Deterministic Riccati Observer

• Deterministic Riccati Observer is a design framework that leverages Riccati differential or algebraic equations to generate innovation gains for deterministic state estimation.
• It applies to both linear (e.g., Kalman–Bucy) and nonlinear systems (e.g., Mortensen type), enabling effective stabilization and sensitivity analysis through structured gain synthesis.
• Practical implementations span navigation, visual SLAM, and unknown-input systems where Riccati-based formulations decouple uncertainties and guarantee exponential convergence under observability conditions.

A deterministic Riccati observer is an observer whose correction law is generated by a Riccati differential or algebraic equation in a deterministic estimation problem. In the classical linear setting, this is the continuous-time Kalman–Bucy covariance equation

$\dot P_t = A P_t + P_t A^\top + R - P_t S P_t,$

with observer gain $K_t=P_tB^\top R_V^{-1}$ when $S=B^\top R_V^{-1}B$ (Bishop et al., 2017). In deterministic minimum-energy formulations, the Riccati variable may instead be an information or certainty matrix, or the Hessian of a value function, and the observer gain is obtained from its inverse rather than from a stochastic covariance interpretation (Bamieh, 10 Jun 2026, Breiten et al., 2023). More generally, the term also covers nonlinear and geometric observers whose exact or induced linear time-varying error dynamics admit gain synthesis through a continuous Riccati equation (Hamel et al., 2016, Bouazza et al., 28 Aug 2025, Boughellaba et al., 12 Jan 2026).

1. Conceptual forms of the deterministic Riccati observer

In deterministic observer theory, the Riccati equation appears in several closely related roles. One standard form is the continuous Riccati equation for an LTV observer,

$\dot{P}=A(t)P+PA^\top(t)-PC^\top(t)Q(t)C(t)P+V(t),$

with gain

$K(t)=k(t)P(t)C^\top(t)Q(t), \qquad k(t)\ge 0.5,$

used in the observer

$$\dot{\hat{x}}=A(t)\hat{x}+B(t)u+K(t)(y-C(t)\hat{x}).$$

The associated error system is

$\dot{\tilde{x}}=(A(t)-K(t)C(t))\tilde{x},$

and the usual Lyapunov function is $\mathcal V(t)=\tilde x^\top P^{-1}(t)\tilde x$ (Hamel et al., 2016).

A second deterministic form arises from minimum-energy estimation. For the linear system

$\dot x(t)=A\,x(t)+w(t),\qquad y(t)=C\,x(t)+v(t),$

the deterministic estimation problem is to minimize

$$J(v,w,x_0)=\int_0^T \bigl(v^*Vv+w^*Ww\bigr)\,dt+x_0^*Nx_0.$$

Using homogeneous coordinates and an LQR-with-final-conditions construction, the resulting matrix $K_t=P_tB^\top R_V^{-1}$0 satisfies

$K_t=P_tB^\top R_V^{-1}$1

and the observer becomes

$K_t=P_tB^\top R_V^{-1}$2

Here $K_t=P_tB^\top R_V^{-1}$3 is interpreted as the curvature of the cost-to-arrive, and the estimate is $K_t=P_tB^\top R_V^{-1}$4 after partitioning the augmented value function (Bamieh, 10 Jun 2026).

A third form appears in deterministic nonlinear minimum-energy observers of Mortensen type. There the Riccati object is not introduced as a covariance or information matrix a priori, but as the Hessian of the value function. For the shifted nonlinear problem in (Breiten et al., 2023), the observer gain is

$K_t=P_tB^\top R_V^{-1}$5

and the nominal Hessian $K_t=P_tB^\top R_V^{-1}$6 satisfies a differential Riccati equation. This suggests that “deterministic Riccati observer” is best read as a structural designation: the observer gain is generated by Riccati-type curvature dynamics, regardless of whether the underlying interpretation is covariance propagation, certainty accumulation, or value-function sensitivity (Breiten et al., 2023).

2. Classical continuous-time Riccati observers and Kalman–Bucy structure

In continuous-time linear-Gaussian filtering, the deterministic Riccati equation is the covariance law of the Kalman–Bucy observer. For the signal-observation model

$K_t=P_tB^\top R_V^{-1}$7

the covariance satisfies

$K_t=P_tB^\top R_V^{-1}$8

and the gain is

$K_t=P_tB^\top R_V^{-1}$9

In observer language, $S=B^\top R_V^{-1}B$0 governs gain adaptation in the optimal linear observer (Bishop et al., 2017).

The same paper treats the deterministic Riccati flow $S=B^\top R_V^{-1}B$1 as a smooth nonlinear semigroup on positive-definite matrices. Its linearization along the Riccati trajectory is encoded by

$S=B^\top R_V^{-1}B$2

and the first Fréchet derivative is

$S=B^\top R_V^{-1}B$3

These formulas make the deterministic Riccati observer a sensitivity object as well as a gain law: perturbations of the initial covariance are propagated exactly through the semigroup generated by $S=B^\top R_V^{-1}B$4 (Bishop et al., 2017).

Under controllability/observability assumptions summarized in (Bishop et al., 2017), there exists a unique positive definite fixed point $S=B^\top R_V^{-1}B$5 solving the algebraic Riccati equation

$S=B^\top R_V^{-1}B$6

and the closed-loop matrix $S=B^\top R_V^{-1}B$7 is asymptotically stable even when $S=B^\top R_V^{-1}B$8 itself is unstable. The Riccati flow is monotone, positive, and bounded in the sense recorded there: $S=B^\top R_V^{-1}B$9 with explicit bounds on $\dot{P}=A(t)P+PA^\top(t)-PC^\top(t)Q(t)C(t)P+V(t),$0 and $\dot{P}=A(t)P+PA^\top(t)-PC^\top(t)Q(t)C(t)P+V(t),$1 (Bishop et al., 2017). In this classical setting, the deterministic Riccati observer is therefore both a covariance propagator and a stabilizing mechanism for the observer error dynamics.

3. Deterministic observer synthesis for LTV and unknown-input systems

For linear time-varying systems with unknown inputs,

$\dot{P}=A(t)P+PA^\top(t)-PC^\top(t)Q(t)C(t)P+V(t),$2

the observer problem is not directly a Kalman–Bucy problem because the error inherits the unknown forcing term $\dot{P}=A(t)P+PA^\top(t)-PC^\top(t)Q(t)C(t)P+V(t),$3. The main result in (Tranninger et al., 2021) is that, under assumptions (A1)–(A3), uniform strong$\dot{P}=A(t)P+PA^\top(t)-PC^\top(t)Q(t)C(t)P+V(t),$4 detectability of $\dot{P}=A(t)P+PA^\top(t)-PC^\top(t)Q(t)C(t)P+V(t),$5 is equivalent to uniform exponential detectability of the auxiliary pair

$\dot{P}=A(t)P+PA^\top(t)-PC^\top(t)Q(t)C(t)P+V(t),$6

where $\dot{P}=A(t)P+PA^\top(t)-PC^\top(t)Q(t)C(t)P+V(t),$7. This reduces unknown-input observer design to a deterministic Kalman-filter-type problem (Tranninger et al., 2021).

The Riccati equation for the auxiliary system is

$\dot{P}=A(t)P+PA^\top(t)-PC^\top(t)Q(t)C(t)P+V(t),$8

with gain components

$\dot{P}=A(t)P+PA^\top(t)-PC^\top(t)Q(t)C(t)P+V(t),$9

From these, one reconstructs the unknown-input observer

$K(t)=k(t)P(t)C^\top(t)Q(t), \qquad k(t)\ge 0.5,$0

whose error satisfies the uniform exponential bound

$K(t)=k(t)P(t)C^\top(t)Q(t), \qquad k(t)\ge 0.5,$1

The deterministic Riccati observer here is not a stochastic filter; it is a constructive decoupling-and-stabilization device (Tranninger et al., 2021).

A related deterministic synthesis appears in position and bias estimation from direction or range measurements. There the nonlinear estimation problems are converted into exact linear or LTV output systems—sometimes after augmenting the state with constant velocity bias and auxiliary monomials—and then observed with a Riccati gain $K(t)=k(t)P(t)C^\top(t)Q(t), \qquad k(t)\ge 0.5,$2 (Hamel et al., 2016). The paper derives global exponential convergence under persistent-excitation conditions stated directly in terms of the measured directions, source geometry, measured velocity, or $K(t)=k(t)P(t)C^\top(t)Q(t), \qquad k(t)\ge 0.5,$3, depending on the sensing configuration. A characteristic example is the single-direction model

$K(t)=k(t)P(t)C^\top(t)Q(t), \qquad k(t)\ge 0.5,$4

with observer

$K(t)=k(t)P(t)C^\top(t)Q(t), \qquad k(t)\ge 0.5,$5

and convergence guaranteed if

$K(t)=k(t)P(t)C^\top(t)Q(t), \qquad k(t)\ge 0.5,$6

This is a deterministic Riccati observer in its most explicit LTV form (Hamel et al., 2016).

4. Minimum-energy and Mortensen formulations

The Mortensen observer is a deterministic minimum-energy observer rather than a stochastic filter. In (Breiten et al., 2023), the nonlinear disturbed system

$K(t)=k(t)P(t)C^\top(t)Q(t), \qquad k(t)\ge 0.5,$7

is associated with the value function

$K(t)=k(t)P(t)C^\top(t)Q(t), \qquad k(t)\ge 0.5,$8

The observer trajectory is defined pointwise by

$K(t)=k(t)P(t)C^\top(t)Q(t), \qquad k(t)\ge 0.5,$9

and the pointwise optimality condition

$$\dot{\hat{x}}=A(t)\hat{x}+B(t)u+K(t)(y-C(t)\hat{x}).$$0

yields the observer equation after differentiation and use of the HJB equation (Breiten et al., 2023).

At the nominal point $$\dot{\hat{x}}=A(t)\hat{x}+B(t)u+K(t)(y-C(t)\hat{x}).$$1, the Hessian

$$\dot{\hat{x}}=A(t)\hat{x}+B(t)u+K(t)(y-C(t)\hat{x}).$$2

satisfies

$$\dot{\hat{x}}=A(t)\hat{x}+B(t)u+K(t)(y-C(t)\hat{x}).$$3

where

$$\dot{\hat{x}}=A(t)\hat{x}+B(t)u+K(t)(y-C(t)\hat{x}).$$4

This Riccati equation is not auxiliary: it is the nominal evolution of the second derivative of the deterministic value function (Breiten et al., 2023).

The resulting observer is

$$\dot{\hat{x}}=A(t)\hat{x}+B(t)u+K(t)(y-C(t)\hat{x}).$$5

with

$$\dot{\hat{x}}=A(t)\hat{x}+B(t)u+K(t)(y-C(t)\hat{x}).$$6

The main local result is that the Hessian is positive definite in a neighborhood of the nominal trajectory, the pointwise minimizing terminal state exists uniquely, and $$\dot{\hat{x}}=A(t)\hat{x}+B(t)u+K(t)(y-C(t)\hat{x}).$$7 is a weak solution of the observer equation for sufficiently small $$\dot{\hat{x}}=A(t)\hat{x}+B(t)u+K(t)(y-C(t)\hat{x}).$$8 (Breiten et al., 2023). In this framework, the deterministic Riccati observer is an observer whose innovation gain is the inverse Hessian of a minimum-energy value function.

The linear minimum-energy case treated in (Bamieh, 10 Jun 2026) is the finite-dimensional analogue of the same philosophy. There the deterministic Kalman filter is obtained from a minimum-energy problem by homogeneous-coordinate embedding and LQR with final conditions, producing the cost-to-arrive

$$\dot{\hat{x}}=A(t)\hat{x}+B(t)u+K(t)(y-C(t)\hat{x}).$$9

the estimate

$\dot{\tilde{x}}=(A(t)-K(t)C(t))\tilde{x},$0

and the observer

$\dot{\tilde{x}}=(A(t)-K(t)C(t))\tilde{x},$1

The data there explicitly interpret $\dot{\tilde{x}}=(A(t)-K(t)C(t))\tilde{x},$2 as a certainty matrix rather than as a stochastic covariance inverse (Bamieh, 10 Jun 2026).

5. Nonlinear and geometric deterministic Riccati observers

A substantial modern use of deterministic Riccati observers is in nonlinear navigation, attitude estimation, visual SLAM, and VIO, where the observer is nonlinear at the state level but admits an exact or induced LTV error subsystem.

For velocity-aided attitude estimation of accelerated vehicles, (Hua et al., 2017) designs nonlinear Riccati observers that the paper explicitly describes as deterministic versions of an EKF. The observer takes the form

$\dot{\tilde{x}}=(A(t)-K(t)C(t))\tilde{x},$3

and the innovation is generated through

$\dot{\tilde{x}}=(A(t)-K(t)C(t))\tilde{x},$4

Local exponential stability follows from uniform observability of the corresponding frozen pair $\dot{\tilde{x}}=(A(t)-K(t)C(t))\tilde{x},$5 under motion and sensing conditions stated explicitly in terms of $\dot{\tilde{x}}=(A(t)-K(t)C(t))\tilde{x},$6, translational motion, and the presence of magnetometer measurements (Hua et al., 2017).

For visual localisation and mapping, (Goor et al., 2019) embeds Riccati gains into an equivariant observer on a symmetry group. For each point landmark $\dot{\tilde{x}}=(A(t)-K(t)C(t))\tilde{x},$7, the per-landmark Riccati equation is

$\dot{\tilde{x}}=(A(t)-K(t)C(t))\tilde{x},$8

with gain

$\dot{\tilde{x}}=(A(t)-K(t)C(t))\tilde{x},$9

The corresponding point-landmark body-frame error satisfies

$\mathcal V(t)=\tilde x^\top P^{-1}(t)\tilde x$0

and the overall observer is shown to be almost globally asymptotically stable and exponentially stable in-the-large on the SLAM manifold (Goor et al., 2019). This is a geometric deterministic Riccati observer with decoupled landmark-wise gains.

In optical-flow-based VIO, (Bouazza et al., 28 Aug 2025) builds a Riccati observer for the translational state $\mathcal V(t)=\tilde x^\top P^{-1}(t)\tilde x$1. The key measurement is only the body-frame direction of velocity,

$\mathcal V(t)=\tilde x^\top P^{-1}(t)\tilde x$2

which is converted into the linear output

$\mathcal V(t)=\tilde x^\top P^{-1}(t)\tilde x$3

for the stacked error state $\mathcal V(t)=\tilde x^\top P^{-1}(t)\tilde x$4. The gain is

$\mathcal V(t)=\tilde x^\top P^{-1}(t)\tilde x$5

and the error system

$\mathcal V(t)=\tilde x^\top P^{-1}(t)\tilde x$6

is globally exponentially stable under the persistent-excitation condition

$\mathcal V(t)=\tilde x^\top P^{-1}(t)\tilde x$7

The full cascaded VIO observer is only almost globally asymptotically stable because the second-stage $\mathcal V(t)=\tilde x^\top P^{-1}(t)\tilde x$8 attitude observer inherits the usual topological obstruction (Bouazza et al., 28 Aug 2025).

A closely related but larger-scale Lie-group construction appears in (Boughellaba et al., 12 Jan 2026). There the translational error state

$\mathcal V(t)=\tilde x^\top P^{-1}(t)\tilde x$9

obeys the exact LTV closed-loop system

$\dot x(t)=A\,x(t)+w(t),\qquad y(t)=C\,x(t)+v(t),$0

with Riccati equation

$\dot x(t)=A\,x(t)+w(t),\qquad y(t)=C\,x(t)+v(t),$1

Under uniform observability of $\dot x(t)=A\,x(t)+w(t),\qquad y(t)=C\,x(t)+v(t),$2, the translational subsystem is globally exponentially stable, and the full VIO observer is almost globally asymptotically stable up to the standard VIO unobservable directions of global translation and rotation about gravity (Boughellaba et al., 12 Jan 2026).

Outside navigation, deterministic Riccati-based design also appears in nonlinear epidemic state estimation. The observer in (Niazi et al., 2022) is synthesized through algebraic Riccati inequalities such as

$\dot x(t)=A\,x(t)+w(t),\qquad y(t)=C\,x(t)+v(t),$3

together with equivalent LMIs. The resulting observer is deterministic and Riccati-based in the nonlinear Lyapunov/ARI sense rather than in the Kalman–Bucy sense (Niazi et al., 2022).

6. Stability theory, perturbations, and scope limitations

A central benefit of deterministic Riccati observers is that their stability can often be quantified sharply. In one dimension, the deterministic Riccati equation

$\dot x(t)=A\,x(t)+w(t),\qquad y(t)=C\,x(t)+v(t),$4

has equilibria

$\dot x(t)=A\,x(t)+w(t),\qquad y(t)=C\,x(t)+v(t),$5

The positive equilibrium $\dot x(t)=A\,x(t)+w(t),\qquad y(t)=C\,x(t)+v(t),$6 is globally attracting on $\dot x(t)=A\,x(t)+w(t),\qquad y(t)=C\,x(t)+v(t),$7, the tangent process satisfies $\dot x(t)=A\,x(t)+w(t),\qquad y(t)=C\,x(t)+v(t),$8, and the asymptotic contraction rate is exactly $\dot x(t)=A\,x(t)+w(t),\qquad y(t)=C\,x(t)+v(t),$9 (Bishop et al., 2017). The associated observer error drift is

$$J(v,w,x_0)=\int_0^T \bigl(v^*Vv+w^*Ww\bigr)\,dt+x_0^*Nx_0.$$0

so even an open-loop unstable scalar system ($$J(v,w,x_0)=\int_0^T \bigl(v^*Vv+w^*Ww\bigr)\,dt+x_0^*Nx_0.$$1) is stabilized by the Riccati observer law (Bishop et al., 2017).

The perturbation theory in (Bishop et al., 2017) shows that the deterministic Riccati flow is also the mean-field limit of stochastic Riccati diffusions arising from ensemble Kalman–Bucy filtering. For the stochastic covariance flow $$J(v,w,x_0)=\int_0^T \bigl(v^*Vv+w^*Ww\bigr)\,dt+x_0^*Nx_0.$$2,

$$J(v,w,x_0)=\int_0^T \bigl(v^*Vv+w^*Ww\bigr)\,dt+x_0^*Nx_0.$$3

and in the EnKF scaling $$J(v,w,x_0)=\int_0^T \bigl(v^*Vv+w^*Ww\bigr)\,dt+x_0^*Nx_0.$$4, this is an $$J(v,w,x_0)=\int_0^T \bigl(v^*Vv+w^*Ww\bigr)\,dt+x_0^*Nx_0.$$5 covariance fluctuation estimate. The same paper proves the negative-bias relation

$$J(v,w,x_0)=\int_0^T \bigl(v^*Vv+w^*Ww\bigr)\,dt+x_0^*Nx_0.$$6

which it explicitly interprets as a negative bias of EnKF sample covariance estimates (Bishop et al., 2017). This suggests that the deterministic Riccati observer is not merely a nominal law but the organizing limit around which finite-ensemble covariance fluctuations can be quantified.

At the opposite end of the theory, not every Riccati differential equation is globally well posed. The matrix Riccati differential equation

$$J(v,w,x_0)=\int_0^T \bigl(v^*Vv+w^*Ww\bigr)\,dt+x_0^*Nx_0.$$7

studied in (Ogura et al., 2022) may escape in finite time, with escape occurring precisely when the block $$J(v,w,x_0)=\int_0^T \bigl(v^*Vv+w^*Ww\bigr)\,dt+x_0^*Nx_0.$$8 in

$$J(v,w,x_0)=\int_0^T \bigl(v^*Vv+w^*Ww\bigr)\,dt+x_0^*Nx_0.$$9

becomes singular. That paper develops an approximation scheme

$K_t=P_tB^\top R_V^{-1}$00

for deterministic escape time. Although it is not an observer-design paper, it underscores a general caveat: Riccati-based observers rely on bounded, well-conditioned Riccati solutions, and finite escape is a structural pathology that must be excluded or controlled (Ogura et al., 2022).

In infinite dimensions, (Wyss, 2019) studies the control-side algebraic Riccati equation

$K_t=P_tB^\top R_V^{-1}$01

for systems with unbounded control and observation operators. The paper itself is not an observer-design paper, but it explicitly frames its observer relevance through duality and records the dual estimation Riccati equation

$K_t=P_tB^\top R_V^{-1}$02

Under quasi-sectoriality, $K_t=P_tB^\top R_V^{-1}$03, and compact-resolvent assumptions, it proves boundedness of the nonnegative stabilizing solution on the control side and exponential stability of the associated feedback generator (Wyss, 2019). A plausible implication is that deterministic Riccati observer theory extends beyond finite-dimensional ODEs, but in this setting the observer interpretation is indirect and domain-sensitive.

The main limitations in the current literature are correspondingly varied. Minimum-energy Mortensen theory in (Breiten et al., 2023) is explicitly local in $K_t=P_tB^\top R_V^{-1}$04 and finite-horizon. The strong-detectability construction in (Tranninger et al., 2021) requires regularity and full-column-rank assumptions on $K_t=P_tB^\top R_V^{-1}$05. The global exponential results in navigation papers depend on persistent excitation or source geometry (Hamel et al., 2016, Bouazza et al., 28 Aug 2025, Boughellaba et al., 12 Jan 2026). Geometric attitude observers remain almost global rather than global because of topological obstructions on $K_t=P_tB^\top R_V^{-1}$06 (Bouazza et al., 28 Aug 2025, Boughellaba et al., 12 Jan 2026). The phrase “deterministic Riccati observer” therefore names a family of designs unified by Riccati-generated gains, but not a single theorem or a single probabilistic interpretation.