text:
User:Asaffell7210grumpysmirf/Collections/Introduction to Computer Information Systems

Introduction to Computer Information Systems

 * Introduction
 * Introduction to Computer Information Systems
 * Introduction to Computer Information Systems/Computers in Your Life
 * Introduction to Computer Information Systems/What is a Computer?
 * Introduction to Computer Information Systems/Computers to Fit Every Need
 * Introduction to Computer Information Systems/Computer Networks and the Internet
 * Introduction to Computer Information Systems/Computers and Society
 * Introduction to Computer Information Systems/Review

 * Hardware
 * Introduction to Computer Information Systems/The System Unit
 * Introduction to Computer Information Systems/Storage
 * Introduction to Computer Information Systems/Input and Output

 * Software
 * Introduction to Computer Information Systems/System Software
 * Introduction to Computer Information Systems/Application Software

 * Networks and the Internet
 * Introduction to Computer Information Systems/Computer Networks
 * Introduction to Computer Information Systems/Internet
 * Introduction to Computer Information Systems/Security

 * Business on the Web
 * Introduction to Computer Information Systems/Multimedia
 * Introduction to Computer Information Systems/E-Commerce

 * Systems
 * Introduction to Computer Information Systems/Information Systems
 * Introduction to Computer Information Systems/Program Development
 * Introduction to Computer Information Systems/Database

 * Computers and Society
 * Introduction to Computer Information Systems/Computer Security
 * Introduction to Computer Information Systems/Ethics


text:
History of wireless telegraphy and broadcasting in Australia/Topical/Publications/Radio in ANZ/Issues/1923 05 02

Link to Issue PDF
WorldRadioHistory.com's scan of Radio in Australia and New Zealand - Vol. I No. 03 - 2 May 1923 has been utilised to create the partial content for this page and can be downloaded at this link to further extend the content and enable further text correction of this issue: Radio in ANZ 1923 05 02

In general, only content which is required for other articles in this Wikibook has been entered here and text corrected. The material has been extensively used, inter alia, for compilation of Biographical articles,  Radio club articles and  Station articles.

Front Cover
Issued every second Wednesday — Sixpence

RADIO IN AUSTRALIA & NEW ZEALAND — incorporating "Sea, Land and Air"

VOL I. — MAY 2, 1923 — No. 3

(Start Graphic Description) SIX-YEAR-OLD RADIO ENTHUSIAST. (See page 63) — Underwood Photo (End Graphic Description)

Registered at G.P.O., Sydney, for transmission by post as a newspaper.

P.49 - Contents Banner
RADIO in AUSTRALIA & NEW ZEALAND Incorporating "Sea, Land and Air"

Managing Editor: S. E. TATHAM       Associate Editor: M. DIXON

CONTENTS

Volume I.  MAY 2, 1923    Number 3

P.49 - Contents
CONTENTS.

 * Radiotorial . . . Page 51
 * Radio Entertainment . . . Page 52
 * Inductance, Capacity and Self Capacity of Coils . . . Page 53
 * How London "Listens-In" . . . Page 56
 * Australian Wireless Development . . . Page 57
 * Wireless Institute of Australia . . . Page 59
 * The Experimenters' Corner . . . Page 60
 * In Radio Land . . . Page 62
 * Call Letters . . . Page 66
 * Movements of Wireless Officers . . . Page 67
 * Low Power Tests . . . Page 68
 * Radiofun . . . Page 69
 * Club Notes and News . . . Page 70
 * Queries and Answers . . . Page 72

P.49 - Publication Notes
Published by: THE WIRELESS PRESS, 97 CLARENCE ST., SYDNEY; 422-24 Lt. Collins St., Melbourne; Australasia Chambers, Wellington, N.Z.

PRICE, 6d. per Copy; Subscription Rate, 10/- per annum (26 issues) throughout Australia and New Zealand; Foreign Rate, 12/6 (26 issues)

OVERSEAS REPRESENTATIVES:

Canada and United States of America: The Wireless Press Inc., 326 Broadway, New York City

Great Britain: The Wireless Press Ltd., 12-13 Henrietta St., London, W.C.2

P.50 - Wireless Press Ad
(Start Photo Caption) (End Photo Caption)

P.51 - Radiotorial
Trans-Pacific Tests

T HE near approach of the Trans-Pacific radio tests directs attention to their importance to the wireless amateurs of .Australia. For some considerable time past a large number of enthusiastic experimenters all over .Australia have been hard at work building apparatus and completing arrangements in order that the best possible results may be achieved. It cannot be too strongly emphasised that these tests have a national as well as a world-wide interest. Therefore, it behoves all who have the progress of radio at heart to do all that lies in their power towards ensuring a large measure of success being recorded to the credit of .Australian experimenters. It can safely be assumed that those engaged in the actual task of attemptjng to receive the.messages transmitted by the .Amer ican amateurs will spare no effort to accomplish the aim they have in view. We trust that those not taking part in the tests will co-operate loyally with them to the rxtent of closing down their stations at the actual times at which it is calculated the messages will reach .Australia. lt is only a small sacrifice they are being asked to make, but non-compliance therewith will certainly have a very detrimental effect on the efforts of those who arc out to ,vin new distinctions for .Australian radio experimenters. There are some thousands of holders of receiving licensefl in the Commonwealth, and it naturally follows that each and every one is anxious to employ all his spare time in  listening in to the signals, music' and speech which, to a more or less extent, fills the air every night. It is these enthusiasts who are asked to call a halt while the tests are on in order that a clear field may be · allowed those who are seeking to establish a new long-distance record for low-power reception. The forthcoming test is easily the most important event that has yet engaged the attention of amateurs in .Australia. A wide appeal was made for entrants, and while the results, numerically comprise only a small percentage of those holding licenses, the standard. of the competitors is exceptionally high. This only emphasises the necessity of non-competitors allowing them a clear field. There is no means of ensuring this other than by appealing to the honour of 1those indirectly concerned in the success of amateur experimentation. "-~ ' We feel that once this fact is realisecl everv radio experimenter who possesses a valve rece'rving ;et will fall into line with · the wish of th Trans-Pacific 'l'est Committee and suspend his activities during the currency of the test. By doing so he will be acting in strict conformity with the high moral standard which has characterised the doings of radio experimenters iu .Australia, and incidentally rendering a distinct service to the cause of radio research.

Growth of Radio Clubs

THE grnclual formation of radio clubs throughout .Australia is a healthy sign. Less than twelve months ago the number in existence was considerably less than it is to-day, and almost every week we hear of new ones being formed. As might be expected, the growth has been much greater in and around the capital cities than in the country, but there is no reason ,Yhy the country districts of the various States should not have a large number of clubs, consistent with area and population, in the very near future. It must be obvious to anyone who cares to give the matter a moments consideration that the clubs are reallv stepping stones on which the great science of radio will rise to the same dominant position in .Australia that it occupies in America and England. That the public here are not yet educated to the valne and possibilities of wireless communication is not at all surprising. The average p erson regards radio as a subject far too complicated for his untechnical mind, and even when it is demonstrated that the science is simplicity itself to a person of ordinary intelligence he still remains more or less unconvinced. It is in cases like this that radio clubs are able to render valuable service. 'l'hc educational influence of a few lectures and demonstrations is beyond question, and the formation of a radio club: in any district is certain to arouse the interest of mai1y who would otherwise remain indifferent to the march of this truly wonderful science. For this reason, if no other, the growth of clubs is to be encouraged, and enthusiasts in country centres, no matter how small they may be numerically, should band themselves together for the advancement of their hobbv. The outback districts offer every facility for exp"erimentation, and a practical demonstration of transmitting and receiving messages between different towns will do more to arouse public interest than the most eloquent word picture of the possibilities of radio communication which it is possib'Je to pa.int. We hope to hear of the inauguration of many country radio clubs in the near future, and those who undertake to form them can rest aHsured that the work thev are doing will stand as a monurnent to their faith and ~nterprise long after radio has overstepped the limits which bound its horizon to-day.

P.52 - Wentworth Club's Successful Effort
'''Radio Entertainment. Wentworth Club's Successful Effort. Large and Appreciative Audience.'''

The members of the Wentworth Radio Club, which is presided over by Mr. Spencer Nolan, have every reason to feel elated at the success which attended their demonstration of wireless music at the Club rooms, Rose Bay, on the night of April 18. The committee went to considerable trouble to ensure success, and the whole-hearted appreciation of the large audience must have convinced them that the effort was well worth while. The first broadcast items to be picked up came from Mr. F. Basil-Cooke's up-to-date station at Clifton Gardens. Punctually at 7.30 p.m. Mr. Cooke commenced his experimental transmission, and for just on an hour the audience in the Club room at Rose Bay, as well as a large number who had gathered on the lawn outside, were delighted and entertained with the vocal and instrumental items which came in loud and clear. At 8.30 p.m. Mr. Colville, another well-known experimenter, commenced transmitting, and half an hour later Mr. C. D. Maclurcan's messages were picked up, although at the time he was conducting experiments with Melbourne amateurs. At the conclusion of the demonstration all present voiced their appreciation of the work of those responsible for the evening's entertainment, and as was his due, Mr. R. C. Marsden, who operated the receiving set, was warmly congratulated. The receiving apparatus consisted of three valves with a detector and two stages of audio frequency amplification. The whole outfit was built by its pardonably proud owner, Mr. Marsden. The Wentworth Radio Club has only been in existence a short time, but its members are determined that no other club in Australia shall outdo it in boosting the possibilities of radio. The fact that its first demonstration was such an unqualified success will spur it on to further efforts, and radio circles generally cannot fail to derive considerable benefit from such progressive moves as the holding of demonstrations designed to educate the general public to the possibilities of entertainment by radio.

(Start Photo Caption) Flashlight photo. of section of the audience at the Wentworth Club's headquarters. On the right is Mr. Spencer Nolan, president of the Club, and near the receiving apparatus is Mr. R. C. Marsden, the operator. Seated next to Mr. Marsden is Mr. Arthur Peters, hon. treasurer. In the centre is Mr. L. Skinner, vice-president, and on the extreme left Mr. Wallace Best, hon. secretary. A large section of the audience listened to the music from the lawn in front of the club house. (End Photo Caption)

P.53 - Inductance, Capacity and Self Capacity of Coils
Inductance, Capacity and Self Capacity of Coils. by E. Joseph

P.56 - How London "Listens-In"
'''How London "Listens-In." Thousands Enjoy High-Class Programmes.'''

P.57 - Australian Wireless Development
'''Australian Wireless Development. Need for Encouragement.''' by George A. Taylor

The following highly-interesting lecture was delivered at the annual meeting of the N.S.W. Division of the Wireless Institute of Australia by Mr. George A. Taylor, whc, enjoys the distinction of being the first Chairman of the Institute-the first to be formed in the British Empire.

T HFJ ·world to-day is alert. For ages it had wearied on. Events happened, and the world woke for a space; then it slept again in its matter-of-fact way. Today it is wide 1:nvake; the crash of the Great War upon history sounded a thump that is still reverberating ronnd the earth in the changing conditions of national governments, in the altered views of human rights, but more in the realm of · scientific achievement; and no where more than in "wireless communication" have better results been achieved. Yet, different from other scientific investigations, wireless development did not want war-pressure to stir it to activ0 ity. On the strength of its own merit its wonders would have risen it to the great place it now holds in human attention. Though the world to-day is aware of wireless wonders of almost daily happening, it is not truly aware of what wireless really is. In fact its development has been so rapid that there has not been time to give the new science a proper name. Since Hertz, in 1887, marshalled electro-magnetic forces, and by means of an oscillator, sent out "beatings" that created waves that were received by a resonator, wireless has swiftly developed, and we note a great procession of clever inventors in the new science, from Branly with his Coherer in 1890, Oliver Lodge, Ducretel, Marconi, Fleming, Fessenden, Lee de Forest, Poulsen, and Armstrong, who have brought the science to be the most wonderful on earth to-day ; in fact, within the brief space of thirty years, wireless bas developed from a spark between two metal balls one inch apart, to t.he spread of a song from Dame folba to listeners-in practically all over the world. The mention of Dame Melba (Australia's soprano) brings to mind the fact that Australia holds an interesting position in the world's wireless achievements, both from an experi-

mental, as well as from other points of view. It is interesiing to look back to the early nineties, when the great liner W aratah, probably with a broken propellor shaft, drifted into obscurity without ever being heard of again, when there arose a general appeal for expediting wireleRs development. At that time very few con.sidered the practirml possibilities. In order to stir attention to its value a party of enthusiaRts joined together to interest the A~istralian Military Authorities, so that at Easter (April, 1910) a party, consisting of Messrs. Kfrkby, Hannam, Wilkinson, and the speaker, arranged the first two military wireless stations in Australia at the Artillery Camp at Heathcote. The aerials were rapidly erected from rough saplings, and what would today be called cumbersome apparatus, was carried over some of the roughest country in Australia. One station was a tent at Headquarters. Debris found by the wayside was used to fit up a cave for the second station, a dilapidated door being made use of as a table. Wireless was to be used for transmitting the result of the artillery action. After a very anxious time, in which difficulties, such as rain water coming and shortening the circuits in the cave, communication was made between the two stations, and success was achieved. In mentioning the ardent enthusiasm of the wireless operators, one cannot refrain from putting on record the encouragement given by Captain Cox Taylor, Lieut.-Col. Wells, Captain

Christian, and Major Rosenthal, the latter now being General Sir Charles Rosenthal. Although that is but a few years ago, it seems a long distance when one studies the heavy apparatus of that time, such as the 6in. coil and other heavy gear, compared with the thermionic valves and other simple fitments of to-day, but one good result of those early experiences was the keen taking up of wireless by the Military Authorities and its general encouragement. There. was, however, one · hindrance to experimenting, and that was a Wireless Act, which had been in operation since 1905, imposed a fine of £500 on unlicensed experimenters, and charged £3/3/- per year for those who desired to experiment. This so crippled wireless development, that, in 1909, this Wireless Institute was formed, and the speaker who had the honour to be its first Chairman had a motion carried by which the Postmaster- General reduced the license fee from £3/3/- to 10/6. This encouragement gave rise to some remarkable achievements. Australia soon led the world in exploding mines and cannon by wireless in 1910 ( vide Melbourne Age, 20/11/1910) ; in the transmission of pictures · by wireless by Wilkinson in 1911, in exchanging messages between express trains in 1911 ( vide Evening News, February, 1911), for showing how sound waves and wireless could be utilised for locating disturbances in 1911 ( vide Commonwealth Military ,T ottrnal, March, 1912), and in the control of airships by wireless, by Roberts, in 1912 ( vi'.de Evening News, 16/5/191:2). It is interesting to note that though these inventions were first exhibited in Australia on the dates mentioned, the older world claimed credit for their discovery some years later. For instance, communication between moving trains was not discovered

by the older world until four years afterwards, when it was chronicled that it had been achieved in America. 'I'he guidance of vessels by wireless waves had been demonstratd in Australia in 1912, yet it was reported in the Sydney Sun on August 28th, 1921, as having been discovered in America; whilst the transmission of pictures by wireless, achieved in Australia by Wilkinson in 1911, has been credited to Denmark in 1921. Another invention which the writer was interested in, viz., that of the 1\fotorless Aeroplane, was :first achieved in Australia in 1909, yet the same construction was last year utilised in Europe for the :first time; an Australian paper, The Melbonrne Herald, of October 19, 1921, stating: '' Germany led the world; would Australia follow,'' whereas the design • utilised in Germany was actually a copy of that made and flown in Australia thirteen years before. With Naval and Military encouragement givei to wireless in Australia, we find success being achieved until the Great War in 1914, when wireless came into a greater sphere of utility. It. was regrettable, however, that wireless and other scientific prewar achievements r.eceived little official encouragement; records that had been made many years before not being recognised by Military Authorities; in fact, the use of 'l'anks was not recognised until the, War was well ahead, although it was an Australian invention of some years before ' ,vhilst sound and wireless. . waves that had been demonstrated m Australia as useful for sound-ranging were not made use of until the final years of the war. Even Gallipoli had been deemed before the War a useless place upon which a landing could be made; yet the pre-war official re:port to that effect was not brought to light until the war had been concluded. 'rhe war; therefore, taught one great lesson, and that was to give increased attention to scientific development, so that the best use could be made of same for all purposes. The lack of recognition of Aus0 tralia 's place in the world's scientific achievements, as in the ignoring of Hargrave in· discovering the secret of human flight, was the incentive to the speaker · to establish a Board to encourage Australian Invention, and in order that Australia could get "RADIO" widest recognition of her scientific achievements, Great Britain was visited, and success was obtained by winning the support of the British Science Guild to act as an Empire centre for encouraging invention. 'l'he Gnild comprises the greatest of British scientists, and branches of it are now being established throughout the Empire. The New South Wales section of the Board of Invention has had its rules and regulations remodelled, so that now it forms the New South Wales Section of the Guild. The Australian Section of the British Science Guild is encouraging inventions generally, and is linking up with various scientific bodies, in order that experimenters in these bodies
 * -;hould benefit by the funds that the

Guild is collecting to encourage invention. Already linked with it is a Branch dealing with Aerial Experiments, and it is hoped that the Wireless Institute will be affiliated with the Guild, iu order that the money the Guild is collecting for experimental work can be at the services of wireless operators. The only fee to be paid for thi;; linking up is a capitation fee of 1 /per member, members of the affiliated bodies having the right to apply for financial assistance to further their experimental work. I put this before the Wireless Institute, as I recognise it as a great body that will link together the best · of wireless enthusiasts, for wireless is in such a position to-day that it needs the keenest attention of a strong Institute such as this, to prevent the science being overdone. The wonders of wireless telephony have been so great that what may be termed a rush for "broadcasting" has set in; In America it became quite a craze, and the authorities were so rushed with applicatiom; that they issued - them without the necessary restrictions, as to wave let1gths and time, causing much jamming. In Great Britain, on the contrary, there ,vas a considerable delay in the issue of licenses, both to  listenersin as well as to transmitters. In fact, Great Britain was noting the mistakes that America was making and profiting by them. 'The speaker happened to be inGreat Britain when sane conditions were being framed for regulating broadcasting," · and he May 2, 1923. is pleased to say that the Australian Government will shortly be ready to put into action what should be sensible regulations for "broadcasting," as a meeting is to be held in Melbourne towards the end of l\Iay for those interested in wireless, in order to discuss same. The temporary delay may have it:-; dra,vbacks, in that it may. make many who have taken up wireless, somewhat impatient, and cause what might be called a  lack of interest,' ' but those who have the science at heart, and who can look ahead and see its possibilities of pleasure and profit, can still wait a few weeks longer, in order that best conditions can be drawn up. In this respect the speaker is happy to report that practically all the firms in Australia connected with wireless have formed an Association for its bm,t protection, and Australia will enter upon its ''broadcasting' ' era, without any of the mistakes of older countries, and with every possibility of achieving something that older countries may be striving for. The value of the linking together of wireless experts in the Wireless Institute is incalculable, for not only ar'e all ardent experimenters in: a science that is the greatest link between the natural and the etherial and with untold possibilities for human comfort and uplift, but in time of National stress the Institute means that Australia has at immediate hand a great band of experts whose apparatus could be used as relay stations, and so be of great National utility.

Note on the Care of Crystals
Note on the Care of Crystals. In receiving outfits employing crystal detectors, the effective range depends a great deal upon the sensitivity of the crystal. Some crystals are naturally more sensitive than others, but even a sensitive crystal may be ruined by improper care. The action of the air on these crvstals sometimes oxidizes. their surfac~ ancl prevents them from functioning properly, but a more_ serious trouble is caused by touching the surface of the crystal with the fingers. Where this has been done and the surface of the crystal is found to be less sensitive after continued use, · it should be scraped lightly with a pen-knife.

P.59 - Australian Wireless Development
'''Wireless Institute of Australia. N.S.W. Presidential Report.'''

The following extracts are taken from the Presidential report furnished by the Vice-President (Mr. Basil Cooke) at the annual meeting of the Institute. The period covered is from September, 1922, to March, 1923.

WING to the recent resignation of our President (Mr. C. P. Bartholomew), the presentation of the Annual Report has devolved upon me. I need hardly say that I greatly regret the necessity for Mr. Bartholomew's action. The period covered by this report has been one of consolidation for the Institute. With the inception of the new Council, in September, 1922, came a feeling of security. Owing to the long absences of Mr. Fisk abroad (who was previously president) matters had been left largely to the discretion of the Honorary Secretary. However, the new Council indicated very clearly that they fully appreciated their responsibility in the election of their President (Mr. C. P . Bartholomew) and Vice-President

(Mr. F. Basil Cooke); also by their regular attendance at all Council meetings. The position of the Council, assuming office at a time when Institute affairs were at a low ebb, was not to be envied. Mueh publicity had been indulged in regarding the load of debt being carried by the Institute, and, as mmal, such reports were greatly exaggerated. Fortunately, the dictates of common sense influenced those governing the affairs of the Institute, and at the

time of this report the indebtedness has been wiped out, and there is a credit balance at the Bank. In addition, it has been possible to raise the prestige of the Institute in such a way as to more than compensate for the trying times through which it has passed. It is now the premier radio society of the Commonwealth. At the beginning of this period a definite move had been made by the N.S.W. Division of the Institute to co-ordinate the interests of the many radio :;ocietie,; which had recently sprung up around Sydney, but ou account of the much-regretted illness of the Honorary Secretary all the negot iations had to be conducted secondhand, with the inevitable result that they were fruitless, and, unfortunately, provoked considerable bitter

feeling between the various societies. Ii is very gratifying that such foclJnj:! s have now been replaced h •.r a cordial fraternisation between the societies, and it is expected that the Radio Association of N.S.W., which has risen from the ashes of last year's negotiations, will soon have full support from all the societies, and it can be confidently expected that if the Association proves a success the Institute will be the first to tender its congratulations. It might be here mentioned that during this period a few members have resigned from the Institute with the specific object of forming local societies in their own districts, and other members of the Institute have been succ·essful in initiating local societies, whilst still preserving their identity with the Institute. The annual subscriptions to the Institute have been increased during this period for two reasons, viz. : to preserve the dignity of the Institute, and in order that the · subscriptions may he commensurate with the standard ad.opted. One feature which deserves mention in this report, seeing that it was primarily responsible for bringing the Division through the dark days of 1922, is the loyalty of its members. Throughout, they have not failed to look for the silver lining, and much credit must be given: to our late President (Mr. C. P. Bartholomew) for his wise counsels when difficulties were encountered. If we can look to the future for progress, as we have progressed during the last seven months have have little to fear, and I have no doubt that it will be a very short

time, indeed, until the Institute in New South Wales will be in regular wireless telephone communication with the Divisions in the other States of the Commonwealth. In this regard it may be said that the N.S.W. Division keeps well in touch with the other Divisions, which are, at all times, whole-hearted in their co-operation in all matters of mutual interest. Indeed, the good fellowship between the respective: Divisions leaves little to be desired. A very pleasing matter is the recent inception of a Division of the Institute in Tasmania, where Mr. L. W. Scanlon has been successful in getting matters on to a satisfactory footing, and a strong following is assured. The establishment of the Tasmanian Division was first mooted in 1919, when Mr. P. Renshaw visited that State, but, unfortunately, the negotiations ended unsuccessfully, and it remained for Mr. H: R Gregory to take this matter up seriously when in Tasmania early this year, when he had the good fortune to find such an enthusiastic wireless experimenter as Mr. Scanlon, who is now Honorary Secretary of that Division. In Queensland there have been many changes, and now the Honorary Secretaryship is in the hands of Mr. W. Finney, a real enthusiast with a live station. Mr. Finney was the last President of that Division, and succeeds Mr. L. 0. Kurlin, who has left Queensland. It will be remembered that Mr. S. V. Colville, who is now a full member of N.S.W. Division, was the inaugural Hon. Secretary in Queensland, and held the position for many years. The Victorian Division has been May 2, 1923. very live indeed, and is looking well to the Institute 's laurels in having undertaken the Trans-Pacific Tests, which promise to be epoch-making as far as experimental wireless is concerned. It is but fair to express our deep interest in the movement, and our appreciation of Mr. Kingsley Love's efforts to bring it to success. Mr. Steane, a member of the Victorian Division Council, recently visited Sydney, and was fortunately enabled to be present at the annual dinner given by this Division. Credit must be given to Mr. Maddick, Hon. Secretary in Melbourne, for the cooperation he extends at all times, and to the Victorian Division generally for their ,,incerity. We can always look to South Australia for concerted action. Indeed, the South Australian Division has always loyally stood up for the rights of the Institute as a whole, and its attitude on many points, notably the reduction of license fees, is commendable. Mr. Clement E. Ames, the Hon. Secretary, is regularly in touch with this Division. From the Far West the voice of the Institute is plainly heard, and we have been glad to note that the Western Australian Division has been upholding the traditions of the Institute by the quality of its lectures and its generally progressive activities. Very cordial relations exist between ourselves and the West and their keen interest in affairs way down east can always be confidently anticipated. Mr. A. E. Stevens, the Honorary Secretary, shows himself as seized with the importance of keeping in touch with New South Wales. Now for a few words regarding the

Interstate position from our own viewpoint. Firstly, we are glad to know that all the Divisions arc aware of the importance of having their proceedings recorded in the one Journal. ''Radio' '1 is the official organ, as it has incorporated Sea, Land and Air, and the feeling from the other States is appreciated. This Division has set a very high standard, and it is confidently hoped that the other Divisions will carefully look to their position as the premier radio bodies in order that a high standard may be maintained. The opinion has freely been expressed in governing circles in Sydney that it would be a wise move for all the Divisions to follow our example and register under the Companies' Act in their respective States, in order that their constitutions become legally binding and their liabilities limited. It is expected that this Division will very shortly be in a position to erect and maintain a transmitting and receiving station under the terms of the license recently granted by the Controller of ·wireless. Then we hope to be in direct communication with all the Divisions, even far-away Perth. To indulge in a little retrospect, I must extend the sincere thanks of the Institute to those members and others who assisted in providing lectures, demonstrations, etc., during the period just ended. Specially would I mention the excellent demonstration and lecture on  Sou.:p.d Ranging, given by Mr. Edgar Booth, M.C., B.Sc., at Sydney University, before our members, on February 1. Our special thanks are also extended to Mr. Newman for his endeavours to provide this Division with suitable headquarters at minimum cost. It is "RADIO" regretted that the negotiations have_ not been finally successful, though we have been indebted to him for arranging our accommodation at the Railway Institute up to date. We regret that Mr. Jack Pike, one of Australia's oldest experimenters, has just resigned for domestic reasons, but hope he will be back with us before long. The Division has been governed strictly in accordance with its Memorandum of Articles of Association up to the wesent, and I look forward to the future government of this Division on similar lines, as it is only by being constitutionally sincere that we can expect the support of our officers and members. The future looks hopeful, and we anticipate great things from many of our members, especially Mr. E. T. Fisk, whose early return is now certain. I cannot conclude without congratulating him on his successful mission abroad, and trust he will long retain his interest in this Society, his many years as President having endeared him to many of us. I must acknowledge the devotion of our Honorary Secretary (Mr. Phil Renshaw, and extend to him the Institute 's warmest thanks. We are glad to -congratulate him on his recovery from a recent illness. To our Treasurer (Mr. Mingay) we must extend our appreciation of the good work he has done, and are glad to know he has successfully piloted us from debit to credit. Mr. Charlesworth, who unselfishly filled the breach in Mr. Renshaw's absence, merits the warmest thanks of the Division and his good work on many occasions has been greatly appreciated. Page 65

TRANS-PACIFIC RADIO TESTS
'''TRANS-PACIFIC RADIO TESTS. Prize List.''' First six prizes to be given to entrants having the most complete log of signals received during the tests. Prizes donated by :-Western Electric Co., Ltd., open order, £10/10/-; Colville & Moore Wireless Suuplies, open order, £5/5/-; Burgin Electric Co., open order, £5/5/-; Wireless Weekly, open order, £5/5/-; Australectric, Limited, open order, £5/5/-; Electricity House, open order, £5/5/-. Four prizes for entrants having the most complete log of signals on the least number of valves. Separate heterodyne not counted as a valve. Prizes donated by :-W. Harry Wiles, open order, £5/5/-; Radio Company, open order, £3/3/-; Sydney Dynamo & Motor Works, open order, £3/3/-; Continental Radio & Electric Co., open order, £3/3/. Two prizes for entrants who receive the greatest number of different American amateur stations. Prizes donated by :-F. E. 0 'Sullivan, open order, £2/2/-; Universal Electric Co., open order, £2/2/-; Two prizes for the most complete log of entrant using a hard valve .as a detector. Prizes donated by:- -Radio House, open order, £1/1/-; Miss Wallace, open order, £1/1/-. One prize for the most complete log of entrant using a soft valve as a detector. Prize donated by J. H. Dewis, open order, 10/-. One prize for the most complete log of entrant using the most original circuit. Prize donated by Malcolm Perry, open order, £3/3/-.

(Start Photo Caption) ~- L, Rothafel, director of the New York Capitol Theatre, sitting in his state room on board the trans-atlantic liner L, "6erengaria," in which he had a complet~ rndio receiving set installed , "Wide World" Photo (End Photo Caption)

P.60 - The Experimenters' Corner
The Experimenters' Corner.

P.62 - In Radio Land
In Radio Land.

Australian Experimenters on Their Mettle. As the time for the commencement of the Trans-Pacific radio test oraw8 near fresh interest is being evidenced in the ranks of experimenters. The entries from N.S.W. total twenty-four, all d them highly proficient men, v:h0 are determined to show the worlJ that Australian experimenters are second to none in the matter of constructing apparatus capable of receiving long distance low-power signals. · Individuals and firms commercially interested in radio have offered liberal encouragement by donating prizes to successful competitors, and there is little doubt that when the results are known more than one Australian experimenter will have covered himself with glory. The American amateurs are 2paring no trouble and expense in their preparations for transmitting the signals, and this, .in itself, places an ·- obligatior! upon Australian experimenters to leave no stone unturned to emmre efficient reception. A distinction will attach to tl1 ·.~ winner, or winners, of this test vvhich will, in itself, prove more than sufficient reward for any trouble or expense amateurs in Australia may incur. Apart from this the national aspect of the matter is of considerable moment, not to mention the benefit likely to be derived from an experimentation point of view. This combination of circumstances invests the Trans-Pacific tests with an importance to Australian experimenters overshadowing any previous event in local radio circles. That they win rise to the occasion there is little <1o11ht1 anif the whole world wiil awriit "RAD"IO" with interest the publication of the results which are expected to be available a couple of weeks after the tr,;ts conclude. Radio Interests a Rich Man. There is probably no more enthusiastic believer in the possibilities of radio telephony than Sir Thomas Lipton, the hero of so many exciting yachting contests. Rir Thomas pays frequent visits to America, and greatly admires the Americans for their enterprise and initiative. ·He was one of those who made an ea.rly attempt to speak from America to England, but the voice failed to carry. Sir Thomas was not at all disheartened, but tried twice more, and again failed. Since then, as all the world knows, the task of speaking across the wide stretch of ocean separating the two countries has been accomplished. Sir Thomas is intensely interested in the possibility of transmittjng power by radio. He does not care to prophesy, but one can easily realise that in the back of his mind he is visualising a time when a race in which radio-driven motor boats will take part will attract wide attention. The great Lipton estate in England boasts several radio sets, and Sir Thomas is determined that wireless will play a part in the next international yacht race. · Ruch is the progressive spirit of this millionaire gentleman who ha,; been aptly described as the merchant prince of London. Healing The Sick. It is refreshing to know that radio has uses outside its commercial asJWCt. Th~ !lverage A 11strnlian doe~ .• - ,.j May 2, 1923. not realise the extent to which wireless telephony has invaded many avenues of life in America, nor the part it is playing in brightening the lives of many who would otherwise be condemned to a cheerless existence. A patient in a Canadian Hospital who contracted consumption during the great war, speaks feelingly aud enthusiastically of the immense benefit he has derived both from a mental and physical point of view, since a radio receiving set has been installed at his bedside. '' For the first year aud a half of rny confinement to the Hospital bed, ' ' said the patient recently, '' I tried reading, and then bead and reed work to pass away the time, but gradually, as my strength failed, I lost all interest in these. I wanted something interesting and entertaining, and which imposed no mental or physical strain on my fast-failing constitution, That 'something' proved to be radio, and despite the scoffs of my friends and the hospital attendants I soon had an outfit installed at my bedside. It is no exaggeration to say that I have improved vastly since the mental strain of always thinking of my trouble has been banished by the soothing and entertaining musical programmes I have been able to hear. I am convinced that the time is not far distant when radio receiving sets will be installed in the majority of hospitals and sanatoriums.'' · '' If readers should ever be called on to contribute to a fund for that pur~ pose let them remember some of their own friends or loved ones may be amongst those cheered and entertained by the magic agency of radio tdephony. ''

"POLICE RADIO" A sound and convincing case for the application of radio to the busi- 11e1'R of aiding the police in locating and apprehending criminals is. made ont in an attractively compiled. booklet, entitled:  Police Radio, issued bv the Melbourne office of Amalgan~ ated Wireless (Aust.), Ltd. The equipment of the police patrol car in · Melbourne with a radio receiving set is unquestionable evidence that the police authorities in Australia recognise the tremendous assistance they will receive by utilising this newest system of communication in tracking clown evil-doers. As the booklet explains, the police authorities all over the world are interesting themselves in radio telephony. Ever since the famous Crippen case in 1910, when that noted criminal and his associate, Miss Le Neve, were apprehended on the Montrose by officers of Scot] and Yard, thanks to the use of wireless telegraphy, it has been conHidered inevitable that the police would sooner or later enlist radio as part of their every clay equipment. As a record of what has been accomplished to date, and a forecast of ffhat mav be done in the future.  Police ·Radio is both interesting and valuable. It will help materially to educate the general public to a realisation of what radio telephony may accomplish in the near future, and it will likewise impress the police anthorities with the wisdom of keeping right up to date in their criminalcatching methods. SOCIAL GATHERING An enjoyable outing, in the form of ~ lannch picnic and gipsy tea, or-

ganised by the Wireless House Socia: l Committee, took place on Thursday evening, April 12, and was liberally patronised by the Head Office Staff of Amalgamated Wireless Ltd. A motor launch conveved the excursionists to Rodd Isl~itd,. where, after refreshments had been served, dancing and card games were indulged in to the enjoyment of all. In welcoming thoRe present the Chairman of the Social Committee (Mr. G. P. AtkinRon) emphasised the pleasure the organisers felt at the s11ccesR which had attended their initial effort. It was particularly gratifying to know that they had the support of the Company's responsible officers. The presence of Mr. and Mrs. Wilson, Mr. Larkins and Mr. Gardill"" harl nut thrm all in good heart. Mr. Perry, in supporting Mr. Atkinson's remarks, stressed the· value of social gatherings as a means of enabling the staff to get to know and understand each other. He supported the chairman's welcome to those already mentioned. Mr. Wilson, in expressing thanks for the kind remarks concerning Mrs. Wilson and himself, spoke of the need for making the most of opportunities for recreation which came all too seldom to the average person immersed in the cares of business life. He could not help expressing how fortunate he considered the Company was in having the services of such an. exa cellent staff as Amalgamated Wireless possessed. He had frequently expressed similar sentiments to Mr. Fisk, prior to the latter'R departure for London. After a thoroughly enjoyable evening the party returned to the city at 11 p.m.

OUR FRONT COVER. The six-year-old boy illustrated on the front cuver of this isslie is known as "Pedro"-otherwise Frank J. Powers, junr.; son of a lawyer in Grand Rapids, Michigan, U.S.A. He is a radio enthusiast, and _when photographed was listening to a lecture on motor cars, in which he is also intensely interested. He knows every make of car, from a Ford to a Rolls Royce, and is credited with being able to distinguish one from another a mile off. ALL-AUSTRALIAN COMPETITION. Some months ago Sea, Land & Air offered a prize of a gold presentation medal to the Australian competitor who records the best log of the messages sent from America during the test. Now that Sea, Land & Air is incorporated in Radio, it has been decided to extend the time for receiving entries for this competition · to. May 12. · Every competitor who fills.in the coupon printrrl. below and returns same to the Editor of Radio not later than May 12 will have a chance of winning this handsome medal.

P.66 - Call Letters
Call Letters.

This is the third list of call letters of Australian and New Zealand ship and land stations. In subsequent issues of "Radio" further lists will appear, all of which should be preserved so that readers will have a complete list of both local and overseas stations.— Ed.

M K A s.s. Ruahine M K B s.s. Ruapehu M K D s.s. Palma M K K s.s. Medic M K R s.s. Beltana M K V s.s. Rernuera M M :b s.s. Malwa M M E s.s. Mantua M M F s.s. Morea M M L s.s. Macedonia M O F s.s. Orsova M O J s.s. Orvieto M O S s.s. Wairnate M O Y s.s. Osterley M Q C s.s. Persic M R F s.s. H ororata M R G s.s. Opawa M R I s.s. Whakatane M R M s.s. Orari M R S s.s. Ifoikoura M R V s.s. W a1:wera M S B s.s. Kararnea M S E s.s. Euripides M S O s.s. Poona M U Z s.s. Z ealandic M V S s.s. Sussex M W C s.s. Runic M W E s.s. Arnwa M W F s.s. Tainni M W I s.s. Ionic M W N s.s. Athenic M Y N s.s. Tahiti N P G San Francisco Radio N P L San Diego, California, Radio N P M Honolulu Radio N P M Pearl Harbour Radio N P N Guam Radio N P O Cavite Radio N P P Peking Radio N P U 'l'utuila Radio ·P M C s.s. Houtman V B B C S.S. Canadian Traveller V G B F S.S. Canadian Mariner V G B K s.s. Canadian Sapper ·V G B M s.s. Canadian Fisher V G B P s.s. Canadian Victor V G B Q · s.s. Canadian Explorer V G B T s.s. Canadian Forester V G B W s.s. Canadian Sk1'rrnisher V G B X s.s. Canad1:an Hnnter V G B Y s.s. Canadian Transporter V G B Z s.s. Canadian HarveSfor· V G D B· S.S. Canadian W1'nner

V G D C s.s. Canadian Highlande1· V G D F s.s. Canadian Freighter V G D K s.s. Canadian Scottish V G D T s.s. Ca.nadian Rirnner V G D Z s.s. Canadian Rover V G J L s.s. Canadian Carnrnander V G J T s.s. Canadian Sqirntter V G J W s.s. Canadian Coaster V G J X s.s. Canadian Leader V G K M s.s. Canadian Carrier V G L B s.s. Canadian Observer V G L F s.s. Canadian Otter V G L N s.s. Canadian Pathfinder V G L Q s.s. Canadian Engineer V G L S s.s. Canadian Logger V G L T s.s. Canadian English V G μ X s.s. Canadian Conqueror V G L Y s.s. Canadian Cha.llenger V G L Z s.s. Cana,d•ia.n Constructor V G N B s.s. Canadian Cruiser V G N C s.s. Canadian Trapper V N P s.s. Apolda. V N W s.s. W onganella V P D Suva Radio V P E Labasa Radio V P F Taveuni Radio V P K Cocos Radio V P S Cape D 'Aguilar Radio V P W Singapore Radio V P X Penang Radio V Q A J esselton Radio V · Q B Sandakan Radio V Q K Ocean Island Radio V Q L Savu Savu Radio V S B Nukualofa Radio V X A s.s. Oonah V X B s.s. Ba.mbra V X C s.s. Period -V X D -s.s. Talawa , V X E s.s. Dilga. V X F s.s. Ashridge V X G. s.s. Enoggera V X H s.s. Xoo·yong V X I s.s. Iron Monarch V X J s.s. Kooringa V X K s.s. Iron Pri1ice V X L s.s. M o-irct V X M s.s. Barwon V X N s.s. Ooma V X O s.s. Corio V X P s.s. Dromana · V X Q S,S. Rona

Book Review
BOOK REVIEW. "Mast and Aerial Construction for Amateurs.'' - By F. J. Ainsley, A.M.1.C.E. 82 VI. p.p. The Wireless Press, Sydney. (Price, 10s. 6d.) 0 NE of the chief problems con. fronting the amateur fitfrng a wireless set is the provision of an aerial. A good, well-elevated aerial is a good investment, as with it satisfactory results can be obtained with simpler and less expensive receiving apparatus. Judging from the type of aerial one usually sees in suburban back gardens, this little book should meet a definite need. The subject matter is divided into eight chapters. Chapters one and two provide general information regarding types of masts and materials. The succeeding chapters are devoted to detailed instructions for the building of various •types of masts. 'I'hese masts include scaffold pole masts, plank masts, stayed · lattice masts, self-supporting lattice towers and tubular steel masts. 'l'he final chapter describes aerial systems suitable for small receiving installations, and gives much useful information on roof attachments, leading-in and soldering. The last few pages contain useful hints on frame aerials. Considering the small size of the masts under· com:ideration, it is doubtful whether, from a cost point of view, any other type but the scaffold pole is justified. On the other hand, the amateur is not wholly influenced by reasons of first cost in regard to a hobby, and the designs would provide much interesting work. Generally the information in this attractive little volume appears to be accurate. The ultimate strength of · yeJlow · deal, given on page 13, appears to be lower than usually assumed by engineers, while on page · 18, by an obvious slip, the breaking strain of iron wire is given as 55 lbs. per square inch. 'l'he book is well printed, and con" tains 65 clear and useful drawings. rt is confidently recommended to all amateurs seeking information on the subject.

P.68 - Low Power Tests
'''Low Power Tests. Victorian Amateur's Performance.'''

Newman succeeds in being heard in Hobart by Watty and others in the Melbourne-Hobart tests '''Low Power Tests. Victorian Amateur's Performance''' Although it is only a few months since the Melbourne experimenters commenced transmission in earnest, some almost record breaking results have already been obtained. Now that the winter months are approaching, it is expected that even greater distances on reduced power will be achieved. A series of tests with C.W. on a wave length of 400 metres was organised by Mr. R. A. Hull with Mr. W. T. Watkins of Hobart, Tasmania, seven stations in all taking part in the tests. Each station was to have been limited to five watts plate input, but as some of the stations showed hardly any radiation on that input, it was decided to extend the power to eight watts. As some of the inputs were not measured at all, it was not definitely known what the plate inputs were, but they were probably not more than 12 watts. However, Mr. Newman's plate input was measured accurately on standard meters, and on one occasion was as low as 3.8 watts, the highest reading being 4.4 watts. On this plate input his radiation was 600 milliamps with a tuned counterpoise. The radiation of the other stations varied from 400 to 800 milliamps. So as to make the tests of a more official nature, each station was allotted a four-letter code word which was used in place of the call letters of the station, so that the receiving operator did not know what was transmitting. In addition, each station sent a sentence of ten words which was different for each night's transmission.
 * STAN-SURE — 3AM — Mr. G. S. Dohrmann and Mr. Dixon.
 * GOAT-GAME — 3BD — Mr. E. H. Cox.
 * JUMP-EAST — 3BM — Mr. H. K. Love.
 * LONG-BEER — 3BQ — Mr. W. F. M. Howden.
 * HIGH-DIAL — 3BY — Mr. H. Holst.
 * FORD-WORD — 3JU — Mr. R. A. Hull.
 * MORE-MOON — 3MC — Mr. S. M. Newman.

The tests commenced at the conclusion of the ocean forecast transmitted by the Melbourne Radio Station at 9 o'clock, and each station continued its transmission for exactly ten minutes. The results of these tests were very satisfactory, all seven stations being heard in Hobart, and at least four had their ten-word sentence copied correctly. Although no telephony was arranged for in the tests, Mr. Watkins and several other amateurs copied some of the telephone conversation between 3BY, 3JU, and 3MC before and after the tests on several occasions. This is now almost a nightly occurrence in spite of the fact that the operators in Hobart are only using single valve receivers. Although the tests were intended for Hobart, Mr. C. D. Maclurcan and Mr. J. H. A. Pike, both of Sydney, also heard the C.W. test signals and were successful in copying the whole sentence of Mr. Newman's transmission and heard the code words of four other stations. Encouraged by the success of the Hobart tests, the seven stations concerned arranged for a similar test with Sydney, Adelaide and Hobart amateurs, on much the same lines as the Tasmanian tests, except that each station finished up with a few "Hullo's" on telephony. These tests were even more successful, as not only were the C.W. signals copied in all three cities, but faint speech was also heard, and it is reasonable to suppose that good speech would have been received had the receiving stations used a stage of high frequency amplification. The outstanding feature of the tests was that Mr. Newman (3MC — MORE" in the first test and "MOON" in the second) was only using an ordinary Marconi-Osram receiving R type Valve with 550 volts on the plate, the plate current being 7-8 milliamps, which is about 3.8-4.4 watts. This is less than half the power of any other station. In spite of this, Mr. Newman's signals were received equal in strength to the other stations at Hobart (360 miles) and much stronger at Sydney (460 miles); even stronger signals were received at Adelaide (400 miles). Mr. Snoswell of Adelaide reports that "MOON'S" C.W. could be heard 8-10 feet from the Receivers. This is possibly accounted for by the directional effect of the inverted L Aerial which Mr. Newman uses, the maximum radiation being towards the west and minimum to the south.

P.69 - Radiofun
Radiofun.

P.70 - Club Notes & News
Club Notes & News.

WIRELESS INSTITUTE OF AUSTRALIA New South Wales Division. 'l'he next General meeting of the XS.W. Division, to be held in the Education Building Lecture Hall, Loftus and Bridge Streets, Sydney, on Thursday, May 10th, at 8 p.m., will be an All-Clubs ' Night. A special lecture, entitled: '' The Co-Relation of Different Forms of Energy,'' wiH be delivered by Mr. A lee. Hector, manager of Messrs. Burroughs, Welcome & Co., ,vho is an eminent physicist. All clubs and societies are specially invited to attend in full strength, and should any club or society not have been notified they are asked to take this a:-, a direct invitation to be present. All experimrnters are also invited to attend, bnt it will be necessary for those not attached_ to a Club or Societv to obtain a eard from Mr. Phil Re1;shaw, Honorary Secretary, :=!rd F loor, 85 P itt Street, Sydney. It is desired to make this gather ing a truly representative one, and all interested are specially requested to be present. South Australian Division. The April meeting of the above Division of the Institute was well attended, and members present had the pleasure of hearing an excellent lecttue on " Automatic Transmission at High Speeds, by Mr. Milne. . . . The lecturer explained the circuits and methods of using valve r eceivers capable of receiving 1,500 words a minute. He exhibited a number of circuits and detailed their functioning in such a clear manner that all present were able to thoroughly grasp the explanations. "RADIO" In eonveying· the thanks of all present to Mr. Milne, the President (Mr. Hambly Clark) mentioned that he had consented to deliver a further lecture on the "Creed System" at the next meeting. A tape reading machine and other apparatus provided by Mr. R. B. Caldwell would enable a dem~ nstration to be given at that meetmg. 'I'he President further announced that in view of the large nnmber of new members who were joining up the Executive Council had decided to form the Society into two divisions. One will consist of a jnnio_r, or associate division, with advancement after a suitable examination to the senior or full member division. It is hoped to have this scheme working shortly, and it will uucloubtedly prove of great benefit to the younger and less experienced members. A committee has been formed to make preliminary arrangement:s for a social meeting and demonstration, with the object of promoting a moi-e sociable Rpirit, and bringing the members closer together. Box Hill Radio Club. The PreRident (Mr. Howden) took the chair at the regular meeting of the above Club, at which there was an excellent attendance. The Secretary gave an interesting lecture on '' Wireless Construction
 * B7 or Beginners,'' in the course of

which he outlined several suitable circuits, and gave details of construction. 'I'he Club transmitting license has been applied fo r, and as soon as granted the set will be assernbled and put into working order. It will be only 5 watt at first, but will later be increase<l to 25. The receiver will he the 3-valve type. May 2, 1923. At the next meeting, Mr. Love, Chairman of the Trans Pacific Test Committ ee, will lecture on the forthcoming tests, which are arousing a lot of interest amongst members. At a later meeting Mr. Howden will lecture on the construction of sets suitable for Short Wave work. All enquiries r elative to the Club's activities should be addressed to · the Hou. Secretary (Mr . H. Hurst), No. .Manly and District Radio Club. An interesting lecture on "Aerials" was delivered by Mr. F. C. Swinlmrne at the third meeting of the above Club, held in the Manlv Literary Institute. The keenness ;f those present to gain information was evidenced by the number of questions asked, and in order that those interested might gain a better knowledg~ of the subject, the lecturer illustrated his remarks by blackboard diagrams wherever possible. · 'l'he Club has now come t o a satisfactory arrangement with the Committee of the Literary Institute regarding accommodation, and an aerial 150 feet long by 50 feet high will shortly be erected in the Institute grounds. It is also proposed to hold a umJL·ber of social entertainments on a big scale in the near future, and the Com, mittee have every expectation of be' ing able to educate the Manly public to the value of radio telephony as a medium of entertainment at an early date. · 'I'he Club meetings are held each second Monday night in the Literary Institute, and buzzer classes every Wednesday night. New members anq visitors a.re always welcome.
 * 3 Wellington Road, Box Hill (Vic.).

P.71 - List of Abbreviations
List of Abbreviations.

P.72 - Queries Answered
Queries Answered.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


text:
LMIs in Control/Click here to continue

LMIs in Control

Table of contents

 * 1) /Preface/
 * 2) Introduction to Control System
 * 3) /Notation/
 * 4) /Fundamentals of Matrix and LMIs/
 * 5) /Software for LMIs and Optimization/
 * 6) /YALMIP/
 * 7) Linear Systems
 * 8) Introduction to Linear System
 * 9) /LMIs in system and stability Theory/
 * 10) /Optimal control systems/
 * 11) /Controller synthesis/
 * 12) /Observer synthesis/
 * 13) /Robust Controls/
 * 14) /Time-Delay Systems/
 * 15) /Integral Quadratic Constraints/
 * 16) /Applications of Linear systems/
 * 17) Non-linear Systems
 * 18) Introduction to Non-Linear systems
 * 19) /Applications of Non-Linear Systems/

title: LMIs in Control/Click here to continue


text:
LMIs in Control/Click here to continue/Preface

Preface

This book will discuss Linear Matrix Inequalities in Control Systems. The recent introduction of Linear Matrix Inequality (LMI) methods in control has dramatically expanded the types and complexity of the systems we can control.

For example, consider the problems: Gain Scheduling for Missile Attitude Control (A switched system); Control of Robots over Noisy Communication Channels (sampled-data systems); Remote Control of Spacecraft Attitude (a delayed system); Behavioral Therapy (A system with binary inputs); or self-driving vehicles (a case of decentralized control). None of these systems can be studied using classical root-locus or PID control methods. Rather, advances in these fields have been made possible through the increased power and flexibility created by the LMI (optimization-based) approach to control.

The objective of this book is to let readers be able to use LMI solvers to synthesize optimal or suboptimal controllers and estimators for multiple classes of state-space systems. This book will require prior knowledge of linear algebra, integral and differential calculus, and at least some exposure to ordinary differential equations. Some background in control systems is also preferrable. In addition, a prior knowledge of integral transforms, specifically the Laplace transforms will be very beneficial. Access to MATLAB is also required, including the robust control toolbox. A guide on the installation and usage of software used in LMI and Optimization is given in this book. Use the links below for the reference on Control systems and transformation of Matrices.


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LMIs in Control/Click here to continue/Notation

Notations

Examples
Consider the square matrix $$ A \in \mathbb{R} ^{n\times n}$$. The eigenvalues of $$A$$ are denoted by $$ \lambda_i (A), i = 1, 2,. . ., n$$. The matrix A is Hurwitz if all of its eigenvalues are in the open left-half complex plane (i.e., Re $$ \lambda_i (A), i = 1, 2, . . ., n$$ ). A matrix is Schur if all of its eigenvalues are strictly within a unit disk centered at the origin of the complex plane (i.e., $$|\lambda_i (A)| < 1, i = 1, . . ., n)$$. If $$A \in \mathbb{S}^n $$, then the minimum eigenvalue of A is denoted by $$\lambda(A)$$ and its maximum eigenvalue is denoted by $$ \bar{\lambda} (A) $$.

Consider the matrix B $$\in \mathbb{R} ^{n\times m}$$. The minimum singular value of B is denoted by $$ \underline{\sigma} $$ (B) and its maximum singular value is denoted by $$ \bar{\sigma} $$(B). The range and nullspace of B are denoted by $$ \mathbb{R} $$(B) and $$ \mathbb{N} $$(B), respectively. The Frobenius norm of B is ||B|| = $$ \sqrt{tr(B ^H B)}$$.

A state-space realization of the continuous-time linear time-invariant (LTI) system

$$ \dot{x}(t) = Ax(t) + Bu(t)$$,

$$ y(t) = Cx(t) + Du(t),$$. is often written compactly as (A, B,C,D) in this document. The argument of time is often omitted in continuous-time state-space realizations, unless needed to prevent ambiguity. A state-space realization of the discrete-time LTI system

$$x_{k+1} = A_d x_k + B_d u_k,$$

$$y_k = C_d x_k + D_d u_k,$$

is often written compactly as $$(A_d, B_d,C_d,D_d)$$.

The $$\mathcal{H}$$∞ norm of the LTI system $$ \mathcal{G}$$ is denoted by ||$$ \mathcal{G} $$||∞ and the $$\mathcal{H}_2 $$ norm of $$\mathcal{G}$$ is denoted by
 * $$\mathcal{G}$$||$$_2$$.

The inner product spaces $$ \mathcal{L}_2 and \mathcal{L}_{2e}$$ for continuous-time signals are defined as follows.

$$ \{\mathcal{L}_2\} = \left\{x: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}^{n} | ||x||^2_2 = \int_{0}^\infty x^T(t)x(t)dt<\infty \right\}, $$

$$ \{\mathcal{L}_{2e}\} = \left\{x: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}^{n} | ||x||^2_{2T} = \int_{0}^T x^T(t)x(t)dt<\infty, \forall T \in \mathbb{R}_{\geq 0} \right\}. $$

The inner product sequence spaces ℓ2 and ℓ2e for discrete-time signals are defined as follows. $$ \{\mathcal{l}_2\} = \left\{x: \mathbb{Z}_{\geq 0} \rightarrow \mathbb{R}^{n} | ||x||^2_{2} = \sum_{k=0}^\infty x^T_k x_k <\infty, \right\}. $$ $$ \{\mathcal{l}_{2e}\} = \left\{x: \mathbb{Z}_{\geq 0} \rightarrow \mathbb{R}^{n} | ||x||^2_{2N} = \sum_{k=0}^N x^T_k x_k <\infty, \forall N \in \mathbb{Z}_{\geq 0} \right\}. $$

Reference

 * LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & Francis Group, 2013
 * LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
 * LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.


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LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs

Fundamentals of Matrix and LMIs

Fundamentals of Matrix and LMIs Properties of LMIs
 * 1) /Basic Matrix Theory/
 * 2) /Notion of Matrix Positivity/
 * 3) /Matrix Inequalities and LMIs/
 * 4) /Convexity of LMIs/
 * 5) /Concatenation of LMIs/
 * 1) /Change of Subject/
 * 2) /Congruence Transformation/
 * 3) /Young’s Relation (Completion of the Squares)/
 * 4) /Young’s Relation-Based Properties/
 * 5) /Special Cases of Young’s Relation/
 * 6) /Iterative Convex Overbounding/
 * 7) /Projection Lemma (Matrix Elimination Lemma)/
 * 8) /Strict Projection Lemma/
 * 9) /Non Strict Projection Lemma/
 * 10) /Reciprocal Projection Lemma/
 * 11) /Projection Lemma-Based Properties/
 * 12) /Ellipsoidal inequality/
 * 13) Continuous Time Properties
 * 14) /Schur Complement/
 * 15) /Strict Schur Complement/
 * 16) /Non-Strict Schur Complement/
 * 17) /Schur Complement Lemma-Based Properties/
 * 18) Eigenvalue related Problems
 * 19) /Matrix Eigen Value Minimization/
 * 20) /Continuous Time/Eigenvalue Problem/
 * 21) /LMI for Generalized eigenvalue problem/
 * 22) /Matrix Norm Minimization/
 * 23) /LMI for Generalized Eigenvalue Problem/
 * 24) /LMI for Linear Programming/
 * 25) /LMI for Feasibility Problem/
 * 26) /Continuous Time/Structured Singular Value/
 * 27) /LMI for Minimizing Condition Number of Positive Definite Matrix/
 * 28) /Continuous time quadratic stability/
 * 29) /Minimizing Norm by Scaling/
 * 30) Discrete Time Properties
 * 31) /Discrete Time Minimum Gain Lemma/
 * 32) /Discrete Time Modified Minimum Gain Lemma/
 * 33) /Finsler’s Lemma/
 * 34) /Dilation/
 * 35) /Tangential Nevanlinna Pick/
 * 36) /Nevanlinna Pick Interpolation with Scaling/
 * 37) /Deduced LMI Conditions for Hinf Index/
 * 38) /Deduced LMI Conditions for H2 Index/
 * 39) /Dissipativity of Systems/
 * 40) /Generalized H2 Norm/
 * 41) /Passivity and Positive Realness/
 * 42) /Non-Expansivity and boundedness Realness/
 * 43) /Variable Reduction Lemma/
 * 44) /D-stability Rise Poles/
 * 45) /D-stability Settling time poles/
 * 46) /D-stability Max Percent Overshoot Poles/
 * 47) /Iterative Convex Overbounding/
 * 48) /Frobenius Norm/
 * 49) /Dualization Lemma/
 * 50) /Submatrix Determinants and Imaginary and Real Parts/


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LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Basic Matrix Theory

Basic Matrix Notation
Consider the complex matrix $$A\in \C^{n\times m} $$.
 * $$ A=\begin{bmatrix} a_{11} & \dots & a_{1m} \\

\vdots & \ddots & \vdots \\ a_{n1} & \dots & a_{nm}\end{bmatrix} \in \C^{n\times m} $$

Transpose of a Matrix

The transpose of $$A$$, denoted as $$A^{T}$$ or $$A'$$ is:
 * $$ A^T=\begin{bmatrix} a_{11} & \dots & a_{n1} \\

\vdots & \ddots & \vdots \\ a_{1m} & \dots & a_{nm}\end{bmatrix} \in \C^{m\times n}. $$

Adjoint of a Matrix

The adjoint or hermitian conjugate of $$A$$, denoted as $$A^{*}$$ is:
 * $$ A^*=\begin{bmatrix} a_{11}^* & \dots & a_{n1} ^*\\

\vdots & \ddots & \vdots \\ a_{1m}^* & \dots & a_{nm}^*\end{bmatrix} \in \C^{m\times n}. $$ Where $$a_{nm}^*$$ is the complex conjugate of matrix element $$a_{nm}$$.

Notice that for a real matrix $$A\in\R^{n\times m}$$, $$A^*=A^T$$.

Important Properties of Matricies
Hermitian, Self-Adjoint, and Symmetric Matricies

A square matrix $$A\in \C^{n\times n} $$ is called Hermitian or self-adjoint if $$A=A^*$$.

If $$A\in \R^{n\times n} $$ is Hermitian then it is called symmetric.

Unitary Matricies

A square matrix $$A\in \C^{n\times n} $$ is called unitary if $$A^*=A^{-1}$$ or $$A^*A=I$$.


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LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Notion of Matrix Positivity

Notation of Positivity
A symmetric matrix $$A\in\R^{n\times n}$$ is defined to be:

positive semidefinite, $$(A\ge 0)$$, if $$x^TAx\ge 0 $$ for all $$x\in\R^n, x\neq \mathbf{0} $$.

positive definite, $$(A>0)$$, if $$x^TAx> 0 $$ for all $$x\in\R^n, x\neq \mathbf{0} $$.

negative semidefinite, $$(-A\ge 0)$$.

negative definite, $$(-A>0)$$.

indefinite if $$A$$ is neither positive semidefinite nor negative semidefinite.

Properties of Positive Matricies

 * For any matrix $$M$$, $$M^TM>0$$.
 * Positive definite matricies are invertible and the inverse is also positive definite.
 * A positive definite matrix $$A>0$$ has a square root, $$A^{1/2}>0$$, such that $$A^{1/2}A^{1/2}=A$$.
 * For a positive definite matrix $$A>0$$ and invertible $$M$$, $$M^TAM>0$$.
 * If $$A>0$$ and $$M>0$$, then $$A+M>0$$.
 * If $$A>0$$ then $$\mu A>0$$ for any scalar $$\mu>0$$.


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LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Matrix Inequalities and LMIs

Matrix Inequality
Definition-1

A Matrix Inequality, $$G : \mathbb{R}^{m} \to \mathbb{S}^{n}$$, in the variable $$x\in\mathbb{R}^{m}$$ is an expression of the form

 * $$\begin{align}

G(x)=G_0+\sum_{i=1}^p f_i(x)G_i\leq0 \end{align}$$,

where $$x^T=[x_1\cdots x_m], G_0\in\mathbb{S}^{n}$$ and $$ G_i \in\mathbb{R}^{n \times n}$$, $$i=1,\ldots,p.$$

Linear Matrix Inequality
Definition-2

A Linear Matrix Inequality, $$F: \mathbb{R}^{m}\to\mathbb{S}^{n}$$, in the variable $$x\in\mathbb{R}^{m}$$ is an expression of the form
 * $$\begin{align}

F(x)=F_0+\sum_{i=1}^mx_iF_i\leq0 \end{align}$$,

where $$x^T=[x_1\ldots x_m]$$ and $$F_i\in\mathbb{S}^{n}$$, $$i=0\ldots, m.$$

Bilinear Matrix Inequality
Definition-3

A Bilinear Matrix Inequality (BMI), $$H:\mathbb{R}^{m}\to \mathbb{S}^{n}$$, in the variable $$x\in \mathbb{R}^{m}$$ is an expression of the form

 * $$\begin{align}

H(x)=H_0+\sum_{i=1}^mx_iH_i+\sum_{i=1}^m\sum_{j=1}^mx_ix_jH_{i,j}\leq0, \end{align}$$

where $$x^T=[x_1\cdots x_m]$$, and $$H_i$$, $$H_{i,j}\in\mathbb{S}^{n},$$ $$i=0,\ldots ,m$$, $$j=0 \ldots,m.$$

Example
Consider the matrices $$A\in\mathbb{R}^{n\times n}$$ and $$Q\in\mathbb{S}^{n}$$, where $$Q>0$$. It is desired to find a symmetric matrix $$P\in\mathbb{S}^{n}$$ satisfying the inequality

 * $$\begin{align}

PA+A^TP+Q<0,   \qquad \qquad \qquad \qquad \qquad  \qquad \qquad \qquad \qquad(1) \end{align}$$

where $$P>0$$. The elements of $$P$$ are the design variables in this problem, and although equation $$(1)$$ is indeed an LMI in the matrix $$P$$, it does not look like the LMI in definition 3. For simplicity, let us consider the case of $$n=2$$ so that each matrix is of dimension $$2 \times 2$$, and $$x=[p_1\quad p_2 \quad p_3]^T.$$ Writing the matrix $$P$$ in terms of a basis $$E_i\in\mathbb{S}^{2},$$ $$i=1,2,3$$, yields

 * $$\begin{align}

P=\begin{bmatrix}p_1&p_2\\p_2&p_3\end{bmatrix} = p_1\underbrace{\begin{bmatrix}1&0\\0&0\end{bmatrix}}_{E_1}+p_2\underbrace{\begin{bmatrix}0&1\\1&0\end{bmatrix}}_{E_2}+p_3\underbrace{\begin{bmatrix}0&0\\0&1\end{bmatrix}}_{E_3} \end{align}$$

Note that the matrices $$E_i$$ are linearly independent and symmetric, thus forming a basis for the symmetric matrix $$P$$. The matrix inequality in equation $$(1)$$ can be written as

 * $$\begin{align}

p_1(E_1A+A^TE_1)+p_2(E_2A+A^TE_2)+p_3(E_3A+A^TE_3). \end{align}$$

Defining $$F_0=Q$$ and $$F_i=E_iA+A^TE_i,$$ $$i=1,2,3,$$ yields

 * $$\begin{align}

F_0+\sum_{i=1}^3 p_i F_i<0, \end{align}$$

which now resembles the definition of LMI given in definition 2. Through out this wiki book, LMIs are typically written in the matrix form of equation $$(1)$$ rather than the scalar form of definition 2.


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LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Convexity of LMIs

Definition
A set, $$\mathcal{S}$$, in a real inner product space is convex if for all $$x, y\in\mathcal{S}$$ and $$\alpha\in\mathbb{R}$$, where $$0\leq\alpha\leq1$$, it holds that $$\alpha x+(1-\alpha)y\in\mathcal{S}$$.

Lemma 1.1
The set of solutions to an LMI is convex.

That is, the set $$\mathcal{S}=\{x\in\mathbb{R}^{m}\mid F(x)\leq0\}$$ is a convex set, where $$F:\mathbb{R}^{m}\longrightarrow\mathbb{S}^{n}$$ is an LMI.

Lemma 1.2
An LMI, $$F:\mathbb{R}^{m}\longrightarrow\mathbb{S}^{n}$$, in the variable $$x\in\mathbb{R}^{m}$$ is an expression of the form

$$F(x)=F_{0}+\sum_{i=1}^m x_{i}F_{i}\leq0$$

where $$x^{T}=[x_{1}\cdots x_{m}]$$ and $$F_{i}\in\mathbb{S}^{n}$$, $$i=0,\ldots,m$$.

Proof
Consider $$x,y\in\mathbb{R}^{m}$$ and $$\alpha\in[0,1]$$, and suppose that $$x$$ and $$y$$ satisfy Lemma 1.2.

The LMI $$F:\mathbb{R}^{m}\longrightarrow\mathbb{S}^{n}$$ is convex, since

$$F(\alpha x+(1-\alpha)y)=F_{0}+\sum_{i=1}^m (\alpha x_{i}+(1-\alpha)y_{i})F_{i}$$

$$\begin{alignat}{2} F(\alpha x+(1-\alpha)y) & = F_{0}+\sum_{i=1}^m (\alpha x_{i}+(1-\alpha)y_{i})F_{i}

\\ & = F_{0}-\alpha F_{0}+\alpha F_{0}+\alpha\sum_{i=1}^m x_{i}F_{1}+(1-\alpha)\sum_{i=1}^m y_{i}F_{i} \\ & = \alpha F_{0}+\alpha\sum_{i=1}^m x_{i}F_{i}+(1-\alpha)F_{0}+(1-\alpha)\sum_{i=1}^m y_{i}F_{i} \\ & = \alpha F(x)+(1-\alpha)F(y) \\ \end{alignat}$$

Convexity of LMI
From Lemma 1.1, it is known that an optimization problem with a convex objective function and LMI constraints is convex.

The following is a non-exhaustive list of scalar convex objective functions involving matrix variables that can be minimized in conjunction with LMI constraints to yield a semi-definite programming (SDP) problem.

 * $$\mathcal{J}(x)=\frac{1}{2}x^{T}PX+q^{T}x+r$$, where $$x,q\in\mathbb{R}^{n}$$, $$P\in\mathbb{S}^{n}$$, $$P>0$$, and $$r\in\mathbb{R}$$.

 * 1) Special case when $$q=0$$ and $$r=0:\mathcal{J}(x)=\frac{1}{2}x^{T}Px$$, where$$x\in\mathbb{R}^{n}$$, $$P\in\mathbb{S}^{n}$$, and $$P>0$$.
 * 2) Special case when $$P=$$2$$\cdot1$$, $$q=0$$, and $$r=0:\mathcal{J}(x)=x^{T}x=\|x\|^{2}_{2}$$, where $$x\in\mathbb{R}^{n}$$.

 * $$\mathcal{J}(X)=tr(X^{T}PX+Q^{T}X+X^{T}R+S)$$, where $$X$$, $$Q$$, $$R\in\mathbb{R}^{n \times m}$$, $$P\in\mathbb{S}^{n}$$, $$S\in\mathbb{R}^{n \times n}$$, and $$P\geq0$$.

 * 1) Special case when $$Q=R=0$$ and $$S=0:\mathcal{J}(X)=tr(X^{T}PX)$$, where $$X\in\mathbb{R}^{n \times m}$$, $$P\in\mathbb{S}^{n}$$, and $$P\geq0$$.
 * 2) Special case when $$P=1$$, $$Q=R=0$$ and $$S=0:\mathcal{J}(X)=tr(X^{T}X)=\|X\|^{2}_{F}$$, where $$X\in\mathbb{R}^{n \times m}$$.
 * 3) Special case when $$P=0$$, $$R=0$$ and $$S=0:\mathcal{J}(X)=tr(Q^{T}X)$$, where $$X$$, $$Q\in\mathbb{R}^{n \times m}$$.
 * 4) Special case when $$P=1$$, $$Q=R=0$$, $$S=0$$, and $$X\in\mathbb{S}^{n}:\mathcal{J}(X)=tr(X^{2})$$, where $$X\in\mathbb{S}^{n}$$.

 * $$\mathcal{J}(X)=\log(\det(X^{ -1}))=\log(\det(X))$$, where $$X\in\mathbb{S}^{n}$$ and $$X>0$$.

Relative Definition of a Matrix
The definiteness of a matrix can be found relative to another matrix.

For example,

Consider the matrices $$A\in\mathbb{S}^{n}$$ and $$B\in\mathbb{S}^{n}$$. The matrix inequality $$A0$$, when we know that $$B>0$$.

This follows from $$0<A<B$$.


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LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Concatenation of LMIs

Introduction
Matrix concatenation is the process of joining one or more matrices to make a new matrix. This process is similar for concatenating the LMIs as well.

Concatenation of LMIs
A useful property of LMIs is that multiple LMIs can be concatenated together to form a single LMI. For example, satisfying the LMIs $$ A < 0$$ and, $$B < 0$$ is equivalent to satisfying the concatenated LMI:

$$ \begin{bmatrix} A && 0 \\ 0 && B \end{bmatrix}$$ $$\begin{align} <0 \end{align}$$,

More generally, satisfying the LMIs $$A_i < 0$$, where $$i = 1,. . ., n $$ is equivalent to satisfying the concatenated LMI:

$$ \begin{bmatrix} A_1 && 0 && 0 &&. && . && 0 \\ 0 && A_2 && 0 &&. && . && 0 \\ 0 && 0 && A_3 &&. && . && 0 \\ . && . && . && . && . && 0\\ . && . && . && . && . && 0 \\ 0 && 0 && 0 && 0 && 0 && A_n\\ \end{bmatrix}$$ $$\begin{align} <0 \end{align}$$,


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LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Change of Subject

LMIs in Control/Matrix and LMI Properties and Tools/Change of Subject

A Bilinear Matrix Inequality (BMI) can sometimes be converted into a Linear Matrix Inequality (LMI) using a change of variables. This is a basic mathematical technique of changing the position of variables with respect to equal signs and the inequality operators. The change of subject will be demonstrated by the example below.

Example
Consider $$A \in \mathbb{R}^{n\times n}, B \in \mathbb{R}^{n\times m},K \in \mathbb{R}^{m\times n}$$, and $$Q \in \mathbb{S}^{n}$$, where $$Q > 0$$.

The matrix inequality given by:

$$QA^{T} + AQ - QK^{T}B{T} - BKQ < 0 $$ is bilinear in the variables $$Q$$ and $$K$$.

Defining a change of variable as $$ F = KQ $$ to obtain

$$QA^{T} + AQ + - F^{T} B^{T} - BF < 0$$,

which is an LMI in the variables $$Q$$ and $$F$$.

Once this LMI is solved, the original variable can be recovered by $$ K = FQ^{-1}$$.

Conclusion
It is important that a change of variables is chosen to be a one-to-one mapping in order for the new matrix inequality to be equivalent to the original matrix inequality. The change of variable $$F = KQ$$ from the above example is a one-to-one mapping since $$Q^{-1}$$ is invertible, which gives a unique solution for the reverse change of variable $$ K = FQ^{-1}$$.


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LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Congruence Transformation

LMIs in Control/Matrix and LMI Properties and Tools/Congruence Transformation

This methods uses change of variable and some matrix properties to transform Bilinear Matrix Inequalities to Linear Matrix Inequalities. This method preserves the definiteness of the matrices that undergo the transformation.

Theorem
Consider $$ Q \in \mathbb{S}^{n}, W \in \mathbb{R}^{n\times n}$$, where $$\operatorname{rank}(W) = n$$. The matrix inequality $$Q < 0$$ is satisfied if and only if $$WQW^{T} < 0$$ or equivalently, $$W^{T}QW < 0$$.

Example
Consider $$ A \in \mathbb{R}^{n\times n}, B \in \mathbb{R}^{n\times m}, K \in \mathbb{R}^{m\times p}, C^{T} \in \mathbb{R}^{n\times p}, P \in \mathbb{S}^{n}$$ and $$ V \in \mathbb{S}^{p} $$, where $$P > 0$$ and $$V > 0 $$. The matrix inequality given by

$$

\begin{align} \ Q &= \begin{bmatrix} \ A^{T}P + PA & -PBK + C^{T}V \\ \ * & -2V \\ \end{bmatrix} \ < 0, \end{align} $$

is linear in variable $$V$$ and bilinear in the variable pair $$(P.K)$$. Choose the matrix $$\operatorname{diag}(P^{-1},V^{-1})$$ to obtain the equivalent BMI given by

$$

\begin{align} \ WQW^{T} &= \begin{bmatrix} \ P^{-1}A^{T} + AP^{-1} & -BKV^{-1} + P^{-1}C^{T} \\ \ * & -2V^{-1} \\ \end{bmatrix} \ < 0, \end{align} $$

Using a change of variable $$X = P^{-1}, U = V^{-1}$$ and $$F = KV^{-1}$$, the above equation becomes

$$

\begin{align} \ WQW^{T} &= \begin{bmatrix} \ XA^{T} + AX & -BF + XC^{T} \\ \ * & -2U \\ \end{bmatrix} \ < 0, \end{align} $$

which is an LMI of variables $$X, U$$ and $$F$$. The original variable $$K$$ is recovered by doing a reverse change of variable $$K = FU^{-1}$$.

Conclusion
A congruence transformation preserves the definiteness of a matrix by ensuring that $$Q < 0$$ and $$W^{T}QW < 0$$ are equivalent. A congruence transformation is related, but not equivalent to a similarity transformation $$TQT^{-1} $$, which preserves not only the definiteness, but also the eigenvalues of a matrix. A congruence transformation is equivalent to a similarity transformation in the special case when $$W^{T} = W^{-1} $$.


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LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Young’s Relation (Completion of the Squares)

This method is used to solve quadratic equations that can't be factorized.

Matrix inequality
Consider $$ X,Y \in \mathbb{R}^{n\times m}$$ and $$ S \in \mathbb{S}^{n\times n}$$ , where $$ S $$ >0, The matrix inequality given by

\begin{align} \ X^{T}Y + Y^{T}X \leq X^{T}S^{-1}X + Y^{T}SY,\\ \end{align}$$ which is named Young’s relation or Young’s inequality.

Derivation
Young’s relation can be derived from a completion of the squares as follows.

\begin{align} 0 \leq (X-SY)^{T}S^{-1}(X-SY) \\ 0 \leq X^{T}S^{-1}X + Y^{T}SY - X^{T}Y - Y^{T}X\\ X^{T}Y + Y^{T}X \leq X^{T}S^{-1}X + Y^{T}SY, \end{align}$$ which is named Young’s relation.

Reformulation of Young’s Relation
Consider $$ X,Y \in \mathbb{R}^{n\times m}$$ and $$ S \in \mathbb{S}^{n\times n}$$ , where $$ S $$ >0, The matrix inequality given by

\begin{align} \ X^{T}Y + Y^{T}X \leq \frac12(X+SY)^{T}S^{-1}(X+SY) ,\\ \end{align}$$ is a reformulation of Young’s relation.


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LMIs in Control/Click here to continue/Applications of Linear systems

Applications of Linear systems

 * 1) /LMI for Attitude Control of Nonrotating Missiles, Pitch Channel/
 * 2) /LMI for Attitude Control of Nonrotating Missiles, Yaw/Roll Channel/
 * 3) /LMI for H2/Hinf Polytopic Controller for Robot Arm on a Quadrotor/
 * 4) /An LMI for the Kalman Filter/
 * 5) /Hinf Optimal Model Reduction/
 * 6) /An LMI for Multi-Robot Systems/
 * 7) /Helicopter Inner Loop LMI/
 * 8) /Hinf LMI Satellite Attitude Control/
 * 9) /H2 LMI Satellite Attitude Control/
 * 10) /Mixed H2-H∞ LMI Satellite Attitude Control/
 * 11) /Mixed H2/Hinf LMI Satellite Attitude Control/
 * 12) /Problem of Space Rendezvous and LMI Approaches/
 * 13) /LMI for Attitude Control of BTT Missiles, Roll Channel/
 * 14) /LMI for Controller Design of Multiple Solar PV Units Connected to Distribution Networks/
 * 15) /Hinf Optimal Stabilization of a Class of Uncertain Impulsive Systems/


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LMIs in Control/Click here to continue/Applications of Linear systems/LMI for Attitude Control of Nonrotating Missiles, Pitch Channel

LMI for Attitude Control of Nonrotating Missles, Pitch Channel

The dynamic model of a missile is very complicated and a simplified model is used. To do so, we consider a simplified attitude system model for the pitch channel in the system. We aim to achieve a non-rotating motion of missiles. It is worthwhile to note that the attitude control design for the pitch channel and the yaw/roll channel can be solved exactly in the same way while representing matrices of the system are different.

The System
The state-space representation for the pitch channel can be written as follows:

$$ \begin{align} \dot{x}(t) &= A(t)x(t) + B_{1}(t)u(t) + B_{2}(t)d(t) \\ y(t) &= C(t)x(t) + D_{1}(t)u(t) + D_{2}(t)d(t) \end{align} $$

where $$ x = [\alpha \quad w_{z} \quad \delta_{z}]^{\text{T}}$$, $$ u = \delta_{zc} $$, $$ y = [\alpha \quad n_{y}]^{\text{T}}$$, and $$ d = [\beta \quad w_{y}]^{\text{T}}$$ are the state variable, control input, output, and disturbance vectors, respectively. The paprameters $$ \alpha $$, $$ w_{z} $$, $$ \delta_{z} $$, $$ \delta_{zc} $$, $$ n_{y} $$, $$ \beta $$, and $$ w_{y} $$ stand for the attack angle, pitch angular velocity, the elevator deflection, the input actuator deflection, the overload on the side direction, the sideslip angle, and the yaw angular velocity, respectively.

The Data
In the aforementioned pitch channel system, the matrices $$ A(t), B_{1}(t), B_{2}(t), C(t), D_{1}(t),$$ and $$D_{2}(t)$$ are given as:

$$ \begin{align} A(t) = \begin{bmatrix} -a_{4}(t) & 1 & -a_{5}(t) \\ -\acute{a}_{1}(t)a_{4}(t) - a_{2}(t) & \acute{a}_{1}(t) - a_{1}(t) & \acute{a}_{1}(t)a_{5}(t) - a_{3}(t) \\ 0 & 0 & -1/\tau_{z} \end{bmatrix} \end{align} $$

$$ \begin{align} B_{1}(t) = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, \quad B_{2}(t) = \frac{w_{x}}{57.3} \begin{bmatrix} -1 & 0 \\ -\acute{a}_{1}(t) & \frac{J_{x}-J_{y}}{J_{z}} \\ 0 & 0 \end{bmatrix} \end{align} $$

$$ \begin{align} C(t) = \frac{w_{x}}{57.3} \begin{bmatrix} 57.3g & 0 & 0 \\ V(t)a_{4}(t) & 0 & V(t)a_{5}(t) \end{bmatrix} \end{align} $$

$$ \begin{align} D_{1}(t) = 0, \quad D_{2}(t) = \frac{1}{57.3g} \begin{bmatrix} 0 & 0 \\ V(t)b_{7}(t) & 0 \end{bmatrix} \end{align} $$

where $$ a_{1}(t) \sim a_{6}(t), \quad b_{1}(t) \sim b_{7}(t), \acute{a}_{1}(t), \acute{b}_{1}(t) $$ and $$ c_{1}(t) \sim c_{4}(t) $$ are the system parameters. Moreover, $$ V $$ is the speed of the missle and $$J_{x}$$, $$J_{y}$$, and $$J_{z}$$ are the rotary inertia of the missle corresponding to the body coordinates.

The Optimization Problem
The optimization problem is to find a state feedback control law $$ u= Kx$$ such that:

1. The closed-loop system:

$$ \begin{align} \dot{x} &= (A+B_{1}K)x + B_{2}d \\ z &= (C + D_{1}K)x + D_{2}d \end{align} $$

is stable.

2. The $$ H_{\infty} $$ norm of the transfer function:

$$G_{zd}(s) = (C + D_{1}K)(sI-(A+B_{1}K))^{-1}B_{2} + D_{2}$$

is less than a positive scalar value, $$ \gamma $$. Thus:

$$ ||G_{zd}(s)||_{\infty} < \gamma $$

The LMI: LMI for non-rotating missle attitude control
Using Theorem 8.1 in [1], the problem can be equivalently expressed in the following form:

$$ \begin{align} &\text{min} \quad \gamma \\ &\text{s.t.} \quad X > 0 \\ &\begin{bmatrix} (AX+B_{1}W)^{T} + AX+B_{1}W & B_{2} & (CX+D_{1}W)^{T} \\ B_{2}^{T} & -\gamma I & D_{2}^{T} \\ CX + D_{1}W & D_{2} & -\gamma I \end{bmatrix} < 0 \end{align}$$

Conclusion:
As mentioned, the aim is to attenuate the disturbance on the performance of the missile. The parameter $$ \gamma $$ is the disturbance attenuation level. When the matrices $$ W $$ and $$ X $$ are determined in the optimization problem, the controller gain matrix can be computed by:

$$ K = WX^{-1} $$

Implementation
A link to Matlab codes for this problem in the Github repository:

https://github.com/asalimil/LMI-for-Non-rotating-Missle-Attitude-Control

Related LMIs
LMI for Attitude Control of Nonrotating Missles, Yaw/Roll Channel

Return to Main Page
LMIs in Control/Tools


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LMIs in Control/Click here to continue/Applications of Linear systems/LMI for Attitude Control of Nonrotating Missiles, Yaw/Roll Channel

LMI for Attitude Control of Nonrotating Missles, Yaw/Roll Channel

Deriving the exact dynamic modeling of a missile is a very complicated procedure. Thus, a simplified model is used to model the missile dynamics. To do so, we consider a simplified attitude system model for the yaw/roll channel of the system. We aim to achieve a non-rotating motion of missiles. Note that the attitude control design for the yaw/roll channel and the pitch channel can be solved exactly in the same way except for different representing matrices of the system.

The System
The state-space representation for the yaw/roll channel can be written as follows:

$$ \begin{align} \dot{x}(t) &= A(t)x(t) + B_{1}(t)u(t) + B_{2}(t)d(t) \\ y(t) &= C(t)x(t) + D_{1}(t)u(t) + D_{2}(t)d(t) \end{align} $$

where $$ x = [\beta \quad w_{y} \quad w_{x} \quad \delta_{x} \quad \delta_{y}]^{\text{T}}$$, $$ u = [\delta_{xc} \quad \delta_{yc} ]^{\text{T}} $$, $$ y = [n_{z} \quad w_{x}]^{\text{T}}$$, and $$ d = \delta_{z}$$ are the state variable, control input, output, and disturbance vectors, respectively. The paprameters $$ \alpha $$, $$ w_{z} $$, $$ \delta_{z} $$, $$ \delta_{zc} $$, $$ n_{y} $$, $$ \beta $$, and $$ w_{y} $$ stand for the attack angle, pitch angular velocity, the elevator deflection, the input actuator deflection, the overload on the side direction, the sideslip angle, and the yaw angular velocity, respectively.

The Data
In the aforementioned yaw/roll channel system, the matrices $$ A(t), B_{1}(t), B_{2}(t), C(t), D_{1}(t),$$ and $$D_{2}(t)$$ are given as:

$$ \begin{align} A(t) = \begin{bmatrix} A_{11}(t) & A_{12}(t)\\ 0    & A_{22}(t) \end{bmatrix} \end{align} $$

where

$$ \begin{align} A_{11}(t) = \begin{bmatrix} -b_{4}(t) & 1 & \frac{\alpha(t)}{57.3} \\ -\acute{b}_{1}(t)b_{4}(t) - b_{2}(t) & -\acute{b}_{1}(t) - \acute{b}_{1}(t) & \frac{J_{y}-J_{z}}{57.3J_{x}}{w}_{z}(t) - \frac{\acute{b}_{1}\alpha(t)}{57.3}\\ c_{2}(t) & \frac{J_{y}-J_{z}}{57.3J_{x}}{w}_{z}(t) & -c_{1}(t) \end{bmatrix} \end{align} $$

$$ \begin{align} A_{12}(t) = \begin{bmatrix} 0 & -b_{5}(t)\\ 0 & -\acute{b}_{1}(t)\acute{b}_{5} - b_{3}(t) \\ -c_{3}(t) & c_{4}(t) \end{bmatrix} \end{align} $$

$$ \begin{align} A_{22}(t) = -\frac{1}{\tau_{x}\tau_{y}} \begin{bmatrix} \tau_{y} & 0\\ 0 & \tau_{x} \end{bmatrix} \end{align} $$

and

$$ \begin{align} B_{1}(t) = \begin{bmatrix} 0 & 0\\ 0 & 0\\ 0 & 0\\ \frac{1}{\tau_{x}} & 0\\ 0 & \frac{1}{\tau_{y}} \end{bmatrix}, \quad B_{2}(t) = \begin{bmatrix} a_{6}(t) \\ -\acute{b}_{1}(t)a_{6}(t) \\ 0\\ 0 \\ 0 \end{bmatrix} \end{align} $$

$$ \begin{align} C(t) = -\frac{1}{57.3g} \begin{bmatrix} V(t)b_{4}(t) & 0 & 0 & 0 & V(t)b_{5}(t)\\ 0 & 0 & -57.3g & 0 & 0 \end{bmatrix} \end{align} $$

$$ \begin{align} D_{1}(t) = 0, \quad D_{2}(t) = -\frac{V(t)}{57.3g} \begin{bmatrix} b_{6}(t) \\ 0 \end{bmatrix} \end{align} $$

where $$ a_{1}(t) \sim a_{6}(t), \quad b_{1}(t) \sim b_{7}(t), \acute{a}_{1}(t), \acute{b}_{1}(t) $$ and $$ c_{1}(t) \sim c_{4}(t) $$ are the system parameters. Moreover, $$ V $$ is the speed of the missle and $$J_{x}$$, $$J_{y}$$, and $$J_{z}$$ are the rotary inertia of the missle corresponding to the body coordinates.

The Optimization Problem
The optimization problem is to find a state feedback control law $$ u= Kx$$ such that:

1. The closed-loop system:

$$ \begin{align} \dot{x} &= (A+B_{1}K)x + B_{2}d \\ z &= (C + D_{1}K)x + D_{2}d \end{align} $$

is stable.

2. The $$ H_{\infty} $$ norm of the transfer function:

$$G_{zd}(s) = (C + D_{1}K)(sI-(A+B_{1}K))^{-1}B_{2} + D_{2}$$

is less than a positive scalar value, $$ \gamma $$. Thus:

$$ ||G_{zd}(s)||_{\infty} < \gamma $$

The LMI: LMI for non-rotating missle attitude control
Using Theorem 8.1 in [1], the problem can be equivalently expressed in the following form:

$$ \begin{align} &\text{min} \quad \gamma \\ &\text{s.t.} \quad X > 0 \\ &\begin{bmatrix} (AX+B_{1}W)^{T} + AX+B_{1}W & B_{2} & (CX+D_{1}W)^{T} \\ B_{2}^{T} & -\gamma I & D_{2}^{T} \\ CX + D_{1}W & D_{2} & -\gamma I \end{bmatrix} < 0 \end{align}$$

Conclusion:
As mentioned, the aim is to attenuate the disturbance on the performance of the missile. The parameter $$ \gamma $$ is the disturbance attenuation level. When the matrices $$ W $$ and $$ X $$ are determined in the optimization problem, the controller gain matrix can be computed by:

$$ K = WX^{-1} $$

Implementation
A link to Matlab codes for this problem in the Github repository:

https://github.com/asalimil/LMI-for-Attitude-Control-Nonrotating-Missle-Yaw-Roll-Channel

Related LMIs
LMI for Attitude Control of Nonrotating Missles, Pitch Channel

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LMIs in Control/Tools


text:
LMIs in Control/Click here to continue/Applications of Linear systems/LMI for H2/Hinf Polytopic Controller for Robot Arm on a Quadrotor

The System:

\begin{align} \dot x(t)&=A x(t)+B_1 w(t)+B_2 u(t) \\ z(t)&=C_1 x(t)+D_{11} w(t)+D_{12} u(t)\\ y(t)&=C_2 x(t)+D_{21} w(t)+D_{22} u(t)\\ \dot x_K(t)&=A_K x_K(t)+B_K y(t)\\ u(t)&=C_K x_K (t)+D_K y(t)\\ \end{align} $$

The Optimization Problem:
Given a state space system of

\begin{align} \dot x(t)&=A x(t)+B_1 w(t)+B_2 u(t) \\ z(t)&=C_1 x(t)+D_{11} w(t)+D_{12} u(t)\\ y(t)&=C_2 x(t)+D_{21} w(t)+D_{22} u(t)\\ \dot x_K(t)=A_K x_K(t)+B_K y(t)\\ u(t)&=C_Kx_K(t)+D_Ky(t)\end{align}$$

where $$A_K,$$ ,$$B_K,$$,$$C_K,$$ and $$D_K,$$ form the K matrix as defined in below. This, therefore, means that the Regulator system can be re-written as:

\begin{bmatrix} \dot x(t)\\

z_1(t)\\z_2(t)\\

y(t) \end{bmatrix} = \begin{bmatrix} \begin{array}{c|c c|c} A&{B}&0&{B}\\ C&D&0&D\\ 0&0&0&I\\ C&D&I&D \end{array} \end{bmatrix} \begin{bmatrix} \dot x(t)\\ w_1(t)\\w_2(t)\\ u(t) \end{bmatrix} $$ With the above 9-matrix representation in mind, the we can now derive the controller needed for solving the problem, which in turn will be accomplished through the use of LMI's. Firstly, we will be taking our $$H_2$$/$$H_\infty$$state-feedback control and make some modifications to it. More specifically, since the focus is modeling for worst-case scenario of a given parameter, we will be modifying the LMI's such that the mixed $$H_2$$/$$H_\infty$$ controller is polytopic.

The LMI:
$$H_2$$/$$H_\infty$$ Polytopic Controller for Quadrotor with Robotic Arm.

Recall that from the 9-matrix framework, $$w_1(t)$$ and $$ {w_2}(t)$$ represent our process and sensor noises respectively and $$u(t)$$ represents our input channel. Suppose we were interested in modeling noise across all three of these channels. Then the best way to model uncertainty across all three cases would be modifying the $$D$$ matrix to $$D_i$$, where ($$i=1,..,k$$ parameters, $$D_i=nI$$, and $$n$$ is a constant noise value). This, in turn results in our $$D_{11}$$-$$D_{22}$$ matrices to be modifified to $$D_{11,i}$$-$$D_{22,i}$$

Using the LMI's given for optimal $$H_2$$/$$H_\infty$$-optimal state-feedback controller from Peet Lecture 11 as reference, our resulting polytopic LMI becomes:

$$ \min\limits_{\gamma_1,\gamma_2,X_1,Y_1,Z,A_n,B_n,C_n,D_n}$$$$ \gamma^2_1$$+$$\gamma^2_2$$

\begin{align} \begin{bmatrix} AA_i&AB^T_i&AC^T_i\\ AB_i&BB_i&BC^T_i\\ AC_i&BC_i&-I \end{bmatrix}&<0\\ \begin{bmatrix} AA_i&AB^T_i&AC^T_i&AD^T_i\\ AB_i&BB_i&BC^T_i&BD^T_i\\ AC_i&BC_i&-I&CD^T_i\\ AD_i&BD_i&CD_i&{-\gamma^2_{2}}I \end{bmatrix}&<0 \\ \begin{bmatrix} Y_1&I&AD^T_i\\ I&X_1&BD^T_i\\ AD_i&BD_i&Z \end{bmatrix}&>0 \\ \end{align}$$ CD=0

$$trace(Z)<\gamma^2_1$$

where i=1,..,k,$$||S(K,P)||_{H_2}$$&$$<\gamma_1$$ and $$||S(K,P)||_{H_\infty}<\gamma_2$$ and:

$$ \begin{align} AA_i=A Y_1+ Y_1 A^T + B_2 C_n+ C^T_n B^T_2\\ AB_i=A^T+A_n+[B_2 D_n C_2]^T\\ AC_i=[B_1+B_2 D_n D_{21,i}]^T\\ AD_i=C_1 Y_1+D_{12,i}C_n\\ BB_i=X_1 A+A^T X_1+B_n C_2+C^T_2 B^T_n\\ BC_i=[X_1 B_1+B_n D_{21,i}]^T\\ BD_i=C_1+D_{12,i} D_n C_2\\ CD_i=D_{11,i}+D_{12,i} D_n D_{21,i} \end{align} $$

After solving for both the optimal $$H_2$$ and $$H_\infty$$ gain ratios as well as $${X_1},{Y_1},Z,{A_n},{B_n},{C_n},{D_n}$$, we can then construct our worst-case scenario controller by setting our $$D$$ matrix (and consequently our $${D_{11}},{D_{12}},{D_{21}},{D_{22}}$$ matrices) to the highest $$n$$value. This results in the controller:
 * $$\begin{align}

K=\begin{bmatrix}\begin{array}{c|c}{A_K}&{B_K}\\ \hline {C_K}&{D_K}\\\end{array}\end{bmatrix}\end{align} $$ which is constructed by setting:

\begin{align} &{D_K}=(I+D_{K_2}D_{22})^{-1}{D_{K_2}}\\ &{B_K}={B_{K_2}}(I+D_{22}D_{K})\\ &{C_K}=(I-D_{K}D_{22}){C_{K_2}}\\ &{A_K}={A_{K_2}}-{B_K}(I-D_{22}D_{K})^{-1}D_{22}{C_K} \end{align} $$

where:

\begin{align} &{X_2}{Y^T_2}=I-{X_1}{Y_1}\\ &\begin{bmatrix}\begin{array}{c|c}{A_{K_2}}&{B_{K_2}}\\ \hline {C_{K_2}}&{D_{K_2}}\end{array}\end{bmatrix}= \begin{bmatrix}{X_2}&{X_1}{B_2}\\0&I\end{bmatrix}^{-1} \begin{bmatrix}{A_n}-{X_1}A{Y_1}&{B_n}\\{C_n}&{D_n}\end{bmatrix} \begin{bmatrix}Y^T_2&0\\{C_2}{Y_1}&I\end{bmatrix} \end{align} $$

Conclusion:
The LMI is feasible and the resulting controller is found to be stable under normal noise disturbances for all states.