UAV control and observability of the Kalman filter

1, observability

1.1, the definition of observability

Refers objective system through the system output can reflect the initial state (state of change can be reflected by the output);

If the form of a series of control inputs and outputs can be within a limited time to uniquely determine the state of the system objective system;

Keywords which there are three definitions: dynamic system status output.

For example:

In Chinese medicine doctor, the system dynamic man, look and smell is the output, and the state is, whether the sick person had what disease, the objective is the look and smell of inferring the human condition;

In large data network, the network is the dynamic system status information is valuable to be observed. The output data can be obtained from a large data network;

Below, very intuitive, states A and C can be obtained is not impressive, B and D abuts on the state output is substantial;

1.2, linear systems observability

(1) the definition of a linear system observability

It is defined as a linear system

Linear system can be observed: if a limited time interval within, the output value y (t) and the input value u (t), can determine the initial system state x ( $t_{0}$ each component), then call this system is completely considerable, referred observable.

(2) linear system observability Sufficient Conditions

Theorem 1 : The system necessary and sufficient conditions observable Rank $O_ {v}$ = n-;

prove:

The linear system is evaluated appeal n-1 order derivative, available

Matrix operation can be constructed:

Wherein the order just turned to n-1 is the cause, according to Kelly - Hamilton theorem:

From the result of $O_ {v}$ rank same.

If Rank $O_ {v}$ = n, x is the unique solution can be determined by the state of inversion, if n is smaller than the infinite number of solutions exist;

So a card was Theorem.

(3) Examples

For example: given two GPS and accelerometer sensors, which it is possible to stably estimate the speed of movement of a trolley-dimensional?

For simplicity, we use the GPS observation position, with the following general model (the reciprocal of the speed position of the guide, is a derivative of the velocity of white noise)

Estimated velocity using accelerometers, the derivative is generally used as a model :( acceleration speed, the acceleration is the derivative of a white noise)

In the embodiment, the acceleration can not know the reason for the speed, acceleration of the unknown object can not be determined only know the initial velocity.

1.3, discrete linear systems

(1) the definition of

For linear systems, the sampling period can $T_{s}$ accurately be converted to a discrete system continuous system. Converting the continuous system model linear system into the following discrete sample

Definition 2: If the time interval is limited $NT_{s}$ within, the output value of $y_{k}$ the input value and the outside world $u_ {k}$ , the system can determine the initial state of $x_{0}$ each component, then call this system is completely significant, observable for short.

(2) direct interpretation observability

Continuous linear system into derivation, available Theorem

2, the Kalman filter

2.1, the Kalman filter is defined

The Kalman filter is a linear system state equation, the observation data through the input-output system, the system state estimation algorithm for the minimum variance. It's best estimate of the following three conditions to be met:

(1) unbiased estimate of :: i.e. true value is equal to a desired state value;

(2) the minimum variance estimate;

(3) real-time.

2.2, the Kalman filter model is assumed to derive

Linear discrete-system model is assumed as follows:

In the formula, process noise $w_{k-1}$ and measurement noise $v_ {k}$ statistical properties of

Wherein represents the process noise and the measurement noise is zero mean, independent and uncorrelated between them;

As it can be seen from the observation noise covariance matrix, observation noise covariance matrix R is a positive definite matrix rather than semi-definite matrix necessarily means that the sensor output includes noise;

The initial state of the $x_{0}$ statistical properties of

Covariance keep up knowledge:

Variance is generally used to describe the one-dimensional data, but in real life we often come across multi-dimensional data set contains data, the simplest of everyone at school will inevitably have to count multiple disciplines test scores. Covariance is such a statistic used to measure the relationship between two random variables. If the result is positive, then the two are directly related (covariance lead from the definition of "correlation coefficient"), the result is negative it means that a negative correlation.

Suppose the initial value of the state $x_{0}$ , $u_ {k}$ and $w_{k-1},v_{k}$ the relevant not, and the noise vector $w_{k-1},v_{k}$ is not relevant, that is

2.3, the Kalman filter derivation ideas

Objective: Suppose filter type

Wherein the current time point k is the minimum variance unbiased estimator (target filter), the right hand side contains three ideas: is a time- optimal linear estimation (minimum variance unbiased estimate), the current input value for the I always enter a value.

Worth for us the model required.

Then derived based on the following:

The first step into the derivation of the results available

From the above derivation,

In the Kalman filter, on the one hand from the predicted, i.e.,

A second aspect of the measurements from:

2.4 Other instructions

(1) In general, the sampling period is reasonably continuous system is considerable, discrete system will be considerable. However, sometimes inappropriate choice of the sampling period, the system may lose controllability and observability;

(2) the Kalman filter is an optimal observer, observer gain $K_{k}$ is time-varying;