Lyapunov equations and stability margins

Let A be a real square matrix and Q a symmetric positive definite matrix. If A is Hurwitz, define P as the integral of e^{Aᵀt}Qe^{At} over all positive time. Exponential decay makes the integral finite. Every nonzero initial vector produces positive integrated Q-energy, so P is positive definite. Differentiating the integrand and using its vanishing limit at infinity gives AᵀP+PA=−Q. The solution is unique: the homogeneous equation would make e^{Aᵀt}Pe^{At} constant, yet its limit is zero.

Conversely, if some P>0 and Q>0 satisfy this equation, V(x)=xᵀPx has derivative −xᵀQx along ẋ=Ax, proving exponential stability. Thus a Hurwitz matrix is characterized by positive-definite Lyapunov solutions for every positive-definite Q. A semidefinite Q alone is not enough without additional observability-type information, since the derivative can vanish on a nondecaying mode. In discrete time the equation is AᵀPA−P=−Q, with A Schur stable; using the continuous-time equation there would analyze the wrong dynamics.

The smallest and largest eigenvalues of P bound V between two multiples of the squared Euclidean norm. Similarly, xᵀQx is at least λ_min(Q) times the squared norm. Hence V̇≤−λ_min(Q)V/λ_max(P). Integrating this scalar differential inequality gives an exponential energy bound and, after taking a square root, a state-norm bound. The prefactor involves the square root of P’s condition number. This prefactor accommodates possible transient growth in Euclidean norm even though V decreases at every instant.

Consider A with diagonal entries −1 and an upper-right coupling 4. Both eigenvalues are −1, but e^{At}=exp(−t)[[1,4t],[0,1]]. Starting from (0,1), the norm at t=1 is √17/e, greater than its initial value one. There is no contradiction with asymptotic stability or a Lyapunov certificate. Different norms and metrics answer different questions. Quoting only the largest real eigenvalue as a bound on every transient state norm would be false for this nonnormal example.

Replace A by A+E, where E is a real additive matrix perturbation and the induced Euclidean norm is used throughout. The derivative gains xᵀ(EᵀP+PE)x, whose magnitude is at most 2||P||₂||E||₂||x||₂². Therefore the same P proves stability whenever 2||P||₂||E||₂<λ_min(Q). This is a uniform sufficient bound for all perturbation directions within that norm ball. It can be much smaller than the actual distance to instability because it insists on preserving one chosen energy function.

Changing state units changes the matrix norm and P, so a numerical margin is meaningful only with its coordinate scaling stated. The bound is also different from classical gain margin or phase margin, which describe structured loop changes rather than an arbitrary additive E. Time-varying perturbations also give uniform exponential decay if there is one fixed positive epsilon with 2||P||₂||E(t)||₂≤λ_min(Q)−epsilon for all t; a merely pointwise strict inequality can lose its decay margin over time, but delays, actuator saturation and neglected dynamics do not automatically fit a static additive matrix model. A certificate’s strength comes from stating exactly the uncertainty class it controls.