Filtered PID Control Loop for Third Order Plants With Delay: Dominant Pole Placement Approach

The paper deals with tuning the filtered PID controller applied to third order plants with delay. This model option is chosen as a representative case where the loop characteristic quasi-polynomial is of higher order than three. Applying the similarity theory for introducing dimensionless parameterization a comparative model of third order plant dynamics is obtained. Four dominant poles – from the infinite spectrum of the control loop – are assigned by means of tuning three controller gains and a filter time constant where a specific argument increment criterion proves their dominance. The pole prescription coordinates are parameterized via damping, root and natural frequency ratios optimized in the space of the introduced similarity numbers according to the IAE criterion with respect to robustness and filtering constraints. Particularly the natural frequency ratio is a new parameter introduced to tune robustly the PID together with its filter. For the constrained IAE optimization the response of disturbance rejection is used as a representative of control loop behavior. In the space of similarity numbers of the plant it is shown that a limited range of plants is suited to be controlled on the PID control principle and the boundaries of this range are outlined. Survey maps of optimum controller parameters are presented and a comparative study on benchmark application example is added.


I. INTRODUCTION
In industry the most frequent output driven control is provided by a PID-type structure controller, even for systems with significant time delay [1]. There are many PI(D) tuning rules for the first-or second-order plants with a delay which is regularly approximated by the Padé or Taylor series [2]- [7]. Predominantly these tunings are based on ultimate cycle identification, various performance index minimizations, gain, phase and jitter margin specifications, magnitude optimum method, pole placement technique or IMC-like tuning (so-called Lambda tuning), which are particularly well suited for non-dominant delay processes, except for the IMC-like tuning, [8], and the dominant pole placement, [9]. The former tuning is suitable for dominant delay processes assuming safe pole-zero cancellation in the open The associate editor coordinating the review of this manuscript and approving it for publication was Zhiguang Feng .
loop [10], in [11] this cancellation is modified to systems with large time delay and in [12] the IMC-like PID with a second-order lead-lag filter compensates for the dominant plant poles and zeros. In [13] the Lambda tuning method is modified for integrating systems by the polynomial approach. The latter tuning is also applicable to the dominant delay processes but assuming no pole-zero cancellation within the control loop. In [14]- [16] the pole-zero matching method is applied to inexact dominant pole placement, which is free of any delay term approximation, and in [17] a universal map of PID tuning for the second-order processes with dominant delay is presented.
Generally, in tuning the PI(D) controllers a trade-off between the robustness and performance is searched, and moreover a trade-off between the reference tracking and disturbance rejection performance/robustness is subject to a controller optimization [18]- [20]. On one hand the robust PID tuning results in conservative setting at the expense of VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ minimization of performance indices like Integral Absolute Error (IAE), Integral Square Error (ISE) etc., [10]. On the other hand the PID controller optimized to the disturbance rejection can be hardly an optimum controller for the reference tracking. The trade-off tuning is tackled by the model matching approach to filtered PID design, [21], and in [20] the disturbance rejection performance with respect to the reference input level is guaranteed by keeping the so-called reference to disturbance ratio index as high as possible. Next, in [22] the proportional gain maximization method is introduced based on the optimizing the damping ratio of the controller zeros which resulted circa three quarters. With this result the control performance is improved without deteriorating the robustness to model uncertainties. To make the disturbance rejection and reference tracking mutually comparable the notion of extreme frequency equivalence is utilized [19], [23]. The reason of unfavorable disturbance rejection and reference tracking consists in a pole-zero cancellation within the control loop. Hence the tuning methods, particularly the Lambda tuning, [10], based on the pole-zero cancellation or methods, like gain and phase margins, [5], assuming the pole-zero cancellation are potentially inappropriate for the (load) disturbance rejection task. Nevertheless, the lambda tuning for given (load) disturbance is obtainable with effective disturbance rejection [12], [24]. To prevent from dominant pole(s) cancellation in case of the reference tracking a two-degree-of-freedom control loop is proposed, [25], or an input prefilter is designed [26]. Measurement noise filter of the second order is designed for mitigating abrupt control actions [27] and in [28] the higher-order noise filters are tuned. In [29] the measurement noise filter tuning is provided for common PID tuning rules in case of reduced order plant model. Based on the filter from [27] the filtered PID control loop is tuned up for the second-order plant with delay by the dominant four-pole placement in [30].
In practice a filter time constant setting obeys a rule of thumb, fixing the derivative time constant ratio to the filter time constant, [8]. As opposed to the rule of thumb, constraining above mentioned ratio due to the loop cut-off frequency and the demand on high-frequency control sensitivity the PID tuning obtains benefits like disturbance rejection capability, phase margin increase and controller frequency band extension [31]. On the other hand a priori tuned filter time constant can contribute to excessive high-frequency control sensitivity to be avoided in practice.
In practice it is a difficult task to identify higher-order plant dynamics hence a model order reduction is imposed on a controller design [24], [32]. In case of higher-order systems with time delay the dominant pole placement in conjunction with the D-partition method is presented in [1]. Comparative studies in [33], [34] compares simple PID tuning rules and methods, where a derivative part filter is in action, provided for higher-order non-oscillating plants or plants with significant delay. As regards the third-order plants with delay the tuning rules are developed in analogous way, [2], that are not frequently applied in literature. One of them is the IMC-like tuning, [35], considered for the third-order plants. In studies [36], [37] all the stabilizing PID controller settings obtained are eligible only to a fixed third-order plant with delay and these settings do not guarantee satisfactory control performance in general. In [38] all the stabilizing PID controllers obtained are extended to more general class of delay systems and a survey on all the stabilizing PID controller designs are presented in [39]. Typically, the plants of the third order are thermal [40], water or wind power plants, [41] or [42] respectively. From application point of view the most difficult is to reject loads or disturbances of plants providing energy conversion, for instance [43], [44]. In [45] a universal adaptive controller design applicable to the first-through third-order plants is presented. Frequently the third-order plant model describes the second-order plant dynamics but with additional, not negligible actuator dynamics, [46], [47]. For higher-order systems the test benchmark models can be found in [48]- [50]. The third-order plants with a delay are apparently present in the industry and therefore any other order reduction means inappropriate approximation.
In the paper the universal PID and filter settings are proposed with the aim to get optimum controller gains and filter time constant with respect to not only the IAE criterion but also to robustness and filtering effect constraints. Besides the damping and root ratio prescription, these settings are based on finding the ratio between two natural frequencies prescribed where the greater one is the ultimate frequency of the plant. The novelty of the dominant pole placement approach consisting in the frequency ratio prescription brings robust PID and filter settings for the sets of dynamically similar third-order plants with delay. Without finding an optimum frequency ratio the four-pole dominance would be barely guaranteed. Finally, the optimum controller and filter parameters are mapped to show varying delay effect.

II. THIRD ORDER PLANT MODEL PARAMETERIZATION
As presented in the Introduction the third-order plants with delay are commonly identified in the industry. Such plants are modelled by the following third-order model with delay where y (t) and u (t), output and input variables respectively, are expressed as dimensionless percentage of the sensor and actuator instrument ranges. The coefficients c 1 , c 2 , c 3 and the delay value υ are supposed positive dimensional constants and K is a nonzero dimensionless steady state gain. From the plant transfer function it follows that the only three poles of the plant are the roots of the characteristic equationM (s) = 0. DividingM (s) by c 3 the characteristic polynomial can be identified with the root factor product form as follows The following two stable options of the M (s) roots are further considered where ω n is the natural frequency and ξ is the damping factor of complex conjugates q 1,2 . Positive constant b is the absolute value of real pole q 1 and q 2,3 = χ 1,2 q 1 are other two distinct real poles. The special case of double real pole and one single real pole belongs to option (5) when ξ = 1, and the case of triple real pole belongs to option (4) with χ 1,2 = 1.
For further modification of the plant model the parameterization principle introduced by Vyshnegradskii, [51], is used. Analogously to [17] the following dimensionless substitution for Laplace transform variable is introduced to achieve the characteristic polynomial (3) in the form Instead of three coefficients c 1 , c 2 , c 3 only two dimensionless parameters 1 determine the character of the plant dynamics. Due to their independence of the time scale of the considered case λ 1 , λ 2 can serve as similarity numbers of the plant dynamics. For the types (4) and (5) First, for aperiodic plants, i.e. for option (4) when D ≤ 0, the coefficients in (3) are given by the poles q 1,2,3 coordinates in the following way 1 Substitution (6) leads in this case to relation s =sb 3 √ χ 1 χ 2 which gives the following form of (8) 1 and the pole option (4) is expressed as q 1,2,3 = q 1,2,3 b 3 √ χ 1 χ 2 . The triple real pole, χ 1,2 = 1, is encountered when D = 0 and this case corresponds to the similarity number option λ 1,2 = 1 3. For the oscillatory plants, i.e. for option (5) when D > 0, the coefficients of M (s) are given by the poles q 1,2,3 coordinates as follows Substitution (6) here results insω n 3 √ χ ξ = s and leads to the coefficients of M (s) determined by the ratios χ , ξ only The pole option (5) is now expressed by q 1,2,3 = q 1,2,3 ω n 3 √ χ ξ . In both types of plants parameters λ 1 , λ 2 are independent of coordinates b and ω n , respectively. In the case of oscillatory plants similarity numbers λ 1 , λ 2 express the degree of oscillability, so they are further referred to as oscillability numbers. Higher values of λ 1 , λ 2 mean less damped natural oscillation.
Plant (1) is considered as stable and therefore coefficients c 1 , c 2 , c 3 are not only positive but are to satisfy the stability condition, c 1 c 2 −c 3 > 0, coming from Hurwitz criterion applied to plant model (2). Expressing the product of two relations in (8) the similarity numbers satisfy for stable plants the following condition Parameters λ 1 , λ 2 describe only the dynamics given by the characteristic polynomial M (s) of plant (1). Nevertheless, it is necessary to be aware of the essential influence of the delay on the plant dynamic properties in closing the feedback loop. From substitution (6) the substitution for time variable, t = t 3 √ c 3 , results. Thus, the delay length is expressed by the ratio and this delay parameter is further referred to as laggardness number of the plant. Using the introduced similarity numbers the plant model (1) is transformed to the form The advantage of this form of model consists in the property that all dynamically similar plants fall into one common point of the λ 1 , λ 2 , ϑ and K parameter space from where steady-state gain K can be still excluded as shown below.

III. DIMENSIONLESS CONTROL LOOP DESCRIPTION
The third-order plant with delay given by model (1) or (16) is controlled by the following ideal PID controller where e f is the filtered control error. The filtering is provided by the solution of the following first-order differential equation where e = w f −y, w f is the prefiltered reference variable. Beside the control error filtering the reference variable is prefiltered to prevent from the excitation of high-frequency unmodeled dynamics. Then the following second-order differential equation is considered where w is the reference variable in percentage again and r P > 0, r D > 0, r I > 0. Controller (17) together with filter (18) corresponds to ideal PID controller in series with a first order lag that is very frequent for the real controller implementation, see [2]. The real controller is specified by three gains r P , r D , r I and time constant T f for the proportional, derivative, integration actions and filtering effect, respectively. To transform the controller description into parameterized form consistent with plant model (16) the relative time variablet is introduced and (17) is multiplied by K in the same way as input u on the right-hand side of (16). Then the following controller description is obtained 20) where the gains are replaced by the dimensionless parameters as follows and gain K is merged with the proportional, derivative and integration gains in (21). Then the ρ P , ρ D and ρ I are the loop gains absorbing the gain K and the filter time constant is in analogy with (15) expressed in dimensionless form The dimensionless plant model (16) can be expressed in the form where K is already absorbed with respect to (20). Beside the plant model (23) let be also introduced the integrating plant model as follows with considered control input (20). Without any necessity of the proof the model (24) is dimensionless model of that model type (1) where the absolute term, y(t), on the left-hand side is missing. Closing the feedback loop of plant (23) or (24) and controller (20), providing also measurement and derivative action filtering, the characteristic quasi-polynomial of the loop is in the form or respectively. In contrast to M (s) the characteristic equation, P(s) = 0, contains the exponential terms with the laggardness number ϑ and therefore has infinite spectrum of roots. This spectrum does not change if (25) or (26) is multiplied by nonzero exp(ϑs), ∀s ∈ C, [52], and thus the characteristic quasi-polynomial is modified to the form +ρ Ds 2 +ρ Ps +ρ I = Q (s) (27) or 1s 2 e ϑs +ρ Ds 2 +ρ Ps +ρ I = Q (s) (28) respectively, where the control loop gains are free of delay term.

A. ULTIMATE FREQUENCY NUMBER AND ULTIMATE LOOP GAIN ASSESSMENT
The ultimate frequency of the plant ω K and ultimate loop gain r K are significant parameters for tuning the controller. Consistently with the control loop model parameterization in (20) and (23) instead of ω K the ultimate frequency number and ultimate loop gain ρ K = Kr K are introduced. Their values corresponding to plant model (23) are given by the following Theorem 1. Theorem 1: Suppose the plant given by the model (23) and close a proportional feedback with the ultimate loop gain, ρ K . The ultimate frequency number ν K is the smallest positive solution to equation Proof: Recall plant model (23) and close proportional feedbackū t = −ρ P y t . Characteristic equation of the loop is not algebraic any morē For the case of ultimate oscillation at ultimate frequency given by ν = ν K the complex variable iss = jν K and proportional gain is ultimate ρ P = ρ K . For this case the ultimate parameters, ν K and ρ K , satisfy the following equality For real and imaginary parts of this equation the following equalities result After expressing the ratio of cos (ϑν K ) and sin (ϑν K ) the relation (30) is obtained. From (33) also the ultimate loop gain results as follows The proof is finished. How to evaluate ν K according to (30) is shown in Sample example in Section VI.A. For integrating plants (24) the following relation for the ultimate frequency number, ν K , is obtained and for the ultimate loop gain the following relation is acquired One can see that relations (36) and (37) are achieved directly from (30) and (35), respectively, by leaving out the stand-alone unit −1 in numerators. Additionally, in (36) the power of ν K is cancelled due to the fact that only non-zero and smallest positive ν K is searched.
The ultimate frequency number, ν K , is given by the plant similarity numbers λ 1 , λ 2 and the laggardness number ϑ. Its value can be displayed over an area of λ 1 , λ 2 for fixed value of ϑ. For the laggardness number ϑ = 0.3 its dependence on λ 1 , λ 2 over the area λ 1 ∈ 0.1, 0.9 and λ 2 ∈ 0.2, 2.4 is plotted in Fig. 1. The intervals of λ 1 , λ 2 correspond with the constraints further specified in Section III.B.
For longer delays, ϑ > 0.3, the shape of the surface does not change only its level is accordingly shifted to lower ν K values. Considering higher values of ϑ than 0.5 turns out to be unsuitable for using the dominant pole placement for tuning PID controller for plants of type (1), as it is discussed in Section V. In Fig. 2 the ultimate loop gain, ρ K , is plotted for the same area of λ 1 , λ 2 and for ϑ = 0.3 as in Fig. 1.

B. CONSTRAINTS ON THE CONSIDERED PLANTS
The PID control principle is not capable to cope with third order plant dynamics in their whole range. Not only unstable plants but also a part of the others is to be excluded from consideration. Some of them are quite incompatible with PID and some can be controlled by PID but their setting by means of common tuning rules does not lead to acceptable results.
The similarity numbers λ 1 , λ 2 and ϑ of the plant introduced by (11) or (13) and (15) serve as generalized parameters in which the suitable range of plants can be effectively displayed. Due to generic meaning of λ 1 , λ 2 and ϑ the range of acceptable plants can be expressed universally with them. The following three constraints are to be respected in the rest of the paper.

1) CONSTRAINT ON PLANT CHARACTER
Unstable plants with λ 1 λ 2 ≥ 1 and the plants approaching the stiff-character cases where are excluded. Also, the too oscillating plants with damping ξ < 0.05 (λ 1 > 0.9, λ 2 > 2.4) are not considered at all. First, the plants free of delay, with laggardness ϑ = 0, are viewed in the plane of plant similarity numbers λ 1 , λ 2 . Their acceptability for applying pole placement in tuning the PID loop is strongly dependent on ϑ, so that the final area of the considered λ 1 , λ 2 options is considerably reduced with growing ϑ, see Section V. The mapping in Fig. 3 shows the area of λ 1 , λ 2 of the delay-free plants with laggardness ϑ = 0, for which the pole placement is applicable. Only a restricted part of this area with λ 1,2 ≤ 1 3 corresponds to aperiodic plants, option (4), the vast majority of the map belongs to oscillatory plants, option (5), with complex conjugate pair of roots.
In this area the isolines of χ = const and ξ = const display the ranges of ratios χ , ξ of plants admissible for consideration. Notice that the mapping in Fig. 3 holds for arbitrary natural frequency ω n of option (5) or for arbitrary value of b for option (4).

2) CONSTRAINT ON PID APPLICABILITY
The applicability of PID to plant type (1) is considerably dependent on the delay, i.e. on the laggardness ϑ, and also the order of differential equation with delayed argument describing the control loop. The higher laggardness the more reduced is the admissible λ 1 , λ 2 area compared with Fig. 3 which was provided for the delay-free case. This reduction involves exclusively the oscillatory plants with lower damping ratio ξ . The higher ϑ the higher lies the lower boundary of ξ . From characteristic quasi-polynomial (27) or (28) it is further apparent that the PID influences mainly the lower derivative terms,s 0 ,s,s 2 whiles 3 ,s 4 ,s 5 are out of any influence, except for filter time constant τ . Therefore in reaching successful PID control application only plants with sufficiently low λ 2 values are applicable (the coefficient is λ −1 2 ). In fact, the upper bound of λ 2 is considerably lower than that in Fig. 3 if ϑ > 0. The higher the laggardness ϑ the lower is the upper bound of λ 2 .

C. ROBUSTNESS CONSTRAINTS IMPOSED ON THE CONTROL LOOP
The robustness to the third-order plants with delay (23) or (24) is characterized by maximum sensitivities, e.g. [27], , ands = jν. Transfer functions G (s) and C (s) coming from Laplace transform of (23) or (24) and (20) together with (18), respectively, are given as follows and respectively. Additionally, to avoid the reference tracking degradation the prefilter due to (19) and with respect to (21) is as follows which is used for the reference variable prefiltering in the classical control loop in Fig. 4. This control scheme also allows for the measurement noise and derivative action filtering.
Besides the maximum sensitivities, M S and M t , the following measure is to be considered, [8]. Quantity M u characterizes high-frequency control sensitivity that in case of the PID control depends particularly on the derivative time constant ratio to the filter time constant where τ D = T D 3 √ c 3 . Ratio N cannot be arbitrarily large and its maximal range is between 5 and 30, see [2]. The value of N is selected from this range in dependence on measurement noise frequency range and level, for more details see [27], [29]. As regards the maximum sensitivities, M S and M t , their default value is 1.4 but can be ranged between 1.2 and 2, for instance [8], [53], [54]. The sensitivity function in (40) results with respect to process transfer function (41) and controller (43) as follows The load disturbance supposed of the same transfer function G (s) as the control variable in (2) its control loop transfer function originates in However, with respect to the control scheme in Fig. 4 the reference transfer function is different from the complementary sensitivity function, T (s), as follows After substituting for G (s), C (s) and F (s) in (49) the reference transfer function is obtained as follows Potential pole-zero cancellation within the delayed control loop results from the numerator in (47) where in fact only the complex conjugate poles of plant option (5) can be cancelled due to four dominant poles prescribed as two complex conjugate pairs, as proposed in Section IV. Since these pairs are prescribed to provide a well-damped disturbance rejection only such plants with ξ > 0.4 could be really cancelled. In addition, the range of these plants is tight as shown in Fig. 3 and controlling the well-damped processes (ξ > 0.5) do not represent any problem for the practice, as a rule. That is why a test benchmark process of the class of the third-order plants with delay is selected a poorly damped process with ξ < 0.4, see Section VI. Next, the measurement filter's pole location is far away to the left in the complex plane from a quadruple of prescribed poles, see Section IV, so that no dominant polezero cancellation takes place. Due to the infinite spectrum of characteristic quasi-polynomial (25) or (26) only the rightmost spectrum is eligible to possible degradation. Cancelling some of the rest of infinite spectrum, i.e. some non-dominant poles, these poles cancellation cannot be responsible for the disturbance rejection deterioration because the disturbance rejection response dynamics is characterized by the dominant pole spectrum assigned. At last, thanks the reference variable prefiltering the reference tracking is free of any pole-zero cancellation as proved by the transfer function structure in (50) or (51) and prefilter F (s) results with damping factor at least 0.5 as gained by optimum values of real controller parameters in Section V.

IV. DOMINANT FOUR-POLE PLACEMENT
Characteristic equation of the loop Q (s) = 0 admits an infinite spectrum of the roots, where only a little group on the rightmost positions determines the dynamic properties of the loop. The potential of four controller parameters is to adjust only four poles of the infinite spectrum as presented in Theorem 2 below. Hence if the four-pole placement technique should fulfil the assignment aim the prescribed poles have to become really dominant for the loop as presented in Theorem 3 following the Theorem 2. Particularly no other pole of the rest of the spectrum may lie to the right from the prescribed quadruple of poles. The more to the left from the placed poles is located the rest of the spectrum the more dominant is their position.

A. FOUR-POLE PLACEMENT TECHNIQUE
The following procedure of dominant four-pole placement is based on that in [30] where the original dominant three-pole placement introduced in [55] is adapted for filtered PID control loop. As novel feature the dominant four-pole placement is extended for the third-order plants with delay characterized by the three parameters λ 1 , λ 2 , ϑ of the plant model (23) or (24). Additionally, the natural frequency ratio, η, is new introduced parameter to meet the robustness requirement by its optimization. As the well-damped and robust control response the following quadruple of poles where ν, δ, κ, η > 0 is prescribed to adjust the proportional, derivative, integration control loop gains ρ P , ρ D , ρ I , and the filter time constant τ , respectively. The prescribed pole locations represent a well-damped and fast oscillatory behavior of control loop characterized by the damping ratio δ, the root ratio κ given by Re p 1,2 > 0 (53) and the ratio between two natural frequencies given by η < 1 where the greater natural frequency is identified with the ultimate frequency, i.e. ν = ν K . This is the rule of thumb to assign the natural frequency as the ultimate frequency which leads in practice to satisfactory disturbance rejection performance, [55], [56]. Thus, the lower natural frequency assigned is given as ην K and η is the key ratio introduced for achieving the robustness. Theorem 2: Consider a set of third order plants characterized by the numbers λ 1 , λ 2 , ϑ as in (23) and close filtered PID control loop according to (20) so that the characteristic quasi-polynomial in (27) is obtained. Four complex numbers p 1 ,p 2 ,p 3 ,p 4 given as in (52) are selected to be placed as the dominant zeros of Q (s). Without guaranteeing the (52) dominance a priori, the following setting of the control loop gains and filter time constant provides the placement of these zeros where the entries, A R , A I , A Rη , A I η and B R , B I , B Rη , B I η , are given as follows A Rη = e −κδϑν a Rη cos (ϑην)−a I η sin (ϑην) with and and respectively. Simultaneously the dominance of placedp 1 ,p 2 , p 3 ,p 4 is checked posteriori applying Theorem 3 below. Proof: Four poles given by (52) are to be the roots of Q (s) = 0 where Q (s) is given by (27). Substituting the first pole,s =p 1 = (−δ+j) ν, into this equation and separating the terms with ρ P , ρ D , ρ I the following equality is obtained By analogous substitutions =p 3 = −κδ η+j ην = (−κδ+jη) ν the following equality results It remains to separate the filter time constant τ in equalities (74) and (75) as follows and respectively. Dividing the equalities for real and imaginary parts of (76) and (77), and expressing them in the matrix equation form AP = B for the vector composed of control loop gains and filter time constant the following matrices, A and B, are obtained where in case of matrix A entries A Rη ν 2 = e −κδϑν a Rη cos (ϑην)−a I η sin (ϑην) (82) A I η ν 2 = e −κδϑν a I η cos (ϑην)+a Rη sin (ϑην) (83) with and in case of matrix B entries Now from equation AP = B the matrix solution P = A −1 B leads to the following results where and Matrix A is non-singular, i.e. det (A) = 0, if at least one of the ratios η, κ is not equal to one and ν > 0, as apparent from (98). Let be remarked that for η = κ = 1 the relations for A R and A I are identical with A Rη and A I η , respectively. From the ratios of determinants (96) the relationships (54), (55), (56), and (57) result wherein the number ν is cancelled either partly or completely. The resulting setting is proved on the four poles dominance as presented in Theorem 3 below. The proof is finished.
Remark 1: For integrating plants (24) the relations for proportional, derivative, integration loop gains and filter time constant given by (54) through (57) are also applicable but with moderate changes in both the coefficient matrix A and the right-hand side matrix B. These changes are such that in (62) and (64) last term δ 2 −1 and κ 2 δ 2 −η 2 , respectively, and in (63) and (65) last term −2δ and −2κδη, respectively, are missing. In case of B in (70) and (72) the stand-alone δ and κδ, respectively, and in (71) and (73) the numbers −1 and −η, respectively, are missing.

B. DOMINANCE PROOF OF THE FOUR PLACED POLES
The four pole placement according to Theorem 2 does not guarantee a priori the dominance ofp 1 ,p 2 ,p 3 ,p 4 in Q (s) infinite spectrum and therefore this dominance is to be proved before accepting the results in (54) through (57). This proof is provided by the following theorem.
Theorem 3: Consider the characteristic quasi-polynomial of the fifth-order PID control loop in the form, P (s), given by (25) or (26) and suppose a straight lines = −β m +jω for either β m > β = κδν, κ > 1, or β m > β = δν, κ < 1, where ω is growing from zero to infinity, ω → ∞. If the following limit is reached Proof: Consider a Jordan curve composed of the circle C,s = R cos ψ +j sin ψ , ψ ∈ −π 2−γ , π 2+γ , and the straight line L,s = −β m +jω, ω ∈ −R cos γ , R cos γ , Fig. 5, and suppose that only four zerosp 1,2,3,4 of quasi-polynomial P s lie inside this curve. By the argument increment rule the total argument increment along this curve run in the clockwise direction (L is oriented downwards) is then 8π. This increment is composed of two increments obtained for the parts C and L, respectively. The argument increment arg C is given by the order five of P (s) and for the limit of R → ∞ and γ → 0 its value is Finally realize that in (99) only one half of the straight line L, which is with ω ≥ 0 and oriented upwards, is considered so that only arg L 2 is to be obtained with opposite sign. Hence by comparing (100) and (101) and dividing by two the argument increment (99) is obtained, and the Theorem 3 is proved.
Let be noted that the application of rule (99) is as follows. For a constant β m > β = max (δν, κδν) quasi-polynomial function P (−β m +jω) is computed for ω from ω = 0 to an enough high ω = ω m , which renders the other terms of the quasi-polynomial negligibly small compared to ω 5 . To obtain the values of arg P from a hodograph inside the unit circle of complex plane the following mapping is applied where the denominator is real and cannot change the argument value of P (s). The greater ratio β m β for which condition (99) is satisfied the stronger is the dominance ofp 1,2,3,4 .
To evaluate the four-pole dominance degree the dominance index from [55] is modified as follows wherep 5 is the rightmost pole from the rest of the P (s) spectrum. This pole results in complex pole for κ < 1 while κ ≥ 1 the fifth pole is real, as a rule. The effect of non-dominant poles on control loop dynamics is negligible if the non-dominant pole spectrum is separated from that dominant spectrum, [57]. Thus, a distinct separation ofp 1,2 , p 3,4 andp 5 all together from the rest of the infinite spectrum is desirable for a marked dominance of them. Remark 2: Three gains ρ P , ρ D , ρ I and time constant τ according to relation (54) through (57) are derived for considered plant model (1) with the premise of prescribed poles dominance. The matrix formulae in these relations originate from the characteristic quasi-polynomial, Q (s), evaluated for prescribed poles. It also includes the delay operator, exp (−τ s), evaluation. Once poles to be prescribed are found at optimum with respect to IAE criterion and maximum sensitivities constraint the dominant pole placement is guaranteed as proved by the argument increment principle due to (99). The compactness of the controller parameter relations would be then lost if instead of all the matrix formulae their determinant evaluations are provided in fact.

V. CONTROL LOOP OPTIMIZATION IN PLANT PARAMETER SPACE
The four-pole placement parameters ν, δ, κ, η are subject to the constrained IAE optimization in the space of plant parameters λ 1 , λ 2 , and ϑ. Due to close relation between the ultimate frequency of the plant (23) or (24) and a desirable natural frequency of the control loop the prescription of ν value in quadruplep 1,2,3,4 is identified due to the ultimate frequency number, ν K , as a reference parameter of the plant. Next, the crucial constraint of the IAE criterion is the maximum sensitivity M S limited to 1.8 as revealed within the control design for benchmark plant in Section VI. This constraint, M S ≤ 1.8, is not peculiar only to the benchmark plant but it is also found suitable for all considered plants characterized by numbers λ 1 , λ 2 , and ϑ. Beside the constraint on M S also the constraint on the ratio N is assumed such that 5 < N < 15 thus N ≈ 10. Only under meeting both constraints the IAE value is minimized. This value is computed from the load disturbance rejection response of the loop that is frequently optimized in practice. Its loop transfer function is obtained in (48). Analogously for the integrating plants given by (24) the corresponding S d (s) is obtained when (42) is substituted for G (s) in (48). The load disturbance rejection is optimized according to the IAE criterion (as also preferred in [10]) As shown in [58], [59], the unconstrained IAE minimization leads to prescribing the natural frequency higher than the plant ultimate frequency. In the assessment according to Theorem 2 the relationship P = A −1 B, (96), provides a mapping of the controller parameters P = ρ P ρ D ρ I τ T in the space of the root coordinates ν, δ, κ, η where primarily the constrained optimization of δ, κ, η is performed. From constraining the IAE optimization it turns out that the natural frequency number should not be prescribed higher than the ultimate one, thus ν ≤ ν K .
The constrained IAE optimization of the control loop is performed for three parameters, ratios δ, κ, η when the natural frequency number ν is fixed to ν K . This optimization is repeated throughout an area of plant parameters λ 1 , λ 2 , ϑ fulfilling the constraints on the plant character, the PID applicability and robustness, see Sections III.B and III.C. The explicit form of the pole placement relationships (54) through (57) from Section IV, makes it possible to map the control loop gains ρ P , ρ D , ρ I and filter time constant τ subject to plant parameters λ 1 , λ 2 , ϑ and optimized ratios δ, κ, η. The optimization results are displayed in the plane of similarity numbers λ 1 , λ 2 and the laggardness number ϑ is considered as a stepwise given parameter. The ultimate frequency number assigned is recorded in Fig. 1 and corresponding ultimate loop gain in Fig. 2 for ϑ = 0.3. The λ 1 , λ 2 area of the maps for ρ P , ρ D , ρ I , τ in Figs. 6,7,8,9 is considerably smaller than that given in Fig. 3 which holds for delay-free plants, ϑ = 0.
For nonzero laggardness, ϑ > 0, this area is strongly reduced into the admissible values of λ 1 , λ 2 at which the afore mentioned constraints are satisfied by the controller parameter settings mapped. Constraining the controller tuning to M S ≤ 1.8 and N ≤ 10(or 15) this tuning results in positive values of the controller parameters as a rule. Also, the varying delay effect is shown in Figs. 6-8 considering both ϑ = 0.3 and ϑ = 0.5. Since both surfaces on area λ 1 , λ 2 are intersected in case of filter time constant τ only the surface for ϑ = 0.3 is displayed in Fig. 9.   The maps of M S and N prove keeping both measures within the given constraints depicted in Fig. 10 and 11, respectively, and in Fig. 12 the resulting IAE is drawn. Particularly the range of λ 2 is considerably reduced in the part of weakly damped oscillatory plants so that only ξ > 0.3 is admissible, in fact, see Fig. 6. The maps for higher ϑ are of very similar shape but shifted down and with more reduced area of λ 1 , λ 2 . The higher ϑ the lower are the control loop gains. On the whole area the plants with laggardness number ϑ > 0.5 are reduced any further when the suitability area of λ 1 , λ 2 for the PID application is too limited to be mapped. For each of the pole placements according to (54) through (57) the dominance ofp 1 ,p 2 ,p 3 ,p 4 is to be checked. To display this check the dominance index σ , (103), is evaluated over the whole λ 1 , λ 2 area and plotted in Fig. 13, where the quadruple dominance for the plants with ϑ = 0.3 is verified. This check is also verified in case of ϑ = 0.5 but not displayed. However, for plants with ϑ = 0.5 and λ 2 > 0.6 irrespective λ 1 value this dominance is already lost originating in the  control loop instability for more oscillatory plants any further (ξ < 0.3). Rising the delay effect any further, ϑ > 0.5, both the quadruple dominance is lost and the delayed control loop becomes unstable for plants with λ 2 > 0.5.
Near-to dominant pole-zero cancellation can appear when the quadruple dominance is lost. Then, however, the cancellation happens between the not prescribed poles and the zeros originating from poorly damped plant spectrum. Hence as ξ → 0 the more the unprescribed rightmost poles are close to the stability margin and this cancellation is more plausible. Let be noted that considering only the suitable constraints and plant parameter ranges the quadruple dominance loss is avoided by the controller setting due to (54) through (57). Out of these constraints and ranges the dominant four-pole placement is not applicable to real PID tuning. Regarding the non-dominant pole spectrum separation from that rightmost spectrum this separation commonly results large enough for plants with ϑ ≤ 0.5 and ξ > 0.3 as proved in Fig. 13 and Sample example below.   The maps with optimum values of ratios δ, κ, η are presented in Figs. 14, 15 and 16, respectively, for ϑ = 0.3. The higher values of λ 2 the less results δ ∈ (0.04, 0.8) and reversely the greater results κ ∈ (0. 4,9). Regarding the   new ratio η an average value across all the considered plants results in 0.35. Due to the delay effect, displayed in range 0 < ϑ ≤ 0.5, and poor plant damping (higher values of λ 2 ) the maps also show boundaries in the space of similarity numbers λ 1 , λ 2 , ϑ wherein PID type control is still acceptable, i.e. real controller gain magnitudes are not negligible (there is still certain amplification at least ρ P ≈ 0.5ρ K ) and the filter time constant is in proper ratio to the derivative time constant. Only for a limited set of plants characterized with λ 1 ≤ 0.2, λ 2 > 0.4 this ratio results N < 5 under keeping robustness constraint M S ≤ 1.8 so that PI type control takes place instead. This limited set of plants is eliminated from the maps recorded. Also the dominance index does not result so strong in this limited plant set, i.e. σ ∈ (2, 5), nonetheless the dominant four-pole placement is guaranteed. All these optimum values and properties are also observed in delay case of ϑ = 0.5. The plants characterized with λ 1 > 0.9, 0.4 < λ 2 ≤ 0.6 and ϑ > 0.5 lead to poor amplification due to ρ P < 10 −2 , hence these plant options are omitted. Be assured that for ϑ > 0.5 and λ 2 > 0.6 irrespective λ 1 value not only the dominance of four-pole placement is not achieved but also the delayed control loop becomes even unstable.
The proposed controller tuning procedure is summarized as follows: 1. Determine controlled plant parameters λ 1 , λ 2 and ϑ due to Section II. 2. Design (adjust) maximum sensitivities M S (M t ) and/or M u (N ). 3. Prescribe natural frequency ν = ν K and optimize ratios δ, κ, η by minimizing criterion (104) with respect to point 2. 4. Compute the real controller parameters by relations (54) through (57). 5. Evaluate the dimensional gains r P , r D , r I and time constant T f in (21) and (22), respectively, based on the knowledge of the dimensionless ones from point 4. Then the PID controller with the measurement noise filter is tuned for (industrial) application. The IAE optimum setting within λ 1 , λ 2 and ϑ boundaries for the dominant four-pole placement applicability provides competitive disturbance rejection performance due to the natural frequency assignment identical with the ultimate one and properly optimized ratios δ, κ, η. Simultaneously the reference tracking performance is not degraded by this setting as presented in Comparative study below.

VI. COMPARATIVE STUDY FOR BENCHMARK OSCILLATORY MODEL
A comparative study is demonstrated on the load disturbance rejection and reference tracking for the real controller (43) tunings based on the proposed dominant four-pole placement, lambda tuning method from [35], Åström-Hägglund tuning method, and gain and phase margin optimization method. All these tuning methods are developed for particular third-order plant with delay, except the Åström-Hägglund and gain and phase margin methods applicable to plants of arbitrary order in general. The latter method for higher-order processes achieves the phase margin up to 80 • , [60]. Particularly the other tuning methods for comparison are selected due to their suitability for poorly damped processes. In this section all the considered methods are optimized for the load disturbance rejection and also tested on reference tracking capability within the control scheme in Fig. 4. As a structurally crucial benchmark model is chosen that model (23) describing the set of oscillatory plants with given similarity numbers λ 1 , λ 2 and ϑ, see Sample example below.

A. SAMPLE EXAMPLE-OSCILLATORY PLANT WITH SIGNIFICANT TIME DELAY
To demonstrate the application of the above introduced dominant four-pole placement the following sample example of poorly damped oscillatory third-order plant with delay (falling into the admissible area of similarity numbers in Fig. 6) where the scaling factor 3 √ c 3 = 3 √ 8 = 2 [s] is applied. From this parameterization the following dimensionless form (23) of the plant model is obtained As in (23) this plant model is considered with the steadystate gain, given as K = 0.8, being the part of the loop gains as defined in (21). The ultimate parameters of this plant are obtained from solving (30) and (35). The solution to (30) is facilitated that, first, ϑν K = K is calculated and then it is got back ν K = K ϑ where K is the ultimate angle. The ultimate frequency number and the ultimate loop gain are obtained as follows To prescribe an adequate natural frequency of the loop the value of ν is identified with ν K from Fig. 1. To prescribe optimum dominant poles the constrained IAE optimization is performed with respect to δ, κ, and η as shown  where the proportional loop gain results as ρ P = 0.46ρ K ≈ 0.5ρ K . As regards the filter's ratio (46) it originates in N = 9.77. Next, the maximum sensitivities, M S and M t , are obtained as 1.79 and 1.09, respectively. In the same way as for the plants with similarity numbers (106) the constrained IAE optimization is performed for the rest of plants (23) as shown in the maps for optimum δ, κ and η in Figs. 14, 15 and 16, respectively, where the pole placement ratios belonging to plant (107) can be found in the point λ 1 λ 2 ϑ = 0.65 0.5 0.3 of corresponding map. In the same way the values of M S (M t ), N and IAE can be found in Figs. 10, 11 and 12, respectively.
The assessment of the controller tuning has been performed in terms of the dimensionless form of the control loop model (48). Now it is necessary to convert these results back into the original dimensional plant (105). Quite simple the conversion is for the proportional controller gain, from (21) ρ P = Kr P and then the controller setting is r P = ρ P K = 0.7769. For the other gains the time scale is given by 3 √ c 3 = 2 [s]. Due to (21) the following setting of the rest of controller gains and filter time constant results In converting the controller parameters back to the specific values also the prescribed natural frequency of the control loop is obtained. Using the scaling factor 3 √ the natural frequency of the control loop is The dominance of the prescribed polesp 1 ,p 2 ,p 3 ,p 4 has been checked by means of increment argument rule (99) in Fig. 19, where the ratio β m β is chosen 1.2. From the argument increment −3π 2 it is confirmed that to the right from line L,s = −β m +jω, with β m = 0.511 does not lie any pole except the quadruplep 1,2,3,4 . Applying the quasi-polynomial root finder from [61] the rightmost pole spectrum is evaluated in Figs. 20 and 21. In Fig. 21    r I result differently from (111), thus In other words the common loop description by dimensionless model (107) together with universal real controller tuning (110) is unique not only for oscillatory plants (105) and (112) but also for the rest of plants with common similarity numbers (106). However, due to various scaling factors the control loop responses in time and frequency domain are different, e.g. disturbance rejection response in case of plant (105) differs from that response belonging to plant (112), additionally to different time delay (despite both plants have common laggardness number ϑ).

B. CONTROL LOOP PERFORMANCE AND ROBUSTNESS FOR BENCHMARK OSCILLATORY MODEL
In Table 1 the tuning methods considered are compared with respect to the performance criterion IAE, constraints on robustness and ratio N kept, and in Fig. 22 the disturbance rejection responses corresponding to the tuning methods compared are recorded for N ≤ 10. The disturbance considered is the input disturbance loading the control loop according to the scheme in Fig. 4 by variable d which is the unit step function starting at origin in Fig. 22. Additionally, in bottom part of Fig. 22 using the control scheme in Fig. 4 the reference tracking responses are added to demonstrate that the dominant four-pole placement method provides quite good performance in comparison with other tuning methods although the proposed method optimizes the control loop preferably for the load disturbance rejection. In all the considered tuning methods for comparison the reference variable prefilter results with the damping factor at least 0.5 and the measurement filter time constant results  Based on the results recorded in Fig. 22 and the data in Table 1 the best PID tuning from the considered ones results the DPD which gives the lowest IAE value under the comparable robustness in the extreme frequency equivalence sense (Fig. 23). Also checking the control signal for the load disturbance rejection in Fig. 24 the DPD superiority is observed. With LM method, = 0.52, the same IAE is achieved but with rather high maximum sensitivity M S = 2.023. According to the GM & PM optimization, see Fig. 26, M S results lower than in case of the proposed DPD, see Table 1, but the IAE results greater by 50%. The reason why the DPD results well in the IAE consists in the highest   Table 1. value of the integration gain obtained over all the considered tuning methods tailored to the third-order plants with delay. Moreover, the high-frequency control sensitivity results quite limited in comparison to other tuning methods, only GM & PM optimization leads to lower sensitivity than the DPD, see Fig. 25. Thus the high-frequency gain, M u , results for tunings from Table 1 and N ≤ 10 as follows: 6.215, 7.546, 6.477, 7.675, and 2.898 for DPD, A-H, LM ( = 0.63),   Table 1. Naturally, the DPD cannot be in competition with LM as regards the reference tracking capability due to arbitrarily short lambda parameter ( ) tuning if no actuator limitations are exceeded. Of course, the LM method then gives better IAE than the DPD but in the expense of robustness, see Table 1.
For other tuning methods, A-H and GM & PM optimization, similarly resulting reference tracking responses can be observed but with rather longer settling times. Notice that in Fig. 26 open loop transfer function L (s) is evaluated for s = jν, ν ∈ 10 −2 , 10 3 . In Table 1 one can also observe analogous results in favor of the DPD where all the tunings, robustness and performance measures are made for N ≤ 15. With rising N under keeping the same M S one can observe gradual decreasing the IAE, in general, while M u is gradually increasing. Additionally increasing N over 20 already leads to implementation issues due to the demand on the sampling period of order less than 10 −3 [s]. Particularly this is the case of A-H and LM tuning methods.
Remark 4: Naturally the notion of extreme frequency equivalence is not applicable to the A-H tuning method and also to the GM & PM optimization that has the capability to minimize the IAE at lower M S than 1.8.

VII. DISCUSSION OF RESULTS
The universal PID and filter settings are proposed giving rise to optimum controller gains and measurement filter time constant with respect to not only the IAE criterion but also to the robustness and filtering effect constraints. The universality of the real controller settings consists in the control loop description via the similarity theory which imparts the control loop responses similar dynamics characterized by the similarity numbers. These responses differ from each other only in their scales of time and input and output variables. Consequently, for all the dynamically similar plants the same controller parameters ρ P , ρ D , ρ I , τ are obtained from (54) through (57) and the four-pole dominance check according to the condition (99) holds for all of them in spite of the significant plant delay effect characterized by varying laggardness number ϑ. These controller and filter parameters are evaluated for optimum values of damping, root, and natural frequency ratios. Particularly the optimum natural frequency ratio, η, results nearly everywhere between 0.3 and 0.5 as shown in Fig. 16 to achieve the robust PID setting with sufficient performance. Additionally, to limit the high-frequency control sensitivity the measurement filter's ratio is constrained as N ≤ 15. Despite the disturbance rejection optimization is preferred to that of the reference tracking the reference tracking capability is saved by prefiltering the reference variable in the classical control loop in Fig. 4. Simultaneously the dominant four-pole placement technique is extended for integrating third-order plants with delay.
The robustness constraint is achieved, as shown in Fig. 10, when the quadruple of dominant poles is assigned as two pairs of complex conjugate poles. Corresponding optimum controller gains and filter time constants are presented in Figs. 6-9. The dominance of the quadruple of assigned poles results abundant because the dominance index, recorded in Fig. 13, is nearly ten times greater than it is in usual dominant four-pole placement (σ = 2, see [30]). Thus, there are two natural frequencies assigned where one of them is the ultimate frequency to guarantee good disturbance rejection performance and the second one is new optimized via prescribing η to meet the robustness constraint as M S ≤ 1.8. The rest of ratios, δ and κ, are prescribed accordingly to keep not only the robustness constraint but also the high-frequency control sensitivity constrained by the filter's ratio as 5 < N < 20. While ratios δ, κ are varying particularly with λ 2 the ratio, η, results in 0.35 as an average value across all the considered plants in case of both ϑ = 0.3, see Fig. 16, and ϑ = 0.5. Keeping N ≤ 10, thus in the aforementioned range, under properly selected δ, κ and η in Fig. 14, 15 and 16, respectively, the filter time constant results in feasible, i.e. positive, number. Due to the enhanced delay effect, ϑ > 0.5, and poor plant damping, λ 2 > 0.5, the loop amplification is vanishing and the PID type control due to the scheme in Fig. 4 loses a potential to compensate for such stringent dynamics. Hence these cases with a poor chance of PID control application are excluded from the survey maps achieved. Finally, the separation of the dominant poles assigned from the rest of the infinite spectrum is considerable guaranteeing the desired control loop dynamics, and simultaneously the measurement filter's pole location falls into this rest preventing the dominant pole-zero cancellation.
The applicability range of the proposed method, DPD, is mainly in the field of the mechanical systems with actuator dynamics (e.g. servomechanism to pull a cart), thermal processes (e.g. heat exchanger), water turbines with long conduit etc. Other applications can be found in steel and energy industry.

VIII. CONCLUSION
The proposed dominant four-pole placement method gives rise to introduce novel tuning method for the real PID controller considered in application to the class of the third-order plants with delay parameterized according to the similarity theory. The delayed control loop results in the retarded system described by the fifth-order differential equation with delayed argument. Nevertheless, the real PID controller tuned due to the dominant four-pole placement is robust with sufficient performance for the practice and the four-pole dominance is guaranteed due to optimally prescribed ratios, particularly the natural frequency ratio. The comparative study, summarized in Table 1, for the benchmark oscillatory model shows the capability of the proposed dominant-four pole placement technique in contrast to other tuning and optimization methods.