工程师手册-过程工业自动化仪表与控制手册(英文版)1-500-副本

工程师手册-过程工业自动化仪表与控制手册(英文版)1-500-副本内容摘要：

nvolved with the autocorrelationfunction of its input. When the input is white noise, its autocorrelation is an impulse function (thederivative of a step function). The cross correlation is then equal to the filter’s impulse response [1]–[3].Similarly, a deterministic disturbance may be considered to be the step or impulse response ofa stable linear filter. A more plicated disturbance may be represented as the filter response to asequence of steps (or integrated impulses). Any delay associated with an unmeasured disturbance isconsidered to determine the timing of the step in order that CU be cancelable (stable zeros).PolePlacement DesignPoleplacement design [1] requires that A, B, C , and D be known and that a suitable H be selected.There may be difficulty in separating the BD product. As an expedient, B may be considered a constantand D allowed to include both stable and unstable zeros. If the degree of all of the polynomials is nogreater than n, Eq. (10) provides 2n + 1 equations, one for each power of s or z −1, to solve for thecoefficients of F and G . If the unmeasured disturbance e is considered an impulse, the degree of theG polynomial should be less than that of A, otherwise the degrees may be equal.A design resulting in a B or F polynomial with a nearly unstable zero, other than one contributingintegral action, should probably be rejected on the grounds that its robustness is likely to be poor. Itmay be necessary to augment H with additional stable factors, particularly if the degree of F or Gis limited. The set point can be prefiltered to shift any undesired poles in the setpoint function D/ Hto a higher frequency. However, the resulting unpensatable unmeasuredload rejection functionF / H may be far from optimal.LinearQuadratic DesignLinearquadratic (LQ) design [1] provides a basis for calculating the H and F polynomials, but isotherwise like the poleplacement approach. In this case D contains both the stable and the unstablezeros and B is a constant. C / A is an impulse response function since e is specified to be an impulse.The H polynomial, which contains only stable zeros, is found from a spectral factorization of thesteadystate Riccati equation,σ H {z} H {z −1} = 181。 A{z} A{z −1} + D{z} D{z −1}orσ H {s } H {−s } = 181。 A{s } A{−s } + D{s } D{−s }(27)The parameter σ is chosen to make the steadystate value of H unity and 181。 is an arbitrary parameterin the criterion function J ,J = E{(r − y)2 + 181。 u 2}(28)E is the expectation operator. The u term imposes a soft constraint on the manipulated variable witha penalty factor 181。 . A polynomial X , satisfyingX {z} A{z −1} + σ H {z}G {z −1} = D{z}C {z −1}X {z} H {z −1} + 181。 A{z}G {z −1} = D{z} F {z −1}(29)CONTROL SYSTEM FUNDAMENTALSand Eq. (27) also satisfies Eq. (10) and minimizes Eq. (28) for an impulse disturbance e. The equationsin s are similar. When the degree of A is n, the first equation of Eqs. (29) provides 2n equations,one for each power of z (or s ). These can be solved for the 2n unknown coefficients of X and G ,after H and σ are found from Eq. (27). G has no nthdegree coefficient and X {z} and D{z} have nozerodegree coefficient. [ X {−s } and D{−s } have no nthdegree coefficient.] None of the polynomialsis more than nth degree. F can then be found by polynomial division from Eq. (10) or the second ofEqs. (29).For the optimization to be valid, the penalty factor 181。 must be large enough to prevent the manipulated variable u from exceeding its limits in responding to any input. However, if 181。 were chosen to betoo large, the closedloop response could be as sluggish as the (stabilized) openloop response. TheLQ problem may be solved leaving 181。 as a tuning parameter. Either experiments or simulations couldbe used to evaluate its effect on performance and robustness.Despite the LQ controller being optimal with respect to J for a disturbance input, a switching(nonlinear) controller that takes into account the actual manipulated variable limits and the load canrespond to a step change in set point r in less time and with less integrated absolute (or squared) error.MinimumTime Switching ControlThe objective for the switching controller is to drive the controlled variable y of a dominantlagprocess from an arbitrary initial value so that it settles at a distant target value in the shortest possibletime. The optimal strategy is to maintain the manipulated variable u at its appropriate limit untily nears the target value r . If the process has a secondary lag, driving u to its opposite limit for ashort time will optimally slow the approach of y to r , where it will settle after u is stepped to theintermediate value q needed to balance the load. Until the last output step, switching control is thesame as “bangbang” control. Determination of the output switching times is sufficient for openloopcontrol. The switching criteria must be related to y and its derivatives (or the state variables) in afeedback controller. Either requires solving a twopoint boundaryvalue problem.As an example, consider a linear integral {T }–lag{τ L }–delay{τ D } process with constant manipulated variable (controller output) u. The general timedomain solution has the formx {t } = at + b exp −tτ L+ c(30)y{t } = x {t − τ D }where x is an unmeasured internal variable and a, b, and c are constants that may have different valuesin each of the regimes. At time zero the controlled variable y is assumed to be approaching the targetvalue r from below at maximum rate,dy{0}dt=d x {0}dt=u M − qT= a −bτ Lx {0} = b + cy{0} = x {0} −(u M − q )τ DT(31)where u M is the maximum output limit and q is the load. At that instant the output is switched to theminimum limit, assumed to be z。