Believe this may be your man:

Francis Smith Robertson

Residence was not listed; a 20 year-old Student.
Enlisted on 6/18/1861 at Washington County, VA as a 2nd Lieutenant.
On 4/17/1861 he mustered into VA Son's Of Liberty
He was Unit disbanded (date not stated)
On 6/18/1861 he was commissioned into "I" Co. VA 48th Infantry
He was discharged for disability on 4/21/1862
(Failed re-election)
He was listed as:
* Served Gen's Stuart And W. H. F. Lee's Staff (date not stated) (As Engineer Officer)
* Hospitalized 9/24/1861 Washington, DC (Sick, camp fever)
* Paroled 5/22/1865 Richmond, VA
Promotions:
* Priv (As of Co. D, 1st VA Cav)
He also had service in:
"D" Co. VA 1st Cavalry (On postwar roster)
Other Information:
born 1/3/1841 in Richmond, VA
died in 1926
Buried: Sinking Spring Cemetery, Abingdon, VA
(Attended Univ of VA 1859-61. Attended VMI for 2 months.
Postwar, Farmer, Abingdon, Washington Co,
VA)

Sources used by Historical Data Systems, Inc.:
- The Virginia Regimental Histories Series
(c) Historical Data Systems, Inc. @ www.civilwardata.com

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From a Goggle search: Confederate soldier Frank S. Robertson

Stuart's Tarheels: James B. Gordon and His North Carolina Cavalry in ...

https://books.google.com/books?isbn=0786486902
Chris J. Hartley - 2011 - ‎History
McRae, J. C. “List of Confederate Soldiers in the General Assembly of North Carolina. ... Robertson, Frank S. “Reminiscence of the Years 1861– 1865.” Edited by ...

https://www.google.com/search?site=&source=hp&q=Confederate+soldier++Frank+S.+Robertson&oq=Confederate+soldier++Frank+S.+Robertson&gs_l=hp.12...3253.3253.0.4973.2.2.0.0.0.0.133.247.0j2.2.0....0...1c.2.64.hp..0.0.0.0.r4oWuSdRE-0