2. CONTROLLER DESIGN FOR PROCESSES WITH ...ncbs.knu.ac.kr/Teaching/MCS_Files/Chapter_2f.pdf3. Controller Design for processes with Difficult Dynamics 2 become unstable. Thus there

3. Controller Design for processes with Difficult Dynamics 1

2. CONTROLLER DESIGN FOR PROCESSES WITH DIFFICULT DYNAMICS

1. Difficult Process Dynamics For normal system, if the input variable were increased from an initial steady-state value of u1 to a new value u1 + ∆u, the dynamic behavior is considered normal if the output variable responds qualitatively as one of the responses depicted in following Figures

These response satisfy:

1. It begins to respond quickly without significant delay.

2. It heads directly for a new steady-state value without first taking an excursion in the opposite direction.

3. It finally settles to a new steady-state value.

The control system structures studied before can control normal processes reasonably well. However, these conventional control systems often do a poor job for the following three classes of processes with difficult dynamics. Thus, we need to analyze the key features of these difficult processes and provide new control system designs for improved closed loop performance of these processes.

Characteristics of Difficult Process Dynamics

The presence of any of difficult characteristics in the dynamic behavior of a process can be identified below:

• Time delay

• Inverse response

• Open loop instability

A process with time delay violates the first condition noted above normal dynamic behavior: it does not respond instantaneously to input change. A substantial number of chemical processes exhibit time-delay behavior.

Processes with time delays have a significant delay before they respond to control action, so that controller with aggressive action (high controller gain) will tend to overcompensate and process

KNU/EECS/ELEC835001 Dr. Kalyana Veluvolu


become unstable. Thus there is a limit on the controller gain that can be used for a process with time delay.

A process with inverse response violates the second condition for normal dynamic behavior. The response, even though its step response eventually ends up heading in the direction of the new steady state, it starts out initially heading in the opposite direction, away from the new steady state, changing direction somewhere during the course of time.

Processes with inverse response will initially move in the wrong direction as they respond to control action. Thus if the controller is tuned too tightly (high controller gain) it will attempt to correct for the movement in the wrong direction and overcompensate. Again there is a limit on the controller gain that can be used for a process having inverse response.

A process for which the step response is unbounded, i.e., the output increases (or decreases) indefinitely with time, is said to be open loop unstable. An open loop unstable process violates the third condition noted above; its output fails to settle to a new steady-state value in response to a step change in the input.

Processes that are open-loop unstable will "run away" without control, so most of the controller tuning procedures cannot be applied. In addition, open loop unstable processes can be unstable for various reasons so that simple PI control may not be enough to stabilize them.

These difficult dynamics translate into unusual phase behavior as non-minimum-phase system.

Non-minimum Phase (NMP) systems

Minimum Phase systems: A normal process with n poles and m zeros, its phase angle approaches (n - m) x (-90) asymptotically at high frequencies.

For a given system

)Im()Re()()( ωωωω jjgsg js +===

with 22 )Im()Re(|)(| ωωω +== jgAR

and

⎥⎦

⎤⎢⎣

⎡== −

)Re()Im(tan)](arg[ 1

ωωωφ jg

For a class of processes (g1(s), g2(s), ..., gn(s)) having same amplitude ratio characteristics (i.e.,



AR1 = AR2 = ... = ARn), and phase angles (φ1, φ2, … φn) of which the minimum is designated φmin. if φi ≠ φmin then the systems are referred to as non-minimum phase systems.

• Time delay system: For a normal system g1(s): sesgsg α−= )()( 12

We have

AR1 = AR2

but

φ2=φ1-αω

• Inverse response system: the two processes having transfer functions:

g1(s) = g0(s)(l + ηs)

and

g2(s) = g0(s)(1- ηs)

We have

AR1 = AR2

but

φ2=φ1-1800

• Open loop unstable system: Consider the first order process

1)(1 +=

sKsg

τ

and

1)(2 −=

sKsg

τ

We have

AR1 = AR2

but

φ2=-1800+φ1

In general, a system which contains any non-minimum phase element: a RHP pole, a RHP zero, or a time delay, is a NMP system.

2. Control of Time-delay Systems Control problems with time-delay elements are:

1. Measuring device delay: control action is based on delayed, obsolete, process information that is not representative of the current situation.

2. Process input delay: the process will not feel the control action immediately.

Delay cause system instability: consider the system



a)

1)(1 +=

sKsg

τ

A normal first-order system, the phase angle asymptotically approaches a limiting value of -900 Thus, the normal system can never be unstable under proportional feedback control since the phase angle can never attain the critical value of -1800.

b) se

sKsg α

τ−

+=

1)(1

A first-order system with time delay: the phase angle decreases monotonically with frequency, a limiting value of proportional controller gain (phase angle crosses 1800) at which the system becomes unstable. It is proportional to the value of the time delay.

Conventional Feedback Controller Design

Consider the general model for a process with time delay: sesgsg α−= )(*)(

where g*(s) has normal dynamics.



Under conventional feedback, the closed-loop system has

0)(*)(1 =+ − sc esgsg α

The increased phase lag of the delay term requires a reduction in the allowable value of the controller gain. The closed-loop system will have to be more sluggish than the corresponding system without delay.

Conventional controllers can be used for time-delay systems, but have to sacrifice speed of response in order to have closed loop stability.

Example: Three Water Tank System

A linear model for the liquid level hi of each tank is

1101

1 hcFdtdhA −=

22112

2 hchcdt

dhA −=

33223

3 hchcdtdhA −=

where Aj, Cj, j=1,2,3, are the cross section area and the outlet valve discharge constant for each



tank. In terms of deviation variables

isii hhy −= , sFFu 00 −=

It becomes

uKydtdy

111

1 +−=τ

2122

2 yyKdt

dy−=τ

3233

3 yykdtdy

−=τ

,,,1,3

23

2

12

11 c

ckcck

ck

cA

j

jj ====τ

The transfer function,

)1)(1)(1( 2213 +++=

sssKy

τττ

Assume 21 =τ , 42 =τ , 63 =τ , 321 kkkK = . Then

)16)(14)(12(6)(

+++=

ssssg

Approximated Model:

ses

sg 31 115

6)( −

+=



PI Control

PID Control

Smith Predictor

Introduce a minor feedback loop around the conventional controller and a model with subscript m

)()()( * suesgsy smm

mα−=

Define:

)()()( ** susgsy =



and

)()()( ** susgsy mm =

Since

)()()( * suesgsy sα−=

y*(s) is the output of un-delayed process output y(s).

Assuming that there are no model errors g*m(s) = g*(s) and αm = α, the signal reaching the controller is a "corrected" error signal.

Then:

))()(()( * sysysyydc −−−=ε

or

)(* syydc −=ε

The equivalent block diagram for the closed loop system is shown as:

The net result of the introduction of the minor loop is therefore to eliminate the time-delay factor from the feedback loop - where it causes stability problems - and "move" it outside of the loop, where it has no effect on closed loop system stability.

The characteristic equation of the equivalent system is

0)(*)(1 =+ sgsgc

which no longer contains the time-delay element and therefore allows the use of higher controller gains without placing the closed-loop stability in jeopardy.

To establish the characteristic equation directly, assuming that there are no model errors. Considering



ε*cgu =

where

)1(1 **

sc

cc egg

gg α−−+=

The overall closed loop transfer function is

dc

sc

s

ygesg

gesgy **

**

)(1)(

α

α

−

−

+=

Using *cg , we obtain:

scc

sc

cs

eggggegggesg α

αα

−

−−

−+= **

***

1)(

and

scc

cc

s

egggggggesg α

α−

−

−++

=+ **

***

11)(1

Substitute both equations into the closed loop transfer function

ds

c

c yegg

ggy α−⎟⎟⎠

⎞⎜⎜⎝

⎛+

= *

*

1

The specialized control scheme is known as time-delay compensation because, the minor loop was introduced to compensate for the presence of the time delay. It is referred to as a compensator. Note:

1. The effective action of the compensator is to feed the signal y* to the controller instead of the actual process output y.

2. By

)()()( ** susgsy =

and

)()()( * suesgsy sα−=

we have

)()(* syesy sα=

and

)()(* α+= tyty

It is clear that y*(t) is a prediction of y(t) exactly α time units ahead, the name "Smith predictor" generally associated with the scheme.

3. The scheme will work perfectly if process model is perfectly known, modeling errors will affect its performance. The most significant criticism of the Smith predictor



technique is its sensitivity to modeling errors.

4. In those processes where the time delays are due to transport of material and/or energy through long pipes, the time delays will vary with the fluid flowrate; an increase in flowrate giving rise to lower time delays, and vice-versa. The Smith predictor scheme is designed for constant time delays and may therefore not perform as well for systems with time delays which vary significantly over time.

Design Procedure

1. Design the Smith Predictor (the minor loop)

The design of the minor loop involves setting up a means by which y* and y are produced from the process model, y is obtained directly from the process model, and y* is obtained from the undelayed version of the process model.

2. Design gc

According to the Smith predictor scheme, the controller is designed for the undelayed system. This permits the use of much higher controller gains than would otherwise be allowable.

However, the Smith predictor requires a perfect model, and real models are never perfect, we must be cautious in choosing the controller parameters. In practice, one would choose controller parameters large enough to achieve much better performance than feedback control alone, but not so large as to cause serious deterioration in performance resulting from inevitable plant/model mismatch.

Assuming there is a variation in dead-time, td can only be approximated. The Smith predictor cannot fully compensate the dead-time effects; there is dead-time residual in the system. The uncompensated dead-time give rise to additional phase lag and leads eventually the reduction of crossover frequency and ultimate gain margin. In this case, too big gain will lead unstable system. Another problem is the disturbance, Smith predictor has no real effect to disturbance rejection improvement.

• Stability

• Disturbance rejection

• Transient characteristic

• Robustness

Improved Smith Predictor-1

Adding a transfer function Gf(s) in feedback loop, input disturbance can be rejected. From the block diagram, we can write the closed loop transfer function from D(s) to Y(s) as:



cf

sscf

scf

sc

s

ggggegegggg

eggggegg

egsdsy

**

***

**

*

*

1)]1(1[

)1(11

)()(

++

−++=

−+++

=

−−

−

−

−

αα

α

α

α

To reject disturbance, it should have

0)1(1 ** =−++ − scf egggg α

we obtain:

*

* )1(1g

eggg

sc

f

α−−+−=

Writing the closed loop transfer function

cf

sc

d ggggegg

sysy

**

*

1)()(

++=

−α

Substitute gf(s) into above equation, we obtain

1)()(

*

*

== −

−

sc

sc

d eggegg

sysy

α

α

It means that the system can exactly following the reference signal and disturbance has been totally rejected.

3. Inverse Response System Control Inverse Response System

Definition: When the initial direction of a process systems step response is opposite to the direction of the final steady state, it exhibits inverse response.



Consider a system

g(s) = g1(s)-g2(s)

g1(s): representing the "main mode";

g2(s): as the "opposition mode";

The second model g2(s) has negative sign, opposite to the first model g1(s).

Example: a system composed of two opposing first-order modes,

sK

sKsg

2

2

1

1

11)(

ττ +−

+=

Assume K1 > K2, the dynamic behavior by transfer function model:

y(s) = g(s) u(s) is given by:

y(s) = g1(s)u(s) - g2(s)u(s) or

y(s) = Y1(s)-Y2(s)

The overall unit step response will be the difference between system 1 and 2.

Steady-State Considerations

The overall steady-state value of the system for unit step response is:

y(∞) = K1-K2

and since K1 > K2 this quantity is positive.



Initial Slope For all t > 0:

dtdy

dtdy

dtdy 21 −=

The net initial slope of the composite system response is the difference between the initial slope of the "main" and "opposition" system. For first-order step responses it is:

2

2

1

10 ττ

KKdtdy

t −==

suppose

2

2

1

1

ττKK

≤

The net initial slope of the overall system response will be negative, if the initial slope of the opposing system response is greater than the initial slope of the "main" system response.

Summary

Inverse response is possible only when the "opposition" mode

1. Has a lower steady-state gain than the "main" mode (K2 <K1), and

2. Responds with a faster initial slope than that of the "main" mode,

To illustrate the characteristics of linear systems that show inverse response, consider the transfer functions of the two processes.

)1)(1()()(

11)(

21

211221

2

2

1

1

ssKKsKK

sK

sKsg

ττττ

ττ

++−+−

=

+−

+=

Note that this is the form of a (2,1)-order system

)1)(1()1()(21 ss

sKsgττ

η+++−

=

With

K = K1-K2

And

)()()(

21

21

2

2

1

1

21

1221

KKKK

KKKK

−⎟⎟⎠

⎞⎜⎜⎝

⎛−=

−−

=−ττ

ττττη

To obtain the inverse response, it required both K1, K2 are positive, K1 > K2, and also K2/ τ2> K1/τ1, Which means K>0, -η<0.

For reverse response system, having taken the proper action, it will eventually yield the desired result. However, the controller is first took the wrong action and is liable to compound the problem further that may affect such a system’s closed-loop stability.



Inverse Response System Control

Consider a system, the Bode diagram for the process whose open-loop transfer function is:

)51)(21()13()(

ssssg

+++−

=

1. Without the RHP zero in the numerator, the transfer function will be second order, the phase angle will asymptotically approach a limiting value of 1800 implying that the closed loop system will always be stable.

2. With the RHP zero, the limiting value for the phase angle has been altered to 2700 and there is now a finite (crossover) frequency at which φ = 1800, and there is a limiting value of Kc for system to be stable.

Conventional Feedback Controller Design

Logic of PID controller: Because the derivative mode of the PID controller, it can anticipate the "wrong way behavior" and appropriately accommodating it by:

1. At the initial stage of the process response, it increases error because of the inversion. However, the derivative of the response is negative during this period, and when this information is incorporated into the controller equation, the result is a net reduction in the magnitude of the control action.

2. After the inversion is over, and the response begins heading in the right direction, the derivative is positive and usually quite large; with the PID controller, this translates to a net increase in control action.

Approximate Time-delay Model for the Inverse-Response System: By reversing the Pade approximation, a RHP zero can also be approximated by a time-delay



element and a mirror image LHP zero

s

se s

21

21

α

αα

+

−=−

then, by the same token: sess ηηη 2)1(1 −+=−

Example: Obtain an approximate time-delay model for the inverse-response system using the reverse Pade approximation and compare the unit step responses.

)51)(21()13()(

ssssg

+++−

=

Solution: Since the model has a RHP zero η=3, using reverse Pade' approximation requires replacing it with (1+ 3s) e-6s. The approximate time-delay model is given by:

sess

ssg 6

)51)(21()31()( −

+++

=

Following Figure shows the unit step response of the two systems, indicating quite reasonable agreement

Classical Techniques: Ziegler-Nichols Designs

Example: Design a PID controller for the following inverse-response system using the frequency-response Ziegler-Nichols technique.

)51)(21()13()(

ssssg

+++−

=

Solution: Figure shows the Bode diagram for the system under proportional-only control: i.e., the open loop transfer function is given by:



)51)(21()13()(ss

sKsg c

+++−

=

From this diagram, we obtain the following critical information required for the Ziegler-Nichols design:

The crossover frequency: ωco= 0.55 radians/time

The magnitude ratio at this point: MR = AR/KKc = 0.5

The ultimate gain and period are given as: Ku =2

and

42112 .Pco

u ==ωπ

By the following table

Controller K τi τD

P 0.5Ku

PI 0.4Ku 0.8Pu

PID 0.6Ku 0.5Pu 0.125Pu

The Ziegler-Nichols recommended values for the PID controller parameters are obtained as:

Kc=1.2; τI=5.7 τD=1.4

Inverse-Response Compensation



Consider the block diagram shown below:

Written conventional feedback control of an inverse-response system as g(s) = g0(s)(1 - ηs); where g0(s) represents the transfer function factoring out the problematic RHP zero element. For example

)51)(21()13()(

ssssg

+++−

=

than

)51)(21(1)(0

sssg

++=

Introducing a minor loop as shown in following Figure with the transfer function g' given by:

g'(s) = g0(s)λs

Objective: choose the quantity λ such that the signal reaching the controller appears to be from a "normal" system.

Define the variable y'

y'(s) = g'(s)u(s)

generated by the minor loop. As a result of this minor loop, the signal reaching the controller is given by:

)(')( sysyydc −−=ε

or

[ ] )()(')( susgsgydc +−=ε



Now, let:

)(')()(* sgsgsg +=

and

)()()( ** susgsy =

then it becomes:

)(* syydc −=ε

Introducing g and g' into g*, we obtain:

g*(s) = g0(s)(I - ηs) + g0(s)λs

or

g*(s) = g0(s)[l + (λ-η)s]

choose λ such that:

λ>η

y* no longer contains a RHP zero. Thus the minor loop provides a corrective signal that eliminates the inverse response from the feedback loop.

In case of plant-model mismatch, choosing λ>η (as opposed to λ=η). Usually select

λ=2η

Example: Design an inverse-response compensator for the inverse-response system

)51)(21()13()(

ssssg

+++−

=

Solution: For the process

)51)(21(1)(0

sssg

++=

and

)51)(21()('

ssssg++

=λ

For this system η = 3, a good value of A to be used is λ =6 so that:

)51)(21(6)('

ssssg++

=

is the transfer function to use in the inverse-response compensator loop. The apparent process transfer function is given by:

)51)(21(31)(*

ssssg++

+=

with no RHP zero.

Example: Investigate the closed loop stability properties of the following system under



proportional-only feedback control, first without any compensation, and then with the inverse response compensator

)51)(21()13()(

ssssg

+++−

=

Solution: Under conventional, proportional feedback control, the characteristic equation for the closed loop system is:

0)51)(21(

)13(1 =+++−

+ss

sKc

which rearranges to:

0)1()37(10 2 =++−+ cc KsKs

The condition for stability is:

With the inverse-response compensator, using the minor loop

ε*cgu =

where

'1*

gggg

c

cc +=

The overall closed loop transfer function is

dc

c ygg

ggy *

*

1+=

and the characteristic equation is:

01 * =+ cgg

the characteristic equation becomes:

0)1()37(10 2 =++++ cc KsKs

which is stable for all positive values of Kc

The inverse response compensation compared to conventional PID control, the response to a unit set-point change with the inverse-response compensator combined with a PI controller (Kc =10, τI=0.167) is shown in the Figure. Note that the inverse-response compensator produces the smallest negative deviation and a rapid response without overshoot.

3/7<cK



Design Procedure: 1. Design the inverse-response compensator loop, this involves obtaining the appropriate

transfer function g'(s) and find λ to use in the minor feedback loop.

2. Design gc: Once the compensator has been designed, it is simply design the controller for g*(s) with absence of the RHP zero permits the use of higher controller.

However, because of process/model mismatch, one must be careful not to increase the controller gains too much.

4. Open-loop Unstable Systems Characteristic

The system transfer function has at least one RHP pole.

1)(

−=

sKsg

τ

Dynamic Behavior Any upset in any direction will result in unstable response.

Difficulties for Control System Design Open loop model identification procedures are impossible, the process must keep under control



while carrying out the modeling experiments.

Determine open-loop model parameters: under feedback control with known controller parameters using a sequence of set-point changes and disturbances.

A process is unstable in the open loop can usually be stabilized by carefully designed conventional feedback control.

Example: Obtain the range of Kc values required to ensure that the closed-loop system involving:

1)(

−=

sKsg

τ

and a proportional controller is stable.

Solution: The characteristic equation for the closed loop system is:

1 + KKc/(τs-1) = 0

which is

τs - 1 + KKc = 0

the one root is located at

s = (1-KKc)/τ

The root will be negative, if

Kc > 1/K

which will stabilize the open-loop unstable system.

Place the closed loop system poles in pre-specified locations in the LHP.

Example: Design a PI controller for above Example with K=1/6 and τ =0.25 that will stabilize the dosed-loop system with the closed-loop system poles located at s = -2 and at s = - 4. Solution: PI Controller is given by

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

sKsg

Icc τ

11)(

The closed-loop characteristic equation

1+gc(s)g(s)=0

becomes:

ττIs2 + (KKc-1)τIs + KKc = 0

solving this equation, gives the required controller parameter values:

6 1cK

Kτ +

= Kc = 15.5

And



8I

cKKττ = τI = 0.775

However, not all open loop unstable processes can be stabilized by P or PI control.

Example: For open-loop transfer function

)15)(12(2)(

+−=

sssg

It not difficult to check that it is impossible to stabilize the process using P or PI control. The characteristic equation of closed loop transfer function is

01 =+ ggc

0)15)(12(

)(21 =+−

++

sssksk cc

ττ

results

0)12(310 23 =+−+− cc kskss τττ

which is unstable. In this case, either a PD or a PID controller is required for stabilization. It means that open loop unstable systems may require special types of controllers to stabilize them.

Open-loop unstable systems can also have conditional closed-loop stability, Kcl < Kc < Kcu which stabilize the process.

Example: Consider the open loop unstable process:

)12)(14()1(2)(+−

+−=

ssssg

under proportional feedback control. Determine the range of controller gains for which the closed-loop system is stable.

Solution: The characteristic equation for the closed-loop system is given by:

8s2+2(1-Kc)s+(2Kc-1) = 0

the stable range of controller gain is 0.5 < Kc <1.

Two step design

For controller design purpose, many of the unstable processes are adequately described by a first-order plus dead-time transfer function:



Lsp e

TsKsG −

−=

1)(

With P controller in the inner feedback loop, the internal closed-loop transfer function )(sGl can be obtained as

Lsl

Ls

pl

pl

eKKTsKe

sGKsG

sG−

−

+−=

+=

1)(1)(

)(

Using Taylor series expansion 225.01 sLLse Ls +−≅−

we obtain

1)(5.0)()(

22'

−+−+=≅

−

lll

Ls

plKKsLKKTsLKK

KesGsG

Since the characteristic equation of )(' sG p should have negative poles to be stable, the following condition must be satisfied from the Routh-Hurwitz stability criterion

maxmin1 K

LKTK

KK l =<<=

The above expression indicates that a condition T/L>1 for unstable processes should be satisfied. That means that the proposed method is suitable for unstable processes with small time delays.

For the optimum gain margin

LT

KKKKl

1maxmin ==

which results

)1(1)(1)5.0()(

2

'

−+−+

=−

LT

KsTLT

KsTL

KL

esGLs

p

As the integrating and unstable processes are stabilized with the P controller in the inner feedback loop, we can design a PID controller for the stabilized processes which have second order plus dead time process structure.



cbsasesG

Ls

p++

=−

2' )(

Writing PID controller transfer function as

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛ ++=

sCBsAsksGc

2

where k

KA d= , k

KB p= and

kKC i= . Let controller zeros to be equal to the poles of model

( )sGp′ , i.e. aA = , bB = , cC = . Hence

( ) ( )s

kesGsGsL

cp

−

=′

where k is determined based on gain and phase-margin specifications. Typical values of gain margin and phase margin range from 2 to 5, and o30 to o60 , respectively. If assign 3=mA , then

o60=Φm and

LLAk

m 62ππ

==

Hence PID settings for unstable processes are given as

( )6p

T TKK L Lπ

= −

)1(6

−=LT

KLKi

π

TLK

Kd 12π

=

Once the model is obtained, we can ignore the inner feedback loop and directly design PID controllers for the unstable time delay processes. The tuning formulae are very simple and straightforward.


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2. CONTROLLER DESIGN FOR PROCESSES WITH ...ncbs.knu.ac.kr/Teaching/MCS_Files/Chapter_2f.pdf3. Controller Design for processes with Difficult Dynamics 2 become unstable. Thus there