Dynamics and Control of Chemical Processes Degree in Chemical Engineering Unit 10. PID Controller Design and Tuning

G784 – Dynamics and Control of Chemical Processes 1. Performance Criteria For Closed-Loop Systems - The function of a feedback control system is to ensure that the closed-loop system has desirable dynamic and steady-state response characteristics

- Ideally, we would like the closed-loop system to satisfy the following performance criteria:

- The closed-loop system must be stable - The effects of disturbances are minimized, providing good disturbance rejection

- Rapid, smooth responses to set-point changes, are obtained, that is, good set-point tracking - Steady-state error (offset) is eliminated

- Excessive control action is avoided - The control system is robust, that is, insensitive to changes in process conditions and to inaccuracies in the process model

- PID controller settings can be determined by a number of alternative techniques: - Direct Synthesis (DS) method; Internal Model Control (IMC) method;

Controller tuning relations; Frequency response techniques; Computer simulation; On-line tuning after the control system is installed

G784 – Dynamics and Control of Chemical Processes 2. Direct Synthesis Method - In the DS method, the controller design is based on a process model and a desired closed-loop transfer function

- The latter is usually specified for set-point changes, but responses to disturbances can also be utilized

- Although these FB controllers do not always have a PID structure, the DS method does produce PI or PID controllers for common process models

- As a starting point for the analysis, consider the block diagram of a FB control system in Figure 1. The closed-loop TF for set-point changes was derived before (Eq. 1):

Km × Gc × Gv × Gp Y = Ysp 1 + G c × G v × G p × G m

Figure 1. Block diagram for a standard FB control system

G784 – Dynamics and Control of Chemical Processes 2. Direct Synthesis Method For simplicity, let G = GvGpGm and assume that Gm = Km. Then we obtain (Eq. 2):

G cG Y = Ysp 1 + G c G Rearranging and solving for Gc gives an expression for the feedback controller (Eq. 3): Y / Ysp 1

Gc =

×

G 1 - Y / Ysp

- Equation 3 cannot be used for controller design because the closed-loop TF Y/Ysp is not known a priori

- Also, it is useful to distinguish between the actual process G and the model, Ĝ, that provides an approximation of the process behavior

- A practical design equation can be derived by replacing the unknown G by Ĝ, and Y/Ysp by a desired closed-loop TF, (Y/Ysp)d (Eq. 4):

- The specification of (Y/Ysp)d is the key design decision

(Y / Ysp ) d 1 Gc = ~ × G 1 - (Y / Ysp ) d

and will be considered later

- Note that the controller TF in Eq. 4 contains the inverse of the process model owing to the 1/Ĝ term

- This feature is a distinguishing characteristic of modelbased control

G784 – Dynamics and Control of Chemical Processes 2. Direct Synthesis Method Desired Closed-Loop Transfer Function For processes without time delays, the first-order model is a reasonable choice (Eq. 5)

Y Ysp

= d

1 τ cs + 1

- By substituting Eq. 5 into 4 and solving for Gc, the controller design equation is (Eq. 6):

1 1 Gc = ~ × G τ cs

- The 1/τIs term provides integral control action and thus eliminates offset - Design parameter τc provides a convenient controller tuning parameter that can be used to make the controller more aggressive (small τc) or less aggressive (large τc)

- If the process TF contains a known time delay θ, a reasonable choice for the desired closed-loop TF is (Eq. 7):

Y Ysp

d

e -θs = τ cs + 1

G784 – Dynamics and Control of Chemical Processes 2. Direct Synthesis Method Desired Closed-Loop Transfer Function

- The time-delay term in Eq. 7 is essential because it is physically impossible for the controlled variable to respond to a set-point change at t = 0 before t = θ

- If the time delay is unknown, θ must be replaced by an estimate - Combining Eqs. 7 and 4 gives (Eq. 8): 1 e -θs Gc = ~ × G τ c s + 1 - e -θs - Although this controller is not in a standard PID form, it is physically realizable - Next, we show that the design equation (Eq. 8) can be used to derive PID controllers for simple process models

- The following derivation is based on approximating the time-delay term in the denominator of Eq. 8 with a truncated Taylor series expansion (Eq. 9):

e - θs ≈ 1 - θ s Substituting Eq. 9 into the denominator of Eq. 8 and rearranging gives (Eq. 10)

1 e -θs Gc = ~ × G ( τ c + θ) × s

Note that this controller also contains integral control action

G784 – Dynamics and Control of Chemical Processes 2. Direct Synthesis Method Desired Closed-Loop Transfer Function First-Order-Plus-Time-Delay (FOPTD) Model Consider the standard FOPTD model (Eq. 11),

Ke -θs ~ G(s) = τ cs + 1 Substituting Eq. 11 into Eq. 10 and rearranging gives a PI controller,

Gc = Kc × 1+

1 τ Is

with the following controller settings (Eq. 12):

1 τ Kc = × K θ + τc

τI = τ

G784 – Dynamics and Control of Chemical Processes 2. Direct Synthesis Method Desired Closed-Loop Transfer Function Second-Order-Plus-Time-Delay (SOPTD) Model Consider the standard SOPTD model (Eq. 13),

~ (s) = G

Ke -θs (τ1s + 1)(τ 2 s + 1)

Substitution into Eq. 10 and rearrangement gives a PID controller in parallel form,

Gc = Kc × 1+

1 + τ Ds τ Is

with the following controller settings (Eq. 14):

1 τ1 + τ 2 Kc = × K θ + τc

τ I = τ1 + τ 2

τ1 τ 2 τD = τ1 + τ 2

G784 – Dynamics and Control of Chemical Processes 3. Internal Model Control (IMC) - A more comprehensive model-based design method, IMC - The IMC method, like the DS method, is based on an assumed process model and leads to analytical expressions for the controller settings

- These two design methods are closely related and produce identical controllers if the design parameters are specified in an consistent manner

- The IMC method is based on the simplified block diagram shown in Figure 2. A

process model Ĝ and the controller output P are used to calculate the model response, Ŷ.

- The model response is subtracted from the actual

response Y, and the difference, Y – Ŷ is used as the input signal to the IMC controller, Gc*

- In general, Y ≠ Ŷ due to modeling errors (Ĝ ≠ G) and unknown disturbances (D ≠ 0) that are not accounted for in the model

- Comparing both control systems, it can be shown

that the two block diagrams are identical if controllers Gc and Gc* satisfy the relation (Eq. 15): *

Gc Gc = ~ 1 - G *c G

G784 – Dynamics and Control of Chemical Processes 3. Internal Model Control (IMC) - Thus, any IMC controller Gc* is equivalent and vice versa

to a standard feedback controller Gc,

- The following closed-loop relation for IMC can be derived from Fig. 2 using the block diagram algebra (Eq. 16):

G *c G 1 - G *c G Y= ~ ) × Ysp + 1 + G * (G - G ~) × D 1 + G *c (G - G c

For the special case of a perfect model, Ĝ = G, so (Eq. 17)

Y = G *c G × Ysp + (1 - G *c G) × D The IMC controller is designed in two steps:

~ =G ~ G ~ - Step 1. The process model is factored as G + time delays and RHP zeros

where Ĝ + contains any

In addition, Ĝ+ is required to have a steady-state gain equal to one in order to ensure that the two factors (Ĝ+ and Ĝ–) are unique

G784 – Dynamics and Control of Chemical Processes 3. Internal Model Control (IMC) - Step 2. The controller is specified as (Eq. 18) 1 G *c = ~ × f Gwhere f is a low-pass filter with a steady-state gain of one. It typically has the form:

f=

1 ( τ c s + 1) r

In analogy with the DS method, τc is the desired closed-loop time constant. Parameter r is a positive integer. The usual choice is r = 1 For the ideal situation where the process model is perfect (Ĝ = G), substituting Eq. 18 into Eq. 17 gives the closed-loop expression

~ f × Y + (1 - f × G ~ )×D Y=G + sp + Thus, the closed-loop TF for set-point changes is

Y ~ = f ×G + Ysp

G784 – Dynamics and Control of Chemical Processes 3. Internal Model Control (IMC) Selection of τc

- The choice of design parameter τc is a key decision in both the DS and IMC design methods

- In general, increasing τc produces more conservative controller because Kc decreases while τI increases

- Several IMC guidelines for τc have been published for the model in Eq. 11 τc/θ > 0.8 and τc > 0.1τ

(Riviera et al., 1986)

τ > τc > θ

(Chien and Fruehauf, 1990)

τ=θ

(Skogestad, 2003)

G784 – Dynamics and Control of Chemical Processes 4. Controller Tuning Relations IMC Tuning Relations The IMC method can be used to derive PID controller settings for a variety of TF models

Table 1. IMC-Based PID Controller Settings for Gc(s)