3.9.2.1 Review of Some Laplace Transform Results

The Laplace transform of a function of time x(t) is defined as

(3.111)

Only behaviour at times t ≥ 0 is considered, so it is implicit in the approach that all time functions are zero up until t = 0.

Apart from the fact that s-domain versions of various functions may be found in tables, one notes that the operator L{·} is linear, so that if , then . Furthermore, Laf(t)} = aF(s).

Now consider the transform of the time derivative:

(3.116)

Integrating by parts

(3.117)

(3.118)

A similar treatment for the second derivative yields

(3.119)

In general, provided for , then

(3.120)

For integration, note that

(3.121)

These results will shortly prove useful for the conversion of ODEs with constant coefficients into transfer functions. However, transport (dead-time) lag (Figure 3.19) cannot be described using ODEs, and warrants a special treatment.

(3.122)

(3.123)

(3.124)

Substituting ,

(3.125)

(3.126)

(3.127)

So the transfer function of a dead-time lag τ_T is

(3.128)

Several useful Laplace transform results are summed up in Table 3.5.

Consider an arbitrarily high-order SISO linear system with constant coefficients similar to the trolley with spring and dashpot in Section 3.5.

(3.129)

Consider the particular circumstance of

(3.130)

(3.131)

This requires that the system starts at t = 0 with both input u and output x at zero, and at a ‘complete' steady state where the indicated time derivatives are all zero. Then Equation 3.120 gives

(3.132)

Thus, in the s-domain, a transfer function G(s) can be used as a multiplier to represent Equation 3.129.

(3.133)

(3.134)

For a system to be physically realisable, it is necessary that n ≥ m. Indeed, in the trolley, spring and dashpot example of Section 3.5, n = 2 and m = 0. Few physical systems would be modelled with the derivative terms on the right-hand side of Equation 3.129. Mathematically, one might propose a system like

(3.135)

but one would be asking for an impossible response, for example if u were a step function.

The standard input function transforms in Table 3.4 suggest that, in general, the input U(s) would be expressed as a ratio of two polynomials in s. Since the transfer function G(s) in Equation 3.133 is similarly a ratio of two polynomials in s, one expects that usually the output X(s) will arise as a ratio of two polynomials in s. These will be more complex than the simple transforms in Table 3.4, so they must be broken down into more elemental pieces using a partial fraction expansion.

Say

(3.136)

Letting

(3.137)

(3.138)

where q_i, i = 1,…, k, are the roots obtained by setting the denominator to zero. If these roots are all distinct, one can write

(3.139)

Otherwise, if a root is repeated – say q₁ occurs three times – write

(3.140)

In the case of distinct roots, one multiplies by each denominator factor in turn, and simultaneously sets s to the root value, for example

(3.141)

For the repeated roots, first multiply by the highest power denominator

(3.142)

Now obtain

(3.143)

(3.144)

(3.145)

In general, s is a complex number, and complex roots q_i certainly can arise in the above procedure. These are associated with oscillation in the response. Such roots will occur in complex conjugate pairs, and the associated coefficients must then also be expected to occur in complex conjugate pairs so that the complex variable does not remain in an expression like Equation 3.139 if a real value of s is substituted.

Say

(3.146)

Then it is required that so that

(3.147)

which is real for real s.

Going a little further, one notes that Tables 3.4 and 3.5 allow the following inversion (L⁻¹{·}) of these first two terms of the expansion to the time domain:

(3.148)

(3.149)

Note that it is implicit in all of these developments that s, a and b have units of inverse time. In the angular sense this is understood as radians per unit time.

The above discussion is based on the premise that the function to be inverted will occur as a ratio of two polynomials in s. One notable exception to this occurs with the transport lag G(s) = e^−τs in Table 3.5. If this is simply a multiplier of the expression, it can be used subsequently to time shift the result. If it is embedded, special procedures such as the Padé approximation will be required (Section 8.2.1.1).

x(t)	Plot	X(s)
δ(t)		1
1
t
e−at
sin(ωt)
cos(ωt)

First derivative
Second derivative
Integral
Transport lag
s associated with a
Complex conjugate partial fractions
Final value theorem

3.9.2.2 Use of Laplace Transforms to Find the System Response

Now consider the use of Laplace transforms in solution of the modelling problem represented by the linear state equation (Equation 3.100).

(3.150)

If the elements of the matrices A and B are constant, transformation using the result (Equation 3.118) yields

(3.151)

(3.152)

The matrix of time functions resulting from the first inversion is known as the state transition matrix (or ‘matrix exponential'– Section 3.9.3.2) and the result of its multiplication with the numerical vector of initial values x(0) will be the complementary solution. One can get an idea of the structure of the state transition matrix by examining a 2 × 2 system:

(3.153)

Recall that

(3.154)

where

(3.155)

where

(3.156)

and the are minors, that is the determinant of what is left after eliminating row i and column j.

(3.157)

where the m_ij are the original elements of M. Applying this to [sI − A] obtain

(3.158)

So the complementary solution x_C(t) (for u(t) = 0), given , requires evaluation of

(3.159)

Each term arises as a ratio of two polynomials in s. It has been noted in reference to Table 3.4 that the terms in the forcing vector U(s) in Equation 3.151 will likewise be ratios of polynomials in s. Thus, the particular solution x_P(t) (for u(t) ≠ 0 but x(0) = 0) will be similar to Equation 3.159 with larger polynomials. The final solution x(t) = x_C(t) + x_P(t) requires inversion of these expressions using the partial fraction expansion methods presented in Section 3.9.2.1.

The idea of a transfer function for linear systems with constant coefficients was built up in Section 3.9.2.1 based on a SISO system. Now one sees that it easily extends to MIMO systems for the case x(0) = 0 (Figure 3.20). Then the state Equation 3.151 becomes the input–output form

(3.160)

that is

(3.161)

Following Equation 3.154, note that a single scalar polynomial

(3.162)

will apply as a denominator throughout this transfer function, and that the adjoint matrix of , multiplied by B, will yield the set of numerator polynomials N(s), expressed here as a constant coefficient matrix for each power of s, that is

(3.163)

that is

(3.164)

where

(3.165)

and the λ_i are clearly the eigenvalues of A.

Since in the above development x contains all of the states, the requirement that x(0) = 0 implies complete steady state at t = 0.

More general linear input–output forms may not involve all of the states (Figure 3.20). These are stated directly as

(3.166)

From Equation 3.71 for linear systems, can effectively arise as some combination of and

(3.167)

(3.168)

(3.169)

Though input–output forms are usually not derived like this, one clearly expects from Equation 3.168 that the transfer function for this state-based system will be a similar matrix of polynomial ratios, with the denominator factors for this state-based system arising similarly from the same root values of (i.e. the characteristic equation of the state open-loop system). However, this is not generally the case for input-output systems, where arbitrary polynomial ratios can occur in G′, requiring more d_i and N_i terms (see Section 7.8.1)

The following examples illustrate the use of Laplace transforms to obtain the output response functions of several systems, for some standard input excitation functions. The response to an oscillating input is dealt with later in Chapter 8, for example Example 8.4.

3.9.2.3 Open-Loop Stability in the s-Domain

The idea of stability becomes very important when controllers are left in charge of a process, and this will be considered in detail later (Chapter 8). The problem is that closing a control loop around a process invites a number of problems, for example overreaction to a disturbance. For the meantime, just the open loop will be considered. Most processes are quite well behaved on their own (in open loop); for example, in all of the tank flow examples in Section 3.9.2.2, if the inflow is stepped up, the levels will rise until the exit flows balance the new inflows, where a new equilibrium is found. However, a few naturally unstable systems exist in the processing industries – notably cooled or heated reactors (exothermic or endothermic). Reaction rates are strongly dependent on temperature according to the Arrhenius relationship for rate constants:

(3.225)

so if the cooling is reduced on an exothermic reactor, the temperature increases, giving higher reaction rates, and thus even more heat is generated, so temperature increases rapidly and a runaway reaction occurs (provided the reagent supply is sufficient). Conversely, an endothermic reaction will die if heating is reduced.

The formal definition of stability is as follows:

The important thing is that if just one finite input function can be found that causes unbounded growth of the output, then the system must be declared unstable. An example of this would be a bridge or chimney that might only resonate in its vortex street at a particular wind speed.

The open-loop MIMO systems considered in Section 3.9.2.2 were of the forms

(3.226)

(3.227)

For these systems derived from the state equation it was noted that both transfer functions involve denominator factors based on roots of the characteristic equation of the state system, namely , that is the eigenvalues λ_i of A. The system itself thus contributes corresponding partial fractions to any response, in the form of factors exp{Re(λ_i) × t}. It is thus clear that the eigenvalues of A must all have negative real parts for open-loop stability (Figure 3.28). More generally, for input–output systems, the denominator factors in the elements of G′(s) in Figure 3.2a can all differ, giving more factors (s − a_i) requiring Re(a_i) < 0 for open-loop stability.

3.9.3.1 Review of Some z-Transform Results

The initial requirement is to develop a formal way of describing a time series of sampled values. This is more or less just a vector of numbers to which a new value is added at each sampling instant. It is conceptually useful to represent these numbers as a series of delayed Dirac impulses of size numerically equal to the original signal values at the time instants. In Figure 3.30, some license has been taken to represent these impulse sizes by the heights of equal-base triangles. So the sampled signal then becomes the impulse-modulated function

(3.232)

To transform this to the s-domain, one makes use of the transport (or dead-time) lag e^−Ts (Table 3.5) to delay the impulses successively. It is also noted in Table 3.4 that the δ(t) transforms to 1.

(3.233)

The idea of z-transforms arises from the substitution

(3.234)

which is just a single forward time shift in the s-domain. Then the notation x(z) is used for the transform of the impulse-modulated signal, so that

(3.235)

A simple example would be a unit step at t = 0 (Figure 3.31).

Here

(3.236)

Some additional useful z-transforms and their corresponding Laplace transforms are included in Table 3.7. Some care must be taken in the use of these functions. The z functions are merely s-domain functions in disguise (where z replaces e^Ts). In particular, the x(z) functions in Table 3.7 are the Laplace transforms of the impulse-modulated signals, and only ‘represent' the smooth functions such as t and e^−at at discrete points in time. The transformation operator Z{·} implies that the argument is to be replaced by the corresponding z function from this table.

x(t)	x(z)	X(s)	Plot
δ(t)	1	1
1
t
e−at
sin(ωt)
cos(ωt)
Delay nT		e−nTs
Zero-order hold of F(s)

To use the z notation to solve for the way continuous signals move through a system, one needs to ‘hold' values between the impulses. The most common way of doing this is by means of the ‘zero-order hold', which keeps the signal constant at the last value, rather than trying to interpolate or extrapolate it in some other way according to higher order holds.

Consider the general signal of Equation 3.235:

(3.237)

that is

(3.238)

The zero-order hold transfer function is

(3.239)

Then

(3.240)

(3.241)

The term in square brackets is seen to be a unit step of +1 delayed until t = iT with an equivalent unit step at t = (i + 1)T subtracted from it after an interval of T. The resultant square pulse is also scaled by its own factor x(iT). All of these functions are added together as in Figure 3.32.

The procedure to convert a given impulse-modulated form to its step function, and feed it to a system G(s), then involves the arrangement in Figure 3.33. Everything between the two sampling switches must be included if a z-domain transfer function G(z) is required to convert u(z) to x(z). Individual transfer functions between the switches G₁(s), G₂(s), G₃(s),…cannot be individually transformed to the corresponding G₁(z), G₂(z), G₃(z),…because the latter are only phased with individual impulses, and do not recognise variations between these values.

In general, the original signal is altered in the process of sampling and holding. Following a zero-order hold, it will vary in a series of steps at interval T. The ramp function t will become a staircase. One exception is of course the step function, so it is interesting to repeat Example 3.6 using the z notation.

It is a mistake to assume that the intermediate values of x can be obtained by substituting intermediate values of t, and in general these must be obtained by dual-rate sampling. In this example, there is no access to x(t) in Figure 3.33 because the discrete transfer function G(z) stretches across it to the next sampling switch. Nevertheless, it is noted in this instance that there is agreement with the complete response in Example 3.6, because the zero-order hold recreates the original step for the single step input.

The response x(z) of a system will generally occur in the form of a ratio of two polynomials as in Equation 3.254. It is worth noting an alternative to seeking inverses in tables of z-transforms, namely long division. For example,

(3.257)

is a response that will have a value of 1 at the end of the first interval T, 3 at the second, 5 at the third and so on.

The idea of an impulse response arises from feeding a unit impulse input δ(t) to a system. In the z-domain, it is seen that the output values at the sampling instants arise as the coefficients of the original transfer function. For example, consider a unit impulse fed to an integrator, giving a finite impulse response which is a series of 1's:

(3.258)

(3.259)

(3.260)

(3.261)

In the case of integrating (or unstable/undamped) systems, one expects an infinite series of nonzero coefficients. However, for non-integrating, stable and damped systems, the coefficients arising from an impulse input become insignificant after a finite number of steps, and one describes this as a finite impulse response (FIR). An example would be the first-order system

(3.262)

where stability and damping require a > 0 and thus .

In Example 3.11, the focus was on the handling of actual signals, subjecting impulses to a zero-order hold to create continuous step functions, and so on. In practice, the concept is often used as a sort of shorthand to represent a sequence of data values. A forward shift operator q, or backward shift operator q⁻¹, is used to shift a time sequence of values in the same way as z or z⁻¹, without invoking the theoretical basis of z-transforms.

3.9.3.2 Use of z-Transforms to Find the System Response

The use of z-transforms here will focus on the time-shifting properties of z or z⁻¹. One notes that

(3.263)

returns the value of x at a time T before the present. If a transport lag from inlet u to outlet x is exactly 3T, then one could write

(3.264)

Consider a linear system which is expressed as a set of first-order differential equations with constant coefficients:

(3.265)

As in Equation 3.152, the Laplace transform yields

(3.266)

The inversion will depend on the time functions in the input vector. A special case will be considered here where the inputs u(t) are held constant at their starting values u(0), so

(3.267)

Recall that the s-domain functions represent zero values up until time zero, so that the input being considered here is a step from zero in each input variable, at time zero. Then

(3.268)

(3.269)

It is useful to expand the inverse matrices of the second term using partial fractions, that is

(3.270)

where α and β are constant matrices.

(3.271)

(3.272)

(3.273)

so

(3.274)

Since is diagonal, and A is square,

(3.275)

Thus,

(3.276)

Taking the inverse Laplace transform

(3.277)

This result involves the state transition matrix , which is known as the matrix exponential, represented symbolically as

(3.278)

It can be evaluated directly in mathematical programs such as MATLAB®, and has similar properties to a normal scalar exponential, for example

(3.279)

(3.280)

(3.281)

There is an obvious resemblance to the SISO case of Equation 3.110 and Table 3.3. The frame of reference so far has been 0 < t < ∞, but the integration can be used in the same way for successive intervals T as follows:

(3.282)

The form of this equation has a strong resemblance to the continuous system case (Equation 3.100). Here one defines similar matrices for the discrete system

(3.283)

(3.284)

so that

(3.285)

In terms of the time-shift operator z, this is

(3.286)

so

(3.287)

(3.288)

The transfer function matrix for this discrete linear state system is thus determined by

(3.289)

with

(3.290)

This transfer function is taking a series of input values at time intervals T and providing the output values at corresponding intervals T. It replaces the integration step of the continuous system in Equation 3.100 over each of these intervals, under the specific condition that the input values vary in a series of steps, remaining fixed at the initial value for each interval. It is quite clear that this formulation replaces the combination of zero-order hold and continuous system G(s) in Figure 3.33, and provides a general MIMO approach to the SISO example (Example 3.11). The successive multiplication of several transfer functions on this basis would imply that the intermediate values are sampled and held between each system. In practice, this is very much how intermediate values from various calculations are only updated periodically in a processing plant computer database.

Following the same treatment as in the s-domain (Section 3.9.2.2)

(3.291)

Again, the eigenvalues λ_i of A_* are the roots of the characteristic equation

(3.292)

(3.293)

(3.294)

and will contribute factors (z − λ₁), (z − λ₂),…in the denominators of the partial fraction expansion of any response. Table 3.7 shows the importance of these denominators in determining the response. Each term in the adjoint matrix shown will also be a polynomial in z, so Equation 3.289 can be expressed as

(3.295)

where the matrices N_i group the coefficients of the relevant powers of z. This relationship is analogous to the s-domain expression (Equation 3.163). The equation may be divided by z successively until the highest power is z⁰. In this form, it provides an alternative recursive predictor for x based on past values of x and u. The transfer function is often expressed using a matrix of polynomial ratios. For this state system it is seen that the matrix G(z) has a common denominator for all elements (but these may all differ for a general input-output system).

(3.296)

(3.297)

(3.298)

3.9.3.3 Evaluation of the Matrix Exponential Terms

Equations 3.283 and 3.284 give the coefficient matrices of the discrete system

(3.321)

(3.322)

The matrix exponential is provided directly in environments such as MATLAB® (‘expm'), but it is worth noting that its Taylor expansion is

(3.323)

since e⁰ = I. It follows then that

(3.324)

This latter result for B_* is particularly useful when the matrix A is singular.

3.9.3.4 Shortcut Methods to Obtain Discrete Difference Equations

The procedure used in Section 3.9.3.2 to obtain an exact discrete equivalent of a continuous system with a piecewise-constant input has been laborious, though one notes that the matrix exponential is readily available in mathematical programs. Useful transfer functions are easily found in the s-domain, so several methods have been devised to convert these directly to approximate discrete equivalent equations. Noting that s represents a derivative, the following substitutions are used:

• Forward difference (implicit Euler):
  (3.325)
• Backward difference (explicit Euler):
  (3.326)
• sTustin (bilinear or trapezoidal):
  (3.327)

  Inverting the Tustin approximation,

  (3.328)

  which is the first-order Padé approximation, sometimes used in the s-domain to deal with transport lags in the quest for polynomial-ratio forms. The inverse form of these approximations (like Equation 3.328: z = fn(s)) is sometimes used for frequency analysis of discrete systems, by substitution of jω for s in the resulting equation, as discussed in Section 8.6.1.

3.9.3.5 Open-Loop Stability in the z-Domain

Equations 3.289–3.295 in the previous section examined the impact of the discrete system transfer function G(z) itself on the output x(z), regardless of the particular input u(z). The full-state representation is

(3.336)

and

(3.337)

So the characteristic equation for this open loop is

(3.338)

(3.339)

It is noted that the factors (z − λ₁), (z − λ₂),…will occur as denominators in the partial fraction expansion of any output x(z).

In Table 3.7, one needs to set λ = e^−αT, and it is seen that

(3.340)

The possibility does exist that λ is complex:

(3.341)

(3.342)

that is

(3.343)

Thus, in the time domain, terms of the following form will occur in the system response:

(3.344)

The presence of a complex conjugate root λ will cause the imaginary values to disappear in the characteristic equation. Whether or not there is oscillation, there will be unbounded growth in the output if ln(a² + b²) is positive (making α negative), meaning that all roots of the characteristic equation of a discrete system must lie within the unit circle for the system to be stable (Figure 3.34). This result is not surprising when it is recalled that z = e^sT, since it has been established in Section 3.9.2.3 that the real part of the solutions of the s-domain characteristic equation must all be negative for stability, that is to ensure that no bounded input can cause an unbounded output.

As in the case of continuous systems (Section 3.9.2.3), discrete systems more generally can be represented in the input–output form

(3.345)

and again one expects the terms in matrix G′(z) to be ratios of polynomials in z, but here the denominators may all differ, giving more factors (see Section 7.8.1). For the same reason as above, the factors of these denominators, (z − a_i), require |a_i| < 1 for stability. The input–output form (Equation 3.345) is often represented as

(3.346)

where the backward shift operator q⁻¹ replaces z⁻¹. This reflects models based directly on data sequences, rather than implying any theoretical relationship to continuous systems.

3.9.4.1 Numerical Solution Using Explicit Forms

An explicit Euler integration formula for Equation 3.352, using a time step of T, is obtained as

(3.354)

The explicit Euler integration method is seen to use a single gradient vector evaluated at the start (or left) of the interval. The effects of this bias can be reduced by means of a smaller integration step T. However, it is worth noting another well-known explicit technique that seeks to eliminate this left-hand bias, the fourth-order Runge–Kutta method. In this technique, the estimates of the gradient are successively updated as new estimates are obtained of the change in x across the interval (Equations 3.361–3.364). It is seen that the final estimate (Equation 3.365) is based on a gradient that is more heavily weighted towards the centre of the interval.

(3.361)

(3.362)

(3.363)

(3.364)

(3.365)

A general observation regarding explicit methods is that the exclusion of x_i+1 from the formulation leaves them prone to overshoot resulting in instability. MIMO systems of ODEs often have widely varying time constants (‘modes') in the equations, so that if a single time step is used, it might have to be very small to deal with this stiffness.

3.9.4.2 Numerical Solution Using Implicit Forms

An implicit form of stepwise integration would appear like Equation 3.353:

(3.366)

The equation might be written like this, but occasionally some manipulation allows extraction of x_i+1. However, the general case will require an iterative solution for x_i+1, for example using the Newton–Raphson method:

(3.367)

where is a Jacobian matrix obtained by differentiating each function in the h vector by each element in the x_i+1 vector, and evaluating the result at the condition. Similarly, represents the vector h evaluated with the estimate (and of course the earlier solutions x_i, x_i−1,…and u_i,…, u_i−m).

An implicit Euler integration for Equation 3.352, using a time step of T, is

(3.368)

However, it is noted that the gradient being used is now biased to the right of the time interval. Since one is already committed to an implicit solution, why not attempt an ‘average' value of the gradient using average values of the variables? Thus, a possibility is

(3.369)

3.9.5.1 Step Response Models

As will be seen in Section 7.8.2, models based on measured process step responses have become very important as they form the basis of common controllers such as the dynamic matrix control algorithm. However, the model part of this model predictive control technique needs to be recognised as a useful open-loop modelling method in its own right.

Consider the two-input, two-output system in Figure 3.36 as being representative of MIMO systems in general. Because the observable outputs are not necessarily all of the states, or indeed the states at all, these will be represented by the vector y instead of x. With the system at steady state, each input is stepped in turn to obtain a matrix of step responses. For example, consider the effect of u₁ on y₁. The step of u₁ from u_1SS to u_1SS + Δu₁₍₀₎ at t = 0 produces y₁ values at subsequent intervals T of y_1SS + Δy₁₍₁₎, y_1SS + Δy₁₍₂₎,…Normalising these with respect to the input step,

(3.377)

one obtains the unit step response function

(3.378)

Here q⁻¹ is being used to represent a backward shift of one time interval, rather than z⁻¹, as is conventional when none of the theoretical z-transform properties are intended. So far, a non-integrating system has been assumed, and the interval T and number of points N in each response have been chosen to both give good definition to the variations, and ensure that the final point is close to the new equilibrium of the system. In Equation 3.378, the final point has been extended indefinitely with a delayed unit step (see Table 3.7).

The important assumption one makes in step response modelling is that the system is linear. So if a system is initially at steady state y₁₍₀₎, a positive or negative step of any size Δu₁₍₀₎ at t = 0 must produce the following x₁ output:

(3.379)

Notice that the product only gives the change in y₁ from its initial value, so a step function is used to add back the offset at each future point in time. If there are now subsequent ‘moves' of u₁ at the intervals T, that is

(3.380)

then the appropriate responses are just delayed in time before being summed:

(3.381)

Now include the effects of moves in u₂, and treat y₂ similarly to obtain

(3.382)

that is

(3.383)

Another way of viewing this is

(3.384)

where

(3.385)

Matching up the time-shift coefficients

(3.386)

This is conveniently represented using a matrix of matrices and several vectors of vectors:

(3.387)

with

(3.388)

Note that the equations developed to this point rely on the system being at steady state at y₍₀₎ at t = 0. The matrix is referred to as the dynamic matrix, and will later form the basis of dynamic matrix control (Section 7.8.2). It is obvious that if there had been input moves Δu prior to the starting time t = 0, these would have an effect extending past t = 0, and would have to be included.

Up to this point only non-integrating systems have been considered, that is systems which reach a steady state within the N times represented in the dynamic matrix (with N = 5 in the preceding 2 × 2 example). Now consider a simple strategy for dealing with integrating systems. A possible coding of the final gradient of a response might be in terms of the last two points (N − 1 and N) of the step response (Figure 3.37). So the integrating gradient of y₁ for a unit step input Δu₁₍₀₎ is .

Then an appropriate delayed ramp function is included in b(z) in Equation 3.384 to obtain

(3.389)

The equivalent dynamic matrix for Equation 3.387 then becomes

(3.390)

Equation 3.387, taken together with the possibility of Equations 3.389 and 3.390 for integrating systems, constitutes the important results of the step response modelling approach. One notes in Equation 3.388 that the output vector values correspond to the end of each interval, whilst the input vector moves are at the start of each interval. The response at the end of an interval T is independent of the input move at that same time, owing to the finite response time required, so the output vector starts at one interval later in time.

The step response modelling approach easily handles transport lag (dead time), since an arbitrary sequence of response values can be specified. In practice, several step response measurements should be done on a plant, to ensure that the features used in the model do not include random and temporary disturbances. Some degree of averaging and smoothing is necessary, and in some installations, a standard response such as first-order plus dead time may be fitted to the measurements for use in a controller. Many industries represent their dynamic matrix as an array of s-domain transfer functions, that is those functions that would convert each input step into the observed output responses.

As mentioned, a limitation of the method is that it assumes linearity. Model validity will thus be improved if the step responses are determined close to the normal operating point. One method used to handle severe process nonlinearity is to superimpose a separate nonlinear model, for example an artificial neural network (Example 3.17), just for the residual nonlinearity.

Another limitation to bear in mind is that the method does not explicitly recognise the relationships between variables in the same way as a mathematical model based on physical principles. For example, consider the two-tank flow system in Figure 3.38.

The separate step responses to each valve do not carry the information that if both valves shut, the levels must equilibrate. The nonlinearity of the level/flow relationship will cause the step response model to find two different tank levels at steady state, with both valves shut (assuming the step response measurements were not made at this state). Even if this relationship were linear, one would have to ensure that the initial output y₍₀₎ used to start the model represented an equilibrium to avoid this situation.

3.9.5.2 Regressed Dynamic Models

Historical plant measurements stored at intervals T constitute an input–output data set of the form

(3.391)

(3.392)

where u_(i) is a vector of the input variables (MVs and DVs) at time iT and y_(i) is a vector of the output variables (PVs or CVs) at time iT. Proper identification of the dynamic behaviour of this system will only be possible if the sampling interval T is not larger than about one-tenth of the shortest time constant in the system, that is one requires about 10 data points to define the shortest transient. A problem arising from too large a sampling interval is that of aliasing (Figure 3.39), where a higher frequency signal manifests at a lower frequency. Usually one is not concerned with frequency signals, but the same effect may cause misinterpretation of any signal.

A form of model must initially be selected, requiring identification of the defining variables and the system order, plus an appropriate interval T for a discrete model.

(3.393)

Here p is a set of constant parameters used in the model. The determination of the model then consists in finding p which minimise some performance criterion, for example a least square deviation:

(3.394)