Kalman Filter

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

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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.

Examples

State Estimation Using Time-Varying Kalman Filter

Estimate states of linear systems using time-varying Kalman filters in Simulink.

• Open Script

Limitations

• The plant and noise data must satisfy these constraints:

  • (C,A) is detectable.

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

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

    $$\begin{array}{l}\overline{Q}=GQ{G}^T\\ \overline{R}=R+HN+N^T{H}^T+HQ{H}^T\\ \overline{N}=G(Q{H}^T+N)\end{array}$$

• The continuous-time Kalman filter cannot be used in Function-Call Subsystems or Triggered Subsystems.

Ports

Input

u — Known inputs

scalar | vector

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.

y — Measured outputs

scalar | vector

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

A — State matrix

real matrix

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

Dependencies

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

B — Input matrix

real matrix

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.

C — Output matrix

real matrix

q-by-p output matrix.

Dependencies

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

D — Feedthrough matrix

real matrix

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.

G — Measured outputs

real matrix

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.

H — Measured outputs

real matrix

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.

Q — Process noise covariance matrix

scalar | vector | matrix

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.

R — Measurement noise covariance matrix

scalar | vector | matrix

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.

N — Process and measurement noise cross-covariance matrix

scalar | vector | matrix

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

Dependencies

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

P0 — Initial state estimation error covariance

scalar | vector | matrix

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.

X0 — Initial state estimates

scalar | vector

Estimated states at the initial time.

Dependencies

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

Enable — Control signal to enable measurement updates

scalar

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.

Reset — Control signal to reset state estimates

scalar

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 parameter to any value other than None.

Output

xhat — Estimated states

scalar | vector

Estimated states of the linear system.

yhat — Estimated outputs

scalar | vector

Estimated outputs of the linear system.

Dependencies

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

Z — State estimation error covariance

matrix

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.

P — State estimation error covariance

matrix

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

Filter Settings

Time domain — Specify whether to estimate continuous-time or discrete-time states

Discrete-Time (default) | Continuous-Time

• 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 measurement y[n] to improve xhat[n] — Update output state estimates using the measure outputs

on (default) | off

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"

Sample Time — Block sample time

-1 (default) | non-negative scalar

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

Model source — Specify how the A, B, C, D matrices are provided to the block

LTI State-Space Variable (default) | Individual A, B, C, D matrices | Input port

• 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 parameters.

• 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 Variable" | "Individual A, B, C, D matrices" | "Input port"

Default: "LTI State-Space Variable"

A — State Matrix

0.95 (default)

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"

B — Input Matrix

1 (default)

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"

C — Output Matrix

1 (default)

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"

D — Feedthrough Matrix

0 (default)

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 — Number of states to be estimated

1 (default) | non-negative scalar

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 inputs — Number of known inputs

1 (default) | non-negative scalar

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 outputs — Number of measured output

1 (default) | non-negative scalar

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

Source — Specify how to enter the initial state estimates and initial state estimation error covariance

Dialog (default) | Input port

• 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[0] and State estimation error covariance P[0], respectively.

Programmatic Use

Block Parameter: InitialEstimateSource

Type: string, character vector

Values: "Dialog" | "Input port"

Default: "Dialog"

Initial states x[0] — Specify the initial state estimate

0 (default) | scalar | vector

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"

State estimation error covariance P[0] — Specify the initial state estimation error covariance

10 (default) | scalar | vector | matrix

Specify the initial state estimation error covariance P[0] 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 Characteristics

Use the Kalman Gain K from the model variable — Specify whether to use pre-identified Kalman Gain contained in state-space plant model

off (default) | on

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 to LTI 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"

Use G and H matrices (default G=I and H=0) — Specify whether to use non-default values for G and H matrices

off (default) | on

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"

G — Specify the G matrix

1 (default) | non-negative scalar | vector | matrix

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

Dependencies

To enable this parameter, 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"

Time-invariant G — Specify the G matrix is time invariant

on (default) | off

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"

H — Specify the H matrix

0 (default) | non-negative scalar | vector | matrix

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 parameter, 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"

Time-invariant H — Specify the H matrix is time invariant

on (default) | off

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"

Number of process noise inputs — Process noise inputs

1 (default) | non-negative scalar

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"

Q — Process noise covariance matrix

0.05 (default) | non-negative scalar | vector | 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 parameter, select the Time-invariant Q parameter.

Programmatic Use

Block Parameter: Q

Type: string, character vector

Values: scalar | vector | matrix

Default: "0.05"

Time-invariant Q — Specify if Q matrix is time invariant

on (default) | off

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"

R — Measurement noise covariance matrix

1 (default) | non-negative scalar | vector | 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 parameter, select the Time-invariant R parameter.

Programmatic Use

Block Parameter: R

Type: string, character vector

Values: scalar | vector | matrix

Default: "1"

Time-invariant R — Specify if R matrix is time invariant

on (default) | off

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"

N — Process and measurement noise cross-covariance matrix

0 (default) | non-negative scalar | vector | matrix

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

Dependencies

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

Programmatic Use

Block Parameter: N

Type: string, character vector

Values: scalar | vector | matrix

Default: "0"

Time-invariant N — Specify if N matrix is time invariant

on (default) | off

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

Additional Inports

Add input port u — Specify if model contains known inputs

on (default) | off

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"

Add input port Enable to control measurement updates — Control the measurement updates

off (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 $$\widehat{x}$$ and $$\widehat{y}$$ 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 $$\dot{\widehat{x}}(t)=A(t)\widehat{x}(t)+B(t)u(t)$$ for a continuous-time Kalman filter and $$\widehat{x}[n+1|n]=A[n]\widehat{x}[n|n-1]+B[n]u[n]$$ 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"

External Reset — Option to reset estimated states and parameter covariance matrix using specified initial values

None (default) | Rising | Falling | Either | Level | Level hold

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 $$\widehat{x}$$ 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"

Additional Outports

Output estimated model output y — Include estimated model outputs

off (default) | on

Add a $$\widehat{y}$$ 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"

Output state estimation error covariance Z — Add Z output port to the block

off (default) | on

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"

Output state estimation error covariance P — Add P output port to the block

off (default) | on

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

Continuous-Time Estimation

Given the continuous plant

$$\begin{array}{l}\dot{x}(t)=A(t)x(t)+\text{}B(t)u(t)+G(t)w(t)\text{ (state equation)}\\ y(t)=C(t)x(t)+D(t)u(t)+H(t)w(t)+v(t)\text{ (measurement equation)}\\ \end{array}$$

with known inputs u, white process noise w, and white measurement noise v satisfying

$$\begin{array}{l}E\left[w(t)\right]=E\left[v(t)\right]=0\\ E[w(t)w^T(t)]=Q(t)\\ E[w(t)v^T(t)]=N(t)\\ E[v(t)v{(}^t]=R(t)\end{array}$$

construct a state estimate $$\widehat{x}$$ that minimizes the state estimation error covariance $$P(t)=E[(x-\widehat{x}){(x-\widehat{x})}^T]$$.

The optimal solution is the Kalman filter with equations

$$\begin{array}{l}L(t)=(P(t)C^T(t)+\overline{N}(t))\overline{R}{(}^t,\\ \dot{P}(t)=A(t)P(t)+P(t)A^T(t)+\overline{Q}(t)-L(t)\overline{R}(t)L^T(t),\\ \dot{\widehat{x}}(t)=A(t)\widehat{x}(t)+B(t)u(t)+L(t)(y(t)-C(t)\widehat{x}(t)-D(t)u(t)),\end{array}$$

where

$$\begin{array}{l}\overline{Q}(t)=G(t)Q(t)G^T(t),\\ \overline{R}(t)=R(t)+H(t)N(t)+N^T(t)H^T(t)+H(t)Q(t)H^T(t),\\ \overline{N}(t)=G(t)(Q(t)H^T(t)+N(t)).\end{array}$$

The Kalman filter uses known inputs u and measurements y to generate state estimates $$\widehat{x}$$. If you want, the block can also output the estimates of the true plant output $$\widehat{y}$$.

The block implements the steady-state Kalman filter when the system matrices (A(t), B(t), C(t), D(t), G(t), H(t)) and noise covariance matrices (Q(t), R(t), N(t)) are constant (specified in the Block Parameters dialog box). The steady-state Kalman filter uses a constant matrix P that minimizes the steady-state estimation error covariance and solves the associated continuous-time algebraic Riccati equation:

$$P=\underset{t\to \infty }{\lim }E[(x-\widehat{x}){(x-\widehat{x})}^T].$$

Discrete-Time Estimation

Consider the discrete plant

$$\begin{array}{l}x\left[n+1\right]=A\left[n\right]x\left[n\right]+B\left[n\right]u\left[n\right]+G\left[n\right]w\left[n\right],\\ y\left[n\right]=C\left[n\right]x\left[n\right]+D\left[n\right]u\left[n\right]+H\left[n\right]w\left[n\right]+v\left[n\right],\end{array}$$

with known inputs u, white process noise w, and white measurement noise v satisfying

$$\begin{array}{l}E[w[n]]=E[v[n]]=0,\\ E[w[n]w^T[n]]=Q[n],\\ E[v[n]v^T[n]]=R[n],\\ E[w[n]v^T[n]]=N[n].\end{array}$$

The estimator has the following state equation

$$\widehat{x}[n+1|n]=A[n]\widehat{x}[n|n-1]+B[n]u[n]+L[n](y[n]-C[n]\widehat{x}[n|n-1]-D[n]u[n]),$$

where the gain L[n] is calculated through the discrete Riccati equation:

$$\begin{array}{l}L[n]=(A[n]P[n]C^T[n]+\overline{N}[n]){(}^C,\\ M[n]=P[n]C^T[n]{(}^C,\\ Z[n]=(I-M[n]C[n])P[n]{(}^I+M[n]\overline{R}[n]M^T[n],\\ P[n+1]=(A[n]-\overline{N}[n]{\overline{R}}^{-1}[n]C[n])Z{(}^A+\overline{Q}[n]-N[n]{\overline{R}}^{-1}[n]{\overline{N}}^T[n],\end{array}$$

where I is the identity matrix of appropriate size and

$$\begin{array}{l}\overline{Q}\text{[n]}=G\text{[n]}Q\text{[n]}{G}^T\text{[n],}\\ \overline{R}\text{[n]}=R\text{[n]}+H\text{[n]}N\text{[n]}+N^T\text{[n]}{H}^T\text{[n]}+H\text{[n]}Q\text{[n]}{H}^T\text{[n],}\\ \overline{N}\text{[n]}=G\text{[n]}(Q\text{[n]}{H}^T\text{[n]}+N\text{[n]}),\\ \text{and}\\ P\text{[n]}=E[(x-\widehat{x}[n|n-1]){(}^x],\\ Z\text{[n]}=E[(x-\widehat{x}[n|n]){(}^x],\end{array}$$

The steady-state Kalman filter uses a constant matrix P that minimizes the steady-state estimation error covariance and solves the associated discrete-time algebraic Riccati equation.

There are two variants of discrete-time Kalman filters:

• The current estimator generates the state estimates $$\widehat{x}[n|n]$$ using all measurements available, including y[n]. The filter updates $$\widehat{x}[n|n-1]$$ with y[n] and outputs:

  $$\begin{array}{l}\widehat{x}[n|n]=\widehat{x}[n|n-1]+M[n](y[n]-C[n]\widehat{x}[n|n-1]-D[n]u[n]),\\ \widehat{y}[n|n]=C[n]\widehat{x}[n|n]+D[n]u[n].\end{array}$$

• The delayed estimator generates the state estimates $$\widehat{x}[n|n-1]$$ using measurements up to y[n –1]. The filter outputs $$\widehat{x}[n|n-1]$$ as defined previously, along with the optional output

  $$\widehat{y}[n|n-1]=C[n]\widehat{x}[n|n-1]+D[n]u[n]$$

The current estimator has better estimation accuracy than the delayed estimator, which is important for slow sample times. However, it has a higher computational cost, so implementing it inside control loops is harder. More specifically, it has direct feedthrough which leads to an algebraic loop if the Kalman filter is used in a feedback loop that does not contain any delays (the feedback loop itself also has direct feedthrough). This algebraic loop can impact the speed of simulation, and you cannot generate code if your model contains algebraic loops.

References

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

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

Extended Capabilities

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

PLC Code Generation
Generate Structured Text code using Simulink® PLC Coder™.

Version History

Introduced in R2014b

R2021a: Kalman Filter block: Numerical changes

Starting in 2021a, numerical improvements in the algorithms used by the Kalman Filter block might produce results that are different from the results you obtained using previous versions.