Documentation

# predict

Predict state and state estimation error covariance at next time step using extended or unscented Kalman filter, or particle filter

## Description

The predict command predicts the state and state estimation error covariance of an extendedKalmanFilter, unscentedKalmanFilter or particleFilter object at the next time step. To implement the extended or unscented Kalman filter algorithms, use the predict and correct commands together. If the current output measurement exists, you can use predict and correct. If the measurement is missing, you can only use predict. For information about the order in which to use the commands, see Using predict and correct Commands.

example

[PredictedState,PredictedStateCovariance] = predict(obj) predicts state estimate and state estimation error covariance of an extended or unscented Kalman filter, or particle filter object obj at the next time step.

You create obj using the extendedKalmanFilter, unscentedKalmanFilter or particleFilter commands. You specify the state transition function and measurement function of your nonlinear system in obj. You also specify whether the process and measurement noise terms are additive or nonadditive in these functions. The State property of the object stores the latest estimated state value. Assume that at time step k, obj.State is $\stackrel{^}{x}\left[k|k\right]$. This value is the state estimate for time k, estimated using measured outputs until time k. When you use the predict command, the software returns $\stackrel{^}{x}\left[k+1|k\right]$ in the PredictedState output. Where $\stackrel{^}{x}\left[k+1|k\right]$ is the state estimate for time k+1, estimated using measured output until time k. The command returns the state estimation error covariance of $\stackrel{^}{x}\left[k+1|k\right]$ in the PredictedStateCovariance output. The software also updates the State and StateCovariance properties of obj with these corrected values.

Use this syntax if the state transition function f that you specified in obj.StateTransitionFcn has one of the following forms:

• x(k) = f(x(k-1)) — for additive process noise.

• x(k) = f(x(k-1),w(k-1)) — for nonadditive process noise.

Where x and w are the state and process noise of the system. The only inputs to f are the states and process noise.

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[PredictedState,PredictedStateCovariance] = predict(obj,Us1,...Usn) specifies additional input arguments, if the state transition function of the system requires these inputs. You can specify multiple arguments.

Use this syntax if your state transition function f has one of the following forms:

• x(k) = f(x(k-1),Us1,...Usn) — for additive process noise.

• x(k) = f(x(k-1),w(k-1),Us1,...Usn) — for nonadditive process noise.

## Examples

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Estimate the states of a van der Pol oscillator using an unscented Kalman filter algorithm and measured output data. The oscillator has two states and one output.

Create an unscented Kalman filter object for the oscillator. Use previously written and saved state transition and measurement functions, vdpStateFcn.m and vdpMeasurementFcn.m. These functions describe a discrete-approximation to a van der Pol oscillator with nonlinearity parameter, mu, equal to 1. The functions assume additive process and measurement noise in the system. Specify the initial state values for the two states as [1;0]. This is the guess for the state value at initial time k, using knowledge of system outputs until time k-1, $\underset{}{\overset{ˆ}{x}}\left[k|k-1\right]$.

obj = unscentedKalmanFilter(@vdpStateFcn,@vdpMeasurementFcn,[1;0]);

Load the measured output data, y, from the oscillator. In this example, use simulated static data for illustration. The data is stored in the vdp_data.mat file.

Specify the process noise and measurement noise covariances of the oscillator.

obj.ProcessNoise = 0.01;
obj.MeasurementNoise = 0.16;

Initialize arrays to capture results of the estimation.

residBuf = [];
xcorBuf = [];
xpredBuf = [];

Implement the unscented Kalman filter algorithm to estimate the states of the oscillator by using the correct and predict commands. You first correct $\underset{}{\overset{ˆ}{x}}\left[k|k-1\right]$ using measurements at time k to get $\underset{}{\overset{ˆ}{x}}\left[k|k\right]$. Then, you predict the state value at next time step, $\underset{}{\overset{ˆ}{x}}\left[k+1|k\right]$, using $\underset{}{\overset{ˆ}{x}}\left[k|k\right]$, the state estimate at time step k that is estimated using measurements until time k.

To simulate real-time data measurements, use the measured data one time step at a time. Compute the residual between the predicted and actual measurement to assess how well the filter is performing and converging. Computing the residual is an optional step. When you use residual, place the command immediately before the correct command. If the prediction matches the measurement, the residual is zero.

After you perform the real-time commands for the time step, buffer the results so that you can plot them after the run is complete.

for k = 1:size(y)
[Residual,ResidualCovariance] = residual(obj,y(k));
[CorrectedState,CorrectedStateCovariance] = correct(obj,y(k));
[PredictedState,PredictedStateCovariance] = predict(obj);

residBuf(k,:) = Residual;
xcorBuf(k,:) = CorrectedState';
xpredBuf(k,:) = PredictedState';
end

When you use the correct command, obj.State and obj.StateCovariance are updated with the corrected state and state estimation error covariance values for time step k, CorrectedState and CorrectedStateCovariance. When you use the predict command, obj.State and obj.StateCovariance are updated with the predicted values for time step k+1, PredictedState and PredictedStateCovariance.

In this example, you used correct before predict because the initial state value was $\underset{}{\overset{ˆ}{x}}\left[k|k-1\right]$, a guess for the state value at initial time k using system outputs until time k-1. If your initial state value is $\underset{}{\overset{ˆ}{x}}\left[k-1|k-1\right]$, the value at previous time k-1 using measurement until k-1, then use the predict command first. For more information about the order of using predict and correct, see Using predict and correct Commands.

Plot the estimated states, using the postcorrection values.

plot(xcorBuf(:,1), xcorBuf(:,2))
title('Estimated States')

Plot the actual measurement, the corrected estimated measurement, and the residual. For the measurement function in vdpMeasurementFcn, the measurement is the first state.

M = [y,xcorBuf(:,1),residBuf];
plot(M)
grid on
title('Actual and Estimated Measurements, Residual')
legend('Measured','Estimated','Residual')

The estimate tracks the measurement closely. After the initial transient, the residual remains relatively small throughout the run.

Load the van der Pol ODE data, and specify the sample time.

vdpODEdata.mat contains a simulation of the van der Pol ODE with nonlinearity parameter mu=1, using ode45, with initial conditions [2;0]. The true state was extracted with sample time dt = 0.05.

tSpan = 0:dt:5;

Get the measurements. For this example, a sensor measures the first state with a Gaussian noise with standard deviation 0.04.

sqrtR = 0.04;
yMeas = xTrue(:,1) + sqrtR*randn(numel(tSpan),1);

Create a particle filter, and set the state transition and measurement likelihood functions.

myPF = particleFilter(@vdpParticleFilterStateFcn,@vdpMeasurementLikelihoodFcn);

Initialize the particle filter at state [2; 0] with unit covariance, and use 1000 particles.

initialize(myPF,1000,[2;0],eye(2));

Pick the mean state estimation and systematic resampling methods.

myPF.StateEstimationMethod = 'mean';
myPF.ResamplingMethod = 'systematic';

Estimate the states using the correct and predict commands, and store the estimated states.

xEst = zeros(size(xTrue));
for k=1:size(xTrue,1)
xEst(k,:) = correct(myPF,yMeas(k));
predict(myPF);
end

Plot the results, and compare the estimated and true states.

figure(1)
plot(xTrue(:,1),xTrue(:,2),'x',xEst(:,1),xEst(:,2),'ro')
legend('True','Estimated')

rmpath(fullfile(matlabroot,'examples','control','main')) % remove example data

Consider a nonlinear system with input u whose state x and measurement y evolve according to the following state transition and measurement equations:

$x\left[k\right]=\sqrt{x\left[k-1\right]+u\left[k-1\right]}+w\left[k-1\right]$

$y\left[k\right]=x\left[k\right]+2*u\left[k\right]+v\left[k{\right]}^{2}$

The process noise w of the system is additive while the measurement noise v is nonadditive.

Create the state transition function and measurement function for the system. Specify the functions with an additional input u.

f = @(x,u)(sqrt(x+u));
h = @(x,v,u)(x+2*u+v^2);

f and h are function handles to the anonymous functions that store the state transition and measurement functions, respectively. In the measurement function, because the measurement noise is nonadditive, v is also specified as an input. Note that v is specified as an input before the additional input u.

Create an extended Kalman filter object for estimating the state of the nonlinear system using the specified functions. Specify the initial value of the state as 1 and the measurement noise as nonadditive.

Specify the measurement noise covariance.

obj.MeasurementNoise = 0.01;

You can now estimate the state of the system using the predict and correct commands. You pass the values of u to predict and correct, which in turn pass them to the state transition and measurement functions, respectively.

Correct the state estimate with measurement y[k]=0.8 and input u[k]=0.2 at time step k.

correct(obj,0.8,0.2)

Predict the state at the next time step, given u[k]=0.2.

predict(obj,0.2)

Retrieve the error, or residual, between the prediction and the measurement.

[Residual, ResidualCovariance] = residual(obj,0.8,0.2);

## Input Arguments

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Extended or unscented Kalman filter, or particle filter object for online state estimation, created using one of the following commands:

Additional input arguments to state transition function, specified as input arguments of any type. The state transition function, f, is specified in the StateTransitionFcn property of the object. If the function requires input arguments in addition to the state and process noise values, you specify these inputs in the predict command syntax.

For example, suppose that your state transition function calculates the predicted state x at time step k using system inputs u(k-1) and time k-1, in addition to the state x(k-1):

x(k) = f(x(k-1),u(k-1),k-1)

Then when you perform online state estimation at time step k, specify these additional inputs in the predict command syntax:

[PredictedState,PredictedStateCovariance] = predict(obj,u(k-1),k-1);

## Output Arguments

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Predicted state estimate, returned as a vector of size M, where M is the number of states of the system. If you specify the initial states of obj as a column vector then M is returned as a column vector, otherwise M is returned as a row vector.

For information about how to specify the initial states of the object, see the extendedKalmanFilter, unscentedKalmanFilter and particleFilter reference pages.

Predicted state estimation error covariance, returned as an M-by-M matrix, where M is the number of states of the system.

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### Using predict and correct Commands

After you have created an extended or unscented Kalman filter, or particle filter object, obj, to implement the estimation algorithms, use the correct and predict commands together.

At time step k, correct command returns the corrected value of states and state estimation error covariance using measured system outputs y[k] at the same time step. If your measurement function has additional input arguments Um, you specify these as inputs to the correct command. The command passes these values to the measurement function.

[CorrectedState,CorrectedCovariance] = correct(obj,y,Um)

The correct command updates the State and StateCovariance properties of the object with the estimated values, CorrectedState and CorrectedCovariance.

The predict command returns the prediction of state and state estimation error covariance at the next time step. If your state transition function has additional input arguments Us, you specify these as inputs to the predict command. The command passes these values to the state transition function.

[PredictedState,PredictedCovariance] = predict(obj,Us)

The predict command updates the State and StateCovariance properties of the object with the predicted values, PredictedState and PredictedCovariance.

If the current output measurement exists at a given time step, you can use correct and predict. If the measurement is missing, you can only use predict. For details about how these commands implement the algorithms, see Extended and Unscented Kalman Filter Algorithms for Online State Estimation.

The order in which you implement the commands depends on the availability of measured data y, Us, and Um for your system:

• correct then predict — Assume that at time step k, the value of obj.State is $\stackrel{^}{x}\left[k|k-1\right]$. This value is the state of the system at time k, estimated using measured outputs until time k-1. You also have the measured output y[k] and inputs Us[k] and Um[k] at the same time step.

Then you first execute the correct command with measured system data y[k] and additional inputs Um[k]. The command updates the value of obj.State to be $\stackrel{^}{x}\left[k|k\right]$, the state estimate for time k, estimated using measured outputs up to time k. When you then execute the predict command with input Us[k], obj.State now stores $\stackrel{^}{x}\left[k+1|k\right]$. The algorithm uses this state value as an input to the correct command in the next time step.

• predict then correct — Assume that at time step k, the value of obj.State is $\stackrel{^}{x}\left[k-1|k-1\right]$. You also have the measured output y[k] and input Um[k] at the same time step but you have Us[k-1] from the previous time step.

Then you first execute the predict command with input Us[k-1]. The command updates the value of obj.State to $\stackrel{^}{x}\left[k|k-1\right]$. When you then execute the correct command with input arguments y[k] and Um[k], obj.State is updated with $\stackrel{^}{x}\left[k|k\right]$. The algorithm uses this state value as an input to the predict command in the next time step.

Thus, while in both cases the state estimate for time k, $\stackrel{^}{x}\left[k|k\right]$ is the same, if at time k you do not have access to the current state transition function inputs Us[k], and instead have Us[k-1], then use predict first and then correct.

For an example of estimating states using the predict and correct commands, see Estimate States Online Using Unscented Kalman Filter or Estimate States Online using Particle Filter.