(555g) Nonlinear Control of a Differential Algebraic Equations System Using Modified POD Method

Nonlinear control of a differential algebraic equations system using modified POD method

In many chemical industry processes, nonlinear PDEs need to be solved due to spatial distributed
nature of the system dynamics. In general, these PDEs canâ??t be solved analytically. One of the methods is to decompose the function into spatial part and temporary part using Galerkinâ??s method. Compared with standard Galerkinâ??s method, which provides optimal basis functions in linear system, in non-linear system, proper orthogonal decomposition method (POD) can provide a better set of basis to approximate the function we need to solve [1]. Basis functions generated by POD are optimal with respect to that the average square of distance between the function and basis is minimized. Data obtained in activating the open loop process is used to approximate the solution of this minimizing problem. This method, which is called a combination of POD and method of snapshots, has been used extensively in analyzing the dynamics distributed system. But this method also has some limitations. Firstly, it requires the process is fully excited so that all the possible spatial modes are contained in the data we use to generate POD basis functions. However, there is no method to properly activate the system [2]. Besides, sampling too few snapshots may also result in basis functions that canâ??t represent the dynamics of the closed-loop system accurately.
To relax the requirements of POD method, we develop a new method to generate basis function. For a
time-varying system, the optimal basis functions to capture the characteristics of the process may change as time goes by [2]. Therefore, the first step is to use new snapshots to recalculate the optimal basis functions. This step will add new important basis function(s) and discard unimportant ones based on what is the energy ratio the present basis has. It will make the result more accurate and reduce unnecessary computational load. This method is called adaptive proper orthogonal decomposition (APOD).
After basis functions (Ï?) are identified using APOD, the inner product ofÏ? and AÏ? is calculated to generate new basis functions, where A is an operator representing the dynamics of the process. In this step, we further decrease the number of modes we keep without losing too much information of the system. Finally, we will use the new basis functions to design nonlinear controller in differential algebraic equations system. By decreasing the number of modes we need, this method is computationally cheaper, which is important for continuous estimation of optimal basis functions in APOD method.
Other than the method proposed here, there are other approaches in literature trying to improve POD method. One of them is the work of Sirovich in which he created a competing method to POD using [3]. Compared to this method, the proposed method has the following advantages: the basis functions
generated by this method are orthonormal and orthogonal to process dynamics; because of this property, subsequent analysis of system dynamics and controller design can be simplified when calculating inner product. Besides, this method reduced the number of modes needed and therefor is more computationally efficient; which is a significant feature when implemented on line.
REFERENCES
[1] Sirovich, L. (1987). Turbulence and the dynamics of coherent structures: part 1 coherent structures.
Quarterly of Applied Mathematics, 561-571.
[2] Pitcliaiah, S., & Armaou, A (2010). Output feedback control of distributed parameter systems using adaptive proper ortliogonal decomposition. Industrial & Engineering Chemistry Research, 49,
10496-10509.
[3] Sirovich, L. (1987). turbulence and the dynamics of coherent structures 3 dynamics and scaling. Quarterly of Applied Mathematics, 583-590.