H-infinity regulation

The H _∞ scheme is a method for system analysis and synthesis controller in the field of robust control techniques . To use 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 area of ​​SISO and MIMO-LTI systems, the expandability to non-linear problems and, with a good design, very robust high-performance control results with guarantee of stability.

In the case of model-based controller design, uncertainties that arise from the creation of the model always flow into the control. A control can then be described as robust if it is insensitive to these model inaccuracies, i.e. the control quality is not severely impaired or the stability is even endangered. The basis of the H _∞ design is the modeling of the known model uncertainties, which leads to an extended transfer function, which is then the basis for the numerical calculation of the H _∞ controller. The designation "H _∞ " comes from the mathematical theory on which the method is based and designates the vector norm of a Hardy function space .

The H _∞ norm

SISO supremum

The norm is a vector norm for the Hardy space with . In the mathematical context, Hardy spaces represent special cases of the -anach spaces , in which holomorphic functions can be examined for their integrability. The space accordingly contains all holomorphic functions (each function value is differentiable in complex) that are bounded in the upper right half of the complex plane ( ) . A mathematical norm that is assigned to a room characterizes the "size" of an object in this room, e.g. B. the length or the amount of a vector. ${\displaystyle {\mathcal {H}}_{\infty }}$ ${\displaystyle {\mathcal {H}}^{p}}$ ${\displaystyle p\to \infty }$ ${\displaystyle {\mathcal {L}}^{p}}$ ${\displaystyle {\mathcal {H}}_{\infty }}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle Re(s)>0}$

In the case of the transfer functions of interest in systems engineering, the standard describes the maximum value of the amplitude response of an investigated transfer function. In the SISO case this simply means: ${\displaystyle {\mathcal {H}}_{\infty }}$

${\displaystyle \|G(s)\|_{\infty }={\underset {\omega }{sup}}\left|G(j\omega )\right|}$

The general calculation rule of the Supremum is:

${\displaystyle {\underset {\omega }{sup}}\left|G(j\omega )\right|=\lim _{p\to \infty }\left(\int _{-\infty }^{\infty }\left|G(j\omega )\right|^{p}\,d\omega \right)^{1/p}}$

In the MIMO case, however, the maximum singular value of the transfer matrix is ​​determined:

${\displaystyle \|{\underline {G}}(s)\|_{\infty }={\underset {\omega }{sup}}\ {\overline {\sigma }}(\left|{\underline {G}}(j\omega )\right|)}$

Modeling the uncertainties

The modeling of the uncertainties that exist when creating the model is the basis for the later controller design. It is important to proceed carefully here because optimizing in the wrong direction can do more damage than synthesizing robust control. Model uncertainties can be divided into parametric and dynamic uncertainties:

Parametric uncertainties

According to their name, parametric uncertainties are fluctuating or generally variant parameters during model identification. Such an uncertainty is represented by a nominal value of the uncertain parameter plus an uncertainty term: ${\displaystyle p}$

${\displaystyle p=p_{nom}\cdot (1+\eta _{p}\delta _{p})}$     with      and  ${\displaystyle |\delta _{p}|\leqq 1\qquad }$ ${\displaystyle \delta _{p},\eta _{p},p,p_{nom}\in \mathbb {R} }$

Here are a dimensionless relative fluctuation, a nominal value of the parameter (usually in the middle of the fluctuation range) and the uncertainty variable . When multiplied, the following can be replaced with the parameter uncertainty: ${\displaystyle \eta _{p}}$ ${\displaystyle p_{nom}}$ ${\displaystyle \delta _{p}}$ ${\displaystyle \eta _{p}\cdot p_{nom}}$ ${\displaystyle W_{p}}$

${\displaystyle p=p_{nom}+W_{p}\delta _{p}}$

Dynamic uncertainties

Additive and multiplicative dynamic uncertainty

Dynamic uncertainties arise from dynamics that are not taken into account in the model identification or that are lost when the model order is reduced. Dynamic uncertainties are frequency dependent and can exist in different ways. The figure on the right shows a system with an additive and a multiplicative uncertainty with the respective uncertainty weight and the uncertainty matrix. ${\displaystyle W}$ ${\displaystyle \Delta }$

Depending on the type of dynamic uncertainty, the unsafe system or the unsafe system matrix with respect to the nominal system is formed as follows (each with ): ${\displaystyle \|{\underline {\Delta }}_{G}\|_{\infty }\leqq 1}$

• Multiplicative uncertainty at the entrance:   ${\displaystyle {\underline {G}}={\underline {G}}_{nom}\cdot ({\underline {I}}+{\underline {\Delta }}_{G}{\underline {W}}_{G})}$

• Multiplicative uncertainty at the exit:   ${\displaystyle {\underline {G}}=({\underline {I}}+{\underline {\Delta }}_{G}{\underline {W}}_{G})\cdot {\underline {G}}_{nom}}$

• Multiplicative uncertainty inversely at the input:   ${\displaystyle {\underline {G}}={\underline {G}}_{nom}\cdot ({\underline {I}}-{\underline {\Delta }}_{G}{\underline {W}}_{G})^{-1}}$

• Multiplicative uncertainty inversely at the output:   ${\displaystyle {\underline {G}}=({\underline {I}}+{\underline {\Delta }}_{G}{\underline {W}}_{G})^{-1}\cdot {\underline {G}}_{nom}}$

Additives can be transformed into multiplicative uncertainties.

Linear Fraction Representation

Complete system in LFC representation

After defining the uncertainties and before using the algorithms for controller synthesis, the system model and the uncertainties must first be transferred to the LFR. With the LFR, virtual inputs and outputs are added to the equations of state in order to eliminate the unknown uncertainty values ​​or to separate them from the known values. At the end there is the following LFC system with the controller matrix (lower LFC) and the uncertainty matrix (upper LFC), see figure on the right. There are: ${\displaystyle {\mathcal {H}}_{\infty }}$ ${\displaystyle \delta _{i}}$ ${\displaystyle {\underline {K}}}$ ${\displaystyle {\underline {\Delta }}}$

• ${\displaystyle w,z}$ added virtual inputs and outputs
• ${\displaystyle d,e}$ external disturbance, weighted virtual error
• ${\displaystyle u,y}$ Control and feedback vector
• ${\displaystyle K}$ Controller matrix

example

The damping of a -link is uncertain: ${\displaystyle PT_{2}}$

${\displaystyle T^{2}{\ddot {x}}+2dT{\dot {x}}+x=Ku}$    With   ${\displaystyle d=d_{nom}\cdot (1+\eta _{d}\delta _{d})}$

${\displaystyle {\ddot {x}}={\frac {Ku}{T^{2}}}-{\frac {2d_{nom}}{T}}{\dot {x}}-{\frac {d_{nom}\eta _{d}}{T}}\underbrace {{\dot {x}}\delta _{d}} _{{\mathrel {\widehat {=}}}w_{d}}-{\frac {x}{T^{2}}}}$

The virtual entrance has been added to eliminate . The output is added to match , so that the return can be closed and no loss of information occurs despite the elimination. ${\displaystyle \delta _{d}}$ ${\displaystyle w_{d}}$ ${\displaystyle z_{d}={\dot {x}}\rightarrow w_{d}=z_{d}\delta _{d}}$

From the insecure system

${\displaystyle \left({\begin{matrix}{\dot {x}}\\{\ddot {x}}\\\hline y\end{matrix}}\right)=\left({\begin{matrix}0&1\\{\frac {-1}{T^{2}}}&{\frac {-2d}{T}}\\\hline 1&0\end{matrix}}\right.\left|{\begin{matrix}0\\{\frac {K}{T^{2}}}\\\hline 0\end{matrix}}\right)\left({\begin{matrix}x\\{\dot {x}}\\\hline u\end{matrix}}\right)=\left({\begin{matrix}{\underline {A}}\\\hline {\underline {c}}\end{matrix}}\right.\left|{\begin{matrix}{\underline {b}}\\\hline d\end{matrix}}\right)\left({\begin{matrix}{\underline {x}}\\\hline u\end{matrix}}\right)}$

the extended system will (also or ): ${\displaystyle {\underline {P}}}$ ${\displaystyle {\underline {G}}_{erweitert}}$ ${\displaystyle {\underline {G}}_{augmented}}$

${\displaystyle \left({\begin{matrix}{\dot {x}}\\{\ddot {x}}\\\hline z_{d}\\y\end{matrix}}\right)=\underbrace {\left({\begin{matrix}0&1\\{\frac {-1}{T^{2}}}&{\frac {-2d_{nom}}{T}}\\\hline 0&1\\1&0\end{matrix}}\right.\left|{\begin{matrix}0&0\\{\frac {-d_{nom}\eta _{d}}{T}}&{\frac {K}{T^{2}}}\\\hline 0&0\\0&0\end{matrix}}\right)} _{\underline {P}}\left({\begin{matrix}x\\{\dot {x}}\\\hline w_{d}\\u\end{matrix}}\right)}$

The procedure is analogous for dynamic uncertainties. If there are several uncertainties, a diagonal matrix is ​​created with the uncertainty values ​​(scalar and / or frequency-dependent) on the diagonal. The unknown uncertainty values ​​are now separated from the route in the upper LFR structure, see figure above. If they cannot be quantified, they have to be neglected when designing the controller, which is usually the case. ${\displaystyle {\underline {\Delta }}}$

H _∞ - controller synthesis

Extended system and regulator in lower LFR

In the lower LFC representation, the extended path with the controller matrix is ​​shown as in the illustration on the right. The aim of the design is to minimize the energy transfer from to or to achieve a suboptimum by falling below a value . In other words, this means that external influences have the least possible impact on the system. The minimization problem expressed in the norm is now: ${\displaystyle {\underline {P}}}$ ${\displaystyle {\underline {K}}}$ ${\displaystyle d}$ ${\displaystyle e}$ ${\displaystyle \gamma }$ ${\displaystyle {\mathcal {H}}_{\infty }}$

${\displaystyle \|{\underline {F}}_{l}({\underline {P}},{\underline {K}})\|_{\infty }<\gamma }$

${\displaystyle {\underline {F}}_{l}}$is the transfer matrix of on and is also referred to as the cost function. ${\displaystyle d}$ ${\displaystyle e}$

With the given extended system matrix

${\displaystyle {\underline {P}}=\left({\begin{matrix}{\underline {A}}\\\hline {\underline {C}}_{1}\\{\underline {C}}_{2}\end{matrix}}\right.\left|{\begin{matrix}{\underline {B}}_{1}&{\underline {B}}_{2}\\\hline 0&{\underline {D}}_{12}\\{\underline {D}}_{21}&0\end{matrix}}\right)}$

the problem can now be solved using various numerical approaches if the following conditions are met:

• ${\displaystyle {\underline {A}},{\underline {B}}_{1}}$can be stabilized, can be observed ${\displaystyle {\underline {C}}_{1},{\underline {A}}}$
• ${\displaystyle {\underline {A}},{\underline {B}}_{2}}$can be stabilized, can be observed ${\displaystyle {\underline {C}}_{2},{\underline {A}}}$
• ${\displaystyle {\underline {D}}_{12}^{T}\left({\begin{matrix}{\underline {C}}_{1}&{\underline {D}}_{12}\end{matrix}}\right)=\left({\begin{matrix}0&{\underline {I}}\end{matrix}}\right)}$
• ${\displaystyle \left({\begin{matrix}{\underline {B}}_{1}\\{\underline {D}}_{21}\end{matrix}}\right){\underline {D}}_{21}^{T}=\left({\begin{matrix}0\\{\underline {I}}\end{matrix}}\right)}$

The two most common options for solving the optimization problem are, on the one hand, the LMI method ( linear matrix inequality ) and, on the other hand, the solution of two algebraic matrix Riccati equations . The sequence of the latter is to be shown briefly, the solution is only possible numerically. The design can be repeated iteratively to make it as small as possible. The resulting controller has the same number of states as the extended system . ${\displaystyle \gamma }$ ${\displaystyle {\underline {P}}}$

To solve the two Riccati equations, the following must apply (all capital letters are matrices):

1. There is such a thing ${\displaystyle X_{\infty }}$ ${\displaystyle X_{\infty }A+A^{T}X_{\infty }+X_{\infty }\left({\frac {1}{\gamma ^{2}}}B_{1}B_{1}^{T}-B_{2}B_{2}^{T}\right)X_{\infty }+C_{1}^{T}C_{1}{\overset {!}{=}}0}$
2. There is such a thing ${\displaystyle Y_{\infty }}$ ${\displaystyle AY_{\infty }+Y_{\infty }A^{T}+Y_{\infty }\left({\frac {1}{\gamma ^{2}}}C_{1}C_{1}^{T}-C_{2}C_{2}^{T}\right)Y_{\infty }+B_{1}^{T}B_{1}{\overset {!}{=}}0}$
3. The greatest eigenvalue of the found or is smaller than : ${\displaystyle X_{\infty }}$ ${\displaystyle Y_{\infty }}$ ${\displaystyle \gamma ^{2}}$ ${\displaystyle \rho \left(X_{\infty }Y_{\infty }\right)<\gamma ^{2}}$

With the found matrices and the controller matrix is ​​finally synthesized . ${\displaystyle X_{\infty }}$ ${\displaystyle Y_{\infty }}$ ${\displaystyle {\underline {K}}=\left({\begin{matrix}{\underline {A}}_{\infty }\\\hline {\underline {F}}_{\infty }\end{matrix}}\right.\left|{\begin{matrix}-{\underline {Z}}_{\infty }{\underline {L}}_{\infty }\\\hline 0\end{matrix}}\right)}$

There are:

${\displaystyle A_{\infty }=A+{\frac {1}{\gamma ^{2}}}B_{1}B_{1}^{T}X_{\infty }+B_{1}F_{\infty }+Z_{\infty }C_{2}}$,

${\displaystyle F_{\infty }=-B_{2}^{T}X_{\infty }}$,

${\displaystyle L_{\infty }=-Y_{\infty }C_{2}^{T}}$,

${\displaystyle Z_{\infty }={\left(I-{\frac {1}{\gamma ^{2}}}X_{\infty }Y_{\infty }\right)}^{-1}}$

See also

• Robust regulation
• Furniture transformation

literature

• Sigurd Skogestad, Ian Postlethwaite: Multivariable Feedback Control: Analysis and Design. Wiley, New York 2005, ISBN 0-470-01167-X .
• Huibert Kwakernaak: Robust Control and H _∞ Optimization Tutorial. (PDF; 1.5 MB) Pergamon Press, 1992.
• Keith Glover, John Doyle: State-space formulas for all stabilizing controllers that satisfy an H _∞ -norm bound and relations to risk sensitivity. (PDF; 531 kB) In: System & Control Letters. 11, 1988, pp. 167-172.