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:0f

Convex optimization

C

Convex optimization

C1x

2x

)(0 xf

Semidefinite programming (SDP)

• Special convex optimization problem:

– is linear, i.e.,

– Domain of optimization is defined via linear matrix inequalities:

0f xcxf T)(0

C

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”)

)()( tPxtx T

P

SDPs in Control

• Stability of LTV system:

Stable if there exists quadratic Lyapunov function

that decays along trajectories, or

No analytical solution! …but SDP

)()( tPxtx T

• 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

• h1(n), …, hN(n) represent the effective channel; assumed fixed and known

• u1(n), …, uN(n) represent noises, assumed independent and white

h1(n)

hN(n)

y(n)

gN(n)

g1(n)

u1(n)

uN(n)

x(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

H1(z)

HN(z)

y(n)

GN(z)

G1(z)

u1(n)

uN(n)

x(n)

• 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 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 variablem R x

n R q, xG q

• Given end-effector velocity, joint velocity generated as

• Joint space to task space:

Mathematical framework

f(q(t)) x(t)

• Joint velocity to end-effector velocity:

q J x

with IJG

• Suppose joint i fails. Then, i th component of is identically zero

• Under perfect servo control:

Control with unidentified failure

aqca q q

• Then actual end-effector velocity is

,ci

a q J x jj0jj J n1i1-i1

i where

ci

a xG J x • 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

cx

Quantifying local performanceMean-square velocity error

cx

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

Performance comparison

Performance comparison

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

Symbol rate 40 KHz

Sampling rate 200 KHz

Symbol alphabet 16-QAM

Symbol waveform Square-root raised cosine,

Number of channels 2

Channel length 3

Equalizer length 3

Equalizer delay 3

Equalizer error bound