zpk

Zero-pole-gain model

Description

Use zpk to create zero-pole-gain models, or to convert dynamic system models to zero-pole-gain form.

Zero-pole-gain models are a representation of transfer functions in factorized form. For example, consider the following continuous-time SISO transfer function:

$$G\left(s\right)=\frac{s^2-3s-4}{s^2+5s+6}$$

G(s) can be factorized into the zero-pole-gain form as:

$$G\left(s\right)=\frac{\left(s+1\right)\left(s-4\right)}{\left(s+2\right)\left(s+3\right)}.$$

A more general representation of the SISO zero-pole-gain model is as follows:

$$h\left(s\right)=k\frac{\left(s-z\left(1\right)\right)\left(s-z\left(2\right)\right)\dots \left(s-z\left(m\right)\right)}{\left(s-p\left(1\right)\right)\left(s-p\left(2\right)\right)\dots \left(s-p\left(n\right)\right)}$$

Here, z and p are the vectors of real-valued or complex-valued zeros and poles, and k is the real-valued or complex-valued scalar gain. For MIMO models, each I/O channel is represented by one such transfer function h_ij(s).

You can create a zero-pole-gain model object either by specifying the poles, zeros and gains directly, or by converting a model of another type (such as a state-space model ss) to zero-pole-gain form.

You can also use zpk to create generalized state-space (genss) models or uncertain state-space (uss (Robust Control Toolbox)) models.

Creation

Syntax

sys = zpk(zeros,poles,gain)

sys = zpk(zeros,poles,gain,ts)

sys = zpk(zeros,poles,gain,ltiSys)

sys = zpk(m)

sys = zpk(___,PropertyName=Value)

sys = zpk(ltiSys)

sys = zpk(ltiSys,Name=Value)

sys = zpk(ltiSys,component)

s = zpk('s')

z = zpk('z',ts)

Description

Create ZPK Model

sys = zpk(zeros,poles,gain) creates a continuous-time zero-pole-gain model with zeros and poles specified as vectors and the scalar value of gain. The output sys is a zpk model object storing the model data. Set zeros or poles to [] for systems without zeros or poles. These two inputs need not have equal length and the model need not be proper (that is, have an excess of poles).

sys = zpk(zeros,poles,gain,ts) creates a discrete-time zero-pole-gain model with sample time ts. Set ts to -1 or [] to leave the sample time unspecified.

sys = zpk(zeros,poles,gain,ltiSys) creates a zero-pole-gain model with properties inherited from the dynamic system model ltiSys, including the sample time.

sys = zpk(m) creates a zero-pole-gain model that represents the static gain, m.

sys = zpk(___,PropertyName=Value) sets Properties of the zero-pole-gain model using one or more property name-value arguments to set additional properties of the model. This syntax works with any of the previous input-argument combinations.

Convert To ZPK Model

sys = zpk(ltiSys) converts the dynamic system model ltiSys to a zero-pole-gain model.

sys = zpk(ltiSys,Name=Value) obtains a truncated zpk representation of the sparse model ltiSys by computing zeros and poles based on one or more specified name-value arguments. Because this method calculates zeros for each input-output pair, it is most suitable for models with small input-output sizes. (since R2025a)

sys = zpk(ltiSys,component) converts the specified component of ltiSys to zero-pole-gain model form. Use this syntax only when ltiSys is an identified linear time-invariant (LTI) model such as an idss or an idtf model.

Create Variable for Rational Expression

s = zpk('s') creates a special variable s that you can use in a rational expression to create a continuous-time zero-pole-gain model. Using a rational expression is sometimes easier and more intuitive than specifying polynomial coefficients.

z = zpk('z',ts) creates special variable z that you can use in a rational expression to create a discrete-time zero-pole-gain model. To leave the sample time unspecified, set ts input argument to -1.

Input Arguments

zeros — Zeros of the zero-pole-gain model

row vector | Ny-by-Nu cell array of row vectors

Zeros of the zero-pole-gain model, specified as:

• A row vector for SISO models. For instance, use [1,2+i,2-i] to create a model with zeros at s = 1, s = 2+i, and s = 2-i. For an example, see Continuous-Time SISO Zero-Pole-Gain Model.

• An Ny-by-Nu cell array of row vectors to specify a MIMO zero-pole-gain model, where Ny is the number of outputs, and Nu is the number of inputs. For an example, see Discrete-Time MIMO Zero-Pole-Gain Model.

For instance, if a is realp tunable parameter with nominal value 3, then you can use zeros = [1 2 a] to create a genss model with zeros at s = 1 and s = 2 and a tunable zero at s = 3.

When you use this input argument to create a zpk model, the argument sets the initial value of the property Z.

poles — Poles of the zero-pole-gain model

row vector | Ny-by-Nu cell array of row vectors

Poles of the zero-pole-gain model, specified as:

• A row vector for SISO models. For an example, see Continuous-Time SISO Zero-Pole-Gain Model.

• An Ny-by-Nu cell array of row vectors to specify a MIMO zero-pole-gain model, where Ny is the number of outputs and Nu is the number of inputs. For an example, see Discrete-Time MIMO Zero-Pole-Gain Model.

Also a property of the zpk object. This input argument sets the initial value of property P.

gain — Gain of the zero-pole-gain model

scalar | Ny-by-Nu matrix

Gain of the zero-pole-gain model, specified as:

• A scalar for SISO models. For an example, see Continuous-Time SISO Zero-Pole-Gain Model.

• An Ny-by-Nu matrix to specify a MIMO zero-pole-gain model, where Ny is the number of outputs and Nu is the number of inputs. For an example, see Discrete-Time MIMO Zero-Pole-Gain Model.

Also a property of the zpk object. This input argument sets the initial value of property K.

ts — Sample time

scalar

Sample time, specified as a scalar. Also a property of the zpk object. This input argument sets the initial value of property Ts.

ltiSys — Dynamic system

dynamic system model | model array

Dynamic system, specified as a SISO or MIMO dynamic system model or array of dynamic system models. Dynamic systems that you can use include:

• Continuous-time or discrete-time numeric LTI models, such as tf, zpk, ss, or pid models.

  If ltiSys is a sparse state-space model (sparss or mechss), the software computes a truncated zero-pole-gain approximation in a specified frequency band of focus. For sparse models, use the name-value arguments to specify computation options. If you do not specify any options, the software computes up to the first 1000 poles and zeros with smallest magnitude. Additionally, obtaining a truncated zero-pole-gain approximation is applicable only for models with a valid sparss representation. (since R2025a)

• Generalized or uncertain LTI models such as genss or uss (Robust Control Toolbox) models. (Using uncertain models requires a Robust Control Toolbox™ license.)

  The resulting zero-pole-gain model assumes

  • current values of the tunable components for tunable control design blocks.

  • nominal model values for uncertain control design blocks.

• Identified LTI models, such as idtf (System Identification Toolbox), idss (System Identification Toolbox), idproc (System Identification Toolbox), idpoly (System Identification Toolbox), and idgrey (System Identification Toolbox) models. To select the component of the identified model to convert, specify component. If you do not specify component, tf converts the measured component of the identified model by default. (Using identified models requires System Identification Toolbox™ software.)

  An identified nonlinear model cannot be converted into a zpk model object. You may first use linear approximation functions such as linearize and linapp (This functionality requires System Identification Toolbox software.)

m — Static gain

scalar | matrix

Static gain, specified as a scalar or matrix. Static gain or steady state gain of a system represents the ratio of the output to the input under steady state condition.

component — Component of identified model

'measured' (default) | 'noise' | 'augmented'

Component of identified model to convert, specified as one of the following:

• 'measured' — Convert the measured component of sys.

• 'noise' — Convert the noise component of sys

• 'augmented' — Convert both the measured and noise components of sys.

component only applies when sys is an identified LTI model.

For more information on identified LTI models and their measured and noise components, see Identified LTI Models.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Example: sys = zpk(sparseSys,Focus=[0 100],Display="off")

UseParallel — Use parallel computing

0 or false (default) | 1 or true

Since R2025a

Use parallel computing during zero-pole computation, specified as a numeric or logical 0 (false) or 1 (true).

When UseParallel is set to true, you can explicitly choose to scale to your preferred parallel environment. Enabling parallel computing may result in improved performance during zero-pole computation. However, even with UseParallel set to false, the algorithm can use built-in multithreading to make best use of the local resources. For more information, see MATLAB Multicore.

This option requires a Parallel Computing Toolbox™ license.

RollOff — Roll-off slope

0 (default) | nonpositive scalar | matrix

Since R2025a

Roll-off slope, specified as a nonpositive scalar or matrix.

• Use a scalar value for SISO models or when the slope is uniform for all input-output pairs for MIMO models.

• Use a matrix when the slope is different for each input-output pair for MIMO models.

This option allows you to specify how the approximation should roll-off past the specified frequency range. For example, Slope of -2 ensures the gain rolls off at a rate of at least –40 dB/decade (the roll-off rate of 1/s²) beyond fmax.

Focus — Frequency range of interest

[0 Inf] (default) | vector

Since R2025a

Frequency range of interest, specified as a vector of form [0,fmax]. When you specify a frequency range of focus, the software computes only the poles with natural frequency in this range. For discrete-time models, the software approximates the equivalent natural frequency through Tustin transform.

Since zpk computes all poles and zeros in the specified frequency range, you typically specify a low-frequency range to limit computing a large number of poles and zeros. By default, the focus is unspecified ([0 Inf]) and the algorithm computes up to MaxNumber poles and zeros.

MaxNumber — Maximum number of poles and zeros to compute

1000 (default) | positive integer

Since R2025a

Maximum number of poles and zeros to compute, specified as a positive integer. This value limits the number of poles and zeros computed by the algorithm and the order of the approximation of the original sparse model.

Shift — Spectral shift

0 (default) | finite scalar

Since R2025a

Spectral shift, specified as a finite scalar.

The software computes poles with the natural frequency in the specified range [0,fmax] using inverse power iterations for A-sigma*E, which obtains eigenvalues closest to the shift sigma. When A is singular and sigma is zero, the algorithm fails as no inverse exists. Therefore, for sparse models with integral action (s = 0 or at z = 1 for discrete-time models), you can use this option to implicitly shift poles or zeros to the value closest to this shift value. Specify a shift value that is not equal to an existing pole or zero value of the original model.

Tolerance — Accuracy of computed poles and zeros

1e-12 (default) | positive finite scalar

Since R2025a

Tolerance for accuracy of computed poles and zeros, specified as a positive finite scalar. This value controls the convergence of computed eigenvalues in inverse power iterations.

Display — Show or hide progress report

"on" (default) | "off"

Since R2025a

Show or hide progress report, specified as either "off" or "on".

Output Arguments

sys — Output system model

zpk model object | genss model object | uss model object

Output system model, returned as:

• A zero-pole-gain (zpk) model object, when the zeros, poles and gain input arguments contain numeric values. sys is always a zpk model object when converting ltiSys to zpk model type.

• A generalized state-space model (genss) object, when the zeros, poles and gain input arguments includes tunable parameters, such as realp parameters or generalized matrices (genmat).

• An uncertain state-space model (uss) object, when the zeros, poles and gain input arguments includes uncertain parameters. Using uncertain models requires a Robust Control Toolbox license.

Properties

Z — System zeros

cell array | Ny-by-Nu cell array of row vectors

System zeros, specified as:

• A cell array of transfer function zeros or the numerator roots for SISO models.

• An Ny-by-Nu cell array of row vectors of the zeros for each I/O pair in a MIMO model, where Ny is the number of outputs and Nu is the number of inputs.

The values of Z can be either real-valued or complex-valued.

P — System poles

cell array | Ny-by-Nu cell array of row vectors

System poles, specified as:

• A cell array of transfer function poles or the denominator roots for SISO models.

• An Ny-by-Nu cell array of row vectors of the poles for each I/O pair in a MIMO model, where Ny is the number of outputs and Nu is the number of inputs.

The values of P can be either real-valued or complex-valued.

K — System gains

scalar | Ny-by-Nu matrix

System gains, specified as:

• A scalar value for SISO models.

• An Ny-by-Nu matrix storing the gain values for each I/O pair of the MIMO model, where Ny is the number of outputs and Nu is the number of inputs.

The values of K can be either real-valued or complex-valued.

DisplayFormat — Specifies how the numerator and denominator polynomials are factorized for display

'roots' (default) | 'frequency' | 'time constant'

Specifies how the numerator and denominator polynomials are factorized for display, specified as one of the following:

• 'roots' — Display factors in terms of the location of the polynomial roots. 'roots' is the default value of DisplayFormat.

• 'frequency' — Display factors in terms of root natural frequencies ω₀ and damping ratios ζ.

  The 'frequency' display format is not available for discrete-time models with Variable value 'z^-1' or 'q^-1'.

• 'time constant' — Display factors in terms of root time constants τ and damping ratios ζ.

  The 'time constant' display format is not available for discrete-time models with Variable value 'z^-1' or 'q^-1'.

For continuous-time models, the following table shows how the polynomial factors are arranged in each display format.

DisplayName Value	First-Order Factor (Real Root $$R$$)	Second-Order Factor (Complex Root pair $$R=a\pm jb$$)
'roots'	$$(s-R)$$	$$(s^2-\alpha s+\beta ),$$ where $$\alpha =2a,\beta =a^2+b^2$$
'frequency'	$$\left(1-\frac{s}{{\omega }_0}\right),$$ where $${\omega }_0=R$$	$$1-2\zeta \left(\frac{s}{{\omega }_0}\right)+{\left(\frac{s}{{\omega }_0}\right)}^2,$$ where $${\omega }_0{}^2=a^2+b^2,\zeta =\frac{a}{{\omega }_0}$$
'time constant'	$$\left(1-\tau s\right),$$ where $$\tau =\frac{1}{R}$$	$$1-2\zeta \left(\tau s\right)+{\left(\tau s\right)}^2,$$ where $$\tau =\frac{1}{{\omega }_0},\zeta =a\tau$$

For discrete-time models, the polynomial factors are arranged similar to the continuous-time models, with the following variable substitutions:

$$s\to w=\frac{z-1}{T_s};R\to \frac{R-1}{T_s},$$

where T_s is the sample time. In discrete-time, τ and ω₀ closely match the time constant and natural frequency of the equivalent continuous-time root, provided that the following condition is fulfilled: $$\left|z-1\right|<<T_s\left({\omega }_0<<\frac{\pi }{T_s}=\text{Nyquist frequency}\right)$$.

Variable — Zero-pole-gain model display variable

's' (default) | 'z' | 'p' | 'q' | 'z^-1' | 'q^-1'

Zero-pole-gain model display variable, specified as one of the following:

• 's' — Default for continuous-time models

• 'z' — Default for discrete-time models

• 'p' — Equivalent to 's'

• 'q' — Equivalent to 'z'

• 'z^-1' — Inverse of 'z'

• 'q^-1' — Equivalent to 'z^-1'

IODelay — Transport delay

0 (default) | scalar | Ny-by-Nu array

Transport delay, specified as one of the following:

• Scalar — Specify the transport delay for a SISO system or the same transport delay for all input/output pairs of a MIMO system.

• Ny-by-Nu array — Specify separate transport delays for each input/output pair of a MIMO system. Here, Ny is the number of outputs and Nu is the number of inputs.

For continuous-time systems, specify transport delays in the time unit specified by the TimeUnit property. For discrete-time systems, specify transport delays in integer multiples of the sample time, Ts. For more information on time delay, see Time Delays in Linear Systems.

InputDelay — Input delay

0 (default) | scalar | Nu-by-1 vector

Input delay for each input channel, specified as one of the following:

• Scalar — Specify the input delay for a SISO system or the same delay for all inputs of a multi-input system.

• Nu-by-1 vector — Specify separate input delays for input of a multi-input system, where Nu is the number of inputs.

For continuous-time systems, specify input delays in the time unit specified by the TimeUnit property. For discrete-time systems, specify input delays in integer multiples of the sample time, Ts.

For more information, see Time Delays in Linear Systems.

OutputDelay — Output delay

0 (default) | scalar | Ny-by-1 vector

Output delay for each output channel, specified as one of the following:

• Scalar — Specify the output delay for a SISO system or the same delay for all outputs of a multi-output system.

• Ny-by-1 vector — Specify separate output delays for output of a multi-output system, where Ny is the number of outputs.

For continuous-time systems, specify output delays in the time unit specified by the TimeUnit property. For discrete-time systems, specify output delays in integer multiples of the sample time, Ts.

For more information, see Time Delays in Linear Systems.

Ts — Sample time

0 (default) | positive scalar | -1

Sample time, specified as:

• 0 for continuous-time systems.

• A positive scalar representing the sampling period of a discrete-time system. Specify Ts in the time unit specified by the TimeUnit property.

• -1 for a discrete-time system with an unspecified sample time.

Note

Changing Ts does not discretize or resample the model. To convert between continuous-time and discrete-time representations, use c2d and d2c. To change the sample time of a discrete-time system, use d2d.

TimeUnit — Time variable units

'seconds' (default) | 'nanoseconds' | 'microseconds' | 'milliseconds' | 'minutes' | 'hours' | 'days' | 'weeks' | 'months' | 'years' | ...

Time variable units, specified as one of the following:

• 'nanoseconds'

• 'microseconds'

• 'milliseconds'

• 'seconds'

• 'minutes'

• 'hours'

• 'days'

• 'weeks'

• 'months'

• 'years'

Changing TimeUnit has no effect on other properties, but changes the overall system behavior. Use chgTimeUnit to convert between time units without modifying system behavior.

InputName — Input channel names

'' (default) | character vector | cell array of character vectors

Input channel names, specified as one of the following:

• A character vector, for single-input models.

• A cell array of character vectors, for multi-input models.

• '', no names specified, for any input channels.

Alternatively, you can assign input names for multi-input models using automatic vector expansion. For example, if sys is a two-input model, enter the following.

sys.InputName = 'controls';

The input names automatically expand to {'controls(1)';'controls(2)'}.

You can use the shorthand notation u to refer to the InputName property. For example, sys.u is equivalent to sys.InputName.

Use InputName to:

• Identify channels on model display and plots.

• Extract subsystems of MIMO systems.

• Specify connection points when interconnecting models.

InputUnit — Input channel units

'' (default) | character vector | cell array of character vectors

Input channel units, specified as one of the following:

• A character vector, for single-input models.

• A cell array of character vectors, for multi-input models.

• '', no units specified, for any input channels.

Use InputUnit to specify input signal units. InputUnit has no effect on system behavior.

InputGroup — Input channel groups

structure

Input channel groups, specified as a structure. Use InputGroup to assign the input channels of MIMO systems into groups and refer to each group by name. The field names of InputGroup are the group names and the field values are the input channels of each group. For example, enter the following to create input groups named controls and noise that include input channels 1 and 2, and 3 and 5, respectively.

sys.InputGroup.controls = [1 2];
sys.InputGroup.noise = [3 5];

You can then extract the subsystem from the controls inputs to all outputs using the following.

sys(:,'controls')

By default, InputGroup is a structure with no fields.

OutputName — Output channel names

'' (default) | character vector | cell array of character vectors

Output channel names, specified as one of the following:

• A character vector, for single-output models.

• A cell array of character vectors, for multi-output models.

• '', no names specified, for any output channels.

Alternatively, you can assign output names for multi-output models using automatic vector expansion. For example, if sys is a two-output model, enter the following.

sys.OutputName = 'measurements';

The output names automatically expand to {'measurements(1)';'measurements(2)'}.

You can also use the shorthand notation y to refer to the OutputName property. For example, sys.y is equivalent to sys.OutputName.

Use OutputName to:

• Identify channels on model display and plots.

• Extract subsystems of MIMO systems.

• Specify connection points when interconnecting models.

OutputUnit — Output channel units

'' (default) | character vector | cell array of character vectors

Output channel units, specified as one of the following:

• A character vector, for single-output models.

• A cell array of character vectors, for multi-output models.

• '', no units specified, for any output channels.

Use OutputUnit to specify output signal units. OutputUnit has no effect on system behavior.

OutputGroup — Output channel groups

structure

Output channel groups, specified as a structure. Use OutputGroup to assign the output channels of MIMO systems into groups and refer to each group by name. The field names of OutputGroup are the group names and the field values are the output channels of each group. For example, create output groups named temperature and measurement that include output channels 1, and 3 and 5, respectively.

sys.OutputGroup.temperature = [1];
sys.OutputGroup.measurement = [3 5];

You can then extract the subsystem from all inputs to the measurement outputs using the following.

sys('measurement',:)

By default, OutputGroup is a structure with no fields.

Name — System name

'' (default) | character vector

System name, specified as a character vector. For example, 'system_1'.

Notes — User-specified text

{} (default) | character vector | cell array of character vectors

User-specified text that you want to associate with the system, specified as a character vector or cell array of character vectors. For example, 'System is MIMO'.

UserData — User-specified data

[] (default) | any MATLAB^® data type

User-specified data that you want to associate with the system, specified as any MATLAB data type.

SamplingGrid — Sampling grid for model arrays

structure array

Sampling grid for model arrays, specified as a structure array.

Use SamplingGrid to track the variable values associated with each model in a model array, including identified linear time-invariant (IDLTI) model arrays.

Set the field names of the structure to the names of the sampling variables. Set the field values to the sampled variable values associated with each model in the array. All sampling variables must be numeric scalars, and all arrays of sampled values must match the dimensions of the model array.

For example, you can create an 11-by-1 array of linear models, sysarr, by taking snapshots of a linear time-varying system at times t = 0:10. The following code stores the time samples with the linear models.

 sysarr.SamplingGrid = struct('time',0:10)

Similarly, you can create a 6-by-9 model array, M, by independently sampling two variables, zeta and w. The following code maps the (zeta,w) values to M.

[zeta,w] = ndgrid(<6 values of zeta>,<9 values of w>)
M.SamplingGrid = struct('zeta',zeta,'w',w)

When you display M, each entry in the array includes the corresponding zeta and w values.

M

M(:,:,1,1) [zeta=0.3, w=5] =
 
        25
  --------------
  s^2 + 3 s + 25
 

M(:,:,2,1) [zeta=0.35, w=5] =
 
         25
  ----------------
  s^2 + 3.5 s + 25
 
...

For model arrays generated by linearizing a Simulink^® model at multiple parameter values or operating points, the software populates SamplingGrid automatically with the variable values that correspond to each entry in the array. For instance, the Simulink Control Design™ commands linearize (Simulink Control Design) and slLinearizer (Simulink Control Design) populate SamplingGrid automatically.

By default, SamplingGrid is a structure with no fields.

Object Functions

The following lists contain a representative subset of the functions you can use with zpk models. In general, any function applicable to Dynamic System Models is applicable to a zpk object.

Linear Analysis

step	Step response of dynamic system
impulse	Impulse response plot of dynamic system; impulse response data
lsim	Compute time response simulation data of dynamic system to arbitrary inputs
bode	Bode frequency response of dynamic system
nyquist	Nyquist response of dynamic system
nichols	Nichols response of dynamic system
bandwidth	Frequency response bandwidth

Stability Analysis

pole	Poles of dynamic system
zero	Zeros and gain of SISO dynamic system
pzplot	Plot pole-zero map of dynamic system
margin	Gain margin, phase margin, and crossover frequencies

Model Transformation

tf	Transfer function model
ss	State-space model
c2d	Convert model from continuous to discrete time
d2c	Convert model from discrete to continuous time
d2d	Resample discrete-time model

Model Interconnection

feedback	Feedback connection of multiple models
connect	Block diagram interconnections of dynamic systems
series	Series connection of two models
parallel	Parallel connection of two models

Controller Design

pidtune	PID tuning algorithm for linear plant model
rlocus	Root locus of dynamic system
lqr	Linear-Quadratic Regulator (LQR) design
lqg	Linear-Quadratic-Gaussian (LQG) design
lqi	Linear-Quadratic-Integral control
kalman	Design Kalman filter for state estimation

Examples

Continuous-Time SISO Zero-Pole-Gain Model

For this example, consider the following continuous-time SISO zero-pole-gain model:

$$sys(s)=\frac{-2s}{(s-1-i)(s-1+i)(s-2)}$$

Specify the zeros, poles and gain, and create the SISO zero-pole-gain model.

zeros = 0;
poles = [1-1i 1+1i 2];
gain = -2;
sys = zpk(zeros,poles,gain)

sys =
 
          -2 s
  --------------------
  (s-2) (s^2 - 2s + 2)
 
Continuous-time zero/pole/gain model.
Model Properties

Discrete-Time SISO Zero-Pole-Gain Model

For this example, consider the following SISO discrete-time zero-pole-gain model with 0.1s sample time:

$$sys(s)=\frac{7(z-1)(z-2)(z-3)}{(z-6)(z-5)(z-4)}$$

Specify the zeros, poles, gains and the sample time, and create the discrete-time SISO zero-pole-gain model.

zeros = [1 2 3];
poles = [6 5 4];
gain = 7;
ts = 0.1;
sys = zpk(zeros,poles,gain,ts)

sys =
 
  7 (z-1) (z-2) (z-3)
  -------------------
   (z-6) (z-5) (z-4)
 
Sample time: 0.1 seconds
Discrete-time zero/pole/gain model.
Model Properties

Concatenate SISO Zero-Pole-Gain Models into a MIMO Zero-Pole-Gain Model

In this example, you create a MIMO zero-pole-gain model by concatenating SISO zero-pole-gain models. Consider the following single-input, two-output continuous-time zero-pole-gain model:

$$sys(s)=\left[\begin{array}{c}\frac{(s-1)}{(s+1)}\\ \frac{(s+2)}{(s+2+i)(s+2-i)}\end{array}\right].$$

Specify the MIMO zero-pole-gain model by concatenating the SISO entries.

zeros1 = 1;
poles1 = -1;
gain = 1;
sys1 = zpk(zeros1,poles1,gain)

sys1 =
 
  (s-1)
  -----
  (s+1)
 
Continuous-time zero/pole/gain model.
Model Properties

zeros2 = -2;
poles2 = [-2+1i -2-1i];
sys2 = zpk(zeros2,poles2,gain)

sys2 =
 
      (s+2)
  --------------
  (s^2 + 4s + 5)
 
Continuous-time zero/pole/gain model.
Model Properties

sys = [sys1;sys2]

sys =
 
  From input to output...
       (s-1)
   1:  -----
       (s+1)
 
           (s+2)
   2:  --------------
       (s^2 + 4s + 5)
 
Continuous-time zero/pole/gain model.
Model Properties

Discrete-Time MIMO Zero-Pole-Gain Model

Create a zero-pole-gain model for the discrete-time, multi-input, multi-output model:

$$sys\left(z\right)=\left[\begin{array}{cc}\frac{1}{(z+0.3)}& \frac{z}{(z+0.3)}\\ \frac{-(z-2)}{(z+0.3)}& \frac{3}{(z+0.3)}\end{array}\right]$$

with sample time ts = 0.2 seconds.

Specify the zeros and poles as cell arrays and the gains as an array.

zeros = {[] 0;2 []};
poles = {-0.3 -0.3;-0.3 -0.3};
gain = [1 1;-1 3];
ts = 0.2;

Create the discrete-time MIMO zero-pole-gain model.

sys = zpk(zeros,poles,gain,ts)

sys =
 
  From input 1 to output...
          1
   1:  -------
       (z+0.3)
 
       - (z-2)
   2:  -------
       (z+0.3)
 
  From input 2 to output...
          z
   1:  -------
       (z+0.3)
 
          3
   2:  -------
       (z+0.3)
 
Sample time: 0.2 seconds
Discrete-time zero/pole/gain model.
Model Properties

Specify Input Names for Zero-Pole-Gain Model

Specify the zeros, poles and gain along with the sample time and create the zero-pole-gain model, specifying the state and input names using name-value pairs.

zeros = 4;
poles = [-1+2i -1-2i];
gain = 3;
ts = 0.05;
sys = zpk(zeros,poles,gain,ts,'InputName','Force')

sys =
 
  From input "Force" to output:
     3 (z-4)
  --------------
  (z^2 + 2z + 5)
 
Sample time: 0.05 seconds
Discrete-time zero/pole/gain model.
Model Properties

The number of input names must be consistent with the number of zeros.

Naming the inputs and outputs can be useful when dealing with response plots for MIMO systems.

step(sys)

Notice the input name Force in the title of the step response plot.

Continuous-Time Zero-Pole-Gain Model Using Rational Expression

For this example, create a continuous-time zero-pole-gain model using rational expressions. Using a rational expression can sometimes be easier and more intuitive than specifying poles and zeros.

Consider the following system:

$$sys(s)=\frac{s}{s^2+2s+10}.$$

To create the transfer function model, first specify s as a zpk object.

s = zpk('s')

s =
 
  s
 
Continuous-time zero/pole/gain model.
Model Properties

Create the zero-pole-gain model using s in the rational expression.

sys = s/(s^2 + 2*s + 10)

sys =
 
         s
  ---------------
  (s^2 + 2s + 10)
 
Continuous-time zero/pole/gain model.
Model Properties

Discrete-Time Zero-Pole-Gain Model Using Rational Expression

For this example, create a discrete-time zero-pole-gain model using a rational expression. Using a rational expression can sometimes be easier and more intuitive than specifying poles and zeros.

Consider the following system:

$$sys(z)=\frac{z-1}{z^2-1.85z+0.9}.$$

To create the zero-pole-gain model, first specify z as a zpk object and the sample time ts.

ts = 0.1;
z = zpk('z',ts)

z =
 
  z
 
Sample time: 0.1 seconds
Discrete-time zero/pole/gain model.
Model Properties

Create the zero-pole-gain model using z in the rational expression.

sys = (z - 1) / (z^2 - 1.85*z + 0.9)

sys =
 
         (z-1)
  -------------------
  (z^2 - 1.85z + 0.9)
 
Sample time: 0.1 seconds
Discrete-time zero/pole/gain model.
Model Properties

Zero-Pole-Gain Model with Inherited Properties

For this example, create a zero-pole-gain model with properties inherited from another zero-pole-gain model. Consider the following two zero-pole-gain models:

$$sys1(s)=\frac{2s}{s(s+8)}andsys2(s)=\frac{0.8(s-1)}{(s+3)(s-5)}.$$

For this example, create sys1 with the TimeUnit and InputDelay property set to 'minutes'.

zero1 = 0;
pole1 = [0;-8];
gain1 = 2;
sys1 = zpk(zero1,pole1,gain1,'TimeUnit','minutes','InputUnit','minutes')

sys1 =
 
    2 s
  -------
  s (s+8)
 
Continuous-time zero/pole/gain model.
Model Properties

propValues1 = [sys1.TimeUnit,sys1.InputUnit]

propValues1 = 1×2 cell
    {'minutes'}    {'minutes'}

Create the second zero-pole-gain model with properties inherited from sys1.

zero = 1;
pole = [-3,5];
gain2 = 0.8;
sys2 = zpk(zero,pole,gain2,sys1)

sys2 =
 
   0.8 (s-1)
  -----------
  (s+3) (s-5)
 
Continuous-time zero/pole/gain model.
Model Properties

propValues2 = [sys2.TimeUnit,sys2.InputUnit]

propValues2 = 1×2 cell
    {'minutes'}    {'minutes'}

Observe that the zero-pole-gain model sys2 has that same properties as sys1.

Static Gain MIMO Zero-Pole-Gain Model

Consider the following two-input, two-output static gain matrix m:

$$m=\left[\begin{array}{cc}2& 4\\ 3& 5\end{array}\right]$$

Specify the gain matrix and create the static gain zero-pole-gain model.

m = [2,4;...
    3,5];
sys1 = zpk(m)

sys1 =
 
  From input 1 to output...
   1:  2
 
   2:  3
 
  From input 2 to output...
   1:  4
 
   2:  5
 
Static gain.
Model Properties

You can use static gain zero-pole-gain model sys1 obtained above to cascade it with another zero-pole-gain model.

sys2 = zpk(0,[-1 7],1)

sys2 =
 
       s
  -----------
  (s+1) (s-7)
 
Continuous-time zero/pole/gain model.
Model Properties

sys = series(sys1,sys2)

sys =
 
  From input 1 to output...
           2 s
   1:  -----------
       (s+1) (s-7)
 
           3 s
   2:  -----------
       (s+1) (s-7)
 
  From input 2 to output...
           4 s
   1:  -----------
       (s+1) (s-7)
 
           5 s
   2:  -----------
       (s+1) (s-7)
 
Continuous-time zero/pole/gain model.
Model Properties

Convert State-Space Model to Zero-Pole-Gain Model

For this example, compute the zero-pole-gain model of the following state-space model:

$$A=\left[\begin{array}{cc}-2& -1\\ 1& -2\end{array}\right],B=\left[\begin{array}{cc}1& 1\\ 2& -1\end{array}\right],C=\left[\begin{array}{cc}1& 0\end{array}\right],D=[\begin{array}{cc}0& 1\end{array}].$$

Create the state-space model using the state-space matrices.

A = [-2 -1;1 -2];
B = [1 1;2 -1];
C = [1 0];
D = [0 1];
ltiSys = ss(A,B,C,D);

Convert the state-space model ltiSys to a zero-pole-gain model.

sys = zpk(ltiSys)

sys =
 
  From input 1 to output:
        s
  --------------
  (s^2 + 4s + 5)
 
  From input 2 to output:
  (s^2 + 5s + 8)
  --------------
  (s^2 + 4s + 5)
 
Continuous-time zero/pole/gain model.
Model Properties

Array of Zero-Pole-Gain Models

You can use a for loop to specify an array of zero-pole-gain models.

First, pre-allocate the zero-pole-gain model array with zeros.

sys = zpk(zeros(1,1,3));

The first two indices represent the number of outputs and inputs for the models, while the third index is the number of models in the array.

Create the zero-pole-gain model array using a rational expression in the for loop.

s = zpk('s');                                                  
for k = 1:3                                                             
    sys(:,:,k) = k/(s^2+s+k);                                          
end
sys

sys(:,:,1,1) =
 
        1
  -------------
  (s^2 + s + 1)
 

sys(:,:,2,1) =
 
        2
  -------------
  (s^2 + s + 2)
 

sys(:,:,3,1) =
 
        3
  -------------
  (s^2 + s + 3)
 
3x1 array of continuous-time zero/pole/gain models.
Model Properties

Extract Zero-Pole-Gain Models from Identified Model

For this example, extract the measured and noise components of an identified polynomial model into two separate zero-pole-gain models.

Load the Box-Jenkins polynomial model ltiSys in identifiedModel.mat.

load('identifiedModel.mat','ltiSys');

ltiSys is an identified discrete-time model of the form: $$y(t)=\frac{B}{F}u(t)+\frac{C}{D}e(t)$$, where $$\frac{B}{F}$$ represents the measured component and $$\frac{C}{D}$$ the noise component.

Extract the measured and noise components as zero-pole-gain models.

sysMeas = zpk(ltiSys,'measured') 

sysMeas =
 
  From input "u1" to output "y1":
            -0.14256 z^-1 (1-1.374z^-1)
  z^(-2) * -----------------------------
           (1-0.8789z^-1) (1-0.6958z^-1)
 
Sample time: 0.04 seconds
Discrete-time zero/pole/gain model.
Model Properties

sysNoise = zpk(ltiSys,'noise')

sysNoise =
 
  From input "v@y1" to output "y1":
            0.045563 (1+0.7245z^-1)
  --------------------------------------------
  (1-0.9658z^-1) (1 - 0.0602z^-1 + 0.2018z^-2)
 
Input groups:        
    Name     Channels
    Noise       1    
                     
Sample time: 0.04 seconds
Discrete-time zero/pole/gain model.
Model Properties

The measured component can serve as a plant model, while the noise component can be used as a disturbance model for control system design.

Zero-Pole-Gain Model with Input and Output Delay

For this example, create a SISO zero-pole-gain model with an input delay of 0.5 seconds and an output delay of 2.5 seconds.

zeros = 5;
poles = [7+1i 7-1i -3];
gains = 1;
sys = zpk(zeros,poles,gains,'InputDelay',0.5,'OutputDelay',2.5)

sys =
 
                      (s-5)
  exp(-3*s) * ----------------------
              (s+3) (s^2 - 14s + 50)
 
Continuous-time zero/pole/gain model.
Model Properties

You can also use the get command to display all the properties of a MATLAB object.

get(sys)

                Z: {[5]}
                P: {[3×1 double]}
                K: 1
    DisplayFormat: 'roots'
         Variable: 's'
          IODelay: 0
       InputDelay: 0.5000
      OutputDelay: 2.5000
        InputName: {''}
        InputUnit: {''}
       InputGroup: [1×1 struct]
       OutputName: {''}
       OutputUnit: {''}
      OutputGroup: [1×1 struct]
            Notes: [0×1 string]
         UserData: []
             Name: ''
               Ts: 0
         TimeUnit: 'seconds'
     SamplingGrid: [1×1 struct]

For more information on specifying time delay for an LTI model, see Specifying Time Delays.

Control Design Using Zero-Pole-Gain Models

For this example, design a 2-DOF PID controller with a target bandwidth of 0.75 rad/s for a system represented by the following zero-pole-gain model:

$$sys(s)=\frac{1}{s^2+0.5s+0.1}$$

Create a zero-pole-gain model object sys using the zpk command.

zeros = [];
poles = [-0.25+0.2i;-0.25-0.2i];
gain = 1;
sys = zpk(zeros,poles,gain)

sys =
 
            1
  ---------------------
  (s^2 + 0.5s + 0.1025)
 
Continuous-time zero/pole/gain model.
Model Properties

Using the target bandwidth, use pidtune to generate a 2-DOF controller.

wc = 0.75;
C2 = pidtune(sys,'PID2',wc)

C2 =
 
                       1              
  u = Kp (b*r-y) + Ki --- (r-y) + Kd*s (c*r-y)
                       s              

  with Kp = 0.512, Ki = 0.0975, Kd = 0.574, b = 0.38, c = 0
 
Continuous-time 2-DOF PID controller in parallel form.
Model Properties

Using the type 'PID2' causes pidtune to generate a 2-DOF controller, represented as a pid2 object. The display confirms this result. The display also shows that pidtune tunes all controller coefficients, including the setpoint weights b and c, to balance performance and robustness.

For interactive PID tuning in the Live Editor, see the Tune PID Controller Live Editor task. This task lets you interactively design a PID controller and automatically generates MATLAB code for your live script.

For interactive PID tuning in a standalone app, use PID Tuner. See PID Controller Design for Fast Reference Tracking for an example of designing a controller using the app.

Compute Truncated ZPK Approximation of Sparse Model

Since R2025a

This example shows how to obtain a truncated zero-pole-gain model of a sparse state-space model. This example uses a sparse model obtained from linearizing a thermal model of heat distribution in a circular cylindrical rod.

Load the model data.

load cylindricalRod.mat
sys = sparss(A,B,C,D,E);
w = logspace(-7,-1,20);
size(sys)

Sparse state-space model with 3 outputs, 1 inputs, and 7522 states.

Analyze the frequency response of the model.

sigmaplot(sys,w)

To obtain a truncated approximation, use zpk and specify the frequency band of focus. For this model, you can use a frequency range from 0 rad/s to 0.01 rad/s to obtain the low-order approximation.

zsys = zpk(sys,Focus=[0 1e-2],Display="off");

Compare the frequency response.

sigmaplot(sys,zsys,w)

This thermal model has a very steep roll-off beyond 0.001 rad/s. By default, the reduced model obtained using zpk does not provide a good match for this roll-off. To mitigate this, you can use the RollOff argument of zpk and specify a minimum roll-off value beyond the frequency band of focus. Specify a roll-off slope value of -45, which corresponds to a rate of at least –900 db/decade.

zsys2 = zpk(sys,Focus=[0 1e-2],RollOff=-45,Display="off");
sigmaplot(sys,zsys2,w)

The reduced model now provides a much better approximation of the roll-off value. However, in this example, readjusting roll-off slope using zpk requires recomputing zeros and poles. This may be computationally expensive in case of large-scale models. As an alternative, you can use the zero-pole truncation method of reducespec and adjust roll-off at no extra computation cost, after the software has computed poles and zeros. For an example, see Zero-Pole Truncation of Thermal Model.

Algorithms

zpk uses the MATLAB function roots to convert transfer functions and the functions zero and pole to convert state-space models.

To convert sparse models, zpk uses the Krylov--Schur algorithm [1] for inverse power iterations to compute poles and zeros in the specified frequency band.

References

[1] Stewart, G. W. “A Krylov--Schur Algorithm for Large Eigenproblems.” SIAM Journal on Matrix Analysis and Applications 23, no. 3 (January 2002): 601–14. https://doi.org/10.1137/S0895479800371529.

Version History

Introduced before R2006a

R2025a: Compute Truncated Zero-Pole-Gain Approximation of Sparse Models

You can now use zpk to convert sparse state-space models and obtain a truncated zero-pole-gain approximation. zpk computes a subset of the zeros and poles of sparse models, typically in a specific low-frequency band [0 f_max]. This can yield better low-frequency approximations than modal truncation model-order reduction at the expense of more computation. zpk also provides direct control over the roll-off slope past the frequency f_max. Use the new syntax zsys = zpk(sparseSys,Name=Value) to obtain an approximation based on specified options, such as frequency range of focus. For an example, see Compute Truncated ZPK Approximation of Sparse Model.