Adaptive control for uncertain nonlinear multi-input multi-output systems
Systems and methods of adaptive control for uncertain nonlinear multi-input multi-output systems in the presence of significant unmatched uncertainty with assured performance are provided. The need for gain-scheduling is eliminated through the use of bandwidth-limited (low-pass) filtering in the control channel, which appropriately attenuates the high frequencies typically appearing in fast adaptation situations and preserves the robustness margins in the presence of fast adaptation.
1. A method of adaptive control for an uncertain nonlinear multi-input multi-output system, the method comprising:
providing a processor operable to execute a control law residing in a memory, the control law including a bandwidth-limited filter configuration, the control law configured to generate a control signal u c as an output of the bandwidth-limited filter configuration;
measuring a system state signal x;
generating a prediction of the system state {circumflex over (x)};
subtracting the prediction of the system state {circumflex over (x)} from the measured system state signal x, and producing an error signal {tilde over (x)};
generating estimates of uncertainties in the multi-input multi-output system; and
generating the control signal u c as an output of the bandwidth-limited filter configuration based on the generated estimates of uncertainties and the measured system state x.
2. The method according to claim 1 further including generating the estimates of uncertainties based on the measured system state signal x and the control signal u c .
3. The method according to claim 1 further including providing a state-predictor configured for mimicking the multi-input multi-output system and for specifying desired system behavior.
4. The method according to claim 1 further including updating the estimates of the uncertainties at a high adaptation rate.
5. The method according to claim 1 wherein the adaptive control system is a closed-loop system.
6. The method according to claim 1 wherein the multi-input multi-output system is in the presence of significant unmatched uncertainty.
7. The method according to claim 1 wherein the bandwidth-limited filter configuration comprises m×m strictly-proper and stable bandwidth-limited filters, wherein m system outputs and m control signals are included in the m×m strictly-proper and stable bandwidth-limited filters.
8. An adaptive control system for controlling a nonlinear multi-input multi-output system with uncertainties, the adaptive control system comprising:
a fast estimation scheme, the fast estimation scheme including a state predictor and an adaptive law, the fast estimation scheme configured to generate estimates of the uncertainties in the multi-input multi-output system; and
a control law, the control law including a bandwidth-limited filter configuration, the control law configured to generate a control signal u c as an output of the bandwidth-limited filter configuration.
9. The system according to claim 8 wherein the estimates of the uncertainties are generated based on a measured system state x and the control signal u c .
10. The system according to claim 8 wherein the state-predictor is configured to mimic the multi-input multi-output system and to specify desired system behavior.
11. The system according to claim 9 wherein the state-predictor is configured to generate a prediction of the system state {circumflex over (x)}, the prediction of the system state {circumflex over (x)} then being subtracted from the measured system state x, to produce an error signal {tilde over (x)}; and
the error signal {tilde over (x)} together with the measured system state signal x and the control signal u c , drives the adaptive process.
12. The system according to claim 9 wherein the control law generates the control signal u c based on the estimates of uncertainties and the measured system state x.
13. The system according to claim 8 wherein the adaptive law is configured to update the estimates of the uncertainties at a high adaptation rate.
14. The system according to claim 8 wherein the adaptive control system is a closed-loop system.
15. The system according to claim 8 wherein the multi-input multi-output system is in the presence of significant unmatched uncertainty.
16. An adaptive control system comprising:
a processor operable to execute a control law residing in a memory, the control law including a bandwidth-limited filter configuration, the control law configured to generate a control signal u c as an output of the bandwidth-limited filter configuration; and
a state predictor configured to generate a prediction of a nonlinear multi-input multi-output system state {circumflex over (x)}, such that when the system state {circumflex over (x)} is subtracted from an actual system state x, an error signal is produced that, together with the actual system state x and the control signal u c , drives an adaptation process.
17. The system according to claim 16 wherein the bandwidth-limited filter comprises a feedback structure in the control law.
18. The system according to claim 16 wherein the control signal u c equals K g (s)r(s)−C 1 (s){circumflex over (η)} 1 (s)−C 2 (s)H m −1 (s)H um (s){circumflex over (η)} 2 (s), wherein
K g (s) is a prefilter,
r(s) is a bounded setpoint,
C 1 (s) and C 2 (s) are bandwidth-limited filters,
{circumflex over (η)} 1 (s) and {circumflex over (η)} 2 (s) are adaptive laws,
H m −1 (s) is an inverse of a matched transfer matrix, and
H um (s) is an unmatched transfer matrix.
19. The system according to claim 18 wherein C 1 (s) and C 2 (s) are stable bandwidth-limited filters.