All Pass and Minimum Phase Systems

In frequency response analysis, understanding whether a system is minimum phase or non-minimum phase determines how much we can infer about the system's phase response from its magnitude response alone. This classification has deep implications for controller design, system identification, and stability margin assessment, and it appears frequently in GATE problems related to Bode plots and transfer function analysis.

Minimum Phase Systems

A minimum phase system is one in which all poles and all zeros lie in the left-half s-plane (LHP), excluding the imaginary axis. The term minimum phase comes from the fact that, among all systems sharing the same magnitude response |G(jw)|, the minimum phase system has the least possible phase lag at every frequency. Any other system with the same magnitude but different phase characteristics will always have more phase lag than the minimum phase counterpart.

This property is important because for minimum phase systems, the phase response is uniquely determined by the magnitude response through the Hilbert transform relationship. This means that if you know the Bode magnitude plot of a minimum phase system, you can calculate the exact phase plot. This is a significant advantage in system identification and controller design, as Bode gain and phase information are not independent for such systems.

All stable systems with no RHP zeros are minimum phase. A simple example is G(s) = (s+2)/((s+1)(s+3)), where the zero at s = -2 and poles at s = -1 and s = -3 all lie in the LHP. The Bode phase plot for this system ranges from -90 degrees (at low frequency) to 0 degrees (at high frequency), displaying the characteristic minimum lag behavior.

Non-Minimum Phase Systems

A non-minimum phase system has one or more zeros located in the right-half s-plane (RHP). The presence of RHP zeros causes additional phase lag beyond what the magnitude response would suggest. A non-minimum phase system can be factored as the product of a minimum phase system and an all-pass factor: G_nmp(s) = G_mp(s) * A(s). Both factors have the same magnitude response, but the all-pass factor A(s) contributes extra phase lag.

The most recognizable physical manifestation of a non-minimum phase system is the undershoot phenomenon in step response. When a non-minimum phase system receives a positive step input, the output initially moves in the negative direction before eventually settling to the positive steady-state value. This behavior is seen in systems with RHP zeros, such as certain aircraft pitch dynamics, chemical process control loops with inverse response, and heat exchanger controls.

Non-minimum phase behavior places fundamental limitations on the achievable control bandwidth. Increasing the controller gain to improve speed of response will cause the system to go unstable earlier than a corresponding minimum phase system would. This is why non-minimum phase plants are considered challenging to control and require careful gain margin management.

All-Pass Systems

An all-pass system is a special case where the magnitude of the transfer function is unity (0 dB) at all frequencies, but the phase varies with frequency. The standard first-order all-pass transfer function is A(s) = (s - a) / (s + a), where a is a positive real number. This places a zero at s = +a (RHP) and a pole at s = -a (LHP), which are mirror images about the imaginary axis.

The phase contributed by this all-pass factor is -2 arctan(w/a), which decreases monotonically from 0 degrees at w = 0 to -180 degrees as w approaches infinity. A higher-order all-pass system is a cascade of such first-order sections, each contributing its own phase lag with no change to the magnitude response. All-pass systems are used in phase equalizers in communication systems to compensate for phase distortion without affecting signal amplitude.

Mathematical Expression

For a first-order all-pass factor A(s) = (s - a)/(s + a), the magnitude and phase are expressed as follows. The magnitude at s = jw is |A(jw)| = |jw - a| / |jw + a| = sqrt(w squared + a squared) / sqrt(w squared + a squared) = 1, confirming unity gain at all frequencies. The phase angle is angle(A(jw)) = arctan(w/(-a)) - arctan(w/a) = pi - arctan(w/a) - arctan(w/a) = pi - 2 arctan(w/a). Accounting for the standard convention this gives phase = -2 arctan(w/a), a purely negative phase contribution.

Solved Numerical Example

Consider a non-minimum phase system G(s) = (s - 1) / ((s + 2)(s + 3)). Factor it into its minimum phase part and all-pass part, and find the phase of the all-pass factor at w = 1 rad/s.

Given:
G(s) = (s - 1) / [(s+2)(s+3)]
NMP zero at s = +1 (RHP)

Why this formula applies:
G_nmp = G_mp * A(s)
Minimum phase part uses mirror of RHP zero: (s+1)
All-pass factor carries the RHP zero: A(s) = (s-1)/(s+1)

Formula:
G_mp(s) = (s+1) / [(s+2)(s+3)]   [all LHP poles and zeros]
A(s) = (s-1)/(s+1)                 [all-pass: unity magnitude]

Verification: G_mp * A = [(s+1)/(s+2)(s+3)] * [(s-1)/(s+1)]
= (s-1)/[(s+2)(s+3)] = G(s)  ✓

Phase of A(jw) at w = 1 rad/s:
Phase = -2 * arctan(w/a) where a = 1
= -2 * arctan(1/1)
= -2 * 45 deg

Final Answer: Phase contribution of all-pass factor at w=1 rad/s = -90 degrees
This extra -90 deg lag is not visible in the magnitude plot but degrades phase margin.

Exam Tip: In GATE problems, to check if a system is minimum phase, look at the sign of the coefficients of the numerator polynomial. A transfer function like (s-2)/denominator has a RHP zero at s=+2 and is non-minimum phase. Any NMP system can be written as a minimum phase system multiplied by an all-pass factor (s-a)/(s+a), where a is the RHP zero.

Mechanism Summary

• Minimum phase systems have all poles and zeros in the LHP. Their phase is uniquely determined by their magnitude via the Hilbert transform.
• Non-minimum phase systems have one or more RHP zeros. They produce more phase lag than a minimum phase system with the same magnitude response.
• Any NMP system can be factored as G_nmp = G_mp * A, where A is the all-pass factor formed from the RHP zeros.
• All-pass systems have unity magnitude at all frequencies but contribute phase lag of -2 arctan(w/a) per RHP zero at s = +a.
• NMP systems exhibit undershoot (initial inverse response) in step response, limiting achievable closed-loop bandwidth.
• Time-delay systems e^(-sT) are non-minimum phase and are equivalent to infinite-order all-pass networks.

Quick Revision

• Minimum phase: all poles and zeros in LHP. Least phase lag for a given magnitude. Phase uniquely linked to magnitude.
• Non-minimum phase: has RHP zeros. Same magnitude as its MP counterpart but extra phase lag added by all-pass factor.
• All-pass transfer function: A(s) = (s - a)/(s + a). |A(jw)| = 1 for all w. Phase = -2 arctan(w/a).
• Factorization: G_nmp(s) = G_mp(s) * A(s). Identifying the MP part simplifies Bode phase analysis.
• Key behavior: NMP systems show undershoot at step response start. Time delay is inherently NMP.
• GATE trap: A transfer function with a term (s - a) in numerator (a > 0) is automatically non-minimum phase. Do not confuse with (s + a) which is LHP and minimum phase.
• Bode magnitude plots of G_nmp and G_mp are identical. Only the phase plots differ by the all-pass contribution.