Probably we can just combine B and Bd, and then separate them after discretizing.

You can combine B and Bd matrices. The size of the D matrix must also change for size fit. You can look at the attached file for a sample system I prepared.

I recommend you to read the following paper:

Conference Paper Can continuous-time disturbance be represented by sampled in...

The main idea of the above paper:

You are dealing with the continuous time disturbance, which may be originated from a physical source. So the physical plant can be represented by (like your notation)

dx(t) = A*x(t) + B*u(t) + B_d*d(t),

y(t) = C*x(t),

and I emphasize the continuous time domain.

After some kind of discretization (ZOH equivalent model is used in the paper), the author represented the discrete time system as

x(k+1) = F*x(k) + G*u(k) + G_d*v(k),

y(k) = H*x(k) .

Main results of the paper say that if we write x(k)=x(kT), u(k)=u(kT), and y(k)=y(kT), then v(k) is not the same as d(kT), but it is calculated from the response y(kT), where y(t) is affected by d(t). Moreover, v(k) is resulted from the system inversion, so v(k) can be unstable signal if the discrete time system has an unstable zero (again, v(k) is not the same as d(kT)).

The paper would be helpful.

Thanks, Selim and Jongsoo. Both your answers were very useful and directly relevant.

My system is an order 6 system and it seems that the c2d command in Matlab is a bit slow for implementation on a real robot (it takes it 20-30ms (milliseconds) to calculate the discrete system). Do you know any other way to discretize the system faster?

I am linearizing my nonlinear system every 1-10ms and I should discretize it every time to use it in my MPC algorithm.

I think you can use Euler's method. It is an approximation but it works in real time applications. See the attached file.

Thanks, Selim. It was very useful.

Selim Sivrioglu

Create a new input matrix BB which consists from both input matrices for control input and also for disturbance BB=[B BD], the input vector is now [u D]T . The same you have to do with the output equation. Afterwards you can use the function c2d.

BR

PN

Thanks, Tian Seng Ng

Dear Mahdi Nabipour,

I suggest you to see links and attached files on topic.

https://math.stackexchange.com/questions/3281767/discrete-state-space-representation-of-a-second-order-system

https://www.springer.com/gp/book/9783319772189

https://folk.ntnu.no/skoge/prost/proceedings/ifac11-proceedings/data/html/papers/1346.pdf

Best regards

Mohamed-Mourad Lafifi Thanks, brother.

Dear Mahdi Nabipour,

You are welcome

Dear Mahdi:

For a simple solution to your problem, please see Chapter 8 Digital Control, in the book

Franklin, G.F., Powell, J. D., and Emami-Naeini, A., Feedback Control of Dynamic Systems, 6th ed., Prentice-Hall 2010

This chapter will help you convert a continuous system to a discrete system. Most importantly, to convert a continuous disturbance to its discrete version, the sampling frequency [equal to 1/(sample period)] should be 30 to 40 times the bandwidth (in Hz) of the disturbance. Note that when we discretize a continuous dynamic system, we basically move from acceleration to incremental velocity form, so the external disturbance in the continuous force (or acceleration) form is discretized to a momentum (linear or angular, depending on the problem in hand) form, and therefore we must ensure that we truthfully capture the momentum imparted by the continuous force (or torque) within a sample period. This would be possible if the sampling frequency is 30-40 times the disturbance bandwidth. In fact, the same is true even for discretizing the natural dynamics of the system. It basically means that the continuous variation of a sinusoid over a 360-degree interval is represented in a staircase manner in 30-40 steps, each step covering a span of 12-9 degrees, respectively. Better yet, the sampling frequency might go up to 60 times the bandwidth if the hardware permits.

For more, please see the book cited above. Also, use Tustin's method for discretizing the natural dynamics of the system. Read the book cited above.

If you are dealing with the angular dynamics of rigid bodies, the situation is much more complicated. Here we use accelerometers and gyros. To learn more about them and how they discretize the dynamics, see

Markley, F. L., and Crassidis, J. L., Fundamentals of Spacecraft Attitude Determination and Control, Springer 2014

I hope this helps. If you wish to communicate with me further, please do so at my personal email: [email protected].

Hari Hablani

Faculty, Space Engineering

Indian Institute of Technology, Indore, India

That's a good question

dear Mahdi Nabipour , I think you have found the exact solution of your problem, if not please let me know. Thanks

Thanks to Hari Hablani and Zahoor Ahmed

I have found the answer.

Mahdi Nabipour ,welcome.