Linear System Homework 2 Due: Oct. 13, 2010 ( in class)

Problem 1 Consider the following linear system

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

where and 0 14 2

A ⎡ ⎤= ⎢ ⎥− −⎣ ⎦

01

B ⎡ ⎤= ⎢ ⎥⎣ ⎦

1. Find the eigenvalues and eigenvectors of the system matrix A (you may use Matlab). Is the system stable ? Explain.

2. Find the modal matrix Am = T −1AT , where columns of the similarity transformation

matrix T are the eigenvectors of the matrix A. Is the modal matrix unique? Explain. 3. Find the exponential matrix eAt using four different methods:

(a) Cayley Hamilton theorem (Finite series representation). (b) Resolvent matrix (Inverse Laplace transform of (sI −A)−1) (c) Modal transformation T where eAt = TeΛtT −1 . (d) Use Maple or Mathematica (if you have accessed and familiar with one of these

tools, note: this part is optional)

4. Find the limit of the exponential matrix eAt as t →∞.

5. Solve analytically for the time responses x(t)to initial conditions x(0) = and no

applied input u(t)=0 for all t ≥0. Plot your time responses.

03⎡ ⎤⎢ ⎥⎣ ⎦

6. (a) Solve analytically for the time responses x(t) to zero initial conditions

x(0) = and with input u(t)=7 for all t ≥0. What are the steady-state values of x(t);

i.e. lim

00⎡ ⎤⎢ ⎥⎣ ⎦

t→∞ x(t)? (see hint below) (b) Find the steady-values using the state-space model given in equation (1) without solving for the time responses?

7. Solve analytically for the time responses x(t) to initial conditions in Part (5) and at the

same time with the applied input in Part (6). Plot your time responses.

Hint: When you solve this part, just try to get an integral form like this:

0

7 sin 33( )

17 (cos 3 sin 3 )3

v

t

v

e vx t d

e v v

−

−

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥−⎢ ⎥⎣ ⎦

∫ v

Once you get the expression of x(t) above, you can use the result as follows directly

7 7 (3cos 3 3 sin 3 )4 12( )

7 sin 33

t

t

e tx t

e t

−

−

⎡ ⎤− +⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

t

Problem 2

Given the following matrix 1 2 32 2 43 2 5

A⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

1. (a) Find a set of orthonormal basis vectors for the range of A. Hint: Use Gram-Schmidt orthogonalization. (b) Can you find a solution x ∈ R3 such that Ax= b where

(i) (ii) 9

1215

b⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

201

b⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

If yes, is the solution unique? If not, explain why.

2. Find a set of orthonormal basis vectors for the null space of A. 3. Use MATLAB to determine the singular value decomposition of A. 4. Using the results from the singular value decomposition in Part 3, answer the following

questions: (a) What are the singular values (σ1,σ2,σ3)of A? Note that σ1≥

σ2≥

σ3.

(b) What is the rank of A? Verify your result using MATLAB rank(A). (c) What is the right singular vector associated with the singular value σ3? Show the

connection of the right singular vector with the null space of A in Part 2. (d) What are the left singular vectors corresponding to the non-zero singular values of

A? Show the connection of the left singular vectors with the range space of A in Part 1.

Problem 3

In this problem, we provide a procedure whereby a singular value decompostion of A can be computed using a method based solely on eigenvalues and eigenvectors. As we know, any real n×m matrix A can always be written according to SVD in the form

A= U Σ VT

(2)

where U and V are unitary matrices and assuming without loss of generality n≥m

1

2

0 00 0

0 0

0

m

σσ

σ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

Σ = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

then ( ) ( ) ( )T T T T T TA A U V U V V V= Σ Σ = Σ Σ (3) where

21

22

2

0 00

0 0

T

m

σσ 0

σ

⎡ ⎤⎢ ⎥⎢Σ Σ =⎢⎢ ⎥⎢ ⎥⎣ ⎦

⎥⎥

mm

0

(m×m diagonal matrix) (4)

with diagonal elements . In other words, the unitary matrix V is a similarity transformation that diagonalizes the matrix A

2 ( 1, 2, )i iσ =TA and are the

eigenvalues of A

2 ( 1, 2, )i iσ =TA.

Namely, we have

VT

(AT

A) V =

21

22

2

0 00

0 0

T

m

σσ

σ

⎡ ⎤⎢ ⎥⎢Σ Σ =⎢⎢ ⎥⎢ ⎥⎣ ⎦

⎥⎥

(5)

On the other hand, AAT

= (U Σ VT

)(U Σ VT

)T= U( ΣΣT

)U T

(6)

21

22

T 2

0 0 00 0 0

0 0 0 00 0 0 0 0

0 0 0 0 0

m

σσ

σ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

ΣΣ = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

00

m

m

00

(n×n diagonal matrix) (7)

with diagonal elements and (n-m) zeros. In other words, the unitary matrix U is a similarity transformation that diagonalizes the matrix AA

2 ( 1, 2, )i iσ =T and

along with (n-m) zeros are the eigenvalues of AA2 ( 1, 2, )i iσ = T. Namely, we have

UT

(AAT) U = (8)

21

22

T 2

0 0 00 0 0

0 0 0 00 0 0 0 0

0 0 0 0 0

m

σσ

σ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

ΣΣ = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

　Let’s apply the above findings to the simple problem below

Given the following matrix

1 42 53 6

A⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

1. Find the eigenvalues and eigenvectors of the matrix ATA using Matlab. The eigenvector matrix of ATA forms the unitary matrix V in the singular value decomposition of A according to equation (5).

2. Find the eigenvalues and eigenvectors of AAT using MATLAB. The eigenvector matrix of AAT forms the unitary matrix U in the singular value decomposition of A according to equation (8).

3. Finally, determine the singular values σi (i =1,2,…r≤ m)of the matrix A from the square root of the eigenvalues of A

T A.

4. Compare the results obtained from the Matlab command svd(A) to the unitary matrices U and V obtained in Parts (1) and (2). The LESSON in this problem is that SINGULAR VALUE DECOMPOSITION can be solved as an EIGENVALUE problem. And in general, singular value decomposition CANNOT be done by hand (except possibly for systems of order less than 3); just like in the EIGENVALUE problem.

Linear System Homework 3 Due: Oct. 29, 2010 ( by 5:00pm)

Problem 1. Consider the nonlinear state equations (which were studied extensively in class)

1 2 1 22 2

2 1 1 2

( ) ( ) 2 ( ) ( )( ) ( ) ( ) ( ) ( )

x t x t x t x tx t x t x t x t u t

−⎡ ⎤ ⎡=⎢ ⎥ ⎢− + + +⎣ ⎦ ⎣

⎤⎥⎦

and the nominal condition defined by u0(t)=0 and 10eqx ⎡ ⎤

= ⎢ ⎥⎣ ⎦

1. Find A and B of the linearized state equations evaluated at the above given nominal

condition, where ( ) ( ) ( )x t A x t B u tδ δ δ= +

Is the system stable?

2. Find the perturbed state responses δx(t) when δu(t)=0 and 0.01

(0)0.01

xδ ⎡ ⎤= ⎢ ⎥⎣ ⎦

3. Find the perturbed state responses δx(t) when δu(t)=0.01sint and 0

(0)0

xδ ⎡ ⎤= ⎢ ⎥⎣ ⎦

4. Under the conditions of Part 2, solve the full nonlinear state equations for x(t) (using numerical integration) for 0≤t≤

20 sec. Compare graphically the results to those obtained

from the linearized state equations x(t)≈ xeq + δx(t). Is the linear state model representative of the nonlinear system? (note: see supplementary reading for writing your own function to solve the nonlinear equation) 5. Under the conditions of Part 3, solve the full nonlinear state equations for x(t) (using numerical integration) for 0 ≤t≤ 20 sec. Compare graphically the results to those obtained from the linearized state equations x(t) ≈

xeq + δx(t). Is the linear state model

representative of the nonlinear system? (note: see supplementary reading for writing your own function to solve the nonlinear equation) 6. Is the linearized system controllable?

Problem 2.

Given the following linear time-invariant system,

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

with initial conditions and where 1

(0)1

x ⎡ ⎤= ⎢ ⎥⎣ ⎦

1 1 0,

0 2 1A B

−⎡ ⎤ ⎡= =

⎤⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦

1. Is the system controllable? Explain.

2. Find a control input u(t) that bring the system states to 1

( )2fx t ⎡ ⎤

= ⎢ ⎥⎣ ⎦

where tf =10 sec.

Confirm your results with simulation of the responses ( )x t for 0 ≤ t≤ 11 sec.

3. Repeat Part 2 where tf =1 sec. Confirm your results with simulation of the responses

( )x t for 0≤ t≤ 11 sec. How does this control input u(t) compare with the previous control

obtained in Part 2? (i.e., higher or smaller control input). 4. Find a totally different solution to the control input u(t) that solves Part 2. That is, the solution u(t)in Part 2 is NOT unique. For example, in digital control let

1 1

2 1

0( )

10u for t t

u tu for t t

≤ ≤⎧= ⎨ ≤ ≤⎩

where u1, u2 and t1 are constant parameters (to be determined) that provide another control input u(t) for the solution of Part 2.

Problem 3

Consider the following linear time-invariant system,

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

where

　

1 2 3 10 2 0 , 10 0 3 0

A B⎡ ⎤ ⎡⎢ ⎥ ⎢= =⎢ ⎥ ⎢⎢ ⎥ ⎢⎣ ⎦ ⎣

⎤⎥− ⎥⎥⎦

1. Is the system controllable? Explain. 2. Is the system stabilizable? Explain. 3. Identify the modes that are uncontrollable. 4. Find a set of orthonormal basis vectors for the controllable subspace. 5. Find a set of basis vectors for the uncontrollable subspace. Are these vectors

orthogonal to the controllable subspace? 6. Determine a state transformation P that separates the system into controllable ( )cx t and uncontrollable ( )cx t subspaces where

( )

( )( )

c

c

x tx t P

x t⎡ ⎤

= ⎢ ⎥⎣ ⎦

7. Find the state model of the system transformed by the similarity transformation P obtained in Part 6. Use this state model to determine which modes in the system are controllable and uncontrollable.

HW #4 Due: Nov. 12, 2010 by 5:00pm Problem 1

Given a linear time-invariant system for the three-mass-spring system described in Example 3 of Section 5.5.3,

( ) ( ) ( )( ) ( ) ( )

x t Ax t Bu ty t Cx t Du t

= += +

with initial conditions , and (0) 0x =

[ ]

0 1 0 0 0 0 01 0 1 0 0 0 1

0 0 0 1 0 0 0, , 0 0 1 0 0 0 , 0

1 0 2 0 1 0 10 0 0 0 0 1 00 0 1 0 1 0 1

A B C

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥

= = =⎢ ⎥ ⎢ ⎥− −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥

−⎣ ⎦ ⎣ ⎦

D =

Note that in this problem we have equal forces u(t) applied simultaneously to masses m1

and m3, and applied on mass m2 in the opposite directions. The measured output y(t)is the position of mass m2.

1. Is the system controllable? Explain. 2. Identify the modes that are controllable and uncontrollable. Provide a physical meaning to each of the controllable modes (if any). 3. Can you find a control input u(t) that bring the system initial states to the following final states

( )fx t ?

(a) (b)

111

( )=1

11

fx t

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

030

( )=2

02

fx t

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(c)

122

( )=2

11

fx t

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

　 　 　

where tf =10 sec. For the case where you can, find the corresponding control u(t) and show the system responses ( )x t for 0≤ t ≤

10 sec.

4. Can you find a similarity transformation that places the system into the controllable form? If no, explain why not.

5. Can you find a similarity transformation that places the system into the controller form ? If no,

explain why not. 6. Is the system observable? Explain. 7. Identify the modes that are observable and unobservable. Provide a physical meaning to the unobservable modes (if any). 8. Can you find a similarity transformation that places the system into the observable form?.If no,

explain why not. 9. Can you find a similarity transformation that places the system into the observer form? If no,

explain why not.

Problem 2

Recall that all the units in Problem 1 are respectively m and m/sec for the mass positions and velocities, and the applied force u(t) in N. For some odd reasons, your high-level manager wants you to present the state-space model of the three-mass-spring system given in Problem 1 to the company executives in the following units for the states x(t), input u(t) and output y(t):

• x1(t)in units of feet, x2(t)in units of feet/min.

•x3(t)in units of inches, x4(t)in units of inches/sec.

• x5(t)in units of miles, x6(t)in units of miles/hr.

u(t)in units of lbs, • y(t)in units of inches.

1. What are your new state model matrices ? , , ,A B C D2. What are the system eigenvalues of your new state model? Are they the same as the original model? 3. Are any of the system controllability properties changed? If yes, explain. 4. Are any of the system observability properties changed? If yes, explain.

HW #5 Due: Nov. 26, 2010 by 5:00pm Problem 1 Find the orthonormal basis vectors “by hand” for the range space of A and the null space of AT for the following matrices:

(a) (b) 1 22 44 8

A⎡ ⎤⎢= ⎢⎢ ⎥⎣ ⎦

⎥⎥

1 2 11 2 3

A ⎡ ⎤= ⎢ ⎥⎣ ⎦

(c) 1 0 2 41 1 3 60 1 1 2

A⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

In each case, 1. Show that the two subspaces Range(A) and Null(AT ) are complementary subspaces (i.e., they are non-overlapping, the basis vectors in one subspace are orthogonal to the basis vectors in the other subspace, and the two subspaces combined would span the whole space Rn where n is the row dimension of A).

2. Verify your answers with Matlab using the commands orth(A) and null(A’).If they are not the same, are your answers still correct ? Explain.

Problem 2

Given the following state model,

0 0 0( ) 0 1 0 ( )

0 0 2x t x

⎡ ⎤⎢ ⎥= −⎢ ⎥⎢ ⎥−⎣ ⎦

t

where x(0)=x0 and the output vector

1 0 1

( ) ( )0 1 0

y t x⎡ ⎤= ⎢ ⎥−⎣ ⎦

t

⎥

1. Is the system observable?

2. From the measurement of the output vector y(t) over the time interval t∈ [0, 10], we

determine that , find x(0).

3. Using your results in Part 2, solve for x(t) for 0≤ t ≤

25( )

3

t

t

ey t

e

−

−

⎡ ⎤+= ⎢ −⎣ ⎦

10.

1/2

Problem 3 Given the linear time-invariant system of Example 4 in Section 5.5.4 of the notes,

( ) ( ) ( )( ) ( ) ( )

x t Ax t Bu ty t Cx t Du t

= += +

with initial conditions x(0) = x0 (unknown) and where

0 1 0 0 0 0 01 0 1 0 0 0 0

0 0 0 1 0 0 0,

1 0 2 0 1 0 00 0 0 0 0 1 00 0 1 0 1 0 1

[0 0 0 0 1 0], 0

A B

C D

⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥

= =⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥

−⎣ ⎦

= =

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Note that in this problem the applied force u(t)ison mass m3. The measured output y(t)is the position of mass m3 1. Let u(t)= 0 and the time history of the output y(t)is provided in the data file

yhw5a.mat (which you need to download from course website) for for 0≤ t ≤ 10.

Can you use the given data of the output y(t) to determine the system initial conditions x(0)? If yes, then find x(0). Check by simulation that your estimation of the initial states x(0) is correct.

2. How would your procedure be different if u(t)is nonzero? Let u(t)=1 and the corresponding time history of the output y(t)is provided in the data file yhw5b.mat (which you need to download from course-website) for 0 ≤ t ≤ 10. Can you use the given data of the output y(t)to determine the system initial conditions x(0)? If yes, then find x(0). Check by simulation that your estimation of the initial states x(0) is correct.

Note: 1. The data y(t) given in yhw5a.mat and yhw5b.mat were measured with Δt=0.1

sec.

2. The provided data yhw5a.mat and yhw5b.mat were generated with x(0)=[-1 -2 0 0 1 2]T; therefore, check if your solution of x(0) is correct (or close to that).

2/2

Linear System Homework #6 Due: Dec. 08, 2010 (in the class)

Problem 1

A linear state model with input u(t) and output y(t)is given below.

1 1 1

2 2 2

3 3 3

( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( )

x t x t x tx t A x t Bu t and y t C x tx t x t x

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= + =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦t

3

where

. [ ]12 23 1 13 6 2 , 9 , 1 21 2 1 4

A B C− −⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= = − =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦

. 1. Transform the above linear state model into the following canonical form:

.(a) Controller form

.(b) Observer form

. 2. Let’s consider a state-feedback controlled system where the control is of the form

[ ]1

1 2 3 2

3

( )( ) ( ) ( )

( )

x tu t Kx t K K K x t

x t

⎡ ⎤⎢ ⎥= = ⎢ ⎥⎢ ⎥⎣ ⎦

Find the gain matrix K such that the closed-loop system eigenvalues are at {-2, -3, -4}; i.e. the eigenvalues of the system matrix (A + BK). Hint: Use the controller form.

3. Let’s consider an observer design using the output y(t) as measurement. The estimator has the

following state model description

ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ( ) ( )) ( ) ( )x t Ax t Bu t L y t y t and y t Cx t= + + − =

with , where L is the gain matrix in the observer design to be selected so that the error in the state estimates decays asymptotically to zero.

ˆ(0) 0x =ˆ( ) ( ) ( )e t x t x t= −

.(a) Find the gain matrix L such that the closed-loop observer eigenvalues are stable and at {-3, -4, -5}; i.e., the eigenvalues of the system matrix (A-

LC).

Hint: Use the observer form.

.(b) Simulate responses ˆ( )x t of your estimator design using the input u(t)= sin(t) and

. Plot the response x1

(0) 23

x⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

i(t) and ˆ ( )ix t together (i=1,2,3) (note: you can use

“lsim()” function in Matlab)

.

ˆ( )x t

System Model Figure 1: Plant System and Estimator Models

Problem 2

Consider the following linear time-invariant system

[ ]

0 1 0 0 0 0 01 0 1 0 0 0 1

0 0 0 1 0 0 0( ) ( ) ( )

1 0 2 0 1 0 00 0 0 0 0 1 00 0 1 0 1 0 1

( ) 1 0 0 0 1 0 ( )

x t x

y t x t

⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥

= +⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥

−⎣ ⎦

=

t u t

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Physically this problem is for a three-mass-spring system (Example 3 in Section 5.5.3) where the input u(t) is a force applied to the first and third masses and the output y(t)is the sum of the first and third mass positions.

(1) Find a set of orthonormal basis vectors for the controllable subspace. Check whether the controllable subspace is A-invariant.

(2) Find a set of orthonormal basis vectors for the uncontrollable subspace which is also orthogonal to the controllable subspace. (a) Check whether the uncontrollable subspace obtained above is A-invariant. (b) Find an uncontrollable subspace that is A-invariant. What is the benefit (if any) of having an uncontrollable subspace that is also A-invariant?

(3) Using results of Parts (1) and (2), find a similarity transformation P that places the

system into the following controllable/uncontrollable form

11 12 1

22

1 2

( )( )( )

( )( ) 0 0

( )( ) ( )

( )

cc

cc

c

c

x tx t A A Bu t

x tx t A

x ty t C C Du t

x t

⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤= +⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥

⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

⎡ ⎤⎡ ⎤= +⎢ ⎥⎣ ⎦ ⎣ ⎦

Verify that the pair 11 1( , )A B is controllable.

(4) From results of Part (3), determine the controllable and uncontrollable modes.

(5) Find a set of orthonormal basis vectors for the unobservable subspace. Check whether the unobservable subspace is A-invariant.

(6) Find a set of orthonormal basis vectors for the observable subspace which is also

orthogonal to the unobservable subspace. .(a) Check whether the observable subspace obtained above is A-invariant. .(b) Find an observable subspace that is A-invariant. What is the benefit (if any) of

having an observable subspace that is also A-invariant? .

(7) Using results of Parts (5) and (6), find a similarity transformation Q that places the system into the following observable/unobservable form

111

221 22

1

( )( ) 0( )

( )( )

( )( ) 0

( )

oo

oo

o

o

x tx t BAu t

x tx t BA A

x ty t C

x t

⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤= +⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥

⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

⎡ ⎤⎡ ⎤= ⎢ ⎥⎣ ⎦

⎣ ⎦

Verify that the pair 11 1( ,CA ) is observable.

(8) From results of Part (7), determine the observable and unobservable modes. (9) Using the results in Parts (1)-(8), determine a similarity transformation T that places

the original system into a Kalman canonical form.

Linear System Homework #7 Due: Dec. 17 (Fri.), 2010 (by 5:00pm)

Problem 1

Consider the linearized equations

[ ]

0 1 0 0 02 0 0 2 1

( ) ( )0 0 0 3 00 2 0 0 0

( ) 0 1 0 0 ( )

x t x t

y t x t

⎡ ⎤⎢ ⎥⎢ ⎥= +⎢ ⎥⎢ ⎥−⎣ ⎦

=

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

)

　

(1) Find a set of orthonormal basis vectors for the uncontrollable subspace which is also orthogonal to the controllable subspace.

(2) Is the uncontrollable subspace obtained above A-invariant ? (3) Find a set of orthonormal basis vectors for the observable subspace which is also orthogonal to the

unobservable subspace. (4) Is the observable subspace obtained above A-invariant ? (5) Transform this system into a Kalman canonical form. You may use the m-file canon_struc.m.

Make sure that you identify the elements of the state vector. (6) Find the SISO transfer function matrix G(s) for the minimal system. What is the order of this

minimal system ? Problem 2

Consider the discrete-time system

( 1) ( ) (( ) ( )

x k Ax k Buy k C k

+ = +=

k

1

where [ ]2 1 1, , 1

1 1 2A B C⎡ ⎤ ⎡ ⎤= = =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

1. Is the system controllable ?

2. If the initial condition is 1

( 0)2

x k ⎡ ⎤= = ⎢ ⎥

⎣ ⎦ , find the control sequence u(k =0) and u(k =1)

required to transfer the initial state x(k=0) to the origin; i.e. 0

( 2)0

x k ⎡ ⎤= = ⎢ ⎥

⎣ ⎦ .

3. Is the system observable ? 4. Given the observation sequence y(k =1)= 8 and y(k =2)= 14 where the input sequence

is u(0) = −4 and u(1) = 2, find the initial state x(k=0).

Problem 3

Given the following MIMO transfer function

2

2

211

3 2( )1 11 2

( 2) 1

ss sG s

s ss ss s

⎡ ⎤⎢ ⎥+⎢ ⎥

−⎢ ⎥= ⎢ ⎥+ +⎢ ⎥− +⎢ ⎥

⎢ ⎥+ +⎣ ⎦

(1) Find a coprime right-polynomial matrix fraction description

(2) Find a observer-form realization of the minimal system

Linear System Homework #8 (last homework) Due: Dec. 29 (Wen.), 2010 by 5:00pm Problem 1 Given the following MIMO transfer function

312( )

2 12 1

sG ss s

s s

⎡ ⎤⎢ ⎥+= ⎢ ⎥

−⎢ ⎥⎢ ⎥+ +⎣ ⎦

(1) A right-polynomial matrix fraction description of is 1( ) ( ) ( )R RG s N s D s−=

The degree of is 4>3 (order of minimal realization of G(s)). Hence, N

1( 1)( 2) 3( 1) ( 1)( 2) 0( )

2 ( 1) ( 1)( 2) 0 ( 1)( 2)s s s s s

G ss s s s s s

−+ + + + +⎡ ⎤ ⎡= ⎢ ⎥ ⎢+ − + + +⎣ ⎦ ⎣

⎤⎥⎦

2

D

2det( ( )) ( 1) ( 2)RD s s s= + +

R(s) and DR(s) are NOT coprime. Find a right-coprime matrix fraction description of

1( ) ( ) ( )R RG s N s D s−=

(2) Using results from Part (1), find a minimal state-space realization (A,B,C,D)of G(s)in controller form.

(3) Find the Smith-McMillan form of G(s). (4) Using results of Part (3), find the system poles and its structure. Problem 2 Consider the following linear time-invariant system (A,B,C,D) where

0 0 0 0 1 00 1 0 0 0 1 0 2 1 3 1 0

, , ,0 0 1 0 1 2 2 0 1 1 1 00 0 0 4 0 0

A B C

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥− ⎡ ⎤ ⎡⎢ ⎥ ⎢ ⎥= = =

⎤=⎢ ⎥ ⎢⎢ ⎥ ⎢ ⎥− − ⎥

⎣ ⎦ ⎣⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦

⎦

(1) Determine the Smith-McMillan form of the transfer function matrix

1( ) ( )G s C sI A B D−= − +

(2) Determine the system eigenvalues. (3) Determine the system poles and its pole structure from the Smith-McMillan form. Are the system eigenvalues the same as the system poles? Explain.