# Parametric uncertainty in conditional term of piecewise nonlinear dynamical system

Consider a Hammerstein nonlinear dynamical system of the form

$\mathbf{\dot{x}} = \mathbf{Ax} + \mathbf{Bu}$,

where the non-linearity is in the control term $\mathbf{u}$, and has a piecewise conditional form, viz.,

$\mathbf{u} = \left\{\begin{array}{cc} \mathbf{x}-\mathbf{\delta}, & x_i \ge \delta_i, \\ 0, & |x_i| < \delta_i \\ \mathbf{x} + \mathbf{\delta}, & x_i \le -\delta_i.\end{array}\right.$

and where the subscript $i$ represent's the $i^{\rm{th}}$ component of the vector (for some fixed $i$).

Such a system represents a symmetric dead-zone nonlinearity and appears often in engineering applications. Now let $\delta_i$ be a random variable with some distribution (on non-negative support), and consequently $\mathbf{x}(t)$ is a process parameterized by a random variable: $\mathbf{x}(t;\delta)$.

My question is this: what is the best way to model the uncertainty in this dynamical system? The random variable appears in the conditional term of the piecewise definition; therefore any treatment must consider the effect of uncertainty not just in the moment created by the $\mathbf{x}-\delta$ term, but also in the location of the breakpoint.

In the past, I have used non-intrusive polynomial chaos with some success to model this effect. However, I am curious if there is another possibly more clever and accurate technique.

-

## 1 Answer

I assume that $i$ is fixed. If it is not fixed, then the problem is not well-posed (as the control law will be a relation, not a function). In other words, is the dead-zone nonlinearity applied only to the $i$-th component, or is it applied to all components?

Note that $\delta$ cannot have just "some distribution". It must take nonnegative values only, otherwise $|x_i| < \delta_i$ makes little sense. For simplicity, let us suppose that $\delta$ is a constant vector. We then have a continuous-time piecewise-affine (PWA) dynamical system, which is already problematic.

If you allow $\delta$ to be a stochastic process, then you have a time-varying CT-PWA system in which the dynamics change stochastically. Even if you're comfortable with stochastic differential equations (SDEs), it appears to be a ridiculously difficult problem.

Jorge Gonçalves did some work on deterministic CT-PWA systems a decade or so ago. You may want to take a look at his PhD thesis and papers.

-
@RodCarvahlo Thanks for the comments; I have edited the question. Suppose that $\delta$ is not a stochastic process w/r.t. time, but rather is a random variable; a single realization of a time-marching solution will hold $\delta$ constant, but multiple realizations will have different values for $\delta$. As a result, one might expect different limit cycle behavior or frequency domain behavior based on values of $\delta$, and it would be ideal to re-frame the problem in terms of the distribution on $\delta$. It still seems complicated, but maybe this simplifies it? – Emily Aug 10 '12 at 16:09
What is your exact goal? You already have a model. You can simulate it in Simulink if you want. But, what is your question? Do you want to know if the system is globally asymptotically stable (under some conditions) for all values of $\delta$ in some interval? Since you have a CT-PWA system, closed-form solutions are kind of out of question, and analytical results will likely be very hard to find. Maybe you can discretize the system in time? Discrete-time PWA systems are actually easier to analyze. – Rod Carvalho Aug 10 '12 at 22:00
Here is an example: a rack-and-pinion steering linkage on a car has some freeplay. This freeplay gap, call it $\delta$, cannot be eliminated, so a regulating agency wishes to minimize it. However, the smaller the acceptable tolerance, the more expensive the manufacturing becomes. In practice, the freeplay gap is not constant, but rather assumes, say, a truncated normal distribution about some mean value, $\delta_0$. Part of this distribution exceeds the the tolerance. We know that the system behavior is acceptable under the tolerated limit. – Emily Aug 10 '12 at 22:06
However, it is desirable to show the regulating agency that the current manufacturing standards should be allowed to pass, because even when the actual freeplay gap is above the tolerance, the probability of encountering undesirable limit cycle behavior is low. To do this, we need to understand the distribution of the system response amplitude/frequency/phase/whatever w/r.t. the distribution of $\delta$. I can do Monte-Carlo analysis, but as you might imagine, that can get time-consuming. I have had some luck in the past using non-intrusive gPC, but I am wondering if there is a better way. – Emily Aug 10 '12 at 22:08
What is an acceptable system behavior? Are you talking about the actual mechanical system, or its mathematical model? – Rod Carvalho Aug 10 '12 at 23:09