State-Space Models, Inputs, and Outputs

How dynamical systems are rewritten in state-space form, why the state summarizes the past for future prediction, and how inputs and outputs organize control problems.
Modified

April 26, 2026

Keywords

state-space, state, input, output, control

1 Role

This is the first page of the Control and Dynamics module.

Its job is to introduce the basic control viewpoint:

a dynamical system is organized around internal state, external input, and measured output

That shift is small in notation, but huge in meaning. It is what turns a differential equation into something we can steer, observe, and eventually optimize.

2 First-Pass Promise

Read this page first in the module, after the ODE bridge into control.

If you stop here, you should still understand:

  • what a state is supposed to represent
  • why control models separate input from state
  • what the output is doing in the model
  • how a second-order physical law becomes a first-order state-space system

3 Why It Matters

Many physical or computational systems come to us as laws of change:

\[ \text{position''} = \text{force law}, \qquad \text{concentration'} = \text{reaction law}, \qquad \text{voltage'} = \text{circuit law}. \]

For analysis alone, that may be enough.

For control, we need a more operational language:

  • what information do we need to predict the future?
  • what quantity can we choose or actuate?
  • what quantity do we actually measure?

That is why state-space models matter.

They separate three roles:

  • the state stores the system’s internal configuration
  • the input is what we can apply from outside
  • the output is what we read out or care about

Once those roles are explicit, later questions about reachability, observability, feedback, and estimation become mathematically clean.

4 Prerequisite Recall

  • a higher-order ODE can be rewritten as a first-order system by introducing state variables
  • linear systems evolve through matrices and, in continuous time, often through e^{tA}
  • sampled systems turn continuous evolution into discrete state updates
  • not every quantity we care about has to be part of the state; some quantities are outputs derived from the state

5 Intuition

5.1 State As The Right Summary Of The Past

If a system is causal, then the future depends on the past.

But carrying around the entire past input and trajectory is too much information.

The idea of state is that we compress the relevant past into a finite vector that is just large enough to predict future behavior once future inputs are specified.

So state is not just “some variables we happen to write down.”

It is:

the memory of the past that the future still needs

5.2 Inputs And Outputs Play Different Roles

The input u is what the outside world can inject:

  • a force
  • a voltage
  • a thrust command
  • a steering angle

The output y is what we measure or care to track:

  • position
  • velocity
  • temperature
  • a sensor reading

State, input, and output are related, but they are not interchangeable.

5.3 The Same Dynamics Can Be Written In Different State Coordinates

There is often more than one valid choice of state.

Different coordinates may make a model easier to simulate, analyze, or control.

So one of the first things to learn is:

the state-space model is not unique, but the dynamical behavior it represents is

5.4 Continuous-Time And Discrete-Time Versions Belong To The Same Family

In continuous time, we usually write

\[ \dot x(t)=f(x(t),u(t),t), \qquad y(t)=g(x(t),u(t),t). \]

In discrete time, we usually write

\[ x_{k+1}=F(x_k,u_k,k), \qquad y_k=G(x_k,u_k,k). \]

The control viewpoint works in both settings. The difference is whether evolution happens continuously or in sampled steps.

6 Formal Core

Definition 1 (Definition: State) At a first pass, the state of a dynamical system at time t_0 is the information needed, together with future inputs after t_0, to determine the future behavior of the system.

This is why the state is often described as a sufficient summary of the relevant past.

Definition 2 (Definition: Continuous-Time State-Space Model) A continuous-time state-space model has the form

\[ \dot x(t)=f(x(t),u(t),t), \qquad y(t)=g(x(t),u(t),t). \]

Here:

  • x(t) is the state
  • u(t) is the input
  • y(t) is the output

Definition 3 (Definition: Discrete-Time State-Space Model) A discrete-time state-space model has the form

\[ x_{k+1}=F(x_k,u_k,k), \qquad y_k=G(x_k,u_k,k). \]

This is the sampled-step viewpoint that becomes important for digital control and simulation.

Definition 4 (Definition: Linear Time-Invariant State-Space Model) The standard continuous-time LTI form is

\[ \dot x = Ax + Bu, \qquad y = Cx + Du. \]

The standard discrete-time LTI form is

\[ x_{k+1}=A_d x_k + B_d u_k, \qquad y_k=C_d x_k + D_d u_k. \]

Theorem 1 (Theorem Idea: State Plus Future Input Determines Future Output) For a well-posed state-space model, once the state at time t_0 and the future input after t_0 are known, the future state and output are determined.

This is the main reason state is the right object for control.

Theorem 2 (Theorem Idea: Linear Responses Split Into Initial-Condition And Forced Parts) For linear state-space systems, the total response can be decomposed into:

  • the response caused by the initial state with zero input
  • the response caused by the input with zero initial state

This split is one of the main advantages of linear control language.

7 Worked Example

Consider the mass-spring-damper equation

\[ m q'' + c q' + k q = u(t), \]

where:

  • q(t) is position
  • u(t) is the applied force

Suppose the measured output is position:

\[ y(t)=q(t). \]

Define the state variables

\[ x_1 = q, \qquad x_2 = q'. \]

Then

\[ \dot x_1 = x_2, \]

and from the original equation,

\[ \dot x_2 = -\frac{k}{m}x_1-\frac{c}{m}x_2+\frac{1}{m}u. \]

So the state-space model is

\[ \dot x = \begin{bmatrix} 0 & 1 \\ -k/m & -c/m \end{bmatrix}x + \begin{bmatrix} 0 \\ 1/m \end{bmatrix}u, \]

with output equation

\[ y= \begin{bmatrix} 1 & 0 \end{bmatrix}x. \]

This example shows the whole control translation in one place:

  • a second-order ODE becomes a first-order system
  • the forcing term becomes an input channel
  • the quantity we care about becomes an output map

The state now carries enough information to evolve the system forward.

8 Computation Lens

When you meet a control model, ask:

  1. what variables are chosen as the state, and why are they enough?
  2. what is the input, and what can actually be actuated in the real system?
  3. what is the output, and is it the same as the full state?
  4. is the model continuous-time, discrete-time, or a sampled version of one into the other?
  5. what parts of the behavior come from the current state versus the external input?

Those questions usually make the model much easier to read than the matrix symbols alone.

9 Application Lens

9.1 Robotics And Mechanical Systems

Position and velocity are natural state variables, while forces or torques become inputs and sensor readings become outputs.

9.2 Circuits And Signals

Voltages and currents often form the state, external sources become inputs, and measured ports become outputs.

9.3 Estimation And Filtering

The output equation is the entry point for observer design and Kalman filtering, because it tells us what part of the hidden state is actually seen.

10 Stop Here For First Pass

If you can now explain:

  • what a state is supposed to summarize
  • why inputs and outputs are distinct from the state
  • how continuous-time and discrete-time state-space models are written
  • how a second-order physical equation becomes a first-order control model

then this page has done its job.

11 Go Deeper

The next natural step in this module is:

The strongest adjacent live pages right now are:

12 Optional Deeper Reading After First Pass

The strongest current references connected to this page are:

13 Sources and Further Reading

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