کنترل یادگیری تکراری قدرتمند بر اساس سیستم های 2D با استفاده از داده فاصله زمانی محدود غیر علی

This paper uses a 2D system setting in the form of repetitive process stability theory to design an iterative learning control law that is robust against model uncertainty. In iterative learning control the same finite duration operation, known as a trial over the trial length, is performed over and over again with resetting to the starting location once each is complete, or a stoppage at the end of the current trial before the next one begins. The basic idea of this form of control is to use information from the previous trial, or a finite number thereof, to compute the control input for the next trial. At any instant on the current trial, data from the complete previous trial is available and hence noncausal information in the trial length indeterminate can be used. This paper also shows how the new 2D system based design algorithms provide a setting for the effective deployment of such information.

Many systems complete the same finite duration operation over and over again, where each execution is known as a trial and the duration the trial length. The exact sequence is that on completion of each trial, the system resets to the starting location and the next one begins. A common application is replicated by a robot undertaking a pick and place task over and over again, i.e., collect an object from a specified location, transfer it over a finite duration, deposit it at a fixed location or onto a moving conveyor, return to the starting location and then repeat this sequence of operations. In this paper, the notation used is yk(p),0≤p≤α−1,k≥0yk(p),0≤p≤α−1,k≥0, where yy is the vector or scalar variable under consideration, α<∞α<∞ is the number of samples along the trial length and kk is the trial number. Also if yref(p),0≤p≤α−1yref(p),0≤p≤α−1, is a given reference trajectory for the output, which is assumed to be a member of the signal space chosen for the output of the controlled system, ek(p)=yref(p)−yk(p)ek(p)=yref(p)−yk(p) is the error on trial kk and in iterative learning control (ILC) the novel feature is the use of the previous trial error in the computation of the control input applied on the next trial, with a generalization to higher-order ILC where the errors from a finite number l>1l>1 of previous trials are used. Since its inception, widely credited to [1], ILC has seen extensive developments from theory through to experimental benchmarking and actual applications. The survey papers [2] and [3] are starting points for the literature and these together with subsequent publications show a wide range of applications from industrial robotics to residual vibration suppression, microelectronics fabrication, process control and recently a technology transfer to healthcare in the form of robotic-assisted upper limb stroke rehabilitation. In this latter application [4] the patient makes repeated attempts to follow a reference trajectory replicating daily living tasks, such as reaching out to a cup across a table top, assisted by a robot and with electrical stimulation applied to the relevant muscles. At the end of each trial, the patient’s arm is returned to the starting location and ILC used to compute the electrical stimulation to be applied on the next trial based on the previous trial error. If the patient is improving with increasing trial number then his/her voluntary effort should increase and the applied stimulation decrease and this has been confirmed in clinical trials. In ILC, all previous trial data are available before the next trial begins and hence there is not a requirement that only causal in pp previous trial data is used in the computation of the current trial input, i.e., the control applied at p∈[0,α]p∈[0,α] on trial kk can use previous trial data at p=p+1,…,αp=p+1,…,α. The ability to use the noncausal information is a novel feature of ILC and many successful implementations use the special case of phase-lead where a term in ek(p+λ),λ>0ek(p+λ),λ>0 is used to form the control input uk+1(p)uk+1(p) on the next trial, where λ=1λ=1 is common. In such cases, one alternative to consider is the inclusion of a weighted sum of previous trial error phase-lead terms or a weighted sum of such terms and ILC lag terms ek(p−β),β>0ek(p−β),β>0. The aim of this paper is to provide the basis of ILC design where the use of such information can be evaluated as a necessary step toward implementation. If the along the trial dynamics are discrete then one setting for ILC design is, since the trial length is finite, to define super-vectors for the variables. For example, let the system be single-input single-output for the ease of presentation with a natural extension to the vector case. Then the error super-vector is View the MathML sourceEk=[ek(0)ek(1)…ek(α−1)]T and the ILC error dynamics can hence be written as a system of linear difference equations in kk of the form Ek+1=HEkEk+1=HEk. Due to the finite trial length, error convergence in kk can occur even if the system has an unstable state matrix. The solution via lifting design is to first design a stabilizing feedback control law and then apply ILC to the resulting controlled system. Also for robust control based on norm bounded or polytopic uncertainty, the entries in the matrix HH will contain products of the matrices describing the uncertainty and this makes the analysis significantly more involved. Moreover, robustness analysis in the frequency domain is approximated as ILC controllers operate on a finite time interval, the trial length. Use of the Fourier transform on the infinite time interval will give a linear time-invariant control law but over the finite trial length errors may result in the initial part of the transient response; an example to support this fact is given in [5]. A robust H∞H∞ based ILC design with noncausal finite trial length is also given in [5] but is somewhat involved and choosing exactly what noncausal data to include is also lacking somewhat in transparency. An alternative to the lifted model analysis is to exploit the natural 2D system structure of ILC where one direction of information is from trial-to-trial, indexed by the subscript kk, and the other along a trial, indexed by pp. The first results on 2D system based ILC analysis is credited to [6] where a Roesser state-space model was used. Repetitive processes [7] are another class of 2D systems where information propagation in pp is over a finite duration and therefore a more obvious setting for analysis. The previous work on ILC laws designed using repetitive process control theory with experimental verification includes [8] and [9]. This setting also extends to differential dynamics and for robust control studies avoids products of matrices describing the uncertainty assumed. This paper will show that the repetitive process setting enables control law design where a weighted sum of noncausal and/or causal in pp is used and hence, for a given example, different combinations of previous trial terms can be considered in the search for a design that meets the performance specifications, with an extension to robust control. The relative merits of this design method are also discussed. Throughout this paper, the null and identity matrices with the required dimensions are denoted by 0 and II respectively. Also View the MathML sourceM≻0(≺0) denotes a real symmetric positive (negative) definite matrix and X⪯YX⪯Y is used to represent the case when X−YX−Y is a negative semi-definite matrix. Finally, View the MathML sourcediag{⋯} denotes a block diagonal matrix.

This paper has developed new results on ILC design in the repetitive process setting, with particular emphasis on the inclusion of noncausal previous pass data in the control law and robustness. The main result is an LMI based design whose performance has been assessed in simulation on a motor model from the literature, demonstrating that the use of previous trial data in ILC can make a significant difference to the performance achieved. Areas for further development include design with disturbance rejection, performance bounds and trade-offs, for which an H∞H∞ setting is a starting point where [7] contains basic results for similar repetitive process problems. As in all other control design algorithms there are parameters that must be chosen based on the knowledge of the application and this is the case for wlwl and whwh in the control law (35). The analysis and case study in this paper have established that the use of noncausal previous trial data beyond a single phase-lead term has merits in ILC design and provides the results necessary for design to begin, i.e., pass-to-pass error convergence and control law design formulas with an extension to robustness. In the latter aspect, the repetitive process setting does not, in comparison to the lifting design, encounter products of matrices defining the uncertainty. A detailed comparison with the lifted approach and other design alternatives can only be attempted after significant further development for which some areas for possible investigation have also been given.