ + L~
where L~ is a positive definite symmetric matrix and L~ is an anti-symmetric matrix. P r o p o s i t i o n 3.1 Suppose that L , is positive definite. P = (V, H) is a process system. P r o o f : We have o s -- x T L x
>_ 0 and the result follows using equation (9) and Proposition 2.3.
[]
A similar result can be established by using a more general expression for the constitutive laws. W h a t is required is that the mapping X ~-* Y is positive real. This codition is satisfied if the the static relations are replaced with dynamic ones, for example the so-called Cattaneo equations. P r o p o s i t i o n 3.2 Suppose that $ s ( t , r) >_ O, l i m s u p , _ o o ~rs(t, r) < oo and L s > O, then f --* O, 0 is a passive state and s <_ O. P r o o f : From Proposition 2.1 and Proposition 3.1 we have f --* f* a passive state. We also have d(t,r)=/~/vXTLXdVdT and lim d(t, r ) = O so that limt~oo X = 0. Hence limt-oo 1~ - 0 according to inequality (14). Suppose now that there exists t so that S(t) > 0. This implies f ( t ) ~ 0 since S(t) = 0 if f = 0. Hence
s(t +r)= S(t)+f f~X TLxdvdT- >0 for all t, which gives X ¢ 0 for all t according to ineqaulity (14) and we get lira s u p r _ ~ ~s(t, r) = cx~ which contradict the assumption, t~ P r o p o s i t i o n 3.3 Suppose that P = (V, II) is a process system. The process x s has a passive action. P r o o f : Causality follows from the fact that the system of differential equations (8) has a concave extension [5]. From Corollary 3.1 it follows that the action ~ s is defined so that
withS=fA(S)'SdV so that S(0) = 0 for f = 0 and the result follows since r s is a Clausius process.
4
[]
Structure and stability in the context of t h e r m o d y n a m i c s
The correspondence between the theory developed in the previous section and the theory developed in [1] is now quite close. From the definition of the Clausius process we have x(t, r) = S(t + r) - S(t) = ds(t, r) + ¢~s(t, r) which we can re-write as A(t + r) - A(t) _ < / _ < - - A ( f ) T , J > s d T ill
where A = E - T S plays the rSle of the storage function. T is here a e o n s t a n t ' " t e m p e r a t u r e " and E is the energy of the system. A is clearly linked to the Helmholtz free energy. It remains to show that we can bound S so that A remains non-negative. The development follows [4], the stability result and zero state detectability are generalizations of the Le Chktelier-Braun principle.
Proposition 4.1 There exists a reference system so that E - T S > O.
European Symposium on Computer Aided Process Engineering--6. Part B
S! 123
P r o o f : Consider an equilibrium system and let A = U - T S be the Legendre transform of U with respect to S. For an equilibrium transformation at constant volume we have the following dA = - S d T ,
S=
//
2 "~cv d T ,
c v = ~OS > O,
T >_ O
1
so that A2 - Aa =
I T 2 / T+
CV d T " d T ' > 0 J T 1 d T 1 T I' --
hence U2 - T2S2 > U1 - T I S z . Choose 7'1 = 0 and reference state so that U1 = 0 and we get U - T S > 0 The result then follows for a non-equlibrium system by using the Clausius inequalty (1) since the non-equlibrium entropy is always bounded by the corresponding equlibrium entropy for an isolated system. [] In classical irreversible thermodynamics linear dependence of fluxes and forces and symmetry is assumed so that y T = X T L . The Onsager theory postulates that the matrix L is positive definite (symmetric) and constant so that Proposition 3.2 applies close to equlibrium. This theory has been extended to stationary non-equlibrium steady states. We define a set of stationary reference systems so that using equation (8) F2={f
:V.J=~}
Non-equilibrium stability theory is concerned with the evolution of f to f* E F2. The following is an adaptation of a result due to Prigogine and is called the theorem of minimum entropy production at stationary states. Proposition 3.1 is a significant generalization since no assumption is made about linearity or symmetry, only positivity. The minimum entropy production theorem states that the entropy production is minimized at a stationary states. Deviations from stationarity are therefore characterized by a higher rate of entropy production and the deviation system P = P1 G P2 is therefore a process system and the all of the results developed above apply to the system P. The Gibbs state is then replaced by a non-equilibrium stationary state.
Proposition 4.2 Let P = P1 0 P2 be a deviation and let Jr3 = 0 and J z l B = O. Suppose f u r t h e r m o r e that the Onsager relations hold. The deviation is positive and f ~ f* where f " E F:. f* is a passive state and S ( t ) < 0 f o r all t.
P r o o f : Prigogine [4] showed that if the Onsager relations hold then we have OtPs < O,
if f ~ F2
where F~ is the set of stationary states compatible with the boundary conditions. We have d(t, r) = f T P s d T
so that d s >__ds~ with equality if and only if f E F2. It follows that d converges to d2 from above and P is a positive deviation. Lemma 3.1 now applies and it follows from Proposition 3.1 that the action ~ s is passive. Moreover E(t) = E(O) for all t since JB = 0 and S ( t + r) - S(t) =- d x ( t , r ) ~ 0 (15) since JsIB = 0. Equality holds if and only if ] E F2. Suppose now that S(0) > 0, then equation (15) implies S ( t ) > 0 for all t which implies ] ~ 0 since ] = 0 gives S(0) = 0 according to Lemma 3.1. S ( t ) --* o¢ violates the inequality T S < E derived in Proposition 3.1. 12 The parallel result in nonlinear control theory is the following.
Proposition 4.3 A process system is zero-state detectable if the Onsager relations hold. Proof: Follows immediately from the result above.
[]
Concavity of s follows from the local equilibrium assumption using Gibbs relation.
5
C h e m i c a l process control via passivity.
The following result is special cases of a more general resul-t obtained in [10]. Let ~ denote the rate of supply so that 4~ = [
~dT
Proposition 5.1 Let f be a conservation process. The mappin9 q) ~ ~ is passive usin9 v 2 as a storage function,
S1124
European Symposium on Computer Aided Process Engineering----6. Part B
P r o o f : From the definition above and equations (4) and (5) we get the differential form of the conservation equation dv = d2dt By multiplying through with v and integrating we get
/~(t+~) /7~dv =
v~dT
Jr(t)
so that by integration by parts v(t + r) 2 - v(t) 2 = 2 f T - v e l d t and the result follows
[]
In [1] a conservation process is called lossless. We now define a deviation system P = P1 0P2: ! = I1 - t 2 , J = J1 - J 2 . P2 is a stationary system. We also define an input-output vector pair so that
Y
-A(t)r )
where E0, T are constants and the energy flux vector J ~ is defined so that CE(t,r)=~JE.dl3dT The definition of passivity is taken from [I]. P r o p o s i t i o n 5.2 The system P with input u and output y is passive when the mapping X s-~ Y is positive. The Onsager relations are suj~cient for positivity. P r o o f : Define the supply rate w = < u,y > z so that from Propositions 2.1 and 5.1 /~r < U, Y >B d T = E(t + r ) Eo
E(t)
Eo - ( , s ( t , r) + ds(t, ~))T
d is positive according to Proposition 4.1 and we can write V(t + r ) -
V(t) < ~
< u,y >B d T
where V = ~ 0 - T S
The result follows using Definition 2.4 in [1] and the fact the V is non-negative with V = 0 for ! = 0.
[]
P r o p o s i t i o n 5.3 Let P be a process system and let k be a smooth function of y so that k(O) = 0 and yTk(y) > 0 for each nonzero y. The control law u = - k ( y ) stabilizes the passive state f = 0 when X ~ Y is positive. P r o o f : Follows from Proposition 2.1.
[]
An alternative proof would be to rely on zero state detectability and then apply Theorem 3.2 in [1]. P r o p o s i t i o n 5.4 The mapping J ~ - A ( ] ) is positive real i] S(O) = 0 proof: Suppose f(0) = 0. Then S(0) = 0. The result follows using A as a storage function with E0 = 0.
O
References [1] Byrnes, C. I., Isidori, A. & J. C. Willems, (1991), Passivity, Feedback equivalence, and the global stabilization of minimum phase nonlinear systems. IEEE Transac. Aurora. Control. 36 (11). 1228 [2] Coleman, B. D. and D. R. Owen (1974), "A Mathematical Foundation for Thermodynamics", Arch. Rational Mech. Anal., Vol 54. pp. 1-104. [3] Dani~ls, H.A.M. (1981), "Passivity and equilibrium for classical Hamiltonian systems", J. Math. Phys., Vol. 22, April, pp. 843-846. [4] deGroot, S.R. and P. Mazur (1962), Non-equilibrium thermodynamics, North Holland Publishing Company, Amsterdam. [5] Friedrichs, K. O. and P. D. Lax (1971), "Systems of Conservations Equations with a Convex Extension", Proc. Nat. Acad. Sci., USA, Vol 68, NO. 6, pp. 1686-1688. [6] Jou, D., J. Casa-Vazquez and G. Lebon, (1993) Extended Irreversible Thermodynamics, Springer Verlag, New York. [7] Liu, I. S. , (1972), "Method of Lagrange multipliers for exploitation of the entropy principle", Arch. Rational Mech. Anal. 46. pp. [6] Muller I. and T. Rugged, (1991). Extended thermodynamics, Springer-Verlag. [9] Willems, J. C. (1974) "Dissipative Dynamical Systems, Part I: General Theory", Arch. Rational Mech. Anal., Vol 45. pp. 321-350. [10] Ydstie, B. E. and K. Visnawath, (1994) "From thermodynamics to a macroscopic theory for process control" , PSE-94, May30-June 3, Kyongju, Korea.
Pergamon
S0098-1354(96)00194.9
Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0098-1354196 $15.00+0.00
Process Systems, Passivity and the Second Law of Thermodynamics A N T O N I O A. A L O N S O and B. ERIK Y D S T I E Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh PA 15213 A b s t r a c t - " In this p a p e r We use the first and second laws of t h e r m o d y n a m i c s to motivate a theory for nonlinear process control. O u r m a i n tenets are: positive entropy production, b o u n d e d n e s s of entropy in t e r m s of energy, concavity of t h e entropy density and Helmholtz free energy as a storage function. These give t h e process s y s t e m a causal i n p u t - o u t p u t description, zero s t a t e detectability and stabilizability. To make the theory apply to practical s y s t e m s we follow ideas from classical irreversible t h e r m o d y n a m i c s and extend the concept of entropy of the non-equilibrium by a s s u m i n g local equilibrium.
1
Introduction
In 1867 R u d o l p h Clausius 1 proposed t h e following inequality for the entropy S of a body undergoing a t r a n s f o r m a t i o n from an equilibrium s t a t e to a n o t h e r via heating: s 2 - s l >_
~
2 dq 7
(1)
dq is t h e i n c r e m e n t a l h e a t i n g and T t h e absolute t e m p e r a t u r e . Inequality (1) is called the second law of t h e r m o d y namics. A r o u n d t h e s a m e time as Clansius developed inequality (1) it also became apparent t h a t h e a t and work are equivalent. T h i s idea is expressed as an equality called the first law of t h e r m o d y n a m i c s : dq = du + dw
(2)
were du s t a n d s for t h e change in t h e internal energy and dw t h e incremental work performed by deformation. By combining t h e first a n d second laws we relate entropy to energy t h r o u g h t the definition of a new t h e r m o d y n a m i c q u a n t i t y referred to as Helmholtz free energy, A = U - TS, where U is the internal energy. For an equlibrium t r a n s f o r m a t i o n we have U2 - T~S2 > V l - T l S ~
(3)
By defining t h e reference s y s t e m at absolute zero so t h a t T1 -- 0"1 = 0 we establish a lower b o u n d for t h e Helmholtz energy and it follows t h a t A can be used a storage function candidate for control s y s t e m design. Internal disspation gives stabilization a n d a connection is m a d e between nonlinear control theory as developed in [9] and [1] and nonequilibrium t h e r m o d y n a m i c theory [4] [2] [8] t h r o u g h the notion of stability of t h e zero dynamics.
2
Process systeins~ potentials and actions
Let Y and T be s u b s e t s of R~. and R repectively. A thermodynamic system is a pair (V, II). T h e elements v • ]2 have t h e form v = ( v l , v 2 , , . . . , ~ , ) where the v~'s are m e a s u r e s on Y and t h e elements r • II have t h e form x = ( r l , *r2,, ..., t n ) where the ,ri's are signed m e a s u r e s on V x T. We will refer to r , as a process on the s y s t e m a n d to v, as t h e potential corresponding to t h e process t i . T h e subscript i m a y be replaced by letters so t h a t M, E, S, A refers to m a s s , energy, entropy a n d Helmholtz free energy respectively. T h e s u b s e t s "V are referred to as volumes and their closures B are referred to as the boundaries. T h e subsets 7- = It, t + r) are intervals of the real line where t is referred to as t i m e a n d r takes values in ( - ~ o , oc). T h e m e a s u r e v is a potential for the process x a n d t h e process performed by a s y s t e m is therefore represented by the transition
• (t, ~) = ~(t + ~) - ~(t)
(4)
A change in potential is initiated by non-zero process. T h e process itself satisfies a unique decomposition
=p+~
(5)
T h e set-function l0 : V x T ~-* R defines production of v in ]) during t h e interval of time 7" and ~ is t h e supply of v to t h e s y s t e m from t h e exterior t h r o u g h t h e b o u n d a r y B. In this way ~ represents an action on t h e s y s t e m (13, II). From t h e theory of integration on p r o d u c t spaces it follows t h a t for every point in R~. x t t we define a unique derivative f of v so t h a t v = f v fd]]" T h e vector function ] = f (x, y, ~, t) is t h e density of v at the point (x, y, z, t) • R 4. An n vector of densities ] = (fa, ..., ]~) defines the s t a t e of a s y s t e m and is referred to as t h e field. From t h e definitions above it t h e n follows t h a t the process v induces a t r a n s f o r m a t i o n on the state space so t h a t ( r , f ) ~-* M ~ f T h e set of processes II0 = {x(t, r ) : ~ = M r f } is referred to cycle is a process which takes t h e s t a t e of t h e s y s t e m back to as t h e s t a t e space. We now enter into t h e main s u b j e c t process control. Existence of such processes are properties
(6)
as t h e set of thermodynamic cycles at f . A t h e r m o d y n a m i c to its s t a r t i n g point. T h e set F = {f : II0 # ~} is referred and define a process pair of e x t r a o r d i n a r y i m p o r t a n c e for of what we call a process system.
1 "Uber ver~chiedene ffir die Anwendung bequeme Forraen der Hauptgleichnungen der mechanischen Warmetheorie," Abhandlungen fiber die mechanische Wgrmetheorie, Vol. II, Viewegm Braumchweig , 1867, pp 1-44. S1119
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European Symposium on Computer Aided Process Engineering--6. Part B
1. The process r is said to be a Clausius process if ~b _< r. 2. A process r is said to be a conservation process if $ = r . On this background we define the dissipation function d for a Clausius process r so that d=r-$ From property 1 it follows that dissipation is non-negative and that the Clausius process r is irreversible in the following sense: d>0and$<0 for a l l r E I I 0 In other words, it is not possible to maintain or return to any state I E F without dissipation. The state set
e'={M,l:d=0} is called the set of passive states. A passive state has minimum dissipation in the sense that d = 0 with equality if and only if I E F*. A description of the thermodynamic system and the topological structure of F is proposed by Coleman and Owen [2]. The notion of a passive state in the context of classical Hamiltonian systems is described by Dani~ls [3] and dissipativity goes at least back to Rayleigh who introduced the dissipation function [4]. Similar notions play i m p o r t a n t rhles in nonlinear control theory [1]. The intuitive basis for these definitions are similar and we will be able to make strong connections. P r o p o s i t i o n 2.1 A Clausius process converges so that limt_oo I = 1", a passive state if l i m s u p ~ _ o o r ( t , r ) < c~ and liminf~_~¢ $(t, r ) > - c ~ . / f l i m i n f ~ _ ~ $(t, r) > 0 then l i m t _ ~ ¢(t, r ) = 0
Proofi • = d + ~b < ~ . It follows that limsup~_~¢ d(t, r ) < ~ . Let r > 0 be real number and n and integer. We have r,-1
~--~d(t +,r, r)=J(t,
nr) <
/=0
Since d(t, r ) is non-negative we must conclude that l i m t _ ~ d(t, r ) = 0 for all r > 0. We get convergence of I since d is positive definite on F*. We have limt_oo(r(t, r ) - $(t, r ) ) = 0. The second part of the result then follows since r is a Clausius process. [] The thermodynamic system P = (V, II) is said to be a process system if there exist processes {rE, r s } so that 1. r E is a conservation process with a potential E called the energy. 2. r s is a Clausius process with potential S called the entropy. 3. The entropy density s = s(1 ) is a concave, homogeneous function (order 1) in f. The first and second statements are referred to as the first and second laws of thermodynamics respectively. The third statement gives structural properties needed to establish passivity and causality. We define an algebra for process systems. The addition of process systems, P = P1 • P2, is defined by adding potentials and processes so that v = vl + v~,r = rl + r2 A multiplication, P = a • P1, is defined so that v = a v l , x = at1. P r o p o s i t i o n 2.2 The algebra of process systems is closed with respect to addition and multiplication with positive a.
Proof: If v is a conservation process then we have r = r l + r2 = ~kl + ~2 = ~b so that ¢ is a conservation process as well. Similarly, if r is a Clausius process r = xl + r2 _> ~bl + ~2 = ~b so that r is a Clausius process. Finally S = f v " ( 1 1 ) + s(12)dl; so that s ( f ) = s(11) + s(12) with f =
I2
and s is concave with respect to t since
O2 s
OfOr T < O. The result for multiplication follows in the same manner.
[]
T h e r m o d y n a m i c stability theory is concerned with deviations from states which often are passive. Of particular importance are the equilibrium, or so called Gibbs states and the stationary non-equilibrium steady states. Recently there has also been interest in studying deviations from non-stationary states. The deviation of a process system PI with respect to a process system P2, P = P1 O P2, is defined so that so that v = vl - v 2 , r = ~rl - r2 The deviation is positive if dl > d2.
Proposition 2.3 The algebra of process systems is closed with respect to positive deviations. Proof: If x is a conservation process then we have r = ,bl - ~2 = ~b It follows that x is a conservation process. If ~r is Clausius process we have x = dl - d2 + ~1 - ~b2 > ~b if and only if d~ - d2 > O. Finally s ( f ) - s ( f l ) - s(f2) so that s ( t ) is concave with respect to 11 since a/t-'~o5 °2~ < 0 . []
European Symposium on Computer Aided Process Engineering--6. Part B
3
A
Clausius
process
with
a passive
S 1121
action
A Clausius process r s is said to have a passive action if the the mapping J ~ A(]) is causal,
.(,,.>= f <
a
with
and S(0) = 0. dB is the unit vector normal to B which by convention is pointing outward from 12. This definition induces a Hilbert space structure on the input output space. The definition of a system with a passive action corresponds to what Byrnes, Isidori and Willems refers to as a passive system [1] and we therefore retain their terminology in all but two respects. They emphasize the passivity of the system and require dissipativity of 7rs. We emphasize the passivity of the action and require that r s is Clausius. This is a subtle, albeit important distinction. The other distinction is that the thermodynamic system is infinite dimensional whereas the paper [1] deals with finite dimensional systems. By combining equations (4) and (5) we get the "macroscopic balance":
~(t + ~) - ~(t) = p(t, ~) + ¢(t, r)
(7)
We define the density of production t, and the density of flow J so that
Substituting these definitions into equation (7) and using the Gauss-Reynolds transport theorems the following system of partial differential equations is obtained: a t / + 0x~ Jk = ~ (8) These equations, which give the local form of the conservation laws and are called field equations and are the infinitessimal generators for the group action (6). We find, in the same way as equation (8) was derived, that the density of dissipation satisfies the equation
Ots+O=kJsk =~rs>O
ds(t,r)=/7./v~SdVdT-
(9)
This equation is not independent of the field equations since s = s(f) is a function of state. Friedrichs and Lax refers to (9) as an extension of the field. The following result is adapted from [7] and [8]. L e m m a 3.1 The density of entropy can be represented as the scalar product s = A ( f ) T f where f satisfies equation (8) and A (t) is concave. P r o o f i We adjoin the field equations (8) to inequality (9) using Lagrange multipliers A so that
Ors + O.~kJsk -- A r (Ot] + 8xk Jk -- ~r) > 0 This we re-arrange so that
(0~, -
:)
o , / + (oj Js~ - : o : J ~ ) ax~/+ A~ > 0
From this we find that the Olausius condition imposes the following restrictions on the field 0fs O.tJs~ Art,
= = >
a AOIJI~ 0
(10) (11) (12)
Differentiating (10), we obtain 0]A = Oi.]js for all i,j and it follows from concavity of s(f) that the vector of Lagrange multipliers A ( / ) defines a concave function. Now, s(f) is a homogeneous function of order 1 so we get s(f) = (Oys)] using Euler' s formula. By multiplying through with IT in equation (10) we get ]TO/s = f T A and the result follows by defining A (1) = A (]). [] The system of partial differential equations (8) are conservation laws whereas the balance equation for entropy (9) is a derived quantity in the sense that s, ~s and Jsk are functions of ] and J. Since s is a concave function in ] we conclude that the initial value problem is well posed and that a unique solution exists in the neighborhood of the initial surface [5]. Hyperbolicity guarantees finite wave speed propagation and causality. C o r o l l a r y 3.1 o's and Jsk are related to the field 1 by the following relations:
O's = Jsk
=
(OqakA) T J~ "4-ATer AT jk
Proof: By using the relation derived for s in proposition 3.1 and the fact that entropy density is a homogeneous function of first order the following relation is obtained after differentiating s over time: 0ts - A T ( l ) 0 t t . Substituting (8) and reordering terms: Ors + Ü~k ( JkA) = (Ox~A) T Jk + ATo " (13)
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European Symposium on Computer Aided Process Engineering--6. Part B
The result follows by comparing equations (9) and (13) and equating terms.
[2
The Corollary above establishes that the entropy production can be written as a bilinear form [4] or = y T x
where X is a vector of thermodynamic forces and Y is a vector of thermodynamic fluxes. The forces satisfy the relation ea[f I < IX[ _< e21Yl (14) where ] is the state of a deviation system chosen using a passive state as a reference and ex and e2 are constants. It follows that the thermodynamic force X vanishes in the passive state. Furthermore, in classical non-equlibrium theory, the thermodynamic flux is a function of the thermodynamic force so that Y = LX
with
L = L~ + L~
where L~ is a positive definite symmetric matrix and L~ is an anti-symmetric matrix. P r o p o s i t i o n 3.1 Suppose that L , is positive definite. P = (V, H) is a process system. P r o o f : We have o s -- x T L x
>_ 0 and the result follows using equation (9) and Proposition 2.3.
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A similar result can be established by using a more general expression for the constitutive laws. W h a t is required is that the mapping X ~-* Y is positive real. This codition is satisfied if the the static relations are replaced with dynamic ones, for example the so-called Cattaneo equations. P r o p o s i t i o n 3.2 Suppose that $ s ( t , r) >_ O, l i m s u p , _ o o ~rs(t, r) < oo and L s > O, then f --* O, 0 is a passive state and s <_ O. P r o o f : From Proposition 2.1 and Proposition 3.1 we have f --* f* a passive state. We also have d(t,r)=/~/vXTLXdVdT and lim d(t, r ) = O so that limt~oo X = 0. Hence limt-oo 1~ - 0 according to inequality (14). Suppose now that there exists t so that S(t) > 0. This implies f ( t ) ~ 0 since S(t) = 0 if f = 0. Hence
s(t +r)= S(t)+f f~X TLxdvdT- >0 for all t, which gives X ¢ 0 for all t according to ineqaulity (14) and we get lira s u p r _ ~ ~s(t, r) = cx~ which contradict the assumption, t~ P r o p o s i t i o n 3.3 Suppose that P = (V, II) is a process system. The process x s has a passive action. P r o o f : Causality follows from the fact that the system of differential equations (8) has a concave extension [5]. From Corollary 3.1 it follows that the action ~ s is defined so that
withS=fA(S)'SdV so that S(0) = 0 for f = 0 and the result follows since r s is a Clausius process.
4
[]
Structure and stability in the context of t h e r m o d y n a m i c s
The correspondence between the theory developed in the previous section and the theory developed in [1] is now quite close. From the definition of the Clausius process we have x(t, r) = S(t + r) - S(t) = ds(t, r) + ¢~s(t, r) which we can re-write as A(t + r) - A(t) _ < / _ < - - A ( f ) T , J > s d T ill
where A = E - T S plays the rSle of the storage function. T is here a e o n s t a n t ' " t e m p e r a t u r e " and E is the energy of the system. A is clearly linked to the Helmholtz free energy. It remains to show that we can bound S so that A remains non-negative. The development follows [4], the stability result and zero state detectability are generalizations of the Le Chktelier-Braun principle.
Proposition 4.1 There exists a reference system so that E - T S > O.
European Symposium on Computer Aided Process Engineering--6. Part B
S! 123
P r o o f : Consider an equilibrium system and let A = U - T S be the Legendre transform of U with respect to S. For an equilibrium transformation at constant volume we have the following dA = - S d T ,
S=
//
2 "~cv d T ,
c v = ~OS > O,
T >_ O
1
so that A2 - Aa =
I T 2 / T+
CV d T " d T ' > 0 J T 1 d T 1 T I' --
hence U2 - T2S2 > U1 - T I S z . Choose 7'1 = 0 and reference state so that U1 = 0 and we get U - T S > 0 The result then follows for a non-equlibrium system by using the Clausius inequalty (1) since the non-equlibrium entropy is always bounded by the corresponding equlibrium entropy for an isolated system. [] In classical irreversible thermodynamics linear dependence of fluxes and forces and symmetry is assumed so that y T = X T L . The Onsager theory postulates that the matrix L is positive definite (symmetric) and constant so that Proposition 3.2 applies close to equlibrium. This theory has been extended to stationary non-equlibrium steady states. We define a set of stationary reference systems so that using equation (8) F2={f
:V.J=~}
Non-equilibrium stability theory is concerned with the evolution of f to f* E F2. The following is an adaptation of a result due to Prigogine and is called the theorem of minimum entropy production at stationary states. Proposition 3.1 is a significant generalization since no assumption is made about linearity or symmetry, only positivity. The minimum entropy production theorem states that the entropy production is minimized at a stationary states. Deviations from stationarity are therefore characterized by a higher rate of entropy production and the deviation system P = P1 G P2 is therefore a process system and the all of the results developed above apply to the system P. The Gibbs state is then replaced by a non-equilibrium stationary state.
Proposition 4.2 Let P = P1 0 P2 be a deviation and let Jr3 = 0 and J z l B = O. Suppose f u r t h e r m o r e that the Onsager relations hold. The deviation is positive and f ~ f* where f " E F:. f* is a passive state and S ( t ) < 0 f o r all t.
P r o o f : Prigogine [4] showed that if the Onsager relations hold then we have OtPs < O,
if f ~ F2
where F~ is the set of stationary states compatible with the boundary conditions. We have d(t, r) = f T P s d T
so that d s >__ds~ with equality if and only if f E F2. It follows that d converges to d2 from above and P is a positive deviation. Lemma 3.1 now applies and it follows from Proposition 3.1 that the action ~ s is passive. Moreover E(t) = E(O) for all t since JB = 0 and S ( t + r) - S(t) =- d x ( t , r ) ~ 0 (15) since JsIB = 0. Equality holds if and only if ] E F2. Suppose now that S(0) > 0, then equation (15) implies S ( t ) > 0 for all t which implies ] ~ 0 since ] = 0 gives S(0) = 0 according to Lemma 3.1. S ( t ) --* o¢ violates the inequality T S < E derived in Proposition 3.1. 12 The parallel result in nonlinear control theory is the following.
Proposition 4.3 A process system is zero-state detectable if the Onsager relations hold. Proof: Follows immediately from the result above.
[]
Concavity of s follows from the local equilibrium assumption using Gibbs relation.
5
C h e m i c a l process control via passivity.
The following result is special cases of a more general resul-t obtained in [10]. Let ~ denote the rate of supply so that 4~ = [
~dT
Proposition 5.1 Let f be a conservation process. The mappin9 q) ~ ~ is passive usin9 v 2 as a storage function,
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European Symposium on Computer Aided Process Engineering----6. Part B
P r o o f : From the definition above and equations (4) and (5) we get the differential form of the conservation equation dv = d2dt By multiplying through with v and integrating we get
/~(t+~) /7~dv =
v~dT
Jr(t)
so that by integration by parts v(t + r) 2 - v(t) 2 = 2 f T - v e l d t and the result follows
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In [1] a conservation process is called lossless. We now define a deviation system P = P1 0P2: ! = I1 - t 2 , J = J1 - J 2 . P2 is a stationary system. We also define an input-output vector pair so that
Y
-A(t)r )
where E0, T are constants and the energy flux vector J ~ is defined so that CE(t,r)=~JE.dl3dT The definition of passivity is taken from [I]. P r o p o s i t i o n 5.2 The system P with input u and output y is passive when the mapping X s-~ Y is positive. The Onsager relations are suj~cient for positivity. P r o o f : Define the supply rate w = < u,y > z so that from Propositions 2.1 and 5.1 /~r < U, Y >B d T = E(t + r ) Eo
E(t)
Eo - ( , s ( t , r) + ds(t, ~))T
d is positive according to Proposition 4.1 and we can write V(t + r ) -
V(t) < ~
< u,y >B d T
where V = ~ 0 - T S
The result follows using Definition 2.4 in [1] and the fact the V is non-negative with V = 0 for ! = 0.
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P r o p o s i t i o n 5.3 Let P be a process system and let k be a smooth function of y so that k(O) = 0 and yTk(y) > 0 for each nonzero y. The control law u = - k ( y ) stabilizes the passive state f = 0 when X ~ Y is positive. P r o o f : Follows from Proposition 2.1.
[]
An alternative proof would be to rely on zero state detectability and then apply Theorem 3.2 in [1]. P r o p o s i t i o n 5.4 The mapping J ~ - A ( ] ) is positive real i] S(O) = 0 proof: Suppose f(0) = 0. Then S(0) = 0. The result follows using A as a storage function with E0 = 0.
O
References [1] Byrnes, C. I., Isidori, A. & J. C. Willems, (1991), Passivity, Feedback equivalence, and the global stabilization of minimum phase nonlinear systems. IEEE Transac. Aurora. Control. 36 (11). 1228 [2] Coleman, B. D. and D. R. Owen (1974), "A Mathematical Foundation for Thermodynamics", Arch. Rational Mech. Anal., Vol 54. pp. 1-104. [3] Dani~ls, H.A.M. (1981), "Passivity and equilibrium for classical Hamiltonian systems", J. Math. Phys., Vol. 22, April, pp. 843-846. [4] deGroot, S.R. and P. Mazur (1962), Non-equilibrium thermodynamics, North Holland Publishing Company, Amsterdam. [5] Friedrichs, K. O. and P. D. Lax (1971), "Systems of Conservations Equations with a Convex Extension", Proc. Nat. Acad. Sci., USA, Vol 68, NO. 6, pp. 1686-1688. [6] Jou, D., J. Casa-Vazquez and G. Lebon, (1993) Extended Irreversible Thermodynamics, Springer Verlag, New York. [7] Liu, I. S. , (1972), "Method of Lagrange multipliers for exploitation of the entropy principle", Arch. Rational Mech. Anal. 46. pp. [6] Muller I. and T. Rugged, (1991). Extended thermodynamics, Springer-Verlag. [9] Willems, J. C. (1974) "Dissipative Dynamical Systems, Part I: General Theory", Arch. Rational Mech. Anal., Vol 45. pp. 321-350. [10] Ydstie, B. E. and K. Visnawath, (1994) "From thermodynamics to a macroscopic theory for process control" , PSE-94, May30-June 3, Kyongju, Korea.