QL regulator

The QL controller or linear-quadratic controller , also called Riccati controller , is a state controller for a linear dynamic system , the feedback matrix of which is determined by minimizing a quadratic cost function. Its synthesis is therefore a sub-problem of optimal regulation .

General

A common method for designing a state controller is by pole placement . The eigenvalues of the closed circuit and thus its dynamics are specifically specified. The disadvantages of this method are that the quality of individual states is not placed in the foreground and manipulated variable limits and the manipulation effort can only be taken into account indirectly. However, both are often desired in practical application and are made possible by the QL controller. A quality functional of the following form is assumed for a finite time horizon : ${\ displaystyle T}$

{\ displaystyle J (u, x_ {0}) = {\ frac {1} {2}} x (T) ^ {\ top} \ mathbf {S} x (T) + {\ frac {1} {2 }} \ int \ limits _ {0} ^ {T} x (t) ^ {\ top} \ mathbf {Q} x (t) + u (t) ^ {\ top} \ mathbf {R} u (t ) \ dt}

The states and manipulated variables are each included as a square. With the weighting matrices , and the final state values as well as the state and manipulated variable trajectories are prioritized. ${\ displaystyle \ mathbf {S}}$ ${\ displaystyle \ mathbf {Q}}$ ${\ displaystyle \ mathbf {R}}$

Meaning of the weighting matrices

${\ displaystyle \ mathbf {Q}}$ : With the diagonal elements of this square matrix the speed with which the individual states are driven towards zero can be determined. The other matrix elements do not allow any direct interpretation of their effect on the system behavior and are therefore usually chosen to be zero. An additional condition is that it must be observable what can be achieved by occupying all diagonal elements with values greater than zero (positive definite matrix). ${\ displaystyle (\ mathbf {A}, \ mathbf {Q})}$

${\ displaystyle \ mathbf {R}}$ : If the system has only one input, it is a scalar, otherwise a symmetrical matrix. In practice, diagonal matrices are often used. The larger the diagonal elements are selected in this case, the smaller the manipulated variables are kept and the slower the regulation becomes. Due to the necessary inversion, it must be positive definite (the diagonal elements must be greater than zero). ${\ displaystyle \ mathbf {R}}$ ${\ displaystyle \ mathbf {R}}$

${\ displaystyle \ mathbf {S}}$ : This matrix is again quadratic and positive semidefinite . When considering a finite time horizon, it serves to minimize the final values of the states if the time is not sufficient to drive the states to zero. When considering an infinite time horizon, this weighting does not apply, since the states for must tend towards zero, since otherwise the integral would not converge. ${\ displaystyle t \ to \ infty}$

In addition, the integrand can be expanded to include the coupled condition in the sense of the first binomial formula . ${\ displaystyle 2x (t) ^ {\ top} \ mathbf {N} u (t)}$

The engineer can therefore set separately for each state and each control input the importance with which it should be driven towards zero or kept small. Although the solution to the problem results in a controller that optimally fulfills the quality requirement, it is still up to the developer to fine-tune the settings to his satisfaction through the selection of the matrix elements. Thus, the design of the QL controller usually remains an iterative process.

Controller synthesis

The basis of the consideration is a linear time-invariant system :

{\ displaystyle {\ dot {x}} = \ mathbf {A} x + \ mathbf {B} u}

To solve the problem, the quality integral has to be solved, which is relatively easy to do through partial integration . The system states are replaced with the initial state and the matrix exponent zu as well as the control input with the states fed back via the controller matrix zu . The integration first leads to the Lyapunov equation , in which the controller matrix must finally be replaced. Since the integrated cost function is quadratic, it has exactly one global minimum at the point where the derivative according to the controller coefficients is zero. This derivation, executed and transformed, finally results in the algebraic Riccati equation for the case of the infinite time horizon : ${\ displaystyle x (t)}$ ${\ displaystyle x_ {0} \ e ^ {\ mathbf {A} t}}$ ${\ displaystyle u}$ ${\ displaystyle - \ mathbf {K} ^ {\ top} x (t)}$

{\ displaystyle \ mathbf {PA + A ^ {\ top} P-PBR ^ {- 1} B ^ {\ top} P + Q} = 0}

After resolving, you get to the optimal QL controller matrix: ${\ displaystyle \ mathbf {P}}$

{\ displaystyle \ mathbf {K = R ^ {- 1} B ^ {\ top} P}}

.

In the case of the finite time horizon , and thus also the resulting controller matrix, are time-dependent and the algebraic Riccati equation becomes the Riccati differential equation : ${\ displaystyle T}$ ${\ displaystyle \ mathbf {P}}$

{\ displaystyle \ mathbf {-P (t) AA ^ {\ top} P (t) + P (t) BR ^ {- 1} B ^ {\ top} P (t) -Q = {\ dot {P }} (t)}}

With

{\ displaystyle \ mathbf {P} (T) = \ mathbf {S}}

Remarks

The QL controller design can be viewed as an automated process that generates a state controller that optimally meets the specified criteria.
The weightings to be defined are clear in their effects and manipulated variable limits or individual states can be observed in a targeted manner.
In contrast to the pole placement, the QL controller design is also clear in the multivariable case .
To solve the Riccati equations, numerical methods are generally required.
The resulting controller is stable , but like all linear, static state controllers , it does not inherently have any steady-state accuracy, which can be compensated for with a pre-filter or a command integrator.

literature

Jan Lunze: Control engineering 2: multi-variable systems, digital control . 5th edition. Springer, 2008, ISBN 978-3-540-78462-3 , pp. 669 .

Holger Lutz, Wolfgang Wendt: Pocket book of control engineering with MATLAB and Simulink . 11th edition. Verlag Europa-Lehrmittel, 2019, ISBN 978-3-8085-5869-0 , pp. 729 .