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Review of Feedback Control JR Boston January 8, 2003 page 1 of 4 System Transfer Functions Consider a linear, time-invariant system with impulse response h(t), input u(t) and output y(t). u(t) h(t) y(t) We often describe the system in terms of its transfer function, which is the ratio of the Laplace transform of the output to the Laplace transform of the input. In general, the transfer function of a system can be written as a ratio of polynomials in s: Y(s) b (s - z1)(s - z2 ) . . . (s - zm ) H(s) = = m U(s) a n (s - p1)(s - p2 ) . . . (s - pn ) where zi are the zeros and pi are the poles of the transfer function H(s). Poles and zeros: are complex numbers determine H(s) and h(t), the impulse response. determine the stability of the system (poles in the LHP). These same ideas can be developed for discrete signals, using z-transforms in place of Laplace transforms. If poles are inside the unit circle in the complex z-plane, the system is stable. Otherwise it is unstable. State Variable Formulation If a (single-input, single-output) system is defined with input u(t), output y(t), and state vector x = (x1 x2 x3 . . . xn)T , then x=Ax+Bu y=Cx+Du where A, B, C, and D are matrices that characterize the system. This system can be solved by direct integration to obtain t x ( t ) = φ ( t ) x ( 0) + ∫ φ ( t − λ ) B u ( λ ) dλ 0 The first term represents the natural response and the second term, which is the convolution of the input with the state transition matrix, represents the forced response. The state transition matrix φ(t) can be evaluated using Laplace transform techniques, yielding L{φ(t)} = (sI – A)-1 We can use this state variable formulation to obtain a good discrete approximation to a linear continuous system. Linear Feedback Consider a negative feedback path H(s) applied to a plant with transfer function G(s) Review of Feedback Control JR Boston January 8, 2003 page 2 of 4 + e(t) u(t) G(s) y(t) - H(s) The closed-loop transfer function is Y (s ) G (s) L(s) = = U (s) 1 + G (s) H (s) The zeros of the closed-loop transfer function L(s) are the same as those of the original plant, but the poles are now the roots of 1 + G(s)H(s). Feedback changes the poles (natural modes) of the system. We often evaluate controllers in terms of their step responses (response to a step input): 1.5 overshoot 1 0.5 settling time risetime 0 0 50 100 150 200 250 300 350 400 450 500 time (dimensionless) Designers typically aim for about 10% overshoot (damping = 0.707 for a second-order system). A common approach to feedback design is to use unity feedback (H(s) = 1) and introduce a compensator network in series with the plant Gp(s) [G(s) = plant + compensator]: + y(t) u(t) compensator Gp(s) - The compensator network (such as a lag-lead network or a PID compensator) can move the poles around in the s-plane to achieve desired responses. An advantage of this approach is that, if 1 + G(s) is large (still assuming that H(s) = 1), the sensitivity of the closed-loop transfer function L(s) to changes in G(s) is reduced. Since ∂L 1 G 1 = − = ∂G 1 + G (1 + G ) 2 (1 + G ) 2 ΔL 1 ΔG we can define sensitivity = . L 1+ G G Review of Feedback Control JR Boston January 8, 2003 page 3 of 4 Comparison of PD and PI control Advantage Disadvantage PD (lead) speed sensitive to noise stabilizes system expensive PI (lag) eliminates steady-state error sluggish smooths data destabilizes system Summary of Discrete Control Essentially nonlinear Discrete control can be superior to linear control (e.g., deadtime control - a form of bang-bang control). Discrete versions of continuous control tend to be less stable. Need to sample at 10-30 samples per period of highest natural mode in the system to avoid problems (stability, hidden oscillations in continuous system between controller sample points). State Feedback Control Can place system poles anywhere in the s-plane. Need to know all states. Costs Need a computer (not much of an issue anymore) Moving poles requires actuator power. State knowledge requires expensive measurements. plant e(t) x(t) + r(t) x = Ax + Be C y(t) - - ^ x KT Observer Steps in Traditional Control Design 1. Determine performance requirements 2. Find plant model Determine appropriate level of detail in model. If model is too complex Too many parameters to measure accurately. High computational requirements. If the model is too simple Can’t describe all performance characteristics. Can’t describe all natural modes of system (especially high-frequency modes). Often involves linearizing about an operating point. 3. Select approach PID can be implemented without a detailed system model PID parameters are usually tuned to achieve acceptable performance. Review of Feedback Control JR Boston January 8, 2003 page 4 of 4 PID parameters often change as operating point changes. State feedback Must determine a “cost function” to select pole positions. Develop an observer (worry about convergence, etc.). Usually doesn’t work (model inadequacy). Intelligent Control Phrase coined by K.S.Fu in 1950’s. It is now an official NSF buzzword. What is intelligent control? “We can’t state precisely what it is, but we know one when we see one.” K.S. Narendra, Center for System Science, Yale Narendra identified 3 categories of difficulties in designing control systems: 1) Computational complexity (environment, sensors) 2) Nonlinear systems (multiple degrees of freedom) 3) Uncertainty The better a system deals with these factors, the more intelligent it is. Some characteristics of intelligent control 1) Sets its own goals 2) Utilizes past experience 3) Assesses its environment 4) Makes decisions in unstructured environments 5) Identifies model limitations / changes 6) Fails gracefully 7) Reasons under uncertainty (shows judgement) 8) Often incorporates hierarchies, as illustrated below: Supervisor Coordinator Executor meta-knowledge control signals Plant sensor signals In intelligent control, concern shifts from dynamic response of system (stability) to 1) Finding set point / reference 2) Robustness to Model errors Measurements errors System failure 3) System identification / fault detection / system reconfiguration Generally assumes a computer-based system.

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transfer function, transfer functions, laplace transform, impulse response, frequency response, time domain, step response, spatial frequency, optical system, control systems, frequency domain, unit step, laplace transforms, state space, order system

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posted: | 3/11/2010 |

language: | English |

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