Applications of Convex Optimization in Systems and Control

1. Applications of Convex Optimization in Systems and Control Venkataramanan Balakrishnan Purdue University

2. Basic idea • Computational methods, esp. convex optimization increasingly relevant to systems and control • Much wider class of problems can now be “solved”

3. Outline • Convex optimization for control • Equalizer design in communications • Fault-tolerant control laws for robots • Conclusion

4. Introduction • Concept of “solution” constantly changing • Often dictates techniques used • Example: Stability of LTI systems • Late 1800s, complex variable techniques • 1900s, numerical linear algebra • Current state of the art: “Reduction to a convex optimization problem constitutes a solution”

5. Convex optimization • is a convex set: • is a convex function:

6. Convex optimization

7. Semidefinite programming (SDP) • Special convex optimization problem: • is linear, i.e., • Domain of optimization is defined via linear matrix inequalities:

8. Solving SDPs • SDPs are “easy” to solve: • Unique global minimum • Polynomial worst-case complexity • Duality theory • Algorithms and software available

9. SDPs in Control • Stability of LTI system: Stable if there exists quadratic Lyapunov function that decays along trajectories, or (Can find suitable by solving linear equations, i.e., can find “analytical solution”)

10. SDPs in Control • Stability of LTV system: Stable if there exists quadratic Lyapunov function that decays along trajectories, or No analytical solution! …but SDP

11. SDPs in control • Lyapunov functions for other uncertain system models • Performance objectives, e.g., bounds on norms • Synthesis of control laws

12. Outline • Convex optimization for control • Equalizer design in communications • Fault-tolerant control laws for robots • Conclusion

13. Communication multi-path

14. h1(n) g1(n)  y(n) x(n)  u1(n) hN(n) gN(n)  uN(n)       A simple block diagram • h1(n), …, hN(n) represent the effective channel; assumed fixed and known • u1(n), …, uN(n) represent noises, assumed independent and white

15. H1(z)  G1(z) y(n) x(n)  u1(n) HN(z) GN(z)  uN(n)       Zero-forcing equalizer design • Design FIR G1(z), …, GN(z) to equalize: • H1(z) G1(z) +    + HN(z) GN(z) = 1 • Mitigate effects of noise

16. Design trade-offs • Equalization error (ISI): • Quantified as • Exactly reformulated as LMI using KYP Lemma • Frequency-windowing possible • Effect of noise: • “Large” G1(z), …, GN(z) amplify noise power • Noise power amplification quantified as • Quadratic in FIR coefficients, another LMI • Tradeoff between e and h via SDP

17. A two-channel example

18. Tradeoff: vs. 

19. Tradeoff: MSEvs. 

20. BER vs SNR ( = 0.1)

21. Outline • Convex optimization for control • Equalizer design in communications • Fault-tolerant control laws for robots • Conclusion

22. Failures in robots • Robots are often used in hostile environments, with an increased likelihood of failures • Some ways of enhancing failure tolerance: • Component redundancy • Kinematic redundancy • Focus here: kinematically redundant robots (more joints than are necessary)

23. Assumptions • Joint failures lead to “locking” of joint • Joint failure is undetected, and controller continues to command motion of the failed joint • No failure detection and identification • Delay in failure detection and identification • Overwhelming number of failures

24. Mathematical framework • Joint space to task space: • Joint velocity to end-effector velocity: • Given end-effector velocity, joint velocity generated as • Joint variable with • Task space variable

25. Control with unidentified failure • Under perfect servo control: • Suppose joint i fails. Then, i th component of is identically zero • Then actual end-effector velocity is where • Thus:

26. Consequences of failures • Global issues: • Does manipulator converge to desired location? • If not, does it converge? • Conditions that guarantee answers can be given • Local issues: • Quantifying local performance measures • Design of G to improve local performance

27. Quantifying local performance

28. Quantifying local performanceMean-square velocity error • Euclidean norm of velocity error, averaged over all single-joint failures • Finding G to minimize MSE( ) is a least-squares problem • Solution is a weighted pseudo-inverse

29. Quantifying local performancePeak-velocity error • Peak norm of velocity error, over all single joint failures: • Finding G to minimize PKE( ) is an SDP: • Can also allow some pre-failure error pre by adding constraint

30. Performance comparison

31. Performance comparison

32. Performance comparison

33. Outline • Convex optimization for control • Equalizer design in communications • Fault-tolerant control laws for robots • Conclusion

34. General conclusions • Convex optimization has become a standard tool in system and control theory • Ideas from system and control theory are effective in many areas of EE

35. Further research directions • Often SDP problems are large, general-purpose solvers inadequate • Need algorithms that take advantage of problem structure • In other applications, data varies with time • Need algorithms that “track” optimal SDP solutions

39. Equalized spectrum ( = 0.1)

40. Simulation parameters