**3. Frequency domain analysis of closed-loop USOPDT processes**

The argument and the magnitude of the loop transfer function (4) are given by

*φL(w)= -3π/2 – dw - atan(w) - atan(τSw) + atan(τIw) + atan(τDw)* (6)

$$A\_L\left(w\right) = \left|\mathcal{G}\_L\left(jw\right)\right| = K\_\mathcal{C} \frac{\sqrt{1 + \left(\tau\_I w\right)^2} \sqrt{1 + \left(\tau\_D w\right)^2}}{\tau\_I w^2 \sqrt{1 + w^2} \sqrt{1 + \left(\tau\_S w\right)^2}}\tag{7}$$

*<sup>U</sup> τU=1 ω w*

In the sequel, our focus of interest is the design of PID-like controllers when applied to

time constants, respectively. In order to simplify the analysis and in order to facilitate

*K* . Thus, the original process and controller parameters are replaced with the dimensionless

Observe now that, the loop transfer function of an USOPDT system in connection with a

ˆˆ ˆ 1 1 exp ( )ˆ ˆˆ ˆ 1 1 *CI D*

 

*I S*

*K s ds G s*

**3. Frequency domain analysis of closed-loop USOPDT processes**  The argument and the magnitude of the loop transfer function (4) are given by

*LL C*

*A w G jw K*

 

while, using the pre-filter *GF, PID(ŝ)=(τΙŝ+1)-1*, the closed-loop transfer function becomes

 

*C D*

*φL(w)= -3π/2 – dw - atan(w) - atan(τSw) + atan(τIw) + atan(τDw)* (6)

1 1

*S*

 

*D*

2 2

 

2 2 2

*w w*

1 1

*ww w*

 

*s s s K s s ds*

<sup>S</sup>

 

*K s s ds*

*ss s*

( 1)exp( ) ˆ ˆ ( )ˆ ˆˆ ˆ ˆ ˆ ˆ ( 1)( 1) ( 1)( 1)exp( ) *C D*

Relations (2) and (5) are next elaborated for the derivation of the tuning methods proposed

comparisons, all system and controller parameters are normalized with respect to

 exp ( ) 1 1 *<sup>P</sup> S U*

*<sup>U</sup>* are the process gain, the time delay and the stable and unstable

*K ds*

 *s s*

Original Parameters

/ *s* ˆ

*<sup>U</sup> K Κ=1* 

*I U* / *KC K KK C C*

(3)

Normalized Parameters

> *U*

> > *<sup>U</sup>* and

(4)

(5)

(7)

*s s U*

Normalized Parameters

 *S SU* 

 *D DU* /

control USOPDT process, with the following transfer function model

*G s*

 

*L*

 

*G s*

Table 1. Normalized vs. original system parameters.

*d d d* /

Original Parameters

*S* 

 

*D*

where *K* , *d* , *<sup>S</sup>*

in this work.

 and 

parameters shown in Table 1.

PID-like controller, is given by

*CL*

It is not difficult to recognize that the Nyquist plot of the *GL(ŝ)* has tow crossover points with the real axis, which determine the critical (or crossover) frequencies *wmin* and *wmax*, and the critical gains *KC,min=1/AL(wmin)* and *KC,max=1/AL(wmax).* These crossover frequencies are obtained as the solutions of the equation *φL(wC)=-π*, or equivalently, of the equation

$$-\pi \mathcal{Q}\text{-d}w\text{-}+\text{atan}(\text{tw}\_{\mathbb{C}}) + \text{atan}(\text{tr}w\_{\mathbb{C}}) + \text{atan}(\text{tr}\_{\mathbb{D}}w\_{\mathbb{C}}) \cdot \text{atan}(\text{tr}\_{\mathbb{S}}w\_{\mathbb{C}}) = 0 \tag{8}$$

when the values of the *atan* function are assigned in the range *(-π/2, π/2)*. Having computed *wmin* and *wmax*, one can determine the acceptable values for the controller gain *KC*, for which the closed-loop system is stable. In particular *KC,min<KC<KC,max*, where, with subscript "*M*" used for either "min" or "max"

$$K\_{C,M} = \frac{\tau\_I w\_M \sqrt{1 + w\_M^2}}{\sqrt{1 + \left(\tau\_I w\_M\right)^2} \sqrt{1 + \left(\tau\_D w\_M\right)^2}} \tag{9}$$

We next define the increasing gain margin *GMinc*, the decreasing gain margin *GMdec* and the gain margin product of the closed-loop system as follows

$$\text{G} \text{M}\_{\text{inc}} \equiv \text{K}\_{\text{C,max}} \text{\prime K}\_{\text{C}} \quad \text{G} \text{M}\_{\text{dec}} \equiv \text{K}\_{\text{C}} \text{\prime K}\_{\text{C,min}} \tag{10}$$

$$\text{G}\,\text{GM}\_{\text{prvd}} = \text{G}\,\text{M}\_{\text{inv}}\,\text{GM}\_{\text{dec}} = \text{K}\_{\text{C,max}}\,\text{\%}\,\text{K}\_{\text{C,min}}\tag{11}$$

Obviously for the closed loop system to be stable *GMinc* and *GMdec* should be grater than one. Note that, the largest the values of *GMprod*, the more robust the system becomes with respect to the gain uncertainty, if the controller gain *KC* is appropriately selected. Furthermore, the phase margin of the closed loop system is defined by *PM=φL(wG)+π*, where *wG* is the frequency at which *AL(wG)=1*. From (7), one can easily conclude that *wG* is given by the maximum real root of the equation

$$
\sigma\_I^2 \tau\_S^2 \alpha\_\odot^6 + \left(\tau\_I^2 \tau\_S^2 + \tau\_I^2 - K\_\text{C}^2 \tau\_I^2 \tau\_D^2\right) \alpha\_\odot^4 + \left[\tau\_I^2 - K\_\text{C}^2 \left(\tau\_I^2 + \tau\_D^2\right)\right] \alpha\_\odot^2 - K\_\text{C}^2 = 0 \tag{12}
$$

In order to obtain the maximum phase margin for given *d*, *τS*, *τI* and *τD*, the controller gain *KC* should be selected as

$$K\_{\mathbb{C}} = \frac{\tau\_I w\_p \sqrt{1 + w\_p^2}}{\sqrt{1 + \left(\tau\_I w\_p\right)^2} \sqrt{1 + \left(\tau\_D w\_p\right)^2}} \tag{13}$$

where *wp* is the frequency at which the argument of the loop transfer function is maximized. From (6), one can easily conclude that *wp* is given by the solution of *dφL/dωw=wp =0*, or equivalently of the equation

$$-d + \frac{1}{1 + w\_p^2} + \frac{\tau\_I}{1 + \tau\_I^2 w\_p^2} + \frac{\tau\_D}{1 + \tau\_D^2 w\_p^2} - \frac{\tau\_S}{1 + \tau\_S^2 w\_p^2} = 0\tag{14}$$

that results in a fourth order linear equation with respect to *wp <sup>2</sup>*, with only one acceptable positive real root. Substituting *wp* in (6), the respective maximum argument *φL(wp)* is calculated.

PID-Like Controller Tuning for Second-Order Unstable Dead-Time Processes 57

order to systematically present the DPC method, we start by selecting the derivative time constant *τD* equal to the lowest value in the range defined by (16). That is, *τD= τS*. With this

> ˆ ˆ 1 exp ( )ˆ ˆ ˆ <sup>1</sup> *C I*

*K ds G s*

*I*

*L*

On the basis of (17), the two ultimate gains are, in this case, given by

1 1( )

*w*

*w w*

min min

,min <sup>2</sup>

 

techniques are proposed here. These approximations are

2 min min

min

*CL*

gains, *KC,min* and *KC,max*, of the closed loop system, that is

*G s*

 

(17)

*K KK C CC* ,min ,max (19)

,max <sup>2</sup>

 

*I I*

 

 

 

 

> ,min ,min

 

 

2 max max

(20)

(21)

max

1 1( )

*w*

*w w*

(18)

*K s ds*

*s s*

exp( )ˆ ( )ˆ ˆˆ ˆ ˆ ( 1) ( 1)exp( ) *C*

*C*

Clearly, in this case, the closed-loop transfer function has no zeroes. Note also that, if initially *τS>>1*, then, the controller parameter *τD* takes very large values, a fact that is not desirable, for reasons of noise amplification. Unfortunately, as suggested by (16), in this case, large values of *τD* are inevitable and an appropriate filtered derivative should be

Let us now select the controller gain *KC* as the geometric middle point of the two ultimate

Note that this selection of *KC* provides the same robustness against both increasing and decreasing parametric uncertainty of the system gain. This is particularly useful for systems with large values of *d* (i.e. *d>0.3*) where the region of stability is reduced significantly

,

In (20), *wmin* and *wmax* are the two critical frequencies given by the two solutions of the equation (8), when *τD=τS* and when the values of the atan function are assigned in the range *(-π/2,π/2)*. For given *d*, the solution of (8), for *τD=τS*, exists only if *τI* is larger than a critical value *τI,min(d)* (Paraskevopoulos et al, 2006). Since there are no analytical solutions for (8), two very accurate approximations for *wmin* and *wmax* that are obtained by using optimization

<sup>1</sup> <sup>ˆ</sup> (, ) (, ) (1 ) *<sup>w</sup>*

 max max 0.9463( 1) <sup>ˆ</sup> (, ) (, ) 2 0.5609( 1) *<sup>w</sup> <sup>d</sup> wd f d d d*

min

0.006+0.03d/(1.14-d) <sup>ˆ</sup> (, ) 1 0.973 0.05 /(1 ) <sup>ˆ</sup> *wf d <sup>d</sup>*

 

 

   

 

*wd f d <sup>d</sup>*

*C*

*K*

*s s K s ds*

 

 

selection, relations (4) and (5) take the forms

considered.

(Paraskevopoulos et al, 2006).

*C*

*K*

When the maximum phase margin is zero, then the closed-loop system (with the appropriate selection of *KC*) is marginally stable. The solution of *max(PM(d,τI,τD,τS))=0*, yields the acceptable values of the controller parameters *τI* and *τD*, which render the close-loop system stable. Obviously these values depend on the rest of the system parameters. From (8) and for *τI→∞*, one can easily verify that *wmin=0* and *φL(0)=-π*. If, at *wmin=0*, the derivative of *φL* is positive, then, it is obvious that the system has a maximum phase margin grater than zero and can be stabilized with the appropriate *KC*. With this observation, using (14), one can easily verify that, for *τD>τD,min≡1-d-τS*, the closed-loop system can be stabilized. Note here that, when *τS≤1*, *τD,min* is also the smallest *τD* that renders the closed-loop system stable, while when *τS>1*, the system can be stabilized with smaller values of *τD*. Moreover, although the function *φL(τD)* is strictly increasing, the function *GMprod(τD)* is not strictly monotonous. In fact, there exists a very large value of *τD* for which *GMprod(τD)=1* and the system is no longer stabilizable. In the case where *τI→∞*, then *KC,min=1*. Solving the equation *KC,min(τD)=1*, one can determine the maximum value of *τD*, say *τD,max*, for which the system can be stabilized. Unfortunately, the solution of *KC,min(τD)=1* involves nonlinear equations that can only be solved using iterative algorithms. A simple and quite accurate approximate solution for *τD,max* has been obtained through fitting, using the optimization toolbox of MATLAB® and is given by

$$
\hat{\tau}\_{D,\text{max}} \approx 0.85 + \tau\_S \left( -0.46 + 1.5 \,/\, d \right) \tag{15}
$$

The maximum normalized error of this approximation is 6%, when *0.1<τD<10* and *0.01<d<0.9*. In general, it is plausible to obtain a stable closed-loop system by selecting *τD,min<τD<τD,max*. In real practice, when *τD* is close to *τD,min* or *τD,max*, the stability region of the closed-loop system is very small. After extensive search, it has been found that a more suitable range for the selection of *τD* is the following

$$
\tau\_{\text{S}} \sphericalangle \mathfrak{r}\_{\text{D}} \\$
\mathbf{r}\_{\text{S}} \\$
\mathbf{r}\_{\text{S}} \\$
\mathbf{d}/\mathbf{2} \tag{16}
$$

When *τD* is selected in the range defined by (16), very large *PMmax* and *GMprod* can be obtained. Moreover, with this selection the functions *max(PM(τΙ))* and *GMprod(τΙ)* are strictly increasing with respect to *τΙ*. This is a very useful property for the design of PID-like controllers for USOPDT processes. It is worth noticing, at this point, that in order to tune PID-like controllers for USOPDT processes one can distinguish three cases depending on the values of *d* and *τS*. In the case where *τS<0.1* the PID-type controllers can be tuned using tuning rules for UFOPDT systems, assuming that the new normalized dead time is equal to *d+τS*. On the other hand, if *τS>10*, then it is possible to tune the PID-type controller assuming that the system is a second order one with no time delay. In this particular case, the inverse of the eigen-frequency of the closed loop system (without delay) must be at least five times larger than the time delay of the USOPDT system. Finally, in the case where *0.1<τS<10*, the above approximate solutions do not provide accurate results, and it is recommended to use the more accurate tuning rules presented in the following Sections.

#### **4. Controller tuning by assigning the closed-loop system dominant poles**

A first method of tuning PID-like controllers for USOPDT processes is based on the appropriate placement of the dominants poles of the closed-loop system. This method is designated here as the DPC method, since it relies on the satisfaction of dominant poles criteria. In

When the maximum phase margin is zero, then the closed-loop system (with the appropriate selection of *KC*) is marginally stable. The solution of *max(PM(d,τI,τD,τS))=0*, yields the acceptable values of the controller parameters *τI* and *τD*, which render the close-loop system stable. Obviously these values depend on the rest of the system parameters. From (8) and for *τI→∞*, one can easily verify that *wmin=0* and *φL(0)=-π*. If, at *wmin=0*, the derivative of *φL* is positive, then, it is obvious that the system has a maximum phase margin grater than zero and can be stabilized with the appropriate *KC*. With this observation, using (14), one can easily verify that, for *τD>τD,min≡1-d-τS*, the closed-loop system can be stabilized. Note here that, when *τS≤1*, *τD,min* is also the smallest *τD* that renders the closed-loop system stable, while when *τS>1*, the system can be stabilized with smaller values of *τD*. Moreover, although the function *φL(τD)* is strictly increasing, the function *GMprod(τD)* is not strictly monotonous. In fact, there exists a very large value of *τD* for which *GMprod(τD)=1* and the system is no longer stabilizable. In the case where *τI→∞*, then *KC,min=1*. Solving the equation *KC,min(τD)=1*, one can determine the maximum value of *τD*, say *τD,max*, for which the system can be stabilized. Unfortunately, the solution of *KC,min(τD)=1* involves nonlinear equations that can only be solved using iterative algorithms. A simple and quite accurate approximate solution for *τD,max* has been obtained through fitting, using the optimization toolbox of MATLAB®

> 

The maximum normalized error of this approximation is 6%, when *0.1<τD<10* and *0.01<d<0.9*. In general, it is plausible to obtain a stable closed-loop system by selecting *τD,min<τD<τD,max*. In real practice, when *τD* is close to *τD,min* or *τD,max*, the stability region of the closed-loop system is very small. After extensive search, it has been found that a more

When *τD* is selected in the range defined by (16), very large *PMmax* and *GMprod* can be obtained. Moreover, with this selection the functions *max(PM(τΙ))* and *GMprod(τΙ)* are strictly increasing with respect to *τΙ*. This is a very useful property for the design of PID-like controllers for USOPDT processes. It is worth noticing, at this point, that in order to tune PID-like controllers for USOPDT processes one can distinguish three cases depending on the values of *d* and *τS*. In the case where *τS<0.1* the PID-type controllers can be tuned using tuning rules for UFOPDT systems, assuming that the new normalized dead time is equal to *d+τS*. On the other hand, if *τS>10*, then it is possible to tune the PID-type controller assuming that the system is a second order one with no time delay. In this particular case, the inverse of the eigen-frequency of the closed loop system (without delay) must be at least five times larger than the time delay of the USOPDT system. Finally, in the case where *0.1<τS<10*, the above approximate solutions do not provide accurate results, and it is recommended to use

**4. Controller tuning by assigning the closed-loop system dominant poles** 

A first method of tuning PID-like controllers for USOPDT processes is based on the appropriate placement of the dominants poles of the closed-loop system. This method is designated here as the DPC method, since it relies on the satisfaction of dominant poles criteria. In

*D S* ,max 0.85 0.46 1.5 / *d* (15)

*τS+d/2* (16)

and is given by

the more accurate tuning rules presented in the following Sections.

suitable range for the selection of *τD* is the following

ˆ

*τSτD* order to systematically present the DPC method, we start by selecting the derivative time constant *τD* equal to the lowest value in the range defined by (16). That is, *τD= τS*. With this selection, relations (4) and (5) take the forms

$$\mathbf{G}\_{L}(\hat{\mathbf{s}}) = \frac{K\_{\mathbb{C}}\left(\tau\_{I}\hat{\mathbf{s}} + 1\right)\exp\left(-d\hat{\mathbf{s}}\right)}{\tau\_{I}\hat{\mathbf{s}}\left(\hat{\mathbf{s}} - 1\right)}\tag{17}$$

$$G\_{\rm CL}(\hat{s}) = \frac{K\_{\rm C} \exp(-d\hat{s})}{\tau\_I \hat{s}(\hat{s} - 1) + K\_{\rm C}(\tau\_I \hat{s} + 1)\exp(-d\hat{s})} \tag{18}$$

Clearly, in this case, the closed-loop transfer function has no zeroes. Note also that, if initially *τS>>1*, then, the controller parameter *τD* takes very large values, a fact that is not desirable, for reasons of noise amplification. Unfortunately, as suggested by (16), in this case, large values of *τD* are inevitable and an appropriate filtered derivative should be considered.

Let us now select the controller gain *KC* as the geometric middle point of the two ultimate gains, *KC,min* and *KC,max*, of the closed loop system, that is

$$K\_{\mathbb{C}} = \sqrt{K\_{\mathbb{C}, \min} K\_{\mathbb{C}, \max}} \tag{19}$$

Note that this selection of *KC* provides the same robustness against both increasing and decreasing parametric uncertainty of the system gain. This is particularly useful for systems with large values of *d* (i.e. *d>0.3*) where the region of stability is reduced significantly (Paraskevopoulos et al, 2006).

On the basis of (17), the two ultimate gains are, in this case, given by

$$K\_{\text{C,min}} = \frac{\tau\_I w\_{\text{min}} \sqrt{1 + w\_{\text{min}}^2}}{\sqrt{1 + \left(\tau\_I w\_{\text{min}}\right)^2}} \quad , \quad K\_{\text{C,max}} = \frac{\tau\_I w\_{\text{max}} \sqrt{1 + w\_{\text{max}}^2}}{\sqrt{1 + \left(\tau\_I w\_{\text{max}}\right)^2}} \tag{20}$$

In (20), *wmin* and *wmax* are the two critical frequencies given by the two solutions of the equation (8), when *τD=τS* and when the values of the atan function are assigned in the range *(-π/2,π/2)*. For given *d*, the solution of (8), for *τD=τS*, exists only if *τI* is larger than a critical value *τI,min(d)* (Paraskevopoulos et al, 2006). Since there are no analytical solutions for (8), two very accurate approximations for *wmin* and *wmax* that are obtained by using optimization techniques are proposed here. These approximations are

$$\begin{aligned} \hat{w}\_{\min}(d,\tau\_I) &= f\_{w\_{\min}}(d,\tau\_I) \sqrt{\frac{1}{\tau\_I - d(1+\tau\_I)}} \\\\ \hat{w}\_{\max}(d,\tau\_I) &= f\_{w\_{\max}}(d,\tau\_I) \frac{\pi}{2d} \frac{\left(\tau\_I - 0.9463(\tau\_I + 1)d\right)}{\left(\tau\_I - 0.5609(\tau\_I + 1)d\right)} \end{aligned} \tag{21}$$

$$f\_{w\_{\min}}(d,\tau\_I) = 1 + \frac{\left(0.006 + 0.03d/\left(1.14 \text{-d}\right)\right)\hat{\tau}\_{I,\min}}{\left(0.973 + 0.05/\left(1 - d\right)\right)\tau\_I - \hat{\tau}\_{I,\min}}$$

PID-Like Controller Tuning for Second-Order Unstable Dead-Time Processes 59

expressions are given in Table 2, together with their maximum normalized error. The response obtained by the DPC method can be distinguished as follows: For *d<0.157* the method gives three real dominant poles (the two slowest are identical) and the response approximates that of a critical second-order system response. For *d>0.157* the method gives

Fig. 2. A typical root locus of (18) for *d=0.5*, *n=25*, *1.1τI,min<τI<10τI,min* and *KC* given by (19).

Real Axis


Fig. 3. A typical closed loop set-point step response of the USOPDT process and the

5

Time in units of τ

Closed-loop USOPDT response 3rd order system response

10 15 25

20

response of the 3rd order system.

0

0

0.2

0.4

0.6

0.8

Output

1

d

1.2

1.4





Imaginary Axis

10

30

50



Dominant Poles

. These approximate

 

maximum normalized error (M.N.E.), defined by ( )/ ˆ

two complex and one real poles all with the same real part (see also Fig. 4).

$$f\_{w\_{\text{max}}}(d,\tau\_I) = \left(1 + 0.22d^4\right) \left[1 + \left(0.1 - 0.3\sqrt{d}\right) \left(\hat{\tau}\_{I,\text{min}} / \left(\tau\_I\right)^2\right)\right] \tag{22}$$

where ,min ˆ *I* is an approximation of *τI,min*, given by

$$
\hat{\tau}\_{I,\min}(d) = \left(0.0029\text{-}0.0682\sqrt{d} + 1.4941\text{d}\right) / \left(1.003\text{-d}\right)^2\tag{23}
$$

The normalized errors of the ultimate gains, defined by ,min ,min ,min ,min <sup>ˆ</sup> *K KK K C CC C* ( ) / and ,max ,max ,max ,max <sup>ˆ</sup> *K KK K C CC C* ( ) / , where ,max <sup>ˆ</sup> *KC* and ,min <sup>ˆ</sup> *KC* are the approximations of *KC,max* and *KC,min*, respectively, obtained using (21), never exceed 2.2% for *d≤0.9* and *τI> 1.2* ,min ˆ *I* . Moreover the normalized error relative to ,min ˆ *I* never exceeds 1.4% for *d≤0.9*.

Since, here *τD=τS*, and *KC* is obtained according to relations (19)-(23) as a function of *τΙ*, in order to tune a PID-like controller it only remains to specify *τΙ*. In the present Section, we propose to select the controller parameter *τI*, in order to maximize the real part of the slowest dominant pole (i.e. the pole with the smallest real part). This way the resulting closed loop system will have a very fast settling time and, at the same time, a very smooth (non-oscillatory) response.

In order to obtain a pole-zero description of (18), the exponential term in (18) is approximated by the relation

$$\exp(-d\hat{\mathbf{s}}) = \lim\_{n \to \infty} \left[ (d \;/\; n)\hat{\mathbf{s}} + 1 \right]^{-n} \tag{24}$$

From (24), it can be easily recognized that the exponential term *exp(-dŝ)* is equivalent to an infinite number of poles at *ŝ=–n/d+j0*. A typical example of the root locus of (18) is shown in Fig. 2 (for *d=0.5*, *n=25*, *KC* given by (19) and *1.1τI,min<τI<10τI,min*). From this figure, it becomes clear that, there exist three dominant poles that are responsible for the shape of the closedloop system response. The rest of the poles contribute only to the delay of the response. Extensive simulation analysis (for *0<d<0.9*, *τI>τI,min* and *KC,min<KC<KC,max*) shows that the step response of an USOPDT system controlled by a PID-like controller (when *τD= τS*) cannot be easily distinguished from that of a 3rd order system with the same dominant poles and the same initial delay, when *n>20* in (24). This fact is illustrated in Fig. 3.


Table 2. Approximate expressions of *τΙ(d)* for the DPC method.

In order to solve the tuning problem presented above, MATLAB® control toolbox was used to estimate the poles of a 27th order closed loop system (*n=25* in (24)). Moreover, a simple algorithm based on the dissection method was used to find the value of *τI* that maximizes the real part of the slowest dominant pole. Since this procedure cannot be applied on-line due to its computational burden, the function *τI(d)* obtained by the DPC method has been approximated by analytical functions ˆ ( ) *d* . The parameters involved in these functions have been estimated using the optimization toolbox of MATLAB®, in order to minimize the

max

,min ˆ ( ) 0.0029-0.0682 d 1.4941d /(1.003-d) *d*

The normalized errors of the ultimate gains, defined by ,min ,min ,min ,min <sup>ˆ</sup> *K KK K*

of *KC,max* and *KC,min*, respectively, obtained using (21), never exceed 2.2% for *d≤0.9* and *τI>* 

Since, here *τD=τS*, and *KC* is obtained according to relations (19)-(23) as a function of *τΙ*, in order to tune a PID-like controller it only remains to specify *τΙ*. In the present Section, we propose to select the controller parameter *τI*, in order to maximize the real part of the slowest dominant pole (i.e. the pole with the smallest real part). This way the resulting closed loop system will have a very fast settling time and, at the same time, a very smooth

In order to obtain a pole-zero description of (18), the exponential term in (18) is

exp( ) lim ( / ) 1 ˆ ˆ *<sup>n</sup> n*

From (24), it can be easily recognized that the exponential term *exp(-dŝ)* is equivalent to an infinite number of poles at *ŝ=–n/d+j0*. A typical example of the root locus of (18) is shown in Fig. 2 (for *d=0.5*, *n=25*, *KC* given by (19) and *1.1τI,min<τI<10τI,min*). From this figure, it becomes clear that, there exist three dominant poles that are responsible for the shape of the closedloop system response. The rest of the poles contribute only to the delay of the response. Extensive simulation analysis (for *0<d<0.9*, *τI>τI,min* and *KC,min<KC<KC,max*) shows that the step response of an USOPDT system controlled by a PID-like controller (when *τD= τS*) cannot be easily distinguished from that of a 3rd order system with the same dominant poles and the

Range of d Estimated τΙ(d) M.N.E. 0<*d*<0.17 <sup>2</sup> 3.06 4.19 12.66 *dd d* 1.5% 0.17<*d*<0.9 2 5 <sup>1</sup> 3.47 -2.9 8.37 18.28 0.95 *dd d d d* 2%

In order to solve the tuning problem presented above, MATLAB® control toolbox was used to estimate the poles of a 27th order closed loop system (*n=25* in (24)). Moreover, a simple algorithm based on the dissection method was used to find the value of *τI* that maximizes the real part of the slowest dominant pole. Since this procedure cannot be applied on-line due to its computational burden, the function *τI(d)* obtained by the DPC method has been

have been estimated using the optimization toolbox of MATLAB®, in order to minimize the

 

*ds d n s* 

,min ( , ) 1 0.22 1 0.1 0.3 / ˆ *wfd d d*

is an approximation of *τI,min*, given by

*C CC C* ( ) / , where ,max <sup>ˆ</sup>

. Moreover the normalized error relative to ,min ˆ

same initial delay, when *n>20* in (24). This fact is illustrated in Fig. 3.

Table 2. Approximate expressions of *τΙ(d)* for the DPC method.

approximated by analytical functions ˆ ( ) *d*

 

and ,max ,max ,max ,max <sup>ˆ</sup> *K KK K*

(non-oscillatory) response.

approximated by the relation

where ,min ˆ *I* 

*1.2* ,min ˆ *I* 

2 4

(23)

*KC* and ,min <sup>ˆ</sup>

*I* 

<sup>2</sup>

(24)

. The parameters involved in these functions

 

*C CC C* ( ) /

never exceeds 1.4% for *d≤0.9*.

*KC* are the approximations

(22)

  maximum normalized error (M.N.E.), defined by ( )/ ˆ . These approximate expressions are given in Table 2, together with their maximum normalized error. The response obtained by the DPC method can be distinguished as follows: For *d<0.157* the method gives three real dominant poles (the two slowest are identical) and the response approximates that of a critical second-order system response. For *d>0.157* the method gives two complex and one real poles all with the same real part (see also Fig. 4).

Fig. 2. A typical root locus of (18) for *d=0.5*, *n=25*, *1.1τI,min<τI<10τI,min* and *KC* given by (19).

Fig. 3. A typical closed loop set-point step response of the USOPDT process and the response of the 3rd order system.

PID-Like Controller Tuning for Second-Order Unstable Dead-Time Processes 61

all values of *d<0.9* and the function *τI(d)* is approximated using the optimization toolbox of

 

estimation of the function *τI(d)* is less than 2.8%, for all these approximations. This error in *τ<sup>I</sup>*

Method Range of d Estimated τΙ(d) M.N.E.

Table 3. Estimates of *τΙ(d)* for the tuning methods based on closed-loop time-domain criteria.

DPC -12.61, -2.502±j0.175 -0.425, -0.412±j1.312 -0.0377, -0.0377±j0.412 FST -12.949, -2.326±j1.641 -0.516, -0.368±j1.302 -0.0550, -0.0291±j0.411 OPOS -12.964, -2.318±j1.675 -0.556, -0.349±j1.299 -0.0609, -0.0262±j0.410 ISE-Sp -14.765, -1.378±j4.231 -0.785, -0.237±j1.298 -0.0883, -0.0129±j0.409

<sup>0</sup> 0.5 <sup>1</sup> <sup>2</sup> 2.5 <sup>3</sup> 3.5 <sup>4</sup> 4.5 <sup>5</sup> <sup>0</sup>

DPC FST

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> <sup>14</sup> <sup>16</sup> <sup>18</sup> <sup>20</sup> <sup>0</sup>

DPC FST

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>0</sup>

DPC FST

Time in units of τ<sup>U</sup>

Fig. 4. Characteristic set-point step responses obtained by the proposed tuning methods.

is used instead of *τI*, to apply the FST method, the maximum

OPOS ISE-Sp

OPOS ISE-Sp

OPOS ISE-Sp

Method *d*=0.1 *d*=0.5 *d*=0.9

Table 4. Locations of dominant poles for some typical examples.

0<*d*<0.17 0.017 0.42 8.08 *d d* 1.5%

2 5 3.26 -1.96 5.55 15.47 0.96

2 5 2.29 +0.69 2.29 15.07 0.96

2 5 0.1 +2.47 2.78 5.59 0.95

 *dd d d d*

 *dd d d d*

 *dd d d d*

does not produce a significant change in the response of the closed loop system.

are given in Table 3. The M.N.E. in the

**d=0.1**

**d=0.5**

**d=0.9**

2%

2.8%

2.7%

MATLAB®. The resulting approximations ˆ ( ) *d*

0.17<*d*<0.9

OPOS 0<*d*<0.9

ISE-Sp 0<*d*<0.9

For example, when ˆ ( ) *d*

 

0.5 1 1.5

0.5 1

System Output

0.5

1

normalized error in the settling time is less than 0.5%.

FST

#### **5. Controller tuning based on closed-loop time-response criteria**

In this Section, we consider again that *τD=τS* as well as that *KC* is obtained through (19)-(23), and we present three alternative methods for the selection of the parameter *τI*. These methods are based on some very useful closed-loop set-point step response criteria.

A first, widely used, criterion for tuning PID-like controllers is the fastest settling time (FST) method. In the case of an oscillatory response, the settling time is usually estimated from the envelope of the response. Since for systems with time delay the closed-loop response is not known in analytical form, to estimate here the envelope of the response, we use the response of a third-order system having the dominant poles of the closed-loop USOPDT system. In particular, the response of a third order system, with two complex poles (*pI,1=a+jb* and *pI,2=ajb*) and one real pole (*pR*), is given by

$$y(t) = 1 - \left[ e^{-\zeta \cdot w\_0 t} \left( A \cos(w\_n t) + B \sin(w\_n t) \right) + C e^{-p\_n t} \right] \tag{25}$$

where 2 2 *w ab* <sup>0</sup> , *ζ=a/w0*, <sup>2</sup> <sup>0</sup> 1 *w w <sup>n</sup>* , *A=pR(-pR+2ζw0)/D*, *B=pRw0(-ζpR+2ζ2w0 w0)/(Dwn)*, <sup>2</sup> *C* w /D 0 and 2 2 R R00 *D* -p +2p ζw -w . The two envelopes (top and bottom) of (25) are given by

$$\mathcal{Y}\_{\mathcal{S}^{1,2}}(t) = \mathbb{1} \pm \left[ e^{-\zeta^\* w\_0 t} \left( \sqrt{A^2 + B^2} \right) + \mathcal{C} e^{-p\_R t} \right] \tag{26}$$

Therefore, for the application of the FST method, a simple algorithm based on the dissection method, is used to estimate the value of parameter *τI* that minimizes the time *tstl* required for obtaining 1 1 ( ) 0.01 *g stl y t* .

A second criterion, on the basis of which the tuning of the PID-like controller is performed, stems from the need to provide the fastest possible set-point step response of the closed loop system with a maximum overshoot of 1% (OPOS method). Also in this case a search algorithm is used to estimate the smallest value of the parameter *τI* (and hence the fastest response) for which the maximum of *y(t)*, given by (25), is smaller than 1.01 for all *t>0*.

Finally, the third method is based on the minimization of the integral of squared errors due to a unit step change in the set point (ISE-Sp method). The first part of the response, for *t<d*, can not be affected by the controller. Hence, for the optimization problem of minimizing the integral of squared errors, one can use the response obtained by (25). The integral of *(1-y(t))2* can then be calculated analytically, and it is given by

$$\begin{split} ISE\_{Sp} &= \int\_0^\infty \left( 1 - y(t) \right)^2 dt = \frac{\mathcal{C}^2}{2p\_R} + 2\mathcal{C} \frac{A(\zeta \, w\_0 + p\_R) + Bw\_n}{p\_R^2 + w\_0^2 + 2\zeta \, w\_0 p\_R} \\ &+ \left[ A^2 \left( 1 + \zeta^2 \right) + B^2 \left( 1 - \zeta^2 \right) + 2AB\zeta \sqrt{1 - \zeta^2} \right] \left( 4w\_0 \zeta \right)^{-1} \end{split} \tag{27}$$

Then, using (27) in combination with a simple search algorithm, the parameter *τI* that minimizes the value of *ISESp* can be estimated.

All three methods presented above cannot be applied on-line because of the excessive computational burden required to calculate the values of the three dominant poles. For this reason, the parameter *τI* obtained by the application of these methods, is next calculated for

In this Section, we consider again that *τD=τS* as well as that *KC* is obtained through (19)-(23), and we present three alternative methods for the selection of the parameter *τI*. These

A first, widely used, criterion for tuning PID-like controllers is the fastest settling time (FST) method. In the case of an oscillatory response, the settling time is usually estimated from the envelope of the response. Since for systems with time delay the closed-loop response is not known in analytical form, to estimate here the envelope of the response, we use the response of a third-order system having the dominant poles of the closed-loop USOPDT system. In particular, the response of a third order system, with two complex poles (*pI,1=a+jb* and *pI,2=a-*

> <sup>0</sup> () 1 cos( ) sin( ) *<sup>R</sup> w t <sup>p</sup> <sup>t</sup> n n y t e A w t B w t Ce*

 <sup>0</sup> 2 2 1,2 () 1 *<sup>R</sup> w t <sup>p</sup> <sup>t</sup> <sup>g</sup> y t e A B Ce* 

Therefore, for the application of the FST method, a simple algorithm based on the dissection method, is used to estimate the value of parameter *τI* that minimizes the time *tstl* required for

A second criterion, on the basis of which the tuning of the PID-like controller is performed, stems from the need to provide the fastest possible set-point step response of the closed loop system with a maximum overshoot of 1% (OPOS method). Also in this case a search algorithm is used to estimate the smallest value of the parameter *τI* (and hence the fastest response) for which the maximum of *y(t)*, given by (25), is smaller than 1.01 for all *t>0*. Finally, the third method is based on the minimization of the integral of squared errors due to a unit step change in the set point (ISE-Sp method). The first part of the response, for *t<d*, can not be affected by the controller. Hence, for the optimization problem of minimizing the integral of squared errors, one can use the response obtained by (25). The integral of *(1-y(t))2*

> 2 2 0

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

2 2

 

*R R R*

*p p w w p*

 

2 22 2 2 1

Then, using (27) in combination with a simple search algorithm, the parameter *τI* that

All three methods presented above cannot be applied on-line because of the excessive computational burden required to calculate the values of the three dominant poles. For this reason, the parameter *τI* obtained by the application of these methods, is next calculated for

 

0 2 2

*<sup>C</sup> <sup>w</sup> <sup>p</sup> Bw ISE y t dt C*

(1 ) (1 ) 2 1 4

*A w*

 

<sup>0</sup> 1 *w w <sup>n</sup>*

(25)

R R00 *D* -p +2p ζw -w . The two envelopes (top and bottom) of

0

 

*R n*

(27)

0 0

, *A=pR(-pR+2ζw0)/D*, *B=pRw0(-ζpR+2ζ2w0-*

(26)

**5. Controller tuning based on closed-loop time-response criteria** 

*jb*) and one real pole (*pR*), is given by

(25) are given by

obtaining 1 1 ( ) 0.01 *g stl y t* .

where 2 2 *w ab* <sup>0</sup> , *ζ=a/w0*, <sup>2</sup>

*w0)/(Dwn)*, <sup>2</sup> *C* w /D 0 and 2 2

can then be calculated analytically, and it is given by

*Sp*

minimizes the value of *ISESp* can be estimated.

 

methods are based on some very useful closed-loop set-point step response criteria.

all values of *d<0.9* and the function *τI(d)* is approximated using the optimization toolbox of MATLAB®. The resulting approximations ˆ ( ) *d* are given in Table 3. The M.N.E. in the estimation of the function *τI(d)* is less than 2.8%, for all these approximations. This error in *τ<sup>I</sup>* does not produce a significant change in the response of the closed loop system.


Table 3. Estimates of *τΙ(d)* for the tuning methods based on closed-loop time-domain criteria.


Table 4. Locations of dominant poles for some typical examples.

Fig. 4. Characteristic set-point step responses obtained by the proposed tuning methods. For example, when ˆ ( ) *d* is used instead of *τI*, to apply the FST method, the maximum normalized error in the settling time is less than 0.5%.

PID-Like Controller Tuning for Second-Order Unstable Dead-Time Processes 63

the derivative term is too large. For this reason, we propose here, to select a priori the derivative term *τD* of the controller, on the basis of the designer's knowledge relative to the process. If there are no restrictions imposed by the process, then it is recommended to select *τD* as large as possible in the range proposed by (16). This way, the resulting closed-loop system has the fastest possible response, for both, the set-point tracking and the load attenuation case, a well as the smallest possible maximum error in the case of regulatory control. Having selected *τD*, as previously mentioned, three methods are then proposed, in

In the case where, the only specification for the closed loop system is the desired phase margin *PMdes*, then it is recommended to tune the PID-like controller in such a way that this single specification is achieved at the maximum phase margin corresponding to the frequency *wp*, namely, when *wG=wp*. This way, the integral reset time *τ<sup>I</sup>* is the smallest possible that satisfies the specification and, hence, the obtained controller provides the fastest possible response, for both set-point tracking and regulatory control. The main steps

*Step 1.* Given the system parameters *d*, *τS*, the controller derivative term *τD* and the phase

*Step 3.* Select the new value of *τI* from the solution of *PMdes=φL(wp)+π,* with respect to *τI*,

*I p w PM dw at w at w at w p Sp p Dp*

*Step 5.* With known *τI,* calculate the corresponding frequency *wp* from (14) and the controller

The above algorithm converges to the correct solution, if such a solution exists, i.e. if for

This method is applicable in the case where the specifications of the closed loop system are described in the form of increasing and decreasing gain margins (*GMinc,des* and *GMdec,des*). To present the method, two iterative algorithms for the calculation of the crossover frequencies

*Step 1.* Start with an initial estimate for *wmin*. An appropriate value for fast convergence is

 min <sup>1</sup> (1 ) *init w d I I* 

 

(29)

<sup>1</sup> tan an( ) an( ) an( ) <sup>2</sup>

(28)

 

*Step 2.* With this value of *τΙ*, calculate *wp* as the maximum real root of (14).

*des*

given *d*, *τS*, *τD* there exists a value of *τI* for which *PM(d,τS,τD,τI)=PMdes*.

*Step 2.* Calculate the error of this approximation using the relation

order to tune the rest of the controller parameters.

**6.1 The Phase Margin (PM) tuning method** 

of this tuning method are the following:

margin specification *PMdes*, set initially *τΙ=0*.

*Step 4.* Repeat Steps 2 and 3 until convergence.

gain *KC* from (13). This completes the method.

**6.2 The Gain Margin (GM) tuning method** 

*wmin* and *wmax* are first presented.

**6.2.1 The wmin algorithm** 

**6.1.1 The PM algorithm** 

which is given by

In Table 4, the locations of the three dominant poles of the closed loop system are given in the case where the normalized dead time takes the values 0.1, 0.5 and 0.9, for all methods presented above. The corresponding closed loop responses obtained from a unit change in the set-point are illustrated in Fig. 4. From these responses and the locations of the dominant poles reported in Table 4, one can easily recognize that the FST and the OPOS methods provide us controllers with similar performance. Moreover, the response obtained when the ISE\_Sp method is used is the fastest, although very oscillatory. Finally, in the case where the DPC method is used, the response obtained is sluggish and smooth. Moreover, since this method yields a large value of *τI*, it provides a very robust controller.

Table 5 presents a stability robustness comparison with other existing PID tuning methods. In particular, the tuning methods presented in Sections 4 and 5 are compared with the R&L method with *λ=2.2* (Rotstein & Lewin, 1991), the P&M method (De Paor and O'Malley, 1989), the H&X method with specifications *Am=1.3* and *φm=10o* (Ho & Xu, 1998), the P&P method based on the ITAE criterion (Poulin & Pomerleau, 1997) and the J&C method based on the IMC tuning rule with *λ=2.5* (Jacob & Chidambaram, 1996), in the special case where *d=0.5* and *τS=1*. Table 5 presents the increasing and decreasing gain margins *GMinc* and *GMdec* as well as the phase margin *PM*. Moreover, it presents the maximum simultaneous multiplicative uncertainty *Aa* of all system parameters (i.e. when the system parameters *d*, *τS*, *K* are increased by *Aa* and *τU* is decreased by *Aa*) and the maximum multiplicative uncertainty *Ad* of the time delay (i.e. when only *d* is increased by *Ad*), for which the closed loop system remains stable. The results presented in Table 5 show that the DPC method provides more robust controllers than most other methods (except the J&C method with *λ=2.5*, that gives a significantly slower response in both set point tracking and regulatory control). The aim of the other three methods, presented in this Section, is to provide faster responses and hence they provide lesser robustness. Finally, it is worth noticing that all the other methods used in robustness comparison are not applicable in cases where *d>0.7*.


Table 5. Robustness performance comparison with other existing tuning methods.

## **6. Controller tuning based on closed-loop stability margins specifications**

When a PID-like controller is used to control an USOPDT process, it is possible, in some cases, to simultaneously satisfy the design specifications *GMdec*, *GMinc*, and *PM* exactly. The PID-like controller sought can be found from the solution of the system of equations (8)-(14). Unfortunately, this system of equations is too complicated to be solved on-line and it is not always solvable. Furthermore, the solution might not be appropriate or useful, especially if the derivative term is too large. For this reason, we propose here, to select a priori the derivative term *τD* of the controller, on the basis of the designer's knowledge relative to the process. If there are no restrictions imposed by the process, then it is recommended to select *τD* as large as possible in the range proposed by (16). This way, the resulting closed-loop system has the fastest possible response, for both, the set-point tracking and the load attenuation case, a well as the smallest possible maximum error in the case of regulatory control. Having selected *τD*, as previously mentioned, three methods are then proposed, in order to tune the rest of the controller parameters.

#### **6.1 The Phase Margin (PM) tuning method**

In the case where, the only specification for the closed loop system is the desired phase margin *PMdes*, then it is recommended to tune the PID-like controller in such a way that this single specification is achieved at the maximum phase margin corresponding to the frequency *wp*, namely, when *wG=wp*. This way, the integral reset time *τ<sup>I</sup>* is the smallest possible that satisfies the specification and, hence, the obtained controller provides the fastest possible response, for both set-point tracking and regulatory control. The main steps of this tuning method are the following:

#### **6.1.1 The PM algorithm**

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

In Table 4, the locations of the three dominant poles of the closed loop system are given in the case where the normalized dead time takes the values 0.1, 0.5 and 0.9, for all methods presented above. The corresponding closed loop responses obtained from a unit change in the set-point are illustrated in Fig. 4. From these responses and the locations of the dominant poles reported in Table 4, one can easily recognize that the FST and the OPOS methods provide us controllers with similar performance. Moreover, the response obtained when the ISE\_Sp method is used is the fastest, although very oscillatory. Finally, in the case where the DPC method is used, the response obtained is sluggish and smooth. Moreover, since this

Table 5 presents a stability robustness comparison with other existing PID tuning methods. In particular, the tuning methods presented in Sections 4 and 5 are compared with the R&L method with *λ=2.2* (Rotstein & Lewin, 1991), the P&M method (De Paor and O'Malley, 1989), the H&X method with specifications *Am=1.3* and *φm=10o* (Ho & Xu, 1998), the P&P method based on the ITAE criterion (Poulin & Pomerleau, 1997) and the J&C method based on the IMC tuning rule with *λ=2.5* (Jacob & Chidambaram, 1996), in the special case where *d=0.5* and *τS=1*. Table 5 presents the increasing and decreasing gain margins *GMinc* and *GMdec* as well as the phase margin *PM*. Moreover, it presents the maximum simultaneous multiplicative uncertainty *Aa* of all system parameters (i.e. when the system parameters *d*, *τS*, *K* are increased by *Aa* and *τU* is decreased by *Aa*) and the maximum multiplicative uncertainty *Ad* of the time delay (i.e. when only *d* is increased by *Ad*), for which the closed loop system remains stable. The results presented in Table 5 show that the DPC method provides more robust controllers than most other methods (except the J&C method with *λ=2.5*, that gives a significantly slower response in both set point tracking and regulatory control). The aim of the other three methods, presented in this Section, is to provide faster responses and hence they provide lesser robustness. Finally, it is worth noticing that all the other methods used in robustness comparison are not applicable in cases where *d>0.7*.

Method *KC τ<sup>I</sup> τD PM(rad) GMinc GMdec aa ad*  DPC 1.618 8.150 1 0.172 1.469 1.462 1.101 1.268 FST 1.622 6.948 1 0.155 1.446 1.436 1.091 1.240 OPOS 1.623 6.539 1 0.148 1.436 1.425 1.088 1.225 ISE-SP 1.632 4.834 1 0.107 1.372 1.353 1.064 1.163 R&L (*λ=*2.2) 2.116 10.24 0.902 0.087 1.173 1.860 1.043 1.103 P&M 1.357 6.960 1 0.133 1.729 1.202 1.103 1.288 H&X 1.518 6.543 1 0.148 1.536 1.332 1.095 1.255 P&P 1.798 8.431 1 0.154 1.325 1.631 1.082 1.204 J&C (*λ=*2.5) 1.573 9.495 1 0.188 1.528 1.443 1.113 1.307

Table 5. Robustness performance comparison with other existing tuning methods.

**6. Controller tuning based on closed-loop stability margins specifications** 

When a PID-like controller is used to control an USOPDT process, it is possible, in some cases, to simultaneously satisfy the design specifications *GMdec*, *GMinc*, and *PM* exactly. The PID-like controller sought can be found from the solution of the system of equations (8)-(14). Unfortunately, this system of equations is too complicated to be solved on-line and it is not always solvable. Furthermore, the solution might not be appropriate or useful, especially if

method yields a large value of *τI*, it provides a very robust controller.

*Step 1.* Given the system parameters *d*, *τS*, the controller derivative term *τD* and the phase margin specification *PMdes*, set initially *τΙ=0*.

*Step 2.* With this value of *τΙ*, calculate *wp* as the maximum real root of (14).

*Step 3.* Select the new value of *τI* from the solution of *PMdes=φL(wp)+π,* with respect to *τI*, which is given by

$$\tau\_I = w\_p^{-1} \tan \left[ PM^{des} + \frac{\pi}{2} + dw\_p + at \tan(\tau\_S w\_p) - at \tan(w\_p) - at \tan(\tau\_D w\_p) \right] \tag{28}$$

*Step 4.* Repeat Steps 2 and 3 until convergence.

*Step 5.* With known *τI,* calculate the corresponding frequency *wp* from (14) and the controller gain *KC* from (13). This completes the method.

The above algorithm converges to the correct solution, if such a solution exists, i.e. if for given *d*, *τS*, *τD* there exists a value of *τI* for which *PM(d,τS,τD,τI)=PMdes*.

#### **6.2 The Gain Margin (GM) tuning method**

This method is applicable in the case where the specifications of the closed loop system are described in the form of increasing and decreasing gain margins (*GMinc,des* and *GMdec,des*). To present the method, two iterative algorithms for the calculation of the crossover frequencies *wmin* and *wmax* are first presented.

#### **6.2.1 The wmin algorithm**

*Step 1.* Start with an initial estimate for *wmin*. An appropriate value for fast convergence is

$$w\_{\min}^{init} = \sqrt{\left[\tau\_I - d(1 - \tau\_I)\right]^{-1}} \tag{29}$$

*Step 2.* Calculate the error of this approximation using the relation

PID-Like Controller Tuning for Second-Order Unstable Dead-Time Processes 65

*Step 1.* For the selected value of *τD*, check if there exists a value of *KC* that is able satisfy all

*Step 2.* Calculate the two controllers obtained by the PM and the GM methods. If the controller with the largest value of *τI* satisfies all three specifications, then this is the

*Step 3.* Assume that *KC,PM* and *τI,PM* are the controller parameters obtained form the application of the PM tuning method and *KC,GM* and *τI,GM* are the controller parameters obtained from the GM tuning method. Then, if none of these two controllers satisfy all specifications, check which controller gives the largest gain *KC*, and distinguish the

1. If *KC,PM>KC,GM*, then in order to satisfy all specifications with the smallest value of *τI*, gradually increase *τI* (starting from the *max(τI,GM,τI,PM)*), while maintaining the same increasing gain margin *GMinc* (by selecting *KC=KC,max(d,τS,τI,τD)/GMinc,des*), until the phase

2. If *KC,PM<KC,GM*, then gradually increase *τI* (starting from the *max(τI,GM,τI,PM)*), while maintaining the same decreasing gain margin *GMdec* (by selecting *KC=* 

Although there are several ways to select the controller parameters in order to satisfy all three specifications (although not exactly), the method presented here is preferred, because it requires the smallest computational effort, since for a given *τI*, the phase margin can be calculated exactly without the use of iterative algorithms (using (12) and *PM=φL(wG)+π*). It is noted here that, in all PID tuning methods presented above, if the response obtained is too oscillatory (due to the small value of *τI*), then, by increasing the value of *τI*, the damping of the closed-loop system increases. From the analysis presented in Section 3, it becomes clear that, when *τI* is increased, the resulting closed-loop system is more robust, and hence all the

The tuning rules presented in the previous sections can significantly be simplified, in the case where *τD=τS*. In this case, the loop transfer function is given by (17), and the solutions of the algorithms presented in Subsections 6.1.1 and 6.2.1-6.2.3, can easily be approximated with satisfactory accuracy for all systems with *0<d<0.9*. In particular, the solutions for *wmin*

approximation of the smallest value of the integral term *τI*, for which (8) has a solution, when *τD=τS*, and when the atan function takes values in the range *(-π/2, π/2)*. Table 6 summarizes useful approximations of some other parameters involved in the aforementioned algorithms. Note that the maximum normalized errors for the parameters *KC,min* and *KC,max*, when their estimates are obtained by (20), using min *w*ˆ and max *w*ˆ as given by (21),

In Table 7, numerical applications of the PM, GM and PGM tuning methods are presented for three processes with normalized dead time 0.1, 0.5 and 0.9. The controller parameters obtained from the application of these tuning methods are presented in the left section of

*d* is an accurate

*KC* , ˆ *I* ). In

*KC,min(d,τS,τI,τD)GMdec,des*), until the phase margin specification is also satisfied.

stability robustness specifications are still satisfied (although not exactly).

and *wmax*, can be approximated by relations (21)-(23). Note that, here, ,min ˆ ( ) *<sup>I</sup>*

*I* .

Table 7 for both the exact *(KC, τI)* and the approximated controller parameters ( ˆ

**6.4 Simplification of the tuning rules for on-line tuning** 

never exceed 2.2% for *d≤0.9* and *τI>1.2* ,min ˆ

**6.3.1 The PGM algorithm** 

following two cases:

This completes the algorithm.

three specifications, when *τI→ ∞*.

controller sought. In the opposite case continue with *Step 3*.

margin specification is also satisfied.

$$e\_r = -\pi \left\langle \text{ $\boldsymbol{\Delta}$ }-d\boldsymbol{w}\_{\text{min}}^{\text{init}} + a \tan(\boldsymbol{w}\_{\text{min}}^{\text{init}}) + at \tan(\tau\_I \boldsymbol{w}\_{\text{min}}^{\text{init}}) + at \tan(\tau\_D \boldsymbol{w}\_{\text{min}}^{\text{init}}) - at \tan(\tau\_S \boldsymbol{w}\_{\text{min}}^{\text{init}}) \right\rangle \tag{30}$$

*Step 3.* Take the new value of *wmin* as min min <sup>1</sup> *new old ww e <sup>r</sup> . Step 4.* Repeat Steps 2 and 3 until a convergence.

#### **6.2.2 The wmax algorithm**

*Step 1.* Start with a very large initial estimate of *wmax*, say max *init w* =104.

*Step 2.* Using (8), calculate the new value of *wmax* as

$$\left[w\_{\text{max}}^{new} = d^{-1}\right] - \frac{\pi}{2} + at\,\text{an}(w\_{\text{max}}^{old}) + at\,\text{an}(\tau\_I w\_{\text{max}}^{old}) + at\,\text{an}(\tau\_D w\_{\text{max}}^{old}) - at\,\text{an}(\tau\_S w\_{\text{max}}^{old})\tag{30}$$

*Step 3.* Repeat Steps 2 and 3 until convergence.

These two algorithms always converge to the correct values of *wmin* and *wmax*, if for given *d*, *τS*, *τD* and *τ<sup>I</sup>* there exists a solution of (8), with respect to *wC*, when the *atan* function takes values in the range *(-π/2,π/2)*. We are now able to present the main steps of proposed GM tuning method.

#### **6.2.3 The GM algorithm**

*Step 1.* Given the system parameters *d*, *τS*, the controller derivative term *τD* and the desired gain matrix product *GMprod,des*, solve *max(PM(d,τI,τD,τS))=0* to obtain *τI,min*.

*Step 2*. Set *τI,1= τI,min* and *τI,2= 103τI,min*.

*Step 3.* Take the new value of *τΙ* as the average of *τI,1* and *τI,2*, i.e. *τΙ=( τI,1+ τI,2)/2*.

*Step 4.* Calculate the values of *wmin* and *wmax* using the wmin Algorithm and the wmax Algorithm, respectively, for the obtained *τI*, and obtain *KC,min* and *KC,max* from (9).

*Step 5.* Calculate the value of *GMprod* from (11).

*Step 6.* If *GMprod<GMprod,des* or *wmin0* or *wmax0*, then *τI,1=τI* or else *τI,2=τI*.

*Step 7.* Repeat Steps 3 to 6 until convergence.

*Step 8.* The controller gain is evaluated from either *KC=KC,max/Ginc,des* or *KC=KC,minGdec,des*. This completes the algorithm.

The above algorithm converges to the correct solution, if such a solution exists, i.e. if for given *d*, *τS*, *τD* there exists a value of *τI* for which *GMprod(d,τS,τD,τI)=GMprod,des*.

#### **6.3 The Phase and Gain Margin (PGM) tuning method**

If the derivative term is a priori selected, then it is not possible, in general, to simultaneously satisfy the specifications on *GMdec*, *GMinc*, and *PM* exactly, with the remaining two free controller parameters. This is due to the fact that, it is not possible to assign three independent specifications with only two independent controller parameters, namely *KC* and *τI*. Indeed, with the controller parameters *KC* and *τI* obtained from the GM Algorithm, in order to satisfy *GMdec* and *GMinc*, then a specific value of the phase margin *PM(d,KC,τI,τD)* is obtained, and, hence, in this case the phase margin cannot be selected independently. Keeping these in mind, we propose here a tuning method, in order to achieve simultaneous, although not exact, satisfaction of all three specifications *PM*, *GMdec* and *GMinc*. This method is based on the tuning methods presented in the previous two subsections. The basic steps, for the selection of the parameters of a PID-like controller that satisfy all three specifications, are the following:
