State Space

Wikipedia · State-space representation · CC BY-SA 4.0

State-space form replaces one high-order differential equation with a system of first-order equations: x' = Ax + Bu (state equation) and y = Cx + Du (output equation). The state vector x captures everything the system remembers. A, B, C, D are matrices.

State variables

A state variable is a quantity whose current value, together with the input, determines the future behavior of the system. For a mass-spring-damper, the state variables are position and velocity. For an electrical circuit, they might be capacitor voltages and inductor currents. The state vector x collects all state variables into a column vector.

Controllability and observability

A system is controllable if you can steer the state from any initial condition to any target using the input. Check: the controllability matrix [B, AB, A^2B, ...] must have full rank. A system is observable if you can determine the full state from the output history. Check: the observability matrix [C; CA; CA^2; ...] must have full rank.

Scheme

; State-space simulation: x' = Ax + Bu, y = Cx
; Mass-spring-damper: m*x'' + c*x' + k*x = u
; State: [position, velocity]
; A = [[0, 1], [-k/m, -c/m]], B = [[0], [1/m]], C = [1, 0]

(define m 1.0) (define c 0.5) (define k 2.0)
(define dt 0.05)

; State: (x1 x2) = (position, velocity)
(define (simulate x1 x2 u steps)
  (if (= steps 0) (begin (display "Done.") (newline))
    (let* ((x1-dot x2)
           (x2-dot (+ (* (- (/ k m)) x1) (* (- (/ c m)) x2) (/ u m)))
           (new-x1 (+ x1 (* x1-dot dt)))
           (new-x2 (+ x2 (* x2-dot dt))))
      (if (= 0 (modulo steps 10))
        (begin
          (display "  pos=") (display new-x1)
          (display "  vel=") (display new-x2)
          (newline))
        'skip)
      (simulate new-x1 new-x2 u (- steps 1)))))

(display "Mass-spring-damper step response:") (newline)
(simulate 0.0 0.0 1.0 200)

Neighbors

📐 Hefferon Ch. 3 — Linear Maps — A is a linear map from state space to state space

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