Lyapunov-Metzler Inequalities for Switched Systems

• Lyapunov–Metzler inequalities are matrix inequalities that combine Lyapunov functions with Metzler matrices to enforce stability in systems with prescribed dwell times.
• They facilitate both state-feedback regulation and performance guarantees by leveraging LMI relaxations for linear and affine switched systems.
• Practical applications include stabilization of unstable dynamics in engineering systems such as boost converters and traffic models.

Lyapunov–Metzler inequalities are matrix inequalities developed for the analysis and synthesis of switching control laws in continuous-time switched linear and affine systems subject to dwell-time constraints. They provide a framework for expressing stability conditions and guaranteed cost bounds via a combined Lyapunov and Metzler-matrix approach, enabling both the rigorous regulation of linear systems to the origin and the practical stabilization of affine systems to a neighborhood around the origin. Their formulation and use leverage irreducible Metzler matrices to encode mode-dependent cost-to-go comparisons over intervals defined by a prescribed minimum dwell time, with matrix inequalities ensuring negativity of aggregated Lyapunov increments at switching events.

1. Formulation of Lyapunov–Metzler Inequalities

Lyapunov–Metzler inequalities are constructed for systems with a finite set of modes $\Omega=\{1,\dots,M\}$, where each mode $i$ is described by dynamics $A_i\in\mathbb{R}^{n\times n}$ and output matrix $C\in\mathbb{R}^{p\times n}$. A dwell time $T>0$ dictates the minimum time spent in each mode before switching. The switching structure is encoded via an irreducible Metzler matrix $\Pi=[\pi_{i,j}]_{i,j=1}^M\in\mathcal{M}$, characterized by $\pi_{i,j}\geq0$ for $i\neq j$ and zero row sums $\sum_j \pi_{i,j}=0$.

For the linear switched case, the algebraic Lyapunov–Metzler inequality for each $i\in\Omega$ is: $$A_i^\top X_i + X_i A_i + \sum_{j\ne i}\pi_{i,j}(Y_{1,j} + Y_{2,j} - X_i) + C^\top C < 0,$$ where $X_i\in\mathbb{S}^n_{++}$ and

$$Y_{1,j} = e^{A_j^\top T} X_j e^{A_j T}, \quad Y_{2,j} = \int_0^T e^{A_j^\top \tau} C^\top C e^{A_j \tau} d\tau.$$

For switched affine systems $\dot{x} = A_\sigma x + b_\sigma$, the system is augmented to a linear form in $\tilde x = [x^\top\;\;1]^\top$, and the inequalities become

$$\tilde A_i^\top\tilde X_i+\tilde X_i \tilde A_i + \sum_{j\ne i}\pi_{i,j}(\tilde Y_{1,j} + \tilde Y_{2,j} - \tilde X_i) + \tilde C^\top\tilde C < \varepsilon \tilde I,$$

with $\tilde I = \mathrm{diag}(0,\dots,0,1)$, and corresponding block matrices: $$\tilde A_i = \begin{pmatrix}A_i & b_i \ 0 & 0\end{pmatrix}, \qquad \tilde C = [C\;\;0], \qquad \tilde X_i = \begin{pmatrix}X_i & 0 \ 0 & 1\end{pmatrix}.$$

2. Differential and Algebraic Structure; Connection to Dwell-Time

The Lyapunov–Metzler inequalities incorporate both algebraic and differential forms. During each dwell interval $[t_k, t_k + T)$, the candidate Lyapunov matrix $P_i(t)$ (or affine extension $\tilde P_i(t)$) evolves according to the backward-flow differential equation: $$-\dot P_i(t) = A_i^\top P_i(t) + P_i(t)A_i + C^\top C, \qquad t\in[t_k, t_k+T)$$ with terminal constraint: $P_i(t_k+T) = X_i.$ This ensures $P_i$ flows backwards in time from $X_i$, resetting at switching points. The transformation between algebraic and integral forms links the differential equation solution to the terms in the Lyapunov–Metzler inequality. The Metzler-summed increments $\sum_{j\ne i}\pi_{i,j}(\cdots)$ enforce the negativity of cost-to-go jumps at switching events, mediating stability across modes and encoding the dwell-time dependence in the comparison principle.

3. Stability Analysis and Guaranteed Cost Bounds

In the linear switched case, the existence of matrices $\{P_i(t), X_i\}$ and a Metzler $\Pi$ satisfying the Lyapunov–Metzler inequalities yields a state-feedback switching law under dwell time $T$ that ensures:

• Global exponential stability of the origin,
• A guaranteed quadratic cost bound:

$$J(x_0,t) = \int_{t_0}^t z(\tau)^\top z(\tau) d\tau \leq x(t_0)^\top P_{\sigma(t_0)}(t_0) x(t_0),$$

where $z = Cx$ is the regulated output.

For switched affine systems, the corresponding construction using $\tilde X_i$, $\Pi$, and $\varepsilon$ produces:

• Practical stability: the state enters and remains in a neighborhood of the origin,
• Cost bound:

$$J(x_0,t) \leq x(t_0)^\top P_{\sigma(t_0)}(t_0) x(t_0) + \varepsilon (t-t_0),$$

with the long-term average cost satisfying $$\limsup_{t\to\infty} \tfrac{1}{t}J(x_0,t) \leq \varepsilon$$.

These results rely on the feasibility of the Lyapunov–Metzler inequalities and enforcement of a fixed minimum dwell time.

4. Computation and Selection of Metzler Matrix and Lyapunov Parameters

The practical application of Lyapunov–Metzler inequalities involves selection or optimization of the Metzler matrix $\Pi$ and Lyapunov matrices $X_i$ (and $\varepsilon$ for the affine case):

• The direct approach jointly optimizes $\{\Pi, X_i\}$ to minimize the cost bound, subject to the inequalities, but this is nonconvex due to bilinear terms in $\pi_{i,j}(Y_{1,j} + Y_{2,j} - X_i)$.
• To address tractability, a line-search/LMI relaxation fixes $\Pi$ to a scalar-diagonal Metzler form (e.g., $\pi_{i,i}=\gamma > 0$, $\pi_{i,j} = -\gamma$ for $i \neq j$). This reduces the inequalities to a set of LMIs in $X_i$ and $\gamma$, solvable with standard LMI solvers and yielding conservative but practical solutions.
• For application-driven choices, if a convex Hurwitz average $A_\lambda = \sum \lambda_i A_i$ exists, $\Pi$ may be chosen so that $\lambda^\top \Pi = 0$. When operating under Markovian switching with dwell times $T_i$, one takes $\pi_{i,j} = 1/T_i$ for preferred transitions.

5. Expressions for Quadratic Cost Bounds

The guaranteed cost bounds resulting from Lyapunov–Metzler inequalities are of quadratic form:

System Type	Cost Bound Expression	Bound Type
Linear switched	$$J(x_0, t) \leq x(t_0)^\top P_{\sigma(t_0)}(t_0) x(t_0)$$	State-dependent
Affine switched	$$J(x_0, t) \leq x(t_0)^\top P_{\sigma(t_0)}(t_0) x(t_0) + \varepsilon (t-t_0)$$	State- and time-dependent

In both cases, $P_{\sigma(t_0)}(t_0)$ is computed as the initial value of the Lyapunov matrix solving the backward-flow differential Lyapunov equation over the dwell interval.

6. Systematic Examples and Numerical Verification

Three illustrative examples demonstrate the feasibility and effect of Lyapunov–Metzler inequalities:

• Unstable-linear system: Two unstable $A_1, A_2$ are stabilized by choosing $$\Pi = \alpha\begin{bmatrix}-1 & 1 \ 1 & -1\end{bmatrix}$$ and $T=0.1$. Numerical simulations show convergence even though individual modes are unstable.
• Boost–boost converter: A 4-mode switched affine representation, with $\lambda = (0.25, 0.25, 0.25, 0.25)$ and $\Pi$ such that $\lambda^\top\Pi=0$, delivers successful regulation at $T=10^{-5}$ s and achieves small $\mathcal{H}_2$-like cost. Comparative analysis indicates performance similar to existing dwell-time stabilization methods.
• Traffic-congestion: A 3-mode switched affine queue model without a Hurwitz convex combination is stabilized by a cyclically-structured Metzler matrix (Markov-chain form) with $T=2.1$. The Lyapunov–Metzler approach produces a stable limit cycle, matching behavior from periodic controllers in established literature.

In all cases, feasibility of the inequalities is validated numerically with LMI solvers or line-search relaxation, and the switching law is observed to enforce dwell-time while maintaining costs within the prescribed bounds.

7. Assumptions, Limitations, and Applicability

Key assumptions are the existence of an irreducible Metzler matrix $\Pi$, feasibility of the Lyapunov–Metzler inequalities (linear or affine), and enforcement of a fixed dwell time $T>0$. The nonconvex optimization for optimal $\Pi$ and $X_i$ may limit general computational tractability; however, line-search/LMI relaxations provide practical conservative solutions. The framework accommodates both systems with Hurwitz convex combinations and those without such structures, extending applicability to switched systems with arbitrary mode dynamics. This suggests that Lyapunov–Metzler inequalities are broadly suitable for engineering systems requiring guaranteed performance under constrained switching, including systems lacking conventional stability in any individual mode.