Feedback linearization with internal dynamics

• For non-linear systems, linearization can be performed at the operating point (Taylor linearization). A linear system is obtained, which (however) only has validity in a small vicinity of the operating point.
• The Harmonic Balance method is mainly used for   system analysis. With this method  important system properties, such as the continuous oscillations, can be determined,  which can be interpreted for a stable behavior of the non-linear control loop.
• We will now describe a method which enables us to design controllers for nonlinear plants directly, not utilizing a linear approximation of the non- linear plant. The basic concept is to design a nonlinear controller so that it completely compensates for the nonlinearity of the plant, thereby creating a linear control loop.

Given a non-linear (input-affine) system

$$\dot{\underline{x}} = a(\underline{x}(t)) + {b}(\underline{x}(t)) \cdot u(t) , \quad y(t)= {c}(\underline{x}(t)).$$

The non-linear system is converted exactly into a linear system by means of a non-linear coordinate transformation and a non-linear feedback law -> Input-Output Linearization.

We start by calculating the time derivative of the output variable y and obtain

$$\dot{y}= \left[\frac{\partial c(\underline{x}(t))}{\partial x_1} \ldots \frac{\partial c(\underline{x}(t))}{\partial x_n}\right]^T \cdot \left[\begin{array}{c} \dot{x}_1\\ \vdots \\ \dot{x}_n\\ \end{array} \right] = \left[ \frac{dc(\underline{x}(t))}{d\underline{x}} \right]^T\dot{\underline{x}} = \frac{\partial c(\underline{x})}{\partial\underline{x}}\dot{\underline{x}}$$

Substituting into the equation $$\dot{\underline{x}}(t) = a(\underline{x}(t)) + {b}(\underline{x}(t))u(t)$$ yields

$$\dot{y} = \underbrace{\frac{\partial c(\underline{x})}{\partial\underline{x}}a(\underline{x}(t))}_{L_a c(\underline{x}(t))} + \underbrace{\frac{\partial c(\underline{x})}{\partial\underline{x}}{b}(\underline{x}(t))}_{L_b c(\underline{x}(t))} \cdot u(t).$$
For most technical systems, in the above equation $$L_b c(\underline{x}(t))=0$$ holds so that we obtain

$$\dot{y} = L_a c(\underline{x}(t))$$

Next, the second time derivative $$\ddot{y}$$  is to be determined. Starting with $$\dot{y} = L_a c(\underline{x}(t))$$, we obtain the
expression

$$\ddot{y} = \underbrace{\frac{\partial L_a c(\underline{x}(t))}{\partial\underline{x}} a(\underline{x}(t))}_{L_aL_ac(\underline{x})} + \underbrace{\frac{\partial L_a c(\underline{x}(t))}{\partial \underline{x}} {b}(\underline{x}(t))}_{L_bL_ac(\underline{x})} \cdot u(t)$$

We refer to $$\delta$$  as the difference degree or mostly as the relative degree of the system. For linear systems, the relative degree is equal to the difference  between the denominator degree n and the numerator degree m of the transfer function, that means $$\delta=n-m$$. In this sequence of time derivatives, the equation

$$L_bL_a^ic(\underline{x}(t))=\frac{\partial L_a^i c(\underline{x}(t))}{\partial \underline{x}}{b}(\underline{x}(t))= 0.$$

holds for all indices  $$i=1,2,\ldots, \delta-2$$. The Lie derivative $$L_bL_a^ic(\underline{x}(t))$$ is not equal to zero only for indices $$i \ge \delta-1$$. We refer to $$\delta$$ as the difference degree or mostly as the relative degree of the system. The relative degree indicates the
degree of the output variable y derivative at which it directly depends on the control variable
u for the first time; that means, for $$i=\delta-1$$  holds $$\frac{\partial L_a^{i=\delta-1} c(\underline{x}(t))}{\partial \underline{x}}{b}(\underline{x}(t))\neq 0$$.

Internal Dynamics

In this chapter, the case $$\delta < n$$ is considered; this is the case when the \textbf{relative degree} $$\delta$$ is reduced compared to the system order n. In this case, too, a nonlinear transformation, here a diffeomorphism $$\underline{z}=\underline{\eta}(\underline{x})$$, a system representation favorable for the controller design  system representation can be found.

Since $$\delta<n$$  holds,only the first $$\delta$$  components $$\eta_1, \ldots, \eta_\delta$$ of the diffeomorphism $$\underline{z}=\underline{\eta}(\underline{x})$$ can be used for the new state variables $$z_1,\ldots, z_n$$ in the same form as in the case in which $$\delta = n$$ holds. It holds that

$$\underline{z}(t)= \left[ \begin{array}{c} \\[-0.5cm] z_1(t)\\ z_2(t)\\ z_3(t)\\ \vdots \\ z_{\delta}(t)\\ z_{\delta+1}(t)\\ \vdots \\ z_n(t) \end{array} \right] = \left[ \begin{array}{} \\[0.1cm] c(\underline{x}(t))\\ L_a c(\underline{x}(t))\\ L^2_a c(\underline{x}(t))\\ \vdots \\ L^{\delta-1}_a c(\underline{x}(t))\\ \eta_{\delta+1}(\underline{x}(t))\\ \vdots \\ \eta_n(\underline{x}) \end{array} \right] =\underline{\eta}(\underline{x}(t)).$$

Using the transformation $$\underline{x}(t)= \underline{\eta}^{-1}(\underline{z}t))$$ the (temporal) derivation

$$\dot{\eta}_i(\underline{x}(t))= \frac{\partial \eta_i(\underline{x})}{\partial \underline{x}}\cdot \dot{\underline{x}} = \frac{\partial \eta_i(\underline{x})}{\partial \underline{x}}\left[ a(\underline{x})+\underline{b}(\underline{x})\cdot u \right]$$

$$\dot{\eta}_i(\underline{x})=L_a\eta_i(\underline{x})+L_b\eta_i(\underline{x})\cdot u = \hat{q}_i(\underline{x}, u)= q_i(\underline{z},u)$$

If the function $$\eta_i$$ is chosen so that
$$L_b\eta_i(\underline{x}) = \frac{\partial \eta_i(\underline{x})}{\partial \underline{x}}\underline{b}(\underline{x}) = 0$$

then above Equation is simplified. Because now the dependence on the input variable u, and the following holds
$$\dot{\eta}_i = L_a\eta_i(\underline{x})=\hat{q}_i(\underline{x})=q_i(\underline{z}).$$

However, the eq. $$L_b\eta_i(\underline{x}) = \frac{\partial \eta_i(\underline{x})}{\partial \underline{x}}\underline{b}(\underline{x}) = 0$$ is a partial differential equation, which can lead to problems when solving it.

• Adamy J.: Nonlinear systems and controls. Springer-Verlag, 1th edition, 2022 (corrected publication 2023). $$\checkmark$$
• Föllinger, O.: Nichtlineare Regelungen, Bd. 2: Harmonische Balance, Popow- und Kreiskriterium. Oldenbourg Verlag München, 7. Auflage, 1993. $$\checkmark$$
• Khalil, H. K: Nonlinear Systems. Prentice Hall, Upper Saddle River, N.J., 3rd edition, 2002. $$\checkmark$$
• Slotine, J.-J. E. and Li, W.: Applied Nonlinear Control. Prentice Hall, Englewood Cliffs, N.J, 1st edition, 1991. $$\checkmark$$

$$\checkmark~~books ~recommendable$$