2.3.2. Nonlinear Optimization

Using linear transfer functions to optimize the plants provide us with an optimized initial parameter set. Although this set might be good enough for primarily linear plants; most brake actuators are nonlinear due to different factors such as friction, plant saturation, and dead-zone. Therefore, to further optimize the plants, we should perform optimization with the nonlinear plants to consider all the nonlinear effects. Since all the plants are going to run in the feedback control environment in practice, the nonlinear optimization is done on the closed-loop systems. A 10 kN step clamp force reference target is chosen for the brake's closed-loop system. The controllers are designed in the same way explained in the Sections 2.2 and 2.3.1. The control parameters are fixed in the same way as the transfer function optimization while the physical parameters of the systems change. Since the physical parameters of the system are changed, we need to recalculate the controllers at each step of function evaluation in the optimization (*Gp* changes, and so does *Y* and *Gc*). Since the optimization is performed on nonlinear plants, a different objective function should be used. The objective function for the nonlinear optimization consists of four parts: energy usage, maximum power, settling time, and overshoot percentage. Energy usage (*Eusage*) is the total amount of energy used by the actuator to follow the target in two seconds (enough for the actuators to reach and hold the target). Maximum power (max P*usage*) is the maximum power used by the actuator during the 2 s that the actuator follows the 10 kN step reference. Settling time (*Ts*) is the time that it takes for the caliper force to build up to near ±2% of the steady-state value. Overshoot percentage (*OS*%) is the percentage that the maximum value of the caliper force deviates from the 10 kN reference target. Power usage for the EMB and EWB is defined as current multiplied by the voltage. For the EHB, we are adding up the amount of power loss (denoted as P*build* and P*dump*) to be equal to the power usage (Figure 2a):

$$\mathbb{P}\_{build} = \varepsilon\_{build} \times f\_{build} = (P\_{in} - \frac{\beta\_{hf}}{V\_{Cyl}} \times q\_{cyl}) \times \{\mathbb{C}\_d \mathbb{S}\_b u\_b \sqrt{\frac{2}{\rho} (P\_{in} - \frac{\beta\_{hf}}{V\_{cyl}} q\_{cyl})}\} \tag{25a}$$

$$\mathbb{P}\_{dump} = \varepsilon\_{dump} \times f\_{dump} = (\frac{\beta\_{hf}}{V\_{\mathbb{C}yl}} \times q\_{cyl}) \times \{\mathbb{C}\_d \mathbb{S}\_d \mu\_d \sqrt{\frac{2}{\rho} (\frac{\beta\_{hf}}{V\_{\mathbb{C}yl}} q\_{cyl})}\} \tag{25b}$$

Taking all of these into account, the cost function for nonlinear optimization is given in Equation (Section 2.3.2). Note that each cost is normalized by its nominal value:

minimize *<sup>x</sup> <sup>f</sup>*(*x*) = *<sup>α</sup>*<sup>1</sup> <sup>×</sup> *Eusage* <sup>+</sup> *<sup>α</sup>*<sup>2</sup> <sup>×</sup> max <sup>P</sup>*usage* <sup>+</sup> *<sup>α</sup>*<sup>3</sup> <sup>×</sup> *Ts* <sup>+</sup> *<sup>α</sup>*<sup>4</sup> <sup>×</sup> *OS*% subject to *x* ∈ [*xmin*, *xmax*] (26)
