2.1 LINEAR MODELS, STABILITY, AND NYQUIST THEOREMS

All forms of commercial power generation involve the controlled manipulation of plant variables to achieve a prescribed contribution to national demand. In fossil and nuclear plants the heat source, boiler feed pumps and turbine control valves are the pertinent items. For wind turbines the rotor-blade angle and generator excitation are the relevant quantities. When several plant variables require control, engineers describe the situation as a multi-input multi-output (MIMO) problem. On the other hand car speed control via fuel-injection rate exemplifies a single-input single-output (SISO) problem. Even over their intact operating regimes fossil and nuclear power plants are materially non-linear1 and distributed.[25] [26] In this framework control design cannot be implemented analytically by existing mathematics. However, for sufficiently small perturbations about a given operating point, plant

1 Consider typical heat transfer correlations; see Refs [63,64,143,219].

2 Necessarily described by partial differential equations.

dynamics can be approximated by a finite set of ordinary linear differential equations that enable methodical insight into the effects of negative feedback (control), variable interaction, and stabilization by means of linear compensating algorithms [79,124]. Aizerman [77,78] conjectures that an ordinary non-linear differential equation that has small signal (linearized) stability about every steady state also has global stability [77,126]. Though no general analytical proof of this exists, it forms a basis for the successful control of fossil and nuclear power plants.

First of all a number of linear models are derived that reasonably characterize plant parameter changes over the normal operating regime. Experience indicates that steps of about 10% of the maximum con­tinuous rating (MCR) are usually sufficient for the purpose. After engineering linear stabilizing algorithms for each power level, a compromise is generally sought that ensures adequate stability (tran­sient damping) for all. This problem is eased for power plants because unlike defense equipment speed of response is not the priority. With no certainty that normal maneuvres can be accomplished, confirmation is imperative using a detailed non-linear plant simulation [117,141] whose individual models have been validated as far as possible against existing plant items. This same non-linear simulation can also provide the required linear models as illustrated by examples in Chapter 3.

Linear control system theory is often couched in the abstract algebras of finite dimensional Banach or Hilbert Spaces [110-112]. However, for engineering design purposes linearized plant dynamics are specified in the state equation format [122,123,134]

x = Ax + Bu and y = Cx + Du (2.1)

where

x(t)—a state vector containing a finite number (n) of Laplace transformable functions

x—temporal derivative of x

A, B, C, D—real matrices with respect to a convenient Cartesian

coordinate system

u(t)—an input vector

y(f)—the output response vector

Defining the Laplace transform [119,120] of a vector of time function as that of each of its components, then from equation (2.1)

x = (xi — A)—1 Bu + (si — A)_1xo and u = Cu + Du (2.2)

where u denotes the Laplace transform of x(t) etc. and xo its initial value. In the present finite dimensional context the spectrum [110,111,121] of an arbitrary matrix L corresponds to values of s for which

det (si — L) = 0 (2.3)

and these roots are termed the eigenvalues of L. A determinant and eigenvalue spectrum are properties of the underlying linear mapping and are independent of the chosen Cartesian coordinate system. Equa­tion (2.3) defines the characteristic polynomial [110,111,121], and for a real transition matrix A it has real coefficients, so the eigenvalues of a linear MIMO system are real or in complex conjugate pairs.

Power series are one method [125] of defining functions and in particular for an arbitrary matrix L

exp (tL) = (tL)k/k! with (tL)° = i (2.4)

k=0

Like its “scalar relative,” the above series is absolutely summable [112,125] for all time t, and therefore

і 1 1 d

dt[eXp (tL)l = £ k! dt(tL)k = L exp (tL)

k=0

It follows that

IL xexp (tL) = L 1 — [exp (tL)| = exp (tL)
dt

or

L xexp (tL) = exp (tL)dt

Application of the above to the Laplace transform of exp (tL) yields exp [-Ф/ — Ц4, = (S, — L)-exp [-f(s; — L)]1 = (S; — L)-

o

so that

exp (tL) and (si — L)-1 are transform pairs (2.6)

For any matrix L it can be shown that a special linear change of Cartesian coordinates transforms L into the diagonal sub-block struc­ture of its Jordan Form J [110,121]. Each block corresponds to a different eigenvalue and Figure 2.1 illustrates a typical sub-block of J and exp (tJ) for an eigenvalue 1k. The literature [77,126] formulates various definitions of dynamic stability based on the mathematical concepts of a bounded variation about, or convergence to, a particular steady state. An appropriate criterion for present purposes is

MIMO system (2.1) is stable if and only if for all xo

and w(f)=0then (2.7)

lim x(t) =0

f! 1

Because dynamic stability is patently independent of the choice of Cartesian coordinates, those creating a transition matrix in the Jordan

Form can be adopted. Accordingly, equations (2.2) and (2.6) with Figure 2.1 translate the above into

MIMO system (2.1) is stable if and only if all eigenvalues of its transition matrix have strictly negative real parts (2.8)

Equation (2.2) provides the output of a linear MIMO system explicitly as

у = G(s)u + C(sI — A) 1xo where G(s) = D + C(sI — A) 1B

(2.9)

and G(s) is termed the Transfer Function Matrix which in practice is usually square. By augmenting the state vectors, transfer function matrices G1(s) and G2(s) in series or parallel combine as those for SISO systems—except for commutivity. Specifically [122,134]

In series : G(s) = G2(s)G1(s) and in parallel G(s) = G1(s) + G2(s)

(2.10)

A Resolvent (sI — A)—1 for finite n-dimensions has the rational form [111]

(sI — A) 1 = X sk XTk Y (s — Ik) with

k=1 k=1

{Tk<n} real matrices (2.11)

So the eigenvalues of state transition matrix are seen to be the poles of G(s) . Furthermore, by definition[27]

s is a zero of G(s) if and only if for some non-zero input

u(t) = exp (~s t) "u, no output response occurs; i. e., y(t) =0

Though these zeros appear to pose an intractable calculation, they are in fact the roots of the zero polynomial [122,123,134]

Z(s) = det (si — A)det G(s) (2.13)

Thus problems of dynamic stability, etc., can be couched in the potent algebra of complex variables [113,114].

For instance, residue calculus implies that the pth output of a stable linear MIMO system for a solitary non-zero input component uk (t) approximates after a “long enough” time to

yp(t) ‘ Residue of estGpk(s)uk at poles of Uk only (2.14)

In particular for

uk(t) = exp (ivt); Uk = (s — iv)—1; and uq(t) =0 forq = k

the pth component of the steady-state response is then evidently

Ур (t) ‘ Gpk (iv)eivt = |Gpk (iv) I exp [ivt + iArgGpk (iv)]

As real or complex pairs of eigenvalues are involved it follows from equations (2.9) and (2.11) that

Gpk(—iv) = IGpk(iv)Iff — ArgGpk (iv)

Consequently, by virtue of system linearity, its steady-state response to

uk(t)=sin vt = — [exp (ivt) — exp (—ivt)]; and uq(t)=0 forq = k is

yp(t) ‘ 1 G^k(iv)1 sin [vt+ArgGpk(iv)] (2.15)

Naturally enough G(iv) is termed the Real Frequency Response of a linear model, and it plays a pivotal role in questions of system stability.

Hc (s)

For design purposes, the controller K(s) in Figure 2.2a is partitioned into two parts

K(s) = Kp(s)F with F a diagonal scalar gain matrix (2.16)

and the feedback system is very often reconfigured [123,134] as in Figure 2.2b. Engineering experience usually matches the number of control inputs to the outputs requiring control, so Kp(s), F and G(s) are usually square (n x n). The overall closed-loop transfer function matrix

Hc (s) is

Hc(s) = [I + Q(s)F]_1Q(s)F where Q(s) = G(s)Kp(s) but Figure 2.2b shows that stability etc., is determined by

H(s) = [I + Q(s)F]-1Q(s)

R Z в

However, for actual MIMO systems in particular, the inverse transfer function relationship [124,135,136]

H(s) , H—1 (s)=F + Q(s) with Q(s) 4 Q"1(s) (2.18)

is evidently more tractable.

If Q(s) for engineering purposes is close enough to diagonal, then design reduces to a number of independent SISO systems. For each SISO system its closed-loop poles and zeros are derived from equation (2.17) as the zeros of 1 + Q(s)F and Q(s) respectively. The contour g in Figure 2.3 encloses all the unstable closed-loop poles, and using the classical Encirclement Theorem [112-114] Nyquist in 1932 showed that

A Unity Feedback SISO system is stable if and only if the mapped contour 1 + Q(g)F encircles the origin Po times anticlockwise, where Po is the number of unstable open loop poles.

A simple vector diagram shows that origin encirclements by1 + Q(g)F equate to encirclements of the critical point (—1,0) by Q(g)F. Also in practice Q(s) ! 0 as |s| !1, so the above reduces to

A Unity Feedback SISO system is stable if and only if its

Real Frequency Response locus Q(irn)F encircles (—1,0) (2.19)

anticlockwise Po times

Though some SISO defense systems have open loop poles at the origin due to kinematic integrations, these can be considered infinitesimally inside the stable region for design purposes.[28] Open loop systems are very often stable, and then the conformal mapping theorem [113,114,129] quantitatively relates transient closed-loop damping to the proximity of a Q(iv)F locus to (—1,0) in terms of Gain and Phase Margins [130,131]. Empirical rules [79,124] then also enable the design of analogue or digital controllers to achieve satisfactory closed-loop transient damping.

Presently MIMO feedback control systems can be designed in the frequency domain only if the pairs of transfer function matrices [Q(s), H(s)] or [Q(s), H(s)] are diagonally dominant [123,134]. Spe­cifically, the magnitudes of their diagonal elements must strictly exceed over the entire g-contour the sum of all others in the corre­sponding row or column. Diagonal dominance of [H(s),Q(s)] can often be contrived and then confirmed by the superposition [82,157] of Gershgorin Discs on all n x n elements of an Inverse Nyquist Array Q (iv) [123,134]. The apparently simplistic replacement of rows or columns by linear combinations with others appears to be an effective first step. Other techniques for achieving diagonal domi­nance are fully described in the literature [123,134]. In essence these procedures correspond to a scalar matrix operating on the control error vector and this matrix is then incorporated in Kp(s). Denoting diagonal elements of F and Q in equation (2.18) by fk and qkk(s), respectively, Rosenbrock’s stability criterion for open loop stable dynamics is

A diagonally dominant Unity Feedback MIMO system is stable if

and only if for allk < n the origin encirclements by the locus

Qkk(iv) equate to similarly orientated encirclements of (—fk, 0)

(2.20)

Thus diagonally dominant MIMO control systems can be engineered by well-established SISO techniques with readily computed Ostrowski Discs to assess residual loop interactions [82,134].

In general linear partial differential equations involve linear map­pings between infinite dimensional vector spaces,[29] and their matrices do not always exist [111]. On the other hand, a finite number of linear ordinary differential equations correspond to linear mappings between finite dimensional spaces whose matrices always exist [110,111] and are computationally tractable. Heat diffusion in power plant metalwork is characterized by linear partial differential equations [224] and so might appear outside the prescribed framework of equation (2.1). However, the vast majority of energy transfer is associated with the smallest eigenvalue [117,118], so that ordinary differential equations become reasonable approximations for suitably sized segments of an Eulerian mesh [117]. Though partial differential equations represent steam generator and reactor dynamics, finite difference equations as a finite number of non-linear ordinary differential equations satisfactorily match practical tests [117]. Their linearization therefore accords with equation (2.1) and with Aizerman’s conjecture, intact plant control via transfer function matrices can be engineered.

Practical studies [133,138-140] indicate that contriving diagonal dominance can require considerable skill: even with transfer function matrices much smaller than those for a complete power plant. Also, when manual intervention becomes imperative in some accidents, the control of an intact plant variable using a linear combination of several inputs appears to be humanly intractable. Moreover personal experience, exemplified by Section 3.4, is that ad hoc accident control cannot be conceived without an essentially one-to-one relationship between controlling and controlled plant variables. However, engi­neered rate constraints on nuclear plant temperatures etc. to ensure economic longevities or the intervention of trip circuits [127] intrin­sically impose such essentially one-to-one relationships. As a result, though fossil and nuclear power plants are MIMO systems, their control can generally be addressed in terms of SISO theory as illustrated in Sections 3.2 and 3.3.

Theoretical concepts necessary to appreciate Chapters 2 and 3 have now been briefly outlined. Their first application is in the control of nuclear reactor power.