the use of mathematica in control engineering linear model descriptions linear model transformations...
TRANSCRIPT
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The Use of Mathematica in
Control Engineering
• Linear Model Descriptions
• Linear Model Transformations
• Linear System Analysis Tools
• Design/Synthesis Techniques
• Pole Assignment• Model-Reference Optimal Control• PID Controller
• Concluding Remarks
Neil MunroControl Systems Centre
UMISTManchester, England.
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Linear Model Descriptions
The Control System Professional currently provides
several ways of describing linear system models; e.g.
1 For systems described by the state-space equations
DuCxy
BuAxx
where y is a vector of the system outputs
u is a vector of the system inputs
and x is a vector of the system state-variables
2 For systems described by transfer-function relationships
)s(u)s(G)s(y
where s is the complex variable
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Examples: -
ss = StateSpace[{{0, 0, 1, 0},{0, 0, 0, 1},
{-(a b), 0, a+b, 0},{0, -(a b), 0, a+b}},
{{0, 0},{0, 0},{1, 0},{0, 1}},{{-b, -a, 1, 1}}]
x11aby
u
10
01
00
00
x
ba0)ba(0
0ba0)ba(
1000
0100
x
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tf = TransferFunction[s,{{1/(s-a),1/(s-b)}}]
sb
1
sa
1)s(G
}}]sb
1sa
1{{nction[s,TransferFu
11abC
10
01
00
00
B
ba0)ba(0
0ba0)ba(
1000
0100
A
viewFormRe//
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New Data Formats have been implemented, for these objects, which are fully editable, as follows: -
ss = StateSpace[{{0, 0, 1, 0},{0, 0, 0, 1},
{-(a b), 0, a+b, 0},{0, -(a b), 0, a+b}},
{{0, 0},{0, 0},{1, 0},{0, 1}},{{-b, -a, 1, 1}}]
now results in the composite data matrix
S
s
0011ab
10ba0)ba(0
010ba0)ba(
001000
000100
DC
BAss
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tfsys = TransferFunction[s,
{{((s+2)(s+3))/(s+1)^2,1/(s+1)^2},
{(s+2)/(s+1)^2,(s+1)/((s+1)^2(s+3))},
{1/(s+2),1/(s+1)}}]
which now yieldsikHs+2LHs+3LHs+1L2 1Hs+1L2
s+2Hs+1L2 1Hs+1LHs+3L1
s+2
1
s+1
y{T
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New Data Objects
Four new data objects have been introduced; namely,
1. Rosenbrock’s system matrix in polynomial form
2. Rosenbrock’s system matrix in state-space form
3. The right matrix-fraction description of a system
4. The left matrix-fraction description of a system.
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u)s(W)s(Vy
u)s(U)s(T
The system matrix in polynomial form provides a compact description of a linear dynamical system described by arbitrary ordered differential equations and algebraic relationships, after the application of the Laplace transform with zero initial conditions; namely
where , u and y are vectors of the Laplace transformed system variables. This set of equations can equally be written as
y
0
u)s(W)s(V
)s(U)s(T
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If the system description is known in state-space form then a special form of the system matrix can be constructed, known as the system matrix in state-space form, as shown below
DC
BAsI)s(P
The system matrix in polynomial form is then defined as
)s(W)s(V
)s(U)s(T)s(P
When a system matrix in polynomial form is being created, it is important to note that the dimension of the square matrix T(s) must be adjusted to be r, where r is the degree of Det[T(s)].
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Matrix Fraction Forms
Given a system description in transfer function matrix form G(s), for certain analysis and design purposes; e.g. the H- approach to robust control system design; it is often convenient to express this model in the form of a left or right matrix-fraction description; e.g.
1. A left matrix-fraction form of a given transfer function matrix G(s) might be
2 A right matrix fraction form of a given transfer function matrix G(s) might be
)s(D)s(N)s(G 1RR
)s(N)s(D)s(G L1
L
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Linear Model Transformations
G(s) [A, B, C, D]
G(s)
System Matrix P(s) in polynomial form
[A, B, C, D]System Matrix P(s) in state-space form
[T(s),U(s),V(s),W(s)]System Matrix P(s) in polynomial form
)s(1RD)s(RNor)s(LN)s(1
LD
P1(s) P2(s)
G(s)
Least Order
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New Data Transformations
All data formats are fully editable
tfsys = TransferFunction[s,
{{((s+2)(s+3))/(s+1)^2,1/(s+1)^2},
{(s+2)/(s+1)^2,(s+1)/((s+1)^2(s+3))},
{1/(s+2),1/(s+1)}}]
ikHs+2LHs+3LHs+1L2 1Hs+1L2
s+2Hs+1L2 1Hs+1LHs+3L1
s+2
1
s+1
y{T
rff = RightMatrixFraction[tfsys]
ikHs+ 2L2Hs + 3Ls +3Hs+ 2L2 s +1Hs+ 1L2 Hs +1LHs + 3L
y{ikHs+ 1L2Hs + 2L 0
0 Hs +1L2Hs +3Ly{- 1¤F
ss = StateSpace[tfsys]i
k
0 1 0 0 0 0 0 00 0 1 0 0 0 0 0
- 2 - 5 - 4 0 0 0 1 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 - 3 - 7 - 5 0 1
10 11 3 3 1 0 1 04 4 1 1 1 0 0 01 2 1 3 4 1 0 0
y
{•
S
rff = RightMatrixFraction[ss,s]
ikHs + 2LHs2 + 5s+ 6Ls +3Hs+ 2L2 s +1Hs+ 1L2 Hs +1LHs + 3L
y{ikHs+ 1L2Hs + 2L 0
0 Hs +1L2Hs +3Ly{- 1¤F
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tfsys = TransferFunction[s, {{(s+1)/(s^2+2s+1)},{(s+2)/(s+1)}}]
ps = SystemMatrix[tfsys,TargetFormRightFraction]
rff = RightMatrixFraction[ps]
TransferFunction[%]
ps = SystemMatrix[tfsys,TargetForm LeftFraction]
lf = LeftFractionForm[ s,s10
0ss21,
s2
s1 2
]
iks+1
s2+2 s+1s+2
s+1
y{T
J1s+ 2NHs +1L- 1¤F
ik1
s+1s+2
s+1
y{T
ik
1 0 0 0
0 s2 + 2s +1 0 s + 10 0 s+ 1 s + 20 - 1 0 00 0 - 1 0
y{M
iks2 +2s + 1 00 s +1
y{- 1Js +1s +2N¤F
ik
1 0 0
0 Hs +1L2 1
0 - s - 1 0
0 -Hs +1LHs +2L0
y{M
New Data Transformations
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New Data Transformations
tfsys =TransferFunction[s, {{(s+1)/(s^2+2s+1)},{(s+2)/(s+1)}]iks+1
s2+2 s+1s+2
s+1
y{T
rff = RightMatrixFraction[tfsys]
J s+1Hs+1LHs+2LNHHs+1L2L- 1ĤÄF
dt = ToDiscreteTime[tfsys,Sampled->20]//Simplifyik
- 1+ã20
ã20 z- 1
ã20 z+ã20- 2
ã20 z- 1
y{20
T
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ik- 1 +ã20
ã20 z+ã20 - 2
y{Hã20 z - 1L- 1ĤÄ20
F
SystemMatrix[dt,TargetForm->RightFraction]ik
ã20 z - 1 1
1 - ã20 0
- ã20 z - ã20 +2 0
y{20
M
SystemMatrix[dt]
ikã20 z - 1 0 - 1+ã20
0 ã20 z - 1 ã20 z +ã20 - 2
- 1 0 00 - 1 0
y{20
M
RightMatrixFraction[%]
New Data Transformations
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A Least-Order form of a System-Matrix in polynomial form is one in which there are no input-decoupling zeros and no output-decoupling zeros, and would yield a minimal state-space realization, when directly converted to state-space form.
For example, the polynomial system matrix
SM,
.
2
2
2232
01000
01000
10)2s(s11s3s)2s(s
10)2s(s2s3s)2s(s
00)1s(s11ss)1s(s
)s(P
is not least order, since T(s) and U(s) have 3
input-decoupling zeros at s = {0, 0, -1}; i.e.
[T(s) U(s)] has rank 4 at these values of s.
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2s
1
)s(W)s(U)s(T)s(V
)s(W)s(U)s(T)s(V)s(G
21
22
11
11
Hence
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System Analysis
Controllable[ss]Observable[ss] ss =[A, B, C, D]
SmithForm[T(s) U(s)]McMillanForm[G(s)]
Controllable[ps]Observable[ps]
DC
BAsI)s(P
)s(W)s(V
)s(U)s(T)s(P
Decoupling Zeros
Controllable[ps]Observable[ps]
DC
BAsI)s(P
Decoupling Zeros
)s(W)s(V
)s(U)s(T)s(P
MatrixLeftGCD[T(s) U(s)]MatrixRightGCD[T(s) V(s)]
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Controllability and Observability
In the same way that the controllability and observability of a system described by a set of state space equations can be determined in the Control System Professional by entering the commands
Controllable[ss] and Observable[ss]
where ss is a StateSpace object.
These tests can now also be directly applied to a system matrix object by entering the commands
Controllable[sm] and Observable[sm]
where sm is a SystemMatrix object in either polynomial form or state space form.
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Preliminary Analysis
• Reduction of State-Space Equations
• Given a system matrix in state-space form
• then an input-decoupling zeros algorithm,
• implemented in Mathematica, reduces P(s) to
• The completely controllable part is then given by
DCBAsI
)s(P n
DCCBAsIA
00AsI)s(P
21
222n21
m,n,11
1
222n B,AsI
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PSINK
TGASPGAS
CVGAS
WCHR
MASSWAIRWSTMWCOLWLS
Gasifier
DuCxy
BuAxx
Model Format
A is 25 x 25 B is 25 x 6C is 4 x 25 D is 4 x 6
Inputs:- 1 char 2 air 3 coal 4 steam 5 limestone 6 upstream disturbance
Outputs:- 1 gas cv 2 bed mass 3 gas pressure 4 gas temperature
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Preliminary AnalysisPreliminary Analysis
The original 25th order system is numerically very ill conditioned. The eigenvalues cover a significant range in the complex plane, ranging from -0.00033 to -33.1252.The condition number is 5.24 x 1019.At = 0 the maximum and minimum singular values are 147500 and 50, respectively.The Kalman controllability and observability tests yield a rank of 1, and the controllability and observability gramians are :-
5
4
3
3
3
2
2
2
0
4
14
o
1011.9
1002.7
1022.1
1078.1
1015.6
1029.2
1029.3
1029.5
1049.4
1084.2
1047.4
W
SM,
SM,
.
2
2232
2
.
2
2
2232
1
01000
01000
00111
10)2s(s2s3s)2s(s
00)1s(s11ss)1s(s
)s(P~
01000
01000
10)2s(s11s3s)2s(s
10)2s(s2s3s)2s(s
00)1s(s11ss)1s(s
)s(P
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Preliminary Analysis
05677.0000000
005677.000000
0005677.00000
0000002426.0000
000005677.000
0000005677.00
00000005677.0
A z
Application of the decoupling zeros algorithm to [sI-A, B] yielded
indicating that the system had 7 input-decoupling zeros,which was confirmed by transforming A and B to spectral form.
Dimensions of )31,29(DC
BA
)24,22(DC
BA
rr
rr
)7,7(A z
Dimensions of
Dimensions of
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Coprime Factorizationsps = SystemMatrix@t, u, v, w, sDtest = MatrixLeftGCD@s, t, uD;Print@"L = ", MatrixForm@test@@1DDDD;Print@"T now = ", MatrixForm@test@@2DDDD;Print@"U now = ", MatrixForm@test@@3DDDD;Expand@test@@1DD.test@@2DDD=== Expand@tDSimplify@test@@1DD.test@@3DDD=== u
Det@test@@2DDDik
s2Hs+ 1Ls3 + s2 - 1 1- s2Hs+ 1L0 0
sHs +2Ls2 + 3s +2 - sHs + 2L0 1
sHs +2Ls2 + 3s +1 1 - sHs +2L0 10 0 0 1 00 0 0 1 0
y{•M , s
L =
ik1 0 0 00 1 0 01 1 s2H1+sL00 0 0 1
y{
T now =
iks2H1+sL- 1+s2+s3 1 - s2 - s3 0
sH2+sL2+3s+s2 - sH2+sL0- 1 - 1 1 00 0 0 1
y{
U now =
ik0100
y{
True
True
2+s
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Smith and McMillan Forms
The Smith form of a polynomial matrix and the McMillan form of a rational polynomial matrix are both important in control systems analysis.
Consider an x m polynomial matrix N(s), then the Smith form of N(s) is defined as
S(s) = L(s)N(s)R(s)
m,0
)}s({diag)s(S
m,)}s({diag)s(S
m,0)}s({diag)s(Swhere
m,m
i
i
m,i
and L(s) and R(s) are unimodular matrices.
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Smith and McMillan Forms
Consider now an x m rational polynomial matrix G(s), and let
G(s) = N(s)/d(s)
where d(s) is the monic least common denominator of G(s), then the McMillan form of G(s) is defined as
m,0
)}s(/)s({diag)s(S
m,)}s(/)s({diag)s(S
m,0)}s(/)s({diag)s(M
m,m
ii
ii
m,ii
where M(s) is the result of dividing the Smith form of N(s) by d(s), and cancelling out all common factors
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Pole Assignment
PID Controller
Nonlinear Systems
Model-Order Reduction
Robust NA
Nyquist Array
Optimal Control
Model Ref.Opt. Control
Design MethodsSynthesis Methods
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Design/Synthesis Methods
Methods implemented are:-
1 Pole Assignment - Some Observations
2 Model-Reference Optimal Control
3 The Systematic Design of PID Controllers
4 Uncertain Nonlinear Systems
5 Robust Direct Nyquist Array Design Method
6 Model-Order Reduction
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Pole Assignment
Control Systems Centre - UMIST
We consider four main types of approaches
• ACKERMANN’S FORMULA
• SPECTRAL APPROACH
• MAPPING APPROACH
• EIGENVECTOR METHODS
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Control Systems Centre - UMIST
Ackermann’s Formula
)(100 1 Apc k
Here is the controllability matrix of [A,b], and pc(s) is the desired closed-loop system characteristic polynomial.
ivk
q
ii
1
where
b.v, iiq
ij
1j jii
q
1j ji
i
Spectral Approach
Here, i and i are the open-loop system and desired closed-loop system poles, respectively, and the vi´ are the associated reciprocal eigenvectors.
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Mapping Approach
Control Systems Centre - UMIST
The state-feedback matrix is given as
11 Xk
where is the controllability matrix of [A, b]
002n2n1n1n
321
1n2n
1n
1n
aaa
1aaa
01aa
001a
0001
X
bAbAb
Here, the ai and i are the coefficients of the open-loop and closed-loop system characteristic polynomials, respectively.
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Control Systems Centre - UMIST
Eigenvector Methods
buic
1 nii IAm
It is also possible to determine the state-feedback pole assignment compensator as
where the mi are randomly chosen scalars and the uci
are the closed-loop system
eigenvectors calculated from
121 2
nnmmm ccc uuuk
1
Selecting mi =1, for example, the state-feedback compensator can be found as
1
2111
nccc uuuk1
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Control Systems Centre - UMIST
-16
-14
-12
-10
-8
-6
-4
-2
0
2
2 4 6 8 10 12 14 16 18 20
Order
Err
or
Spectral
Mapping
Ackermann's
EVAssign
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
2 4 6 8 10 12 14 16 18 20
Order
Tim
e [s
]
Comparison of Dyadic Methods Comparison of Dyadic Methods under Numeric Considerationsunder Numeric Considerations
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Control Systems Centre - UMIST
Comparison of Dyadic Methods Comparison of Dyadic Methods under Symbolic Considerationsunder Symbolic Considerations
-5
15
35
55
75
95
115
135
2 4 6 8 10 12 14
Order
Tim
e [s
]
Mapping
Ackermann's
EV Assign
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Model Reference LQR System
yM
y u
-
+
e
System Model
Reference Model
x~]CC[
CxxC
yye
M
MM
M
o
tt dt]RuuQee[J
Model-Reference Optimal Control
The resulting optimal feedback controller Ko can be partitioned as
2221
1211
oKK
KKK
where K11 and K21 operate on the reference-model state vector xM
and K12 and K22 operate on the system state vector x.
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Model Reference LQR feedback paths.
e
-
-
+ +
+
-
+
+
+ - -
+
r
y B dtI
A
B M
C
K 2 1
K 2 2
K 1 2
y C M
A M
K 11
dtI
-
+
-
+ y r System x
K22
xM K21 Model
Model Reference LQR System Closed-Loop.
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)s41(
1
)s51(
7.0
)s51(
3.0
)s51(
2.0
)s51(
6.0
)s41(
1
)s51(
4.0
)s51(
35.0
)s51(
35.0
)s51(
4.0
)s41(
1
)s51(
6.0
)s51(
2.0
)s51(
3.0
)s51(
7.0
)s41(
1
)s(G
)s1(
1000
0)s5.01(
100
00)s5.01(
10
000)s1(
1
)s(M
10000000000
01000000000
00100000000
00010000000
000001.0000
0000001.000
00000001.00
000000001.0
Q
01.00000000
001.0000000
0001.000000
00001.00000
00001000000
00000100000
00000010000
00000001000
R
![Page 38: The Use of Mathematica in Control Engineering Linear Model Descriptions Linear Model Transformations Linear System Analysis Tools Design/Synthesis Techniques](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649e3b5503460f94b2d5f9/html5/thumbnails/38.jpg)
Unit-step on reference input 1 Unit-step on reference input 2
1 2 3 4 5
0.2
0.4
0.6
0.8
1 2 3 4 5
0.2
0.4
0.6
0.8
1
Unit-step on reference input 3 Unit-step on reference input 4
1 2 3 4 5
0.2
0.4
0.6
0.8
1
1 2 3 4 5
0.2
0.4
0.6
0.8
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• In recent years, several researchers have been re-examining the PID controller to determine the limiting Kp, Ki, and Kd parameter values to guarantee a stable closed-loop system; namely,
• Keel and Bhattacharyya
• Ho, Datta, and Bhattacharyya
• Shafei and Shenton
• Astrom and Hagglund
• Munro and Soylemez
The PID Controller
![Page 40: The Use of Mathematica in Control Engineering Linear Model Descriptions Linear Model Transformations Linear System Analysis Tools Design/Synthesis Techniques](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649e3b5503460f94b2d5f9/html5/thumbnails/40.jpg)
The PID Controller
+-
uer yPID Plant
+
+
+e u
pK
dt
dKd
dtKi
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- + +
-
u e r y Plant
2-D Test Compensator k
Kx
Test compensator arrangement
60s3s15s29s3s
1s2s6s)s(g
2345
23
Ki
Kp
+1.0
+1.0 -1.0
Test compensator space
![Page 42: The Use of Mathematica in Control Engineering Linear Model Descriptions Linear Model Transformations Linear System Analysis Tools Design/Synthesis Techniques](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649e3b5503460f94b2d5f9/html5/thumbnails/42.jpg)
The Nyquist plot for Kp = 0.5 and Ki = 0.5
The admissible PI compensator space
![Page 43: The Use of Mathematica in Control Engineering Linear Model Descriptions Linear Model Transformations Linear System Analysis Tools Design/Synthesis Techniques](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649e3b5503460f94b2d5f9/html5/thumbnails/43.jpg)
0
5
10Kd
0
25
50
75
100Ki
0
2.5
5
7.5
10
Kp
0
5
10Kd
0
25
50
75
100Ki
27s54s36s10s
27
)3s)(1s(
27)s(g
234
3
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Design Requirements
• Stability
• Performance
• Robustness
• Simplicity
• Transparency
increasing
difficulty
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Acknowledgements
My thanks to Dr Igor Bakshee of Wolfram Research for his interest and help in carrying out this work.
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D-Stability
Im
Re
d
D
= Sin()
Control Systems Centre
UMIST
Control Systems Centre
UMIST
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The Nyquist Plot Approach
• Here, we detect 5 axis crossings, (-2,+2,+2,-1,-1), where the last is due to the infinite arc, on the right, due to the pole at the origin.
Control Systems Centre
UMIST
Control Systems Centre
UMIST
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The resulting stability boundary is
The Nyquist Plot Approach
Note that the origin is not included in the region because the basic system is unstable.
0 5 10 15 20
0
5
10
15
20
25
30
Kp
Ki
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The Nyquist Plot Approach
• Here, with Kp = 5 and Ki = 18 the system is stable, even with an additional gain of k =1.3134
• yielding closed-loop poles =
• -0.2519 ± 5.4879i
• -1.2320 ± 1.5258i
• -0.0161 ± 0.4510i
Control Systems Centre
UMIST
Control Systems Centre
UMIST
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Diagonal Dominance Concepts
Various definitions of Diagonal Dominance exist, namely :-
Rosenbrock’s row/column form ~ R Limebeer’s Generalised Diagonal Dominance ~ L Bryant & Yeung’s Fundamental Dominance ~ Y
where the conservatism of the resulting dominance criterion reduces as
Y < L < RMathematica Code Code
tfsys = TransferFunction@s,883 �Hs+2L, 1 �Hs+2L, 1 �Hs+2L<,81 �Hs+1L, 8 �Hs+4L, 1 �Hs+2L<,81 �Hs+2L, 3 �Hs+2L, 10 �Hs+5L<<Dfreqs = [email protected], 30, 50D;NyquistArray@tfsys, freqs,DominanceCriterion - > Column,
CircleCriterion - > Ostrowski,
DominanceSteps - > 1,
FeedbackGains - >81, 0.5, 2<D
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Nyquist Array Example
ik
3s+2
1s+2
1s+2
1s+1
8s+4
1s+2
1s+2
3s+2
10s+5
y{T
Ostrowski circles are shown in red for gains =81, 0.5, 2<Nyquist Array with Ostrowskicircles
0.1 0.2 0.3 0.4 0.5
-0.25-0.2-0.15-0.1-0.05 0.20.40.60.8 1 1.21.4
-0.7-0.6-0.5-0.4-0.3-0.2-0.1
-0.5 0.5 1 1.5 2 2.5 3
-1
-0.5
0.5
1
0.2 0.4 0.6 0.8 1
-0.5-0.4-0.3-0.2-0.1
-2 -1 1 2 3 4
-2
-1
1
2
0.1 0.2 0.3 0.4 0.5
-0.25-0.2-0.15-0.1-0.05
-1 1 2 3
-1.5
-1
-0.5
0.5
1
1.5
0.1 0.2 0.3 0.4 0.5
-0.25-0.2-0.15-0.1-0.05 0.1 0.2 0.3 0.4 0.5
-0.25-0.2-0.15-0.1-0.05
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PSINK
TGASPGAS
CVGAS
WCHR
MASSWAIRWSTMWCOLWLS
Gasifier
DuCxy
BuAxx
Model Format
A is 25 x 25 B is 25 x 6C is 4 x 25 D is 4 x 6
Inputs:- 1 char 2 air 3 coal 4 steam 5 limestone 6 upstream disturbance
Outputs:- 1 gas cv 2 bed mass 3 gas pressure 4 gas temperature
![Page 53: The Use of Mathematica in Control Engineering Linear Model Descriptions Linear Model Transformations Linear System Analysis Tools Design/Synthesis Techniques](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649e3b5503460f94b2d5f9/html5/thumbnails/53.jpg)
Combined Sequential Loop Closing and Diagonal Dominance Method
Combined Sequential Loop Closing and Diagonal Dominance Method
• This approach is a new combination of Bryant’s
Sequential Loop Closing Approach with MacFarlane
and Kouvaritakis’ ALIGN Algorithm, Edmunds’
Scaling and Normalization Technique, and
Rosenbrock’s Diagonal Dominance.
• It is particularly appropriate in cases where a simple
controller structure is desired.
• Advantages:
– It can be implemented by closing one loop at a
time.
– Usually, the resulting control scheme is quite
simple and can be easily realized in practice.
![Page 54: The Use of Mathematica in Control Engineering Linear Model Descriptions Linear Model Transformations Linear System Analysis Tools Design/Synthesis Techniques](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649e3b5503460f94b2d5f9/html5/thumbnails/54.jpg)
Achieving Diagonal DominanceAchieving Diagonal Dominance
• Normalization :
– Generates the input-output scaling to be applied to the
system in order to minimize interaction.
– Determines the best input-output pairing for control
purposes.
– Produces good diagonal dominance properties at low and
intermediate frequencies.
– Results are obtained by using simple, wholly real
permutation matrices.
• High frequency decoupling :
– Aims at improving the transient response of the system.
– Emphasis is on frequencies close to the bandwidth,
around which interaction is most severe.
– Results are obtained by making use of wholly real
matrices.
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Preliminary AnalysisPreliminary Analysis
The original 25th order system is numerically very ill conditioned. The eigenvalues cover a significant range in the complex plane, ranging from -0.00033 to -33.1252.The condition number is 5.24 x 1019.At = 0 the maximum and minimum singular values are 147500 and 50, respectively.The Kalman controllability and observability tests yield a rank of 1, and the controllability and observability gramians are :-
Wc
107 10
7 64 10
7 29 10
4 63 10
8 08 10
317 10
159 10
3 09 10
9 35 10
0
0
16
5
7
15
22
31
33
43
69
.
.
.
.
.
.
.
.
.
Wo
4 47 10
2 84 10
4 49 10
5 29 10
3 29 10
2 29 10
615 10
178 10
122 10
7 02 10
9 11 10
14
4
0
2
2
2
3
3
3
4
5
.
.
.
.
.
.
.
.
.
.
.
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Preliminary Analysis
05677.0000000
005677.000000
0005677.00000
0000002426.0000
000005677.000
0000005677.00
00000005677.0
A z
Application of the decoupling zeros algorithm to [sI-A, B] yielded
indicating that the system had 7 input-decoupling zeros,which was confirmed by transforming A and B to spectral form.
Dimensions of )31,29(DC
BA
)24,22(DC
BA
rr
rr
)7,7(A z
Dimensions of
Dimensions of
![Page 57: The Use of Mathematica in Control Engineering Linear Model Descriptions Linear Model Transformations Linear System Analysis Tools Design/Synthesis Techniques](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649e3b5503460f94b2d5f9/html5/thumbnails/57.jpg)
H* GEC PI Controller *Lkp = - 0.003; ki = - 0.00001;
pPlusi = Together@kp+ki � sDki � kpipcol = TakeColumns@b4,81, 1<D;oprow = TakeRows@c,82, 2<D;ss21 = StateSpace@a, ipcol, oprowD;clss = GenericConnect@TransferFunction@1D, TransferFunction@s, pPlusiD,ss21,882, 1,83, Negative<<,83, 2<<,81<,83<D;
tfrl = TransferFunction@s, pPlusighigh@@2, 1DDD;RootLocusPlot@tfrl,8k, 0, 0.25<, PlotPoints - >1000,
PlotRange - >88- 0.001, 0.0001<,8- 0.0005, 0.0005<<,PoleStyle - > [email protected],Epilog - > Line@880, 0<,8- zeta wn, wnSqrt@1 - zeta^2D<� .8wn - > 0.002, zeta - > .690107<<DD
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-0.001 -0.0008 -0.0006 -0.0004 -0.0002Re
-0.0004
-0.0002
0.0002
0.0004
Im
H* GEC PI Controller *LSimulationPlot@clss, UnitStep@tD,8t, 4500<,Sampled - > Period@5D, PlotRange - > All,
GridLines® AutomaticD� � Timing
1000 2000 3000 4000
0.2
0.4
0.6
0.8
1
1.2
87.481Second, … Graphics …<
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H* Moving the PI zero closer to the origin more gain *Lkp = - 0.003; ki = - 0.00001;
kp = - 0.01; ki = - 0.000002;
pPlusi = Together@kp+ki � sDki � kpipcol = TakeColumns@b4,81, 1<D;oprow = TakeRows@c,82, 2<D;ss21 = StateSpace@a, ipcol, oprowD;clss = GenericConnect@TransferFunction@1D, TransferFunction@s, pPlusiD,ss21,882, 1,83, Negative<<,83, 2<<,81<,83<D;
tfrl = TransferFunction@s, pPlusighigh@@2, 1DDD;RootLocusPlot@tfrl,8k, 0, 1<, PlotPoints - >200,
PlotRange - >88- 0.003, 0.0001<,8- 0.0015, 0.0015<<,PoleStyle - > [email protected], Epilog - > Line@880, 0<,8- zeta wn, wnSqrt@1 - zeta^2D<� .8wn - > 0.002, zeta - > .690107<<DD
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H* Moving the PI zero closer to the origin *LSimulationPlot@clss, UnitStep@tD,8t, 4500<,Sampled - > Period@5D, PlotRange - > All,
GridLines® AutomaticD� � Timing
1000 2000 3000 4000
0.2
0.4
0.6
0.8
1
87.871Second, … Graphics …<
-0.003 -0.0025 -0.002 -0.0015 -0.001 -0.0005Re
-0.0015
-0.001
-0.0005
0.0005
0.001
0.0015
Im
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Design Procedure - 1Design Procedure - 1
• The Nyquist Array after an initial output scaling of diag{0.00001 , 0.001 , 0.001 , 0.1} looks like :
![Page 62: The Use of Mathematica in Control Engineering Linear Model Descriptions Linear Model Transformations Linear System Analysis Tools Design/Synthesis Techniques](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649e3b5503460f94b2d5f9/html5/thumbnails/62.jpg)
Design Procedure - 2Design Procedure - 2
• The Nyquist Array after swapping the first two outputs (calorific value of fuel gas and bedmass) and closing the bedmass/char off-take loop is :
![Page 63: The Use of Mathematica in Control Engineering Linear Model Descriptions Linear Model Transformations Linear System Analysis Tools Design/Synthesis Techniques](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649e3b5503460f94b2d5f9/html5/thumbnails/63.jpg)
Design Procedure - 3Design Procedure - 3
• The Nyquist Array of the 3 x 3 subsystem after normalisation and high frequency decoupling at = 0.001 rad/sec is (where the outputs are pressure, temperature and calorific value of fuel
gas) :
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Design SummaryDesign Summary
• Implement PI controller on bedmass/char-extraction loop.• Scale inputs and outputs, to normalize them.• Use ALIGN Algorithm for the remaining 3-input 3-output
subsystem.• Design a PI controller for the fast Calorific Value Loop.• Design a PI controller for the fast Pressure Loop.
• Design a Lag-Lead controller for the remaining slow
Temperature Loop.
Output Scaling
GasifierInput
Scalingconstant
decoupling
bedmass PI
temperaturecontrol
pressure PI
cv PI
The control scheme resulting from this approach is as follows :
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Controller
29.178.006.0
22.319.175.0
66.054.065.0
1
Constant Pre-compensator Constant Post-compensator
Dynamic Controller
)001.0s(s
)0005.0s(0015.0s
02.01
s
01.01
s
0001.0100
05.0
00001.0
0001.0
001.0
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Model Simplification
20 40 60 80 100
50
100
150
200
250
2500 5000 7500 10000 12500 15000
-4000
-3000
-2000
-1000
1000
2000
Green = high - order
80.991Second, Null<II3499.26s7 +238.081s6 +0.885487s5 +
0.00135158s4 +1.05604´ 10- 6 s3 + 4.42642´ 10- 10 s2 + 9.42066´ 10- 14 s+ 7.95078´ 10- 18M‘Is8 +0.455558s7 +0.0239052s6 + 0.0000735781s5 +
1.00076´ 10- 7 s4 + 7.42698´ 10- 11 s3 +3.1183´ 10- 14 s2 + 7.0044´ 10- 18 s + 6.55111´ 10- 22MM̂T80, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1<80, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1<
264 � Abs@ghigh@@3, 2DD� . s - > 0.1707ID0.0330784
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-0.001 -0.0005 0.0005Re
-0.0004
-0.0002
0.0002
0.0004
Im
Root-locus diagram of g1,1(s)
Model Simplification
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Theorem: By using just the first input of a givenMIMO system, it is almost always possible toarbitrarily assign 1 self conjugate poles of thesystem, and make these poles uncontrollable from the other inputs, provided that the system[A, b1,C] has 1 controllable and observablepoles, where b1 is the first column of the input matrix (B), where
m1
This result can be compared with a previous resultdeveloped by Munro and Novin-Hirbod (1979) for the case of dynamic output feedback, where the degree r of the necessary compensator is given by
),mmax(
)1m(nr