Gain scheduling

Under gain scheduling (English, approximately:. " Operating point dependent gain setting") is meant in the control theory, an approach to control of nonlinear systems using linear mathematical models . The term gain scheduling is used for historical reasons and today no longer simply refers to the amplification of signals, but also to the regulation of other parameters of a process .

principle

Here, a classical gain scheduling by LPV gain scheduling ( linear parameter-varying control ) differed. Gain scheduling is assigned to the class of adaptive controls . The gain scheduling controller design can be summarized as follows:

1. Linearization of the non-linear system,
2. Determination of a scheduling parameter vector that maps the non-linear behavior of the process,
3. Parameterization of linear systems and design of linear controllers ,
4. Interpolation of the linear controller.

Classic gain scheduling

The classical gain scheduling method makes it possible, based on linear system theory , to design simple, tried and tested controls for non-linear control systems . Typical applications are flight controls, controls in the chemical process industry and controls in mechatronic systems. In a classical gain scheduling the starting point of the design is the linearization of the nonlinear system at a fixed point of equilibrium (engl. Equilibrium point ). The linearization at any number of fixed process points leads to a family of locally linear models. With the introduction of a scheduling parameter vector , parameterized linear models are obtained, the so-called linearization family ( family of frozentime linear time-invariant systems ). A general class of nonlinear time-invariant multivariable systems can be used with ${\displaystyle {\underline {\phi }}^{g}}$

${\displaystyle {\dot {\underline {x}}}(t)={\tilde {f}}({\underline {x}}(t),{\underline {u}}(t))}$

${\displaystyle {\underline {y}}(t)={h}({\underline {x}}(t))}$

to be discribed. Here, the state vector, the manipulated vector and the output vector and the images , are in the working area continuously differentiable. The linearization of the nonlinear system takes place with the aid of the first order Taylor series expansion around a fixed equilibrium point ( operating point ). If the nonlinear system is at an equilibrium point, one obtains for the state vector differential equation ${\displaystyle {\underline {x}}(t)\in {\mathcal {R}}^{n}}$ ${\displaystyle {\underline {u}}(t)\in {\mathcal {R}}^{p}}$ ${\displaystyle {\underline {y}}(t)\in {\mathcal {R}}^{q}}$ ${\displaystyle {\tilde {f}}:{\mathcal {R}}^{n}\times {\mathcal {R}}^{p}\rightarrow {\mathcal {R}}^{n}}$ ${\displaystyle {h}:{\mathcal {R}}^{n}\rightarrow {\mathcal {R}}^{q}}$ ${\displaystyle {\mathcal {R}}^{n\times p}}$ ${\displaystyle {\tilde {f}}\left({\underline {x}}^{g},{\underline {u}}^{g}\right)={\underline {0}}.}$

The non-linear system is linearized around a fixed equilibrium point . The linearized time-invariant system results from the Taylor series expansion ${\displaystyle ({\underline {x}}^{g},{\underline {u}}^{g})}$

${\displaystyle \Delta {\dot {\underline {x}}}(t)=A^{g}\Delta {\underline {x}}(t)+B^{g}\Delta {\underline {u}}(t)}$

${\displaystyle \Delta {\underline {y}}(t)=C^{g}\Delta {\underline {x}}(t),}$

where and are the respective Jacobi matrices at the equilibrium point. Gain scheduling controllers are adaptive controllers in which the controller parameters change as a function of the internal process variables and / or external variables. The (selected) time-dependent process variables that characterize the dynamic behavior of the non-linear system then represent the scheduling parameters or scheduling variables. These parameters are combined in the scheduling vector. Through analytical considerations at the equilibrium point, generalizations can be formulated for the determination of scheduling parameters. In the steady state (rest position) the following applies ${\displaystyle A^{g},B^{g}}$ ${\displaystyle C^{g}}$

${\displaystyle {\underline {0}}={\tilde {f}}\left({\underline {x}}^{g},{\underline {u}}^{g}\right).}$

This nonlinear system of equations is composed of n equations with ( n + p ) unknowns. The maximum number of parameters of a scheduling vector is limited to ( n + p ). The parameterization of the set of equilibrium points results in the scheduling vector . The defined set of equilibrium points can be parameterized using a scheduling vector: ${\displaystyle {\tilde {\mathcal {G}}}}$ ${\displaystyle {\underline {\phi }}^{g}\subset ({\underline {x}}^{g},{\underline {u}}^{g})\in {\mathcal {R}}^{n+p}\quad mit\quad p\leq n_{\phi ^{g}}\leq (n+p)}$ ${\displaystyle {\underline {\phi }}^{g}}$

${\displaystyle {\tilde {\mathcal {G}}}=\left\{\ ({\underline {x}}^{g},{\underline {u}}^{g})\in {\mathcal {R}}^{n+p}\quad |{\underline {x}}^{g}={\underline {z}}_{x}^{g}({\underline {\phi }}^{g}),{\underline {u}}^{g}={\underline {z}}_{u}^{g}({\underline {\phi }}^{g})\right\}}$.

Here and are images that describe the analytical dependencies between or and the parameter vector for the set of equilibrium points. The system parameterized with the scheduling vector is obtained for the linearized system ${\displaystyle {\underline {z}}_{x}^{g}:{\mathcal {R}}^{n_{\phi ^{g}}}\rightarrow {\mathcal {R}}^{n}}$ ${\displaystyle {\underline {z}}_{u}^{g}:{\mathcal {R}}^{n_{\phi ^{g}}}\rightarrow {\mathcal {R}}^{p}}$ ${\displaystyle {\underline {x}}^{g}}$ ${\displaystyle {\underline {u}}^{g}}$ ${\displaystyle {\underline {\phi }}^{g}}$

${\displaystyle \Delta {\underline {\dot {x}}}(t)=A({\underline {\phi }}^{g})\Delta {\underline {x}}(t)+B({\underline {\phi }}^{g})\Delta {\underline {u}}(t)}$

${\displaystyle \Delta {\underline {y}}(t)=C({\underline {\phi }}^{g})\Delta {\underline {x}}(t)}$.

If you choose a finite number of different equilibrium points, you get the linearization family ${\displaystyle n_{\rho }^{g}}$

${\displaystyle \Delta {\underline {\dot {x}}}(t)=A({\underline {\phi }}_{i}^{g})\Delta {\underline {x}}(t)+B({\underline {\phi }}_{i}^{g})\Delta {\underline {u}}(t)}$

${\displaystyle \Delta {\underline {y}}(t)=C({\underline {\phi }}_{i}^{g})\Delta {\underline {x}}(t),\qquad i=0,1,2,\ldots ,n_{\rho }^{g}}$.

Linear state controller design

Among the various controller structures, the status controllers with complete status feedback have also gained significant importance among users. One reason for this is the relatively simple structure. In the process, the state variables are measured, then fed back and processed in the controller. Another advantage is that the measurement of the state variables also provides more information about the process than the controlled variable . Because of the requirement for steady-state accuracy, state controllers with an I component are a practicable option. This controller can be implemented relatively easily in computer systems. Furthermore, design problems , such as those with a structure-restricted state feedback, are avoided. As an example, the design of a linear state controller with an I component using the design method pole specification can be described for a member of the linearization family . For the i linear model ${\displaystyle {\underline {y}}(t)}$

${\displaystyle \Delta {\underline {\dot {x}}}(t)=A({\underline {\phi }}_{i}^{g})\Delta {\underline {x}}(t)+B({\underline {\phi }}_{i}^{g})\Delta {\underline {u}}(t)}$

${\displaystyle \Delta {\underline {y}}(t)=C({\underline {\phi }}_{i}^{g})\Delta {\underline {x}}(t)}$

is a linear state controller with an I component ${\displaystyle \Delta {\underline {\dot {x}}}_{c}(t)=\Delta {\underline {w}}(t)-\Delta {\underline {y}}(t)}$

${\displaystyle \Delta {\underline {u}}(t)=-F({\underline {\phi }}_{i}^{g})\Delta {\underline {x}}(t)+V({\underline {\phi }}_{i}^{g})\Delta {\underline {x}}_{c}(t)=-[F({\underline {\phi }}_{i}^{g}),-V({\underline {\phi }}_{i}^{g})]\left({\begin{array}{c}\Delta {\underline {x}}(t)\\[5pt]\Delta {\underline {x}}_{c}(t)\end{array}}\right)}$

${\displaystyle \Delta {\underline {u}}(t)=-F^{e}({\underline {\phi }}_{i}^{g})\left({\begin{array}{c}\Delta {\underline {x}}(t)\\[5pt]\Delta {\underline {x}}_{c}(t)\end{array}}\right)\quad mit\quad F^{e}({\underline {\phi }}_{i}^{g})=[F({\underline {\phi }}_{i}^{g}),-V({\underline {\phi }}_{i}^{g})]}$

to design. For the controlled system expanded by the I component of the controller , this results

${\displaystyle \underbrace {\left({\begin{array}{c}\Delta {\dot {\underline {x}}}(t)\\[5pt]\Delta {\dot {\underline {x}}}_{\!\!c\,}(t)\end{array}}\right)} _{\Delta {\dot {\underline {x}}}^{e}}=\underbrace {\left[{\begin{array}{cc}A({\underline {\phi }}_{i}^{g})&{\underline {0}}\\[5pt]-C({\underline {\phi }}_{i}^{g})&{\underline {0}}\end{array}}\right]} _{A^{e}({\underline {\phi }}_{i}^{g})}\left({\begin{array}{c}\Delta {\underline {x}}(t)\\[5pt]\Delta {\underline {x}}_{\!\!c\,}(t)\end{array}}\right)+\underbrace {\left({\begin{array}{c}B({\underline {\phi }}_{i}^{g})\\[5pt]{\underline {0}}\end{array}}\right)} _{B^{e}({\underline {\phi }}_{i}^{g})}\Delta {\underline {u}}(t)+\underbrace {\left({\begin{array}{c}{\underline {0}}\\[5pt]{\underline {I}}\end{array}}\right)} _{E}\Delta {\underline {w}}(t)}$

${\displaystyle \Delta {\underline {y}}^{e}(t)=\underbrace {[C({\underline {\phi }}_{i}^{g})\quad {\underline {0}}]} _{C^{e}({\underline {\phi }}_{i}^{g})}\left({\begin{array}{c}\Delta {\underline {x}}(t)\\[5pt]\Delta {\underline {x}}_{c}(t)\end{array}}\right)}$.

The following then applies to the closed control loop

${\displaystyle \Delta {\dot {\underline {x}}}^{e}(t)=A^{e}({\underline {\phi }}_{i}^{g})\Delta {\underline {x}}^{e}(t)-B^{e}({\underline {\phi }}_{i}^{g})F^{e}({\underline {\phi }}_{i}^{g})\Delta {\underline {x}}^{e}(t)+E\Delta {\underline {w}}(t)}$

${\displaystyle \Delta {\underline {y}}^{e}(t)=C^{e}({\underline {\phi }}_{i}^{g})\Delta {\underline {x}}^{e}(t)}$.

In order to be able to guarantee (asymptotic) stability of the i-th closed control loop at the equilibrium point , the eigenvalues ​​of the system matrix of the closed control loop must lie in the left half of the complex plane, i.e. H. . The eigenvalues ​​are the zeros of the characteristic polynomial . If this polynomial is to have the (desired) eigenvalues , then must ${\displaystyle A_{cl}^{e}({\underline {\phi }}_{i}^{g})=A^{e}({\underline {\phi }}_{i}^{g})-B^{e}({\underline {\phi }}_{i}^{g})F^{e}({\underline {\phi }}_{i}^{g})}$ ${\displaystyle \Re \lbrace \lambda _{j}(A_{cl}^{e}({\underline {\phi }}_{i}^{g}))\rbrace <0\qquad \forall j}$ ${\displaystyle det[s{\underline {I}}-A_{cl}^{e}({\underline {\phi }}_{i}^{g})]}$ ${\displaystyle \lambda _{j}}$

${\displaystyle det[sI-A_{cl}^{e}({\underline {\phi }}_{i}^{g})]~{\stackrel {!}{=}}~\prod _{j=1}^{\tilde {n}}(s-\lambda _{j})}$ be valid.

If you think of calculating the determinant and multiplying the product, you get a complex (nonlinear) equation. However, in particular for the multivariable case considered here, the calculation of the elements of the controller matrix is very complex. Only in the single- variable case is it relatively easy to determine the elements of the controller vector , that is, by comparing coefficients . ${\displaystyle F_{i}^{e}({\underline {\phi }}^{g})}$ ${\displaystyle f_{i}^{e}({\underline {\phi }}^{g})}$

Interpolation of the linear state controller

The last step of the gain scheduling controller design, i.e. the interpolation of the linear controllers, leads directly to the non-linear overall controller. The choice of the interpolation method is therefore of particular importance for the design of a gain scheduling controller. The following interpolation variants are favored:

• Analytical interpolation
• Linear interpolation of two controllers
• Local controller network based on standardized Gaussian radial basic functions .

Interpolation by means of a controller network on the basis of standardized Gaussian radial basis functions

In a controller network, the local controller parameter matrices with the weighting functions, i. H. the so-called validity functions , linked (multiplied). The controller parameter matrices of the locally designed controllers are weighted depending on the distance from the respective design process point using the validity functions. A gain scheduling state controller with an I component designed in this way can be specified as follows ${\displaystyle F({\underline {\phi }}_{i}^{g}),V({\underline {\phi }}_{i}^{g})}$ ${\displaystyle {\rho }_{1},\ldots ,{\rho }_{n_{\rho }^{g}}}$

${\displaystyle {\dot {\underline {x}}}_{c}(t)={\underline {w}}(t)-{\underline {y}}(t)}$

${\displaystyle {\underline {u}}(t)=-\sum _{i=1}^{n_{\rho }^{g}}\rho _{i}({\underline {\phi }}({\underline {x}}(t)))F({\underline {\phi }}_{i}^{g}){\underline {x}}(t)+\sum _{i=1}^{n_{\rho }^{g}}\rho _{i}({\underline {\phi }}({\underline {x}}(t)))V({\underline {\phi }}_{i}^{g}){\underline {x}}_{c}(t).}$

The validity functions must meet the condition

${\displaystyle \sum _{i=1}^{n_{\rho }^{g}}\rho _{i}({\underline {\phi }}({\underline {x}}(t)))=1\qquad \forall {\underline {\phi }}({\underline {x}}(t))\in \Phi }$

suffice. The validity functions are values ​​between

${\displaystyle \rho _{i}({\underline {\phi }}({\underline {x}}(t)))\in [0;1]}$

assigned. Continuous, convex functions are primarily used as validity functions . An example of this is the Gaussian function (Gaussian distribution curve), which is also referred to as the normal distribution and generally specifies the probability distribution for certain randomly occurring events. The Gaussian radial basis function can be specified as follows

${\displaystyle \mu _{i}({\underline {\phi }}({\underline {x}}(t)))=exp\left[-\left({\underline {\phi }}({\underline {x}}(t))-{\underline {c}}_{i}\right)^{T}{\tilde {\xi }}_{i}^{-2}\left({\underline {\phi }}({\underline {x}}(t))-{\underline {c}}_{i}\right)\right],\qquad {\underline {\phi }}(t)\in \Phi .}$

${\displaystyle {\tilde {\xi }}_{i}}$is a positive definite diagonal matrix

${\displaystyle {\tilde {\xi }}_{i}={\begin{bmatrix}{\sigma _{i,1}}&0&\ldots &0\\0&{\sigma _{i,2}}&\ldots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\ldots &{\sigma _{i,n_{\phi }^{g}}}\end{bmatrix}}.}$

The standard deviations to be determined in the diagonal matrix can be determined with ${\displaystyle {\tilde {\xi }}_{i}}$ ${\displaystyle \sigma _{i,j},\quad j=1,2\ldots ,n_{\phi }^{g}}$

${\displaystyle \sigma _{ij}=\vartheta _{\sigma }{\frac {1}{N}}\sum _{k=1}^{N}|{c}_{ij}-{\tilde {c}}_{ik}|}$

determine. They are the neighboring points of the respective center (equilibrium point) . Since the proportionality factor has an active influence on the dimensions and shapes of the individual basic functions, it should be in the range . So that the sum of all validity functions fulfills requirement one , the radial basis functions have to be normalized ${\displaystyle {\tilde {c}}_{ik}}$ ${\displaystyle N}$ ${\displaystyle {c}_{ij}}$ ${\displaystyle \vartheta _{\sigma }}$ ${\displaystyle 0<\vartheta _{\sigma }<1}$

${\displaystyle \rho _{i}({\underline {\phi }}({\underline {x}}(t)))={\frac {\mu _{i}({\underline {\phi }}({\underline {x}}(t)))}{\sum _{i\,=1}^{n_{\rho }^{g}}\mu _{i}({\underline {\phi }}({\underline {x}}(t)))}}.}$

literature

• Jürgen Adamy: Nonlinear Systems and Controls. SpringerVieweg Verlag, 2nd edition 2014, pp. 273ff. ISBN 978-3-642-45013-6 .
• Sebastian Engell: Design of non-linear controls . Oldenbourg Wissenschaftsverlag, 2002, ISBN 978-3-486-23065-9 .

Individual evidence

1. ↑ DJ Leith and WE Leithead: Survey of gain-scheduling analysis and design. International journal of control , 73 (11), pp. 1001-1025, November 2000 (English).
2. ↑ J. Adamy: Nonlinear Systems and Controls . SpringerVieweg Verlag, 2nd edition 2014, pp. 273 ff. ISBN 978-3-642-45013-6 .
3. ↑ G. Roppenecker: Pole specification through state feedback . Control engineering, 29 (7), 1981, pp. 228-233.
4. ↑ P. Korba: A Gain-Scheduling Approach to Model Fuzzy Control. Dissertation, VDI Series 8 No. 837, 1999 (English).
5. ↑ G. Roppenecker: time-domain design of linear arrangements . 1st edition. Oldenbourg Verlag, Munich 1990.
6. ↑ DA Lawrence and WJ Rugh: Gain scheduling dynamic linear controllers for a nonlinear plans . automatica, 31 (3), March 1995, pp. 381-390 (English).
7. ^ JH Kelly and JH Evers: An interpolation strategy for scheduling dynamic compensators . American institute of aeronautics and astronautics, 1997, pp. 1682-1690 (English).
8. ^ RA Hyde and K. Glover: Vstol aircraft flight control system design using controllers and a switching strategy . Proceedings of the 29th conference on decision and control Honolulu, Hawaii (USA), December 1990, pp. 2975-2980 (English). ${\displaystyle h_{\infty }}$
9. ^ KJ Hunt and TA Johansen: Design and analysis of gain-scheduling control using local controller networks . International journal of control, 66 (5), May 1997, pp. 619-651 (English).
10. ^ R. Murray-Smith and TA Johansen: Local Learning in Local Model Networks. In Multiple Model Approaches to Modeling and Control , pp. 185-210. Murray-Smith, R. and Johansen, TA, Taylor and Francis, 1997 (English).
11. ↑ O. Nelles and M. Fischer: Local linearization of fuzzy models. at-Automatisierungstechnik , 47 (5), 1999, pp. 217-223.
12. ^ R. Murray-Smith and TA Johansen: Local Learning in Local Model Networks. In Multiple Model Approaches to Modeling and Control , pp. 185-210. Murray-Smith, R. and Johansen, TA, Taylor and Francis, 1997 (English).
13. ↑ U. Korn and D. Döring: PI gain scheduling controller based on a local controller network - an overview . Otto von Guericke University, Faculty for Electrical Engineering and Information Technology, Magdeburg 2001, p. 10 f.
14. ↑ KJ Hunt, R. Haas and M. Brown: Extending the functional equivalence of radial basis function networks and fuzzy inference systems . IEEE transactions on neural networks, 7 (3), May 1996, pp. 776-781 (English).
15. ↑ JSR Jang and CT Sun: Functional Equivalence Between Radial Basis Function Networks and Fuzzy Inference Systems . IEEE transactions on neural networks, 4 (1), January 1993, pp. 156-158 (English).