SigSys General Properties

8/9/2019 SigSys General Properties

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2 General Properties of

Linear and Nonlinear

Systems

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, lets

briefly consider standard sources of nonlinearities.

Key points

Linear systems satisfy the properties of superposition and homogeneity. Anysystem that does not satisfy these properties is nonlinear.

Linear systems have one equilibrium point at the origin. Nonlinear systems mayhave many equilibrium points.

Stability needs to be precisely defined for nonlinear systems.

The principle of superposition does not necessarily hold for forced response fornonlinear systems.

Nonlinearities can be broadly classified.


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Many physical quantities, such as a vehicles velocity, or electrical signals, have an upper

bound. When that upper bound is reached, linearity is lost. The differential equationsgoverning 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 (forexample, 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 oftendescribed 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 domainof the function f,

)()()( yfxfyxf +=+

The property of homogeneity states that for a given input, x, in the domain of the function

f, and for any real number k,

)()( xkfkxf =

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 notsatisfying 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:

BuAxx +=&

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 equationdetermines 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.


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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:

A point x0in the state space is an equilibrium point of the autonomous system Axx =& 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:

00 =Ax

If A has rank n, then x0= 0. Otherwise, the solution lies in the null space of A.

Stability

Axx =&

The system is stable if ],1[0]Re[ nifori


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The output sinusoids 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. ForLTIS, it is unique and single-valued.

Example of a Bode plot. The horizontal axis is frequency, . The vertical axis of the topplot 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:

A point x0in the state space is an equilibrium point of the autonomous system )(xfx =&

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:

0)( =exf

One has to solve n nonlinear algebraic equations in n unknowns. There might be between

0 and infinity solutions.


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Example: Pendulum

m

Lg

k

0sin2 =++ mgLbmL &&&

If we plot the angular position against the angular speed, we obtain a phase-plane plot.

Example: Mass with Coulomb friction


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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 state-

space.

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.


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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. Rememberthat in nonlinear systems, stability, about an equilibrium point:

Is dependent on initial conditions

Local vs. global stability is important

Possibility of limit cycles

Forced response

The principle of superposition does not hold in general. For example for initial conditionsx0, the system may be stable, but for initial conditions 2x0, the system could be unstable.


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Classification of Nonlinearities

Single-valued, time invariant

Memory or hysteresis

Example:


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Single-input vs. multiple input nonlinearities


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SUMMARY: General Properties of Linear and Nonlinear Systems

LINEAR SYSTEMS

Axx=&

NONLINEAR SYSTEMS

)(xfx=&

EQUILIBIUM POINTS

A point where the system can stay forever

without moving.

UNIQUE

If A has rank n, then xe=0, otherwise the

solution lies in the null space of A.

MULTIPLE

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:

Dependent on IC

Local vs. Global stabilityimportant

Possibility of limit cycles

LIMIT CYCLES

A unique, self-excitedoscillation

A closed trajectory in the statespace

Independent of IC

FORCED RESPONSE BuAxx +=&

The principle of superpositionholds.

I/O stabilitybounded input,bounded output

Sinusoidal inputsinusoidaloutput of same frequency

),( uxfx=&

The principle of superpositiondoes not hold in general.

The I/O ratio is not unique ingeneral, may also not be single

valued.

CHAOS

Complicated steady-state behavior, may

exhibit randomness despite the

deterministic nature of the system.

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