طراحی کنترل یادگیری تکراری مقاوم محدوده فرکانس محدود و تأیید تجربی

Iterative learning control is an application for two-dimensional control systems analysis where it is possible to simultaneously address error convergence and transient response specifications but there is a requirement to enforce frequency attenuation of the error between the output and reference over the complete spectrum. In common with other control algorithm design methods, this can be a very difficult specification to meet but often the control of physical/industrial systems is only required over a finite frequency range. This paper uses the generalized Kalman–Yakubovich–Popov lemma to develop a two-dimensional systems based iterative learning control law design algorithm where frequency attenuation is only imposed over a finite frequency range to be determined from knowledge of the application and its operation. An extension to robust control law design in the presence of norm-bounded uncertainty is also given and its applicability relative to alternative settings for design discussed. The resulting designs are experimentally tested on a gantry robot used for the same purpose with other iterative learning control algorithms.

Many industrial systems execute a task over a finite duration, reset to the starting location and then completes this task over and over again. Each execution is known as a trial or pass and the duration the trial length. Once a trial has been completed all data generated is available to update the control signal for the next trial and thereby improve performance from trial-to-trial. This area is known as Iterative Learning Control (ILC) and since the initial work, widely credited to Arimoto, Kawamura, and Miyazaki (1984), has been an established area of control systems research and application, where one starting point for the literature is the survey papers (Ahn et al., 2007 and Bristow et al., 2006). Major application areas include robotics, with recent work in, for example (Barton & Alleyene, 2011), flexible valve actuation for non-throttled engine load control (Heinzen, Gillella, & Sun, 2011) and also a transfer from engineering to next generation healthcare for robotic-assisted upper limb stroke rehabilitation with supporting clinical trials (Freeman et al., 2009 and Freeman et al., 2012). A significant part of the currently published ILC research starts from a linear time-invariant discrete model of the system dynamics in either state-space or shift operator/transfer-function form. In this case, one way to do control law design is, since the trial duration is finite, to define super-vectors for the variables. For example, let y k(p ) be the scalar, for ease of presentation with a natural extension to the vector case, output on trial k , which is of length α<∞α<∞. Then the super-vector, for linear dynamics and systems with a nonzero first Markov parameter, is View the MathML sourceYk=[yk(0)yk(1)⋯yk(α-1)]⊤, Turn MathJax on and the ILC problem can hence be written as a system of linear difference equations with updating in k. The current trial error is the difference between the supplied reference signal and the trial output and, since the trial length is finite, trial-to-trial error convergence is independent of the state matrix. This, in turn, could lead to unacceptable along the trial dynamics. In the lifting approach, this problem can be addressed by applying a feedback control law to stabilize the system and/or improve transient performance and then design the ILC law for the controlled system. An alternative is to use a two-dimensional (2D) systems setting where ILC can be represented in this form with one direction of information propagation from trial-to-trial and the other along the trial. Given that the trial length is finite, ILC fits naturally into the repetitive process setting for analysis, where these processes (Rogers, Gałkowski, & Owens, 2007) have their origins in the mining and metal rolling industries and a substantial body of systems theory and control law design algorithms exists for them. Using the repetitive process setting, it is possible to simultaneously design a control law for trial-to-trial error convergence and along the trial performance. The method is to use a form of stability for repetitive processes that demands a bounded-input bounded-output property independent of the trial length. Control laws designed in this setting have been experimentally tested on a gantry robot replicating a robotic pick and place operation that often arises in industrial applications to which ILC is applicable (Hładowski et al., 2010 and Hładowski et al., 2012). This previous ILC design in a repetitive process setting includes a stability condition that requires frequency gain attenuation over the complete frequency range and hence, by analogy with the standard linear systems case, is a very strict condition, especially as the reference signal many only have significant frequency content over a finite range. This paper develops a new control law design method where frequency attenuation is enforced over a finite range to be decided by frequency decomposition of the reference signal. The design makes use of the generalized Kalman–Yakubovich–Popov (KYP) lemma to establish the equivalence between frequency domain inequalities over finite and/or semi-finite frequency ranges for a transfer-function and a linear matrix inequality (LMI) defined in terms of its state-space realization (Iwasaki & Hara, 2005). The resulting design algorithm is experimentally applied to the gantry robot used in Hładowski et al., 2010 and Hładowski et al., 2012, including the extension to robust design using a norm bounded uncertainty representation that offers advances not possible for the same problem in the lifted setting. The following notation is used throughout this paper. For a matrix X , X⊤X⊤ and X ⁎ denote its transpose and complex conjugate transpose respectively. The null and identity matrices with appropriate dimensions are denoted by 0 and I respectively. Moreover, the notation X≽YX≽Y (respectively X≻YX≻Y) means that the matrix X−YX−Y is positive semi-definite (respectively, positive definite). Also sym{X}sym{X} is used to denote the symmetric matrix X+X⊤X+X⊤ and X⊥X⊥ denotes the orthogonal complement of the matrix X , that is, a matrix whose columns form a basis of the nullspace of X . The symbol (⋆)(⋆) denotes block entries in symmetric matrices and ρ(·)ρ(·) denotes the spectral radius of its matrix argument, that is, if hihi, 1≤i≤h1≤i≤h, is an eigenvalue of the h×hh×h matrix H then ρ(H)=max1≤i≤h|λi|ρ(H)=max1≤i≤h|λi|.

This paper has developed new results on the design and experimental verification of ILC laws for discrete linear systems in the repetitive process setting. The new design algorithm enforces a required frequency attenuation over a finite frequency range in comparison to previous results that demanded this attenuation over the entire frequency range. The results have been established by using the generalized KYP lemma to transform frequency domain inequalities over finite and/or semi-finite frequency ranges for a transfer-function to LMIs. The resulting control law has a well defined physical basis and the analysis extends to allow norm-bounded uncertainty. Experimental verification of the control law design algorithm on a gantry robot has also been undertaken with very good agreement between predicted and measured data. In comparison to design using stability along the pass (Hładowski et al., 2010) no optimization is required, i.e., maximizing the value of K2 in the control law to increase the speed of pass-to-pass error convergence. Also in this previous design a low-pass filter is added to attenuate high frequencies but is not required in this new design. It is, however, possible to include a low-pass filter with pass-band edge above 5 Hz with the aim of increasing the pass-to-pass error convergence. The previous design was completed over the entire frequency range and then the pass-band limited by adding a low-pass filer, whereas the new design limits the frequency range within the design procedure. In the new design there exists possibility that the upper frequency can be maximized, that is, extend the bandwidth and hence the resulting control law will ensure better performance by trading-off between robustness and fast convergence (performance). Also in experimental verification of the design in Hładowski et al. (2010) it was necessary to zero-phase filter the error on each trial. The robust control design in this paper avoids an issue that arises in lifted model approaches by avoiding the product of matrices that define the uncertainty. These results require much further development and comparison with alternative, such as van de Wijdeven, Donkers, and Bosgra (2011) robust control design algorithms. Further work is also required to develop the option of directly maximizing the range of frequencies over which the learning is performed. This can be done by formulating the problem as the minimization of a linear objective function under LMI constraints.