Unifying Input-Output and State-Space Perspectives of Predictive Control
In this work we present a predictive control formulation that bridges the input-output and state-space approaches of predictive control. A key element in this bridge is the relationship between state-space and input-output models. Through a special interaction matrix which can be viewed as a generalization of the standard Cayley-Hamilton theorem, the relationship between the two kinds of models becomes extremely simple. The present predictive control formulation is neither input-output model based nor state-space model based, but rather it is both due to the transparent connection provided here. From the input-output perspective, the formulation clearly explains and justifies the structures of various input-output models. From the state-space perspective, it shows how the coefficients of various predictive models and the predictive controller gains are related to those of the original state-space model. The experimental results reported in this work also show among other things the benefit of designing a controller directly from input-output data as opposed to working through an intermediate identification model. The general philosophy here is to identify the needed parameters for a particular controller design instead of identifying a model of the system first from which the needed parameters for the controller are computed. This general philosophy can be further extended to the problem of state-space system identification with accurate order estimation, the problem of designing an optimal state estimator directly from input-output data, and the problem of periodic disturbance cancellation without explicit disturbance or system identification. For each of these problems, an appropriate interaction matrix arises that helps provide a considerably concise solution. More details can be found in Phan, Lim, and Longman (1998).
llustration: A series of experiments on SPINE is carried out to compare the performance of a predictive controller designed directly from a set of experimental input-output data versus that designed from an intermediate identification model (extracted from the same data). A laser-based optical sensing system detects the torsional motion of the third rod, and the tenth rod is clamped to a rigid support. The system is very lightly damped and flexible, making it difficult to control. Some results are shown in the next set of figures below.
Illustration: The left 4 figures show a comparison between simulated and actual experimental results using a predictive controller designed from an intermediate identification model. The lighter curves show uncontrolled responses and the black curves show controlled responses. The right 4 figures show the corresponding results using a predictive controller designed from input-output data directly. The performance of the controller designed by the direct approach is better in that the control signal of the indirectly designed controller continues to persist in the steady-state whereas the control signal of the directly designed controller subsides (compare the bottom right figure on the left group of 4 versus that in the right group of 4).
Simultaneous Feedback Stabilization and Disturbance Rejection by Predictive Control in Structural Acoustics
We show the design of a predictive controller that is capable of both feedback stabilization and disturbance rejection. The design is direct in that the controller gains are derived from a set of disturbance-corrupted input-output data directly without having to estimate a disturbance-free model or the disturbance itself. The disturbance rejection feature is built in automatically as long as such disturbances are present in the data. Experimental results illustrate this direct predictive controller design approach.
To put the above result in perspectives, we briefly describe some background information leading to this design. First, it is clear that a predictive controller can be designed to handle the feedback stabilization problem, Phan and Juang (1998), Juang and Phan (1998). Second, in the area of periodic disturbance rejection, it is shown in Goodzeit and Phan (1997) that from a set of disturbance-corrupted input-output data alone, one can identify (a) the system disturbance-free dynamic model, (b) the disturbance frequencies -- even if they coincide with the system flexible mode frequencies, (c) the contribution of each disturbance frequency on the output, and (d) the corresponding feedforward control signal to cancel it. This can be accomplished without knowledge of the disturbance locations, frequencies, amplitudes, or phases. Thus prior to applying control, one can determine by identification alone which disturbance frequencies should be targeted and which should be ignored to make efficient use of limited control resources. In simpler situations, we look for the possibility of designing a predictive feedback controller that can stabilize the system, and at the same time, provide disturbance rejection without explicit knowledge of a disturbance-free model or the disturbance frequencies. Particularly, we look for a way to design such a controller directly from a set of input-output data, and if the output data happens to be corrupted by unknown harmonic disturbances, then the designed controller would automatically reject these disturbances as well. In other words, the disturbance rejection capability is automatic in this design without explicit knowledge of a disturbance-free model or the disturbance frequencies. More details can be found in Phan, Juang, and Eure (1999) and Phan, Lim, and Longman (1998).
Illustration: The described controller is applied to regulate the vibrations of a plate as shown in the left figure. The aluminum plate is 0.040 inches thick. Bonded to the plate are piezoceramic actuators. The actuators are 2.25 inches by 1.25 inches and 0.010 inches thick. An accelerometer (feedback sensor) is mounted on the top center of the piezoceramic patch that is used as the control actuator. The piezoceramic patch shown in the lower right hand corner is driven by band-limited white noise, and is used as the "unknown" disturbance source. To collect data for this controller design, the system is excited, and 5 seconds of input-output data is collected in the presence of this "unknown" disturbance. The right figure shows the autospectrum of the uncontrolled (gray) and controlled (black) accelerometer signals. It is seen that the controller significantly reduces the disturbance contents of the accelerometer signal. Nearly a dozen disturbance frequencies are automatically suppressed by this controller. This experiment was carried out at NASA Langley Research Center.
