Nyquist Stability Criterion ANITEX Interactive Control-System Demonstration

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Purpose

The Nyquist stability criterion determines closed-loop stability from the open-loop frequency response L(s)=G(s)H(s). Instead of solving the closed-loop characteristic polynomial directly, the locus of L(j), -<< is drawn on the complex plane and its encirclement of the critical point -1+j0 is examined.

Example feedback system

Consider a negative-feedback system with G(s)=Ks(1+sT_1)(1+sT_2), H(s)=1. Therefore L(s)=Ks(1+sT_1)(1+sT_2). For the interactive example, use T_1=1, T_2=0.2, so that L(s)=Ks(1+s)(1+0.2s). The closed-loop transfer function is T(s)=L(s)1+L(s), and its characteristic equation is 1+L(s)=0. Hence the critical Nyquist point is L(s)=-1+j0.

Frequency-domain vector

Putting s=j gives L(j)=Kj(1+j)(1+j0.2). Let (1+j)(1+j0.2) =(1-0.2^2)+j1.2. Then the denominator becomes j =-1.2^2+j(1-0.2^2). Thus the real and imaginary components used by the animation are \L(j)\ =-1.2K^2 (1.2^2)^2+[(1-0.2^2)]^2, \L(j)\ =-K(1-0.2^2) (1.2^2)^2+[(1-0.2^2)]^2. The vector L()=\L(j)\x +j\L(j)\y rotates and changes length as increases. Its endpoint traces the Nyquist locus.

Nyquist criterion

Define P: number of open-loop poles in the right-half s plane, N: net encirclements of -1+j0 by the Nyquist contour, using the engine's signed convention, Z: number of closed-loop poles in the right-half s plane. With the signed convention used in this ANITEX scene, Z=P+N. Closed-loop stability requires Z=0. The engine must display its clockwise/counter-clockwise sign convention explicitly so that the visual count and equation cannot be confused.

For this demonstration the open-loop transfer function has no right-half-plane pole; the pole at the origin is treated by the standard indented Nyquist contour. Therefore P=0. The important visual question is whether the complete Nyquist locus causes the critical point -1+j0 to acquire a nonzero net encirclement.

Oscillation boundary

The phase-crossover condition occurs when \L(j)\=0. For >0, 1-0.2^2=0, therefore _180=52.236\ rad/s. At this frequency, L(j_180)=-K6. Consequently the locus passes exactly through -1+j0 when K_crit=6. Hence this example provides three especially useful animation states: 0<K<6: stable, K=6: marginal stability / sustained-oscillation boundary, K>6: unstable closed loop. At the boundary, the corresponding oscillation frequency is approximately _o=5\ rad/s, f_o=520.356\ Hz.

Interpretation

The important rule is not simply ``right side = stable'' and ``left side = unstable.'' Nyquist stability is topological: the complete contour, the critical point -1+j0, the number of right-half-plane open-loop poles, and the signed number of encirclements must be considered together.

For the selected example, increasing K expands the Nyquist locus radially. At K=6 it touches the critical point. Increasing K further changes the closed-loop stability and produces a right-half-plane closed-loop pole pair.

ANITEX interactive scene

The implemented plot displays finite positive and negative frequency branches. It does not draw the infinite-frequency closure or the indentation around the pole at the origin, and must not be used to count encirclements from a clipped picture. Here clockwise encirclements are positive in Z=P+N. The status is obtained analytically from the characteristic polynomial, not by counting the displayed finite curve. The unwrapped Bode phase is minus 90 degrees minus atan(omega T1) minus atan(omega T2).

Recommended viewer behavior

The ANITEX viewer should show the feedback block diagram and Nyquist plane simultaneously. Pressing Play sweeps logarithmically from low to high frequency. A moving vector from the origin to L(j) should trace the positive-frequency branch while the conjugate branch is displayed to complete the locus. The current point should also move synchronously on the Bode magnitude and phase plots.

Changing K should redraw all frequency-response graphics immediately. Preset buttons should demonstrate K=3, K=5.8, K=6, and K=8. At K=6 and =5, the viewer should emphasize the event L(j)=-1+j0 and display ``Oscillation Boundary.''

Teaching summary

Nyquist: frequency-response locus closed-loop stability The critical point is -1+j0. For this particular system, K<6\;stable, K=6\;oscillation boundary, K>6\;unstable. For a general system, however, stability must be determined from the complete Nyquist contour together with the open-loop right-half-plane pole count and the signed encirclement count.