6.3 LMI-based Anti-windup Scheme for Stable LTI Systems
6.3.2 Cart-Spring-Pendulum System
The other example used to demonstrate the effectiveness of the anti-windup scheme pro-posed in [128] is an experimental cart-spring-pendulum system, described by (6.19),
satu-−1 0 1 2 3 4 5
Figure 6-9: Step response for F8 aircraft longitudinal dynamics, with large step input.
ration limits umax=−umin = 5 (V), and matrices
The outputs y = [y1, y2]T= Cx represent the cart’s displacement (in meters) from equilib-rium and pendulum angle (in radians) from the downward vertical respectively. The input u represent the motor input in Volts. The nominal controller is described by (6.20) and matrices
In contrast to Section 6.3.1, the LMI-based dynamic anti-windup compensator (6.22) defined by matrices
in [128], does reproduce the results in [128, Figs. 9 and 10]. As mentioned in [128], the conditions to construct static anti-windup compensators are infeasible.
GPAW-Compensated System
It can be verified that both the open-loop plant (6.19), (6.27), and the nominal control-ler (6.20), (6.28), are strictly stable. However, condition (5.10) was found to be numerically infeasible, so that Theorem 5.2.1 cannot be applied. This again suggests conservatism of Theorem 5.2.1. Without a systematic mechanism to determine the GPAW parameter Γ, we use an ad hoc method as before.
We will be optimizing some measure of system performance over Γ for a specified distur-bance input, similar to the approach in Section 6.3.1. First, we show that we can completely define the GPAW-compensated controller (6.24) using only 3 elements of Γ = ΓT ∈ R4×4. Using a modified Gram-Schmidt orthonormalization process [124, pp. 15 – 16], a nonsingular transformation matrix
was found, that transforms the nominal controller (6.20), (6.28), into
˙˜xc= ˜Acx˜c+ ˜Bcyy = T AcT−1x˜c+ T Bcyy,
uc= ˜Ccx˜c= CcT−1x˜c=1 0 0 0 ˜xc, (6.29) with the special form of ˜Cc (see also Section 5.5.1). Then from the closed-form expres-sions (A.7) (in Appendix A) for the GPAW-compensated controller derived from the trans-formed nominal controller (6.29), we see that Γ = [γij] appears only in the projection matrix, which, due to the special structure of ˜Cc, can be simplified to
I−Γ ˜CcTC˜c elements of Γ can always be defined in a way to ensure Γ is symmetric positive definite, so that no restrictions need to be imposed on ˜γ.
The GPAW-compensated controller derived from the transformed nominal controller (6.29) is given by
˙˜xg = RI∗(˜xg, y)( ˜Acx˜g+ ˜Bcyy),
ug = ˜Ccx˜g. (6.30)
As before, we call the resultant closed-loop system comprising (6.19) and (6.30) with u = ug the GPAW-compensated system and denote it by Σg. To determine ˜γ (and hence Γ), we
optimize (over ˜γ) the time response of Σg to the disturbance input (see (6.19))
w =
(1.588, if t∈ [0, 0.01],
0, otherwise. (6.31)
Specifically, we solve the unconstrained minimization problem min from simulation. The preceding unconstrained optimization problem was solved using the nonlinear program solver fminunc of the MATLAB R Optimization Toolbox [186], which yields a satisfactory solution
˜
γ =−0.21853 −4.9972 −3.9991T, when started with the initial guess ˜γig = [−0.2, −5, −4]T.
Remark 6.4. Numerical experience indicates that the posed optimization problem is poorly scaled, i.e. the objective is highly sensitive to ˜γ1 while being relatively insensitive to ˜γ2 and
˜
γ3. The optimization was solved repeatedly with numerous initial guesses, and the initial guess ˜γig = [−0.2, −5, −4]T was found to yield a satisfactory solution.
Numerical Results
The systems Σu, Σn, Σaw, and Σg are first simulated for the nominal disturbance in-put (6.31). The responses are shown in the left plot of Fig. 6-10. It can be seen that
0 2 4 6 8 10
Figure 6-10: Response of cart-spring-pendulum system to disturbances. The left plot shows the case for nominal disturbance while the right plot shows the case where the nominal disturbance is magnified by 50%.
without anti-windup compensation, the response of system Σn is highly oscillatory with a large settling time of approximately 10 s. The response of the GPAW-compensated sys-tem Σg is comparable to that of the LMI-based anti-windup compensated system Σaw, although Σaw exhibits a marginally superior response. Both responses of systems Σg and
Σaw are significantly superior to the response of the uncompensated system Σn, reaching steady state at approximately 5 s. Observe also that controller state-output consistency (see Theorem 2.5.3) holds for the GPAW-compensated system Σg, but not for system Σaw. Since the GPAW parameter Γ was determined by optimizing for a specific disturbance input, namely (6.31), we check its performance against a disturbance which is 50% larger in magnitude. The responses to the increased excitation are shown in the right plot of Fig. 6-10. Clearly, the GPAW-compensated system Σg exhibits satisfactory responses to disturbance inputs for which it was not optimized for.
6.4 Chapter Summary
We compared the GPAW scheme against three state-of-the-art anti-windup schemes using examples available from the literature. Because the stability results obtained thus far are too conservative to be applied, stability of the GPAW-compensated systems are not established.
Even in the absence of stability results, ad hoc methods can be devised to design the GPAW-compensated controller. We showed that the GPAW scheme achieves comparable performance against these state-of-the-art anti-windup schemes in these examples. Where current stability results are not applicable, the GPAW scheme provides practitioners with a candidate anti-windup scheme where no candidates may be available otherwise.
Chapter 7
Conclusions and Future Work
Windup induced by control saturation remains one of the major problems affecting vir-tually all practical control systems, with adverse consequences. The gradient projection anti-windup (GPAW) scheme proposed in this dissertation is a general purpose anti-windup scheme constructed for saturated nonlinear systems driven by nonlinear controllers, a topic recognized as a largely open problem. The GPAW-compensated controller achieves control-ler state-output consistency, possesses clear geometric properties, and is characterized by a passive projection operator. It is defined by either the online solution to a combinatorial optimization subproblem, a convex quadratic program, or a projection onto a convex poly-hedral cone problem. When the controller output has dimension one or two, closed-form expressions for the GPAW-compensated controller are available.
Strong results were obtained when GPAW compensation is applied to saturated first-order LTI plants driven by first-first-order LTI controllers. This simple system illustrates nu-merous attractive features of the GPAW scheme. For this system, GPAW compensation can only maintain/enlarge the ROA of the uncompensated system, a result independent of any Lyapunov function. Qualitative weaknesses of some existing anti-windup results were demonstrated. This motivates a new paradigm to address the anti-windup problem, where results relative to the uncompensated system are sought. Numerical results further suggest a need to consider asymmetric saturation constraints for general saturated systems, which has been largely ignored in the literature.
We derived some ROA comparison and stability results for GPAW-compensated MIMO nonlinear and MIMO LTI systems. These ROA comparison results represent the first steps consistent with the new anti-windup paradigm. We note that these ROA comparison results state explicit advantages of adopting GPAW compensation. This is in contrast to some existing anti-windup results where the purported advantages offered by the anti-windup scheme may be achieved by the uncompensated system.
Stability results obtained thus far are still fairly conservative. By means of non-trivial examples available in the literature, we showed that even in the absence of applicable sta-bility results, ad hoc methods can still be devised to design GPAW-compensated controllers with performance comparable to some state-of-the-art anti-windup schemes. Where current stability results are not applicable, the general purpose GPAW scheme provides practition-ers with a candidate anti-windup scheme endowed with some attractive properties.
The significance of the research presented herein has to be seen in a larger context. Con-sider the standard anti-windup structure depicted in Fig. 1-1. This anti-windup structure is essentially a generalization of that adopted in [187], which in turn is inspired by the
back-calculation method [53]. Apart from the fact that this structure preserves the unconstrained response when the saturation constraints are not triggered,1 it has no attractive inherent properties. All performance and stability properties of the anti-windup compensated sys-tem are intimately dictated by the anti-windup gains. Numerous anti-windup schemes have since been developed based on this anti-windup structure, differing only in the assump-tions imposed and method of determining the anti-windup gains [14]. Even state-of-the-art anti-windup schemes for nonlinear systems like [24, 65] adopted variants of the standard anti-windup structure of Fig. 1-1.
In contrast to the standard anti-windup structure, the GPAW scheme has several addi-tional “built-in” features (mentioned above) induced by the projection operator, and is de-fined by a single symmetric positive definite matrix parameter. In essence, the projection op-erator has endowed the GPAW-compensated controller much of its inherent properties, and the GPAW parameter is only meant to allow some “fine tuning”.2 The fact that the single unified structure of the GPAW scheme can achieve comparable performance as three state-of-the-art anti-windup schemes adopting variants of the standard anti-windup structure shows the versatility of the GPAW scheme. Moreover, these state-of-the-art anti-windup schemes cannot be applied to the simple system (4.42) used in Section 4.5.2 to demonstrate the application of the ROA comparison result, Theorem 4.4.1 (see Remark 4.11). These show the potential for the GPAW scheme to be developed into a truly general purpose anti-windup scheme with stability guarantees.
The research presented herein represents the first steps in the study of GPAW compen-sation. We invite the reader to join us in this quest to solve one of the most prevalent problems affecting control systems, in a journey that promises to be theoretically rich and practically important.
7.1 Future Work
Given the conservativeness of current stability results, there is a need to develop less con-servative results for the GPAW scheme.3 Apart from this obvious line of research, potential work that have been identified are listed below.
7.1.1 Robustness Issues due to Presence of Noise, Disturbances, Time