Is this process an AR(1)? The following process $X$ is generated as the sum of a structure $y$ and an $AR(1)$ process with $\phi=0.4$ and $\sigma_{\epsilon}^2=1$:
N=200 # length of time series
t=1:N
A=20 # amplitude of the structure
y = rep(0,N)    
a = 30
b = 170
y[a:b] = A * (t[a:b] - a)/ (b-a) # the structure (see the plot below)

X=y+arima.sim(list(order = c(1,0,0),ar=0.4),n=N,sd=1) # the process

model=ar.ols(X) # fit AR process

Box.test(model$resid,lag=round(log(N)),type="Ljung-Box",fitdf=1) # white noise test for the model residuals

par(mfrow=c(1,3))
plot(y,type='l',ylab='y',xlab='')
plot(X,type='l',ylab='X',xlab='')
plot(model$resid,xlab='')


In the end, the test tells the model residuals are independent with $p=0.8327$ (please click here), which means $X$ can be modelled as $AR(1)$ process?
 A: Note that the question in the title is not the same question as the question you ask at the end of your text.  I'll answer them both.
1) The process is not an AR(1) process, it is the sum of a deterministic process and an AR(1) process.  
2) However, you CAN model it as an AR(1) process (you can model almost anything as an AR(1) process.)  A better question is, is such a model any good?  (I suspect that this is what you meant by your question.)  The rest of this answer addresses that question.
If the AR(1) model is supposed to represent the end result of a  model-building process, I'd say no; a simple plot of the data (your middle plot) reveals clearly the potential for a much better model; even the first try at a better model would probably get the underlying deterministic component almost exactly right (give or take a little uncertainty about where the upwards trend started), and a more sophisticated model would contain terms to fit that too (rather than have the modeler specify it).  My attempt at doing so gave me a residual variance of about 0.87, compared to roughly 3.3 for your model and 46 for the original series.  Even ignoring the AR(1) component and using linear regression with two breaks gives a residual variance of about 1.0.  Missing such an obvious way of greatly improving the model implies the modeling process has fallen well short of the ideal.
On the other hand, if all you want to know is whether this model is much better than assuming no structure and using the sample mean, I'd say yes.  In my simulation, the variance of X was roughly 46; the residual variance was roughly 3.3.  This is quite a significant improvement.  However, this question is not really a very good one.  What would you do with this model that you couldn't do by eyeballing the original data?  How has developing an AR(1) model helped you along the road to understanding the underlying process, or prediction for that matter?  
