Control Systems: State-Space Methods

State-space methods are the backbone of modern control theory. They describe a dynamic system using vectors and matrices, which makes them well suited to multi-input multi-output (MIMO) plants, systems with coupled dynamics, and designs that must handle internal constraints and measured or unmeasured states. Unlike classical transfer-function approaches that focus mainly on input-output behavior, state-space modeling explicitly represents what is happening inside the system.

This article introduces state variables and state equations, then develops two central structural properties of state-space models: controllability and observability.

Why state-space methods matter

Many real systems are inherently multivariable. An aircraft has multiple control surfaces affecting multiple motion axes. A chemical process may have several actuators and several quality variables that interact. In these cases, treating each input-output pair independently can hide coupling and lead to fragile controllers.

State-space methods address this directly:

• They model the system with vectors, so multiple signals are handled naturally.
• Internal variables can be included, allowing analysis of stability, transients, and energy storage.
• The framework supports systematic design techniques such as state feedback and observers, which are difficult to express purely with transfer functions.

State variables and the idea of “state”

A system’s state is a minimal set of variables that, together with the inputs, determines the future evolution of the system. If you know the state at time $t_0$ and you know the future input $u(t)$ for $t\ge t_0$, then you can predict the output and internal behavior for all future times.

Choosing state variables

State variables are not unique. For a mechanical system, common states include positions and velocities. For an electrical network, capacitor voltages and inductor currents are natural choices because they store energy and describe dynamic behavior.

The practical goal is to choose state variables that are:

• Physically meaningful when possible
• Sufficient to capture the dynamics
• Convenient for measurement or estimation

The state-space model

For a continuous-time, linear time-invariant (LTI) system, the standard state-space form is:

$$\dot{x}(t)=Ax(t)+Bu(t)$$

$$y(t)=Cx(t)+Du(t)$$

where:

• $x(t)$ is the state vector (dimension $n$)
• $u(t)$ is the input vector (dimension $m$)
• $y(t)$ is the output vector (dimension $p$)
• $A$ is the state matrix ($n\times n$)
• $B$ is the input matrix ($n\times m$)
• $C$ is the output matrix ($p\times n$)
• $D$ is the feedthrough matrix ($p\times m$)

This model cleanly separates the dynamics (in $A$ and $B$) from how the system is measured (in $C$ and $D$). In many physical systems, $D=0$, meaning the output does not respond instantaneously to the input.

Discrete-time form

For sampled-data systems (digital control), the corresponding discrete-time model is:

$$x_{k+1}=A{x}_k+B{u}_k$$

$$y_k=C{x}_k+D{u}_k$$

The same concepts of controllability and observability apply, with closely related tests.

Interpreting the matrices in real systems

A useful way to read the state equation is to see $A$ as describing how the system evolves on its own, while $B$ describes how inputs push the state around. Similarly, $C$ tells you which combinations of internal states show up in sensors, and $D$ captures any direct input-to-output path.

In MIMO systems, each input can affect multiple states, and each output can depend on multiple states. The matrix form makes that coupling explicit, which is essential for correct analysis and controller structure.

Controllability: can you drive the state where you want?

Controllability is the property that determines whether an input can move the state throughout the state space. Intuitively, a system is controllable if you can steer the state from any initial condition to any desired final condition in finite time (for LTI systems under standard assumptions).

This matters because state feedback control relies on the ability to influence the states. If some modes are uncontrollable, no controller using the available actuators can reshape those dynamics.

The controllability matrix and test

For continuous-time LTI systems, the controllability matrix is:

$$\mathcal{C}=\left[\begin{array}{ccccc}B& AB& A^{2}B& \cdots & A^{n-1}B\end{array}\right]$$

The system is controllable if $\mathcal{C}$ has full rank $n$.

Practical meaning

• If rank is full, every dynamic mode of the system can be influenced through the inputs.
• If rank is deficient, some part of the dynamics is “out of reach” of the actuators. You might still be able to control certain outputs, but you cannot arbitrarily place all closed-loop poles through state feedback.

A common real-world cause is an actuator that does not couple into a particular energy storage element. For example, if a motor torque does not affect a flexible mode due to mechanical isolation, that mode may be weakly controllable or effectively uncontrollable in a simplified model.

Observability: can you infer the state from measurements?

Observability is the dual concept: whether you can reconstruct the internal state from the measured outputs over time. This is critical because many states are not directly measured. In practice, control algorithms often require state estimates, produced by an observer (or estimator) that fuses measurements and a model.

A system is observable if, given knowledge of the inputs and outputs over a finite interval, the initial state can be uniquely determined.

The observability matrix and test

The observability matrix is:

$$\mathcal{O}=\left[\begin{array}{c}C\\ CA\\ C{A}^2\\ ⋮\\ C{A}^{n-1}\end{array}\right]$$

The system is observable if $\mathcal{O}$ has full rank $n$.

Practical meaning

• If the rank is full, the outputs contain enough information to reconstruct all states (in principle).
• If not, some internal modes never show up in the measurements, so no observer can estimate them reliably.

A simple example is a sensor placed where a particular vibration mode has a node. The mode exists physically, but the measurement is insensitive to it, resulting in poor observability for that mode.

Controllability and observability in design

These properties are not abstract. They guide concrete decisions:

• Actuator placement and selection affects controllability. Adding an actuator, moving it, or changing its coupling can improve rank and influence key modes.
• Sensor selection and placement affects observability. Better measurement channels can reveal hidden dynamics and improve estimation quality.
• Model order and state choice can change the representation. Different state variables can lead to different matrices, but controllability and observability are properties of the underlying system behavior, not of one particular coordinate choice.

In many workflows, engineers first confirm controllability and observability, then proceed to design:

• State feedback (often written $u=-Kx$) to shape dynamics when states are available.
• Observers to estimate $x$ from $y$ when states are not measured, enabling output-based implementation.

While the detailed synthesis methods go beyond this introduction, the key point is structural: without controllability you cannot fully shape the system, and without observability you cannot fully know what the system is doing internally from your measurements.

Common pitfalls and how to avoid them

Confusing input-output behavior with internal properties

A transfer function might look well behaved while the state-space realization hides an uncontrollable or unobservable mode. That hidden mode can still matter, especially for disturbance response, internal stability, or when interconnections introduce feedback paths.

Overlooking scaling and numerical issues

Rank tests can be sensitive to numerical conditioning. In practice, engineers use robust numerical routines and consider “effective rank” when a mode is extremely weakly controllable or observable due to tiny couplings.

Ignoring the role of disturbances and noise

Even a theoretically observable system may be hard to estimate accurately if measurements are noisy and the relevant dynamics are lightly excited. Good modeling and sensible sensor bandwidth choices matter as much as the algebraic tests.

Summary

State-space methods represent control systems using state variables and matrix equations, making them ideal for MIMO systems and for designs that require insight into internal dynamics. The state equations

$$\dot{x}=Ax+Bu,y=Cx+Du$$

provide a unified framework for modeling, analysis, and design. Controllability answers whether inputs can drive the full state, tested via the rank of $\mathcal{C}$. Observability answers whether outputs reveal the full state, tested via the rank of $\mathcal{O}$. Together, these concepts form the foundation for modern control design, guiding actuator and sensor choices and enabling systematic feedback and estimation approaches.