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    Modern Anti-windup Synthesis: Control Augmentation for Actuator Saturation
    Modern Anti-windup Synthesis: Control Augmentation for Actuator Saturation
    Modern Anti-windup Synthesis: Control Augmentation for Actuator Saturation
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    Modern Anti-windup Synthesis: Control Augmentation for Actuator Saturation

    By Luca Zaccarian and Andrew R. Teel

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    This book provides a wide variety of state-space--based numerical algorithms for the synthesis of feedback algorithms for linear systems with input saturation. Specifically, it addresses and solves the anti-windup problem, presenting the objectives and terminology of the problem, the mathematical tools behind anti-windup algorithms, and more than twenty algorithms for anti-windup synthesis, illustrated with examples. Luca Zaccarian and Andrew Teel's modern method--combining a state-space approach with algorithms generated by solving linear matrix inequalities--treats MIMO and SISO systems with equal ease. The book, aimed at control engineers as well as graduate students, ranges from very simple anti-windup construction to sophisticated anti-windup algorithms for nonlinear systems.


    • Describes the fundamental objectives and principles behind anti-windup synthesis for control systems with actuator saturation

    • Takes a modern, state-space approach to synthesis that applies to both SISO and MIMO systems

    • Presents algorithms as linear matrix inequalities that can be readily solved with widely available software

    • Explains mathematical concepts that motivate synthesis algorithms

    • Uses nonlinear performance curves to quantify performance relative to disturbances of varying magnitudes

    • Includes anti-windup algorithms for a class of Euler-Lagrange nonlinear systems

    • Traces the history of anti-windup research through an extensive annotated bibliography

    LanguageEnglish
    PublisherPrinceton University Press
    Release dateJul 11, 2011
    ISBN9781400839025
    Modern Anti-windup Synthesis: Control Augmentation for Actuator Saturation

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      Modern Anti-windup Synthesis - Luca Zaccarian

      terms.

      PART 1

      Preparation

      Chapter One

      The Windup Phenomenon and Anti-windup Illustrated

      1.1 INTRODUCTION

      Every control system actuator has limited capabilities. A piezoelectric stack actuator cannot traverse an unlimited distance. A motor cannot deliver an unlimited force or torque. A rudder cannot deflect through an unlimited angle. An amplifier cannot produce an unlimited voltage level. A hydraulic actuator cannot change its position arbitrarily quickly. These actuator limitations can have a dramatic effect on the behavior of a feedback control system.

      In this book, the term windup refers to the degradation in performance that occurs when a saturation nonlinearity is inserted, at the plant input, in an otherwise linear feedback control loop. Usually the term is reserved for the situation where this degradation is severe. The term has its origins in the fact that, among the simple analog control architectures that were used in the early days of electronic control, feedback loops with controllers that contained an integrator were the most likely to experience a severe performance degradation due to input saturation. Windup, as the term is use here, was said to occur because the saturation nonlinearity would slow down the response of the feedback loop and thus cause the integrator state to wind up to excessively large values.

      Anti-windup refers to augmentation of a controller in a feedback loop that is prone to windup so that:

      1. the closed-loop performance is unaltered when saturation never occurs, in other words, the augmentation has no effect for small signals;

      2. acceptable performance is achieved, to the extent that it is possible, even when actuator saturation occurs.

      Anti-windup synthesis refers to the design of such augmentation. This book provides principles, guidelines, and algorithms for anti-windup synthesis.

      In order to motivate anti-windup synthesis, the rest of this chapter contains examples where windup occurs. In each of these examples, alternatives to anti-windup synthesis include investing in actuators with more capabilities, or redesigning the controller from scratch to account for input saturation directly. These strategies should be considered when the control system's actuators are continuously trying to act beyond their limits. On the other hand, suppose that hitting the actuator limits is the exception rather than the rule. In addition, suppose the operating budget or some physical constraint does not permit more capable actuators. Moreover, suppose the small signal performance is highly desirable and very difficult to reproduce with control synthesis tools that account for saturation directly. In this case, anti-windup synthesis becomes a very appealing design tool: it is uniquely qualified to address saturation with potentially dramatic performance improvement using the existing actuators without sacrificing the small signal performance for the sake of guaranteeing acceptable large signal behavior. The examples will illustrate these capabilities of anti-windup synthesis, without going into the synthesis details yet. The examples will be revisited after the anti-windup synthesis algorithms have been described.

      1.2 ILLUSTRATIVE EXAMPLES

      1.2.1 A SISO academic example

      Figure 1.1 An integrator plant in negative feedback with a PID controller (a) without input saturation, (b) with input saturation, (c) with input saturation and anti-windup augmentation.

      Consider the closed-loop system resulting from using a PID (proportional + integral + derivative) controller with unity gains to control a single integrator plant, as shown in Figure 1.1a. When the force applied to the object is not limited, the closed-loop system is linear and the related response to a unitary step reference corresponds to the dashed curves in Figure 1.2. During the initial transient, the applied force exhibits a large peak. Its maximum value, which exceeds the lower plot's range, is one unit. If the maximum force that the actuator can deliver is ±0.1 units, then undesirable input and output oscillations occur, as shown by the dotted curves in Figure 1.2. Although the velocity eventually converges to the desired steady-state value, the response is very sluggish: it takes approximately 60 seconds to recover the linear performance. The output oscillations consist of rising and falling ramps that correspond to large time intervals where the force sits on either its positive or negative limit.

      Figure 1.2 Responses of the integrator control system to a unitary step reference: unconstrained response (dashed curves), response with input saturation (dotted curves), and response of the anti-windup augmented system according to Algorithm 11 (solid curves).

      Although the limits on the allowable input force imposed by saturation must cause some deviation from the ideal linear response, the large oscillations indicated by the dotted curve in Figure 1.2 are unacceptable. Since this undesirable response is induced by the large step reference input, in principle it could be avoided by shaping the reference signal so that it does not feed large and sudden changes to the control system. This solution does not address the core of the problem, however, because similar behavior will also occur whenever large enough disturbances affect the control system. Indeed, the response after 75 seconds in Figure 1.2 is due to an impulsive disturbance acting at the integrator's input, as drawn in Figure 1.1a. This impulsive disturbance resembles the action of an external element hitting the object being controlled and remaining in contact with it for a very short time interval. Mathematically, this is modeled as a very large pulse acting for a very short time.

      The effect of the impulsive disturbance on the closed loop is essentially the same as that of the step reference input. However, the reference can be shaped to avoid input saturation and its undesired consequences, while the disturbance input cannot be changed. It is therefore desirable to insert extra compensation into the control scheme, aimed at eliminating the undesirable oscillatory behavior occurring after the actuator reaches its magnitude limit, regardless of the reason for actuator saturation. For this example, Algorithm 11, which appears on page 185, has been used to illustrate the capabilities of anti-windup augmentation. On page 186, Example 7.2.4 provides details of this construction for the current example. With anti-windup augmentation, after the initial, inevitable deviation, the resulting closed-loop velocity and force signals, corresponding to the solid curves in Figure 1.2, converge rapidly to the unconstrained linear response after both the large reference change and the impulsive disturbance. In each case, the linear response is recovered after about 10 seconds. Thus, the PID controller's anti-windup augmentation, which has no effect for small signals, can induce a dramatic improvement for signals that cause input saturation.

      1.2.2 A MIMO academic example

      The simulations in this section are for a closed-loop system where the plant is a multi-input/multi-output (MIMO), two-state system with lightly damped modes in feedback with a MIMO PI controller. For details on the plant and controller, see Example 7.2.1 on page 178. An important feature of the plant model is that each input has a significant effect on both of the plant's states, which also correspond to the plant's outputs.

      Figure 1.3 Output responses for the MIMO academic example: unconstrained (dashed), saturated (dotted), and anti-windup (solid).

      Figure 1.3 shows the responses of the control system to a step reference of [0.6,0.4] for the two outputs. The dashed curves represent the response of the closed loop when no limitation is imposed on the control input. When the two inputs are limited to values between ±3 and ±10, respectively, the closed-loop response is such that the plant outputs converge to values that are far from the reference values. In turn, the driving signals to the controller's integrators approach nonzero constant values, causing the controller states to diverge, as shown in the long simulation reported in Figure 1.5. This behavior belies the fact that it is possible to almost exactly reproduce the unconstrained closed-loop response even with the given input constraints. Indeed, synthesizing anti-windup augmentation by using Algorithm 10, given on page 176, results in the anti-windup augmented closed-loop response represented by the solid line in Figure 1.3. There is very little difference between the unconstrained response and the anti-windup augmented response, while the constrained (non augmented) response is completely unacceptable. Without anti-windup augmentation, the MIMO PI controller would need to be abandoned for reference values leading to input saturation. Anti-windup design permits retaining the MIMO PI controller without modification for small reference signals and with assistance for larger reference signals.

      Figure 1.4 Input responses for the MIMO academic example: unconstrained (dashed), saturated (dotted), and anti-windup (solid).

      Figure 1.5 Diverging input and output responses for the MIMO academic example when the input is saturated.

      Figure 1.6 Response of F8 aircraft without input constraints. Output: pitch angle (thick) and flight path angle (thin). Input: elevator angle (thick) and flaperon angle (thin). Input amplitude limits NOT imposed (dotted).

      Figure 1.7 Response of F8 aircraft with physically limited elevator and flaperon angles. Output: pitch angle (thick) and flight path angle (thin). Input: elevator angle (thick) and flaperon angle (thin). Input amplitude limits (dotted).

      1.2.3 The longitudinal dynamics of an F8 airplane

      Consider a fourth-order linear model describing the longitudinal dynamics of an F8 aircraft with two inputs, elevator angle and flaperon angle, both measured in degrees, and two outputs, pitch angle and flight path angle, also measured in degrees. Additional details about this example can be found in Example 4.3.5 on page 90.

      Figure 1.8 Partial performance recovery of the F8 aircraft by doubling the amplitude limits. Output: pitch angle (thick) and flight path angle (thin). Input: elevator angle (thick) and flaperon angle (thin). Previous input amplitude limits (dotted).

      Figure 1.9 Response of F8 aircraft with anti-windup augmentation (Algorithm 4). Output: pitch angle (thick) and light path angle (thin). Input: elevator angle (thick) and laperon angle (thin). Input amplitude limits (dotted).

      Suppose a controller has been designed following an LQG/LTR methodology, so that, in the absence of input amplitude limits, the resulting closed loop has a desirable response. In particular, in the absence of input constraints, the system response to a step reference change of 10 degrees in pitch angle and flight path angle is as shown in Figure 1.6. The input plot in Figure 1.6 shows that the controller is attempting to use large input angles, especially flaperon angle, to effect this step change. If elevator and flaperon angles are limited in magnitude to 25 degrees, then the response deteriorates to what is shown in Figure 1.7. Even with the limits set to 50 degrees in magnitude, there is still some potentially undesirable oscillations in the step response, as shown in Figure 1.8.

      Nevertheless, the situation is not hopeless. The trajectory shown in Figure 1.9 corresponds to limiting the actuator angles to 25 degrees in magnitude and enhancing the original controller with anti-windup augmentation, synthesized using Algorithm 4 given on page 114, so that the small signal response is not altered and the response for large step changes is improved. In particular, the response in Figure 1.9 shows no undesirable oscillations in the pitch and flight path angles.

      Figure 1.10 A servo-positioning system.

      1.2.4 A servo-positioning system

      Consider controlling the position of a mass, an autonomous vehicle, for example, constrained to a one-dimensional path of variable elevation. The forces acting on the mass are gravity and a motor force that serves as a control input. The gravity force is state dependent, as shown in Figure 1.10, but can be modeled as an unknown external disturbance, especially when the elevation of the path is not known ahead of time.

      Figure 1.11 A small signal response of the servo-positioning system.

      The objective is to design a control system that quickly drives the mass to a given reference value with minimal overshoot and zero steady-state tracking error. For small step changes, the transition from one position to another should take no more that 0.5 seconds. Moreover, this behavior should occur for all reasonable path elevation profiles. To accomplish this control objective, a third-order linear control system containing integral action and a double lead network has been designed. See Example 7.2.5 for details. A resulting step response is shown in Figure 1.11.

      Figure 1.12 The unpredictable effects of actuator limitations on the servo-positioning system.

      Figure 1.13 The closed-loop responses of the servo-positioning system when increasing the actuator force capability by a factor of five.

      The behavior for some larger references changes is shown in Figure 1.12. The upper plot shows a step reference change from p0 to pA while the lower plot shows a step reference change from p0 to pB. When moving to the position pA, the system exhibits only a small oscillation and rapidly settles to the desired steady-state value. However, when moving to the position pB, persistent oscillations occur. In both cases, the control system asks for more force than the motors can deliver. However, the effect of the force limitations is much more severe when moving toward pB. The problem in moving to pB would not occur if using a motor with five times the force capability. Indeed, Figure 1.13 shows what would happen with such a motor. Of course, the stronger motor might still have a problem with even larger step changes. It may also be judged to be prohibitively expensive, especially when it becomes clear that, to compensate for input saturation, there is a relatively simple control software fix produced by an anti-windup augmentation algorithm.

      Figure 1.14 The closed-loop responses of the servo-positioning system after the augmentation of Algorithm 11.

      Figure 1.15 Closed-loop responses of the servo-positioning system with force limits and no additional compensation (dotted curves), without force constraints (dashed curves), and with force limits and anti-windup compensation per Algorithm 11 (solid curves).

      Using the anti-windup recipe given in Algorithm 11 on page 185 results in the response shown in Figure 1.14 for the transition from p0 to pB. The details of the synthesis for this particular system are described in Example 7.2.5 on page 187. As usual, the augmentation is such that the response for small reference changes, like in Figure 1.11, is unchanged.

      Figure 1.15 shows the transition from p0 to pB for all three scenarios considered. The dotted curve in Figure 1.15 corresponds to using the original motor and the original controller without augmentation, as in the lower plot of Figure 1.12. The dashed curve corresponds to the ideal response without input constraints, as in Figure 1.11. The solid curve corresponds to using the original motor and the original controller with anti-windup augmentation, as in Figure 1.14. Anti-windup augmentation has induced an adequate response for large reference changes without having to buy a more expensive (and heavier) motor and without compromising the response for small step changes.

      1.2.5 The damped mass-spring system

      Consider a damped mass-spring system, as shown in the diagram of Figure 1.16, where the input up is the force exerted on the mass and the output is the mass position q. See Example 7.2.6 on page 190 for additional details about this example.

      Figure 1.16 The damped mass-spring system.

      Suppose a two degrees of freedom linear controller is designed such that the mass follows a reference input while rejecting constant force disturbances d acting at the plant input. Without input force limits, the plant output and input responses would be as shown by the dashed curves in Figure 1.18. However, when the input is constrained the asymptotic tracking is lost completely as the system converges to a limit cycle with large amplitude shown in Figure 1.17. This response is also shown by the dotted curves in Figure 1.18. Applying Algorithm 12, which appears on page 189, the controller is enhanced with anti-windup augmentation and stability is recovered, while the tracking performance is only slightly deteriorated, as shown by the solid curves in Figure 1.18.

      Figure 1.17 The mass-spring response with input saturation converges to a very large limit cycle.

      Figure 1.18 The closed-loop responses of the damped mass-spring system without input constraints (dashed curves), with input constraints (dotted curves), and with anti-windup augmentation coming from Algorithm 12 (solid curves).

      Figure 1.19 A picture of the experimental spring-gantry system.

      1.2.6 The experimental spring-gantry system

      Consider the experimental spring-gantry system shown in Figure 1.19, where a pendulum hangs from a cart constrained to linear motion and the cart is attached to a fixed point via a spring. The input is the voltage applied to a DC motor that drives the cart and the outputs are the pendulum angle and cart position. A linear controller has been designed based on an LQG construction to regulate the pendulum angle to zero quickly and to attenuate small forces. For larger forces, which lead to a requested control input that substantially exceeds the voltage of the power supply, the resulting output and input trajectories are highly oscillatory. This behavior is shown in Figure 1.20, both for simulations (thick curves) and for experiments (thin curves).

      Figure 1.20 The saturated closed-loop trajectories of the spring-gantry system: simulated (thick) and experimental (thin).

      Using the anti-windup augmentation Algorithm 4, given on page 114, the performance of the system with input constraints can be improved without altering the response to small forces exerted on the pendulum. The result of such a modification is shown in Figure 1.21, both in simulation and in experiment.

      Figure 1.21 The response of the spring-gantry control system augmented with the anti-windup compensation of Algorithm 4: simulated (thick) and experimental (thin).

      1.2.7 A robot manipulator

      Consider the selective compliance assembly robot arm (SCARA). The SCARA is a common workhorse for industrial assembly tasks, typically combining components that are located on a horizontal working surface. For this task, the SCARA uses its first two rotational joints to position the tip of the robot at a desired coordinate in the horizontal plane. It uses its vertical translational joint to impose a desired tilt to the robot tip. Its last rotational joint, located at the tip, is used to impose a desired orientation angle to the robot gripper.

      Figure 1.22 Small signal response of the computed torque control scheme for the SCARA robot.

      The nonlinear coupling between the different joints in the SCARA is very strong. Thus it is difficult to design effective feedback controllers using linear control design techniques. However, since the SCARA has a motor on each of its joints, in principle it can be controlled effectively using the so-called computed torque algorithm, which is a model-based nonlinear control strategy that ignores motor torque constraints. When combined with PID feedback, the control approach can enforce a desirable linear and decoupled behavior on all the robot's joints, at least when the commanded torques do not exceed the capabilities of the motors. By suitably selecting the PID gains of this controller (see Section 10.3.3 on page 258 for details), the linear decoupled response of Figure 1.22 is obtained for motions that do not approach the limits of the robot's motors. This figure represents the four joint positions when the desired position reference step changes from(0 deg, 0 deg, 0 cm, 0 deg) to (3 deg, -2 deg, 2 cm, 2 deg).

      Figure 1.23 Unpredictable effects of actuator limitations on the SCARA robot control system.

      When commanding step reference changes that are double in size, the controller torque commands exceed the maximum torque values attainable by the motors and, as shown in Figure 1.23, the rotational joints experience persistent oscillations that could damage the robot. In Figure 1.23 only the first two joint responses are shown because they are sufficient to illustrate the windup phenomenon. It is once again evident that the effects of input limitations on an otherwise desirable control system can be unpredictable. Indeed, there is a very small threshold over which the robot's response changes from a decoupled linear response to a highly oscillatory nonlinear response.

      The possibilities for eliminating the undesired behavior seen in Figure 1.23 are the same as they were for the previous examples. One possibility is to increase the size and capabilities of the motors. For example, with actuators twice as big as those employed in Figure 1.23, the response would once again be desirable, linear, and decoupled.

      Figure 1.24 Response of the SCARA control system when increasing two times the actuators’ maximum torque.

      While increasing the size of the actuators may seem to be a reasonable approach to eliminate the behavior shown in Figure 1.23, this might result in overly high costs or weight of the robotic structure. Moreover, even with larger actuators, the same oscillatory behavior occurs when the reference is twice as big as the reference used in Figure 1.24. Another option for eliminating the large signal oscillations is to reduce the controller gains. However, this compromises the rate of convergence reported in Figure 1.22. Another possibility is to use anti-windup augmentation. Using Algorithm 25, the original computed torque plus PID control law is augmented with extra model-based dynamical elements that aim at preserving the small signal behavior of Figure 1.22 while eliminating the behavior shown in Figure 1.23. The resulting response for small signals is linear and decoupled and exactly matches the response in Figure 1.22. Moreover, the response to the same reference that caused the undesired oscillations of Figure 1.23 is well behaved and only deviates slightly from the ideal response of Figure 1.24, which corresponds to employing larger actuators. This can be verified in Figure 1.25, where the dotted curves, which are largely out of range in the plot, reproduce the response of Figure 1.23. The dashed curves represent the ideal response of Figure 1.24, and the solid curves reproduce the response of the control system augmented according to Algorithm 25. The solid curve is almost indistinguishable from the ideal, dashed curve. Once again, anti-windup augmentation solves the problem of preserving the small signal response while eliminating the deleterious behavior for large signals that results from the interaction of motor torque limitations with the original controller.

      Figure 1.25 Response of the SCARA robot control system augmented with the construction proposed in Algorithm 25 (solid) compared to the ideal linear response (dashed) and to the undesired response of Figure 1.23 (dotted).

      1.2.8 A disturbance rejection problem

      Consider a second-order plant containing an integrator subjected to both low-frequency input-matched disturbances and high-frequency output measurement noise. A schematic block diagram of the plant is shown in Figure 1.26.

      Figure 1.26 A disturbance rejection problem.

      For this plant, a controller has been designed following standard loop shaping techniques (see Example 4.4.1 reported on page 102 for details about this example) so that the closed loop guarantees asymptotic rejection of constant disturbances. Moreover, the loop gain is sufficiently large at low frequencies to guarantee a -60 dB attenuation of disturbances below wl = 0.5 rad/s, and the loop gain is sufficiently small at high frequencies to make the control system insensitive to measurement noise above wh = 100 rad/s acting at the plant output. See Example 4.4.1 reported on page 102 for additional information about the controller parameters.

      Figure 1.27 Responses of the disturbance attenuation system: response without saturation (dashed), response with saturation and no anti-windup (dotted), response with saturation and anti-windup constructed following Algorithm 3 on page 99.

      The dashed line of Figure 1.27 shows the system steady-state response to specific selections of the input disturbance and output noise. From the upper plot, which shows the plant output, it is evident that disturbance attenuation is obtained. The lower plot shows that the plant input exhibits an oscillatory behavior caused, in part, by the controller reaction to the input disturbance and, in part, by the controller reaction to the output noise.

      On that same figure, the dotted curves correspond to the response of the system when the plant control input is saturated between +15 and −15. In particular, to simulate the closed-loop response when the saturation limits are exceeded, a pulse is added to the input disturbance at time t = 1. The resulting response diverges to infinity. The closed-loop system is then augmented with anti-windup compensation using Algorithm 3 on page 99. The resulting response corresponds to the solid curves in Figure 1.27, which show that stability

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