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054414 PROCESS CONTROL SYSTEM DESIGNLECTURE ELEVEN

Daniel R. Lewin, Technion1

Systematic Discrete DesignPROCESS CONTROL SYSTEM DESIGN - (c) Daniel R. Lewin1 11-

054414 – Process Control System Design

LECTURE 11:SYSTEMATIC DISCRETE

CONTROL DESIGN

Daniel R. LewinDepartment of Chemical Engineering

Technion, Haifa, Israel


Objectives

Be able to implement the IMC design procedure on discrete systems.Be able to compare common discrete control algorithms, namely:

The discrete PID algorithm (plus filter)Dead-beat AlgorithmDahlin’s Algorithm

using a systematic frameworkBe able to identify situations where adaptive controllers are needed, and be able to conceptually design such a system.

On completing this section, you should:




IMC Design Principles

( ) ( ) (perfect model)If p z p z=

y S y −

− +

+ + d

( )p z

( )p z( ) q z

q(s) p(s)yS y−

+ +

d

d

y = pq⋅ ys + (1 – pq)⋅d ⇒ q = p-1f

( ) p z( ) q z

“Open-loop” control design method – Simple!


Basic IMC Design ProcedureIdeally, we require y = ys, implying that: 1q p −=Two problems can arise:Problem 1:

( ) ( )( )

3

1 2 21.5e.g. or

0.5 0.5kz zp z p z

z z

− −= =

− −then if we were to design a controller such that :1q p −=

( ) ( ) ( ) ( )23

1 20.5 0.5

and 1.5

z z zq z q zk z

+− −= =

−

Non-causal Unstable

If contains non-minimum phase components:( )p z




Basic IMC Design ProcedureSolution to Problem 1:1. Partition the process model into its minimum phase and

non-minimum phase (all-pass) components:( ) ( ) ( )A Mp z p z p z=

( )Note that 1 i TAp e ω = ∀ω

Invertible Non-invertible

such that (stable, causal, and strictly proper) and the “all-pass” component, , is defined as:

1 1MAp p p− −=

( )Ap z

( )( ) ( )( ) ( )

1

11

1.

1

hi iN

Ai i i

zp z z

z

−−

−=

− φ − φ=

− φ − φ∏

N is a time delay, selected so that will be strictly proper, and h is the number of zeros in outside the unit circle. ( )p z

( )Mp z


Basic IMC Design ProcedureSolution to Problem 1 (Cont’d):

2. Select the controller as the inverse of the invertible partof the process model:

( ) ( )( )1 1 n.b. 1 1M

q p q p− −= =

Note that as defined above is proper.( )q zIn the two examples:

( )3

1 0.5

kzp zz

−

=−

( )( )2 2

1.50.5

zp zz

−=

−

( ) ( )41 1,

0.5A Mkzp z z p z

z−= =

−⇒ ( ) 0.5 zq z

kz−

⇒ =

( )( )

( ) ( )( )

12

2 2

1.50.6670.667

0.6671.5

0.5

A

M

zp z zz

z zp zz

−

⇒

−= −

−−

= −−

( ) ( )( )

20.667 0.5

0.667zq z

z z− −

=−




Basic IMC Design ProcedureRobustness Filter.

For step-like inputs (Type I)

Since the true process may be different from the model used to design the controller, the IMC controller is augmented with a low-pass filter, f(z), to enable the detuning the controller to impart robustness:

( )q z

( ) ( ) ( ), where:q z q z f z=

( ) −

− α=

− α 11

1f z

z Note: To guarantee offset-free response, the filters must

satisfy the set of conditions:

( )( )=

− = ≤ <1

1 0, 0k

k z

d f z k mdz input type (1 or 2)

( )( )=

−=

− α− = − =− α 1

111e.g. for Type 1, 1 1 0

1 zz

f zz


Example IMC Design

Solution:( )

( )3

21.50.5

zp z zz

−−=

−Design an IMC controller for the process:

η = = = Apq p f

If ys = z/(z – 1) (unit step): ( ) ( )( )( )( )

-30.667 1.5 1z0.667k z 1

lim lim =1 zz zy kT − − −α

− −α→∞ →⎡ ⎤= ⎣ ⎦

( ) ( )1i.e. design with 1 1 guarantees offset-free responseq p −=




Example IMC DesignResponse of the controlled system for decreasing values of the filter constant.


Basic IMC Design Procedure Just when we think we have solved all the problems, here is

another one, particular to discrete systems.

Problem 2: “Ringing” and the placement of poles.

( ) ( ) ( )1M

q z p z f z−= ⋅

The optimal design of the IMC controller is:

As seen previously, any negative poles in q(z) lead to “ringing”(the controller output will exhibit a sign change in each sample). The closer these negative poles are to –1, the more severe will be the “ringing.”




Example of “Ringing”Consider the process sampled at T=3.( ) ( )( )+ += 1

10 1 25 1s sp s The equivalent discrete process as seen by a discrete controller

(after ZOH) is (see Lecture 8):

( ) ( )( ) ( )

+=

− −0.016 0.87

0.90 0.74zp z

z z IMC design procedure gives:

( ) ( )( ) ( )

+=

− −0.016 0.87

0.90 0.74Mz zp z

z z( ) −= 1

Ap z z

( ) ( ) ( ) ( ) ( ) ( )( ) ( )

− − − − α= ⋅ =

+ − α1 63.7 0.90 0.74 1

0.87Mz zq z p z f z

z z

( ) ( )( ) ( )

( ) ( ) ( )( ) ( )− α − −

= =− + −

63.7 1 0.90 0.741 0.87 1

z zq zc zp z q z z z


Example of “Ringing”

( ) ( )( ) ( )

( ) ( ) ( )( ) ( )− α − −

= =− + −

63.7 1 0.90 0.741 0.87 1

z zq zc zp z q z z z

k

k

The closed loop servo response is shown below (for α = 0.5).

Note that while the output at the sample intervals does not show ringing, the true response does, which is a result of the ringing exhibited by the input, due to the pole at z = -0.87.




Basic IMC Design ProcedureSolution to Problem 2:

( ) ( )

( )=

′=

− φ∏1

P

ii

q zq zz

In cases where optimal design gives controller poles close to –1, these are removed using the following procedure:

Let the optimal controller (obtained by the IMC design procedure, be:

where q’(z) is the portion of the controller transfer function without the undesirable poles, φi are the undesirable pole locations (p altogether). We modify the controller:

( ) ( )( )

( )

( )

( )−=

= =

− φ ′= =

− φ − φ

∏

∏ ∏1

mod

1 11 1

P

iPi

P PP

i ii i

z q zq z q z zz


Basic IMC Design ProcedureSolution to Problem 2 (Cont’d):

( ) ( )( )

( )

( )

( )−=

= =

− φ ′= =

− φ − φ

∏

∏ ∏1

mod

1 11 1

P

iPi

P PP

i ii i

z q zq z q z zz

The negative poles zi = φi , -1 ≤ φi ≤ 0, have been substituted by poles at the origin, eliminating the “ringing”phenomenon. The P poles need to be added to keep the controller proper. The steady-state gain of the IMC controller has been conserved (why is this important?)

The modified controller, qmod(z), has the following properties:




k

k

Back to the Example of “Ringing” The original IMC controller is: ( ) ( ) ( )

( ) ( )− −

=+ −

31.8 0.90 0.740.87 0.5

z zq zz z

( ) ( ) ( )( ) ( )

− −=

+ −mod31.8 0.90 0.74

1 0.87 0.5z zq z

z z( ) ( ) ( )( )

− −=

−mod17.0 0.90 0.74

0.5z zq z

z z Using the procedure:

The response with the modified controller now no longer satisfies the desired trajectory (note the overshoot!). However, we have eliminated the ringing. That’s life…


Accounting for Model Uncertainty The same representation for uncertainty used for continuous

systems is used here:

+ ( )p z

( )m z

Multiplicative uncertainty

Process model

( )= + mp(z) p(z) 1 (z)

Re(z)

( )iωTp e Im(z)

( )iωTp e

( )ω ω ω= +i T i T i Tmp(e ) p(e ) 1 (e )

ω ωω

ω

−=

i T i Ti T worst

m i Tp (e ) p(e )(e )

p(e )

depends on Tm




worstm

p (z) p(z) (z)p(z)

−=

Computing - Example

( )skp(s) e , 1 ,k 1 k, 1s 1

−θ= θ = θ ± ∆θ = ± ∆ τ = ∆ττ +

∓ Compute, ,the multiplicative uncertainty for the uncertain

process:m(z)

Solution:

m(z)

( ) ( )( )( ) ( )( )

− −∆τ

θ +∆θ − −∆τ

+ ∆ −=

−

T / 1

worst 1 T T / 1

1 k 1 ep (z)

z z e

( )( )

−

θ −

−=

−

T

T T

1 ep(z)

z z e

iωTz e

⎫⎪⎪⎬⎪⎪

=

⎭


T = 0.5

T = 0.05

1, k 0.2θ = ∆τ = ∆θ = ∆ =0.1, k 0.2θ = ∆τ = ∆θ = ∆ =

T = 0.5

T = 0.05

Computing - Examplem(z) Note the effect of sample time of result…

The results for T = 0.05 are close to those one would expect for a continuous system. For T = 0.5, the effect of uncertainty may be different, depending on the ratio of θ/τ.

0 T< ω < π 0 T< ω < π




Designing for Robust Stability The procedure is just as before, but using z instead of s.


Systematic Discrete Control DesignThe first step is the definition of the desired closed loop transfer function:

ys(z)

Since the closed loop transfer function is given by:( )( )

( ) ( )( ) ( ) { }1

s

y z G z C z , where G(z) H(s)P(s)y z 1 G z C z

Z L−= = ⎡ ⎤⎣ ⎦+

We can compute the controller to achieves the desired performance directly:

( )( )

( )s

y z zy z

= η

( ) ( ) ( )( )

-1 zC z G z1 z

η=

− η

ys(z) ( )zη




Systematic Discrete Control DesignAt this point, we recall the IMC design procedure and note that where Q(z) is the IMC controller.( )z G(z)Q(z),η =

-1 -1M MQ(z) G (z)F(z), where G (z) =Note that: is the MP

component of the process model, and F(z) is a low-pass robustness filter. Substituting for in the equation forC(z) gives:

( )zη

( ) ( ) ( )( )

( )( )

( )( ) ( )

-1 z Q z Q zC z G z1 z 1 z 1 G z Q z

η= = =

− η − η −

This is the same equation used to translate an IMC controller to its classical equivalent! We shall use this interpretation toanalyze several common discrete controllers:

Discrete PID algorithmDead-beat algorithmDahlin algorithm


Discrete PID Algorithm Continuous theoretical PID: ( ) ( )c

1u s K 1 s ss D

Ie⎛ ⎞

= + + τ⎜ ⎟τ⎝ ⎠ Discrete equivalent (position form):

( )k

c si 0

Tu(k) K e(k) e(i) e(k) - e(k-1) uTD

I =

⎛ ⎞τ= + + +⎜ ⎟τ⎝ ⎠

∑ Velocity form:

( )k-1

c si 0

Tu(k-1) K e(k-1) e(i) e(k-1) - e(k-2) uTD

I =

⎛ ⎞τ= + + +⎜ ⎟τ⎝ ⎠

∑

( )c

u(k) u(k) u(k-1)

TK e(k) e(k-1) e(k) e(k) 2e(k-1) e(k-2)TD

I

∆ = −

⎛ ⎞τ= − + + − +⎜ ⎟τ⎝ ⎠

cTK 1 e(k) 1 2 e(k-1) e(k-2)

T T TD D D

I

⎛ ⎞⎛ ⎞τ τ τ⎛ ⎞= + + − + +⎜ ⎟⎜ ⎟⎜ ⎟τ ⎝ ⎠⎝ ⎠⎝ ⎠




Discrete PID Algorithm (Cont’d)

c

-1 -2

-1

Tu(k) K 1 e(k) 1 2 e(k-1) e(k-2)T T T

u(k) z ze(k) 1 z

D D D

I

⎛ ⎞⎛ ⎞τ τ τ⎛ ⎞∆ = + + − + +⎜ ⎟⎜ ⎟⎜ ⎟τ ⎝ ⎠⎝ ⎠⎝ ⎠α + β + γ

= =−

Advantages:The velocity form is independent of initial condition, us.It is not subject to reset wind-up.

Disadvantages:It responds aggressively to step changes in setpoint.The differential operator is sensitive to noise.


Discrete PID Algorithm (Cont’d)The two disadvantages are eliminated by implementing the configuration shown below.

ys(k)




Dead-beat AlgorithmIn dead-beat control, the desired response is to ensure zero-error from the n+1 sample, where n = θ/T. The desired closed loop transfer function is: ( )

( )( ) -(n+1)

s

y z z zy z

= η =

( ) ( ) ( )( )

( )-(n+1)

-1 -1DB -(n+1)

z zThus, C z G z G z1 z 1 z

η= =

− η −Example.

( )0.5 1 1

0.5 11

1 z10 8.6zT 1,P(s) ,G(z) HP(z) 100.5s 1 1 0.14z1 z

T

T

ee

− − −

− −−

−= = = = =

+ −−

Since the process has no dead-time, n = 0, and ( ) -1z zη =

( ) ( )-1 1 -1 1

-1DB -1 1 -1 -1

z 1 0.14z z 1 0.14zThus, C z G z 0.1161 z 8.6z 1 z 1 z

− −

−

− −= = =

− − −


Dead-beat Algorithm (Cont’d)

1. The equivalent IMC controller, Q(z) is:

Example (Cont’d).

( )1

DB -11 0.14zC z 0.116

1 z

−−=

−

( )( ) ( ) ( )1DB

DBDB

C z 0.14Q (z) 0.116 1 0.14z 0.1161 G z C z z

z− −= = − =

+2. The output is specified at the sample instances only. The

actual process could have large inter-sample overshoots and/or be highly oscillatory.




Dahlin’s AlgorithmRecall that the IMC design for a delay process gives:

( )( )

( )s

y s 1sy s 1

ses

−θ= η =λ +

( ) ( ) ( )( )

( )

-1DA

-(n+1)-1

-1 -(n+1)

zThus, C z G z1 z

(1 b)zG z ,b1 bz (1 b)z

Te − λ

η=

− η

−= =

− − −

This approach enables us to target not only the desired delay of the response, but also define how fast we would like it to settle (as determined by the value of λ). Taking this idea into the discrete domain:

( )( )- n+1

-1(1 b)zz ,b , nT

1 bzTe − λ−

η = = θ =−


Example: Comparing DB with DADesign both Dead-beat (DB) and Dahlin’s (DA) controllers for the process:

( ) ( ) ( )1 s1 kP s ,T 0.5min (Thus, n 2 ).

s 1e −θ +∆θ+ ∆

= = = θ+

In the above, ∆k and ∆θ are fractional uncertainties. We shall compare designs with no uncertainty and then check the effect of uncertainties. In the following, we study the case θ = 1 min.Dead-beat Controller.

( ) ( ) ( )( )

( ) ( )-(n+1) -3

-1 -1 -1DB -3-(n+1)

z z zC z G z G z G z1 z 1 z1 z

η= = =

− η −−( ) 3 3

1 1

1 z 0.39zG(z)1 z 1 0.61z

T

T

ee

− − −

− − −

−= =

− −

( ) ( )-3 1

-1DB -3 -3

z 1 0.61zC z G z 2.541 z 1 z

−−= =

− −

11 0.61zQ(z)0.39

−−≡ =




Example: Comparing DB with DADahlin’s Controller.

( ) ( )-3

-1DA -1 -3

1 -3

3 -1 -3

(1 b)zHence, C z G z ,b1 bz (1 b)z

1 0.61z (1 b)z0.39z 1 bz (1 b)z

Te − λ

−

−

−= =

− − −

− −=

− − −

( )-3

-1(1 b)zTarget response is z ,b1 bz

Te − λ−η = =

−

1

-11 0.61z 1 bQ(z)

0.39 1 bz

−− −≡ =

−

Thus, DA and DB are both equivalent to IMC control with and without and a filter. In DA, selection of the filter parameter, b, allows robustness to be guaranteed.


DB (λ = 0 min)

DA (λ = 0.5 min)

DA (λ = 2.0 min)

Example: Comparing DB with DA Response of DB (λ = 0), and DA (λ = 0.5 and 2 min), ∆θ = 0.

Here,

Note that the fastest response is obtained with DB, with the filter time constant provided a back-off on performance for DA.

3

10.39zG(z)

1 0.61z

−

−=−




Example: Comparing DB with DA Response of DB (λ = 0), and DA (λ = 0.5 and 2 min), ∆θ = 0.5

DB (λ = 0 min)

DA (λ = 0.5 min)

DA (λ = 2.0 min)

Here,

With no filter, DB is unstable with delay uncertainty. DA with λ = 0.5 is close to the theoretical stability limit, and with λ = 2 gives a similar to nominal response.

1

40.39zG(z)1 0.61z

−

−=−


Adaptive Control

Large, frequent disturbances (feed composition, feed quality)Batch operation (no steady-state)Inherent nonlinear behavior (pH control, highly nonlinear kinetics, catalyst decay, heat exchanger fouling, etc).

In cases where parameter changes are expected to be large, robust control may lead to poor performance. Such cases are:

A variety of adaptive control techniques are often used in these situations, where:

The process changes are largely known and can be anticipated – this calls for programmed adaptation, which is essentially a feedforward strategy. The process changes are largely unknown – this calls for on-line adaptive control or self-tuning control, and involves feedback.




Programmed Adaptation (PA)

Gain Scheduling – a different set of controller tuning parameters are prepared for each anticipated operating point of the process. A look-up-table is used to change the controller tuning as appropriate.

Several alternatives for programmed adaptation are often used:

PA based on physical process knowledge – For example, suppose the static gain, time delay and time constant of a process are twice as large at 50% of flow as at 100% of flow. The PID tuning could then be:

0 0 0, , ,where is the fraction of flow

cc I I D DK K= ω τ = τ ω τ = τ ωω


Programmed AdaptationGain Scheduling based on knowledge of nonlinearity – If the process gain is known to vary significantly, we can use a controller with a nonlinear gain: C(s) = f(e) x g(s):

Options for f(e):a. Continuous gain –

If α→0, this leads to offset because the controller is insensitive to small e.

( ) ( )1 e ,0 1100

f e − α ⋅= α + ≤ α ≤

b. 3-piece nonlinear controller –

( ) c band

c,low band

K , e eK , ee ef

≥⎧= ⎨ <⎩




Self-tuning ControllersWhen process changes can be neither measured nor anticipated, programmed adaptation cannot be used. Instead, a feedback strategy is employed, in which the controller is retuned on-line.

Controller computation

Process model parameter estimation

controller settings

parameter estimates

input

output


Case Study: pH Control Ref: Kulkarni et al, “Nonlinear pH Control,” Chem. Eng. Sci., 46(4), 995 (1991)

The figure shows a process in which a continuous flow of HCl, F2, at concentration CCl1 is neutralized using NaOH, F1. Both F2 and CCl1 are disturbances.

The pH of the exiting stream, F, must be kept at 7. The figure on the right shows the pH curve for this strong acid-strong base process.





Assuming perfect mixing, reaction at equilibrium, constant volume and density, the system is described by:

( ) ( ) ( )

( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )

( ) ( )( ) ( ) ( ) ( )

1 2

1 1

2 2

14 2 2

160

160log , , 10 gmol /li

ClCl Cl

NaNa Na

H H W OH W

OH Na H Cl

F t F t F tdC F t C t F t C tdt V

dC F t C t F t C tdt VpH C C t K C t K

C t C t C t C t

−

= +

= −

= −

= − = =

= + −

1 2

2 2 1 1

60 , 60, ,H W n Na W c Wh Cl

P F V Q F VC C K C C K C C K

= =

= = =

Defining variables:



( )

( ) ( )

2 312

2 32

11

, 0 1

h hh

h h

hch

nh h

dC C C C C Pdt C

C C C C Q C

⎡= + −⎣+

+ + − =

Combining equations and substituting dimensionless variables, we obtain the single ODE:

The nominal values of inputs are:6 5

120.15, 1.5, 10 , 10cnQ P C C= = = =




Case Study: pH Control SIMULINK model for the pH control system.

P: 1.5 →1.6 at t = 5 s

4 5I c,lowParameters: 5 10 , 2 s,pH_band 2, 2.5 10cK K− −= × τ = = = ×


Case Study: pH Control Comparing linear PI to nonlinear PI.




Case Study: pH Control Change in Kc in the nonlinear pH controller


Summary

Be able to implement the IMC design procedure on discrete systems.

Much of the material covered in the continuous part of the course carries over – with some exceptions.

Be able to compare common discrete control algorithms, using a systematic framework (IMC):

The discrete PID algorithm (plus filter)Dead-beat AlgorithmDahlin’s Algorithm

Be able to identify situations where adaptive controllers are needed, and be able to conceptually design such a system.

On completing this section, you should:

lecture 11 ho - icessecice.weebly.com/uploads/1/0/1/3/10132821/lecture_… · ·...

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