- Padhi R
- Balakrishnan S
- Unnikrishnan N

IET Control Theory & Applications (2007) 1(6) 1650-1661

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Combining the principles of dynamic inversion and optimisation theory, two stabilising state-feedback control design approaches are presented for a class of nonlinear distributed par-ameter systems. One approach combines the dynamic inversion with variational optimisation theory and it can be applied when there is a continuous actuator in the spatial domain. This approach has more theoretical significance in the sense that it does not lead to any singularity in the control computation and the convergence of the control action can be proved. The other approach, which can be applied when there are a number of discrete actuators located at distinct places in the spatial domain, combines dynamic inversion with static optimisation theory. This approach has more relevance in practice, since such a scenario appears naturally in many practical problems because of implementation concern. These new techniques can be classified as 'design-then-approximate' techniques, which are in general more elegant than the 'approximate-then-design' techniques. However, unlike the existing design-then-approximate techniques, the new techniques presented here do not demand involved mathematics (like infinite-dimensional operator theory, inertial manifold theory and so on). To demonstrate the potential of the proposed techniques, a real-life temperature control problem for a heat transfer application is solved, first assuming a continuous actuator and then assuming a set of discrete actuators. 1 Introduction There are wide class of problems (e.g. heat transfer, fluid flow, flexible structures and so on) for which a lumped par-ameter modelling is inadequate and a distributed parameter system (DPS) approach is necessary. In contrast to lumped parameter systems, which are governed by ordinary differ-ential equations (ODEs), distributed parameter systems are governed by a set of coupled partial differential equations (PDEs) in general. Control design for distributed parameter systems is often more challenging when compared with lumped parameter systems. Control of DPSs has been studied both from math-ematical as well as engineering point of view. An interest-ing brief historical perspective of the control of such systems is found in [1]. In a broad sense, existing control design techniques for distributed parameter systems can be attributed to either 'approximate-then-design (ATD)' or 'design-then-approximate (DTA)' categories. An inter-ested reader can refer to [2] for discussions on the relative merits and limitations of the two approaches. In the ATD approach, the idea is to first design a low-dimensional reduced (truncated) model, which retains the dominant modes of the system. This truncated model (which is often a finite-dimensional lumped parameter model) is then used to design the control action. One such potential approach, which has become fairly popular, first comes up with problem-oriented basis functions using the idea of proper orthogonal decomposition (POD) (through the 'snap-shot solutions') and then uses those in a Galerkin procedure to arrive at a low-dimensional reduced lumped parameter approximate model (which usually turns out to be a fairly good approximation). Out of numer-ous literatures published on this topic and its use in control system design, we cite [3– 11] for reference. For linear systems, such an approach of designing the POD-based basis function leads to the optimal representation of the PDE system in the sense that it captures the maximum energy of the system with least number of basis functions when compared with any other set of orthogonal basis func-tions [8]. For nonlinear systems, however, such a useful result does not exist. Even though the POD-based model reduction idea has been successfully used for numerous linear and nonlinear DPS in both linear as well as nonlinear problems, there are a few important shortcomings, which include (i) the technique is problem-dependent and not generic; (ii) there is no guarantee that the snap-shot solutions will capture all dominant modes of the system and; most important, (iii) it is usually difficult to have a set of 'good' snap-shot solutions for the closed-loop system prior to the control design. This is a serious limiting factor for applying this technique in the closed-loop control design. Because of this reason, some attempts are being made in recent litera-ture to adaptively redesign the basis functions (and hence the control action) in an iterative manner. An interested reader can see [3, 4, 10] for a few ideas in this regard. An interesting ATD approach has appeared in [12], where the authors have proposed an innovative model

- Constraints
- Optimization devices
- Predictive control
- Robust control
- Uncertain systems

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