A practical method is developed for limit-cycle predictions in the nonlinear multivariable feedback control systems with large transportation lags. All nonlinear elements considered are linear independent. It needs only to check maximal or minimal frequency points of root loci of equivalent gains for finding a stable limit-cycle. This reduces the computation effort dramatically. The information for stable limit-cycle checking can be shown in the parameter plane also. Sinusoidal input describing functions with fundamental components are used to find equivalent gains of nonlinearities. The proposed method is illustrated by a simple numerical example and applied to one

In current literature, for nonlinear multivariable systems the Nyquist, inverse Nyquist and numerical optimization, methods are usually used to predict the existence of limit cycles. These methods are based upon the graphical or numerical solutions of the linearized harmonic-balance equations [

In general, real and imaginary parts of the characteristic equation are used as two simultaneous equations to find the solution of the limit cycle for single-input single-output (SISO) nonlinear feedback control systems [

However, nonlinearities in multivariable feedback systems are usually independent of each other. Therefore, infinite number of solutions of limit cycles satisfy the characteristic equation for phase shifts are not in the characteristic equation and the number of parameters to be found is always greater than two. The number of parameters to be found are

The proposed method is based on the parameter-plane analyses method [

This merit of the work rather than the previous work [

This paper is organized as follows: (1) in Section

Consider the nonlinear multivariable feedback system shown in Figure

A general block diagram of nonlinear multivariable feedback control systems.

Note that nonlinearities in the off-diagonal and on-diagonal terms are dependent for they have same input signal. For instance, nonlinearities (

For illustration, assume that a

Block diagram of a

A useful equivalent gain expression of nonlinearity is the sinusoidal-input describing function (SIDF)

The above statement will be illustrated by a simple numerical example. Consider a

Root loci of limit cycles in the parameter plane.

The criteria for checking existence of a limit cycle will be explained by use of the illustrating example discussed above, and applied to several

Every point on the root loci evaluated by (

A limit cycle may exist only if the values of

If the root loci separate the stable and unstable regions, then a stable limit cycle may exist at the root loci. The reason is that the system will become stable (unstable) when amplitude

The stable and unstable regions are identified by the root loci found for ^{+} represents the solution found with ^{−} represents the solution found with ^{+}/0^{−} root-locus classifies the stability of the system in the parameter plane.

The descriptions of a stable limit cycle can be expressed mathematically by the following equation:

A stable limit cycle exists only for phase angles found by (

A stable limit cycle exists only for magnitudes found by (

The unique solution of a stable limit cycle is at the unique frequency point of the root locus; that is, the solutions of (

From the root loci shown in Figure

If the solution satisfies all six criteria for a stable limit cycle, then a stable limit cycle will exist. Table

Calculated results of a stable (Point

Point | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

0.5719 | 0.5691 | 0.7888 | 1.904 | 1.889 | −0.185 | -0.1849 | −70.39° | −70.39° | 0.93 | |

1.175 | 1.179 | +0.788 | +0.762 | −70.39° | −70.39° | 0.92 | ||||

0.6366 | 0.5313 | 0.8002 | 1.414 | 1.136 | 0 | +1.149 | −88.49° | −88.49° | 1.12 | |

1.414 | 2.112 | 0 | −0.179 | −88.49° | −88.49° | 0.61 |

Time responses of the illustrating example.

If the loop gain

The gains

Methods | Gain |
---|---|

Proposed method | 1.7915 |

Aizerman conjecture | 1.79 |

Hirsch plot | 1.25 |

Mee plot | 1.50 |

Digital simulation | 1.787 |

Root locus analyses for

The procedures for finding a stable limit cycle have been developed and illustrated by a simple example with two “single-valued nonlinearities”. If the nonlinearities are “two double-valued nonlinearities”; that is,

The double-valued nonlinearity.

Root loci of limit cycles with two doubled-valued nonlinearities.

Time responses with two double-valued nonlinearities.

Six criteria for finding a stable limit cycle have been developed for nonlinear multivariable feedback control systems with single- and double-valued nonlinearities. The same analyzing and design procedures with six checking criteria will be applied to following

Consider a nonlinear multivariable system with transfer function matrix [

Calculated and simulated results of Example

Osci. Freq. (rad/s) | Channel no. 1 | Channel no. 2 | ||||
---|---|---|---|---|---|---|

Calculation | 0.4875 | 0.4541 | 0.2929 | −53.3 | 0.95 | |

1.0961 | 2.1390 | |||||

Simulation | 0.4836 | 1.0607 | 2.2454 | −54.4 |

Nonlinearities of Example

Root loci analyses of limit cycles of Example

Time responses of Example

Consider a

The maximal frequency point

1.273 | 1.20 | 1.15 | 1.10 | 0.9968 | 0.90 | 0.70 | 0.30 | 0.10 | 0.00 | |

2.040 | 2.049 | 2.054 | 2.058 | 2.061 | 2.059 | 2.054 | 2.044 | 2.041 | 2.039 |

(a) Root loci analyses of limit cycles of Example

Time responses of Example

Consider a

Nonlinearities of Example

(a) Root loci analyses of limit cycles of Example

Time responses of Example

From analyses and simulated results of Examples

plotting root loci from the characteristic equation in the parameter plane (or space);

finding the maximal and minimal frequency points (

checking points (

In this paper, a practical method for limit-cycle predictions in nonlinear multivariable feedback control system has been presented and found to be much simpler than other methods given in the current literature. It has been shown that calculated results give accurate predictions for consider nonlinear multivariable feedback control systems.