Controllability - Need a Controls Guru

[Iawia]

Hi,

I am investigating a typical 'active' 1/4 car suspension system (one actuator). Along my studies of controls, I was told along the way that mechanical systems (or electrical) with symmetry sometimes end up being 'uncontrollable', (i.e. the matrix M = [B AB A^2B ... A^(n-1)B] is not full rank).

Conceptually, an active suspension with only one actuator should be uncontrollable. The states can not all be controlled by one actuator. So, is my concept of controllability incorrect? Or is it that all the states (x1-x4) can be taken to the next desired state in finite time, but not all states simultaneously??? When I perform rank(M), it equals 4, which means it is full rank. Shouldn't it be 3 revealing that it is not controllable?

I am not a state space expert (obviously!), but I am writing a paper that contains controllability and would like to get this bit correct. I have search all over the web for info on symmetrical controllable 2 Degree of Freedom Systems examples but had no luck. I figured if anyone can help it would be someone here.

Thanks!

--t

 

[MrAl]

Hi,

Take a look and see if this helps:

https://en.wikipedia.org/wiki/Controllability_Gramian

 

[Iawia]

Hi Mr. Al,
The gramian is not helpful. It describes the general solution to the controllability problem in a very abstract manner. I need a more discrete explanation.

I have a similar question about observability in that for a multiple input system the observability matrix O_M = [C CA CA^2 ... CA^n-1]' the matrix results in 2*n rows. Do we still apply the rank(O_M) = n to confirm observability (because most of the rows are all zeros), I am used to SISO (single input single output) systems and not MIMOs (multiple input multiple output).

 

[MrAl]

Hi,

Oh ok sorry, i know some of these abstract concepts are so abstract sometimes that we cant always be sure they are giving us the correct result. And there are catches which are NOT represented at all in the formulations such as system limitations like saturation...ie the formulas assume the system can respond according to the describing equations perfectly no matter what the signal levels may have to be to achieve that performance...something i have always disliked.

I'll look around and see what else i can find. In the mean time a simple example is a balanced bridge circuit where we apply a signal to the top node of the bridge with bottom node grounded (the usual drawing depicts the bridge circuit as a diamond shape with elements taking the place of each side of the diamond and each vertex a node). and the output taking from across the left and right nodes. No matter what signal we apply to the top node, we can never control the output of the bridge. If however we make the bridge an unbalanced bridge, we can then force a change in output with a change in the input and so suddenly it works. That's a very simple example maybe too simple so i hope it helps a little anyway.

 

[misterT]

Conceptually, an active suspension with only one actuator should be uncontrollable. The states can not all be controlled by one actuator.

It depends on the selection of the state variables. You can design the state space matrices to be controllable or observable etc. A system does not have a unique state space model. You can have one actuator controlling the speed, position and acceleration of an object. That is one actuator controlling three state variables.

Or is it that all the states (x1-x4) can be taken to the next desired state in finite time, but not all states simultaneously???

The state is the vector x which includes all state variables (x1-x4). If the system is controllable, then it means that you can drive all those variables (the vector) to a desired state as you like.

 

[MrAl]

Hi again,

Yes i agree but it must also be able to get to any state from any OTHER state, and in a finite amount of time to be called completely controllable.

The algebraic controllability theorem definition is that the rank of the controllability test matrix must be the same as the order of the system.

The more general definition is that a system is controllable if and only if by the inputs the system can be taken from any initial state to any other state in a finite amount of time.

The general definition can be hard to prove in simulation so perhaps either the algebraic theorem or the controllability grammian can be used as a test, then a simulation can be run to test for extremes that go over the physical limitations of the system that are not represented in the state equations.