# Kalman Filter

Estimate states of discrete-time or continuous-time linear system

• Libraries:
Control System Toolbox / State Estimation
System Identification Toolbox / Estimators

## Description

Use the Kalman Filter block to estimate states of a state-space plant model given process and measurement noise covariance data. The state-space model can be time-varying. A steady-state Kalman filter implementation is used if the state-space model and noise covariance matrices are all time-invariant, and a time-varying Kalman filter is used otherwise.

A Kalman filter provides the optimal solution to the continuous or discrete estimation problems in Continuous-Time Estimation and Discrete-Time Estimation.

The Kalman Filter block differs from the `kalman` (Control System Toolbox) command in the following ways:

• When you call `kalman(sys,...)`, it assumes that `sys` includes the `G` and `H` matrices. Specifically, `sys.B` is of the form `[B G]` and `sys.D` is of the form `[D H]`. When you provide a LTI variable to the Kalman Filter block, it does not assume that the LTI variable provided contains `G` and `H`. They are optional and separate.

• The filter created by the `kalman` command outputs `[yhat;xhat]` by default. The block outputs only `xhat` by default.

• The `Kalman` command can output both `P` and `Z` covariance matrices for discrete-time systems. The block can only output `P` or `Z` for such systems.

## Limitations

• The plant and noise data must satisfy these constraints:

• (C,A) is detectable.

• $\overline{R}>0$ and $\overline{Q}-\overline{N}{\overline{R}}^{-1}{\overline{N}}^{T}\ge 0$.

• $\left(A-\overline{N}{\overline{R}}^{-1}C,\overline{Q}-\overline{N}{\overline{R}}^{-1}{\overline{N}}^{T}\right)$ has no uncontrollable mode on the imaginary axis (or unit circle in discrete time), where

`$\begin{array}{l}\overline{Q}=GQ{G}^{T}\\ \overline{R}=R+HN+{N}^{T}{H}^{T}+HQ{H}^{T}\\ \overline{N}=G\left(Q{H}^{T}+N\right)\end{array}$`
• The continuous-time Kalman filter cannot be used in Function-Call Subsystems or Triggered Subsystems.

## Ports

### Input

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Known inputs `u(t)` or `u[k]`.

#### Dependencies

To enable this port, select the Add input port u parameter. This parameter is selected by default.

Measured output `y[n]` to update the estimated states.

n-by-n state (or system) matrix.

#### Dependencies

To enable this port, set the Model source parameter to Input Port.

n-by-p input matrix.

#### Dependencies

To enable this port, set the Model source parameter to Input Port and select the Add input port u parameter.

q-by-p output matrix.

#### Dependencies

To enable this port, set the Model source parameter to Input Port.

q-by-p feedthrough (or feedforward) matrix. In cases where the system model does not have a direct feedthrough, D is the zero matrix.

#### Dependencies

To enable this port, set the Model source parameter to Input Port and select the Add input port u parameter.

Noise transformation in the state-space equation.

#### Dependencies

To enable this port, select Use G and H matrices (default G=I and H=0) parameter.

Noise transformation in the state-space equation.

#### Dependencies

To enable this port, select Use G and H matrices (default G=I and H=0) parameter.

Process noise covariance matrix, specified as one of the following:

• Real nonnegative scalar. Q is an Nw-by-Nw diagonal matrix with the scalar on the diagonals. Nw is the number of process noise inputs in the model.

• Vector of real nonnegative scalars. Q is an Nw-by-Nw diagonal matrix with the elements of the vector on the diagonals of Q.

• Nw-by-Nw positive semi-definite matrix.

#### Dependencies

To enable this port, deselect the Time-invariant Q parameter.

Measurement noise covariance matrix, specified as one of the following:

• Real positive scalar. R is an Ny-by-Ny diagonal matrix with the scalar on the diagonals. Ny is the number of measured outputs in the model.

• Vector of real positive scalars. R is an Ny-by-Ny diagonal matrix with the elements of the vector on the diagonals of R.

• Ny-by-Ny positive-definite matrix.

#### Dependencies

To enable this port, deselect the Time-invariant R parameter.

Process and measurement noise cross-covariance matrix, specified as a Nw-by-Ny matrix. The matrix [Q N; NT R] must be positive definite.

#### Dependencies

To enable this port, deselect the Time-invariant N parameter.

P matrix at the initial time.

#### Dependencies

To enable this port, set the Model source parameter to Input Port and set the Source parameter to Input Port.

Estimated states at the initial time.

#### Dependencies

To enable this port, set the Source parameter to Input Port.

This port controls the measurement updates and takes a scalar signal.

#### Dependencies

To enable this port, select Add input port Enable to control measurement updates parameter.

Control signal to reset estimated states and the parameter covariance matrix using specified initial values. See External Reset for more information on when a reset is triggered.

#### Dependencies

To enable this port, set the External reset paramter to any value other than None.

### Output

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Estimated states of the linear system.

Estimated outputs of the linear system.

#### Dependencies

To enable this port, select Output estimated model output y parameter.

Add Z output port to the block.

To enable this port, select Output state estimation error covariance Z parameter.

#### Dependencies

To enable this port, set the Time domain parameter to Discrete-Time and select Use the current measurement y[n] to improve xhat[n] parameter.

Add P output port to the block.

To enable this port, select Output state estimation error covariance P parameter.

#### Dependencies

To enable this port, set the Time domain parameter to Continuous-Time or set the Time domain parameter to Discrete-Time and deselect Use the current measurement y[n] to improve xhat[n] parameter.

Note

• All input ports except Enable and Reset must have the same data type (single or double).

• Enable and Reset ports support `single`, `double`, `int8`, `uint8`, `int16`, `uint16`, `int32`, `uint32`, and boolean data types.

## Parameters

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### Filter Settings

• `Discrete-Time` (default) — Block estimates discrete-time states.

• `Continuous-Time` — Block estimates continuous-time states.

When the Kalman Filter block is in a model with synchronous state control (see the State Control (HDL Coder) block), you cannot select Continuous-Time.

#### Programmatic Use

 Block Parameter: `TimeDomain` Type: string, character vector Values: `"Discrete-Time"` | `"Continuous-Time"` Default: `"Discrete-Time"`

Use the current estimator variant of the discrete-time Kalman filter. When this parameter is not selected, the delayed estimator (variant) is used.

This parameter is available only when Time domain is Discrete-Time.

#### Programmatic Use

 Block Parameter: `UseCurrentEstimator` Type: string, character vector Values: `"off"` | `"on"` Default: `"on"`

Block sample time, specified as `-1` or a positive scalar.

The default value is `-1`, which implies that the block inherits its sample time based on the context of the block within the model. All block input ports must have the same sample time.

#### Dependencies

This parameter is available only when Time domain is Discrete-Time and Model source is Individual A, B, C, D matrices or Input port. When Model source is LTI State-Space Variable, the block takes its sample time from the LTI state-space variable.

#### Programmatic Use

 Block Parameter: `Ts` Type: string, character vector Values: `"-1"` | scalar Default: `"-1"`

### Model Parameters

System Model
• ```LTI State-Space Variable``` — Use the model specified in Variable. The default value is `ss(0.95,1,1,0)`. The sample time of the model must match the Time domain parameter; that is, the model must be discrete-time if Time domain is discrete-time.

• `Individual A, B, C, D matrices` — Specify the A, B, C, and D in the block paramaters.

• `Input port` — Specify the A, B, C, D matrices as input signals to the Kalman Filter block. If you select this option, the block includes additional input ports A, B, C, D. You must also specify Number of states, Number of inputs, and Number of outputs in the block parameters.

#### Programmatic Use

 Block Parameter: `ModelSource` Type: string, character vector Values: `"LTI State-Space Varaible"` | `"Individual A, B, C, D matrices"` | `"Input port"` Default: `"LTI State-Space Variable"`

Specify the A matrix. It must be real and square. The default value is `0.95`.

#### Dependencies

To enable this port, set the Model source parameter to Individual A, B, C, D matrices.

#### Programmatic Use

 Block Parameter: `A` Type: string, character vector Values: `"real matrix"` Default: `"0.95"`

Specify the B matrix. It must be real and have as many rows as the A matrix. The default value is `1`.

#### Dependencies

To enable this port, set the Model source parameter to Individual A, B, C, D matrices.

#### Programmatic Use

 Block Parameter: `B` Type: string, character vector Values: `"real matrix"` Default: `"1"`

Specify the C matrix. It must be real and have as many columns as the A matrix. The default value is `1`.

#### Dependencies

To enable this port, set the Model source parameter to Individual A, B, C, D matrices.

#### Programmatic Use

 Block Parameter: `C` Type: string, character vector Values: `"real matrix"` Default: `"1"`

Specify the D matrix. It must be real and must have as many rows as the C matrix and as many columns as the B matrix. The default value is `0`.

#### Dependencies

To enable this port, set the Model source parameter to Individual A, B, C, D matrices.

#### Programmatic Use

 Block Parameter: `D` Type: string, character vector Values: `"real matrix"` Default: `"0"`

Number of states to be estimated, specified as a positive integer. The default value is `1`.

#### Dependencies

To enable this port, set the Model source parameter to Input port.

#### Programmatic Use

 Block Parameter: `NumberOfStates` Type: string, character vector Values: `"1"` | `scalar` Default: `"1"`

Number of known inputs in the model, specified as a positive integer. The default value is `1`.

#### Dependencies

To enable this port, set the Model source parameter to Input port.

#### Programmatic Use

 Block Parameter: `NumberOfInputs` Type: string, character vector Values: `"1"` | `scalar` Default: `"1"`

Number of measured outputs in the model, specified as a positive integer. The default value is `1`.

#### Dependencies

To enable this port, set the Model source parameter to Input port.

#### Programmatic Use

 Block Parameter: `NumberOfOutputs` Type: string, character vector Values:`"1"` | `scalar` Default: `"1"`
Initial Estimates
• `Dialog` — Specify the values directly in the dialog boxes.

• `Input port` — Inherit the values from input ports. The default is `10`. The block includes an additional input port X0. A second additional input port P0 is added when time-varying Kalman filter is used. X0 and P0 must satisfy the same conditions as the parameters Initial states x and State estimation error covariance P, respectively.

#### Programmatic Use

 Block Parameter: `InitialEstimateSource` Type: string, character vector Values: `"Dialog"` | ```"Input port"``` Default: `"Dialog"`

Specify the initial state estimate as a real scalar or vector. If you specify a scalar, all initial state estimates are set to this scalar. If you specify a vector, the length of the vector must match with the number of states in the model. The default is `0`.

#### Dependencies

To enable this port, set the Source parameter to Dialog.

#### Programmatic Use

 Block Parameter: `X0` Type: string, character vector Values: `"0"` | `scalar` | `vector` Default: `"0"`

Specify the initial state estimation error covariance P for a discrete-time Kalman filter or P(0) for continuous-time. This parameter must be specified as one of the following:

• Real nonnegative scalar. P is an Ns-by-Ns diagonal matrix with the scalar on the diagonals. Ns is the number of states in the model.

• Vector of real nonnegative scalars. P is an Ns-by-Ns diagonal matrix with the elements of the vector on the diagonals of P.

• Ns-by-Ns positive semi-definite matrix.

#### Dependencies

To enable this port, set the Model source parameter to Input port and Source parameter to Dialog.

#### Programmatic Use

 Block Parameter: `P0` Type: string, character vector Values: `"10"` | `scalar` | `vector` | `matrix` Default: `"10"`
Noise Characterstics

Specify whether to use the pre-identified Kalman Gain present in the state-space model specified by Variable.

#### Dependencies

To enable this parameter, you must meet the following conditions:

• The Model source is set toLTI State-Space Variable and Variable is an identified state-space model (`idss`) with a nonzero K matrix.

• Select the Time Invariant Q, Time Invariant R , and Time Invariant N parameters.

• If the Use G and H matrices (default G=I and H=0) parameter is selected, the Time Invariant G and Time Invariant H parameters must also be selected.

#### Programmatic Use

 Block Parameter: `UseK` Type: string, character vector Values: `"off"` | `"on"` Default: `"off"`

By default `G=I` and `H=0`. If you select this option, you must specify G and H parameter.

#### Programmatic Use

 Block Parameter: `UseGH` Type: string, character vector Values: `"off"` | `"on"` Default: `"off"`

It must be a real matrix with as many rows as the A matrix. The default value is `1`.

#### Dependencies

To enable this paramter, select Use G and H matrices (default G=I and H=0) parameter.

#### Programmatic Use

 Block Parameter: `G` Type: string, character vector Values: scalar | vector | matrix Default: `"1"`

If you unselect this option, the block includes an additional input port G.

#### Programmatic Use

 Block Parameter: `TimeInvariantG` Type: string, character vector Values: `"off"` | `"on"` Default: `"on"`

It must be a real matrix with as many rows as the C matrix and as many columns as the G matrix. The default value is `0`.

#### Dependencies

To enable this paramter, select Use G and H matrices (default G=I and H=0) parameter.

#### Programmatic Use

 Block Parameter: `H` Type: string, character vector Values: scalar | vector | matrix Default: `"0"`

If you unselect this option, the block includes an additional input port H.

#### Programmatic Use

 Block Parameter: `TimeInvariantH` Type: string, character vector Values: `"off"` | `"on"` Default: `"on"`

Specify the number of process noise inputs in the model. The default value is `1`.

#### Dependencies

This parameter is available only when Time-invariant G and Time-invariant H are deselected. Otherwise, this information is inferred from the G or H matrix.

#### Programmatic Use

 Block Parameter: `NumberOfProcessNoiseInputs` Type: string, character vector Values: scalar Default: `"1"`

Specified as one of the following:

• Real nonnegative scalar. Q is an Nw-by-Nw diagonal matrix with the scalar on the diagonals. Nw is the number of process noise inputs in the model.

• Vector of real nonnegative scalars. Q is an Nw-by-Nw diagonal matrix with the elements of the vector on the diagonals of Q.

• Nw-by-Nw positive semi-definite matrix.

#### Dependencies

To enable this paramater, select the Time-invariant Q parameter.

#### Programmatic Use

 Block Parameter: `Q` Type: string, character vector Values: scalar | vector | matrix Default: `"0.05"`

If you deselect this parameter, the block includes an additional input port Q.

#### Programmatic Use

 Block Parameter: `TimeInvariantQ` Type: string, character vector Values: `"off"` | `"on"` Default: `"on"`

Specified as one of the following:

• Real positive scalar. R is an Ny-by-Ny diagonal matrix with the scalar on the diagonals. Ny is the number of measured outputs in the model.

• Vector of real positive scalars. R is an Ny-by-Ny diagonal matrix with the elements of the vector on the diagonals of R.

• Ny-by-Ny positive-definite matrix.

#### Dependencies

To enable this paramater, select the Time-invariant R parameter.

#### Programmatic Use

 Block Parameter: `R` Type: string, character vector Values: scalar | vector | matrix Default: `"1"`

If you deselect this parameter, the block includes an additional input port R.

#### Programmatic Use

 Block Parameter: `TimeInvariantR` Type: string, character vector Values: `"off"` | `"on"` Default: `"on"`

Specify this parameter as a Nw-by-Ny matrix. The matrix [Q N; NT R] must be positive definite.

#### Dependencies

To enable this paramater, select the Time-invariant N parameter.

#### Programmatic Use

 Block Parameter: `N` Type: string, character vector Values: scalar | vector | matrix Default: `"0"`

If you deselect this parameter, the block includes an additional input port N.

#### Programmatic Use

 Block Parameter: `TimeInvariantN` Type: string, character vector Values: `"off"` | `"on"` Default: `"on"`

### Options

Select this option if your model contains known inputs `u(t)` or `u[k]`. The parameter is selected by default. Deselecting this parameter removes the input port u from the block and removes the B, D and Number of inputs parameters from the block dialog box.

#### Programmatic Use

 Block Parameter: `AddInputPort` Type: string, character vector Values: `"off"` | `"on"` Default: `"on"`

Select this option if you want to control the measurement updates. The block includes an additional inport Enable. The Enable input port takes a scalar signal. This parameter is not selected by default.

By default the block does measurement updates at each time step to improve the state and output estimates $\stackrel{^}{x}$ and $\stackrel{^}{y}$ based on measured outputs. The measurement update is skipped for the current sample time when the signal in the Enable port is `0`. Concretely, the equation for state estimates become $\stackrel{˙}{\stackrel{^}{x}}\left(t\right)=A\left(t\right)\stackrel{^}{x}\left(t\right)+B\left(t\right)u\left(t\right)$ for a continuous-time Kalman filter and $\stackrel{^}{x}\left[n+1|n\right]=A\left[n\right]\stackrel{^}{x}\left[n|n-1\right]+B\left[n\right]u\left[n\right]$ for discrete-time.

Note

Enabling the Enable port allows measurement updates to be controlled. By default, Kalman Filter does the measurement updates.

#### Programmatic Use

 Block Parameter: `AddEnablePort` Type: string, character vector Values: `"off"` | `"on"` Default: `"off"`

This parameter helps control when the block is reset. Suppose you reset the block at a time step, `t`. If the block is enabled at `t`, the software uses the initial parameter values specified either in the block dialog or the input ports P0 and X0 to estimate the states. In other words, at `t`, the block performs a time update and, if it is enabled, a measurement update after the reset. The block outputs these updated estimates.

Specify one of the following:

• `None` (Default) — Estimated states $\stackrel{^}{x}$ and state estimation error covariance matrix P values are not reset.

• `Rising` — Triggers a reset when the control signal rises from a negative or zero value to a positive value. If the initial value is negative, rising to zero triggers a reset.

• `Falling` — Triggers a reset when the control signal falls from a positive or zero value to a negative value. If the initial value is positive, falling to zero triggers a reset.

• `Either` — Triggers a reset when the control signal is either rising or falling.

• `Level` — Triggers a reset in either of these cases:

• The control signal is nonzero at the current time step.

• The control signal changes from nonzero at the previous time step to zero at the current time step.

• `Level hold` — Triggers reset when the control signal is nonzero at the current time step.

When you choose an option other than `None`, a Reset input port is added to the block to provide the reset control input signal.

#### Programmatic Use

 Block Parameter: `ExternalReset` Type: string, character vector Values: `"None"` | `"Rising"` | `"Falling"` | `"Either"` | `"Level"` | ```"Level hold"``` Default: `"None"`

Add a $\stackrel{^}{y}$ output port to the block to output the estimated model outputs. The parameter is not selected by default.

#### Programmatic Use

 Block Parameter: `OutputEstimatedY` Type: string, character vector Values: `"off"` | `"on"` Default: `"off"`

Add a Z output port to the block. The Z matrix is provided only when Time domain is Discrete-Time and the Use the current measurement y[n] to improve xhat[n] parameter is selected. Otherwise, the P matrix, as described in the Algorithms section, is provided.

This parameter is not selected by default.

#### Programmatic Use

 Block Parameter: `OutputZ` Type: string, character vector Values: `"off"` | `"on"` Default: `"off"`

Add a P output port to the block. This parameter is not selected by default.

#### Dependencies

To enable this port, set the Time domain parameter to Continuous-Time or set the Time domain parameter to Discrete-Time and deselect Use the current measurement y[n] to improve xhat[n] parameter.

#### Programmatic Use

 Block Parameter: `OutputP` Type: string, character vector Values: `"off"` | `"on"` Default: `"off"`

## Algorithms

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 Franklin, Gene F., J. David Powell, and Michael L. Workman. Digital Control of Dynamic Systems. 2nd ed. Reading, Mass: Addison-Wesley, 1990.

 Lewis, Frank L. Optimal Estimation: With an Introduction to Stochastic Control Theory. New York: Wiley, 1986.