Post on 27-Dec-2015
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.
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
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
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//
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
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
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.
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
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)].
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
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
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
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
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
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
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.
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
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)]
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.
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
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
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
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
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
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.
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
Pole Assignment
PID Controller
Nonlinear Systems
Model-Order Reduction
Robust NA
Nyquist Array
Optimal Control
Model Ref.Opt. Control
Design MethodsSynthesis Methods
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
Pole Assignment
Control Systems Centre - UMIST
We consider four main types of approaches
• ACKERMANN’S FORMULA
• SPECTRAL APPROACH
• MAPPING APPROACH
• EIGENVECTOR METHODS
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.
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.
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
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
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
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.
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.
)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
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
• 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
The PID Controller
+-
uer yPID Plant
+
+
+e u
pK
dt
dKd
dtKi
- + +
-
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
The Nyquist plot for Kp = 0.5 and Ki = 0.5
The admissible PI compensator space
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
Design Requirements
• Stability
• Performance
• Robustness
• Simplicity
• Transparency
increasing
difficulty
Acknowledgements
My thanks to Dr Igor Bakshee of Wolfram Research for his interest and help in carrying out this work.
D-Stability
Im
Re
d
D
= Sin()
Control Systems Centre
UMIST
Control Systems Centre
UMIST
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
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
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
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 = logList@0.01, 30, 50D;NyquistArray@tfsys, freqs,DominanceCriterion - > Column,
CircleCriterion - > Ostrowski,
DominanceSteps - > 1,
FeedbackGains - >81, 0.5, 2<D
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
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
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.
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.
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
.
.
.
.
.
.
.
.
.
.
.
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
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 - > PointSize@0.01D,Epilog - > Line@880, 0<,8- zeta wn, wnSqrt@1 - zeta^2D<� .8wn - > 0.002, zeta - > .690107<<DD
-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 …<
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 - > PointSize@0.01D, Epilog - > Line@880, 0<,8- zeta wn, wnSqrt@1 - zeta^2D<� .8wn - > 0.002, zeta - > .690107<<DD
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
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 :
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 :
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) :
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 :
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
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
-0.001 -0.0005 0.0005Re
-0.0004
-0.0002
0.0002
0.0004
Im
Root-locus diagram of g1,1(s)
Model Simplification
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