Controllability is a key concept in control system design, determing whether ther a system 's states can be contron to desired values using input signals. Calculating thee controllability matrix helps controlters assess this concuritte for a given system modeled in state space form.

State Space Requiretion

A system in state space form is contrited by the equations:

Xi1; Xi1; FLT: 0 Xi3; Xi3; Xix (t) = Ax (t) + Bu (t) Xi1; Xi1; FLT: 1 Xi3; Xi3;

where message 1; Xi1; FLT: 0 message 3; Xi3; x (t) message 1; Xi1; FLT: 1 message 3; Xi3; is the state vector, Xi1; FLT: 2 message 3; FLT: 1; Xi3; FLT: 3 message; Xi3; Is the system matrix, and begas1; FLT: 4 message 3; Xi3; B message 1; FLT: 5 message 3; X3; is the input matrix.

Kontrollability Matrix Calculation

Thee controllability matrix, aspect 1; head1; FLT: 0 head3; head3; head1; fLT: 1 head3; head3;, is construted by concatenating thee matrices:

BEL1; BEL1; FLT: 0 BEL3; BEL1; B, AB, A ² B, BEL3; BEL1; FLT: 1 BEL3; BEL3; BEL3; EL3;

where beh1; Xi1; FLT: 0 Xi3; Xi3; n Xi1; FLT: 1 Xi3; is the number of states in thee system. This matrix indicates whether thee states can be controlled through the input.

Etap to Kalkulacja

Follow these steps to compute the controllability matrix:

  • Identyfikator macierzy: 1; 1; 1; FLT: 0; 3; A; 1; FLT: 1; 3; FLT: 1; 1; FL3; and; 1; FLT: 2; 3; B; 1; FLT: 3; 3; FLT; 3; FLT; FLM; FLM; 3; FLM; FLM; FLT; FLT: 3; FL3; FL3; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM; FLM;
  • Obliczenie tej macierzy jest 1; 1; FLT: 0; AB: 3; AB: 1; AX1; FLT: 1 X3; FLT: 1 XI3; BY multipliing XI1; FLT: 2 XI3; FLT: 3; FLT: 3 XI3; FL3; With XI1; FLT: 4 XI3; FLT: 3; B XI1; FLT: 5 XI3; FLT: 5 XI3; FLS; 3; FLT: 3; FLT: 4 XIXIXIX3; FL1; BL: 3; B XIX1; FLT: 5 XIXIXIXIX3; 3; FLT: 5 XIXIXIX3; 3;
  • Compute higher powers of prevent 1; Xi1; FLT: 0 presenta3; Xi3; A presenta1; FLT: 1 presenta3; Xi3; (A ², A ³, etc.) and multiply each by presentation 1; Xi1; FLT: 2 presenta3; Xi3; B pretendation 1; FLT: 3 preventable 3; Xi3; FLT:.
  • Concatenate all matrices horizontally to form prevent 1; Xi1; FLT: 0 presentation 3; Xi3; Xion1; Xion1; FLT: 1 presentally 3; Xion3;.

Kontrola kontroli

Thee system is controllable if thee controllability matrix eng1; ing1; FLT: 0 contri3; ing3; FLT: 1 contribul3; engy3; has full rank, equal to the number of states. This can be verified by y calculating thee e rank of eng1; FLT: 2 contribul 3; eng. 3; eng. 1; FLT: 3 contribuild 3; eng.