The behavior of nonlinear systems and the control of them is of interest for a variety of reasons. Linear control methods are very well developed and are applicable in most cases but are still superseded by nonlinear control methods in many areas.

  • Linear control assumes small ranges of motion: The most common reason for using nonlinear control methods is the significant nonlinearity of the system thus that the linear methods no longer produce an effective or stable result.
  • Discontinuities are not handled by Linear Methods: Most nonlinear methods do not allow for effective control of a discontinuous system. The sudden addition of a large load or backlash in mechanical components can be unhandled by linear methods.
  • Uncertancies in model Parameters: Many applications can have slowly changing or rapidly changing parameters that effect the behavior of the system, temperature changes in an oil line will effect the hydraulic behavior, and the llifting of an unknown load will have to be adapted to for controllers to function effectively.
  • Nonlinear control methods can be simpler: Nonlinear control designs can be much easier to modify and adapt to other applications as they are mostly model dependant and can be adapted easily to fit a similar but different application.

Nonlinear System Behavior

Nonlinearities come in two forms, intentional and inerrant nonlinearities. Inerrant nonlinearities are those that naturally come from the system buildup, specifically the geometry, motion, and boundary conditions. Intentional nonlinearities are those introduced to the system for a purpose, be it improved system behavior, control, etc.

Linear System Behavior

Linear Systems are where most work in controls happens, most systems that are controlled can be effectively controlled using linear approximations of the nonlinear system or assumptions that make the system behave linearly.

The typical LTI system takes the form:

$$\dot{\pmb{x}}=\pmb{A}\pmb{x}$$

Where $\pmb{x}$ is the state vector and $\pmb{A}$ is the linear Jacobian matrix.

A linear system will have a unique equilibrium point if $\pmb{A}$ is non-singular and that point will be stable if all eigenvalues of $\pmb{A}$ have negative real parts. The behavior of a linear system can thus be solved analytically.

If an external input vector $\pmb{u}$, is introduced, the system can be represented as:

$$\dot{\pmb{x}}=\pmb{A}\pmb{X}+\pmb{B}\pmb{u}$$

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Nonlinear System Behavior

In nonlinear systems, the behavior when the magnitude of system inputs or states are changed differs compared to an LTI system. If an input is doubled for an LTI system, the result should be doubled. however in a nonlinear system, the resulting value could be higher or lower depending on the system. It's also possible for the direction of the result to change, and for the rate of convergence as well.

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Phase Plane Analysis

Phase plane analysis is a graphical study of time invariant systems. It is typically used for second order systems however could theoretically be used for as high order as can be visually represented.

What Are Phase Portraits

For analysis of phase planes, a model is formulated as:

$$\dot x_1 = f_1(x_1,x_2)\\ \dot x_2 = f_2(x_1,x_2) $$

where $x_1$ and $x_2$ are system states and the functions $f_1$ and $f_2$ are the linear or nonlinear autonomous describing equations of the system. The phase-plane is the plane made up of all values of $\pmb{x}$ where $x_1$ and $x_2$ are on each axis.

From any position on the phase plane where $x(t=0)=x_0$, a solution can be found as $t\rightarrow \infty$ which can be represented as a curve on the phase plane.

Mass Spring Damper System

A typical LTI mass spring damper system can be represented by:

$$\dot x_1 = x_2\\     \dot x_2 = \frac{-K\cdot x_1-B\cdot x_2}{M} $$

With $M=1$, $K=1$, and $B=0$, the representing no damping, the phase portrait can be seen to be:

As there is no damping, the states do not converge and for any initial condition, the states will oscillate in displacement and velocity for all time.

If the damping parameter $B$ is increased to $1$, the phase portrait would thus be:

As the damping reduces the energy in the system, the states will converge to $x_1=x_2=0$ as time approaches infinity.

If the damping factor is reduced to $0.1$, the phase portrait will oscillate about the equilibrium point. The number of rotations about the origin represent the number of oscillatory periods of the system before settling.

It is important to note that though the phase portrait shows how the system behaves over time, it does not say anything about how fast the system converges. finite time is not considered in the analysis.

Singular Points

The equilibrium points in the phase plane where $\dot x_1=0$ and $\dot x_2=0$ are referred to as singular points.

The behavior of the states in the region near the singular points determines the type of the equilibrium point. If the states diverge from the point it is characterized as unstable if the states converge, it is characterized as stable. There are additional sub-types such as saddle points however they will not be discussed here.

To demonstrate singular points, a model of a damped pendulum is used.

$$\dot \theta =\dot x_1= x_2\\ \ddot \theta =\dot x_2= \frac{MgLsin(\theta)-B\cdot dx}{I} $$

In this case, the values are set as: $M = 1, \ g=-9.81,\ L=1,\ B=1,\ I=1$.

In this phase portrait, there are two singular points shown, one at $x_1=0, \ x_2=0$ and one at $x_1=\pi, \ x_2=0$. The singular point at $0$ is considered to be stable as all states in the region converge to the point. This point corresponds to the upward positioning of the pendulum. The singular point at $\pi$ is considered to be unstable as the states diverge in the region. This corresponds to the upright position of the pendulum which is obviously unstable.

The singular points repeat with convergent points at $x_1=n\cdot 2\pi$ and divergent points at $x_2=n\cdot 2\pi+\pi$ where $n$ is all integers. This behavior is expected with the periodic nature of the sine function.

Drawing Phase Portraits

In order to construct phase portraits, a couple methods are available for drawing by hand or with software.

Pplane8

The pplane8 tool allows for matlab to generate phase portraits easily and additionally plot trajectories based on some initial condition.

Depending on the version of matlab used, there may be corrections needed to some methods used by the pplane8 function.

https://www.mathworks.com/matlabcentral/answers/uploaded_files/69954/pplane8.m

Isoclines

The method of isoclines can be used to graphically generate phase portraits by hand from a model of a system.

An isocline is defined as a curve of the points with a given tangent slope. As such, the isocline for slope $\alpha$ would be defined as:

$$\frac{dx_2}{dx_1}=\frac{f_2(x_1,x_2)}{f_1(x_1,x_2)}=\alpha$$

Using this knowledge, the curve for a given slope can be calculated and drawn on to a plot. With many different values of $\alpha$,  a relatively accurate representation can be generated and, through interpolation, trajectories plotted.

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Phase Plane Examples

Linear Systems

Nonlinear Systems

Limit Cycles

Sources:

[1] Slotine, J. and Li, W. (1991). Applied nonlinear control. Upper Saddle River, NJ: Prentice Hall Inc.