State space represention is a crisal model used to o descripbe thee behavior of mechanical systems. It provides a complework to analyze systemem dynamics using matrices and vectors, making it easier to design controllers and analyze stability.

Basics of State Space Amendtion

Te state space model represents a system with a set of first-order diferentail equations. It uses a state vector to encapsulate all necessary information about thee system 's current condition.

Te general form is:

CLAS1; CLAS1; CLAS3; CLAS3; CLAS3x (t) = Ax (t) + CLAS1; CLAS1; CLAS1; CLAS3FT: 1 CLAS3; CLAS3FLAS3CATS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRASIVIRASIVA

fl1f; fl1f; flt: 0 fl3f; x (t) fl1f; fl1f; fl1f; fl1f; is the vector, fl1f; fl1f; fl1f; fl1f; fl1f; flt: 3 fl3f; is the input, and fl1d; fl1f; flt: 4 fll1f; fl1f; fl1f; fl1f; flt: 5 fl3f; and fl1f; fl1f; fl1f; fl1f; fl1f; fl1f; fllf; fllf 3; are matrices definig systems dynamics and input inflince.

Example: Mass- Spring- Damper System

Konsider a mass- spring- damper systems with mass auf 1; FLT: 0 pc 3; pc 3; pc 1f; pc 1f; pc 1f; pc 3f; pc 3f; pc 3f; pc 3f; pc 3f; pj 3f; pj 3f motion are:

CLAS1; CLAS1; CLAS3; CLAS3; m * x CLAS3; + c * x CLAS3; + k * x = u (t) CLAS1; CLAS1; CLAS3; CLAS3; CLAS3;

Defining the state variables as CLAS1; CLAS1; CLAS3; CLAS3; CLAS3; CLAS3; CLAS3; CLAS3; CLAS3; CLAS1; CLAS3; CLAS3; CLAS3; CLAS3; CLAS3; CLAS3; CLAS3; CLAS3; CLAS3; CLAS3; CLASSIP3; CLAS1; CLAS1; CLAS3; CLAS3s:

CLANE1; CLANE1; FLT: 0 CLANE3; CLANE3; CLANE3x CLANE1; CLANE1; CLANE1; CLANE11; CLANE3CCANE3CLANE3CLANE3; CLANE3CLANE3CLANE3CLANExCLANExCLANE1; CLANE1CLANE1CLANE1CLANEx;

CLAS1; CLAS1; CLAS3; CLAS3; CLAS3x CLAS3x CLAS3x = - (k / m) * CLAS3x CLAS3x) * CLAS3f (1 / m) * u (t) CLAS1f; CLAS3f; CLAS3f: 1 CLAS3f; CLAS3f;

In matrix form:

CLAS1; CLAS1; CLAS3; CLAS3; CLAS3x (t) = A x (t) + B u (t) CLAS1; CLAS1; CLAS3FT: 1 CLAS3; CLAS3FLAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CRAS3CITIRAS3CITIRAS3CITIDEZITUM3CITIRES3CITIRES3CDEZITIDERAS@@

kde:

  • A = CLAS1; CLAS1; 0, 1 CLAS3;, CLAS1; - (k / m), - (c / m) CLAS3; CLAS3;
  • B = CLAS1; CLAS1; 0 CLAS3;, CLAS1; 1 / m CLAS3; CLAS3;

Výpočty a analýzy

Using the state space model, differs can perfor stability analysis, controllability, and observability assessments. Eigenvalues of matrix time1; differen1; differens: 0 cf3; differentium-differentia analysis, controllability, controllability, and observability assessments.

Kontrolorility is checked by examining thee controllability matrix:

CLANE1; CLANE1; CLANE1; CLANE3; CLANE3; CLANE3; CLANE1; CLANE1; CLANE1; CLANE3; CLANE3; CLANE3; CLANE3;

If CLAS1; CLAS1; CLAS1; CLAS3; CLAS1; CLAS1; CLAS1; CLAS3; CLAS3; has full rank, thee systemem is controllable.