The undertaking involves developing a simple intuitive methodological analysis for planing a accountant that would stabilise an open-loop unstable works. Two attacks, viz. ; Youla Parametrization and Feedback Stabilization ( two-degree of freedom ) have been identified through the feasibleness surveies by which the job can be tackled. The Internal Model Control construction is modified in the feedback stabilisation technique and simulations were implemented to corroborate the information gotten from the literatures. A undertaking program and methodological analysis as to how the undertaking will be executed so as to accomplish the purposes and aims within the allocated thesis period every bit good as the undertaking hazard and Health & A ; Safety Assessment involved were highlighted. The undertaking is disputing, but from the feasibleness surveies study, it is executable.

## 1.1 Introduction:

The undertaking is titled Simple Control for Open loop unstable workss. Unstable workss can be defined as: workss that yield boundless response when given a delimited input signal. For additive systems, they have a pole located in the unfastened right-half plane ( RHP ) ( Dorf and Bishop 1998, p.295 ) . The purpose of the undertaking is to develop a accountant through a simple intuitive methodological analysis that would stabilise such workss in closed cringle.

Two methods have been identified in developing a accountant to stabilise such unstable systems, they are ; ( I ) Feedback Stabilization or Two-degree of freedom ( Modified Internal Model Control ) . ( three ) Youla Parametrization

These methods will be evaluated in this survey and would organize a major activity during the thesis to detect their restrictions and benefits with the possibility of accomplishing control synthesis of the different methods to acquire the best control scheme.

## 1.2 Backgrounds:

As mentioned earlier on, an unstable works would give an boundless response when given a delimited input signal and for additive systems they have a pole in the unfastened right-half plane ( RHP ) . Examples of such systems are systems whose transportation maps are given as ;

; , etc..

For the intent of control action, the type of unstable system that would be used in this survey and in the existent thesis work is considered to be additive, uninterrupted, time-invariant and Single-input Single-output ( SISO ) .

Modified Internal Model Control ( IMC ) construction would be studied during the undertaking to see how it was applied to stabilise unstable systems. But before so a brief debut of the normal IMC construction for stable workss is being introduced in the following subdivision so as to heighten a better apprehension of its construction.

1.2.1 Internal Model Control: Internal theoretical account control ( IMC ) was developed by Garcia and Morari and they established its relationship with other control strategies, such as the Smith Predictor, LQG, etc. ( Garcia and Morari, 1982 ) .

G ( s )

Q ( s )

R u Y vitamin D

Gm ( s )

ym

Fig1.1 IMC construction by Garcia and Morari ( 1982, p.310 )

The diagram above shows the schematic of IMC for stable workss, where ; Q ( s ) – the IMC accountant, G ( s ) – the works, Gm ( s ) – the works Model, r – the mention input, y – the works end product, u – the control input, ym – the theoretical account end product, d – end product perturbations, – feedback signal. Garcia and Morari ( 1982, p.310 )

The feedback signal is given as ;

aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦.. ( 1 )

If the theoretical account is the same with the works i.e. and there are no perturbations ( d = 0 ) , so the theoretical account and the works would hold the same end product which means the feedback signal will be zero. Therefore, if all the inputs of the works are known, the control system is virtually unfastened cringle in the absence of perturbations, plant-model mismatch or uncertainness. The feedback signal is a step of the uncertainness and perturbations in the works. ( Morari and Zafiriou, 1989, p.41 )

Harmonizing to Garcia and Morari ( 1982 ) , IMC merely allows stabilisation of unfastened cringle stable workss, but our mark is to see how it could be applied to open cringle unstable workss.

The closed cringle transportation map is given as:

aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦ . ( 2 ) Q ( s ) = 1/G ( s ) , which means that the IMC accountant is given as the opposite of the works transportation map, but to guarantee that the map is proper, a filter map is added. Therefore a more general IMC accountant is given as:

aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦ . ( 3 ) Where G_ ( s ) is the invertible portion of the works, – is the tuning parametric quantity, r – positive whole number to guarantee properdness.

The closed cringle authoritative feedback accountant is parametrized in footings of the stable IMC accountant as:

aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦ . ( 4 )

which is stabilising for any stable Q ( s ) .

Besides the IMC accountant can be expressed in footings of the closed cringle feedback accountant as:

aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦ . ( 5 )

which makes the two constructions interchangeable.

## 1.2.2 Advantages of IMC:

Harmonizing to Kou Yamada ( 1999 ) , IMC has the undermentioned advantages ;

The stableness of the Internal theoretical account control system is merely dependent on the stableness of the works G ( s ) and the accountant Q ( s ) . Which makes it easy to plan, but for this undertaking the works is unstable.

Closed cringle response is easy adjusted by tuning ( tuning parametric quantity ) from the filter.

The IMC construction includes hardiness in an expressed mode through its feedback signal given in equation ( 1 ) . Garcia and Morari ( 1982 ) .

It provides for online tuning which makes it attractive to the operator.

In this undertaking, alterations of IMC accountant on how it could be applied to open cringle unstable systems will be studied due to its advantages listed supra.

## 1.3 AIMS AND OBJECTIVES:

The purpose of this undertaking is to develop a simple, intuitive methodological analysis of obtaining a accountant that would stabilise an unfastened cringle unstable system in closed cringle, guarantee hardiness to perturbations, easy estimable and besides easy to tune. Already bing methods will be studied, analyzed and evaluated to compare their schemes and come up with a control synthesis of the selected methods. The control action will be investigated through simulations and its public presentation will be observed. Two methods used to develop such accountants would be studied during the undertaking period, they are:

Youla Parametrization: The Parametrization of all stabilising accountants for unstable systems developed by Youla et Al ( 1976 ) , popularly called Youla Parametrization will be studied and simulations will be carried out besides to see how effectual the accountant is.

Feedback stabilisation ( Two-degree of freedom ) : This involves planing the accountant such that servo-tracking and perturbation rejection is done individually. It involves modifying the IMC construction and using it to open cringle unstable systems. Further surveies will be done on this attack every bit good as simulations as illustrated in the literatures.

The Deliverable gettable from this undertaking is a control synthesis of both attacks every bit good as comparing, so as to develop a accountant that would stabilise an unfastened cringle unstable works in closed cringle. The designed accountant should be easy to plan, guarantee stableness, good perturbation rejection and less equivocal to the operator.

## 2.1 LITERATURE REVIEW:

Harmonizing to Garcia and Morari ( 1982 ) , IMC does non back up the stabilisation of unfastened cringle unstable workss, but could merely be achieved if the if the accountant naturals precisely the unstable poles of the system which they said was non possible in pattern. This could be true, because in most instances the works kineticss might non be really accurate and therefore the accountant might non call off out the unstable pole.

Chiu et Al. ( 1990 ) besides confirmed that IMC can non be used for unstable workss, but suggested that the conventional feedback must be used with Parametrization through the IMC construction.

But Morari and Zafiriou, ( 1989 ) , Kou Yamada ( 1999 ) and Tan et Al ( 2003 ) all had the same position of using IMC to unstable workss and proposed certain conditions must be satisfied to accomplish stabilisation.

The Parametrization of all stabilising accountants derived from the IMC is given as

aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦ . ( 6 )

Where Q ( s ) is the IMC accountant, C ( s ) is the closed cringle accountant G ( s ) is the works. This accountant would be stabilising if certain conditions are fulfilled, which would be considered in the subdivisions below.

2.1.1 Pole – Zero Cancellation with Internal Model Control: As stated earlier, to stabilise an unstable works utilizing IMC, the accountant needs to call off the unstable pole of the works. In the paper by Tan et Al ( 2003 ) and Morari and Zafiriou, ( 1989 ) , the accountant given in equation ( 6 ) is stabilising if and merely if the undermentioned looks hold: ( I ) Q ( s ) must be stable ( two ) G ( s ) Q ( s ) must be stable, this would intend that the nothing of Q ( s ) must call off the right-half plane poles ( RHP ) of G ( s ) . ( three ) ( 1 – G ( s ) Q ( s ) ) G ( s ) must be stable, which means the RHP poles of P must be canceled by the nothing of ( 1 – G ( s ) Q ( s ) ) .

In the illustration in the paper by Tan et Al ( 2003 ) , implementing this for an unstable system given as ;

G aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦ ( 7 )

The IMC accountant is designed as ;

aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦ . ( 8 )

Where I± is a parametric quantity that guarantees ( 1-G ( s ) Q ( s ) ) cancels the RHP pole of G ( s ) , I» is the tuning parametric quantity.

Choosing I± = 3 and I»=1,

The closed cringle accountant designed from the IMC is given as ;

== aˆ¦ . ( 9 )

The simulation is given below:

Fig2.1 Pole zero cancelation utilizing IMC as derived from Tan et Al ( 2003, p. 204 )

Fig2.2 Plot of Plant Output which shows the simulation of the accountant designed in the paper Tan et Al ( 2003, p. 204 )

From the figure above, it can be observed that the designed accountant was able to stabilise the works and do it track set point but there was a big wave-off of approximately 40 % and besides a big subsiding clip which can be adjusted by tuning the parametric quantities decently. But for this survey, it was non tuned because the purpose was merely to look into stableness.

This method is merely considered in the feasibleness surveies as a an illustration, but will non be studied in the thesis.

2.1.2 Feedback Stabilization ( Modified Internal Model Control ) : This is besides referred to as a two-degree of freedom accountant. Huang and Chen ( 1997 ) proposed a three-element construction method that can be used to deduce a two-degree of freedom accountant and besides the conventional PID systems. Their method is the same with that of T. Liu et Al ( 2004 ) and Tan et Al ( 2003 ) who did similar work and proposed the application of modified IMC for unstable procedures with clip holds. This attack has the advantage of work outing control jobs such as stabilisation of the works, servo-tracking and perturbation rejection independently. Wang et Al ( 2004 ) proposed H2 optimum accountant for the perturbation rejection accountant, while Tan et Al ( 2003 ) proposed a PD accountant for the perturbation rejection accountant. The Modified IMC construction harmonizing to Tan et Al ( 2003 ) is shown below for unstable procedure with clip hold.

Fig2.3 Simplified Modified IMC Structure for Unstable works with clip hold ( Tan et al, ( 2003 ) )

From the figure above, it can be seen that that the construction has three compensators, viz. , K0, K1, and K2 and they each have influence on the overall closed cringle response as follows:

K0 is used to stabilise Gm ( s ) , the unstable works theoretical account internally without the clip hold, so as to acquire a stable system. K0 can be any stabilising accountant chosen by the interior decorator, e.g Proportional accountant.

K1 is an IMC accountant for the stabilised works

K2 is used to stabilise the original unstable works with hold, and is considered as the perturbation rejection accountant which is really of import for the internal stableness of the construction.

This method was tried out utilizing a simple system and taking the parametric quantities as outlined in the paper every bit good as the tuning regulations provided in the paper, and the consequence of the simulation is shown below.

The stabilised works is given as:

, taking K0 = 2,

K1 is the chosen as the IMC accountant and is given as:

K2 is designed utilizing tuning regulations provided in the paper and is designed as a PD accountant given as:

K2 = KC ( TCs+1 ) = 2.079 ( 0.156s+1 )

The system is given below

Fig2.4 Modified IMC construction with the designed parametric quantities ( illustration 1: Tan et Al ( 2003 ) , p.209 )

Fig2.5 Simulation Result retroflexing the simulation from illustration 1 in the paper: Tan et Al ( 2003 ) , p.209

From the figure above it can be seen that the accountant was able to do the system path set point, although attempts were non made to tune the accountant to accomplish a better response because of the purpose was merely to look into stableness.

In other to hold a better apprehension of the construction, alterations were made on the construction to use the same rule to workss without hold and besides go forthing out the perturbation rejection accountant K2, and the construction every bit good as the response is shown below:

Fig2.6 Feedback stabilisation with K0 and K1 entirely with works hold, Modified from illustration 1 in the paper Tan et Al ( 2003 )

Fig2.7 Simulation response of the Modified construction above

From the figure above it can be observed that the feedback stabilisation and IMC modified accountant was able to stabilise the system and do it to track set point, which shows that K2 is for perturbation rejection. But the feedback stabilising accountant K0 value affects the IMC accountant value and the overall response, such that if it ‘s changed, the IMC accountant has to be recalculated to accommodate the new value.

## 2.1.3 Youla Parameterization:

Youla Parametrization besides called the Q-Parametrization is the word picture of all stabilising accountants that stabilize a given procedure.

Harmonizing to Maciejowski ( 1989, p.315 ) , Youla Parametrization was applied in surveies of optimum control in the 1950 ‘s. Kucera and Youla et Al made much betterment on the Parametrization which has found its importance in the topic of linear-feedback systems. Harmonizing to him, “ Kucera and Youla et Al. applied the Parametrization for quadratic ( H2 ) optimisation, while Zames applied it to Ha?z optimisation job ” ( Maciejowski ( 1989 p.315 ) . The systems considered in the stuff were Multiple-input Multiple-output ( MIMO ) systems.

The construct of Youla Parametrization, harmonizing to ( Astrom and Murray, 2008 ) , is the word picture of all accountants that stabilize a given procedure. The construct was applied to Single-input Single-output ( SISO ) systems. The diagram demoing the parametrization application to unstable workss is shown below:

Fig2.8 Block Diagram of Youla Parametrization for Unstable works ( Astrom and Murray, 2008, p. 357 )

In its application to unstable procedure, the system is expressed in a rational multinomial transportation map signifier, such as ;

, where a ( s ) and B ( s ) are multinomials

A stable multinomial degree Celsius ( s ) is chosen to stabilise the procedure, and so the procedure is given as ;

, where A ( s ) = a ( s ) /c ( s ) and B ( s ) = B ( s ) /c ( s )

A accountant is introduced given as ;

, where G0 ( s ) and F0 ( s ) are stable rational maps

But the Errata supplied by S. Livingstone ( 15 Nov 2010 ) shows that there are some mistakes in the expression. The corrections are:

= and

Besides from the diagram of the construction, there seem to be something incorrect about the location of the set point input which will all be investigated during the thesis.

For any stable rational map Q ( s ) , the word picture of the stabilising accountant for unstable works harmonizing to Astrom and Murray ( 2008 ) is given as ;

aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦ ( 10 )

The Sensitivity map is given as:

aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦ . ( 11 )

While the Complimentary Sensitivity map is given as:

aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦aˆ¦.. ( 12 )

Equation 10 gives the Parametrization of all stabilising accountants for unstable workss and the sensitiveness and complementary sensitiveness maps are stable if the rational map AF0 + BG0 does non hold any nothings in the RHP. Therefore the accountant is stabilising for any stable Q. ( Astrom and Murray, 2008, p. 357, 358 )

Much work was non done on this method during this feasibleness survey but it constitute a major portion of the thesis work and hence would be studied in inside informations and simulations carried out to hold a better apprehension of the construct and how it is used.

## 3.1 METHODOLOGY

The undertaking purpose is to develop a simple, intuitive methodological analysis of planing a accountant for unfastened cringle unstable workss. Some methods have been identified among which is the Youla Parametrization and the feedback stabilisation methods.

In other to accomplish this purpose, the undermentioned actions will be taken with regard to the undertaking program, the undertaking hazard and alternate attacks may be considered where the demand may originate.

Literature reappraisal of stuffs, diaries and documents refering to Youla Parametrization and feedback stabilisation ( two-degree of freedom accountant ) would be studied in deepness in line with the allocated period of the in the undertaking program, so as to hold a better apprehension and background of the methods.

Planing accountants based on the different methods listed supra.

Hands – on simulation of the accountants and their application to unstable systems will be carried out in line with the undertaking program. The Simulation package to be used is Matlab of which the section has license.

Analysis of stableness and hardiness issues every bit good as the accountant restrictions will so be carried out.

Control synthesis of both schemes to look into if there is a possibility of unifying them.

Recommendations based on findings after the synthesis and perchance implementing alternate attacks to the topic.

Alternatively if the control synthesis between the Youla Parametrization and the feedback stabilisation does non give the appropriate consequences needed, so one of the methods may hold to be used. Both methods will be compared in footings of public presentation standards such as set point trailing, hardiness and good perturbation rejection as good fulfilling the purposes and aims of the undertaking. The same theoretical account will be used to prove the methods and the design method with a better response to the chosen theoretical account will be chosen as the control scheme. If they both do non give satisfactory responses needed, so hardiness would be used as the standards for taking the accountant. The accountant that has better hardiness would be chosen.

## 4.1 PROJECT PLANNING AND MILESTONE CHART

The undertaking is scheduled to get down the on the 13th of June, but I will be taking a four yearss holiday so I would get down exactly on the 20th of June. The undertaking through until the first hebdomad of September with the entry deadline being 5th of September. The Gantt chart provided shows the way with which the undertaking will be implemented and how each activity is connected. A float is provided between 3rd and the 15th of August so as to suit any exceeded deadline for the undertaking. The Gantt chart is shown below:

## 5.1 Undertaking Hazard:

The undertaking work will chiefly affect reappraisals of literatures, planing of accountants, simulation of the accountants on unstable system and transporting out analysis on certain public presentation standards such as hardiness, set point trailing and perturbation rejection. Most of which will be done utilizing my personal laptop. The hazard involved in put to deathing this undertaking is outlined as follows:

There is a hazard that the attack i.e. the control synthesis of both the Youla Parametrization and the feedback stabilisation might non give a accountant that is simple and intuitive.

Extenuation Scheme: Then one of the methods might hold to be used which produces a better public presentation response.

Simulations were non done on the Youla parametrization method during this survey ; hence there might be a hazard of the method non fulfilling certain conditions in the purposes and aims of the undertaking.

Extenuation Scheme: Recommendations would be made, and perchance concentrate on the 2nd method which some simulations had already been done during this period.

The accountant designed based on the Youla Parametrization may non be really efficient on the selected theoretical account construction ( SISO ) and therefore might impact the original aims of the undertaking.

Extenuation Scheme: Agenda a meeting with my supervisor to reexamine the range and discourse possible options.

There might be a possibility of clip convergence between undertakings, because there are sub-tasks which might be running at the same time and some may depend on the other which may do me transcend certain deadlines and non run into up the overall thesis deadline.

Extenuation Scheme: A float has been added to the undertaking Gantt chart, so as to take attention of such state of affairss.

## 6.1 Decision:

The feasibleness study has shown that it is executable to stabilise an unfastened cringle unstable works utilizing the feedback stabilisation method. Besides it shows that the literatures can be sourced from the school ‘s library.

The major package to be used for the undertaking is Matlab of which the section has license. The Gantt chart shows the undertaking can be executed within agenda and can therefore conclude that the undertaking irrespective of being disputing is executable.

## 7.1 Mentions

[ Astrom and Murray ( 2008 ) ] Astrom K. J. and Murray R. M. ( 2008 ) “ Feedback Systems: An Introduction for Scientist and Engineers ” , Princeton University Press, p. 356-358

[ Chiu et Al. ( 1990 ) ] Chiu, Min-Sen and Arkun, Yaman ( 1990 ) ‘Parametrization of all stabilising IMC accountants for unstable workss ‘ , International Journal of Control, 51: 2, 329 – 340

[ Dorf and Bishop ( 1998 ) ] Dorf R. C. and Bishop R. H. ( 1998 ) “ Modern Control Systems ” ( Eight edition ) Addison Wesley Longman Inc. p. 295

[ Garcia and Morari ( 1982 ) ] Garcia, C.E. and M. Morari ( 1982 ) “ Internal Model Control-1 A consolidative reappraisal and some new consequences, ” Ind. Eng. Chem. Process Des. & A ; Dev. , 21, pp.308-323

[ Huang and Chen ( 1997 ) ] Huang H. P and Chen C. C. ( 1997 ) “ Control-Synthesis for Open-loop Unstable Process with Time-delay ” , IEE proc. Control Theory Appl, Vol. 144, No. 4, 334-346

[ Kou Yamada ( 1999 ) ] Kou Yamada ( 1999 ) “ Modified Internal Model Control for Unstable Systems ” , Proceedings of the seventh Mediterranean Conference on Control and Automation Hiafa, Isreal-June 28-30 ”

[ Maciejowski, 1989 ] Maciejowski, J. M ( 1989 ) “ Multivariable Feedback design ” , Addison-Wesley Publishers Ltd, p. 315

[ Morari and Zafiriou ( 1989 ) ] Morari, M. ; Zafiriou, E. ( 1989 ) “ Robust Procedure Control ” ; Prentice-Halk Englewood Cliffs, NJ ; pp 85-112.

[ Tan et Al. ( 2003 ) ] Wen, Tan, Horacio J. Marquez. and Tongwen, Chen ( 2003 ) “ IMC design for unstable procedures with clip holds ” Journal of procedure Control,13,203-213.

[ T. Liu et Al. ( 2004 ) ] Tao Liu, Xing He, Danying Gu, Wei Wang ( 2004 ) “ A fresh control strategy for typical unstable procedures with time-delay ” Proceeding of the 2004 American Control Conference Boston, Massachusetts