EFFICIENT LINEAR MIMO ADAPTIVE INVERSE CONTROL Dr. Gregory L. Plett Electrical and Computer Engineering Department University of Colorado at Colorado Springs P.O. Box 715, Colorado Springs, CO 8933 715 +1 719 262 3468; glp@eas.uccs.edu 29 August 21 Efficient Linear MIMO Adaptive Inverse Control G. L. Plett [1]
Overview What is Adaptive Inverse Control (AIC)? How does linear SISO AIC wor? The problem with linear MIMO AIC. Overcoming the problem using filter transposition. Illustrative example. 29 August 21 Efficient Linear MIMO Adaptive Inverse Control G. L. Plett [2]
u u ũ ũ û ũ What is AIC? y y Adapt P(z) ŷ to model P(z). (Adaptive system ID) ŷ Adapt C(z) ˇy to control P(z). (Adaptive feedforward control) ŷ ỹ Adapt X (z) to cancel disturbances. (Adaptive disturbance d canceling) d r ε ε C Plant P P Dist. w y r P X z 1 I ŵ 29 August 21 Efficient Linear MIMO Adaptive Inverse Control G. L. Plett [3]
ŷ ŷ ˇy ŷ ỹ An d Adaptive Filter d An adaptive filter has: an input x, an output y, and a special input d called the desired response. An error signal e ε is computed: e = d y. The error signal is used to ε adjust the internal parameters of the filter. r P Many algorithms (e.g. LMS, RLS) may be used. RLS in particular is very fast. Input x Adaptive Filter Error e Output y Desired Response d 29 August 21 Efficient Linear MIMO Adaptive Inverse Control G. L. Plett [4]
û ũ y y ŷ ŷ Adaptive System Identification ˇy ŷ A plant model ỹ P(z) is adapted to minimize the mean-square difference between d the plant and model outputs for the d same input. The model is unbiased by disturbance if u is uncorrelated with w. ε ε Dist. w r P series-parallel parallel u Plant P P y (Modeling error plus dist.) 29 August 21 Efficient Linear MIMO Adaptive Inverse Control G. L. Plett [5]
ˇy ŷ ỹ d d e SISO (mod) Feedforward Control ε ε r P No desired response exists for C(z). Desired response only available at plant output. Note, however, that P(z)C(z) = C(z)P(z). Therefore, r Copy C(z) Plant P(z) y n Copy P(z) x C(z) M(z) d 29 August 21 Efficient Linear MIMO Adaptive Inverse Control G. L. Plett [6]
u ũ ũ û SISO System with Disturbance Canceling y y ŷ Integrated ŷ system with disturbance canceling. No bias in plant model. ˇy ŷ ỹ d d ε ε r P r C u ũ X P Plant P P COPY Dist. w z 1 I ŵ y 29 August 21 Efficient Linear MIMO Adaptive Inverse Control G. L. Plett [7]
û ũ y y y ŷ ŷ Training a Disturbance Canceler, SISO ˇy ŷ ỹ Want z 1 P(z)X d (z) = 1 in order to cancel disturbance d completely. This would give non-causal X (z). However, by minimizing the MSE between ε both sides of the equation, we get the optimal causal ε disturbance canceler. r P ŵ z 1 Copy P(z) X (z) d 29 August 21 Efficient Linear MIMO Adaptive Inverse Control G. L. Plett [8]
The Problem with MIMO AIC MIMO systems may be represented as matrices of SISO transfer functions (e.g., [H(z)]). Matrix multiplications do not commute. [P(z)][C(z)] = [C(z)][P(z)]. Therefore, the simple method for adapting SISO controllers and disturbance cancelers will not wor. However, {[P(z)][C(z)]} T = [C(z)] T [P(z)] T. The novel approach of this paper is to use the transposition identity to adapt a MIMO controller and disturbance canceler. 29 August 21 Efficient Linear MIMO Adaptive Inverse Control G. L. Plett [9]
[h 13 ] 2 [h 13 ] 3 [h 13 ] 4 [h 13 ] 2 [h 13 ] 3 [h 13 ] 4 Filter Transposition [h 21 ] 2 [h 21 ] 3 [h 21 ] 4 How do we transpose a filter? Consider a 3-input 2-output filter [H(z)] and its transpose. [h 22 ] 2 [H(z)] = [h 22 ] 3 [h 22 ] 4 [h 23 ] 2 [h 23 ] 3 H 11(z) H 12 (z) H 13 (z) H 21 (z) H 22 (z) H 23 (z) [h 21 ] 2 [h 21 ] 3 [h 21 ] 4, [H(z)] T = [h 22 ] 2 [h 22 ] 3 [h 22 ] 4 H 11 (z) H 12 (z) H 13 (z) H 21 (z) H 22 (z) H 23 (z) In terms of impulse-response matrices, this might loo lie [h 11 ] [h 12 ] [h 13 ] [h 11 ] 1 [h 12 ] 1 [h 13 ] 1 [h 21 ] [h 22 ] [h 23 ] [h 21 ] 1 [h 22 ] 1 [h 23 ] 1 [h 23 ] 4 and. [h 11 ] [h 21 ] [h 11 ] 1 [h 21 ] [h 1 23 ] 2 [h 12 ] [h 22 ] [h 12 ] 1 [h 22 ] [h 1 23 ] 3. [h 13 ] [h 23 ] [h 13 ] 1 [h 23 ] [h 1 23 ] 4 29 August 21 Efficient Linear MIMO Adaptive Inverse Control G. L. Plett [1]
ˇy ŷ ỹ d d e MIMO (mod) Feedforward Control ε ε r P To perform MIMO feedforward control, we want [P(z)][C(z)] = [M(z)]. Equivalently, we want [C(z)] T [P(z)] T = [M(z)] T. r Copy C(z) Plant P(z) y n Copy P(z) T x C(z) T M(z) T d 29 August 21 Efficient Linear MIMO Adaptive Inverse Control G. L. Plett [11]
y y y ŷ ŷ ˇy MIMO ŷ Disturbance Canceling ỹ d d To perform MIMO disturbance canceling, we want z 1 [P(z)][X (z)] = I. ε Equivalently, ε we want z 1 [X (z)] T [P(z)] T = I. r P ŵ z 1 I Copy P(z) T X (z) T d 29 August 21 Efficient Linear MIMO Adaptive Inverse Control G. L. Plett [12]
Boeing 747 Example Problem Two inputs (rudder & aileron angles), one through three outputs (yaw-rate, roll-rate & ban-angle). [M(z)] = z 1 I. A d = B d =.8876.381.415.198.22.3973.46.24, 1.2515.516.7617.139.3313.151.447.9976.486.13 1.589.3887 1, C d =.599 4.839 1 1.39 1.2585 29 August 21 Efficient Linear MIMO Adaptive Inverse Control G. L. Plett [13].
u u ũ ũ û ũ y y y ŷ ŷ ˇy ŷ ỹ d 3 2 2-Input 1-Output Result Single-output tracing, MSE= 2.44 1 7 d ε ε r P 1 1 2 2 4 6 8 1 29 August 21 Efficient Linear MIMO Adaptive Inverse Control G. L. Plett [14]
u u ũ ũ û ũ y y y ŷ ŷ ˇy ŷ ỹ d 5 2-Input 2-Output Result Two-output tracing, MSE= 2.25 1 3 d ε ε r P 5 2 4 6 8 1 5 5 2 4 6 8 1 29 August 21 Efficient Linear MIMO Adaptive Inverse Control G. L. Plett [15]
u u ũ ũ û ũ y y y ŷ ŷ ˇy ŷ ỹ d d ε ε r P 5 2-Input 3-Output Result 5 2 4 6 8 1 5 5 2 4 6 8 1 5 Three-output tracing, MSE= 1.34 5 2 4 6 8 1 29 August 21 Efficient Linear MIMO Adaptive Inverse Control G. L. Plett [16]
ũ û ũ y y y ŷdisturbance Canceling Result ŷ ˇy ŷ ỹ d d ε ε r P Amplitude Square.7.6.5.4.3.2.1 System error 5 1 15 2 Time (s) 29 August 21 Efficient Linear MIMO Adaptive Inverse Control G. L. Plett [17]
u ũ ũ û ũ y ŵ ŷ ŵ ŷ E COPYF F COPYzI z 1 I Dist. w Dist. w ˆv Sensor ˇy Noise, ŷv ỹ r u d u d ũ ũ û ũ ε y ε ŷ r ŷ P 1log 1 ( e 2 ) ˇy ŷ ỹ d d ε ε r P 1log 1 ( e 2 ) 2 4 6 8 1 12 Learning curve, one output 14 2 4 6 8 1 6 5 4 3 2 1 1 Learning curve, three outputs u ũ ũ û ũ y y ŷ ŷ ˇy Learningŷ Curves ỹ d d ε ε r P 2 2 4 6 8 1 1log 1 ( e 2 ) 4 2 2 4 Learning curve, two outputs 6 2 4 6 8 1 29 August 21 Efficient Linear MIMO Adaptive Inverse Control G. L. Plett [18]
Conclusions AIC has been used to effectively control stable linear SISO systems. Known methods for adapting controllers for MIMO systems are very slow. The approach presented here uses MIMO filter transposes to give an architecture for very fast adaptation. It wors well! 29 August 21 Efficient Linear MIMO Adaptive Inverse Control G. L. Plett [19]