Nonlinear Control Lecture Just what constitutes nonlinear control? Control systems whose behavior cannot be analyzed by linear control theory. All systems contain some nonlinearities, most are small and can be ignored. Frequently, these effects will manifest themselves in the loop performance. The most common effect of the nonlinearity is that the loop experiences a limit cycle. What is a Limit Cycle? Some definitions: Continuous oscillations of nearly equal amplitude that persists for a variety of controller gains (resolution limit or non linearity slip stick) McMillan Condition of sustained oscillations in a closed loop system caused by nonlinearities in the loop Phillips, Nagle Periodic oscillations whose amplitude and frequency depend only on properties of the system and not initial state Coughanowr uniform oscillation having a fixed amplitude repeated indefinitely & changes in controller tuning typically affect only amplitude and periods Shinskey Another term for this behavior is hunting These nonlinear effects are outside the mainstream control theory instruction and often ignored. I maintain that in industrial applications, they are very common. In most cases where the effect manifests itself, the user simply reduces the gain until the effect diminishes. This is done at the expense of good robust control. In industrial process control these nonlinearities are typically one of the following four:
Note that they are described as to how the output waveform behaves in comparison to an input sin wave, frequently called a driving force. This gives rise to the term describing function, which is how these effects are analyzed. What causes these nonlinearitites? Some of these nonlinearities are shown in the above figure. Variable coefficient (a) is present in many control loops. A control valve with nonlinear installed characteristics is a frequent occurrence. But with a low controller gain, the effect is mitigated. An orifice plate used to measure flow by way of differential pressure exhibits a square root relationship due to Bernoulli s energy law. We saw this effect when draining the tank in experiment 1. The flow rate out of the tank was greater when the level was higher. Deadband (b) is the most common nonlinearity found in chemical process control loops. Sloppy linkages on shaft guided control valves can be a cause. The problem is also observed with rack and pinion valve actuators. This is a mechanical/hardware problem. Deadband is also called backlash by some authors. The reader should be conscious that there is no universal rule governing the definitions. Deadband occurs is when the driving force changes direction the load does not move right away. There is a time of no movement before the load begins to move in the other direction with the driver. This time is proportional to the deadband distance and the driver velocity. The picture above shows the effect when gears are employed. Please note that there are good reasons for the clearances. Some gear manufactures use spring loaded gearing to
mitigate the problem. It can be concluded that deadband or backlash is inherent in some designs and not caused by a malfunctioning component. Control valves exhibit an effect called stick-slip. The valve will not stroke, that is the stem will not move exactly as the controller output signal dictates. Friction on the valve packing will cause it to stick, then slip to the point the output sets. This effect is even shown with a positioner. Without a positioner, the valve shows a deadband of up to 50%
Rotating stem valves, ball, butterfly or e-disk, have more deadband than stem guided valves. Deadband is more apt to be introduced when linear travel is converted to rotational travel. Position saturation (c) can occur when the final control element is operating close to its physical limit. Reduction in controller gain will usually alleviate the problem. Velocity Saturation (d) this effect is caused by the velocity travel time it takes to move a valve position. Large valves show this effect. We don t see this problem in our lab because the valves are small. This is usually defined in terms of control valve travel time. Adjustable speed drives for motor speed control have Ramp Times that can exhibit the same behavior, called acceleration, and deceleration ramps. Ramps limit the motor starting current. They are integrators and the default setting is typically 5 to 10 seconds. This is a typical flow loop reset value. If the integrator in the PI or PID controller is set faster than the ramps, the loop will have a limit cycle. The controller integrator is integrating faster than the load can respond. Many controllers offer a feature that allows the reset action to follow actual valve travel. An auxiliary speed drive output signal proportional to the actual load can be configured to the controller function block to eliminate the limit cycle. For control valves and cascade loops, this can be corrected by using BKCAL input in our DeltaV system. But we must make use of the dynamic reset limit to eliminate the problem. CONTROL OPTIONS (FRSIPID_OPTS) Dynamic Reset Limit must be selected in the controller block in DeltaV. How does the controller do that, limit the reset term? Recall from Experiment 2 we discussed the two forms a PID controller may have, Series or Standard. See your text ref. p. 183 to 185 and the DeltaV help files. The series form implements the reset term as one term in a series. This allows the user to limit the reset term by an external signal.
The velocity-limit or rate-limit property of a final element can pose a danger to a control loop, if the integral time of the controller is shorter that the stroking time of that element. However, the danger is hidden while the loop is operating around set point because the actuator takes little time to move only a short distance. When a sufficiently large disturbance strikes, the actuator may fall behind the controller output enough to cause more integral action and further falling behind. External-reset feedback of the measured position of the final element can eliminate the danger without compromising controller performance. Shinskey The power of external-reset feedback ControlGlobal.com 5/26/2006. (Dynamic Reset Limit) prevents windup caused by limiting of the downstream block or from dynamic performance because of tuning. The downstream block must have Use PV for BKCAL_OUT set as an I/O option for Dynamic Reset Limit option to be used in the upstream block. from DeltaV help files. Other sources on nonlinearities may be present if the control system controls that have mechanical loads that have inertia. These are no normally seen in chemical or process control systems. The Dynamic Reset Limit function removes limit cycle. This function clamps the reset term in the controller based on the BKCAL signal. The result is the controller only changes the reset term as fast as the output function can move.
Describing Functions What is the Describing Function? Nonlinear function blocks are defined as a transfer function, but only for a specific input signal, a sin wave. The describing function (DF) is a Fourier series. It is usually designated as N. A Fourier series contains the fundamental frequency as well as the harmonics. In most control systems, the higher order frequencies are ignored because the other transfer functions in the system act as filters and attenuate them. Y ( jω) FundamentalOutput N = = The input is a sin wave. Most DFs are independent X ( jω) Input of frequency. Therefore: A DF is a complex fundamental-harmonic gain of a nonlinear in the presence of a driving sinusoidal. The principals of separation do not apply with DF. Two DFs in series require the calculation of a separate DF. How is the DF calculated? The DF is calculated by a Fourier integral, but there are various forms of the integral depending on the nature of the nonlinearity. We will not calculate these integrals in this lab, but we will use them to show system behavior. For details on the various forms see the referenced Gelb text. The important aspect of a DF is if it is single valued. A single valued DF only changes gain. Variable gains can be linearized with little difficulty and these functions do not inhibit robust control. Where a particular nonlinearity provides both gain and phase shift, control is more difficult if not impossible. For the four nonlinearities we defined: a) Variable gain, this has no phase shift. b) Deadband has both gain and phase c) Position saturation has only gain change d) Velocity Saturation has both gain and phase. Phase change is very detrimental for control. Phase change is what occurs with deadtime. So the DFs that have variable gain and phase cause the most difficulties for control. We will demonstrate two of these nonlinearities in the lab, deadtime and velocity saturation.
Deadband For the deadband DF the model is taken in a horizontal plane The motor in this sketch is the valve actuator signal. In this case the friction around the valve packing is strong enough to prevent further travel when the actuator changes direction. This shows how the load M will move to a sin wave motor driver. The sin wave peak is A. Note how the output peak is clipped.
The DF is calculated as: Note that N(A) is a complex number. It can be plotted as a magnitude and phase as a ratio of the gap to the magnitude of the driver.
How do we analyze control stability with a nonlinear element? Recall the control ratio, we add the nonlinear element, N: C R NG = 1+ NG + Σ g1 N g2 - For the stability analysis, we assume the condition of instability that is 1+ NG = 0. Rewriting: G = - 1/N The locus of 1/N is the critical point. G = g1 * g2. If there is an intersection of the linear transfer function with N, an oscillation will occur. The analysis is conducted in reverse; we perform the calculations assuming an oscillation (instability). If there is not an intersection, the system is stable. There can be multiple intersections, which further complicates the problem. For our lab experiment, we will use the level controller in experiment 1 and assume we will simulate the control valve with 1.5% deadband. We will do this by inserting a calculation block between the controller output and the level control valve. This calculation block will simulate the deadband.