Almost every morning these days, the black limousines pull up to Washington's Madison Hotel to collect their Saudin Arabian passengers.

Their destination, very often, is Capitol Hill, where the battle of the F15s unfolds - the struggle over the sale of warplanes to Saudi Arabia, Egypt and Israel.

The presence here of Saudi dignitaries operating out of the Madison and their embassy on 18th Street, is one more new and intriguing dimension in the history of grappling to win hearts and minds in Congress.

The Saudis have thrown some of their most personable and articulate representatives, guided by ring-wise American political and public relations advisers, into this lobbying struggle.

Among them are Price Bandar Bin Sultan, member of the royal family and military pilot; Prince Turki Faisal, chief intelligence officer; Ghazi al Gosaibi, minister of industry and electricity; Ambassador Alireza; Minister of Commerce Sulayman Sulaym - all Western-educated, sophisticated, low-key.

They make the rounds on Capitol Hill, touching base with know sympathizers, trying to make their case for buying the F15s when they meet with skeptics and, apparently, mostly avoiding the legistors they know to be the warmest friends of Irael.

On the one hand, the Saudis are talking of their need to have the 60 jet fighters the Carter administration has promised.

But for the longer range, their presence and their suasive efforts fit into a general campaign aimed at raising American consciousness on the friendship strategic and economic interests that bind this country to oil-rich Saudi Arabia.

Time and the coincidence of interests have conspired to project the Saudis onto center stage of a congressional theater where a larger sort of drama has been building.

For good or for ill, one of the legacies of Vietnam is the emergence of Congress as a parther with the White House - be it Nixon's, Ford's, Carter's - in making foreign poilicy decisions.

Inevitably, this means lobbying by foreign interests. Israel's cause has been nurtured by an eggective American Jewish lobby. The South Koreans and Tongsum Park tried still another approach.

That same pattern was followed even before Vietnam when Congress was handling out sugar quotas. The town swarmed with foreign agents, unsparing in their solicitousness.

But with Congress hefting more muscle on foreign policy in the 1970s, the agents are back. America agents and foreign agents are back. American agents and foreign nationals - Arabs, Turks, Greek, Fillipinos Japanese - are here to tell their story and make their case with the Congress.

Bryan Atwood, a Department of State liaison man with Congress, noted, "Every embassy in this town these days has someone assigned to the Congress. As a result of Vietnam and Watergate, Congress has created laws that force it to get involved in issues."

One of the more obvious results is that unfolding battle over the F15.

In recent weeks, a stream of Saudi officials and members of the royal family have been in Washington. This week, Prince Saud, the foreign minister, comes in.

As they make their personal visits on Capitol Hill, they carry out a second phase of their mission - taking their story to the American people. It is a campaign that is having tagible effect, if the public opinion surveys are an indication.

They issue formal statement to The New York Times. They took full-page ads in major newspapers on the occasion of Sun Day. They have paid for big advertising spreads in Time and Newsweek. The magazines give cover play to the development boom in Saudi Arabia.

American journalists have been issued visas and treated to the grand tour of the Arabian peninsula. The Saudis have hire lawyers and publicity experts to carry their torch here.

The mails are full of literature promoting the Saudi cause, the television channels carry interviews with Saudi officials - all in all, a remarkable change in attitude in a remote nation that until recently has followed a policy of isolation and secrecy.

Most immediate in their eyes is the F15. And there is yet another dimension in this effort to win support on Capitol Hill. American busineses with now and future interests in Saudi Arabia - Bechtel Corp., Computer Sciences Corp., to name a pair - have sent their people around to quietly whisper to legislators about the urgency of approving the airplane sale to the Saudis.

"It is a full court press," sommented one awed staff member of the Senate Foreign Relations Committee.

"There is a changed atmosphere, no doubt about it," said another. "You are seeing an evolution. There is an enhanced perception up here about the Arab cause...And they're doing a helluva lot better public relations job. They're learning very well."

Coincidental or not, even the sense of timing shows. Last Monday afternoon, for example, Sen. Horward H. Baker Jr. (R-Tenn.), the minority leader, went to the White House to help receive Israeli Prime Minister Menanchem Begin.

When Baker returned to his office, waiting for him by appointment was Prince Bandar bin Sultan, a jet pilot in the Saudi air force and the son of the defenee minister. He quietly made his case to Baker for sale of the F15s.

On Bandar's heel was Ambassador Alireza, with another Saudi cabinet official in town, stopping in for a fisit with Frank Church (D-Idaho), one of the Senate's so-called "doves," to plump for the sale of the planes.

The scene was repeated again and again around the Hill. The engaging Bandar found willing and receptive ears except in an occasional office, such as that of Rep. Anthony C. Beilenson (D-Calif.), who had refused to meet personally with either supporters or opponents of the jet sale.

Michael Polyac, a Beilenson aide on foreign affairs, said that the congress-man a member of the International Relations Committee, will talk to administration representatives only in the committee format - not in private.

Commented another foreign relations aide: "The Arabs have learned their lesson very well. The Israeli lobby, or example, floated some memos detailing why the Arabs should not be sold F15s and F5 fighters.

This week, we got some simiral memos from the Saudis, detailing their side of it. They have observed the Israeli-American lobby and they've learned from it. They are quite legitimate, not all pushy, doing everything above board and very tastefully."

Two Americans, adherents to the Arab cause, see that sort of comment and the generally warm reaction to the Saudis on Capitol Hill as symbolic of the effectiveness of the Arabs' effort to get their story to the public.

Dr. Malcom Peck of the Middle East Institute here put it this way:

"I think much of this stems from the visit of President Sadat to Israel.That was watershed event. The polls show that Americans perceive Sadat as more of a man of peace than Begin. A year ago, you would have been dismissed as a crank if you thought that would be so."

"In Congress there is a more favorable perception of Sadat and the Arab world. It has also to do with the large number of American delegations going out there for visits. This is having its due effect, too...More and More Americans are seeing the Arabs in a human perspective."

John Richardson of the National Association of Arab Americans picked o. idtd ett ahtadm deupa eth

"I would say the titillation up that theme and added to it."

"I would say the titillation potential of any well-placed Saudi coming to this city becomes legion. The Saudis are both love and hate objects and there is a bit of prurient interest - people waiting for them to drop their spoons, as it were," Richardson said.

"But there is a new political vocabulary being developed here and it included the idea that Arabs are people who are important to Americans and not just objects...I think it will sort out. It is long overdue," he said.