Nonlinear systems performance

One of the big distinctions that one does when working in the field of systems and control theory is whether one is dealing with linear or nonlinear systems. The distinction is not academic at all, as the class of nonlinear systems is definitely a lot more difficult to deal with than linear systems.

In short, the differences between linear and nonlinear system behaviours can be summarised as follows. A linear system with a given constant input can have an equilibrium point. If the input changes to a different value, the equilibrium point can also change, and an infinite number of equilibrium points are possible in this way; all of these equilibrium points have the same stability properties (either all stable, either all unstable, either all critically stable): for this reason, one can talk of stability of a system instead of stability of an equilibrium point (as usually done in physics or mechanics). If a system is stable, and if different initial conditions are set, that does not matter, as the state will eventually converge to the same equilibrium point regardless of the initial conditions. Last but not least, a periodic input to a stable system will cause a steady-state periodic motion with the same period.

All of this is in general not true for nonlinear systems. Nonlinear systems can have different equilibrium points for a given constant input. Each of these equilibrium points can be stable (it attracts trajectories starting a from nearby “region of attraction”) or not. Nonlinear systems can also feature limit cycles (equilibrium trajectories) or chaotic trajectories (impossible to predict).

For the sake of argument, let us see a simple (yet famous) example of mind-blowing  nonlinearities. First, start with the following linear state-space system:

For  some values of the constants, this system is asymptotically stable and quite uninteresting: for any initial condition x(0), we have the limit of x for time going to infinity is 0. Let us now make a very small change by introducing a nonlinearity, we just set ξ = x1 so that in fact two entries of the state matrix are now depending on the state itself:

This looks like no big deal, but it turns the uninteresting system above in to the Lorenz attractor, which features state-space trajectories as in the figure below.

This example shows how difficult it can be to predict and evaluate the behaviour of nonlinear systems. Some tools based on convex optimisation, like robust control or the sum of squares, can help sort out the unpredictability of the chaos.

References

P. Massioni, L. Bako, G. Scorletti, A. Trofino, Analysis of pulse width modulation controlled systems based on a piecewise-affine description. International Journal of Robust and Nonlinear Control Volume 30, Issue 15 (Special Issue: New Trends in Modelling and Control of Hybrid Systems), October 2020, pages 5917–5935.

P. Massioni, N. Salnikov, G. Scorletti, Ellipsoidal state estimation based on sum of squares for nonlinear systems with unknown but bounded noise. IET Control Theory & Applications, Volume 13, Issue 12, 13 August 2019, pages 1955–1961.

S. Waitman, P. Massioni, L. Bako, G. Scorletti, Incremental L2-gain stability of piecewise-affine systems with piecewise-polynomial storage functions. Automatica,  Volume 107, September 2019, Pages 224-230.

P. Massioni, G. Scorletti, Guaranteed systematic simulation of discrete-time systems defined by polynomial expressions via convex relaxations. International Journal of Robust and Nonlinear Control, Volume 28, Issue 3, February 2018, pages 1062–1073.

H. Ben-Talha, P. Massioni, G. Scorletti, Robust Simulation of Continuous-Time Systems with Rational Dynamics. International Journal of Robust and Nonlinear Control, Volume 27, Issue 16, November 2017, pages 3097–3108.

O. Ameur, P. Massioni, G. Scorletti, X. Brun, M. Smaoui Lyapunov stability analysis of switching controllers in presence of sliding modes and parametric uncertainties with application to pneumatic systems IEEE Transactions on Control Systems Technology, Volume 24, Issue 6, November 2016, pages 1953-1964.