# Sliding Mode Control Algorithms for Anti-Lock Braking Systems with Performance Comparisons
## 3 Sliding Mode Control Design
### Reaching Law Sliding Mode Control Design with Partial Process Model Uncertainty
On the sliding manifold \(g(\mathbf{x})=0\), the closed-loop dynamics have reduced order, since \(g(\mathbf{x})=0\) leads to the control \(u(t)\) from (13) substituted in (5) and to the simplified dynamics \(\dot{x}_{2}=(f_{1}(x_{1},x_{2})g_{2}(x_{1},x_{2})-f_{2}(x_{1},x_{2})g_{1}(x_{ 1},x_{2}))/[(1-\lambda_{d})g_{2}(x_{1},x_{2})-g_{1}(x_{1},x_{2})]\) constrained by \(x_{1}=(1-\lambda_{d})x_{2}\). The simplified dynamics lead -after some calculations- to a (family of) linear differential equations in \(x_{2}\), of the form \(\dot{x}_{2}=k_{1}(\lambda_{d})x_{2}+k_{2}(\lambda_{d})\) with \(x_{2}(0)>0\), \(k_{1}<0\), \(k_{2}<0\). This family of \(\lambda_{d}\)-dependent equations is shown in Figure 4 below for \(\lambda_{d}\) in \(\{0,0.05,0.1,\ldots,1\}\). This reduced order system is again asymptotically stable for all \(S(\lambda_{d})<1.5\) where \(0\leq\lambda_{d}\leq 1\), with \(x_{2}\to 0\) since \(x_{2}\) is being bounded below by zero: because of braking, the car must come to a full stop. Then again, Then Theorem 4.2 from [34] claimed by Theorem 2.1 from [35] implies asymptotic stability of the closed-loop system, whenever the reduced order system on the sliding surface if asymptotically stable and when the evolution of the closed-loop on the sliding surface is preserved. Consequently, \(\lim_{t\to\infty}\lambda(t)=\lambda_{d}(t)\) which completes the proof. \(\quad\Box\)  
**Theorem 1**: Both situations related to the implementation of the command in the form (13) suggest choosing a larger value of the parameter \(k\). On the other hand, a larger \(k\) value introduces high amplitude switching at the control input level, due to the switch element \(sgn(g)\). A compromise is needed: the more accurate the model (2) is, the smaller \(k\) parameter value is admitted. **Theorem 2**: Towards the braking process' end, the lower wheel speed \(x_{2}(t)\) tends to zero. To ensure good numerical conditioning, the denominator \(x_{2}^{2}(t)\) occurring in both \(f(t),b(t)\) in (12) and in both \(F(t)\), \(G(t)\) in (6), is replaced with \(x_{2}^{2}(t)+\xi\), where \(\xi\) is a small positive number. In practice this is less problematic since, at low car speeds, it is common to deactivate the ABS control. **Theorem 3**: The sign function of the control law (13) is practically implemented using the modified sign function \(\mathrm{sgn}_{\Delta}(\bullet)\) defined in (8). **Theorem 4**: For the computed control laws (8) and (13) to avoid deep control input saturation that could lead to significant windup followed by high-amplitude oscillations, the control inputs calculated as in (8) and as in (13) are manually saturated inside [-1;1] (also observing that the control input domain \((-\infty;0)\cup(1,\infty)\) is not actually usable from the viewpoint of the DC motor actuator, but it is preserved to further show that the SMC compute the actual control inputs in this domains also). On the other hand, this saturation could restrict the domain of the sliding manifold and lead to a longer reaching phase.