Controller design procedures

G

cs

G

m e^!ms

G

cd

Ge^! s

r y

"

"

! !

"

"

d

! u

e

Figure 2.3: New Smith predictor structure

and θ_m=θ, (2.6) and (2.7) reduce to

Yr(s) =GcsGe^−θs

1 +G (2.8)

Y_d(s) = Ge^−θs

1 +G_cdGe^−θs (2.9)

It is apparent from (2.8) and (2.9) that the new Smith predictor has decoupled the load disturbance response from the setpoint response. (2.9) shows that the load disturbance response will be unstable when G_cd= 0 and the plant is unstable.

2.3 Controller design procedures

The dynamics of a large number of industrial processes with time delay can be represented by the following typical transfer function models:

• pure integrating process plus time delay,

G_p = ke^−θs

s (2.10)

• first order plus time delay,

G_p = ke^−θs

τ s+ 1 (2.11)

• integrating second order plus time delay,

G_p = ke^−θs

s(τ s+ 1) (2.12)

• unstable first order plus time delay,

G_p = ke^−θs

τ s−1 (2.13)

• stable second order plus time delay,

G_p = ke^−θs

(τ₁s+ 1)(τ₂s+ 1) (2.14)

• unstable second order plus time delay,

G_p = ke^−θs

(τ₁s−1)(τ₂s+ 1) (2.15)

Again when a process is of higher order, it can be reduced to any of the above mentioned process models by using the relevant model reduction method.

2.3.1 Design of G_cs

The setpoint tracking controller (Gcs) is designed based onH2optimal control specification method, minkek²2. The system performance objective is achieved by implementing

minkW(s)(1−Y_r(s))k²2 for the proposed structure, where W(s) is a setpoint input weight function.

W(s) is selected as ¹_s for the step changes of the setpoint input and load disturbances in industrial and chemical practice.

Consider a processG_p=ke^−θs

(τ₁s±1)(τ₂s+ 1). Using the n/norder all-pass Pad´e approxima- tion of the time delay e^−θs, it becomes

kP (−θs)

2.3 Controller design procedures

where

P_nn(θs) =

n

X

j=0

(2n−j)!n!

(2n)!j!(n−j)!(θs)^j (2.17)

and n is chosen to be an integer large enough to guarantee that the approximation error can be neglected in comparison with the process model mismatch in practice. Again, using (2.8) when there is no model mismatch, it follows that

kW(s)(1−Y_r(s))k²2 = 1 s

1− kG_cs(s)P_nn(−θs) [(τ₁s±1)(τ₂s+ 1) +k]P_nn(θs)

2

2

=

P_nn(θs)

sP_nn(−θs)− kG_cs(s)

s[(τ₁s±1)(τ₂s+ 1) +k]

2 2

(2.18) It is noted thatP_nn(0) = 1 and all zeros ofP_nn(−θs) are in the right hand side of thesplane. Using the orthogonality property of theH₂ norm, we get

kW(s)(1−Y_r(s))k²2 =

P_nn(θs)−P_nn(−θs) sP_nn(−θs)

2

2

+

[(τ₁s±1)(τ₂s+ 1) +k]−kG_cs(s) s[(τ1s±1)(τ2s+ 1) +k]

2 2

(2.19) In order to obtain a optimal controller, minimizing the right hand side of the above expression (equating its second term to zero) i.e.

[(τ1s±1)(τ2s+ 1) +k]−kGcso(s)

s[(τ₁s±1)(τ₂s+ 1) +k] = 0 (2.20) we obtain the optimal controller

G_cso= (τ₁s±1)(τ₂s+ 1) +k

k (2.21)

For ease in implementation a low pass filter of the form f_cs(s) = 1

(λ_css+ 1)² (2.22)

is introduced where λ_cs is the filter time constant. Then, the setpoint tracking controller becomes G_cs(s) =G_csof_cs(s) = (τ1s±1)(τ2s+ 1) +k

k(λ_css+ 1)² (2.23)

When the tuning parameter λ_cs tends to zero, G_cs(s) becomes optimal. λ_cs which is an adjustable closed-loop design parameter is always positive. A faster speed of response is achievable by a small

the tuning parameter λ_cs. Thus, λ_cs should be selected such that there is a good compromise between faster speed of the responses and improved robustness. The guidelines to choose such a parameter given in Majhi and Atherton [27], is adopted here. The setpoint tracking controllers obtained using similar analysis for different processes are summarized in Table 2.1.

Table 2.1: Setpoint tracking controllerGcs

Process Gcs

FOPTD _k(λ^{τ s+k+1}

css+1) τ1=τ, τ2= 0

IPTD _k′(λ^s+kcss+1)^′ τ1→ ∞, k^′ =_τ^k₁ →f inite, τ2= 0

ISOPTD _k(λ^{τ s2+s+k}

css+1)² τ1→ ∞, τ2=τ

UFOPTD _k(λ^{τ s+k−1}

css+1) τ1=τ, τ2= 0

SOPTD ^τ1τ2s

2+(τ1+τ2)s+k+1 k(λcss+1)²

USOPTD ^τ1τ2s

2+(τ1−τ2)s+k−1

k(λcss+1)² τ1> τ2

2.3.2 Design of G_cd

The disturbance rejection controller (G_cd) is designed based on IMC-PID approach [91]. The IMC controller design involves the following two steps.

Step 1: The process model G_m is factored as G_m(s) =G_m+(s)G_m−(s), where G_m+ is the invertible portion of the model and G_m− is the portion of the model having non-minimum phase characteristics and generally contains dead times and/or right half plane zeros and givingGm−(0) = 1.

Step 2: The idealized IMC controller is the inverse of the invertible portion of the process model i.e.

Q_m =G⁻¹_m+

To make the IMC controller proper, it is mandatory to add a low-pass filter f_cd with a steady state gain of 1. The filter is introduced for physical realisability of the IMC controller. Thus, the IMC controller is designed as

Q=Q_mf_cd=G⁻¹_m+f_cd (2.24)

The ideal feedback controller that is equivalent to the IMC controller can be expressed in terms of the internal model, G_m and the IMC controller, Q as

G_cd= Q

1−G Q (2.25)

2.3 Controller design procedures

Since the resulting controller does not have a standard PID controller form, the remaining issue is to design the PID controller that approximates the equivalent feedback controller most closely. Therefore, the mathematical Maclaurin series expansion formula is employed to produce the desired disturbance estimator in a simple way. In practice the desired disturbance estimator should possess an integral characteristic to eliminate any system output deviation arising from load disturbances. Therefore, let

G_cd= f(s)

s (2.26)

Expanding G_cd in Maclaurin series in sgives G_cd = 1

s(f(0) +f^′(0)s+f^′′(0)

2! s²+...) (2.27)

The first three terms of (2.27) can be interpreted as exactly a standard PID controller given by G_cd=K_p+ 1

T_is+T_ds (2.28)

whereK_p =f^′(0) ,T_i = _f(0)¹ andT_d= ^f′′₂⁽⁰⁾.

Now consider a FOPTD process modelG_p = ^ke_{τ s+1}^−θs wherekis the process gain,τ is the time constant and θis the time delay. The optimum IMC filter structure is given by

f_cd = (αs+ 1)²

(λ_cds+ 1)³ (2.29)

The resulting IMC controller becomes

Q= (τ s+ 1)(αs+ 1)²

k(λ_cds+ 1)³ (2.30)

Therefore, the disturbance rejection controller, which is equivalent to the IMC controller, is G_cd = (τ s+ 1)(αs+ 1)²

k[(λ_cds+ 1)³−e^−θs(αs+ 1)²] (2.31) whereα calculated by solving

1−(αs+ 1)²e^−θs

(λ_cds+ 1)³

s=−1/τ = 0, becomesα=τ

"

1−

1−^λ_τ^cd3

e^{−θ/τ 1}/2#

When the process model is IPTD, it is not possible to apply the aforementioned IMC procedure as the terms including α disappear ats = 0. Usually, the controller based on the model with a stable pole near zero can give a more robust closed-loop response than that based on the model with an

to FOPTD as given below,

G_p = ke^−θs

s = ke^−θs

s+¹_φ = φke^−θs

φs+ 1 (2.32)

where φis a constant with sufficiently large value. The optimum filter structure for IPTD is same as that for the FOPTD. Following a similar calculation procedure, the disturbance rejection controller can be obtained in the form

G_cd= (φs+ 1)(αs+ 1)²

kφ[(λ_cds+ 1)³−e^−θs(αs+ 1)²] (2.33) where α is calculated by solving

1−(αs+ 1)²e^−θs

(λ_cds+ 1)³ s=−1/φ

= 0, i.e. α=φ

1−

(1−λ_cd/φ)³e^−θ/φ1/2

The ISOPTD process model can be approximated as SOPTD i.e. G_p = ^ke−θs

(s+1_/

φ^{)(τ s+1)}

= (φs+1)(τ s+1)^φke−θs , where φis a constant with sufficiently large value. The optimum IMC filter is considered as

f_cd = (α₂s²+α₁s+ 1)

(λ_cds+ 1)⁴ (2.34)

The disturbance rejection controller is obtained in the form

G_cd = (τ s+ 1)(φs+ 1)(α2s²+α1s+ 1)

kφ[(λ_cds+ 1)⁴−e^−θs(α₂s²+α₁s+ 1)] (2.35) The value of α₁, α₂ are calculated by solving

1−(α₂s²+α₁s+ 1)e^−θs

(λ_cds+ 1)⁴ s=−1

φ ,−1 τ

= 0, i.e.

α₁ =

φ²

1−^λcd_φ 4

e⁻θ_/φ−1

−τ²

1−^λcd_τ 4

e⁻θ_/τ−1

τ−φ andα₂ =τ²

1−^λ_τ^cd4

e^−θ/τ −1

+τ α₁.

For UFOPTD, the optimum IMC filter is considered as that of FOPTD. The disturbance rejection controller becomes

G_cd = (τ s−1)(αs+ 1)²

k[(λ_cds+ 1)³−e^−θs(αs+ 1)²] (2.36) whereαis calculated by solving

1−(αs+ 1)²

(λ_cds+ 1)³ s=1

τ

= 0,α=τ

"

1 +^λ_τ^cd3

e^{θ/τ 1}/2

−1

#

For the SOPTD process model type (2.14), the optimum IMC filter is same as that of ISOPTD and the disturbance rejection controller becomes

G_cd= (τ₁s+ 1)(τ₂s+ 1)(α₂s²+α₁s+ 1)

k[(λ_cds+ 1)⁴−e^−θs(α₂s²+α₁s+ 1)] (2.37)

2.3 Controller design procedures

Table 2.2: Disturbance Rejection controllerGcd

Process Parameters ofGcd

FOPTD

Kp= _T¹_i

"

τ+ 2α−^3λ

2

cd−θ²/2^+2αθ−α2

3λ_cd−2α+θ

#

Ti=k(3λcd−2α+θ) Td= _T¹_i

"

2τ α+α²−^λ

3

cd+θ³/6^{−αθ2+α2θ}

3λ_cd−2α+θ

#

−^Kp(3λ

2

cd−θ²/2^{+2αθ−α2)}

3λ_cd−2α+θ

IPTD

Kp=^φ+2α_T

i −^3λ

2 cd−θ²

2 +2αθ−α² T_i(3λ_cd−2α+θ)

Ti=φk(3λcd−2α+θ) Td=_T¹_i

"

(2φα+α²)−^λ

3 cd+θ³

6−αθ²+α²θ 3λ_cd−2α+θ

#

−^Kp(3λ

2 cd−θ²

2 +2αθ−α²) 3λ_cd−2α+θ

ISOPTD

Kp= _T¹

i

"

φ+τ+α1−^6λ

2

cd−θ²/2^+θα1−α2

4λcd−α1+θ

#

Ti=φk(4λcd−α1+θ) Td= _T¹

i

"

φτ + (τ+φ)α1+α2−^4λ

3

cd+θ³/6^−α1θ²/2^+θα2

4λ_cd−α1+θ

#

−^Kp(6λ

2

cd−θ²/2^{+θα1−α2)}

4λ_cd−α1+θ

UFOPTD

Kp= _T¹

i

−τ+ 2α−^3λ

2

cd−θ²/^{2+2αθ−α2}

3λ_cd−2α+θ

Ti=−k(3λcd−2α+θ) Td= _T¹_i

−2τ α+α²−^λ

3

cd+θ³/^{6−αθ2+α2θ}

3λ_cd−2α+θ

−^Kp(3λ

2

cd−θ²/^{2+2αθ−α2)}

3λ_cd−2α+θ

SOPTD

Kp= _T¹

i

τ1+τ2+α1−^6λ

2

cd−θ²/^{2+θα1−α2}

4λcd−α1+θ

Ti=k(4λcd−α1+θ) Td= _T¹

i

τ1τ2+ (τ1+τ2)α1+α2−^4λ

3

cd+θ³/^6−α1θ2/^2+θα2

4λcd−α1+θ

−^Kp(6λ

2

cd−θ²/^{2+θα1−α2)}

4λcd−α1+θ

USOPTD

Kp=_T¹

i

τ2−τ1+ 2α−^6λ

2

cd−θ²/^{2+2αθ−α2}

4λcd−2α+θ

Ti=−k(4λcd−2α+θ) Td=_T¹

i

−τ2τ1−2α(−τ2+τ1) +α²−^4λ

3

cd+θ³/^{6−αθ2+α2θ}

4λcd−2α+θ

−^Kp(6λ

2

cd−θ²/^{2+2αθ−α2)}

4λcd−2α+θ

Here α₁=

τ₁²

1−^λcd_τ

1

4

e−θ_/τ1−1

−τ₂²

1−^λcd_τ

2

4

e−θ_/τ2−1

τ2−τ1 andα₂=τ₂²

1−^λ_τ^cd₂ 4

e^−θ/τ2−1

+τ₂α₁.

Analogously, the optimum IMC filter for USOPTD is considered as f_cd= (αs+ 1)²

(λ_cds+ 1)⁴ (2.38)

and the disturbance rejection controller obtained in the form G_cd= (τ1s−1)(τ2s+ 1)(αs+ 1)²

k[(λ_cds+ 1)⁴−e^−θs(αs+ 1)²] (2.39) where solving the inequality

1−e^−θs(αs+ 1)²

(λ_cds+ 1)⁴ s=1

τ1

= 0,α =τ₁

"

λcd

τ1 + 14

e^{θ/τ1 1}/2

−1

# .

Expanding G_cd in a Maclaurin series in s, enables one to express the PID tuning rules as given in Table 2.2.