Here's useful code I use on one of my project with a multi-shop setup PRESTASHOP (v1.5.6) to automatically import product for each shops. First i generate the CSV file with the product data i want to insert or update, then the script log in like you would do on the prestashop admin area and POST the data on the AdminImport controller.

Here is the background/cron script and below the curl wrapper class (You can replace the php curl wrapper of course)

require_once ('class.PSRequest.php'); //see class source code below

$shopID = "myfirstshop"; // ID or unqiue name of the target shop
$products = ...; // your set of product object or array

// CSV first line for products
$csv = "id;Active (0/1);Name*;Categories (x,y,z,...);Price tax excluded;Tax rules id;Wholesale price;On sale (0/1);Discount amount;Discount percent;Discount from (yyy-mm-dd);Discount to (yyy-mm-dd);Reference #;Supplier reference #;Supplier;Manufacturer;EAN13;UPC;Ecotax;Weight;Quantity;Short description;Description;Tags (x,y,z,...);Meta-title;Meta-keywords;Meta-description;URL rewritten;Text when in-stock;Text if back-order allowed;Available for order (0 = No, 1 = Yes);Product creation date;Show price (0 = No, 1 = Yes);Image URLs (x,y,z,...);Delete existing images (0 = No, 1 = Yes);Feature (Name:Value:Position);Available online only (0 = No, 1 = Yes);Condition (new,used,refurbished);ID / Name of shop".PHP_EOL;

foreach ($products as $prod) {
        // fill in the csv string with your datas
       $csv .= $prod->id.";1;".html_entity_decode($prod->name)......$shopID.PHP_EOL;
}

$now=time();
$csvname = $now."-PRODUCTS-".$shopID.".csv";
$file = YOUR_PATH.$csvname;
file_put_contents($file, $csv);

echo "Write file : ".$file.PHP_EOL;

echo "Login on Prestashop Admin area...".PHP_EOL;   
$adminUrl = "http://YOUR_PS_ADMIN/adminXXXX/"; // change that with your admin url of course
$request = new PSRequest();
$request->setCookiFileLocation(__DIR__.'/PScookie.txt');

$request->setPost(array("email" => "YOUR_ADMIN_EMAIL","passwd" => "YOUR_ADMIN_PASS", "submitLogin" => "Connexion")); // you must be a super admin
$request->call($adminUrl."index.php?controller=AdminLogin");

echo "Get token...".PHP_EOL;

$request->call($adminUrl."index.php?controller=AdminImport");
list(,$response) = explode("\r\n\r\n", $request->_webpage, 2);
preg_match("/token=([a-z0-9]+)/", $response, $matches);
$token = $matches[1];

echo "Token : ".$token.PHP_EOL;

// Send POST datas just like the admin form would do it, those datas depends on what you want to do : check the import admin page.
$request->setPost(array(
    "controller"               => "AdminImport",
    "token"                    => $token,
    "skip"                     => 1,
    "csv"                      => $csvname,
    "convert"                  => '',
    "regenerate"               => '',
    "entity"                   => 1, //1 is for products import
    "iso_lang"                 => "fr",
    "truncate"                 => 0,
    "forceIDs"                 => 1,
    "match_ref"                => 1,
    "separator"                => ";",
    "multiple_value_separator" => ",",
    "import"                   => 1,
    "type_value" => array( 0 => 'id', 1 => 'active', 2 => 'name', 3 => 'category', 4 => 'price_tex', 5 => 'id_tax_rules_group', 6 => 'wholesale_price', 7 => 'on_sale', 8 => 'reduction_price', 9 => 'reduction_percent', 10 => 'reduction_from', 11 => 'reduction_to', 12 => 'reference', 13 => 'supplier_reference', 14 => 'supplier', 15 => 'manufacturer', 16 => 'ean13', 17 => 'upc', 18 => 'ecotax', 19 => 'weight', 20 => 'quantity', 21 => 'description_short', 22 => 'description', 23 => 'tags', 24 => 'meta_title', 25 => 'meta_keywords', 26 => 'meta_description', 27 => 'link_rewrite', 28 => 'available_now', 29 => 'available_later', 30 => 'available_for_order', 31 => 'date_add', 32 => 'show_price', 33 => 'image', 34 => 'delete_existing_images', 35 => 'features', 36 => 'online_only', 37 => 'condition', 38 => 'shop')
    )
);

echo "call AdminImport and POST datas...".PHP_EOL;

$request->call($adminUrl."index.php?controller=AdminImport&token=".$token);

echo "-- END --".PHP_EOL;

$request = null;

Here's the wrapper curl class

class PSRequest {

    protected $_useragent          = 'Mozilla/5.0 (Windows NT 6.1; WOW64; rv:10.0.2) Gecko/20100101 Firefox/10.0.2';
    protected $_cookieFileLocation = './cookie.txt';
    protected $_referer            = "http://www.google.com";

    protected $_url;
    protected $_followlocation;
    protected $_timeout;
    protected $_maxRedirects;
    protected $_post;
    protected $_postFields;

    protected $_session;
    protected $_includeHeader;
    protected $_noBody;
    protected $_status;
    protected $_binaryTransfer;

    public $_webpage;
    public $authentication = 0;
    public $auth_name      = '';
    public $auth_pass      = '';

    protected $ch; // curl handler

    public function __construct($url = '', $followlocation = true, $timeOut = 30, $maxRedirecs = 4, $binaryTransfer = false, $includeHeader = true, $noBody = false)
    {
        $this->_url                = $url;
        $this->_followlocation     = $followlocation;
        $this->_timeout            = $timeOut;
        $this->_maxRedirects       = $maxRedirecs;
        $this->_noBody             = $noBody;
        $this->_includeHeader      = $includeHeader;
        $this->_binaryTransfer     = $binaryTransfer;

        $this->_cookieFileLocation = dirname(__FILE__).'/cookie.txt';

        $this->ch = curl_init();
    }

    public function __destruct() {
        curl_close($this->ch);
    }

    public function useAuth($use){
        $this->authentication = 0;
        if($use == true) $this->authentication = 1;
    }

    public function setName($name){
        $this->auth_name = $name;
    }

    public function setPass($pass){
        $this->auth_pass = $pass;
    }

    public function setReferer($referer){
        $this->_referer = $referer;
    }

    public function setCookiFileLocation($path)
    {
        $this->_cookieFileLocation = $path;
    }

    public function setPost($postFields)
    {
        $this->_post = true;

        if (is_array($postFields)) {
            $fields_string = http_build_query($postFields);
        }
        else {
            $fields_string = $postFields;
        }

        $this->_postFields = $fields_string;
    }

    public function setUserAgent($userAgent)
    {
        $this->_useragent = $userAgent;
    }

    public function call($url = null)
    {
        if ($url) {
            $this->_url = $url;
        }

        if (!$url) {
            throw new Exception('You should set an URL to call.');
        }

        curl_setopt($this->ch,CURLOPT_URL,$this->_url);
        curl_setopt($this->ch,CURLOPT_HTTPHEADER, array('Content-Type: application/x-www-form-urlencoded'));
        curl_setopt($this->ch,CURLOPT_TIMEOUT,$this->_timeout);
        curl_setopt($this->ch,CURLOPT_MAXREDIRS,$this->_maxRedirects);
        curl_setopt($this->ch,CURLOPT_RETURNTRANSFER,true);
        curl_setopt($this->ch,CURLOPT_FOLLOWLOCATION,$this->_followlocation);
        curl_setopt($this->ch,CURLOPT_SSL_VERIFYPEER, false);
        curl_setopt($this->ch,CURLOPT_COOKIESESSION, true );
        curl_setopt($this->ch,CURLOPT_COOKIEJAR,$this->_cookieFileLocation);
        curl_setopt($this->ch,CURLOPT_COOKIEFILE,$this->_cookieFileLocation);

        if ($this->authentication == 1) {
            curl_setopt($this->ch, CURLOPT_USERPWD, $this->auth_name.':'.$this->auth_pass);
        }

        if ($this->_post) {
            curl_setopt($this->ch,CURLOPT_POST,true);
            curl_setopt($this->ch,CURLOPT_POSTFIELDS,$this->_postFields);

        }

        if ($this->_includeHeader) {
            curl_setopt($this->ch,CURLOPT_HEADER,true);
        }

        if ($this->_noBody) {
            curl_setopt($this->ch,CURLOPT_NOBODY,true);
        }

        curl_setopt($this->ch,CURLOPT_USERAGENT,$this->_useragent);
        curl_setopt($this->ch,CURLOPT_REFERER,$this->_referer);

        $this->_webpage = curl_exec($this->ch);
        $this->_status = curl_getinfo($this->ch,CURLINFO_HTTP_CODE);

        return $this->_webpage;
    }

    public function getHttpStatus()
    {
        return $this->_status;
    }

    public function __tostring(){
        return $this->_webpage;
    }

}

and that's it. Just setup your cron like this for example :

@daily /usr/bin/php /path/to/my_script.php > /dev/null 2>&1

You can add a log to file logic to the script to check everything went well !