# Converting nonlinear system into equivalent nonlinear of the Byrnes-Isidori normal form

I have a nonlinear system (Ball & Beam) which is described by the following equations of motion:

$$\ddot{y} + \frac{mg}{a} \sin(θ) -\frac{m}{a}y\dot{θ}^2 = 0$$

$$\ddot{θ} + \frac{2m}{b}y\dot{y}\dot{θ}+\frac{mg}{b}y\cos(θ) = u$$

where the letters stand for:

• $$y \rightarrow$$ Ball Position
• $$θ \rightarrow$$ Angle of Beam
• $$u \rightarrow$$ External Torque Applied
• $$b = my^2+J$$
• $$a, m, g, J \rightarrow$$ Positive Constants

Now, my final goal is to convert this nonlinear system into an equivalent nonlinear system of the Byrnes-Isidori normal form. The state-space representation ($$\dot{x} = f(x)+g(x)u)$$ of the system by considering the state variables $$x_1 = y$$, $$x_2 = \dot{y}$$, $$x_3 = θ$$ and $$x_4 = \dot{θ}$$ is:

$$\begin{gather*} \dot{x_1} = x_2 \\ \dot{x_2} = \frac{m}{a}x_1x_4^2-\frac{mg}{a}\sin(x_3) \\ \dot{x_3} = x_4 \\ \dot{x_4} = -\frac{2m}{b}x_1x_2x_4-\frac{mg}{b}x_1\cos(x_3)+\frac{1}{b}u \\ y = x_1 \rightarrow Output \end{gather*}$$

The relative degree of the system is $$r=3 < n=4$$ (resticted to the region where $$D_o = \{x\epsilon R^4 \ | \ x_1 \neq 0 \ \& \ x_4 \neq 0\}$$ ), so we have the internal states $$μ_1$$, $$μ_2$$ and $$μ_3$$, which are:

• $$μ_1 = y \Rightarrow \dot{μ_1} = μ_2$$
• $$μ_2 = \dot{y} \Rightarrow \dot{μ_2} = μ_3$$
• $$μ_3 = \ddot{y} \Rightarrow \dot{μ_3} = \dddot{y}$$

Now, in order to find the internal dynamics we need a function $$ψ$$ which validates the expression:

$$\nabla{ψ}\cdot g(x) = 0 \Rightarrow \frac{\partial ψ}{\partial x_4}\cdot \frac{1}{mx_1^2+J} = 0 \Rightarrow$$

$$\frac{\partial ψ}{\partial x_4} = 0, \ \text{since} \ \frac{1}{mx_1^2+J} > 0$$

where $$g = [0 \ \ 0 \ \ 0 \ \ \frac{1}{mx_1^2+J}]^T$$. I chose the function to be $$ψ(x) = x_1 + x_2 + x_3$$. And now, the internal dynamics by differentiating this function is:

$$\dot{ψ} = \dot{x_1}+\dot{x_2}+\dot{x_3} \Leftrightarrow \dot{ψ} = y + \dot{y} + x_4$$

$$\dot{ψ} = μ_1 + μ_2 + x_4$$

Now, what's left to do is to express the term $$x_4$$ as a function of $$ψ$$ and $$μ$$. In order to do so, I followed the below procedure:

$$μ_3 = \ddot{y} \Rightarrow μ_3 = \dot{x_2} = \frac{m}{a}x_1x_4^2-\frac{mg}{a}\sin(x_3)$$

The term $$x_3 = ψ - x_1 - x_2 = ψ - μ_1 - μ_2$$ and combibning these I obtained an expression for $$x_4$$:

$$x_4 = [\frac{a}{m}\frac{μ_3}{μ_1}+g\frac{\sin(ψ-μ_1-μ_2)}{μ_1}]^{\frac{1}{2}}$$

My question is regarding the zero dynamics that can be found by $$\dot{ψ} = w(0,ψ)$$ meaning that I should set $$μ_1=μ_2=μ_3=0$$ while $$μ_1 \neq 0$$ (which also validates restriction for the relative degree to be well-defined). I am stuck at this point and considering what to do. I also considered to take a simpler $$ψ$$ function such as:

$$ψ = x_1 + x_2 \Rightarrow \dot{ψ} = μ_2 + μ_3$$

but this means that $$\dot{ψ}$$ only depends on $$μ_1$$ & $$μ_2$$ resulting the zero dynamics to be $$\dot{ψ}(0,ψ) = 0$$. Do I miss something regarding the procedure or maybe am I somewhere wrong ? Could use any help.

• Check Theorem 13.1 in H. K. Khalil, Nonlinear Systems: Pearson New International Edition, vol. Third editof Pearson New International Edition. Essex: Pearson, 2014. May 13 '20 at 16:57
• I have only the old version and I assume the theorem is the one regarding the diffeomorphism $T(x)$. Well, haven't I followed the same procedure ? May 13 '20 at 17:33
• Yes and if you follow the proposed procedure you'll arrive at the desired. I did not check your answer. May 13 '20 at 18:23
• You should pay attention to some terms. $\mu_1,\mu_2,$ and $\mu_3$ are not internal states. At least they are not related to the internal dynamics. They form the feedback-linearized part of the system and the $\Psi$ is your "internal" state. You should check if the nonlinear transformation $\chi=[\mu_1,\mu_2,\mu_3 \Psi]$ is, in fact, a diffeomorphism which you validate with the Jacobian that must have full rank (i.e. $n$). It should be sufficient to choose $\Psi=x_3$. If you find that the zero dynamics don't change, i.e. the rate of change is 0, you can apply the control law