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, let’s briefly consider standard sources of nonlinearities.

Key points

• 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.

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,

)()()( 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 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:

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 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.

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 x0 in 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 ∈<λ . 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

BuAxx +=& 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.

)sin()sin( 00000 φωω +=⇒= tyytuu

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: A point x0 in 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.

Example: Pendulum

m

Lθg

k

0sin2 =−++ θθθ mgLbmL &&& If we plot the angular position against the angular speed, we obtain a phase-plane plot.

Example: Mass with Coulomb friction

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.

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 •

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.

Classification of Nonlinearities Single-valued, time invariant

“Memory” or hysteresis

Example:

Single-input vs. multiple input nonlinearities

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 stability

important • Possibility of limit cycles

LIMIT CYCLES

• A unique, self-excited

oscillation • A closed trajectory in the state

space • Independent of IC

FORCED RESPONSE

BuAxx +=&

• The principle of superposition

holds. • I/O stability → bounded input,

bounded output • Sinusoidal input → sinusoidal

output of same frequency

),( uxfx=&

• 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.