ELEC4631 Chap.11 Controller and Observer Synthesis Using LMIs
Controller and Observer Synthesis Using LMIs
Direct state-feedback stability, (A-BK)^TP+P(A-BK)<0 with P>0, is correct but not jointly convex because P and K multiply. Congruence with X=P^-1 and the substitution Y=KX yield the affine condition AX-BY+XA^T-Y^TB^T<0 with X>0. A feasible solution recovers K=YX^-1 and must be checked in the original closed-loop dynamics. For a fixed decay rate alpha, adding 2alpha X creates a rate certificate; optimising alpha simultaneously with X would reintroduce a bilinear product and often needs an outer search. Observer synthesis follows the dual construction and, under the convention here, recovers L=X^-1Y^T. Ackermann, LQR and LMIs answer different prompts: exact spectrum, quadratic optimality and convex certificate feasibility. None should be selected merely because it was the most recently memorised method.
What this chapter covers
- 01Direct feedback BMI
- 02Congruence transformation
- 03X=P^-1 and Y=KX
- 04State-feedback stability LMI
- 05Fixed decay-rate constraint
- 06Observer dual LMI and L recovery
- 07Numerical conditioning and original-model verification
Recover a scalar stabilising gain from X and Y
- XThe scalar plant has A=B=1. X=1 is strictly positive, satisfying the certificate side condition.
- LMIThe transformed symmetric expression is AX-BY+XA-YB=2-4=-2<0.
- strictnessBecause the inequality is strict, the candidate lies inside the feasible stability region rather than on its boundary.
- recoveryRecover K=Y/X=2.
- returnThe original closed-loop coefficient is A-BK=1-2=-1.
- conclusionThe closed-loop pole is -1, so the origin is globally asymptotically stable; the original property agrees with the transformed certificate.
Key terms
- Congruence transformation
- Mapping M to X^TMX with invertible X; it preserves definiteness and enables the feedback variable change.
- X variable
- The inverse Lyapunov variable X=P^-1, constrained symmetric positive definite.
- Y variable
- The product Y=KX used so feedback terms become affine; for state feedback Y is m-by-n.
- Gain recovery
- Returning from transformed decisions to the physical controller, K=YX^-1, preferably by a linear solve in computation.
- Decay-rate LMI
- A strengthened Lyapunov inequality including 2alpha X for fixed alpha to certify exponential convergence at the requested rate.
- Observer LMI
- The dual synthesis condition for A-LC, with gain recovered in the dimensionally correct order L=X^-1Y^T under this convention.
Controller and Observer Synthesis Using LMIs FAQ
Why is the original Lyapunov condition a BMI?
Expanding it produces K^TB^TP and PBK. When P and K are both decisions, their products are bilinear. The stability theorem is correct, but its direct joint design model is nonconvex.
Why does the X/Y substitution work?
Congruence by P^-1 preserves definiteness and converts the products into KX. Declaring Y=KX makes every term affine in X and Y. X>0 ensures invertibility so the gain can be recovered.
Can alpha be optimised inside the rate LMI?
For fixed alpha, 2alpha X is affine in X. If alpha and X are both decisions, their product is bilinear. A standard approach fixes alpha, solves feasibility and searches over alpha outside the LMI, unless another supported formulation is supplied.
What should I check after a solver returns feasible?
Symmetrise X, inspect its positive eigenvalue margin, evaluate the strict LMI residual, recover K or L with a stable solve, form A-BK or A-LC and verify the original spectrum or Lyapunov residual. A badly conditioned X can magnify numerical error.
What is the full state-feedback LMI derivation chain?
Start from u=-Kx and A_cl=A-BK. Write A_cl^TP+PA_cl<0 with P>0 and expand the products to show why P,K form a BMI. Set X=P^-1, congruence-transform by X and define Y=KX. Obtain He(AX-BY)<0 with X>0 and verify every term is affine in X,Y. Solve, then recover K=YX^-1 in the original state coordinates. Evaluate the original A-BK spectrum and a Lyapunov residual. If a fixed rate alpha is required, include 2alpha X and state that alpha is fixed during the feasibility solve. These steps separate theorem, convexification, numerical decision and physical controller; omitting any one can leave a correct-looking formula with no verified closed-loop meaning.
How does observer recovery avoid transpose mistakes?
Begin from the observer error matrix A-LC and either derive its Lyapunov design inequality directly or apply controller synthesis to the dual pair (A^T,C^T). Track dimensions: L is n-by-p. Under this chapter’s variable convention the dual decision Y has dimensions p-by-n, and recovery is L=X^-1Y^T. Confirm X^-1Y^T is n-by-p and that LC is n-by-n. A formula YX^-1 would generally have the wrong orientation. After recovery, form A-LC and check the original error dynamics, not merely the transposed solver residual. Writing dimensions beside each variable is the fastest defence against an apparently symmetric but physically impossible observer gain.
How should I practise gain recovery without trusting a solver blindly?
Begin with a known stabilising K and a convenient X>0, set Y=KX and verify the transformed inequality; then reverse the exercise by hiding K and recovering it from X,Y. Perturb X toward ill-conditioning and observe how numerical recovery becomes sensitive even though the symbolic relation remains valid. Repeat with the observer dual and write every dimension. For a rate constraint, fix alpha and test feasibility at several values, explaining why the outer search preserves convexity. An AI tutor can supply fresh stable or unstable plants, but require it to report only residuals after you have derived the substitution and recovered the gain. The educational target is the equivalence chain, not the solver interface.
Exam move
Always begin with the original closed-loop Lyapunov inequality and expand it to identify the bilinear product. Then write the congruence and substitutions explicitly; do not jump to a memorised sign pattern. Check dimensions of X, Y, K and L before recovery. Practise one fixed-alpha rate problem and explain why making alpha a decision changes convexity. For observers, derive the dual form from A-LC and verify the recovery order through dimensions. Compare the three design languages on one plant: Ackermann for exact poles, LQR for a declared quadratic cost and LMIs for feasibility with convex certificate constraints. Treat numerical solver output as intermediate evidence and always return to the original dynamics for the engineering conclusion.
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