Brief paperControllers for linear systems with bounded actuators: Slab scheduling and anti-windup☆
Introduction
Unavoidable actuator limitation has made controller design for systems with bounded actuators an important area of research for decades. Early attempts led to the development of the two-step Anti-Windup schemes, in which first a nominal linear controller is designed for the small signal region; then the nominal controller is augmented with an anti-windup loop to address the undesirable behavior that saturation creates. Recently, in Grimm, Hatfield, Teel, Turner, and Zaccarian (2003) and Mulder, Kothare, and Morari (2001), for example, methodical approaches with rigorous stability and performance guarantees have become available for anti-windup synthesis (see also Bernstein and Michel, 1995, Kapila and Grigoroadis, 2002, Stoorvogel and Saberi, 1999).
In recent years, a number of approaches in which the saturation nonlinearity is taken into account, explicitly, at the controller design stage have been developed (e.g., see Cao et al., 2002, da Silva and Tarbouriech, 2005, Hu and Lin, 2002, Lin, 1998). In the explicit approach, resulting (often nonlinear) controllers typically provide performance guarantees that are better than those of the open-loop and are applicable to open-loop unstable systems, as well. These approaches can be used for a variety of objectives (reducing or peak-to-peak gains, etc.), though the nominal or small signal controller is often not nearly as desirable as the one used in the anti-windup approach (since the latter is obtained by ignoring saturation constraints with a focus on high performance and small signals). This particular disadvantage becomes more critical if saturation is expected to be infrequent. To alleviate this undesirable characteristic, a family of controllers in some form of scheduling, in response to the closed-loop behavior to disturbances or commands, is often used (e.g. Jabbari and Kose, 2004, Kose and Jabbari, 2003, Wu et al., 2000).
The motivation for this paper are situations where a high performance nominal controller is available but the periods and severity of saturation could be significant and varied, during which guaranteed performance—better than that of the open-loop—is desired. Examples include wind hazard, secondary structural elements in earthquake engineering, or systems with an unstable open-loop.
In the first step, we follow Kose and Jabbari (2003) to obtain a continuous family of controllers with increasing levels of aggressiveness or performance which will avoid saturation for a given bound on the worst case exogenous input. For simplicity, we use the same technique to obtain the family of controllers as in Kose and Jabbari (2003). The main difference is the way this controller is implemented. Instead of reliance on ellipsoids, which tend to result in significant conservatism, we select the controller based on (state in case of state feedback) or (compensator state in case of dynamic output feedback) and seek the most aggressive controller that can be implemented among the family of controllers. Graphically, the resulting scheduling scheme relies on slab regions in the phase plane. We also discuss new insights, from establishing performance guarantees or (local) Input-to-State Stability (ISS) to the role played by the parameter that controls how fast the controller can be made more aggressive.
We use a mild assumption: an upper bound to the peak disturbance is known. Given the scheduling used, this bound can be chosen with a great deal of safety margin. Since cases where disturbances are truly unbounded are relatively rare, we consider this assumption mild.
Next, we show the most aggressive controller of the continuous family can be set to be the nominal controller designed separately without any regard to actuator bounds. Then, the rest of the family of controllers ensure stability and performance once the nominal controller is saturated, resulting in an anti-windup scheme with a scheduled structure. Scheduling has been attempted in the anti-windup approach before. For example, in Zaccarian and Teel (2004) scheduling is used to improve the system performance (transients) after it re-enters the small signal domain. However, here the proposed approach uses scheduling during nominal controller saturation.
Throughout the paper, given a matrix, we use to denote , whenever space does not allow the full expression. Similarly, in symmetric matrices some of the off diagonal terms are replaced with ‘’.
Section snippets
Preliminaries and problem definition
Consider an open-loop plant with plant state , control input with bounds , exogenous external input , and measured and controlled outputs and : The objective here is to design controllers that make the closed-loop system internally stable with a guaranteed disturbance attenuation level without violating saturation bounds. Since the aim is a method that also applies to open-loop unstable systems, such
Continuous family of controllers to avoid saturation
Here, we focus on designing a controller which avoids saturation for disturbances satisfying Assumption 1. To reduce the effects a conservative estimation for , we use a scheduling scheme that results in guaranteed bounds for disturbances smaller than , automatically, as shown below.
Before stating the technical discussion, a clarification on notation might be helpful. Here, we use parameter ‘’ as the index to denote the continuous family of Lyapunov functions, ellipsoids, and
Anti-windup via overriding controllers
In addition to reduced conservatism, a key benefit of the slab-based scheduling is the possibility of using it in a fashion similar to the anti-windup augmentation, even for unstable systems. Here, we can use the nominal controller as the most aggressive controller, corresponding to , hence establishing an anti-windup scheme. However, instead of an augmentation loops often seen in the traditional anti-windup, this technique replaces the nominal controller with a family of controllers via
Example 1: Slab versus ellipsoidal scheduling
The numerical example here, taken from Kose and Jabbari (2003), shows the improvement due to the slab conditions, as opposed to using ellipsoids of Kose and Jabbari (2003). To be consistent with Kose and Jabbari (2003), we implement the -gain version of Lemma 1. The MIs can be obtained from Kose and Jabbari (2003), only the algorithm to select the controller— in (40) of Appendix—is different. Consider an active suspension control system model of automobile shown in Fig. 1. The
Solmaz Sajjadi-Kia is a University of California President Postdoctoral fellow at the Mechanical and Aerospace Engineering department of UC San Diego. She received her B.Sc. and M.Sc., both with honors, in Aerospace Engineering from Sharif University of Technology, Iran, in 2001 and 2004, respectively. She received her Ph.D. in Mechanical and Aerospace Engineering from UC Irvine in 2009. Before starting her postdoc in UC San Diego in 2012, she was a senior research engineer in SySense Inc. Her
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Solmaz Sajjadi-Kia is a University of California President Postdoctoral fellow at the Mechanical and Aerospace Engineering department of UC San Diego. She received her B.Sc. and M.Sc., both with honors, in Aerospace Engineering from Sharif University of Technology, Iran, in 2001 and 2004, respectively. She received her Ph.D. in Mechanical and Aerospace Engineering from UC Irvine in 2009. Before starting her postdoc in UC San Diego in 2012, she was a senior research engineer in SySense Inc. Her main research interests include nonlinear control, distributed control/optimization/estimation in networked multi-agent systems, and flight dynamics and control. She was the recipient of the Graduate Dean’s Dissertation Fellowship and the Holmes Endowed Fellowship in UC Irvine, the Zonta International Amelia Earhart Fellowship for Women in Aviation in 2008, and the Irene Goldsmith scholarship when she was briefly a Ph.D. student in the Aerospace Engineering department of the University of Kansas in 2004–2005.
Faryar Jabbari obtained his Ph.D. in 1986 (UCLA, Mechanical Engineering) and has been with the Department of Mechanical and Aerospace Engineering of the University of California, Irvine (UCI). His research interests are in control systems (e.g., saturation) and applications such as civil engineering and energy systems.
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The material in this paper was partially presented at the 18th IFAC World Congress, Aug. 28–Sept. 2, 2011, Milano, Italy. This paper was recommended for publication in revised form by Associate Editor Zongli Lin under the direction of Editor Andrew R. Teel.
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