Closed-loop circuits improve control of optical components

John Ardini

Circuits designed to control optical components such as MEMS switches demand a properly shaped dynamic response and the ability to reject system disturbances. The author makes the case that closed-loop controllers, used in conjunction with frequency domain analysis, best meet the challenge.

In designing electronic control systems for optical components, closed-loop controllers offer improvements in dynamic response and disturbance rejection over open-loop controllers (see Fig. 1). The analysis and design of such closed-loop controllers is facilitated by understanding the frequency domain representation of the signals encountered in the system.

A review of the basics of the root-locus and bode-plot design techniques, and how these techniques can be applied to a servo-controller design for a single axis of a microelectromechanical systems (MEMS) mirror can provide an understanding of these design issues.

TRANSFER FUNCTION
For a given system, it is convenient to work with a transfer function, which represents the frequency response of the system to its input. The transfer function is formed by taking the Laplace transform of the differential equation that describes the system. For simple systems, we can also easily see the poles and zeros of the transfer function. These are the roots of the transfer function denominator and numerator, and they directly relate to the time-domain response of the system.

These roots are typically plotted in the complex frequency plane, and their positions in that plane tell us how they affect the dynamic response. For example, a spring-mass system is a system frequently encountered in engineering and the describing differential equation is the familiar

Taking the Laplace transform of both sides gives

And, forming the transfer function gives

The physical system being controlled is referred to as the plant. In this discussion, we will use the above transfer function to model our MEMS mirror plant. Our plant has no zeros, two poles, and is a second-order system.

PID CONTROLBy far the most commonly used control algorithm in a closed-loop design is the proportional + integral + derivative (PID) controller (see Fig. 2). It is relatively easy to implement in hardware or software, and can be intuitively explained—that is, mechanical analogues such as an automotive suspension system exist for the P and D terms. The integral term helps achieve zero steady-state error by continually increasing the control output as long as an error exists.

A PID filter can be described by the following transfer function in the s-domain:

Where u is the controller output and e is the error sensed by the controller. Even when using simple control structures such as PID, it is important to have an understanding of the frequency-domain representation of the controller.

Each gain in the PID controller has a specific effect on performance and stability of the system, and in some cases a simplified controller is possible (PI, PD, or P). Also, the optimal coefficients of the PID filter for a given application can result in imaginary zeros in the above transfer function. With this description of the PID controller, we see that the PID structure gives us the ability to compensate our plant dynamics with the addition of two zeros and one pole located at the complex plane origin.

DIGITAL IMPLEMENTATION Digital control systems offer all the benefits that come with any software-based system: algorithms that can be upgraded or can change on the fly, and self-calibration of the system. The main disadvantage over an analog implementation is that there will be a practical limit on the rate at which the control law can be updated and, thus, a practical limit on the bandwidth of the controller that can be achieved.

Typically, a control-law sample rate is chosen to be 10 to 25 times the desired bandwidth of the controller. This allows most of the design to be performed in the continuous domain and transferred to the discrete domain prior to implementation. The conversion of a filter from continuous to discrete time is typically done by pole-zero mapping or by numerical integration. In the special case of a PID filter, the conversion to discrete time can be done by simple application of numerical integration and derivative approximation. The difference equation implementation of a discrete PID control law is

where the sampling interval factor T has been incorporated into the gains k_i and k_d.

POLE-ZERO MAPS, ROOT-LOCUS PLOTS
Pole-zero maps are plots of a system's poles and zeros in the complex frequency plane. In this context, a "system" may refer to an optical device or a hardware or software-based controller algorithm. We know that the location of these roots dictate the dynamic response of our system.

Purely real, nonoscillatory poles would be located on the [-∞,0-] part of the x-axis. Purely oscillatory poles would be located on the y-axis as complex conjugates. Pole pairs with varying levels of damping are located in the rest of the left-hand plane. Poles with a damping ratio of 0.7071 are located along the 45° lines emanating from the origin into the left-hand plane.

Based on the location of our plant's poles and zeros and some experience in designing compensators, we can pick a control structure that can give us good closed-loop system performance. For example, we might pick a PID controller, or decide that we need to use some other number of poles and zeros for our compensator.

For a given structure, the root locus shows us how the closed-loop pole locations move in the plane for varying controller gains. We note that a pole located in the right half of the complex plane will be unstable and is thus unacceptable in our design. Stable poles are located in the left-half plane. This understanding allows us to select a controller gain that is optimum in some sense for our application.

The power of this technique lies in the fact that we can work with the open-loop transfer function of our system vs. the closed-loop transfer function. Closed-loop system roots are more difficult to explicitly solve for.

BODE PLOTS
Bode plots are frequency response plots of a dynamic system. These plots show the gain and phase response of a system to a sinusoidal input over a range of frequencies of interest. A log scale is used for the frequency axis and the magnitude is shown in decibels. Traditionally, the plot is used to assess frequency response and stability, particularly for control systems. As a design aid, these plots can be used to easily construct a graphical representation of the combined open-loop system response of a plant and controller to sinusoidal inputs.

It can be shown that if the open-loop phase crosses 180° when the magnitude response is 0 dB, or unity gain, our close-loop system will be unstable. The gain in excess of 0 dB at the 180° crossover point is referred to as the gain margin. Similarly, the phase short of 180° at the 0-dB crossing (the crossover frequency) is referred to as the phase margin.

Both these numbers give some indication of the relative stability of our closed-loop system. Acceptable margins depend on the application, but typically 6-dB gain margin and 40° phase margin are reasonable targets.

Bode's gain-phase relationship tells us that gain and phase are directly related. On a Bode plot, this gives the approximate relationship:

where n is the slope of the magnitude line in multiples of 20 dB/decade. Thus, a plot with a magnitude slope of -20-dB/decade will have a phase of -90°. This gives a phase margin of 90°, and if we can keep the slope of our magnitude line at -20 dB/decade for at least a decade at the crossover frequency, we will have a stable system.

The effects of additional poles and zeros in a compensator can easily be visualized on a Bode plot since the effects of each pole and zero are additive in the shape of the graph. The bandwidth of a system is defined as the point at which the magnitude response drops to -3 dB. The open-loop crossover frequency for a system with good phase margin is typically close to the closed-loop bandwidth. As with the root-locus method, the power in this approach is the ability to work with the open-loop transfer function.

DESIGN EXAMPLES
The following closed-loop design examples use a single axis of a servo-controlled MEMS mirror; one uses a root-locus design approach, while the other uses a Bode-plot design approach. Some designers may prefer one or the other, but both are suited to similar classes of problems.

A MEMS mirror can be mechanically oscillatory. A closed-loop controller can be used to attenuate these oscillations, improve the response time of the structure, and improve its performance in the face of variations in manufacturing and operating conditions.

Let's start by looking at the root locus for a sample MEMS mirror (see Fig. 3). Each pole location of the plant is marked with an x (the one at the origin is added by the PID controller). Our map shows that the poles are located above the 45° line and will be oscillatory—a damping ratio of 0.3 was chosen for this example. Also, the larger the radial distance from the origin, the larger the natural frequency of the poles—in this example, 500-Hz (3142 rad/s) was chosen.

ROOT-LOCUS METHOD
If we could cancel the effect of our plant poles and add a pole on the real axis, the response would be critically damped. Algebraically, the zeros in our PID controller can be chosen to "cancel" these poles. It is difficult to perfectly cancel poles, but if we put the zeros close to the under-damped poles of the mirror, the time-domain contribution of the closed-loop poles to the magnitude of the response would be greatly reduced. Thus, while we may not be able to eliminate the oscillations altogether, we may be able to reduce them to a small, acceptable level. The PID controller also adds a pole at the origin.

In our root-locus plot, all closed-loop poles will move toward open-loop zeros or infinity as the gain is increased. The two zeros we inserted with the PID compensator attract the under-damped poles of the mirror. The pole at the origin that our PID controller added moves to the left (faster response) with increasing gain and this pole tends to dominate the closed-loop response.

From this plot, we select an overall gain on the compensator to give us our desired response. In this case, we choose a relatively large gain to show the ability to improve the response time of a system. In an actual system, improved response time must be traded off against the resulting size and power of the actuator required to provide the commanded control action for the mirror.

The measurement of response time of a MEMS mirror to a step input in position reference using open- and closed-loop controllers indicates that the open-loop response takes approximately 5 ms to settle at its final position and has a final position error of 10% (see Fig. 4). The closed-loop controller not only eliminates the oscillations, but "servos out" the steady-state error. The simulated structure includes a 10% deviation in system gain due to manufacturing tolerances.

BODE-PLOT METHODIn the Bode plot for the MEMS mirror plant the response is typical of a second-order system (see Fig. 5). Recognizing this, the design task becomes very similar to the root-locus approach.

If we were to have a pair of complex zeros in our compensator, we could cancel the effect of the plant's complex poles and the result would be a flat magnitude curve and flat (0°) phase curve. However, our goal is to get a large DC gain and a -90° phase response at our desired crossover frequency. This can be done with the addition of one extra pole in our compensator. Fortunately, the structure of the PID controller provides just what we need: two zeros and one pole.

Finally, an overall gain factor is incorporated into the PID terms to position the -20 dB/decade curve to give us the crossover frequency we desire.

John Ardini is a systems architect at Cadence Design Systems' Embedded Systems Design Group, 270 Billerica Road, Chelmsford, MA 01824. He can be reached at [email protected].