Inversion-based Feedforward

Issue 1: Inversion of Nonlinear Non-minimum Phase Systems

Background: Research in the 1960s and 1970s showed that a system's dynamics can be inverted to find inputs that exactly track a desired output trajectory. For example, system inversion can be used to find control inputs that enable an aircraft to track a desired maneuver. These standard inversion methods led, however, to unbounded inputs for nonminimum phase systems -- such systems can be characterized as having "unstable" inverse systems. Thus, the standard inversion method with unbounded inputs could not be applied to practical nonminimum phase systems such as flexible structures with non-collocated sensors and actuators. In the past, other researchers sought to alleviate this inability to invert nonminimum phase systems by relaxing the problem, e.g., (a) inverting a modified system that is not nonminimum phase, i.e., minimum phase, (b) changing the desired output trajectory, i.e., tracking a different output to enable inversion or (c) tracking the output asymptotically if it belongs to a special class of exogenous signals.

Our early research (Ref 1) provided a breakthrough by directly solving the inversion problem for nonlinear nonminimum phase systems. It developed and proved the convergence of a Picard-like iteration to obtain bounded inverses for nonlinear nonminimum phase systems. The use of stable (but possibly non-causal) output-to-input maps enabled us to find bounded solutions to "unstable" inverse systems. (MATLAB Code used in Ref 1 can be downloaded here.)

Issue 2: Inversion under Plant Uncertainty

Background: In general, any inversion-based approach, e.g., the approach in Ref 1, to precision tracking is a model-based approach; hence, its performance depends on the accuracy of the model. For example, model-based inversion can be used to achieve high-precision, output tracking if the models are relatively accurate. Anecdotal experimental evidence showed, however, that large errors in computing the inverse input (due to errors in the models) can adversely affect the output-tracking performance. These observations led to the basic question "Should model-based inversion be used for precision output-tracking in the presence of modeling uncertainties?"

The above fundamental question was answered by our research (Ref 2), which found bounds on size of model uncertainties so that the inversion-based input together with a feedback input beats the output-tracking performance of the feedback alone. Moreover the research developed algorithms to find inverses when the modeling uncertainty varies with frequency. This development is important in practical applications because modeling uncertainty typically tends to grow and become large at high frequencies.

Issue 3: Managing Non-causality of the Inverse

Background: An important challenge in implementing the inversion approach (Ref 1) is that the inputs are non-causal for nonminimum phase systems; hence the input needs be changed before the output is modified. While it is intuitive that preview information of the desired output can be used to find the current value of the inverse input, the challenge is to address the question, "how much of the future output trajectory information is needed to achieve a desired precision in output tracking?"

This question was addressed by our research group (see Ref 3); we quantified the amount of preview information needed to achieve a desired precision in output tracking and developed an algorithm for online implementation of the non-causal inverse input. In Ref 4, we also experimentally verified the preview-based method and achieved sub-nano-scale positioning in a Scanning Tunneling Microscope when imaging carbon atoms in graphite. The theory was extended to nonlinear systems in Ref 5.

Issue 4: Inversion for Non-Square Systems and To Handle Actuator Limits

Background: Standard inversion methods are applicable to square systems with the same number of outputs as inputs. A challenge is to extend the approach to nonsquare systems, say, with actuator redundancy. Similarly, in some situations, a systematic approach is needed to trade off the tracking requirement to manage system limitations such as actuator bandwidth restrictions.

Our research developed optimal inversion methods to account for actuator saturation and bandwidth limitations (e.g., Ref 6). While the formulation of the problem is similar to standard optimal control theory, the optimal inverse has an explicit solution. The same optimal inversion method was extended to develop inverses for non-square systems, e.g., to handle actuator redundancy, Ref 7.

Issue 5: Output Transition Versus State Transition

Background: The minimum-time, state transition problem with bounds on the input magnitude is a classical problem that leads to a bang-bang-type input for the fastest state transition. However, the transition time can be reduced further if only the system output needs to be transitioned from one value to another (in time interval [0,T]) rather than the entire system state. The time-optimal output transition problem is to change the system output from an initial value y1 (for all time t before t=0) to a final value y2 (for all time t after t=T).

The main contribution of our work is to solve the optimal output transition problem (e.g., Ref 8 and Ref 9). It uses inversion-based feedforward to maintain the output (using pre- and post-actuation) at the desired values, y1 and y2 respectively, outside of the output-transition interval [0, T]. The original approach minimized time and energy for optimizing the transition. The minimum-time transition problem was solved with extension to nonlinear systems in Ref 10 and dual-stage systems in Ref 11.

Our theoretical developments on inversion-based output transitions were applied to Disk Drive servo systems in Ref 12 – this work was partially funded by INSIC, which is a consortium of Disk Drive companies. A description of the feedforward control to nano-positioning for high-density storage systems can be found in this presentation.

Application to Nanopositioning

A review of feedforward control and its application to nanopositioning can be found in Ref 13. This article reviews inversion-based feedforward approaches used for high-speed nanopositioning such as optimal inversion that accounts for model uncertainty and inversion-based iterative control for repetitive applications.

The article establishes connections to other existing methods such as zero-phase-error-tracking feedforward and robust feedforward.

Nanopositioning is also reviewed in the following presentation slides.