ARTICLE pubs.acs.org/IECR
Analysis and Tuning of RTD-A Controllers Antonius Yudi Sendjaja, Zhen Fu Ng, Si Si How, and Vinay Kariwala* School of Chemical and Biomedical Engineering, Nanyang Technological University, Singapore ABSTRACT: RTD-A controller has recently been proposed as an alternative to the proportional-integral-derivative (PID) controller for control of single-input single-output systems. The RTD-A controller maintains the simplicity like the PID controller, has predictive capabilities like model predictive controller, and allows easier tuning of the controller parameters to achieve desired closed-loop performance. The available tuning rule for RTD-A controllers are aggressive, and their application to practical problems can be difficult. In this paper, a block diagram representation of the RTD-A controller is developed. It is shown that the RTD-A controller can be viewed as a generalized analytical predictor augmented with noise and set point filters. On the basis of the block diagram representation, semianalytical tuning rules are proposed for the RTD-A algorithm. It is shown that the RTD-A controller tuned using the proposed rule provides reasonable performance for a wide class of processes including inverse response and delaydominant processes.
1. INTRODUCTION The proportional-integral-derivative (PID) controller is the most widely used control algorithm for regulation purposes in industrial plants.1,2 The PID controller has many advantages such as simple controller structure, ease of implementation, and robustness to model errors.3 The PID control algorithm, however, also has several weaknesses. For example, the tuning parameters of the PID controller affect the performance for both set point tracking and disturbance rejection. In such cases, achieving good performance for both set point tracking and disturbance rejection simultaneously can be difficult.4 Furthermore, the controller parameters can take any values up to infinity, which makes transparent tuning difficult. Lastly, a PID controller does not take the long-term process behavior into account as it only reacts to reduce the error at the next time step. A more advanced algorithm used in process industries is the model predictive controller (MPC).5-7 MPC requires solving an optimization problem at every time step, which makes MPC computationally expensive. Thus, the application of MPC has largely been limited to medium- and large-scale multivariable processes in supervisory mode. In addition, despite the availability of some guidelines,7-9 the selection of tuning parameters for MPC can be difficult due to its complex structure. Thus, many researchers have focused on finding alternate control algorithms, which can overcome the drawbacks of the PID controller, while maintaining simplicity in design and algorithm.4,10,11 Recently, a new control algorithm, namely the RTD-A controller, has been proposed by Ogunnaike and Mukati.4 The term “RTD-A” stands for (R)obustness, set point (T)racking, (D)isturbance rejection, and overall (A)ggressiveness. The RTD-A controller has four tuning parameter (θR, θT, θD, and θA), which are directly related to the corresponding properties of the closed-loop system. All these parameters are normalized between 0 and 1, where a value closer to 0 implies that the corresponding property of the closed-loop system is more pronounced and vice versa. Among these parameters, θR, θT, and θD can be tuned independently of each other,4 e.g. improving r 2011 American Chemical Society
the set point tracking ability does not decrease its disturbance rejection and robustness abilities. The RTD-A controller only requires a first-order plus time delay (FOPTD) model, hence, maintains the simplicity like a PID controller, and has predictive ability like MPC. With these advantages, the RTD-A controller has recently been applied for control of a vapor deposition process12 and hypnosis regulation.13,14 Although the RTD-A controller has been shown to have distinct advantages over the PID controller,4 its properties are not well understood. Recently, tuning guidelines for the RTD-A controller have been proposed based on robust stability analysis,12 where a trade-off between robust stability and controller aggressiveness is sought. This tuning rule is referred to as Ogunnaike’s tuning rule in this paper. This tuning rule, however, only provide bounds on the allowable tuning parameters for stability of the closed-loop system and requires the selection of an uncertainty parameter, which can be difficult in practice. In this paper, a block diagram representation of an RTD-A controller is proposed for easier interpretation of the controller. Subsequently, the block diagram is analyzed to understand the role of each tuning parameter in the overall controller structure. It is shown that the RTD-A controller can be viewed as a generalized analytical predictor (GAP)15,16 augmented with set point and noise filters. On the basis of the block diagram representation, a semianalytical tuning rule is developed. This rule is derived by drawing analogies between the various blocks in the RTD-A structure with the established concepts in process control literature. The performance of the proposed tuning rule is evaluated using a wide class of numerical examples, including delay-dominant, higher-order, and inverse response processes. It is shown that the RTD-A controllers tuned using the proposed rule provide reasonable performance for these processes with the performance being comparable to the PI controller tuned using Skogestad’s internal model control (SIMC) rules.17 On the Received: October 24, 2010 Accepted: January 20, 2011 Revised: January 20, 2011 Published: February 22, 2011 3415
dx.doi.org/10.1021/ie102154y | Ind. Eng. Chem. Res. 2011, 50, 3415–3425
Industrial & Engineering Chemistry Research
ARTICLE
The reference trajectory y*(k) is determined as yðk þ 1Þ ¼ θT yðkÞ þ ð1 - θT Þysp ðkÞ
where θT is the tuning parameter for set point tracking. Assuming that the set point remains the same over the entire prediction horizon, i.e. ysp(k þ i) = ysp(k), i = 1, 2, ...,N, the future reference trajectory can be expressed as
Figure 1. Block diagram of closed loop system.
other hand, the RTD-A controller tuned by Ogunnaike’s tuning rule provides aggressive response, which leads to instability, even for processes, which can be accurately represented by FOPTD model. The rest of this paper is organized as follows: In section 2, the RTD-A control algorithm is revisited and Ogunnaike’s tuning rule is presented. A block diagram representation of the RTD-A controller is developed in section 3. In section 4, a semianalytical tuning rule for the RTD-A controller is proposed based on the analysis of the block diagram. The closed-loop performance of the proposed RTD-A controller is analyzed using numerical examples in section 5, and conclusions are drawn in section 6.
2. RTD-A CONTROLLER In this section, a brief overview of the RTD-A control scheme is given; see the work of Ogunnaike and Mukati4 for further details. Subsequently, Ogunnaike’s tuning rule for tuning the RTD-A controllers12 is described. 2.1. RTD-A Algorithm. The block diagram of the closed-loop system with the RTD-A controller is shown in Figure 1, where y, u, d, and ysp denote the output, input, disturbance, and set point, respectively. In this algorithm, it is assumed that the process behavior can be adequately described by an FOPTD model given as Kp expð - RsÞ ð1Þ GM ðsÞ ¼ τs þ 1 where Kp, τ, and R are the steady-state gain, time constant, and time delay of the model, respectively. The continuous-time model can be discretized with sampling time ΔT to obtain the equivalent discretetime representation of the FOPTD model given as bz-ðm þ 1Þ ð2Þ GM ðz-1 Þ ¼ 1 - az-1 where ΔT R ð3Þ a ¼ exp ; b ¼ Kp ð1 - aÞ; m ¼ τ ΔT Note that in comparison to the work of Ogunnaike and Mukati,4 where m is taken as m = round(R/ΔT), we assume that R is an integer multiple of ΔT. In the numerical examples considered in section 5, this assumption is satisfied through appropriate selection of ΔT. In the RTD-A algorithm, the input u(k) is updated by minimizing the deviation between the predicted process output ̂y(k) and the reference trajectory y*(k) over the prediction horizon N. In particular, the following optimization problem is solved N X ð4Þ ðyðk þ iÞ - ^y ðk þ m þ iÞÞ2 min uðkÞ
i¼1
1 uðkÞ ¼ b
PN
ð5Þ
yðk þ iÞ ¼ θiT yðkÞ þ ð1 - θiT Þysp ðkÞ; i ¼ 1, 2, :::, N
On the basis of the process model in eq 2, the predicted output can be equivalently expressed by the following difference equation ^yðk þ 1Þ ¼ a^yðkÞ þ buðk - mÞ
ð7Þ
The RTD-A algorithm assumes that the input remains the same for the next N steps, i.e. u(k þ i) = u(k); i = 1, 2, ..., N. Thus, the predicted output can be written as m X ai uðk - iÞ ^yðk þ m þ iÞ ¼ am þ i^yðkÞ þ ai - 1 b i¼1
þ bηi uðkÞ þ ^eD ðk þ m þ iÞ; i ¼ 1, 2, :::, N
ð8Þ
where ηi ¼
1 - ai 1-a
ð9Þ
In eq 8, ^eD denotes the nonbiasing prediction error, which is determined by the robustness tuning parameter θR as ^eD ðk þ 1Þ ¼ θR^eD ðkÞ þ ð1 - θR ÞeðkÞ
ð10Þ
where the current error e(k) is the difference between the measured output y(k) and its prediction ̂y(k) obtained using eq 7, i.e. e(k) = y(k) - ̂y(k). In addition, future estimates of the disturbance are determined using the disturbance rejection tuning parameter θD as 1 - θD ½1 - ð1 - θD Þm þ i θD ð11Þ ½^eD ðkÞ - ^eD ðk - 1Þ
^eD ðk þ m þ iÞ ¼ ^eD ðkÞ þ
By solving the optimization problem in eq 4, the following explicit expression for u(k) can be obtained
P ðk þ iÞ - am þ i^yðkÞ - ai - 1 b mi¼ 1 ai uðk - iÞ - ^e ðk þ m þ iÞÞ D PN 2 η i¼1 i
i ¼ 1 ηi ðy
Equation 12 is the control law for the RTD-A controller, based on which the input u(k) is updated at every time step.
ð6Þ
ð12Þ
2.2. Ogunnaike’s Tuning Rule. In RTD-A algorithm, θR determines controller’s ability to handle plant-model 3416
dx.doi.org/10.1021/ie102154y |Ind. Eng. Chem. Res. 2011, 50, 3415–3425
Industrial & Engineering Chemistry Research
ARTICLE
where
Table 1. Ogunnaike’s RTD-A Tuning Rule composite parameter
θR
θT
θD
θA
-1
GT, i ðz Þ ¼
θiT
low noise processes F0.95
0.8
g1 - θR
0.995
0.9
0.8
g0.37
0.9F
1 e F e 1.6
0.9
0.8
g0.37
1.25(1 - e-F)
F g 1.6
>0.95
0.8
g1 - 0.7θR
0.995
N ¼ 1-
τ lnð1 - θA Þ ΔT
ð18Þ
1 - θR 1 - θR z-1
ð19Þ
where
As before, the following expression of ^eD(k þ m þ i) is obtained by substituting eq 18 into eq 11 ^eD ðk þ m þ iÞ ¼ GR ðz-1 ÞGD, i ðz-1 ÞeðkÞ;
ð13Þ
To select these tuning parameters, Ogunnaike et al.12 proposed a set of generic rules. This tuning rule is derived based on robust stability analysis performed using hundreds of FOPTD models with different parameters and assumed plant-model mismatch. The tuning rule (referred to as Ogunnaike’s tuning rule in this paper) is shown in Table 1. Here F is a composite variable defined as 1þλ R ð14Þ F¼λ 1-λ τ where λ is the multiplicative uncertainty parameter. The recommended value for λ is 0.1 for processes, which can be closely represented by an FOPTD model and higher otherwise. In addition, Ogunnaike et al.12 recommended that ΔT be chosen to be 0.1τ or lower. RTD-A controllers tuned using this rule have been used for control of quadruple tank process and vapor deposition process.12 A shortcoming of this rule, however, is that it often leads to aggressive closed-loop behavior; see Section 5 for details. Furthermore, this rule only provides bounds on θR and θD, and unambiguous selection of λ can be difficult in practice.
3. BLOCK DIAGRAM REPRESENTATION The derivation of a simple tuning rule for the RTD-A controller requires a proper understanding of the role of each tuning parameter. For this purpose, a block diagram representation of the RTD-A controller is developed in this section. 3.1. Derivation. The central idea of the block diagram representation is to simplify the different terms in the control law given in eq 12 and express them in terms of set point ysp(k) and error e(k). First, we note that based on eq 5, the expression for y*(k þ 1) can be equivalently written in transfer function form as 1 - θT ysp ðkÞ ð15Þ yðk þ 1Þ ¼ 1 - θT z-1 The following expression for y*(k þ i) is found by substituting eq 15 in eq 6 yðk þ iÞ ¼ GT, i ðz-1 Þysp ðkÞ; i ¼ 1, 2, :::, N
^eD ðkÞ ¼ GR ðz-1 ÞeðkÞ
GR ðz-1 Þ ¼
mismatch, while θT and θD determine future reference trajectory and disturbance prediction, respectively. The prediction horizon N is related to the overall aggressiveness tuning parameter θA as
ð16Þ
ð17Þ
Similar to eq 15, the expression for current estimated error^eD can also be written in transfer function form as
noisy processes F
