[[Category:method]]
[[Category:non-linear method]]
[[Category:parametric method]]

The model reduction method we present here is applicable to linear parameter-varying (LPV) systems of the form

<center>
<math>
\dot{x}(t)=Ax(t) + \sum_{i=1}^d p_i(t)A_i x(t)+B_0u_0(t),\quad y(t)=Cx(t),
</math>
</center>

where
<math>
A,A_i \in \mathbb R^{n\times n}, B_0 \in \mathbb R^{n\times m}
</math>
and
<math> C \in \mathbb R^{p\times n}.
</math>

The main idea is that the structure of the above type of systems is quite similar to so-called bilinear control systems. Although belonging to the class of nonlinear control systems, the latter exhibit many features of linear time-invariant systems. In more detail, a bilinear control system is given as follows

<center>
<math>
\dot{x}(t)=Ax(t) + \sum_{i=1}^m N_i x(t) u_i(t) + B u(t),
</math>
</center>

where
<math>
A,N_i \in \mathbb R^{n\times n}, B \in \mathbb R^{n\times m}
</math>
and
<math> C \in \mathbb R^{p\times n}.
</math>

As one can see, there seems to be a close connection between LPV and bilinear systems which may be advantageous. To be more precise, following [2], we can interpret LPV systems as special bilinear system by simply setting

<math>
\tilde{A}=A,\;\; \tilde{N}_i=0,\;\;i=1,\dots,m,\;\; \tilde{N}_i=A_i,\;\; i=m+1,\dots,m+d, \;\; \tilde{B}=\begin{bmatrix} B_0 & 0\end{bmatrix},\;\; \tilde{C}=C, \;\; \tilde{u}=\begin{bmatrix} u_0 \\ p_1(t) \\ \vdots \\ p_d(t)\end{bmatrix} .
</math>

It is important to note that in this way the parameter dependency has been hidden in the structure of a bilinear system and if we use a bilinear model reduction method, we automatically end up with a structure-preserving model reduction method for LPV systems as well. Due to the above mentioned similarity of bilinear and linear control systems, one can generalize some useful and well-known linear concepts. For example, it is possible to extend the method of balanced truncation for bilinear systems. Here, we have to solve generalized Lyapunov equations of the form

<center>
<math>
AX+XA^T +\sum_{i=1}^m N_i X N_i^T + BB^T = 0, \qquad A^TY+YA +\sum_{i=1}^m N_i^T Y N_i + C^TC = 0.
</math>
</center>

For a detailed analysis of this approach, we refer to e.g. [3]. Another possibility is to aim at a model reduction method which is optimal with respect to a certain system norm. In [1], the authors investigate a possible extension of the linear <math>H_2</math>-norm and propose an algorithm which extends the well-known iterative rational Krylov algorithm (IRKA) to the bilinear setting specified above.

Both methods have been tested on several LPV control systems (e.g., [[Microthruster_Unit|Microthruster Unit]] and [[Scanning_Electrochemical_Microscopy|Scanning Electrochemical Microscopy]]) and seem to be a reasonable alternative to existing parametric model reduction methods.

==References==

[1] P. Benner and T. Breiten, Interpolation-based H2-model reduction of bilinear control systems, 2011, Preprint MPIMD/11-02.

[2] P. Benner and T. Breiten, On H2-model reduction of linear parameter-varying
systems, In Proceedings in Applied Mathematics and Mechanics. Wiley InterScience,
2011.

[3] P. Benner and T. Damm, Lyapunov Equations, Energy Functionals, and Model Order Reduction
of Bilinear and Stochastic Systems, SIAM J. Cont. Optim., 49 (2011), pp. 686-711.

Contact information:

'' [[User:breiten|Tobias Breiten]]''

2012-11-20T16:15:03Z

Breiten:

FitzHugh-Nagumo System

2012-11-20T16:13:51Z

Breiten:

[[Category:benchmark]]
[[Category:non-linear system]]
[[Category:time invariant]]
[[Category:first order system]]

==Description==

The FitzHugh-Nagumo system describes a prototype of an excitable system (e.g., a neuron).
If the external stimulus <math>i_0(t)</math> exceeds a certain threshold value, the system will exhibit a characteristic excursion in phase space, before the variables <math>v</math> and <math>w</math> relax back to their rest values. This behaviour is typical for spike generations (=short elevation of membrane voltage <math>v</math>) in a neuron after stimulation by an external input current.

Here, we present the setting from [1], where the equations for the dynamical system read

<math>
\epsilon v_t(x,t)=\epsilon^2v_{xx}(x,t)+f(v(x,t))-w(x,t)+g,
</math>

<math>
w_t(x,t)=hv(x,t)-\gamma w(x,t)+g,
</math>

with <math>f(v)=v(v-0.1)(1-v)</math> and initial and boundary conditions

<math>
v(x,0)=0,\quad w(x,0)=0, \quad x\in [0,1],
</math>

<math>
v_x(0,t)=-i_0(t), \quad v_x(1,t)\quad =0, \quad t \geq 0,
</math>

where <math>\epsilon=0.015,\;h=0.5,\;\gamma=2,\;g=0.05,\;i_0(t)=5\cdot
10^4t^3 \exp(-15t).</math> In [1], the previous system of coupled nonlinear PDEs is spatially discretized by means of a finite difference scheme with <math>k=512 </math> nodes for each PDE. Hence, one obtains a nonlinear (cubic) system of ODEs with state dimension <math>n=1024. </math> The following picture shows the typical limit cycle behaviour described above.

[[File:FHN.png]]

==Reformulation as a quadratic-bilinear system==

Instead of the cubic system of ODEs, one can alternatively study a so-called quadratic-bilinear control system of the form

<math>
E\dot{x}(t) = A x(t) + H x(t) \otimes x(t) + N x(t) u(t) + b u(t),
</math>

with <math>E,A,N \in \mathbb R^{n\times n},\ H \in \mathbb R^{n\times n^2} </math> and <math> b \in \mathbb R.</math> The idea relies on artificially introducing a new state variable defined as <math>z(t)=v(t)^2</math> and subsequently computing the dynamics of the new variable, i.e., specifying <math>\dot{z}(t).</math> The technique goes back to [2], where it is successfully applied to to several smooth nonlinear control-affine systems. As it is discussed in [3], the previously mentioned introduction of <math>z</math> yields a quadratic-bilinear control of dimension <math> N = 3\cdot k</math> with state vector <math>x = [v,w,z]^T.</math> The increase of the state dimension has the advantage of reducing the nonlinearity from cubic to quadratic. This however opens up the possibility to reduce the system by the generalized moment-matching approach from [2], see also [3] for more details on the implementation.

==Data==

[[File:FHN.tar]]

All matrices of the quadratic-bilinear formulation discretized with <math>k=512</math> are in the Matrix Market format (http://math.nist.gov/MatrixMarket/). The matrix name is used as an extension of the matrix file. For more information on the discretization details, see [1].

==References==

[1] S. Chaturantabut and D.C. Sorensen, Nonlinear Model Reduction via Discrete Empirical Interpolation, SIAM J. Sci. Comput., 32 (2010), pp. 2737-2764.

[2] C. Gu, QLMOR: A Projection-Based Nonlinear Model Order Reduction Approach Using Quadratic-Linear Representation of Nonlinear Systems, IEEE T. Comput. Aid. D., 30 (2011), pp. 1307-1320.

[3] P. Benner and T. Breiten, Two-sided moment matching methods for nonlinear model reduction, 2012, Preprint MPIMD/12-12.

Contact information:

'' [[User:Breiten]]''

FitzHugh-Nagumo System

2012-11-20T16:10:56Z

2011-12-06T07:28:33Z

Breiten:

The model reduction method we present here is applicable to linear parameter-varying (LPV) systems of the form

<center>
<math>
\dot{x}(t)=Ax(t) + \sum_{i=1}^d p_i(t)A_i x(t)+B_0u_0(t),\quad y(t)=Cx(t),
</math>
</center>

where
<math>
A,A_i \in \mathbb R^{n\times n}, B_0 \in \mathbb R^{n\times m}
</math>
and
<math> C \in \mathbb R^{p\times n}.
</math>

The main idea is that the structure of the above type of systems is quite similar to so-called bilinear control systems. Although belonging to the class of nonlinear control systems, the latter exhibit many features of linear time-invariant systems. In more detail, a bilinear control system is given as follows

<center>
<math>
\dot{x}(t)=Ax(t) + \sum_{i=1}^m N_i x(t) u_i(t) + B u(t),
</math>
</center>

where
<math>
A,N \in \mathbb R^{n\times n}, B \in \mathbb R^{n\times m}
</math>
and
<math> C \in \mathbb R^{p\times n}.
</math>

As one can see, there seems to be a close connection between LPV and bilinear systems which may be advantageous. To be more precise, following [2], we can interpret LPV systems as special bilinear system by simply setting

<math>
\tilde{A}=A,\;\; \tilde{N}_i=0,\;\;i=1,\dots,m,\;\; \tilde{N}_i=A_i,\;\; i=m+1,\dots,m+d, \;\; \tilde{B}=\begin{bmatrix} B_0 & 0\end{bmatrix},\;\; \tilde{C}=C, \;\; \tilde{u}=\begin{bmatrix} u_0 \\ p_1(t) \\ \vdots \\ p_d(t)\end{bmatrix} .
</math>

It is important to note that in this way the parameter dependency has been hidden in the structure of a bilinear systems and if we use a bilinear model reduction method, we automatically end up with a structure-preserving model reduction method for LPV systems as well. Due to the above mentioned similarity of bilinear and linear control systems, one can generalize some useful and well-known linear concepts. For example, it is possible to extend the method of balanced truncation for bilinear systems. Here, we have to solve generalized Lyapunov equations of the form

<center>
<math>
AX+XA^T +\sum_{i=1}^m N_i X N_i^T + BB^T = 0, \qquad A^TY+YA +\sum_{i=1}^m N_i^T Y N_i + C^TC = 0.
</math>
</center>

For a detailed analysis of this approach, we refer to e.g. [3]. Another possibility is to aim at a model reduction method which is optimal with respect to a certain system norm. In [1], the authors investiate a possible extension of the linear <math>H_2-</math>norm and propose an algorithm which extends the well-known iterative rational Krylov algorithm (IRKA) to the bilinear setting specified above.

Both methods have been tested on several LPV control systems and seem to be a reasonable alternative to existing parametric model reduction methods.

==References==

[1] P. Benner and T. Breiten, ``Interpolation-based H2-model reduction of bilinear control systems," 2011, Preprint MPIMD/11-02.

[2] P. Benner and T. Breiten, ``On H2-model reduction of linear parameter-varying
systems," In Proceedings in Applied Mathematics and Mechanics. Wiley InterScience,
2011.

[3] P. Benner and T. Damm, ``Lyapunov Equations, Energy Functionals, and Model Order Reduction
of Bilinear and Stochastic Systems," SIAM J. Cont. Optim., 49 (2011), pp. 686-711.

Bilinear PMOR method

2011-12-06T07:18:07Z

Breiten:

Bilinear PMOR method

2011-12-06T07:17:32Z

Breiten: