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

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

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

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” Introduction

is a convex set: is a convex function: Convex optimization

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

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

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”)

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

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

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

Communication multi-path

A simple block diagram h 1 (n), …, h N (n) represent the effective channel; assumed fixed and known u 1 (n), …, u N (n) represent noises, assumed independent and white h1(n)h1(n) hN(n)hN(n)    y(n)y(n) gN(n)gN(n) g1(n)g1(n)  u1(n)u1(n) uN(n)uN(n)      x(n)x(n)

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

Equalization error (ISI): –Quantified as –Exactly reformulated as LMI using KYP Lemma –Frequency-windowing possible Effect of noise: –“Large” G 1 (z), …, G N (z) amplify noise power –Noise power amplification quantified as –Quadratic in FIR coefficients, another LMI Tradeoff between  and  via SDP Design trade-offs

A two-channel example

Tradeoff:  vs. 

Tradeoff: MSE vs. 

BER vs SNR (  = 0.1)

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

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)

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

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

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

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

Quantifying local performance

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 Quantifying local performance Mean-square velocity error

Quantifying local performance Peak-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

Performance comparison

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

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

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 Further research directions

Equalized spectrum (  = 0.1)

Simulation parameters