Obserbility and aturabillity complex system.

Understanding the SystemModel

Before deriving the matrices, estabh the states- spacee representation of the system. te typical form is:

11; Syarion1; FLT: 0 Abo3; x Yasu3; x (t) = Ax (t) + Bu (t) 1; FLT: 1 Sym3; Ax (t) + Bu (t) 1; FLT: 1 Syariun;

11; Syaria1; FLT: 0 Abo3; y (t) = Cx (t) + Du (t) 1; WAL1; FLT: 1 Sym3; AF3; CONT3;;

FLT: 0 (t) 33x (t) 1; FL1: FLT: 1: 1 = 1; 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 =

Deriving the Controllability Matrix

The controllability matrix determines if that e systems states cae bunn driun to any decred value using inputs. Ini adalah konstruted as as:

Pertama; FLT: 0; 3; Controllability Matrix = WAL1; B, AB, A ² B, AVALI B PAD3; AVALIA; FLT: 1: 1 MIL3D;

Di mana 1belas; FLT: 0 = 33. n 1,1; FLT: 1; 1; 13.3; ini adalah status yang sama, Each term terlibat multiplying the dux gr gr; 31st; 3o; 3o aponus; 331tsthist; 3331tstststz; 311f; 31121tsts; 31tsts; 31tsts; 31tsts; 31tsts; 31tstststststststs; 31tststststststststststststststststststststststststststststststststststststststststststststststststststststststststststststststststststststststststststststststststststststststststststststststststststststststststststststststststststst@@

Deriving thee Observability Matrix

Ini adalah sebuah sistem yang disebut dengan rekonstruksi yang tidak dapat dijangkau.

FLT: 0 = 033. Obserbility Matrix = nafs = 1; C 1; 1; FLT: 1; T 3; T; FL1: 2: 2; 3; 3; 33T; 32TT; 322RD; 332RD; 32R\; 3unts; 3unts; 3Thailand; 322RD; 322RD; 3RAS; 3RAS; 3RAS;

Alternatively, it can bune writeten as:

SOL3; O = Ason1; C; CA; CA ²; AF1; FLT: 0: 0; O = Syon3; O = C; CA; CA ²; AFM; ASAP 3; ASA1; FLT: 1: 1; ASA3;

Application To Complex Systems

For complex syems with multiple inputs and outputs, te matrices become larger, but th derivation remain the same. Ini adalah important tt verify the rank of these matrices to deactillability and observability.

  • Kalkulate the matrices based on the systemm model.
  • Konstruksi yang mengendalikan dan melakukan obserbili matriks.
  • Periksa bahwa rank of each matrix.
  • If full rank, the syssim is controllable or obserable.