Increasing eigenstructure assignment design degree of freedom using lifting

Abstract

This paper presents the exposition of an output-lifting eigenstructure assignment (EA) design framework, wherein the available EA design degrees of freedom (DoF) is significantly increased, and the desired eigenstructure of a single-rate full state feedback solution can be achieved within an output feedback system. A structural mapping is introduced to release the output-lifting causality constraint. Additionally, the available design DoF can be further enlarged via involving the input-lifting into the output-lifting EA framework. The newly induced design DoF can be utilised to calculate a structurally-constrained, causal gain matrix which will maintain the same assignment capability. In this paper, the robustification of the output-lifting EA is also proposed, which allows a trade-off between performance and robustness in the presence of structured model uncertainties to be established. A lateral flight control benchmark in the EA literature and a numerical example are used to demonstrate the effectiveness of the design framework.Keywords: lifting; eigenstructure assignment; causality constraint IntroductionEigenstructure assignment (EA) is a mature technique for the design of control systems, especially for flight control system (Alireza & Batool, 2012;B. Chen & Nagarajaiah, 2007;Clake, Ensor, & Griffin, 2003;Farineau, 1989;Kshatriya, Annakkage, Hughes, & Gole, 2007;G. P. Liu & Patton, 1998;Y. Liu, Tan, Wang, & Wang, 2013;Moore, 1976;Ouyang, Richiedei, Trevisani, & Zanardo, 2012;Piou & Sobel, 1994, 1995Pomfret & Clarke, 2009;Shi & Patton, 2012;Wahrburg & Adamy, 2013;White, 1995;White, Bruyere, & Tsourdos, 2007). EA facilitates control system design by synthesizing a feedback gain matrix that exactly places the closed loop eigenvalues whilst matching the closed loop eigenvectors as closely as possible to a desired set. Some useful properties EA imbues on a system are: stability of response, appropriateness of transient response, decoupling of state or output response and disturbance rejection. Compared with many other competitive approaches exist to manipulate the eigenvalues of the closed-loop system and do not takes into account the role of the eigenvector, EA clearly exploits how system inputs affect mode dynamics and how these mode dynamics will be assigned to system states. Through defining a set of ideal closed-loop eigenstructure (e.g. eigenstructure which represents the realistic handling qualities of the flight control system), the realistic control effect will be guaranteed. In addition, the algorithm itself and the expression of the available design DoF can be highly visible. However, due to the lack of design DoF, e.g. the Kimura condition is not satisfied (Kimura, 1975), output feedback EA usually cannot fully assign the desired eigenstructure. This is an open problem which has been widely discussed in the literature (Andry, Shapiro, & Chung, 1983;L. Chen & Clarke, 2009;Clarke & Griffin, 2004;Pomfret, Clarke, & Ensor, 2005;Roppenecker & O'Reilly, 1989;Srinathkumar, 1978;Zhao & Lam, 2016a, 2016b.Mul...

Multistage output-lifting eigenstructure assignment: a multirate ball and plate example

“…In this section, the output-lifting EA approach (L. Chen et al, 2017) will be introduced briefly. Consider a discrete-time, controllable and observable linear time-invariant systeṁ…”

“…Remark 3.1: Clearly from (24), the dimension of the left allowable subspace is pm − s 1 and hence the maximum number of exactly assignable elements of each eigenvector becomes pm − s 1 .T h i s implies another potential benefit of using output-lifting if pm − s 1 >r, which was not exploited by L. Chen et al (2017).…”

“…Remark 3.2: As argued in L. Chen et al (2017), the term I + D L N c a na l w a y sb ee n s u r e dt ob e non-singular via choosing a set of suitable desired eigenvectors (L. Chen et al, 2017).…”

“…Remark 3.3: With the causality constraint induced by output-lifting, any fully-populated gain matrix, calculated from (29), may be non-causal. As in L. Chen et al (2017), a structural mapping can be used to relax the causality constraint.…”

“…The lifting framework and EA were first time combined by Patton et al (1995) to enlarge DoF, however this approach only lifted system inputs and generated inter-sample ripple. Recently, L. Chen, Pomfret, and Clarke (2017) developed output lifting EA which largely improved DoF and the single-rate full state feedback eigenstructure can be assigned in an output feedback framework, particularly when the Kimura condition (Kimura, 1975) is not satisfied.…”