Automatic Control LaboratoryD-ITET

Prof. Dr. Roy Smith Spring Term 2015

CONTROL SYSTEMS II (REGELSYSTEME II)

PROBLEM SET 8Objectives:

• MIMO Poles and Zeros

• RHP-poles and RHP-zeros

• State space representation

• Norms for MIMO systems

Background: Lecture slides, Skogestad book: chapters 3 and 4

Exercise 1 MIMO poles and zeros

Determine the poles and zeros of the following plant:

G(s) =

[s+2

s(s+1)(s−5)0

5(s+2)(s+1)(s−5)

s+1(s+2)(s−5)

]

How many poles does G(s) have?

Exercise 2 Multivariable RHP-zeros

Consider a plant G(s) given in the state space form:

x(t) = Ax(t) +Bu(t) (1)

y(t) = Cx(t) +Du(t), (2)

where the matrices A, B, C and D are given as:

A =

[10 00 −1

], B = I, C =

[10 c110 0

], D =

[0 00 1

]. (3)

a) For what values of c1 does the plant have RHP-zeros?

b) Compute the transfer function G(s) = C(sI −A)−1B +D and show that for c1 = 0, G(s) does not have zeros.

c) In case c1 = 0, is it possible to arbitrarily place system poles by output feedback? Is it possible to place the polesarbitrarily by measuring the states and using state feedback?

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Exercise 3 SVD for MIMO systems

For a closed loop system consider the following complementary sensitivity transfer function matrix:

T3(s) =

[1 − 0.001

s+1

− s+1(s+10)2

1

]

a) Say whether the given closed loop system can have steady state tracking error that is newer bigger than 1% of thereference signal amplitude.

b) Determine the reference signal with amplitude equal to 1 that results in the biggest possible tracking error.

Exercise 4 Sate space representation – poles

Consider the control system shown by the block diagram in Figure 1. For this system do the following:

1s+1

1s+1

1s+1

x1

x2 x3

u1

u2

y1

y2

+

+

Figure 1: Control system block diagram

a) Find the transfer function matrix G(s) from the input u(t) = [u1(t), u2(t)]T to the output y(t) = [y1(t), y2(t)]

T .

b) Find the MIMO poles of the transfer function matrix, G(s). How many poles does the minimal representation ofG(s) have?

c) Taking the state space vector to be x(t) = [x1(t), x2(t), x3(t)]T find the matrices A, B and C of the state space

representation of the system:dx(t)

dt= Ax(t) +Bu(t)

y(t) = Cx(t)

(4)

d) Calculate the controllability matrix for (4) and say whether the system is controllable.

e) Calculate the observability matrix for (4) and say whether the system is observable.

f) Find the poles for the system representation (4). How many poles are there? If the number is different than the resultin b) say what is the cause for this difference.

Exercise 5 System norms for MIMO systems (exam question 2010)

Consider the one degree-of-freedom control loop in Figure 2, where the blocks K(s) and G(s) denote the controller and theplant, respectively. Both K(s) and G(s) are MIMO transfer function matrices.

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-r c -+

- K(s) - c++?

d

u- G(s) -y

6

Figure 2: Block diagram of the system.

The feedback system is physically realizable, if all closed-loop transfer function matrices are proper. This condition is calledthe well-posedness property. One can show that the feedback system in Figure 2 is well-posed if and only if

det(I +G(∞)K(∞)) 6= 0, (5)

where K(∞) = lims→∞K(s) and G(∞) = lims→∞G(s).Assume that the minimal state-space realizations of the system G(s) and the controller K(s) are:

Gs=

[A B

C D

], K

s=

[Ak BkCk Dk

]. (6)

a) Express the well-posedness condition in (5) in terms of the state-space realization matrices in (6).

b) Assume the feedback system in Figure 2 is well-posed and the controller K(s) is a simple proportional gain, namelyK(s) = Dk.

(i) Derive the state-space representation of the closed-loop transfer function matrix from r to y in terms of thestate-space realization matrices of G(s) and K(s).

(ii) Assume that the closed-loop system in Figure 2 is internally stable. Derive the necessary and sufficient condi-tion in terms of the closed-loop state-space realization matrices for finiteness of theH2 norm of the closed-loopsystem from r to y.

c) Assume that the complementary sensitivity function T (s) of the feedback system in Figure 2 is the transfer functionmatrix

T (s) =

[0 1

s+αe−s−e−αs

s0

],

where α > 0 is a controller parameter. For all values of α compute the induced (worst case) 2-norm in the timedomain:

max‖r(t)‖2=1

‖y(t)‖2 = maxr(t)6=0

‖y(t)‖2‖r(t)‖2

.

You may use Figure 3 for your computation.

−25 −20 −15 −10 −5 0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

x

(1−

cos(

x))/

x2

Figure 3: Plot of (1−cos(x))x2 versus x

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