1. Introduction

74 Introduction to PID Controllers – Theory, Tuning and Application to Frontier Areas

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Today, most tuning rules for PID controllers are based either on the process step response or else on relay-excitation experiments. Tuning methods based on the process step response are usually based on the estimated process gain and process lag and rise times (Åström & Hägglund, 1995). The relay-excitation method is keeping the process in the closed-loop configuration during experiment by using the on/off (relay) controller. The measured data is the amplitude of input and output signals and the oscillation period.

The experiments mentioned are popular in practice due to their simplicity. Namely, it is easy to perform them and get the required data either from manual or from automatic experiments on the process. However, the reduction of process time-response measurement into two or three parameters may lead to improperly tuned controller parameters.

Therefore, more sophisticated tuning approaches have been suggested. They are usually based on more demanding process identification methods (Åström et al., 1998; Gorez, 1997; Huba, 2006). One such method is a magnitude optimum method (MO) (Whiteley, 1946). The MO method results in a very good closed-loop response for a large class of process models frequently encountered in the process and chemical industries (Vrančić, 1995; Vrančić et al., 1999). However, the method is very demanding since it requires a reliable estimation of quite a large number of process parameters, even for relatively simple controller structures (like a PID controller). This is one of the main reasons why the method is not frequently used in practice.

Recently, the applicability of the MO method has been improved by using the concept of 'moments', which originated in identification theory (Ba Hli, 1954; Strejc, 1960; Rake, 1987). In particular, the process can be parameterised by subsequent (multiple) integrals of its input and output time-responses. Instead of using an explicit process model, the new tuning method employs the mentioned multiple integrals for the calculation of the PID controller parameters and is, therefore, called the "Magnitude Optimum Multiple Integration" (MOMI) tuning method (Vrančić, 1995; Vrančić et al., 1999). The proposed approach therefore uses information from a relatively simple experiment in a time-domain while retaining all the advantages of the MO method.

The deficiency of the MO (and consequently of the MOMI) tuning method is that it is designed for optimising tracking performance. This can lead to the poor attenuation of load disturbances (Åström & Hägglund, 1995). Disturbance rejection performance is particularly

Magnitude Optimum Techniques for PID Controllers 77

forward and the feedback paths are generally different, the PID controller (2) is a twodegrees-of-freedom (2-DOF) controller. Note that controller (2) becomes a 1-DOF

The PID controller in a closed-loop configuration with the process is shown in Figure 1.

+

+

Signals e, d and ur denote the control error, disturbance and process input, respectively. The

Ys G sG s

Ys G sG s

The deficiency of 1-DOF controllers is that they usually cannot achieve optimal tracking and disturbance rejection performance simultaneously. 2-DOF controllers may achieve better overall performance by keeping the optimal disturbance rejection performance while

One possible means of control system design is to ensure that the process output (y) follows the reference (r). The ideal case is that of perfect tracking without delay (y=r). In the frequency domain, the closed-loop system should have an infinite bandwidth and zero phase shift. However, this is not possible in practice, since every system features some time

delay and dynamics while the controller gain is limited due to physical restrictions.

+ +

<sup>F</sup> 1+ sT 1

( ) ( ) 1 () () R P

Rs G sG s = = <sup>+</sup> . (4)

Rs G sG s = = <sup>+</sup> . (5)

C P

( ) ( ) 1 () () C P

C P

PID controller

d

<sup>+</sup> <sup>u</sup>

u y <sup>r</sup>

GP(s) process

controller when choosing b=c=1.

r e - +

improving tracking performance.

3. Magnitude Optimum (MO) criteria

c

b K<sup>P</sup>

Fig. 1. The closed-loop system with the PID controller



s KI

closed-loop transfer function with the PID controller is defined as follows:

CL

CL

G s

G s

( ) ( ) ( )

( ) ( ) ( )

For the 1-DOF PID controller (b=c=1), the closed-loop transfer function becomes:

<sup>D</sup>sK

decreased for lower-order processes. This is one of the most serious disadvantages of the MO method, since in process control disturbance rejection performance is often more important than tracking performance.

The mentioned deficiency has been recently solved by modifying the original MO criteria (Vrančić et al., 2004b; Vrančić et al., 2010). The modified criteria successfully optimised the disturbance rejection response instead of the tracking response. Hence, the concept of moments (multiple integrations) has been applied to the modified MO criteria as well, and the new tuning method has been called the "Disturbance Rejection Magnitude Optimum" (DRMO) method (Vrančić et al., 2004b; Vrančić et al., 2010).

The MOMI and DRMO tuning methods are not only limited to the self-regulating processes. They can also be applied to integrating processes (Vrančić, 2008) and to unstable processes (Vrančić & Huba, 2011). The methods can also be applied to different controller structures, such as Smith predictors (Vrečko et al., 2001) and multivariable controllers (Vrančić et al., 2001b). However, due to the limited space and scope of this book, they will not be considered further.

#### 2. System description

A stable process may be described by the following process transfer function:

$$\mathcal{G}\_P(\mathbf{s}) = K\_{PR} \frac{1 + b\_1 \mathbf{s} + b\_2 \mathbf{s}^2 + \dots + b\_m \mathbf{s}^m}{1 + a\_1 \mathbf{s} + a\_2 \mathbf{s}^2 + \dots + a\_n \mathbf{s}^n} \mathbf{e}^{-sT\_{delay}},\tag{1}$$

where KPR denotes the process steady-state gain, and a1 to an and b1 to bm are the corresponding parameters (m≤n) of the process transfer function, whereby n can be an arbitrary positive integer value and Tdelay represents the process pure time delay. Note that the denominator in (1) contains only stable poles.

The PID controller is defined as follows:

$$\mathcal{L}\mathcal{L}(s) = \mathcal{G}\_R(s)\mathcal{R}(s) - \mathcal{G}\_\mathcal{C}(s)\mathcal{Y}(s) \tag{2}$$

where U, R and Y denote the Laplace transforms of the controller output, the reference and the process output, respectively. The transfer functions GR(s) and GC(s) are the feed-forward and the feedback controller paths, respectively:

$$\begin{aligned} G\_R(s) &= \frac{K\_I + bK\_P s + cK\_D s^2}{s\left(1 + sT\_F\right)}\\ G\_C(s) &= \frac{K\_I + K\_P s + K\_D s^2}{s\left(1 + sT\_F\right)}\end{aligned} \tag{3}$$

The PID controller parameters are proportional gain KP, integral gain KI, derivative gain KD, filter time constant TF, proportional reference weighting factor b and derivative reference weighting factor c (Åström & Hägglund, 1995). Note that the first-order filter is applied to all three controller terms instead of only the D term in order to reduce noise amplitude at the controller output and to simplify the derivation of the PID controller parameters. The range of parameters b and c is usually between 0 and 1. Since the feed-

decreased for lower-order processes. This is one of the most serious disadvantages of the MO method, since in process control disturbance rejection performance is often more

The mentioned deficiency has been recently solved by modifying the original MO criteria (Vrančić et al., 2004b; Vrančić et al., 2010). The modified criteria successfully optimised the disturbance rejection response instead of the tracking response. Hence, the concept of moments (multiple integrations) has been applied to the modified MO criteria as well, and the new tuning method has been called the "Disturbance Rejection Magnitude Optimum"

The MOMI and DRMO tuning methods are not only limited to the self-regulating processes. They can also be applied to integrating processes (Vrančić, 2008) and to unstable processes (Vrančić & Huba, 2011). The methods can also be applied to different controller structures, such as Smith predictors (Vrečko et al., 2001) and multivariable controllers (Vrančić et al., 2001b). However, due to the limited space and scope of this book, they will not be

> 2 1 2

2 1 2

where KPR denotes the process steady-state gain, and a1 to an and b1 to bm are the corresponding parameters (m≤n) of the process transfer function, whereby n can be an arbitrary positive integer value and Tdelay represents the process pure time delay. Note that

where U, R and Y denote the Laplace transforms of the controller output, the reference and the process output, respectively. The transfer functions GR(s) and GC(s) are the feed-forward

( ) ( )

+ + <sup>=</sup> <sup>+</sup>

K bK s cK s G s s sT

R

C

1

IP D

IP D

( ) ( )

+ + <sup>=</sup> <sup>+</sup>

K Ks Ks G s s sT

1

The PID controller parameters are proportional gain KP, integral gain KI, derivative gain KD, filter time constant TF, proportional reference weighting factor b and derivative reference weighting factor c (Åström & Hägglund, 1995). Note that the first-order filter is applied to all three controller terms instead of only the D term in order to reduce noise amplitude at the controller output and to simplify the derivation of the PID controller parameters. The range of parameters b and c is usually between 0 and 1. Since the feed-

as as as + + ++ <sup>−</sup> <sup>=</sup> + + ++

⋯

delay

<sup>⋯</sup> , (1)

. (3)

<sup>m</sup> <sup>m</sup> sT

U s G sRs G sY s ( ) = − R C ( ) ( ) ( ) ( ) , (2)

2

2

F

F

A stable process may be described by the following process transfer function:

1 1

P PR <sup>n</sup> <sup>n</sup> bs bs b s Gs K <sup>e</sup>

important than tracking performance.

considered further.

2. System description

(DRMO) method (Vrančić et al., 2004b; Vrančić et al., 2010).

( )

the denominator in (1) contains only stable poles.

and the feedback controller paths, respectively:

The PID controller is defined as follows:

forward and the feedback paths are generally different, the PID controller (2) is a twodegrees-of-freedom (2-DOF) controller. Note that controller (2) becomes a 1-DOF controller when choosing b=c=1.

The PID controller in a closed-loop configuration with the process is shown in Figure 1.

Fig. 1. The closed-loop system with the PID controller

Signals e, d and ur denote the control error, disturbance and process input, respectively. The closed-loop transfer function with the PID controller is defined as follows:

$$G\_{\mathbb{C}\mathcal{L}}\left(s\right) = \frac{\mathcal{Y}\left(s\right)}{\mathcal{R}\left(s\right)} = \frac{G\_{\mathcal{R}}\left(s\right)G\_{\mathcal{P}}\left(s\right)}{1 + G\_{\mathcal{C}}\left(s\right)G\_{\mathcal{P}}\left(s\right)}.\tag{4}$$

For the 1-DOF PID controller (b=c=1), the closed-loop transfer function becomes:

$$G\_{\rm CL} \left( s \right) = \frac{Y(s)}{R \left( s \right)} = \frac{G\_{\rm C} \left( s \right) G\_{P} \left( s \right)}{1 + G\_{\rm C} \left( s \right) G\_{P} \left( s \right)}. \tag{5}$$

The deficiency of 1-DOF controllers is that they usually cannot achieve optimal tracking and disturbance rejection performance simultaneously. 2-DOF controllers may achieve better overall performance by keeping the optimal disturbance rejection performance while improving tracking performance.

#### 3. Magnitude Optimum (MO) criteria

One possible means of control system design is to ensure that the process output (y) follows the reference (r). The ideal case is that of perfect tracking without delay (y=r). In the frequency domain, the closed-loop system should have an infinite bandwidth and zero phase shift. However, this is not possible in practice, since every system features some time delay and dynamics while the controller gain is limited due to physical restrictions.

Magnitude Optimum Techniques for PID Controllers 79

2 2 20 2 0

f f e ee f n <sup>+</sup> − −

Before calculating the parameters of the 1-DOF PID controller, according to the given MO criteria, the pure time delay in expression (1) has to be developed into an infinite Taylor

sT delay delay delay

1 1 1 <sup>2</sup> <sup>2</sup> 2 2! 2 ! lim

k k <sup>n</sup> delay delay delay delay

Then, the closed-loop transfer function (5) is calculated from expressions (1), (3) and (10) or else (11). The closed-loop parameters ei and fi can be obtained by comparing expressions (8) and (5). The PID controller parameters are then obtained by solving the first three equations

The expressions (12)-(14) are not explicitly given herein, since they would cover several pages. In order to calculate the three PID controller parameters – according to the given MO tuning criteria – only the parameters KPR, a1, a2, a3, a4, a5, b1, b2, b3, b4, b5, and Tdelay of the process transfer function (1) are required, even though the process transfer function can be of a higher-order. However, accurately estimating such a high number of process parameters from real measurements could be very problematic. Moreover, if one identifies the fifth-order process model from the actually higher-than-fifth-order process, a systematic error in the estimated process parameters would be obtained, therefore leading to the calculation of non-optimal controller parameters. Accordingly, the accuracy of the estimated

Note that the actual expressions (12)-(14) remain exactly the same when the process with pure time-delay is developed into a Taylor (10) or Padé (11) series (Vrančić et al., 1999).

The problems with original MO tuning method just mentioned can be avoided by using the concept of 'moments', known from identification theory (Ba Hli, 1954; Preuss, 1991).

4. Magnitude Optimum Multiple Integration (MOMI) tuning method

k sT

<sup>n</sup> <sup>k</sup> <sup>e</sup> sT sT s T s T

<sup>1</sup> <sup>1</sup> <sup>2</sup> <sup>2</sup> 2 2! 2 !

<sup>+</sup> + + ++ +

n k

sT sT s T s T

<sup>−</sup> − + − +− +

1 0; 1,2,

∑ − − == … (9)

k

. (11)

k

( )

k

… ⋯

… ⋯

( ) ( ) ( ) ( ) 2 3 <sup>k</sup> <sup>1</sup>

sT sT sT

2! 3! !

n k k delay delay delay k delay

K f K aa abb bT T P PR <sup>=</sup> 1 12 512 5 ( ,,,,,,,,, , … … delay F ) (12)

K f K aa abb bT T I PR <sup>=</sup> 2 12 512 5 ( ,,,,,,,,, , … … delay F ) (13)

K f K aa abb bT T D PR <sup>=</sup> 3 12 512 5 ( ,,,,,,,,, , … … delay F ) (14)

<sup>−</sup> <sup>−</sup> =− + − + + ⋯ ⋯+ (10)

( ) ( )

i ni i ni

2

i

1

→∞

delay

delay

−

e sT

series:

or Padé series:

=

0

<sup>n</sup> i n

delay

= =

(n=1, 2 and 3) in expression (9) (Vrančić et al., 1999):

process parameters in practice remains questionable.

The new design objective would be to maintain the closed-loop magnitude (amplitude) frequency response (GCL) from the reference to the process output as flat and as close to unity as possible for a large bandwidth (see Figure 2) (Whiteley, 1946; Hanus, 1975; Åström & Hägglund, 1995; Umland & Safiuddin, 1990). Therefore, the idea is to find a controller that makes the frequency response of the closed-loop amplitude as close as possible to unity for lower frequencies.

Fig. 2. The amplitude (magnitude) frequency response of the closed-loop system These requirements can be expressed in the following way:

$$G\_{CL} \begin{pmatrix} 0 \end{pmatrix} = 1 \tag{6}$$

$$\left. \frac{d^{2k} \left| \mathbf{G}\_{\rm CL}(j\rho) \right|^{2}}{d\rho^{2k}} \right|\_{\rho=0} = 0 \; ; \; k = 1, 2, \cdots, k\_{\rm max} \tag{7}$$

for as many k as possible (Åström & Hägglund, 1995).

This technique is called "Magnitude Optimum" (MO) (Umland & Safiuddin, 1990), "Modulus Optimum" (Åström & Hägglund, 1995), or "Betragsoptimum" (Åström & Hägglund, 1995; Kessler, 1955), and it results in a fast and non-oscillatory closed-loop time response for a large class of process models.

If the closed-loop transfer function is described by the following equation:

$$\mathbf{G}\_{\rm CL}\left(\mathbf{s}\right) = \frac{f\_0 + f\_1\mathbf{s} + f\_2\mathbf{s}^2 + \cdots}{\mathbf{e}\_0 + \mathbf{e}\_1\mathbf{s} + \mathbf{e}\_2\mathbf{s}^2 + \cdots},\tag{8}$$

then expression (7) can be met by satisfying the following conditions (Vrančić et al., 2010):

$$\sum\_{i=0}^{2n} (-1)^{i+n} \left( f\_i f\_{2n-i} e\_0^{\; 2} - e\_i e\_{2n-i} f\_0^{\; 2} \right) = 0; \quad n = 1, 2, \ldots \tag{9}$$

Before calculating the parameters of the 1-DOF PID controller, according to the given MO criteria, the pure time delay in expression (1) has to be developed into an infinite Taylor series:

$$\mathrm{s}e^{-sT\_{delay}} = 1 - \mathrm{s}T\_{delay} + \frac{\left(\mathrm{s}T\_{delay}\right)^2}{2!} - \frac{\left(\mathrm{s}T\_{delay}\right)^3}{3!} + \dots + \frac{\left(-1\right)^k \left(\mathrm{s}T\_{delay}\right)^k}{k!} + \dotsb \tag{10}$$

or Padé series:

78 Introduction to PID Controllers – Theory, Tuning and Application to Frontier Areas

The new design objective would be to maintain the closed-loop magnitude (amplitude) frequency response (GCL) from the reference to the process output as flat and as close to unity as possible for a large bandwidth (see Figure 2) (Whiteley, 1946; Hanus, 1975; Åström & Hägglund, 1995; Umland & Safiuddin, 1990). Therefore, the idea is to find a controller that makes the frequency response of the closed-loop amplitude as close as possible to unity for

Fig. 2. The amplitude (magnitude) frequency response of the closed-loop system

( )

0

This technique is called "Magnitude Optimum" (MO) (Umland & Safiuddin, 1990), "Modulus Optimum" (Åström & Hägglund, 1995), or "Betragsoptimum" (Åström & Hägglund, 1995; Kessler, 1955), and it results in a fast and non-oscillatory closed-loop time

01 2

f fs fs G s

then expression (7) can be met by satisfying the following conditions (Vrančić et al., 2010):

01 2

e es es ++ + <sup>=</sup> ++ +

=

dG j k k

2 max

0 ; 1,2, ,

2

⋯

2

GCL (0 1 ) = , (6)

= = ⋯ (7)

<sup>⋯</sup> , (8)

These requirements can be expressed in the following way:

k CL k

for as many k as possible (Åström & Hägglund, 1995).

response for a large class of process models.

2 2

ω

<sup>d</sup> <sup>ω</sup>

If the closed-loop transfer function is described by the following equation:

( )

CL

ω

lower frequencies.

$$e^{-sT\_{delay}} = \lim\_{n \to \infty} \left( \frac{1 - \frac{sT\_{delay}}{2n}}{1 + \frac{sT\_{delay}}{2n}} \right)^n = \frac{1 - \frac{sT\_{delay}}{2} + \frac{s^2 T\_{delay}}{2^2 2!} - \dots + (-1)^k \frac{s^k T\_{delay}}{2^k k!} + \dots}{1 + \frac{sT\_{delay}}{2} + \frac{s^2 T\_{delay}}{2^2 2!} + \dots + \frac{s^k T\_{delay}}{2^k k!} + \dots}. \tag{11}$$

Then, the closed-loop transfer function (5) is calculated from expressions (1), (3) and (10) or else (11). The closed-loop parameters ei and fi can be obtained by comparing expressions (8) and (5). The PID controller parameters are then obtained by solving the first three equations (n=1, 2 and 3) in expression (9) (Vrančić et al., 1999):

$$K\_P = f\_1\left(K\_{PR}, a\_1, a\_2, \dots, a\_5, b\_1, b\_2, \dots, b\_5, T\_{delay}, T\_F\right) \tag{12}$$

$$K\_{I} = f\_{2}\left(K\_{\text{PR}}, a\_{1}, a\_{2}, \dots, a\_{5}, b\_{1}, b\_{2}, \dots, b\_{5}, T\_{\text{delay}}, T\_{\text{F}}\right) \tag{13}$$

$$K\_D = f\_3\left(K\_{PR}, a\_1, a\_2, \dots, a\_5, b\_1, b\_2, \dots, b\_5, T\_{\text{delay}}, T\_F\right) \tag{14}$$

The expressions (12)-(14) are not explicitly given herein, since they would cover several pages. In order to calculate the three PID controller parameters – according to the given MO tuning criteria – only the parameters KPR, a1, a2, a3, a4, a5, b1, b2, b3, b4, b5, and Tdelay of the process transfer function (1) are required, even though the process transfer function can be of a higher-order. However, accurately estimating such a high number of process parameters from real measurements could be very problematic. Moreover, if one identifies the fifth-order process model from the actually higher-than-fifth-order process, a systematic error in the estimated process parameters would be obtained, therefore leading to the calculation of non-optimal controller parameters. Accordingly, the accuracy of the estimated process parameters in practice remains questionable.

Note that the actual expressions (12)-(14) remain exactly the same when the process with pure time-delay is developed into a Taylor (10) or Padé (11) series (Vrančić et al., 1999).

#### 4. Magnitude Optimum Multiple Integration (MOMI) tuning method

The problems with original MO tuning method just mentioned can be avoided by using the concept of 'moments', known from identification theory (Ba Hli, 1954; Preuss, 1991). Namely, the process transfer function (1) can be developed into an infinite Taylor series around s=0, as follows:

$$\mathbf{G}\_P(\mathbf{s}) = A\_0 - A\_1 \mathbf{s} + A\_2 \mathbf{s}^2 - A\_3 \mathbf{s}^3 + \cdots \tag{15}$$

where parameters Ai (i=0, 1, 2, …) represent time-weighted integrals of the process impulse response h(t) (Ba Hli, 1954; Preuss, 1991; Åström & Hägglund, 1995):

$$A\_k = \frac{1}{k!} \Big| \int\_0^\infty t^k h(t) \, dt \,\,. \tag{16}$$

Magnitude Optimum Techniques for PID Controllers 81

Fig. 3. Graphical representation of the moment (area) A1 measured from the process steady-

Therefore, in practice the process can be easily parameterised by the moments Ai from the

On the other hand, the moments can also be obtained directly from the process transfer

2 22 1 1 1

 = −− + + 

AK baT b A a

1

i

Let us now calculate the 1-DOF PID controller parameters by using the process transfer function parameterised by moments (15). In order to simplify derivation of the PID controller parameters, the filter within the PID controller (3) is considered to be a part of the

> ( ) ( ) \* 1 P

G s

F

 = − −+ − <sup>+</sup>

∑

<sup>i</sup> <sup>k</sup> <sup>k</sup> k i delay k i

+ + − =

2

!

⋮ . (20)

sT <sup>=</sup> <sup>+</sup> . (21)

i

T b

2!

delay

T

process step-response or else from any other change of the process steady-state.

( )

PR delay

PR delay

( ) ( ) ( )

iki

P

G s

−

A a

1 1

1

function (1), as follows (Vrančić et al., 1999; Vrančić et al., 2001a):

PR

1 11

AK abT

= −+

( )

<sup>k</sup> k i

1

<sup>−</sup> + −

1

+ −

∑

i

process (1):

=

<sup>1</sup> <sup>1</sup>

AK ab

k PR k k

A1

0 y

t

A0

<sup>00</sup>uA

state change time response (see shadowed area).

0

A K

=

However, the process impulse response cannot be obtained easily in practice since – due to several restrictions – we cannot apply an infinite impulse signal to the process input. Fortunately, the moments Ai can also be obtained by calculating repetitive (multiple) integrals of the process input (u) and output (y) signals during the change of the process steady-state (Strejc, 1960; Vrančić et al., 1999; Vrančić, 2008):

$$\begin{aligned} u\_0(t) &= \frac{u(t) - u(0)}{u(\infty) - u(0)} & y\_0(t) &= \frac{y(t) - y(0)}{u(\infty) - u(0)} \\ I\_{L11}(t) &= \begin{bmatrix} u\_0(\tau) \, d\tau & I\_{Y1}(t) = \int y\_0(\tau) \, d\tau \\ 0 & 0 \end{bmatrix} & \cdot \\ I\_{L12}(t) &= \begin{bmatrix} I\_{L11}(\tau) \, d\tau & I\_{Y2}(t) = \int I\_{Y1}(\tau) \, d\tau \\ 0 & \vdots \end{bmatrix} & \cdot \end{aligned} \tag{17}$$

The moments (integrals, areas) can be calculated as follows:

$$\begin{aligned} A\_0 &= y\_0(\infty) \; ; \; y\_1 = A\_0 I\_{l11}(t) - I\_{Y1}(t) \\ A\_1 &= y\_1(\infty) \; ; \; y\_2 = A\_1 I\_{l11}(t) - A\_0 I\_{l12}(t) + I\_{Y2}(t) \\ A\_2 &= y\_2(\infty) \; ; \; y\_3 = A\_2 I\_{l11}(t) - A\_1 I\_{l12}(t) + A\_0 I\_{l13}(t) - I\_{Y3}(t) \\ &\vdots \end{aligned} \tag{18}$$

It is assumed that:

$$
\dot{\mathbf{y}}\left(\mathbf{0}\right) = \ddot{\mathbf{y}}\left(\mathbf{0}\right) = \ddot{\mathbf{y}}\left(\mathbf{0}\right) = \dots = \mathbf{0} \tag{19}
$$

Given that in practice the integration horizon should be limited, there is no need to wait until t=∞. It is enough to integrate until the transient of y0(t) in (17) dies out. Note that the first impulse (A0) equals the steady-state process gain, KPR.

In order to clarify the mathematical derivation, a graphical representation of the first moment (area) is shown in Figure 3. Note that u0 and y0 represent scaled process input and process output time responses, respectively.

Namely, the process transfer function (1) can be developed into an infinite Taylor series

where parameters Ai (i=0, 1, 2, …) represent time-weighted integrals of the process impulse

0

However, the process impulse response cannot be obtained easily in practice since – due to several restrictions – we cannot apply an infinite impulse signal to the process input. Fortunately, the moments Ai can also be obtained by calculating repetitive (multiple) integrals of the process input (u) and output (y) signals during the change of the process

1 ! <sup>k</sup> A t h t dt <sup>k</sup> <sup>k</sup> ∞

( )

( ) () ( ) ( ) ()

0 0 0 0

τ τ τ τ

τ τ τ τ

() ( )

() ( )

response h(t) (Ba Hli, 1954; Preuss, 1991; Åström & Hägglund, 1995):

steady-state (Strejc, 1960; Vrančić et al., 1999; Vrančić, 2008):

The moments (integrals, areas) can be calculated as follows:

 

⋮

process output time responses, respectively.

It is assumed that:

; ; ;

first impulse (A0) equals the steady-state process gain, KPR.

( ) () ( ) ( ) ()

0 0

U Y

10 10 0 0

= =

t t

− − = = ∞ − ∞ −

It u d It y d

∫ ∫

uu uu

ut u yt y u t y t

t t UU YY

It I d It I d

∫ ∫

21 21 0 0

( ) () () ()

=∞ = − + −

U Y

2 2 3 21 12 03 3

A y y AI t AI t AI t I t

Given that in practice the integration horizon should be limited, there is no need to wait until t=∞. It is enough to integrate until the transient of y0(t) in (17) dies out. Note that the

In order to clarify the mathematical derivation, a graphical representation of the first moment (area) is shown in Figure 3. Note that u0 and y0 represent scaled process input and

( ) () () () ()

U U UY

yyy ɺ ɺɺ ɺɺɺ () () () 000 0 = = == <sup>⋯</sup> . (19)

U UY

= =

() ( )

() ( )

⋮ ⋮

( ) () ()

=∞ = − +

1 1 2 11 02 2

A y y AI t AI t I t

0 0 1 01 1

=∞ = −

A y y AI t I t

( ) 2 3 G s A As As As <sup>P</sup> =− + − + 01 2 3 <sup>⋯</sup> , (15)

<sup>=</sup> ∫ . (16)

. (17)

. (18)

around s=0, as follows:

Fig. 3. Graphical representation of the moment (area) A1 measured from the process steadystate change time response (see shadowed area).

Therefore, in practice the process can be easily parameterised by the moments Ai from the process step-response or else from any other change of the process steady-state. On the other hand, the moments can also be obtained directly from the process transfer function (1), as follows (Vrančić et al., 1999; Vrančić et al., 2001a):

$$\begin{aligned} A\_0 &= K\_{PR} \\ A\_1 &= K\_{PR} \left( a\_1 - b\_1 + T\_{delay} \right) \\ A\_2 &= K\_{PR} \left[ b\_2 - a\_2 - T\_{delay} b\_1 + \frac{T\_{delay}}{2!} \right] + A\_1 a\_1 \\ &\vdots \\ A\_k &= K\_{PR} \left( \left( -1 \right)^{k+1} \left( a\_k - b\_k \right) + \sum\_{i=1}^k \left( -1 \right)^{k+i} \frac{T\_{delay} i b\_{k-i}}{i!} \right) + \\ &+ \sum\_{i=1}^{k-1} \left( -1 \right)^{k+i-1} A\_i a\_{k-i} \end{aligned} \tag{20}$$

Let us now calculate the 1-DOF PID controller parameters by using the process transfer function parameterised by moments (15). In order to simplify derivation of the PID controller parameters, the filter within the PID controller (3) is considered to be a part of the process (1):

$$\text{G}\_P^\*(\text{s}) = \frac{\text{G}\_P(\text{s})}{\text{1} + \text{s}T\_F}. \tag{21}$$

Therefore, GC(s) (3) simplifies into the "schoolbook" PID controller without a filter:

$$\mathbf{G}\_{\mathbf{C}}^{\*}\left(\mathbf{s}\right) = \left(\mathbf{K}\_{\mathrm{I}} + \mathbf{K}\_{\mathrm{P}}\mathbf{s} + \mathbf{K}\_{\mathrm{D}}\mathbf{s}^{2}\right) / \left|\mathbf{s}\right.\tag{22}$$

Since a filter is considered as a part of the process, the measured moments (18) should be changed accordingly. One solution to calculate any new moments is to filter the process output signal:

$$Y\_F(\mathbf{s}) = \frac{Y(\mathbf{s})}{\mathbf{1} + \mathbf{s}T\_F} \tag{23}$$

Magnitude Optimum Techniques for PID Controllers 83

Note that the calculation of the filtered PID controller parameters is based on the fact that the filter time constant is given a priori. In practice this is often not entirely true, since the usual way is rather to define the ratio (N) between the derivative time constant (TD=KD/KP)

> D D F PF

The controller parameters can be calculated iteratively by first choosing TF=0 (or any relatively small positive value) and then calculating the controller parameters by using expression (25). In the second iteration, the filter time constant can be calculated from (26),

D

P

The moments are recalculated according to expression (24) and the new controller parameters from (25). By performing a few more iterations, quite accurate results can be

The PI controller parameters can be calculated in a similar manner to those of the PID controller by choosing KD=0. Since a filter is usually not needed in a PI controller (TF=0), the original moments (Ai) are applied in the calculation. Repeating the same procedure as before and solving the first two equations in (9), the following PI controller parameters are

> 1 0 3 2

Note that the vectors and matrices in (28) are just sub-vectors and sub-matrices of

The proportional (P) controller gain can be obtained by fixing KI=0 and KD=0, repeating the

2 2 P

entirely fulfil the MO conditions and will not be used in any further derivations.

AA A <sup>K</sup>

1

( )

A A AA <sup>−</sup> <sup>=</sup>

However, condition (6) is not satisfied, since proportional controllers cannot achieve closedloop gain equal to one at lower frequencies. Therefore the proportional controller does not

In some cases, the controller parameters have to be re-tuned for certain practical reasons. In particular, when tuning the PID controllers for the first-order or the second-order process, the controller gain is theoretically infinite. In practice (when there is process noise), the

2 02 1 2 0 1 02

<sup>−</sup> − − <sup>=</sup> <sup>−</sup>

K AA K AA

expression (25). Similarly, the I (integral-term only) controller gain is the following:

I P 1

0.5 0

F

<sup>K</sup> <sup>T</sup>

T KT = = . (26)

K N <sup>=</sup> . (27)

0.5 <sup>K</sup><sup>I</sup> <sup>A</sup> <sup>=</sup> . (29)

<sup>−</sup> . (30)

. (28)

T K <sup>N</sup>

Typical values of N are 8 to 20 (Åström & Hägglund, 1995).

and the filter time constant:

obtained for the a priori chosen ratio N.

procedure and solving the first equation in (9):

obtained (Vrančić et al., 2001a):

as follows:

and use signal yF(t) instead of y(t) in expression (17). However, a much simpler solution is to recalculate the moments as follows:

$$\begin{aligned} A\_0^\star &= A\_0\\ A\_1^\star &= A\_1 + A\_0 T\_F\\ A\_2^\star &= A\_2 + A\_1 T\_F + A\_0 T\_F^2\\ \vdots \end{aligned} \tag{24}$$

where Ai\* denote the moments of the process with included the filter (21).

The parameters ei and fi in expression (8) can be obtained by placing expressions (22) and (15) (by replacing moments Ai with Ai\*) into (5). By solving the first three equations in (9), the following PID controller parameters are obtained (Vrančić et al., 2001a):

$$
\begin{bmatrix} K\_I \\ K\_P \\ K\_D \end{bmatrix} = \begin{bmatrix} -A\_1^\* & A\_0^\* & 0 \\ -A\_3^\* & A\_2^\* & -A\_1^\* \\ -A\_5^\* & A\_4^\* & -A\_3^\* \end{bmatrix}^{-1} \begin{bmatrix} -0.5 \\ 0 \\ 0 \\ 0 \end{bmatrix}. \tag{25}
$$

The expression for the PID controller parameters is now much simpler when compared to expressions (12)-(14). There are several other advantages to using expression (25) instead of expressions (12)-(14) for the calculation of the PID controller parameters.

First, only the steady-state process gain A0=KPR and five moments (A1 to A5) instead of the 12 transfer function parameters (KPR, a1..a5, b1..b5, and Tdelay) are needed as input data.

Second, the expression for KI, KP, and KD is simplified, which makes it more transparent and simpler to handle.

Third, the moments A1 to A5 can be calculated from the process time-response using numerical integration, whilst the gain A0=KPR can be determined from the steady-state value of the process steady-state change in the usual way. This procedure replaces the much more demanding algorithm for the estimation of the transfer function parameters.

In addition, it is important to note that the mapping of expressions (12)-(14) into expression (25) results in exact (rather than approximate) controller parameters. This means that the frequency-domain control criterion can be achieved with a model parameterised in the timedomain. Thus the proposed tuning procedure is a simple and very effective way for controller tuning since no background in control theory is needed.

Since a filter is considered as a part of the process, the measured moments (18) should be changed accordingly. One solution to calculate any new moments is to filter the process

> ( ) ( ) 1 <sup>F</sup>

and use signal yF(t) instead of y(t) in expression (17). However, a much simpler solution is to

\* 2 221 0

The parameters ei and fi in expression (8) can be obtained by placing expressions (22) and (15) (by replacing moments Ai with Ai\*) into (5). By solving the first three equations in (9),

<sup>1</sup> \* \*

0 0.5

−

0 0

1 0 \*\* \* 32 1 \*\* \* 54 3

<sup>−</sup> <sup>−</sup> =− − − −

The expression for the PID controller parameters is now much simpler when compared to expressions (12)-(14). There are several other advantages to using expression (25) instead of

First, only the steady-state process gain A0=KPR and five moments (A1 to A5) instead of the 12 transfer function parameters (KPR, a1..a5, b1..b5, and Tdelay) are needed as input data. Second, the expression for KI, KP, and KD is simplified, which makes it more transparent and

Third, the moments A1 to A5 can be calculated from the process time-response using numerical integration, whilst the gain A0=KPR can be determined from the steady-state value of the process steady-state change in the usual way. This procedure replaces the much more

In addition, it is important to note that the mapping of expressions (12)-(14) into expression (25) results in exact (rather than approximate) controller parameters. This means that the frequency-domain control criterion can be achieved with a model parameterised in the timedomain. Thus the proposed tuning procedure is a simple and very effective way for

K AA A K AA A

A A AT AT

=+ +

F

F F

Y s

sT <sup>=</sup> <sup>+</sup>

F

( ) ( ) \* 2 G s <sup>C</sup> =+ + K K s K s /s IP D . (22)

(23)

, (24)

. (25)

Therefore, GC(s) (3) simplifies into the "schoolbook" PID controller without a filter:

Y s

110

A A AT

= +

\* 0 0 \*

A A

=

⋮

where Ai\* denote the moments of the process with included the filter (21).

the following PID controller parameters are obtained (Vrančić et al., 2001a):

K A A

expressions (12)-(14) for the calculation of the PID controller parameters.

demanding algorithm for the estimation of the transfer function parameters.

controller tuning since no background in control theory is needed.

I P D

output signal:

simpler to handle.

recalculate the moments as follows:

Note that the calculation of the filtered PID controller parameters is based on the fact that the filter time constant is given a priori. In practice this is often not entirely true, since the usual way is rather to define the ratio (N) between the derivative time constant (TD=KD/KP) and the filter time constant:

$$N = \frac{T\_D}{T\_F} = \frac{K\_D}{K\_P T\_F} \,\,\,\,\tag{26}$$

Typical values of N are 8 to 20 (Åström & Hägglund, 1995).

The controller parameters can be calculated iteratively by first choosing TF=0 (or any relatively small positive value) and then calculating the controller parameters by using expression (25). In the second iteration, the filter time constant can be calculated from (26), as follows:

$$T\_F = \frac{K\_D}{K\_P N} \,\,\,\,\,\,\tag{27}$$

The moments are recalculated according to expression (24) and the new controller parameters from (25). By performing a few more iterations, quite accurate results can be obtained for the a priori chosen ratio N.

The PI controller parameters can be calculated in a similar manner to those of the PID controller by choosing KD=0. Since a filter is usually not needed in a PI controller (TF=0), the original moments (Ai) are applied in the calculation. Repeating the same procedure as before and solving the first two equations in (9), the following PI controller parameters are obtained (Vrančić et al., 2001a):

$$
\begin{bmatrix} K\_I \\ K\_P \end{bmatrix} = \begin{bmatrix} -A\_1 & A\_0 \\ -A\_3 & A\_2 \end{bmatrix}^{-1} \begin{bmatrix} -0.5 \\ 0 \end{bmatrix}. \tag{28}
$$

Note that the vectors and matrices in (28) are just sub-vectors and sub-matrices of expression (25). Similarly, the I (integral-term only) controller gain is the following:

1 0.5 <sup>K</sup><sup>I</sup> <sup>A</sup> <sup>=</sup> . (29)

The proportional (P) controller gain can be obtained by fixing KI=0 and KD=0, repeating the procedure and solving the first equation in (9):

$$K\_P = \frac{2A\_0A\_2 - A\_1^2}{2A\_0\left(A\_1^2 - A\_0A\_2\right)}.\tag{30}$$

However, condition (6) is not satisfied, since proportional controllers cannot achieve closedloop gain equal to one at lower frequencies. Therefore the proportional controller does not entirely fulfil the MO conditions and will not be used in any further derivations.

In some cases, the controller parameters have to be re-tuned for certain practical reasons. In particular, when tuning the PID controllers for the first-order or the second-order process, the controller gain is theoretically infinite. In practice (when there is process noise), the calculated controller gain can have a very high positive or negative value. In this case, the controller gain should be limited to some acceptable value, which would depend on the controller and the process limitations (Vrančić et al., 1999). Note that the sign of the proportional gain is usually the same to the sign of the process gain:

$$\text{sgn}\left(K\_P\right) = \text{sgn}\left(K\_{PR}\right). \tag{31}$$

Magnitude Optimum Techniques for PID Controllers 85

• The PI or I parameters can be calculated from expressions (28) or (29), respectively. The proposed tuning procedure will be illustrated by the following process models:

( )

=

=

P

G s

P

G s

P

( )

( )

( )

<sup>e</sup> G s

4

P

0 10 20 30 40

time [s]

Process GP3: Open−loop response

0 5 10 15 20

time [s]

Process GP1: Open−loop response

0 0.2 0.4 0.6 0.8 1

−1.5 −1 −0.5 0 0.5 1

experiment for processes GP1 to GP4.

remaining parameters according to expressions (33)-(36).

the process gain (KPR=A0), set KP manually to some desired value (32) and recalculating

( )( )

+ +

s s

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

Fig. 4. The process input (--) and the process output (\_\_) signals during an open-loop

0 10 20 30

time [s]

Process GP4: Open−loop response

0 5 10 15 20

time [s]

Process GP2: Open−loop response

(37)

1 2 2

( )

+

s

s

2 6

3 2

<sup>−</sup> <sup>=</sup> +

<sup>s</sup> G s

1

applied to the process inputs. The process open-loop responses are shown in Figure 4.

<sup>=</sup> <sup>+</sup>

( )

5

s

s −

The process models have been chosen in order to cover a range of different processes, including higher-order processes, highly non-minimum phase processes and dominantly delayed processes. The models have the same process gain (A0=1) and the first moment A1=6. If the process transfer function is not known in advance, the moments (areas) can be calculated according to the time-domain approach given above. The ramp-like input signal has been

The recommended values of the proportional gain are:

$$\left|\frac{1}{A\_0}\right| \le \left|K\_P\right| \le \left|\frac{10}{A\_0}\right|.\tag{32}$$

The remaining two controller parameters can now be calculated according to the limited (fixed) controller gain from expression (25). If the chosen controller gain is:

$$K\_P > \frac{1}{\frac{2A\_1^\* A\_2^\*}{A\_3^\*} - 2A\_0^\*} \, ' \tag{33}$$

then:

$$K\_I = \frac{0.5 + K\_P A\_0^\*}{A\_1^\*} \tag{34}$$

and:

$$K\_D = \frac{A\_3^\*}{A\_1^{\*2}} \left[ \frac{A\_1^\* A\_2^\* K\_P}{A\_3^\*} - 0.5 - A\_0^\* K\_P \right]. \tag{35}$$

If expression (33) is not true:

$$K\_D = \mathbf{0} \,. \tag{36}$$

When limiting the proportional gain of the PI controller, only Eq. (34) is used. Note that proposed re-tuning can also be used in cases when a slower and more robust controller should be designed (by decreasing KP), or if a faster but more oscillatory response is required (by increasing KP).

The PID controller tuning procedure, according to the MOMI method, can therefore proceed as follows:


calculated controller gain can have a very high positive or negative value. In this case, the controller gain should be limited to some acceptable value, which would depend on the controller and the process limitations (Vrančić et al., 1999). Note that the sign of the

> 0 0 1 10 <sup>K</sup><sup>P</sup> A A

The remaining two controller parameters can now be calculated according to the limited

\* \* 1 2 \* \* 0 3

K A <sup>K</sup> A

\* \* \* <sup>3</sup> 1 2 \* 0 \*2 \*

 = −− 

When limiting the proportional gain of the PI controller, only Eq. (34) is used. Note that proposed re-tuning can also be used in cases when a slower and more robust controller should be designed (by decreasing KP), or if a faster but more oscillatory response is

The PID controller tuning procedure, according to the MOMI method, can therefore proceed

• If the process model is not known a priori, modify the steady-state process by changing

• Find the steady-state process gain KPR=A0 and moments A1-A5 by using numerical integration (summation) from the beginning to the end of the process time response according to expressions (17) and (18). If the process model is known, calculate the

• Fix the filter time constant TF to some desired value and calculate the PID controller parameters from (25). If needed, change the filter time constant and recalculate the PID controller parameters. If the proportional gain KP is too high or has a different sign to

0.5 <sup>P</sup> D P <sup>A</sup> AAK <sup>K</sup> A K

1 <sup>2</sup> <sup>2</sup> <sup>K</sup><sup>P</sup> A A <sup>A</sup> A

−

\* 0 \* 1 0.5 <sup>P</sup>

sgn sgn (K K <sup>P</sup> ) = ( PR ) . (31)

≤ ≤ . (32)

<sup>+</sup> <sup>=</sup> (34)

K<sup>D</sup> = 0 . (36)

, (33)

. (35)

proportional gain is usually the same to the sign of the process gain:

(fixed) controller gain from expression (25). If the chosen controller gain is:

>

I

1 3

A A

The recommended values of the proportional gain are:

then:

and:

If expression (33) is not true:

required (by increasing KP).

the process input signal.

moments from expression (20).

as follows:

the process gain (KPR=A0), set KP manually to some desired value (32) and recalculating remaining parameters according to expressions (33)-(36).

• The PI or I parameters can be calculated from expressions (28) or (29), respectively.

The proposed tuning procedure will be illustrated by the following process models:

$$\begin{aligned} G\_{P1}\left(s\right) &= \frac{1}{\left(1+2s\right)^2 \left(1+s\right)^2} \\ G\_{P2}\left(s\right) &= \frac{1}{\left(1+s\right)^6} \\ G\_{P3}\left(s\right) &= \frac{1-4s}{\left(1+s\right)^2} \\ G\_{P4}\left(s\right) &= \frac{e^{-5s}}{1+s} \end{aligned} \tag{37}$$

The process models have been chosen in order to cover a range of different processes, including higher-order processes, highly non-minimum phase processes and dominantly delayed processes. The models have the same process gain (A0=1) and the first moment A1=6. If the process transfer function is not known in advance, the moments (areas) can be calculated according to the time-domain approach given above. The ramp-like input signal has been applied to the process inputs. The process open-loop responses are shown in Figure 4.

Fig. 4. The process input (--) and the process output (\_\_) signals during an open-loop experiment for processes GP1 to GP4.

Magnitude Optimum Techniques for PID Controllers 87

in Table 2. It can be seen that the values are practically equivalent, so the closed-loop

 A1 A2 A3 A4 A5 KI KP KD TF KI KP K<sup>I</sup> GP1 6 23 72 201 522 0.31 1.44 1.76 0.2 0.17 0.55 0.08 GP2 6 21 56 126 252 0.22 0.87 0.96 0.2 0.15 0.4 0.08 GP3 6 11 16 21 26 0.12 0.25 0.13 0.2 0.11 0.16 0.08 GP4 6 18.5 39.3 65.4 91.4 0.16 0.49 0.45 0.2 0.13 0.27 0.08 Table 2. The values of moments and controller parameters for processes (37) by using direct

The MOMI tuning method will be illustrated by the three-water-column laboratory setup shown in Figure 6. It consists of two water pumps, a reservoir and three water columns. The water columns can be connected by means of electronic valves. In our setup, two water columns have been used (R1 and R2), as depicted in the block diagram shown in

Fig. 6. Picture of the laboratory hydraulic setup (taken in stereoscopic side-by-side format). The selected control loop consists of the reservoir R0, the pump P1, an electronic valve V<sup>1</sup> (open), a valve V3 (partially open) and water columns R1 and R2. The valve V2 is closed and the pump P2 is switched off. The process input is the voltage on pump P1 and the process output is the water level in the second tank (h2), measured by the pressure to voltage transducer. The actual process input and output signals are voltages measured by an A/D

and a D/A converter (NI USB 6215) via real-time blocks in Simulink (Matlab).

Moments (areas) PID PI I

responses are the same to those shown in Figure 5.

calculation from the process model.

Figure 7.

The moments are calculated by using expressions (17) and (18) and the controller parameters by using expressions (25), (28) and (29). The calculated parameters are given in Table 1.


Table 1. The values of moments and controller parameters for processes (37) when using a time-domain approach (by applying multiple integration of the process time-response).

The closed-loop responses for all the processes, when using different types of controllers tuned by the MOMI method, are shown in Figure 5. As can be seen, the responses are stable and relatively fast, all according to the MO tuning criteria.

Fig. 5. Closed-loop responses for processes GP1 to GP4 when using PID controller (\_\_), PI controller (--) and I controller (-.-) tuned by the MOMI method.

The results can be verified by calculating the moments and controller parameters directly from the process transfer functions (37). The moments can be calculated from expression (20). The controller parameters are calculated as before. The obtained parameters are given

The moments are calculated by using expressions (17) and (18) and the controller parameters by using expressions (25), (28) and (29). The calculated parameters are given in

 A1 A2 A3 A4 A5 KI KP KD TF KI KP K<sup>I</sup> GP1 6 23 72 201 521 0.31 1.45 1.76 0.2 0.17 0.55 0.08 GP2 6 21 56 126 252 0.22 0.87 0.96 0.2 0.15 0.4 0.08 GP3 6 11 16 21 26 0.12 0.25 0.13 0.2 0.11 0.16 0.08 GP4 6 18.5 39.3 65.4 91.3 0.16 0.49 0.45 0.2 0.13 0.27 0.08 Table 1. The values of moments and controller parameters for processes (37) when using a time-domain approach (by applying multiple integration of the process time-response).

The closed-loop responses for all the processes, when using different types of controllers tuned by the MOMI method, are shown in Figure 5. As can be seen, the responses are stable

and relatively fast, all according to the MO tuning criteria.

Process GP1: Closed−loop responses

0 10 20 30

time [s]

Process GP3: Closed−loop responses

0 20 40 60

time [s]

controller (--) and I controller (-.-) tuned by the MOMI method.

0 0.2 0.4 0.6 0.8 1 1.2 1.4

−1

−0.5

0

0.5

1

1.5

Moments (areas) PID PI I

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Fig. 5. Closed-loop responses for processes GP1 to GP4 when using PID controller (\_\_), PI

The results can be verified by calculating the moments and controller parameters directly from the process transfer functions (37). The moments can be calculated from expression (20). The controller parameters are calculated as before. The obtained parameters are given

0 20 40 60

time [s]

Process GP4: Closed−loop responses

0 20 40 60

time [s]

Process GP2: Closed−loop responses

Table 1.


in Table 2. It can be seen that the values are practically equivalent, so the closed-loop responses are the same to those shown in Figure 5.

The MOMI tuning method will be illustrated by the three-water-column laboratory setup shown in Figure 6. It consists of two water pumps, a reservoir and three water columns. The water columns can be connected by means of electronic valves. In our setup, two water columns have been used (R1 and R2), as depicted in the block diagram shown in Figure 7.

Fig. 6. Picture of the laboratory hydraulic setup (taken in stereoscopic side-by-side format).

The selected control loop consists of the reservoir R0, the pump P1, an electronic valve V<sup>1</sup> (open), a valve V3 (partially open) and water columns R1 and R2. The valve V2 is closed and the pump P2 is switched off. The process input is the voltage on pump P1 and the process output is the water level in the second tank (h2), measured by the pressure to voltage transducer. The actual process input and output signals are voltages measured by an A/D and a D/A converter (NI USB 6215) via real-time blocks in Simulink (Matlab).

Table 2. The values of moments and controller parameters for processes (37) by using direct calculation from the process model.

Magnitude Optimum Techniques for PID Controllers 89

Process input [V]

<sup>0</sup> <sup>200</sup> <sup>400</sup> <sup>600</sup> <sup>800</sup> <sup>1000</sup> <sup>1200</sup> <sup>1400</sup> <sup>1</sup>

t [s]

Process output [V]

<sup>0</sup> <sup>200</sup> <sup>400</sup> <sup>600</sup> <sup>800</sup> <sup>1000</sup> <sup>1200</sup> <sup>1400</sup> 0.5

t [s] Fig. 8. The process input and process output responses over the entire working region.

The calculated PID controller parameters, for an a priori chosen filter parameter TF=1s, were

Process input [V]

<sup>200</sup> <sup>250</sup> <sup>300</sup> <sup>350</sup> <sup>400</sup> <sup>450</sup> 0.4

<sup>200</sup> <sup>250</sup> <sup>300</sup> <sup>350</sup> <sup>400</sup> <sup>450</sup> 0.8

t [s]

t [s]

Process output [V]

0.305, 19.7, 264 K KK I PD = == (40)

the following (the proportional gain has been limited to the value KP=10/A0):

1.5 2 2.5 3 3.5 4 4.5

1

0.5 0.6 0.7 0.8 0.9 1 1.1

0.9 1 1.1 1.2 1.3 1.4

Fig. 9. Process open-loop response.

1.5

2

2.5

Fig. 7. Block diagram of the laboratory hydraulic setup.

First, the linearity of the system was checked by applying several steps at the process input. The process input and output responses are shown in Figure 8. It can be seen that both – the process steady-state gain and the time-constants – change according to the working point. In order to partially linearise the process, the square-root function has been placed between the controller output (u) and the process input (ur) signals:

10 <sup>r</sup> u u = ⋅ , (38)

The control output signal u is limited between values 0 and 10. The pump actually starts working when signal ur becomes higher than 1V.

Note that artificially added non-linearity cannot ideally linearise the non-linearity of the process gain. Moreover, the process time constants still differ significantly at different working points.

After applying the non-linear function (38), the open-loop process response has been measured (see Figure 9). The moments (areas) have been calculated by using expressions (17) and (18):

$$A\_0 = 0.507, \ A\_1 = 33.9, \ A\_2 = 1.76 \cdot 10^3, \ A\_3 = 8.44 \cdot 10^4, \ A\_4 = 3.9 \cdot 10^6, \ A\_5 = 1.78 \cdot 10^8 \tag{39}$$

LT 1

First, the linearity of the system was checked by applying several steps at the process input. The process input and output responses are shown in Figure 8. It can be seen that both – the process steady-state gain and the time-constants – change according to the working point. In order to partially linearise the process, the square-root function has been placed between the

The control output signal u is limited between values 0 and 10. The pump actually starts

Note that artificially added non-linearity cannot ideally linearise the non-linearity of the process gain. Moreover, the process time constants still differ significantly at different

After applying the non-linear function (38), the open-loop process response has been measured (see Figure 9). The moments (areas) have been calculated by using expressions

 3 46 8 0 12 3 4 5 A AA A A A = = = ⋅ = ⋅ =⋅ = ⋅ 0.507, 33.9, 1.76 10 , 8.44 10 , 3.9 10 , 1.78 10 (39)

P1

h1

R0

Fig. 7. Block diagram of the laboratory hydraulic setup.

controller output (u) and the process input (ur) signals:

working when signal ur becomes higher than 1V.

working points.

(17) and (18):

V3

P2

<sup>h</sup><sup>2</sup> <sup>h</sup><sup>3</sup>

V<sup>1</sup> V<sup>2</sup>

R<sup>1</sup> R<sup>2</sup> R<sup>3</sup>

LT 2

10 <sup>r</sup> u u = ⋅ , (38)

LT 3

Fig. 8. The process input and process output responses over the entire working region.

The calculated PID controller parameters, for an a priori chosen filter parameter TF=1s, were the following (the proportional gain has been limited to the value KP=10/A0):

$$K\_I = 0.\text{505, } K\_P = 1\text{9.7, } K\_D = 264\tag{40}$$

Fig. 9. Process open-loop response.

Magnitude Optimum Techniques for PID Controllers 91

( ) ( )

<sup>=</sup> <sup>+</sup>

2 2

=

=

3 6

1

Two of them (GP3 and GP4) are the same as in the previous section (37) while we added two lower-order processes in order to clearly show the degraded disturbance-rejection performance. The moments and controller parameters for the chosen processes are given in

 A1 A2 A3 A4 A5 KI KP KD TF KI KP K<sup>I</sup> GP1 6 36 216 1296 7776 1.75 10 0 0 1.75 10 0.08 GP2 6 27 108 405 1458 1.69 10 14.5 0.2 0.25 1 0.08 GP3 6 21 56 126 252 0.22 0.87 0.96 0.2 0.15 0.4 0.08 GP4 6 18.5 39.3 65.4 91.4 0.16 0.49 0.45 0.2 0.13 0.27 0.08 Table 3. The values of the moments and controller parameters for processes (41) using the

A step-like disturbance (d) has been applied to the process input (see Figure 1). The process output responses are shown in Figure 11. It is clearly seen that the closed-loop responses of the processes GP1 and GP2, when using the PI and the PID controllers, are relatively slow

It is obvious that the MO criteria should be modified in order to achieve a more optimal disturbance rejection. The closed-loop transfer function between the disturbance (d) and the

Ys G s

However, the function GCLD (42) cannot be applied instead of GCL in expressions (6) and (7), since GCLD has zero gain in the steady-state (s=0). However, by adding integrator to function (42) and multiplying it with KI, it complies with the MO requirements (Vrančić et al., 2004b;

( ) ( ) ( )

Therefore, in order to achieve optimal disturbance-rejection properties, the function GCLI

I I P

K KG s

s s G sG s = = <sup>+</sup>

Ds G sG s = = <sup>+</sup>

( ) 1 () () P

( ) 1 () ()

C P

C P

( ) ( ) ( )

<sup>=</sup> <sup>+</sup>

( )

+

s

s

(41)

(42)

(43)

( )

+

s

5

s

s −

Moments (areas) PID PI I

Let us observe the disturbance-rejection performance of the following process models:

( )

1

G s

P

P

G s

P

G s

( )

( )

<sup>e</sup> G s

4

P

Table 3. Note that the proportional gain has been limited to 10 for GP1 and GP2.

with visible "long tails" (exponential approaching to the reference).

CLD

G s

CLI CLD

should be applied instead of GCL in the MO criteria (6) and (7).

Gs G s

MOMI method.

2010):

process output (y) is the following:

The closed-loop response of the process with the controller was calculated in the previous step, as shown in Figure 10. At t=300s, the set-point has been changed from 1.2 to 1.5 and at t=900s it is returned back to 1.2. A step-like disturbance has been added to the process input at t=700s and t=1300s. It can be seen that the closed-loop response is relatively fast (when compared to the open-loop response) and without oscillations.

Fig. 10. The process closed-loop response in the hydraulic setup when using the PID controller tuned by the MOMI method.

## 5. Disturbance-Rejection Magnitude Optimum (DRMO) tuning method

The efficiency of the MOMI method has been demonstrated on several process models (Vrančić, 1995). The MO criteria, according to expressions (6) and (7), optimises the closedloop transfer function between the reference (r) and the process output (y). However, this may lead to the poor attenuation of load disturbances (Åström & Hägglund, 1995). The disturbance-rejection performance is particularly degraded when controlling lower-order processes.

The closed-loop response of the process with the controller was calculated in the previous step, as shown in Figure 10. At t=300s, the set-point has been changed from 1.2 to 1.5 and at t=900s it is returned back to 1.2. A step-like disturbance has been added to the process input at t=700s and t=1300s. It can be seen that the closed-loop response is relatively fast (when

<sup>400</sup> <sup>600</sup> <sup>800</sup> <sup>1000</sup> <sup>1200</sup> <sup>1400</sup> <sup>1</sup>

<sup>400</sup> <sup>600</sup> <sup>800</sup> <sup>1000</sup> <sup>1200</sup> <sup>1400</sup> <sup>0</sup>

Fig. 10. The process closed-loop response in the hydraulic setup when using the PID

5. Disturbance-Rejection Magnitude Optimum (DRMO) tuning method

The efficiency of the MOMI method has been demonstrated on several process models (Vrančić, 1995). The MO criteria, according to expressions (6) and (7), optimises the closedloop transfer function between the reference (r) and the process output (y). However, this may lead to the poor attenuation of load disturbances (Åström & Hägglund, 1995). The disturbance-rejection performance is particularly degraded when controlling lower-order

t [s]

t [s]

Controller output [V]

Process ouput and set−point [V]

process output set−point

compared to the open-loop response) and without oscillations.

1.1 1.2 1.3 1.4 1.5 1.6 1.7

controller tuned by the MOMI method.

processes.

Let us observe the disturbance-rejection performance of the following process models:

$$\begin{aligned} G\_{P1}(s) &= \frac{1}{\left(1 + 6s\right)}\\ G\_{P2}(s) &= \frac{1}{\left(1 + 3s\right)^2}\\ G\_{P3}(s) &= \frac{1}{\left(1 + s\right)^6}\\ G\_{P4}(s) &= \frac{e^{-5s}}{1 + s} \end{aligned} \tag{41}$$

Two of them (GP3 and GP4) are the same as in the previous section (37) while we added two lower-order processes in order to clearly show the degraded disturbance-rejection performance. The moments and controller parameters for the chosen processes are given in Table 3. Note that the proportional gain has been limited to 10 for GP1 and GP2.


Table 3. The values of the moments and controller parameters for processes (41) using the MOMI method.

A step-like disturbance (d) has been applied to the process input (see Figure 1). The process output responses are shown in Figure 11. It is clearly seen that the closed-loop responses of the processes GP1 and GP2, when using the PI and the PID controllers, are relatively slow with visible "long tails" (exponential approaching to the reference).

It is obvious that the MO criteria should be modified in order to achieve a more optimal disturbance rejection. The closed-loop transfer function between the disturbance (d) and the process output (y) is the following:

$$G\_{\rm CLD}\left(s\right) = \frac{Y\left(s\right)}{D\left(s\right)} = \frac{G\_P\left(s\right)}{1 + G\_\mathbb{C}\left(s\right)G\_P\left(s\right)}\tag{42}$$

However, the function GCLD (42) cannot be applied instead of GCL in expressions (6) and (7), since GCLD has zero gain in the steady-state (s=0). However, by adding integrator to function (42) and multiplying it with KI, it complies with the MO requirements (Vrančić et al., 2004b; 2010):

$$\mathbf{G\_{CLI}}\left(\mathbf{s}\right) = \frac{K\_I}{\mathbf{s}} \mathbf{G\_{CLD}}\left(\mathbf{s}\right) = \frac{K\_I \mathbf{G\_P}\left(\mathbf{s}\right)}{\mathbf{s}\left(1 + \mathbf{G\_C}\left(\mathbf{s}\right) \mathbf{G\_P}\left(\mathbf{s}\right)\right)}\tag{43}$$

Therefore, in order to achieve optimal disturbance-rejection properties, the function GCLI should be applied instead of GCL in the MO criteria (6) and (7).

However, the expression for the PID controller parameters – due to higher-order equations – is not analytic and the optimisation procedure should be used (Vrančić et al., 2010). Initially, the derivative gain KD is calculated from expression (25). As such, the proportional and integral term gains are calculated as follows (Vrančić et al., 2010):

$$\begin{aligned} K\_P &= \frac{\beta - \sqrt{\beta^2 - \alpha \gamma}}{\alpha} \\ K\_I &= \frac{\left(1 + K\_P A\_0^\*\right)^2}{2\left(K\_D A\_0^{\*2} + A\_1^\*\right)} \end{aligned} \tag{44}$$

Magnitude Optimum Techniques for PID Controllers 93

• The PI controller parameters can be calculated by fixing KD=0 and using expression (44). If the value α=0 or if the proportional gain KP is too high or has a different sign to the process gain (KPR=A0), set KP manually to some more suitable value and then

and then recalculate KI from (44).

0 10 20 30 40

time [s]

Process GP3: Closed−loop responses

0 10 20 30

time [s]

(29). The parameters for all of the controllers are given in Table 4.

Process GP1: Closed−loop responses

recalculate KI from (44).

−0.2

−0.5

MOMI method.

DRMO method.

0

0.5

1

1.5

0

0.2

0.4

0.6

0.8

different sign to the process gain (KPR=A0), set KP manually to some more suitable value

−0.2

−0.2 0 0.2 0.4 0.6 0.8 1 1.2

Moments (areas) PID PI I

Fig. 11. Closed-loop responses to step-like input disturbance (d) for processes GP1 to GP4 when using a PID controller (\_\_), a PI controller (--) and an I controller (-.-) tuned by the

The proposed DRMO tuning procedure will be illustrated by the same four process models (41), as before. The PID and PI controllers' parameters are calculated by the procedure given above. Note that the I controller parameters remain the same as with the MOMI method

 A1 A2 A3 A4 A5 KI KP KD TF KI KP K<sup>I</sup> GP1 6 36 216 1296 7776 10.1 10 0 0 1.75 10 0.08 GP2 6 27 108 405 1458 2.92 10 14.5 0.2 0.25 1 0.08 GP3 6 21 56 126 252 0.27 0.97 0.96 0.2 0.17 0.43 0.08 GP4 6 18.5 39.3 65.4 91.4 0.18 0.52 0.45 0.2 0.14 0.29 0.08

Table 4. The values of moments and controller parameters for processes (41) using the

0

0.2

0.4

0.6

0.8

0 10 20 30 40 50

time [s]

Process GP4: Closed−loop responses

0 10 20 30 40

time [s]

Process GP2: Closed−loop responses

where

$$\begin{aligned} \alpha &= A\_1^{\*3} + A\_0^{\*2} A\_3^\* - 2 A\_0^\* A\_1^\* A\_2^\* \\ \beta &= A\_1^{\*} A\_2^\* - A\_0^\* A\_3^\* + K\_D \left( A\_0^{\*} A\_1^{\*2} - A\_0^{\*2} A\_2^\* \right) \\ \gamma &= K\_D^3 A\_0^{\*4} + 3 K\_D^2 A\_0^{\*2} A\_1^\* + K\_D \left( 2 A\_0^{\*} A\_2^\* + A\_1^{\*2} \right) + A\_3^\* \end{aligned} \tag{45}$$

The optimisation iteration steps consist of modifying the derivative gain KD and recalculating the remaining two parameters from (44) until the following expression becomes true (Vrančić et al., 2010):

$$\begin{aligned} &-4A\_0A\_4K\_IK\_D - 2A\_3K\_D + 2A\_4K\_P - 2A\_5K\_I + 2A\_0A\_4K\_P^2 - 2A\_0A\_2K\_D^2 - \\ &-2A\_1A\_3K\_P^2 - 2A\_2^2K\_IK\_D + A\_1^2K\_D^2 + A\_2^2K\_P^2 + 4A\_1A\_3K\_DK\_I = 0 \end{aligned} \tag{46}$$

Any method that employs an iterative search for a numeric solution – that solves the system of nonlinear equations – can be applied. However, in Vrančić et al. (2004a) it was shown that the initially calculated parameters of the PID controller are usually very close to optimal ones. Therefore, a simplified (sub-optimal) solution is to use only the initial PID parameters. In the following text, the simplified version will be applied and denoted as the DRMO tuning method.

Note that the PI controller parameters do not require any optimisation procedure. The derivative gain is fixed at KD=0 and the PI controller parameters are then calculated from expression (44).

The PID controller tuning procedure, according to the DRMO method, can therefore proceed as follows:


However, the expression for the PID controller parameters – due to higher-order equations – is not analytic and the optimisation procedure should be used (Vrančić et al., 2010). Initially, the derivative gain KD is calculated from expression (25). As such, the proportional and

> ( ) ( )

P

KA A

3 \*4 2 \*2 \* \* \* \*2 \* 0 0 1 02 1 3

K A K A A K AA A A

+

K A

1

\* \* \* \* \* \*2 \*2 \* 12 03 01 0 2

D

AA AA K AA A A

3 2

= + + ++

The optimisation iteration steps consist of modifying the derivative gain KD and recalculating the remaining two parameters from (44) until the following expression becomes

0 4 3 4 5 04 02

AAKK AK AK AK AAK AAK

− − + −+ − −

Any method that employs an iterative search for a numeric solution – that solves the system of nonlinear equations – can be applied. However, in Vrančić et al. (2004a) it was shown that the initially calculated parameters of the PID controller are usually very close to optimal ones. Therefore, a simplified (sub-optimal) solution is to use only the initial PID parameters. In the following text, the simplified version will be applied and denoted as the DRMO

Note that the PI controller parameters do not require any optimisation procedure. The derivative gain is fixed at KD=0 and the PI controller parameters are then calculated from

The PID controller tuning procedure, according to the DRMO method, can therefore

• If the process model is not known a priori, modify the process steady-state by changing

• Find the steady-state process gain KPR=A0 and moments A1-A5 by using numerical integration (summation) from the beginning to the end of the process step response according to expressions (17) and (18). If the process model is defined, calculate the gain

• Fix the filter time constant TF to some desired value and calculate moments and the derivative gain KD from (24) and (25). Calculate the remaining controller parameters from expression (44). If the value α=0 or if the proportional gain KP is too high or has a

P ID D P D I

ID D P I P D

4 2 2 22 2 2 2 4 0

AAK AKK AK AK AAK K

− − +++ =

D

+

2

P

K

I

\*3 \*2 \* \* \* \* 1 0 3 012

A A A AAA

=−+ −

DD D

2 2 22 22 13 2 1 2 13

2

=

K

2

β β αγ α − − <sup>=</sup>

> <sup>2</sup> \* 0 \*2 \* 0 1

( )

( )

, (44)

2 2

. (45)

. (46)

integral term gains are calculated as follows (Vrančić et al., 2010):

=+ −

α β

γ

true (Vrančić et al., 2010):

tuning method.

expression (44).

proceed as follows:

the process input signal.

and moments from expression (20).

where

different sign to the process gain (KPR=A0), set KP manually to some more suitable value and then recalculate KI from (44).

• The PI controller parameters can be calculated by fixing KD=0 and using expression (44). If the value α=0 or if the proportional gain KP is too high or has a different sign to the process gain (KPR=A0), set KP manually to some more suitable value and then recalculate KI from (44).

Fig. 11. Closed-loop responses to step-like input disturbance (d) for processes GP1 to GP4 when using a PID controller (\_\_), a PI controller (--) and an I controller (-.-) tuned by the MOMI method.

The proposed DRMO tuning procedure will be illustrated by the same four process models (41), as before. The PID and PI controllers' parameters are calculated by the procedure given above. Note that the I controller parameters remain the same as with the MOMI method (29). The parameters for all of the controllers are given in Table 4.


Table 4. The values of moments and controller parameters for processes (41) using the DRMO method.

Magnitude Optimum Techniques for PID Controllers 95

0 10 20 30 40

Process GP4: Closed−loop responses

MOMI method DRMO method

0 10 20 30 40

time [s]

Process GP2: Closed−loop responses MOMI method DRMO method

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

−0.2 0 0.2 0.4 0.6 0.8 1 1.2

Fig. 13. A comparison of process output disturbance rejection performance for processes GP1 to GP4 when using a PI controller tuned the by MOMI (\_\_) and DRMO (--) tuning methods.

0 10 20 30 40

Process GP3: Closed−loop responses

0 10 20 30 40

MOMI method DRMO method

Process GP1: Closed−loop responses MOMI method DRMO method

−0.02

−0.5

0

0.5

1

1.5

0

0.02

0.04

0.06

0.08

A step-like disturbance (d) has been applied to the process input. The process output responses, when using the PID and the PI controllers, are shown in Figures 12 and 13. It can be clearly seen that the closed-loop performance for processes GP1 and GP2 is now improved when compared with the original MOMI method.

However, improved disturbance-rejection has its price. Namely, the optimal controller parameters for disturbance-rejection are usually not optimal for reference following. Deterioration in tracking performance, in the form of larger overshoots, can be expected for the lower-order processes. A possible solution for improving deteriorated tracking performance, while retaining the obtained disturbance-rejection performance, is to use a 2- DOF PID controller, as shown in Figure 1. Namely, it has been shown that tracking performance can be optimised by choosing b=c=0 (Vrančić et al., 2010). The closed-loop responses on a step-wise reference changes and input disturbances (at the mid-point of the experiment) are shown in Figures 14 and 15. It can be seen that the overshoots are reduced when using b=c=0 while retaining disturbance-rejection responses.

Fig. 12. A comparison of process output disturbance-rejection performance for processes GP1 to GP4 when using a PID controller tuned by the MOMI (\_\_) and DRMO (--) tuning methods.

A step-like disturbance (d) has been applied to the process input. The process output responses, when using the PID and the PI controllers, are shown in Figures 12 and 13. It can be clearly seen that the closed-loop performance for processes GP1 and GP2 is now improved

However, improved disturbance-rejection has its price. Namely, the optimal controller parameters for disturbance-rejection are usually not optimal for reference following. Deterioration in tracking performance, in the form of larger overshoots, can be expected for the lower-order processes. A possible solution for improving deteriorated tracking performance, while retaining the obtained disturbance-rejection performance, is to use a 2- DOF PID controller, as shown in Figure 1. Namely, it has been shown that tracking performance can be optimised by choosing b=c=0 (Vrančić et al., 2010). The closed-loop responses on a step-wise reference changes and input disturbances (at the mid-point of the experiment) are shown in Figures 14 and 15. It can be seen that the overshoots are reduced

0 10 20 30 40

Process GP4: Closed−loop responses

MOMI method DRMO method

0 10 20 30 40

time [s]

Process GP2: Closed−loop responses

MOMI method DRMO method

−0.02

−0.2 0 0.2 0.4 0.6 0.8 1 1.2

Fig. 12. A comparison of process output disturbance-rejection performance for processes GP1 to GP4 when using a PID controller tuned by the MOMI (\_\_) and DRMO (--) tuning methods.

0

0.02

0.04

0.06

0.08

when compared with the original MOMI method.

when using b=c=0 while retaining disturbance-rejection responses.

0 10 20 30 40

Process GP3: Closed−loop responses

MOMI method DRMO method

0 10 20 30 40

Process GP1: Closed−loop responses MOMI method DRMO method

−0.02

−0.5

0

0.5

1

1.5

0

0.02

0.04

0.06

0.08

time [s]

Fig. 13. A comparison of process output disturbance rejection performance for processes GP1 to GP4 when using a PI controller tuned the by MOMI (\_\_) and DRMO (--) tuning methods.

Magnitude Optimum Techniques for PID Controllers 97

0

2

0

Fig. 15. Process output tracking and disturbance-rejection performance for processes GP1 to GP4 when using a PI controller tuned by the DRMO tuning method for the controller

The DRMO tuning method will be illustrated on the same three-water-column laboratory setup, described in the previous section. According to the previously calculated values of moments (39), the PID controller parameters are the following (the proportional gain has

The closed-loop responses, when setting the parameter b=c=0.1, are shown in Figure 16. Similarly, as with the MOMI method, the set-point has been changed from 1.2 to 1.5 at t=300s and is returned to 1.2 at t=900s. A step-like disturbance has been added to the process input at t=700s and t=1300s. The disturbance rejection performance is now improved when compared with Figure 10. A comparison of responses obtained by the MOMI and the DRMO methods with PID controllers is shown in Figure 17. It is clear that the tracking response is slower and with a smaller overshoot, while the disturbance-rejection is

0.5

1

1.5

0.5

1

1.5

0 20 40 60

Process GP4 output: Closed−loop response

0 20 40 60 80

time [s]

0.59, 19.7, 264 KK K IPD === (47)

b=0 b=1

b=0 b=1

Process GP2 output: Closed−loop response

0 5 10 15 20

Process GP3 output: Closed−loop response

0 20 40 60 80

been limited to value KP=10/A0) for the chosen TF=1s:

b=0 b=1

b=0 b=1

Process GP1 output: Closed−loop response

0 0.2 0.4 0.6 0.8 1 1.2 1.4

2

0

parameters b=c=0 and b=c=1.

significantly improved.

0.5

1

1.5

Fig. 14. Process output tracking and disturbance-rejection performance for processes GP1 to GP4 when using a PID controller tuned by the DRMO tuning method for the controller parameters b=c=0 and b=c=1.

0 5 10 15 20

Process GP3 output: Closed−loop response

0 20 40 60 80

b=0 b=1

b=0 b=1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

2

0

Fig. 14. Process output tracking and disturbance-rejection performance for processes GP1 to GP4 when using a PID controller tuned by the DRMO tuning method for the controller

0.5

1

1.5

0 20 40 60

Process GP4 output: Closed−loop response

0 20 40 60 80

time [s]

b=0 b=1

b=0 b=1

Process GP2 output: Closed−loop response

Process GP1 output: Closed−loop response

0 0.2 0.4 0.6 0.8 1 1.2 1.4

2

0

parameters b=c=0 and b=c=1.

0.5

1

1.5

Fig. 15. Process output tracking and disturbance-rejection performance for processes GP1 to GP4 when using a PI controller tuned by the DRMO tuning method for the controller parameters b=c=0 and b=c=1.

The DRMO tuning method will be illustrated on the same three-water-column laboratory setup, described in the previous section. According to the previously calculated values of moments (39), the PID controller parameters are the following (the proportional gain has been limited to value KP=10/A0) for the chosen TF=1s:

$$K\_I = 0.59, \ K\_P = 19.7, \ K\_D = 264 \tag{47}$$

The closed-loop responses, when setting the parameter b=c=0.1, are shown in Figure 16. Similarly, as with the MOMI method, the set-point has been changed from 1.2 to 1.5 at t=300s and is returned to 1.2 at t=900s. A step-like disturbance has been added to the process input at t=700s and t=1300s. The disturbance rejection performance is now improved when compared with Figure 10. A comparison of responses obtained by the MOMI and the DRMO methods with PID controllers is shown in Figure 17. It is clear that the tracking response is slower and with a smaller overshoot, while the disturbance-rejection is significantly improved.

Magnitude Optimum Techniques for PID Controllers 99

Tracking response

300 350 400 450 500 550 600

DRMO method MOMI method

DRMO method MOMI method

t [s]

Disturbance rejection response

t [s]

Fig. 17. A comparison of the process closed-loop responses in the hydraulic setup with PID

The purpose of this Chapter is to present tuning methods for PID controllers which are based on the Magnitude Optimum (MO) method. The MO method usually results in fast and stable closed-loop responses. However, it is based on demanding criteria in the frequency domain, which requires the reliable estimation of a large number of the process

It was shown that the same MO criteria can be satisfied by performing simple time-domain experiments on the process (steady-state change of the process). Namely, the process can be parameterised by the moments (areas) which can be simply calculated from the process steady-state change by means of repetitive integrations of time responses. Hence, the method is called the "Magnitude Optimum Multiple Integration" (MOMI) method. The measured moments can be directly used in the calculation of the PID controller parameters without making any error in comparison with the original MO method. Besides this, from the time domain responses, the process moments can also be calculated from the process transfer function (if available). Therefore, the MOMI method can be considered to be a universal method which can be used either with the process model or the process time-

650 700 750 800 850

parameters. In practice, such high demands cannot often be satisfied.

controllers tuned by the MOMI and DRMO methods.

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7

1.48

6. Conclusion

responses.

1.49

1.5

1.51

1.52

1.53

Fig. 16. The process closed-loop response in the hydraulic setup when using the PID controller tuned by the DRMO method.

400 600 800 1000 1200 1400

400 600 800 1000 1200 1400

t [s]

Fig. 16. The process closed-loop response in the hydraulic setup when using the PID

t [s]

Controller output [V]

Process ouput and set−point [V]

process output set−point

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7

0

controller tuned by the DRMO method.

2

4

6

8

10

Fig. 17. A comparison of the process closed-loop responses in the hydraulic setup with PID controllers tuned by the MOMI and DRMO methods.
