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 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210  # Mathematical Symbols and Notation ([[See also our page on coordinate systems|CoordinateSystem]]) ---- ## Notation used in State Space control and Kalman filtering The basic state space formulation of a dynamic system expresses the system as a linear first order differential equation with respect to time - $$\\bf\{x'\} = \\bf\{F\\ x\} + \\bf\{G\} \\pmatrix\{\\bf\{u\} \\cr \\bf\{w\}\}$$ or - $$\\bf\{x'\} = \\bf\{\\Phi\\ x\} + \\bf\{\\Gamma\} \\pmatrix\{\\bf\{u\} \\cr \\bf\{w\}\}$$ The first form is used for continuous time systems, the second form is usually reserved for systems modeled in discreet time. **Summary of State Space Symbols**

Symbol

Dimension

Name

x

n×1

state vector

F

n×n

state transition matrix

Φ

n×n

discreet state transition matrix

G

n×q

input matrix

Γ

n×q

discreet input matrix

u

qu×1

input vector

w

qw×1

process noise vector

The vector lengths of (**u**) and (**w**) are given as (qu) and (qw) respectively. The total length of the combined input vector is (q = qu + qw), which conforms correctly to the dimensions of the input matrix (**G**) or (Γ). Perhaps awkwardly, in this formulation we have both distinguished and combined the input vector (**u**) and the process noise vector (**w**). The distinction is that (**u**) is usually treated as a controlled input, like a knob on a machine or the rudder of a ship. The process noise vector (**w**), though still an input to the system, is not under the control of any part of the system. The process noise input represents such things as random disturbances in the movement of a vehicle, or errors in modeling of the system. An alternate approach is to write the two inputs as separate terms. This requires the introduction of a new matrix like so - **x'** = **F x + G u + L w** or - **x'** = Φ **x** + Γ **u** + Λ **w** The new matrices (**L**) and (Λ) might as well be called _process noise input matrices_. They are exactly analogous to (**G**) and (Γ) but operating on the process noise rather than the control input. ## Estimates, Measurements, and Updates State space systems commonly make use of a _predictor-corrector_ loop. Basically this means that the information available at the present time is used to predict the state of the system in the future. When new measurements of the system become available they are compared to the predicted values and any difference forms an error-feedback signal which is used to adjust the system model so that the predicted outputs are closer to the latest measurements. As a notational side effect the predictor-corrector loop creates many distinctions between similar variables. For example the state vector (**x**) now comes in a predicted form, a measured form, and extrapolated form, among others. To keep track of the various forms we attempt to follow the most common convention for decorating the symbols, like so: **Update decorations**

Example

Decoration

Meaning

x

none

exact or "truth" value

$\bf{\hat x}$

hat

estimate or measurement update

x-

minus

time update or extrapolation

$\bf{\tilde x}$

tilde

measurement

There's no doubt that using the update decorations can be confusing. As an aid we humbly suggest these mnemonics: The truth value is pure and perfect, needing no decoration. The estimate is the best value we can realistically get. It's the value that the pointy headed smart people calculate, the ones with the wizard hats. The extrapolation is not as good a value as the estimate, thus the minus sign. The minus sign also signifies that the extrapolation is the value that is used _before_ the measurements come in. Once there are new measurements they can be used to generate estimates. This is why the estimates are also called "measurement updates". Extrapolations are called "time updates" because they are generated by running the system model forward in time using the current estimate as a starting point. Finally, the measurements are "tilde"(~) because, although they're good, they're only approximate, being corrupted by noise and error, they're somewhat wishy-washy. Note: Another common convention is to denote extrapolations with over-bars. This is visually similar to the superscript-minus used here. We prefer the minus because it seems easier to follow, and the over-bar conflicts with several other common conventions, particularly for the average value. Note: The sign convention for error terms is rarely consistent between authors. To us it seems preferable to write the measured value as the sum of the true value and the measurement error. Thus measurements which are too large have a positive error, and the true value is the measured value minus the error. Symbolically we define the error as - $$\\bf \\epsilon\_x \\equiv \\bf\{ \\tilde x - x\}$$