{ "node": { "id": "term:state-space-model", "type": "Term", "label": "State-space model", "properties": { "id": "state-space-model", "name": "State-space model", "aka": [ "state-space representation" ], "section": "topic-transport", "definition": "A model written as first-order equations for a set of internal states — such as state of charge and the voltages across RC pairs — plus an output equation for the measured terminal voltage. Kalman filters and other observers need this form, so equivalent circuit and reduced physics models are cast this way for a BMS.", "formula": "ẋ = A·x + B·u; y = C·x + D·u", "related": [ "kalman-filter", "equivalent-circuit-model", "rc-pair", "reduced-order-model", "state-of-charge" ], "example": { "title": "Read a one-state thermal model", "text": "Let x be temperature rise and u heating power. With ẋ = −0.01x + 0.002u, the time constant is 100 s and a steady 10 W input gives x = 2 K. The output y = x directly reports temperature rise." }, "url": "https://lemonergy.com/learn/glossary/state-space-model/", "mentions": [ "voltage", "bms" ] } }, "edges": [ { "id": "edge:4788", "source": "term:state-space-model", "target": "topic:topic-transport", "type": "belongsTo", "properties": { "sourceFile": "terms.json" } }, { "id": "edge:4789", "source": "term:state-space-model", "target": "term:kalman-filter", "type": "relatedTo", "properties": { "sourceFile": "terms.json", "meaning": "Related context; not a causal claim" } }, { "id": "edge:4790", "source": "term:state-space-model", "target": "term:equivalent-circuit-model", "type": "relatedTo", "properties": { "sourceFile": "terms.json", "meaning": "Related context; not a causal claim" } }, { "id": "edge:4791", "source": "term:state-space-model", "target": "term:rc-pair", "type": "relatedTo", "properties": { "sourceFile": "terms.json", "meaning": "Related context; not a causal claim" } }, { "id": "edge:4792", "source": "term:state-space-model", "target": "term:reduced-order-model", "type": "relatedTo", "properties": { "sourceFile": "terms.json", "meaning": "Related context; not a causal claim" } }, { "id": "edge:4793", "source": "term:state-space-model", "target": "term:state-of-charge", "type": "relatedTo", "properties": { "sourceFile": "terms.json", "meaning": "Related context; not a causal claim" } }, { "id": "edge:mentions/state-space-model/voltage", "source": "term:state-space-model", "target": "term:voltage", "type": "mentions", "properties": { "sourceFile": "terms.json", "method": "build_glossary.mention_index + mentions", "field": "definition", "meaning": "Text mention; not a causal claim" } }, { "id": "edge:mentions/state-space-model/bms", "source": "term:state-space-model", "target": "term:bms", "type": "mentions", "properties": { "sourceFile": "terms.json", "method": "build_glossary.mention_index + mentions", "field": "definition", "meaning": "Text mention; not a causal claim" } }, { "id": "edge:4796", "source": "term:state-space-model", "target": "visual:state-response", "type": "illustratedBy", "properties": { "sourceFile": "visual-pages.json" } }, { "id": "edge:4797", "source": "term:state-space-model", "target": "reference:astrom-murray-feedback", "type": "cites", "properties": { "sourceFile": "wiki-references.json", "scope": "direct technical reference" } } ], "neighbors": [ { "id": "topic:topic-transport", "type": "Topic", "label": "Transport and modelling", "properties": { "id": "topic-transport", "name": "Transport and modelling", "lede": "How ions move, and the equations and models that describe a cell.", "context": "Specify phase, direction and boundaries.", "example": "The same concentration difference across 10 μm gives ten times the gradient across 100 μm; at equal D, first-law flux magnitude is ten times greater.", "lesson_url": "https://lemonergy.com/learn/transport/ficks-first-law/", "lesson_label": "Open diffusion" } }, { "id": "term:kalman-filter", "type": "Term", "label": "Kalman filter", "properties": { "id": "kalman-filter", "name": "Kalman filter", "section": "topic-state", "definition": "A recursive estimator that blends a model prediction with noisy measurements according to how much each is trusted. In a BMS an extended Kalman filter combines coulomb counting with a voltage model to track SOC and sometimes resistance.", "related": [ "coulomb-counting", "equivalent-circuit-model", "state-of-charge", "bms", "fuzzy-logic", "data-driven-soc-estimation" ], "url": "https://lemonergy.com/learn/glossary/kalman-filter/", "mentions": [ "coulomb" ] } }, { "id": "term:equivalent-circuit-model", "type": "Term", "label": "Equivalent circuit model", "properties": { "id": "equivalent-circuit-model", "name": "Equivalent circuit model", "abbr": "ECM", "aka": [ "Thevenin model" ], "section": "topic-transport", "definition": "A cell model built from a voltage source for OCV, a series resistor and one or more RC pairs for the slower dynamics. Its few parameters are fitted from pulse tests, and it is what most battery management systems run in real time.", "related": [ "rc-pair", "open-circuit-voltage", "hppc", "kalman-filter", "single-particle-model" ], "url": "https://lemonergy.com/learn/glossary/equivalent-circuit-model/", "mentions": [] } }, { "id": "term:rc-pair", "type": "Term", "label": "RC pair", "properties": { "id": "rc-pair", "name": "RC pair", "aka": [ "RC element", "RC branch" ], "section": "topic-transport", "definition": "A resistor and capacitor in parallel, the building block of an equivalent circuit model. Each pair reproduces one time constant of the cell’s voltage response, from the seconds of charge transfer to the minutes of diffusion.", "formula": "τ = R × C", "related": [ "equivalent-circuit-model", "relaxation", "charge-transfer" ], "example": { "title": "Calculate a response time", "text": "An illustrative 0.02 Ω resistance and 500 F capacitance have τ = RC = 10 s. A first-order step response reaches about 63% of its final change after one time constant." }, "url": "https://lemonergy.com/learn/glossary/rc-pair/", "mentions": [ "diffusion", "capacitor" ] } }, { "id": "term:reduced-order-model", "type": "Term", "label": "Reduced-order model", "properties": { "id": "reduced-order-model", "name": "Reduced-order model", "abbr": "ROM", "aka": [ "model order reduction" ], "section": "topic-transport", "definition": "A simplified version of a detailed model that keeps its dominant behaviour with far fewer states. The single-particle model is one reduction of the Newman model; others use polynomial profiles, Padé approximation or proper orthogonal decomposition. A ROM gives a BMS internal quantities, such as lithium concentration in the electrodes, that an equivalent circuit model cannot.", "related": [ "newman-model", "single-particle-model", "state-space-model", "equivalent-circuit-model", "bms" ], "url": "https://lemonergy.com/learn/glossary/reduced-order-model/", "mentions": [ "reduction" ] } }, { "id": "term:state-of-charge", "type": "Term", "label": "State of charge", "properties": { "id": "state-of-charge", "name": "State of charge", "abbr": "SOC", "section": "topic-state", "definition": "How full a cell is, as a fraction of its available capacity. State of charge, like temperature, affects how fast a stored cell ages.", "unit": "%", "related": [ "depth-of-discharge", "ocv-soc-curve", "coulomb-counting", "state-of-health", "bms" ], "lesson": "fundamentals/battery-aging/#storage-soc", "example": { "title": "Interpret a charge inventory", "text": "If available capacity is 5 Ah and 3 Ah remains by the chosen estimator, SOC is 60%. A different estimate of available capacity changes this percentage, so compare SOC values only with the capacity convention recorded." }, "url": "https://lemonergy.com/learn/glossary/state-of-charge/", "mentions": [ "usable-capacity" ] } }, { "id": "term:voltage", "type": "Term", "label": "Voltage", "properties": { "id": "voltage", "name": "Voltage", "aka": [ "terminal voltage" ], "section": "topic-units", "definition": "The potential difference between a cell’s two terminals. It comes from the chemical difference between the electrodes, moves with state of charge, and sags under load because of internal resistance.", "unit": "V", "related": [ "open-circuit-voltage", "nominal-voltage", "internal-resistance", "ir-drop" ], "lesson": "fundamentals/cell-anatomy/#voltage", "url": "https://lemonergy.com/learn/glossary/voltage/", "mentions": [ "state-of-charge" ] } }, { "id": "term:bms", "type": "Term", "label": "Battery management system", "properties": { "id": "bms", "name": "Battery management system", "abbr": "BMS", "section": "topic-pack", "definition": "The electronics and software that keep a battery inside its safe window: measuring every cell voltage and many temperatures, estimating SOC and SOH, balancing cells, computing current limits, controlling the contactors and reporting over the vehicle network.", "related": [ "state-of-charge", "cell-balancing", "state-of-power", "cell-supervision-circuit", "contactor", "can-bus" ], "lesson": "fundamentals/battery-basics/#cell-module-pack", "url": "https://lemonergy.com/learn/glossary/bms/", "mentions": [ "state-of-health", "cell-voltage" ] } }, { "id": "visual:state-response", "type": "Visual", "label": "Follow an internal state after an input step", "properties": { "title": "Follow an internal state after an input step", "summary": "A one-state linear model separates the stored state from its observed output.", "readout": "At τ = 10 s, output reaches 63.2% after 10 s and 95.0% after 30 s.", "formula": "dx/dt = −x/τ + u/τ; y = x; x(0) = 0", "note": "Scalar continuous-time, stable linear model with no saturation, delay or measurement noise.", "software": { "status": "reference", "label": "Reference concept", "body": "This is an analytical one-state example. It does not implement an observer or identify state matrices for a battery.", "href": "../chapter-1/" }, "html": "
First-order state-space step response
First-order state-space step responseTime (s) versus Normalized input and output. Unit input u; Output y = 1 − exp(−t/τ). Interpretation follows the graphs.000.275250.55500.825751.1100Time (s)Normalized input andoutput
Unit input uOutput y = 1 − exp(−t/τ) (dashed)
" } }, { "id": "reference:astrom-murray-feedback", "type": "Reference", "label": "Feedback Systems: An Introduction for Scientists and Engineers, second edition", "properties": { "title": "Feedback Systems: An Introduction for Scientists and Engineers, second edition", "publisher": "Princeton University Press; author-maintained FBSwiki", "url": "https://fbswiki.org/wiki/index.php/Feedback_Systems%3A_An_Introduction_for_Scientists_and_Engineers", "type": "Textbook", "scope": "State-space modeling, feedback and PID control. These mathematical foundations do not establish a particular software implementation.", "year": 2021, "authors": [ "Karl J. Åström", "Richard M. Murray" ], "accessed": "2026-10-27" } } ] }