Sensitivity Functions in Feedback Systems

Understanding how a control system reacts to the inevitable imperfections of the real world is the core of robust design. Sensitivity functions provide the precise mathematical language for this analysis, quantifying how closed-loop performance is affected by internal parameter drifts and external disturbances. Mastering these functions is essential for moving beyond naive stability and achieving systems that perform reliably despite component aging, manufacturing tolerances, and unpredictable operating environments.

Defining Sensitivity and Complementary Sensitivity

At the heart of feedback analysis are two fundamental transfer functions derived from the standard unity-feedback block diagram. The first is the sensitivity function, denoted as $S$. For a system with forward path transfer function $G(s)$ and feedback path transfer function $H(s)$, the sensitivity function is defined as the transfer function from a reference input to the tracking error, or equivalently, how the output responds to a disturbance at the output. Its mathematical form is:

$$S(s)=\frac{1}{1+G(s)H(s)}$$

The second key function is the complementary sensitivity function, denoted as $T$. It is called "complementary" for a reason that will soon become clear. This function represents the closed-loop transfer function from the reference input to the system output. Its definition is:

$$T(s)=\frac{G(s)H(s)}{1+G(s)H(s)}$$

A simple but profound algebraic truth emerges from these definitions: for any frequency $s=j\omega$, they sum to unity.

$$S(s)+T(s)=1$$

This equation, $S+T=1$, is not just a curiosity; it is an absolute constraint that governs all feedback design, forcing critical trade-offs between different performance objectives.

Interpreting Sensitivity: Robustness to Variations and Disturbances

The sensitivity function’s primary role is to measure robustness. Specifically, it quantifies how sensitive the closed-loop system's performance is to variations in the plant's internal parameters. Consider a plant whose true transfer function is $G(s)$. If a particular parameter (like a motor's inertia or a circuit's resistance) changes, $G(s)$ becomes $G(s)+\Delta G(s)$. The sensitivity function $S(s)$ directly relates the relative change in the overall closed-loop transfer function to the relative change in the plant.

More intuitively, $S(s)$ also dictates disturbance rejection. A disturbance signal entering at the plant output is attenuated by the factor $S(s)$. Therefore, to achieve good disturbance rejection—for example, maintaining a room's temperature despite an open window—the magnitude of $S(j\omega )$ must be small at the frequencies where the disturbances are significant. Typically, disturbances are low-frequency phenomena (like slow thermal changes or constant load forces), which leads to a fundamental design rule: Achieve low sensitivity ($|S|\ll 1$) at low frequencies.

This is accomplished by ensuring the loop gain $L(s)=G(s)H(s)$ is large in magnitude at those low frequencies. Since $S(s)\approx 1/L(s)$ when $|L(s)|\gg 1$, a large loop gain directly translates to small sensitivity and excellent disturbance rejection.

Interpreting Complementary Sensitivity: Tracking and Noise Response

While $S(s)$ tells the story of robustness and disturbance rejection, the complementary sensitivity function $T(s)$ governs reference tracking and sensitivity to measurement noise. As the closed-loop transfer function, the ideal $T(s)$ is 1 across all frequencies, which would mean the output perfectly follows the reference. In practice, we strive for $T(j\omega )\approx 1$ at low frequencies for good tracking of slow commands.

However, $T(s)$ also describes how sensor noise propagates to the system output. High-frequency noise, inherent in all physical sensors, is fed back through the controller. The transfer function from measurement noise to the output is precisely $T(s)$. Therefore, to prevent the controller from frantically reacting to meaningless noise, we require $|T(j\omega )|$ to be small at high frequencies. This is achieved by ensuring the loop gain $L(s)$ is small (rolling off) at those high frequencies, since $T(s)\approx L(s)$ when $|L(s)|\ll 1$.

The Fundamental Design Trade-off: The $S+T=1$ Constraint

The algebraic constraint $S(s)+T(s)=1$ manifests as a critical, inescapable trade-off in the frequency domain. You cannot make both $|S(j\omega )|$ and $|T(j\omega )|$ small at the same frequency. If you design for perfect disturbance rejection ($S\approx 0$) at a specific frequency, then you inherently have $T\approx 1$ at that same frequency, meaning sensor noise is fully passed to the output.

This trade-off shapes the entire control design process. The standard design paradigm is:

1. Shape $|S|$ to be small at low frequencies for disturbance rejection and tracking.
2. Shape $|T|$ to be small at high frequencies for noise attenuation and robustness to high-frequency modeling errors.
3. Manage the crossover region (where $|L({\omega }_c)|=1$) carefully, as this is where both $S$ and $T$ have magnitudes around 0.7. The slope and phase of $L(s)$ here determine stability margins.

The performance limits are visualized on a Bode magnitude plot. The difference between the low-frequency gain and the high-frequency roll-off of the loop gain defines the bandwidth of the control system and highlights the region where the designer must balance the competing requirements dictated by $S$ and $T$.

Common Pitfalls

1. Ignoring the Noise Implications of High Bandwidth. A common mistake is pushing for extremely high bandwidth (making $S$ small over a very wide frequency range) to achieve aggressive tracking. This inevitably makes $T$ near unity over that same wide range, amplifying any high-frequency sensor noise. The result can be a system with chattering actuators, wasted energy, and premature wear, even if its simulation with perfect sensors looks excellent. Always consider the practical noise spectrum of your sensors when specifying bandwidth.

2. Overlooking the "Waterbed Effect" for Non-Minimum Phase Systems. For stable, minimum-phase plants, you can roll off $S$ and $T$ as described. However, if the plant has right-half-plane zeros (non-minimum phase), a profound limitation called the waterbed effect occurs. The Bode Integral Theorem states that if you suppress sensitivity ($|S|<1$) over a range of frequencies, it will necessarily peak ($|S|>1$) at other frequencies. You cannot "push down" sensitivity everywhere; pushing it down in one band makes it bulge up elsewhere. Failing to account for this can lead to unexpected fragility at frequencies you weren't testing.

3. Confusing Output Disturbance Rejection with Input Disturbance Rejection. The sensitivity function $S$ directly applies to disturbances entering at the plant output. However, disturbances often enter at the plant input (e.g., a load torque on a motor shaft). The transfer function for input disturbance rejection is $G(s)S(s)=\frac{G(s)}{1+L(s)}$, not just $S(s)$. A design that achieves small $S(s)$ at low frequencies will also reject low-frequency input disturbances if $G(s)$ has sufficient gain at those frequencies. Forgetting this distinction can lead to a system robust to one type of disturbance but vulnerable to another.

4. Designing Based on $T(s)$ Alone. While achieving a desired $T(s)$ shape is important for tracking, designing a controller by focusing solely on $T(s)$ ignores the robustness picture. A system can have a beautiful $T(s)$ but a poorly shaped $S(s)$ that has large peaks, indicating very poor robustness to parameter variations and poor disturbance rejection at specific frequencies. A complete design always analyzes both $S(s)$ and $T(s)$ to verify all performance and robustness specifications are met.

Summary

• The sensitivity function $S=1/(1+GH)$ quantifies a closed-loop system's robustness to parameter variations and its ability to reject output disturbances. Small $|S|$ at low frequencies is a primary control objective.
• The complementary sensitivity function $T=GH/(1+GH)$ defines the closed-loop tracking response and the amplification of sensor noise. Small $|T|$ at high frequencies is necessary for noise attenuation.
• The fundamental algebraic constraint $S+T=1$ creates an unavoidable trade-off: you cannot make both functions small at the same frequency. This forces the standard design pattern of high low-frequency gain and high-frequency roll-off.
• Effective control design involves shaping the loop gain $L=GH$ to simultaneously achieve desirable $S$ and $T$ characteristics, managing the trade-off across the frequency spectrum.
• Always consider practical limitations like sensor noise spectra and the waterbed effect in non-minimum phase systems, as these impose hard limits on achievable performance.