Technology / Technical guide
Dynamic States & Observability: What Outputs Can Reveal
Here a state means an internal variable of a dynamic model, rather than a logic label such as a permitted operating mode. Observability asks whether the defined output history can distinguish the model’s possible initial states when its inputs are known. Counting sensors alone cannot answer that question.
Internal variables can affect observations over time.
A variable need not be measured directly to influence an output, but a mathematical model must explain how its information reaches that output.
01 / State meaning
The state carries the model’s internal memory.
A continuous linear model can be written dx/dt = Ax+Bu and y = Cx+Du.
A continuous linear model can be written dx/dt = Ax+Bu and y = Cx+Du. The vector x contains dynamic states, u known inputs and y represented observations. Matrix entries have the units needed to map their associated quantities.
Different coordinate choices can describe the same model. A state can be a physical quantity or a mathematical combination; its name is not automatically a measured channel. This differs from the logical-state guide, where transitions define modes and permissions.
02 / Distinguishability
Identical outputs can hide different initial states.
If two initial states produce the same complete output history under the same known input, the selected observations cannot distinguish them.
If two initial states produce the same complete output history under the same known input, the selected observations cannot distinguish them. The ambiguity belongs to the defined model and endpoints, not merely to a display’s decimal precision.
Conversely, output variation over time can reveal information absent from one instantaneous value. Observability uses that dynamic relationship. Repeatedly measuring the same output more precisely cannot create a coupling that the model does not contain.
03 / Toy relationship
One output can carry two states over time.
Hypothetical continuous model: let dx1/dt = x2/(1 second), dx2/dt = 0, and y = x1, with both states in arbitrary units.
Hypothetical continuous model: let dx1/dt = x2/(1 second), dx2/dt = 0, and y = x1, with both states in arbitrary units. Then y(t) = x1(0)+(t/1 second)x2(0) under these ideal equations.
The initial output identifies x1(0), while its change over a nonzero interval distinguishes x2(0). This is an ideal information argument with exact observations, not a procedure to infer real turbine speed from noisy heading data. The coefficient is invented.
04 / Rank condition
A full-rank condition concerns the chosen linear model.
For an n-state linear time-invariant model, the observability matrix stacks C, CA, …, C A^(n−1).
For an n-state linear time-invariant model, the observability matrix stacks C, CA, …, C A^(n−1). Full column rank n is the ideal observability criterion. MathWorks presents this model-level rank condition.
In the toy relationship, the independent rows corresponding to C and CA distinguish both states. If both state derivatives were zero and y remained x1, x2 would no longer affect the observation; the corresponding matrix would lack full rank.
05 / Practical evidence
Theoretically distinct outputs can be hard to separate.
Noise, short observation duration and poorly conditioned relationships can make different states produce outputs that are practically difficult to distinguish.
Noise, short observation duration and poorly conditioned relationships can make different states produce outputs that are practically difficult to distinguish. An ideal rank calculation does not specify the uncertainty of a numerical estimate from finite measured data.
Unknown inputs and incorrect dynamics introduce further ambiguities. The estimation guide describes a declared noise model, while the voting guide concerns agreement between comparable observations rather than hidden-state reconstruction.
06 / Scope
Observability is different from control authority.
Being able to infer a state does not mean an input can drive it to an arbitrary value.
Being able to infer a state does not mean an input can drive it to an arbitrary value. Observability and controllability describe different model relationships. Likewise, an available sensor does not independently establish a complete, accurate estimator.
No MARSTON state model, observer or estimation accuracy is assigned. This explanation supplies no sensor substitution, differentiation of live data, algorithm implementation or control changes; it identifies the information that a defined model can represent.
Related technical guides: Control Transfer Functions: Poles, Zeros & Initial State · Kalman Estimation: Prediction, Innovation & Covariance · Frequency Coherence: Shared Content & Evidence Limits.
Review parameter identifiability and model ambiguity as a separate model-information topic alongside the state-observability discussion.
General background: MathWorks: State-space observability and full-rank condition. These references explain industry concepts; they are separate from the applicable MARSTON product and project documents.
Key concepts and conditions.
Read each quantity with its reference, units and stated conditions. A general relation does not establish a model-specific value.
| Concept / quantity | Meaning / relation | Conditions / limits |
|---|---|---|
| x | Dynamic internal-state vector | Different from a logic operating label. |
| y | Defined output observations | One instantaneous value may omit dynamic information. |
| A and C | Dynamics and state-to-output maps | Their quantity units and coordinates matter. |
| Observability matrix | Stack of C A^k for k=0…n−1 | Full column rank is the ideal LTI criterion. |
| Practical estimate | Finite-data state inference | Noise and conditioning remain separate from rank. |
Engineering questions.
Does every dynamic state have a sensor?
No. A model may infer it from how outputs evolve.
Does one output imply one observable state?
No. A time history can contain several independent state contributions.
Are logical modes the same as dynamic states?
No. Their mathematical roles differ.
Does full rank guarantee accurate noisy estimation?
No. Duration, conditioning and model validity still matter.
Does observability prove control authority?
No. Controllability is a separate property.
Is an observer confirmed for MARSTON?
No actual model, estimator or accuracy is declared.
Technical enquiry
Discuss the technical question.
Here a state means an internal variable of a dynamic model, rather than a logic label such as a permitted operating mode. Share the model and applicable operating or design information for an engineering discussion.