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Exploring the secrets of H∞ technology: Why does it play such a powerful role in multivariable systems?
In control theory, the H∞ (or "H-infinity") method is widely used to synthesize controllers to achieve stability and ensure performance. This technology began in the late 1970s and early 1980s, led by some scholars such as George Zames, J. William Helton and Allen Tannenbaum. Compared with traditional control technology, H∞ technology performs well in multivariable systems, especially in When dealing with cross-coupling issues between channels.
H∞ control emphasizes providing an optimized solution in complex systems to achieve optimal stability and performance.
When using the H∞ method, the control designer first needs to express the control problem as a mathematical optimization problem. Next, control design is completed by finding a controller that enables a solution to this optimization problem. This process requires deep mathematical understanding and relatively high requirements for the model of the system being controlled.
Advantages and challenges of H∞ method
The main advantage of the H∞ method over traditional control techniques is that it can be easily applied to problems involving multivariable systems. In multivariable systems, the coupling effect between the channels of the system is particularly significant, which requires a more flexible and powerful control strategy to deal with it. The disadvantage of the H∞ method is that the controller it generates is only optimal for the specified cost function and does not necessarily achieve the best results in the performance criteria commonly used to evaluate controllers (such as settling time, energy consumption, etc.) .
Nonlinear constraints (such as saturation effects) are usually not handled well in H∞ technology, which requires special caution when designing the controller.
In addition, H∞ technology is critical to the ability to minimize the effects of perturbation on closed loops. Depending on the problem formulation, the impact can be assessed in terms of stability or performance. However, it is a relatively difficult challenge to simultaneously optimize robust performance and robust stability. The H∞ loop-shaping method is a strategy that comes close to achieving the above goals. Designers can obtain good robust performance through traditional loop-shaping concepts and optimize around the system bandwidth to achieve good robust stability. .
The process of problem formulation
Before implementing H∞ control design, it is first necessary to express the system process according to the standard configuration. In this configuration, plant P has two inputs, where the exogenous input w includes the reference signal and disturbance, and the manipulated variable u is generated by the control signal K. The feedback output z of the system is the error signal we wish to minimize, while the measured output v is used to control the operation of the system.
When expressing the system, we need to consider the properties of all variables. These variables are usually vectors, and P and K are matrices.
Therefore, the dependence of z on w can be expressed as: z = Fℓ(P, K) w, where Fℓ is called the lower linear fraction transformation. The goal of H∞ control design is to find a controller K that minimizes the H∞ norm of Fℓ(P, K).
Application and implementation
There are currently many commercial software on the market that support the synthesis of H∞ controllers. The emergence of these tools makes the process of designing H∞ controllers more efficient. Designers can use these software to perform progressive optimization to obtain controllers with superior performance.
However, designers using these tools still need to have a solid foundation in mathematics and a deep understanding of system behavior.
Different from simple single-input single-output (SISO) systems, the H∞ method is particularly suitable for multiple-input multiple-output (MIMO) systems, which makes it increasingly important in modern control system design. Although this technology faces many challenges in its application, such as the required mathematical model accuracy and mastery of complexity, its potential and flexibility are undoubtedly a distinctive feature of current control technology.
Looking at the various characteristics of H∞ technology, its importance in control theory cannot be underestimated. With the advancement of technology, will more intuitive and easy-to-use H∞ control design tools appear in the future to change the current design method?
