A Novel Process-Reaction Curve Method for Tuning PID Controllers

A novel process-reaction curve method for tuning PID controllers for (possible) higher order processes/models is presented. The proposed method is similar to the Ziegler-Nichols process reaction curve method, viz. only the maximum slope and lag need to be identified from an open loop step response. The relative time delay error (relative delay margin), δ is the tuning parameter. The proposed method is verified through extensive numerical simulations and is found close to optimal in many of the motivated process examples. In order to handle the wide set of process models, two model reduction modes are presented.


Introduction
The main focus of this paper and previous work in Dalen andDi Ruscio (2017, 2018) is to approximate step responses from (possible) higher order models/systems with Double Integrating Plus Time Delay (DIPTD) models such that the (ideal/parallel) PID controllers, may be tuned to archive some kind of optimality, e.g.minimising the Integrated Absolute Error (IAE) index.In Eq. (1) K is the gain acceleration and τ is the time delay.In Eq. (2) K p is the proportional gain, T i is the integral time constant and T d is the derivative time constant.Two of the first and most used PID controller tuning methods are presented in the work of Ziegler (1941); Ziegler andNichols (1942, 1943), viz. the Ziegler-Nichols (ZN) Process-Reaction Curve (PRC) method which is based on an open loop response, and an ultimate gain method which is based on a closed loop response.We note the ZN PRC PID controller settings as follows: K p = 1.2 R1L , T i = 2L and T d = L 2 , where, R 1 is the unit reaction rate (maximum slope) and L is known in this paper as Ziegler's lag (see Sec. 4 and Figure 2 for details).Note the statement that identifying process dynamics with only two parameters is insufficient; see Åström and Hägglund (2004).Note, in general, that the ZN PID controllers demonstrate poor robustness, see e.g.Åström and Hägglund (2004).One advantage with this PRC method is that the user does not need to wait for the process to reach steady state, as is usually needed for methods based on e.g.first order plus time delay model approximations.
• The δ-PRC method proposed in Dalen and Di Ruscio (2018) is further developed and proven on motivated process model examples.
• Model reduction modes are introduced.
• The δ-PRC method is compared to the modelbased tuning methods, SIMC and K-SIMC on motivated time constant models.
• The δ-PRC method is compared to the heuristic optimisation tuning method in Dalen and Di Ruscio (2018) on motivated process models containing complex poles.The ZN PRC PID controller tuning method is also included in this comparison.
• A possible ζ-PRC tuning variant is demonstrated.
In this variant the main tuning parameter is present in the model reduction step, i.e. the gain acceleration is proportionally varied.
All numerical calculations and plotting facilities are provided by using the MATLAB software, MATLAB (2016).The rest of this paper is organised as follows.In Sec. 2 the preliminary definitions are given.In Sec. 3 the PO PID controller is presented.The δ-PRC method including the model reduction modes are presented in Sec. 4. The numerical results are presented for a wide range of examples in Sec. 5. Lastly, the discussion and concluding remarks are given in Sec. 6.

Preliminary Definitions
Definition 2.1 (System) The underlying systems/models are assumed to be described by the following transfer function form, where, n > 1, the gain, k = 0, time constant T z , time constants, T 1 ≥ T 2 ≥ . . .≥ T j ≥ 0, the "speed" of response, 0 ≤ τ 0 , and the relative damping, ξ in the range, 0 < ξ < 1.In Eq. (3) we assume that T j and τ 0 are not both zero at the same time.Assuming only deterministic systems/models.
In the case of a pure single time constant process, we obtain an on-off controller, which is not a part of the topic of this paper, hence the reason for setting n > 1 if τ 0 = 0.One motivation for approximating models as in Eq. (3) with DIPTD models (Eq.( 1)) is that for a short time interval the approximation is fairly good, which is illustrated in Figure 2. However the main motivation is that it gives close to PO PID controllers which is documented in this and previous work in Dalen and Di Ruscio (2018).In this paper we are only interested in tuning PID controllers (for models as Eq. ( 3)) based on DIPTD models (Di Ruscio and Dalen (2017)).
Consider the standard feedback system with disturbances as illustrated in Figure 1.In order to compare the different controllers against each other we will consider indices such as defined in Åström and Hägglund (1995), Seborg et al. (1989) and Skogestad (2003).We define these in the following.

Definition 2.2 (Performance)
For measuring performance in a feedback system, the IAE is defined as where, e = r − y, is the control deviation error and r is the reference.Furthermore, the following is defined: • IAE vu evaluates the performance in the case of a step input disturbance (H v (s) = H p (s)), v = 1, with the reference, r = 0.
• IAE vy evaluates the performance in the case of a step output disturbance (H v (s) = 1), v = 1, with the reference, r = 0. (3)), PID controller, H c (s), (Eq.( 2)) and disturbance model, H v (s), where, step disturbance, v, at the input when H v (s) = H p (s) and at the output when H v (s) = 1.
Robustness (i.e.allowing for inaccuracies in the acceleration gain and time delay in the DIPTD model in Eq. ( 1)) may be quantified in various ways, and in this work we define it according to Garpinger and Hägglund (2008).
To evaluate the amount of input usage we include the following measure.
Definition 2.4 (Input Usage) Input usage is defined as Total input Value (TV) where, ∆u k = u k − u k−1 , is the control rate of change.

Pareto-Optimal PID Controller
For quantifying multiple performances, i.e. indices IAE vu , and, IAE vy , we define the following Pareto performance objective, where s r is the servo-regulator parameter chosen in the range 0 ≤ s r ≤ 1 (originally introduced in Di Ruscio ( 2012)) for trade-off weighting between the output disturbance (servo) weighting s r = 1 and input disturbance (regulator) weighting s r = 0.In this work, and as in earlier papers, we will set s r = 0.5 (Skogestad and Grimholt (2012)).The following main performance objective is defined in a mean square error sense, where x is a tuning method and M = length(J).

δ-PRC Controller Tuning
The δ-PRC PID controller tuning method is defined as Algorithm 2.1 in Dalen and Di Ruscio (2018), where steps 1-2 are substituted with the following DIPTD model (Eq.( 1)) approximation rules for gain acceleration and time delay, where R = max / min t dy dt (i.e.min if y(t f inal ) < 0), is the reaction rate (maximum slope), R 1 = R ∆u , is the unit reaction rate, ∆u = 1 (default), is the input step change, L = t 1 − y1 R1 , is defined as Ziegler's lag, t 1 = arg R 1 , and, y 1 = y(t 1 ).See Figure 2 for an illustration of the model reduction technique.Based on extensive simulations in this and previous work (Dalen and Di Ruscio (2018)), and since we usually encounter relatively low order models, we recommend choosing ζ in the range 0 < ζ ≤ 10.Furthermore, the main performance objective V M (Eq.( 8)) is observed to be relatively insensitive to small changes around η = 1 2π (holds at least for pure multiple pole models), hence we propose to keep this constant as in Dalen and Di Ruscio (2018).We take a shot at covering a broad set of possible models/systems and, at the same time, make the method practical for the user.We present a couple of model reduction modes in the following.
Mode 1 corresponds to the method given in Dalen and Di Ruscio (2018) where a possible proof was given therein.Note that mode 1 works well for most processes (satisfying Eq. ( 3)), however, arguably, not for processes where the time constants are equal or approximately equal with order n > 3.For such processes we would suggest mode 2. It would be useful to have some information on the process before assigning a mode, but this is not necessary.Given that we only have two modes, the user may perform a trial-and-error approach.However, one might find the appropriate mode by fixing the main tuning parameter, δ, (e.g.δ = 2.12) and observing the closed loop response, input step response, or by calculating the M s (Eq.( 5)) directly, subject to changing between modes.Some comments regarding optimal method product settings, c, and integral-derivative ratio, γ, are given for step 3 in Algorithm 2.1 (Dalen and Di Ruscio (2018)), i.e. δ-PID controller tuning.Consider the DIPTD model (Eq.( 1)) where the gain acceleration K = 1 and the time delay τ = 1.In the incoming we define Alg.3.1 as Alg.2.1 and Eq. ( 27) in Di Ruscio and Dalen (2017).
The first setting is obtained by solving the following optimisation problem where Alg.3.1 (c, γ, δ i ) and Alg.3.1 o (δ i ) are precalculated as follows Notice that γ = 2.25 was found to be optimal in Kristiansson and Lennartson (2006).
The second setting which is used in this work and was originally proposed in Dalen and Di Ruscio (2017), is found by where Alg.3.1 (c, γ, δ (M i s )) and PO (M i s ) are precalculated as follows Notice that the conventional ratio, γ = 4, is larger than the couple presented above, see e.g.Ziegler and Nichols (1942), Astrom and Hagglund (1984), Mantz and J. Tacconi (1989) and Skogestad (2003).
The algorithm for the δ-PRC method is presented as follows.
Algorithm 4.1 (δ-PRC PID Controller Tuning) 1. Find Ziegler's lag, L, and the unit reaction rate, R 1 based on the open loop step response (reaction curve) of the (possible) higher order model/system.
2. Choose one of the two model reduction modes (proposed in Eq. ( 11)) based on trial-and-error.Find the gain acceleration, K, and time delay, τ , in the DIPTD model, using Eqs.( 9) and (10).
The above method is implemented in a MATLAB m-file function shown in App. A.

Numerical Results
A set of PO PID controllers is obtained for each process model example (Es1-12) using the exact gradient optimisation method in Grimholt and Skogestad (2016b).
Note that the PID controller tuning methods of SIMC and K-SIMC are based on second order plus time delay models.Furthermore the SIMC and K-SIMC tuned PID controllers are on cascade form, hence they need to be converted to the ideal/parallel form.
For Es1-6, the δ-PRC tuning method is compared to SIMC and K-SIMC in terms of trade-off curves shown in Figure 4, where the corresponding V M measures are given in Table 3.The corresponding time-domain output and input step responses and input usage, for a prescribed robustness, M s = 1.59, are illustrated in Figures 7 and 8, i.o.For Es7-12 (i.e.complex pole examples), the δ-PRC tuning method is compared to the Opt-PRC method (Dalen and Di Ruscio (2018)) in terms of trade-off curves shown in Figure 5, where the corresponding V M measures are given in Table 4. Notice that the ZN PRC PID controller tuning is included as a point in the trade-off plots.We present Figure 6 which is 'zoomed out' version of Figure 5.The corresponding time-domain output and input step responses and input usage, for a prescribed robustness, M s = 1.59, are illustrated in Figures 9 and 10, i.o.See Table 5 for the choice of model reduction modes in the δ-PRC method for Es1-12.The prescribed PID controller parameters, including the Pareto performances J and the margins, are shown in App.B in Tables 6 and 7.

Discussion and Concluding Remarks
The discussion and concluding remarks are itemised as follows.
• The δ-PRC method in Dalen and Di Ruscio (2018) is further developed and proven.Model reduction modes have been presented.These are demonstrated through Es1-12.
• It is seen in Es1-6 that the proposed method has an edge over the other model-based methods SIMC and K-SIMC, viz.δ-PRC is seen to be at the minimum VK-SIMC V δ−PRC = 1.6 times better than the runner-up method (K-SIMC), and at the maximum, VSIMC V δ−PRC = 70.1 times better than the runnerup method (SIMC).See Table 3.
• These simple heuristic modes give PID controller

C. ζ-PRC PID Controller Tuning
Note that in some examples we may possibly use ζ as the tuning parameter, i.e. for the incoming example we set δ = c = γ = 2.12.
The main performance objective is V M = 4.2250e-4.Interestingly, in this case, the ζ-tuning is 17 times better than mode 1 and 148 times better than SIMC in Dalen and Di Ruscio (2018).The trade-off curves are illustrated in Figure 11.Pareto performance J (Eq. ( 7)) vs. robustness M s trade-off curves.

Figure 1 :
Figure 1: Control feedback system.The plant model is described by the process model H p (s) (Eq.(3)),PID controller, H c (s), (Eq.(2)) and disturbance model, H v (s), where, step disturbance, v, at the input when H v (s) = H p (s) and at the output when H v (s) = 1.

Figure 3 :
Figure 3: E1.The figure illustrates the model reduction technique.Shows the magnitude responses for the higher order process model in column 2 inTable 2 and the DIPTD model, H p (s) = K e −τ s s 2 , approximation where the gain acceleration K = ζ R1 L varies, i.e. ζ ∈ {0.1, 1, 10}.

Figure 4 :
Figure 4: Es1-6.The figure shows the Pareto performance J (Eq. (7)) and M s (Eq.(5)) tradeoff curves for the methods δ-PRC (δ), SIMC (T c ), K-SIMC (λ) and PO PID (M s ), where δ is the prescribed relative time delay error, T c and λ are the prescribed set point response time constants.The circles illustrates controllers with prescribed robustness M s = 1.59.

Figure 6 :
Figure 6: Es7-12.The figure shows the Pareto performance J (Eq. (7)) and robustness M s (Eq.(5)) trade-off curves for the methods δ-PRC (δ), Opt-PRC (δ 1 ) and PO PID (M s ).This figure illustrates the ZN PRC PID tuning as a single point per example.This figure is a 'zoomed out' version of Figure 5.

Figure 7 :Figure 8 :
Figure 7: Es1-6.The figure illustrates the output and input step time-domain responses for a prescribed robustness, M s = 1.59, for the following methods: δ-PRC, SIMC, K-SIMC and PO PID.

Figure 9 :Figure 10 :
Figure 9: Es7-12.The figure illustrates the output and input step time-domain responses for a prescribed robustness, M s = 1.59, for the following methods: δ-PRC, SIMC, K-SIMC and PO PID.

Table 1 :
The controller arguments are structured as p = [K p , T i , T d ] T .IAE o vy = min p IAE vy (p, M s ) and IAE o vu = min p IAE vu (p, M s ), are the optimal output and input disturbance indices, i.o, where M s = 1.59.See Table 1 for details of E1.The table shows the optimal input and output disturbance controllers for prescribed robustness, M s = 1.59.* means that this value is not important and is not given.

Table 2 :
Shows the test batch, i.e. the motivated process model examples (Es1-12) used in the numerical simulations. )

Table 3 :
Es1-6.The table shows the main performance objective V M (Eq.(8)) measures for the δ-PRC, K-SIMC and SIMC tuned PID controllers, i.e. corresponding to the trade-off curves in Figure4.

Table 5 :
The table shows the chosen model reduction modes (Eq.(11)) in the δ-PRC method for the examples Es1-12.

Table 7 :
. The table contains the PID controller parameters for fixed robustness M s = 1.59 for the methods; δ-PRC, Opt-PRC and PO PID.Also include the Pareto performance J and the input usage T V .The following margins are included: Gain Margin (GM), Phase Margin (PM) and Delay Margin (DM).