ELEC4631 Chap.10 Linear Matrix Inequalities: Convexity, Feasibility and Schur Complements
Linear Matrix Inequalities: Convexity, Feasibility and Schur Complements
An LMI is a symmetric matrix inequality whose matrix depends affinely on decision variables. F(x)=F0+x1F1+...+xmFm is positive semidefinite when z^TF(x)z>=0 for every z; strict definiteness excludes zero for nonzero z. Affinity makes the feasible set convex because every segment between feasible decisions remains feasible. Multiple constraints can be stacked in block-diagonal form. The Schur complement replaces certain inverse-containing inequalities with affine block inequalities, but only with the correct definiteness side condition and strictness. For fixed A, P>0 and A^TP+PA<0 are LMIs in symmetric P. Controller synthesis is harder because P and K multiply; recognising that bilinear matrix inequality before transforming it is a central modelling skill. A solver’s feasible status must be followed by eigenvalue margins, residuals and an engineering interpretation of the certificate actually posed.
What this chapter covers
- 01Symmetric matrix order
- 02Positive semidefinite versus definite
- 03Affine dependence on decision variables
- 04Convex feasible sets
- 05Block-diagonal stacking
- 06Schur-complement equivalence and side conditions
- 07Lyapunov inequality as an analysis LMI
Solve a scalar-parameter LMI exactly
- modelF is symmetric and affine in the single decision theta.
- conditionsFor a diagonal matrix, positive semidefiniteness requires theta+1>=0 and 3-theta>=0.
- closed setThus the semidefinite feasible set is -1<=theta<=3.
- strict setStrict positive definiteness requires both diagonal entries strictly positive, giving -1
- convexityBoth sets are intervals and therefore convex; equivalently affinity preserves feasibility under convex combinations.
Key terms
- LMI
- A positive/negative semidefinite constraint on a symmetric matrix that is affine in all decision variables.
- Decision variable
- A scalar or matrix chosen by the optimisation. Convexity is judged with respect to these unknowns, not fixed plant data.
- Convex set
- A set containing every line segment between any two of its points.
- Strict feasibility
- Satisfying definite inequalities with nonzero margin; numerically represented using tolerances rather than an exact open constraint.
- Schur complement
- A block-matrix equivalence that trades an inverse expression for a larger affine inequality under a definite/invertible side condition.
- BMI
- A bilinear matrix inequality containing products of decision variables, such as P and K in direct feedback design; it is not an LMI.
Linear Matrix Inequalities: Convexity, Feasibility and Schur Complements FAQ
How can I tell whether a matrix condition is an LMI?
Box every unknown. Each may appear only affinely: fixed offset plus fixed matrices multiplied by scalar decisions, or equivalent linear matrix operations. Products between unknowns, decision inverses and squared decisions break affinity. Repeat the audit after any substitution.
Why does symmetry matter?
Positive semidefinite order and eigenvalue sign tests are defined for symmetric or Hermitian matrices. Expressions such as A^TP+PA are symmetric when P is symmetric. If a derived residual is not symmetric, a transpose or modelling term is probably missing.
What side condition accompanies a Schur complement?
It depends on which diagonal block is complemented. That block must be positive or negative definite and hence invertible with the appropriate sign. Write the side condition beside the equivalence; the block inequality alone cannot silently supply it.
Does infeasibility prove no controller exists?
Not necessarily. An LMI may be only a sufficient certificate or may impose a particular common quadratic structure. Infeasibility proves no solution exists within the modelled certificate and numerical tolerance, not impossibility across all controller classes.
What is a complete LMI modelling answer?
List the fixed data and box every decision variable. Rewrite the matrix map as fixed offset plus fixed coefficient matrices times decisions and confirm symmetry. State positive/negative and strict/non-strict inequalities separately. If using a Schur complement, name the block being complemented and its definiteness side condition; if stacking, explain that every diagonal block must satisfy its constraint. Prove or cite convexity only after affinity has been established. For a numerical solution, report feasibility margins through eigenvalues or residuals rather than solver status alone. Finally translate the feasible matrix into the exact control statement—stability, a bound or a rate. Do not call it an optimum unless an objective was included, and do not treat failure of one sufficient certificate as universal impossibility.
How are strict LMIs handled in computation?
A numerical solver cannot represent an open set with exact symbolic strictness, so P>0 is implemented as P>=epsilon I and a negative inequality as F<=-epsilon I for a chosen tolerance/scale. The reported epsilon must be meaningful relative to matrix magnitudes and solver tolerances. After solving, symmetrise matrices, inspect smallest positive and largest negative eigenvalue margins and evaluate every original constraint. A nearly singular positive matrix may formally clear a tiny tolerance yet make later inversion or gain recovery unreliable. Rescaling variables or data can improve conditioning, but it must not change the mathematical property silently. Written analytical problems should keep exact strict symbols and distinguish boundary feasibility from interior feasibility.
What practice best separates LMI from BMI?
Make a table of expressions and underline decisions: A^TP+PA for fixed A; PK+K^TP; F0+xF1; alpha X with alpha fixed and then with alpha also unknown; and a Schur block with P and a fixed M. Classify each before transforming it. For the nonlinear cases, propose a substitution only when you can state how the physical variable is recovered and why definiteness is preserved. Ask an AI tutor to generate five classifications but hide the labels until you justify each one. Then use a one-parameter diagonal LMI to solve the feasible interval exactly and compare its strict and closed boundaries with numerical epsilon implementations.
Exam move
Take ten matrix expressions and classify each as affine or not before doing any control design. For valid LMIs, list fixed matrices, scalar/matrix decisions, symmetry and strictness. Prove convexity once from the affine-combination identity, then use the geometry to interpret feasible intervals or regions. Practise both directions of a Schur complement, always writing the complemented block’s side condition. For numerical results, inspect the smallest eigenvalue of positive blocks and largest eigenvalue of negative blocks relative to tolerance. End every feasibility answer by translating the matrix certificate back into stability, rate or another stated property—never call a feasible point optimal unless an objective was actually included.
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