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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.

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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.

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    Acceptable system behavior would be defined in terms of, say, the loading on the steering linkage, which can be derived the from system response amplitude, and also in the frequency of cycles above that amplitude, with regards to fatigue life reduction. I am also interested in the problem from a controlability standpoint; one approach to the problem involves adaptive inversion of the dead-zone nonlinearity. Understanding the stochastics allows for intelligent selection of gain, etc., as well as understanding the effect due to actuator rate limits.2012-08-12