PID Controller Design for Specified Performance

„How can proper controller adjustments be quickly determined on any control application?” The question posed by authors of the first published PID tuning method J.G.Ziegler and N.B.Nichols in 1942 is still topical and challenging for control engineering community. The reason is clear: just every fifth controller implemented is tuned properly but in fact:  30% of improper performance is due to inadequate selection of controller design method,  30% of improper performance is due to neglected nonlinearities in the control loop,  20% of improper closed-loop dynamics is due to poorly selected sampling period. Although there are 408 various sources of PID controller tuning methods (O ́Dwyer, 2006), 30% of controllers permanently operate in manual mode and 25% use factory-tuning without any up-date with respect to the given plant (Yu, 2006). Hence, there is natural need for effective PID controller design algorithms enabling not only to modify the controlled variable but also achieve specified performance (Kozáková et al., 2010), (Osuský et al., 2010). The chapter provides a survey of 51 existing practice-oriented methods of PID controller design for specified performance. Various options for design strategy and controller structure selection are presented along with PID controller design objectives and performance measures. Industrial controllers from ABB, Allen&Bradley, Yokogawa, FischerRosemont commonly implement built-in model-free design techniques applicable for various types of plants; these methods are based on minimum information about the plant obtained by the well-known relay experiment. Model-based PID controller tuning techniques acquire plant parameters from a step-test; useful tuning formulae are provided for commonly used system models (FOPDT – first-order plus dead time, IPDT – integrator plus dead time, FOLIPDT – first-order lag and integrator plus dead time and SOPDT – second-order plus dead time). Optimization-based PID tuning approaches, tuning methods for unstable plants, and design techniques based on a tuning parameter to continuously modify closed-loop performance are investigated. Finally, a novel advanced design technique based on closed-loop step response shaping is presented and discussed on illustrative examples.

is zero if in the open-loop L(s)=G(s)G R (s), the integrator degree  L = S + R is greater than the degree q of the reference signal w(t)=w q t q , i.e.
where  S and  R are integrator degrees of the plant and controller, respectively, K L is openloop gain and w q is a positive constant (Harsányi et al., 1998).

Principles of controller structure selection based on the plant type
Industrial process variables (e.g.position, speed, current, temperature, pressure, humidity, level etc.) are commonly controlled using PI controllers.In practice, the derivative part is usually switched off due to measurement noise.For pressure and level control in gas tanks, using P controller is sufficient (Bakošová & Fikar, 2008).However, adding derivative part improves closed-loop stability and steepens the step response rise (Balátě, 2004).
where K c and  c are critical gain and frequency of the plant, respectively.Normed time delay  j and parameter  j can be used to select appropriate PID control strategy.According to Tab. 1 (Xue et al., 2007), the derivative part is not used in presence of intense noise and a PID controller is not appropriate for plants with large time delays.
In practical cases N8;16 (Visoli, 2006).The PID controller design objectives are: 1. tracking of setpoint or reference variable w(t) by y(t), 2. rejection of disturbance d(t) and noise n(t) influence on the controlled variable y(t).The first objective called also "servo-tuning" is frequent in motion systems (e.g.tracking required speed); techniques to guarantee the second objective are called "regulator-tuning".

Model-free PID controller design techniques with guaranteed performance
Model-free tuning PID controller techniques are used if plant dynamics is not complicated (without oscillations, vibrations, large overshoots) or if plant modelling is time demanding, uneconomical or even unfeasible.To find PID controller coefficients, instead of a full model usually 2-4 characteristic plant parameters are used obtained from the relay test.

Tuning rules based on critical parameters of the plant
Consider the closed-loop in Fig. 1 with proportional controller.If the controller gain K is successively increased until the process variable oscillates with constant amplitudes, critical parameters can be specified: the period of oscillations T c and the corresponding gain K c .If the controller (4a) is considered, coefficients of P, PI and PID controllers are calculated according to Tab. 2, where  c =2/T c is critical frequency of the plant.

Specification of critical parameters of the plant using relay experiment
To quickly determine critical parameters K c and T c , industrial autotuners apply a relay test (Rotach, 1984) either with ideal relay (IR) or a relay with hysteresis (HR).In the loop in Fig. 1 when adjusting the setpoint w(t) in manual mode and switching SW into "3", a stable limit cycle around y() arises.Due to switching between the levels -M, +M, G(s) is excited by a periodic rectangular signal u(t), (Fig. 3a).Then,  c and K c can be calculated from where the period and amplitude of oscillations T c and A c , respectively, can be obtained from a record of y(t) (Fig. 3b);  DB is the width of the hysteresis.Relay amplitude M is usually adjusted at 3%10% of the control action limit.A relay with hysteresis is used if y(t) is corrupted by measurement noise n(t) (Yu, 2006); the critical gain is calculated using (6c).
Fig. 3.A detailed view of u(t) and y(t) to determine critical parameters K c and T c

Model-based PID controller design with guaranteed performance
Steday-state and dynamic properties of real processes are described by simple FOPDT, IPDT, FOLIPDT or SOPDT models.Model parameters further used to calculate PID controller coefficients can be found e.g. from the plant step responses (Fig. 4 and 5).

PID controller design for specified performance
These methods provide tuning rules are based on a single tuning parameter that enables to systematically affect closed-loop performance by step response shaping.

Performance measures used as a PID tuning parameter
Most frequent parameters for tuning PID controllers are following performance measures (Åström & Hägglund, 1995):   M and G M : phase and gain margins, respectively,  M s and M t : maximum peaks of sensitivity S(j) and complementary sensitivity T(j) magnitudes, respectively,  : required closed-loop time constant.
If a controller G R (j) guarantees that S(j) or T(j) do not overrun prespecified values M s or M t , respectively, defined by over 0,), then the Nyquist plot L(j) of the open-loop L(s)=G(s)G R (s) avoids the respective circle M S or M T , each given by the their center and radius as follows If L(j) avoids entering the circles corresponding to M S or M T , a safe distance from the point C S is kept (Fig. 6a).Typical S(j) and T(j) plots for properly designed controller are plotted in Fig. 6b.The disturbance d(t) is sufficiently rejected if M s (1,2;2).The reference w(t) is properly tracked by the process output y(t) if M t (1,3;2,5).With further increasing of M t the closed-loop tends to be oscillatory.
Relations between them are given by following inequalities The point at which the Nyquist plot L(j) touches the M T circle defines the closed-loop resonance frequency  Mt .

Tuning formulae with performance specification
Table 8 shows open formulae for PID controller design.The coefficients tuning is carried out with respect to closed-loop performance specification.Rules No. 47 -49 consider tuning of ideal PID controller (4a).To apply the Rotach method, knowledge of the plant magnitude G(j) is supposed as well as of the roll-off of argG() at = Mt , where the maximum peak M t of the complementary sensitivity is required.Method No. 50 is based on so-called -tuning, with the resulting closed-loop expressed as a 1 st order system with time constant ; this rule considers a real PID controller (4b) with filtering constant in the derivative part T f =T d /N=0,5D 1 /(1+D 1 ) where  is to be chosen to meet following conditions: >0,25D 1 ; >0,25T 1 (Morari & Zafiriou, 1989).The -tuning technique is used also in the rule No. 51 to design interaction PI controller.
The engineering practice is persistently demanding for PID controller design methods simultaneously guaranteeing several performance criteria, especially maximum overshoot η max and settling time t s .However, we ask the question: how to suitably transform the above-mentioned engineering requirements into frequency domain specifications applicable for PID controller coefficients tuning?The response can be found in Section 3 where a novel original PID controller design method is presented.

Advanced PID controller design method based on sine-wave identification
The presented method is applicable for linear stable SISO systems even with unknown mathematical model.The control objective is to provide required maximum overshoot  max and settling time t s of the process variable y(t).The method enables the designer to prescribe  max and t s within following ranges (Bucz et al., 2010b(Bucz et al., , 2010c)), (Bucz, 2011)   max 0%; 90% and t s 6,5/ c ; 45/ c  for systems without integrator,   max 9,5%; 90% and t s 11,5/ c ; 45/ c  for systems with integrator, where  c is the plant critical frequency.The PID controller design provides guaranteed phase margin  M .The tuning rule parameter is a suitably chosen point of the plant frequency response obtained by a sine-wave signal with excitation frequency  n .The designed controller then moves this point into the gain crossover with the required phase margin  M .With respect to engineering requirements, the pair ( n ; M ) is specified on the closed-loop step response in terms of η max and t s according to parabolic dependencies in Fig. 11 and Fig. 14-16.A multipurpose loop for the proposed sine-wave method is in Fig. 7. Fig. 7. Multipurpose loop for identification and control using the sine-wave method

Plant identification by a sinusoidal excitation input
By switching SW into "4", the loop in Fig. 7 opens; a stable plant with unknown model G(s) is excited by a persistent sinusoid u(t)=U n sin( n t) (Fig. 8a) where U n denotes the amplitude and  n excitation frequency.The plant output y(t)=Y n sin( n t+) is also a persistent sinusoid with the same frequency  n , amplitude Y n and phase shift  with respect to the input excitation sinusoid (Fig. 8b).From the stored records of y(t) and u(t) it is possible to read-off the amplitude Y n and phase shift  n and thus to identify a particular point of the plant frequency response G(j) under excitation frequency  n with coordinates G≡G(j n ) arg () a r g () where =argG( n ).The point G(j n ) can be plotted in the complex plane (Fig. 8c).
The output sinusoid amplitude Y n can be affected by the amplitude U n of the excitation sinusoid generated by the sine wave generator; it is recommended to use U n =37%u max .
Identified plant parameters are represented by the triple  n ,Y n ( n )/U n ( n ),φ( n ).In the SW position "4", identification is performed in the open-loop.Hence, this method is applicable only for stable plants.The excitation frequency  n is to be adjusted prior to identification and taken from the empirical interval (29) (Bucz et al., 2010a(Bucz et al., , 2010b(Bucz et al., , 2011)).

Sine-wave method tuning rules
In the control loop in Fig. 7, let us switch SW in "5"and put the PID controller into manual mode.The closed-loop characteristic equation 1+L(j)=1+G(j)G R (j)=0 at the gain crossover frequency  a * can be broken down into the amplitude and phase conditions as follows where  M is the required phase margin, L(j n ) is the open-loop transfer function.Denote =argG R ( a * ).We are searching for K, T i and T d of the ideal PID controller (4a).Comparing frequency transfer functions of the PID controller in parallel and polar forms 1 () coefficients of PID controller can be obtained from the complex equation at = a * using the substitution G R (j a * )=1/G(j a * ) resulting from the amplitude condition (20a).The complex equation ( 22) is solved as a set of two real equations * cos where ( 23a) is a general rule for calculation of the controller gain K. Using (23a) and the ratio of integration and derivative times =T i /T d in (23b), a quadratic equation in T d is obtained after some manipulations A positive solution of (24) yields the rule for calculating the derivative time T Substituting ( 27a) and ( 27b) into (23a) and (23b), respectively, and ( 26) into (25a), tuning rules in Table 9 are obtained (Bucz et al., 2010a(Bucz et al., , 2010b(Bucz et al., , 2010c(Bucz et al., , 2011)), (Bucz, 2011).Resulting PID controller coefficients guarantee required phase margin  M for =4.
No  9. PI, PD and PID controller tuning rules according to the sine-wave method Note that PI controller tuning rules were derived for T d =0, and PD tuning rules for T i  in (21a).The excitation frequency is taken from the interval (Bucz et al., 2011), (Bucz, 2011) 0,2 ;0,95 obtained empirically by testing the sine-wave method on benchmark examples (Åström & Hägglund, 2000).Shifting the point G(j n )=G(j n )e j  into the gain crossover L A (j n ) on the unit circle M 1 is depicted in Fig. 9a.

Controller structure selection using the "triangle ruler" rule
The argument Θ appearing in tuning rules in Tab. 9 indicates, what angle is to be contributed to the identified phase φ by the controller at  n to obtain the resulting open-loop phase (-180°+ M ) needed to provide the required phase margin  M .The working range of PID controller argument is the union of PI and PD controllers phase ranges symmetric with respect to 0 Re The working range (30) can be interpreted by means of an imaginary transparent triangular ruler turned as in Fig. 9b; its segments to the left and right of the axis of symmetry represent the PD and PI working ranges, respectively.Put this ruler on Fig. 9a, the middle of the hypotenuse on the complex plane origin and turn it so that its axis of symmetry merges with the ray (0,G).Thus, the ruler determines in the complex plane the cross-hatched area representing the full working range of the PID controller argument.The controller type is chosen depending on the situation of the ray (0,L A ) forming the angle  M with the negative real halfaxis: situation of the ray (0,L A ) in the left-hand-sector suggests PD controller, and in the right-hand sector the PI controller.The case when the phase margin  M is achievable using both PI or PID controller is shown in Fig. 9b (Bucz et al., 2010b(Bucz et al., , 2011)), (Bucz, 2011).( 1)(0, 5 1)(0, 25 1)(0,125 1) for three different phase margins  M =40,60,80 each on three excitation levels  1 = n1 / c =0,2;  3 = n3 / c =0,5 and  5 = n5 / c =0,8.Qualitative effect of  nk and  Mj on closed-loop step response is demonstrated.Achieving t s and η max was tested by designing PID controller for a vast set of benchmark examples (Åström & Hägglund, 2000) at excitation frequencies and phase margins expressed by a Cartesian product  Mj × nk of ( 31) and (32) for j=1...8, k=1...6.Acquired dependencies η max =f( M , n ) and t s =( M , n ) are plotted in Fig. 11 (Bucz et al., 2010b(Bucz et al., , 2011)).Phase margin  M [°]   Maximum overshoot Phase margin  M [°]   Relative settling time Dependencies max=f(M,n), for systems without integrator, =4 Dependencies τs=f(M,n), for systems without integrator, =4 Considering (26) resulting from the assumptions of the engineering method, the settling time can be expressed by the relation similar to (17c) (Hudzovič, 1989),  is the curve factor of the step response.In (17c) valid for a 2 nd order closed-loop,is from the interval (1;4) and depends on the relative damping (Hudzovič, 1989).In case of the proposed sine-wave method,  varies in a considerably broader interval (0,5;16) found empirically, and strongly depends on  M , i.e. =f( M ) at the given excitation frequency  n .To examine closed-loop settling times of plants with various dynamics, it is advantageous to define the relative settling time (Bucz et al., 2011) Substituting  n = c into (35), the following relation for the relative settling time is obtained where t s is related to the critical frequency  c .By substituting  c in (37) its left-hand side is constant for the given plant, independent of  n .Fig. 11b depicts (37b) empirically evaluated for different excitation frequencies  nk ; it is evident that at every excitation level  k with increasing phase margin  M the relative settling time τ s first decreases and after achieving its minimum  s_min it increases again.Empirical dependencies in Fig. 11 were approximated by quadratic regression curves and called B-parabolas.B-parabolas are a useful design tool to carry out the transformation :( max ,t s )( n , M ) that enables choosing appropriate values of phase margin and excitation frequencies  M and  n , respectively, to provide performance specified in terms of maximum overshoot  max and settling time t s (Bucz et al., 2011).Note that pairs of B-parabolas at the same level (Fig. 11a, Fig. 11b) are always to be used.
Procedure 1. Specification of  M and  n from  max and t s from B-parabolas prior to designing the controller 1. Set the PID controller into manual mode.Find the plant critical frequency  c using the multipurpose loop in Fig. 7 (SW in position "3").2. From the required settling time t s calculate the relative settling time τ s = c t s .3. On the vertical axis of the plot in Fig. 11b find the value of τ s calculated in Step 2. 4. Choose the excitation level  (e.g. 5 = n5 / c =0,8). 5.For τ s , find the corresponding phase margin  M on the parabola τ s =f( M , n ) with the chosen excitation level found in Step 4. 6. Find  M from Step 5 on the horizontal axis of the plot in Fig. 11a.7.For  M , find the corresponding maximum overshoot η max on the parabola η max =f( M , n ) with the chosen excitation level found in Step 4. 8.If the found η max is inappropriate, repeat Steps 4 to 7 for other parabolas τ s =f( M , n ) and η max =f( M , n ) corresponding to other levels  k = nk / c (related with the choice  5 = n5 / c =0,8 for  k =0,2;0,35;0,50;0,65;0, 95, k=1...4,6).Repeat until both the required performance measures η max and t s are satisfied.
Introduction to PID Controllers -Theory, Tuning and Application to Frontier Areas 20 9. Calculate the excitation frequency  n according to the relation  n = c using the critical frequency  c (from Step 1) and the chosen excitation level  (from Step 4).

Discussion
When choosing  M =40 on the B-parabola corresponding to the excitation level  5 = n5 / c =0,8 (further denoted as B 0,8 parabola), maximum overshoot  max =40% and relative settling time τ s 10 are expected.Point  corresponding to these parameters is located on the left (falling) portion of B 0,8 yielding oscillatory step response (see response  in Fig. 10c).If the phase margin increases up to  M =60, the relative settling time decreases up to the point  on the right (rising) portion of the B 0,8 parabola; the corresponding step response  in Fig. 10c is weakly-aperiodic.For the phase margin  M =80 the B 0,8 parabola indicates a zero maximum overshoot, the relative settling time τ s =20 corresponds to the position  on the B 0,8 parabola with aperiodic step response  (Fig. 10c).If the maximum overshoot  max =20% is acceptable then  M =53 yields the least possible relative settling time τ s =6,5 on the given level  5 =0,8 ("at the bottom" of B 0,8 ) (Bucz et al., 2011), (Bucz, 2011).
Procedure 2. PID controller design using the sine-wave engineering method 1.From the required values (η max ,t s ) specify the couple ( n ; M ) using Procedure 1. 2. Identify the plant using the sinusoidal excitation signal with frequency  n specified in Procedure 1.The switch SW is in position "4".3. Specify =argG( n ), andG(j n ).Calculate the controller argument  by substituting  and  M into (27c); if  is within the range shown in the last column of Tab. 9, go to Step 4, if not, change ( n ; M ) and repeat Steps 1-3. 4. Substitute the identified values =argG( n ), G(j n ) and specified  M into the tuning rules in Tab. 9 to calculate PID controller parameters. 5. Adjust the resulting PID controller values, switch into automatic mode and complete the controller by switching SW into position "5".

Example 1
Using the sine-wave method, ideal PID controller (4a) is to be designed for the operating amplifier modelled by the transfer function G A (s) 33 11 () ( 1) (0,01 1) The controller has to be designed for two values of the maximum overshoot of the closedloop step response  max1 =30% (Design No. 1) and  max2 =5% (Design No. 2) and maximum relative settling time τ s =12 in both cases.

Systems with time delay
The sine-wave method is applicable also for plants with time delay considered as difficultto-control systems.It is a well-known fact, that the time delay D turns the phase at each frequency  n 0,) by  n D with respect to the delay-free system.For time delayed plants, phase condition of the sine-wave method (20b) is extended by additional phase where φ ´ is the phase of the delay-free system and is the identified phase of the plant including the time delay.The added phase φ D =- n D can be associated with the required phase margin The only modification in using the PID tuning rules in Tab. 9 is that increased required phase margin is to be specified (Bucz, 2011) Ḿ and the controller working angle Θ is computed using the relation The phase delay  n D increases with increasing frequency of the sinusoidal signal  n .
To lessen the impact of time delay on closed-loop dynamics, it is recommended to use the smallest possible added phase φ D =- n D.

Discussion
Time delay D can easily be specified during critical frequency identification as the time D=T y -T u , that elapses since the start of the test at time T u until time T y , when the system output starts responding to the excitation signal u(t).A small added phase φ D =- n D due to time delay can be secured by choosing the smallest possible  n attenuating effect of D in ( 43) and subsequently in the PID controller design.Therefore, when designing PID controller for time delayed systems according to Procedure 1, in Step 4 it is recommended to choose the lowest possible excitation level on the performance B-parabolas (most frequently  n / c =0,2 resp.0,35) and corresponding couples of B-parabolas in Fig. 11.Procedure 2 is used for plant identification and PID controller design. M is specified from the given couple ( max ;t s ) using the chosen couple of Bparabolas, however its increased value  M ´ given by ( 42) is to be supplied in the design algorithm thus minimizing effect of the time delay on closed-loop dynamics.

Example 2
Using the sine-wave method, ideal PID controllers (4a) are to be designed for the distillation column modelled by the transfer function G B (s)   13a).Note that L B2 has the same location in the complex plane as L A2 in Fig. 12a  3.4.3Systems with 1 st order integrator By testing the sine-wave method on benchmark systems with 1 st order integrator, the B-parabolas in Fig. 14 -16 were obtained (for Cartesian product  Mj × nk of sets ( 31) and (32), j=1...8, k=1...6 and three various ratios T i /T d : =4, 8 and 12).

Discussion
Inspection of Fig. 14a, 15a and 16a reveals, that increasing  results in decreasing of the maximum overshoot  max , narrowing of the B-parabolas of relative settling times τ s =f( M , n ) for each identification level  n / c , and consequently settling time increasing.Consider e.g. the B 0,95 parabolas in Fig. 14b, Fig. 15b and Fig. 16b: if  M =70 and =4, relative settling time is τ s =30, for =8 it grows to τ s =40, and for =12 even to τ s =45.If a 10% maximum overshoot is acceptable, then the standard interaction PID controller can be used with no need to use a setpoint filter; however a larger settling time is to be expected.Procedure 1 is used to specify the performance in terms of ( M , n ) from ( max, t s ) using pertinent B-parabolas in Fig. 14 -16.Procedure 2 is used for plant identification and PID controller design.

Discussion
All data necessary to design two PID controllers of all three plants G A (s), G B (s) and G C (s) along with specified and achieved performance measure values are summarized in Tab. 10 where  max and t s in the last two columns marked with "*" indicate closed-loop performance complying with the required one.

Conclusion
The proposed new engineering method based on the sine-wave identification of the plant provides successful PID controller tuning.The main contribution has been construction of empirical charts to transform engineering time-domain performance specifications (maximum overshoot and settling time) into frequency domain performance measures (phase margin).The method is applicable for shaping the closed-loop response of the process variable using various combinations of excitation signal frequencies and required phase margins.Using B-parabolas, it is possible to achieve optimal time responses of processes with various types of dynamics and improve their performance.When applying digital PID controller, it is recommended to set the sampling period T s from the interval 02 06 where  c is the critical frequency of the controlled plant (Wittenmark, 2001).By applying appropriate PID controller design methods including the above presented 51+3 tuning rules for prescribed performance, it is possible to achieve cost-effective control of industrial processes.The presented advanced sine-wave design method offers one possible way to turn the unfavourable statistical ratio between properly tuned and all implemented PID controllers in industrial control loops.

Acknowledgment
This research work has been supported by the Scientific Grant Agency of the Ministry of Education of the Slovak Republic, Grant No. 1/1241/12.

Fig. 4 .
Fig. 4. Typical step responses of a) FOPDT; b) IPDT and c) FOLIPDT models From the read-off parameters, transfer functions of individual models have been obtained

Fig. 6 .
Fig. 6. a) Definition and geometrical interpretation of  M and G M in the complex plane; b) Sensitivity and complementary sensitivity magnitudes S(j), T(j) and performance measures M s , M t From Fig. 6a results, that increasing open-loop phase margin  M causes moving the gain crossover L(j a * ) lying on the unit circle M 1 away from the critical point (-1,j0).Increasing open-loop gain margin G M causes moving the phase crossover L(j f * ) away from (-1,j0).Therefore, parameters  M or G M given by * 180 arg ()

Fig. 8 .
Fig. 8. Time responses of a) u(t); b) y(t), and c) location of G(j n ) in the complex plane

Fig. 9
Fig. 9. a) Graphical interpretation of  M ,  a * and shifting G into L A at  a * = n ; b) controller structure selection with respect to location of G and L A using the "triangle ruler" rule

Fig. 10 .
Fig. 10.Closed-loop step responses of G 1 (s) under PID controllers designed for various  M and  n

Table 1 .
Controller structure selection with respect to plant model parameters: A: forward compensation suggested, B: forward compensation necessary, C: dead-time compensation suggested, D: dead-time compensation necessary, E: set-point weighing necessary, F: pole-placement2.2PIDcontroller design objectivesConsider the following most frequently used PID controller types: ideal PID (4a), real interaction PID with derivative filtering (4b) and ideal PID in series with a first

Table 2 .
Controller tuning based on critical parametres of the plant ) use various weighing of critical parameters thus allowing to vary closed-loop performance requirements.Methods (No. 1 -10) are applicable for various plant types, easy-to-use and time efficient.

Table 3 .
PID tuning rules based on FOPDT model;  1 =K 1 D 1 /T 1 is the normed process gain

Table 5 .
Tuning rules based on SOPDT model parameters

2.6 PID controller design based on optimization techniques
ISTE) for n=1, and integral squared time-squared weighed error (IST 2 E) for n=2.Some tuning formulae for PID controller in form (4a) are shown in Tab. 6. Settling time t s in rules No. 40 and 41 is affected by D 2 .

Table 6 .
Tuning rules based on minimizing performance indices tuning rules for PID controller (4a) (No. 42 -44 in Tab. 7).Tuning rules No. 45 and 46 for PID controller (4c) show that settling time t s increases with growing normed time delay  1 =D 1 /T 1 of the FOPDT model (12).

Table 8
. PID design formulae for specified performance based on tuning parameters  M , G M , M t and 2.8.3 Performance evaluationPhase margin  M is the most wide-spread performance measure in PID controller design.Maximum overshoot  max and settling time t s of the closed-loop step response are related with  M according to Reinisch relations Introduction to PID Controllers -Theory, Tuning and Application to Frontier Areas 16 where =argG R ( a * ) is found from the phase condition (20b).Thus, using the PID controller with coefficients {K;T i =T d ;T d }, the identified point G(j n ) of the plant frequency response with coordinates (19) can be moved on the unit circle M 1 into the gain crossover L A ≡L(j a * ); the required phase margin  M is guaranteed if the following identity holds between the excitation and amplitude crossover frequencies  n and  a * , respectively Control objective is to provide the maximum overshoots of the closed-loop step response  max1 =30%,  max2 =20% and a maximum relative settling time τ s =20.