16.513 control systems - faculty server contactfaculty.uml.edu/thu/controlsys/note03.pdf–...
TRANSCRIPT
1
1
16.513 Control Systems
Last Time:Matrix Operations -- Fundamental to Linear Algebra– Determinant– Matrix Multiplication– Eigenvalue– Rank
Math. Descriptions of Systems ~ Review– LTI Systems: State Variable Description– Linearization
2
Today: Modeling of Selected Systems– Continuous-time systems (§2.5)
• Electrical circuits• Mechanical systems• Integrator/Differentiator realization• Operational amplifiers
– Discrete-Time systems (§2.6)• Derive state-space equations – difference equations• Two simple financial systems
Linear Algebra, Chapter 3– Linear spaces over a field– Linear dependence
2
3
2.5 Modeling of Selected Systems
We will briefly go over the following systems– Electrical Circuits– Mechanical Systems– Integrator/Differentiator Realization– Operational Amplifiers
For any of the above system, we derive a state space description:
)t(Du)t(Cx)t(y)t(Bu)t(Ax)t(x
+=+=&
Different engineering systems are unified into the sameframework, to be addressed by system and control theory.
4
Electrical CircuitsState variables?
– i of L and v of C
DuCxyBuAxx
+=+=&
++- - -
u(t) R C
L+
y
i
v
• How to describe the evolution of the state variables?
vuvdtdiL L −==
Rvii
dtdvC C −== RC
viC1
dtdv
uL1v
L1
dtdi
−=
+−= State Equation: Two first-order differential equations in terms of state variables and input
Output equation:
u0L1
vi
RC1
C1
L10
dtdvdtdi
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡+⎥
⎦
⎤⎢⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−
−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡In matrix form:
y = v [ ] 0uvi
10 +⎥⎦
⎤⎢⎣
⎡=
x& x
3
5
Steps to obtain state and output equations:Step 1: Pick {iL, vC} as state variablesStep 2:
u),v,(ifidt
dvC
u),v,(ifvdtdiL
CL2CC
CL1LL
==
== Linear functions
By using KVL and KCL
Step 3:L
1 C L
C2 C L
di (1/L)f (i ,v ,u)dtdv (1/C)f (i ,v ,u)dt
=
=
Step 4: Put the above in matrix formStep 5: Do the same thing for y in terms of state variables and
input, and put in matrix form
6
Example• State variables?
– i1, i2, and v,
++- - -
u(t) R2 C
L1
+ y
L2 R1i1
v
• State and output equations?
1L1
1 vdtdiL =
CidtdvC = 21
22
2
22
111
1
11
iCvi
C1
dtdv
vL1i
LR
dtdi
uL1v
L1i
LR
dtdi
−=
+−=
+−−=
u00L1
vii
0C1
C1
L1
LR0
L10
LR
dtdvdtdidtdi
1
2
1
22
2
11
1
2
1
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
+⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−
−−
=
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
i2
2L2
2 vdtdiL =
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡==
vii
0R0iRy 2
1
222
viRu 11 −−=
22iRv −=
21 ii −=
x& x DuCxyBuAxx
+=+=&
4
7
• Elements: Spring, dashpot, and mass– Spring:
u(t)y(t)
M y(t), position,
positive direction
fS = Ky, opposite direction ~ Hooke’s law, K: Stiffness
– Dashpot: y(t), y'(t)
fD = Dy', opposite direction ~ D: Damping coefficient
– Mass: M, Newton’s law of motionNfyM =&& ~ Net force– LTI elements, LTI systems
– Linear differential equations with constant coefficients
Mechanical Systems
fN(t) y(t)
M
8
• Number of state variables? Which ones?– 2 state variables: x1 ≡ y, x2 ≡ x1'
u(t)y(t)
M
K
D
• How to describe the system? • Free body diagram:
u(t) y(t)
M Ky
Dy'
yDKyuyM &&& −−=
=1x& x2 =2x& =y&&M
DxKxuM
yDKyu 21 −−=
−− &
uM10
xx
MD
MK
10
dtdxdt
dx
2
1
2
1
⎥⎥⎦
⎤
⎢⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡
⎥⎥⎦
⎤
⎢⎢⎣
⎡−−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡1xy = [ ] 0u
xx
012
1 +⎥⎦
⎤⎢⎣
⎡=
uM1y
MKy
MDy =++ &&&
~ Input/Output description
5
9
• Steps to obtain state and output equations:Step 1: Determine ALL junctions and label the displacement
of each oneStep 2: Draw a free body diagram for each rigid body to obtain
the net force on itStep 3: Apply Newton's law of motion to each rigid body Step 4: Select the displacement and velocity as state variables,
and write the state and output equations in matrix form• For rotational systems: τ = Jα
• τ: Torque = Tangential force⋅arm• J: Moment of inertia = ∫r2dm • α: Angular acceleration
– There are also angular spring/damper
10
u(t)M
K D y1(t)
u(t)y1(t)
M
D(y1'-y2')(relative motion)
( )211 yyDuyM &&&& −−=
D(y2'-y1')
y2(t)
Ky2
y2(t)
( )122 yyDKy0 && −−−=
• Number of state variables? How to select the state variables?
231211 yx;yx;yx ≡≡≡ &
211 xyx =≡ &&
( )3212 xDDxuM1yx &&&& +−=≡
323 xDKxx −=&
uM1x
MK
3 +−=
u
0M10
x
DK10MK00
010x
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
−
−=&
=⎥⎦
⎤⎢⎣
⎡=2
1yy
y u00
x100001
⎥⎦
⎤⎢⎣
⎡+⎥⎦
⎤⎢⎣
⎡
• How to describe the system?• How many junctions are there?
6
11
Example: an axial artificial heart pump
AMB Motor
AMB PMB Motor PMB
Indu
cer
Dif
fuse
r
PMB PMBAMB
AMB PM
PM
PM
PM
Illustration
12
PM
PM PM
PM
1F 2F3F
F1: the active force that can be generated as desired(F =k I1
2 / (c+y1)2 – kI22/(c-y1)2
F2, F3: passive forces, F2 = -k1y2, F3= -k3y3, similar to springs
I1
I2
y1y2y3
The forces acting on the rotor:
7
13
Modeling
1 11 1 12 2 1 1
2 21 1 22 2 2 1
y a y a y b Fy a y a y b F
= + += + +
&&
&&
1 2 3 1 2 2 1
1 1 2 2 3 3 2 1
, ( ) /, ( ) /
c cMy F F F y l y l y lJ l F l F l F y y lα α
= + + = += − + − = −
&&
&&
The motion of the rotor:
l1
y1
y2
yc
l2
F1 F3
F2
F2 and F3 depend on y1 and y2. Equation can be expressed in terms of y1 and y2
1 1 2 1 3 2 4 2Let , , ,x y x y x y x y= = = =& &
α
Center of massl3
14
1 1 2 1 3 2 4 2Let , , ,x y x y x y x y= = = =& &
1 1 2
2 1 11 1 12 3 1 1
3 2 4
4 2 21 1 22 3 2 1
x y xx y a x a x b Fx y xx y a x a x b F
= == = + += == = + +
& &
& &&
& &
& &&
11 12 11
21 22 2
0 1 0 0 00 0
0 0 0 1 00 0
=
a a bx x F
a a bAx Bu
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥
= +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
+
&
1 11 1 12 2 1 1
2 21 1 22 2 2 1
y a y a y b Fy a y a y b F
= + += + +
&&
&&
In matrix form?
8
15
Integrator/Differentiator Realization• Elements: Amplifiers, differentiators, and integrators
Differentiator
f(t) y(t) = df/dts
Amplifier
f(t) y(t) = af(t)a
Integratorf(t)
y(t) =
1/s )t(yd)(f 0t
t0+∫ ττ
=τ
• Are they LTI elements? Yes• Which one has memory? What are their dimensions?
– Integrator has memory. Dimensions: 0, 0, and 1, respectively• They can be connected in various ways to form LTI
systems– Number of state variables = number of integrators– Linear differential equations with constant coefficients
16
• What are the state variables?
• Select output of integrators as SVs
2
1/s 1/s+
-u(t)
y(t)
x1x2
• What are the state and output equations?21 xx =&
12 x2ux −=& y = x2
u10
xx
0210
xx
2
1
2
1⎥⎦
⎤⎢⎣
⎡+⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡−
=⎥⎦
⎤⎢⎣
⎡&
& [ ] u0xx
10y2
1 +⎥⎦
⎤⎢⎣
⎡=
A B C D
• Linear differential equations with constant coefficients
12 xx &&
9
17
• Steps to obtain state and output equations:Step 1: Select outputs of integrators as state variablesStep 2: Express inputs of integrators in terms of state
variables and input based on the interconnection of the block diagram
Step 3: Put in matrix formStep 4: Do the same thing for y in terms of state
variables and input, and put in matrix form
18
a
b
u y
+ +
+
- 1/s 1/s
Exercise: derive state equations for the following sys.
2x1x
10
19
Operational Amplifiers (Op Amps)
• Usually, A > 104
– Ideal Op Amp:• A → ∝ ~ Implying that (va - vb) → 0, or va → vb
• ia → 0 and ib → 0 – Problem: How to analyze a circuit with ideal Op Amps
-Vcc, -15V
Non-inverting terminal, va
Inverting terminal, vb
Output, vo
ioia ib
Vcc, 15V
+
-
v0 = A (va - vb), with -VCC ≤ v0 ≤ VCC
-Vcc
va - vb
Slope = A
Vcc
Vo
20
• Key ideas:– Make effective use of ia = ib = 0 and va = vb
– Do not apply the node equation to output terminals of op amps and ground nodes, since the output current and power supply current are generally unknown
+ u1(t)
R2R1
y u2(t)
+ +
-
- -
-
+i1
i2 i1 + i2 = 0
0R
uyR
uu2
2
1
21 =−
+−
21
21
1
2 uRR1u
RRy ⎟
⎠
⎞⎜⎝
⎛++−=
Delineate the relationship between input and outputInput/Output description
Pure gain, no SVs
u2
11
21
R2
R1
+ +
-+
u(t) -
-
y -
+
- + C1
C2
v1
v2 State and output equations
=dt
dvC 11
1vy =
v1 + v2What are the state variables?
State variables: v1 and v2
( )2
2
2
121Rv
Rvvv
=−+
=dt
dvC 22
( )2
2
1
21Rv
Rvvu
−+−
v1
uCR10
vv
R1
R1
C1
CR1
CR10
dtdvdt
dv
212
1
11221
122
1
⎥⎥⎦
⎤
⎢⎢⎣
⎡+⎥⎦
⎤⎢⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎦
⎤⎢⎣
⎡+
−−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
[ ] u0vv
012
1 +⎥⎦
⎤⎢⎣
⎡=
i1
ic2
ic1
22
Today: Modeling of Selected Systems– Continuous-time systems (§2.5)
• Electrical circuits• Mechanical systems• Integrator/Differentiator realization• Operational amplifiers
– Discrete-Time systems (§2.6): • Derive state-space equations – difference equations• Two simple financial systems
Linear Algebra, Chapter 3– Linear spaces over a field– Linear dependence
12
23
2.6 Discrete-Time Systems• Thus far, we have considered continuous-time
systems and signals
t
y(t)
• In many cases signals are defined only at discrete instants of time
k
y[k]
T
– T: Sampling period– No derivative and no differential equations– The corresponding signal or system is described by
a set of difference equations
24
y[k] = u[k-1]
k
Elements: Amplifiers, delay elements, sources (inputs) Amplifiers:
u[k] y[k] = a u[k]a
~ LTI and memoryless
• Delay Element: u[k] y[k] = u[k-1]
z-1
~ LTI with memory (1 initial condition)They can be interconnected to form an LTI system
u[k]
k
13
25
Example
3
y[k]
Delay Delay
K
+ + - -
u[k]
• How to describe the above mathematically?– I/O description:
( ) or,]1k[Ky]1k[u]k[u3]k[y]1k[y −−−++−=+
]1k[u]k[u3]1k[Ky]k[y]1k[y −++−−−=+
y[k+1]
– A linear difference equation with constant coefficient
26
3
y[k]
Delay Delay
K
+ + - -
u[k]
State space description: Select output of delay elements as state variables
]k[u3]k[x]k[x]1k[x 211 ++−=+
]k[x]k[y 1=
]k[u]k[xK]1k[x 12 +−=+
x1[k]x1[k+1]x2[k]x2[k+1]
14
27
Two state variables, x1[k], x2[k]x1[k+1] = u[k] + x2[k]x2[k+1] = -bx1[k] + ax2[k]y[k] = x1[k]
u[k] +
+ z-1
a
y[k]+
-
z-1
b
x1[k]x1[k+1]
]k[u01
]k[xab10
]1k[x ⎥⎦
⎤⎢⎣
⎡+⎥⎦
⎤⎢⎣
⎡−
=+
x2[k]
[ ] ]k[x01]k[y =
x2[k+1]
Exercise:Derive state equations for the discrete-time system.
28
Example 1: Balance in your bank account
A bank offers interest r compounded every day at 12am– u[k]: The amount of deposit during day k
(u[k] < 0 for withdrawal)– y[k]: The amount in the account at the beginning
of day k What is y[k+1]?
y[k+1] = (1 + r) y[k] + u[k]
15
29
Example 2: Amortization
How to describe paying back a car loan over four years with initial debt D, interest r, and monthly payment p? Let x[k] be the amount you owe at the
beginning of the kth month. Thenx[k+1] = (1 + r) x[k] − p
Initial and terminal conditions: x[0] = D and final condition x[48] = 0 How to find p?
30
x[k+1] = (1 + r) x[k] + (−1) p The system:
Solution: A B u
pr
1r)(1Dr)(1pr)(1Dr)(1
1)p(r)(1x[0]r)(1
Bu[m]Ax[0]Ax[k]
kk
1k
0m
1mkk
1k
0m
1mkk
1k
0m
1mkk
−+−+=⎟
⎠
⎞⎜⎝
⎛+−+=
−+++=
+=
∑
∑
∑
−
=
−−
−
=
−−
−
=
−−
Given D=20000; r=0.004; x[48]=0;
p0.004
1)004.0(100002)004.0(10 48
48 −+−+= p=458.7761
Your monthlypayment
16
31
Today: Modeling of Selected Systems– Continuous-time systems (§2.5)
• Electrical circuits• Mechanical systems• Integrator/Differentiator realization• Operational amplifiers
– Discrete-Time systems (§2.6):• Derive state-space equations – difference equations• Two simple financial systems
Linear Algebra, Chapter 3– Linear spaces over a field– Linear dependence
32
Our modeling efforts lead to a state-space description of LTI system
)t(Du)t(Cx)t(y)t(Bu)t(Ax)t(x
+=+=&
]k[Du]k[Cx]k[y]k[Bu]k[Ax]1k[x
+=+=+
Analysis problems: stability; transient performances;potential for improvement by feedback control, …
Linear Algebra: Tools for System Analysis and Design
17
33
• Consider an LTI continuous-time system
)t(Du)t(Cx)t(y)t(Bu)t(Ax)t(x
+=+=&
• For a practical system, usually there is a natural way to choose the state variables, e.g.,
⎥⎦
⎤⎢⎣
⎡=
2
1
xx
x iL
vc• However, the natural state selection may not be the
best for analysis. There may exist other selection tomake the structure of A,B,C,D simple for analysis
• If T is a nonsingular matrix, then z = Tx is also the state and satisfies
Du(t)z(t)CTy(t)TBu(t)z(t)TAT(t)z
1
1
+=+=
−
−&
u(t)D~z(t)C~y(t)u(t)B~z(t)A~(t)z
+=+=&
34
• Two descriptions
)t(Du)t(Cx)t(y)t(Bu)t(Ax)t(x
+=+=&
u(t)D~z(t)C~y(t)u(t)B~z(t)A~(t)z
+=+=&
are equivalent when I/O relation is concerned. • For a particular analysis problem, a special form of
D~,C~,B~,A~ may be the most convenient, e.g.,
[ ]001C~
,100
B~,aaa100010
A~,λ000λ000λ
A~
3213
2
1
=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
• We need to use tools from Linear Algebra to get a desirable description.
18
35
The operation x → z = Tx is called a lineartransformation. It plays the essential role in obtaining a desired
state-space description
u(t)D~z(t)C~y(t)u(t)B~z(t)A~(t)z
+=+=&
Linear algebra will be needed for the transformationand analysis of the system– Linear spaces over a vector field– Relationship among a set of vectors: LD and LI– Representations of a vector in terms of a basis– The concept of perpendicularity: Orthogonality– Linear Operators and Representations
36
3.1 Linear Vector Spaces and Linear Operators
Rn: n-dimensional real linear vector spaceCn: n-dimensional complex linear vector spaceRn×m: the set of n×m real matrices (also a vector space)Cn×m: the set of n×m complex matrices (a vector space)
Notation:
A matrix T∈Rn×m represents a linear operation fromRm to Rn: x∈Rm → Tx∈Rn.
All the matrices A,B,C,D in the state space equationare real
19
37
Linear Vector Spaces Rn and Cn
R: The set of real numbers; C: The set of complex numbersIf x is a real number, we say x∈R; If x is a complex number, we say x∈CRn: n-dimensional real vector spaceCn: n-dimensional complex vector space
1
n 21 2R : , , , R ,n
n
xxx x x x
x
⎧ ⎫⎡ ⎤⎪ ⎪⎢ ⎥⎪ ⎪= = ∈⎨ ⎬⎢ ⎥⎪ ⎪⎢ ⎥
⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭
LM
1
n 21 2C : , , , Cn
n
xxx x x x
x
⎧ ⎫⎡ ⎤⎪ ⎪⎢ ⎥⎪ ⎪= = ∈⎨ ⎬⎢ ⎥⎪ ⎪⎢ ⎥
⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭
LM
If x,y∈Rn, a,b∈R, then ax + by∈Rn Rn is a linear space.
If x,y∈Cn, a,b∈C, then ax + by∈Cn Cn is a linear space.
38
SubspaceConsider Y ⊂ Rn. Y is a subspace of Rn iff Y itself is a linear space– Y is a subspace iff α1y1 + α2y2 ∈ Y for all y1, y2 ∈ Y
and α1, α2 ∈ R (linearity condition)– Subspace of Cn can be defined similarly
11 2 1 2
2: 2 1 0, , RxY x x x xx
⎧ ⎫⎡ ⎤= − + = ∈⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
Is the linearity condition satisfied?
Example: Consider R2. The set of (x1,x2) satisfying x1 - 2x2 + 1 = 0 can be written as
20
39
Then how about the set of (x1,x2) satisfying x1 - 2x2 = 0?
– Yes. In fact, any straight line passing through 0 form a subspace
What would be a subspace for R3 ?– Any plane or straight line passing through 0
{(x1,x2,x3): ax1+bx2+cx3=0} for constants a,b,c denote a plane. How to represent a line in the space?
The set of solutions to a system of homogeneous equation is a subspace: {x ∈Rn: Ax=0}. How about {x ∈Rn: Ax=c} ?
Y: x1-2x2=0
x1
x2
11 2 1 2
2: 2 0, , RxY x x x xx
⎧ ⎫⎡ ⎤= − = ∈⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
40
Consider Rn, – Given any set of vectors {xi}i=1 to n, xi ∈Rn. – Form the set of linear combinations
⎭⎬⎫
⎩⎨⎧
∈≡ ∑=
Rα :xαY i
n
1iii
– Then Y is a linear space, and is a subspace of Rn.– It is the space spanned by {xi}i=1 to n
21
41
Linear Independence
A set of vectors {x1, x2, .., xm} in Rn is linearly dependent (LD)iff ∃ {α1, α2, .., αm} in R, not all zero, s.t.
α1x1 + α2x2 + .. + αnxm = 0 (*)– If (*) holds and assume for example that α1 ≠ 0, then
x1 = −[α2x2 + .. + αnxm]/α1
i.e., x1 is a linear combination of {αi}i=2 to m
If the only set of {αi}i=1 to m s.t. the above holds isα1 = α2 = .. = αm = 0
then {xi}i=1 to m is said to be linearly independent (LI )– None of xi can be expressed as a linear combination of the rest
Relationship among a set of vectors.
42
– A linearly dependent set ~ Some redundancy in the setExample. Consider the following vectors:
x1
x2
x3x4
• For the following sets, are they linearly dependent (LD) or independent (LI)?– {x1, x2}– {x1, x3}– {x1, x3, x4}– {x1, x2, x3, x4}
If you have a LD set, {x1,x2,…xm}, then {x1,x2,…xm,y} is LDfor any y.
22
43
• A general way to detect LD or LI:– {x1, x2, .., xm} are LD iff ∃ {α1, α2, .., αm}, not all
zero, s.t. α1x1 + α2x2 + .. + αmxm= 0
[ ] ,0
α:αα
x...xxxαxαxα
m
2
1
m21mm2211 =⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=+++ L
• Given a set of vectors, {x1,x2,…,xm}⊂ Rn, how to findout if there are LD or LI?
= A∈Rn×m = α∈Rm
Aα = 0
– {x1, x2, .., xm} are LD iff Aα =0 has a nonzero solution
Need to understand the solution to a homogeneous equation. There is always a solution α = 0. Question: under what condition is the solution unique?
44
Given {x1, x2, .., xm}, form [ ]1 2 mA x x ... x=
Aα = 0
• If n=m and A is singular, det(A)= 0, rank(A) < m ∃ α≠0 s.t. Aα=0, hence LD
Detecting LD and LI through solutions to linear equations
Consider the equation
If the equation has a unique solution, LI;If the equation has nonunique solution, LD.
This is related to the rank of A.
If rank(A)=m, (A has full column rank), the solution is unique;If rank(A)<m, the solution is not unique.
• If n=m and A is nonsingular, det(A)≠ 0, rank(A)=monly α=0 satisfies. Aα=0, hence LI
23
45
• Are the following vectors LD or LI?
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
741
x,521
x,432
x 321
LI010754423112
A)det( ≠−==
• How about
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
654
x,432
x,321
x 321
643532421
)Adet( =
0202436323018
145622433442352163
=−−−++=
××−××−××−××+××+××=
LD
det(A)=?
46
• All depends on the uniqueness of solution for Aα=0• If m> n, A is a wide matrix, rank(A)<m, always has a
nonzero solution, e.g.,
⎥⎦
⎤⎢⎣
⎡−
=⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡−
=211101
A,21
x,10
x,1
1x 321
If m < n and rank(A) = m, LI;If m < n and rank(A)<m, LD;
,1111
00,
110110
21⎥⎥⎦
⎤
⎢⎢⎣
⎡
−−=
⎥⎥⎦
⎤
⎢⎢⎣
⎡= AA
rank(A1)=2=m, LI rank(A2)=1<m, LD
A m
n
m
n A
24
47
• Examples: determine the LD/LI for the following group of vectors
sin cos, ,cos sin
sin cos,cos sin
θ θθ θ
θ θθ θ
⎧ ⎫−⎡ ⎤ ⎡ ⎤⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭⎧ ⎫⎡ ⎤ ⎡ ⎤⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭
1 1 1, , ,0 1 3
1 01 , 11 3
1 0 21 , 1 , 31 3 5
⎧ ⎫⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎨ ⎬⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎩ ⎭⎧ ⎫⎡ ⎤ ⎡ ⎤⎪ ⎪⎢ ⎥ ⎢ ⎥⎨ ⎬
⎢ ⎥ ⎢ ⎥⎪ ⎪⎣ ⎦ ⎣ ⎦⎩ ⎭⎧ ⎫⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎪ ⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎨ ⎬
⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎪ ⎪⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎩ ⎭
48
Dimension
• For a linear vector space, the maximum number of LI vectors is called the dimension of the space, denoted as D
• Consider {x1,x2,…,xm}⊂ Rn
If m>n, they are always dependent, D ≤ n For m=n, there exist x1,x2,…xn such that
with A=[x1,x2,…xn], |A|≠0, xi’s LI, D ≥ n Hence D = n
25
49
Today: Modeling of Selected Systems– Continuous-time systems (§2.5)
• Electrical circuits, Mechanical systems• Integrator/Differentiator realization• Operational amplifiers
– Discrete-Time systems (§2.6): • Derive state-space equations – difference equations• Two simple financial systems
Linear Algebra, Chapter 3– Linear spaces over a field– Linear dependence
Next time: Linear algebra continued.
50
C
RL
Input u=Vin Output: y =Vo,
-+
R
2. Derive state-space description for the diagram:
1. Derive state-space description for the circuit:
u y
−+
+- 1/s1/s 1/s
a
b
c+
+
Homework Set #3
+ vc −
iL
+
−
+
−
26
51
4. Are the following sets subspace of R2 ?
.,0:01
11
,,:01
11
,,:11
10
10
2
2
1
⎭⎬⎫
⎩⎨⎧
∈≥⎥⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡−
=
⎭⎬⎫
⎩⎨⎧
∈⎥⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡−
=
⎭⎬⎫
⎩⎨⎧
∈⎥⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡−
+⎥⎦
⎤⎢⎣
⎡=
RbabaY
RbabaY
RbabaY
52
5. Are the following groups of vectors LD or LI?
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡−⎥
⎦
⎤⎢⎣
⎡
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−
1cossin
,100
,2
sin2cos
)4,12
,1
,1
)3
110
,0
,11
1)2,
422
,313
,11
1)1
θθ
θθ
aa
ba
,
1211
,
111
2
,
1001
,
1101
)6,
1111
,
021
1
,
2110
,
1201
)5
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡