SoMMME
Department of Mechanical Engineering
EMG 2410: Control Engineering II
Nonlinear Systems Control
James K. Kimotho
SoMMME
Department of Mechanical Engineering
Introduction
Nonlinear control theory is the area of control theory which
deals with systems that are nonlinear, time-variant, or both.
Properties of nonlinear systems:
– They do not follow the principle of superposition (linearity and
homogeneity).
– They may have multiple isolated equilibrium points.
– They may exhibit properties such as limit cycle, bifurcation, chaos.
– Finite escape time: Solutions of nonlinear systems may not exist
for all times.
– For a sinusoidal input, the output signal may contain many
harmonics and sub-harmonics with various amplitudes and phase
differences (a linear system's output will only contain the sinusoid
at the output)
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EMG 2410: Control Engineering II / 11/11/2022
SoMMME
Department of Mechanical Engineering
Linearization of Nonlinear Functions
The behavior of a nonlinear system, described by 𝑦 = 𝑓(𝑥) , in
the vicinity of a given operating point, 𝑥 = 𝑥0 , can be
approximated by plotting a tangent line to the graph of 𝑓(𝑥) at
that point.
Analytically, linearization of a nonlinear function involves firstorder Taylor series expansion about the operative point.
Let 𝛿𝑥 = 𝑥 − 𝑥0 represent the variation from the operating
point; then the Taylor series of a function of single variable is
written as:
The resulting first order model is described by:
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EMG 2410: Control Engineering II / 11/11/2022
SoMMME
Department of Mechanical Engineering
Linearization of State Variable Models
Example 1
The model of a simple pendulum is described by the dynamic
equation:
where 𝜃(𝑡) is the pendulum angle, 𝑇(𝑡) is the applied
torque; 𝑚, 𝑙 represent the mass and the length of the
pendulum, and 𝑔 is the gravitational constant.
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EMG 2410: Control Engineering II / 11/11/2022
SoMMME
Department of Mechanical Engineering
Linearization of Nonlinear Functions
Example 1
By using (𝜃, 𝜔) as the state variables for the pendulum, the
nonlinear model is expressed as:
The Jacobian matrix for the simple pendulum is expressed as:
Two equilibrium points in the case of a simple pendulum can
be identified: 𝜃𝑒 = 0°, 180°.
– 𝜃𝑒 = 0°. – For disturbance rejection
– 𝜃𝑒 = 180°. – For balancing.
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EMG 2410: Control Engineering II / 11/11/2022
SoMMME
Department of Mechanical Engineering
Linearization of Nonlinear Functions
Example 1
The linearized models defined at the equilibrium points are
given as:
The output equation in both cases is given as:
Applications
– Walking robots
– Segway
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EMG 2410: Control Engineering II / 11/11/2022
SoMMME
Department of Mechanical Engineering
Linearization of Nonlinear Functions
Example 2
Linearize the nonlinear equation, 𝑧 = 2𝑥𝑦, in the region 4 ≤ 𝑥 ≤
6, 8 ≤ 𝑦 ≤ 10. Find the error if the linearization is used to
calculate the value of z when 𝑥 = 4, 𝑦 = 8.
Using Taylors Series expansion of the non-linear equation and
neglecting higher order terms, we have
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EMG 2410: Control Engineering II / 11/11/2022
SoMMME
Department of Mechanical Engineering
Lyapunov Stability
Lyapunov’s stability analysis technique is very common and
dominant.
The main deficiency, which severely limits its utilization is the
complication linked with the development of the Lyapunov
function which is needed by the technique.
The system dynamics must be described by a state-space
model. It is a description in terms of a set of first-order
differential equations.
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EMG 2410: Control Engineering II / 11/11/2022
SoMMME
Department of Mechanical Engineering
Lyapunov Stability
Theorem: Consider a continuously differentiable function 𝑉(𝑥)
such that 𝑉(𝑥) > 0 for all 𝑥 ≠ 0 and 𝑉(0) = 0 (𝑉 is positive
definite).
We then have the following conditions for the various notions
of stability.
ሶ
1) If 𝑉(𝑥)
≤ 0 for all 𝑥, then 𝑥 = 0 is stable.
ሶ
2) In addition, if 𝑉(𝑥)
< 0 for all 𝑥 ≠ 0, then 𝑥 = 0 is
asymptotically stable.
3) In addition to (1) and (2) above, if 𝑉 is radially unbounded,
i.e., 𝑥 → ∞ ⇒ 𝑉(𝑥) → ∞ , then 𝑥 = 0 is globally
asymptotically stable (GAS).
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EMG 2410: Control Engineering II / 11/11/2022
SoMMME
Department of Mechanical Engineering
Lyapunov Stability
𝑉 is called a Lyapunov function
𝑉 must be positive definite.
Stability means that the Lyapunov function decreases along
the trajectory of 𝑥(𝑡).
Case (1) means that 𝑉ሶ is negative semi-definite
Case (2) means that 𝑉ሶ is negative definite
This is only sufficient condition! Constructing Lyapunov
functions is basically by trial-and-error.
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EMG 2410: Control Engineering II / 11/11/2022
SoMMME
Department of Mechanical Engineering
Lyapunov Stability
Example: Nonlinear Spring-Mass-Damper System
A spring-mass-damper system for which the damping force is
proportional to the third power of the velocity is described by
the differential equation
Let mass position 𝑦 and velocity 𝑦ሶ be chosen as state
variables:
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EMG 2410: Control Engineering II / 11/11/2022
SoMMME
Department of Mechanical Engineering
Lyapunov Stability
𝑉ሶ is negative semi-definite, so the system is globally stable.
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EMG 2410: Control Engineering II / 11/11/2022
SoMMME
Department of Mechanical Engineering
Control Design of Non-Linear Systems
Gain scheduling
Adaptive control
Feedback linearization
Lyapunov based methods
Lyapunov Redesign
Back-stepping
Sliding mode control
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EMG 2410: Control Engineering II / 11/11/2022
SoMMME
Department of Mechanical Engineering
Gain Scheduling
It is an approach to control of non-linear systems that uses a
family of linear controllers, each of which provides satisfactory
control for a different operating point of the system
One or more observable variables, called the scheduling
variables, are used to determine what operating region the
system is currently in and to enable the appropriate linear
controller
Aircraft example:
– flight control system, the altitude and Mach number might be the
scheduling variables, with different linear controller parameters
available for various combinations of these two variables.
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EMG 2410: Control Engineering II / 11/11/2022
SoMMME
Department of Mechanical Engineering
Gain Scheduling
Example of scheduling variables
– Production rate
– Machine speed, e.g., DVD player
– Mach number and dynamic pressure
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EMG 2410: Control Engineering II / 11/11/2022
SoMMME
Department of Mechanical Engineering
Adaptive Control
Modifying the controller parameters to compensate slow time–
varying behavior or uncertainties
For example, as an aircraft flies, its mass will slowly decrease
as a result of fuel consumption; we need a control law that
adapts itself to such changing conditions
Adaptive control does not need a priori information about the
bounds on these uncertain or time varying parameters
Robust control guarantees that if the changes are within given
bounds the control law need not be changed while adaptive
control is precisely concerned with control law changes
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EMG 2410: Control Engineering II / 11/11/2022
SoMMME
Department of Mechanical Engineering
Adaptive Control
The main feature of the adaptive controller is that it achieves
the control objective (regulation) despite large-scale
uncertainty
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EMG 2410: Control Engineering II / 11/11/2022
SoMMME
Department of Mechanical Engineering
Thank You
for
Your Kind Attention!
Dr.-Ing. James Kuria Kimotho
jkuria@eng.jkuat.ac.ke
Tel: +254791872389
Department of Mechanical Engineering
JKUAT
P.O. Box 62000-00200
Nairobi