User:Fletli/Dynamic TP model

The dynamic Tensor Product model (TP model) is a type of polytopic model representations of quasi Linear Parameter Varying (qLPV) state-space models. It is proposed by Baranyi et. al. \cite{}. The widely utilized TS fuzzy models belong to the class of TP models. The TP model transformation is capable of generating various special TP model form and the HOSVD based canonical form of qLPV models.

Consider the following linear parameter-varying (LPV) state-space model:

$$ \mathbf{\dot{x}}(t)=\mathbf{A}(\mathbf{p}(t))\mathbf{x}(t)+\mathbf{B}(\mathbf{p}(t))\mathbf{u}(t)$$

$$ \mathbf{y}(t)=\mathbf{C}(\mathbf{p}(t))\mathbf{x}(t)+\mathbf{D}(\mathbf{p}(t))\mathbf{u}(t) $$

with input $$\mathbf{u}(t)\in\mathbb{R}^k$$, output $$\mathbf{y}(t)\in\mathbb{R}^l$$ and state vector $$\mathbf{x}(t)\in\mathbb{R}^m$$. The system matrix

$$ \mathbf{S}(\mathbf{p}(t))=\begin{pmatrix} \mathbf{A}(\mathbf{p}(t)) & \mathbf{B}(\mathbf{p}(t)) \\ \mathbf{C}(\mathbf{p}(t)) & \mathbf{D}(\mathbf{p}(t)) \end{pmatrix}\in\mathbb{R}^{(m+k)\times(m+l)} $$

is a parameter-varying object, where $$\mathbf{p}(t)\in\Omega$$ is a time-varying $$N$$-dimensional parameter vector, and is an element of the closed hypercube $$\Omega= [a_1,b_1]\times[a_2,b_2]\times\cdots\times[a_N,b_N]\subset\mathbb{R}^N$$. Parameter $$\mathbf{p}(t)$$ can also include some elements of $$\mathbf{x}(t)$$, in this case (\ref{eq:LPValap}) is termed as quasi LPV (qLPV) model. Therefore this type of model is considered to belong to the class of non-linear models.

Note that, for the shake of simplicity we define (\ref{eq:LPValap}), but without the loss of generality we can define the TP model (and the TP model transformation) form in the same way for models, where the system matrix describes a multi-channel system:

$$ \left(\begin{array}{c} \dot{x}(t) \\ z_1(t) \\ \vdots \\ z_q(t) \\ y(t) \\ \end{array} \right)=\left( \begin{array}{ccccc} \mathbf{A} & \mathbf{B}_1 & \cdots & \mathbf{B}_q & \mathbf{B} \\ \mathbf{C}_1 & \mathbf{D}_{1,1} & \cdots & \mathbf{D}_{1,q} & \mathbf{E}_1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ \mathbf{C}_q & \mathbf{D}_{q,1} & \cdots & \mathbf{D}_{q,q} & \mathbf{E}_q \\ \mathbf{C} & \mathbf{F}_1 & \cdots & \mathbf{F}_q & 0 \\ \end{array} \right)\left( \begin{array}{c} x(t) \\ w_1(t) \\ \vdots \\ w_q(t) \\ u(t) \\ \end{array} \right) $$

where $$ w_j(t)\rightarrow z_j(t) $$ are the channels on which we want to impose certain robustness and/or performance objectives.

For the sake of simplicity, we drop the $$t$$ from the equations from now on.

$$ \mathbf{S}(\mathbf{p})=\sum_{r=1}^R{w}_r(\mathbf{p})\mathbf{S}_r $$
 * Finite element polytopic model

$$\mathbf{S}(\mathbf{p})$$ in (\ref{eq:systemmatrix}) is given for any parameter vector $$\textbf{p}(t)$$ as the parameter-varying combination of linear time-invariant (LTI) system matrices $$\mathbf{S}_r\in\mathcal{R}^{(m+k)\times(m+l)}$$ also called vertex systems. The combination is defined by the multi-variable weighting functions $$w_r(\mathbf{p})\in [0,1]$$. Finite element means that R is bounded $$(R<\infty)$$.

The TP model
belongs to the class of polytopic models. In case of the TP model the multi variable weighting functions $${w}_r(\mathbf{p})$$ are decomposed to the product of one variable weighing functions $${w}_n(\mathbf{p}_n)$$. Having the one variable weighting functions we may have various advantages during manipulating with the convex hull defined by the LTI systems as it is necessary in case of optimisation of the control performances (see TP model based control).

We say TP model for brevity. $$\mathbf{S}(\mathbf{p})$$ in (\ref{eq:systemmatrix}) is given for any parameter $$\mathbf{p}$$ as the parameter varying combination of bounded number of linear time-invariant (LTI) system matrices $$\mathbf{S}_{i_1i_2\ldots i_N}\in \mathcal{R}^{(m+k)\times(m+l)}$$ as: $$ \mathbf{S}(\mathbf{p})=\sum_{i_1=1}^{I_1} \sum_{i_2=1}^{I_2} \ldots \sum_{i_N=1}^{I_N} \prod_{n=1}^N w_{n,i_n}(p_n)\mathbf{S}_{i_1,i_2,\ldots,i_N} $$
 * Finite element TP type polytopic model

that is with compact tensor notation (In order to use tensor algebra we refer to the works \cite{Lath00}.): $$ \mathbf{S}(\mathbf{p})=\mathcal{S}\boxtimes_{n=1}^N\mathbf{w}_n(p_n), $$

where the (N+2)-dimensional coefficient tensor $$\mathcal{S}\in \mathcal{R}^{I_1\times I_2\times \ldots I_N\times(m+k) \times (m+l)}$$ is constructed from LTI vertex systems $$\mathbf{S}_{i_1 i_2 \ldots i_N}$$ and row vector $$\mathbf{w}_n(p_n)\in [0,1], (i_n=1 \ldots I_n)$$ contains one variable and continuous weighting functions $$w_{n,i_n}(p_n)\in [0,1],(i_n=1 \ldots I_n)$$. The function $$w_{n,i_n}(p_n)$$ is the $$i_n$$-th weighting function defined on the $$n$$-th dimension of $$\Omega$$, and $$p_n$$ is the $$n$$-th element of vector $$\mathbf{p}$$. The dimensions of $$\Omega$$ are respectively assigned to the elements of the parameter vector $$\mathbf{p}$$.

Note that, the widely spread Takagi-Sugeno (TS) fuzzy model applied for state-space qLPV models belongs to the class of TP models. In case of TS fuzzy models the weighting function $$w_{n,i_n}(p_n)$$ varies in the interval of $$[0,1]$$ and represents the membership function of the antecedent fuzzy set $A_{n,i_n}$, and the LTI matrix $$\mathbf{S}_{i_1,...,i_N}$$ is the consequent part of fuzzy rule:

$$ \text{IF} \quad A_{i,i_1} \quad \text{AND} \quad A_{2,i_2} \quad \text{AND} \cdots \text{AND} \quad A_{N,i_N} \quad \text{THEN} \quad \mathbf{S}_{i_1,i_2,...,i_N}. $$

For Linear Matrix Inequality based design, especially for Parallel Distributed Compensation design (works by Tanaka \cite{Tanaka01}), the convexity of the TP model is required. Furthermore the resulting control performance and the feasibility of LMIs are very sensitive for the type of the convexity. Therefore let us define the following types of TP models:

The TP model (\ref{eq:TPmodel}) is NN (Non-Negativeness ) type if its all weighting functions $$\mathbf{w}(p)\geq 0$$ for all $$\mathbf{p}\in\Omega$$.
 * NN type TP model

The TP model (\ref{eq:TPmodel}) is SN (Sum Normalized) type if the weighting functions satisfy $$	\forall n,p_n : \sum_{i=1}^{I_n}w_{n,i}(p_n)=1. $$
 * SN type TP model

The TP model (\ref{eq:TPmodel}) is convex if it is SN and NN type, namely, if its weighting functions satisfy $$\forall n,i,p_n : w_{n,i}(p_n)\in[0,1]$$
 * Convex TP model

$$\forall n,p_n : \sum_{i=1}^{I_n}w_{n,i}(p_n)=1.$$

This actually means that the system matrix $$\mathbf{S}(\mathbf{p})$$ is within the convex hull, defined by the vertices LTI systems, for all $$\mathbf{p\in\Omega}$$.

Note that, in case of TS fuzzy models Definition \ref{def:convexTPmodel} actually means the Ruspini-partition of the antecedent fuzzy sets. The convex TP model or TS fuzzy model with Ruspini partitioned antecedent fuzzy sets are used in PDC control design.

A tight convex hull of the LTI systems is defined by:

The TP model is a normal (NO) type, if it is convex and and the largest value of all weighting functions $$\mathbf{w}(p)$$ is 1. Also, it is \textsl{Close to NOrmal} if it is convex and the largest value of all weighting functions is 1 or close to 1.
 * NO/CNO type TP model

Various further types of convex TP models are defined in papers \cite{JGDC2}.

A TP model is termed \textsl{Exact TP model}, if for all $\mathbf{p}(t)\in \Omega$: $$ \mathbf{S}(\mathbf{p})=\mathcal{S}\boxtimes_{n=1}^N\mathbf{w}_n(p_n) $$ holds.
 * Exact/Non-exact TP model

A TP model is termed \textsl{Non-Exact TP model} if: $$ \hat{\mathbf{S}}(\mathbf{p})=\mathcal{S}\boxtimes_{n=1}^N\mathbf{w}_n(p_n) $$ holds, where $$\hat{\mathbf{S}}(\mathbf{p})$$ is only an approximation of $$\mathbf{S(\mathbf{p})}$$, where the error $$\gamma$$ is defined as:

$$ \max_{\mathbf{p}}\|{\mathbf{S}(\mathbf{p})-\hat{\mathbf{S}}(\mathbf{p})}\|_{\mathcal{L}_2}=\gamma. $$