The state-space model of a lumped process system obeying the “general mod- eling assumptions” in Section 4.1.1 can be obtained by substituting the al- gebraic constitutive equations into the conservation balance equations. This fact is used in this section to highlight the special structural properties which characterize a nonlinear state-space model derived from conservation balance equations.
4.2.1 System Variables
A possiblestate vectorxˆ for the nonlinear state-space model is:
ˆ x= [
E(j),(m(j)k , k= 1, . . . , K)
, j= 1, . . . ,C]T (4.8) dim(ˆx) =n= (K+ 1)ã C
withK being the number of components andC being the number of balance volumes. Note that the linear relationship
4.2 State-space Models of Process Systems 47
m(j)= XK k=1
m(j)k (4.9)
enables us to choose the complete set of component masses to be present in the state vector.
Thepotential input variables(including both manipulable input variables and disturbances) are also fixed by the state equations (4.3). They are the time-dependent variables (signals) appearing on the right-hand side of the equations, and not being state variables, that is
ˆ
u= [(v(j), φ(j)inv(j)), j= 1, . . . ,C]T (4.10) Note that we have formed a composite input vector (φ(j)inv(j)) from two sig- nals so that the corresponding term in the balance equation (4.3) has a ho- mogeneous form. Any external signal present in the source relationsqφ,source(j) should also be included in the set of potential input variables.
4.2.2 State Equations in Input-affine Form
In order to transform the general form of conservation balance equations into canonical nonlinear state equation form, the state and input variables above need to be centered using an arbitrary steady state as reference. Note that this step is not needed if there is no source term or if all sources are homogeneous functions. There can be process systems with no steady state at all, such as batch or fed-batch processes. Their variables cannot be centered, therefore their conservation balance equations cannot be transformed into a nonlinear state-space model in input-affine form.
For the reference input–state pair (x∗, u∗) the left-hand sides of the balance equations (4.3) are zero.
Definition 4.2.1 (Centered variable)
A centered variable is then the difference between its actual and reference value, that is
ϕ=ϕ−ϕ∗
The centered state variables and the centered input variables are then as follows:
x= [
E(j),(m(j)k , k= 1, . . . , K)
, j = 1, . . . ,C]T, |x|=n= (K+ 1)∗ C (4.11) u= [(v(j), φ(j)inv(j)), j= 1, . . . ,C]T (4.12)
With the centered state (4.11) and input vectors (4.12) the general form of lumped dynamic models of process systems can be transformed into the standard input-affine form of nonlinear concentrated parameter state-space equations:
˙
x=f(x) + Xm i=1
gi(x)ui, u∈Rm, f(0) = 0 (4.13) wherex= [x1, . . . , xn]T are local coordinates of a state-space manifold M. It is important to note thatthe state equations of process systems are always in an input-affine form above, because of the structure of the general state equation (4.3), if the system possesses a steady state.
4.2.3 Decomposition of the State Equations Driven by Mechanisms
The state equations in a nonlinear input-affine state-space model of a process system are derived from the general conservation balance equation (4.3).
This equation has four terms in its right-hand side, corresponding to the four principal mechanisms we take into account when constructing process models:
• input convection (inflow term),
• output convection (outflow term),
• (inter-phase) transfer,
• sources including both generation and consumption.
In addition, the concrete mathematical form of the state equations depends on the selection of the actual input variables from the set of potential ones.
Let us assume that:
1. The flow ratesv(j) (and the conserved extensive quantities at the inlet) are the input variables (manipulable input variables or disturbances).
2. There are no external sources to the system to be included in the set of potential input variables, that is
qφ,source(j) =Q(j)φ (T(j), c(j)1 , . . . , c(j)K−1) (4.14) whereQ(j)φ is a given nonlinear function.
It can be shown that, under the above conditions, one can decompose the nonlinear vector–vector functionsf(x) andg(x) in the nonlinear state equa- tion (4.13) into structurally different additive parts with clear engineering meaning:
˙
x=Atransf erx+Qφ(x) + Xm
Nixui+Bconvu (4.15)
4.2 State-space Models of Process Systems 49
The first term in the above equation originates from the transfer, the sec- ond from the sources, while the last two correspond to the output and in- put convection respectively. The coefficient matrices Atransf er, Bconv and (Ni, i = 1, . . . , m) are constant matrices, with Atransf er depending on the non-negative transfer coefficientsKφ,transf er(j,k) in Equation (4.7), whileNi is a matrix with non-negative elements that depends on the connections between the balance volumes.
It is important to note that Ni is a matrix where only itsi-th column is different from zero. More on the connection matrixNi will follow in Section 4.2.4.
It is easy to see from relation (4.7) that the linear constant (i.e. time- invariant) state matrix Atransf er is always negative semi-definite and also has zero eigenvalues. Moreover, the nonlinear source function Qφ(x) is of block diagonal form with the blocks joining the state variables belonging to the same balance volume.
Thus, the decomposed state equation contains a linear state term for the transfer, a general nonlinear state term for the sources, a bilinear input term for the output, and a linear input term for the input convection respectively.
4.2.4 Balance Volumes Coupled by Convection
Until now we have only taken into account that the balance volumes are coupled by the transfer terms (4.7), giving rise to the linear constant (i.e.
time-invariant) state matrixAtransf er. Now we consider the effect of convec- tive flows joining balance volumes to find the form of the input and output convection matrices (Ni, i = 1, . . . , m) and Bconv. In order to describe the general case let us assume that the outlet flow of the balance volume j is divided into parts and fed into other balance volumes, giving rise to the equation:
XC
`=0
α(j)` = 1, j = 0, . . . ,C (4.16)
where α(`)j is the fraction of the total outlet flow v(`) of balance volume ` that flows into balance volumej. The mass inflow of balance volumej then consists of the outflows from all the connected balance volumes, including the balance volume itself, together with a flow from the environment which is described as a pseudo-balance volume with index 0:
vin(j)= XC
`=0
α(`)j vout(`) , j= 0, . . . ,C (4.17)
Finally we collect the ratios above into a convection matrix Cconv as follows:
Cconv=
−(1−α(1)1 )α(2)1 ... α(C)1 ... ... ... ...
α(1)C α(2)C ...−(1−α(C)C )
(4.18)
Observe that only the ratios belonging to the internal (that isnotenviron- mental) flows are collected in the matrix above. It is important to note that because of Equation (4.16) the convection matrixCconvis a column conserva- tion matrix and therefore it is a stability matrix (see Section 7.4.2 in Chapter 7).
From theoretical and practical viewpoints there are two cases of special in- terest:
• free output convection,
• passive (controlled) mass convection.
These are described below and will be used throughout the book.
Free Mass Convection Network. The first special case of interest is when the output mass flow of any of the balance volumes is proportional to the overall mass in the balance volume, that is
vout(j) =κ(j)m(j), κ(j)>0 (4.19)
The inlet mass flow of the balance volumej can then be written as vin(j)=
XC
`=1
α(`)j κ(`)m(`)+α(0)j vout(0), j= 0, . . . ,C (4.20) where α(0)j v(0)out = v(j)IN is the mass inflow of the balance volume from the environment, that is, the real inflow. With the free convection equation above, the overall mass balance of the balance volume j (4.2) takes the following form:
dm(j) dt =
XC
`=1
α(`)j κ(`)m(`)−κ(j)m(j)+v(j)IN, j= 0, . . . ,C (4.21) Let us collect the overall mass and the real inlet mass flow of every balance volume into vectorsM andVIN respectively:
M = [m(1) . . . m(C)]T, VIN= [vIN(1) . . . v(C)IN]T
Equations (4.21) then can be written in the following matrix-vector form:
dM
dt =CconvKM+VIN (4.22)
where K = diag[κ(j) | j = 1, . . . ,C] is a diagonal matrix with positive ele- ments.