An LMI approach to synthesis subject to almost asymptotic regulation constraints
Introduction
Asymptotic tracking/rejection of infinite-energy references/disturbances is commonly referred to as the regulation problem and has been studied since the seventies with various extensions (see [19] for an in-depth treatment of the subject for linear systems together with references). The fundamental result in the regulation theory, which is known as the internal model principle [6], states that, for exact asymptotic rejection of infinite-energy disturbances in the measured outputs, the dynamics of their generating systems need to be replicated in the feedback loop (see also [18], [19] for extensions to the regulation of outputs different than the measurements). This typically requires a structured controller composed of two parts: a suitably constructed copy of the exo-system that generates the considered reference/disturbance signals, and an accompanying controller with which the system is stabilized. Thanks to the freedom in choosing the accompanying controller, it is possible and in fact desirable to consider other performance objectives (, etc.) in addition to stability (see e.g. [2], [1], [20], [8], [22], [4]). A common approach in solving such multi-objective problems is based on extending the dynamics of the plant by merging it with the dynamics of the exo-system that is replicated in the feedback loop as a part of the controller. The accompanying part of the controller is then designed to stabilize the extended plant and achieve the specified performance objectives. When step or sinusoidal signals are considered, this approach leads to non-standard design problems for plants having poles/zeros on the boundary of the stability domain. These problems can of course be rendered standard by simply perturbing the boundary poles slightly away from the boundary of the stability domain. This approach is, however, vulnerable to the troubles related with the lightly-damped plant poles and hence special care has to be taken in problem formulations as discussed in [11], [21], [24], [5], [3]. Moreover, the controllers synthesized in this fashion will typically have higher orders than actually required. Alternatively the overall problem can be formulated within the modern controller design framework by the use of unstable weighting filters and solved via the available techniques [15], [16], [17].
In this paper we consider a relaxed version of the asymptotic regulation problem with guaranteed performance, as formulated precisely in the next section, and employ the linear matrix-inequality (LMI) optimization approach to modern controller design [10], [7], [20]. The relaxed versions of the regulation problem for sinusoidal disturbances has been formulated in some papers [13], [12] as the minimization of a quadratic cost that penalizes the average power of the output as well as the control effort, whereas the notion of almost asymptotic regulation adopted in this paper is inspired by the recent work [9] and is defined in terms of the steady-state peak gain from the disturbance to the output to be regulated (cf. [19, Chapter 17]). We first characterize in Section 3 the controllers that guarantee almost asymptotic regulation by favor of their particular structure, which reduces to one that replicates the dynamics generating the infinite-energy disturbances when exact regulation of the measured output is considered. Along parallel lines to [22], we then reformulate the overall problem as a design to be performed for a suitably extended plant in Section 4. The main result derived in this fashion relies on semi-definite programming for the synthesis of a suitable controller whose order is equal to the order of the plant plus the order of the exogenous system. Ways to improve the transient response of the output to be regulated are also discussed. The special version of the solution for the case of synthesis with exact asymptotic regulation also contributes to the existing literature, the most relevant of which are [20], [8], [22]. The solution obtained when the measured output is to be exactly regulated is applicable without having to modify the extended plant dynamics thanks to the fact that the LMI approach does not require absence of any plant poles on the boundary of the stability domain. In fact the solution in this case simplifies that of [20] by reducing the number of variables during feasibility analysis, in addition to extending it such that exact regulation of the outputs that are not contained in the measured one is possible and the controllers obtained through a particular procedure have—generically—reduced order. The analysis of the degradation in the best achievable performance due to the exact regulation constraint as considered by [22] can be performed easily irrespective of whether the regulator equations admit a unique solution or not. The solution of the problem in its most general form allows to explore the trade-off between almost regulation and performance as illustrated by an example in Section 5. Possible extensions of and relevant research motivated by the main result are discussed as concluding remarks.
Section snippets
Problem statement
Problem 1 Consider a linear time-invariant (LTI) plant whose dynamics are described bywhere is the state, is the control input, is the input to the performance channel, is the disturbance to be rejected, is the output to be regulated, is the performance output and is the measured output. The disturbance is generated by an LTI exogenous system of the formAssume that: is anti-Hurwitz (i.e. has all its eigenvalues with non-negative real parts
The controller structure for almost asymptotic regulation
As a first step towards the solution of Problem 1, we establish in this section that a controller of particular structure guarantees almost asymptotic regulation provided that it also stabilizes the closed-loop. To this end, we first introduce a state transformation of the form ; , by which we obtain an alternative description of the closed-loop dynamics for asThis alternative representation leads us to the required controller
Almost asymptotic regulation with guaranteed -performance
Having described the structure of the controller that guarantees almost asymptotic regulation, we will provide in this section the main result of the paper that outlines a solution to Problem 1 in the form of semi-definite programming. With the candidate controllers parameterized as in (8), let us first make the following crucial observation with the help of Fig. 1: the dynamics of the closed-loop formed by K in (8) and G are identical to the dynamics of the closed-loop formed by and an
Illustrative example
In this section we consider an example derived from a mass-spring damper system for purposes of illustrating the trade-off between asymptotic regulation and performance. The dynamics of the system are described bywhere is a multi-sinusoidal disturbance with three different harmonics generated as with initial conditions . We are interested in the worst-case energy gain from to represented by , under an almost regulation constraint on e
Concluding remarks
We have presented in this paper an LMI solution to the synthesis problem subject to exact as well as almost asymptotic regulation constraints. The solution not only allows to investigate the trade-offs between the asymptotic regulation and performance objectives but also makes it possible to synthesize reduced order controllers without modifying the original problem formulation for purposes of tractability. Considering the synthesis problem with unstable weighting filters is aimed at
Acknowledgments
Supported by the Technology Foundation STW, applied science division of NWO and the technology programme of the Ministry of Economic Affairs.
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