TUTORIAL
Sliding-Mode Control ​
This example demonstrates how to model a system with a discrete-time sliding-mode controller (SMC) in Dyad. The system consists of a second-order plant with a disturbance and a SuperTwistingSMC controller. The controller is implemented as a discrete-time system, and the plant is modeled as a continuous-time system.
Background ​
When designing an SMC controller, we must choose a switching function
Since the dynamics in this example is of relative degree
which yields the switching variable
Implementation ​
The sliding variable is computed from the sampled signals using block diagram components. The plant position is sampled (with AD effects for realism), filtered through an ExponentialFilter, and differentiated using a DiscreteDerivative to estimate velocity. The position and velocity errors relative to the reference signals are summed to form the sliding variable SuperTwistingSMC controller. Download as a Dyad projectSMC.zipOpen in Dyad Studio
dyad
"""
Test component that validates the SuperTwistingSMC sliding mode controller.
A second-order plant tracks a sinusoidal reference `qr` under a disturbance `d` that
consists of a constant and two sinusoids. The plant position is sampled and differentiated
to estimate velocity. The sliding variable `s = (xd - qdr) + (x - qr)` is computed from the
sampled signals and is the input of the super-twisting controller. The controller output is held by a ZOH
and summed with the disturbance before driving the plant.
"""
test component TestSlidingModeControl
"Plant"
secondorder = BlockComponents.Continuous.SecondOrder() {^secondorder}
"Velocity estimation via discrete derivative of sampled position"
dd_xd = DiscreteComponents.DiscreteDerivative(initialization = DiscreteComponents.InitialCondition.InitialOutput(y0 = 0.01)) {^dd_xd}
"Position reference qr"
sine_ref = BlockComponents.Sources.Sine(amplitude = 1, frequency = 2 / (2 * pi)) {^sine_ref}
"Velocity reference qdr, the time derivative of qr"
cosine_refd = BlockComponents.Sources.Cosine(amplitude = 2, frequency = 2 / (2 * pi)) {^cosine_refd}
ps_qr = DiscreteComponents.Sampler() {^ps_qr}
ps_qdr = DiscreteComponents.Sampler() {^ps_qdr}
zoh = DiscreteComponents.ZeroOrderHold() {^zoh}
"Constant component of the disturbance"
constant_dist = BlockComponents.Sources.Constant(k = 2) {^constant_dist}
"First sinusoidal component of the disturbance"
sine_dist1 = BlockComponents.Sources.Sine(amplitude = 2, frequency = 3 / (2 * pi)) {^sine_dist1}
"Second sinusoidal component of the disturbance"
sine_dist2 = BlockComponents.Sources.Sine(amplitude = 1, frequency = 5 / (2 * pi)) {^sine_dist2}
"Position error x - qr"
err_pos = BlockComponents.Math.Add(k2 = -1) {^err_pos}
"Velocity error xd_est - qdr"
err_vel = BlockComponents.Math.Add(k2 = -1) {^err_vel}
"Sliding variable s, the sum of the position and velocity errors"
sliding_surface = BlockComponents.Math.Add() {^sliding_surface}
"Control + disturbance → plant"
add_ctrl = BlockComponents.Math.Add() {^add_ctrl}
add3 = BlockComponents.Math.Add3() {^add3}
smc = DiscreteComponents.SuperTwistingSMC(k = 50) {^smc}
periodicclock = DiscreteComponents.PeriodicClock(dt = 0.01) {^periodicclock}
samplewithadeffects = DiscreteComponents.SampleWithADEffects(y_min = -3, y_max = 3, bits = 12, sigma = 0.01) {^samplewithadeffects}
exponentialfilter = DiscreteComponents.ExponentialFilter(a = 0.2, initialization = DiscreteComponents.InitialCondition.InitialOutput(y0 = -1)) {^exponentialfilter}
variable disturbance::Real
relations
initial secondorder.x = -1
initial secondorder.xd = 0
# Sample references
connect(sine_ref.y, ps_qr.u) {^id20}
connect(cosine_refd.y, ps_qdr.u) {^id21}
connect(zoh.y, add_ctrl.u1) {^id22}
connect(add_ctrl.y, secondorder.u) {^id23}
connect(dd_xd.y, err_vel.u1) {^id24}
connect(ps_qdr.y, err_vel.u2) {^id25}
connect(err_pos.y, sliding_surface.u1) {^id26}
connect(ps_qr.y, err_pos.u2) {^id27}
connect(sine_dist1.y, add3.u2) {^id28}
connect(add3.y, add_ctrl.u2) {^id29}
connect(err_vel.y, sliding_surface.u2) {^id30}
connect(smc.s, sliding_surface.y) {^id31}
connect(smc.y, zoh.u) {^id32}
connect(constant_dist.y, add3.u1) {^id33}
connect(sine_dist2.y, add3.u3) {^id34}
connect(samplewithadeffects.u, secondorder.y) {^id35}
connect(samplewithadeffects.y, periodicclock.y, exponentialfilter.u) {^id36}
connect(exponentialfilter.y, dd_xd.u, err_pos.u1) {^id37}
metadata {
"Dyad": {
"tests": {
"case1": {"stop": 6, "expect": {"signals": ["secondorder.y", "smc.y"]}}
},
"doc": {"behavior": false}
},
"_links": {
"secondorder": {
"Dyad": {
"placement": {
"diagram": {"iconName": "default", "x1": 1020, "y1": 660, "x2": 1120, "y2": 760, "rot": 0}
},
"tags": []
}
},
"dd_xd": {
"Dyad": {
"placement": {
"diagram": {"iconName": "default", "x1": 140, "y1": 850, "x2": 40, "y2": 950, "rot": 90}
},
"tags": []
}
},
"sine_ref": {
"Dyad": {
"placement": {
"diagram": {"iconName": "default", "x1": -200, "y1": 480, "x2": -100, "y2": 580, "rot": 0}
},
"tags": []
}
},
"cosine_refd": {
"Dyad": {
"placement": {
"diagram": {"iconName": "default", "x1": -200, "y1": 690, "x2": -100, "y2": 790, "rot": 0}
},
"tags": []
}
},
"ps_qr": {
"Dyad": {
"placement": {
"diagram": {"iconName": "default", "x1": -70, "y1": 480, "x2": 30, "y2": 580, "rot": 0}
},
"tags": []
}
},
"ps_qdr": {
"Dyad": {
"placement": {
"diagram": {"iconName": "default", "x1": -70, "y1": 690, "x2": 30, "y2": 790, "rot": 0}
},
"tags": []
}
},
"zoh": {
"Dyad": {
"placement": {
"diagram": {"iconName": "default", "x1": 690, "y1": 690, "x2": 790, "y2": 790, "rot": 0}
},
"tags": []
}
},
"constant_dist": {
"Dyad": {
"placement": {
"diagram": {"iconName": "default", "x1": 520, "y1": 280, "x2": 620, "y2": 380, "rot": 0}
},
"tags": []
}
},
"sine_dist1": {
"Dyad": {
"placement": {
"diagram": {"iconName": "default", "x1": 520, "y1": 410, "x2": 620, "y2": 510, "rot": 0}
},
"tags": []
}
},
"sine_dist2": {
"Dyad": {
"placement": {
"diagram": {"iconName": "default", "x1": 520, "y1": 540, "x2": 620, "y2": 640, "rot": 0}
},
"tags": []
}
},
"err_pos": {
"Dyad": {
"placement": {
"diagram": {"iconName": "default", "x1": 150, "y1": 660, "x2": 250, "y2": 560, "rot": 0}
},
"tags": []
}
},
"err_vel": {
"Dyad": {
"placement": {
"diagram": {"iconName": "default", "x1": 150, "y1": 820, "x2": 250, "y2": 720, "rot": 0}
},
"tags": []
}
},
"sliding_surface": {
"Dyad": {
"placement": {
"diagram": {"iconName": "default", "x1": 360, "y1": 690, "x2": 460, "y2": 790, "rot": 0}
},
"tags": []
}
},
"add_ctrl": {
"Dyad": {
"placement": {
"diagram": {"iconName": "default", "x1": 880, "y1": 760, "x2": 980, "y2": 660, "rot": 0}
},
"tags": []
}
},
"add3": {
"Dyad": {
"placement": {
"diagram": {"iconName": "default", "x1": 690, "y1": 410, "x2": 790, "y2": 510, "rot": 0}
},
"tags": []
}
},
"smc": {
"Dyad": {
"placement": {
"diagram": {"iconName": "default", "x1": 520, "y1": 690, "x2": 620, "y2": 790, "rot": 0}
},
"tags": []
}
},
"periodicclock": {
"Dyad": {
"placement": {
"diagram": {"iconName": "default", "x1": 880, "y1": 800, "x2": 980, "y2": 900, "rot": 0}
},
"tags": []
}
},
"samplewithadeffects": {
"Dyad": {
"placement": {
"diagram": {"iconName": "default", "x1": 980, "y1": 940, "x2": 880, "y2": 1040, "rot": 0}
},
"tags": []
}
},
"exponentialfilter": {
"Dyad": {
"placement": {
"diagram": {"iconName": "default", "x1": 530, "y1": 940, "x2": 430, "y2": 1040, "rot": 0}
},
"tags": []
}
},
"id20": {"Dyad": {"edges": [{"S": 1, "M": [], "E": 2}], "renderStyle": "standard"}},
"id21": {"Dyad": {"edges": [{"S": 1, "M": [], "E": 2}], "renderStyle": "standard"}},
"id22": {"Dyad": {"edges": [{"S": 1, "M": [], "E": 2}], "renderStyle": "standard"}},
"id23": {"Dyad": {"edges": [{"S": 1, "M": [], "E": 2}], "renderStyle": "standard"}},
"id24": {
"Dyad": {
"edges": [{"S": 1, "M": [{"x": 90, "y": 800}], "E": 2}],
"renderStyle": "standard"
}
},
"id25": {"Dyad": {"edges": [{"S": 1, "M": [], "E": 2}], "renderStyle": "standard"}},
"id26": {
"Dyad": {
"edges": [{"S": 1, "M": [{"x": 280, "y": 610}, {"x": 280, "y": 710}], "E": 2}],
"renderStyle": "standard"
}
},
"id27": {
"Dyad": {
"edges": [{"S": 1, "M": [{"x": 68, "y": 530}, {"x": 68, "y": 580}], "E": 2}],
"renderStyle": "standard"
}
},
"id28": {"Dyad": {"edges": [{"S": 1, "M": [], "E": 2}], "renderStyle": "standard"}},
"id29": {
"Dyad": {
"edges": [{"S": 1, "M": [{"x": 830, "y": 460}, {"x": 830, "y": 680}], "E": 2}],
"renderStyle": "standard"
}
},
"id30": {"Dyad": {"renderStyle": "standard", "edges": [{"S": 1, "E": 2, "M": []}]}},
"id31": {"Dyad": {"edges": [{"S": 1, "M": [], "E": 2}], "renderStyle": "standard"}},
"id32": {"Dyad": {"edges": [{"S": 1, "M": [], "E": 2}], "renderStyle": "standard"}},
"id33": {
"Dyad": {
"renderStyle": "standard",
"edges": [{"S": 1, "M": [{"x": 657.5, "y": 330}, {"x": 657.5, "y": 430}], "E": 2}]
}
},
"id34": {
"Dyad": {
"renderStyle": "standard",
"edges": [{"S": 1, "M": [{"x": 657.5, "y": 590}, {"x": 657.5, "y": 490}], "E": 2}]
}
},
"id35": {
"Dyad": {
"edges": [{"S": 1, "M": [{"x": 1160, "y": 990}, {"x": 1160, "y": 710}], "E": 2}],
"renderStyle": "standard"
}
},
"id36": {
"Dyad": {
"edges": [
{"S": 1, "M": [], "E": -1},
{"S": -1, "M": [{"x": 800, "y": 990}, {"x": 800, "y": 850}], "E": 2},
{"S": 3, "M": [], "E": -1}
],
"junctions": [{"x": 800, "y": 990}],
"renderStyle": "standard"
}
},
"id37": {
"Dyad": {
"edges": [
{"S": 1, "M": [], "E": -1},
{"S": -1, "M": [{"x": 90, "y": 990}], "E": 2},
{"S": 3, "M": [{"x": -250, "y": 640}, {"x": -250, "y": 990}], "E": -1}
],
"junctions": [{"x": 90, "y": 990}],
"renderStyle": "standard"
}
}
}
}
end
analysis SlidingModeControlDemoAnalysis
extends TransientAnalysis(stop = 10.0)
model = TestSlidingModeControl()
endjulia
using Plots
result = SlidingModeControlDemoAnalysis()
using DyadInterface
model = artifacts(result, :SimplifiedSystem)
figy = plot(result; idxs = model.secondorder.y, label = "Plant position")
plot!(result; idxs = model.sine_ref.y, label = "Reference")
figu = plot(result; idxs = model.smc.y, label = "Control signal")
plot!(result; idxs = model.add3.y, label = "Disturbance")
plot(figy, figu, layout = (2, 1))
The simulation indicates that the controller is able to track the reference signal despite the presence of the disturbance. The control signal exhibits high-frequency chattering, a common characteristic of sliding-mode controllers. The use of SampleWithADEffects and ExponentialFilter adds realistic measurement imperfections: 12-bit quantization, small Gaussian noise, and low-pass filtering of the position signal.