The name Thyssen is synonymous with escalators, elevators and great art. August Thyssen founded the dynasty in the 19th century, when he turned a chicken-wire business into an steel and iron empire. He bought Flemish Old Masters and six sculptures from his friend Rodin. His son Heinrich continued the tradition. Heinrich bought Hans Holbein's portrait of Henry VIII from the Spencers, while his son, Baron Hans Heinrich Thyssen-Bornemisza, added more than 1,000 Old (and modern) Masters to amass the world's greatest private collection (apart from the Queen's). Hans Heinrich's daughter, Francesca, is the fourth generation in the family's tradition of collecting. However, Francesca has taken a divergent path: she champions contemporary art that is anything but traditional.

Francesca's foundation, Thyssen-Bornemizsa Art Contemporary (also known as T-B A21), is housed within a Viennese palace. It is the antithesis of the family collection. Eschewing the style of her father's patronage – he acquired paintings privately from dealers, restorers and priests – Francesca, with great leaps of faith, commissions wildly ambitious artworks. She works with artists, nurturing projects often destined for the public domain from inception to completion. Whether it's a colossal 3D sound pavilion structure consisting of 20 tons of black aluminum and 50 loud speakers conceived by Matthew Ritchie to sit on the banks of the Golden Horn; or a barge carrying a cargo of art up the Danube, no matter how ambitious the artist's proposals appear to be, Francesca realizes artist's dreams.

Recently she bankrolled a full-length feature film, The Artist is Present, about Marina Abramovic who takes performance art to painful extremes. The film charts Abramovic's life's work and her MoMA show in New York in 2010. After watching a clip of this film about the courageous Marina – she certainly suffers for her art – I am led upstairs to Francesca's apartment on the floor above. It is surprisingly modest for someone so grand and rich, and yet the décor is theatrical and playful, interspersed with pots of wild orchids. Despite fighting the flu, Francesca appears, regally towering in platform shoes. As we sit down to a Viennese brunch in her Pop Art kitchen inspired by the sets of A Clockwork Orange, Francesca jumps straight into a discussion about art.

Francesca is, she declares, "more interested in the creative process than the product". It all started, apparently, when she joined forces with Artangel in London. James Lingwood, who runs the acclaimed arts organization that mounts projects outside traditional gallery spaces, introduced her to Kutlug Ataman's Kuba, a 40-monitor video piece interviewing impoverished Turks. Francesca took the artwork a stage further and commissioned an 'art barge' to cruise from Istanbul's ghetto, Kuba, across the Black Sea, along the Danube to Vienna, arriving at the former imperial city in a blaze of bonfires that were lit along the river banks. It was a royal entry.

But then Francesca is royal. Known as 'Chessy' to her friends, she is the Archduchess of Austria, Francesca von Habsburg. In 1993, she married Karl, the great-grand nephew of the Archduke whose assassination sparked off World War I. Karl proposed to her in the family's Capuchin crypt where the Holy Roman Emperors lie entombed. "His idea of proposing to me was to ask, 'How would you like to be buried here?'" They have three children, Eleonore, 18, Ferdinand, 14, and Gloria, 12. She and her husband now lead fairly separate lives.

In 2001, Francesca moved to Vienna, where the tempo is slow, and there are only ten contemporary art galleries. In spite of referring to the capital as 'little Vienna' – alluding to
its pervading provincialism – she finds that the city "gives me sanity, distance and privacy" away from the frenetic art world.

Once in the grip of contemporary art, Francesca was intent on acquiring large 'hard core' unwieldy installations. Frustrated by her thwarted attempts to buy Janet Cardiff's Forty-Part Motet – because she was merely a bona fide collector and not a museum or institution and thus allegedly not equipped to show the piece – she sprung into action and simply established her own foundation in 2002.

When Cardiff asked her about what she was going to do with the work, it dawned on Francesca that "art is not only an investment, it is a commitment".

T-B A21 opened with Cardiff's astonishing sound installation, brought to life with a choir of 40 singers, each singing their own part. The fruits of Francesca's tireless tours of art biennales, fairs and myriad studios, resulted in T-B A21 going from strength to strength and staging exhibitions and interdisciplinary commissions, called 'pavilions', that combine not just art, but music, architecture and science. The pavilions are specially commissioned spaces that are erected all over the world. One such work by Olafur Eliasson and David Adjaye – Your Black Horizon Art Pavilion – was a windowless corridor of vertical slats allowing for shafts of sunlight to filter through. It was sited on Lopud Island, a few sea miles from Dubrovnik.

But then Francesca makes bold purchases. With a combination of philanthropic zeal and a keen eye for spotting talent, Francesca commissions and collects world class and groundbreaking artists such as Marina Abramovic, Douglas Gordon, Darren Almond, Fiona Banner, Mike Kelly, Ai Weiwei and Olafur Eliasson. But this is a fraction of the foundation's burgeoning collection that includes some 300 digital and video pieces. Last year she bought a $60,000 sound installation by little-known Icelandic artist Finnbogi Petursson.

In May, T-B A21 will, in collaboration with the Belvedere, relocate to a lush corner of Augarten Park in a vast 1950s studio that will allow greater flexibility and space to expand with ever-more ambitious art projects commencing with Simon Starling, followed by Marina Abramovic and Ernesto Neto.

It did, however, take Francesca some time to arrive at her role as a pioneering and philanthropic patron. When I first knew her in the early 1980s, she was living in London. A quasi punk, she was sharing a flat in Chelsea with New Romantic singer Steve Strange for whom she sang backing vocals, while also attending St Martin's School
of Art. Often outrageous and wild, she was once famously snapped entering Count Volpi's ball in Venice when a sudden gust of wind swept her gown up to reveal her callipygian bare bottom. When I was editing Inside Art, she interviewed the German dealer and collector Bruno Bischofberger for me and it was plain that the art world excited her. She was a devoted friend of the late Robert Fraser, and flew with 'Groovy Bob' to New York in 1983. "We slept on people's floors and met Keith Haring, Julian Schnabel, Jean-Michel Basquiat and Robert Mapplethorpe", who then photographed her.

But throughout these years, there was a hidden seriousness. Although her formal education was not a success – "I hated Le Rosey" – when she was 17, Francesca became fascinated with the process of installing art when she saw a survey show of the Minimalists at the Whitney Museum in New York. "I went with my father when I lost him in the lift. He went back to the Pierre [Hotel], saying it was rubbish while I spent hours looking at the works of Carl Andre; I was more excited than I have ever been before about art. Andre's simplicity appealed to me."

She fixes me with a steely gaze, then smiles: "It was the most powerful reaction I had to art. I had been brought up to revere art, I had an intimate appreciation of art; every now and then a new Goya or Matisse would arrive at the villa, but it was not like a museum. I hated anything that got in between me and art."

Francesca ushers me into her sitting room dotted with purple sofas, Olafur Eliasson's boreal Icelandic tundra hanging on the walls. Her spiritual catharsis was meeting the Dalai Lama in 1985. "My father donated $100,000 for a cultural institute to preserve the traditions of Tibetan art and he asked me to organize the project." This took her on an extended tour of the Soviet Union while Perestroika was gaining momentum with an expert iconographer, Heather Stoddard. Francesca took 3,000 polaroids of Tibetan art. "In 1991 I went on tour with his Holiness the Dalai Lama, who filled football stadiums with his talks. Buddhists came out of the woodwork, many of whom had been hiding sacred art objects under the floorboards and taking huge risks in doing so. It was then that I learnt the true appraise of heritage.

It made her focus on conservation. "I realized that conservation needed a charismatic figure at the helm to make it more fun," she says, a touch disarmingly. "I did fund raise for the World Monument Fund, which, of course, being a Thyssen posed problems." During the 1990s, she spent time restoring Croatian monasteries (while the war was still raging), before eventually returning to the family fold. She spent five years at the Thyssen family home, Villa Favorita in Lugano, curating and archiving her father's vast collection.

Now Francesca channels her energies into her modern art projects. T-B A21 looks destined to become one of the most innovative and comprehensive collections of 21st century art, while the new Augurten space opens next month. This space – an artist's studio built in the 1950s for the sculptor Gustinus Ambrosi that is set in a corner of Vienna's Augarten Park – will be used to mount site-specific pieces and will give T-B A21 an even greater gravitas.

I suspect this is what Francesca has always been searching for. As she says, "I don't lead a 'glam' lifestyle – not like these Russian oligarchs on yachts who are too mean to pay for the wine at their openings. I have never been back to Russia; I could not stand the nouveau richness." Gazing over to a table football stand that looks as if it has been carved out of huge gold bar in the adjacent salon, she says emphatically, "in this climate, cash is king. Real estate and art are the best investments."

The King Midas gold football table dominates the room and at first one scarcely notices a Man Ray and the famous Lucian Freud portrait of her father. The billionaire Baron looks worn down by his fortune, his tension and introspection conveyed by his hands and fingers splayed on his knees. As she bids me goodbye, she says wistfully, "The National Portrait Gallery is coming to take my father away for an exhibition." And one senses that paintings still remain true to her heart, despite the transient and ephemeral nature of some of the sound and digital art works she has commissioned. Her father would be proud of her if he were still alive.

Celia Lyttelton's most recent book is The Scent Trail: A Journey of the Senses.

Thyssen-Bornemisza Art Contemporary.

The Thyssen-Bornemisza Augarten Contemporary in Vienna will open
in May in collaboration with the Belvedere. Celia Lyttelton flew to Vienna courtesy of Austrian Airlines.

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