Linear Matrix Inequalities in Robust Control Venkataramanan (Ragu) Balakrishnan School of ECE, Purdue University MTNS 2002
Objective A brief introduction to LMI techniques for Robust Control Emphasis on robust control, not LMIs Only techniques that use LMIs Tutorial, no new results General ideas, little detail
What is robust control? Modeling often first step in engineering Models often inexact: Exact models difficult, expensive, sometimes impossible Simpler models often desirable Need to analyze and design for inexact models Good designs, designed for one model, may fail for even a nearby model Hence robust control
Control system analysis and design framework w u Inexact system model z y w is vector of exogenous inputs z is vector of outputs of interest u is vector of control inputs y is vector of observed outputs Analysis problems: Study of properties from w to z Design problems: Design control law u = K(y, t) such that map from w to z is good
Polytopic models S_2 S_1...... S_n... Actual model lies inside a polytope Vertices S 1,..., S n represent extremes of operating conditions or parameter values, or linear models from linearization of a nonlinear system Goals of robust control: Analyze worst-case behavior over the polytope Design for worst-case behavior over the polytope
Feedback uncertainty models w u p Linear model y z q models uncertainties, nonlinearities, parameters Assumed to have additional properties: bounded in some norm, passive, etc structured (diagonal, block diagonal, etc) Goals of robust control: Analyze worst-case behavior over all Design for worst-case behavior over all
Robust stability Question: Inputs identically zero. Does state go to zero for all ICs irrespective of? Sufficient condition: There exists V (ζ) = ζ T P ζ, P > 0, s.t. dv (x(t))/dt < 0 a.k.a. quadratic stability condition yields LMIs for a number of robust control models Thus: Search for quadratic Lyapunov functions for robust stability is an LMI
LMIs: A very brief introduction Linear Matrix Inequality (LMI) F (x) = F 0 + x 1 F 1 + + x m F m 0, ( ) where x R m is the variable, and F i s are given symmetric matrices A convex constraint on x R m Feasibility problem: Find x such that ( ) holds Optimization problem: Minimize c T x over ( ) Fundamental complexity is low, polynomial-time algorithms available Efficient solvers available
Example: Robust stability of polytopic systems ẋ(t) = A(t)x(t), A(t) Co {A 1,..., A L }. There exists V (ζ) = ζ T P ζ, P > 0, s.t. dv (x(t))/dt < 0 iff P = P T > 0, A(t) T P + P A(t) < 0, A(t) Co {A 1,..., A L }, or P = P T > 0, A T i P + P A i < 0, i = 1,..., L. (An LMI) Technique can be generalized to more sophisticated Lyapunov functions ( parameter-dependent ) Can account for rate of parameter variations
Example system: Example: Robust stability of feedback-uncertain system ẋ(t) = Ax(t)+Bp(t), q(t) = Cx(t), p(t) = (t)q(t), (t) 1. There exists V (ζ) = ζ T P ζ, P > 0, s.t. dv (x(t))/dt < 0 iff x(t) T (A T P + P A)x(t) + 2x(t) T P Bp(t) < 0 whenever p(t) T p(t) q(t) T q(t). Equivalent to x(t) T (A T P + P A)x(t) + 2x(t) T P Bp(t) + q(t) T q(t) p(t) T p(t) < 0
Equivalent to P > 0, λ 0, AT P + P A + C T C B T P (An LMI) P B I < 0. Stability condition has connections with small-gain theorem Technique can be generalized to more sophisticated Lyapunov functions Can handle a much larger class of Can handle structure information on
Robust performance analysis Question: What is worst-case value, over uncertainties, of a performance measure 0 φ(x(t), w(t), z(t)) dt? Approach: Suppose V (x(t)) = x(t) T P x(t), P > 0, and for all trajectories d V (x(t)) φ(x(t), w(t), z(t)) dt ( ) Then, x(0) T P x(0) 0 φ(x(t), w(t), z(t)) dt, or x(0) T P x(0) an upper bound on robust performance Condition ( ) is an LMI for a large class of functions φ Approach applicable to both polytopic systems and feedback-uncertain systems Approach can be extended to handle more sophisticated Lyapunov functions
Robust control design w u Uncertain System y z u=k(y,t) Problem: Find the optimal control strategy, u = K(y, t) s.t. closed-loop system enjoys good robustness properties Some feedback possibilities: y = x, i.e., state feedback Only a linear combination of state components available, i.e., output feedback The uncertainties are measurable in real-time, and can be used for feedback, i.e., gain-scheduled
Constant state feedback Feedback strategy is u(t) = Kx(t) Yields LMI problems in several important cases For example, polytopic systems: ẋ(t) = A(t)x(t)+Bu(t), u(t) = Kx(t), A(t) Co {A 1,..., A L } Closed-loop system stable if P = P T > 0, (A i + BK) T P + P (A i + BK) < 0, i = 1,..., L. With Q = P 1, Y = KP 1, Q = Q T > 0, QA T i + Y T B T + A i Q + BY < 0, i = 1,..., L. (An LMI)
Output-feedback design with LMIs Feedback strategy y = K(y(t), t) In general much harder than state feedback, get LMIs only in a handful of cases Underlying difficulty illustrated with constant state-feedback design for polytopic systems: ẋ(t) = A(t)x(t) + Bu(t), y(t) = Cx(t), u(t) = Ky(t), Closed-loop system stable if A(t) Co {A 1,..., A L } P = P T > 0, (A i +BKC) T P +P (A i +BKC) < 0, i = 1,..., L. (No convexification possible)
Gain-scheduled-feedback design with LMIs Feedback scheduled by the measured uncertainties Example: feedback-uncertain systems (t) q y Linear system p u v Klti f (t) Determining K lti and a quadratic Lyapunov function for the closed-loop system is an LMI problem
Concluding remarks LMIs offer numerical solution of many robust control problems Interest began in the 1990s when efficient LMI algorithms became available, and continues to grow Rich literature available, many specialized results for special robust control problems
Challenges Many practical problems yield large-scale LMI problems, current solvers often inadequate. Special-purpose algorithms and solvers desirable Study of applications of LMIs to robust control fairly mature, other applications remain to be explored
Some references SIAM monograph Linear Matrix Inequalities in System and Control Theory, by Boyd et al. Kluwer Handbook on Semidefinite Programming, Wolkowicz et al. (editors) SIAM collection Recent Advances on LMI Methods in Control, El Ghaoui et al. (editors) Most recent control conferences ACC and CDC