# Closed loop stability

Regarding the Lyapunov stability, we check if a nonlinear system stays near the equilibrium point or approaches to e.p. as time goes to infinity, when it is disturbed.

Let's assume that we have a nonlinear system (an automobile) and designed an optimal controller. The controller executes the given driver input for safely driving by controlling the braking forces and steering angle. The case might be a lane change at high velocities.

In this case, or in similar cases like control of an airplane/AUV/ship/etc., there is a moving (not at rest) system; the controller takes the system from a system of states to another system of states. How can we talk about the stability in this case? If I designed a controller by using an unknown method, say, my method, how can I check that the closed loop system is stable?

Edit: It is an autonomous system. My point is how I can prove that the stability is guaranteed for that controlled system. How will I know that my controller will not make the dynamic system unstable?

• I edited it and hope I am clear now. – Siha Aug 1 '13 at 17:04
• The answer still heavily depends on the task that is given to you to solve. The desired behaviour of system must have some interpretation in terms of phase space that you've chosen. To be clear, take the example of inverted pendulum. The desired behaviour is keeping pendulum in the vertical position, making it stable. Interpretation — the stability of $(0, 0)$ if you choose phase space as angle from vertical line and angular velocity. – Evgeny Aug 1 '13 at 17:13
• You mean I have to study different cases and plot the phase space for each case? For instance, yaw angle vs. yaw rate; lateral displacement vs. lateral velocity, etc. for speeds, say between 10-140 km/h. – Siha Aug 1 '13 at 18:36

Firstly, you have to define what you mean with stability.

Nonlinear systems are different to linear systems since the former can have more than one equilibrium point, whereas for the linear system, with a change of coordinates which always exists, is always the origin.

In the literature you can find such definitions of stability such as globally, almost-globally, semi-globally, locally stability. Input to state stability, Input to output stability, asymptotically or exponentially stability, etc, etc.

Also note that with Lyapunov you can not conclude stability, you need the help of LaSalle's principle if your equilibria is time-invariant, or Barballat's lemma if the equilibria is time-variant. In addition, there are more techniques to show the stability of a system such as the Centre Manifold theorem.

One starting point to show if one equilibrium point for some equilibrium set is stable in a non-linear system is to look at the linearized dynamics of the system at this equilibrium point. Then apply linear techniques to conclude that your system is locally stable.

• Are you sure about your comment "...Lyapunov you cannot conclude stability ..."? I think this is wrong because you can show asymptotic stability, stability and instability by using Lyapunov's theorems. I also think that you should correct the comment "whereas for the linear system, with a change of coordinates which always exists, is always the origin...". I think this statement is also not correct e.g. $\dot{x}=0$ has also infinitely many equilibria and is linear. – MrYouMath Dec 11 '17 at 11:47
• You are correct about the statement about Lyapunov. It should be "you cannot always conclude stability from Lyapunov" and this is why sometimes we need of LaSalle's, Barballat's lemma, etc, in addition to Lyapunov. – user51196 Dec 12 '17 at 14:02
• About the equilibrium of a linear system. It is true that I should say $\dot x = Ax$ with $A \neq 0$ (trivial case). – user51196 Dec 12 '17 at 14:09

Let me recollect the above pieces of comments to an answer.

There is a difference between stability of a control, the stability of an autonomous system and the stability of a non-autonomous system. If you have a controller coupled to an autonomous system, then you have as result a non-autonomous system and the stability you look for will be about the whole non-autonmous system.

In general this is about an own stack of theory see here >>>

And on StackExchange here >>>

• It's not necessarily so. There is a class of controllers that are governed by states of the autonomous system and which usage don't make coupled system non-autonomous. – Evgeny Aug 1 '13 at 17:56
• yes that is true. But then you have an autonomous system where the Lyapunov Stability can be applied as applied to non-linear autonomous systems. This is for instance a nonlinear self-organising master/slave system. The above description however connects to autonavigation systems where state transitions are controlled often non-linear and non-autonomous. That is a complex case to which I referred above. It remains still unclear to me Siha which one is your focus. – al-Hwarizmi Aug 1 '13 at 18:12
• The nonlinear system and the nonlinear controller are connected in a feedback system. It is not clear for me how to combine both of them to obtain the system (1) as given in the paper referenced above: $\dot{x}=f(t,x(t))$. – Siha Aug 1 '13 at 18:47
• Could you post up the equations for both nonlinear system and the nonlinear controller. We can see then whether that is explicitly or implicitly possible? – al-Hwarizmi Aug 2 '13 at 5:46
• I highly appreciate your comments. Thank you. I will post it at my earliest convenience. – Siha Aug 2 '13 at 22:54