5 Torque and Slip Limiter
5.1 Anti-Windup Compensator - Overview
The problem of windup for proportional-integral (PI) controllers was practically recog-nised in the 1930s, and has been theoretically discussed since the 1950s. One solution to the windup problem is to reconfigure the controller, to make the controller less aggres-sive. The control input udoes not reach physical saturation and the closed-loop system behaves as designed. This solution is acceptable for torque vectoring if the vehicle drives from one place to another with constant, low performance requirements. However, per-sonal automobiles are driven in various conditions, and sometimes the operator wants to drive with high power consumption, which is impossible with a smooth controller.
5.1.1 Anti-windup compensator - the classical scheme
The classical solution to the PI controller windup problem is the so-called anti-windup compensator (AWC) [144]. The AWC uses the control input difference ∆u between the
69
70 5 Torque and Slip Limiter desired control input u and the measured control input um. The difference ∆u forms the input to the AWC and is used to suppress the windup of the integral controller part.
The general assembly is given in Figure 5.1, showing the signal interconnection between the controller, the plant and the actuator limit. The actuator saturation is modelled as
Controller u um Plant
∆u
y
r e
-Figure 5.1: Anti-windup compensator PID
um=
u foru > u u foru≤u≤u u foru < u,
(5.1) where the maximum control inputu and the minimum control input u are constant.
The AWC achieves the following properties:
• If the desireduand the applied control inputum are equal, the AWC has no effect on the control-loop system. This is an important property for maintaining all controller specifications, such as performance and robustness, in the unsaturated case.
• If the actuator is saturated, the real control input is different from the desired control input. The AWC then limits the control error of the integral controller part, and the integrator does not windup. This operation suppresses overshoots and oscillations. The system is stabilised in case of a saturated control input.
5.1.2 Anti-windup compensator - modern control theory
In modern control theory, such as optimal H∞, H2 or LPV control, every controller is calculated on the basis of a generalised plant. The controller synthesis guarantees stability and performance properties that are configured using shaping filters. When the actuator is saturated, none of these properties are guaranteed, because the plant model assumptions are no longer correct. Several methods [142], [145], [146], [147] have been developed to regain stability and performance.
In general, modern anti-windup compensators are divided into two classes [142]. The first class is based on a one-step procedure, where the input saturation is part of the controller, and the AWC is included in the controller synthesis. The controller and the anti-windup compensator are designed simultaneously. The second class is based on a two-step approach, where in the first step the controller is calculated on the basis of
5.1 Anti-Windup Compensator - Overview 71 unlimited actuators, and in the second step, the AWC is calculated on the basis of the plant, the designed controller, and the saturation limit. In the two-step approach the controller itself is not modified. The AWC is an additional system that modifies the input, the output, and sometimes the states of the controller. Modern anti-windup
de-r
ylim
ζu
ζy
∆u
u um y
K(s) G(s)
Γ(s) ulim
-Figure 5.2: Anti-windup compensator for LTI MIMO-systems, see [148]
signs are mostly based on full order AWCs, where the order of the AWC is equal to the sum of the plant and controller orders. With the limited resources of an automotive microcontroller, it is advantageous to use zero- or first-order AWCs to keep the com-putational effort as low as possible. During the eFuture project, several anti-windup concepts, such as [79], [106], [108], have been tested. Modified versions of the approach in [148] obtained the most promising results in simulations.
Weston and Postlethwaite [149] showed that the closed-loop system in Figure 5.2 can be transformed into a system with nominal linear dynamics and a non-linear control-loop. Turner and Postlethwaite [148] designed AWCs for LTI systems on the basis of this transformed system, as shown in Figure 5.3. The detailed definition of the matrix
Nominal linear system
K(s) G(s)
Gf bM M−I
ζu
u ∆u ζy
ylin y Non-linear system
r
ulin
-Figure 5.3: Dead-zone based representation of the actuator saturation, see [148]
M and the plant feedback matrix Gf b is given in [148], and the saturation is converted
72 5 Torque and Slip Limiter to a dead-zone function using
dz(u) =u−um, (5.2)
see Figure 5.2 and Figure 5.3. The transformed representation emphasises the fact that the system operates as a nominal system without saturation. When the actuator limits are reached, the non-linear loop is activated. Turner and Postlethwaite [148] minimise the L2 gain of the non-linear system to improve stability and performance in case of saturation. [148] considers: "purely static anti-windup compensators, which are, from a practical point of view, most desirable. Then these ideas are extended to the sub-optimal ’low-order’ compensators, which are often feasible for problems for which static compensators are not." Turner and Postlethwaite [148] propose the anti-windup problem as solved if:
1. when no saturation occurs, the non-linear control-loop has no effect,
2. when saturation occurs, the non-linear system isLp gain bounded forp∈[1,∞).
Furthermore, Turner and Postlethwaite [148] consider the anti-windup in Figure 5.3 as "strongly" solved by the anti-windup compensator if the operator τ : ulin 7→ ζy is well-defined.
The design [148] deals with the feedback parts of the plant G and the controller K.
Also, it assumes that the plantGis asymptotically stable, the controllerKis stabilisable and detectable, and the saturation inputum is given. The closed-loop feedback system, containingG and K, must be stable and mathematically well-posed.
Turner and Postlethwaite [148] present LMI conditions for the synthesis of the anti-windup compensator Γ(s). In their design, they partition Γ(s) as Γ(s) = Γd(s)Γs, where Γd(s) is a heuristically tuned low-pass filter. Γs is a static gain and found by the solution of a set of LMIs. Here, the LTI anti-windup compensator is extended to a gain-scheduled approach to use the advantages of the LPV design. With the LPV concept, the compensator achieves improved results for non-linear vehicle behaviour. The AWC is designed for every vertexθias an LTI-AWC. The LTI-AWCs are gain-scheduled with the scheduling parameters α from the polytopic LPV controller. From a theoretical point of view, the system does not guarantee stability or performance for varying parameters, but this configuration works well in practice and is based on the idea of gain-scheduled control.