10.2 H-infinity control and linear matrix inequalities (LMIs)

H-infinity control is a powerful method for designing robust controllers in nonlinear systems. It minimizes the worst-case impact of disturbances on outputs, making it ideal for handling uncertainties and external disturbances in control systems.

Linear Matrix Inequalities (LMIs) are key tools in solving H-infinity control problems. They allow complex control design specifications to be formulated as convex optimization problems, which can be efficiently solved using modern optimization techniques and software tools.

H-infinity Optimization for Nonlinear Control

Formulating Nonlinear Control Problems as H-infinity Optimization

• H-infinity control minimizes the worst-case gain from disturbances to outputs in the presence of model uncertainties and external disturbances, providing a robust control technique
• The H-infinity norm of a transfer function represents the worst-case gain from disturbances to outputs, defined as the maximum singular value of the transfer function matrix over all frequencies
• Formulate nonlinear control problems as H-infinity optimization problems by defining the system dynamics, performance objectives, and uncertainty descriptions in the frequency domain
• The standard H-infinity control problem involves finding a controller that:
  • Stabilizes the closed-loop system
  • Minimizes the H-infinity norm of the transfer function from disturbances to outputs
• The generalized H-infinity control problem includes additional constraints on the closed-loop system:
  • Robust stability
  • Robust performance
  • Mixed sensitivity specifications (weighted sensitivity and complementary sensitivity functions)

Frequency-Domain Representation and Uncertainty Modeling

• Represent nonlinear systems in the frequency domain using transfer functions or frequency response data obtained from linearization or system identification techniques
• Model uncertainties in the frequency domain using:
  • Additive uncertainty (unmodeled dynamics)
  • Multiplicative uncertainty (parameter variations)
  • Coprime factor uncertainty (structured uncertainty in the plant model)
• Specify performance objectives in the frequency domain using weighting functions that shape the desired closed-loop behavior (bandwidth, disturbance rejection, robustness margins)
• Formulate the H-infinity control problem as a minimization of the weighted sensitivity and complementary sensitivity functions subject to the uncertainty and performance constraints

LMI-based Solutions for H-infinity Control

Linear Matrix Inequalities (LMIs) for Convex Optimization

• Linear matrix inequalities (LMIs) allow the formulation of control design specifications as convex optimization problems, providing a powerful tool for solving H-infinity control problems
• An LMI is an inequality of the form $F(x) > 0$, where:
  • $F(x)$ is a symmetric matrix that depends affinely on the decision variables $x$
  • The inequality means that $F(x)$ is positive definite
• H-infinity control problems can be transformed into equivalent LMI problems using the bounded real lemma, which relates the H-infinity norm of a transfer function to the solution of an LMI
• Convex optimization techniques efficiently solve LMI problems and obtain the optimal controller parameters:
  • Interior-point methods
  • Semidefinite programming (SDP)

Software Tools for LMI-based H-infinity Control Design

• MATLAB's Robust Control Toolbox provides built-in functions for formulating and solving H-infinity control problems using LMIs and convex optimization:
  • hinfstruct for structured H-infinity synthesis
  • mixsyn for mixed-sensitivity H-infinity synthesis
• CVX optimization package is a MATLAB-based modeling system for convex optimization that supports LMI-based control design:
  • Allows the specification of LMIs using a high-level programming language
  • Interfaces with various convex optimization solvers (SeDuMi, SDPT3)
• Other software tools for LMI-based control design include:
  • YALMIP (MATLAB toolbox for optimization modeling)
  • LMI Lab (MATLAB toolbox for LMI-based control design)
  • Python-Control (Python package for control system analysis and design)

Robust Nonlinear Control for Disturbance Minimization

Weighting Function Selection for Performance and Robustness

• Select appropriate weighting functions to shape the frequency response of the closed-loop system and specify the desired performance and robustness requirements for robust nonlinear controllers based on H-infinity optimization
• Choose weighting functions to reflect:
  • Relative importance of different frequency ranges
  • Expected magnitude of disturbances
  • Acceptable level of control effort
• Weighting functions can be designed as:
  • Low-pass filters to emphasize low-frequency performance (tracking, disturbance rejection)
  • High-pass filters to emphasize high-frequency robustness (noise attenuation, uncertainty suppression)
  • Bandpass filters to specify frequency-dependent performance and robustness trade-offs

Controller Implementation and State Estimation

• The resulting H-infinity optimal controller is typically a dynamic state-feedback controller that depends on the system states and the weighting functions
• Implement the controller using observer-based techniques to estimate the system states from noisy measurements:
  • Kalman filter for optimal state estimation in the presence of Gaussian noise
  • H-infinity filter for robust state estimation in the presence of bounded disturbances
• Design the observer to have a faster response than the controller to ensure accurate state estimation and closed-loop stability
• Consider the computational complexity and real-time implementation aspects when selecting the observer and controller order

H-infinity Control Applications for Practical Systems

Application Process and Linearization Techniques

• Apply H-infinity control to a wide range of practical nonlinear systems:
  • Aerospace systems (aircraft, satellites, missiles)
  • Robotics (manipulators, mobile robots, humanoids)
  • Power systems (generators, converters, microgrids)
  • Process control (chemical reactors, heat exchangers, distillation columns)
• Model the nonlinear system dynamics, identify the sources of uncertainties and disturbances, and formulate the control design specifications as an H-infinity optimization problem
• Use linearization techniques to obtain a linear approximation of the nonlinear system around an operating point for H-infinity controller design:
  • Jacobian linearization (first-order Taylor series expansion)
  • Feedback linearization (exact input-output linearization using nonlinear feedback)

Validation and Practical Considerations

• Validate the designed H-infinity controller through simulations and experimental testing to assess its performance and robustness under various operating conditions and disturbance scenarios
• Consider practical aspects when implementing the H-infinity controller on a real-world system:
  • Actuator saturation and rate limits
  • Sensor noise and quantization effects
  • Computational complexity and real-time implementation constraints
• Implement anti-windup techniques (integrator clamping, conditional integration) to handle actuator saturation and maintain closed-loop stability
• Use gain scheduling or adaptive control techniques to extend the operating range of the H-infinity controller and accommodate system nonlinearities and parameter variations