# Transfer function

A transfer function is the ratio of the [Laplace transform](https://www.edgepedia.org/laplace-transform) of a system's output to the Laplace transform of its input, with all initial conditions assumed to be zero. For a linear time-invariant (LTI) system it takes the form of a rational function in the complex variable \( s = \sigma + j\omega \), a ratio of two polynomials in \( s \) whose coefficients come from the system's differential equation<sup>[1](https://web.mit.edu/2.14/www/Handouts/PoleZero.pdf)</sup><sup> • </sup><sup>[2](https://eng.libretexts.org/Bookshelves/Introductory_Engineering/Mechatronics%3A_Fundamentals_Design_Integration_and_Validation_%28Zhu%29/02%3A_Mechatronic_System_Modeling/2.01%3A_Laplace_Transforms)</sup>. The transfer function lets an engineer determine important response characteristics, including stability, without solving the complete differential equation<sup>[1](https://web.mit.edu/2.14/www/Handouts/PoleZero.pdf)</sup>.

| Key fact | Detail |
|---|---|
| Definition | \( H(s) = N(s)/D(s) \), a rational function in \( s = \sigma + j\omega \), with zero initial conditions assumed<sup>[1](https://web.mit.edu/2.14/www/Handouts/PoleZero.pdf)</sup><sup> • </sup><sup>[2](https://eng.libretexts.org/Bookshelves/Introductory_Engineering/Mechatronics%3A_Fundamentals_Design_Integration_and_Validation_%28Zhu%29/02%3A_Mechatronic_System_Modeling/2.01%3A_Laplace_Transforms)</sup> |
| Poles and zeros | Zeros are the roots of the numerator \( N(s) \), where \( H(s) \to 0 \); poles are the roots of the denominator \( D(s) \), where \( H(s) \to \infty \)<sup>[1](https://web.mit.edu/2.14/www/Handouts/PoleZero.pdf)</sup> |
| Stability criterion | A continuous-time system is stable if all its poles lie strictly in the left half of the complex plane, \( \operatorname{Re}(s) < 0 \)<sup>[3](https://ocw.mit.edu/courses/18-03sc-differential-equations-fall-2011/a8b0706dd2f7d811deb867bce13ad3d6_MIT18_03SCF11_s31_3text.pdf)</sup> |
| Interconnections | Series: \( H = H_2 H_1 \); parallel: \( H = H_1 + H_2 \); feedback (Black's formula): \( H_{r\to y} = P/(1 + KP) \)<sup>[4](https://introcontrol.mit.edu/spring26/prelabs/prelab4/tf_interconnect)</sup> |
| State-space conversion | \( G(s) = C(sI - A)^{-1}B + D \) converts a state-space model to a transfer function<sup>[5](https://ocw.mit.edu/courses/2-154-maneuvering-and-control-of-surface-and-underwater-vehicles-13-49-fall-2004/2609f8b1bdebe4566891080afe007f51_1349_notes.pdf)</sup> |
| Discrete-time form | The Z-domain transfer function is a ratio of polynomials in \( z \); the stability region is the interior of the unit disk<sup>[6](https://www.mathworks.com/help/signal/ug/discrete-time-system-models.html)</sup><sup> • </sup><sup>[7](https://introcontrol.mit.edu/spring24/labs/lab05)</sup> |
| MIMO generalization | The transfer function matrix \( G(s) \) has entries \( g_{ij}(s) \) relating actuator \( j \) to sensor \( i \); analysis uses the singular values of \( G(s) \)<sup>[8](https://web.mit.edu/16.31/www/Fall06/New_topic10.pdf)</sup> |

## Poles, zeros, and the pole-zero plot

Writing the numerator and denominator polynomials in factored form gives

\[ H(s) = K \frac{(s - z_1)(s - z_2)\cdots(s - z_m)}{(s - p_1)(s - p_2)\cdots(s - p_n)}, \]

where the \( z_i \) are the system zeros, the \( p_i \) are the system poles, and \( K \) is the gain. When \( s = z_i \) the transfer function vanishes; when \( s = p_i \) it becomes unbounded<sup>[1](https://web.mit.edu/2.14/www/Handouts/PoleZero.pdf)</sup>. Because the polynomial coefficients are real, every complex pole or zero appears with its complex conjugate<sup>[9](https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/aa46a1127460310c7506654176a72ec0_lecture_05.pdf)</sup>.

Plotting these locations on the complex \( s \)-plane, with zeros marked by circles and poles by crosses, gives the pole-zero plot. The poles and zeros characterize the system except for the overall gain constant, and they allow reconstruction of the input/output differential equation<sup>[1](https://web.mit.edu/2.14/www/Handouts/PoleZero.pdf)</sup><sup> • </sup><sup>[9](https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/aa46a1127460310c7506654176a72ec0_lecture_05.pdf)</sup>. The system poles directly define the homogeneous response: the unforced response is a sum of terms \( C_i e^{\lambda_i t} \) whose exponents are the roots of the characteristic equation, which coincide with the poles<sup>[1](https://web.mit.edu/2.14/www/Handouts/PoleZero.pdf)</sup>.

## Stability

The poles determine stability. A continuous-time LTI system is stable if every pole has negative real part, that is, if all poles lie strictly in the left half of the complex plane, \( \operatorname{Re}(s) < 0 \); this is equivalent to every solution of the unforced system decaying to zero<sup>[3](https://ocw.mit.edu/courses/18-03sc-differential-equations-fall-2011/a8b0706dd2f7d811deb867bce13ad3d6_MIT18_03SCF11_s31_3text.pdf)</sup>. In discrete time the corresponding region is the interior of the unit disk<sup>[7](https://introcontrol.mit.edu/spring24/labs/lab05)</sup>.

**Feedback changes the criterion.** The stability of \( H(s) \) alone depends only on its denominator polynomial, but once \( H(s) \) is placed in a feedback loop the closed-loop natural frequencies are the roots of \( d_h(s) + K_0 n_h(s) \), so closed-loop stability depends on both the denominator and the numerator of \( H(s) \), in proportions set by the feedback gain<sup>[10](https://introcontrol.mit.edu/spring25/prelabs/prelab3/sss)</sup>.

**Unstable cancellations.** A pole of the open-loop system can be hidden by a zero in the closed-loop reference transfer function while remaining visible in other input-output relationships. In a loop with controller \( K(s) = s/(s+1) \) and plant \( P(s) = 1/s^2 \), Black's formula gives \( H_{r\to y}(s) = 1/(s^2+s+1) \), whose poles \( s = -\tfrac{1}{2} \pm j\tfrac{\sqrt{3}}{2} \) look stable, but the transfer function from a plant disturbance to the output is \( (s+1)/(s(s^2+s+1)) \), which has a pole at \( s = 0 \). The controller zero at \( s = 0 \) canceled the plant pole at \( s = 0 \), masking the instability. Cancelling any zero or pole with \( \operatorname{Re}(s) \ge 0 \) is guaranteed to produce an unstable closed-loop system<sup>[4](https://introcontrol.mit.edu/spring26/prelabs/prelab4/tf_interconnect)</sup>.

## Combining transfer functions

Three standard interconnections build complex systems from subsystems<sup>[4](https://introcontrol.mit.edu/spring26/prelabs/prelab4/tf_interconnect)</sup>:

- **Series.** The output of the first subsystem feeds the input of the second, and the overall transfer function is the product \( H(s) = H_2(s) H_1(s) \).
- **Parallel.** Both subsystems share the same input and their outputs are summed, giving \( H(s) = H_1(s) + H_2(s) \).
- **Feedback.** A portion of the output is fed back and compared with the reference. With forward path \( P(s) \) and feedback path \( K(s) \), Black's formula gives the closed-loop transfer function \( H_{r\to y}(s) = P(s)/(1 + K(s)P(s)) \), memorized as forward gain divided by one plus total loop gain.

For systems described by state equations \( \dot{x} = Ax + Bu \), \( y = Cx + Du \), the transfer function is \( G(s) = C(sI - A)^{-1}B + D \), where \( I \) is the identity matrix of the same size as \( A \)<sup>[5](https://ocw.mit.edu/courses/2-154-maneuvering-and-control-of-surface-and-underwater-vehicles-13-49-fall-2004/2609f8b1bdebe4566891080afe007f51_1349_notes.pdf)</sup>.

## Frequency response

Evaluating the transfer function along the imaginary axis, \( s = j\omega \), gives the frequency response. For a sinusoidal input \( u(t) = U\cos(\omega t) \), the steady-state output is \( y(t) = U|H(j\omega)|\cos(\omega t + \angle H(j\omega)) \), so the magnitude scales the amplitude and the angle shifts the phase<sup>[10](https://introcontrol.mit.edu/spring25/prelabs/prelab3/sss)</sup>. Because \( H(s) \) is a rational function with a finite set of real coefficients, its values on this single line of the \( s \)-plane completely characterize the function<sup>[10](https://introcontrol.mit.edu/spring25/prelabs/prelab3/sss)</sup>. In the factored form, \( H(j\omega) \) is a product of vectors from each pole and zero to the test point \( j\omega \), which supports graphical evaluation methods such as Bode plots<sup>[9](https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/aa46a1127460310c7506654176a72ec0_lecture_05.pdf)</sup>.

## Discrete-time and MIMO transfer functions

For sampled-data and digital systems the same idea applies in the [Z-transform](https://www.edgepedia.org/z-transform) domain: the transfer function is a ratio of two polynomials in \( z \), \( H(z) = b(z)/a(z) \), and it is the principal discrete-time model in signal processing tools<sup>[6](https://www.mathworks.com/help/signal/ug/discrete-time-system-models.html)</sup>. A discrete-time transfer function is specified together with its sample time; for example \( H(z) = (z-1)/(z^2 - 1.85z + 0.9) \) with a sample time of 0.1 seconds<sup>[11](https://www.mathworks.com/help/control/ug/creating-discrete-time-models.html)</sup>.

For systems with multiple inputs and multiple outputs (MIMO), the transfer function generalizes to a transfer function matrix \( G(s) \), whose entry \( g_{ij}(s) \) relates the input of actuator \( j \) to the output of sensor \( i \)<sup>[8](https://web.mit.edu/16.31/www/Fall06/New_topic10.pdf)</sup>. Examining the individual entries is not sufficient for analysis; the system is instead assessed as a whole using the singular values of \( G(s) \)<sup>[8](https://web.mit.edu/16.31/www/Fall06/New_topic10.pdf)</sup>.

## Design limitations from right-half-plane poles and zeros

Poles and zeros in the right half plane impose limits that no stabilizing controller can remove. For a stable plant \( P \) with controller \( K \), the sensitivity function \( S = (1 + PK)^{-1} \) satisfies \( S(z_i) = 1 \) at every right-half-plane zero \( z_i \) of \( P \), so disturbances cannot be uniformly attenuated over the entire frequency range when the plant has right-half-plane zeros. Dually, for a plant with right-half-plane poles in an internally stabilized feedback loop, the complementary sensitivity \( T = PK(1+PK)^{-1} \) equals one at each such pole, which restricts uniform attenuation of noise<sup>[12](https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/996025f6db0d90b00f11c44fc49b85f9_MIT6_241JS11_textbook.pdf)</sup>.

## Software representation

Control software stores transfer functions in two equivalent forms. In MATLAB's Control System Toolbox, the `tf` model object holds the polynomial form \( G(s) = N(s)/D(s) \) as coefficient vectors, while the `zpk` object holds the factorized zero-pole-gain form \( K (s-z_1)\cdots/(s-p_1)\cdots \)<sup>[13](https://www.mathworks.com/help/control/ug/transfer-functions.html)</sup>. The Python `control` module deliberately avoids canceling poles and zeros when multiplying models, so an unstable cancellation remains visible in the closed-loop result<sup>[4](https://introcontrol.mit.edu/spring26/prelabs/prelab4/tf_interconnect)</sup>.

## References

1. [Understanding Poles and Zeros, MIT 2.14 Analysis and Design of Feedback Control Systems](https://web.mit.edu/2.14/www/Handouts/PoleZero.pdf)
2. [Laplace Transforms, Engineering LibreTexts (Mechatronics, Zhu)](https://eng.libretexts.org/Bookshelves/Introductory_Engineering/Mechatronics%3A_Fundamentals_Design_Integration_and_Validation_%28Zhu%29/02%3A_Mechatronic_System_Modeling/2.01%3A_Laplace_Transforms)
3. [Poles and Stability, MIT 18.03SC Differential Equations (Fall 2011)](https://ocw.mit.edu/courses/18-03sc-differential-equations-fall-2011/a8b0706dd2f7d811deb867bce13ad3d6_MIT18_03SCF11_s31_3text.pdf)
4. [Transfer Functions of Interconnections, MIT 6.310 Prelab 4 (Spring 2026)](https://introcontrol.mit.edu/spring26/prelabs/prelab4/tf_interconnect)
5. [Control Fundamentals, MIT 2.154 Maneuvering and Control of Surface and Underwater Vehicles (Fall 2004)](https://ocw.mit.edu/courses/2-154-maneuvering-and-control-of-surface-and-underwater-vehicles-13-49-fall-2004/2609f8b1bdebe4566891080afe007f51_1349_notes.pdf)
6. [Discrete-Time System Models, MathWorks Signal Processing Toolbox](https://www.mathworks.com/help/signal/ug/discrete-time-system-models.html)
7. [MIT 6.310 Spring 2024 Lab 05](https://introcontrol.mit.edu/spring24/labs/lab05)
8. [Multivariable Control, MIT 16.31 Lecture Notes, Topic 10](https://web.mit.edu/16.31/www/Fall06/New_topic10.pdf)
9. [Frequency Response and the Pole-Zero Plot, MIT 2.161 Signal Processing (Fall 2008), Lecture 5](https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/aa46a1127460310c7506654176a72ec0_lecture_05.pdf)
10. [Sinusoidal Steady State, MIT 6.310 Prelab 3 (Spring 2025)](https://introcontrol.mit.edu/spring25/prelabs/prelab3/sss)
11. [Creating Discrete-Time Models, MathWorks Control System Toolbox](https://www.mathworks.com/help/control/ug/creating-discrete-time-models.html)
12. [Limitations Due to RHP Zeros and Poles, Lectures on Dynamic Systems and Control, MIT 6.241J (Spring 2011)](https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/996025f6db0d90b00f11c44fc49b85f9_MIT6_241JS11_textbook.pdf)
13. [Transfer Functions, MathWorks Control System Toolbox](https://www.mathworks.com/help/control/ug/transfer-functions.html)

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