S4-3.5 State Space Analysis

What this is and why it exists

Classical control watches one input and one output and hides everything between. State space opens the box: model the system as a vector of internal states evolving under matrix dynamics, and suddenly multi-input multi-output systems, time-varying systems and computer-based design all become tractable. Two yes-or-no questions — can the inputs steer every state, can the sensors see every state — plus pole placement by feedback form the core. This formulation is also the direct ancestor of the state representations running through Kalman filters, robotics and reinforcement learning, which makes it the bridge from this course to modern AI work.

The vocabulary

• State — the smallest set of variables whose present values plus future inputs determine all future behaviour.
• State equations — first-order matrix form: x-dot equals A x plus B u; y equals C x plus D u.
• State transition matrix — e to the A t: the operator that carries the state forward through time with no input.
• Controllability — whether the inputs can drive the state anywhere in finite time; tested by the rank of the controllability matrix.
• Observability — whether the state can be reconstructed from watching the outputs; tested by the rank of the observability matrix.
• State feedback — control as u equals minus K x: every state weighed and fed back, placing the closed-loop poles where design demands.

The mental model

The state is the system's complete memory. For a moving mass: position and velocity — with those and the future forces, the entire future is fixed; nothing else about the past matters. Writing dynamics as x-dot equals A x plus B u trades one high-order equation for a first-order VECTOR equation, and the payoff is that matrix tools now do dynamics: eigenvalues of A are the system's poles, and the state transition matrix e to the A t — computable via the Laplace route as the inverse transform of (sI minus A) inverse — replays the transient response as pure linear algebra. The full solution stacks the free motion and the driven motion: transition matrix times initial state, plus the convolution of the transition matrix with the input.

Controllability and observability are rank tests with practical meaning. Assemble B, AB, A-squared-B and onward: full rank means the inputs reach every internal mode — an actuator placement question in matrix form. Assemble C, CA, CA-squared likewise: full rank means no internal motion is invisible at the sensors — a sensor placement question. A transfer function silently discards any uncontrollable or unobservable mode through pole-zero cancellation; the state model refuses to hide it, which is a safety property when the hidden mode is unstable.

State feedback is the design reward. Measure (or estimate) all states, feed back u equals minus K x, and the closed-loop matrix becomes A minus B K — whose eigenvalues, when the system is controllable, K can place ANYWHERE. Choose pole locations for the response you want and solve for K: pole placement. Where classical design nudged a locus with one gain knob, state feedback has as many knobs as states. Verifying a placement in simulation — poles where you asked, response matching — closes the loop on the whole course.

What you should now be able to explain or do

Write state equations for a physical system and identify A, B, C, D. Compute a state transition matrix via the Laplace route and solve the state equation. Run both rank tests and interpret failure physically. Design K to place poles for a controllable system and confirm by simulation.

Check yourself