NONLINEAR SYSTEMS
AUTOMATIC CONTROL, 2015
IVO HERMAN
TYPES OF NONLINEAR SYSTEMS
 Nonlinear systems
 Mechanical systems with rotations - sines, cosines
 Electric circuits – diodes, AC drives
} Usually can be dealt with using linearization
 Nonlinear control
 Bang-bang control
 Relay
} Simple control, complicated analysis
 Linear(ized) systems with static nonlinearity
 Saturation, deadzone
 Hysteresis, backlash
 Nonlinear input characteristic
}
Cannot be linearized
SIMPLE EXAMPLE – LIMITED CONTROL EFFORT AND PI
WIND-UP EFFECT
 Solution
 Clamping – saturate the state of the integrator
 Feedback – back-calculation
 Always check your control effort!
 Design the P part to match the saturation.
STABILITY FOR STATIC NONLINEARITIES
 Linear dynamics, static nonlinear function
 Eigenvalue criterion does not work – often system is not linearizable
 Two typical types of behavior:
 Stability/instability – Popov criterion, circle criterion
G(s)
 Limit cycles – harmonic analysis
Ψ(y)
SYSTEMS WITH SECTOR NONLINEARITIES
 Memoryless function 𝑢 = 𝜓(𝑦) is in the sector [𝛼, 𝛽] if
𝛼𝑦 ≤ 𝜓 𝑦 ≤ 𝛽𝑦 for 𝑦 > 0
𝛽𝑦 ≤ 𝜓 𝑦 ≤ 𝛼𝑦 for 𝑦 ≤ 0
u
G(s)
β(y)
ψ(y)
α(y)
y
Ψ(y)
EXAMPLE
 Other examples:

Gain scheduling

Saturation + deadzone
 Does not include

Higher-order polynomials,

Exponentials, …
POPOV CRITERION
 Frequency based – modified frequency response
 Conditions for use:
G(s)

Time invariant nonlinearity 𝜓 in sector [0, 𝛽]

𝜓 0 =0

𝐺 𝑠 = 𝑠𝑛 𝑞

The poles 𝐺(𝑠) are in CLHP – left half plane or on imaginary axis

Marginally stable in singular case
1 𝑝 𝑠
𝑠
with deg 𝑝 𝑠
< deg 𝑞 𝑠
Ψ(y)
 Popov criterion: The closed loop is absolutely stable if for 𝜓 in [0, 𝛽] there exists a constant 𝑞 such that
ℜ 𝐺 𝑗𝜔
− 𝑞𝚥𝜔ℑ 𝐺 𝑗𝜔
≻
1
𝛽
POPOV CRITERION – NOTES
Slope 1/q
 Only for time-invariant nonlinearity
 Strong result – holds for arbitrary nonlinearity in the sector
-1/β
 If sector condition holds on finite interval – finite domain of stability
 Only sufficient condition!
ω
CIRCLE CRITERION
 Suitable for time-varying nonlinearity 𝜓[𝛼, 𝛽]
 Uses disk 𝐷 𝛼, 𝛽 defined over the line segment on the real line between −1/𝛽 and −
1
-1/α
-1/β
𝛼
 Only sufficient condition
 The system is absolutely stable if either of the following holds:
1.
0 < 𝛼 < 𝛽: The Nyquist plot of G(s) does not enter the disk and encircles it m-times. m is the number of ORHP poles
2. 0 = 𝛼 < 𝛽: For stable G(s) the Nyquist plot stays to the right of −1/𝛽
3. 𝛼 < 0 < 𝛽: For stable G(s) the Nyquist plot stays within 𝐷(𝛼, 𝛽)
We cannot tell
Stable for sector [α, β ]
-1/α
G(jω)
-1/β
ω
-1/α
-1/β
G(jω)
ω
DESCRIBING FUNCTION METHOD
 Also known as harmonic balance
 Tests a presence of a limit cycle
 Assumes harmonic signal in the system
 Assumptions

Single nonlinearity, time invariant

System has low-pass properties

Nonlinearity symmetric to the origin
DESCRIBING FUNCTIONS
e(t)
A sin(ωt)
Ψ(e)
u(t)
Ψ(A sin(ωt))
G(s)
y(t)
-A sin(ωt)
 If harmonic oscillations 𝑒 𝑡 = 𝐴𝑠𝑖𝑛(𝜔𝑡), then 𝜓(𝑒) is a periodic signal with Fourier series
𝑎0
𝑢 𝑡 =
+
2
2
𝑎𝑘 =
𝑇
 Then 𝑦 𝑡 =
∞
𝑘=1 𝐺𝑘
𝑏
𝑎
∞
𝑎𝑘 sin 𝑘𝜔𝑡 + 𝑏𝑘 cos 𝑘 𝜔𝑡
𝑘=1
2
𝑢 𝑡 cos 𝑘𝜔𝑡 𝑑𝑡 , 𝑏𝑘 =
𝑇
𝑏
𝑢 𝑡 sin 𝑘𝜔𝑡 𝑑𝑡, 𝑎0 = 0
𝑎
𝑎𝑘 sin 𝑘𝜔𝑡 + 𝜙𝑘 + 𝑏𝑘 cos 𝑘 𝜔𝑡 + 𝜙𝑘
with 𝐺𝑘 = |𝐺(𝑗𝜔𝑘)| and 𝜙𝑘 = arg(𝐺(𝑗𝜔𝑘))
HARMONIC BALANCE
 Due to low-pass property of G(s), 𝑦 𝑡 = 𝐺1 (𝑎1 cos 𝜔𝑡 + 𝜙1 + 𝑏1 sin(𝜔𝑡 + 𝜙1 ))
 With 𝑒 𝑡 = 𝐴 𝑒 𝑗𝜔𝑡 we get 𝑦 𝑡 = 𝐺1 𝑒 𝑗𝜙1 𝑒 𝑗𝜃1 𝑒 𝑗𝜔𝑡 𝑀1 and 𝑀1 =
𝑎12 + 𝑏12 , 𝜃1 = atan
𝑎1
𝑏1
 For harmonic balance e 𝑡 = −𝑦(𝑡), from which follows
𝐴𝑒 𝑗𝜔𝑡 = −𝐺1 𝑒 𝑗𝜙1 𝑒 𝑗𝜃1 𝑒 𝑗𝜔𝑡 𝑀1 ⟹ 1 + 𝐺1 𝑒 𝑗𝜃1
𝑀1 𝑗𝜙
𝑒 1 =0
𝐴
 Harmonic balance equation
1 + 𝐺 𝑗𝜔 𝑁 𝐴, 𝜔 = 0
1
𝐴
with 𝑁 𝐴, 𝜔 = (𝑏1 + 𝑗𝑎1 )
 Graphical test using 1 /𝑁(𝐴, 𝜔)
ω
G(jω)
A
1/N(A,ω)
DESCRIBING FUNCTIONS
 Signum function – Relay 𝑁 𝐴, 𝜔 =
 Saturation – 𝑁(𝐴) =
2𝑘
𝜋
arcsin
𝑀
𝑘𝐴
4𝑀
𝐴𝜋
+
𝑀
𝑘𝐴
1−
𝑀 2
𝑘𝐴
for 𝐴 >
𝑀
𝑘
or 𝑁 𝐴 = 𝑘 for A ≤
ω
G(jω)
𝑀
𝑘
A
1/N(A,ω)
STABILITY OF LIMIT CYCLES
 Modified Nyquist criterion
1/N(A,ω)
1 + 𝐺 𝑗𝜔 𝑁 𝐴, 𝜔 = 0
A
1 1'
 Points 1’, 1’’ can be viewed as critical points to 𝐺 𝑗𝜔 .
 Then apply Nyquist criterion.
2' 2
 1 is unstable: if 1 is perturbed to 1’ (amplitude decreased), 𝐺 𝑗𝜔 does not
encircle 1’ -> system is stable -> amplitude decreases further. If perturbed
to 1’’, system gets unstable and amplitude increases.
 2 is stable: if 2 is perturbed to 2’ (amplitude decreased), 𝐺 𝑗𝜔 encircles 2’
-> system is unstable -> amplitude increases and system gets back to 2. If
perturbed to 2’’, 𝐺 𝑗𝜔 does not encircle 2’’ -> system is stable ->
amplitude decreases back to 2.
ω
G(jω)
1'
2'
REFERENCES
1.
http://control.ee.ethz.ch/~apnoco/Lectures2009/04-Popov%20and%20Circle%20Criterion.pdf
2.
http://stanford.edu/class/ee363/lectures/nonlin-fdbk.pdf
3.
http://www.diee.unica.it/~eusai/didattica/AnalisiSistemi2/Describing_Function_Analysis.pdf