Uppsala universitet


Signals and Systems, Uppsala University

Researchers: J Öhr , M Sternad and A Ahlén

Anti-Windup Design for Controllers with Actuator Limitations

Anti-windup compensation modifies the dynamics of a control loop when control signals saturate. The aim is to attain a good transient behaviour after desaturation, while nonlinear oscillations and repeated saturations are avoided.

It is hard to address these issues with presently available methods. It would not be adequate to take just the states of the controller into account, as many methods do. The whole loop around the saturation must be considered. Some more advanced schemes, such as the conditioning technique of Hanus, lack adjustable parameters. Others, such as the observer-based method of Åström and Wittenmark do have adjustable parameters, but there exist no tools for adjusting them in a systematic way.

A methodology aimed at overcoming these difficulties was developed for scalar problems by Stefan Rönnbäck and Mikael Sternad. Based on these results, we are presently developing model-based anti-windup compensation strategies for regulators with multiple control signals.

We consider discrete-time multivariable, stable or marginally stable systems with m inputs and p outputs, parameterized in rational fractional form as
 y(k) = B(q)A<sup>-1</sup>(q)v(k) ;
              v(k) = sat[u<sub>w</sub>(k)]
Here, "sat[ . ]" denotes a set of m saturations, B is a stable rational matrix in the forward shift operator q and A is assumed to be a diagonal stable rational matrix. We assume A-1 to be stable or marginally stable. For anti-windup compensation, it is crucial that the saturated control signals, or models thereof, are fed back into the controller. We introduce a general linear controller structure with three degrees of freedom, with feedback from v(k) as well as from y(k),
u<sub>w</sub>(k) = [I-W(q)R(q)]v(k) 
                + W(q)[T(q)r(k) - S(q)y(k)]
where r(k) is the reference vector for y(k) and uw(k) is the control signal before the saturation elements. The above formulation can be used to represent state-space models and controllers as well as input-output descriptions. Above, W, R, S and T are stable rational matrices in the forward shift operator q, of appropriate dimension. In particular, W is the "anti-windup filter". It affects the properties of the system during and after control saturation events, but has no effect if saturation does not occur.

Now, the dynamics of the desaturation transient should be fast. This can be achieved by appropriate choices of W, but if it is made too fast, then repeated saturations and limit cycles may occur.

We reduce this tradeoff to a problem of shaping m scalar loops: The loop gain around the bank of saturations is made diagonal. The properties of the loop around each saturation can then be tuned separately. The aim is to adjust each Nyquist curve so that it keeps an appropriate distance from the describing function of the saturation nonlinearity. We propose a systematic way of doing this, by means of solving a set of m separate scalar H2 problems, using scalar spectral factorizations.

To use this strategy for handling nonlinear actuators in multivariable systems, one only needs to master the standard conceptual tools for scalar feedback design, which are explained in undergraduate courses. This is important in an industrial setting.

The PhD thesis by Jonas Öhr, see below, develops the method outlined above further, and compares it with other proposed methods.

PhD Thesis by Jonas Öhr, 2003.
Scalar case, described in a report by Rönnbäck and Sternad 1993.
Conference paper on the performance during saturation.
Conference paper on the corresponding design for multivariable control systems.
Internal Report, with more details on the multivariable case.
Simulation tool in Simulink for the multivariable design (Master thesis)

2. Bumpless Transfer among Different Controllers

The problem of switching between different control strategies with well-controlled transients is related to the antiwindup problem. Safe switching is important e.g. when changing between PID control and more advanced schemes, and also in the implementation of gain scheduling controllers.

Together with Stefan Graebe, we have studied this problem in the scalar case , and are presently developing a design tool for taking care of both anti-windup and bumpless transfer in one design step.

3. Robust Filtering and Control

We study probabilistic approaches for obtaining performance robustness in observers and multivariable controllers. Click here for more details.