Balanced Truncation & System Lifting

• Balanced truncation is a model order reduction technique that balances reachability and observability before truncating weakly contributing states.
• System lifting reformulates nonlinear or quadratic-output dynamics into quadratic-bilinear forms, enabling the application of balanced truncation.
• Together, these methods reduce computational cost while preserving key system dynamics and ensuring robust error bounds for large-scale problems.

Balanced truncation is a well-established model order reduction (MOR) technique for linear systems, and system lifting refers to the process of reformulating nonlinear or nonstandard problems (e.g., those with quadratic outputs) into forms amenable to balanced truncation. When a system possesses outputs that are nonlinear or quadratic in the state, or when dynamics are nonlinear but polynomial, system lifting to a quadratic-bilinear (QB) or bilinear form is a key step prior to applying balanced truncation. These methodologies are fundamental in reducing computational complexity for large-scale systems while preserving essential dynamical characteristics.

1. Fundamental Concepts: Balanced Truncation and System Lifting

Balanced truncation for a standard linear time-invariant (LTI) system centers on computing reachability and observability Gramians, transforming the system into a balanced realization where states are equally controllable and observable, and truncating the least controllable/observable states. For general nonlinear systems or systems with quadratic outputs, direct application is not possible.

System lifting addresses this by introducing auxiliary variables to encapsulate nonlinearities or quadratic outputs, resulting in an augmented (typically higher-dimensional but structured) system—often quadratic-bilinear (QB) or bilinear with linear output—that can then be tackled using generalized versions of the Gramians and Lyapunov equations underpinning balanced truncation (Pulch et al., 2017, Kramer et al., 2019, Faßbender et al., 4 Jul 2025).

2. Lifting Linear and Nonlinear Systems

For a linear system with quadratic output,

$\dot x = Ax + Bu,\qquad y = x^T M x,$

lifting proceeds by augmenting the state with the quadratic output. The lifted state $\tilde{x} = [x^T, y]^T$ evolves according to a quadratic-bilinear system with a single linear output: $$\dot{\tilde{x}} = \tilde{A}\tilde{x} + \tilde{B}u + \sum_j u_j \tilde{N}_j \tilde{x} + \tilde{H}(\tilde{x} \otimes \tilde{x}),\qquad \tilde{y} = \tilde{c}^T \tilde{x},$$ where $\tilde{A}$, $\tilde{B}$, $\tilde{N}_j$, and $\tilde{H}$ encode the dynamics and output mappings, with only one nonzero row in $\tilde{H}$ corresponding to the quadratic output structure. This lifted system, however, is generally not stable and thus requires stabilization (e.g., shifting $\tilde{A}$ by $-\varepsilon$ on the new auxiliary state) before balanced truncation can be applied (Pulch et al., 2017).

For general nonlinear ODEs,

$\tilde{x} = [x^T, y]^T$0

lifting is achieved by introducing auxiliary variables representing monomials or nonlinear functions in $\tilde{x} = [x^T, y]^T$1, rewriting the dynamics in terms of the original and auxiliary variables, and differentiating to obtain a full quadratic-bilinear system of the form

$\tilde{x} = [x^T, y]^T$2

where the block structure ensures that the original coordinates have linear self-dynamics, and auxiliary states are slaved to the original via quadratic or bilinear coupling (Kramer et al., 2019).

For bilinear systems with quadratic (and linear) outputs,

$\tilde{x} = [x^T, y]^T$3

lifting uses the second-moment state, $\tilde{x} = [x^T, y]^T$4, to obtain a lifted system in $\tilde{x} = [x^T, y]^T$5 with a linear output in the lifted variables, facilitating direct application of bilinear balanced truncation techniques (Faßbender et al., 4 Jul 2025).

3. Gramians and Lyapunov Equations for Lifted Systems

Balanced truncation for linear systems relies on standard (linear) Lyapunov equations. For lifted quadratic-bilinear or bilinear systems, computable Gramians satisfy generalized Lyapunov equations incorporating cubic, quadratic, or Volterra terms.

• In the quadratic-bilinear lifted case, the Gramians $\tilde{x} = [x^T, y]^T$6 obey coupled quadratic Lyapunov equations: $\tilde{x} = [x^T, y]^T$7

$\tilde{x} = [x^T, y]^T$8

However, exploiting the problem's structure, these reduce to just two linear Lyapunov equations for the main blocks, specifically for $\tilde{x} = [x^T, y]^T$9 and $$\dot{\tilde{x}} = \tilde{A}\tilde{x} + \tilde{B}u + \sum_j u_j \tilde{N}_j \tilde{x} + \tilde{H}(\tilde{x} \otimes \tilde{x}),\qquad \tilde{y} = \tilde{c}^T \tilde{x},$$0 as follows: $$\dot{\tilde{x}} = \tilde{A}\tilde{x} + \tilde{B}u + \sum_j u_j \tilde{N}_j \tilde{x} + \tilde{H}(\tilde{x} \otimes \tilde{x}),\qquad \tilde{y} = \tilde{c}^T \tilde{x},$$1

$$\dot{\tilde{x}} = \tilde{A}\tilde{x} + \tilde{B}u + \sum_j u_j \tilde{N}_j \tilde{x} + \tilde{H}(\tilde{x} \otimes \tilde{x}),\qquad \tilde{y} = \tilde{c}^T \tilde{x},$$2

with explicit algebraic formulas for the scalar auxiliary parts (Pulch et al., 2017).

• For general lifted bilinear systems, reachability and observability Gramians $$\dot{\tilde{x}} = \tilde{A}\tilde{x} + \tilde{B}u + \sum_j u_j \tilde{N}_j \tilde{x} + \tilde{H}(\tilde{x} \otimes \tilde{x}),\qquad \tilde{y} = \tilde{c}^T \tilde{x},$$3 are defined via Volterra series and satisfy coupled generalized Lyapunov equations: $$\dot{\tilde{x}} = \tilde{A}\tilde{x} + \tilde{B}u + \sum_j u_j \tilde{N}_j \tilde{x} + \tilde{H}(\tilde{x} \otimes \tilde{x}),\qquad \tilde{y} = \tilde{c}^T \tilde{x},$$4 and its adjoint for $$\dot{\tilde{x}} = \tilde{A}\tilde{x} + \tilde{B}u + \sum_j u_j \tilde{N}_j \tilde{x} + \tilde{H}(\tilde{x} \otimes \tilde{x}),\qquad \tilde{y} = \tilde{c}^T \tilde{x},$$5 (Faßbender et al., 4 Jul 2025). Truncated Gramians can be computed by approximating the series using the leading terms, significantly reducing computational cost.
• For lifted QB systems with zero eigenvalues (from auxiliary variables), Lyapunov equations are singular. Artificial stabilization (shifting the spectrum of the auxiliary block) is introduced to ensure Gramians exist and decay rates are favorable for low-rank projection (Kramer et al., 2019).

4. Balanced Truncation Procedure for Lifted Systems

The reduction procedure consists of the following steps:

1. Gramian Computation: Solve linear or generalized Lyapunov equations for the reachability ($$\dot{\tilde{x}} = \tilde{A}\tilde{x} + \tilde{B}u + \sum_j u_j \tilde{N}_j \tilde{x} + \tilde{H}(\tilde{x} \otimes \tilde{x}),\qquad \tilde{y} = \tilde{c}^T \tilde{x},$$6) and observability ($$\dot{\tilde{x}} = \tilde{A}\tilde{x} + \tilde{B}u + \sum_j u_j \tilde{N}_j \tilde{x} + \tilde{H}(\tilde{x} \otimes \tilde{x}),\qquad \tilde{y} = \tilde{c}^T \tilde{x},$$7) Gramians of the lifted system, employing low-rank or iterative solvers (e.g., ADI, MESS toolbox) where possible. For QB and bilinear systems, truncated Gramians using only first- or second-order terms can drastically reduce computational demand (Pulch et al., 2017, Faßbender et al., 4 Jul 2025).
2. Balancing Transformation: Factor Gramians as $$\dot{\tilde{x}} = \tilde{A}\tilde{x} + \tilde{B}u + \sum_j u_j \tilde{N}_j \tilde{x} + \tilde{H}(\tilde{x} \otimes \tilde{x}),\qquad \tilde{y} = \tilde{c}^T \tilde{x},$$8, $$\dot{\tilde{x}} = \tilde{A}\tilde{x} + \tilde{B}u + \sum_j u_j \tilde{N}_j \tilde{x} + \tilde{H}(\tilde{x} \otimes \tilde{x}),\qquad \tilde{y} = \tilde{c}^T \tilde{x},$$9, compute SVD $\tilde{A}$0, and form balancing/projecting matrices. In lifted QB systems, the SVD structure captures the original system's Gramian singular values plus an auxiliary term.
3. Projection and Truncation: Restrict the lifted system to the $\tilde{A}$1 dominant balanced modes to obtain a reduced order quadratic-bilinear or bilinear system.
4. Recovery: The resulting reduced system, typically of order much smaller than the full lifted dimension, preserves key input-output behavior with strong error bounds in the linear regime ($\tilde{A}$2) or energy-based local bounds in the QB case.
5. Stabilization and Structure Preservation: Any artificial stabilization introduced during lifting is removed or minimized in the final model. Block-structured projections can retain physical or algebraic structure of the original model (Pulch et al., 2017, Kramer et al., 2019).

5. Numerical Performance and Applications

Empirical studies highlight several advantages and tradeoffs:

• Lifting quadratic-output or nonlinear systems to QB/bilinear form allows balanced truncation with controllable computational cost and error.
• Direct (unlifted) approaches for linear systems with quadratic outputs suffer from a “many-output penalty”: the number of outputs in the reduced-order model (ROM) can be prohibitively large. Lifting circumvents this by encoding all quadratic outputs into a single linear output for the lifted auxiliary state (Pulch et al., 2017).
• In benchmark problems (e.g., 5000-dimensional linear systems, stochastic-Galerkin mass-spring-damper, tubular-reactor PDEs), the lifted approach matches or surpasses the direct multi-output approach in accuracy while drastically reducing CPU time (speed-ups of 6x–90x depending on the chosen Lyapunov solver and lift) (Pulch et al., 2017, Kramer et al., 2019, Faßbender et al., 4 Jul 2025).
• The stabilization parameter or artificial damping introduced for auxiliary states does not noticeably affect ROM accuracy within practical ranges ($\tilde{A}$3 relative error for $\tilde{A}$4 over several orders of magnitude) (Pulch et al., 2017).
• Consistent findings across linear, bilinear, and nonlinear test cases indicate that truncated Gramians provide near-identical accuracy to full nonlinear Gramians at a fraction of the cost, validating their use for large-scale systems (Faßbender et al., 4 Jul 2025).

In comparisons with alternative data-driven MOR approaches such as POD-DEIM, balanced truncation of lifted systems demonstrates superior robustness to noise and mismatch between training data and test scenarios. Unlike snapshot-based methods, the balanced truncation approach for lifted systems yields a priori error control and captures input-driven dynamics even in the absence of representative training data (Kramer et al., 2019).

6. Connections and Recent Developments

The lifting–balanced truncation paradigm has catalyzed advances in high-fidelity MOR for systems with polynomial or quadratic outputs and for classes of weakly nonlinear PDEs. Multi-stage lifting (introducing several layers of auxiliary variables) extends applicability to higher-order polynomial and even certain transcendental nonlinearities. The Kronecker product and tensor-matricization structures inherent in the lifted system dynamics facilitate efficient exploitation of sparsity and symmetry in both Gramian computation and numerical solution of generalized Lyapunov equations (Pulch et al., 2017, Kramer et al., 2019, Faßbender et al., 4 Jul 2025).

Recent contributions include primal-dual frameworks for formulating Gramians of bilinear–quadratic-output systems, rigorous existence and uniqueness guarantees for the corresponding Gramians under small-gain conditions, and systematic construction of computationally efficient truncated Gramians via Volterra expansions (Faßbender et al., 4 Jul 2025).

A plausible implication is that further algorithmic improvements in low-rank and tensor-structured Lyapunov solvers, as well as extensions to even broader classes of nonlinear systems, can be expected to synergize with the lifting–balanced truncation methodology, widening its applicability in control, estimation, and simulation of high-dimensional complex systems.