Shifted Passivity in Control Systems

• Shifted passivity is a dissipation property that measures energy balance incrementally relative to a nonzero operating point or reference trajectory.
• It employs shifted storage functions—such as Bregman-type Hamiltonian shifts or quadratic error energies—to properly reflect nonzero steady states.
• The concept is pivotal in analyzing forced equilibria, output regulation, and digital control, with applications in power electronics and networked energy systems.

Shifted passivity is a dissipation property formulated with respect to a nonzero operating point, or more generally a reference trajectory, rather than the origin. For a system with equilibrium $(\bar x,\bar u,\bar y)$, it replaces the standard supply rate $u^\top y$ by the incremental supply $(u-\bar u)^\top (y-\bar y)$ and uses a shifted storage that vanishes at the reference state. In port-Hamiltonian settings this storage is typically the Bregman-type shift of the Hamiltonian, while in quadratic-energy models it reduces to a quadratic error energy. The concept is used to analyze forced equilibria, output regulation, and interconnections when desired steady states are nonzero, and it has been extended to discrete-time systems, time-varying references, periodic motions, and passivizing input-output transformations (Monshizadeh et al., 2017, Kawano et al., 2019, Kawano et al., 2022, Jiang, 28 Apr 2026).

1. Definition and conceptual scope

For the port-Hamiltonian system

$$\dot x = \bigl(J(x)-R(x)\bigr)\nabla H(x)+Gu,\qquad y = G^\top \nabla H(x),$$

shifted passivity about an equilibrium $(\bar x,\bar u,\bar y)$ means that there exists a nonnegative shifted storage

$$S(x)=H(x)-(x-\bar x)^\top \nabla H(\bar x)-H(\bar x)$$

such that

$\dot S(x)\le (u-\bar u)^\top (y-\bar y).$

This formulation explicitly measures energy balance relative to the forced equilibrium and not relative to the origin (Monshizadeh et al., 2017).

For a general nonlinear system $\dot x=f(x,u)$, $y=h(x)$ with equilibrium $(x^*,u^*)$, shifted passivity is similarly defined by the existence of a $u^\top y$0 storage $u^\top y$1 with $u^\top y$2 such that

$u^\top y$3

In this sense, shifted passivity is standard passivity of the shifted system with input $u^\top y$4 and output $u^\top y$5 (Kawano et al., 2019).

The same idea extends beyond constant equilibria. Along a bounded reference trajectory $u^\top y$6 with output $u^\top y$7, strict shifted passivity requires a storage $u^\top y$8 and a positive-definite function $u^\top y$9 such that

$(u-\bar u)^\top (y-\bar y)$0

where $(u-\bar u)^\top (y-\bar y)$1, $(u-\bar u)^\top (y-\bar y)$2, and $(u-\bar u)^\top (y-\bar y)$3. This formulation is used when the consensus or regulation target is disturbance-dependent and time-varying (Kawano et al., 2022).

A further generalization appears in trajectory-based settings. For the single-machine infinite-bus system, shifted passivity is formulated with respect to a periodic synchronous motion rather than an equilibrium. There the storage depends on the current state and the periodic reference and yields a local inequality of the form

$(u-\bar u)^\top (y-\bar y)$4

thus preserving the periodic structure of the rotor angle rather than reducing the problem to an equilibrium in a rotating frame (Jiang, 28 Apr 2026).

This suggests that the core invariant across the literature is not the particular coordinate representation but the replacement of absolute energy balance by incremental energy balance around a chosen operating regime.

2. Shifted storage functions and structural conditions

The shifted storage used in many Hamiltonian-like models is the second-order Taylor remainder of the energy around the reference state. For an energy function $(u-\bar u)^\top (y-\bar y)$5, the incremental storage is

$(u-\bar u)^\top (y-\bar y)$6

When $(u-\bar u)^\top (y-\bar y)$7 with $(u-\bar u)^\top (y-\bar y)$8, this becomes

$(u-\bar u)^\top (y-\bar y)$9

The same construction appears in discrete-time converter models, in continuous-time port-Hamiltonian analysis, and in output-consensus problems, where it is explicitly described as a Bregman-type shift of the Hamiltonian (Moreschini et al., 26 Aug 2025, Monshizadeh et al., 2017, Kawano et al., 2022).

For general port-Hamiltonian systems with strictly convex Hamiltonian, sufficient conditions for shifted passivity can be stated in co-energy variables. Writing $$\dot x = \bigl(J(x)-R(x)\bigr)\nabla H(x)+Gu,\qquad y = G^\top \nabla H(x),$$0, $$\dot x = \bigl(J(x)-R(x)\bigr)\nabla H(x)+Gu,\qquad y = G^\top \nabla H(x),$$1, $$\dot x = \bigl(J(x)-R(x)\bigr)\nabla H(x)+Gu,\qquad y = G^\top \nabla H(x),$$2, and $$\dot x = \bigl(J(x)-R(x)\bigr)\nabla H(x)+Gu,\qquad y = G^\top \nabla H(x),$$3, shifted passivity follows under the monotonicity condition

$$\dot x = \bigl(J(x)-R(x)\bigr)\nabla H(x)+Gu,\qquad y = G^\top \nabla H(x),$$4

with $$\dot x = \bigl(J(x)-R(x)\bigr)\nabla H(x)+Gu,\qquad y = G^\top \nabla H(x),$$5 and $$\dot x = \bigl(J(x)-R(x)\bigr)\nabla H(x)+Gu,\qquad y = G^\top \nabla H(x),$$6 (Monshizadeh et al., 2017).

In the important quadratic-affine case

$$\dot x = \bigl(J(x)-R(x)\bigr)\nabla H(x)+Gu,\qquad y = G^\top \nabla H(x),$$7

with $$\dot x = \bigl(J(x)-R(x)\bigr)\nabla H(x)+Gu,\qquad y = G^\top \nabla H(x),$$8, $$\dot x = \bigl(J(x)-R(x)\bigr)\nabla H(x)+Gu,\qquad y = G^\top \nabla H(x),$$9, and $(\bar x,\bar u,\bar y)$0, the general monotonicity condition reduces to the constant LMI

$(\bar x,\bar u,\bar y)$1

This reduction is significant because it converts a state-dependent condition into a numerical matrix test (Monshizadeh et al., 2017).

Specific nonlinear models yield explicit storage functions and parameter conditions. For the nonlinear synchronous machine in the $(\bar x,\bar u,\bar y)$2-frame, the storage is the incremental kinetic-plus-magnetic energy

$(\bar x,\bar u,\bar y)$3

and shifted passivity with supply $(\bar x,\bar u,\bar y)$4 is obtained under the Schur-complement condition

$(\bar x,\bar u,\bar y)$5

Equivalently,

$(\bar x,\bar u,\bar y)$6

with strict inequality yielding strict shifted passivity (Khodabakhshloo et al., 6 May 2026).

For district heating systems, shifted passivity is established separately for hydraulic and thermal subsystems. The hydraulic storage

$(\bar x,\bar u,\bar y)$7

relies on monotonicity of the friction map $(\bar x,\bar u,\bar y)$8, while the thermal storage

$(\bar x,\bar u,\bar y)$9

is identified with the shifted total-ectropy of the network and uses the negative semi-definiteness of the Kirchhoff-convection matrix $$S(x)=H(x)-(x-\bar x)^\top \nabla H(\bar x)-H(\bar x)$$0 (Machado et al., 2020).

3. Discrete-time shifted passivity

A central complication in digital control is that passivity is generally not preserved by standard discretization. For a continuous-time passive system, a naive forward-Euler discretization of the dynamics and of $$S(x)=H(x)-(x-\bar x)^\top \nabla H(\bar x)-H(\bar x)$$1 typically does not yield a discrete-time inequality of the form

$$S(x)=H(x)-(x-\bar x)^\top \nabla H(\bar x)-H(\bar x)$$2

even when the sampling period $$S(x)=H(x)-(x-\bar x)^\top \nabla H(\bar x)-H(\bar x)$$3 is very small. This directly obstructs practical implementations of passivity-based control built from continuous-time designs (Moreschini et al., 26 Aug 2025).

In discrete time, shifted passivity of

$$S(x)=H(x)-(x-\bar x)^\top \nabla H(\bar x)-H(\bar x)$$4

around $$S(x)=H(x)-(x-\bar x)^\top \nabla H(\bar x)-H(\bar x)$$5 requires a nonnegative storage $$S(x)=H(x)-(x-\bar x)^\top \nabla H(\bar x)-H(\bar x)$$6 with $$S(x)=H(x)-(x-\bar x)^\top \nabla H(\bar x)-H(\bar x)$$7 such that

$$S(x)=H(x)-(x-\bar x)^\top \nabla H(\bar x)-H(\bar x)$$8

For quadratic storage, $$S(x)=H(x)-(x-\bar x)^\top \nabla H(\bar x)-H(\bar x)$$9 (Moreschini et al., 26 Aug 2025).

For averaged port-Hamiltonian power-converter models,

$\dot S(x)\le (u-\bar u)^\top (y-\bar y).$0

with $\dot S(x)\le (u-\bar u)^\top (y-\bar y).$1, $\dot S(x)\le (u-\bar u)^\top (y-\bar y).$2, and $\dot S(x)\le (u-\bar u)^\top (y-\bar y).$3, the implicit midpoint rule is used:

$\dot S(x)\le (u-\bar u)^\top (y-\bar y).$4

Because the vector field is evaluated at the midpoint, the method preserves the underlying symplectic, energy, and dissipation structure to second order in $\dot S(x)\le (u-\bar u)^\top (y-\bar y).$5, allowing discrete-time energy-balance arguments that fail for Euler or typical explicit Runge-Kutta schemes (Moreschini et al., 26 Aug 2025).

The resulting discrete shifted output is not simply the continuous-time passive output sampled at $\dot S(x)\le (u-\bar u)^\top (y-\bar y).$6. Instead it is constructed as

$\dot S(x)\le (u-\bar u)^\top (y-\bar y).$7

With storage $\dot S(x)\le (u-\bar u)^\top (y-\bar y).$8, the increment satisfies

$\dot S(x)\le (u-\bar u)^\top (y-\bar y).$9

hence

$\dot x=f(x,u)$0

This is a constructive discrete-time shifted-passivity certificate for the incremental model of the converter (Moreschini et al., 26 Aug 2025).

The same work shows that the midpoint-discretized PID controller

$\dot x=f(x,u)$1

$\dot x=f(x,u)$2

is output strictly passive with storage

$\dot x=f(x,u)$3

Interconnecting the shifted-passive converter and the shifted-passive PID yields a combined Lyapunov function

$\dot x=f(x,u)$4

whose variation is

$\dot x=f(x,u)$5

If

$\dot x=f(x,u)$6

then global asymptotic convergence follows, and no small-$\dot x=f(x,u)$7 assumption appears in the stability proof (Moreschini et al., 26 Aug 2025).

A common misconception is that sufficiently fast sampling preserves passivity “for practical purposes.” The discrete-time converter result states the opposite in precise terms: passivity is not in general preserved by standard discretizations, even for very small $\dot x=f(x,u)$8.

4. Passivity indices, passive-short systems, and input-output transformations

A second line of work treats shifted passivity through passivity indices and input-output transformations. For a system with equal input and output dimension, I/O $\dot x=f(x,u)$9-passivity means that there exists a storage $y=h(x)$0 such that

$y=h(x)$1

with $y=h(x)$2. Here $y=h(x)$3 correspond to strict passivity, while $y=h(x)$4 represent passive-short systems (Sharf et al., 2019).

For SISO systems, the supply rate

$y=h(x)$5

defines a symmetric double cone

$y=h(x)$6

An invertible linear transformation

$y=h(x)$7

passivizes the system precisely when it maps the original cone into a target cone $y=h(x)$8. The classification theorem states that all such transformations are of the form

$y=h(x)$9

where $(x^*,u^*)$0 and $(x^*,u^*)$1 is invertible with nonnegative entries (Sharf et al., 2019).

For MIMO systems, the same question leads to an S-lemma and LMI characterization. Writing the quadratic form as $(x^*,u^*)$2, the condition that $(x^*,u^*)$3 map $(x^*,u^*)$4 into $(x^*,u^*)$5 is equivalent to

$(x^*,u^*)$6

or, in the factorized form,

$(x^*,u^*)$7

This gives a complete parameterization of passivizing I/O maps (Sharf et al., 2019).

A related equilibrium-independent viewpoint appears in the theory of equilibrium-independent passive-short systems. There the shifted dissipation inequality is written with respect to every forced equilibrium:

$(x^*,u^*)$8

Evaluating this inequality at two distinct equilibria yields the projective quadratic inequality

$(x^*,u^*)$9

whose solution set is again a symmetric double cone (Sharf et al., 2019).

The geometric method then constructs an invertible matrix $u^\top y$00 that maps this cone to the monotonicity cone $u^\top y$01. The same $u^\top y$02 both monotonizes the steady-state input-output relation and passivizes the dynamics. Any such $u^\top y$03 can be realized as a composition of output-feedback, post-gain, input-feedthrough, and pre-gain factors (Sharf et al., 2019).

This suggests that one strand of the literature reserves “shifted passivity” for incremental dissipation around nonzero equilibria, whereas another uses geometric shifting of the supply-rate cone under I/O transformations. The two views are not contradictory: both are organized around modifying the supply rate so that passivity becomes compatible with the relevant operating regime.

5. Stability and controller synthesis

Shifted passivity is primarily a stability tool for forced equilibria and interconnections. For port-Hamiltonian systems, setting $u^\top y$04 in the shifted dissipation inequality gives $u^\top y$05, so the shifted storage acts as a Lyapunov function. If a stronger inequality

$u^\top y$06

holds locally, then $u^\top y$07 and LaSalle’s argument yields local asymptotic stability; if $u^\top y$08 is globally strongly convex and the condition holds globally, the same argument yields global asymptotic stability of the forced equilibrium (Monshizadeh et al., 2017).

When the system is not shifted passive but satisfies the relaxed estimate

$u^\top y$09

a proportional output feedback

$u^\top y$10

renders the closed loop shifted-passive from $u^\top y$11 to $u^\top y$12. This is an explicit passivity-enforcement mechanism (Monshizadeh et al., 2017).

Dynamic output feedback also follows directly from shifted passivity. For a shifted-passive system with output $u^\top y$13, the controller

$u^\top y$14

with $u^\top y$15 and $u^\top y$16, yields the closed-loop storage

$u^\top y$17

satisfying

$u^\top y$18

With $u^\top y$19, LaSalle’s invariance principle gives convergence to $u^\top y$20 and, under detectability, asymptotic stability of $u^\top y$21 (Kawano et al., 2019).

Interconnection results are equally central. For the nonlinear synchronous machine, once shifted passivity from current input to voltage output is established, passive droop control preserves the same supply rate. A droop-PI torque controller with storage

$u^\top y$22

satisfies $u^\top y$23, and the sum $u^\top y$24 is a Lyapunov function that guarantees asymptotic stability of the shifted equilibrium by the standard passivity-theorem argument (Khodabakhshloo et al., 6 May 2026).

In distributed output-consensus problems, strict shifted passivity along a disturbance-dependent reference allows the controller

$u^\top y$25

to drive the weighted outputs to consensus:

$u^\top y$26

If $u^\top y$27 is constant, exact output consensus follows (Kawano et al., 2022).

For periodic references the stability claim is necessarily local. In the single-machine infinite-bus problem, a shifted storage made of an error Hamiltonian plus a correction term produces

$u^\top y$28

Local shifted passivity holds when $u^\top y$29, and sufficient conditions are given in terms of damping, inertia, field excitation magnitude, resistance, torque, and steady-state current. Under an additional bound, a sublevel set of $u^\top y$30 yields a region-of-attraction estimate (Jiang, 28 Apr 2026).

6. Representative application domains, assumptions, and limitations

Power electronics provide one of the clearest demonstrations of the method. In power-converter control, the main issue is that the reference output is nonzero, so the relevant property is passivity of the incremental model, currently known as shifted passivity. The discrete-time converter result further shows that global stability can be retained under digital implementation if both the plant output and the PID discretization are redesigned to preserve passivity, rather than obtained by direct discretization of a continuous-time controller (Moreschini et al., 26 Aug 2025).

Electric-machine models show a broader range of shifted-passivity formulations. In the nonlinear $u^\top y$31-frame synchronous machine, the shifted passivity property is equilibrium-based and supports passive droop interconnection (Khodabakhshloo et al., 6 May 2026). In the stationary-frame single-machine infinite-bus model, the reference is a periodic synchronous steady state, the storage preserves the periodic geometry of the rotor angle, and the stability guarantee is local rather than global because multiple synchronous solutions exist (Jiang, 28 Apr 2026).

Networked energy systems provide further examples. In islanded DC power networks, shifted passivity around a disturbance-dependent equilibrium underlies output-consensus control for current sharing (Kawano et al., 2022). In district heating systems, shifted passivity is proved separately for hydraulic and thermal subsystems under structural assumptions such as constant density and specific heat, one-dimensional cylindrical pipes, neglected gravitational effects, no flow reversal, and time-scale separation between fast hydraulics and slow thermal dynamics (Machado et al., 2020).

Several limitations recur across the literature. First, shifted passivity is not automatic for nonzero equilibria; explicit monotonicity, convexity, or matrix-inequality conditions are required (Monshizadeh et al., 2017, Khodabakhshloo et al., 6 May 2026). Second, shifted passivity may fail under naive discretization, so continuous-time proofs do not directly transfer to sampled-data implementations (Moreschini et al., 26 Aug 2025). Third, when the reference is periodic or the state lives on a manifold with angular periodicity, only local attractivity can generally be expected (Jiang, 28 Apr 2026). Fourth, in two-time-scale models such as district heating, the passivity proof may depend on freezing the fast subsystem at equilibrium (Machado et al., 2020).

A common misconception is that shifted passivity is merely a cosmetic rewrite of standard passivity. The literature indicates otherwise. The storage is changed, the supply rate is changed, the admissible equilibria are changed, and in several cases the passive output itself must be reconstructed, as in the midpoint-discretized converter model. A plausible implication is that shifted passivity should be regarded not as a trivial coordinate translation but as a structural reformulation of dissipation suited to nonzero operating regimes.