1. 2
General Properties of
Linear and Nonlinear
Systems
Key points
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Linear systems satisfy the properties of superposition and homogeneity. Any
system that does not satisfy these properties is nonlinear.
Linear systems have one equilibrium point at the origin. Nonlinear systems may
have many equilibrium points.
Stability needs to be precisely defined for nonlinear systems.
The principle of superposition does not necessarily hold for forced response for
nonlinear systems.
Nonlinearities can be broadly classified.
Here we consider general properties of linear and nonlinear systems. We consider the
existence and uniqueness of equilibrium points, stability considerations, and the
properties of the forced response. We also present a broad classification of nonlinearities.
Where Do Nonlinearities Come From?
Before we start a discussion of general properties of linear and nonlinear systems, let’s
briefly consider standard sources of nonlinearities.
2. Many physical quantities, such as a vehicle’s velocity, or electrical signals, have an upper
bound. When that upper bound is reached, linearity is lost. The differential equations
governing some systems, such as some thermal, fluidic, or biological systems, are
nonlinear in nature. It is therefore advantageous to consider the nonlinearities directly
while analyzing and designing controllers for such systems. Mechanical systems may be
designed with backlash – this is so a very small signal will produce no output (for
example, in gearboxes). In addition, many mechanical systems are subject to nonlinear
friction. Relays, which a re part of many practical control systems, are inherently
nonlinear. Finally, ferromagnetic cores in electrical machines and transformers are often
described with nonlinear magnetization curves and equations.
Formal Definition of Linear and Nonlinear Systems
Linear systems must verify two properties, superposition and homogeneity.
The principle of superposition states that for two different inputs, x and y, in the domain
of the function f,
f ( x + y ) = f ( x) + f ( y )
The property of homogeneity states that for a given input, x, in the domain of the function
f, and for any real number k,
f (kx) = kf ( x)
Any function that does not satisfy superposition and homogeneity is nonlinear. It is worth
noting that there is no unifying characteristic of nonlinear systems, except for not
satisfying the two above-mentioned properties.
A Brief Reminder on Properties of Linear Time Invariant Systems
Linear Time Invariant (LTI) systems are commonly described by the equation:
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x = Ax + Bu
In this equation, x is the vector of n state variables, u is the control input, and A is a
matrix of size (n-by-n), and B is a vector of appropriate dimensions. The equation
determines the dynamics of the response. It is sometimes called a state-space realization
of the system. We assume that the reader is familiar with basic concepts of system
analysis and controller design for LTI systems.
3. Equilibrium point
An important notion when considering system dynamics is that of equilibrium point.
Equilibrium points are considered for autonomous systems (no control input)
Definition:
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A point x0 in the state space is an equilibrium point of the autonomous system x = Ax if
when the state x reaches x0, it stays at x0 for all future time.
That is, for an LTI system, the equilibrium point is the solutions of the equation:
Ax0 = 0
If A has rank n, then x0 = 0. Otherwise, the solution lies in the null space of A.
Stability
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x = Ax
The system is stable if Re[λi ] < 0
for i ∈ [1, n] .
A more formal statement would talk about the stability of the equilibrium point in the
sense of Lyapunov. There are many kinds of stability (for example, bounded input,
bounded output) and many kinds of tests.
Forced response
&
x = Ax + Bu
The analysis of forced response for linear systems is based on the principle of
superposition and the application of convolution.
For example, consider the sinusoidal response of LTIS.
u = u 0 sin(ω 0 t ) ⇒ y = y 0 sin(ω 0 t + φ 0 )
4. The output sinusoid’s amplitude is different than that of the input and the signal also
exhibits a phase shift. The Bode plot is a graphical representation of these changes. For
LTIS, it is unique and single-valued.
Example of a Bode plot. The horizontal axis is frequency, ω. The vertical axis of the top
plot represents the magnitude of |y/u| (in dB, that is, 20 log of), and the lower plot
represents the phase shift.
As another example, consider the sinusoidal response of LTIS.
If the input into the system is a Gaussian, then the output is also a Gaussian. This is a
useful result.
Nonlinear System Properties
Equilibrium point
Reminder:
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A point x0 in the state space is an equilibrium point of the autonomous system x = f (x )
if when the state x reaches x0, it stays at x0 for all future time.
That is, for a nonlinear system, the equilibrium point is the solutions of the equation:
f ( xe ) = 0
One has to solve n nonlinear algebraic equations in n unknowns. There might be between
0 and infinity solutions.
5. Example: Pendulum
k
g
θ
L
m
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mL2θ& + bθ + − mgL sin θ = 0
If we plot the angular position against the angular speed, we obtain a phase-plane plot.
Example: Mass with Coulomb friction
6. Stability
One must take special care to define what is meant by stability.
• For nonlinear systems, stability is considered about an equilibrium point, in the
sense of Lyapunov or in an input-output sense.
• Initial conditions can affect stability (this is different than for linear systems), and
so can external inputs.
• Finally, it is possible to have limit cycles.
Example:
A limit cycle is a unique, self-excited oscillation. It is also a closed trajectory in the statespace.
In general, a limit cycle is an unwanted feature in a mechanical system, as it causes
fatigue.
Beware: a limit cycle is different from a linear oscillation.
7. Note that in other application domains, for example in communications, a limit cycle
might be a desirable feature.
In summary, be on the lookout for this kind of behavior in nonlinear systems. Remember
that in nonlinear systems, stability, about an equilibrium point:
• Is dependent on initial conditions
• Local vs. global stability is important
• Possibility of limit cycles
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Forced response
The principle of superposition does not hold in general. For example for initial conditions
x0, the system may be stable, but for initial conditions 2x0, the system could be unstable.
11. SUMMARY: General Properties of Linear and Nonlinear Systems
LINEAR SYSTEMS
NONLINEAR SYSTEMS
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x= Ax
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x= f(x)
EQUILIBIUM POINTS
UNIQUE
MULTIPLE
A point where the system can stay forever
without moving.
If A has rank n, then xe=0, otherwise the
solution lies in the null space of A.
f(xe)=0
n nonlinear equations in n unknowns
0 → +∞ solutions
ESCAPE TIME
x → +∞ as t → +∞
The state can go to infinity in finite time.
STABILITY
The equilibrium point is stable if all
eigenvalues of A have negative real part,
regardless of initial conditions.
About an equilibrium point:
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Dependent on IC
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Local vs. Global stability
important
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Possibility of limit cycles
LIMIT CYCLES
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x= Ax + Bu
FORCED RESPONSE
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The principle of superposition
holds.
I/O stability → bounded input,
bounded output
Sinusoidal input → sinusoidal
output of same frequency
A unique, self-excited
oscillation
A closed trajectory in the state
space
Independent of IC
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x= f(x,u)
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The principle of superposition
does not hold in general.
The I/O ratio is not unique in
general, may also not be single
valued.
CHAOS
Complicated steady-state behavior, may
exhibit randomness despite the
deterministic nature of the system.
