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

Подождите немного. Документ загружается.

8.6 Analysis of State Feedback with Observer and Polynomial Pole Placement 341

u(t)=−Kx(t) (8.260)

where K is a constant matrix. This controller implements proportional ac-

tion and it is not able to guarantee the zero steady-state control error. This

drawback can be circumvented by introducing a state feedback of the form

u(t)=−Kx(t)+ ˜w(t) (8.261)

However, this state feedback control law cannot usually be implemented as

not all states are measurable. A possible remedy is to implement the state

feedback as

u(t)=−K ˆx(t)+ ˜w(t) (8.262)

where ˆx is the deterministic state estimate. The realisation of the feed-

back (8.262), i. e. state feedback with state observer is shown in Fig. 8.13.

It is important to observe that even if the control law of the form (8.262) is

realised and x(t)andˆx(t) are diﬀerent, the stability of the closed-loop system

is still assured. Combined state feedback with state observer can be separated

into two independent designs as the following theorem states.

State observer

Fig. 8.13. State feedback with observer

Theorem 8.2 (Separation principle). Consider state the feedback control

with observer for a completely controllable and observable system (8.211),

(8.212), with observer (8.213) and state feedback (8.262). Closed-loop eigen-

values consist of eigenvalues of matrix A − BK (state feedback without ob-

server) and of eigenvalues of matrix A − LC (observer dynamics).

Proof. Substituting (8.262) and (8.215) into (8.211) yields

342 8 Optimal Process Control

˙x(t)=(A −BK) x(t)+BKe(t)+B ˜w(t), x(0) = x

(8.263)

Equations (8.263), (8.219) give the complete closed-loop dynamics



˙x(t)

˙e(t)





A −BK BK

0 A − CL



x(t)

e(t)







˜w(t), (8.264)



x(0)

e(0)







y(t)=



C 0





x(t)

e(t)



(8.265)

Eigenvalues of this upper triangular matrix are given by the eigenvalues at

the main diagonal. As the closed-loop dynamics is governed by such a matrix,

the proof is complete. 

We note that the estimation error is not reachable from the reference input

˜w.

The separation principle thus makes it possible to design the state feedback

independently on the fact whether all states are measurable or not. If the

controlled system is fully controllable, eigenvalues of the closed-loop system

can be placed arbitrarily. We illustrate this fact on a singlevariable system.

Consider a singlevariable fully controllable system

˙x(t)=Ax(t)+Bu(t), x(0) = x

(8.266)

y(t)=Cx(t) (8.267)

It can be transformed using

x = Tx

(8.268)

to the controllable canonical form

˙x

(t)=A

(t)+B

u(t) (8.269)

y(t)=C

(t) (8.270)

where

= T

−1

AT (8.271)

= T

−1

B (8.272)

= CT (8.273)

Matrix T is non-singular and we will not go into details of its construction

here. x

is a new state vector with dimension n. Matrices A

, B

, C

the controllable canonical form are

8.6 Analysis of State Feedback with Observer and Polynomial Pole Placement 343

⎛

⎜

⎝

010... 0

001... 0

000··· 1

−a

··· −a

n−1

⎞

⎟

⎠

(8.274)

⎛

⎜

⎝

⎞

⎟

⎠

(8.275)



··· b

n−1



(8.276)

The state feedback applied to the system (8.266)

u(t)=−K

−1

x(t)+ ˜w(t) (8.277)

corresponds to the state feedback (cf. (8.268))

u(t)=−K

(t)+ ˜w(t) (8.278)



··· k



(8.279)

and the state feedback (8.278) is applied to the system (8.269) then the closed-

loop state representation is given by

˙x

(t)=

⎛

⎜

⎝

01··· 0

00··· 1

−a

− k

−a

− k

··· −a

n−1

− k

⎞

⎟

⎠

(t)+

⎛

⎜

⎝

⎞

⎟

⎠

˜w(t)

(8.280)

The characteristic polynomial of the controlled system is

(s) = det (sI −A

)=s

+ a

n−1

+ ···+ a

s + a

(8.281)

and the characteristic polynomial of the closed-loop system is given as

Rcl

(s) = det (sI −A

Rcl

) (8.282)

where

344 8 Optimal Process Control

Rcl

⎛

⎜

⎝

010··· 0

001... 0

000··· 1

−a

− k

−a

− k

−a

− k

··· −a

n−1

− k

⎞

⎟

⎠

(8.283)

and after a manipulation

Rcl

(s)=s

+ a

Rn−1

n−1

+ ···a

s + a

(8.284)

where

= a

+ k

= a

+ k

Rn−1

= a

n−1

+ k

(8.285)

The last row of the matrix A

contains coeﬃcients of the characteristic

polynomial of the closed-loop system c

Rcl

(s). Therefore, the choice of elements

of matrix K

places arbitrarily eigenvalues of the closed-loop system.

If a

, a

,..., a

Rn−1

are prescribed then the feedback is given as



··· a

Rn−1



−



··· a

n−1



(8.286)

We can notice that parameters of a state feedback controller applied to

a system in controllable canonical form are given as the diﬀerence between

desired coeﬃcients of the closed-loop characteristic polynomial and coeﬃcients

of the characteristic polynomial of the controlled system.

Of course, the same holds for the closed-loop system consisting of the

original system (8.266) and feedback controller (8.277) where

K = K

−1

(8.287)

Therefore, the ﬁrst step in combined state feedback control with observer

is to determine matrix K. This speciﬁes eigenvalues of the closed-loop system

A −BK.

The second step is to design an observer. The design consists in placing its

eigenvalues provided the controlled system is fully observable. The eigenvalues

of the observer are eigenvalues of matrix A − LC and are chosen so that the

estimation error e(t) is removed as fast as possible. Consequence of this is high

gain matrix L implying that noise ξ(t) has a great inﬂuence on the estimation

error e(t). This can be seen in

˙e(t)=(A − LC) e(t) −Lξ(t) (8.288)

where

ξ(t)=y(t) − Cx(t) (8.289)

8.6 Analysis of State Feedback with Observer and Polynomial Pole Placement 345

8.6.2 Input-Output Interpretation of State Feedback

with Observer

We analyse the same problem as before using the input-output representation

of the controller and the controlled process. Mainly singlevariable systems

will be investigated but some results will also be presented for multivariable

systems.

Polynomial interpretation of the state feedback with observer is governed

in the following theorem.

Theorem 8.3 (Pole placement design). Consider a state feedback

u(t)=−K ˆx(t)+ ˜w(t) (8.290)

and an observer

ˆx(t)=Aˆx(t)+Bu + L (y − C ˆx) (8.291)

with a completely reachable and observable system

˙x(t)=Ax(t)+Bu (8.292)

y(t)=Cx(t) (8.293)

The state feedback (8.290) with observer can be written as

p(s)

o(s)

u = −

q(s)

o(s)

y +˜w (8.294)

where p(s) is a polynomial with degree n and q(s) is a polynomial with degree

n −1. Polynomials p(s) and q(s) satisfy the equation

a(s)p(s)+b(s)q(s)=o(s)f(s) (8.295)

and a(s), b(s) are deﬁned as

C (sI − A)

−1

B =

b(s)

a(s)

(8.296)

f(s) is the characteristic polynomial of the matrix A −BK and o(s) is char-

acteristic polynomial of the matrix A −LC.

Proof. To prove the theorem we make use of the following properties of poly-

nomials.

a(s) is the characteristic polynomial of matrix A. If the controlled system

is reachable and observable then polynomials a(s)andb(s) are coprime. This

is an important condition related to the solution of equation (8.295).

If the controlled system is reachable then a(s)andB

(s) given by

346 8 Optimal Process Control

(sI −A)

−1

B =

(s)

a(s)

(8.297)

are right coprime. Equation (8.297) can be written as

(sI −A)B

(s)=Ba(s) (8.298)

Adding BKB

(s) to both sides of equation gives

(sI −A) B

(s)+BKB

(s)=Ba(s)+BKB

(s) (8.299)

(sI −(A − BK)) B

(s)=B (a(s)+KB

(s)) (8.300)

(s)

a(s)+KB

(s)

adj (sI − (A − BK))

det (sI − (A − BK))

B (8.301)

From (8.301) follows that the feedback gain can be calculated from equation

a(s)+KB

(s)=f(s) (8.302)

where f(s) is the characteristic polynomial of the closed-loop matrix A−BK.

From stability of the closed-loop system follows that the polynomial a(s)+

(s) is stable as well.

The observer equations can be written as

sˆx =(A −LC) ˆx + Bu + Ly (8.303)

(sI −(A − LC)) ˆx = Bu + Ly (8.304)

ˆx =(sI −(A − LC))

−1

Bu +(sI −(A − LC))

−1

Ly (8.305)

ˆx =

(s)

o(s)

u +

(s)

o(s)

y (8.306)

where

(s)=adj(sI −(A −LC)) B (8.307)

(s)=adj(sI −(A −LC)) L (8.308)

o(s) = det (sI −(A −LC)) (8.309)

Equation (8.306) demonstrates that the observer can be deﬁned as in

Fig. 8.14. If the output of the system in Fig. 8.14 would be the variable

˜u = K ˆx then

K ˆx =

r(s)

o(s)

u +

q(s)

o(s)

y (8.310)

where

r(s)=KT

(s) (8.311)

8.6 Analysis of State Feedback with Observer and Polynomial Pole Placement 347

()

Fig. 8.14. Block diagram of state estimation of a singlevariable system

()

Fig. 8.15. Polynomial state estimation

()

()()

1−

Fig. 8.16. Interpretation of a pseudo-state

q(s)=KT

(s) (8.312)

Equation (8.310) is shown in the block diagram in Fig. 8.15.

If we deﬁne the pseudo-state as in Fig. 8.16 then

y = b(s)x

(8.313)

u = a(s)x

(8.314)

x = B

(s)x

(8.315)

Kx = k(s)x

(8.316)

where

k(s)=KB

(s) (8.317)

Substitution of (8.313) and (8.314) into (8.310) and using (8.316) gives

348 8 Optimal Process Control

o(s)˜u −o(s)Kx = r(s)a(s)x

+ q(s)b(s)x

− o(s)k(s)x

(8.318)

o(s)(˜u − Kx)=(r(s)a(s)+q(s)b(s) −o(s)k(s)) x

(8.319)

Asymptotic observer is described by (8.310) if and only if o(s) is a stable

polynomial and r(s), q(s) satisfy equation

r(s)a(s)+q(s)b(s)=o(s)k(s) (8.320)

From (8.309) follows that the degree of the polynomial o(s)isn. This

corresponds to a full order observer. If the state estimation has been realised by

a reduced order observer, polynomial o(s) would be a stable monic polynomial

with degree n −1.

Let us return to the feedback (8.290). If K ˆx from (8.310) is substituted

into (8.290) then

u = −

r(s)

o(s)

u −

q(s)

o(s)

y +˜w (8.321)

p(s)=o(s)+r(s) (8.322)

then equation (8.321) reduces to (8.294).

Adding a(s)o(s) to either side of equation (8.320) gives

r(s)a(s)+q(s)b(s)+a(s)o(s)=o(s)k(s)+a(s)o(s) (8.323)

a(s)(r(s)+o(s)) + q(s)b(s)=o(s)(k(s)+a(

s)) (8.324)

Taking into account equations (8.302), (8.317), (8.322) then (8.324) is of the

form (8.295). This concludes the proof. 

()

Fig. 8.17. Realisation of a state feedback with an observer

Pole placement theorem thus shows that the state feedback with observer

can be implemented as shown in Fig. 8.17. Transfer function between input

˜w and output y is given as

y =

b(s)o(s)

a(s)p(s)+b(s)q(s)

˜w (8.325)

8.6 Analysis of State Feedback with Observer and Polynomial Pole Placement 349

uBAxx +=

Fig. 8.18. State feedback control

y =

b(s)

f(s)

˜w (8.326)

Based on Fig. 8.18 this further gives

y = C (sI −(A −BK))

−1

B ˜w (8.327)

y = C

adj (sI − (A − BK))

det (sI − (A −BK))

B ˜w (8.328)

From (8.301) follows

(s)=adj(sI −(A −BK)) B (8.329)

and from (8.296), (8.297), (8.301), (8.302) implies

(s)=b(s) (8.330)

det (sI − (A − BK)) = f(s) (8.331)

The relation between the closed-loop input ˜w and output y in ﬁgures 8.17 and

8.18 is thus the same. Hence, the pole placement design is entirely the same in

both state-space and input-output representations. However, the closed-loop

structures in ﬁgures 8.17, 8.18 are diﬀerent.

From (8.307) and (8.308) follows that T

(s)andT

(s)aren-dimensional

vectors with elements being polynomials with degree n −1. As o(s) is a poly-

nomial with degree n, the polynomial form of the observer consists of strictly

proper transfer functions. The right-hand side polynomial in equation (8.295)

has degree 2n.

The fact that derivation of (8.324) involves cancellation of o(s) implies

that estimated states are not controllable by the input ˜w.

The realisation of state feedback with observer in Fig. 8.17 takes the form

of a closed-loop system with the two-degree-of-freedom controller.

If the closed-loop input in Fig. 8.17 is given as

˜w =

q(s)

o(s)

w (8.332)

then (8.294) can be written as

u = −

q(s)

p(s)

y +

q(s)

p(s)

w (8.333)

350 8 Optimal Process Control

u = −

q(s)

p(s)

(w − y) (8.334)

The choice of ˜w from (8.332) changes the closed-loop system in Fig. 8.17 from

a two-degree-of-freedom to an one-degree-of-freedom system (Fig. 8.19).

The transfer function between w and y is given as

y =

b(s)q(s)

o(s)f(s)

w (8.335)

()

Fig. 8.19. Feedback control with one-degree-of-freedom controller

The design procedure of the polynomial pole placement controller as an

alternative to the state feedback controller with observer is as follows:

1. Given are state-space matrices of the controlled process A, B, C. Gain

matrices K and L are also given.

2. Polynomials a(s)andb(s) are calculated from

C (sI − A)

−1

B =

b(s)

a(s)

(8.336)

3. Controller polynomials p(s)andq(s) are found from the equation

a(s)p(s)+b(s)q(s)=o(s)f(s) (8.337)

where

o(s) = det (sI −(A −LC)) (8.338)

f(s) = det (sI − (A −BK)) (8.339)

are stable polynomials. Polynomial c(s)=o(s)f (s) has the degree 2n (full

order observer) or 2n −1 (reduced order observer).

Example 8.4:

www

Consider a controlled process of the form

G(s)=

s +6

+5s +6