1 Background and Motivation

1.1 State-space modeling and internal dynamics

In control and systems theory, many dynamical systems are modeled using a state-space representation. The model tracks how internal variables (the “state”) evolve over time and how those variables determine outputs in response to inputs. Although different state choices can describe the same physical behavior, the internal structure of a state-space model may contain redundancies—internal motions that do not influence outputs, or internal modes that cannot be driven by admissible inputs.

1.2 Why minimality matters (simplicity, identifiability, computation)

Minimal realization addresses the problem of removing such redundancies. A minimal model is typically simpler: it uses fewer state variables, which can reduce computational cost in simulation, estimation, and control synthesis. It can also support identifiability, since redundant internal degrees of freedom can obscure how parameters relate to measured input-output behavior. In system identification, insisting on minimality helps ensure the fitted model captures only the dynamics that actually appear in the data.

1.3 Behavior equivalence: input-output preservation

Two realizations are treated as equivalent if they generate the same input-output behavior. In the standard linear time-invariant setting, this often means they share identical transfer characteristics; more generally, equivalence can be expressed through matching system responses to inputs. Minimal realization aims to find the realization with the smallest internal dimension among those equivalent descriptions, without altering how the system behaves externally.

2 Definitions

2.1 Realizations of a dynamical system

A realization is a particular mathematical model that represents a dynamical system using internal state variables. In the linear time-invariant case, a realization typically has the form of a state equation and an output equation. Different realizations can correspond to different coordinate descriptions of the same underlying dynamics, or to models that include additional internal structure not required to reproduce the observed behavior.

2.1.1 Realization equivalence (conceptual criteria)

Realization equivalence is defined by agreement in input-output behavior. For linear time-invariant systems, transfer matrices or equivalent frequency-domain descriptions provide common criteria; for broader contexts, equality of external behavior can be formulated in terms of impulse responses, convolution operators, or matching trajectories produced by the same inputs from consistent initial conditions. The essential idea is that internal coordinates may vary, but the externally visible behavior does not.

2.2 Minimal realization

A minimal realization is an equivalent realization with the smallest possible number of state variables.

2.2.1 Controllability and observability as minimality conditions

For linear time-invariant state-space models, minimality is characterized by two properties:

  • Controllability: every relevant internal mode can be driven by the system inputs.
  • Observability: every relevant internal mode can be inferred from the outputs.

A realization that is both controllable and observable contains no redundant internal dynamics. Any additional states beyond this pair of properties correspond to internal behavior that cannot be reached by inputs and/or cannot be detected at the outputs.

2.3 Reachable/observable subspaces (intuition)

Controllability and observability correspond geometrically to subspaces associated with the system. The reachable (or controllable) subspace describes which components of the state can be generated from an initial condition using admissible inputs. The observable subspace describes which components can affect output measurements over time. Minimal realizations are those whose state space coincides with both: no part of the state lies outside the reachable dynamics, and no part is invisible to the output mapping.

3 Controllability and Observability

3.1 Controllability in state-space terms

Controllability formalizes the ability of the inputs to influence the state. In state-space language, it concerns whether the system’s internal directions can be reached from the origin in finite time (or, equivalently, whether the span of input-driven directions covers the intended state subspace).

3.1.1 Controllable subspace and redundant states

If some directions in the state are not reachable, they correspond to redundant internal states: they may evolve but cannot be affected by the chosen inputs. Such components do not contribute to the system’s ability to respond to inputs, and they can often be eliminated without changing input-output behavior, provided the model reduction respects the behavior equivalence criterion.

3.2 Observability in state-space terms

Observability formalizes the ability to reconstruct the state—or at least distinguish its effects—from output data. It addresses whether different internal states can produce distinct output trajectories under admissible inputs.

3.2.1 Observable subspace and hidden dynamics

If parts of the state cannot be detected from the outputs, they represent hidden dynamics. These unobservable components may still be internally dynamic, but they do not leave a signature in the measured output. Eliminating them can preserve the input-output relationship while yielding a reduced model with fewer states.

3.3 Relationship between minimality and state elimination

Redundant dynamics can be removed systematically:

  • Unreachable (not controllable) components can be pruned because no input can excite them from the considered initial condition.
  • Unobservable components can be pruned because output measurements cannot distinguish their influence.

A model that is both controllable and observable is the one with no such eliminable parts, hence minimal in state dimension.

4 Algorithms for Minimal Realization

4.1 Minimal realization from a given state-space model

When a state-space model is available, minimal realization can be computed by identifying and discarding unreachable and/or unobservable parts.

4.1.1 Constructing minimal coordinates via decomposition

A common approach uses decompositions of the state space into structured subspaces aligned with controllability and observability properties. By transforming the model into coordinates where reachable/unreachable and observable/unobservable components are separated, one can obtain a reduced system by retaining only the blocks corresponding to the controllable and observable dynamics. This procedure typically involves finding a similarity transformation that reveals the internal structure, followed by truncation of the redundant portions.

4.1.2 Eliminating unreachable and unobservable components

After the state space is partitioned, minimal reduction proceeds by deleting states associated with the unreachable subspace and states associated with the unobservable subspace. The result is a smaller state-space model that remains equivalent in input-output behavior to the original, while having fewer state variables and improved compactness.

4.2 Minimal realization from transfer data (high-level workflow)

Sometimes the system is specified indirectly through input-output descriptions rather than a complete state-space model. Minimal realization then means constructing the smallest state-space model consistent with that data.

4.2.1 Choosing a realization representation for reduction

A practical workflow begins by selecting an initial realization method that converts the available input-output data (such as impulse response samples, Markov parameters, or a transfer description) into some state-space form. From there, reduction techniques identify the portion corresponding to minimal internal dynamics. The goal is to ensure the starting realization is general enough to contain the true system behavior so that minimal reduction can subsequently remove spurious degrees of freedom introduced by modeling choices.

4.3 Numerical considerations (stability and conditioning)

In computation, identifying controllable and observable subspaces can be sensitive to numerical errors, especially when dynamics are nearly redundant or when the model has poorly conditioned matrices. Methods that use structured matrix factorizations, balanced coordinate transformations, or robust rank-revealing computations help mitigate issues. Additionally, reduction should be performed in a way that respects numerical stability, since truncated modes that are small but not exactly zero can lead to approximation errors if handled naively.

5 Canonical Forms and Their Role

5.1 Controllable canonical form (overview)

Canonical forms provide structured coordinates in which system parameters appear in specific patterns. The controllable canonical form emphasizes the controllable structure, arranging the state so that input reachability corresponds to a clear companion-like structure. While canonical forms are often not the most numerically convenient, they can clarify how minimality is reflected in the model’s structure.

5.2 Observable canonical form (overview)

Observable canonical form similarly arranges coordinates to highlight observability. In this representation, the output mapping and internal dynamics are structured so that the observable portion becomes explicit. Like the controllable form, it offers theoretical transparency, though practical implementations may favor other transformations for robustness.

Balanced realization is a method that transforms the model into coordinates where the controllability and observability properties are matched, typically through balancing the relative importance of different states. Although “balanced” does not automatically imply minimality, it is closely related: for minimal systems, balancing can highlight which modes are both controllable and observable. For model reduction tasks, balanced coordinates often enable truncation of weakly contributing dynamics.

5.3.1 Model order reduction connections (intuition)

In model reduction, one seeks an approximation with fewer states. Balanced realization provides a basis for separating strongly retained modes from weak modes using quantities like Hankel singular values. This is conceptually connected to minimal realization because modes that are either unreachable or unobservable are associated with zero or negligible contribution in the input-output mapping, and balanced methods make those contributions easier to assess.

6 Properties and Uniqueness

6.1 Existence of a minimal realization

A minimal realization exists for standard linear time-invariant models when an equivalent realization is available. Conceptually, among all equivalent models, one can identify those with the smallest internal dimension. In well-posed settings, the minimal state dimension is determined by intrinsic system behavior rather than by the particular coordinates used in a starting model.

6.2 Uniqueness up to similarity transforms

Although coordinates can differ, minimal realizations of the same input-output behavior are equivalent in a structural sense.

6.2.1 What “same behavior, different coordinates” means

Two minimal realizations may have different state vectors and different matrices, but they describe identical external behavior. The relationship between them is given by a similarity transformation: one realization can be converted into the other via a change of state coordinates. This expresses that minimality removes redundant dynamics, leaving only the essential internal degrees consistent with the input-output map.

6.3 Order (minimal dimension) as a system invariant

The minimal dimension, often called the system order, is an invariant associated with the input-output behavior. If two systems (or two descriptions of the same system) share identical behavior, their minimal orders coincide. This provides a way to interpret minimal realization as capturing a fundamental structural characteristic rather than an artifact of modeling choices.

7 Applications in Software and Engineering

7.1 Model reduction in simulation pipelines

In simulation-based workflows, large state-space models can be expensive. Minimal realization is a baseline simplification technique: by removing unreachable and unobservable parts, simulation can proceed with fewer states while preserving the same response to inputs. This is especially relevant when models are obtained from intermediate design steps or from system composition where redundancies can accumulate.

7.2 System identification workflows

Identification methods produce fitted models from data. These fitted realizations may include redundant dynamics due to overparameterization, noise, or imperfect constraints. Applying minimal realization as a post-processing step helps ensure the model reflects only the essential input-output dynamics supported by the data, improving interpretability and potentially reducing the burden of later tasks.

7.2.1 Validating minimality after parameter estimation

Minimality can be checked by evaluating controllability and observability ranks (in numerical settings, using tolerances). If the model is found to be non-minimal, reduction can yield a smaller equivalent system. In practice, identification uncertainty means the “true” minimal structure may be obscured by numerical thresholds; careful selection of tolerances and validation against held-out data is therefore important.

7.3 Integration with control design toolchains

Many control design tools assume or benefit from compact state representations. Minimal realization can improve compatibility with algorithms for stabilization, observer design, or state feedback by ensuring the design operates on an essential state basis. In automated pipelines, reducing models before design can help avoid issues tied to redundant modes and can streamline controller/observer order.

8 Common Pitfalls and Edge Cases

8.1 Degenerate systems (e.g., static gain only)

Some systems have no genuine internal dynamics and reduce to static input-output maps. In such cases, the minimal realization may have zero dynamic states. Algorithms must handle these degenerate cases correctly by returning an appropriate reduced model rather than forcing unnecessary state structure.

8.2 Non-minimal models produced by identification or fitting

Overfitting can yield models with extra modes that do not affect measured outputs or cannot be excited by inputs. Reduction addresses these, but identification noise can make the distinction between true and spurious modes subtle. As a result, minimal realization might reduce too aggressively if numerical rank decisions are not tuned to the noise level.

8.3 Discrete vs. continuous-time modeling differences

Minimality concepts extend across continuous and discrete time, but the mechanics of reachability/observability computations can differ. The state transition behavior, sampling, and numerical discretization steps influence how dynamics appear in the model.

8.3.1 Sampling effects on perceived dynamics

When a continuous-time system is sampled, the discrete-time model can exhibit dynamics that differ in apparent order from the continuous-time description, especially when sampling is slow relative to system time scales. Minimality in the sampled model reflects what the sampled input-output data can support, which may not match the continuous-time internal structure in a one-to-one manner.

9 Example Walkthroughs (Conceptual)

9.1 Reducing a non-minimal state-space model

Consider a state-space model provided with more states than necessary. The reduction process identifies which components are unreachable and which are unobservable.

9.1.1 Identifying unreachable states

One checks which state directions can be influenced by inputs. Those directions that cannot be driven from the chosen initial condition correspond to an unreachable subspace. In the transformed coordinates, these appear as blocks that do not affect the reachable dynamics, enabling their removal without changing the input-output response.

9.1.2 Identifying unobservable states

Next, one examines which internal directions influence outputs. States whose effects on output trajectories cannot be detected form the unobservable subspace. In coordinates aligned with this partition, the reduced output behavior depends only on the observable component, so the unobservable states can be truncated.

9.2 Verifying that the reduced model matches behavior

After truncation, the reduced model should reproduce the original input-output behavior. Verification is typically carried out by comparing transfer characteristics or by confirming that the impulse/step response generated by the reduced model matches the original within numerical tolerance. Consistent agreement indicates that eliminated states were indeed redundant under the chosen behavior equivalence criterion.

10 Further Reading and References

10.1 Standard textbooks in systems and control

Classical treatments of state-space modeling, controllability, observability, and realization theory are found in standard control textbooks covering linear system theory, matrix methods, and observer/controller design. These resources typically introduce minimality early and then discuss canonical forms and algorithms.

10.2 Survey articles on realization and minimality methods

Survey literature addresses computational approaches for minimal realization, balanced methods, and extensions to broader realization problems. Such articles often emphasize practical issues such as rank determination, numerical stability, and connections to system identification and model reduction.