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- 1. Adaptive Decentralized Control with Nonminimum-Phase Closed-Loop Channel Zeros 55th IEEE Conference on Decision and Control Syed Aseem Ul Islam, Yousaf Rahman, and Dennis S. Bernstein University of Michigan 1
- 2. RCAC: Retrospective Cost Adaptive Control RCAC is a direct, digital, adaptive control technique that • Applies to stabilization, command following, and disturbance rejection • Uses minimal modeling information • Works on plants with nonminimum-phase zeros Tutorial session on RCAC at ACC 2016 Forthcoming article in IEEE Control Systems Magazine Developed for centralized control This talk: Extensions to decentralized control 2
- 3. Standard problem in state space As transfer functions (time domain!) The objective is to minimize 𝑧 in the presence of 𝑤 using minimal modeling information Standard Problem Framework 3
- 4. Controller Structure We use the dynamic compensator of order 𝑛c where The transfer function of the controller is given by 4
- 5. We define the retrospective performance variable where 𝐺f is defined by The rational behind is to replace with , where is obtained through optimization It can be seen that minimizing , yields which best matches to Retrospective Performance Variable 5
- 6. We define the cumulative retrospective cost function where 𝑅 𝑧, 𝑅 𝑢, and 𝑅 𝜃 are weighting matrices We use recursive least squares (RLS) minimization of the cumulative retrospective cost to obtain the unique global minimizer given by where Cumulative Retrospective Cost Function 6
- 7. Consider the SISO case The modeling information needed is • The relative degree of • The sign of • The NMP zeros of Modeling Information Needed by RCAC 𝑧 𝑘, 𝜃∗ = 𝐺𝑧𝑤 ∗ 𝐪 𝑤 𝑘 + 𝐺 𝑧 𝑢 ∗ 𝐪 − 𝐺f 𝐪 𝑢(𝑘) RCAC Cost Decomposition Closed-loop performance Target model matching Alternatively, the retrospective performance can be interpreted as the sum of the actual performance and an adaptation term. 7
- 8. NMP zeros RCAC will attempt to cancel NMP zeros that are not zeros of 𝐺f Cancelling NMP zeros leads to unstable pole-zero cancellation Therefore, we use the NMP zeros of to construct 𝐺f 8
- 9. RCXX talks at CDC 2016 “Retrospective Cost Adaptive Control with Concurrent Closed-Loop Identification of Time-Varying Nonminimum-Phase Zeros,” F. Sobolic and D. S. Bernstein 10:00-10:20 MoA11 Regular Session, Starvine 11 “Adaptive Control of Plants That Are Practically Impossible to Control by Fixed-Gain Control Laws,” Y. Rahman and D. S. Bernstein 10:40-11:00 MoA11 Regular Session, Starvine 11 “Adaptive Input Estimation for Nonminimum-Phase Discrete-Time Systems,” A. Ansari and D. S. Bernstein 13:30-13:50 MoB09 Regular Session, Starvine 9 “Combined State and Parameter Estimation and Identifiability of State Space Realizations,” M.-J. Yu and D. S. Bernstein 11:00-11:20 TuA13 Regular Session, Starvine 13 9
- 10. Effective Plants in Decentralized Control Each subcontroller “sees” a different effective plant The NMP zeros of this effective plant are the NMP channel zeros We define 10
- 11. Effective Plants in Decentralized Control Gc1 sees If If For Gc1 the zeros of effective plant are a combination of poles of the other subcontroller and the zeros of * *similarly for effective plant seen by Gc2 11
- 12. Adaptation Schemes for Decentralized RCAC 1CAT or 1 controller-at-a-time Subcontroller 1 adapts and Controller 2 is frozen; several steps later Subcontroller 2 adapts and Subcontroller 1 is frozen; this is repeated NMP zeros of effective plant need to be calculated only at switch Concurrent adaptation Subcontroller 1 and Subcontroller 2 are adapted at the same time NMP zeros of effective plant need to be calculated at each step (ideally) 12
- 13. RCAC is a Discrete-Time Method The following examples are given in terms of continuous-time plants with a sampled-data interface RCAC treats each system as a discrete-time plant 13
- 14. Example 1: Disturbance Rejection for a Two-Mode Oscillator Consider the continuous-time plant where , and the discretization is at 1 Hz The objective is to reject the disturbance using a decentralized controller structure 14
- 15. Gf For each subcontroller, we construct Gf at each time step by putting any and all NMP zeros of the effective plant in and choosing d so that the relative degree of Gf matches the relative degree of For Gc1, Nf is the polynomial containing the NMP zeros of For Gc2 , Nf is the polynomial containing the NMP zeros of 15
- 16. We apply RCAC with 1CAT Adaptation Gc1 first, then Gc2 Gc1 Gc2 Rθ 10,00 0 10 Ru 0 0 nc 4 4 Example 1: Disturbance Rejection for a Two-Mode Oscillator 16
- 17. Gc1 Gc2 Rθ 10 0.1 Ru 0 0 nc 4 4 We apply RCAC with 1CAT Adaptation Gc2 first, then Gc1 Example 1: Disturbance Rejection for a Two-Mode Oscillator 17
- 18. Example 2: Position and Shape Control of 2 DoF Flexible Body The continuous-time dynamics are which are sampled at 100 Hz, where , and This plant is unstable due to rigid body mode 18
- 19. The objective is to control the average velocity and separation of the two constituent masses in the presence of disturbance Achieved by appropriate ramp commands at r1 and r2 with concurrent RCAC We set Gc1 Gc2 Rθ 10 10 Ru 200 200 nc 8 8 Example 2: Position and Shape Control of 2 DoF Flexible Body 19
- 20. We can express a discrete-time transfer function as a Laurent expansions 𝐻𝑖 ≜ 𝐸0 𝐴𝑖−1 𝐵 is the 𝑖th Markov parameter of A finite but sufficiently large number of Markov parameters yields an FIR target model with • Correct relative degree • Correct first Markov parameter • Zeros that approximate the NMP zeros of •We construct Gf for each subcontroller this way using the first 10 Markov parameters at time 0 and use this Gf throughout the simulation Markov Parameter-Based Gf 20
- 21. Example 2: Position and Shape Control of 2 DoF Flexible Body 21
- 22. Summary For the two-mode oscillator we show two RCAC subcontrollers with 1CAT adaptation are able to reject broadband disturbance better than either subcontroller operating alone For the 2 DoF flexible body we show that two RCAC subcontrollers are able to achieve shape and average velocity objectives, with each subcontroller having access to only one mass each, for observation and control 22
- 23. Future Work Decentralized control with MIMO subcontrollers Further investigation of adaptation schemes 1CAT vs Concurrent 23