DiscreteLQEstimatorGains

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`DiscreteLQEstimatorGains`

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DiscreteLQEstimatorGains[ssm,{w,v},τ]

gives the optimal discrete-time estimator gain matrix with sampling period τ for the continuous-time StateSpaceModel ssm, with process and measurement noise covariance matrices w and v.

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DiscreteLQEstimatorGains[{ssm,sensors},{w,v},τ]

specifies sensors as the noisy measurements of ssm.

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DiscreteLQEstimatorGains[{ssm,sensors,dinputs},{w,v},τ]

specifies dinputs as the deterministic inputs of ssm.

Details and Options

The standard state-space model ssm can be given as StateSpaceModel[{a,b,c,d}], where a, b, c, and d represent the state, input, output, and transmission matrices of the continuous-time system .
The descriptor continuous-time state-space model ssm defined by can be given as StateSpaceModel[{a,b,c,d,e}].
The input can include the process noise , as well as deterministic inputs .
The argument dinputs is a list of integers specifying the positions of in .
The output consists of the noisy measurements , as well as other outputs.
The argument sensors is a list of integers specifying the positions of in .
DiscreteLQEstimatorGains[ssm,{…},τ] is equivalent to DiscreteLQEstimatorGains[{ssm, All,None},{…},τ].
The noisy measurements are modeled as , where and are the submatrices of and associated with , and is the noise.
The process and measurement noises are assumed to be white and Gaussian:
, process noise

, measurement noise
The estimator with the optimal gain minimizes , where is the estimated state vector.
DiscreteLQEstimatorGains computes the estimator gains based on the discrete equivalent of the noise matrices.
The state-space model ssm is discretized using the zero-order hold method.

Examples

open allclose all

Basic Examples (1)Summary of the most common use cases

Compute the discrete LQ estimator gains for a continuous-time state-space model:

In[1]:=1

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https://wolfram.com/xid/0ejrq8bz9uwp102keduac-b8u09u

Out[1]=1

Scope (3)Survey of the scope of standard use cases

Compute the discrete-time Kalman gains for a state-space model:

In[1]:=1

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https://wolfram.com/xid/0ejrq8bz9uwp102keduac-b0ghau

In[2]:=2

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https://wolfram.com/xid/0ejrq8bz9uwp102keduac-nf6be

The gains based on the measurement of just the second output:

In[3]:=3

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https://wolfram.com/xid/0ejrq8bz9uwp102keduac-3tucws

The gains for a system in which all inputs except the first are stochastic:

In[1]:=1

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https://wolfram.com/xid/0ejrq8bz9uwp102keduac-el6ya

In[2]:=2

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https://wolfram.com/xid/0ejrq8bz9uwp102keduac-6ko45l

Out[2]=2

Find the optimal gains for a descriptor state-space model:

In[1]:=1

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https://wolfram.com/xid/0ejrq8bz9uwp102keduac-r1g9i

Out[1]=1

Properties & Relations (1)Properties of the function, and connections to other functions

Find estimator gains using DiscreteLQEstimatorGains:

In[1]:=1

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https://wolfram.com/xid/0ejrq8bz9uwp102keduac-i2y59e

In[2]:=2

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https://wolfram.com/xid/0ejrq8bz9uwp102keduac-bn6wl

Out[2]=2

Create a discrete-time Kalman estimator with the gains and a discretized model:

In[3]:=3

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https://wolfram.com/xid/0ejrq8bz9uwp102keduac-h7hl39

This is different from that obtained by discretizing a continuous-time estimator:

In[4]:=4

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https://wolfram.com/xid/0ejrq8bz9uwp102keduac-yhnf4p

Response of the first estimator in the presence of process and measurement noises:

In[5]:=5

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https://wolfram.com/xid/0ejrq8bz9uwp102keduac-utllod

In[6]:=6

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https://wolfram.com/xid/0ejrq8bz9uwp102keduac-r76zpx

In[7]:=7

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https://wolfram.com/xid/0ejrq8bz9uwp102keduac-xnjvt9

In[8]:=8

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https://wolfram.com/xid/0ejrq8bz9uwp102keduac-y2s8k9

Response of the discretized Kalman estimator:

In[9]:=9

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https://wolfram.com/xid/0ejrq8bz9uwp102keduac-oo3p31

The responses are different:

In[10]:=10

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https://wolfram.com/xid/0ejrq8bz9uwp102keduac-80czbn

Out[10]=10

Possible Issues (1)Common pitfalls and unexpected behavior

The system must be detectable:

In[1]:=1

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https://wolfram.com/xid/0ejrq8bz9uwp102keduac-onpkkj

Out[1]=1

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