M5-T4 — Sliding-Mode Control

Learning objectives

After this lesson, you should be able to:

• Choose a sliding variable so that motion on it is a first-order decay with a chosen rate.
• Derive the switching gain that guarantees the surface is reached against a bounded disturbance.
• Explain chattering and relate it to the actuator servo and rate limit.
• Choose a boundary-layer width and state the resulting error bound.

The intuition

UAV M5-T16: Disturbance rejection and wind compensation cancelled a disturbance by estimating it. Sliding-mode control takes the opposite stance: the disturbance is not estimated, only bounded, and the controller is made strong enough to beat any disturbance inside the bound.

A surface to live on

For Hopper's pitch, $\ddot{\theta} = \lambda^2\theta + b\delta + d$, define the sliding variable $s = q + c\,\theta$ with $c = 3$ rad/s. If $s$ is held at zero the pitch obeys $\dot{\theta} = -c\theta$: a first-order decay with a pole at −3, whatever the plant or the disturbance does, because they never enter. The second-order problem has been reduced to two steps: bring $s$ to zero, then let the pitch decay on its own.

Switching to stay on it

The control that drives $s$ to zero in finite time is a hard switch: push as hard as the largest disturbance whenever $s$ is positive, and the opposite way whenever it is negative. If the push exceeds the disturbance bound, $s$ cannot stay on either side, and slides along zero. With a gimbal that has a 40 rad/s servo and a 3 rad/s rate limit, though, a hard switch is a command that flips sign thousands of times per second, and the gimbal tries to follow: chattering.

A boundary layer

The remedy is to replace the switch by a saturation: full push when $|s|$ exceeds a thickness $\varphi$, proportional inside. The gimbal then moves smoothly. The surface is no longer held exactly: the loop stays within $|s| \le \varphi$, and the pitch within $\varphi/c$ of zero. A thicker layer gives a gentler gimbal and a larger error, and the width is chosen from the rate limit and the accuracy needed.

Sliding mode buys independence from a bounded disturbance with a hard switch. The switch is paid for in actuator activity, and a boundary layer pays it back in accuracy.

Definitions

Sliding variable $s$

A combination of the states, $s = q + c\,\theta$, whose zero set is the sliding surface.

Reaching and sliding

The first phase brings $s$ to zero; the second keeps it there while the state moves along the surface.

Matched disturbance

A disturbance that enters through the same channel as the control, as $d$ does in $\ddot{\theta}$. Only matched disturbances are rejected completely.

Boundary layer $\varphi$

The thickness of the band around $s = 0$ inside which the switch is replaced by a linear segment.

Derivation: the reaching condition and the layer bound

Hopper's pitch with a matched disturbance $|d| \le D$ and the law with $\hat{\lambda} = \lambda,\ \hat{b} = b$.

1. $$\dot{s} = \ddot{\theta} + c\,q = \lambda^2\theta + b\delta + d + c\,q$$

   Differentiate the sliding variable and substitute the plant.

2. $$\dot{s} = d - \eta\,\mathrm{sgn}(s)$$

   Substitute the sign form of the law ($\varphi \to 0$): the model terms cancel and what remains is the disturbance and the switching term.

3. $$s\,\dot{s} = s\,d - \eta\,|s| \le -(\eta - D)\,|s|$$

   Multiply by $s$ and use $|d| \le D$. This is $\tfrac{1}{2}\tfrac{d}{dt}s^2$.

4. $$t_{\mathrm{reach}} \le \dfrac{|s(0)|}{\eta - D}$$

   With $\eta > D$ the function $s^2/2$ is a Lyapunov function (the method of M5-T1: Lyapunov Stability) that decreases at a rate no smaller than $(\eta - D)|s|$, so $|s|$ reaches zero in finite time.

The boundary layer

1. $$\dot{s} = d - \eta\,s/\varphi \quad (|s| \le \varphi)$$

   Inside the layer the saturation is linear, so the sliding variable obeys a first-order stable equation driven by the disturbance.

2. $$|s|_{ss} \le \dfrac{\varphi D}{\eta}\ \text{(for a constant } d), \qquad |s| \le \varphi \text{ always}$$

   At a steady state $s = \varphi d/\eta$, which is smaller than $\varphi$ when $\eta > D$.

3. $$|\theta|_{ss} \le \dfrac{|s|_{\max}}{c}$$

   On the band the pitch obeys $\dot{\theta} = -c\theta + s$, a stable first-order system driven by a bounded input, so its steady value is bounded by $|s|/c$.

The switching term overpowers any matched disturbance within a known bound, so the error dynamics on the surface are those the designer chose. The price is a command that switches faster than the actuator can follow; the boundary layer trades exactness for a smooth command.

A worked example

Hopper's pitch from 0.3 rad with $c = 3$, switching gain $\eta = 1.5$ against a disturbance $d = 1.0\sin 3t$ rad/s² (so $D = 1$), a 40 rad/s gimbal servo, a 0.15 rad limit and a 3 rad/s rate limit that is only monitored.

1. Pure switching. The sliding variable is held to |s| ≤ 0.002 and the pitch to 3·10⁻⁴ rad after 5 s: near-perfect rejection. The gimbal rate has an rms of 2.71 rad/s and exceeds the 3 rad/s limit 30% of the time.
2. Boundary layer 0.05. |s| ≤ 0.033 and |θ| ≤ 0.0079 rad; the gimbal rate has an rms of only 0.12 rad/s and never exceeds the limit.
3. Wider layers. φ = 0.2, 0.5, 1.0 give |θ| ≤ 0.030, 0.058, 0.074 rad, and the same gimbal activity (rms 0.12 to 0.13 rad/s).
4. Bound check. The bound $\varphi/c$ is 0.017, 0.067, 0.17 and 0.33 rad for the layers 0.05 to 1.0: always above the measured values, and conservative because the disturbance is not at its worst the whole time.
5. Reading. The pure switch is the most accurate and the only one that violates the actuator. The thinnest layer already removes the chatter and costs an error of 0.008 rad, small against a 0.3 rad start.

import math
import gnckit as g

lam2, b = 0.49, 19.62
c = 3.0                                    # sliding surface s = q + c theta : on it, theta decays with pole -c
D = 1.0                                    # bound on the unknown disturbance acceleration
eta = 1.5                                  # switching gain > D
wservo, dmax = 40.0, 0.15
dt = 0.0005

def run(phi, T=10.0):
    th, q, d = 0.3, 0.0, 0.0
    ths, qs, ss, ds, drate = [], [], [], [], []
    for k in range(int(T / dt)):
        t = k * dt
        s = q + c * th
        sat = 1.0 if phi == 0 else max(-1.0, min(1.0, s / phi))
        sgn = (1.0 if s > 0 else -1.0 if s < 0 else 0.0) if phi == 0 else sat
        u = -(lam2 * th + c * q + eta * sgn) / b
        u = max(-dmax, min(dmax, u))
        dist = D * math.sin(3.0 * t) 
        dd = wservo * (u - d)
        th, q, d = th + dt * q, q + dt * (lam2 * th + b * d + dist), d + dt * dd
        ths.append(th); qs.append(q); ss.append(q + c * th); ds.append(d); drate.append(dd)
    return ths, qs, ss, ds, drate

print('sliding mode, s = q + 3 theta, switching gain 1.5 against a disturbance of 1.0 sin(3 t) rad/s^2, 40 rad/s gimbal servo:')
n0 = int(5.0 / dt)
for phi in (0.0, 0.05, 0.2, 0.5, 1.0):
    th, q, s, d, dr = run(phi)
    tail = slice(n0, None)
    maxth = max(abs(x) for x in th[tail]); maxs = max(abs(x) for x in s[tail])
    rms = math.sqrt(sum(x * x for x in dr[tail]) / len(dr[tail]))
    # fraction of time the gimbal rate exceeds the 3 rad/s limit (the actuator would be rate limited)
    frac = sum(1 for x in dr[tail] if abs(x) > 3.0) / len(dr[tail])
    label = 'sign (phi = 0)' if phi == 0 else f'phi = {phi:4.2f}'
    print(f'  {label:14s}: |theta| <= {maxth:.4f} rad after 5 s, |s| <= {maxs:.3f}, gimbal rate rms {rms:5.2f} rad/s, rate above 3 rad/s {100 * frac:4.0f}% of the time')

sliding mode, s = q + 3 theta, switching gain 1.5 against a disturbance of 1.0 sin(3 t) rad/s^2, 40 rad/s gimbal servo:
  sign (phi = 0): |theta| <= 0.0003 rad after 5 s, |s| <= 0.002, gimbal rate rms  2.71 rad/s, rate above 3 rad/s   30% of the time
  phi = 0.05    : |theta| <= 0.0079 rad after 5 s, |s| <= 0.033, gimbal rate rms  0.12 rad/s, rate above 3 rad/s    0% of the time
  phi = 0.20    : |theta| <= 0.0298 rad after 5 s, |s| <= 0.127, gimbal rate rms  0.13 rad/s, rate above 3 rad/s    0% of the time
  phi = 0.50    : |theta| <= 0.0581 rad after 5 s, |s| <= 0.246, gimbal rate rms  0.13 rad/s, rate above 3 rad/s    0% of the time
  phi = 1.00    : |theta| <= 0.0743 rad after 5 s, |s| <= 0.315, gimbal rate rms  0.12 rad/s, rate above 3 rad/s    0% of the time

In robotics

Sliding-mode control is used where the disturbances are large, bounded and poorly modelled, and where the actuator is fast.

• Power electronics and motor drives. The switches are the actuator, so hard switching costs nothing: the original home of the method.
• Vehicles. Attitude and trajectory loops with large aerodynamic or thrust uncertainty use a boundary layer (smoothed sliding mode) or a higher-order variant. Hopper's thrust-vector actuator, with a rate limit, is on the unfavourable side of this trade.
• Choosing the layer. Pick the width from the rate limit first (the gimbal must follow the command), and read the error bound $\varphi/c$ from it. If the error is too large, a faster actuator or a lower switching gain is what is needed, not a thinner layer.
• Unmatched and dynamic effects. A disturbance that does not enter through the control channel, a servo lag or a delay is not rejected, and can excite the chatter. The 40 rad/s servo above is already enough to make the pure switch violate the rate limit.
• Higher-order sliding modes. Variants that switch the derivative of the control keep finite-time convergence and give a continuous command. They are in the references.

Design the same loop with the tools of module 3 and compare delay margins: the boundary-layer law is a saturated high-gain PD loop, and its margins are those of the linear segment, as the lab M5-L1: Schedule the Gains Through the Burn shows when the schedule is added.

Common misconceptions

Key takeaways

1. A sliding variable such as $s = q + c\theta$ reduces the second-order problem to a first-order decay with pole −c on the surface.
2. A switching gain larger than the matched-disturbance bound brings s to zero in a time at most $|s(0)|/(\eta - D)$.
3. The hard switch rejects the disturbance almost perfectly but chatters: with a 40 rad/s servo Hopper's gimbal exceeds its rate limit 30% of the time.
4. A boundary layer φ removes the chatter and leaves $|s| \le \varphi$, so $|\theta| \lesssim \varphi/c$.
5. With φ = 0.05 the error is 0.008 rad and the gimbal rate rms is 0.12 rad/s.
6. Unmatched disturbances, lag and delay are outside the guarantee; the layer is chosen from the actuator first.

References and further reading

Each source is marked free or paid. Free to read does not mean free to reuse: the course resources page lists every source with its licence.

• K. J. Åström and R. M. Murray, Feedback Systems, 2nd edition, Princeton University Press, 2021 , the nonlinear control sections (Lyapunov analysis of switching). Free to read as the authors' PDF; the book is copyrighted.
• MIT 6.243J Dynamics of Nonlinear Systems (OpenCourseWare) , the Lyapunov background used in the reaching argument. Free; licence CC BY-NC-SA 4.0.
• J.-J. E. Slotine and W. Li, Applied Nonlinear Control, Prentice Hall, 1991, chapter 7 on sliding control. The standard text. Paid.
• Y. Shtessel, C. Edwards, L. Fridman and A. Levant, Sliding Mode Control and Observation, Birkhäuser, 2014, on higher-order sliding modes. Paid.

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What’s next

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This course leaves the following to the pages that already teach them:

• Control barrier functions and safety filters
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• Learned controllers and residual policies