Delayed Model Approximation and Control Design for

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Delayed Model Approximation and Control Design for Under-Damped Systems Juan F. Márquez-Rubio David F. Novella-Rodríguez Basilio del Muro-Cuéllar German Hernández-Hernández Instituto Politécnico Nacional Instituto Tecnológico de Estudios Superiores de Monterrey

1 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Table of Contents. Introduction Background Approximation for oscillating systems Reduction procedure Controller Tuning Example Conclusions

2 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Outline Introduction Background Approximation for oscillating systems Reduction procedure Controller Tuning Example Conclusions

3 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

High-Order Under-Damped Systems Examples

4 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Features

◮

Amplification of the motion at a specific frequency. ◮

◮

Large mechanical structures can be represented as systems with complex conjugated poles. ◮ ◮

◮

Natural frequency.

Real part typically small. The distance from the origin is the natural frequency.

A large mechanical or civil structure consists on several under-damped systems connected among them.

They represent a challenge for the design of suitable control laws.

5 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Outline Introduction Background Approximation for oscillating systems Reduction procedure Controller Tuning Example Conclusions

6 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Model Reduction

There are different works dealing with the problem of characterize a large order system by means a simpler description. ◮

Truncated balanced realizations.

◮

Singular values decomposition.

◮

Frequency domain methods.

◮

Half-rule method.

7 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Half-rule method The empirical methodology of Skogestad is based on the time response of the open-loop high order system. G(s) = Where

Q  j



−Tjinv + 1

i (τi0 s

Q

τ1 = τ10 + τ20

+ 1)

and

e θ0 s

⇒

θ = θ0 +

X i≥3

e −θs ˆ G(s) = τ1 s + 1

τi0 +

X

inv Tj0 .

j

The method was developed for typical process. Oscillating systems has not been took into account.

8 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Outline Introduction Background Approximation for oscillating systems Reduction procedure Controller Tuning Example Conclusions

9 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Aims

◮

To obtain a second-order approximation for high-order oscillating structures. ◮

◮

◮ ◮

◮

Typical models on mechanical and civil engineering.

To consider the frequency domain characteristics of the high-order system. Natural frequency. Damping coefficient.

To simplify the controller tuning by means the model reduction.

10 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

The aim of this work is to compute a descriptive approximation for the high order system: K

Qm

j=1 (s 2 i=1 (s

GHO (s) = Qn

2

+ 2ζj ωnj s + ωn2j )

+ 2ζi ωni s + ωn2i )

,

(1)

where m < n, ζi is the damping coefficient and ωni is the natural frequency of the corresponding under damped subsystem.

11 / 37

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Three important characteristics of the frequency domain analysis to point out: ◮ Resonance frequency ωr , ◮ Resonance peak Mr , ◮ Phase margin θr . Bode Diagram 50

Mr Magnitude (dB)

0

−50

−100

ω

r

−150 0

Phase (deg)

Title

−180

θ

r

−360

−540 −2

10

−1

10

0

10 Frequency (rad/s)

1

10

2

10

Figure 1: Oscillating system frequency analysis 12 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Proposed Approximation For the high-order system given by (1), it is proposed the second order process with time delay given by: GLO (s) =

¯ e −τ s K , (s 2 + 2ζωn s + ωn2 )

(2)

Every couple of complex conjugated poles of (1) adds a lag of 180◦ on the high order frequencies region. In order to compensate the mentioned phase difference, a time-delay is introduced in to the approximation.

13 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Outline Introduction Background Approximation for oscillating systems Reduction procedure Controller Tuning Example Conclusions

14 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Damping coefficient ζ

◮

The relation between the resonance peak of the high order system and the damping coefficient of a second order process is given by: Mr =

1 2ζ 1 − ζ 2 p

(3)

30

25

r

From the Bode diagram of the high-order open-loop system it is possible to compute the resonance peak.

Resonance peak M

◮

20

15

10

5

0

0

0.1

0.2

0.3 0.4 0.5 Damping coefficient ζ

0.6

0.7

0.8

Figure 2: Relation: Damping coefficient and resonance peak.

15 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Natural frequency ωn

it is possible to relate the resonant frequency of the high order system with the natural frequency of the reduced order system by means of the following equation: ωr ωn = p , 1 − 2ζ 2

(4)

16 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

¯ Proportional gain K

The proportional gain should be determined taking into account the DC gains of the high order system and the reduced model, as follows: ¯ = K limω→0 GHO (jω) . K limω→0 GLO (jω)

(5)

17 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Time delay τ As a parameter of compensation, the gain margin of the high order system is considered, then, the crossover frequency ωgm is taking into account. 30

High-order system Reduced model

25

Magnitude (dB)

20 15 10 5 0 -5 -10 0

Phase (deg)

-90

φ LO -180

Phase to be compensated

-270

φ HO -360 3

4

5 Frequency (rad/s)

6

7

8

9

10

11

12

13

Figure 3: Phase compensation 18 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Time delay τ

The difference between the phase of the reduced model φLO (ωgm ) and the phase original model φHO(ωgm ) at this frequency is related with the delay term as follows: τ=

φHO − φLO 57.3ωgm

(6)

whit the phase given in degrees.

19 / 37

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Bode Diagram Gm = -3.64 dB (at 0.953 rad/s) , Pm = -97.7 deg (at 1.34 rad/s) 10

Magnitude (dB)

0

-10

-20

-30

High-Order System Second order plus time-delay system

-40 0

-90

Phase (deg)

Title

-180

-270

-360 10 -1

10 0

Frequency (rad/s)

Figure 4: Frequency response: delayed system

20 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Outline Introduction Background Approximation for oscillating systems Reduction procedure Controller Tuning Example Conclusions

21 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

PID Strucure The PID controller is given for the following equation: CPID (s) = KP +

KI + KD s. s

(7)

The closed loop transfer function is: Y (s) GLO (s)C (s) = . R(s) 1 + GLO (s)C (s)

(8)

Figure 5: Closed Loop System 22 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

PID Gains Let us consider a second order desired transfer function: 

Y (s) R(s)



=

d

2 ωnd −τ s , 2 e s 2 + 2ζd ωnd s + ωnd

(9)

The PID controller gains are computed as follows: KD = KP

=

KI

=

2 ωnd , ¯ K 2 ωnd 2ζωn , ¯ K 2 ωnd ωn2 , ¯ K

(10)

and the first-order filter has a pole located at s = −(2ζd ωnd + τ ). 23 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Outline Introduction Background Approximation for oscillating systems Reduction procedure Controller Tuning Example Conclusions

24 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Application Example

Figure 6: Mass spring damper fourth order system

The system parameters are: m1 = m2 = 1kg, k1 = k2 = 1N/m, b1 = 0.65Ns/m and b2 = 1Ns/m

X2 (s) b1 s + k1 = , F (s) m1 m2 s 4 + αs 3 + βs 2 + (b1 k1 + b2 k1 )s + k1 k2

(11)

with α = m1 (b1 + b2 ) + m2 b1 and β = m1 k1 + m2 (k1 + k2 ) + b2 b1 .

25 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Model Approximation Applying the proposed model reduction, a second order approximation is computed, obtaining the following result: GLO (s) =

0.3853 e −0.319s . s 2 + 0.3129s + 0.3853 Step Response

Amplitude

1.5 1 0.5

High-Order System Approximation

0 0

5

10

15

20

25

30

35

40

Time (seconds) Linear Simulation Results Amplitude

4 2 0 -2 0

5

10

15

20

25

30

Time (seconds)

Figure 7: Response of the open-loop system.

26 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

PID Controller The PID controller is designed in order to regulate the output X2 (s), considering the tuning methods given previously. The tuning parameters are ωnd = 4ωn , with ωn being the reduction natural frequency, and ζd = 1.2. Under these conditions, the closed loop performance shown in the Fig. 8. 1.2

1

0.8

0.6

0.4 Original system Reduced system

0.2

0 0

5

10

15

20

25

30

35

40

Figure 8: Closed loop Performance

27 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Disturbance Rejection Disturbance rejection is one of the main issues of concern in mechanical and civil engineering. In order to obtain a satisfactory performance with respect to the disturbance rejection, the feed-forward control structure, shown in Fig. 9, could be used, [1].

Figure 9: Feedforward action for load disturbance rejection task.

28 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Disturbance Rejection

Taking into account the model reduction the proposed feed-forward controller is: 2 ωnh −τ s , 2 e s 2 + 2ζh ωnh s + ωnh

(12)

2 s 2 + 2ζωn s + ωn2 K¯h ωnh 2 , ¯ ωn2 s 2 + 2ζh ωnh s + ωnh K

(13)

H(s) = K¯h and T (s) =

29 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Disturbance Rejection The feedforward controller gains are: ζh = 2ζ, ωnh = 2ωn and K¯h = 0.1 The obtained response are shown in Fig. 10. 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 PID controller Feedforward PID controller

0.2 0 0

10

20

30

40

50

60

70

80

90

100

Figure 10: Closed-loop response: Feedforward effect on the disturbance rejection task. 30 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Evaluation of the reduction model with different tuning control methods

In order to validate the reduced order model, it is evaluated using different tuning methods for PID controllers found in the available literature [2]. The tuning rules considered have been taken from different authors, [3, 4, 5, 6]. The computed controller is applied to the original high order system.

31 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Evaluation of the reduction model with different tuning control methods In order to numerically evaluate the output control performance, it is possible to compute the Integral Absolute Error (IAE) of the error signal, e(t) = r (t) − y (t): IAE =

Z

∞

|e(t)|dt,

0

10 1. Proposed Method 2. Shaedel 3. Rivera and Jun 4. Ziegler−Nichols 5. Huang et al 6. Jahanmiri and Fallali

9

Integral Absolute Error IAE

8 7 6 5 4 3 2 1 0

1

2

3 4 Tuning Method

5

6

32 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Outline Introduction Background Approximation for oscillating systems Reduction procedure Controller Tuning Example Conclusions

33 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Conclusions ◮

A reduced order delayed approximation for high order systems with oscillating dynamics is proposed.

◮

The reduction methodology is simple but it takes into account frequency domain characteristics of the high order original system and uses them to obtain an accurate model reduction.

◮

In order to match the phase characteristics of the high order systems and the low order simplification a time-delay is introduced to the model reduction.

◮

Following the IMC tuning rules it is possible to synthesize PID controller parameters in a simple and efficient way and use this controllers in the original high order process.

34 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Further Work

◮

To use numerical methods to optimize the proposed model approximation.

◮

To propose a methodology to obtain more efficient controllers.

35 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Antonio Visioli. PID Control in the Third Millennium. Lessons Learned and New Approaches. Advances in Industrial Control. Springer, 2006. Aidan O’Dwyer. Handbook of PI and PID Controller Tuning Rules. Imperial College Press, 2009. Herbert M. Schaedel. A new method of direct PID controller design based on the principle of cascaded damping ratios. In 1997 European Control Conference (ECC), pages 1265–1271, 1997. D. E. Rivera and K. S. Jun. An integrated identification and control design methodology for multivariable process system applications. IEEE Control Systems, 20(3):25–37, June 2000. 36 / 37

Title

Introduction

Background

Approximation

Reduction

PID Control

Example

Conclusions

Hsiao-Ping Huang, Jyh-Cheng Jeng, and Kuo-Yuan Luo. Auto-tune system using single-run relay feedback test and model-based controller design. Journal of Process Control, 15(6):713 – 727, 2005. A. Jahanmiri and H.R. Fallahi. New methods for process identification and design of feedback controller. Chemical Engineering Research and Design, 75(5):519 – 522, 1997.

37 / 37