Extended State Observer

Estimate states and disturbances of a system

Since R2024a

Libraries:
Simulink Control Design / Disturbance Observer

Description

The Extended State Observer block lets you design an extended state observer (ESO) algorithm to estimate and compensate for disturbances of linear time-invariant or linear time-varying systems. These disturbances represent total disturbances, including unknown dynamics and internal and external disturbances. Use this block to estimate the states and disturbances of a given discrete-time or continuous-time system based on the system input and output.

The block requires you to specify system dynamics in the following form.

$\begin{matrix} \dot{x} / x^{+} = A x + B_{u} u + B_{d} d \\ y_{m} = C_{m} x \end{matrix}$

Here:

x is the plant states
$\dot{x}$ is the state derivative in continuous time and x⁺ is the state update x[k+1] in discrete time.
A, B_u, and C_m are the plant dynamics.
y_m is the measurable plant output.
B_d is the disturbance input matrix.
d is the total disturbances in the model, including unknown dynamics and internal and external disturbances.

The block does not require you to provide an actual model of the plant. You can specify the dynamics obtained from a physical representation or an approximation of the plant. Therefore, the accuracy of estimation will depend on the accuracy of the plant approximation you provide.

For more information, see Control Design and Disturbance Compensation Using Extended State Observers.

Examples

Apply Extended State Observer for Reference Tracking of DC Motor

Improve the disturbance rejection performance of a PID controller using the Extended State Observer block.

Since R2024a
Open Live Script

Ports

Input

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u — Known inputs
scalar | vector

Specify the known inputs of the model as an N_u-by-1 signal, where N_u is the number of inputs in the model.

y — Measured outputs
scalar | vector

Specify the measured outputs of the model N_y-by-1 signal, where N_y is the number of outputs in the model.

A — State matrix
scalar | vector | matrix

Specify the state matrix as an N_x-by-N_x signal, where N_x is the number of states in the model.

Dependencies

To enable this port, select the Use external source for model matrices (A,Bu,Cm) and Use external source for L parameters.

Bu — Input matrix
scalar | vector | matrix

Specify the input matrix as an N_x-by-N_u signal, where N_x is the number of states and N_u is the number of inputs in the model.

Dependencies

To enable this port, select the Use external source for model matrices (A,Bu,Cm) and Use external source for L parameters.

Cm — Output matrix
scalar | vector | matrix

Specify the output matrix as an N_y-by-N_x signal, where N_x is the number of states and N_y is the number of outputs in the model.

Dependencies

To enable this port, select the Use external source for model matrices (A,Bu,Cm) and Use external source for L parameters.

Bd — Disturbance matrix
scalar | vector | matrix

Specify the disturbance matrix as an N_x-by-N_d signal, where N_x is the number of states and N_d is the number of disturbances in the model.

Dependencies

To enable this port, select the Use external source for matrix Bd and Use external source for L parameters.

S0 — Initial conditions
scalar | vector

Specify the initial conditions for the extended states as a scalar or vector with length N_x + N_d, where N_x is the number of states and N_d is the number of disturbances int the model.

Dependencies

To enable this port, select Use external source for initial conditions S0 (x0,d0).

L — Observer matrix
scalar | vector | matrix

Specify the observer matrix as an (N_x+N_d)-by-N_y signal.

Dependencies

To enable this port, select Use external source for matrix L.

Output

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dhat — Estimated disturbances
vector

Estimated disturbances in the model, returned as an N_d-by-1 signal, where N_d is the number of disturbances in the model.

xhat — Estimated states
vector

Estimated states of the model, returned as an N_x-by-1 signal, where N_x is the number of states in the model.

Dependencies

To enable this port, select Output estimated model states xhat.

Parameters

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To edit block parameters interactively, use the Property Inspector. From the Simulink^® Toolstrip, on the Simulation tab, in the Prepare gallery, select Property Inspector.

Parameters Tab

Time domain — Controller time domain
discrete-time (default) | continuous-time

Specify the controller time domain.

When you select discrete-time, specify the sample time using the Sample time parameter.

LTI state-space variable — Known plant dynamics
`ss` object | `idss` object

Specify the known plant dynamics as a state-space model object. You can obtain this model from a physical representation or an approximation of the plant.

Use external source for model matrices (A,Bu,Cm) — Specify model matrices at block input ports
`off` (default) | `on`

Select this option to specify the model matrices A, B_u, and C_m using the block input ports A, Bu, and Cm, respectively.

Dependencies

To enable this parameter, you must also select the Use external source for L parameter.

Number of states — Number of states in the model
`1` (default) | positive integer

When you specify model matrices as block inputs, use this parameter to specify the number of states in the model.

Dependencies

To enable this parameter, select the Use external source for model matrices (A,Bu,Cm) and Use external source for L parameters.

Number of inputs — Number of inputs in the model
`1` (default) | positive integer

When you specify model matrices as block inputs, use this parameter to specify the number of inputs in the model.

Dependencies

To enable this parameter, select the Use external source for model matrices (A,Bu,Cm) and Use external source for L parameters.

Number of outputs — Number of outputs in the model
`1` (default) | positive integer

When you specify model matrices as block inputs, use this parameter to specify the number of outputs in the model.

Dependencies

To enable this parameter, select the Use external source for model matrices (A,Bu,Cm) and Use external source for L parameters.

**Specify gain k (such that Bd = k*Bu)** — Gain value for computing disturbance input matrix
`1` (default) | nonzero scalar

Specify the gain value k for computing disturbance matrix B_d such that B_d = k×B_u.

Use external source for matrix Bd — Specify disturbance matrix at block input port
`off` (default) | `on`

Select this option to specify the disturbance matrix B_d using the block input port Bd.

Dependencies

To enable this parameter, you must also select the Use external source for L parameter.

Number of disturbances — Number of disturbances in the model
`1` (default) | positive integer

When you specify disturbance matrices as a block input, use this parameter to specify the number of inputs in the model.

Dependencies

To enable this parameter, select the Use external source for matrix Bd and Use external source for L parameters.

Initial conditions — Initial conditions for the extended states
`0` (default) | scalar | vector

Specify the initial conditions for the extended states as a scalar or vector with length N_x + N_d, where N_x is the number of states and N_d is the number of disturbances int the model.

Use external source for initial conditions S0 (x0,d0) — Specify initial conditions at block input port
`off` (default) | `on`

Select this option to specify the initial conditions using the block input port S0.

Tuning Tab

Bandwidth (rad/s) — Observer bandwidth
`1` (default) | positive scalar

Specify the observer bandwidth ω_o as a positive scalar.

Spacing (rad/s) — Observer bandwidth spacing
`0.01` (default) | positive scalar

Specify the spacing for the closed-loop poles. For given observer bandwidth ω_o and pole spacing δ, the block places the closed-loop poles at $[\begin{matrix} ω_{o} & ω_{o} + δ & ω_{o} + δ \times 2 & \dots & ω_{o} + δ \times n \end{matrix}]$ , where n is the number of closed-loop poles.

Use external source for matrix L — Specify observer matrix at block input port
`off` (default) | `on`

Select this option to specify the observer matrix using the block input port L.

Block Tab

Data type — Floating point precision
`double` (default) | `single`

Specify the floating-point precision based on simulation environment or hardware requirements.

For continuous-time models, the block only supports double data type.

Time delay — Specify time delays
`1` (default) | positive scalar | vector

Specify time delays in the model as a scalar or vector of length N_u, where N_u is the number of inputs in the model.

For discrete-time models, specify the delay as an integer multiple k of the sample time. The delay in discrete time is given by T_d = k×T_s.

Dependencies

To enable this parameter, select the Specify input delays parameter.

Output estimated model states xhat — Output estimated states at block output port
`on` (default) | `off`

Select this option to output estimated model states at the block output xhat.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2024a

Extended State Observer

Description

Examples

Apply Extended State Observer for Reference Tracking of DC Motor

Ports

Input

u — Known inputs scalar | vector

y — Measured outputs scalar | vector

A — State matrix scalar | vector | matrix

Dependencies

Bu — Input matrix scalar | vector | matrix

Dependencies

Cm — Output matrix scalar | vector | matrix

Dependencies

Bd — Disturbance matrix scalar | vector | matrix

Dependencies

S0 — Initial conditions scalar | vector

Dependencies

L — Observer matrix scalar | vector | matrix

Dependencies

Output

dhat — Estimated disturbances vector

xhat — Estimated states vector

Dependencies

Parameters

Parameters Tab

Time domain — Controller time domain discrete-time (default) | continuous-time

LTI state-space variable — Known plant dynamics ss object | idss object

Use external source for model matrices (A,Bu,Cm) — Specify model matrices at block input ports off (default) | on

Dependencies

Number of states — Number of states in the model 1 (default) | positive integer

Dependencies

Number of inputs — Number of inputs in the model 1 (default) | positive integer

Dependencies

Number of outputs — Number of outputs in the model 1 (default) | positive integer

Dependencies

Specify gain k (such that Bd = k*Bu) — Gain value for computing disturbance input matrix 1 (default) | nonzero scalar

Use external source for matrix Bd — Specify disturbance matrix at block input port off (default) | on

Dependencies

Number of disturbances — Number of disturbances in the model 1 (default) | positive integer

Dependencies

Initial conditions — Initial conditions for the extended states 0 (default) | scalar | vector

Use external source for initial conditions S0 (x0,d0) — Specify initial conditions at block input port off (default) | on

Tuning Tab

Bandwidth (rad/s) — Observer bandwidth 1 (default) | positive scalar

Spacing (rad/s) — Observer bandwidth spacing 0.01 (default) | positive scalar

Use external source for matrix L — Specify observer matrix at block input port off (default) | on

Block Tab

Data type — Floating point precision double (default) | single

Time delay — Specify time delays 1 (default) | positive scalar | vector

Dependencies

Output estimated model states xhat — Output estimated states at block output port on (default) | off

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using Simulink® Coder™.

Version History

See Also

Topics

u — Known inputs
scalar | vector

y — Measured outputs
scalar | vector

A — State matrix
scalar | vector | matrix

Bu — Input matrix
scalar | vector | matrix

Cm — Output matrix
scalar | vector | matrix

Bd — Disturbance matrix
scalar | vector | matrix

S0 — Initial conditions
scalar | vector

L — Observer matrix
scalar | vector | matrix

dhat — Estimated disturbances
vector

xhat — Estimated states
vector

Time domain — Controller time domain
discrete-time (default) | continuous-time

LTI state-space variable — Known plant dynamics
`ss` object | `idss` object

Use external source for model matrices (A,Bu,Cm) — Specify model matrices at block input ports
`off` (default) | `on`

Number of states — Number of states in the model
`1` (default) | positive integer

Number of inputs — Number of inputs in the model
`1` (default) | positive integer

Number of outputs — Number of outputs in the model
`1` (default) | positive integer

**Specify gain k (such that Bd = k*Bu)** — Gain value for computing disturbance input matrix
`1` (default) | nonzero scalar

Use external source for matrix Bd — Specify disturbance matrix at block input port
`off` (default) | `on`

Number of disturbances — Number of disturbances in the model
`1` (default) | positive integer

Initial conditions — Initial conditions for the extended states
`0` (default) | scalar | vector

Use external source for initial conditions S0 (x0,d0) — Specify initial conditions at block input port
`off` (default) | `on`

Bandwidth (rad/s) — Observer bandwidth
`1` (default) | positive scalar

Spacing (rad/s) — Observer bandwidth spacing
`0.01` (default) | positive scalar

Use external source for matrix L — Specify observer matrix at block input port
`off` (default) | `on`

Data type — Floating point precision
`double` (default) | `single`

Time delay — Specify time delays
`1` (default) | positive scalar | vector

Output estimated model states xhat — Output estimated states at block output port
`on` (default) | `off`

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.