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What are the challenges and limitations of LQR in the presence of uncertainties and disturbances?

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Linear quadratic regulator (LQR) is a popular technique for designing optimal feedback controllers for linear systems. It minimizes a quadratic cost function that balances the tracking performance and the control effort. However, real systems are often nonlinear, uncertain, and subject to disturbances. How does LQR cope with these challenges and limitations? In this article, you will learn about some of the common issues and solutions related to LQR in the presence of uncertainties and disturbances.

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Germán García

MSc in Mechanical Engineering at EPFL. BSc in Mechatronics Engineering at ITESM

1 Nonlinearity and Linearization

One of the main assumptions of LQR is that the system is linear, or can be approximated by a linear model around an operating point. However, many systems exhibit nonlinear behavior, especially at large deviations from the equilibrium. This can lead to poor performance, instability, or even divergence of the LQR controller. To overcome this problem, one can use linearization techniques, such as Taylor series expansion, to obtain a local linear model for each operating point. Then, one can design a LQR controller for each linear model, and switch between them according to the state of the system. This is known as gain scheduling, and it can improve the robustness and adaptability of LQR.

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Germán García

MSc in Mechanical Engineering at EPFL. BSc in Mechatronics Engineering at ITESM
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Being such a simple controller to set up, LQR is still relevant when studying nonlinear dynamics because it can offer a local approximation of the optimal control solution for the system. Yet, it is important to wage when it is best to resort to linearizing the model around an equilibrium point, or to consider other techniques like MPC. Linearization, as achieved by i.e. Taylor series expansion, typically stabilizes the system only locally around an equilibrium point. Nonlinear MPC can handle nonlinear dynamics but it may involve solving non-convex optimization problems, which can be extremely difficult depending on the system.

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Mitja Echim

Co-Founder & CEO bei TOPAS Industriemathematik | Dr.-Ing.
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Interesting points on LQR and nonlinearity. Modern computing allows real-time recalibration of Q and R matrices, adapting to system deviations in <20ms. This is crucial for large state domains, avoiding storage of numerous control matrices. Additionally, where computational resources permit, consider Model Predictive Control (MPC). MPC's forward-looking approach and nonlinearity handling make it a robust alternative to traditional LQR in complex systems.

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2 Uncertainty and Robustness

Another challenge of LQR is that it requires an accurate knowledge of the system parameters, such as the mass, inertia, friction, or stiffness. However, these parameters may vary due to manufacturing tolerances, aging, wear and tear, or environmental changes. This can introduce uncertainty in the system model, and affect the performance and stability of the LQR controller. To deal with this issue, one can use robust control techniques, such as H-infinity or loop shaping, to design a LQR controller that is less sensitive to the uncertainty. Alternatively, one can use adaptive control techniques, such as model reference adaptive control (MRAC) or self-tuning regulator (STR), to update the LQR controller online based on the observed system behavior.

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Mitja Echim

Co-Founder & CEO bei TOPAS Industriemathematik | Dr.-Ing.
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Great points on LQR's challenges with parameter uncertainties. Adaptive LQR is key in such scenarios. It uses parametric sensitivity analysis in nonlinear optimization to dynamically adjust Q and R matrices. This is especially useful when system parameters vary due to factors like manufacturing tolerances. By measuring and identifying these deviations in real-time, adaptive LQR allows for timely, albeit suboptimal, adjustments. This enhances controller robustness against uncertainties, maintaining system stability and control efficacy.

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3 Disturbance and Noise

A third limitation of LQR is that it assumes that the system is free of external disturbance and measurement noise. However, these factors are inevitable in real systems, and can degrade the tracking performance and the control quality. To cope with this problem, one can use disturbance rejection techniques, such as feedforward control, integral action, or Kalman filter, to estimate and compensate for the disturbance and noise. Moreover, one can use stochastic LQR (SLQR), which extends the quadratic cost function to include the covariance of the disturbance and noise. SLQR can optimize the trade-off between the expected performance and the control effort under uncertainty.

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4 Constraints and Saturation

A fourth drawback of LQR is that it does not account for the physical constraints and limitations of the system and the controller. For example, the actuators may have limited range, bandwidth, or power, and the state variables may have safety or operational limits. These constraints can cause saturation or violation of the LQR controller, and result in poor performance, oscillations, or instability. To solve this problem, one can use constrained LQR (CLQR), which incorporates the constraints into the optimization problem. CLQR can ensure the feasibility and optimality of the control solution under the given constraints.

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Ahmed Mahmoud

Lecturer at Minia University
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Although CLQR guarantees stability, it might lead to abrupt signals around constraint boundaries. Of course, solving constrained optimization problems will increase complexity which might be a problem with higher-order systems.

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5 Complexity and Computation

A final challenge of LQR is that it can be computationally demanding and complex to implement, especially for large-scale, high-dimensional, or nonlinear systems. The LQR controller requires solving a matrix Riccati equation, which can be time-consuming and numerically unstable. Furthermore, the LQR controller may need to be updated or redesigned frequently to adapt to the changes in the system dynamics, uncertainty, disturbance, or constraints. This can increase the computational burden and the implementation difficulty. To address this issue, one can use approximation or simplification techniques, such as model reduction, discretization, or diagonalization, to reduce the complexity and computation of the LQR controller.

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Mitja Echim

Co-Founder & CEO bei TOPAS Industriemathematik | Dr.-Ing.
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Indeed, the computational demands of LQR, particularly in the context of complex systems, present a significant challenge. Balancing the computational power against energy consumption and cost is crucial in practical applications. This is especially pertinent in autonomous systems like robots, where energy efficiency is paramount due to their limited power reserves. The need for frequent updates to the LQR controller, in response to dynamic changes and uncertainties, further amplifies these constraints. It's essential to consider not just the mathematical feasibility but also the operational sustainability of such systems, making energy-efficient computation a key factor in the design and implementation of LQR controllers.

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6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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What are the challenges and limitations of LQR in the presence of uncertainties and disturbances?

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1 Nonlinearity and Linearization

2 Uncertainty and Robustness

3 Disturbance and Noise

4 Constraints and Saturation

5 Complexity and Computation

6 Here’s what else to consider

Control Systems Design

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