Apparatus for relay based multiple point process frequency response estimation and control tuning

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to closed loop systems having a regulator for control of a process, and more particularly to an apparatus for estimating the process frequency response and using the estimated frequency response to design the regulator.

2. Brief Description of the Prior Art

Controllers for tuning the response of a system or a process have been the subject of attention in the prior art. A typical block diagram of such a controller and process is shown in

FIG. 1

wherein the output y of a process

10

is fed back through a feedback network

12

, the output y′ of which is subtracted from a reference signal r by a comparitor

14

which outputs an error signal e. A control circuit

16

is designed to respond to the error signal e by outputting a process control signal u for directing the process

10

to output a corrected y such that the error e is zero.

Achieving automatic tuning involves identification of the process

10

, and the design of the controller

16

. The control output u as a function of time is applied to the process

10

to assure the desired output y. It is generally convenient in analysis of such processes and control circuitry to mathematically transform the parameters from the time domain to a frequency domain. This process is done in electrical circuit analysis by means of the well known Fourier Transform. Manipulation of design parameters is much more conveniently achieved in a frequency domain. The frequency range of interest for such applications is usually from zero up to the process critical frequency ω

c

i.e. the frequency at which the phase of a process frequency response crosses −180°. This critical frequency and the process frequency response at this point are useful for control analysis and design. For traditional frequency response identification, the frequencies of the exciting signals u fed into the process/system

10

should be carefully selected based on the process bandwidth. The use of relay feedback can automatically excite an unknown stable process around this frequency ω

c

. Such a method is described in U.S. Pat. No. 4,549,123 by Hagglund and Astrom. Furthermore, traditional frequency response identification usually involves an open loop test, while the use of relay feedback involves a closed-loop controller. A closed-loop test is preferred to an open-loop test in control applications, since it keeps the process close to the set point so that the process operates in a linear region where the frequency response is of interest.

Relay based process identification and control tuning have received a great deal of attention. The method of Astrom and Hagglund is useful in many process control applications, but it also faces two major problems. First, due to the use of a describing function approximation, the estimation of the critical point is not accurate enough for some kinds of processes. Second, the method of Astrom and Hagglund depends on the single critical point ω

c

, and only crude controller settings can be obtained based on this single point. Attempts have been made to solve this problem, including several modified identification methods using relay-based feedback control systems in order to identify more than one point on the process frequency response. In order to accomplish this, additional linear components (or varying hysteresis width) have been connected into the system requiring additional relay tests to be performed. These methods are time consuming and the resultant estimations are still approximate in nature since they rely on repeated use of the standard method as described in Astrom and Hagglund.

It is therefore clear that a method is needed whereby more points on the process frequency response can be accurately identified from a single relay test.

SUMMARY OF THE INVENTION

It is therefore an object of the present invention to provide an improved apparatus for process frequency response estimation and controller tuning.

It is a further object of the present invention to provide an apparatus for process frequency response estimation and controller tuning for accurately identifying multiple points on the process frequency response with a single relay test.

It is a still further object of the present invention to provide an apparatus for process frequency response estimation and controller tuning for accurately identifying multiple points on the process frequency response with a single relay test, whereby the controller regulator is tuned with respect to the identified points on the process frequency response, whereby the controller system achieves an improved, uniform response.

Briefly, a preferred embodiment of the present invention includes an apparatus for tuning a regulator in a process controller. The digitized output from a standard relay and a parasitic relay are fed through a digital to analog converter providing signals at 0.5 ω

c

, ω

c

and 1.5 ω

c

, to the process, where ω

c

is the process critical frequency. The apparatus records the digitized relay output u′ and the digitized process output y′ until stationary oscillations are reached. The outputs u′ and y′ are then used to calculate the process frequency response at 0.5 ω

c

, ω

c

and 1.5 ω

c

. This data is then used to design the regulator, the output of which is converted to analog form and inputted to control the process.

An advantage of the auto-tuner of the present invention is that it can estimate multiple points on a process frequency response simultaneously with one single relay test, resulting in time savings.

A further advantage of the present invention is that it gives accurate results since no approximations are made, and the computations involved are simple so that it can be easily implemented on microprocessors.

A still further advantage of the present invention is that the method is insensitive to noise and step-like load disturbances, and non-zero initial conditions.

Another advantage of the present invention is that the controller tuning methods employed work well for the general class of linear processes with different dynamics, achieving consistent satisfactory responses.

IN THE DRAWING

FIG. 1

shows a prior art process control system;

FIG. 2

is a block diagram of the preferred embodiment of the present invention;

FIG. 3A

shows the digitized modified relay output for the process frequency response estimation relay test

FIG. 3B

shows the process output during the relay test;

FIG. 4

illustrates a relay with hysteresis for noise reduction;

FIG. 5

is a Nyquist plot;

FIG. 6

shows a regulator using an IMC systems configuration; and

FIG. 7

shows the results of PID and IMC regulator designs.

DESCRIPTION OF THE PREFERRED EMBODIMENT

Referring now to

FIG. 2

of the drawing, a preferred embodiment of the system of the present invention

18

is shown to include an auto-tuner

20

which is responsive to a digitized reference signal r at input

22

and a digitized process output signal y′ at input

24

to provide signals u′ at output

26

.

The digitized signals u′ are converted to analog signals u by digital to analog converter

28

and outputted at

30

to the process

32

. The output y at

34

of the system

18

is shown schematically to include the sum of the output at

36

of process

32

, and noises n and disturbances w, the addition schematically illustrated through use of summing networks

38

and

40

. The output y is converted by analog to digital converter

42

to provide the digitized input y′ at

24

.

The functions performed by the auto-tuner

20

are schematically illustrated inside the auto-tuner

20

block.

The objective of the auto-tuner

20

is to automatically identify i.e., estimate the frequency response/dynamics of the process

32

, and then to automatically design/configure a regulator to control the output y of the process

32

according to the applied reference signal r.

A more detailed description of the functions performed by the auto-tuner

20

will now be given in reference to the circuitry illustrated in

FIG. 2

inside the auto-tuner

20

block. In order to identify the frequency response of the process

32

, the switch

44

is set to connect the output

46

of summing network

48

to the input

26

of the digital to analog converter

28

. In this mode, the signal output at

46

, provided by the relays

50

and

52

, controls the process

32

. The digitized process input u′ at

26

and digitized process output y′ at

24

are then recorded by conventional circuitry (not shown) until stationary oscillations are reached i.e., until there are no variations in amplitude or period of oscillations in the output y. The output at

46

includes process excitation frequencies including 0.5 ω

c

, ω

c

and 1.5 ω

c

, and in response, the output y of the process also contains these frequencies. When the stationary oscillation condition is reached, the recorded/saved u′ and y′ are then used to compute the process

32

frequency response data at the frequencies 0.5 ω, ω

c

and 1.5 ω

c

using equations 2 and 3, which will now be described in detail. The auto-tuner

20

circuitry then uses the calculated process frequency response data to design the regulator

54

. When the regulator

54

is determined, the switch

44

is activated to connect the regulator output

56

to the input

26

of the digital to analog converter

28

. At this point the regulator

54

responds to the regulator input e at

57

, e being the difference between the digital reference r and digitized process output y′ by applying a signal u′, which converted by digital to analog converter

28

, provides the signal u to the processor to drive the processor to output ay at

34

corresponding to the reference r.

The procedure described above and with the following text as performed by the auto-tuner

20

can be implemented in a computer, the inputs being the digital reference r and the digitized process output y′, and the output being the digital control signal u′. In addition to the specific blocks noted within block

20

, block

20

also represents the digital apparatus responsive to r and y′ for calculating the process frequency response, and for calculating the required regulator parameters and setting the regulator as required. Such operations performed according to the equations disclosed will be understood by those skilled in the art.

As discussed above,

FIG. 2

is an arrangement of apparatus for automatically estimating a process frequency response and designing a process regulator. The details of the process frequency estimating apparatus will now be described in detail.

Assume that initially, process

32

runs in open loop. Then a standard relay

50

and the novel parasitic relay

52

of the present invention are applied to process

32

. This results in input and output time responses u(t) and y(t) of process

32

. u(t) and y(t) are sampled at a sampling period of T and are recorded until stationary limit cycles of u(t) and y(t) have been reached.

The key of the present invention is the modified relay which consists of the standard relay

50

and parasitic relay

52

, as shown in FIG.

2

. The standard relay operates as usual with the amplitude of the sampled output u

1

(k) being h, where u

1

(k) is the k-th sample of u

1

(t). It is well known that the standard relay can excite process

32

mainly at frequency ω

c

. In order to provide excitation of the process

32

at frequencies other than ω

c

, for use in control while maintaining the process output oscillation under such an arrangement, a parasitic relay

52

with output amplitude αh and twice the period of u

1

(k) is introduced and superimposed on u

1

(k). This implies that the output u

2

(k) of the parasitic relay

52

flip-flops once every period of oscillation in u

1

(k). The parasitic relay

52

output u

2

(k) is described by the following equations,

\begin{matrix} {\begin{matrix} u_{2} (0) = α h; \\ u_{2} (k) = - α h \cdot sign (u_{2} (k - 1)), if u_{1} (k - 1) > 0 and u_{1} (k) < 0; \\ u_{2} (k) = u_{2} (k - 1), otherwise; \end{matrix} & (1) \end{matrix}

wherein the term “sign” is defined by

y=sign(x)=1 for x≧0

y=sign(x)=−1 for x<0

where α is a constant coefficient. α should be large enough to have sufficient stimulation on the process

32

and it should also be small enough such that the parasitic relay

52

will not change the period of oscillation generated by the main relay

50

too much. The preferred range of values for α is from 0.1˜0.3. The output at

46

of the modified relay apparatus is thus given by u(k)=u

1

(k)+u

2

(k), and is sent as the input to process

32

.

In this way, the process is stimulated by two different excitations whose periods are T

c

and 2T

c

respectively. The digital input u′ at

26

resulting from the modified relay apparatus is shown in FIG.

3

A. The output y at

34

from the process

32

will appear as shown in

FIG. 3B

, and will reach a stationary oscillation with the period being 2T

c

. Due to two excitations in u, y consists of frequency components at

\frac{2 π}{T_{c}}, \frac{π}{T_{c}}

and their odd harmonics

\frac{6 π}{T_{c}}, \frac{10 π}{T_{c}}, \dots and \frac{3 π}{T_{c}}, \frac{5 π}{T_{c}}, \dots respectively .

For linear process

32

, the process frequency response can be obtained by

\begin{matrix} G ({jω}_{i}) = \frac{\int_{0}^{2 T_{c}} y_{s} (t) ⅇ^{- {jω}_{i} t} ⅆ t}{\int_{0}^{2 T_{c}} u_{s} (t) ⅇ^{- {jω}_{i} t} ⅆ t}, i = 1, 2, \dots, & (2) \end{matrix}

where

ω_{i} = \frac{(2 i - 1) 2 π}{2^{l} T_{c}}, l = 0, 1,

are the basic frequency and its odd harmonic frequencies in u

s

and y

s

. u

s

and y

s

have a period (2T

c

) related to the stationary oscillations of u(k) and y(k) respectively. G(jω

i

) in (2) can be computed using the FFT (fast fourier transform) algorithm as

\begin{matrix} G ({jω}_{i}) = \frac{FFT (y_{s})}{FFT (u_{s})} & (3) \end{matrix}

The spectrum analysis method used by the auto-tuner

20

in estimating the process frequency response gives a more accurate result than methods relying on the describing function. The method of the present invention employs the FFT only once and the required computation burden is modest. It can identify multiple points on the process frequency response from a single test using the modified relay of the present invention. Moreover, the method can be easily extended to find additional points on the frequency response. One can flip-flop the parasitic relay every 3

rd

or 4

th

period of the main oscillations generated by the standard relay and get other frequency points. One can also use more than one parasitic relay in a relay test and find more points on the frequency response in one relay test. To estimate the static gain of a process, either a small set-point change at reference may be introduced or a bias may be added to the modified relay.

In a realistic environment, the major concerns for any identification method are disturbance and noise. It should be noted that the identification method is unaffected by the step-like load disturbance w referred to in

FIG. 2

, which is a common case in practice. This can easily be shown from (2) as

\begin{matrix} \begin{matrix} \frac{\int_{0}^{2 T_{c}} \tilde{y} s (t) ⅇ^{- {jω}_{i} t} ⅆ t}{\int_{0}^{2 T_{c}} {\tilde{u}}_{s} (t) ⅇ^{- {jω}_{i} t} ⅆ t} = \frac{\int_{0}^{2 T_{c}} [y_{s} (t) + w (t)] ⅇ^{- {jω}_{i} t} ⅆ t}{\int_{0}^{2 T_{c}} (u_{s} (t) + \overline{u}) ⅇ^{- {jω}_{i} t} ⅆ t} \\ = \frac{\int_{0}^{2 T_{c}} y_{s} (t) ⅇ^{- {jω}_{i} t} ⅆ t + w}{\int_{0}^{2 T_{c}} u_{s} (t) ⅇ^{- {jω}_{i} t} ⅆ t + \overline{u}} \frac{\int_{0}^{2 T_{c}} ⅇ^{- {jω}_{i} t} ⅆ t}{\int_{0}^{2 T_{c}} ⅇ^{- {jω}_{i} t} ⅆ t} \\ = \frac{\int_{0}^{2 T_{c}} y_{s} (t) ⅇ^{- {jω}_{i} t} ⅆ t}{\int_{0}^{2 T_{c}} u_{s} (t) ⅇ^{- {jω}_{i} t} ⅆ t} = G ({jω}_{i}), i = 1, 2, \dots \end{matrix} & (4) \end{matrix}

where ω

i

has the same definition as in (2). The modified relay estimating apparatus of the auto-tuner

20

of the present invention is also insensitive to noise. A preferred method of reducing the noise in the process output y(t) is by introducing a hysteresis in the standard relay

50

. This is illustrated in FIG.

4

. The width of hysteresis should be bigger than the noise band and is usually chosen to be 2 times larger than the noise band. Filtering is another possibility. To reduce the effect of noise further, especially in the case of large noise-to-signal ratios, the estimator uses the average of the last 2-4 periods of oscillation as the stationary oscillation period, depending on the noise level. With these anti-noise measures, the method of the present invention can reject noise very effectively, and provide accurate frequency response estimation at frequencies of 0.5 ω

c

, ω

c

and 1.5 ω

c

. It should be also noted that a non-zero initial condition of the process at the start of a relay test has no effect on the estimation because only stationary oscillations u

s

and y

s

after transient are used in the estimation, where u

s

and y

s

are independent of the initial condition.

EXAMPLE.

In order to test the method in a realistic environment, extensive real-time relay testing was performed using the Dual Process Simulator KI 100 from KentRidge Instruments, Singapore. For illustration, a process was configured on the Simulator as

G (s) = \frac{1}{5 s + 1} ⅇ^{- 5 s}

In the test, the standard relay amplitude was chosen as 0.5 and the parasitic relay height was set to 20%×0.5. Without additional noise, the noise-to-signal ratio of the inherent noise in the test environment was measured at N

1

=0.025%, where N

1

is defined as

N_{1} = \frac{mean power spectrum density of noise}{mean power spectrum density of signal}

or as N

2

=4%, where the noise-to-signal ratio N

2

is

N_{2} = \frac{mean (abs (noise))}{mean (abs (signal))}

The identification error ERR is 2.5%, where

ERR = \max_{i} {&LeftBracketingBar; \frac{\hat{G} ({jω}_{i}) - G ({jω}_{i})}{G ({jω}_{i})} &RightBracketingBar; \times 100 %, i = 1, 2, 3}

in which G(jω

i

) and Ĝ(jω

i

) are the actual and the estimated process frequency responses respectively. To see noise effects, extra noise was introduced with the noise source in the Simulator. Time sequences of y(t) and u(t) in a relay test under N

1

=10% (N

2

=31%) are shown in

FIGS. 3A and 3B

. The first part of the test in

FIGS. 3A and 3B

(t=0˜12) was the “listening period”, in which the noise bands of y(t) and u(t) at the steady state were measured. Under this noise, the hysteresis was chosen as 0.3. With averaging 4 periods of stationary oscillations, the estimated frequency response points under this noise level are shown in FIG.

5

.

Upon completion of the process frequency response estimation, the auto-tuner uses the estimated data to automatically design the regulator

54

. This process will now be described in detail.

The three estimated points of frequency response are used for tuning the regulator

54

in FIG.

2

. The three points are first converted into a second order plus dead time model with the following structure

\begin{matrix} \hat{G} (s) = \frac{ⅇ^{- sL}}{{as}^{2} + bs + c}, & (5) \end{matrix}

where a, b, c and L are unknowns to be determined. It follows that

\begin{matrix} \hat{G} ({jω}_{i}) = \frac{1}{{({jω}_{i})}^{2} a + {jω}_{i} b + c} ⅇ^{- {jω}_{i} L}, i = 1, 2, 3. & (6) \end{matrix}

The magnitude of both sides of (6) is taken as

\begin{matrix} [ω_{i}^{4} ω_{i}^{2} 1] θ = \frac{1}{{&LeftBracketingBar; G ({jω}_{i}) &RightBracketingBar;}^{2}}, i = 1, 2, 3. & (7) \end{matrix}

where

θ=[α

2

b

2

−2ac c

2

]

τ

.

Then,

[a b c] = [\sqrt{θ_{1}} \sqrt{θ_{2} + 2 \sqrt{θ_{1} θ_{3}}} \sqrt{θ_{3}}]

In addition, the phase relation in (6) gives

\begin{matrix} ω_{i} L = - \arg [G ({jω}_{i}^{*})] - \tan^{- 1} (\frac{b ω_{i}}{c - a ω_{i}^{2}}), i = 1, 2, 3. & (8) \end{matrix}

It will be clear to those skilled in the art that L can be obtained with the least squares method.

With the model in (5) for the process available, a controller can be designed. A PID controller can be written in the form:

\begin{matrix} K (s) = k (\frac{{As}^{2} + B + C}{s}), & (9) \end{matrix}

where

\begin{matrix} A = \frac{K_{D}}{k}, B = \frac{K_{P}}{k}, C = \frac{K_{I}}{k} & (10) \end{matrix}

The controller zeros are chosen to cancel the model poles, i.e. A=a, B=b, C=c, where a, b and c are the model parameters determined from (7) and (8). Then the resultant open-loop transfer function G(s) K(s) is approximated by

\begin{matrix} \hat{G} (s) k (s) = \frac{k ⅇ^{- sL}}{s} . & (10) \end{matrix}

Its closed-loop poles can be selected from the root locus of the loop by assigning a proper value for k.

For the transfer function in (10), real or complex roots may be obtained for the closed-loop. In the method of the present invention, the controller zeros have been chosen to cancel the model poles. Exact cancellation is never expected since the process can be of any order while the model is only of second order. For highly oscillatory processes, it is possible that the uncanceled dynamics will drive the system to heavy oscillation, and hence it is reasonable not to create additional oscillatory dynamics by having complex close-loop poles. Real closed-loop poles are chosen for the system instead. On the other hand, for non-oscillatory or lightly oscillatory processes, the uncanceled dynamics will not bring the system to severe oscillation, and hence it is advisable to introduce some overshoot by selecting complex closed-loop poles so as to speed up the response.

For ease of presentation, some terms will now be defined that will be used in the following description. The equivalent time constant τ

o

of a process is inversely proportional to its speed of response. For monotonic processes, the speed of response is reflected by the locations of the dominant poles. For oscillatory ones, it is related to the real part of the complex poles which determines the system attenuation and hence serves as a measure of the process speed. According to equivalent time principles, we have

\begin{matrix} \frac{1}{τ_{o}} = {\begin{matrix} \frac{c}{\sqrt{b^{2} - 2 a c}}, & b^{2} - 4 a c \geq 0; \\ \frac{b}{2 a} & b^{2} - 4 a c < 0 \end{matrix} & (11) \end{matrix}

where a, b and c are model parameters that can be obtained from (7). Another variable of interest to the design method is the damping ratio ζ

o

of the open-loop plant which is defined as

\begin{matrix} Ϛ_{o} = {\begin{matrix} \frac{b}{2 \sqrt{a c}}, & b^{2} - 4 a c < 0; \\ 1, & b^{2} - 4 a c \geq 0. \end{matrix} & (12) \end{matrix}

with a, b and c being the model parameters. The solution will now be presented by separating it into three cases.

Case I. ζ>0.7071 or

\frac{L}{τ_{o}} < 0.15 or \frac{L}{τ_{o}} > 1.

Complex closed-loop poles on the root locus are chosen based on the reasoning given earlier. In order for a pair of the desired poles

s=−ω

n

ç

n

±jω

n

{square root over (1−ç

n

2

+L )}

where ζ

n

is the closed-loop damping ratio, to be on the root locus for the system in (10), the phase condition

−ω

n

{square root over (1−ζ

n

2

+L )}L−(π−cos

−1

ζ

n

)=−π (13)

has to be satisfied, giving

\begin{matrix} ω_{n} = \frac{\cos^{- 1} Ϛ_{n}}{L \sqrt{I - Ϛ_{n}^{2}}} . & (14) \end{matrix}

The magnitude condition then assigns the value of k to

k=ω

n

e

−ω

n

Lζ

n

. (15)

Simulation results show that the feedback system gives satisfactory responses if closed-loop poles of damping ratio ζ

n

=0.7071 are chosen. Substituting ζ

n

=0.7071 into eq. (14) and (15) results in

\begin{matrix} k = \frac{0.5064}{L} & (16) \end{matrix}

Case II. ζ

o

≦0.7071 or

0.15 \leq \frac{L}{τ_{o}} \leq 1.

In this case, we choose the closed-loop poles as real double poles on the root-locus. Its location is selected to have a similar response speed to that of the open-loop one if it is achievable, i.e. it is before the breakaway point. Otherwise the breakaway point will be used. In order for

\begin{matrix} s_{1, 2} = - \frac{l}{τ_{o}} \\ to be on the root locus, \\ k = \frac{l}{τ_{o}} ⅇ^{- \frac{L}{τ_{o}}} . & (17) \end{matrix}

To find the breakaway point, it is noted from eq. (14) that

\begin{matrix} \lim_{Ϛ_{n} \to 1} ω_{n} = \lim_{Ϛ \to I} (\frac{\cos^{- 1} Ϛ_{n}}{L \sqrt{1 - Ϛ_{n}^{2}}}) = \frac{1}{L} & (18) \end{matrix}

Hence, the breakaway point is

s_{1, 2} = - ω_{n} Ϛ_{n} \pm {jω}_{n} \sqrt{1 - Ϛ_{n}^{2}} = - \frac{1}{L}

and the corresponding gain is

\begin{matrix} k_{break} = - s ⅇ^{sL} |_{s = - \frac{1}{L}} = \frac{1}{eL} & (19) \end{matrix}

Therefore, the value of k should be taken as

\begin{matrix} k = \min {\frac{l}{τ_{o}} ⅇ^{- \frac{L}{τ_{o}}}, \frac{l}{ⅇ L}} & (20) \end{matrix}

Case III.

\frac{L}{τ_{o}} > 2.

Processes with long dead time tend to produce very slow responses. In fact, the single-loop structure is inadequate in handling such processes. One possible solution is to apply the Internal Model Control (IMC) strategy to compensate for the long dead time. The controller structure is shown in FIG.

6

. The model of the process is denoted as Ĝ(s) in the figure, and

\hat{G} (s) = \frac{ⅇ^{- sL}}{{as}^{2} + bs + c}

where the parameters a, b ,c and L are given in (7)-(8). The controller is now modified to

\begin{matrix} K (s) = \frac{{as}^{2} + bs + c}{{(\frac{1}{α} s τ_{o} + 1)}^{2}} & (21) \end{matrix}

where τ

o

is the equivalent time constant defined in (11) and α is a design parameter to adjust the speed of response. The larger the value of α, the faster the response becomes. The typical range of α is αε[0.5,5]. If the model is exact, the system will have an equivalent transfer function of

\begin{matrix} H_{y r} (s) = \frac{1}{{(\frac{1}{α} s τ_{o} + 1)}^{2}} ⅇ^{- sL} & (22) \end{matrix}

The dead time is therefore brought out of the loop, and the speed can be chosen to be as fast as desired by selecting a large a value. Since this inevitably increases the size of the control effort, the limit is set by the saturation of the control signal and the accuracy of the model.

When the regulator

54

in

FIG. 2

is designed,the switch

44

is connected to the lower node

56

, and the auto-tuning is completed.

EXAMPLE

Consider the following long dead time process given by the transfer function

G (s) = \frac{1}{(s^{2} + 2 s + 3) (s + 3)} ⅇ^{- 2 s}

The model of the process is found to be

\hat{G} (s) = \frac{1}{2.921 s^{2} + 6.536 s + 9.031} ⅇ^{- 2.289 s}

The PID controller is designed to be

K (s) = 1.446 + \frac{1.998}{s} + 0.646 s .

Since the process is dead time dominated, the IMC approach can be employed to speed up the response. Choosing the value of α=2 in eq. (21), the controller is computed to be

K (s) = \frac{2.921 s^{2} + 6.536 s + 9.031}{{(\frac{1}{2} \times 0.894 s + 1)}^{2}}

The control performance of the IMC and PID controllers is also shown in FIG.

7

. The performance of the IMC controller is an improvement over the PID controller. The speed is much increased, and the settling time and overshoot are significantly reduced.

Although a preferred embodiment of the present invention has been described above, it will be appreciated that certain alterations and modifications thereof will be apparent to those skilled in the art. It is therefore intended that the appended claims be interpreted as covering all such alterations and modifications as fall within the true spirit and scope of the invention.

Number	Name	Date	Kind
4549123	Hagglund et al.	Oct 1985	A
5283729	Lloyd	Feb 1994	A
5453925	Wojsznis et al.	Sep 1995	A
5537310	Tanake et al.	Jul 1996	A
5587899	Ho et al.	Dec 1996	A
5691896	Zou et al.	Nov 1997	A

Apparatus for relay based multiple point process frequency response estimation and control tuning

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