Daniel L. Laughlin and Manfred Morari
Abstract
Regions on the complex plane are employed in a versionof the Nyquist stability test for control systems when the system model contains parameter uncertainties.
Generally, the uncertainparameters are real and bounded and appear as numerator and denominator coefficients in a rational function of frequency. Through the func
tion, theparameters decribe a setof model points at each frequency. Theproblemof locating such sets is defined for a modeluseful in describing the frequency response of physical processes. A new algorithm for locating region boundaries enclosing the set of models is presented. Boundary points connected by straight line segments are determined by the algorithm. The algorithm offers two principal advantages over previously available methods: 1) The region boundaries are guaranteed to en close the set of models, and 2) the algorithm preserves concave sides on the region enclosing the set.
part of a robust-stability analysistest for single-input, single-output (SISO) control systems.
1.1. Problem Definition
The continuous frequency responseof a physical process is conveniently mod elledby the set definedin 1. In 1, the terms x(s), y(s), π,(s), and ⅛(s) are assumed to be exact functions, while real numerator coefficients aj-, denominator coefficients bι, gain k, and time-delay θ are bounded by some minimum and maximum values.
Since the real parameters in 1 are inexactly known, π represents a set of process models. Through the uncertain parameters, the set π can capture uncertainty, or error, in models.
π = <p(s)∣p(s) = x{s) + y(s) k ajnAs} + ∙∙∙ + aιnι(g) + αono(∙s)
δ∕d∕(s) + ... + 6ιdι(s) + 0o<⅛(s) —0« ( (1) a3 e iαim⅛∙flymαJι ¼ ∈ [⅛mi.>⅛mκ]
∈ [kmin» kmax], θ ∈ [0minι ^max]
Uncertain real parameters in 1 are assumed to be uncorrelated; that is, each may take on allvalues within its specified range irrespective of values takenonby other parameters.
point in their allowed range.
π(ιω) - the exact region containing the set of frequency response models π evaluated at one frequency s = iω, including no extra points for all possible combinations of bounded real parameters in 1.
Λr(ω,αj) - the convex hull (the smallest convex polygon containing a set of points) defined by the numerator in 1 with bounded real coefficients αj.
N(ω,a3∙,k) - the convex hull containing N(ω,a3) after multiplication by an uncertain gain k.
D(ω, bι) - the convex hull defined by the denominator in 1 with bounded real coefficients 6j.
D(ω,bι,θ) - the boundary containing P(ω,δ∕) after multiplication by the in
verse ofan uncertain delay eθiu.
D~1(ω,bιiθ) - the inverse of boundary D(ω,b[,θ).
di 1(ω, bι,0) -linesegments andtriangles defined to enclose the entire boundary D~i(u,bi,θ).
πi(ω,a,f,b[,θ,k) - convex hulls resulting from the multiplication of∙7V(ω,θy,fc) by each d^1(ω,bι,θ).
and maximum values, π(zω) is a connected region on the complex plane at each frequency. Use of the regions π(iω) in SISO control-system robust-stabilityanalysis is described by Analysis Test 1 based on the familiar Nyquist stability test.
Nyquist Stability Test - SISO Nominal Stability: The SISO system in Figure 1 with controller c(s) and nominal model p(s) is stable if and only if the number of clockwise (positive) encirclements of (—1,0), by p(s)c(s) as s encircles (clockwise) the Nyquist contour (the right-halfplane excluding singularities on the imaginary axis), is equal to the negative of the number of open-loop unstable poles.
Analysis Test 1 - SISO Robust Stability: The SISO system in Figure 1 with con
troller c(s) and model p(s) ∈ π is robustly stable if and only if the system is nominally stable for one p(s) ∈ jγ and regions 7r(ιω)c(zω) exclude (—1,0) for all frequencies α>.
(See Laughlin et al. [1986] for more information about this type of control-system robust-stability analysis). If boundaries resulting from an algorithm are used in Analysis Test 1, it is important that they enclose the set of models 7r(tω) in 1 so that the test is not indeterminate. Additionally, they must not contain any extra models in orderto avoid unwanted conservativeness in the robust-stability analysis test.
The new algorithm presented in this paper locates a boundary consisting of points connected by straight line segments. In Section 3, proof is given that both
1.2.1. Factorial Methods
The use of factorial methods for locating region boundaries is reported by several authors (e.g. Chen [1984], East [1981, 1982], Saeki [1986]). Each of n real parameters defining aset ofmodels through a function as in 1 is chosen to take on d values within its bounds. The function is then evaluated at each of dn possible combinations of the parameters. The result of these evaluations is a “shotgun” pattern ofpoints on the complexplane. A boundary is then usually defined as the set of line segments connecting points on the outer edge of the “shotgun” pattern.
In general, such factorial methods cannot guarantee that π(iω) will be interior to the located boundary. For example, if 7r(iω) is a disk, some of the points in the shotgun pattem may be on the edge of the disk, but segments connecting these points will always be on the interior of π(tω). The factorial method can take into account correlation between parameters in thefunction. For example,ifα3∙ = bι in1, p(iω) canbe evaluated at each of the dn points whilemaintaining this equality. The factorial method converges on 7r(iω) as d is increased, but the number of required computations grows exponentially with n.
1.2.2. Region Arithmetic Methods
The result of a mathematical operation between two regions can be formally defined to contain all possible results ofthe same operation between points ineach region. For example, Bolton [1981] and Henrici [1974] define operations between disks on the complex plane. A set of such operations between regions is a “region
Concave edges on π(tω) are universally approximated by straight lines. Since these region-arithmetic methods are in general one-step operations, no sequence of im
proving approximations to π(tω) is possible. Therefore, the region-arithmeticmeth
ods as defined do not converge to the actual set π(tω).