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ECE5580: Multivariable Control in the Frequency Domain 6–1
Introduction to Multivariable Control
! In this chapter, we introduce the idea of multi-input multi-output(MIMO) systems
! Specifically, we’ll develop the following ideas: transfer functionmatrices, multivariable frequency response analysis, singular valuedecomposition (SVD), the relative gain array (RGA), multivariablecontrol and multivariable zeros
Introduction
! Consider a MIMO plant having m inputs and l outputs
! Basic transfer function model is:y.s/ D G.s/u.s/
where we have used bold-face symbols to make the vector-matrixstructure more apparant
! In general form, we can write266664
y1.s/
y2.s/:::
yl.s/
377775
D
266664
g11.s/ g12.s/ " " " g1m.s/
g21.s/ g22.s/ " " " g2m.s/::: ::: : : : :::
gl1.s/ gl2.s/ " " " glm.s/
377775
266664
u1.s/
u2.s/:::
um.s/
377775
where it is easy to see that each of the elements of G.s/ is itself aSISO transfer function gij .s/ relating input uj .s/ to output yi.s/
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–2
! A change in the first input u1 will generally affect all outputs,y1; y2; : : : ; yl – this means there is interaction between inputs and
outputs
! The main difference between SISO and MIMO systems is theexistence of directions in MIMO systems – these will be explored ingreater detail in this chapter
Transfer function matrices for MIMO systems
! Transfer functions for MIMO systems are matrices as describedabove, and as such must follow the rules of linear algebra
! Consider the following block diagrams:
zG1u G2
G
Cascade system
yu
Positive feedback system
G2
G1+
+v
z
! CASCADE RULE
For the cascade (series) interconnection of G1 and G2 , the overalltransfer function matrix is G D G2G1.
– Note that the order of the transfer function matrices is reversedfrom how they appear
! FEEDBACK RULE
With reference to the positive feedback system, we havev D .I $ L/$1u where L D G2G1 is the transfer function around theloop.
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–3
! PUSH-THROUGH RULE
For matrices of appropriate dimension,G1 .I $ G2G1/
$1 D .I $ G1G2/$1 G1
! From these rules, we may state the following MIMO RULE:
Start from the output and write down the blocks as you meet themwhen moving backwards (against signal flow) towards the input. Ifyou exit from a feedback loop then inlude a term .I $ L/$1 for positivefeedback (or .I C L/$1for negative feedback) where L is the looptransfer function. Parallel branches should be treated independentlyand their contributions added together.
Example 6.1
! Find the transfer function from w to z in the block diagram below.
zw
P12
+
+
P22
P21
P11
K
+
+
SOLUTION
! Start at the output z and move backward towards w.
– One branch beyond the summing junction goes to P11 directly
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–4
– At the other branch, continue backward to encounter P12 and thenK
! Exit the feedback loop and get term .I $ L/$1(positive feedback) withL D P22K
! Finally, encounter term P21
! The complete transfer function is given by:
z D!P11 C P12K .I $ P22K/$1 P21
"w
Negative feedback control systems
! Consider the negative feedback system shown below.
G+ +
K+++
!
d1d2
ru
y
! We define L to be the loop transfer function when breaking the loopat the plant output,
L D GK
! Accordingly, the sensitivity and complementary sensitivity are definedas
S , .I C L/$1
T , I $ S D L .I C L/$1
– To make it clear we’re defining these transfer function at the plantoutput, we further define
LO D L SO D S TO D T
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–5
! We next define LI to be the loop transfer function at the input to theplant,
LI D KG
! We then defineSI , .I C LI /$1
TI , I $ SI D LI .I C LI /$1
! Some useful relationships follow:.I C L/$1 C .I C L/$1 L D S C T D I
G .I C KG/$1 D .I C GK/$1 G
GK .I C GK/$1 D G .I C KG/$1 K D .I C GK/$1 GK
L .I C L/$1 D!I C L$1
"$1 D .I C L/$1 L D T
– A simple rule to remember: “G comes first and then G and K
alternate in sequence”.
Multivariable frequency response analysis
! G.s/ is a transfer function matrix which is a function of the complexvariable s
– For s D s0, G.s0/ becomes a constant complex-valued matrix
! G.j!/ is a complex matrix function representing the response tosinusoidal signals of frequency !
Obtaining the frequency response from G .s/
! Consider the system G.s/ with input vector d.s/ and output vectory.s/:
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–6
G(s)d y
– Input-output dynamics are given byy.s/ D G.s/d.s/
! A sinusoidal input to channel j is given bydj .t/ D dj 0 sin
!!t C ˛j
"
! The output in channel i is a sinusoid with the same frequencyyi.t/ D yi0 sin .!t C ˇi/
– Amplification gain is given by:yi0
dj 0D
ˇ̌gij .j!/
ˇ̌
– Phase shift is given by:ˇi $ ˛j D †gij .j!/
! We can represent the result in phasor notation as:yi .!/ D gij .j!/ dj .!/
where dj .!/ D dj 0ej˛j and yi .!/ D yi0e
jˇi .
– In matrix form, we havey .!/ D G .j!/ d .!/
Example 6.2
! Consider a 2 % 2 multivariable system where we simultaneously applysinusoidal signals of the same frequency ! to the two input channels:
d.t/ D"
d1.t/
d2.t/
#D
"d10 sin .!t C ˛1/
d20 sin .!t C ˛2/
#
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–7
! The output signal,
y.t/ D"
y1.t/
y2.t/
#D
"y10 sin .!t C ˇ1/
y20 sin .!t C ˇ2/
#
can be computed by multiplying the complex matrix G .j!/ by thecomplex vector d .!/:
y .!/ D G .j!/ d .!/
where
y .!/ D"
y10ejˇ1
y20ejˇ2
#and d .!/ D
"d10e
j˛1
d20ej˛2
#
Directions in multivariable systems
! For a SISO system, y D Gd , the gain is given byjy .!/jjd .!/j D jG .j!/ d .!/j
jd .!/j D jG .j!/j
– The gain depends on !, but is otherwise independent of themagnitude jd .!/j
! For MIMO systems, things are very different, because input andoutput are vectors
– So we need to use a measure appropriate for vectors – norm!
! Using the vector 2-norm, we have
kd .!/k2 DsX
j
ˇ̌dj .!/
ˇ̌2 Dq
d 210 C d 2
20 C " " "
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–8
ky .!/k2 DsX
i
jyi .!/j2 Dq
y210 C y2
20 C " " "
! The gain of the system transfer function matrix G.s/ isky .!/k2
kd .!/k2
D kG.j!/d.!/k2
kd.!/k2
! In the MIMO case, the gain depends on frequency !, but isindependent of the norm kd .!/k2
– However, the gain depends on the direction of the input vector d
! The MIMO gain given above is recognized as the induced 2-norm –its maximum is computed as the maximum singular value of G:
maxd¤0
kGdk2
kdk2
D maxkdk2D1
kGdk2 D N! .G/
– Conversely, the minimum gain is the minimum singular value of G:
mind¤0
kGdk2
kdk2
D minkdk2D1
kGdk2 D ! .G/
Example 6.3
! Consider the five different inputs shown below (kdk2 D 1):
d1 D"
1
0
#; d2 D
"0
1
#; d3 D
"0:7071
0:7071
#
d4 D"
0:7071
$0:7071
#; d5 D
"0:6
$0:8
#
! For the 2 % 2 system
G1 D"
5 4
3 2
#
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–9
! The five inputs dj lead to the following corresponding output vectors:
y1 D"
5
3
#; y2 D
"4
2
#; y3 D
"6:36
3:54
#
y4 D"
0:7071
0:7071
#; y5 D
"$0:2
0:2
#
! The corresponding 2-norms are:
ky1k2 D 5:83
ky2k2 D 4:47
ky3k2 D 7:30
ky4k2 D 1:00
ky5k2 D 0:28
! The figure below shows a functional plot of the matrix gain as afunction of a parameterized input direction
– It is easy to see that maximum and minimum gains are achieved atdistinct input vector directions
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–10
! = 7.34
! = 0.27
! Another way to visualize the singular value decomposition ispresented below where the domain and corresponding image of theinput and output map are shown.
Eigenvalues are a poor measure of gain
! Consider the system y D Gd with
G D"
0 100
0 0
#
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–11
– Note that both eigenvalues are equal to zero, but the gain is equalto 100
! PROBLEM: Eigenvalues measure the gain for the special case whenthe inputs and the outputs are in the same direction (that of theeigenvectors)
! For generalizations of jGj when G is a matrix, we need the concept ofa matrix norm, denoted kGk
! Two important properties of norms:
– Triangle inequalitykG1 C G2k & kG1k C kG2k
– Multiplicative propertykG1G2k & kG1k " kG2k
! Note that the spectral radius" .G/ , j#max .G/j
does not satisfy the properties of a matrix norm
Singular value decomposition
! Any matrix G may be decomposed into its singular valuedecomposition,
G D U˙V '
– ˙ is an l % m matrix with k D min fl; mg non-negative singularvalues !i arranged in descending order along its main diagonal,where
!i .G/ Dp
#i .G'G/
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–12
– U is an l % l unitary matrix of output singular vectors, ui
– V is an m % m unitary matrix of input singular vectors, vi
Example 6.4
! The SVD of a real 2 % 2 matrix can always be written in the form
G D"
cos $1 $ sin $1
sin $1 cos $1
# "!1 0
0 !2
# "cos $2 ˙ sin $2
$ sin $2 ˙ cos $2
#T
– Note that U and V involve rotations and their columns areorthonormal
! The singular values are sometimes called the principal values orprincipal gains, and the associated directions are called principaldirections
Input and output directions
! The column vectors of U , denoted by ui , represent the outputdirections of the plant
– They are orthogonal and of unit length (orthonormal), that is
kuik2 D 1
u'i ui D 1
uiuj D 0; i ¤ j
! The column vectors of V , denoted by vi , are orthogonal and of unitlength, and represent the input directions.
! Input and output directions are related through the singular values,Gvi D !iui
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–13
– If we consider an input in the direction vi ; then the output is in thedirection ui
! Since kvik2 D 1 and kuik2 D 1, !i gives the gain of the matrix G in thisdirection
!i .G/ D kGvik2 D kGvik2
kvik2
Maximum and minimum singular values
! The largest gain for any input direction is equal to the maximumsingular value
N! .G/ D !1 .G/ D maxd¤0
kGdk2
kdk2
D kGv1k2
kv1k2
! The smallest gain for any input direction is
! .G/ D !k .G/ D mind¤0
kGdk2
kdk2
D kGvkk2
kvkk2
where k D minn
l; mo
! Therefore, for any vector d we have the general relationship
! .G/ & kGdk2
kdk2
& N! .G/
! Define u1 D Nu, and v1 D Nv,
– Then it follows,G Nv D N! Nu
– Nv is the maximum principal input direction, and corresponds to theinput direction with the largest amplification
– Nu is the maximum principal output direction, and is the outputdirection in which the inputs are most effective
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–14
! Next define uk D u , and vk D v
– Then we can writeGv D !u
– v is the minimum principal input direction, and corresponds to theinput direction with the smallest amplification
– u is the minimum principal output direction, and is the outputdirection in which the inputs are least effective
! These important vectors are associated with the “highest gain” and’lowest gain” directions respectively
Example 6.5
! Consider the system of Example 6.3,
G1 D"
5 4
3 2
#
! The singular value decomposition of G1 is
G1 D"
0:872 0:490
0:490 $0:872
# "7:343 0
0 0:272
# "0:794 $0:608
0:608 0:794
#'
– The largest gain of 7:343 is for an input in the direction
Nv D"
0:794
0:608
#
– The smallest gain of 0:272 is for an input in the direction
v D"
$0:608
0:794
#
! Note from the form of G1 that both inputs affect both outputs – thusthe system exhibits interaction
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–15
! Note also that the system is ill-conditioned, meaning that somecombinations of inputs have a strong effect on the outputs, whereasothers have a weak effect
– CONDITION NUMBER: % .G1/ D N! .G1/
! .G1/D 7:343
0:272D 27:0
Example 6.6: Distillation Process
! Consider the following steady-state model of a distillation column:
G D"
87:8 $86:4
108:2 $109:6
#
– Since the element magnitudes are much larger than 1, thereshould be no problem with input constraints – however, thelow-gain direction could be problematic
G D"
0:625 $0:781
$:781 0:625
# "197:2 0
0 1:39
# "0:707 $0:708
$0:708 $0:707
#'
– The distillation process is clearly ill-conditioned, with conditionnumber % .G/ D 197:2=1:39 D 141:7
– The system is such that the two inputs counteract each other in thedirection of v in steady-state
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–16
Singular values for performance
! The maximum singular value is very useful in terms of frequencydomain performance and robustness
! For SISO systems we found that jS .j!/j D je.!/j=jr.!/j evaluated overfrequency gives useful information for feedback control
! Generalizations for MIMO systems can be obtained if we consider theratio ke.!/k2=kr.!/k2 where we write
! .S .j!// & ke .!/k2
kr .!/k2
& N! .S .j!//
! For performance we want the gain ke.!/k2=kr.!/k2 small for any directionof r .!/
N! .S .j!// <1
jwP .j!/j; 8!
, N! .wP S/ < 1; 8!
, kwP Sk1 < 1
where the H1 norm is the peak of the maximum singular value of thefrequency response:
kM .s/k1 , max!
N! .M .j!//
! The singular values of S .j!/ may be plotted versus frequency –typically they are small at low frequencies where feedback is effective,and approach 1 at high frequencies because any real system isstrictly proper
! BANDWIDTH, !B : frequency where N! .S/ crosses 1=p
2 D 0:7071 frombelow
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–17
– The bandwidth is at least !B for any direction of the input signal
– Since S D .I C L/$1, the singular value inequality
! .A/ $ 1 & 1
N! .I C A/$1& ! .A/ C 1
gives
! .L/ $ 1 & 1
N! .S/& ! .L/ C 1
– So at low frequencies,
! .L/ ( 1 ) N! .S/ ) 1
! .L/
– And at high frequencies,N! .L/ * 1 ) N! .S/ ) 1
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–18
Relative Gain Array (RGA)
! The RGA is a useful technique applied to multivariable systems toassess interaction
! Mathematically, the RGA of a non-singular square complex matrix G
is defined asRGA .G/ D & .G/ , G %
!G$1
"T
where here ’%’ denotes the Hadamard product (element-by-elementmultiplication)
– In Matlab we write RGA = G.*pinv(G).’
! The RGA of a transfer matrix is normally computed as a function offrequency
RGA as an interaction measure
! Let uj and y i denote a particular input-output pair for plant G.s/ – wewish to use uj to control y i
! There will be two extreme cases:
– All other loops are open: uk D 0; 8k ¤ j
– All other loops are closed with perfect control: yk D 0; 8k ¤ i
! Evaluating gain @y i=@uj for the two cases
– Other loops open:#
@y i
@uj
$ukD0; k¤j
D gij
– Other loops closed:#
@y i
@uj
$ykD0; k¤i
, Ogij
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–19
! Here, gij D ŒG'ij is the ij th element of G, whereas Ogij is the inverse ofthe j i th element of G$1:
Ogij D 1
ŒG$1'j i
! The ratio between the “loops open” and “loops closed” gains turns outto be a useful measure of interaction – we define the ij th relative gainas
#ij , gij
OgijD ŒG'ij
%G$1
&j i
– The RGA is the corresponding matrix of these relative gains
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–20
Example 6.7: RGA for 2 % 2 system
! Consider the system given by"y1
y2
#D
"g11 g12
g21 g22
# "u1
u2
#
where our task is to control ouput y1 using input u1.
! For the ouputs, we writey1 D g11u1 C g12u2
y2 D g21u1 C g22u2
! We consider two cases:
– Case 1: u2 D 0y1 D g11u1
– Case 2: y2 D 0
u2 D $g21
g22u1
! Substituting, we get
y1 D#
g11 $ g21
g22g12
$u1
y2 D 0
where we defineOg11 D g11 $ g21
g22g22
! The interpretation here is that the gain changes from g11 to Og11 as weclose the other loop
! Practically speaking, we prefer to pair input-output varables so thatthe corresponding #ij is close to 1
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–21
Example 6.8
! Consider a blending process where we mix sugar (u1 ) and water (u2)to make a given amount of a soft drink (y1 D F ) with a given sugarfraction (y2 D x)
! Mass balance gives:F1 C F2 D F
F1 D xF
! Linearization yields (note that the process has no dynamics):dF1 C dF2 D dF
dF1 D x'dF C F 'dx
! Assigning u1 D dF1, u2 D dF2, y1 D dF , and y2 D dx gives the model,
y1 D u1 C u2
y2 D#
1 $ x'
F '
$u1 $ x'
F 'u2
where x' D 0:2 is the nominal steady-state sugar fraction andF ' D 2 kg=s is the nominal amount
! We the obtain the corresponding transfer function matrix,
G.s/ D
24 1 1
1 $ x'
F ' $ x'
F '
35 D
"1 1
0:4 $0:1
#
! Constructing the RGA we obtain (at all frequencies)
& D"
x' 1 $ x'
1 $ x' x'
#D
"0:2 0:8
0:8 0:2
#
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–22
! For de-coupled control, we should pair on the off-diagonal elements;i.e., use u1 to control y1.
– Physically, this corresponds to using the largest stream (water, u2)to control the amount (y1 D F )
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–23
Example 6.9: Frequency-dependent RGA
! The following model describes a large pressurized vessel (e.g., asused in offshore oil-gas separations)
– Inputs are: (i) valve positions for liquid (u1), and (ii) vapor (u2) flow
– Outputs are: (i) liquid volume (y1), and (ii) pressure (y2)
! Transfer function matrix, G.s/ D0:01e$5s
.s C 1:72 10$4/ .4:32s C 1/
"$34:54 .s C :0572/ 1:913
$30:22s $9:188!s C 6:95 10$4
"#
! The RGA matrix depends on frequency
– At steady-state (s D 0), we see that g21 D 0; hence ƒ.0/ D I ,suggesting diagonal pairing should be used
– However, examining other frequencies we see that off-diagonalterms can approach 1
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–24
Introduction to MIMO robustness
! An example is presented to motivate the need for a deeperunderstanding of robustness
Example 6.10: Spinning satellite
! Consider the following model of angular velocity control of a satellitespinning about one of its principal axes:
G.s/ D 1
s2 C a2
"s $ a2 a .s C 1/
$a .s C 1/ s $ a2
#I a D 10
– A minimal state-space realization is given by:
"A B
C D
#D
266664
0 a 1 0
$a 0 0 1
1 a 0 0
$a 1 0 0
377775
! The plant has a pair of j!-axis poles at s D ˙ja , so it needs to bestabilized
! For stabilization, try the simple diagonal constant controller K D I ;this gives
T .s/ D GK .I C GK/$1 D 1
s C 1
"1 a
$a 1
#
! NOMINAL STABILITY. Two closed-loop poles are located at s D $1, soit is stable
– Examining the closed-loop system,
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–25
Acl D A $ BKC D"
0 a
$a 0
#$
"1 a
$a 1
#D
"$1 0
0 $1
#
! NOMINAL PERFORMANCE. The frequency-dependent singular valuesof L D GK D G are shown below.
– Note that ! .L/ D 1 at low frequencies and starts dropping offaround ! D 10 rad=sec
– Since ! .L/ never exceeds 1 , we do not have tight control in thelow-gain direction
– Also, the large off-diagonal elements in T .s/ show that we havestrong interaction in the closed-loop system
! ROBUST STABILITY. Check stability one loop at a time – here we havebroken the loop at the first input
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–26
G
K
+
+
!
!
z1 w1
! Loop transfer function –z1
w1, L1.s/ D 1
s) GM D 1; PM D 90ı
! Good robustness? Let’s check.
– Consider input gain uncertainty, and let (1 and (2 denote therelative gain error in each input channel.
u01 D .1 C (1/ u1; u
02 D .1 C (2/ u2
– The resulting change in the state-space B-matrix is
B 0 D"
1 C (1 0
0 1 C (2
#
– The corresponding closed-loop A-matrix is
A0cl D A $ BKC D
"0 a
$a 0
#$
"1 C (1 0
0 1 C (2
# "1 a
$a 1
#
– And characteristic polynomial –
det!dI $ A0
cl
"D s2 C .2 C (1 C (2/ s
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–27
C1 C (1 C (2 C!a2 C 1
"(1(2
– Stability conditions are:$1 < (1 < 1
(2 D 0
and$1 < (2 < 1
(1 D 0
– But only small simultaneous changes in the two channels: e.g., let(1 D $(2, then the system is unstable (a0 < 0) for
j(1j >1p
a2 C 1) 0:1
! Summary:
– Checking single-loop margins is inadequate for MIMO problems
– Large values of N!.T / or N!.S/ indicate robustness problems
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–28
Characteristic Loci
Introduction
! Recall that for a square system,
det ŒG .s/' D ˛z .s/
p .s/
where z.s/ and p.s/ are the zero and pole polynomial of G.s/ after thedeterminant has been adjusted to have p.s/ as its denominator.
! For a unity feedback system, the return difference matrix is defined asF .s/ D I C G .s/ K .s/ D I C L .s/
! In terms of the closed-loop transfer function matrix relating r to y , wehave
T .s/ D .I C L .s//$1 L .s/ D L .s/ .I C L .s//$1 D L .s/ F .s/$1
! This expression for T .s/ gives a multivariable generalization of thescalar relationship,
t .s/ D L .s/
1 C L .s/D L .s/ .1 C L .s//$1 D L .s/ .1 C L .s//$1
! We’ve seen that the closed-loop poles are determined by the poles of.I C L.s//$1and hence the zeros of I C L.s/ D F.s/
– Consider the free response of the closed-loop output, y.s/, (withr.s/ D 0)
y.s/ D L.s/e.s/ D $L.s/y.s/
.I C L.s// y.s/ D F.s/y.s/ D 0
– Clearly this expression is only satisfied for values of s for whichdet fF.s/g D 0; thus, closed-loop poles are described by the zerosof F.s/
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–29
– Since the poles of F.s/ are the same as the poles of L.s/; we maywrite
det fF.s/g D pc.s/
po.s/D closed loop characteristic polynomial
open loop characteristic polynomial
! Consider the scalar relationships,
g.s/ D n.s/
d.s/
t.s/ D g.s/
1 C g.s/D n.s/
n.s/ C d.s/
! Then the open- and closed-loop polynomials, po.s/ D d.s/, andpc.s/ D n.s/ C d.s/ satisfy
pc.s/
po.s/D n.s/ C d.s/
d.s/D 1 C n.s/
d.s/D 1 C g.s/ D f .s/
– Intuitively we expect det fF.s/g to be the vehicle for thegeneralization of the Nyquist criterion to the MIMO case
! We define the CHARACTERISTIC GAIN FUNCTION, g.s/ of G.s/ to bethe natural extension of the graphical Nyquist criterion
– The eigenfunctions of G.s/; denoted gi.s/, define the branches ofg.s/ and form the generalization of the scalar loop gain g.s/
– As such, g.s/ obeys the fame fundamental open- to closed-looprelationship, namely ti .s/ D gi .s/=1Cgi .s/
! The frequency response plots of the gi.s/; gi.j!/, are called the“characteristic loci” of G.s/
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–30
! Using these functions, multivariable analysis and design can becarried out employing well-established SISO techniques ofgain-phase margins, bandwidth, dc gain, etc....
Physical motivation
! Extending the idea of scalar transfer functions to transfer matrices, wehave for a 2 % 2 system,
y1 D g11 .j!/ u1 C g12 .j!/ u2
y2 D g21 .j!/ u1 C g22 .j!/ u2
– In general, each component of u is stretched and rotated by adifferent amount – so it is not possible to use any arbitrary inputvector phasor to define a multivariable gain
– We need a special input phasor u D w that produces an outputphasor y that is a scalar multiple of the input phasor, i.e.,
G .j!/ w D gw
– Thus the concept of gain we are after emerges as the n
eigenvalues of G.j!/; gi.j!/ for i D 1; 2; : : : ; n
– The corresponding phasors are the eigenvectors wi .j!/
associated with the gi.j!/
! Repeated eigenvalue calculations at a number of frequency pointsresult in n plots called the characteristic loci (CL) of G.s/
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–31
Example 6.11
! Consider a two-tank system with transfer function matrix
G.s/ D
266664
3
s C 3
1:5
s C 3
1
s C 6
$1
s C 6
377775
! To calculate the CL, we select values of s D j!, compute G.j!/,perform an eigenvalue-vector decomposition and plot in polar form:
Open- and closed-loop characteristic loci
! We may derive closed-loop characteristic loci directly from theopen-loop form by considering the spectral decomposition of G.j!/
and T .j!/
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–32
! It is easy to show that under unity feedback,G .j!/ D W .j!/ ƒG .j!/ V .j!/
T .j!/ D W .j!/ ƒT .j!/ V .j!/
and thatƒT .j!/ D .I C ƒG .j!//$1
D
2666666664
g1 .j!/
1 C g1 .j!/g2 .j!/
1 C g2 .j!/: : :
gn .j!/
1 C gn .j!/
3777777775
! This gives rise to two important results:
– The closed-loop CL, ti .j!/, are related to the open-loop CL,gi.j!/, in the same simple way as for SISO systems
– The eigenframes of G.j!/ remain invariant under unity feedback
! A corollary to this is that a MIMO system is stable under unityfeedback if its open-loop CL satisfy the classical Nyquist stabilitycriterion
Characteristic gain and the characteristic locus
Example 6.12
! Consider the following transfer function matrix,
G .s/ D 10
.s C 1/ .s C 2/
"3s C 4 $ .4s C 6/
2s C 3 $ .3s C 5/
#
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–33
! Analytically computing the eigenfunctions, we get
det fgI $ G.s/g D 0
D g2 C 10
.s C 1/ .s C 2/g $ 100
.s C 1/ .s C 2/D 0
! Solving gives the two eigenfunctions,
g1.s/ D 10
s C 2
g2.s/ D $10
s C 1
! Computing the eigenvector functions, we obtain
w1.s/ D"
g1.s/ $ g22.s/
g21.s/
#D 20s C 30
.s C 1/ .s C 2/
"2
1
#
w2.s/ D"
g2.s/ $ g22.s/
g21.s/
#D 20s C 30
.s C 1/ .s C 2/
"1
1
#
which gives,
W D"
2 1
1 1
#
! The system of Example 6.12 was especially constructed to giveeigenfunctions which turn out to be rational functions of s – but this isnot the case in general
! Indeed, if we perturb the matrix just slightly, we get a different result:
G 0 .s/ D 10
.s C 1/ .s C 2/
"3s C 4 C ( $ .4s C 6/
2s C 3 $ .3s C 5/
#
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–34
– Solving for the eigenfunctions,
det fgI $ G.s/g D 0
D g2 C 10 C 10(
.s C 1/ .s C 2/g $ 100 $ ( .3s C 5/
.s C 1/2 .s C 2/2D 0
– So that,
g1;2.s/ D $ .10 C 10(/
2 .s C 1/2 .s C 2/2
˙
q%.10 C 10(/2 C 4 .100 $ ( .3s C 5//
&.s C 1/2 .s C 2/2
2 .s C 1/2 .s C 2/2
! The discriminant (under the square root) is no longer a perfect square– thus the equation is irreducible over the field of rational functions
– Because of this, g defined on the complex plane is multi-valued –i.e., closed curves on the s-plane may not map to closed curves onthe complex g-plane
– A full treatment of this effect requires the introdution of Riemannsurfaces and will not be addressed here
Generalized Nyquist Stability Criterion
! The forgoing development leads to a multivariable generalization ofthe SISO Nyquist criterion and may be stated as follows:
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–35
A multivariable system is closed-loop stable (under unityfeedback) if and only if the net sum of critical point encir-clements by the CL of the transfer function matrix G.s/ isin a counterclockwise sense and is equal to the number ofopen-loop unstable poles. For open-loop stable systemsthe net sum of critical point encirclements must be zero.
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–36
Multivariable interaction
! In general a signal being applied to the i th input will cause all theoutputs (j D i and j ¤ i ) to respond – this defines interaction
! It is generally desirable to suppress interaction so that the i th inputexcites mostly the i th output while the response of the j ¤ i inputsremain below acceptable levels
! In principal, it is possible to supress interaction by ensuring themoduli of the CL be sufficienly large, because then ti ) 1
– But this cannot normally be done due to stability concerns
! An alternative approach involves the characteristic direction(eigenvector function) set
– Low interaction requires the standard basis directions ei to bereproduced at the outputs – which demands that vector ei shouldbe an eigenvector of T .s/ , or equivalently of L.s/
– Thus a sufficient condition for low interaction is that thecharacteristic direction set of L.j!/ approach the standard basisset
– One may take as a convenient measure of interaction the anglebetween the vectors wk.j!/ and the standard basis directions ei
cos $i .j!/ Dˇ̌w'
k .j!/ ei
ˇ̌wk'.j!/w
'k .j!/
where the $i are termed the “misalignment angles”.
ı The convention is adopted that k is chosen so that wk is theeigenvector that makes the smallest angle $i with ei atfrequency !
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
ECE5580, Introduction to Multivariable Control 6–37
! In summary,
– Interaction is determined at low frequencies by $i and/or moduli ofthe CL of the open-loop operator
– Interaction is determined at high frequencies by the $i
Lecture notes prepared by M. Scott Trimboli. Copyright c# 2016, M. Scott Trimboli
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