The Use of Mathematica in Control Engineering Linear Model Descriptions Linear Model Transformations...

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



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- 1Ä¤ÄF

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- 1Ä¤Ä20

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[%]


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


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


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


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.


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


-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


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


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.


UMIST


UMIST

The resulting stability boundary is


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


• 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


UMIST


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 = [email protected], 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

.

.

.

.

.

.

.

.

.

.

.


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 - > [email protected],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 - > [email protected], 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 :


• 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 :


• 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

The Use of Mathematica in Control Engineering Linear Model Descriptions Linear Model Transformations...

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