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

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8.6 Analysis of State Feedback with Observer and Polynomial Pole Placement 361

Non-Proper PID Controller for the Second Order System

If the structure of the classical PID controller is desired, the characteristic

polynomial of the closed-loop systems c(s) needs to be of the second degree.

The pole placement controller has the structure with the transfer function

(see Example 8.7 – case 2)

R(s)=

+ q

s + q

(8.381)

This transfer function is equivalent to the structure of the PID controller of

the form

PID

(s)=Z

+ T

s (8.382)

where

(8.383)

This controller was also derived in Chapter 7.4.6 (see page 286).

We can see that controllers R

PID

from (8.382) and R

PIDF

from (8.374)

have similar behaviour if the ﬁlter time constant T

PIDF

is suﬃciently small.

8.6.6 Polynomial Pole Placement Design for Multivariable Systems

The state feedback control with observer for multivariable systems shown in

Fig. 8.21 can be interpreted as a closed-loop system with the two-degree-of-

freedom controller.

The observer dynamics is given as

ˆx =(sI −A + LC)

−1

Bu +(sI − A + LC)

−1

Ly (8.384)

The controller output is

u = −K ˆx + ˜w (8.385)

and after substituting for ˆx from (8.384) yields

u = −K (sI − A + LC)

−1

Bu − K (sI − A + LC)

−1

Ly + ˜w (8.386)

After some manipulations the controller can be written as



I + K (sI −A + LC)

−1



u = −K (sI − A + LC)

−1

Ly + ˜w (8.387)

The feedforward part is given as

(s)=



I + K (sI −A + LC)

−1



−1

(8.388)

362 8 Optimal Process Control

Cxy

()

xCyLBuxAx

ˆˆˆ

−++=

Fig. 8.21. State feedback control with state observation and reference input

The feedback part is given as

R(s)=−



I + K (sI −A + LC)

−1



−1

K (sI − A + LC)

−1

(8.389)

The controller in Fig. 8.21 has two degrees of freedom and its description using

matrix transfer functions R

(s)andR(s)isoftheform

u =



(s) R(s)





˜w



(8.390)

Using the matrix inversion lemma (Theorem 6.1)



I + K (sI −A + LC)

−1



−1

= I − K (sI −A + LC + BK)

−1

(8.391)

and thus

(s)=I −K (sI −A + LC + BK)

−1

B (8.392)

R(s)=



I −K (sI −A + LC + BK)

−1



K (sI − A + LC)

−1



= K



(sI −A + LC)

−1

− (sI − A + LC + BK)

−1

BK (sI −A + LC)

−1



= K (sI − A + LC + BK)

−1

L (8.393)

8.6 Analysis of State Feedback with Observer and Polynomial Pole Placement 363

As it was shown in the singlevariable case, pole locations of the closed-loop

system are given by the feedback part of the state controller with observer

with the matrix transfer function (8.393). If left or right matrix fraction of

the pole placement controller is to be determined (8.393), it suﬃces to study

the case with the zero reference input.

Consider state the feedback

u(t)=−K ˆx(t) (8.394)

with observer

ˆx(t)=Aˆx(t)+Bu(t)+L (y(t) −C ˆx(t)) (8.395)

applied to a controllable and observable system

˙x(t)=Ax(t)+Bu(t) (8.396)

y(t)=Cx(t) (8.397)

Matrix transfer functions of the controlled system are given as

(sI −A)

−1

B = B

(s)A

−1

(s) (8.398)

where A

, B

are right coprime polynomial matrices and A

is column

reduced.

C (sI − A)

−1

= A

−1

(s)B

(s) (8.399)

where A

, B

are left coprime polynomial matrices and A

is row reduced.

C (sI − A)

−1

B = A

−1

(s)B

(s)=B

(s)A

−1

(s) (8.400)

where A

, B

are left coprime polynomial matrices, A

, B

are right co-

prime polynomial matrices and

(s)=B

(s)B (8.401)

(s)=CB

(s) (8.402)

Matrix K is chosen such that the closed-loop matrix A − BK reﬂects

desired dynamics. From (8.398) follows

(s)=(sI −A)B

(s) (8.403)

Adding BKB

to either side of (8.403) gives

B(A

(s)+KB

(s)) = (sI −A + BK)B

(s) (8.404)

(sI −A + BK)

−1

B = B

(s)(A

(s)+KB

(s))

−1

(8.405)

(sI −A)andB are left coprime because the controlled system is controllable.

This implies that (sI − A + BK), B are left coprime as well. Similarly, it

follows from the deﬁnition that A

, B

are right coprime and thus (A

364 8 Optimal Process Control

), B

are right coprime polynomial matrices. This also implies that

(sI −A + BK) and (A

(s)+KB

(s)) have the same determinants.

If the gain matrix K exists, it is unique and of the form

K = X

−1

(8.406)

Then X

, Y

are solution of the equation

(s)+Y

(s)=F

(s) (8.407)

(s) is a stable polynomial matrix with det F

(s) =0.F

(s) is a matrix

with dimensions m ×m, it is column reduced and has the same degree as the

matrix A

(s).

The observer gain matrix L is chosen such that the matrix A − LC cor-

responds to the desired dynamics of the observer. From (8.399) follows

(s)C = B

(s)(sI −A) (8.408)

Adding B

(s)LC to either side of (8.408) gives

(s)+B

(s)L)C = B

(s)(sI −A + LC) (8.409)

C(sI −A + LC)

−1

=(A

(s)+B

(s)L)

−1

(s) (8.410)

C and (sI −A) are right coprime as the controlled system is observable. This

implies that C and (sI −A+LC) are right coprime. Similarly, it follows from

the deﬁnition that A

and B

are left coprime and thus (A

+ B

L)and

are left coprime. This also implies that (sI − A + LC) and (A

(s)+

(s)L) have the same determinants.

If the gain matrix L exists, it is unique and of the form

L = Y

−1

(8.411)

Then X

a Y

are solution of the equation

(s)X

+ B

(s)Y

= O

(s) (8.412)

(s) is a stable polynomial matrix with det O

(s) =0.O

(s) is a matrix

with dimensions r × r, it is row reduced and has the same degree as matrix

(s).

The matrix transfer function of the feedback controller corresponding to

the state feedback controller with observer is given as

K(sI −A + BK + LC)

−1

L = P

−1

(s)Q

(s)=Q

(s)P

−1

(s) (8.413)

Matrices P

(s)andQ

(s) of the controller left matrix fraction are solution

of the matrix Diophantine equation

(s)A

(s)+Q

(s)B

(s)=

(s)F

(s) (8.414)

8.7 The Youla-Kuˇcera Parametrisation 365

Matrices P

(s)andQ

(s) of the controller right matrix fraction are solution

of the matrix Diophantine equation

(s)P

(s)+B

(s)Q

(s)=O

(s)

(s) (8.415)

Matrices

(s)and

(s) are calculated using auxiliary factorisations of

matricesO

(s)andF

(s). From the practical point of view it is advanta-

geous to use equations (8.393), (8.407) and (8.412) for calculation of the pole

placement controller.

If the singlevariable case is considered i. e. m = r = 1, then

(s)=A

(s)=a(s) (8.416)

(s)=B

(s)=b(s) (8.417)

(s)=o(s) (8.418)

(s)=f(s) (8.419)

and equations (8.414) and (8.415) are replaced by (8.295) that determines

transfer function of the singlevariable pole placement controller.

8.7 The Youla-Kuˇcera Parametrisation

Previous sections of control design presented the optimal feedback control as

the pole placement problem. This section deals with all feedback control laws

that guarantee some desired properties of the closed-loop system.

−

Fig. 8.22. Feedback control system

The control problem shown in Fig. 8.22 consists in ﬁndig such a controller

R(s) for the controlled system G(s) that the closed-loop system satisﬁes de-

sired speciﬁcations. This problem can be divided into two tasks. The ﬁrst is

to stabilise the closed-loop system and the second is to guarantee additional

performance speciﬁcations.

The Youla-Kuˇcera parametrisation employed in the second step uses frac-

tional models for all systems. It is possible to parametrise feedback controllers

366 8 Optimal Process Control

from a set of fractions of stable rational functions. Another use of the Youla-

Kuˇcera parametrisation is in the dual parametrisation when a ﬁxed controller

is known and a set of plants that can be stabilised by it is searched. The

dual parametrisation can be used in identiﬁcation of processes. Finally, the

Youla-Kuˇcera parametrisation can be employed simultaneously for both the

controller and the plant.

The Youla-Kuˇcera parametrisation is based on assumption that a stabilis-

ing controller is known. For its purposes the bounded-input bounded-output

(BIBO) stability deﬁnition is suitable.

The closed-loop system is stable if arbitrary bounded input causes results

in a bounded output at any place of the closed-loop system. A system with

transfer function F (s) is BIBO stable if and only if F (s) is proper and Hurwitz-

stable.

From this deﬁnition follows that all transfer functions of the closed-loop

system have to be stable and proper.

A system is asymptotically stable if its characteristic polynomial is stable.

We will at ﬁrst investigate singlevariable systems and then the problem will

be generalised to multivariable systems. Finally, parametrisation of discrete-

time systems will be treated.

8.7.1 Fractional Representation

Fractional representation of systems (plants, controllers) consists in expressing

transfer functions as a fraction of two stable transfer functions. Consider for

example a controlled process of the form

G(s)=

b(s)

a(s)

(8.420)

where a(s), b(s) are coprime polynomials with deg b(s) ≤ deg a(s), and deg

denotes the degree of a polynomial. This transfer function can be rewritten

into a Hurwitz-stable and proper fractional representation as

G(s)=

B(s)

A(s)

(8.421)

where B(s)andA(s) are stable transfer functions of the form

B(s)=

b(s)

c(s)

,A(s)=

a(s)

c(s)

(8.422)

and c(s) is a monic Hurwitz polynomial, i. e. 1/c(s) is asymptotically stable,

and deg c(s) ≥ deg a(s). A monic polynomial contains unit coeﬃcient at the

maximum power of s.

Consider for example a transfer function

G(s)=

s + b

+ a

s + a

(8.423)

8.7 The Youla-Kuˇcera Parametrisation 367

Its fractional representation can be of the form

G(s)=

s + b

+ c

s + c

+ a

s + a

+ c

s + c

(8.424)

where s

+ c

s + c

is Hurwitz polynomial.

Let F

denotes a set of stable proper rational transfer functions in the

above deﬁned sense. Two transfer functions B(s),A(s) ∈ F

are coprime

if there exist transfer functions X(s),Y(s) ∈ F

satisfying the Diophantine

equation

A(s)X(s)+B(s)Y (s) = 1 (8.425)

Let us now consider the feedback control system shown in Fig. 8.22 and

apply the concept of the BIBO stability. Input variables of the closed-loop

system are w, d

and output variables are u, y. We express the plant and the

controller as ratios of proper stable restional functions. The controlled system

can be written as

G(s)=

B(s)

A(s)

(8.426)

where B(s),A(s) ∈ F

are coprime. The controller transfer function is of the

form

R(s)=

Q(s)

P (s)

(8.427)

where Q(s),P(s) ∈ F

are coprime.

The closed-loop system in Fig. 8.22 is BIBO stable if and only if all transfer

functions between inputs w, d

and outputs u a y are proper and Hurwitz-

stable.

The transfer functions are given as

(s)=

R(s)

1+G(s)R(s)

A(s)Q(s)

A(s)P (s)+B(s)Q(s)

(8.428)

(s)=

G(s)R(s)

1+G(s)R(s)

B(s)Q(s)

A(s)P (s)+B(s)P (s)

(8.429)

(s)=

1+G(s)R(s)

A(s)P (s)

A(s)P (s)+B(s)Q(s)

(8.430)

(s)=

G(s)

1+G(s)R(s)

B(s)P (s)

A(s)P (s)+B(s)Q(s)

(8.431)

From equations



u(s)

y(s)



C(s)



A(s)Q(s) A(s)P (s)

B(s)Q(s) B(s)P (s)



w(s)

(s)



(8.432)

368 8 Optimal Process Control

A(s)P (s)+B(s)Q(s)=C(s) (8.433)

follows that the closed-loop system in Fig. 8.22 is BIBO stable if and only if

1/C(s) is proper and Hurwitz-stable.

8.7.2 Parametrisation of Stabilising Controllers

A controller with transfer function R(s) that guarantees the BIBO closed-loop

stability for a controlled system G(s) is called a stabilising controller for the

given controlled system.

Suppose that the controlled system gives rise to the transfer function

G(s)=

B(s)

A(s)

(8.434)

and B(s),A(s) ∈ F

are coprime. If some controller with Q



(s),P



(s) ∈ F

R(s)=



(s)



(s)

(8.435)

stabilises the controlled system (8.434) then

C(s)=A(s)P



(s)+B(s)Q



(s) (8.436)

and 1/C(s) is proper and Hurwitz-stable.

Equation (8.436) can be divided by a stable factor C(s) and yields

A(s)



(s)

C(s)

+ B(s)



(s)

C(s)

= 1 (8.437)

Introducing the notation

P (s)=



(s)

C(s)

,Q(s)=



(s)

C(s)

(8.438)

follows that P (s), Q(s) are a solution of the equation

A(s)P (s)+B(s)Q(s) = 1 (8.439)

where Q(s),P(s) ∈ F

Stabilising controllers from (8.436) and (8.439) are the same as

R(s)=



(s)



(s)

Q(s)

P (s)

(8.440)

where P (s) = 0. Equation (8.439) is a Diophantine equation that has inﬁnity

of solutions. These solutions can be easily parametrised.

We will now show that all controllers that stabilise the controlled system

are solutions of equation (8.439) of the form

8.7 The Youla-Kuˇcera Parametrisation 369

P (s)=X(s) − B(s)T (s),Q(s)=Y (s)+A(s)T (s) (8.441)

where X(s)andY (s) is a particular solution of the equation and T (s) ∈ F

This follows from the substitution of P (s)andQ(s) from (8.441) into (8.439)

A(s)(X(s) − B(s)T (s)) + B(s)(Y (s)+A(s)T(s)) = 1 (8.442)

A(s)X(s)+B(s)Y (s) = 1 (8.443)

This is a very important result as it gives parametrisation of the set of all

stabilising controllers for the controlled system G(s)oftheform

R(s)=

Y (s)+A(s)T(s)

X(s) − B(s)T (s)

(8.444)

where Y (s),X(s) ∈ F

is arbitrary particular solution of the equation

A(s)X(s)+B(s)Y (s) = 1 (8.445)

and T (s) ∈ F

under the condition that X(s) −B(s)T (s) = 0 holds.

Substracting (8.439) from (8.445) gives

A(s)(X(s) − P(s)) + B(s)(Y (s) − Q(s)) = 0 (8.446)

As A(s), B(s) are coprime, A(s) divides Y (s) −Q(s)andB(s) divides X(s) −

P (s). Denote

Q(s) −Y (s)

A(s)

= T (s) (8.447)

Then from (8.446) follows

P (s)=X(s) − B(s)T (s) (8.448)

Similarly, denote

X(s) − P(s)

B(s)

= T (s) (8.449)

Then from (8.446) follows

Q(s)=Y (s)+A(s)T(s) (8.450)

This shows that any solution of (8.439) is of the form (8.448) and (8.450) for

some T (s) ∈ F

The detailed block diagram of the parametrised controller in the closed-

loop system is shown in Fig. 8.23.

370 8 Optimal Process Control

)(sB )(sA

)(

)(sY

)(sT

−

)(

Fig. 8.23. Block diagram of the parametrised controller

8.7.3 Parametrised Controller in the State-Space Representation

Consider a minimal realisation of the controlled system

G(s)=C(sI −A)

−1

B (8.451)

State feedback controller with observer is then given as

ˆx(t)=(A − BK −LC)ˆx(t)+Ly(t) (8.452)

u(t)=−K ˆx(t) (8.453)

This controller is shown in Fig. 8.24. It is not diﬃcult to show that this

diagram is the same as the controller in Fig. 8.25.

−

Fig. 8.24. State feedback controller with observer