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Control and Estimation Co-Design via Envelope-Theorem Gradients

Mohammad S. Ramadan, Philip Dinenis, Mihai Anitescu

arXiv:2609.36090v1eess.SYcs.SYmath.OC

TL;DR

Sequential design can leave performance constrained by earlier choices, while existing control co-design often fixes sensing and estimation. ContEst jointly optimizes actuation and sensing/estimation using envelope-theorem gradients from inner dual variables, and the paper reports significant design cost reductions across realistic studies.

  • Problem

    Sequential plant, sensing, controller, and estimator design is generally not optimal, while control co-design typically leaves sensing and estimator design fixed or postponed.

    Source: §1 · p. 1

  • Method

    ContEst poses actuation and sensing/estimation design as a two-stage stochastic optimal-control problem, using inner-solver duals for envelope-theorem gradients across SDP and approximate eKF–MPC formulations.

    Source: §1 · p. 1, §7 · p. 8, §10 · p. 13

  • Results

    The studies report significant design cost reductions, with sensing a major and sometimes dominant contributor, across four realistic co-design applications.

    Source: §1 · pp. 1–2

  • Takeaways & Limitations

    Jointly treating sensing and estimation with actuation lets the design optimize against the information state rather than leaving sensing outside control co-design.

    Source: §1 · p. 1, §10 · p. 13

    For nonlinear systems, the envelope theorem applies to the convex eKF–MPC surrogate, not the underlying stochastic-control value, and the eKF and local linearizations introduce approximation gaps.

    Source: §7 · p. 8

Abstract

from arXiv · show

Instead of the sequential plant--control--estimator design pipeline, which is in general not optimal, we propose the Control and Estimation Co-Design (ContEst) framework, which poses the entire system design as a single two-stage problem solved by first-order methods. A key enabler is that the gradient of each inner control/estimation optimal cost with respect to the design parameters is available from the dual variables the inner solver already returns, by the envelope theorem: exactly under mild regularity (a unique inner minimizer), and as a subgradient under more relaxed conditions. That is, no differentiation through the optimizer or its optimality conditions is needed. We first present the framework in a linear quadratic Gaussian (LQG) regime, where the control and estimation costs become two semidefinite programs (SDPs) whose design gradients are read from the duals of the Lyapunov constraints. We extend the results to robust ($H_\infty$) formulation to cover worst-case control and filtering problems. We also present an extension to constrained and nonlinear systems, using an extended Kalman filter (eKF) and a local linearization of the information-state dynamics to yield an approximate but convex model predictive control (MPC) problem whose design gradients are read directly from the costates. We apply our methods to a variety of realistic design examples from different fields and report significant design improvements.

1 Introduction

Sequential design can constrain later choices and is generally not optimal, while existing control co-design typically leaves sensing and estimation fixed. ContEst jointly optimizes actuation and sensing/estimation, using inner-solver duals to obtain design gradients.

1 Introduction

  • Sequential plant, sensing, controller, and estimator design restricts later stages and is generally not optimal.A sequential solution is a feasible point of the larger joint problem, not necessarily an optimum or stationary point.Source: §1 · p. 1
  • Control co-design usually optimizes plant and controller together while leaving sensing configuration and estimator design fixed or postponing them.For systems without full, noiseless measurements, achievable control performance is limited by sensing and estimation quality.Source: §1 · p. 1
  • Across four realistic co-design studies, ContEst yields significant design cost reductions, with sensing a major and sometimes dominant contributor.The studies span spacecraft, process, and power systems.Source: §1 · pp. 1–2
  • The framework places actuation and sensing/estimation parameters in one two-stage optimization whose inner objective is stochastic optimal-control cost.The output-feedback cost depends on the information state, coupling the output equation and state dynamics.Source: §1 · p. 1
  • Dual variables returned by the inner solver provide exact design gradients under a unique inner minimizer and valid subgradients under more relaxed conditions.This avoids differentiating through the optimizer or its optimality conditions.Source: §1 · pp. 1–2
  • The framework also covers H∞ design, an approximate eKF-based MPC extension, sensor allocation, and flexible SDP modeling, and is evaluated in four studies.Sensor selection can be represented by mixed-integer decisions or an ℓ1 relaxation; the nonlinear extension uses costates for approximate gradients.Source: §1 · p. 2

2 Related work

Prior co-design methods commonly restrict the scope to plant and controller, or treat sensing and estimation through fixed-plant or fixed-class formulations. ContEst’s distinction is its joint stochastic objective, dual-based gradients across several regimes, and SDP modeling flexibility.

2 Related work

  • Earlier CCD work couples plant and controller, and dynamic-system CCD was cast as MDO; estimator and sensing layers are typically fixed.The related-work discussion also identifies as systematizing nested and simultaneous solution strategies.Source: §2 · p. 2
  • Prior structural and information-architecture methods use alternating LMIs or fixed-plant gain designs, whereas ContEst obtains a joint design gradient for actuation, sensing, and estimation.The cited lines include,,.Source: §2 · p. 2
  • ContEst differs from plant-only dual-based co-design by jointly including actuation, sensing, and estimator design in a stochastic objective across H2, H∞, and eKF–MPC variants.For [33], the design vector enters only the plant, the output map is fixed, and estimation error is absent from its deterministic worst-case objective.Source: §2 · p. 2
  • Sensor-selection methods include convex relaxation, greedy approaches [45,47], and sparsity promotion; closest LQG co-design work selects from a discrete sensor menu.That closest work is restricted to the LQG regime.Source: §2 · p. 2
  • For the optimal value, ContEst reads gradients from already-computed duals rather than differentiating the KKT system or the minimizer.The compared KKT approach solves an m × m linear system per design direction; the passage’s complexity expressions are truncated.Source: §2 · p. 2
  • ContEst’s SDP path supports structured gain patterns, sparsity-promoting regularizers, coupled sensing-actuation budgets, and quadratic stability over convex uncertainty sets.For unconstrained LQR cases, Riccati or adjoint routes can be cheaper, and ContEst can use them when possible.Source: §2 · pp. 2–3

3 Problem formulation

The problem formulation treats plant, actuator, and sensing choices as design parameters in a two-stage problem, with stochastic optimal control value in the inner stage. The inner controller acts on the information state, and the formulation can accommodate multiple scenarios and design considerations.

3 Problem formulation

  • The design stage chooses parameters governing plant, actuation, and sensing, accounting for design cost and feasible-set considerations.The design cost and set can encode price, manufacturability, placement difficulty, and mechanical or other considerations.Source: §3 · p. 3
  • The information available at time k comprises the prior, measurements through k, and controls through k; causal controls use information available no later than k−1.The stated expectation is over the initial state and process and measurement disturbances.Source: §3 · p. 3
  • Multiple operating scenarios can be represented by replacing the inner value with a weighted average of costs across scenarios.The formulation is introduced for a single prior, with scenario extension described as an option.Source: §3 · pp. 3–4
  • The inner problem is stochastic dual control with the filtered density as its sufficient information state; its value, rather than its argmin, enters the outer problem.The envelope theorem supplies the outer gradient from duals returned by the inner solver.Source: §3 · pp. 3–4

4 The LQG case: LQR and Kalman SDPs

In the LQG setting, separation splits the inner cost into control and estimation components, each expressible as an SDP. Design parameters couple the two components, and their sensitivities are obtained from SDP dual variables; parametrized cost weights and noise covariances also support sensor-precision co-design.

4 The LQG case: LQR and Kalman SDPs

  • The LQG inner cost separates into control and estimation terms, independently optimized for fixed design parameters, although design choices couple their gains.The control uses a certainty-equivalent state-feedback law and the estimation component uses an observer gain.Source: §4 · p. 4
  • The control and estimation halves are formulated as dual-form SDPs, exposing the control and sensing operators, respectively.The control SDP uses a linearizing change of variables; the estimation SDP uses an observability-gramian formulation and an LMI relaxation.Source: §4 · p. 4
  • Under the stated regularity conditions, gradients with respect to system matrices are read from optimal LMI dual variables and combined by the chain rule.The control and estimation problems have transposed-dual forms, enabling corresponding sensitivity calculations.Source: §4 · p. 5

4.1 Sensor precision: noise level co-design

  • Parametrizing quadratic-cost weights and process or measurement noise covariances makes sensor and actuator precision available as design choices.Both LQG subproblems are affine in these data, and the SDP formulation also permits convex regularizers, sensor-selection constraints, and polytopic quadratic-stability constraints.Source: §4.1 · p. 5

5 Exact design gradients via the envelope theorem

The envelope theorem reads the inner optimal-value gradient from the solver’s primal-dual solution, without differentiating through the optimizer. Under stated regularity this is an exact gradient; when uniqueness conditions weaken, the returned dual can still provide a subgradient.

Exact design gradients via the envelope theorem

  • Under Assumption 1, the inner optimal value is differentiable and its design gradient is the Lagrangian’s partial derivative with the optimal primal and dual variables held fixed.The conditions include strong duality, a unique primal minimizer, a unique dual, and strict complementarity.Source: §5 · p. 6
  • The framework retains differentiability when added convex, conic-representable regularizers or constraints preserve Assumption 1.Exactness can be lost at a degenerate active face or when a constraint binds at its achievable floor, making the optimal dual set-valued.Source: §5 · p. 6
  • If regularity fails but either the primal or dual optimum remains unique, the returned dual constructs a Clarke subgradient; a unique dual also supplies a descent direction.Without uniqueness of either solution, the inclusion can fail.Source: §5 · p. 6
  • For the LQG control and estimation SDPs, stabilizability, detectability, positive-definite weights, and C1 data ensure unique primal and dual solutions with strict complementarity.Under these conditions, the envelope-gradient theorem applies to both SDPs.Source: §5 · p. 6

6 The H∞analog

The H∞ extension replaces average stochastic performance with worst-case closed-loop gain for bounded disturbances. Its minimal-attenuation formulation generally yields subgradients, whereas fixing attenuation strictly above its optimum restores exact dual-based gradients for control and filtering.

The H∞ analog

  • For unknown-but-bounded ℓ2 disturbances, the inner objective is squared worst-case closed-loop gain rather than stationary variance.The plant includes an explicit disturbance channel and performance output, with the objective weighting the same state and input signals as in the LQG cost.Source: §6 · pp. 6–7
  • At minimal attenuation, the returned dual expression is generally only a subgradient, not the exact gradient, but still supplies a descent direction and stationarity certificate.At the optimum the BRL LMI loses rank, so strict complementarity generically fails.Source: §6 · p. 7
  • Fixing attenuation strictly above its optimal level makes the control SDP strictly feasible with a nondegenerate, strictly complementary dual, yielding exact gradients from the third-LMI dual.The resulting expression has the same form as the minimal-attenuation expression, but is exact.Source: §6 · p. 7
  • The fixed-attenuation estimation SDP follows by transposing the control formulation, with the returned control gain transposed to obtain the optimal filter gain.The corresponding Jacobians transpose the control-case expressions for Aθ and Bθ.Source: §6 · p. 7
  • A Riccati counterpart to the fixed-attenuation SDP avoids an interior-point solve, but restricts which regularizers and constraints can be added.The passage presents this route as useful at scale.Source: §6 · p. 7

7 The locally linearized case: MPC and the eKF

For nonlinear constrained systems, the paper uses an eKF and locally linearized information-state dynamics to form a convex MPC inner problem. Its costate-based gradients support design optimization, but are approximate for the underlying stochastic-control value.

The locally linearized case: MPC and the eKF

  • The eKF approximates the infinite-dimensional Bayesian filter by propagating the state estimate and covariance, then its prediction and measurement updates are linearized into an information-state MPC model.The information state stacks the estimated mean and vectorized upper-triangular covariance, and the dynamics are linearized about a nominal state and input.Source: §7 · pp. 7–8
  • The envelope gradient for the convex MPC problem is read from its dynamics costates, since the explicit design dependence enters through Aθ and Bθ.The underlying stochastic-control value is not covered by the theorem in this nonlinear regime.Source: §7 · p. 8
  • Because separation fails in the nonlinear regime, the controller uses the information state and can exhibit probing or caution effects through the covariance.Fixing the covariance, for example to λ_probing I, turns λ_probing into an operator-selected probing knob.Source: §7 · p. 8
  • The nonlinear gradient is approximate: the eKF is exact only for affine dynamics and measurements, prediction-only mode drops innovation feedback, and the model uses a current-time linearization.The paper reports that this approximate gradient can nevertheless be effective in design optimization.Source: §7 · p. 8
  • Sensor allocation can use binary deployment variables at the outer level without changing inner gradients, but this yields an NP-hard mixed-integer problem; continuous gains with an ℓ1 penalty promote sparse sensing.An epigraph lift removes the ℓ1 kink, and thresholding the optimized gains recovers a binary deployment.Source: §7 · p. 8
  • The outer design uses projected BFGS with exact gradients, while piecewise-smooth regimes use a nonsmooth BFGS variant with a suitable line search.Under usual smoothness and local strong-convexity assumptions, full BFGS attains a local superlinear rate.Source: §7 · p. 8

8 On the computational complexity of ContEst with interior-point methods

Envelope gradients add only a matrix multiplication to an inner solve, but dense interior-point covariance problems scale steeply with state dimension. The reported timings and scaling motivate exploiting structure, sparsity, and Krylov methods for larger systems.

On the computational complexity of ContEst with interior-point methods

  • Each design gradient requires one inner solve plus a matrix multiplication, so the outer BFGS iteration inherits the inner solver’s complexity.Both inner problems include a covariance matrix among their decision variables.Source: §8 · p. 9
  • ≈2.6 s at r_x = 20 and ≈30 s at r_x = 30 were measured for the LQR covariance program, an ≈11× increase consistent with O(r_x^6) scaling.The estimate reflects dense Schur-complement factorization in the interior-point method.Source: §8 · p. 9
  • Naive dense covariance SDPs therefore cap ContEst at a few tens of states; exploiting sparsity or using Krylov methods is suggested to scale further.The eKF–MPC inner problem encounters the same scaling wall, while nonlinear and constrained studies are run at modest state dimension.Source: §8 · p. 9

9 Simulation studies

Four studies evaluate ContEst across different physical systems, using projected BFGS with multiple initial designs; linearized cases also check whether optimized designs improve realized closed-loop cost. The section reports co-designed allocations, placements, and operating-point-dependent tuning, alongside the cost of dual-based gradients.

9 Simulation studies

  • Across four studies, projected BFGS ran from five starts; linearized cases used Monte Carlo checks, while dual gradients cost essentially one inner solve versus rθ + 1 or 2rθ finite-difference solves.The analytic gradient matched central finite differences to relative error below 10^-4 in a one-time sanity check.Source: §9 · p. 9

9.1 Spacecraft attitude determination and control (ADCS)

  • Five starts converge to the same θ: co-design raises star-tracker precision 1 →1.70, holds wheel authority near 1, and lowers gyro precision 1 →0.30 under the effort cap.The hard effort cap saturates the return on additional actuation authority.Source: §9.1 · p. 10, Table 2 · p. 10

9.2 Binary distillation column

  • Three distinct minima emerge; random restarts recover placements that spread feeds toward lower and upper trays and move sensors across mid- and upper-column trays.The baseline start alone converges to an inferior minimum.Source: §9.2 · p. 10, Table 3 · p. 11

9.3 PLL/DSE tuning in a low-inertia grid

  • In peacetime, PLL bandwidths rise 6 →10.6, cutting Jest 12.1% (38.0 →33.4) and Jdes 5.6% (121.5 →114.7); post-fault, b1 rises 10.6 →13.7 and damping authority is unlocked.All five starts converge to the same minimizer in each trigger.Source: §9.3 · p. 11, Table 4 · p. 12

9.4 Multi-terminal HVDC: sparse sensing

  • 33.0% lower Jsc (3.126 →2.093) and 21.6% lower Jdes (3.351 →2.626) accompany retaining 12 of 15 sensors; with four sensors, co-designed Jest 0.653 nearly matches fixed-droop 15-sensor Jest 0.650.The sensor-count comparison uses the same sensor sets for fixed-droop pruning and droop co-design.Source: Table 5 · p. 12, §9.4 · pp. 12–13

10 Conclusion

ContEst jointly optimizes system and sensing parameters against a stochastic objective expressed through the information state. In convex LQG and H∞ inner problems, envelope-theorem gradients come from solver duals; the nonlinear extension uses approximate gradients from costates.

10 Conclusion

  • ContEst obtains design gradients from inner dual variables via the envelope theorem, avoiding implicit differentiation of the solution map and KKT system.For sensor selection, the framework uses a mixed-integer problem with a sparse ℓ1 relaxation whose duals price essential sensors.Source: §10 · p. 13
  • The inner value is convex in LQG and H∞ regimes, represented respectively by two SDPs and a bounded-real-lemma SDP.Sensor selection enters as a mixed-integer problem with a sparse ℓ1 relaxation.Source: §10 · p. 13
  • The nonlinear extension uses an eKF and further information-state linearizations to form a convex MPC problem, with design gradients constructed from returned costates.The authors report that this constructed gradient is helpful in numerical examples but is not exact because of the multiple approximation and linearization layers.Source: §10 · p. 13
  • Future work includes filters beyond the eKF, less-approximate nonlinear handling, gradient-error bounds, chance-constrained and scenario-based inner problems, and decomposition for large interconnected systems.Source: §10 · p. 13

A The information state and its LQG and eKF specializations

The section treats the filtering density as the sufficient information state for stochastic control and describes its Bayesian-filter evolution. For linear systems it specializes to certainty-equivalent LQG control; for nonlinear systems it uses an approximate deterministic eKF evolution.

A The information state and its LQG and eKF specializations

  • A causal control law depends on available information Z_k, whose sufficient statistic is the filtering density p(x_k | Z_k; θ) governed by the Bayesian filter.Source: A The information state and its LQG… · p. 13
  • The Bayesian filter alternates time and measurement updates, and both updates involve the output equation and thus sensing parameters in θ.Source: A The information state and its LQG… · p. 13
  • In the linear regime, separation permits infinite-horizon certainty-equivalent state-feedback control because the state-error covariance evolution does not depend on the control input.Source: A The information state and its LQG… · p. 14
  • For nonlinear systems, the eKF tracks the first two state moments; dropping its future measurement dependence yields a deterministic information-state evolution.The approximation is described as plausible when the measurement correction is nearly independent of the process noise and the dynamics are not highly nonlinear in the state.Source: A The information state and its LQG… · p. 14

B Proof of the LQG regularity Proposition

The proof establishes regularity for the control SDP under convexity, smoothness, strict feasibility, and uniqueness conditions. It constructs a unique dual and verifies strict complementarity using the Lyapunov structure and rank conditions.

B Proof of the LQG regularity Proposition

C Existence of a subgradient under (A.I) and a unique primal or dual solution

The returned Lagrangian partial gradient is a subgradient of the optimal value when either the primal or dual solution is unique. If neither is unique, the returned quantity need not be a subgradient.

C Existence of a subgradient under (A.I) and a unique primal or dual solution

  • The solver-returned pair defines the candidate design gradient as the partial gradient of the Lagrangian at the primal-dual optimum.Source: C Existence of a subgradient under… · p. 15
  • If either the primal or dual solution is unique, the returned gradient is a Clarke subgradient of the optimal value.When the dual is unique and the gradient is nonzero, its negative is a descent direction; primal uniqueness alone does not ensure this.Source: C Existence of a subgradient under… · p. 16
  • If neither solution is unique, the returned gradient need not be a subgradient, even when Slater’s condition holds and the data are affine in θ.The passage gives an example where the optimal value is identically zero but the returned gradient at θ = 0 is 1.Source: C Existence of a subgradient under… · p. 16
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