Passive Matrix Control Scheme

• Passive matrix control scheme is a framework that uses LMIs and AREs to enforce system passivity, ensuring energy dissipation and stability without full controllability.
• The methodology integrates state-feedback synthesis and optimal energy extraction through explicit matrix inequalities, providing robust control even under constrained conditions.
• This control strategy is applied in networked and underactuated systems, offering practical solutions for robust stabilization and efficient energy harvesting.

A passive matrix control scheme refers to a class of control strategies characterized by the interconnection or synthesis of controllers for systems whose dynamics exhibit passivity or non-expansiveness. Passivity, in this context, ensures that the system cannot generate energy and is fundamental to stability in networked, constrained, or underactuated systems. Matrix control typically involves methods based on state-space and behavioral system matrices or operator-valued transfer functions. The passive matrix control paradigm generalizes classical control constructions (positive-real and bounded-real lemmas) and provides new solutions for optimal energy extraction, robust control design, and stability analysis without relying on strong controllability or observability assumptions.

1. Foundations of Passive Matrix Control: Passivity and Non-Expansiveness

Passivity is defined for a system with state-space equations $dx/dt = Ax + Bu$, $y = Cx + Du$ via the existence of a storage function $S(x)$ that satisfies an energy dissipation inequality. The quadratic supply rate

$\sigma_p(u, y) = y^\top y - u^\top u$

is used to formalize the "available energy" after an initial condition $x_0$, given by

$$S_a^{(\sigma_p)}(x_0) = \sup \left\{ -\int_{t_0}^{t_1} \sigma_p(u, y)(t) \, dt \mid x(t_0)=x_0 \right\}.$$

Passive matrix control hinges on finding a symmetric positive semi-definite matrix $X \geq 0$ such that the linear matrix inequality (LMI)

$$\Omega(X) = \begin{bmatrix} - A^\top X - X A & C^\top - X B \ C - B^\top X & D + D^\top \end{bmatrix} \geq 0$$

holds (Hughes, 2017). Non-expansiveness is similarly characterized, but via the storage of the $\mathcal{H}_\infty$ norm and a more complicated LMI (

$$\Lambda(X) = \begin{bmatrix} - A^\top X - X A - C^\top C & -C^\top D - X B \ - D^\top C - B^\top X & I - D^\top D \end{bmatrix} \geq 0$$

). Uniquely, these formulations do not require the system to be controllable or observable.

2. Algebraic Riccati Equations and Feedback Synthesis

When $D + D^\top$ (resp., $I - D^\top D$) is strictly positive definite, the minimal solution $X_-$ to the LMI also solves an associated algebraic Riccati equation (ARE): $$\Gamma(X) = -A^\top X - X A - (C^\top - X B) (D + D^\top)^{-1} (C - B^\top X) = 0$$ with spectral conditions on the closed-loop matrix

$$A_\Gamma(X) = A - B (D + D^\top)^{-1} (C - B^\top X),$$

requiring its spectrum to lie in the left-half complex plane. The optimal state-feedback law that extracts maximal available energy is

$u = - (D + D^\top)^{-1} (C - B^\top X_-) x$

and achieves $$S_a^{(\sigma_p)}(x_0) = \frac{1}{2} x_0^\top X_- x_0$$. For non-expansive systems and robust control, similar ARE and feedback constructions apply with corresponding modifications. This approach generalizes classical energy-based control but is robust to violations of the usual "niceness" assumptions.

3. Bounded-Real and Positive-Real Pairs: Behavioral System Representations

The classical bounded-real and positive-real conditions relate to transfer function norms ($\|H\|_{\infty} \leq 1$) but are insufficient for uncontrollable or unobservable systems. The paper introduces the notion of "bounded-real pairs" for polynomial matrices: a pair $(P, Q)$ is bounded-real if for all $\lambda \in \mathbb{C}_+$,

$$Q(\lambda)Q(\bar{\lambda})^\top - P(\lambda)P(\bar{\lambda})^\top \geq 0,$$

and the concatenated matrix $[P, -Q](\lambda)$ has full rank. These algebraic conditions, alongside the LMIs, ensure the external behavior of the system (the set of admissible input-output trajectories) is non-expansive and passive even when internal modes are not accessible to control. This generalizes the understanding of passive and non-expansive behaviors beyond classical transfer function analysis.

4. Methods of Energy and Storage Characterization

Available energy (for passive systems) and available storage (for non-expansive systems) are proven to be quadratic forms in the initial state, computable via the minimal $X_-$ solution to the relevant LMIs and AREs. The optimization over system trajectories is cast as determining the supremum in the energy supply rate integral, reducible to solving the matrix inequality. In physically relevant scenarios (e.g., passive electric circuits), these formulas provide explicit and maximal bounds on extractable energy—unlike standard synthesis methods that may be overly conservative or inapplicable due to lack of controllability.

5. Practical Implementation and Implications

Passive matrix control is applicable in domains where strict actuation and sensing assumptions are violated:

• Passive electrical circuits may have uncontrollable components; the LMI/ARE-based control design still achieves maximum energy extraction.
• In networked or interconnected systems (where system matrices naturally arise), controllers can be synthesized to exploit passivity, optimize damping or energy harvesting, and guarantee closed-loop stability by design.
• For robust control and $\mathcal{H}_\infty$ synthesis, especially in non-expansive systems, the framework ensures that performance specifications tied to energy or storage are satisfied, even when not all system modes can be excited or measured.

Controllers constructed via the passive matrix scheme are expressed as explicit state-feedback, directly computable from the system matrices and critical for applications where hardware limitations preclude full accessibility or controllability.

6. Extensions and Generalizations

The presented non-standard passive matrix scheme extends positive-real and bounded-real theory using a behavioral viewpoint. It allows synthesis of optimal controllers and energy extraction algorithms for systems with unobservable or uncontrollable dynamics, relevant for heterogeneous physical systems, integrated energy harvesting circuits, and robust stabilization in mechanical or electrical networks.

These advances can be leveraged in the broader context of matrix-centric control engineering frameworks, enhancing both the rigor and applicability of passive control theory in modern cyberphysical systems.

7. Summary Table of Key Concepts and Characterizations

Concept	Matrix Condition / Formula	Applicability
Passivity	$\Omega(X) \geq 0$	Finite available energy
Feedback Synthesis	$u = - (D+D^\top)^{-1}(C-B^\top X_-) x$	Optimally extract energy
Non-expansiveness	$\Lambda(X) \geq 0$	Finite available storage
Bounded-real pair	$$Q(\lambda)Q(\bar{\lambda})^\top - P(\lambda)P(\bar{\lambda})^\top \geq 0$$	Robust $\mathcal{H}_\infty$ design
ARE for passivity	$\Gamma(X) = 0$ with closed-loop stability	Existence of feedback law
Energy Characterization	$S_a(x_0) = \frac{1}{2} x_0^\top X_- x_0$	Maximal energy/store computable

This scheme advances the theory and implementation of passive matrix control by providing LMI/ARE-based explicit characterizations, relaxing pervasive linear system assumptions, and opening rigorous pathways for energy/robustness-optimal control in networked, underactuated, and physically constrained systems (Hughes, 2017).