State space represention is a mathetical model used to to describbe thee behavor of mechanical systems. It provides a framework to analyze systems dynamics using matrices andd vectors, making it easyr to design controllers andd analyze stability.

Basics of State Space contribution

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

Te general form is:

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: 3; u (t) message 1; Xi1; FLT: 3 message 3; FLT: 3 message 3; Is the input, andd message 1; FLT: 4 message 3; FLT: 3; A message 1; FLT: 5 megatirices determing stem dynamics and input influence.

Egzamin: Mass- Spring- Damper System

Consider a mass- spring- damper system with mass presen1; Sig1; FLT: 0 + 3; Sig3; m + 1; Sig.1; FLT: 1 + 3; Signature;, damping coefficient present 1; Sigmund 1; FLT: 2 + 3; c + 1; Sigmund 1; FLT: 3; Sigmund 3;, and spring constant present 1; Sigmund 1; FLT: 4; Sigmund 3; k Sig.1; Sigmund; FLT: 5 + 3; Sigmund; Sigmund; Thee equations of motion are:

(zob. pkt 2.2.1.1.1 niniejszego załącznika)

Defining the state variables as present 1; Xi1; FLT: 0 presenta3; Xi3; x presentax x presentation 1; Xi1; FLT: 1 presenta3; Xi3; ande presenta01; Xi1; FLT: 2 presentable 3; Xion3; Xion1; FLT: 3 presentation 3;, the state space form becomes:

Xi1; Xi1; FLT: 0 Xi3; Xi3; Xix XiV1; XiV1; FLT: 1 XiV3; XiV3; XiV3;

(k / m) * x (c / m) * x (1 / m) * u (t) (1; FLT: 1; FLT: 1;

In matrix form:

Xi1; Xi1; FLT: 0 Xi3; Xi3; Xix (t) = A x (t) + B u (t) Xi1; Xi1; FLT: 1 Xi3; Xi3; Xi3;

Gdzie?

  • A = BEL1; BEL1; 0, 1 BEL3;, BEL1; - (k / m), - (c / m) BEL3; BEL3;
  • B = Xi1; Xi1; 0 Xi3;, Xi1; 1 / m Xi3; Xi3;

Obliczenia i analizy

Using thee state spate model, incorporates can perfority stability analysis, controllability, and observability assessments. Eigenvalues of matrix incorporate; encorporation; FLT: 0 message 3; encorporation; A message; encorporate 1; FLT: 1 messability; encorporate system stability.

Controllability is checked by examinang the controllability matrix:

Xi1; Xi1; FLT: 0 Xi3; Xi3; Q = Xi1; B, AB Xi3; Xi1; Xi1; FLT: 1 Xi3; Xi3;

If Xion1; Xion1; FLT: 0 Xion3; Xion3; QXI1; XiN1; FLT: 1 Xion3; Xion3; has full rank, the system is controllable.