How does isstable(sys) and step response plot in all toolsbox on MATLAB work?

We can check for stability of a system using isstable() or tuning PID and see step response using controlsystemdesigner() or just build the model in simulink and plot the step response. However, how does MATLAB consider a system stable or unstable or plot the step response to infinity or to constant 1? I have been doing an inverted pendulum project, and according from the 2 documents from the kit producer, and externally 2 more documents, including the document that the papers from the kit company referenced, they all stated:

To my understanding, they are basically saying you can have a stable system, even with positive poles, as long as you have an even number of positive real poles between every pair of real zeros, all >=0. Youla (1974) added these and the same conclusion:

I framed MATLAB to have 1 real zero at 1, 2 real poles at 2 and 4, and MATLAB still state the plant transfer function is unstable (top left stable case). Earlier cases it gave me responses that go to infinity with zeros at 0 and 10, 2 poles at 6 and 7. Why is this? Am I wrong, or the documents are wrong? Or MATLAB is wrong? If MATLAB should not be used to plot the step response for this, then what should I use?

Thank you

[Les Beckham]

Keep in mind that there is a difference between a stable system and a stabilizable system.

Answer 1

[Paul]

In the Control System Toolbox, isstable

returns a logical value of 1 (true) for stability of a dynamic system if:

• In continuous-time systems, all the poles lie in the open left half of the complex plane.

But, as @Les Beckham rightly stated, stable (stability) and stabilizable (stabilizability) are not the same.

AFAIK, the Control System Toolbox does not offer any functions to test if a system is stabilizable or strongly stabilizable, though it wouldn't be difficult to write your own function for either property, at least for SISO systems (is strongly stabilizable defined for MIMO systems?).