Model-based control uses information about the dynamics of the system's structure and its behavior in time to obtain a better control result regarding stability and performance of the controlled system. Take the following simple example:

Let d^2 x /dt^2=f(x) + u                                                                           (1)

be the system to be controlled, where x - state vector, f(x) - nonlinear vector function, u - control vector. Suppose we have some estimation f_est(x) than we use a control law like 

u= -f_est(x) + u_st                                                                                 (2)

where u_st  be some maybe linear PID control law

u_st=K_p*x + K_d*dx/dt + K_I*S x dt                                                 (3)

So we get

d^2 x /dt^2=Delta f+  K_p*x + K_d*dx/dt + K_I*S x dt

where Delta f= (f(x) -f_est(x)), S - integral sign

If the model uncertainties are small enough we get a linear system with a disturbance Delta f which can be stabilized using appropriate control gains K_p, K_d, KI.This is for many real systems like robots  an appropriate  way to control the nonlinear system dynamics. However, for other systems  this is not so easy as it looks like, but this example should only demonstrate what a model-based control approach is good for.

Besides the above mentioned exact linearization technique, there is the very popular approach of modelpredictive control (MPC), which uses in its standard form an online optimization. The corresponding optimization problem  can contain an explicit model of the system dynamics as constaint. 

But in summary, it is hard to distinguish between model-based and standard feedback control, if the feedback controller was designed with the knowledge of a model of the plant. Because then the feedback controller contains the model knowledge implicitly. If you look at it from an intuitive point of view, the controller has to "know" how the system behaves anyway in order to control it.

Rainer,

Thanks. What is the model in this case, is it f_est(x)? And how does it inform the control equation? 

Also where do the values of x come from in the control equation? And the K values?

Actually, I am not understanding what is different here from a standard PID controller?

Fabian,

Thanks. When you say knowledge of the plant is required for a standard controller do you mean to work out the gain, for example?

If you had a process for the auto-tuning of the gain value would you still regard that as a model-based, or having knowledge of the plant?

Hi Rupert,

f_est(x) is indeed the system model which can be a more or less good approximation of the real nonlinear system. This method is often used in robotics called as "computed torque" method. In my example are the outputs y equal to the states x.

If the model is good enough than one can freely design the standard PID gains as if there was a linear system present. In addition, one may introduce an additional sliding mode control term to  cope better with model uncertainties Delta_f .

Hi Rainer,

I'd like to set up a simulation of a standard controller and model-based controller to compare the two, of the same plant. Could you suggest a suitable function to use for f(x)? 

What is "y", I didn't see that in your original post? Also how do you know Delta f if you don't know f(x)?

Hi Rupert,

I wrote: "In my example are the outputs y equal to the states x".So, y is the measured output.

You are right saying that Delta f  is unknown but usually  one can find an upper bound for  |Delta f| so that one can design a controller that is robust enough to cope with this uncertainty.

Hi Rainer,

Could you point me to an actual example that I could implement in simulation? That is, a simple process model I could use to build a simulation, and compare standard and model-base controllers.

Hi Rupert,

please find my example in the attachment (pdf-file). I know that this is a very simple example for using models to compensate for nonlinear system's behavior but it was my intention to illustrate the approach. To simulate the whole thing you just have to discretize the continuous differential equation.

Best

                  Rainer

Hi Rainer,

Thanks for that. An inverted pendulum on a cart can be controlled without incorporating a model. But before we come to that let me see if I understand the system you posted.

It is a stationary pivot point with a pendulum  on a fixed length rod moved by a motor at the pivot point. And the issue is to move the rod to a particular angle.

Why not just rotate the motor until the measured angle matches the desired angle? Why do you need to know gravity or the rod length?

I must be misunderstanding something.

Regards,

Rupert

Hi Rupert,

it's a dc motor (not a stepping motor) that has to generate an appropriate  force/torque  to cope with the moment of inertia of the pendulum both to move it and to hold it in a defined position/at a well defined angle. Try to understand the dynamics of rigid bodies as an equilibrium of forces and torques.

see any textbook on robot kinematics, dynamics, and control (also look at Wikipedia)

Hi Rainer,

The point I am trying to get to is why the dynamics need to be modeled as part of the control equations when it can be done without them?

Would you mind clarifying some points and queries? I've attached to display the symbols.  

Regards,

Rupert

Hi Rupert,

1. Sorry, I made a mistake with u_comb, the RHS should be negative :                u_comb= -g/l^*sin phi

2. The desired velocity can  also be a function of time

3.  The acceleration is usually written on the LHS of the system equation.                   The output is  the angle phi.

4. The power to apply to the motor is u=u_comb + u_PID

5. The length of the rod is never exactly known. So,  Delta  has always a finite value.

6. You can, of course, try to control the nonlinear system without a compensation term (the model), but u_comp makes it easier to design a linear PID control law because of the linearized system equation.

Regards,

Rainer

Hi Rainer, (just seen your reply)

This model-based approach seems  a complicated way of doing things. As the environment gets more complex surely it would get impractical to determine the models. A general robot would need a multitude of specific and highly complex models.

Linked is a way of designing the control systems for the inverted pendulum without the requirement for a model. And a video of my implementation on a real system.

What is the model (u_comp) term doing?

Regards,

Rupert

http://www.livingcontrolsystems.com/demos/inverted_pendulum/inverted_pendulum.pdf

https://youtu.be/WeMuqZ4Mpg4