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Linear quadratic control for discrete-time systems with stochastic and bounded noises

Xuehui Ma, Shiliang Zhang, Xiaohui Zhang, Jing Xin, Hector Garcia de Marina

TL;DR

This work addresses linear quadratic control for discrete-time systems affected by both stochastic and bounded noises. It introduces a mixed-noise model and a hybrid MIX state estimator that fuses Kalman filter and ellipsoidal set-membership filtering to produce balanced, non-overconservative state estimates. A robust, finite-horizon LQC design is then formulated as an SDP using the S-procedure, yielding a control law derived from the first block of a transformed decision vector. Numerical results show improved estimation accuracy and control performance over standard KF/ESM baselines, indicating the method extends LQC applicability to real-world sensing with mixed noise characteristics. The approach offers a practical path to robust performance in systems with diverse sensing modalities and disturbances, though it notes the computational cost of the SDP.

Abstract

This paper focuses on the linear quadratic control (LQC) design of systems corrupted by both stochastic noise and bounded noise simultaneously. When only of these noises are considered, the LQC strategy leads to stochastic or robust controllers, respectively. However, there is no LQC strategy that can simultaneously handle stochastic and bounded noises efficiently. This limits the scope where existing LQC strategies can be applied. In this work, we look into the LQC problem for discrete-time systems that have both stochastic and bounded noises in its dynamics. We develop a state estimation for such systems by efficiently combining a Kalman filter and an ellipsoid set-membership filter. The developed state estimation can recover the estimation optimality when the system is subject to both kinds of noise, the stochastic and the bounded. Upon the estimated state, we derive a robust state-feedback optimal control law for the LQC problem. The control law derivation takes into account both stochastic and bounded-state estimation errors, so as to avoid over-conservativeness while sustaining stability in the control. In this way, the developed LQC strategy extends the range of scenarios where LQC can be applied, especially those of real-world control systems with diverse sensing which are subject to different kinds of noise. We present numerical simulations, and the results demonstrate the enhanced control performance with the proposed strategy.

Linear quadratic control for discrete-time systems with stochastic and bounded noises

TL;DR

This work addresses linear quadratic control for discrete-time systems affected by both stochastic and bounded noises. It introduces a mixed-noise model and a hybrid MIX state estimator that fuses Kalman filter and ellipsoidal set-membership filtering to produce balanced, non-overconservative state estimates. A robust, finite-horizon LQC design is then formulated as an SDP using the S-procedure, yielding a control law derived from the first block of a transformed decision vector. Numerical results show improved estimation accuracy and control performance over standard KF/ESM baselines, indicating the method extends LQC applicability to real-world sensing with mixed noise characteristics. The approach offers a practical path to robust performance in systems with diverse sensing modalities and disturbances, though it notes the computational cost of the SDP.

Abstract

This paper focuses on the linear quadratic control (LQC) design of systems corrupted by both stochastic noise and bounded noise simultaneously. When only of these noises are considered, the LQC strategy leads to stochastic or robust controllers, respectively. However, there is no LQC strategy that can simultaneously handle stochastic and bounded noises efficiently. This limits the scope where existing LQC strategies can be applied. In this work, we look into the LQC problem for discrete-time systems that have both stochastic and bounded noises in its dynamics. We develop a state estimation for such systems by efficiently combining a Kalman filter and an ellipsoid set-membership filter. The developed state estimation can recover the estimation optimality when the system is subject to both kinds of noise, the stochastic and the bounded. Upon the estimated state, we derive a robust state-feedback optimal control law for the LQC problem. The control law derivation takes into account both stochastic and bounded-state estimation errors, so as to avoid over-conservativeness while sustaining stability in the control. In this way, the developed LQC strategy extends the range of scenarios where LQC can be applied, especially those of real-world control systems with diverse sensing which are subject to different kinds of noise. We present numerical simulations, and the results demonstrate the enhanced control performance with the proposed strategy.

Paper Structure

This paper contains 6 sections, 3 theorems, 77 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. State Estimation incorporating Kalman Filter and Set-membership Filter
  4. Optimal Controller Design
  5. Simulations and results
  6. Conclusion

Key Result

Proposition 1

Consider the discrete-time linear system described by systemdynamic and observationequation, which is disturbed by stochastic and bounded noises as in (noise_w)-(bound_v). Let the initial state be set as state_x0 and state_x0_prior, with initial mixed state center $\bar{\bm{x}}_0$, initial covarianc (ii) Update where the scalar parameters $p_k$ and $q_k$ are calculated by and

Figures (4)

  • Figure 1: State estimation using Kalman filter (KF).
  • Figure 2: State estimation using ellipsoid set-membership filter (ESM).
  • Figure 3: State estimation using mixed estimator (MIX).
  • Figure 4: State regulating under RCKF, RCESM and RCMIX.

Theorems & Definitions (6)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Theorem 1
  • proof