PID control is the gold standard for most moving machines, and quite adequate in most applications that require closed-loop control. However, it's vulnerable to disturbances, and tuning can be tedious. For these reasons, some industrial drives go beyond traditional PID structures and servo algorithms for position control — with sliding mode algorithms that increase stability over a wider range of conditions, to make tuning easier for design engineers.

Sliding mode control was first defined and expanded in the 1950s by Russian mathematicians. Since then, it has been developed into a general design method. Though it's still relatively uncommon in industrial applications, sliding mode control has been applied in some cutting-edge positioning machines, and promises to benefit more motion systems in the future. In fact, the algorithms are particularly well suited to embedded control designs with limited resources and budgets.

Sliding-mode theory

Sliding-mode tuning is robust in that it can handle large inertia variation — so then only a few parameters require adjustment to optimize system response. The control design process is divided into two stages:

1. Design an asymptotically stable hyperplane that contains the destination of the control — for which any points on the plane automatically converge to the destination. In short, the hyperplane is a mathematical set of distinct machine conditions.

2. Design an enforcing control so that the system state starting from anywhere in the state space is forced to reach the hyperplane.

For second-order motor control:

Where θ and ω = Motor position and velocity, and their dots denote derivatives with respect to time

T_e = Motor torque

b_p = Motor damping/friction coefficient

ΔT = Summation of load torque and disturbance torque acting on the motor shaft.

Assuming that motor motion follows a target trajectory:

Where θ_d and ω_d = Target motor position and velocity

f(t) = Time-dependent function describing target motor acceleration

Defining e₁ = θ_d - θ and e₂ = ω_d - ω, we then subtract our first equation from the second to get:

Based on this equation, we define the target hyperplane:

After the sliding mode is designed, we can design an enforcing control:

Where K_sliding = Constant to be selected

ε = A small number

SIGN (S) = Sign function of S

We must select a proper K_sliding to make the system asymptotically stable. This can be achieved by selecting an appropriate Lyapunov function based on specific application conditions.