ADVANCED ANALOGIC TECHNOLOGIES INCORPORATED AMENDMENT TO

Exhibit 10.13.1

ADVANCED ANALOGIC TECHNOLOGIES INCORPORATED

AMENDMENT TO

AMENDED AND RESTATED INVESTORS’ RIGHTS AGREEMENT

This Amendment is made as of May 18, 2005, by and between Advanced Analogic Technologies Incorporated, a Delaware corporation (the “Company”), and the persons and entities identified on Exhibit A hereto (each a “Holder” and collectively, the “Holders”).

WHEREAS the parties entered into an Amended and Restated Investors’ Rights Agreement dated October 27, 2003 (the “Original Agreement”);

WHEREAS, Section 4.8 of the Original Agreement provides that Holders holding a majority of the shares of outstanding Registrable Securities (as defined in the Original Agreement) held by all Holders under the Original Agreement may, with the consent of the Company, amend, on behalf of all the Holders, any term of the Original Agreement;

WHEREAS the Company and the undersigned Holders have mutually agreed to amend Section 1.13 of the Original Agreement in order (i) to revise the market stand-off provision in such section to substantially match the market stand-off agreement required by the Company’s managing underwriters of its anticipated initial public offering and (ii) thereby to treat all Holders equally with respect to such provision, regardless of whether such Holder has separately entered such an agreement with the managing underwriters; and

WHEREAS, the undersigned Holders are holders of at least a majority of the shares of outstanding Registrable Securities under the Original Agreement.

NOW THEREFORE, the parties agree as follows:

1. The text of Section 1.13 of the Original Agreement shall be amended and restated in its entirety to read as follows:

“1.13 “Market Stand-Off” Agreement. The Holder hereby agrees that, without the prior written consent of the Company, it will not, during the period commencing on the date hereof and ending 180 days after the date of the final prospectus (the “Prospectus”) included in the first registration statement of the Company filed under the Securities Act which covers securities to be sold to the public in an underwritten offering (the “Public Offering”), (1) offer, pledge, sell, contract to sell, sell any option or contract to purchase, purchase any option or contract to sell, grant any option, right or warrant to purchase, lend, or otherwise transfer or dispose of, directly or indirectly, any shares of Common Stock or any securities convertible into or exercisable or exchangeable for Common Stock; or (2) enter into any swap or other arrangement that transfers to another, in whole or in part, any of the economic consequences of ownership of Common Stock, whether any such transaction described in

clause (1) or (2) above is to be settled by delivery of Common Stock or such other securities, in cash or otherwise. The foregoing sentence shall not apply to (a) the sale of Shares to the underwriters by the Holder pursuant to an underwriting agreement in connection with the Public Offering, if the Holder sells such shares as a selling stockholder named as a party to such underwriting agreement; (b) transactions relating to shares of Common Stock or other securities acquired in open market transactions after the completion of the Public Offering, provided that no filing under Section 16(a) of the Securities Exchange Act of 1934 (the “1934 Act”) shall be required or shall be voluntarily made in connection with subsequent sales of Common Stock or other securities acquired in such open market transactions; (c) transfers of shares of Common Stock or any security convertible into Common Stock as a bona fide gift; (d) distributions of shares of Common Stock or any security convertible into Common Stock to limited partners, members or stockholders of the Holder; (e) transfers of shares of Common Stock or any security convertible into Common Stock to any trust for the direct or indirect benefit of the Holder and/or the immediate family of the Holder; or (f) transfers of shares of Common Stock or any security convertible into Common Stock by will or intestate succession to the Holder’s immediate family; provided that in the case of any transfer or distribution pursuant to clause (c), (d), (e) or (f), (i) each donee, distributee or transferee shall sign and deliver a lock-up agreement substantially in the form of this lock-up agreement; and (ii) no filing under Section 16(a) of the 1934 Act, reporting a reduction in beneficial ownership of shares of Common Stock, shall be required or shall be voluntarily made during the restricted period referred to in the foregoing sentence. Notwithstanding the foregoing, the Holder shall be permitted to enter into a written sales plan designed to comply with Rule 10b5-1(c) under the 1934 Act, so long as such plan does not allow for any sales thereunder to be made until the expiration or termination of the restricted period referred to in this lock-up agreement. In addition, the Holder agrees that, without the prior written consent of the Company, it will not, during the period commencing on the date hereof and ending 180 days after the date of the Prospectus, make any demand for or exercise any right with respect to, the registration of any shares of Common Stock or any security convertible into or exercisable or exchangeable for Common Stock. The Holder also agrees and consents to the entry of stop transfer instructions with the Company’s transfer agent and registrar against the transfer of the Holder’s shares of Common Stock except in compliance with the foregoing restrictions.

If (1) during the last 17 days of the 180-day restricted period the Company issues an earnings release or material news or a material event relating to the Company occurs; or (2) prior to the expiration of the 180-day restricted period, the Company announces that it will release earnings results during the 16-day period beginning on the last day of the 180-day period, then the restrictions imposed by this lock-up agreement shall automatically continue to apply until the expiration of the 18-day period beginning on the issuance of the earnings release or the occurrence of the material news or material event, as applicable. In addition, the Holder shall not engage in any transaction that may be restricted by this lock-up agreement during the 34-day period beginning on the last day of the initial 180-day restricted period unless the Holder requests and receives prior written confirmation from the Company that the restrictions imposed by this lock-up agreement have expired.

2

To enforce the foregoing covenant, the Company may impose stop-transfer instructions with respect to the Registrable Securities of the Holder (and the shares or securities of every other person subject to the foregoing restriction) until the end of such period. Each Holder agrees to execute a market stand-off agreement with said underwriters in customary form consistent with the provisions of this Section 1.13.”

2. Miscellaneous.

(i) Governing Law. THIS AMENDMENT SHALL BE GOVERNED IN ALL RESPECTS BY THE INTERNAL LAWS OF THE STATE OF CALIFORNIA AS APPLIED TO AGREEMENTS ENTERED INTO AMONG CALIFORNIA RESIDENTS TO BE PERFORMED ENTIRELY WITHIN CALIFORNIA, WITHOUT REGARD TO CONFLICT OF LAWS RULES.

(ii) Successors and Assigns. Except as otherwise provided herein, the provisions hereof shall inure to the benefit of, and be binding upon, the successors, assigns, heirs, executors and administrators of the parties hereto.

(iii) Entire Agreement; Amendment. This Amendment and the Original Agreement constitute the full and entire understanding and agreement between the parties with regard to the subject matter hereof and thereof.

(iv) Counterparts. This Amendment may be executed in any number of counterparts, each of which shall be enforceable, and all of which together shall constitute one instrument.

[The remainder of this page is left blank intentionally.]

3

IN WITNESS WHEREOF, the parties have duly executed this Amendment as of the day and year first set forth above.

ADVANCED ANALOGIC TECHNOLOGIES INCORPORATED

AMENDMENT TO AMENDED AND RESTATED INVESTORS’ RIGHTS AGREEMENT

SIGNATURE PAGE

ADVANCED ANALOGIC TECHNOLOGIES INCORPORATED

AMENDMENT TO AMENDED AND RESTATED INVESTORS’ RIGHTS AGREEMENT

SIGNATURE PAGE

EXHIBIT A

SCHEDULE OF HOLDERS

Series A Holders

Battery Investment Partners VI, L.L.C.

Battery Ventures VI, L.P.

Chun-Peng Tsai

Ho Hung Yang

Series B Holders

A-Tech Electronics Co., Ltd.

Hsiu-Chih Chang

Mu-Tien Chang

Tong-Long Chang

Wei-Jung Chang

Wen-Jung Chen

Hsin-Chia Cheng

Ming-Feng Chiang

Cheng-Wei Chiu

Feng-Chao Huang Chen

Li-Tzen Huang

Hwa Chung Venture Capital Co., Ltd.

Hwa Hsin Venture Capital Co., Ltd.

Hwa Nan Venture Capital Co., Ltd.

International Master Technologies LTD

Wen-Jung Kao

Chien-Chih Kuo

Cheng-Han Lai

Land Mark Venture Capital Corp.

Shih-Kuang Lue

Vision 2000 Venture Limited

Jui-Chang Wang

Long-Chia Wang

Chiu-Chi Wei

Po-Chuan Wu

Fu-Cheng Yang

Series C Holders

A-Tech Electronics Co., Ltd.

CCT Investment LLC

Kam Hang Chan and Shao Ying Chan JTWROS

Wei-Jung Chang

Yu-Chan Chang

Chuang Hsien Chen

Cheney S.H. Cheng

Yi-Chian Chiang

Kuo-Chin Chiou

Hui-Ying Fu

John Fulton

GAINS Investment Corporation

Goldman Development Limited

Harwin Resources Limited

Yuan-Chen Hsiao

Chih Gung (Chih-Kung) Huang

Hwa Chung Venture Capital Co., Ltd.

Hwa Nan Venture Capital Co., Ltd.

Hwa Yi Venture Capital Co., Ltd.

Karmting Chiu And Alice Chiu JTWROS

Chih-Hung Kuo

Shih-Chi Lai

Lin Li

Pei Hsin Lin

Kun-Lin Liu

Chin-Hsiang Lu

Maton Fund I L.P.

Chi-Hsuan Shen

Shi Jar Construction Co., Ltd.

Sunsino International Development Associate, Inc.

Tsuyoshi Taira

Chung-Hsing Tzu

Nu-Ting Wei

Sun Yao Wong

Agnes Wu

Chun Chung Wu

Jouanna Yeh

Series C Holders

Tien-Min Yuan

Series D Holders

Above VC Consulting, Inc.

Ascentek Venture Capital Corporation

A-Tech Electronics Co., Ltd.

Bai-Ho Investment Co., Ltd.

Baotung Venture Capital Corporation

Chung-Ming Chang

Chih-Wen Chang

Mu-Tien Chang

Wei-Jung Chang

Bor-Po Chen

Jui-Hua Chen

Wen-Jung Chen

Hsin-Chia Cheng

Yao-Hui Cheng

Nai-Rong Chi

Tsai-Ting Chiang

Ming-Feng Chiang

Rong-Kuo Chiang

Cheng-Wei Chiu

Chen-Chiang Chiu

Ping-Ying Chiu

Tung-Lin Chiu

Wen-Ying Chiu

Chin-Te Chou

Hsing-Tsun Chou

Ming-Ta Chou

Su Ping Chung (Ping-Chung Su)

e-Sunsino Venture Co., Ltd.

Fairchild Semiconductor Corporation

Chao-Jui Fong

Hui-Ying Fu

GAINS Investment Corporation

Su-Chuan Ho

Series D Holders

Mei-Shin Ho

Shu-Yun Ho

Holmsgreen International Co., Ltd.

Ching-Yuan Hsieh

Chang-Hua Hsieh

Ching-Lian Hsieh

Ju-Fen Hsieh

Mei-Jung Hsieh

Ting-Lung Hsieh

Feng-Tseng Hsu

Chun-Ta Huang

Jui-Ling Huang

Kuo-Cheng Huang

Hwa Chung Venture Capital Co., Ltd.

Hwa Hsin Venture Capital Co., Ltd.

Hwa Nan Venture Capital Co., Ltd.

Hwa Yi Venture Capital Co., Ltd.

InveStar Burgeon Venture Capital, Inc.

InveStar Excelsus Venture Capital (Int’l) Inc., LDC

InveStar Semiconductor Development Fund, Inc. (II) LDC

Wen-Shyong (James) Jan

Chien-Chih Kuo

Hsiu-Chin Kuo

Cheng-Han Lai

Jenn Chyou Lai

Land Mark Venture Capital Corp.

Po Yen Lee

Tsung Lieh Lee

Tsan Hsun Li

Chun-Luan Liao

Mei-Feng Liao

Kwo-Feng Lin

Ching-Yuan Lin

Pei Hsin Lin

Pi-Lien Lin

Shu-Lan Lin

Yu-Fen Lin

Yu-Chia Lin

Series D Holders

Yue-Din Lin

Lite-On Semiconductor Corporation

Litung Venture Capital Corporation

Yu-Mei Liu Hsieh

June Cheng Liu

Kuei-Ying Liu

Chin-Hsiang Lu

Chia-Ping Lu

Maton Fund II L.P.

Multi-Program Inc.

Nutek Corporation

Lee Chang Pin

Lee Tung Ping

Maritje Rusli

Semiconductor Components Industries, LLC

Shi Jar Construction Co., Ltd.

Sunsino International Development Associate, Inc.

Fu-Kuo Teng

Bing-Min Tian

Tien-Chien Investment Co., Ltd.

Li-Ping Tsai

Ting-I Tsai

Fang-Ying Tseng

Li-Kuo Tseng

Hsiao- Yen Fan Tung

Vision 2000 Venture Limited

Ju-Hsiang Wang

Li-Chung Wang

Long-Chia Wang

Chiu-Chi Wei

Chun Chung Wu

Po-Chuan Wu

Su-Ching Wu

Cheng-Wen Yeh

Series E Holders

A&D Capital Corp.

Baotung Venture Capital Corporation

Battery Investment Partners VI, L.L.C.

Battery Ventures VI, L.P.

CAM-CID Asia Pacific Investment Corp.

CHINA Century Venture Capital Co., Ltd.

China International Investment Co., Ltd.

China International Venture Co., Ltd.

China Power Venture Co., Ltd.

e-Sunsino Venture Co., Ltd.

Fuhwa I Venture Capital Co., Ltd.

Global Vision Venture Capital Co., Ltd.

Hwa Chung Venture Capital Co., Ltd.

Hwa Nan Venture Capital Co., Ltd.

InveStar Excelsus Venture Capital (Int’l) Inc., LDC

J&D Capital Corp

Gregory (Gwe Hyun) Kim

Litung Venture Capital Corporation

Maton Fund II L.P.

MC Capital B.V.

Nien Hsing International Investment Co.

Sycamore Ventures Capital, L.P.

VIS Associates Inc.

Vision 2000 Venture Limited

Way-Han Investment Corp.

SVIC No. 4 New Technology Business Investment L.L.P.