Linear Algebra
Matrix Equations: Sylvester and Lyapunov
How can the stability of an aircraft control system be verified without running a 1000-year simulation? The Lyapunov equation gives the answer in a single algebraic step.
- **Aviation:** MATLAB lyap() underlies stability analysis of flight control systems in all commercial Airbus and Boeing aircraft
- **Model order reduction:** Hankel singular values from Lyapunov Gramians drive the balanced truncation method used in automotive ECU design
- **Kalman filter:** the Riccati equation P*A^T + A*P + Q - P*C^T*R^{-1}*C*P = 0 is the nonlinear generalization of the Lyapunov equation for optimal state estimation
- **Quantum systems:** the Lindblad equation for open quantum systems has a Lyapunov structure over complex matrices
Предварительные знания
- Schur decomposition
- Eigenvalues and spectrum
- Positive definiteness
Sylvester equation
The Lyapunov equation AX + XA^T = -Q (Q > 0) has a unique solution X > 0 if and only if A is Hurwitz (all Re(lambda_i) < 0). This is the fundamental stability test. MathWorks sells lyap() licenses at $2000/year; the Bartels-Stewart algorithm underlies stability analysis for the entire aviation industry.
The Sylvester equation AX + XB = C is the fundamental linear matrix equation generalizing the scalar equation ax + xb = c. The existence criterion - A and -B share no eigenvalues - follows directly from vectorization via Kronecker products. The Lyapunov equation is the special case B = A^T and C = -Q, with deep connections to stability theory.
When does the Sylvester equation AX + XB = C have a unique solution?
Sylvester-Rosenblum theorem: AX + XB = C has a unique solution X iff A and -B share no eigenvalues. Proof via diagonalisation or the integral representation through exp(At) C exp(Bt).
Lyapunov equation and stability
The Lyapunov equation is a special case of Sylvester with B = A^T. It appears in stability analysis of linear systems: dx/dt = Ax is stable iff for every Q > 0 there exists P > 0 satisfying AᵀP + PA = -Q.
Stability connection: the system dx/dt = Ax is asymptotically stable (Re(lambda_i) < 0) if and only if for every Q > 0 the Lyapunov equation AP + PA^T + Q = 0 has a solution P > 0. The Lyapunov function V(x) = x^T P x decreases strictly along every trajectory.
What condition on A guarantees stability of dx/dt = Ax?
The linear system dx/dt = Ax is asymptotically stable iff Re(lambda_i) < 0 for all eigenvalues of A. Equivalently: the Lyapunov equation AᵀP + PA = -Q has a positive-definite solution P for every Q > 0.
Riccati equation and LQR
The Riccati equation AᵀP + PA - PBR^{-1}BᵀP + Q = 0 is the non-linear extension of Lyapunov. Its solution defines the LQR (Linear Quadratic Regulator) controller minimising J = ∫(xᵀQx + uᵀRu)dt. Used in aircraft control, missile guidance, portfolio optimisation.
The Sylvester criterion has a geometric meaning: AX + XB = C defines a linear operator on the space of matrices. Its nonsingularity is equivalent to A and -B having disjoint spectra, which ensures the operator is invertible.
How does the Riccati equation differ from the Lyapunov equation?
The Lyapunov equation AᵀP + PA = -Q is linear in P. The Riccati equation AᵀP + PA - PBR^{-1}BᵀP + Q = 0 has a quadratic-in-P term and is non-linear. Solved by Schur decomposition or Newton-Kleinman iteration.
Connections to control theory and numerical methods
Lyapunov and Riccati equations are the cornerstone of modern control theory.
- Kalman filter — Related topic
- LQR optimal control — Related topic
- Model order reduction — Related topic
- Spectral theory — Related topic
Итоги
- Sylvester equation AX+XB=C has a unique solution iff the spectra of A and -B are disjoint
- Bartels-Stewart algorithm: Schur decomposition plus back substitution in O(n^3)
- Lyapunov equation AP+PA^T+Q=0 (Q>0) has P>0 iff A is Hurwitz (all Re(lambda)<0)
- Lyapunov function V(x)=x^TPx decreases along trajectories dx/dt=Ax iff AP+PA^T<0
- Controllability and observability Gramians are Lyapunov solutions that determine model balancing
- The Riccati equation is the nonlinear Lyapunov generalization at the core of Kalman filtering and LQR
What is the physical meaning of the matrix P > 0 solving the Lyapunov equation AP + PA^T + Q = 0?
dV/dt = d/dt(x^T P x) = x^T(AP + PA^T)x = -x^T Q x < 0. This is direct proof of stability via the Lyapunov function. P > 0 ensures V(x) > 0 for x != 0, and AP+PA^T = -Q < 0 ensures V decreases along every trajectory.