H-infinity methods in control theory
The H ∞ control is a method for system analysis and controller synthesis in the field of robust control technology. To apply the method, the control task must be formulated as an optimization problem, which requires a relatively high mathematical effort. The advantages of the method lie in the broad applicability in the field of SISO and MIMO LTI systems, the extensibility to nonlinear problems and, with good design, very robust performance control results while ensuring the stability.
In the model-based controller design always flow uncertainties that arise from the modeling in the control. A control can be described as robust if it is insensitive to these model inaccuracies, so do not strongly affect the control performance or even the stability is at risk. The basis of the H ∞ draft is the modeling of the known model uncertainties, which leads to an extended transmission function, which is then the basis for the computation of the H ∞ control. The term " H ∞ " comes from the mathematical theory, which is the method to reason and referred to the vector norm of a Hardy function space.
- 3.1 Example
The H ∞ - norm
The norm is a vector norm for the Hardy space with. In the mathematical context represent special cases of the Hardy spaces, Banach spaces in which holomorphic functions can be examined for their integration. The room includes corresponding all holomorphic functions ( each function value is complex differentiable ), which are restricted in the upper right half of the complex plane (). A mathematical norm that is assigned to a room that characterizes the "size" of an object of that space, eg the length or magnitude of a vector.
In the case of interest in the technology transfer functions, the standard describes the maximum value of the amplitude response of an examined transfer function. In the SISO case, this simply means:
The general rule for converting the supremum is:
In the MIMO case, however, the maximum singular value of the transfer matrix is determined:
Modeling of uncertainties
The modeling of uncertainties exist in modeling is the basis for the subsequent controller design. It is important to proceed carefully here, because an optimization in a wrong way can cause more damage than that robust control is synthesized. Model uncertainties can be divided into parametric and dynamic uncertainties:
Parametric Uncertainties
Parametric uncertainties are named after fluctuating in the model identification or general variant parameters. Shown is such uncertainty by a nominal value of the uncertain parameter uncertainty plus a term:
Here are a dimensionless relative fluctuation, a nominal value of the parameter (usually in the middle of Schwankungsbreichs ) and the uncertainty variable. Multiplied out can be replaced with the parameter uncertainty:
Dynamic uncertainties
Dynamic uncertainty caused by not taken into account in the model identification or lost during a previous model order reduction dynamics. Dynamic uncertainties are frequency dependent and can be present in various ways. In the right figure, a system with an additive and a multiplicative uncertainty with the respective uncertainty weight and the uncertainty matrix is shown.
Depending on the type of dynamic uncertainty, the uncertain system and the uncertain system matrix is formed with respect to the nominal system as follows (each with ):
- Multiplicative uncertainty at the input:
- Multiplicative uncertainty at the output:
- Multiplicative inverse uncertainty at the input:
- Multiplicative inverse uncertainty at the output:
Additives can be transformed into multiplicative uncertainties.
Linear Fraction Representation
According to the definition of uncertainty and in the application of the algorithms for controller synthesis first the system model with the uncertainties in the SAR to be transferred. In LFR the state equations are added to virtual inputs and outputs in order to eliminate or separate from the known values of the unknown uncertainty values . At the end, the following LFR system with the controller matrix (lower LFR ) and the uncertainty matrix (upper LFR), see right figure. Where:
- Added virtual inputs and outputs
- External disturbance, weighted virtual error
- Setting and return Vector
- Knob Matrix
Example
The attenuation of a gate is uncertain:
For elimination of the virtual input has been added. Matching the output is added, so that the feedback can be closed and despite elimination no information loss.
For the uncertain system
Is the extended system ( or even ):
For dynamic uncertainties, the process is analogous. If there are several uncertainties is a diagonal matrix with the uncertainty values (scalar and / or frequency- dependent) on the diagonal arises. The unknown uncertainty values are now separated into upper- LFR structure of the track as shown in above figure. Can not you be quantified, they must be neglected in the controller design, which is usually the case.
H ∞ - controller synthesis
In lower- LFR representation, the extended route introduces the controller matrix as in the right figure Represents the aim of the design is to minimize the energy transfer from or to the achievement of a Suboptimums by entering a value below. In other words, this means that external influences have little impact on the system as possible. Expressed The minimization problem in the standard is now:
Is the matrix of transfer, and is also referred to as a cost function.
With the extended given system matrix
Can now be solved the problem through various numerical approaches if the following conditions are met:
- Be stabilized, is observable
- Be stabilized, is observable
The two most common ways of solving the optimization problem are firstly the LMI method (linear matrix inequality) and on the other the solution of two algebraic matrix Riccati equations. The sequence of the latter will be shown shortly, the solution is only possible numerically. The design can be repeated iteratively to achieve an as small as possible. The resulting controller has the same number of states as the extended system.
To solve the two Ricatti equations must hold (all capital letters are matrices):
With the found matrices and the controller matrix synthesized to the end.
Where: