Reduced‐order suboptimal control design for a class of nonlinear distributed parameter systems using POD and θ–D techniques

Abstract

A new computational tool is presented in this paper for suboptimal control design of a class of nonlinear distributed parameter systems (DPSs). In this systematic methodology, first proper orthogonal decomposition-based problem-oriented basis functions are designed, which are then used in a Galerkin projection to come up with a low-order lumped parameter approximation. This technique has evolved as a powerful model reduction technique for DPSs. Next, a suboptimal controller is designed using the emerging -D technique for lumped parameter systems. This time domain control solution is then mapped back to the distributed domain using the same basis functions, which essentially leads to a closed form solution for the controller in a state-feedback form. We present this technique for the class of nonlinear DPSs that are affine in control. Numerical results for a benchmark problem as well as for a more challenging representative real-life nonlinear temperature control problem indicate that the proposed method holds promise as a good optimal control design technique for the class of DPSs under consideration. 192 R. PADHI, M. XIN AND S. N. BALAKRISHNAN equations (PDEs). DPSs appear naturally in various application areas such as chemical processes, thermal processes, vibrating structures, fluid flow systems, etc. They inherently have an infinite number of system modes, and hence, are also known as infinite-dimensional systems. Since it is impossible to deal with all the modes, some sort of approximation technique is usually applied for the analysis and synthesis procedures related to DPS. Control of DPS has been studied both from a mathematical and an engineering point of view. An interesting brief historical perspective of the control of such systems can be found in Reference [1].There exist infinite-dimensional operator-theory-based methods for the control of DPSs. While there are many advantages, these operator-theory-based approaches are mainly limited to linear systems [2] and some limited class of problems like spatially invariant systems [3]. Moreover, for implementation purpose, the infinite-dimensional control solution needs to be approximated (e.g. truncating an infinite series, reducing the size of feedback gain matrix, etc.). Such a control design approach is known as 'design then approximate'. Eventhough the resulting controller from this approach retains most of the information of the associated full-order controller, the technique is mainly confined to linear DPS. Besides, it is not completely free from errors because of the approximations involved for its implementation. Another control design approach is 'approximate then design'. Here, the PDEs describing the system dynamics are first approximated to yield a finite-dimensional approximate model. This approximate system is then used for control design. In this approach, it is relatively easy to design controllers using various concepts of finite-dimensional control design. This approach has more engineering relevance, since it can be applied to both linea...