$db =PDO baru(mysql:host=$host;db_name=$db_name, $user, $pass);

Seperti kata Hobo Sapiens, host yang Anda sediakan tidak ada atau salah eja, saya tidak tahu struktur $request Anda obyek. Struktur objek Anda harus seperti berikut ini.

class Request 
{
private $host;
private $db_name;
private $db_username;
private $db_pwd;

public function __construct($host,$db_name,$db_username,$db_pwd) {
    $this->host = $host;
    $this->db_name = $db_name;
    $this->db_username = $db_username;
    $this->db_pwd = $db_pwd;
}
public function getHost(){
    return $this->host;
}
public function getDBName(){
    return $this->db_name;
}
public function getDBUsername(){
    return $this->db_username;
}
public function getDBPwd(){
    return $this->db_pwd;
}
}

Kemudian Anda dapat melakukan sesuatu seperti berikut

$request = new Request('localhost', 'TestingPDO', 'root', '');
$modelinstall = new modelInstall();
$modelinstall->main($request);