Image copyright Getty Images

The last state visit by a leader of China to the UK was a decade ago. In the space of a mere 10 years, China has changed dramatically.

Its economy now is over three times what it was when Hu Jintao visited in 2005. While the UK and most other developed countries suffered during the great financial crisis from 2008, China forged ahead.

The country that exists now has an emerging middle class, working in the services sectors, living in the vast new cities being renovated or simply built from scratch, whose future patterns of consumption will be crucial for global growth.

Per capita GDP - the amount of goods and services produced per person - is moving quickly into the middle-income bracket, around $9,000 (£6,000). And while Chinese growth today is not in the double digits as it was during the golden years of the mid-2000s, it is still a solid 7%.

As an outward investor, and an intellectual partner, China simply matters more to partners like the UK than it did a decade before.

Xi Jinping, the President of China, will come to the UK in October as a leader with a markedly different style to his predecessor.

President Hu was reticent, anonymous, and low-profile. Few will even remember his visit. His dislike of large public events and big speech occasions was legendary.

President Xi is a more extrovert character and his language, at least while travelling abroad since he became leader in 2012, has been characterised by grand new descriptions - the US and China are "major power status" players, the EU and China are "civilisational partners".

Image copyright AP Image caption President Xi has signed huge trade deals with many countries

Even the vast area around China into Central Asia, the Middle East, and right out into South and South East Asia has been captured by the "one belt, one road" Silk Road idea.

Will Xi seek to capture UK-China relations in a neat catchphrase? And what features of the relationship is he likely to pay attention to?

Long history

The major characteristic of bilateral relations, and the reason why anything but a very platitudinous description is unlikely, is that they are, and are always likely to be, complicated.

Image copyright Getty Images Image caption The UK and China's intertwined history includes long negotiations over Hong Kong, which returned to Chinese rule in 1997

There is ironically a very simple reason for this. The UK's relations with China are mired in history, from the first act of colonial meddling (in Chinese eyes at least) during the First Opium Wars of 1839, to the long negotiations over Hong Kong and its reversion to Chinese sovereignty from British rule in 1997.

And while there have been restarts over the last few decades British leaders, when they speak to their Chinese counterparts about human rights, Tibet, Taiwan or issues in Hong Kong, always arouse particularly passionate responses.

David Cameron, after all, was in effect barred from Beijing for over a year from July 2012 when he met the Dalai Lama in London.

British policy towards China has always been ambiguous, and that is unlikely to change. Underneath the lectures on values and the assertions of China needing to change politically to adopt a British-style democratic system, there runs simultaneously a hard core of self-interested pragmatism.

Trade between the two powers actually increased, even during the Beijing cold shoulder to Cameron.

Image copyright PA Image caption Mr Cameron's meeting with the Dalai Lama in 2012 did not go down well with Beijing

And the UK was among the first European economies to embrace the China-initiated Asian Infrastructure Investment Bank (AIIB) earlier this year, earning it a sharp rebuke from its ostensibly most important diplomatic ally, the US.

While it is unlikely at the moment that the UK feels that stronger economic links with China are worth jettisoning its strong security relations with the US, allies in the EU, and elsewhere, it says far less about issues like the South China Sea disputes involving China.

Europe question

Perhaps the most pressing geopolitical issue for China when President Xi visits will be understanding Britain's place in the EU and what an exit might mean for relations with China.

Like it or not, China supports the unity the EU offers, at least financially and economically, and while it would never publicly state anything on this, it would almost certainly regard a UK exit negatively.

The likelihood is that this mixture of the pragmatic and the more idealistic will feature during Xi's visit, but the main priority, driven from the Treasury under George Osborne, will be to secure as much new, high-quality investment and finance co-operation from China as possible.

The UK is currently the main investment destination for Chinese companies, receiving $5.1bn (£3.3bn) in 2014.

Image copyright Getty Images Image caption Last year, BP signed a deal with China National Offshore Oil Corporation (CNOOC) worth about $20bn (£12bn)

But that is still underwhelming in view of the overall size of China's economy and its potential investment capacity. When Chinese Premier Li Keqiang visited the UK last June a target of £18bn was set.

So the Xi visit has to be seen to contributing to this lofty aim.

Xi has signed up for multi-billion dollar trade deals almost everywhere he has gone.

He joked in Canberra during his state visit to Australia late last year that there were so many new agreements being signed that he felt he could stand all day watching accords being inked in front of him.

Intellectual partners?

The UK will want to equal this theatre. They will want to lay claim to as much Chinese largesse as possible.

So while Mr Cameron will be stuck with the unenviable task of whispering as quickly and discreetly as he can concerns about human rights issues in China, appeasing various constituencies here, trade will be the main issue of the visit. That at least will be the hope of politicians from both sides, whatever they say for public consumption.

There may be attempts at something more imaginative.

The UK as an intellectual partner of China, as it attempts to build a new kind of economic structure and deal with some of its huge environmental and sustainability issues, is a new kind of role, and one of the few where it can be liberated from the shadow of its historic issues with the People's Republic.

Image copyright Getty Images Image caption President Hu was the last Chinese leader to make a state visit to the UK, meeting-then PM Tony Blair in 2005

Using the visit to encourage more British to learn Mandarin is another (currently only 350 students a year take the option of studying Chinese at university, a figure that has not risen in the last decade or so). As is defining a better narrative within which the 100,000-plus Chinese students currently studying here might be able to contribute more to relations between the two countries.

UK relations with China have often been characterised by a lack of imagination and deep cautiousness.

Maybe the Xi visit can challenge the UK to be more creative about the relationship, and take some risks from time to time.

In the end, Britain has to use this visit to raise awareness of its service, attributes and finance expertise back among the vast new middle class emerging in China.

For it is they, not President Xi, who are going to decide the fate of the world's largest emerging economy. And for this reason, the UK has to find a better way to speak to them than it currently has.

Kerry Brown is associate fellow on the Asia Programme at Chatham House, prof of Chinese Politics and director of the China Studies Centre at the University of Sydney, and author of What's Wrong with Diplomacy: The Case of the UK and China.