A Heating Systems Application of Feedback Linearization for MTI Systems in a Tensor Framework

Abstract

Input-affine nonlinear systems are known to be feedback linearizable, i.e. the closed loop shows a predefined linear behaviour. The nonlinear control law can be computed symbolically by Lie derivatives calculus with the help of a state space model of the plant. In applications, this can lead to problems, because the complexity of the controller is not predefined by the order of the system, but depends on the structure of the nonlinear terms of the model. It was shown that this is not true for the multilinear subclass of input-affine nonlinear systems, where only the order of the model determines the complexity of the controller. Moreover, models and controllers can be represented as parameter tensors, which makes modern tensor decomposition methods also applicable. The paper shows how this tensor framework of Multilinear Time-Invariant (MTI) Systems can be used for the design of a feedback linearizing controller for a heating system.

This work was partly sponsored by the project OBSERVE of the Federal Ministry of Economic Affairs and Energy, Germany (Grant No.: 03ET1225B).

Appendix

Proof (Lie Bracket)

The Lie bracket of two vector functions is defined by (48). With Theorem 1 the Lie derivatives of a vector function $\mathbf g (\mathbf{{x}})$ along a vector field $\mathbf h (\mathbf{{x}})$ and vice versa are written as

$$\begin{aligned} L_\mathbf{h }{} \mathbf g (\mathbf{{x}})&= \sum \limits _{i=1}^{n} h_i(\mathbf{{x}})\frac{\partial }{\partial x_i} \mathbf g (\mathbf{{x}}) = \left\langle \,\mathsf{{L}}_{\mathsf{{H}},\mathsf{{G}},1}\,\left| \,\mathsf{{M}}_p^{2N}\left( \mathbf{{x}}\right) \,\right. \right\rangle _{},\\ L_\mathbf{g }{} \mathbf h (\mathbf{{x}})&= \sum \limits _{i=1}^{n} g_i(\mathbf{{x}})\frac{\partial }{\partial x_i} \mathbf h (\mathbf{{x}}) = \left\langle \,\mathsf{{L}}_{\mathsf{{G}},\mathsf{{H}},1}\,\left| \,\mathsf{{M}}_p^{2N}\left( \mathbf{{x}}\right) \,\right. \right\rangle _{}. \end{aligned}$$

Inserting this to (48) and rearranging results in

$$\begin{aligned} \left[ \mathbf h ,\mathbf g \right]&= \left\langle \,\mathsf{{L}}_{\mathsf{{H}},\mathsf{{G}},1}\,\left| \,\mathsf{{M}}_p^{2N}\left( \mathbf{{x}}\right) \,\right. \right\rangle _{} - \left\langle \,\mathsf{{L}}_{\mathsf{{G}},\mathsf{{H}},1}\,\left| \,\mathsf{{M}}_p^{2N}\left( \mathbf{{x}}\right) \,\right. \right\rangle _{}= \left\langle \,\mathsf{{L}}_{\mathsf{{H}},\mathsf{{G}},1} - \mathsf{{L}}_{\mathsf{{G}},\mathsf{{H}},1}\,\left| \,\mathsf{{M}}_p^{2N}\left( \mathbf{{x}}\right) \,\right. \right\rangle _{}. \end{aligned}$$

   $\square$

Proof (Feedback Linearizability)

The conditions for an affine SISO system to be feedback linearizable with relative degree of n are given for nonlinear systems in [10]. The matrix

$$\begin{aligned} \begin{pmatrix}\mathbf {b}&ad_{\mathbf {a}} \mathbf {b}(\mathbf {x_0})&\cdots&ad_{\mathbf {a}}^{n-1} \mathbf {b}(\mathbf {x_0})\end{pmatrix} \end{aligned}$$

must have rank n and can be expressed in the case of an MTI model given in tensor form by $\left\langle \,\mathsf{{T}}\,\left| \,\mathsf{{M}}_p^{n}\left( \mathbf{{x}}_0\right) \,\right. \right\rangle _{}$. Each repeated Lie bracket $\left\langle \,\mathsf{{L}}_{ad_\mathbf{a }{} \mathbf b }^{k-1}\,\left| \,\mathsf{{M}}_p^{n}\left( \mathbf{{x}}\right) \,\right. \right\rangle _{}$ is described with respect to the monomial tensor of order n and concatenated resulting in

$$\begin{aligned} \mathsf{{T}}(:,\ldots ,:,k) = \mathsf{{L}}_{ad_\mathbf{a }{} \mathbf b }^{k-1},\ k = 1,\ldots ,n. \end{aligned}$$

The evaluation of $\mathsf{{T}}$ with the monomial tensor of order n gives the matrix of the Lie brackets

$$\begin{aligned} \left\langle \,\mathsf{{T}}\,\left| \,\mathsf{{M}}_p^{n}\left( \mathbf{{x}}_0\right) \,\right. \right\rangle _{} = \begin{pmatrix} \mathbf {b}&ad_{\mathbf {a}} \mathbf {b}(\mathbf {x_0})&\cdots&ad_{\mathbf {a}}^{n-1} \mathbf {b}(\mathbf {x_0})\end{pmatrix}. \end{aligned}$$

The matrix should have rank n, which can be easily checked, by evaluation of the contracted product at $\mathbf{{x}}_0$. The second condition is the tensor formulation of the condition that the distribution $\mathrm {span}\left\{ \mathbf b ,ad_\mathbf{a } \mathbf b , \ldots , ad_\mathbf{a }^{n-2} \mathbf b )\right\}$ is involutive at $\mathbf{{x}}_0$. The distribution is involutive if and only if

$$\begin{aligned} \mathrm {rank}\left( \begin{pmatrix}\mathbf {b}(\mathbf {x}_0)&\cdots&ad_{\mathbf {a}}^{n-2} \mathbf {b}(\mathbf {x_0})&\left[ ad_{\mathbf {a}}^{i} \mathbf {b}(\mathbf {x_0}),ad_{\mathbf {a}}^{j} \mathbf {b}(\mathbf {x_0})\right] \end{pmatrix}\right) = \mathrm {rank}\left( \begin{pmatrix}\mathbf {b}(\mathbf {x}_0)&\cdots&ad_{\mathbf {a}}^{n-2} \mathbf {b}(\mathbf {x_0})\end{pmatrix}\right) \end{aligned}$$

is fulfilled for all i and $j = 0,\ldots ,n-2$. Describing the two matrices in terms of parameter tensors, like it was done for the rank condition before, leads to the second condition.$\square$