I am able to upgrade my kubernetes cluster from 1.10.0 to 1.10.11.

[root@ centos]# kubectl get nodes
NAME                         STATUS    ROLES     AGE       VERSION
ip-.ec2.internal             Ready     master    2m        v1.10.0

Here is the step by step instructions:

export VERSION="1.10.11"
export ARCH=amd64
wget https://storage.googleapis.com/kubernetes-release/release/v1.10.11/bin/linux/amd64/kubeadm > /usr/bin/kubeadm

chmod a+rx /usr/bin/kubeadm

Now check the kubeadm upgrade plan

[root@ centos]# kubeadm upgrade plan
[preflight] Running pre-flight checks.
[upgrade] Making sure the cluster is healthy:
[upgrade/config] Making sure the configuration is correct:
[upgrade/config] Reading configuration from the cluster...
[upgrade/config] FYI: You can look at this config file with 'kubectl -n kube-system get cm kubeadm-config -oyaml'
[upgrade/plan] computing upgrade possibilities
[upgrade] Fetching available versions to upgrade to
[upgrade/versions] Cluster version: v1.10.11
[upgrade/versions] kubeadm version: v1.10.11
[upgrade/versions] Latest stable version: v1.13.1
[upgrade/versions] Latest version in the v1.1 series: v1.1.8
[upgrade/versions] WARNING: No recommended etcd for requested kubernetes version (v1.13.1)

Components that must be upgraded manually after you have upgraded the control plane with 'kubeadm upgrade apply':
COMPONENT   CURRENT       AVAILABLE
Kubelet     1 x v1.10.0   v1.13.1

Upgrade to the latest stable version:

COMPONENT            CURRENT    AVAILABLE
API Server           v1.10.11   v1.13.1
Controller Manager   v1.10.11   v1.13.1
Scheduler            v1.10.11   v1.13.1
Kube Proxy           v1.10.11   v1.13.1
Kube DNS             1.14.8     1.14.8
Etcd                 3.1.12     N/A

You can now apply the upgrade by executing the following command:

kubeadm upgrade apply v1.13.1

Note: Before you can perform this upgrade, you have to update kubeadm to v1.13.1.

Now instead of upgrade apply to v1.13.1 you should use:

kubeadm upgrade apply 1.10.11

[preflight] Running pre-flight checks.
[upgrade] Making sure the cluster is healthy:
[upgrade/config] Making sure the configuration is correct:
[upgrade/config] Reading configuration from the cluster...
[upgrade/config] FYI: You can look at this config file with 'kubectl -n kube-system get cm kubeadm-config -oyaml'
[upgrade/version] You have chosen to change the cluster version to "v1.10.11"
[upgrade/versions] Cluster version: v1.10.11
[upgrade/versions] kubeadm version: v1.10.11
[upgrade/confirm] Are you sure you want to proceed with the upgrade? [y/N]: y
.......
[bootstraptoken] Configured RBAC rules to allow the csrapprover controller automatically approve CSRs from a Node Bootstrap Token
[bootstraptoken] Configured RBAC rules to allow certificate rotation for all node client certificates in the cluster
[addons] Applied essential addon: kube-dns
[addons] Applied essential addon: kube-proxy

[upgrade/successful] SUCCESS! Your cluster was upgraded to "v1.10.11". Enjoy!

Now follow the official documentation of upgrade here and you will be able to get the system up and running:

Drain the master nodes using:

kubectl drain ip-.ec2.internal --ignore-daemonsets

Upgrade the kubelet and kubeadm package using:

yum upgrade kubelet-1.10.11 kubeadm-1.10.11 --disableexcludes=kubernetes

Restart kubelet and uncordon the host:

systemctl restart kubelet
kubectl uncordon $HOST

And now you will be able to get you system up and running:

[root@ip- centos]# kubectl get nodes
NAME                         STATUS    ROLES     AGE       VERSION
ip-.ec2.internal             Ready     master    32m       v1.10.11
[root@ip- centos]# kubectl get pods
NAME                                READY     STATUS    RESTARTS   AGE
nginx-deployment-75675f5897-pmmkc   1/1       Running   0          9m
nginx-deployment-75675f5897-wshgz   1/1       Running   0          9m