410 likes | 588 Views
Applications of Convex Optimization in Systems and Control. Venkataramanan Balakrishnan Purdue University. Basic idea. Computational methods, esp. convex optimization increasingly relevant to systems and control Much wider class of problems can now be “solved”. Outline.
E N D
Applications of Convex Optimization in Systems and Control Venkataramanan Balakrishnan Purdue University
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
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”
Convex optimization • is a convex set: • is a convex function:
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
SDPs in control • Lyapunov functions for other uncertain system models • Performance objectives, e.g., bounds on norms • Synthesis of control laws
Outline • Convex optimization for control • Equalizer design in communications • Fault-tolerant control laws for robots • Conclusion
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
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
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
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
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
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:
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 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
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
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
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