DE3225811A1 - Self-adjusting simulation and control system - Google Patents

Abstract

The self-adjusting simulation and control system (AMCS) represents a complete operating system for simulation, determination and self-adjusting control, which can be constructed as a closed control system. The self-adjusting simulation system (AMS) is the core of the system in this arrangement. This system consists of a simulation determination algorithm (MIA) and a disturbance variable determination algorithm (DIA). The AMCS operates in an outer and an inner loop. The outer loop tunes the PID controller in a feedback loop, the inner loop determines and compensates for model errors or external load disturbance variables.

Description

Self-adjusting simulation and control system

I. Introduction to the AMCS

The self-adjusting simulation and control system (AMCS abbreviation) represents a complete operating system for simulation, determination and self-adjusting control that can be implemented as a closed control system. This control system is shown in FIG. The functional structure of the AMCS is shown in FIG. It can be seen that the self-adjusting simulation system (AMS) is the heart of the whole system.

The AMS consists of two main algorithms, the simulation determination algorithm (MIA) and the disturbance variable determination algorithm (DIA). These algorithms use dynamic data to simulate the process dynamics and to determine the immeasurable load sizes. It is important to emphasize that the dynamic data used in the AMS can be collected during closed loop control operations.

The implementation of the AMCS takes place in two loops. The outer loop A works in batch mode so that when the dynamic data is sufficient to determine the process, an up-to-date model is obtained by the MIA in order to tune the PID controller in the conventional feedback loop. The inner loop B works in real-time operation, so that model errors or external charge disturbance variables can be determined and compensated for immediately.

The conventional feedback control loop that is the subject of self-adjusting tuning loop A is referred to as the self-adjusting feedback control loop (AFBC). On the other hand, the inner loop B, which compensates for detected load in a forward mode, is referred to as the self-adjusting feed forward loop (AFFC).

It can be seen that the AMCS is basically a self-learning system. By sequential implementation of the loops A and B in the AMCS, the control activity can be updated from time to time, so that the system always forces itself to its optimal conditions.

Conventional feedback control loop

Figure 1: Self-adjusting simulation and control system

Process data

Figure 2: Functional structure of the AMCS

II. Self-adjusting simulation model (AMS)

1. Definition of the M [deep] A (n, m) and M [deep] B (n, m, s) models

The models used in the AMS can be divided into two categories. One is the continuous deterministic model called the M [deep] A (n, m) model. The other is the discrete stochastic model, which includes deterministic processes caused by stochastic system disturbances or measurement disturbances. This model is referred to as the M [deep] B (n, m, s) model. An M [deep] B (n, m, s) model not only solves the deterministic input-output dynamics in terms of a differential equation, but also solves a discrete stochastic differential equation, which takes into account the influences of the system disturbances or the measurable observation error.

Despite the difference in the forms of development of the models, both models can be used to describe a system or a process that is characterized by the following equation: w (t) = u (tD)

and

y (t) = x (t) + v (t) (II.1)

whereby

x = process variable to be checked or controlled

y = observed value of the variable x

u = control input

w = delayed control input

small zeta = process noise or disturbance, assumed in normal distribution

small beta i, (i = 0,1,2, ..., m) = coefficients in the dynamics of the process input small alpha i, (i = 1,2, ..., n) = coefficients in the dynamics of the process variables

D = time delay

v = measurement noise, assumed in normal distribution

The detailed description of the M [deep] A (n, m) and M [deep] B (n, m, s) models is given below:

(1) M [deep] A (n, m) models

The models cataloged as M [deep] A (n, m) are given in the form: (II.2)

whereby

= the model value for x

= the estimated output for y

Note that n and m in the M [deep] A (n, m) models are used to denote the degree of differentiation of or w.

In the M [low] A (n, m) models, the nominal parameter vector can be defined as the subset of large omega [high] N [low] MA, where large omega [high] N [low] MA is given as : (II.3)

In other words, the nominal parameter vector P [deep] N can be defined as:

P [low] N is a subset of large omega [high] N [low] MA (II.4)

whereby

= Initial value of the i-th derivative of x

a [deep] i, i = 1,2, ...., n = coefficients for the variable x

b [deep] i, i = 0,1,2, ...., m = coefficients for w

The M [deep] A (n, m) models given by equation (II.2) can be used to describe the deterministic process dynamics, whereby both the system noise (system interference) and measurement noise (measurement interference) in comparison with the system input and -Output are insignificant.

(2) The M [deep] B (n, m, s) models

The M [deep] B (n, m, s) models are given in the form of stochastic differential equations: and (II.5)

The dynamic part of

(k) and w (k) in the absence of the small epsilon (k) in equation (II.5), that is, (II.6)

is directly related to a continuous differential equation: (II.7)

The discrete quantities in the above equations, for example y (k), w (k), small epsilon (k) denote draw that these values are queried at the instant which k large delta is away from the origin (large delta = sampling period).

The symbols used in equations (II.5) to (II.7) are as follows:

= estimated output for y 0 [deep] i (i = 1, ..., s) = coefficients of small epsilon

y = current output variable

w = delayed process input, i.e. u (t-D), so that w (k) stands for u (k-small Eta), where small Eta = integer [D / large delta]

small epsilon = error between the current and the estimated output

small Phi i, (i = 1,2, ...., n) = coefficients for y in the differential equation

F [deep] i, (i = 1,2, ...., n) = coefficients for W in the differential equation

a [deep] i, (i = 1,2, ...., n) = coefficients for b [deep] i, (i = 0,1, ...., m) = coefficients for w

n = degree of differentiation of in equation (II.7)

m = degree of differentiation of w in equation (II.7)

s = degree of deviation for small epsilon in equation (II.5)

Unlike the model indices n, m in the M [deep] A (n, m) models, n, m and s in the M [deep] B (n, m, s) models refer to both equations (II .5) and (II.7).

Because of the connection of equation (II.7) in the model of equation (II.5), the coefficients of small Phi [deep] i, (i = 1,2, ..., n) and Fi, (i = 1,2, ..., n) the coefficients of a [deep] j and b [deep] j in the form of:

small Phi [deep] i = small Phi [deep] i (a [deep] 1, a [deep] 2, ......, a [deep] n), i = 1,2, .... .., n

F [deep] I = F [deep] i (a [deep] 1, a [deep] 2, ......, a [deep] n, b [deep] 0, b [deep] 1,, b [deep] m), I = 1,2,, n

(II.8)

The derivatives of equation (II.8) for each value of n and m in equation (II.7) are lengthy and unnecessary. Later, the relationships of equation (II.8) are determined implicitly according to their state expressions.

Because of the co-existence of equation (II.6) and equation (II.7), the nominal parameter vector for the M [low] B (n, m, s) models is a subset of large omega [high] N [ deep] MB given, where

large omega [high] N [low] MB = [a [deep] 1, a [deep] 2, ......, a [deep] n, b [deep] 0, b [deep] 1 ,. ....., b [deep] m, O [deep] 1, O [deep] 2, ......, O [deep] S]

(II.9)

(3) The current parameters and the nominal parameters of the M [deep] A (n, m) and M [deep] B (n, m, s) models

In order to calculate the model reactions of the M [deep] A (n, m) or M [deep] B (n, m, s) models, the state representations for equation (II.2) are required to carry out the integration. Therefore the current parameters that are shown in the state representations are not the same as the nominal parameters. However, the latter can be calculated from the former. It is also known that the state representations of equations (II.2) or (II.7) are ambiguous. The equations of state that use the fewest parameters are usually desirable.

In other words, either equation (II.2) or equation (II.7) is first formulated in a state representation as follows: and

y = CZ (II.10)

whereby

Z = [Z [low] 1, Z [low] 2, ......, Z [low] n] [high] T

A (P [deep] A), B (P [deep] A) are coefficient matrices and P [deep] A is the current parameter vector. Then the current parameters can be subsets of large omega [high] A [low] MA, i.e.

large omega [high] A [low] MA = [small Ny [low] 1, small Ny [low] 2, ..., small Ny [low] t, Z [low] 1 (0), Z [low] 2 (0), ..., Z [deep] n (0)] (II.11)

where small Ny [deep] i (i = 1,2, ..., t) = parameters in the state representation of equation (II.10), so that

P [low] A is a subset of large omega [high] A [low] MA.

The relationship of the actual parameters to the nominal parameters used in equation (II.2) can be found elsewhere. For example

(i) M [deep] A (1.0) models or M [deep] B (1.0, s) models

The state representation for these models can be given as:

A = [- a [deep] 1], B = [b [deep] o]

In this formulation one can find that

P [deep] A = P [deep] N and y = Z (1) = x

(ii) M [deep] A (2.0) models or M [deep] B (2.0, s) models

The A, B matrices for the state representation of equation (II.2) or equation (II.7) can be given as:

In this formulation it is found that

P [deep] A = P [deep] N and Z (1) =

(iii) M [deep] A (2.1) models or M [deep] B (2.1, s) models

The A, B matrices for the state representation of equation (II.2) or equation (II.7) can be given as:

Then one can conclude that C [deep] 1 and C [deep] 2 can be used to compute b [deep] 0 and b [deep] 1 in equation (II.2) or equation (II.7), i.e.

b [deep] 0 = c [deep] 1

b [deep] 1 = c [deep] 2 + a [deep] 1c [deep] 1

In this case the current parameters are subsets of large omega [high] A [low] MA, where

large omega [high] A [deep] MA = [a [deep] 1, a [deep] 2, c [deep] 1, c [deep] 2, z [deep] 1 (0), z [deep] 2 (0)]

There are many other representations of states that lead to the same M (2,1) models.

(4) Difference equations, which represent continuous (steady) state models.

In the M [deep] B (n, m, s) models, the dynamics in equation (II.6) correspond to those in equation (II.7). Therefore the coefficients small Phi [deep] i and F [deep] i are determined from the relationships in equation (II.8). These relationships can be derived from their state representations.

H = [h [deep] ij] [deep] nxn = e [high] large delta large delta

G = [g [low] i] [low] n = -A [high] -1 [I-e [high] large delta large delta] B (II.12)

where A and B are such matrices from equation (II.10) and

h [deep] ij = element of the matrix H

g [deep] i = element of the matrix G

large delta = integration sub-interval.

Then the relationships in equation (II.8) are given as follows:

(i) Equation of state with n = 1 (II.13)

(ii) Equation of state with n = 2

small Phi [deep] 1 = h [deep] 11 + h [deep] 22

small Phi [deep] 2 = h [deep] 12h [deep] 21-h [deep] 22h [deep] 11

F [deep] 1 = g [deep] 1F [deep] 2 = g [deep] 2h [deep] 12-g [deep] 1h [deep] 22 (II.14)

(iii) Equation of state with n = 3

small Phi [deep] 1 = t [deep] r [H]

small Phi [deep] 2 = - (large delta [deep] 11 + large delta [deep] 22 + large delta [deep] 35)

small Phi [deep] 3 = Det [H]

F [deep] 1 = g [deep] 1

F [deep] 2 = (h [deep] 12 g [deep] 2-g [deep] 1 h [deep] 22) + (h [deep] 12 g [deep] 3-g [deep] 1 h [deep ] 33)

F [deep] 3 = g [deep] 3 large delta [deep] 31 + g [deep] 2 large delta [deep] 21 + g [deep] 1 large delta [deep] 11

(II.15) whereby

t [deep] r [] = line of a matrix in brackets

large delta [deep] ij = adjuncts of h [deep] ij

Det [] = determinant of a square matrix in brackets.

(5) Significance of the M [deep] B (n, m, s) models

The M [deep] B (n, m, s) models can be used to represent the following continuous processes in which system noise and measurement noise have an influence.

y (t) = x (t) + v (t)

and

w (t) = u (t-D) (II.16)

It follows that the continuous (continuous) process of equation (II.16) leads to the M [deep] B (n, m, s) models under three different circumstances.

(i) The noise-free case

With this ratio, both the small epsilon and v have no influence on the process output. Therefore, if W is considered constant during small comparison intervals, equation (II.16) can be converted into a difference equation in terms of y.

y (k) = small Phi [deep] 1 y (k-1) + small Phi [deep] 2 y (k-2) + ... + small Phi [deep] ny (kn) + F [deep] 1 w (k-1) + ... + F [deep] nw (kn)

with

y (k) = x (k) (II.17)

The coefficients small Phi [deep] i and F [deep] i, which follow the definitions in equations (II.13) - (III.15), correlate the difference model with a sampled continuous model.

(II.18)

Therefore the result model will be an M [deep] B (n, m, 0) model.

(ii) The case in which small Xi matters

In the case where small Xi in equation (II.16) has an influence, equation (II.16) can be viewed as: (II.19-1) (II.19-2)

x = x [deep] 1 + x [deep] 2 (II.19-3)

Equation (II.19-1) can be converted into a difference equation as in the case free of interference or noise. Consequently

x [deep] 1 (k) = small Phi [deep] 1 x [deep] 1 (k-1) + small Phi [deep] 2 x [deep] 1 (k-2) + ... + small Phi [ deep] nx [deep] 1 (kn) + F [deep] 1 w (k-1) + ... + F [deep] nw (kn) (II.20)

equation (II.19-2) is a stochastic differential equation. A lot of experiments (Wu, Journal of Engineering Industrial; ASME Trans., Vol. 99-Ser. B, No. 3, pp. 708-714, August 1977) have shown that such an equation can be replaced by an ARMA (
<notreadable>
n-1) model can be represented: x [deep] 2 (k) = small Phi [deep] 1 x [deep] 2 (k-1) + ... + small Phi [deep] nx [deep] 2 ( kn) + small theta [deep] 1 small epsilon (k-1) + ... + small theta [deep] n-1 small epsilon (k-n + 1) + small epsilon (k) (II.21)

so that the combination of equation (II.20) and equation (II.21) leads to the equation:

x (k) = small Phi [deep] 1 x (k-1) + ... + small Phi [deep] nx (kn) + F [deep] 1 w (k-1) + ... + F [ deep] nw (kn) + small theta [deep] 1 small epsilon (k-1) + ... + small theta [deep] n-1 small epsilon (k-n + 1) + small epsilon (k) (II .22)

If only small Xi is of influence, then y (k) = so that

y (k) = small Phi [deep] 1 y (k-1) + ... + small Phi [deep] ny (kn) + F [deep] 1 w (k-1) + ... + F [ deep] nw (kn) + small theta [deep] 1 small epsilon (k-1) + ... + small theta [deep] n-1 small epsilon (k-n + 1) + small epsilon (k)

Therefore the case for the process in which small Xi is of influence leads to an M [deep] B (n, m, n-1) model.

(iii) The case where observation noise is an influence.

In these circumstances only v is of any influence.

Therefore the process can be described as: (II.23)

and

y = x + v (II.24)

Equation (II.23) can in turn be converted into a difference model:

x (k) = small Phi [deep] 1 x (k-1) + ... + small Phi [deep] nx (kn) + F [deep] 1 w (k-1) + ... + F [ deep] nw (kn) (II.25)

Since y (k) = x (k) + v (k), (II.26)

the substitution of equation (II.26) in equation (II.25) leads to:

y (k) = small Phi [deep] 1 y (k-1) + ... + small Phi [deep] ny (kn) + F [deep] 1 w (k-1) + ... + F [ deep] nw (kn)

-small Phi [deep] 1 small Ny (k-1) -small Phi [deep] 2 small Ny (k-2) -...- small Phi [deep] n small Ny (kn) + small Ny (k) (II.27)

The result model can be referred to as the M [deep] B (n, m, n) model.

It is therefore of great importance to use the M [deep] B (n, m, s) model to represent the continuous processes as in equation (II.16), in which noise is an influence.

2. Parameter estimation for M [deep] A (n, m) models

(1) Parameter estimation algorithm

In the M [deep] A (n, m) models the model equations are given as: (II.28)

with

The nominal parameter vector P [deep] N, which has to be estimated, is also defined as:

P [low] N is a subset of large omega [high] N [low] MA (II.29)

whereby

In order to estimate the nominal parameter P [deep] N, equation (II.28) must first be converted into a canonical state equation: (II.30)

so that P [deep] A can be determined in terms of P [deep] N.

With the definition P = P [deep] A to simplify the notation, equation (II.30) gives the estimated output reaction as:

Accepted (II.31)

Y = [y (t [low] 0), y (t [low] 1), ..., y (t [low] N)] [high] T (II.32)

where y is the current observed output.

Still written (II.33)

Then is the sensitivity matrix (II.34)

where the subscript r denotes the total number of parameters that are determined.

then (II.35)

assuming that small delta P is small.

If Equation (II.28) [or Equation (II.30)] is the appropriate model, it is expected that (II.36)

so that (II.37)

This results from equation (II.37) (II.38) provided that is not singular.

The values of in equation (II.34) y (t) and u (t) are calculated from the input-output data.

The time series of y (t [deep] i), u (t [deep] i), i = 1, 2, ......, N can be obtained either from an open-loop attempt or from a closed-loop -Settlement operation are collected. The calculations are based on the equations: (II.39)

Alternatively can can be calculated from the approximation: (II.40)

whereby and can be obtained from the integration of equation (II.30) with the possible input w.

Equation (II.38) is then applied iteratively to to determine which deviates from Y in terms of the smallest sum of the squared error.

To begin determining P with the aid of equation (II.38), an initial value for the parameter P is required desirable. The initial value of P, labeled P [high] o, can be obtained by some existing methods. The integration method according to Edward Moore and the block pulse function method, which are given in the appendix, are recommended.

The estimation procedure is as follows:

(i) Determine the M [deep] A (n, m) model in the form of equation (II.30) and thus the current parameter P [deep] A or P.

(ii) Use of an initial estimation method such as Moore's integration or the block impact function method to obtain the first estimate for P, i.e. for P [high] o.

(iii) Start with P [high] o and use the integration method for M [low] A (n, m) models to calculate the small delta P from equation (II.38).

(iv) Replace P [high] o with P [high] o + small delta P and go back to level (iii) until small delta P is found within a given tolerance.

(v) Calculation of the nominal parameters from the result of stage (iv), i.e. the current parameters.

A flow chart for the above method is given in FIG. 3.

The procedural steps listed for determining the parameters in the M [deep] A (n, m) models have proven to be successful for the data either from an open-loop

Method or from a closed-loop control, whereby the application of the closed-loop data for determining the M [deep] A (n, m) model is to be emphasized. The schematic diagram is shown in FIG.

Many other methods of using the closed-loop data for estimating the parameters fail and they are dependent on the design of the controller which is used in the closed-loop. It has been found that applying the procedure given here for lower order models, such as for M [deep] A (1.0), M [deep] A (2.0), and M [deep] A (2.1 ) delivers reliable results that do not depend on the controller used. Some simulation results are given and compared in section (II-5).

(2) Integration of M [deep] A (n, m) models

In the previous section it is necessary to and to calculate. Although existing numerical methods such as Euler, Runge-Kutta methods can be used, the application of these methods requires initial values for which exempt from are usually unknown. To see the first derivatives of to exclude from nominal parameters, a difference equation approximation is proposed. The M [deep] A (n, m) model is given in the form of equation (II.28), ie

(II.28)

with W = u (td) and

In the first step, equation (II.28) can be transformed into its state form like equation (II.10) and the corresponding current parameters can be defined. Here the current parameters will be similar to those in equation (II.11), with such initial values of Z [deep] 1 (0), Z [deep] 2 (0), ..., Z [deep] n (0 ) can be excluded.

Then, using the equation (II.13), the equation (II.14) or the equation (II.15), a difference model is obtained, i.e.

3 flow diagram for estimating the parameters for the M [deep] A (n, m) model

Fig. 4 Simulation of the M [deep] A (n, m) model under closed-loop operations

(II.41)

If for the first n initial values y (k) = is assumed, the following values of be calculated. So, for the case where the initial estimate of initial derivatives is not available, the integration procedure becomes:

(i) Bring equation (II.28) into the state form of equation (II.10)

(ii) Assign the values of the current parameters from equation (II.10).

(iii) Calculate small Phi '[deep] i s and F' [deep] i s from equations (II.13) - (II.15).

(iv) Start with y (0), y (1), ..., y (n-1), for calculate

The flow diagram for this method is shown in FIG.

For the case in which initial estimates for the initial derivatives are available, the integration methods can be carried out according to the existing numerical integration methods, which result from equation (II.28) and are based on equation (II.10).

3. Parameter estimation for M [deep] B (n, m, s) models

For an M [deep] B (n, m, s) model two equations are necessary. One is the said deterministic differential equation and the other is the explicit difference equation for the M [deep] B (n, m, s) models, i.e.

(II.42)

and

y (k) = small Phi [deep] 1y (k-1) + small Phi [deep] 2y (k-2) + ... + small Phi [deep] ny (kn) + F [deep] 1w (k -1) + ... + F [deep] nw (kn) + small theta [deep] 1 small epsilon (k-1) + small theta [deep] 2 small epsilon (k-2) + ... + small theta [ deep] small epsilon (ks) + small epsilon (k) (II.43)

The independent parameters to be estimated are: a [deep] 1, a [deep] 2, ..., a [deep] n, b [deep] 0, b [deep] 1, ..., b [deep] m, small theta [deep] 1, small theta [deep] 2, ..., small theta [deep] s.

These are all nominal parameters, i.e.

Large Omega [high] N [low] MB = [a [deep] 1, a [deep] 2, ..., a [deep] n, b [deep] 0, b [deep] 1, ..., b [deep] m, small theta [deep] 1, small theta [deep] 2, ..., small theta [deep] s]

To estimate the nominal parameters in equation (II.42) and equation (II.43),

Fig. 5 Flow diagram for the integration of M [deep] A (n, m) models when no initial derivative is available. Equation (II.42) is to be converted into an equation of state with the same procedural steps as this for the M [deep] A (n, m) models. The state expression is in the same form as in equation (II.30), namely (II.44)

The current parameters are then determined from equation (II.44) and equation (II.43), i.e .:

P [low] A is a subset of large Omega [high] A [low] MB

where large omega [high] A [low] MB is the set of all parameters that result from equation (II.44) and equation (II.43).

The estimation begins with an estimated set of parameters for a [deep] 1, ..., a [deep] n, b [deep] o, ..., b [deep] m and small theta [deep] 1 ,. .., small theta [deep] s. This data set is referred to as P [high] o [low] N.

Then by determining the current parameters from equation (II.44) and equation (II.43) and using equation (II.13), equation (II.14) or equation (II.15) a set of small Phi [deep] i and F [deep] i are calculated so that an estimation model like equation (II.43) results. By calculating the error in each observation, the sum of the square errors can be obtained. A non-linear Newton method is used to minimize the sum of the squared error by tuning the current parameters. The sum of the square error is denoted by large phi, i.e.

(II.45) where the small epsilon (k) is calculated from equation (II.46).

Small epsilon (k) = y (k) -small Phi [low] 1 [high] 0y (k-1) -...- small Phi [low] n [high] 0y (kn) -F [low] i [high] 0w (k-1) -...- F [low] n [high] 0w (kn) -small theta [low] 1 [high] 0small epsilon (k-1) -...- small theta [low] s [high] 0 small epsilon (ks) (II.46)

where the exponent "0" indicates that all coefficients have been calculated or given for their initial parameter P [high] o.

The new parameter P can then be obtained from:

P = P [high] 0 + small delta P (II.47)

where the small delta P is calculated from: (II.48)

whereby (II.49)

and (II.50)

The matrix is the Hessian matrix for large Phi and is the gradient vector. It should be noted that (II.51)

and results approximately as:

(II.52)

Thus, the estimates are iteratively continued by repeating equation (II.48) and replacing P [high] o with P [high] o + small delta P until small delta P is within a given tolerance.

The procedure can be completed as follows:

(i) Determine equation (II.42) as an equation of state like equation (II.44).

(ii) Give an initial set of estimated current parameters.

(iii) Compute H [high] o = e [high] large delta large delta and G [high] o = -A [high] -1 [I-e large delta large delta] B from P [high] o.

(iv) Compute small Phi [low] 1 [high] 0, small Phi [low] 2 [high] 0, ..., small Phi [low] n [high] 0, F [low] 1 [high] 0 , F [low] 2 [high] 0, ..., F [low] n [high] 0 from equations (II.13), (II.14) or (II.15) from the resulting H [ high] o and G [high] o.

(v) Compute small epsilon (k), where

small epsilon (k) = y (k) -small Phi [low] 1 [high] 0 y (k-1) -...- small Phi [low] n [high] 0 y (kn) -F [low ] 1 [high] 0 w (k-1) -...- F [low] n [high] 0 w (kn) -small theta [low] 1 [high] 0 small epsilon (k-1) -. ..- small theta [low] s [high] 0 small epsilon (ks), k = 1,2, ..., N

(II.53)

Note that small epsilon (k) = 0 k is less than or equal to 0

(vi) Compute large Phi from equation (II.45).

(vii) Compute and form the Hessian matrix and the gradient vector

(viii) Compute small delta P from equation (II.48) with the result of and for respectively.

(ix) Replace P [high] o with P [high] o + small delta P and check whether small delta P is within a given tolerance or not.

(x) If the small delta P is still outside the tolerance, repeat steps (iii) to (ix).

The flow diagram for this method is shown in FIG. 6.

4. Criteria for model construction

(1) Criteria for the M [deep] A (n, m) models

To simulate the M [deep] a (n, m) models, the M [deep] A (1,0), M [deep] A (2,0) and M [deep] A ( 2.1) considered model. In order to select an appropriate model from these possible models, an index called ID (n, m) is defined.

(II.54)

where P * = optimal parameters for each of the possible structures

SSE = sum of the squared error

The matrix H (P *) in equation (II.54) is given as: (II.55)

whereby the value of which is calculated from P *.

The appropriate model among these possible models is then determined to be the one whose value of ID (n, m) is the smallest.

There are two reasons to use this index as a criterion for the model structure. First, if the model is perfect, the sum of the squared error becomes zero. If the model is not perfect, but suitable compared to others, the sum of the squared error will also be small. Second is the output sensitivity matrix have a rank that is equal to the number of parameters to be identified if these are optimal parameters that are identifiable for each of the possible models. So the matrix H has the same rank as that of

The determinant value of H thus becomes a reference point for the parameter identifiability. A compromise between the accuracy of the model and the identifiability of the given parameters leads to an index ID (n, m) as indicated above.

6 shows a flow diagram for the parameter estimation for the M [deep] B (n, m, s) models

Some results from using this ID (n, m) to identify the simulation process or real processes are shown in Tables 7 and 8. The results of the identified model justify the use of the ID (n, m) as a criterion for the model structure (see FIGS. 7 and 8).

(2) Criterion for the M [deep] B (n, m, s) models

Since the method of parameter determination for the formation of the M [deep] B (n, m, s) models is of non-linear least squares, the criterion for selecting the model structure can follow the F quotient test which is usually used to test the Significance of some existing parameters in a given model.

If M [deep] B (n [deep] 1, m [deep] 1, s [deep] 1) and M [deep] B (n [deep] 2, m [deep] 2, s [deep] 2) the possible models for a procedure are, and if (n [deep] 1 + m [deep] 1 + s [deep] 1) is greater than (n [deep] 2 + m [deep] 2 + s [deep] 2 ], target

J [deep] 1 = sum of the squared errors for M [deep] B (n [deep] 1, m [deep] 1, s [deep] 1) model

J [deep] 2 = sum of the squared errors for M [deep] B (n [deep] 2, m [deep] 2, s [deep] 2) model

small Ny [deep] 1 = degree of freedom for M [deep] B (n [deep] 1, m [deep] 1, s [deep] 1)

and

small Ny [deep] 2 = degree of freedom for M [deep] B (n [deep] 2, m [deep] 2, s [deep] 2

then determine (II.56)

Compare the calculated f with the F [deep] small alpha (small Ny [deep] 2-small Ny [deep] 1, small Ny [deep] 1) from the F distribution table. Where F [deep] small alpha (small Ny [deep] 2-small Ny [deep] 1, small Ny [deep] 1) is determined as:

Prob. [F <F [deep] small alpha] = small alpha%

If f <F [deep] small alpha, then M [deep] B (n [deep] 2, m [deep] 2, s [deep] 2) as a suitable model of the M [deep] B (n [deep ] 1, m [deep] 1, s [deep] 1) and M [deep] B (n [deep] 2, m [deep] 2, s [deep] 2) models.

5. Explanations for the identification and parameter estimation procedures

(1) Results for the M [deep] A (n, m) models

In order to show that the estimation method for M [deep] A (n, m), which produces precise results regardless of the control device used in the closed loop, some simulation results are given in Table 1.

Table 1

Table 2 shows the simulation results, with other integration methods for parameter estimation being completely dependent on the control device that is used in the closed loop. In the same table, the results from the parameter sensitivity method are superior to all others.

Table 2

where BPF = block impact function method

MI = integration method according to Moore

PS = parameter sensitivity method (parameter sensitivity method)

Table 3 shows the results in which the integration methods other than the parameter sensitivity fail to identify the parameters, while the parameter sensitivity method still gives accurate estimates.

Table 3

The results of Tables 1, 2 and 3 clearly show the reliability of the parameter sensitivity method for parameter estimation from the closed-loop data (data from closed-loop data).

(2) Results for the M [deep] B (n, m, s) models

Table 4 shows the simulation results for the identification of a second order method by the M [deep] B (n, m, s) models. The F quotient test shows that the M [deep] B (2.0.0) model is a suitable model compared to the M [deep] B (2.0.2) model. The result is exactly the same as expected from theory, i.e. the M [deep] B (2,0,0) model.

Table 4

Kc = 4.0, T [deep] I = 5.0, T [deep] D = 0.1

large delta = 0.1

SP = 5.0

NOB = 100

* Confidence interval contains zero

* NOB is the number of observations.

Table 5 shows the results for identifying a second order method in which system noise is an influence. The F-quotient test between M [deep] B (2,0,1) and M [deep] B (2,0,2) shows that the M [deep] B (2,0,1) model is the most suitable model for the procedure is. The result is also the same as expected, i.e. M [deep] B (2,0,1).

Table 5

Kc = 4.0, T [deep] I = 5.0, T [deep] D = 0.1

large delta = 0.1

SP = 5.0

NOB = 100

small Xi similar to NID (0.0, 5.0)

Table 6 shows the simulation results in which both small Xi and v are of influence. The F-quotient test reveals that the M [deep] B (2,0,2) model is more important than the M [deep] B (2,0,1) model. That is also as expected.

Table 6

y = x + v

Kc = 4.0, T [deep] I = 5.0, T [deep] D = 0.1, SP = 5.0

NOB = 100, large delta = 0.1

small Xi similar to NID (0.0, 5.0)

v similar to NID (0.0, 0.05)

(3) Results using the ID (n, m)

The ID (n, m) index was used to check the model structure, using both the simulation data and the real data. Table 7 shows the results for the simulation.

Table 7

Table 8 shows the results from the real data.

The data was collected from experiments for a CSTR process and from experiments for a gas fire heater process. Both processes are under closed-loop control.

Table 8

The results from the models were shown in FIGS. 7 and 8 together with the real data. According to these figures, the use of the ID (n, m) for simulation is justified.

7 results of the models of M [deep] A (1.0) and M [deep] A (2.0) for the real data from the CSTR process

Fig. 8 Results of the models of M [deep] A (1.0) and M [deep] A (2.0) for the real data from a gas fire heater

6. Simulation strategies and methods

In the method where no further information is available except for a set of input-output data, the simulation and identification problem is even more complicated. The simulation problem includes determining the model structure, estimating the parameters, and determining the process delay. Therefore, simulation strategies that integrate these simulation problems are desirable.

So far, the simulation of the discrete time models usually starts with the lowest order model. The order of the model is then successively increased until some criteria are satisfactory. As a typical example, a flow chart is given by Hsia (see Figures 5-5 in "System Identification", T.C. Hsia, Lexington Books, 1979).

In AMCS, the models for the continuous processes can be limited to the simple first and second order with or without delay. Therefore, possible models for M [deep] A (n, m) models are M [deep] A (1.0), M [deep] A (2.0) and M [deep] A (2.1) models. While for M [deep] B (n, m, s) models the possible models on M [deep] B (1.0, s), M [deep] B (2.0, s) and M [deep] B (2,1, s) Models can be constrained, where s is less than or equal to n.

These simulation strategies used in the AMCS are established in a different way. As a result of the limitation of the possible models, the simulation begins with a second-order model with n = 2 and m = 0. The simulation strategies for M [deep] A (n, m) and M [deep] B (n, m, s ) Models are shown in Figs. 9, 10 and 11.

Thus, according to the simulation strategies as shown in the figures, the simulation method for M [deep] A (n, m) models is as follows:

(i) Obtain an estimate of the time lag from the input-output data.

(ii) Start with the M [deep] A (2.0) model to preserve the estimation parameters for the estimation methods outlined above.

(iii) Apply the parameter sensitivity estimation method to estimate the parameters in the model.

(iv) Perform the one-dimensional search at time delay D by repeating step (iii) after replacing D with D + small delta D until a minimum of the sum of the squared errors is obtained.

(v) Calculate the ID (2.0) value from the result from step (iv).

(vi) Change the model structure to M [deep] A (1,0) and repeat (iii) and (iv).

Simulation strategy for M [deep] A (n, m) models

9 simulation strategy for M [deep] A (n, m) models

Fig. 10 Strategy for determining the time delay in M [deep] A (n, m) models

11 simulation strategy for M [deep] B (n, m, s) models

The initial value for the time delay used in the M [deep] A (1,0) model can be calculated from (II.57)

whereby

D [high] (1) = estimate for time delay

D [high] (2) = time lag value in the resulting M [low] A (2.0) model

T [deep] 1 and T [deep] 2 = time constants for the M [deep] A (2.0) model (T [deep] 2 <T [deep] 1)

(vii) Calculate the value of ID (1,0) and check if the M [deep] A (1,0) model is sufficient.

(viii) If the M [deep] A (1,0) model is not sufficient, use the M [deep] A (2,1) structure and repeat steps (ii) - (iv) and compute ID (2 ,1).

(ix) Check whether the M [deep] A (2,0) model is sufficient.

(x) If the M [deep] A (2.0) model is not sufficient, M [deep] A (2.1) is assumed to be proven.

The simulation procedures for the M [deep] B (n, m, s) models are as follows:

(i) Estimate the value for the time lag using the cross-correlation between the input and output data. This estimate of the time delay is then taken and used in the simulation procedure that follows.

(ii) Form M [deep] B (2.0.0), M [deep] B (2.0.1), and M [deep] B (2.0.2) using the estimation methods given in Section 3 .

(iii) Use the F-test to determine the M [deep] B (2.0, s) model under the M [deep] B (2.0.0), M [deep] B (2.0.1 ) and M [deep] B (2,0,2) models.

(iv) Repeat steps (ii) and (iii) for the M [deep] B (1,0, s') model from the M [deep] B (1,0,0) and M [deep] B ( 1,0,1) models.

(v) Use the F quotient test to test whether the M [deep] B (1,0, s') model is sufficient.

(vi) If M [deep] B (1,0, s') is not sufficient, proceed with the simulation procedure for the M [deep] B (2,1, s' ') model from the M [deep] B (2,1,0), M [deep] B (2,1,1) and M [deep] B (2,1,2) models.

(vii) Check whether the M [deep] B (2,0, s) model is sufficient using the F quotient test.

(viii) If the M [deep] B (2.0, s) model is not sufficient, then the M [deep] B (2.1, s ") model is accepted.

7. Identification of the immeasurable charge

The above simulation and identification methods use the entire batch of data that has been collected in a balancing period that results from a setpoint change during the closed-loop operation. The resulting continuous process is of the form: (II.58)

The output is of the form and

(II.59)

for the M [deep] B (n, m, s) models.

It is then expected that the errors between the expected output and the current output y must have zero and finite variants at every sampling moment. However, this assumption can be violated because of the constant bias in the continuous process. This constant or at least the very small system. Error in time change is treated as a charge disturbance.

To consider this charge disturbance, the continuous process in equation (II.58) must be written in the form: (II.60)

whereby denotes the charge disruption.

A repeatable real-time algorithm is desirable for identifying the loading disturbance for purposes of controlling the process. The following is the algorithm taking values of for real-time estimation, provided based on the identified parameters from the last batch, ie a [deep] 1, ..., a [deep] n, ..., b [deep] o, ..., b [deep ] m.

First, a right angle window filter is used to hold back a constant length of data needed for estimation. Assume that the rectangular window has a length of l, i.e. y (t [deep] i), y (t [deep] i-1), ...., y (t [deep] il), u (t [deep] i), u (t [deep] i-1), ...., u (t [deep] il) are available at every point in time t [deep] i.

(1) Estimate of with the help of parameter sensitivity

The only parameter to be estimated in this case is

Other constants like a [deep] i and b [deep] j (i = 1,2, ..., n and j = 0,1, ..., m) are considered given values.

So is where P is the parameter vector.

Determine

Y [deep] i = [y (t [deep] i-small iota), y (t [deep] i-small iota + [deep] 1), ..., y (t [deep] i)] (II .61)

Then, starting with an initial value of d, for example d [high] o, the same algorithm as that of the M [low] A (n, m) model estimate can be used to compute the new value of at the time of t [deep] i, ie

(II.62)

whereby (II.63)

When proceeding to compute from t [deep] i to t [deep] i + 1 etc., a series of obtain. This series of can be provided for forward compensation.

The calculation of requires the integration of equation (II.60). However, the initial conditions are usually not available. Therefore the calculation can be carried out in an alternative way.

First, by inserting equation (II.13) to equation (II.15), it is possible to determine small Phi [deep] i, F [deep] i (i = 1,2, ..., n) for a difference equation model a continuous model like equation (II.60), where the discrete model is formed as:

(II.64)

Here µ is the term of the systematic error that results from the constant perturbation of equation (II.60). Then, assuming the first n-values of equals y the integration can be continued using equation (II.60) with the correct values of a [deep] i and b [deep] j (i = 1,2, ..., n; j = 0,1, ..., m).

The procedure is as follows:

(i) Collect the first l theorems of y (t) and u (t).

(ii) Assign the values of a [deep] i and b [deep] j from the last identified results.

(iii) Wise the initial value of from the last result too.

(iv) Calculate the parameter sensitivity.

The integration of equation (II.60) does not yield an initial value of if equation (II.64) and equations (II.12) to (II.15) are used.

(v) Compute from equation (II.62).

(vi) Replace by

(vii) Go to the next point in time with the right-angled window and repeat steps (iv) to (vi).

Some simulation results are shown in FIGS. 12 and 13.

(2) estimate of with the help of non-linear optimization

The estimation can be carried out using the procedures for the M [deep] B (n, m, s) estimation. The procedure is described below.

(i) Compute with

(ii) Compute

(iii) Compute

(iv) Let

(v) Replace

(vi) Admit the latest data and drop the oldest data from y (t) and u (t), then repeat from (i) to (v).

Figure 12 Results of the identification for the unknown disorder

Figure 13 Results of the identification for the unknown disorder

III. The self-adjusting voting algorithm

The dynamic models that result from the AMS can provide the basic information for tuning the conventional PID controller, which is used extensively today for chemical process control. Coordination relationships for such PID control systems are given from various sources. In this section, some specific voting results are suggested. A schematic diagram for the self-adjusting voting system is shown in FIG.

1. Coordination relationships for an interactive PID controller (mutually influencing PID controllers)

Interactive PID controllers are those that are now used in industry. However, the existing voting relationships are usually made for the ideal non-interactive PID controllers. An interactive PID controller considered here has given its algorithm either in digital or in analog form as follows:

(i) Analog algorithm (III.1)

(ii) Digital algorithm (III.2)

where P = output of the analog controller

E = error input, M = controller output

Q [deep] n-1 = previously calculated value of Q

Q [deep] n = latest calculated value of Q

Kc = proportional constant

Ti = integral time

T [low] D = Derived time

large delta = interval for sampling.

If the process in the PID control system can be treated as a first order process with a delay transfer function, the reconciliation relationships that minimize some integral execution indices such as ITAE, IAE, ITSE and ISE can be put into the form:

P = A times [D / small rope] [high] B (III.3)

or

P = A times [D / small rope] + B (III.4)

whereby

P = scaled tuning parameter

D = time delay in the first order method

small tau = time constant

A and B = constant

The values of A and B are shown in Table 9.

Note that the tuning relationships for the smallest T (small tau, assume 0.005, are proposed for tuning the analog interactive controllers, which have the same transfer function as equation (III.1). For digital controllers with a smaller T / small tau, the Voting results of Table 9 also proposed as the convergence of the voting relationships cannot distinguish the values given in Table 9.

2. Voting relationships using the quadratic execution index (quadratic effective parallel resistance)

Reconciliation relationships using the quadratic execution index can also be put into the form of equation (III.3) or equation (III.4). The quadratic execution index is given as (III.5)

If the process in the loop is also considered to be a first order process with a delay transfer function, the voting relationships are as follows:

(i) PI controller

K [low] P K [low] C = 0.6347 [D / small dew] [high] -0.6576 (III.6)

T [low] small iota / small rope = 1.2113 [D / small rope] [high] 0.4617 (III.7)

(ii) PID controller

K [low] P K [low] C = 0.975 [D / small dew] [high] -0.5724 (III.8)

T [low] small iota / small rope = 1.2113 [D / small rope] [high] 0.4872 (III.9)

T [low] D / small rope = (D / small rope) [2,092 + 1,449 (D / small rope)] [high] -1

(III.10)

Table 9

The numbers without * are to be used in equation (III.3) and the numbers with * in equation (III.4)

3. Coordination relationships for regulated procedures of the second order with supercritical damping

Assume T [low] 1, T [low] 2 are time constants of a second order supercritical damping method, and D [high] (2) is the time delay in the second order method. Then there is an equivalent (in the sense of least squares) first order method with a time constant and time delay like this: whereby

D [high] (1) = time lag in the equivalent first order method

small tau = time constant in the equivalent first order method.

Thus, the existing voting relationships including equations (III.6) to (III.10) and those given in Table 9 can be used.

IV. Self-adjusting regulation for the immeasurable charge

The immeasurable charge, which is characterized by the equation (II.60) using the algorithm AMS, can be viewed schematically with the structure of Fig. 14. The simulation results in Fig. 12 and Fig. 13 show that it is possible beautiful obtain close-up estimates for such an immeasurable charge disturbance in real-time calculations. It is therefore expected to compensate for such an identified charge by a forward compensator without fully resorting to the feedback control system. The resulting control system is then a combination of feedback and feedforward control. This means:

u (t) = u [deep] F (t) + u [deep] B (t) (IV.1)

Note that U [deep] F, the forward rule input, results from the calculated charge, which is immeasurable. The forward control input U [low] F can be set in the following two different ways.

1. Zero order approximation

This approximation assumes that is roughly the same whereby

= the estimated value of at t [deep] j based on the information up to t [deep] i.

Then U [deep] F is given as: (IV.2)

whereby

b [deep] o, a [deep] n = constant in the M [deep] A (n, m) or M [deep] B (n, m, s) models

= desired value of ie setting point

K [deep] I = constant of integration, usually given as Kc / T [deep] I

small lambda = a constant tuning parameter so that the rule input can be written as: (IV.3)

2. Leading / lagging approximation

if and are assumed to be following a leading / lagging relationship then is (IV.4)

then U [deep] F is given as: (IV.5)

where Q (t [deep] i) is calculated from equation (IV.6).

(IV.6)

wherein

D = time delay in the process

small alpha = tuning constant

3. Explanation of the feedforward control algorithm

The effectiveness of the feedforward control algorithm proposed above can be illustrated by some simulation results.

Figure 14 is a block diagram for the control system in which immeasurable charge exists

(i) Example 1:

Consider the block diagram of the current system in Figure 15-a:

The resulting M [deep] A (2.0) and M [deep] A (1.0 models) from the AMS are:

M [deep] A (2.0) model: (IV.7)

M [deep] A (1.0) model: (IV.8)

The designated block diagram of the AMS is given in Figure 15-b.

The resulting designs using the feedback control algorithm and the proposed control algorithm are shown in Table 10.

Table 10

Case 1 and Case 3 use the Kc, T [deep] I, T [deep] D values that are matched to the M [deep] A (1,0) model

Case 2 and Case 4 use the Kc, T [deep] I, T [deep] D values that are matched with the M [deep] A (2.0) model.

(ii) Example 2:

Consider the block diagram of the current system in Figure 16-a

The resulting M [deep] A (2.0) and M [deep] A (1.0) models are: M [deep] A (2.0) model: (IV.9)

M [deep] A (1.0) model: (IV.10)

The designated block diagram is shown in Figure 16-b.

The execution indices from the simulations are listed in Table 11.

Table 11

Cases 1, 2, and 3 use the Kc, T [deep] I, and T [deep] D values that are matched with the M [deep] A (1,0) model.

** The charge estimate uses the exponential window approximation with
<notreadable>

= 0.74 (exponential section approximation)

* The charge estimate uses the square window approximation with l = 7 (square cutout approximation.

Fig. 15a Current control system

15b Identified control system

Fig. 16a Current control system

16b Identified control system

V. Appendix

1. Integration procedure according to Moore:

This integration procedure is based on the integration formula: (A-1)

Consider a given differential equation (A-2)

Then (A-3)

if t = t [deep] 1 equation (A-3) becomes: (A-4)

Further, if t = t [deep] 2, equation (A-3) becomes: (A-5)

Thus, the small alpha and small beta estimates can be obtained by solving Equation (A-4) and Equation (A-5) at the same time.

2. Estimation method using the block pulse function

This procedure is based on the integration formula: (A-6)

whereby

X [high] T = [x [low] 1, x [low] 2, ..., x [low] 8] with x [low] i = x (t [low] i) = x (i large delta ) and t [deep] small iota = k large delta

large Phi [high] T (t) = [small Phi [low] 0 (t), small Phi [low] 1 (t), ..., small Phi [low] k (t)]

with

small Phi [deep] 1 (t) = 1 for t [deep] i-1 <t <t [deep] i

small Phi [deep] i (t) = 0 for t is not an element of [t [deep] i-1, t [deep] i]

and (A-7)

Consider a given differential equation:

then (A-8)

so that

X [high] T large Phi (t) - X [low] 0 [high] T large Phi (t) + small alpha X [high] TH large Phi (t) = small beta U [high] TH large phi (A -9)

whereby

X [low] 0 [high] T = [x [low] 0, 0, ......, 0] (A-10)

U [high] T = [u (t [low] 1), u (t [low] 2), ......, u (t [low] k)] (A-11)

In order to

X - X [low] 0 = [H [high] T U - H [high] T X] [[high] small beta [low] small alpha] = V [[high] small beta [low] small alpha]

(A-12)

whereby

V = H [high] T U - H [high] T X (A-13)

In the end

[[high] small beta [low] small alpha] = [V [high] T V] [high] -1 [V [high] T (X - X [low] 0)].

(A-14)

Equation (A-14) can give the initial estimate of small alpha and small beta.

15. Digital computer programs:

Table on page 103 to page 119

Self-adjusting simulation and control system

contents

I. Introduction to the AMCS

II. Self-adjusting simulation system

1. Definition of the M [deep] A (n, m) and M [deep] B (n, m, s) models

2. Parameter estimation for M [deep] A (n, m) models

3. Parameter estimation for M [deep] B (n, m, s) models

4. Criteria for model construction

5. Explanations for the identification and parameter estimation procedures

6. Simulation strategies and methods

7. Identification of the immeasurable charge

III. Self-adjusting voting algorithm

1. Matching relationships for an interactive PID controller

2. Reconciliation relationships using the quadratic execution index

3. Coordination relationships for regulated procedures of the second order with supercritical damping

IV. Self-adjusting regulation for the immeasurable charge

1. Zero order approximation

2. Leading / lagging approximation

3. Explanation of the feedforward rule algorithm

V. Appendix

1. Integration process according to Moore

2. Estimation method using the block shock function

Claims (14)

1. The self-adjusting simulation and control system (AMCS abbreviation) provides for the simulation, identification and self-adjusting control operations that can be carried out under closed-loop. The operating system is shown in FIG. The functional structure of the AMCS is shown in FIG. The AMCS has the self-adjusting simulation system (AMS) as a core part. The AMS consists of two main algorithms: the simulation and identification algorithm (MIA) and the fault identification algorithm (DIA). These two algorithms use the dynamic data to simulate the process dynamics and to identify the immeasurable charge disturbances. The required dynamic data is generated from closed-loop operations. The AMCS is carried out in two loops. The outer loop A (referred to as AFBC) works in batch, so the temporary results resulting from a change in the level of operation (such as set point adjustment during normal operation or manual adjustment during the start-up period) are used to enter through MIA get updated model. Then follows the tuning PID controller, based on this new model, so that the self-adjusting feedback control loop (AFBC) is formed. The inner loop B (referred to as AFFC) works on the real-time basis so that the model errors or the outer charge disturbances are identified and compensated for can be operated in a forward sense in real time. The feedforward control loop is referred to as self-adjusting feedforward control (AFFC). Therefore, the AMCS is basically a self-learning and self-adjusting system, which always directs the control operations against their optimal conditions, regardless of the occurrence of immeasurable disturbances or model errors. 2. Definition of the M [deep] A (n, m) models The M [deep] A (n, m) models are given in the form: (2.1) W (t) = u (t-D) whereby = Model output = estimated observed output n = highest order of differentiation of x m = highest order of differentiation of W u = process input D = true time delay w = delayed input The nominal set of parameters for M [deep] A (n, m) model, large omega [high] N [low] MA is defined as: (2.2) Conventional feedback control loop Fig. 1 Self-adjusting simulation and control system Process data Fig. 2 Functional structure of the AMCS whereby a [deep] i (i = 1,2, ..., n) = the coefficient constant for b [deep] i, (i = 0,1, ..., m) = the coefficient constant for W = initial value of the ith derivative of So that P [low] N is a subset of large omega [high] N [low] MA (2.3) whereby P [low] N = nominal estimated parameters. The set of current parameters for M [deep] A (n, m) models is determined from their equivalent canonical state representation, i.e. = A Z + B w (2.4) with large omega [high] A [low] MA = [small Ni [low] 1, small Ni [low] 2, ..., small Ni [low] 1, z [low] 1 (0), z [low] 2 (0), ..., z [deep] n (0)] (2.5) and P is a subset of large omega [high] A [low] MA (2.6) whereby z [deep] i (i = 1,2, ..., n) = state variable small Ni [deep] i, (i = 1,2,, l) = coefficient constant in the canonical equation of state P = estimated parameter vector. 3. Definition of the M [deep] B (n, m, s) models The M [deep] B (n, m, s) models are given in the form of a stochastic differential equation that is connected with a hidden continuous differential equation model, i.e. (3.1) with (3.2) and (3.3) with small Phi [deep] i = small Phi [deep] i (a [deep] 1, ..., a [deep] n) and F [deep] i = F [deep] i (a [deep] 1 ,. .., a [deep] n, b [deep] o, ..., b [deep] m), i = 1,2, ..., n The nominal set of parameters in the M [low] B (n, m, s) model is defined as large omega [high] N [low] MB, d-h- large omega [high] N [low] MB = [a [deep] 1, a [deep] 2, ......, a [deep] n, b [deep] o, b [deep] 1 ,. ....., b [deep] m, small theta [deep] 1, small theta [deep] 2, ......, small theta [deep] s] (3.4) The symbols used in equations (3.1) to (3.4) mean the following: = estimated output for y y = current output variable w = the delayed process input, i.e. u (t-D), so that w (k) stands for u (k-small Eta), where small Eta = integer [D / large delta] small epsilon = error between the current and the estimated output small Phi [deep] i (i = 1,2, ..., n) = coefficients for y in the differential equation F [deep] i, (i = 1,2, ..., n) = coefficients for w in the differential equation a [deep] i, (i = 1,2, ..., n) = coefficients for b [deep] i, (i = 0,1, ..., m) = coefficients for w n = order of differentiation of m = order of differentiation of w s = order of differentiation for small epsilon small theta [deep] i (i = 1, ..., s) = coefficients for small epsilon The set of current parameters for M [deep] B (n, m, s) models is determined from their equivalent canonical state representation for equation (3.3), so that = A Z + B w (3.5) with large omega [high] A [low] MB = {small Ny [low] 1, small Ny [deep] 2, ......, small Ny [deep] 1, small theta [deep] 1, small theta [deep] 2, ......, small theta [deep] s} (3.6) and P is a subset of large omega [high] A [low] MB (3.7) where P = current estimation parameter vector [deep] small Ny [deep] i, (i = 1,2, ..., 2) = coefficient constant in the state representation for equation (3.3) Note that the initial values of the state variables Z [low] 1 (0), Z [low] 2 (0), ..., z [low] n (0) do not contain MB in large omega [high] A [low] are. 4. Uniform sampled discrete model for a continuous state dynamics system. The assumed continuous state dynamics system is given as: Z = A Z + B w y = z [deep] 1 (4.1) The discrete model sampled uniformly from the system in equation (4.1) is given as: y (k) = small Phi [deep] 1 y (k-1) + small Phi [deep] 2 y (k-2) + ... + small Phi [deep] ny (kn) + F [deep] 1 w (k-1) + ... + F [deep] nw (kn) (4.2) The relationship between small Phi [deep] i, F [deep] i and equation (4.1) are given as follows: Define H = [h [deep] ij] [deep] nxn = exp [A times large delta] and G = [g [low] i] [low] nxI = -A [high] -1 [I-exp (A times large delta)] B (4.3) whereby A, B = coefficient matrices in equation (4.1) h [deep] ij = i-, j-th element of matrix H g [deep] i = i-th element of matrix G large delta = integration sub-interval for the selection interval (1) For the equation of state with n = 1 small Phi [deep] 1 = h [deep] 11 F [deep] 1 = g [deep] 1 (4.4) (2) For the equation of state with n = 2 small Phi [deep] 1 = h [deep] 11 + h [deep] 22 F [deep] 1 = g [deep] 1 small Phi [deep] 2 = (h [deep] 12 h [deep21-h [deep] 22 h [deep11) F [deep] 2 = g [deep] 2 h [deep] 12-g [deep] 1 h [ deep] 22 (4.5) (3) For the equation of state with n = 3 small Phi [deep] 1 = h [deep] 11 + h [deep] 22 + h [deep] 33 F [deep] 1 = g [deep] 1 small Phi [deep] 2 = - (large delta [deep] 11 + large delta [deep] 22 + large delta [deep] 33) F [deep] 2 = (h [deep] 12 g [deep] 1 - g [ deep] 1 h [deep] 22) + (h [deep] 13 g [deep] 3 - g [deep] 1 h [deep] 33) small Phi [deep] 3 = Det [H] F [deep] 3 = g [deep] 3 large delta [deep] 31 + g [deep] 2 large delta [deep] 21 + g [deep] 1 large delta [deep] 11 (4.6) whereby Det [] = determinant of the square matrix large delta [deep] ij = factor of h [deep] ij. 5. Parameter estimation for M [deep] A (n, m) models The flow diagram of the parameter estimation for M [deep] A (n, m) is shown in FIG. The process steps result as follows: (1) Define the M [deep] A (n, m) model and the nominal parameter vector P [deep] N. (2) Transform the M [deep] A (n, m) model from equation (2.1) into the state representation of equation (2.4) and define the current parameter vector P. (3) Estimate the initial value of P [high] o [low] N from the estimation routines, for example from Moore's integration method or the block impact function method, and then calculate the corresponding current parameters P [high] o. (4) Start with P [high] o, compute and The calculation of or requires the integration of equation (2.1). The integration of equation (2.1) can be carried out in two different ways. (i) When all the estimates for the initial derivatives are available, the integration is performed by the known numerical integration methods. (ii) If all or part of the initial derivative estimates are not available, integration methods that do not use initial derivatives should be used. This integration process is as follows; (a) Transform the M [deep] A (n, m) model into the form of equation (4.1) (b) Use P [high] o [low] N to calculate the corresponding P [high] o (c) Use P [high] o and the equations (4.4) to (4.6) to calculate small Phi [deep] i, (i = 1,2, ..., n) and F [deep] i, ( i = 1,2, ..., n) (d) Assume y (0), y (1), ..., y (n-1) as and the equation: (5.1) to calculate for k = n, n + 1, ..., N. The flow chart for the method presented above is shown in FIG. 4. (5) Establishment of the (5.2) where r is the total number of parameters in P. (6) Compute small delta [deep] P by (5.3) whereby Y [high] small tau = [y (t) [low] 1), y (t [low] 2), ......, y (t [low] N)] small delta Y = [small delta y (t [deep] 1), small delta y (t [deep] 2), ......, small delta y (t [deep] N)] (5.4) and small delta y (t [deep] i) = y (t [deep] i) - (t [deep] i) (7) Check whether small delta P is within a given tolerance limit; if so, stop. (8) Otherwise replace P [high] o with P [high] o + small delta P and repeat from level (4) to level (7). 6. Parameter estimation for the M [deep] B (n, m, s) models This estimation procedure gives the estimates for the parameters in two connected equations of the M [deep] B (n, m, s) models, i.e. ..... h. (6.1) and Y (k) = small Phi [deep] 1 y (k-1) + small Phi [deep] 2 y (k-2) + ... + small Phi [deep] ny (kn) + F [deep] 1 w (k-1) + ... + F [deep] nw (kn) + small theta [deep] 1 small epsilon (k-1) + small theta [deep] 2 small epsilon (k-2) + .. . + small theta [deep] s small epsilon (ks) + small epsilon (k) (6.2) 3 is a flow diagram of the parameter estimation for the M [deep] A (n, m) model Fig. 4 Flow diagram for the integration of M [deep] A (n, m) models when no initial derivatives are available. The flow diagram for parameter estimation in M [deep] B (n, m, s) models is shown in FIG. The procedure is as follows: (1) Define equation (6.1) in the form of equation (4.1) (2) Get an initial estimate for the values of a [deep] i (i = 1,2, ..., n) and b [deep] i (i = 0,1, ..., m) and small theta [low] i (i = 1,2,, s), which are referred to as P [high] o. (3) Use P [high] o and apply equation (4.3) to compute the matrices H and G. (4) Use equation (4.4) or equation (4.5) or equation (4.6) to calculate small Phi [deep] i (i = 1,2, ..., n) and F [deep] i (i = 1 , 2, ..., n (5) Calculate the small epsilon (k) for k = 1,2, ..., N where small epsilon (k) = y (k) -small phi [deep] 1y (k-1) -......- small phi [deep] ny (k-n) -F [deep] 1w (k-1) -......- F [deep] nw (k-n) -small theta [deep] 1 small epsilon (k-1) -......- small theta [deep] s small epsilon (k-s) (6.3) Note that small epsilon (k) = 0 for k less than or equal to 0 (6) Compute large Phi (P) = (7) Compute large pi and large sigma where (6.4) and (6.5) (6.6) (6.7) 5 shows a flow diagram for the parameter estimation for the M [deep] B (n, m, s) models (8) Compute small DeltaP as small DeltaP = -small lambda large Pi [high] -1 large sigma (6.8) (9) Compute large Phi (P [high] o + small delta P) and check if it is within a tolerance of large Phi (P [high] o); if so, stop. (10) If no, replace P [high] o with P [high] o + small delta P and repeat from level (3) to level (9). 7. Criteria for the sufficiency of the M [deep] A (n, m) and M [deep] B (n, m, s) models. (1) Criterion for the sufficiency of the M [deep] A (n, m) models. This criterion tests the sufficiency for a given M [deep] A (n, m) model against the other possible M [deep] A (n ', m') model. A test index ID (l [deep] 1, l [deep] 2) is defined as follows: (7.1) whereby P * = optimal parameter for an M [deep] A (n, m) model with n as l [deep] 1 and m as l [deep] 2 SSE = sum of the quadratic errors, connected with P * for the given M [deep] A (l [deep] 1, l [deep] 2) model Then the test criterion for the adequacy of the given M [deep] A (n, m) model versus the M [deep] A (n ', m') model is to find out whether ID (n, m) is considerably less than ID (n ', m'). In particular, if ID (n, m) is smaller than ID (n ', m'), then the M [deep] A (n, m) model is sufficiently adequate sufficiency. (2) Criterion for the sufficiency of M [deep] B (n, m, s) models. If the sufficiency of an M [deep] B (n [deep] 2, m [deep] 2, s [deep] 2) model is checked against an M [deep] B (n [deep] 1, m [deep] 1, s [deep] 1) Model, where (n [deep] 1 + m [deep] 1 + s [deep] 1) is greater than (n [deep] 2 + m [deep] 2 + s [deep] 2), the F quotient test is carried out: Be it J [deep] 1 = sum of the squared errors for the M [deep] B (n [deep] 1, m [deep] 1, s [deep] 1) model J [deep] 2 = sum of the squared errors for the M [deep] B (n [deep] 2, m [deep] 2, s [deep] 2) model small Ny [deep] 1 = degree of freedom for the M [deep] B (n [deep] 1, m [deep] 1, s [deep] 1) model and small Ny [deep] 2 = degree of freedom for the M [deep] B (n [deep] 2, m [deep] 2, s [deep] 2) model then define (7.2) Compare the calculated f with the value of F [deep] small alpha (small Ny [deep] 2-small Ny [deep] 1, small Ny [deep] 1) from the F distribution table, where F [deep] small alpha ( small Ny [deep] 2-small Ny [deep] 1, small Ny [deep] 1) is defined as: Prob [F less than F [deep] small alpha] = small alpha% If f is smaller than F [deep] small alpha, then M [deep] B (n [deep] 2, m [deep] 2, s [deep] 2) as a sufficient model compared to the M [deep] B ( n [deep] 1, m [deep] 1, s [deep] 1) Model assumed. 8. Simulation strategy for M [deep] A (n, m) models The simulation strategy for M [deep] A (n, m) models is shown in FIGS. The strategy takes advantage of the fact that the time delay in the The M [deep] A (2.0) model will be smaller than that in the M [deep] A (1.0) model, so fewer iterations in Fig. 3 will be required. In addition, the estimate of the delay and time constants in M [deep] A (1.0) from the identified M [deep] A (2.0) model is available in the form: (8.1) and (8.2) whereby T [deep] 1, T [deep] 2 = equivalent time constants of M [deep] A (2.0) model small tau = time constant in the M [deep] A (1,0) model D [high] (1) = equivalent time delay in the M [low] A (1,0) model D [high] (2) = identified time lag in the M [low] A (2,0) model 9. Simulation strategy for M [deep] B (n, m, s) models The strategy for simulating M [deep] B (n, m, s) models is shown in FIG. 10. Identification of the immeasurable charge (1) Parameter sensitivity approximation The continuous process is believed to be: (10.1) Simulation strategy for M [deep] A (n, m) models Fig. 6 Simulation strategy for M [deep] A (n, m) models Fig. 7 Strategy for determining the time delay in M [deep] A (n, m) models 8 simulation strategy for M [deep] B (n, m, s) models In equation (10.1) the coefficients a [deep] i (i = 1,2, ..., n), b [deep] i (i = 0,1, ..., m) are assumed as constants. Just is the parameter to be identified. A rectangular window (section) is used to handle the input-output data, so that l points of input and output data are available for parameter estimation at any point in time. Use the given values of a [deep] i (i = 1,2, ..., n) and b [deep] i (i = 0,1, ..., m) to form the differential equation: (10.2) where µ is the constant systematic error resulting from in equation (10.1) results. Define Y [deep] i = [y (t [deep] i-1), y (t [i-1 + 1), ......, y (t [deep] i)] (10.3) W [deep] i = [w (t [deep] i-1), w (t [deep] i-1 + 1), ......, w (t [deep] i)] (10.4) and small delta Y [deep] i = [small delta y (t [deep] i-1), small delta y (t [deep] i-1 + 1), ......, small delta y (t [ deep] i)] (10.5) whereby (10.6) Then you can start with be calculated. (10.7) whereby h = int. [D / large delta] (10.8) = Estimate of d at t [deep] j with the data up to t [deep] i The expression is given below. The process steps are as follows: (i) Put Y [deep] i, W [deep] i, at t [deep] i (ii) Determine the values of a [deep] i, (i = 1,2, ..., n) and b [deep] i, (i = 0,1, ..., m) from the last identified ones Results (iii) Find the value of at the last identified value, ie (iv) Calculate the parameter sensitivity (10.9) whereby (10.10) (v) Compute by: (10.11) (vi) Compute by (10.12) (vii) Increase i by 1 and repeat from step (iii). (2) Non-linear optimization approximation With this approximation, the process model under consideration is an M [deep] B (n, m, s) model, i.e. (10.13) and whereby small Phi [deep] i = small Phi [deep] 1 (a [deep] 1, a [deep] 2, ......, a [deep] n) and F [deep] i = F [deep] 1 (b [deep] 0, ......, b [deep] m, a [deep] 1, ......, a [deep] n = constant charge disturbance and small epsilon = the white noise of the system L = the constant systematic error resulting from d Define Y [deep] i and W [deep] i as in equation (10.3) and equation (10.4). Then the process steps are as follows: (i) Set the required data window (data section) of Y [deep] i and W [deep] i (ii) Assign (iii) Compute (10.15) whereby (10.16) (iv) Compute (10.17) (v) Compute (vi) Compute (10.18) (vii) Compute (10.19) (viii) Repeat from step (iii) to step (vii) several times to converges (ix) Increase i by 1 and repeat from level (i) 11. Tuning relationship for an interactive digital PID controller The digital PID control algorithm is given as: (11.1) with Qn, calculated from: whereby Q = dummy variable Q [low] n-1 = the value of Q at a previous point in time Q [low] n = the value of Q at the current time M [deep] n-1 = previous control output M [low] n = current rule output e [deep] n-1 = previous deviation between the output and the setting point e [low] n = current deviation between the output and the set point large delta = sample interval The voting relationship is based on some integral criteria, such as ITAE, IAE, ITSE and ISE and are expressed in the form: P = A [D / small rope] [high] B (11.2) or O = A [D / small rope] + B (11.3) whereby P = scaled tuning parameter D = time lag in the M [deep] A (1,0) model used to represent the dynamic process small Tau = time constant in the M [deep] A (1,0) model which is used to represent the dynamic process A and B = constants for the correlation of the voting relationships. The values of A and B for the voting relationships are listed in Table 1. 12. Tuning relationship for the ideal PID controller with a quadratic function index The quadratic function index used is given as: (12.1) The reconciliation relationships obtained from this: (1) PI controller K [low] C K [low] P = 0.635 (D / small rope) [high] -0.658 (12.2) T [low] 1 / small rope) = 1.211 (D / small rope [high] 0.462 (12.3) (2) PID controller K [low] C K [low] r = 0.975 (D / small dew) [high] -0.572 (12.4) T [low] 1 / small rope = 1.211 (D / small rope) [high] 0.487 (12.5) T [deep] D / small rope = (D / small rope) / [2.092 + 1.449 (D / small rope)] (12.6) Table 1 The numbers without * are in Eq. (II.2) and the numbers with * in Eq. (II.3) to be used. with Kc = proportional control constant Kp = process goal T [low] i = integral time T [low] D = derivative time small Tau = time constant of the approximate M [deep] A (1,0) model D = time delay of the approximate M [deep] A (1.0) model 13. Tuning relationship for controlled methods of supercritical damping of the second order. Let T [low] 1, T [low] 2 be the time constants of the second order supercritical damping method and D [high] (2) be the time delay in the second order method. Then there is an equivalent (in terms of least squares) first order process that has the following time constant and time delay: (13.1) (13.1) where D [high] (1) = time lag in the equivalent first order process small tau = time constant in the equivalent first order process. Existing tuning relationships, which contain the equations (11.2), (11.3), (12.2), (12.3), (12.4), (12.5) and (12.6) and Table 1, can be used to tune the PID controller. 14. Self-adjusting feedforward control algorithm (self-adjusting forward control algorithm) Let t [deep] i be the current point in time and let the identified charge disturbance at t [deep] i - D of the data up to t [deep] i. (1) Zero order approximation for The feedforward rule input is given as: (14.1) whereby b [deep] o = the coefficient of W in the M [deep] A (n, m) model a [deep] n = the coefficient of y in the M [deep] A (n, m) model = the desired setting point for y K [deep] I = the integral factor = K [deep] c / T [deep] I small lambda = tuning constant, range from 0 to 1 so that the total rule input is u (t): (14.2) (2) Lead / lag approximation for In the lead / lag approximation, the estimated value is given with: (14.3) and the forward control is given by: this makes the entire rule input: