4 Arithmetic of Feedback Loops

Transcription

1 ME 132, Spring 2005, UC Berkeley, A. Packard 18 4 Arithmetic of Feedback Loops Many important guiding principles of feedback control systems can be derived from the arithmetic relations, along with their sensitivities, that are implied by the figure below. Process to be controlled Controller d H r C G e u y S y f F Filter y m Sensor n The analysis in this section is oversimplified, and at a detail-oriented level, not realistic. Nevertheless, the results we derive will reappear throughout the course (in more precise forms) as we introduce additional realism and complexity into the analysis. In this diagram, lines represent variables, and rectangular block represent operations that act on variables to produce a transformed variable. Here, r, d and n are independent variables. The variables e, u, y, y m, y f are dependent, being generated (caused) by specific values of (r, d, n). The blocks with upper-case letters represent multiplication operations, namely that the input variable is transformed into the output variable via multiplication by the number represented by the upper case letter in the block. For instance, the block labled Filter indicates that the variables y m and y f are related by y f = F y m. Each circle represents a summing junction, where variables are added (or subtracted) to yield an output variable. Addition is always implied, with subtraction explicitly denoted by a negative sign ( ) next to the variable s path. For example, the variables r, e and y f are related by e = r y f.

2 ME 132, Spring 2005, UC Berkeley, A. Packard 19 Each summing junction and/or block can be represented by an equation which relates the inputs and outputs. Writing these (combining some steps) gives e = r y f generate the regulation error u = Ce control strategy y = Gu + Hd process behavior y m = Sy + n sensor behavior y f = F y m filtering the measurement (11) Each block (or group of blocks) has a different interpretation. Process: The process-to-be-controlled has two types of input variables: a freely manipulated control input variable u, and an unknown disturbance input variable d. These inputs affect the output variable y. As users of the process, we would like to regulate y to a desired value. The relationship between (u, d) and y is y = Gu + Hd. G is usually considered known, but with some modest petential error. Since d is unknown, and G is slightly uncertain, the relationship between u and y is not perfectly known. Controller: The controller automatically determines the control input u, based on the difference between the desired value of y, which is the reference input r, and the filtered measurement of the actual value of y. Sensor: Measured variable is the noisy output of another system, called the sensor. Filter: Electrical/Mechanical/Computational element to separate (as best as possible) the noise n from the signal y m. The goal (unattainable) of feedback (the S, F and C) is: for all reasonable (r, d, n), make y r, independent of d and n, and this behavior should be resilent to modest/small changes in G (once C is fixed). Note that there is a cycle in the cause/effect relationships - specifically, starting at y f have r, y f cause e e causes u u, d cause y y, n cause y m y m causes y f This is called a feedback loop, and can be beneficial and/or detrimental. For instance, we Note that d (and u) also affects y, and through the the feedback loop, ultimately affects u, which in turn again affects y. So, although u explicitly only depends on e, through the feedback loop the control action, u, may in actuality compensate for disturbances d.

3 ME 132, Spring 2005, UC Berkeley, A. Packard 20 However, through the feedback loop, y is affected by the imperfection to which it is measured, n. Eliminating the intermediate variables (such as e, y f and y m ) yields the explicit dependence of y on r, d, n. This is called the closed-loop relationship. y = }{{} (r y) CL r + H }{{} (d y) CL d + F }{{} (n y) CL n (12) Note that y is a linear function of the independent variables (r, d, n), but a nonlinear function of the various component behaviors (the G, H, F, C, etc). Each term which multiplies one of the external variables is called a closed-loop gain, and the notation for a closed-loop gain is given. Now, in more mathematical terms, the goals are: 1. Make the magnitude of (d y) CL significantly smaller than the uncontrolled effect that d has on y, which is H. 2. Make the magnitude of (n y) CL small, relative to 1 S. 3. Make (r y) CL gain approximately equal to 1 4. Generally, behavior should be insensitive to G. Implications Goal 1 implies which is equivalent to H << H, 1 << 1 This, in turn, is equivalent to F S >> 1. Goal 2 implies that any noise injected at the sensor output should be significantly attenuated at the process output y (with proper accounting for unit changes by S). This requires F << 1 S. This is equivalent to requiring F S << 1.

4 ME 132, Spring 2005, UC Berkeley, A. Packard 21 So, Goals 1 and 2 are in direct conflict. Depending on which is followed, Goal 3 is accomplished in different manners. By itself, goal 3 requires 1. If Goal 1 is satisfied, then F S is large (relative to 1), so Therefore, the requirement of Goal 3 is that F S = 1 F S. 1 F S 1, >> 1. (13) On the other hand, if Goal 2 is satisfied, then F S is small (relative to 1), so Therefore, the requirement of Goal 3 is that F S << 1, 1 (14) The requirements in (13) and (14) are completely different. Equation (13) represents a feedback strategy, and equation (14) an open-loop, calibration strategy. 4.1 Tradeoffs Several of the implications are at odds with each other. Let T (G, C, S, F ) denote the factor that relates r to y T (G, C, S, F ) = Use T to denote T (G, C, F, S) for short, and consider two sensitivities: sensitivity of T to G, and sensitivity of T to the product F S (since F and S only enter T as a product). Simple calculation gives Note that (always!) S T G = 1, ST F S = F S S T G = 1 + S T F S Hence, if one of the sensitivity measures is very small, then the other sensitivity measure will be approximately 1. So, if T is insensitive to G, it will be sensitive to F S.

5 ME 132, Spring 2005, UC Berkeley, A. Packard Signal-to-Noise ratio Ignore r, and compute the fraction y/d, as a function of d and n. Noise-to-Signal ratio. After simple division, we get Denote n/d as the y d = P C + P C n 1 + P C d If Noise-to-Signal ratio is large, the quotient can be made only so small, namely about 1, by choosing C 0 (no feedback). If Noise-to-Signal ratio is small, then by using high-gain ( P C >> 1), the quotient made about equal to the Noise-to-Signal. 4.3 What s missing? The most important thing missing from the analysis above is that the relationships are not in fact constant multiplications, and the special nature of the types of signals. Nevertheless, many, if not all, of the ideas presented here will be applicable even in the more general setting. 4.4 Problems 1. Use elementary algebra to derive (12) from equation (11). 2. We have seen many advantages of feedback, particularly the ability to reduce the effect of external disturbances and to reduce the sensitivity of the of the overall system to changes in the plant itself. Feedback also has a linearizing effect, which is used beneficially in the design of amplifiers and actuators. In this problem, we study a non-dynamic analysis of this property. Suppose that you can build a high-power amplifier, whose input/output characteristic is y = φ(u), as graphed below φ(u) u

6 ME 132, Spring 2005, UC Berkeley, A. Packard 23 The slope of φ varies from approximately 0.3 to 1.6. Now, because of power considerations, assume that you can build a low-power amplifier that is much more linear, in fact, assume that its input/output relationship is u = Ke, where K is a constant. Consider the closed-loop system shown below. r + e K φ( ) u y (a) Write down the relationship between y and r. Because we don t have an exact formula for φ, you cannot solve for y in terms of r - instead, you can only write down the relationship, which implicitly determines how r affects y. Make sure your equation only involves φ, r, y and K. It should not involve u or e. (b) Take K = 5. On graph paper, make a graph of φ(u) versus u (as above). Also on the graph, for many different values of r, plot r 1 K u versus u Each one of these graphs (straight lines) should intersect the φ(u)-versus-u graph at one point. Explain why the intersection point gives the value y(r) (for the specific r). On a separate graph, graph this function y(r) versus r. (c) Using the relationship you derived in part 2a, and treating it as an implicit description of the dependence that y has on r, differentiate it with respect to r to get y (r). (d) What should be K be chosen as so that the slope of the closed-loop behavior between y and r varies at most between 0.9 and 1.1.