2.2.2. Stability

When analyzing the stability of a dynamic linear system two cases arise. The first is related to the zero state-response, where the output is expected to be bounded if the input is also bounded and the initial condition is equal to zero, meaning that the system is BIBO stable. The second case is about the zero-input response, where the system has no input and has nonzero initial condition. In this way, the system will be stable in the sense of Lyapunov if the output response is bounded and will be asymptotically stable if the output response approaches zero as *t* → ∞. Both cases can be assessed through the roots of the characteristic polynomial associated with matrix **A** of the state space model representation. Thus, if the roots of such a characteristic polynomial have negative real part then the system is completely stable.

Regarding the full-bridge Buck inverter–DC motor system, the state space representation of its model (21)–(24) is given by

$$\begin{aligned} \dot{\mathbf{x}} &= \mathbf{A}\mathbf{x} + \mathbf{B}u\_{\text{av}}, \\ y &= \mathbf{C}\mathbf{x}, \end{aligned} \tag{33}$$

where

$$\mathbf{x} = \begin{bmatrix} i \\ v \\ i\_a \\ \omega \end{bmatrix}, \qquad \mathbf{A} = \begin{bmatrix} 0 & -\frac{1}{L} & 0 & 0 \\ \frac{1}{C} & -\frac{1}{RC} & -\frac{1}{C} & 0 \\ 0 & \frac{1}{L\_a} & -\frac{R\_a}{L\_a} & -\frac{k\_c}{L\_a} \\ 0 & 0 & \frac{k\_m}{\parallel} & -\frac{b}{\uparrow} \end{bmatrix}, \qquad \mathbf{B} = \begin{bmatrix} \frac{E}{L} \\ 0 \\ 0 \\ 0 \end{bmatrix}, \qquad \mathbf{C} = \begin{bmatrix} 0 \ 0 \ 0 \ 1 \end{bmatrix}. \tag{34}$$

While the characteristic polynomial associated with **A** is

$$P(s) = a\_0s^4 + a\_1s^3 + a\_2s^2 + a\_3s + a\_4. \tag{35}$$

with

$$\begin{aligned} a\_{0} &= 1, \\ a\_{1} &= \frac{bL\_{a}R\mathcal{C} + fR\_{a}R\mathcal{C} + fL\_{a}}{fL\_{a}R\mathcal{C}}, \\ a\_{2} &= \frac{fL\_{a}R + fRL + bR\_{a}R\mathcal{C}L + k\_{c}k\_{m}R\mathcal{C}L + bL\_{a}L + fR\_{a}L}{fL\_{a}R\mathcal{C}L}, \\ a\_{3} &= \frac{bL\_{a}R + bRL + fR\_{a}R + bR\_{a}L + k\_{c}k\_{m}L}{fL\_{a}R\mathcal{C}L}, \\ a\_{4} &= \frac{bR\_{a} + k\_{c}k\_{m}}{fL\_{a}\mathcal{C}L}. \end{aligned}$$

By using the following Routh array,

*s*4 *a*0 *a*2 *a*4 *s*3 *a*1 *a*3 *s*2 *b*1 *b*2 *s*1 *c*1 *s*0 *d*1 (36)

where

$$\begin{aligned} b\_1 &= \frac{a\_1 a\_2 - a\_0 a\_3}{a\_1}, \\ b\_2 &= \frac{a\_1 a\_4 - a\_0 a\_5}{a\_1} = a\_4, \\ c\_1 &= \frac{b\_1 a\_3 - a\_1 b\_2}{b\_1} = \frac{a\_1 a\_2 a\_3 - a\_0 a\_3^2 - a\_1^2 a\_4}{a\_1 a\_2 - a\_0 a\_3}, \\ d\_1 &= \frac{c\_1 b\_2 - b\_1 b\_3}{c\_1} = b\_2 = a\_{4,2} \end{aligned}$$

It can be demonstrated that the roots of (35) have negative real part if *a*0, *a*1, *a*2, *a*3, *a*4, *b*1, *c*1, and *d*1 are positive. As all system parameters associated with (21)–(24) are positive and after computing (36), it is concluded that the full-bridge Buck inverter–DC motor system is completely stable.
