With the aim of overcoming this deficiency, the second type of control-scheduling co-design employs state equations to describe the control system. The performance is measured by using the control plant information, such as plant states [18], control inputs [19], and a cost function for the states and inputs [20]. Various schedulers have been designed to guarantee the system stability. For example, a feedback scheduler that periodically assigns new sampling periods was developed based on estimates of the current plant states and noise intensities for a linear stochastic system [20]. The proof of the schedulability condition also serves as a systematic method to design a scheduler for a networked linear time-invariant system [19]. By designing an exponential stable fault detector, a real-time feedback scheduler was developed for a linear discrete system, considering time delay and data dropouts of the network transmission [21]. An integrated design for a hybrid scheduling strategy, an adaptive quantizer and a feedback controller for a linear networked control system was developed to maintain asymptotic stability by using the multiple-Lyapunov function and switched system theory [18]. The co-design of control law, bandwidth scheduling and event generator to guarantee globally uniformly asymptotic stability was formulated as a linear matrix inequality problem [22]. The sufficient stability condition was derived for each plant by the simultaneous stabilization of a collection of plants remotely controlled via a wireless network [23]. Nevertheless, no previous literature has provided the effect of the scheduling period on the critical boundary with regard to stability and on the control performance once the stability condition is satisfied. Besides, most of the studies in the literature refer to linear systems. Their methods and conclusions cannot be directly applied to the piecewise nonlinear system used as the powertrain during clutch engagement. In this study, a two-step control-scheduling co-design framework is proposed for clutch engagement, as depicted in Figure 1. Since each segment is linear in a piecewise nonlinear system, the powertrain model can be regarded as linear when the clutch is slipping. Thus, in the first step, the stability analysis can be conducted according to the pole locations of the integrated system model that envelops the clutch-employed powertrain model, the feedback control law and the zero-order holder. In this way, the critical period regarding stability can be determined, and factors influencing the critical period can be investigated. In the second step, the transition between adjacent segments, from the slipping to the being locked phase of the clutch-employed powertrain dynamics is investigated. The dynamic performance is evaluated in terms of vehicle jerk and frictional loss during clutch engagement. The dynamic responses are obtained by real-time hardware-in-the-loop experiments and the performances of different scheduling periods within the stable region can be compared. The major contributions of this paper are summarized as:  
(1) A two-step framework of control-scheduling co-design for selecting the task scheduling period is proposed for clutch engagement control. (2) The sensitivity analysis of the critical period with regard to stability based on five key parameters--the natural frequency, damping coefficient, proportional gain, integral gain, and time delay--is illustrated. (3) The effect of scheduling period on dynamic performance during clutch engagement is exposed by real-time hardware-in-the-loop experiments. A sub-boundary of the scheduling period within the stable region is found to guarantee the control performance. This paper is organized as follows. The system model under closed-loop control is described in Section 2. In Section 3, the model is linearized and discretized for the slipping phase, and the stability condition is derived according to pole locations. Thereafter, a sensitivity analysis is conducted.