Analysis of mechatronic systems second class by the matrix method

Jerzy Smyczek

Department of Electronics and Computer Science, Koszalin University of Technology

Abstract: In the paper an analysis of mechatronic systems by using matrix method has been described. On the base a real matrix method system is presented a model the member: electrics, electronics, mechanics, hydraulics and others in connections with feedback and without them has been examined. In the end an example at a control bus door for this purpose obtaining minimum time control has been presented.

Keywords: matrix method, mechatronics

1. Introduction

Investigating of dynamics in mechatronics systems which contain the members: electrics, electronics, mechanics, hydraulics, thermals, and others is important matter because the system has to be stable with regard for same parameters. In general, members of mechatronic systems are multipoles. In technical applications the system may be presented as two-port networks. The one is assumed as linear.

Fig. 1. A two-port network in general shape
Rys. 1. Czwórnik w postaci ogólnej

It is meaning that

is linear function. A separate important problem is defining an amplitude range on surrounding at working point. The signals

are Laplace or Fourier transform.

, or , (1)

The two-port networks are described in form of differential or integrated equations. After Laplace (or Fourier) transformation the couple of linear equations have been got. In works on two-port networks are presented formulas between different forms of matrix. To consider the cascade connection of matrix has been got, as:

; (2)

2. Members of mechatronic systems and their connections

In the tab. 1 has been shown a quantity of mechatronic members.

With a progress of technique the new converters are being application, as for ex. ultrasonic, optics. In connection with it following mechatronic members may be presented:

an electric-electronic member

(3)

a member as generator

(4)

a member as motor

(5)

a member as electromagnetic

(6)

a member as hydraulic (or pneumatic) converter

(7)

a member as thermal converter

(8)

In connections of members the output signals at a previous member and input signals at a following member have to get the same physical character.

Tab. 1. Quantity of mechatronic memebers
Tab. 1. Wielkości członów mechatronicznych

System

Electric

Pneumatic

Thermal

Mechanic

Mechanic (rotatable)

Potential
R

Voltage
U [V]

Pressure
P [N/m2]

Temperature
T [K]

Velocity
V [m/s]

Angular velocity
w [rd/s]

Flow
X

Current
I [A]

Flow (volume) V [m3/s]

Flow (mass)
V [kg/s]

Force
F [N]

Moment
M [Nm]

3. Input and output impedance of a member

Knowing a four-terminal member the impedance of members has been defined. Analogical to definition using in electrics

and (9)

The definition has been extended for different mechatronics members

and (10)

When the above values to present in frequency

(11)

Then, it may be calculation in frequency band of a work member.

a) A cascade connection of members

Fig. 2. Cascade connection of members
Rys. 2. Połączenie kaskadowe członów

The connection presented in the fig. 3 may be represented by transmittance

(12)

When a following member do not load a previous member. Meaning, that

(13)

Fig. 3. A cascade connection of matrices
Rys. 3. Połączenie kaskadowe macierzy

Result matrix of the system is

(14)

If the condition (13) is not satisfy or impossible to estimation, then the matrix method should be applying in order to avoid a errors [7 ].

b) A system with feedback

, (15)

Fig. 4. A system with feedback at parallel. The arrows are meaning of signal at direction
Rys. 4. System ze sprzężeniem zwrotnym równoległym

If the relation (15) is satisfied, then a block diagram may be presented as one-thread diagram.

c) The connection of parallel members

The mutual loading should be satisfying the conditions

Fig. 5. The connection of parallel members
Rys. 5. Połączenie równoległe członów

, (16)

If the relations (16) are satisfying then diagram may be presented in shape

Fig. 6. The one-thread block diagram
Rys. 6. Jednonitkowy schemat blokowy

4. Matrix of systems with negative feedback

4.1. Connection with feedback of parallel-series

Fig. 7. A block’s diagram with feedback of parallel-series
Rys. 7. Schemat blokowy ze sprzężeniem zwrotnym równoległo-szeregowym

Equations on input

(17)

where

(18)

and

(19)

It means negative feedback.

Output equations

(20)

and

(21)

Now, the vector

is

(22)

The vector’s components in (22) having form

(23)

and

(24)

For instance a connection between D and G = H is the following:

If for (25)

Into consideration (23) and (24) in (22) we are having

(26)

In the result of matrix

(27)

H – type is as follows (24).

4.2. Connection with feedback at series-series

Fig. 8. A block’s diagram with feedback of series-series
Rys. 8. Schemat blokowy ze sprzężeniem zwrotnym szeregowo-szeregowym

Equations on input

(28)

where

(29)

and

(30)

It means negative feedback.

Output equations

(31)

and

(32)

Now, the vector

is

(33)

The vector’s components in (33) having form

(34)

and

(35)

For instance a connection between Z and C is the following:

If for (36)

Into consideration (34) and (35) in (33) we are having

(37)

In connection with it, the result matrix Z-type of system having form

(38)

4.3. A connection with feedback of parallel-parallel

Equations on input

Fig. 9. A block’s diagram with feedback of parallel-parallel
Rys. 9. Schemat blokowy ze sprzężeniem zwrotnym równoległo-równoległym

(39)

where

(40)

and

(41)

It means negative feedback.

Output equations

(42)

and

(43)

Now, the vector

is calculated

(44)

The vector’s components in (44) are having form

(45)

and

(46)

For instance, a connection between Z and C is the following:

If for (47)

Then the expression (44) has a form

(48)

In connection with it, the result matrix Y-type of system is having a formula

(49)

4.4. A connection with feedback of series-parallel

Fig. 10. A block’s diagram with feedback of series-parallel
Rys. 10. Schemat blokowy ze sprzężeniem zwrotnym szeregowo-równoległym

Equations on input

(50)

where

(51)

and

(52)

It means negative feedback.

Output equations

(53)

and

(54)

Now, it will be calculated

(55)

It notice, that

(57)

and

(58)

For instance a connection between of components of matrix G and D is:

If

for (59)

The expression (55) with regard to (57) and (58) having form

(60)

In connection with it, the result matrix H-type of system is

(61)

5. Concluding remarks

A presentation of systems in shape at a block diagram where members are two-port networks and describing by matrix is making possible a resultant matrix of system. By using at computer base of matrix transformation two-port networks the algorithm of calculation the matrix is quite simple.

6. Example

It should calculate a time constant at a integral circuit in feedback path at a control bus door like that a settling time will be minimum.

Fig. 11. A block diagram of matrix the control system door
Rys. 11. Schemat blokowy macierzy systemu sterowania drzwiami

A1 – matrix of electronic amplifier

A2 – matrix of power converter electric-hydraulic

A3 – matrix of load

A4 – matrix of shift-voltage converter

A5 – matrix of integral circuit

The input parameters are voltage-current and output parameters are force and velocity. The scheme in fig. 11 may be reduction for the shape of fig. 12.

Fig. 12. A connection of feedback a series-parallel
Rys. 12. Macierzowy schemat blokowy ze sprzężeniem zwrotnym szeregowo-równoległym

in which

(62)

In the ex ample a response is Xout for unit step is Rout = 1/s. Using with the matrix Hres we have

(63)

Where L(s), M(s) are polynomials with regard for s.

If (64)

Is a oscillation type, then settling time (with accuracy 2 %) getting

(65)

where a – coefficient by the s1.

For higher order of systems the same formula is applied then is estimation.

In connection with that

,

Tc – time constant. It should minimize that value

(66)

Algorithm of calculating Tc for presented system is the following.

START

¯

Calculate resultant matrix of main line

Calculate resultant matrix of feedback line

¯

Change of chain matrix to hybrid matrix

Change of chain matrix to hybrid matrix

¯

Calculate resultant hybrid matrix

¯

Change of resultant hybrid matrix to chain matrix

¯

Determine parameter a11(s) of resultant chain matrix

¯

Calculate transfer function

¯

END

7. Conclusion

In the paper the analyses of systems in which may be presented as two-port networks have been considered. To calculation the matrix method has been used. The conditions affording possibilities reduction at systems have been expressed.

The matrix of systems with feedback at different making possible to calculate a result matrix of systems in a quite simple. An example of control system algorithm has been presented in purpose a reckoning of a parameter that secure of minimum control time.

References

1. Distefano J.J., Feedback and control system, Powells Books, 2005.

2. Gruz J.B., Układy ze sprzężeniem zwrotnym, PWN, Warszawa 1977.

3. Kaczorek T., Teoria wielowymiarowych układów dynamicznych liniowych, WNT, Warszawa 1983.

4. Kaczorek T., Wektory i macierze w automatyce i elektrotechnice, WNT, 1998.

5. Misra P., Patel R., Computation of transfer function matrice’s of linear multivariable systems, “Automatica”, 1997.

6. Ramey R.L., White E.J., Zastosowanie macierzy w maszynowej analizie układów, PWN, Warszawa 1974.

7. Smyczek J., Ostapko B., Macierzowa analiza transmitancji – wpływ obciążenia członów skończoną impedancją, Polioptymalizacja i Komputerowe Wspomaganie Projektowania. Prace XXIV Ogólnopolskiej Konferencji, Mielno 2006.

8. Smyczek J., Ostapko B., Macierzowa analiza transmitancji – algorytm dla układów ze sprzężeniem zwrotnym, XII Konferencja „Zastosowanie Komputerów w Elektrotechnice”, Poznań 2007.

9. Vargay A., Computation of transfer function matrices of periodic systems, “International Journal of Control”, Vol. 76, No. 17, 2003, 1712–1723.

Appendix 1 – two port’s matrixes

Two port’s matrixes are:

(A1)

Define next original matrices

(A2)

(A3)

(A4)

and

(A5)

(A6)

(A7)

This type of matrixes are using in matrix systems with feedback.

Appendix 2 – two port’s matrixes with negative feedback

Appendix 2 – Matrices of two-port networks with negative feedback

Prof. Jerzy Józef Smyczek, DSc, PhD

Electronics and telecommunication lecturer; MSc, Lodz University of Technology 1962, PhD 1970, DSc 1990. Professor at Koszalin Univ. of Technology, Dept. of Electronics and Informatics. Research interest: Theory of Signal and Circuits, Filtration and Identification of Nonlinear Systems, Systems of Transmission Information, Matrix Analysis of the Feedback Systems.

e-mail: jerzy.smyczek@tu.koszalin.pl