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

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3.3 Input-Output Process Models 107

−a

- - - - - -



u(t)

y(t)

Fig. 3.26. Block scheme of controllable canonical form of a second order system

Observable Canonical Form

Consider again the system with transfer function given by (3.175) and assume

n = m. Observable canonical form of this system is given by

dx(t)

= A

x(t)+B

u(t) (3.187)

y(t)=C

x(t)+D

u(t) (3.188)

where x =(x

,...,x

)

⎛

⎜

⎝

−a

n−1

10... 0

−a

n−2

01... 0

−a

00... 1

−a

00... 0

⎞

⎟

⎠

, B

⎛

⎜

⎝

n−1

− a

n−1

n−2

− a

n−2

− a

⎞

⎟

⎠



10 ... 0



, D

= b

The corresponding block scheme is shown in Fig. 3.27.

3.3.7 I/O Models of MIMO Systems - Matrix Fraction

Descriptions

Transfer function matrices provide a way to describe I/O ( external) models of

MIMO linear continuous systems. An alternate way is to give descriptions of

such systems in polynomial matrix fractions that provide a natural generalisa-

tion of the singlevariable concept. A fundamental element of such descriptions

is the polynomial matrix or matrix polynomial.

Real polynomial matrix is a matrix

108 3 Analysis of Process Models

−a

n−1

− a

n−1

− a

n−1

eeee



- - - - - - -

? ?

u(t)

y(t)

...



Fig. 3.27. Block scheme of observable canonical form of a system

P (s)=

⎛

⎜

⎝

(s) p

(s) ... p

(s)

(s) p

(s) ... p

(s)

(s) p

(s) ... p

(s)

⎞

⎟

⎠

(3.189)

with elements being polynomials

(s)=p

ij0

+ p

ij1

s + ···+ p

ijd

(3.190)

where p

ijk

are real numbers and i =1...n, j =1...m, k =0...d

An element p

(s) identically equal to zero has according to deﬁnition a

degree equal to minus one.

Row degree of i-th row of P is denoted by r

and it is the maximum

degree of all polynomials in the i-th row (r

= max

). Analogously are

deﬁned column degrees c

as max

Degree of polynomial matrix P is denoted by deg P and is deﬁned as the

maximum degree of all polynomials of P (max

An alternate way of writing (3.189) is as matrix polynomial

P (s)=P

+ P

s + ···+ P

(3.191)

where

3.3 Input-Output Process Models 109

⎛

⎜

⎝

11k

12k

... p

1mk

21k

22k

... p

2mk

n1k

n2k

... p

nmk

⎞

⎟

⎠

k =0,...,d

d = deg P

A square polynomial matrix P (s) is nonsingular if det P (s) is not identi-

cally equal to zero.

Roots of a square polynomial matrix are roots of the determinant of the

polynomial matrix. A polynomial matrix is stable if all its roots lie in the open

left half plane of the complex plane.

An unimodular polynomial matrix is a square polynomial matrix with a

determinant equal to a nonzero constant. P (s) is unimodular if its inverse is

also unimodular.

Rank of polynomial matrix is the highest order of a nonzero minor of this

matrix.

For any P (s) there exist unimodular matrices U (s)andV (s) such that

U(s)P (s)V (s)=Λ(s) (3.192)

where

Λ(s)=

⎛

⎜

⎝

(s)0... 00

0 λ

(s) ... 00

00... λ

(s)0

00... 00

⎞

⎟

⎠

are unique monic polynomials (i.e. polynomials with a unit ﬁrst coeﬃcient)

for which holds λ

divides λ

for i<j≤ r and r is equal to rank P (s). The

matrix Λ(s)[n ×m] is called the Smith form of polynomial matrix.

Elementary row (column) operations on P (s)are:

1. interchange of any two rows (columns) of P (s),

2. multiplication of any row (column) of P (s) by any nonzero number,

3. addition to any row (column) of a polynomial multiple of any other row

(column).

For any polynomial matrix we can ﬁnd elementary row and column oper-

ations transforming the matrix into the Smith form.

A[r × m] transfer function matrix G(s) can be of the form

G(s)=

M(s)

d(s)

(3.193)

where d(s) is the least common multiplier of denominators of elements of G(s)

and M (s) is a polynomial matrix. The matrix G(s) from Eq. (3.193) can also

be written as

110 3 Analysis of Process Models

G(s)=B

(s)A

−1

(s) (3.194)

where B

(s)=M(s), A

(s)=d(s)I

. It is also possible to write

G(s)=A

−1

(s)B

(s) (3.195)

where B

(s)=M(s), A

(s)=d(s)I

The description of G(s) given by (3.194) is called right matrix fraction

description (RMFD). The degree of determinant of A

(s)is

deg det A

(s)=d

m (3.196)

where d

is a degree of d(s)andm is a dimension of vector u.

Analogously, the description of (3.194) is called left matrix fraction de-

scription (LMFD). The degree of determinant of A

(s)is

deg det A

(s)=d

r (3.197)

where r is the dimension of vector y.

Given G(s) there are inﬁnitely many LMFD’s and RMFD’s.

Minimum degree of determinant of any LMFD (RMFD) of G(s)isequal

to the minimum order of some realisation of G(s).

If some RMFD of G(s)isoftheform

G(s)=B

(s)A

−1

(s) (3.198)

and some other RMFD of the form

G(s)=B

(s)A

−1

(s) (3.199)

then B

(s)=B

(s)W (s), A

(s)=A

(s)W (s), and W (s) is some poly-

nomial matrix and it is common right divisor of B

(s), A

(s). Analogously,

the common left divisor can be deﬁned.

Deﬁnition of relatively right (left) prime (RRP-RLP) polynomial matrices:

Polynomial matrices B(s), A(s) with the same number of columns (rows) are

RRP (RLP) if their right (left) common divisors are unimodular matrices.

Matrix fraction description of G(s) given by A(s), B(s)isright (left) ir-

reducible if A(s), B(s) are RRP (RLP).

The process of obtaining irreducible MFD is related to greatest common

divisors.

Greatest right (left) common divisor (GRCD-GLCD) of polynomial ma-

trices A(s), B(s) with the same number of columns (rows) is a polynomial

matrix R(s) that satisﬁes the following conditions:

• R(s) is common right (left) divisor of A(s), B(s),

• if R

(s) is any common right (left) divisor of A(s), B(s) then R

(s)is

right (left) divisor of R(s).

3.3 Input-Output Process Models 111

Lemma: relatively prime polynomial matrices: Polynomial matrices B

(s),

(s) are RRP if and only if there exist polynomial matrices X

(s), Y

(s)

such that the following Bezout identity is satisﬁed

(s)B

(s)+X

(s)A

(s)=I

(3.200)

Polynomial matrices B

(s), A

(s) are RLP if and only if there exist poly-

nomial matrices X

(s), Y

(s) such that the following Bezout identity is

satisﬁed

(s)Y

(s)+A

(s)X

(s)=I

(3.201)

For any polynomial matrices B

(s)[r ×m]andA

(s)[m ×m] an unimod-

ular matrix V (s) exists such that

V (s)=



(s) V

(s)

(s) V

(s)



(s) ∈ [m ×r] V

(s) ∈ [m ×m]

(s) ∈ [r × r] V

(s) ∈ [r × m]

(3.202)

and

V (s)



(s)





R(s)



(3.203)

R(s)[m × m]isGRCD(B

(s), A

(s)). The couples V

, V

and V

, V

are RLP. An analogous property holds for LMFD: For any polynomial matrices

(s)[r × m]andA

(s)[r × r] an unimodular matrix U (s) exists such that

U(s)=



(s) U

(s)

(s) U

(s)



(s) ∈ [r × r] U

(s) ∈ [r × m]

(s) ∈ [m ×r] U

(s) ∈ [m ×m]

(3.204)

and

(s) B

(s)) U(s)=(L(s) 0) (3.205)

L(s)[r × r]isGLCD(B

(s), A

(s)). The couples U

, U

and U

, U

are RRP.

Equations (3.203), (3.205) can be used to obtain irreducible MFD of G(s).

When assuming RMFD the G(s) is given as

G(s)=B

(s)A

−1

(s) (3.206)

where B

(s)=−V

(s)andA

(s)=V

(s).

Lemma: division algorithm:LetA(s)[m ×m] be a nonsingular polynomial

matrix. Then for any B(s)[r × m] there exist unique polynomial matrices

Q(s), R(s) such that

112 3 Analysis of Process Models

B(s)=Q(s)A(s)+R(s) (3.207)

and R(s)A

−1

(s) is strictly proper.

The previous lemma deals with right division algorithm. Analogously, the

left division can be deﬁned.

This lemma can be used in the process of ﬁnding of a strictly proper part

of the given R(L)MFD.

Lemma: minimal realisation of MFD: A MFD realisation with a degree

equal to the denominator determinant degree is minimal if and only if the

MFD is irreducible.

Lemma: BIBO stability: If the matrix transfer function G(s) is given by

Eq. (3.194) then it is BIBO stable if and only if all roots of det A

(s) lie in

the open left half plane of the complex plane. (analogously for LMFD).

Spectral factorisation: Consider a real polynomial matrix B(s)[m×m] such

that

(−s)=B(s) (3.208)

B(jω) > 0 ∀ω ∈ R (3.209)

Right spectral factor of B(s) is some stable polynomial matrix A(s)[m × m]

that satisﬁes the following relation

B(s)=A

(−s)A(s) (3.210)

3.4 References

The use of the Laplace transform in theory of automatic control has been

treated in large number of textbooks; for instance,

B. K.

Cemodanov et al. Mathematical Foundations of Automatic Control I.

Vyˇsˇsaja ˇskola, Moskva, 1977. (in Russian).

K. Reinisch. Kybernetische Grundlagen und Beschreibung kontinuericher

Systeme. VEB Verlag Technik, Berlin, 1974.

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

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

(in Slovak).

State-space models and their analysis are discussed in

C. J. Friedly. Dynamic Behavior of Processes. Prentice Hall, Inc., New Jersey,

1972.

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

(in Slovak).

H. Unbehauen. Regelungstechnik II. Vieweg, Braunschweig/Wiesbaden, 1987.

L. B. Koppel. Introduction to Control Theory with Application to Process

Control. Prentice Hall, Englewood Cliﬀs, New Jersey, 1968.

3.4 References 113

L. A. Zadeh and C. A. Desoer. Linear System Theory - the State-space Ap-

proach. McGraw-Hill, New York, 1963.

A. A. Voronov. Stability, Controllability, Observability. Nauka, Moskva, 1979.

(in Russian).

H. P. Geering. Mess- und Regelungstechnik. Springer Verlag, Berlin, 1990.

H. Kwakernaak and R. Sivan. Linear Optimal Control Systems. Wiley, New

York, 1972.

W. H. Ray. Advanced Process Control. McGraw-Hill, New York, 1981.

M. Athans and P. L. Falb. Optimal Control. McGraw-Hill, New York, 1966.

R. E. Kalman, Y. C. Ho, and K. S. Narendra. Controllability of linear dy-

namical systems in contributions to diﬀerential equations. Interscience

Publishers, V1(4):189 – 213, 1963.

P. L. Kalman, R. E. Falb and M. Arib. Examples of Mathematical Systems

Theory. Mir, Moskva, 1971. (in Russian).

E. D. Gilles. Systeme mit verteilten Parametern, Einf

uhrung in die Regelungs-

theorie. Oldenbourg Verlag, M

unchen, 1973.

D. Chm´urny, J. Mikleˇs, P. Dost´al, and J. Dvoran. Modelling and Control of

Processes and Systems in Chemical Technology. Alfa, Bratislava, 1985.

(in Slovak).

A. M´esz´aros and J. Mikleˇs. On observability and controllability of a tubular

chemical reactor. Chem. prumysl, 33(2):57 – 60, 1983. (in Slovak).

P. Dost´al, J. Mikleˇs, and A. M´esz´aros. Theory of Automatic Control. Exercises

II.ESSV

ST, Bratislava, 1983. (in Slovak).

A. M. Lyapunov. General Problem of Stability of Motion. Fizmatgiz, Moskva,

1959. (in Russian).

R. E. Kalman and J. E. Bertram. Control system analysis and design via the

second method of Lyapunov. J. Basic Engineering, 82:371 – 399, 1960.

J. Mikleˇs. Theory of Automatic Control of Processes in Chemical Technology,

Part II.ESSV

ST, Bratislava, 1980. (in Slovak).

The subject of input-output models is considered as a classical theme in

textbooks on automatic control; for example,

G. Stephanopoulos. Chemical Process Control, An Introduction to Theory and

Practice. Prentice Hall, Englewood Cliﬀs, New Jersey, 1984.

Y. Z. Cypkin. Foundations of Theory of Automatic Systems. Nauka, Moskva,

1977. (in Russian).

C. A. Desoer and M. Vidyasagar. Feedback Systems: Input-Output Properties.

Academic Press, New York, 1975.

C. T. Chen. Linear System Theory and Design. Holt, Rinehart and Winston,

New York, 1984.

114 3 Analysis of Process Models

Further information about input-output models as well as about matrix

fraction descriptions can be found in

W. A. Wolovich. Linear Multivariable Systems. Springer Verlag, New York,

1974.

H. H. Rosenbrock. State-space and Multivariable Theory. Nelson, London,

1970.

T. Kailaith. Linear Systems. Prentice Hall, Englewood Cliﬀs, New Jersey,

1980.

V. Kuˇcera. Discrete Linear Control: The Polynomial Equation Approach.

Wiley, Chichester, 1979.

J. Jeˇzek and V. Kuˇcera. Eﬃcient algorithm for matrix spectral factorization.

Automatica, 21:663 – 669, 1985.

J. Jeˇzek. Symmetric matrix polynomial equations. Kybernetika, 22:19 – 30,

1986.

3.5 Exercises

Exercise 3.1:

Consider the two tanks shown in Fig. 3.10. A linearised mathematical model

of this process is of the form

= a

+ b

= a

+ a

y = x

where

= −



= −



Find:

1. state transition matrix of this system,

2. if x

(0) = x

(0) = 0 give expressions for functions

(t)=f

(u(t))

(t)=f

(u(t))

y(t)=f

(u(t))

Exercise 3.2:

Consider CSTR shown in Fig. 2.11 and examine its stability. The rate of

reaction is given as (see example 2.6)

3.5 Exercises 115

r(c

,ϑ)=kc

= k

−

Rϑ

Suppose that concentration c

and temperatures ϑ

,ϑ

are constant. Perform

the following tasks:

1. deﬁne steady-state of the reactor and ﬁnd the model in this steady-state

so that dc

/dt =dϑ/dt =0,

2. deﬁne deviation variables for reactor concentration and temperature and

ﬁnd a nonlinear model of the reactor with deviation variables,

3. perform linearisation and determine state-space description,

4. determine stability conditions according to the Lyapunov equation (3.82).

We assume that if the reactor is asymptotically stable in large in origin

then it is asymptotically stable in origin.

Exercise 3.3:

Consider the mixing process shown in Fig. 2.15. The task is to linearise the

process for the input variables q

and output variables h, c

and to deter-

mine its transfer function matrix.

Exercise 3.4:

Consider a SISO system described by the following diﬀerential equation

¨y(t)+a

˙y(t)+a

y(t)=b

˙u(t)+b

u(t)

Find an observable canonical form of this system and its block scheme.

Exercise 3.5:

Assume 2I/2O system with transfer function matrix given as LMFD (3.195)

where

(s)=



1+a

s 1+a



(s)=





By using the method of comparing coeﬃcients, ﬁnd the corresponding

RMFD (3.194) where

(s)=



+ a



(s)=





Elements of matrix





can be chosen freely, but A

must be nonsingular.

Dynamical Behaviour of Processes

Process responses to various simple types of input variables are valuable for

process control design. In this chapter three basic process responses are stud-

ied: impulse, step, and frequency responses. These characteristics are usu-

ally investigated by means of computer simulations. In this connection we

show and explain computer codes that numerically solve systems of diﬀeren-

tial equations in the programming languages BASIC, C, and MATLAB.

The end of this chapter deals with process responses for the case of stochas-

tic input variables.

4.1 Time Responses of Linear Systems to Unit Impulse

and Unit Step

4.1.1 Unit Impulse Response

Consider a system described by a transfer function G(s) and for which holds

Y (s)=G(s)U(s) (4.1)

If the system input variable u(t) is the unit impulse δ(t) then

U(s)=L{δ(t)} = 1 (4.2)

and the system response is given as

y(t)=g(t) (4.3)

where g(t)=L

−1

{G(s)} is system response to the unit impulse if the system

initial conditions are zero, g(t) is called impulse response or weighting function.

If we start from the solution of state-space equations (3.47)