64 Standard feedback structure with commands r(s), . d) The magnitude of the sensitivity function must be less than ∣W2(ιω)∣ for all frequencies to achieve robust performance. The sum of these convex hulls are the uncertainty regions corresponding to the rational part of the transfer function. An important insight of Doyle's was the realization that robust performance of the system in Figure 1 corresponds to robust stability of the system in Figure 2.
Therefore, the structured singular value μ is a major measure of the robust performance of a control system. When σmαic(∆) ≤ 1 ∀ω, the robust performance condition given by 1 can also be defined as nominal stability and μ[Λf(s)] < 1 Vs = iω, where μ is calculated according to the structure associated with Δ. The beauty of the structured-singular-value framework is that all control structures can be written.
Model Parameter Uncertainties and Region-Based Analysis
The method of Doyle [1982] is used to compute an upper bound on μ[Λf(s)] when Δ consists of complex blocks ∆l∙. Methods are being developed that allow the calculation of μ when Δ contains real scalar blocks [Sideris and de Gaston, 1987]. To compute the structured singular value μ for systems with a set of process models π, the new algorithm locates regions on the complex plane containing π(fω) at each frequency u>.
For the SISO systems in Figure 1 with controller c(s), the robust performance test 1 with W2 equal to the scalar W2 is equivalent to the requirement that the distance of regions π(fω)c(ιω) from (—1,0) of pass jw2. (ιω) ∣ for all frequencies ω. It is clearly important that the regions π(iω)c(ιu>) contain all the patterns in 5 without having any extra patterns to avoid conservativeness. The robust SISO performance test based on the uncertainty regions π(iω)c(iω) motivates graphical techniques for control system analysis.
Internal Model Control and Smith Predictors
Note that the gain characteristics, time constants, and delay uncertainties are readily identifiable in the figures—they are transparent to the control system designer. When the process model is open-loop stable with time delay, the IMC structure is a parameterization of the control structure of the Smith predictor introduced by O. In the absence of modeling error, the Smith predictor has been shown to lead to an optimal response to step disturbances.
The proposed IMC controller design procedure using graphical robust performance analysis is a powerful tool for control system design in the face of model parameter uncertainties.
CD Response Control in Paper Manufacturing
Successful application of the proposed controller design procedure to CD-paper response control therefore has the potential for a significant, beneficial impact on a larger industry. The trade-off between performance and robustness associated with choosing the IMC filter tuning parameter is illustrated. The performance of the Smith prediction resulting from the design procedure is compared with the performance of a PID controller for different levels of uncertainty.
The Smith predictive controller derived from the IMC design procedure is compared with the μ-optimal controller in Section ΓV. In Section V, the robust IMC controller design process is applied to the CD response control problem in paper manufacturing. The robust stability, robust performance, and robust fault tolerance of the controllers despite the uncertainty of the interaction parameters are demonstrated.
Introduction
Regions on the complex plane are used in a version of the Nyquist stability test for control systems when the system model contains parameter uncertainties. The algorithm offers two principle advantages over previously available methods: 1) The region boundaries are guaranteed to close the set of models, and 2) the algorithm preserves concave sides on the region enclosing the set. Use of the regions π(iω) in SISO control system robust-stability analysis is described by Analysis test 1 based on the well-known Nyquist stability test.
If the bounds derived from the algorithm are used in the analysis test 1, it is important to include a set of models 7r(tω) in 1 so that the test is not indeterminate. A boundary is then usually defined as a set of line segments connecting points on the outer edge of the "gun" pattern. For example, if 7r(iω) is a disk, some points in the rifle pattern may be at the edge of the disk, but the segments connecting these points will always be inside π(tω).
Algorithm for Locating Uncertainty Region Boundary
If I0max - θmi∏∖ ≥ 2π∕c√, the region -D(α>,δi,0) will be an annulus or circle centered at the origin, depending on whether the origin is contained in D(ω,bι or not }. If ∣θmaχ — 0min∣ < 2π∕ω, the product region D(ω,bι,θ} will be composed of sections of P(cu,bι)eθminiω, sections of D(ω,bl)eθmaχiω, and arcs of circles centered at the origin . As parameter r, the size of line segments and triangles d71(ω,δ∣,0) used to approximate P-1(ω,δ∕,0) is increased. The actual product area that results by multiplying a so small convex hull with N{ω,a3,k) converges to N(ω,a3∙,k) multiplied by a single complex number - a convex re.
In the discrete version of theNyquistStability test, the surrounds of (—1,0)byh(z)c(z}) are defined, since z surrounds the discrete Nyquist contour (the unit disk excluding the positive real axis and singularities on the unit circle). If the spectrum of the perturbation is a narrow band centered near ω (i.e. the perturbation looks almost like sin ωt) the perturbation power (amplitude) is limited to ∣wd(∕ω)∣. The weight w2(iω) in (15) is given by (5) The ability to guarantee strong stability and strong performance if and only if μ{M) < 1, ∀ ω, defines the strength of structured singular-valued analysis.
The union of these convex bodies is the uncertainty regions corresponding to the rational part of the transfer function. The union of these convex bodies is the uncertainty region corresponding to the rational part of the transfer function. Since 0min = 0 in (24) the union of the convex bodies is one of these regions.) The final uncertainty region π(iω) is closed when the two regions are connected by circular arcs as shown.
The effect of different filter parameters ε on the maximum magnitude of the sensitivity function is illustrated in Fig. In this study the robust performance will be defined mathematically by placing a limit on the magnitude of the sensitivity function s(iω) = [1 + p( ι'ω)c(z'ω)]^1. The performance requirement (13) indicates that the weight w2 (s) specifies a limit on the maximum peak of the sensitivity function.
Using tuning method A it can be verified that this is the best level of robust performance achievable with nominal model parameters at their mean values as in (7) - the system would fail the robust performance test (14) if parameter a had a value smaller than that indicated in Table 5. However, the shape of the response of s*(ω) may not resemble the real system response, as different models in ∏ contribute to s*(ω) at different frequencies. The extent to which s*(ω) provides a conservative estimate of the two-norm of the error highlights the inherent conservatism introduced when transfer functions with real parameter uncertainties are represented by uncertainty regions on the complex plane: the parametric structure of the process model is lost.
Model Development for Cross-Directional Response
Each real parameter in the scalar dynamics in equation 7 is allowed to vary between the specified upper and lower bounds, independently of the other real parameters. Negative CD response elements are found in many of the models, reflecting the observation that attempts to increase the basis weight downstream of one actuator position can actually decrease it on either side of that position. A high condition number of the CD response model means that strong control action is needed to weaken perturbations entering the process in the direction of the minimum singular vector.
The plot has been truncated to 7 = 50.0 to improve scale - 7 increases to ∞ above each of the “flat” peaks in the plot. Values of r at which singularities occur change dramatically as the dimension of the system changes. For values of r < 0.25, the segment of the real axis containing the eigenvalues excludes the origin for all dimensions n (Pci? is positive-definite for these values of r).
The design of the paper machine disc has a significant effect on the interactions in the CD response. Since the disk is symmetrical about the center of the paper machine, half of it can be modeled as a cantilever beam acting on both a moment and a force at (0,0,0) in Figure 4. The expression for the deflection of the beam 5( x) caused by a force Fa acting at a distance x = a from the origin in figure 4, is given by equation 9 [Crandall et.
The displacement of the slice at the actuator positions is given in dimensionless form by. The effect of this is to flatten the peak of the wave exiting at the center of the slice opening in Figure 5. If a paper machine were constructed symmetrically with respect to a vertical plane through the center of the sheet, then the physical CD response would be exactly centrosymmetric. .
Centrosymmetric models can account for edge effects observed in the CD response, i.e. small differences in response observed at different distances from the center of the leaf.
Robust Stability Result
Robust Performance Result
Robust Failure Tolerance Result
Controller Synthesis Methods
In order to calculate the bounds used in the design of the responsive CD controller with d inputs and outputs, another structure of d × d uncertainty blocks is added to the structured singular value analysis problem described above. Since μ(u>) < 1 for all ω, the sufficient conditions in Theorem 3 for the robust operation of the ΜΙΜΟ system are fulfilled. An attempt was made to calculate μ(ω) for this case (with regulator c(s), interaction uncertainties, parameter uncertainties in scalar dynamics, and performance requirements kept the same) using structured singular value analysis.
Perturbation di0b enters with a large component in the direction of the vector corresponding to the minimum singular value and is therefore difficult to reject. Limits based on Pc ' can therefore be used in the diagonal regulator design procedure - 85 percent uncertainty in the gain k is required. Robust performance analysis of the ΜΙΜΟ system with Toeplitz symmetric controller and uncertain Toeplitz symmetric in.
CW = c(s)(j where scalar c(s) = g(s)∕[l — p(s)g(s)], and g(s) is tuned for robustness with respect to scalar dynamic uncertainty (usually with a single tuning parameter) as in the diagonal controller design procedure above. The response with diagonal controller, labeled 1 in Figure 13, is for a step change in the direction of the maximum singular vector of P^°'3. The response with diagonal controller , labeled 3 in Figure 13, is for a step change in the direction of the minimum singular vector.
Robust performance is addressed in the spirit of Doyle's [1982] new structured singular value theory. Robust performance of a given SISO operating system design based on singular value bounds of the CD interaction matrix implies robust per. The paper machine slice actuator was shown to have a significant effect on interactions in the CD response model.
Ostman, »Principles and Potentials of CD-Basis Weight Control,« Proceedings of the EUCEPA Sympo sium on Control Systems in the Pulp and Paper Industry, Stock Holm, Švedska, maj pp.
