Mikles J., Fikar M. Process Modelling, Identification, and Control

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290 7 The Control Problem and Design of Simple Controllers

Consider the fourth order system described by

G(s)=

(s +1)

with the PID controller. The step response gives T

=1.425, T

=4.463

and thus Z

=3.76, T

=2.85, T

=0.71. Simulation results are shown in

Fig. 7.24. The closed-loop response is fairly oscillatory. This is a common

drawback of Ziegler-Nichols methods.

0 5 10 15 20 25 30

−2

−1

w,y,u

Fig. 7.24. PID control tuned with the Ziegler-Nichols method

Astr

om-H

agglund Methods

A possible disadvantage of Ziegler-Nichols methods is that they are based on

only two process characteristics: either T

for a step response or Z

for permanent oscillations. Generally speaking, Ziegler-Nichols methods have

the following in common:

• transient responses are fairly oscillatory,

• not suitable for tume-delay systems,

• diﬃcult tuning.

Based on these observations,

Astr

om a H

agglund have developed a new

method that uses three process characteristics. The third one is optional –

maximum of the sensitivity function M

that speciﬁes robustness of the con-

troller. M

is tunable and two choices exist: 1.4 and 2.0. The higher value of

characterises a faster controller.

7.4 PID Controller 291

The PID controller is assumed with setpoint weighting with the parame-

ters: proportional gain Z

, integral time constant T

, derivative time constant

, and setpoint weighting b

u(t)=Z



bw(t) −y(t)+



e(s)ds + T

de(t)



(7.85)

Similarly to the Ziegler-Nichols method, the authors have developed con-

troller tuning based on frequency or step response.

Step Response To calculate the controller parameters, three process charac-

teristics are used: process gain Z, time constant T , and time delay T

found by

step response identiﬁcation. For the identiﬁcation procedure of these param-

eters see Section 6.2.1. For the further derivation of the method, normalised

parameters are introduced

a = Z

(7.86)

τ =

+ T

(7.87)

If the controlled process is unstable with a pure integrator, it is recommended

to measure its impulse response by applying a short pulse on the process input.

From such a stable response ﬁctitious values of Z



can be determined

and normalised parameters from them

a = Z



+ T



) (7.88)

τ = T



+ T



(7.89)

The controller parameters are normalised in the similar way to the process

and are of the form aZ

, T

(or T

/T for stable processes), and T

Controller parameters have been obtained empirically from numerous sim-

ulations with diﬀerent process models and interpolating the best results. The

authors have shown that it is advantageous to relate the controller parameters

to the normalised time delay τ in the functional form

f(τ )=a

exp(a

τ + a

) (7.90)

PI and PID controller tuning for stable processes are given in Tables 7.4,

7.5, respectively. In the same manner, integrating processes are covered by

Tables 7.6 (PI) and 7.7 (PID).

Example 7.15:

Astr

om-H

agglund method

www

Consider the fourth order system with the transfer function of the form

G(s)=

(s +1)

controlled by PID controller. The process step response gives T

=1.425,

T =4.463. For the PID controller follow from Table. 7.5 parameter values

292 7 The Control Problem and Design of Simple Controllers

Table 7.4. PI controller tuning based on step response for stable processes. Param-

eters are given as functions of τ of the form (7.90)

=1.4 M

aZ 0.29 -2.7 3.7 0.78 -4.1 5.7

8.9 -6.6 3.0 8.9 -6.6 3.0

/T 0.79 -1.4 2.4 0.79 -1.4 2.4

b 0.81 0.73 1.9 0.44 0.78 -0.45

Table 7.5. PID controller tuning based on step response for stable processes. Pa-

rameters are given as functions of τ of the form (7.90)

=1.4 M

aZ 3.8 -8.4 7.3 8.4 -9.6 9.8

5.2 -2.5 -1.4 3.2 -1.5 -0.93

/T 0.46 2.8 -2.1 0.28 3.8 -1.6

0.89 -0.37 -4.1 0.86 -1.9 -0.44

/T 0.077 5.0 -4.8 0.076 3.4 -1.1

b 0.40 0.18 2.8 0.22 0.65 0.051

Table 7.6. PI controller tuning based on step response for processes with pure

integrator. Parameters are given as functions of τ of the form (7.90)

=1.4 M

aZ 0.41 -0.23 0.019 0.81 -1.1 0.76

5.7 1.7 -0.69 3.4 0.28 -0.089

b 0.33 2.5 -1.9 0.78 -1.9 1.2

Table 7.7. PID controller tuning based on step response for processes with pure

integrator. Parameters are given as functions of τ of the form (7.90)

=1.4 M

aZ 5.6 -8.8 6.8 8.6 -7.1 5.4

1.1 6.7 -4.4 1.0 3.3 -2.3

1.7 -6.4 2.0 0.38 0.056 -0.60

b 0.12 6.9 -6.6 0.56 -2.2 1.2

=2.39, T

=3.58, T

=0.87, and b =0.49. Fig. 7.25 shows two

simulations. The ﬁrst uses the calculated controller, the second shows

the inﬂuence of the setpoint weighting parameter b = 1. In contrast to

the Ziegler-Nichols tuning control performance is better and oscillations

are more heavily damped. When no setpoint weighting is used, larger

overshoot results. However, compared to Example 7.14 on page 289, it is

still smaller.

Frequency Response Like above, there are empirical procedures for PID con-

troller tunung in the frequency domain. In this case it is necessary to ﬁnd

7.4 PID Controller 293

0 5 10 15 20 25 30

0.5

1.5

w,y

y(b=1)

0 5 10 15 20 25 30

u(b=1)

Fig. 7.25. PID control using the

Astr

om-H

agglund method

values of T

, that are approximated as functions of the parameter

κ =1/Z Z

Results for stable processes are given in Table 7.8 (PI controller) and in

Table 7.9 (PID controller). When integrating processes are considered, it can

be possible to ﬁnd the parameters by setting κ = 0 for the PI controller. The

case of the PID controller is more complex and the settings cannot be used.

Table 7.8. PI controller tuning based on frequency response for stable processes.

Parameters are given as functions of κ of the form a

exp(a

κ + a

)

=1.4 M

Z/Z

0.053 2.9 -2.6 0.13 1.9 -1.3

0.90 -4.4 2.7 0.90 -4.4 2.7

b 1.1 -0.0061 1.8 0.48 0.40 -0.17

Table 7.9. PID controller tuning based on frequency response for stable processes.

Parameters are given as functions of κ of the form a

exp(a

κ + a

)

=1.4 M

Z/Z

0.33 -0.31 -1.0 0.72 -1.6 1.2

0.76 -1.6 -0.36 0.59 -1.3 0.38

0.17 -0.46 -2.1 0.15 -1.4 0.56

b 0.58 -1.3 3.5 0.25 0.56 -0.12

294 7 The Control Problem and Design of Simple Controllers

7.5 References

This chapter is more or less a standard part of the majority books dealing

with automatic control. At this place we mention particularly:

J. Mikleˇs and V. Hutla. Theory of Automatic Control. Alfa, Bratislava, 1986.

(in Slovak).

H. Unbehauen. Regelungstechnik I. Vieweg, Braunschweig/Wiesbaden, 1986.

J.-P. Corriou. Process Control. Theory and Applications. Springer Verlag,

2004.

S. Kub´ık, Z. Kotek, V. Strejc, and J.

Stecha. Theory of Automatic Control I.

SNTL/ALFA, Praha, 1982. (in Czech).

PID controllers, their design, tuning, and implementation are covered by

the works:

J. G. Ziegler and N. B. Nichols. Optimum settings for automatic controllers.

Trans. ACME, 64(8):759–768, 1942.

K. J.

Astr

om and T. J. H

agglund. PID Controllers: Theory, Design, and

Tuning. Instrument Society of America, Research Triangle Park, 2 edition,

1995.

K. K. Tan, Q.-G. Wang, C. C. Hang, and T. J. H

agglund. Advances in PID

Control. Springer Verlag, London, 1999.

V. Bob´al, J. B

ohm, and J. Fessl. Digital Self-tuning Controllers. Springer

Verlag, London, 2005.

C.-C. Yu. Autotuning of PID Controllers. Relay Feedback Approach. Springer

Verlag, London, 1999.

M. Huba. Theory of Automatic Control 3. Constrained PID Control.STU

Press, Bratislava, 2006b. (in Slovak).

M. Huba. Constrained pole assignment control. In L. Menini, L. Zaccarian,

and C. T. Abdallah, editors, Current Trends in Nonlinear Systems and

Control, pages 395 – 398. Birkh

auser, Boston, 2006a.

7.6 Exercises

Exercise 7.1:

Consider a system with the transfer function

G(s)=

1 −2s

+ s +1

Find the following frequency indices:

1. gain and phase margins,

2. bandwidth,

7.6 Exercises 295

3. resonant frequency and resonance peak.

Exercise 7.2:

Consider a system with the transfer function

G(s)=

(s +1)

The task is:

1. to control the process and to ﬁnd the critical P control gain, as well as

the critical period,

2. to calculate both parameters analytically,

3. to design PI and PID controllers by the Ziegler-Nichols and

Astr

om-

agglund methods,

4. to compare the control performance.

Exercise 7.3:

Consider a controlled system with the transfer function

G(s)=

1 −2s

1. ﬁnd the PID controller by the pole placement design method. The closed-

loop poles should be placed using various standard forms with ω

=1,

2. compare the performance of the closed-loop system for all cases.

Exercise 7.4:

A continuous-time PID controller can be transformed to its discrete-time coun-

terpart using the following substitutions:

1. backward approximation:

s =

1 −z

−1

2. forward approximation:

s =

1 −z

−1

3. the Tustin approximation:

s =

1 −z

−1

1+z

−1

Derive the discrete-time controllers.

Exercise 7.5:

Consider a closed-loop system consisting of the controlled process

G(s)=

+2s

+3s +4

and a controller with the structures P, I, PI. Find such intervals of controller

parameters that the closed-loop system is stable.

Optimal Process Control

This chapter explains design of optimal feedback control based on state-space

and input-output representations of linear systems. In the ﬁrst part, dynamic

programming and principle of minimum are presented. Then the pole place-

ment feedback control design is derived. This serves as a basis for all advanced

design methods. Based on the pole placement, stability of the closed-loop sys-

tem is guaranteed. In the next parts it is shown how optimal pole locations

can be found. This can be done either for minimisation of an integral cost or

of H

norm. The Youla-Kuˇcera parametrisation of all stabilising controllers is

derived. Included are also state reconstruction and estimation.

The chapter deals mainly with continuous-time systems. However, discrete-

time equivalents are also presented and diﬀerences in both domains are ex-

plained. In addition, dead-beat control is also presented.

In all cases, state-space control design is completed with the input-output

design based on Diophantine equations.

8.1 Problem of Optimal Control and Principle

of Minimum

There are many possible formulations of optimal control design in process con-

trol. One of them that leads to optimal feedback control of lumped continuous-

time systems can be formulated as follows.

Consider a controlled process with mathematical model of the form

dx(t)

= f(x(t), u(t)), x(t

)=x

(8.1)

where x is n-dimensional vector of state variables and u is m-dimensional

vector of manipulated variables. f is n-dimensional vector function and it is

assumed that it is continuous-time and continuously diﬀerentiable with respect

to all its variables.

298 8 Optimal Process Control

The optimal control design is to ﬁnd such u(t) that the cost function

I(u(t)) = G

(x(t

)) +



F (x(t), u(t))dt (8.2)

is minimal. It is assumed that there are no constraints on x, u. Functions G

F are scalars and continuously diﬀerentiable.

Variations of this problem can cover additional requirements and con-

straints that can be included. This covers constraints on input and state vari-

ables, constraints on ﬁnal state vector x(t

). Some of the problems deal with

ﬁxed end time t

, sometimes ﬁnal time is also minimised.

Let us assume that the optimal control trajectory u

∗

(t) exists. If any other

control trajectory u(t) is considered, it holds:

I [u(t)] ≥ I [u

∗

(t)] (8.3)

Equation (8.3) holds in general. However, it can happen that u

∗

(t) sat-

isfying (8.3) does not exist. It is very diﬃcult to prove the existence of op-

timal control. Therefore, applications of optimal control often use intuition

and based on it assume that the optimal control trajectory exists. Its proof is

based on the Hamilton-Jacobi equation.

Let us therefore assume that the optimal solution u

∗

(t) exists. We will

derive conditions that this solutions has to satisfy. These are only necessary

conditions and are based on the variational calculus. We assume a small change

of u

∗

(t) leading to a new trajectory u(t) and study change of the cost function

Let u

∗

(t) be the optimal control and x

∗

(t) the corresponding optimal

system response given by (8.1). Indeﬁnitely small control variation δu(t) then

produces state variation δx(t). The system response is then given as

x(t)=x

∗

(t)+δx(t) (8.4)

that is caused by the control

u(t)=u

∗

(t)+δu(t) (8.5)

Variation of the time derivative of state can be rewritten as





−

∗

(8.6)

∗

d(δx)

−

∗

(8.7)

d(δx)

(8.8)

It follows from (8.8) that linear operators d/dt and δ are commutative. The

Taylor expansion in the neighbourhood of the optimal state gives

8.1 Problem of Optimal Control and Principle of Minimum 299

f(x, u)=f (x

∗

, u

∗



∂f

∂x



∗

δx +



∂f

∂u



∗

δu (8.9)

where only linear terms were taken into account. Partial derivatives in (8.9)

are evaluated for optimal trajectories u

∗

(t), x

∗

(t).

The terms



∂f

∂x



⎛

⎜

⎝

∂f

∂x

∂f

∂x

···

∂f

∂x

∂f

∂x

∂f

∂x

···

∂f

∂x

. ···

∂f

∂x

∂f

∂x

···

∂f

∂x

⎞

⎟

⎠

(8.10)



∂f

∂u



⎛

⎜

⎝

∂f

∂u

∂f

∂u

···

∂f

∂u

∂f

∂u

∂f

∂u

···

∂f

∂u

. ···

∂f

∂u

∂f

∂u

···

∂f

∂u

⎞

⎟

⎠

(8.11)

are Jacobi matrices. Superscripts (·)

∗

denote optimal trajectories of u

∗

, x

∗

Equation (8.9) can be rewritten when considering (8.1) and (8.8) as

d(δx)



∂f

∂x



δx +



∂f

∂u



δu (8.12)

The cost function I(u) attains the minimum for the function u

∗

= u

∗

(t)

from a class of permitted functions if an arbitrary function u(t) satisﬁes in-

equality (8.3).

In general, an increment of the cost function I(x) can be deﬁned as

ΔI = I(x + δx) − I(x) (8.13)

and thus it can be written as

ΔI =ΔI(x,δx) (8.14)

The ﬁrst variation δI is the part of ΔI that is linear in the variation δx.

The fundamental theorem of the variational calculus says that if x

∗

is on the

extreme then necessary condition for extremum of the cost function is

δI(x

∗

,δx) = 0 (8.15)

for all possible δx.

It can be noted that there exists a formal similarity between extremes of

a function y = y(x)oftheform(dy/dx) = 0 and relation (8.15).

300 8 Optimal Process Control

It can be shown that if the increments ΔI and its variation δI are con-

tinuous functions and the variation δu can be arbitrary, then the equality

δI = 0 (8.16)

is the necessary condition of extremum. If only single-sided variations (for

example δu ≥ 0) are permitted then δI = 0 does not constitute the necessary

condition.

The variation of the cost (8.2) can be written as

δI =



∂G

∂x(t

)



δx(t





∂F

∂x



δx +



∂F

∂u



δu

dt (8.17)

Partial derivatives in (8.17) are calculated in optimal trajectories.

Equation (8.17) holds if the ﬁnal time t

is ﬁxed. The ﬁnal state of the

system x(t

) is considered to be free.

Let us deﬁne an adjoint vector λ(t) and manipulate equation (8.12) as

follows

d(δx)

= λ



∂f

∂x



δx + λ



∂f

∂u



δu (8.18)

Integrating (8.18) from t = t

to t = t

yields





d(δx)

− λ



∂f

∂x



δx −λ



∂f

∂u



δu



dt = 0 (8.19)

Adding (8.17) to (8.19) gives for δI

δI =



∂G

∂x (t

)



δx(t

)





∂F

∂x



+ λ



∂f

∂x



δx +



∂F

∂u



+ λ



∂f

∂u



δu

−



d(δx)

dt (8.20)

Derivative of the term λ

δx gives

d(δx)



δx



−

dλ

δx (8.21)

and thus equation (8.20) is of the form

δI =



∂G

∂x (t

)



δx (t





∂F

∂x



+ λ



∂f

∂x



dλ

δx





∂F

∂u



+ λ



∂f

∂u



δudt + λ

δx|

t=t

− λ

δx|

t=t

(8.22)