The quintessential requirement of a closed-loop dynamic system is stability:
the ability of the system to operate under a variety of conditions without self-destructing [1].

From the previous post, we show that the eigenvalues determine the characteristic of the system’s response. Here we provide a detailed definition of stability, which is one of the most crucial concepts in the field of control. Stability is defined w



Equilibrium

An equilibrium state of the system is when the \(\dot{\mathbf{x}}=\mathbf{0}\). This implies that once the system is at the equilibrium,

For sure, the reason why we are focused on the origin \(\mathbf{0}\) is because the system is invariant with respect to the transfer of coordinates.

For that, it is necessary to discuss in great detail about the eigenvalues of the state matrix \(\mathbf{A}\).1



Characteristic Equation of the System

Now consider a case where we use full-state feedback \(\mathbf{u=-Kx}\), or simply the initial response of the system. \[ \dot{\mathbf{x}} = \mathbf{Ax} \] Then the eigenvalues of matrix \(\mathbf{A}\) is determined by the characteristic equation of matrix \(\mathbf{A}\): \[ \det({\lambda\mathbf{I-A}})=\lambda^n + a_{n-1}\lambda^{n-1} =0 \]

Note that the fundamental theorem of Gauss shows that we definitely have \(n\) eigenvalues, which can be complexed value too.

Why always the origin?

One might wonder why we only care about the origin point \(\mathbf{0}\)

Algebraic and Geometric Multiplicity



Quick Example 1



Quick Example 2



As you can see from equation (6), the eigenvalues of matrix \(\mathbf{A}\) fully determines the how the system. Quoting from:

  • (Remark 1) As shown in equation 6, the eigenvalues of matrix \(\mathbf{A}\) determine the destiny of system’s response
    • If at least one of the eigenvalues of matrix \(\mathbf{A}\) is strictly on the right-half plane of the complex plane (i.e., positive real value), then there exists an unbounded \(z_i\) value, which means the system response is unbounded and therefore unstable.
    • If all the eigenvalues are on the left-half plane of the complex plane (i.e., non-positive real values), the system response is bounded and therefore stable.
    • If all the eigenvalues are strictly on the left-half plane of the complex plane (i.e., negative real values), the system is stable AND the response \(\mathbf{x}\) converges to \(\mathbf{0}\). We call this the system is asymptotically-stable.

References

[1]
B. Friedland, Control system design: An introduction to state-space methods. Courier Corporation, 2012, p. 124.

  1. For nonlinear systems, the definition of stability is more mathematically exotic.↩︎