Adaptive Inverse Control

6.0 INTRODUCTION

An adaptive inverse control system is diagrammed in Fig. 6.1. If the controller were ideal, its transfer function would be

Figure 6.1 An adaptive inverse control system that works well but adapts slowly.

The adaptive controller will generally not be ideal; its transfer function can therefore be designated as

The controller weight vector can be expressed accordingly as

The LMS algorithm cannot be used to adapt the controller of Fig. 6.1. Many other adaptive algorithms can be used, however, to automatically adjust the weights of Ĉ(z) of Fig. 6.1. Examples are the differential steepest-descent (DSD) algorithm and the linear random search algorithm (LRS) of reference [1]. When using these algorithms, changes in the controller weights are made to minimize the measured mean square error. Each time the controller weights are changed, time must be allowed for statistical equilibrium to develop in the plant before measuring MSE.

The DSD algorithm is based on the method of steepest descent. It uses a gradient vector which is obtained one component at a time, and each gradient component is obtained by measuring MSE with the corresponding weight increased and held for some time, then decreased and held for some time. The LRS algorithm tries random changes in the weight vector. After each trial change, the MSE is measured and compared with the measured MSE before the trial change. The actual weight vector change is made equal to the trial change multiplied by the MSE difference (before and after the trial change). If the trial change causes an improvement in performance, a lowering of MSE, the actual weight change will be in the trial direction and proportion to the improvement. If the trial change causes a reduction in performance, then the actual change will be opposite in direction to the trial direction and proportional to the reduction in performance. The DSD and LRS algorithms perform similarly, except that LRS converges twice as slowly as DSD when both adapt with the same level of misadjustment. Both of these algorithms converge extremely slowly compared to LMS when they are all set to adapt with the same level of misadjustment.

It would be desirable to use the LMS algorithm because it is much faster than LRS and DSD. It cannot be used directly because the available error є_k of Fig. 6.1 is an error referred to the plant output.¹ LMS really needs an error referred to the plant input, that is, to the adaptive controller output. To get an appropriate error for LMS implementation, one would need to apply є_k to the inverse of the plant P(z), thus requiring the solution in order to get the solution. The system of Fig. 6.1 is not our system of choice.

In order to be able to make use of the LMS algorithm and other high-speed adaptive processes, the inverse modeling configuration of Fig. 6.2 has been devised. The plant and its inverse model are commuted, so that the error є_kis directly available for the adaptation of Ĉ(z). Once Ĉ(z) is obtained, an exact digital copy can be used as a controller for the plant. This adaptive control system concept was first proposed in reference [2].

The system of Fig. 6.2 works very well as long as there is no plant disturbance. If plant disturbance is present, its effect is to bias the Wiener solution so that Ĉ(z) will not be a proper controller. The disturbance that appears at the plant output adds a component to the covariance of the input signal of the adaptive inverse model, directly affecting the Wiener solution for Ĉ(z). So what should one do? There are a number of choices, and the approach indicated in Fig. 6.3 offers the possibility of rapid adaptation and proper control even in the presence of plant disturbance.

The control system of Fig. 6.3 is based on the inverse modeling scheme of Fig. 5.13. It works in the following way. A model _k(z) of the plant P(z) is formed, using in this case dither scheme A. An offline process can be used to obtain controller Ĉ_k(z) from a digital copy of _k(z), and the reference model M(z). The offline process, illustrated in Fig. 6.3, adapts Ĉ_k(z) so that the output of the cascade of _k(z) and Ĉ_k(z)becomes a best least squares match to the output of the reference model M(z).² Both the cascade and the reference model are driven simultaneously by the same modeling signal. This signal is synthesized to have an appropriate spectral character, like that anticipated for the command input.

The process for finding Ĉ(z) could also be nonadaptive. Fundamentally, Ĉ_k(z) is deterministically related to _k (z) and M(z) for any specified modeling signal spectrum.³ Now given Ĉ_k(z), an exact digital copy of it can be used as a controller, as shown in Fig. 6.3. The result is a controller and plant having an overall dynamic response which closely approximates the optimal dynamic response of the reference model M(z).

The offline process of Fig. 6.3 forms a model-reference inverse of the plant model _k (z). We have used the model _k{z) rather than the plant P(z) because the output of the real P(z) is generally corrupted by plant disturbance. Since _k (z) does not perfectly match P(z) at all times, use of (z) in determination of Ĉ(z) causes errors in Ĉ(z). Also, even if (z) were perfect, there would generally be limitations preventing the perfect realization of C(z) = M(z)/P(z). These limitations will be explored next. We will assume at first a perfect _k(z) = P(z) and proceed to find best inverses of (z). Errors in the inversion process will be analyzed. Then the additional effects of errors in _k(z) will be considered.

Figure 6.2 An adaptive inverse control system with commuted plant and inverse model. It works well only when plant disturbance has low level.

Figure 6.3 An adaptive inverse control system with offline inverse modeling adapts rapidly and works well even with plant disturbance.

¹ Nor can any of the exact least squares lattice algorithms be used in this application for the same reasons.