Included in at least one reviewed higher-education syllabus.
S4-3.5 State Space Analysis
You can write state-space models, compute the state transition matrix, test controllability and observability, and design state-feedback control.
State space trades the single input-output transfer function for a vector of internal states — the price is matrix algebra, the payoff is systems with many inputs and outputs that classical methods cannot touch. Controllability and observability are the unit's twin questions: can the inputs steer every state, and can the sensors reconstruct every state — each a rank test with real engineering meaning. It closes the subject and deliberately opens a door: this formulation is the direct ancestor of the state representations in Kalman filters, robotics and reinforcement learning. The common confusion is treating the state vector as physical; it is a choice of coordinates, and different choices describe the same system.
Work through these
State, state variable, state vector, state space: the concepts
State space replaces one input-output relationship with a vector of internal variables that together capture everything the system remembers. The price is matrix algebra; the payoff is systems that classical methods cannot touch.
State-space representation of LTI systems
Writing a linear time-invariant system as four matrices relating states, inputs and outputs. Once a system is in this form, the same solution machinery applies whatever the physical domain was.
State transition matrix and its properties
The state transition matrix is the operator that moves the state forward in time with no input applied, and its properties are what make the general solution possible. It is the exponential of a matrix, and it behaves largely as you would hope.
Solution of the state equation
Solving the state equation gives the response to any input, splitting cleanly into the part that comes from the initial state and the part that comes from the input. That separation is one of the formulation's real advantages.
Controllability and observability tests
Controllability asks whether the inputs can steer every internal state, observability asks whether the sensors can reconstruct every one. Both are rank tests with blunt practical meaning: fail either and no controller design will rescue you.
Design of control via state-variable feedback
Feeding the measured states back through a gain matrix lets you place the closed-loop poles wherever you want, provided the system is controllable. This is the modern counterpart of tuning a gain by root locus.
Lab: simulate a state-feedback design and verify pole placement
Simulate a state-feedback design and confirm the poles landed where you placed them. This unit is also the bridge to Kalman filters and reinforcement learning, which use the same state description.
Sign in to keep your progress.
Free resources
We haven't checked most of these for screen reader use yet.
Links last checked 29 Aug 2026.
Stuck here?
Ask a mentor. A real person answers, and they can see exactly which topic you're on. Usually within a couple of working days.
Checking your session…
Topics shown in module order.