Engine Limit Management with Linear Regulators

Abstract

This chapter describes the min–max logic arrangement used in standard engine control systems to maintain critical variables within the permissible bounds. A thorough analysis of this arrangement is conducted using the concept of positive invariance. The shortcomings of the min–max approach are made evident in simulations. A brief description of an acceleration-limiting approach is also included.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The control techniques presented so far do not address the need to maintain critical engine variables within permissible limits. As described in Sect. 1.3, engine outputs such as stall margins and turbine temperatures must be kept between safe limits at all times. The values chosen for safety limits depend on particular engine characteristics, and typically reflect a tradeoff between high engine performance and engine durability and operational safety. Indeed, large shaft accelerations are desirable for improved maneuverability, since they produce fast thrust responses to pilot commands. Large accelerations lead to transient reductions in stall margin, however, increasing the danger of compressor surge. Large accelerations also correlate with large turbine temperature transients. Since blade wear rate increases with temperature, transient peak temperature suppression should be included as an objective when designing fan speed controllers.

Constraints can be placed on system inputs and outputs. Temperatures, stall margins, and pressure ratios are examples of output variables that may be subject to constraints as part of a particular design process. Fuel flow rates and VSV and VBV openings are inputs whose ranges are constrained. Input constraints can arise from the same considerations as output constraints, that is, to address engine durability or safety, or from the presence of physical limits in actuator systems. An example of a safety-related input constraint is a minimum fuel flow rate requirement, imposed to maintain the combustion chamber away from lean blowout conditions. An example of a physical actuator constraint is given by the fact that a valve may not be more than 100% open or less than fully closed. When such actuator saturation effects are ignored during controller design, the implemented closed-loop controller may oscillate or even become unstable. Designs which do not explicitly address saturation must be validated via simulation studies to evaluate the possibility and effects of saturation modes during real-time operation.

In this chapter, a widely used control architecture intended to achieve limit protection is examined. The min–max approach, presented in Sect. 7.1, is used in many GTE control systems. CMAPSS-1 and CMAPSS-40k are distributed with min–max limit management implementations, representative of the actual control systems used in commercial engines such as General Electric’s GE90 and Pratt and Whitney’s PW2000.

The concept of set invariance is introduced in Sect. 7.2. This concept is instrumental to the analysis of constrained control systems in general. Some basic invariant set constructions are presented to illustrate the application of the theory to GTE control. In Sect. 7.3, min–max architecture based on state feedback controllers is analyzed for its invariance properties. Detailed examples using the CMAPSS-1 model are given throughout the chapter.

7.1 The Min–Max Limit Management Logic

The min–max selector arrangement is representative of actual aircraft engine control systems. Similar arrangements have also been used in industrial processes where a number of system outputs must be kept between prescribed limits as a main output is controlled between setpoints. The premise is that a single control input is available for the tasks of controlling the main output and maintaining a set of outputs between desired limits. Figure 7.1 shows the min–max arrangement with linear compensators. A version of the min–max arrangement with state feedback controllers is used for analysis in Sect. 7.3, and a version replacing the linear compensators with sliding mode controllers is presented in Chap. 8.

Fig. 7.1

Min–max limit management logic with linear compensators and integral control

As shown in Fig. 7.1, the technique uses input integration. The fuel flow control input W _F is applied to the engine, resulting in a set of h outputs characterized by their linearized transfer functions from W _F. Outputs are divided into two groups: a set of upper-limited outputs, numbered as y _i , for i = 1, 2. . . l and a set of lower-limited outputs, numbered with $i = l + 1,l + 2,...h$. The reader should note that transfer functions G _i (s) for i = 1, 2. . . h share the same poles, since they arise from the same linearized state-space matrix A. The static gains G _i (0) and zero locations will differ, due to differences in matrices C and D defining the outputs. It is also useful to remember that GTE dynamics are such that neither the transfer functions G _i (s) nor A have poles at the origin.

The min–max system is conceived to prevent selected variables from crossing their limits, by activating their regulators as needed. If a variable approaches its limit, its regulator should take over and attempt to drive the output to the prescribed limit without overshooting it. The diagram of Fig. 7.1 shows that upper-limited variables y _i (s) each have a controller K _i (s) with a classical error feedback structure. The setpoint corresponds to the desired output limit, r _i . Each upper-limited controller produces a “candidate” control rate u _ri . The minimum rate among upper-limit regulator outputs is preliminarily chosen. A separate set of lower-limit controllers produce their own candidate control rates. The maximum control rate among lower-limit regulator rates and the “winner” from the min stage is selected as the rate to be integrated, producing the fuel flow input command applied to the plant.

The above description cannot be expected to satisfy the average controls engineer. Unfortunately, few works can be found in the open literature that conduct a detailed analysis of this system. Is the system always stable? Could it produce endless switching among regulators? Why are upper-limited variables associated with the min selector and lower-limited variables to the max selector? Under which conditions is limit preservation guaranteed? It turns out that some of these questions have simple and thorough answers, while others are either unresolved or require analysis beyond the scope of this book.

The question of stability of min–max implementations using linear regulators, for instance, has been treated with nonlinear techniques applicable to sector-bounded nonlinearities, such as Popov’s criterion and the famous Small Gain Theorem (see Glattfelder and Schaufelberger [58], for instance). These tools provide only sufficient conditions, resulting in very conservative stability assessments. Johansson [33] has analyzed a conceptually similar system using the tools of piecewise-linear systems and multiple Lyapunov functions.

The author of this book proposed that the linear regulators be replaced by sliding mode controllers. Although analysis becomes more complex due to the nonlinearity inherent to sliding mode control, a full global asymptotic stability proof was achieved and simple design guidelines generated, see Richter [59]. This is presented in Chap. 8.

The question of limit protection effectiveness is difficult to establish for the transient regime when dynamic compensators (control transfer functions) are used. When static feedback laws of the form $u = -Kx + Pr$ are used instead, some simplifications occur, facilitating analysis. In the following sections, the dynamic compensation case is studied for its limit protection properties at steady-state only, while the static feedback case is analyzed for both transient and steady-state limit preservation properties.

7.1.1 Default Index Assumptions: Min and Max Operators

Although the generic mathematical operations of taking the minimum or maximum require no explanations, assumptions must be made regarding their behavior in the event of non-unique minimum or maximum values among operands. For the remainder of the book, we adopt the assignment rules made by the min and max operations implemented in many computer languages. Let {N(i)} represent an indexed collection of real numbers, for i = 1, 2, . . . n and let $\bar{N}$ and $\overline{N}$ denote, respectively, the maximum and minimum values found in the collection. If there exists a single index $\bar{i}$ such that $N(\bar{i})=\bar{N}$, $\bar{i}$ is chosen as the outcome of the max selection process. In general, if a set of maximizing indices exists, that is set {I(j)} is such that $N(I(j))=\bar{N}$, then the lowest of such indices is chosen: $\bar{i}=$min(I(j)). For example, take the set $N=\{3,\;5,\;-2,\;5,\;5,\;-2\}$. Here, $\bar{N}=5$ and I = {2,  4,  5}, thus $\bar{i}=2$. A similar convention is adopted for the min selector. In the same example, \underline{i_} = 3. This convention will be referred to as default index assumption in what follows.

7.1.2 Static Properties of the Min–Max Arrangement with Dynamic Compensators

Three fundamental questions pertaining to the operation of the min–max arrangement are decided next: determining the regulator that becomes active at the initial time (t = 0), determining the regulator that remains active during a steady-state regime (t → ∞), and establishing conditions for limits to be preserved at steady-state. As shown later in the chapter, a min-only or a max-only arrangement is sufficient in some cases. Because of this and to facilitate understanding, analysis is separated into min-only, max-only, and min–max cases.

7.1.2.1 Initial Regulator: Min-Only Case

Suppose only the min selector is used to select a control rate u _r among candidate rates u _ri , i ∈ L. The min selection law is expressed as

$${u}_{r} ={ \min }_{{}_{i\in L}}\left \{{u}_{ri}\right \}.$$

(7.1)

Suppose i = i ₀ is the regulator at the initial time. Then the following inequality must hold for all j:

$${u}_{r{i}_{0}}(0) \leq {u}_{rj}(0).$$

If there exists j≠i ₀ such that ${u}_{rj} = {u}_{r{i}_{0}}$, then it is necessary that i ₀ < j due to the default index assumptions. The control rates are expressed in terms of transfer functions as

$$\begin{array}{rcl}{ U}_{r{i}_{0}}(s)& = \frac{{K}_{{i}_{0}}(s)} {1+{G\prime}_{{i}_{0}}(s){K}_{{i}_{0}}(s)}{R}_{{i}_{0}}(s),&\end{array}$$

(7.2)

$$\begin{array}{rcl}{ U}_{rj}(s)& = {K}_{j}(s)({R}_{j}(s) - {G}_{j}(s){U}_{ri}(s)),&\end{array}$$

(7.3)

where ${G\prime}_{{i}_{0}}(s) = {G}_{{i}_{0}}(s)/s$. Recalling the initial value theorem [26] and noting that ${R}_{{i}_{0}}(s) = {r}_{{i}_{0}}/s$ and ${R}_{j}(s) = {r}_{j}/s$, we have

$${u}_{r{i}_{0}}(0) ={ \lim }_{{}_{s\rightarrow \infty }}s{U}_{r{i}_{0}}(s) ={ \lim }_{{}_{s\rightarrow \infty }} \frac{{r}_{i}{K}_{{i}_{0}}(s)} {1 + {G\prime}_{{i}_{0}}(s){K}_{{i}_{0}}(s)}.$$

(7.4)

Note that ${G\prime}_{{i}_{0}}$ has a pole at s = 0 and that it is strictly proper, since ${G}_{{i}_{0}}$ is proper and does not have zeroes at s = 0. Therefore, if ${K}_{{i}_{0}}(s)$ is proper, we have ${u}_{r{i}_{0}}(0) = {K}_{{i}_{0}}(\infty ){r}_{{i}_{0}}$. When ${K}_{{i}_{0}}(s)$ is an improper PD transfer function, the derivative term is impulsive when ${r}_{{i}_{0}}$ is constant. The initial regulator observed in simulations or real-time deployments will depend on the approximation involved in the implementation of the derivative term. Therefore, we restrict our formulas for the initial regulator to the case when all controllers are proper.

Now, using the initial value theorem on u _rj , we have

$${u}_{rj}(0) ={ \lim }_{{}_{s\rightarrow \infty }}s{U}_{rj}(s) ={ \lim }_{{}_{s\rightarrow \infty }}\left \{{r}_{j}{K}_{j}(s) - s{G}_{j}(s){U}_{r{i}_{0}}(s)\right \}.$$

(7.5)

The reader can verify that if K _j (s) is proper u _rj (0) has a well-defined constant value:

$${u}_{rj}(0) = {K}_{j}(\infty ){r}_{j}.$$

Therefore, the initial regulator is calculated as the smallest index i ₀ satisfying

$${K}_{{i}_{0}}(\infty ){r}_{{i}_{0}} \leq {K}_{j}(\infty ){r}_{j}$$

(7.6)

for all j ∈ L.

7.1.2.2 Initial Regulator: Max-Only Case

Now consider that only the max selector is used to select a control rate u _r among candidate rates u _ri , i ∈ H. The max selection law is expressed as

$${u}_{r} ={ \max }_{{}_{i\in H}}\left \{{u}_{ri}\right \}.$$

(7.7)

Suppose i = i ₀ is the regulator at the initial time. The initial values of the rates u _ri are the same regardless of the type of selector, max or min. The initial selection is therefore given by the smallest index i ₀ satisfying

$${K}_{{i}_{0}}(\infty ){r}_{{i}_{0}} \geq {K}_{j}(\infty ){r}_{j}$$

(7.8)

for all j ∈ H.

7.1.2.3 Initial Regulator: Min–Max Case

The min–max case has a subtlety that requires careful attention. Referring back to Fig. 7.1, the min–max selection law is given by

$${u}_{r} ={ \max }_{{}_{k\in H}}\left \{{\min }_{{}_{j\in L}}\left \{{u}_{rj}\right \},\;\;{u}_{rk}\right \},$$

(7.9)

where u _rj are the min-selected regulator outputs and u _rk are the max-selected regulator outputs. Again, the values of the initial rates are the same regardless of the selection mechanism. Suppose i ₀ is the index of the regulator selected at t = 0. If i ₀ ∈ L, the following inequalities must be true:

$$\begin{array}{rcl}{ u}_{r{i}_{0}}& \leq {u}_{rj}\mbox{ for all }j \in L,&\end{array}$$

(7.10)

$$\begin{array}{rcl}{ u}_{r{i}_{0}}& \geq {u}_{rk}\mbox{ for all }k \in H.&\end{array}$$

(7.11)

If i ₀ ∈ H, however, the following inequalities apply:

$$\begin{array}{rcl}{ u}_{r{i}_{0}}& > {u}_{rk}\mbox{ for all }k \in L,&\end{array}$$

(7.12)

$$\begin{array}{rcl}{ u}_{r{i}_{0}}& >{ \min }_{{}_{j\in L}}\left \{{u}_{rj}\right \}.&\end{array}$$

(7.13)

Note that strict inequality must be used, since the winner of the min stage is applied to the first port of the max selector. Thus, equality between the winner of the max stage and the winner of the min stage would result in i ₀ ∈ L due to the default index assumptions. Also note that ${u}_{r{i}_{0}}$ is required to be greater than the minimum of the rates produced by regulators in L, but not necessarily greater than each. This implies that a two-step iterative process must be followed to determine i ₀. Assume first that i ₀ ∈ L and find the smallest index satisfying inequalities (7.10) and (7.11). If no such index can be found, it must be that i ₀ ∈ H. Inequalities (7.12) and (7.13) are then tested using the known value of the winner of the min stage.

The reader should observe that the calculations for the initial regulator in the min, max and min–max cases are not influenced by whether a variable is upper- or lower-limited. Only the type of selector must be taken into account.

7.1.2.4 Steady Regulator: Min-Only Case

Here, we assume that the closed-loop system is asymptotically stable, with a fixed steady-state regulator selection denoted as i ^∗. Applying the final value theorem [26] to the Laplace expression for the control rate of (7.2), we obtain

$${u}_{r{i}^{{_\ast}}}(\infty ) ={ \lim }_{{}_{s\rightarrow 0}}s{U}_{r{i}^{{_\ast}}}(s) ={ \lim }_{{}_{s\rightarrow 0}} \frac{{r}_{{i}^{{_\ast}}}{K}_{{i}^{{_\ast}}}(s)} {1 + {G\prime}_{{i}^{{_\ast}}}(s){K}_{{i}^{{_\ast}}}(s)}.$$

(7.14)

The reader can verify that ${u}_{r{i}^{{_\ast}}} = 0$ whenever ${K}_{{i}^{{_\ast}}}$ and ${G}_{{i}^{{_\ast}}}$ are proper and do not have zeroes at the origin. Furthermore, the same is true if ${K}_{{i}^{{_\ast}}}$ is a PD transfer function and ${G}_{{i}^{{_\ast}}}$ is proper with no zeroes at the origin. This is expected, since the active loop is of type I, and the steady-state error to constant inputs is zero. Since ${K}_{{i}^{{_\ast}}}$ is driven by the error, its output will be zero at steady-state.

Applying the final value theorem to U _rj (s) from (7.3), we obtain

$${u}_{rj}(\infty ) ={ \lim }_{{}_{s\rightarrow 0}}s{K}_{j}(s)\left [\frac{{r}_{j}} {s} - {G}_{j}(s){U}_{r{i}^{{_\ast}}}(s)\right ].$$

If G _j (s) and K _j (s) are proper with no poles at the origin, this reduces to

$${u}_{rj}(\infty ) = {K}_{j}(0)\left [{r}_{j} - {r}_{i} \frac{{G}_{j}(0)} {{G}_{{i}^{{_\ast}}}(0)}\right ].$$

(7.15)

Therefore, i ^∗ is the smallest index satisfying

$${K}_{j}(0)\frac{{r}_{{i}^{{_\ast}}}{G}_{j}(0)} {{G}_{{i}^{{_\ast}}}(0)} \leq {K}_{j}(0){r}_{j},$$

(7.16)

for all j ∈ L. Note that

$$\frac{{r}_{{i}^{{_\ast}}}{G}_{j}(0)} {{G}_{{i}^{{_\ast}}}(0)} =\bar{ {y}}_{j/{i}^{{_\ast}}},$$

where $\bar{{y}}_{j/{i}^{{_\ast}}}$ is the steady value attained by y _j when i ^∗ is the steady regulator. If K _j (0) > 0, inequality (7.17) reduces to $\bar{{y}}_{j/{i}^{{_\ast}}} \leq {r}_{j}$. This is interpreted verbally as follows: “i ^∗ is the regulator for which outputs do not exceed their setpoints at steady-state”. If K _j (0) < 0, the inequality becomes $\bar{{y}}_{j/{i}^{{_\ast}}} \geq {r}_{j}$. This corresponds to: “i ^∗ is the regulator for which outputs do not become smaller than their setpoints at steady-state”.

7.1.2.5 Steady Regulator: Max-Only Case

The final regulator index for this case is the smallest index i ^∗ satisfying

$${K}_{j}(0)\frac{{r}_{{i}^{{_\ast}}}{G}_{j}(0)} {{G}_{{i}^{{_\ast}}}(0)} \geq {K}_{j}(0){r}_{j}$$

(7.17)

for all j ∈ H. If K _j (0) > 0, inequality (7.17) reduces to $\bar{{y}}_{j/{i}^{{_\ast}}} \geq {r}_{j}$. This is interpreted verbally as follows: “i ^∗ is the regulator for which outputs do not become smaller than their setpoints at steady-state”. If K _j (0) < 0, the inequality becomes $\bar{{y}}_{j/{i}^{{_\ast}}} \leq {r}_{j}$. This corresponds to: “i ^∗ is the regulator for which outputs do not exceed their setpoints at steady-state”.

The steady properties of the isolated min and max selectors provide a basic guideline for design. If all outputs are upper-limited and K _j (0) > 0 for all j, a max regulator provides steady-state limit protection. If all outputs are lower-limited and K _j (0) > 0 for all j, a max selector provides steady-state limit protection. This seems to be the assumption justifying the assignments found in the standard min–max arrangements used in the aerospace industry.

As the reader may appreciate, the more general case where outputs have different signs of K _j (0), some being lower-limited and some being upper-limited are not covered by the min-only or max-only arrangements. The min–max arrangement was introduced as an attempt to cover these cases.

7.1.2.6 Steady Regulator: Min–Max Case

The same logic used for determining the initial regulator is followed, using the appropriate formulas for the steady rates. If i ^∗ ∈ L, the following inequalities hold:

$$\begin{array}{rcl} 0\!& \leq & \!{u}_{rj}\mbox{ for all }j \in L,\end{array}$$

(7.18)

$$\begin{array}{rcl} 0\!& \geq & \!{u}_{rk}\mbox{ for all }k \in H.\end{array}$$

(7.19)

If i ^∗ ∈ H, however, the following inequalities apply:

$$\begin{array}{rcl} 0\!& >& \!{u}_{rk}\mbox{ for all }k \in L,\end{array}$$

(7.20)

$$\begin{array}{rcl} 0\!& >& \!{\min }_{{}_{j\in L}}\left \{{u}_{rj}\right \}.\end{array}$$

(7.21)

7.1.3 Example: CMAPSS-1

Consider the transfer functions from incremental fuel flow to incremental fan speed (rpm), incremental HPT temperature (^∘R) and HPC stall margin (%) near FC01:

$$\begin{array}{rlrlrl} {G}_{1}(s) & = \frac{\Delta {N}_{\mathrm{f}}(s)} {\Delta {W}_{\mathrm{F}}(s)} = \frac{230.9s + 2000} {{s}^{2} + 8.504s + 17.16}, & & \\ {G}_{2}(s) & = \frac{\Delta {T}_{48}(s)} {\Delta {W}_{\mathrm{F}}(s)} = \frac{146.2{s}^{2} + 1027s + 1528} {{s}^{2} + 8.504s + 17.16}, & & \\ {G}_{3}(s) & = \frac{\Delta SmHPC(s)} {\Delta {W}_{\mathrm{F}}(s)} = \frac{-4.405{s}^{2} - 28.81s - 20.49} {{s}^{2} + 8.504s + 17.16}. & & \end{array}$$

Suppose that a set of controllers is selected that independently stabilize the above transfer functions under an integral control loop (disregarding transient response qualities), as follows:

$$\begin{array}{rlrlrl} {K}_{1}(s) & = 0.21\frac{s + 3.715} {s + 20}, & & \\ {K}_{2}(s) & = 0.1, & & \\ {K}_{3}(s) & = \frac{-s - 1} {s + 2}. & & \end{array}$$

Suppose r ₁ = 10, r ₂ = 20, and ${r}_{3}\,=\, - 10$. In this example, G ₁(∞) = 0, G ₂(∞) =  146. 2 and ${G}_{3}(\infty ) = -4.405$. Also, K ₁(∞) = 0. 21, K ₂(∞) =  0. 1, and ${K}_{3}(\infty ) = -1$. Similarly, G ₁(0) = 116. 55, G ₂(0) = 89. 044, ${G}_{3}(0) = -1.194$ and K ₁(0) = 0. 039, K ₂(0) = 0. 1 and ${K}_{3}(0) = -0.5$.

If only a min selector is used, (7.6) can be used to determine that the initial regulator is i ₀ = 2 and the final regulator is i ^∗ = 1. If only a max selector is used, ${i}_{0} = {i}^{{_\ast}} = 3$. Suppose now that u _r1 and u _r2 are associated with a min selector, while the min-preselection and u _r3 are associated with a max selector. It can be readily verified that ${i}_{0} = {i}^{{_\ast}} = 3$ as well.

The concept of positive invariance is introduced next as an essential tool to study limit regulator behavior.

7.2 Basic Set Invariance Concepts

Set invariance theory is a unifying umbrella under which many techniques for constraint handling have been developed. Although a detailed account of set invariance and derived techniques is out of the scope of this book, some definitions and results will be included. For an excellent introduction to the topic, readers are referred to the survey by Blanchini [60].

Given a dynamical system with state vector x, a positively invariant set is a subset ${\mathcal{S}}_{o}$ of the state-space formed by all initial conditions resulting in trajectories, which remain in $\mathcal{S}$ for all subsequent times. That is, ${\mathcal{S}}_{o}$ is positively invariant if $x(0) \in {\mathcal{S}}_{o}$ implies $x(t) \in {\mathcal{S}}_{o}$ for all t > 0, justifying the positive invariance qualification.

A result by Nagumo [61] establishes a necessary and sufficient condition for a set to be positively invariant with respect to a given dynamical system. For the purposes of this book, invariant sets of simple descriptions are considered, such as ellipsoids, half-spaces, and real intervals. Roughly, Nagumo’s condition is equivalent to the requirement that the vector $\dot{x}$ evaluated along the boundary of ${\mathcal{S}}_{o}$ be directed toward its interior, as illustrated in Fig. 7.2. Nagumo’s condition is rather intuitive, and becomes more so when x is a scalar, so that the invariant set is an interval, as shown next.

Fig. 7.2

Illustration of Nagumo’s invariance condition. ${\mathcal{S}}_{o}$ is the invariant set and $\partial {\mathcal{S}}_{o}$ its boundary

7.2.1 Positive Invariance of an Interval

An interval ( − ∞, b] is invariant for a generic real variable z(t) if \dot{z}(t) ≤ 0 at z = b. Similarly, an interval [a, ∞) is invariant if \dot{z}(t) ≥ 0 at z = a. When an interval is invariant and z(t ₁) belongs to the interval for some t ₁ > 0, then z(t) will remain in the interval for t ≥ t ₁. The concept is readily applied to limit protection: the interval ( − ∞, 0] must be invariant for the error $e = r - y$ when y is a lower-limited output. Conversely, [0, ∞) must be invariant for the error of upper-limited variables.

7.2.2 Ellipsoidal Invariant Sets for Linear Systems

An n-dimensional ellipsoid is described by the inequality

$${x}^{\mathrm{T}}Px \leq 1,$$

where P is a symmetric, positive-definite matrix. Consider first an autonomous linear state-space system described $\dot{x} = Ax$ and an ellipsoidal set ${\mathcal{S}}_{o}$ described as

$${\mathcal{S}}_{o} = \mathcal{E} =\{ x\; :\; {x}^{\mathrm{T}}Px \leq 1\}.$$

For ℰ to be PI, Nagumo’s theorem requires that $\dot{x} = f(x) = Ax$ be directed toward the interior of ℰ, when x is taken at its boundary, defined by $h(x) = {x}^{\mathrm{T}}Px\,=\,1$. This can be ensured by the condition that the projection of f(x) onto a vector normal to the boundary be negative, as shown in Fig. 7.3. The invariance condition is expressed as

$$\nabla h(x).f(x) < 0.$$

For an ellipsoidal boundary we have ∇ h(x) = 2x ^T P, thus the condition becomes 2x ^T PAx < 0. The reader may recall that any square matrix can be decomposed into a symmetric component and an antisymmetric component. Decompose PA as follows:

$$PA = \frac{1} {2}(PA + {A}^{\mathrm{T}}P) + \frac{1} {2}(PA - {A}^{\mathrm{T}}P).$$

The first term is symmetric, while the second is antisymmetric. Thus,

$$2{x}^{\mathrm{T}}PAx = {x}^{\mathrm{T}}(PA + {A}^{\mathrm{T}}P)x + {x}^{\mathrm{T}}(PA - {A}^{\mathrm{T}}P)x.$$

Recalling that the cross-terms of a quadratic form xY ^T x cancel out when Y is anti-symmetric, the invariance condition reduces to the following Lyapunov inequality

$$ PA + {A}^{\mathrm{T}}P < 0. $$

(7.22)

The above condition can be applied to evaluate the invariance of an ellipsoid relative to the closed loop system $\dot{x} = (A - BK)x$ resulting from using the state feedback control law $u = -Kx$:

$$ P(A - BK) + {(A - BK)}^{\mathrm{T}}P < 0. $$

(7.23)

Note that inequality (7.23) is satisfied for someP when K stabilizes A − BK, but that a predetermined P need not satisfy the inequality for all stabilizing K. A family of invariant ellipsoids associated with a given stabilizing K can be found by turning inequality (7.23) into equality, using an arbitrary positive-definite, symmetric matrix Q:

$$ P(A - BK) + {(A - BK)}^{\mathrm{T}}P = -Q. $$

(7.24)

All ellipsoids of the form x ^T Px = a, with a ≥ 0 are then PI relative to the closed-loop dynamics.

Fig. 7.3

Invariance enforcement by negative gradient projection

7.2.3 Invariance of a Half-Space

State and output constraints can be formulated as a set of linear inequalities in the state variables: $\mathcal{G} = \cap {\mathcal{G}}_{i}$ for i = 1, 2. . m, where ${\mathcal{G}}_{i} = \left \{x\; :\; {G}_{i}x \leq 1\right \}$. Application of Nagumo’s result to a system $\dot{x} = f(x)$ and an individual linear constraint G _i x ≤ 1 results in the following condition for invariance:

$${G}_{i}f(x) \leq 0\mbox{ along }{G}_{i}x = 1.$$

(7.25)

In the case of a linear state-space system (A, B) under state feedback $u = -Kx$, this reduces to

$${G}_{i}(A - BK)x \leq 0\mbox{ along }{G}_{i}x = 1.$$

(7.26)

In general, condition (7.26) cannot be satisfied for all points belonging to the boundary defined by G _i x = 1. Geometrically, if G _i is not parallel to G _i (A − BK), the boundary will be divided into three subsets: a subset where ${G}_{i}(A - BK)x = 0$, a subset where G _i (A − BK) > 0 and a subset where the condition is satisfied. When several constraints exist, methodologies have been developed to ensure that points belonging to the set, where G _i (A − BK)x > 0 do not satisfy the other constraints. Polyhedral invariant set theory provides means to construct invariant sets using linear segments. Given a constraint set defined by design requirements, polyhedral invariant sets can be constructed with little conservativeness. An invariant set construction is conservative if it excludes points of the constraint set which actually lead to permissible trajectories. Polyhedral sets require a large number of vertices to achieve low conservativeness. The interested reader is referred to Blanchini’s survey for an introduction [60].

7.2.4 Ellipsoidal Operating Sets

Invariant sets are used to determine the range of allowable initial conditions so that the ensuing trajectories do not violate the constraints. When error dynamics are used as a basis, an invariant set description can be used to determine the allowable range of the setpoints so that no limit violations will occur in the transient or steady regimes. Such operating sets are thus invariant subsets of the constraint set. A simple approach to finding an operating set is to find the largest invariant ellipsoid contained in the constraint set. For this, ellipsoids which are tangent to individual constraints are found. The smallest of such ellipsoids will be tangent to one of the constraints and tangent or interior to the remaining ones, in addition to being invariant.

Given a linear constraint G _i x ≤ 1, and an invariant ellipsoid family defined by matrix P, the ellipsoid which is tangent to the boundary G _i x = 1 can be found by solving the following constrained optimization problem:

$$\begin{array}{rlrlrl} \max V & = {x}^{\mathrm{T}}Px\mbox{ subject to} & & \\ {G}_{i}x & = 1. & & \end{array}$$

This is readily solved using Lagrange multipliers, yielding the following formula for the maximum value of x ^T Px:

$${V }_{i} = \frac{1} {{G}_{i}{P}^{-1}{G}_{i}^{\mathrm{T}}}.$$

Thus, the largest invariant ellipsoid tangent to the i-th constraint is given by

$${\mathcal{E}}_{i} =\{ x\; :\; {x}^{\mathrm{T}}Px \leq {V }_{ i}\},$$

where P satisfies inequality (7.24). The overall invariant ellipsoid can be found by taking the smallest V _i and matrix P. As an introductory example, consider the double-integrator plant $G(s) = 1/{s}^{2}$ with state-space matrices A and B as follows:

$$A = \left [\begin{array}{l@{\quad }l} 0\quad &1\\ 0\quad &0\end{array} \right ],\;\;\;\;B = \left [\begin{array}{l} 0\\ 1 \end{array} \right ].$$

Suppose the constraints are given by | x ₁ | ≤ 1 and | x ₂ | ≤ 1. 5. The plant is stabilized with a state-feedback law of the form $u = -Kx$, where K is chosen so that ${A}_{\mathrm{c}} = A - BK$ has eigenvalues with negative real parts. Suppose a K is designed using an LQR approach, with Q = I and R = 1. This yields $K = [1\;\;\sqrt{3}]$, which places the poles of A _c at $-\sqrt{3}/2 \pm i/2$. The following P satisfies Lyapunov inequality (7.23):

$$P = \left [\begin{array}{l@{\quad }l} 1.5847\quad &0.5489\\ 0.5489\quad &0.6339 \end{array} \right ].$$

The constraints are expressed as G ₁ x = [1 0]x ≤ 1, ${G}_{2}x = [-1\;0]x \leq 1$, ${G}_{3}x = [0\;1/1.5]x \leq 1$ and ${G}_{4} = [0\; - 1/1.5]x \leq 1$. The values of V _i are calculated as ${V }_{1} = {V }_{2} = 1.109$ and ${V }_{3} = {V }_{4} = 1$. Thus, the ellipsoid described by x ^T Px ≤ 1 is tangent the third and fourth constraints and interior to the first and second constraints, in addition to being positively invariant. Figure 7.4 shows the ellipsoidal boundary in relation to the constraints. A few trajectories have also been plotted.

Fig. 7.4

Ellipsoidal invariant set and constraints: double-integrator system

The dotted line represents a trajectory which satisfies the constraints but whose initial condition is not captured by the ellipsoid. In contrast, the dashed line is a trajectory whose initial point belongs to the constraint set but which produces constraint violation, and is correctly excluded by the ellipsoidal set. An example of the ellipsoidal construction as applied to the GTE problem is given in Sect. 7.5.2. A less conservative invariant set construction which retains the simplicity of ellipsoids is given by the semiellipsoidal approach of O’Dell [62]. In this approach, an invariant ellipsoid which may exceed the constraints is sought under the condition that its intersection with the constraints retains the positive invariance property. The methodology regards K and P as free parameters, optimized to yield an ellipsoid of maximum volume. The resulting semiellipsoidal set is an approximation to the maximal recoverable set under state feedback, i.e., the largest set of initial conditions yielding trajectories which proceed to the origin without constraint violation. The maximizing argument K does not offer any performance guarantees, however.

The operating set approach amounts only to “passive” constraint validation. If invariance cannot be established without unreasonable conservativeness, a different control law must be found or the constraints relaxed. In contrast, approaches such as the min–max arrangement constitute active means to enforce the constraints as the system operates. The use of a reference governor allows to exploit set invariance properties in an active way. The key idea is to filter reference inputs to force the state to remain in an invariant set, see, for instance, Bemporad [63].

7.3 Min–Max Limit Management with Integral State Feedback Controllers

Analysis of the min–max arrangement is somewhat simplified when state feedback controllers are used instead of dynamic compensators. Consider a linear plant with input integration, described by the following augmented state-space model:

$$\begin{array}{rcl} \dot{{x}}_{\mathrm{a}}& = {A}_{\mathrm{a}}{x}_{\mathrm{a}} + {B}_{\mathrm{a}}{u}_{r},&\end{array}$$

(7.27)

$$\begin{array}{rcl}{ y}_{i}& = {C}_{ai}{x}_{\mathrm{a}},&\end{array}$$

(7.28)

where x _a ^T = [x ^T u] is the augmented state vector and A _a and B _a are defined as in (4.31). Outputs ${y}_{i} = {C}_{i}x + Du$ are defined by their augmented matrices C _ai  = [C _i   D _i ]. Consider the static state feedback law

$${u}_{ri} = -{K}_{i}{x}_{\mathrm{a}} + {P}_{i}{r}_{i},$$

(7.29)

where K _i is such that A _a − B _a K _i is stable. Although K _i could be chosen to be constant across regulators to guarantee stability and achieve desirable transient properties for the main controlled output, the min–max arrangement would become useless. To see this, note that u _ri  − u _rj is constant for a given pair i, j when a constant K is used. Thus, the regulator selected by min and max at t = 0 becomes permanent and no switching occurs. Assume, then, that the min, max, and min–max arrangements involve distinct feedback gains.

The value of the prefilter gainP _i is chosen to ensure that y _i  = r _i in the steady-state, and must therefore vary with i (see (7.35)).

7.3.1 Closed-Loop Behavior Under a Fixed Regulator

Suppose the i-th regulator is active at all times. The closed-loop system becomes

$$\dot{{x}}_{\mathrm{a}} = {A}_{ci}{x}_{\mathrm{a}} + {B}_{\mathrm{a}}{P}_{i}{r}_{i},$$

(7.30)

where ${A}_{ci} = {A}_{\mathrm{a}} - {B}_{\mathrm{a}}{K}_{i}$. Since K _i is designed so that A _ci has eigenvalues with negative real parts, the augmented state reaches a steady-state value of $\bar{{x}}_{ai}$ satisfying

$$\begin{array}{rcl} 0& = {A}_{\mathrm{a}}\bar{{x}}_{ai} + {B}_{\mathrm{a}}\bar{{u}}_{ri},&\end{array}$$

(7.31)

$$\begin{array}{rcl} 0& =\bar{ {u}}_{ri} = -{K}_{i}\bar{{x}}_{ai} + {P}_{i}{r}_{i},&\end{array}$$

(7.32)

where (7.32) is obtained from the requirement that the last component of the augmented state derivative be zero at steady-state, that is, $\dot{u}\,=\,\bar{{u}}_{ri}\,=\,0$. Equations (7.31) and (7.32) imply that ${A}_{\mathrm{a}}\bar{{x}}_{ai} = 0$ and that P _i must be related to $\bar{{x}}_{ai}$ as follows:

$${P}_{i} = {K}_{i}\frac{\bar{{x}}_{ai}} {{r}_{i}}.$$

(7.33)

The value of $\bar{{x}}_{ai}$ is determined by the requirement that y _i  = r _i at steady-state. Separating the first n components of (7.31), we have:

$$0 = A\bar{x} + B\bar{u},$$

where $\bar{{x}}_{ai}^{\mathrm{T}} = [\bar{{x}}^{\mathrm{T}}\;\;\bar{u}]$. Since A has no poles at the origin, $\bar{x} = -{A}^{-1}B\bar{u}$. The steady output is then

$$\bar{{y}}_{i} = {C}_{i}\bar{x} + {D}_{i}\bar{u} = (-{C}_{i}{A}^{-1}B + {D}_{ i})\bar{u} = {G}_{i}(0){r}_{i}.$$

For $\bar{{y}}_{i} = {r}_{i}$, we must have $\bar{u} = {r}_{i}/{G}_{i}(0)$ and $\bar{x} = -{A}^{-1}B{r}_{i}/{G}_{i}(0)$, thus

$$\begin{array}{rcl} \bar{{x}}_{ai}& = \frac{{r}_{i}} {{G}_{i}(0)}\left [\begin{array}{l@{\quad }l} - {A}^{-1}\quad &B\\ \quad &1 \end{array} \right ],&\end{array}$$

(7.34)

$$\begin{array}{rcl}{ P}_{i}& = \frac{{K}_{i}} {{G}_{i}(0)}\left [\begin{array}{l@{\quad }l} - {A}^{-1}\quad &B\\ \quad &1 \end{array} \right ].&\end{array}$$

(7.35)

7.3.2 Closed-Loop Behavior Relative to a Fixed Index

It is convenient to shift the augmented state variable by the steady-state value corresponding to an arbitrary index i. That is, define

$$\tilde{{x}}_{ai} = {x}_{ai} -\bar{ {x}}_{ai}.$$

(7.36)

Relevant quantities are now expressed in terms of $\tilde{{x}}_{ai}$. The i-th and j-th outputs become

$$\begin{array}{rcl}{ y}_{i} = {C}_{ai}{x}_{\mathrm{a}}& = {C}_{ai}(\tilde{{x}}_{ai} +\bar{ {x}}_{ai}) = {r}_{i} + {C}_{ai}\tilde{{x}}_{ai},&\end{array}$$

(7.37)

$$\begin{array}{rcl}{ y}_{j} = {C}_{aj}{x}_{\mathrm{a}}& = {C}_{aj}(\tilde{{x}}_{ai} +\bar{ {x}}_{ai}) =\bar{ {y}}_{j/i} + {C}_{aj}\tilde{{x}}_{ai},&\end{array}$$

(7.38)

where $\bar{{y}}_{j/i}$ is the steady value of y _j when i is active at steady-state. The tracking errors are expressed in terms of the new state variable as

$$\begin{array}{rcl}{ e}_{i} = {r}_{i} - {y}_{i}& = -{C}_{ai}\tilde{{x}}_{ai},&\end{array}$$

(7.39)

$$\begin{array}{rcl}{ e}_{j} = {r}_{j} - {y}_{j}& =\bar{ {e}}_{j/i} - {C}_{aj}\tilde{{x}}_{ai},&\end{array}$$

(7.40)

where \bar{e} _j ∕ i denotes the steady-state error in y _j when i is active. The control rates are expressed as

$$\begin{array}{rcl}{ u}_{ri}& = -{K}_{i}(\tilde{{x}}_{ai} +\bar{ {x}}_{ai}) + {P}_{i}{r}_{i} = -{K}_{i}\tilde{{x}}_{ai},&\end{array}$$

(7.41)

$$\begin{array}{rcl}{ u}_{rj}& = -{K}_{j}(\tilde{{x}}_{ai} +\bar{ {x}}_{ai}) + {P}_{j}{r}_{j},&\end{array}$$

(7.42)

where (7.32) has been used.

The reader should observe that (7.37)–(7.42) use an arbitrary index i as a reference, but are valid regardless of the active regulator. The next two expressions for the derivatives of tracking errors e _i and e _j , however, assume that i is the active regulator, and this is reflected in the notations \dot{e} _i ∕ i and \dot{e} _j ∕ i .

$$\begin{array}{rcl} \dot{{e}}_{i/i}& = -{C}_{ai}{A}_{ci}\tilde{{x}}_{ai},&\end{array}$$

(7.43)

$$\begin{array}{rcl} \dot{{e}}_{j/i}& = -{C}_{aj}{A}_{ci}\tilde{{x}}_{ai}.&\end{array}$$

(7.44)

Equations (7.30) and (7.31), and the definition of $\tilde{{x}}_{ai}$ from (7.36) have been used in the derivation of (7.43) and (7.44). Finally, the following expression for the difference between control rates will be useful in subsequent developments:

$${u}_{ri} - {u}_{rj} = ({K}_{j} - {K}_{i})\tilde{{x}}_{ai} + {K}_{j}\bar{{x}}_{ai} - {P}_{j}{r}_{j}.$$

(7.45)

7.3.3 Static Properties of the Min–Max Arrangement with State Feedback

The static analysis conducted for the min–max arrangement with dynamic compensators is repeated for the state feedback case. The same fundamental issues are discussed: determining the regulator that becomes active at the initial time (t = 0), determining the regulator that remains active during a steady-state regime (t → ∞), and establishing conditions for limits to be preserved at steady-state. Unlike min–max systems with control transfer functions, initial conditions have an effect in the initial regulator selection. Since linear systems are considered, initial conditions do not determine steady-state properties, however.

7.3.3.1 Initial Regulator: Min-Only Case

Using (7.45) directly, the initial regulator i ₀ must satisfy

$$({K}_{j} - {K}_{{i}_{0}})\tilde{{x}}_{a{i}_{0}}(0) \leq -{K}_{j}\bar{{x}}_{{a}_{{i}_{ 0}}} + {P}_{j}{r}_{j}$$

for all j ∈ L. Using identity (7.34), i ₀ is determined as the smallest index satisfying

$$({K}_{j} - {K}_{{i}_{0}})\tilde{{x}}_{a{i}_{0}}(0) \leq -{M}_{j} \frac{{r}_{{i}_{0}}} {{G}_{{i}_{0}}(0)} + {P}_{j}{r}_{j}$$

(7.46)

for all j ∈ L where the scalar M _j is defined as

$${ M}_{j} = {K}_{j}\left [\begin{array}{l@{\quad }l} - {A}^{-1}\quad &B \\ \quad &1 \end{array} \right ].$$

(7.47)

7.3.3.2 Initial Regulator: Max-Only Case

Simply reversing inequality (7.46), we see that initial regulator i ₀ is the smallest index satisfying

$$({K}_{j} - {K}_{{i}_{0}})\tilde{{x}}_{a{i}_{0}}(0) \geq -{M}_{j} \frac{{r}_{{i}_{0}}} {{G}_{{i}_{0}}(0)} + {P}_{j}{r}_{j}$$

(7.48)

for all j ∈ H.

7.3.3.3 Initial Regulator: Min–Max Case

The logic is the same as for the dynamic compensator case: if i ₀ ∈ L, the following inequalities must hold:

$$\begin{array}{rcl} ({K}_{j} - {K}_{{i}_{0}})\tilde{{x}}_{a{i}_{0}}(0)& \leq -{K}_{j}\bar{{x}}_{{a}_{{i}_{ 0}}} + {P}_{j}{r}_{j}\mbox{ for all }j \in L,&\end{array}$$

(7.49)

$$\begin{array}{rcl} ({K}_{j} - {K}_{{i}_{0}})\tilde{{x}}_{a{i}_{0}}(0)& \geq -{M}_{k} \frac{{r}_{{i}_{0}}} {{G}_{{i}_{0}}(0)} + {P}_{k}{r}_{k}\mbox{ for all }k \in H.&\end{array}$$

(7.50)

If i ₀ ∈ H, however, the following inequalities apply:

$$\begin{array}{rcl} ({K}_{j} - {K}_{{i}_{0}})\tilde{{x}}_{a{i}_{0}}(0)& \geq -{M}_{k} \frac{{r}_{{i}_{0}}} {{G}_{{i}_{0}}(0)} + {P}_{k}{r}_{k}\mbox{ for all }k \in H,\ &\end{array}$$

(7.51)

$$\begin{array}{rcl} ({K}_{j} - {K}_{{i}_{0}})\tilde{{x}}_{a{i}_{0}}(0)& >{ \min }_{{}_{j\in L}}\left \{{u}_{rj}(0)\right \}.&\end{array}$$

(7.52)

A guess must be made regarding whether i ₀ ∈ L or i ₀ ∈ H and the corresponding inequalities verified.

7.3.3.4 Steady Regulator: Min-Only Case

Suppose i ^∗ is the index of the steady regulator. Then $\dot{u} = {u}_{r{i}^{{_\ast}}} = 0$ and $\tilde{{x}}_{a{i}^{{_\ast}}} = 0$. Equation (7.45) is used directly to determine that i ^∗ is the smallest index satisfying

$${M}_{j} \frac{{r}_{{i}^{{_\ast}}}} {{G}_{{i}^{{_\ast}}}(0)} \leq {P}_{j}{r}_{j}$$

for all j ∈ L. If P _j is designed so that y _j  = r _j in steady-state (i.e., according to (7.35)), then the reader can verify that i ^∗ is the smallest index satisfying

$$\frac{1} {{G}_{j}(0)}{M}_{j}\left (\bar{{y}}_{j/{i}^{{_\ast}}}- {r}_{j}\right ) \leq 0,$$

(7.53)

where $\bar{{y}}_{j/{i}^{{_\ast}}} = {G}_{j}(0){r}_{{i}^{{_\ast}}}/{G}_{{i}^{{_\ast}}}(0)$ is the steady value attained by y _j when i ^∗ is active at steady state. This result is interpreted in a similar way as done for the dynamic compensator case, where M _j  ∕ G _j (0) plays the role of the compensator’s low-frequency gain K(0): when M _j  ∕ G _j (0) > 0, i ^∗ is the regulator for which outputs do not exceed their setpoints at steady-state. If M _j  ∕ G _j (0) < 0, i ^∗ is the regulator for which outputs do not become smaller than their setpoints at steady-state. Note from (7.35) that M _j  ∕ G _j (0)≠0, otherwise the tracking task would not be possible.

7.3.3.5 Steady Regulator: Max-Only Case

In this case, i ^∗ is the smallest index satisfying

$$\frac{1} {{G}_{j}(0)}{M}_{j}\left (\bar{{y}}_{j/{i}^{{_\ast}}}- {r}_{j}\right ) \geq 0$$

(7.54)

for all j ∈ H. When M _j  ∕ G _j (0) > 0, i ^∗ is the regulator for which outputs do not become smaller than their setpoints at steady-state. When M _j  ∕ G _j (0) < 0, i ^∗ is the regulator for which outputs do not exceed their setpoints at steady-state.

In summary, when M _j  ∕ G _j (0) > 0, the min selector protects upper-limited outputs at steady-state and the max selector protects lower-limited outputs at steady-state. When M _j  ∕ G _j (0) < 0, the reverse steady-state protections are afforded.

7.3.3.6 Steady Regulator: Min–Max Case

If i ^∗ ∈ L, the following inequalities must hold:

$$\begin{array}{rcl}{ M}_{j} \frac{{r}_{{i}^{{_\ast}}}} {{G}_{{i}^{{_\ast}}}(0)}& \leq {P}_{j}{r}_{j},&\end{array}$$

(7.55)

$$\begin{array}{rcl}{ M}_{k} \frac{{r}_{{i}^{{_\ast}}}} {{G}_{{i}^{{_\ast}}}(0)}& \geq {P}_{k}{r}_{k}&\end{array}$$

(7.56)

for all j ∈ L and all k ∈ H. If there exist values of j or k for which the above inequalities fail, then it must be that i ^∗ ∈ H and the following inequalities must hold:

$$\begin{array}{rcl}{ M}_{k} \frac{{r}_{{i}^{{_\ast}}}} {{G}_{{i}^{{_\ast}}}(0)}& \geq {P}_{k}{r}_{k},&\end{array}$$

(7.57)

$$\begin{array}{rcl} \mbox{ min }\left \{-{M}_{j} \frac{{r}_{{i}^{{_\ast}}}} {{G}_{{i}^{{_\ast}}}(0)} + {P}_{j}{r}_{j}\right \}& < 0&\end{array}$$

(7.58)

for all k ∈ H, with the minimum in inequality (7.58) is taken over j ∈ L. These conditions can be interpreted following a similar process as done for the isolated min and max cases. Suppose first that i ^∗ ∈ L. Inequalities (7.55) and (7.56) can be written as

$$\begin{array}{rcl} \frac{{M}_{j}} {{G}_{j}(0)}(\bar{{y}}_{j/{i}^{{_\ast}}}- {r}_{j})& \leq 0,&\end{array}$$

(7.59)

$$\begin{array}{rcl} \frac{{M}_{k}} {{G}_{k}(0)}(\bar{{y}}_{k/{i}^{{_\ast}}}- {r}_{k})& \geq 0&\end{array}$$

(7.60)

for all j ∈ L and all k ∈ H. Thus, min-linked outputs y _j such that M _j  ∕ G _j (0) > 0 and max-linked y _k outputs such that M _k  ∕ G _k (0) < 0 will remain below their setpoints at steady-state. Similarly, min-linked outputs y _j such that M _j  ∕ G _j (0) < 0 and max-linked y _k outputs such that M _k  ∕ G _k (0) > 0 will remain above their setpoints at steady-state. Now suppose design parameters are chosen so that i ^∗ ∈ H. Inequality (7.60) holds for all k ∈ H, but min-linked outputs must satisfy the following:

$$0 > \mbox{ min}\left \{ \frac{{M}_{j}} {{G}_{j}(0)}({r}_{j} -\bar{ {y}}_{j/{i}^{{_\ast}}})\right \}.$$

When M _j  ∕ G _j (0) < 0 we have

$$\mbox{ min }\left \{ \frac{{M}_{j}} {{G}_{j}(0)}({r}_{j} -\bar{ {y}}_{j/{i}^{{_\ast}}})\right \} = \frac{{M}_{j}} {{G}_{j}(0)}\mbox{ max }\left \{{r}_{j} -\bar{ {y}}_{j/{i}^{{_\ast}}}\right \}.$$

It follows that ${r}_{j} <\bar{ y}j/{i}^{{_\ast}}$ for all j ∈ L, indicating that min-linked variables are lower-limited by r _j , matching the min-only result. However, if M _j  ∕ G _j (0) > 0, we can only say:

$$0 > \mbox{ min}\left \{{r}_{j} -\bar{ {y}}_{j/{i}^{{_\ast}}}\right \}$$

and no bounds can be guaranteed for $\bar{{y}}_{j/{i}^{{_\ast}}}$. Thus, if the design requires i ^∗ ∈ H for some reason, only lower-limited outputs having M _j  ∕ G _j (0) < 0 should be linked to the min-selector.

The preceding analysis can be regarded as a design guideline for the association of outputs to selectors. This is done solely on the basis of steady-state characteristics. The guidelines have been summarized in Table 7.1.

Table 7.1 Guidelines for the association of outputs to selectors based on steady-state characteristics. When using the min–max arrangement, i ^∗ ∈ L is required

7.3.4 Example: CMAPSS-1 Linearized Model

Consider the same transfer functions of Example 7.1.3. Matrices A and B of the state-space realization are as follows:

$$A = \left [\begin{array}{r@{\quad }r} - 3.8420\quad & 1.4125\\ 0.5310\quad & - 4.6623 \end{array} \right ]\;,\;\;\;B = \left [\begin{array}{r} 230.9226\\ 653.7255 \end{array} \right ].$$

For y ₁ = ΔN _f, C ₁ = [1  0] and D ₁ = 0. The values of matrices C and D for ΔT ₄₈ are $C\,=\,[-0.1022\;\; - 0.2952]$ and D = 146. 24. Finally, a state-space realization of the transfer function to ΔSmHPC using the same A and B has C = [0. 0189  0. 0066] and $D = -4.4052$. Note that this flight condition is close, but not equal to FC01 listed in Appendix B

Consider first that y ₁ must be driven to a setpoint r ₁ = 100 rpm while preventing y ₂ = ΔT ₄₈ from exceeding r ₂ = 200^∘R. That is, y ₂ is an upper-limited output. The individual regulators are designed using an LQR approach for the augmented plant, with Q = C _a1 ^T C _a1 and Q = C _a2 ^T C _a2, where C _a1 = [C ₁ D ₁] and C _a2 = [C ₂ D ₂] are the augmented output matrices. Using R = 1 for both designs, the resulting state-feedback gains are

$$\begin{array}{rlrlrl} {K}_{1} & = \left [0.7098\;\;0.0840\;\;20.9204\right ], & & \\ {K}_{2} & = \left [-0.1006\;\; - 0.2870\;\;144.7929\right ]. & & \end{array}$$

The required values of P ₁ and P ₂ for perfect tracking when the regulators are used independently are P ₁ = 1 and P ₂ = 1, with $\bar{{x}}_{a1}^{\mathrm{T}} = [100\;131.7106\;0.8581]$, as determined from formulas (7.35) and (7.34). Suppose that a min selector is used and, for illustrative purposes, that the input integrator has initial condition u(0) = 1. Then $\tilde{{x}}_{a1}(0) = [-100\;\; - 131.7106\;\;0.1419]$. The initial regulator is then i ₀ = 2 by application of condition (7.46). Similarly, condition (7.53) predicts that i ^∗ = 1. This is confirmed by the simulation results of Fig. 7.5. If the min selector is not used and a feedback loop is established to control y ₁ to its setpoint, y ₁ has a settling time of about 0.4 s and zero steady-state error by design. Output y ₂, however, exceeds its intended limit by far during the transient regime and settles at $\bar{{y}}_{2/1} = 76.39 < {r}_{2}$. If the min selector is used, u _r2 is active from t = 0 until t = 0. 22, which causes y ₂ to be regulated toward its limit. A regulator switching occurs near t = 0. 22 and i = 1 becomes active for all subsequent times. Note that y ₁ overshoots its limit by about 2%. Thus, this example shows that the min selector preserved limits during the transient regime for y ₂ but not for y ₁, motivating further analysis of limit protection properties. This is done in Sect. 7.3.5. Now suppose ΔT ₄₈ is not a concern, but that the HPC stall margin must be kept above certain limit to minimize the risk of stall or surge. Define y ₂ = Δ SmHPC and suppose that the stall margin must not decrease more than 10% from its steady value at FC01, that is y ₂ must be lower-limited by ${r}_{2} = -10$. References to the standard min–max arrangement used in the GTE industry assume a priori that the regulators of lower-limited outputs are applied to the ports of a max selector. As this example demonstrates, the choice of selector must take additional considerations into account. Suppose K ₂ is designed using an LQR approach with Q = C _a2 ^T C _a2 and R = 1, and K ₁ and r ₁ are maintained. Now,

$${K}_{2} = \left [-0.0097\;\; - 0.0045\;\;3.0135\right ]$$

and ${P}_{2} = -1$. If, according to conventional wisdom, the max selector is used, it can be verified that i ₀ = 1 and i ^∗ = 2. As shown in Fig. 7.6, the results are disastrous. Since i ^∗ = 2, the lower limit imposed on y ₂ is preserved at steady-state, but $\bar{{y}}_{1/2} = 975.98$, far above r ₁. Also, y ₂ becomes smaller than its limit during the transient. If a min selector is used instead, the results are acceptable if fan speed is allowed to overshoot. In this case, i ₀ = 2 and i ^∗ = 1. Since $\bar{{y}}_{2/1} = -1.02$, the lower limit imposed in y ₂ is still preserved at steady-state, and moreover, transient limit protection is observed. To illustrate the initial and steady properties of the min–max arrangement, consider the following output definitions: y ₁ = ΔN _f, y ₂ = ΔT ₄₈, y ₃ = Δ Ps30 and ${y}_{4} = \Delta ({W}_{\mathrm{F}}/$Ps30). The last two outputs are the incremental static pressure at HPC outlet, and the incremental ratio between fuel flow and Ps30. To prevent lean blowout conditions in the combustor, y ₃ is lower-limited. A low limit is usually imposed on y ₄ to prevent the LPC from stalling.

Fig. 7.5

Response of state-feedback system with min selector: CMAPSS-1 linearized model with upper-limited output y ₂ = ΔT ₄₈

Fig. 7.6

Response of state-feedback system with max and min selectors: CMAPSS-1 linearized model with lower-limited output y ₂ = Δ SmHPC

The linearized C and D parameters for y ₃ and y ₄ can be found in Appendix B. The state feedback gains are designed using an LQR approach with Q = C _ai ^T C _ai and R = 1 as before. The resulting gains are

$$\begin{array}{rlrlrl} {K}_{3} & = \left [0.0639\;\;0.1534\;\;25.2464\right ], & & \\ {K}_{4} & = \left [-0.0039\;\; - 0.0085\;\;3.5781\right ] & & \end{array}$$

with corresponding values of P ₃ = 1 and P ₄ = 1. The two upper-limited outputs are linked to the min selector, while the lower-limited outputs are linked to the max selector. Consider setpoints as follows: r ₁ = 100, r ₂ = 200, ${r}_{3} = -50$, and ${r}_{4} = -20$. Formulas (7.49)–(7.52) can be used to determine that i ₀ = 1. Likewise, formulas (7.55)–(7.58) indicate that i ^∗ = 1. The response to this case is shown in Fig. 7.7. Since y ₃ and y ₄ have DC gains G ₃(0) = G ₄(0) with the same sign as G ₁(0), their DC value increases when r ₁ > 0. Therefore, they move away from their

Fig. 7.7

Response of state-feedback system with min–max selectors: CMAPSS-1 linearized model with upper-limited output y ₂?=??T ₄₈ and lower-limited outputs y ₃?=?? Ps30 and y ₄?=??W _F/Ps30 (fan speed increase demand)

negative limits and their regulators are never active. As in the previous example, the T ₄₈ limit regulator becomes active in the transient regime. Now suppose ${r}_{1} = -100$ rpm. In this case, y ₃ and y ₄ decrease from their initial values. It can be verified that i ₀ = 4 and i ^∗ = 1. As shown in Fig. 7.8, the 4th regulator remains active for some time after t = 0 and y ₃ and y ₄ are maintained above their lower limits at steady state, as well as during the transient.

Fig. 7.8

Response of state-feedback system with min–max selectors: CMAPSS-1 linearized model with upper-limited output y ₂ = ΔT ₄₈ and lower-limited outputs y ₃ = Δ Ps30 and y ₄ = ΔW _F/Ps30 (fan speed decrease demand)

7.3.5 Transient Limit Protection Analysis

The invariance condition for intervals presented in Sect. 7.2.1 is now applied to study the limit protection properties of the min–max approaches with static state feedback. Analysis is divided into three cases corresponding to min-only, max-only, and min–max selectors. For upper-limited outputs, the error $e = r - y$ must not become negative. That is, [0, ∞) must be positively invariant. For lower-limited outputs, e must not become positive, that is, ( − ∞, 0] must be positively invariant. The sign of \dot{e} must be examined along the boundary e = 0 to determine invariance. Given a fixed i and j≠i, five sets are relevant:

• 1. ${\mathcal{U}}_{ij}$: the set where u _ri  = u _rj

• 2. ℰ _i : the set where e _i  = 0

• 3. ℰ _j : the set where e _j  = 0

• 4. $\dot{{\mathcal{E}}}_{i}$: the set where $\dot{{e}}_{i/i} = 0$

• 5. $\dot{{\mathcal{E}}}_{j/i}$: the set where $\dot{{e}}_{j/i} = 0$

Set ${\mathcal{U}}_{ij}$ represents the boundary between the two half-spaces which are the regions of activity of each regulator, and the roles of the other sets are clear from their definitions. For the min-only or max-only cases, the sets are hyperplanes, as represented in Figs. 7.9 and 7.10. Sets $\dot{{\mathcal{E}}}_{i}$ and $\dot{{\mathcal{E}}}_{j/i}$ divide the space into two halves, one where the error increases and the other where the error decreases, and this has been indicated with plus or minus signs.

Fig. 7.9

Schematic of the regions corresponding to sets ${\mathcal{U}}_{ij}$, ℰ _i and $\dot{{\mathcal{E}}}_{i}$

Fig. 7.10

Schematic of the regions corresponding to sets ${\mathcal{U}}_{ij}$, ℰ _j and $\dot{{\mathcal{E}}}_{j/i}$

For upper-limited variables, the invariance condition requires that the derivative of the error be nonnegative whenever the error is zero. Now, as seen in Figs. 7.9 and 7.10, ℰ _i and ℰ _j ∕ i are divided into two regions, corresponding to positive and negative error derivatives. Assuming that i is the active regulator, limit protection is ensured by requiring that the regions ofℰ _i andℰ _j ∕ i where, the error increases lie outside the region whereiis active. That is, the intersection of $\dot{{\mathcal{E}}}_{i}$ and ℰ _i must be outside the half space where i is active, and the intersection of $\dot{{\mathcal{E}}}_{j/i}$ and ℰ _j must likewise be outside the half space where i is active.

Equations (7.39) and (7.43) are combined to determine the set $\dot{{\mathcal{E}}}_{i} \cap {\mathcal{E}}_{i}$, resulting in the following system of linear equations on $\tilde{{x}}_{ai}$:

$$\left [\begin{array}{r} - {C}_{ai} \\ - {C}_{ai}{A}_{ci} \end{array} \right ]\tilde{{x}}_{ai} = \left [\begin{array}{r} 0\\ 0 \end{array} \right ].$$

(7.61)

The matrix characterizing the above system of equations is full-rank, unless C _ai and C _ai A _ci are parallel vectors. For this to happen, there would have to exist a scalar λ such that C _ai A _ci  = λC _ai . Taking the transpose of this equation, we get

$${A}_{ci}^{\mathrm{T}}{C}_{ ai}^{\mathrm{T}} = \lambda {C}_{ ai}^{\mathrm{T}}$$

that is, C _ai ^T would have to be an eigenvector of A _ci ^T. Assuming that K _i has been verified not to produce this exception, the only solution to system (7.61) is $\tilde{{x}}_{ai} = 0$.

Likewise, (7.40) and (7.44) are combined to determine the set $\dot{{\mathcal{E}}}_{j/i} \cap {\mathcal{E}}_{j}$, resulting in the following system of linear equations on $\tilde{{x}}_{ai}$:

$$\left [\begin{array}{r} - {C}_{aj} \\ - {C}_{aj}{A}_{ci} \end{array} \right ]\tilde{{x}}_{ai} = \left [\begin{array}{r} \bar{{e}}_{j/i} \\ 0 \end{array} \right ].$$

(7.62)

System (7.62) does not have a unique solution in general. In this case, it becomes necessary to find particular ones that assist in arriving at invariance conditions. This is done separately for the min-only, max-only, and min–max cases.

7.3.5.1 Transient Limit Protection: Min-Only Case

Suppose all variables are upper-limited. To guarantee that y _i will not exceed its upper limit, we require that u _ri  − u _rj  > 0 along solutions to (7.61). If C _ai ^T is not an eigenvector of A _ci ^T, $\tilde{{x}}_{ai} = 0$ is the only solution. Using (7.45), we see that the required inequality reduces to ${K}_{j}\bar{{x}}_{ai} > {P}_{j}{r}_{j}$ for all j ∈ L. Using (7.34) and the definition of M _j from (7.47), this becomes:

$${M}_{j} \frac{{r}_{i}} {{G}_{i}(0)} > {P}_{j}{r}_{j}$$

(7.63)

for all j ∈ L. Noting that inequality (7.63) is the opposite of inequality (7.53), we see that transient limit protection cannot be guaranteed for i = i ^∗, even for intervals of time where i ^∗ is active! This can be confirmed in the min-only case of Example 7.3.4, where i ^∗ = 1 and y ₁ exhibits overshoot while its own regulator is active. In the max-only case where i ^∗ = 2, y ₂ violates its limit in the transient regime, even when its regulator is active. These shortcomings are removed by using sliding mode controllers instead of linear regulators (See Chap. 8).

Now consider the condition for y _j not to exceed its upper limit while i is active. For this, we require that u _ri  − u _rj  > 0 along solutions to (7.62). Since there is no explicit solution in this case, the solution that minimizes u _ri  − u _rj is sought, followed by a requirement that the minimum solution be positive. Mathematically, the invariance condition can be expressed as

$$({K}_{j} - {K}_{i})\tilde{{x}}_{ai}^{{_\ast}} > -{K}_{ j}\bar{{x}}_{ai} + {P}_{j}{r}_{j},$$

where $\tilde{{x}}_{ai}^{{_\ast}}$ is the solution to the following constrained minimization problem:

$$\begin{array}{rcl} \tilde{{x}}_{ai}^{{_\ast}} =& \mbox{ arg min }({K}_{ j} - {K}_{i})\tilde{{x}}_{ai} & \\ & \mbox{ subject to} & \\ \left [\begin{array}{r} - {C}_{aj} \\ - {C}_{aj}{A}_{ci} \end{array} \right ]\tilde{{x}}_{ai} =& \left [\begin{array}{r} \bar{{e}}_{j/i} \\ 0 \end{array} \right ].&\end{array}$$

(7.64)

A negative test for invariance can be conducted by finding any solution to system (7.62) and checking whether u _ri  ≤ u _rj is satisfied for all j ∈ L for that particular value of $\tilde{{x}}_{ai}$. If so, the set $\dot{{\mathcal{E}}}_{j/i} \cap {\mathcal{E}}_{j}$ contains at least one point for which the active regulator is i and invariance for y _j cannot be concluded.

7.3.5.2 Transient Limit Protection: Max-Only Case

The corresponding invariance conditions for y _i and y _j are obtained by reversing the inequalities in (7.63) and (7.64), and are the same whether upper- or lower-limited variables are involved. That is, invariance for y _i is guaranteed if:

$${K}_{j}\bar{{x}}_{ai} < {P}_{j}{r}_{j}$$

(7.65)

for all j ∈ H. Similarly, invariance for y _j is guaranteed if:

$$({K}_{j} - {K}_{i})\tilde{{x}}_{ai}^{{_\ast}} < -{K}_{ j}\bar{{x}}_{ai} + {P}_{j}{r}_{j},$$

where $\tilde{{x}}_{ai}^{{_\ast}}$ is the solution to the following constrained minimization problem:

$$\begin{array}{rcl} \tilde{{x}}_{ai}^{{_\ast}}& =& \mbox{ arg min }({K}_{ j} - {K}_{i})\tilde{{x}}_{ai} \\ & & \mbox{ subject to} \\ \left [\begin{array}{r} - {C}_{aj} \\ - {C}_{aj}{A}_{ci} \end{array} \right ]\tilde{{x}}_{ai}& =& \left [\begin{array}{r} \bar{{e}}_{j/i} \\ 0 \end{array} \right ].\end{array}$$

(7.66)

Although the invariance conditions for the min-only and max-only cases were developed used two regulators, their validity extends to more than two regulators by considering all possible pairs of regulators.

7.3.5.3 Transient Limit Protection: Min–Max Case

In this case, analysis must be made in groups of three regulators, an arbitrary fixed regulator i assumed active, a regulator j in the min group, and a regulator k in the max group. Two cases must be considered: i ∈ L and i ∈ H. Assuming i ∈ L, the region where i is active is given by ${\mathcal{U}}_{L} \cap {\mathcal{U}}_{H}$, where:

$${\mathcal{U}}_{L} = \left \{\tilde{{x}}_{ai}\; :\; {u}_{ri} - {u}_{rj} \leq 0\mbox{ for all }j \in L\right \},$$

$${\mathcal{U}}_{H} = \left \{\tilde{{x}}_{ai}\; :\; {u}_{ri} - {u}_{rk} \geq 0\mbox{ for all }k \in H\right \}.$$

Invariance will be guaranteed if $\dot{{\mathcal{E}}}_{i} \cap {\mathcal{E}}_{i}$ lies outside ${\mathcal{U}}_{L} \cap {\mathcal{U}}_{H}$. As before, if $\tilde{{x}}_{ai} = 0$ is the only element of $\dot{{\mathcal{E}}}_{i} \cap {\mathcal{E}}_{i}$, the invariance conditions reduce to

$$\begin{array}{rcl}{ K}_{j}\bar{{x}}_{ai}& > {P}_{j}{r}_{j},&\end{array}$$

(7.67)

$$\begin{array}{rcl}{ K}_{k}\bar{{x}}_{ai}& < {P}_{k}{r}_{k}&\end{array}$$

(7.68)

for all j ∈ L and for all k in H. For a regulator i ∈ H, the region where it is active is given by ${\mathcal{U}}_{H} \cap {\mathcal{U}}_{l}$, where ${\mathcal{U}}_{l}$ is defined as

$${\mathcal{U}}_{l} = \left \{\tilde{{x}}_{ai}\; :\; {u}_{ri} > \mbox{ min }\left \{{u}_{rj}\right \},j \in L\right \}.$$

The reader will observe that this set has a complex description. Establishing the invariance of y _j while i is active is also rather difficult. First, j must be assumed to begin to either L or H. For each case, subcases corresponding to i ∈ L and i ∈ H must be contemplated. This process is best handled numerically, with the aid of a computer program.

7.4 Example: CMAPSS-1 Linearized Model

Continuing with Example 7.3.4, consider first the problem of establishing transient limit protection when only the min selector is used. It can be verified that C _a1 is not parallel to C _a1 A _ci and therefore $\tilde{{x}}_{a1} = 0$ is the only solution to system (7.61). Thus, no transient limit protection can be guaranteed for i ^∗. Although (7.63) is only a sufficient condition, simulation showed that y ₁ overshoots its limit in the transient regime, even for intervals when its own regulator is active. To see whether invariance can be established for y ₂ while i ^∗ = 1 is active, note that $\bar{{y}}_{2/1} = 76.39$, so $\bar{{e}}_{2/1} = 123.61$. An exact solution to system  (7.62) can be readily found using the pseudoinverse as $\tilde{{x}}_{a1}^{\mathrm{T}} = [25.4488\;\;4.2484\;\; - 0.8189]$. Evaluating u _r1 − u _r2 at this value using (7.45) gives − 247.25, indicating, again, that there is a point where ${e}_{2} =\dot{ {e}}_{2/1} = 0$ with i = 1 being active. Therefore, invariance for y ₂ cannot be concluded during the transient regime for periods where i = 1 is active. Simulation shows, however, that y ₂ remains below its limit. The conservativeness of the sufficient condition can be traced to the fact that the solution found for $\tilde{{x}}_{a1}$ does not belong to the actual state trajectory followed by the system. A refined version of the invariance condition would require taking into account such trajectories, which depend on the previous switching history.

Now consider the max-only case of Example 7.3.4. Again, no transient limit protection can be guaranteed for i ^∗ = 2 for periods where this regulator is active. Condition (7.63), is satisfied for i = 1, however. Therefore, y ₁ will not exceed its limit when its own regulator is active. In this example i ₀ = 1, and i = 1 remains active for a short time, where y ₁ < r ₁, verifying transient invariance for this case. A regulator switching occurs and the active regulator becomes i ^∗ = 2 for all future times. Invariance cannot be concluded for y ₂ while i = 2 is active, and simulation confirms that y ₂ becomes smaller than its limit.

The preceding steady and transient limit protection analysis, as well as the examples, suggest that the standard min–max arrangement may be ill-conceived, at least when used in conjunction with linear regulators. Although steady limit preservation is achieved in a simple fashion, there is no clear way to guarantee that outputs will remain within their prescribed bounds in the transient regime. When sliding mode controllers are used, stability and limit protection are guaranteed under most cases, as described in Chap. 8.

7.5 Alternative Minimum-Interaction Design: ℋ _∞ Approach

A reasonable alternative to using limit regulators is to design the state feedback gain so that one of the outputs (say, fan speed) has a suitably fast transient response while the other outputs are suppressed during transients, minimizing the possibility of limit violations. This problem can be formulated using the multiobjective ℋ _∞ state feedback synthesis discussed in Chap. 4. Adequate fan speed response is obtained by the built-in robust stability property, together with a regional eigenvalue placement constraint. Limited outputs are regarded as performance outputs z ₂ or ${z}_{\infty }$. The latter choice is convenient when output disturbances are included, since the ℋ _∞ norm remains finite even for nonstrictly proper transfer functions.

7.5.1 Example: CMAPSS-1

The constant feedback gain ℋ _∞ design is now applied to the fan speed control problem near FC01. System parameters are given in Appendix B. Recalling the developments of Chap. 4, the robust state feedback synthesis approach handles the system description given below:

$$\begin{array}{rcl} \dot{{x}}_{\mathrm{a}}& = {A}_{\mathrm{a}}{x}_{\mathrm{a}} + {B}_{\mathrm{a}}{u}_{r} + {\Gamma }_{a}w,&\end{array}$$

(7.69)

$$\begin{array}{rcl}{ z}_{\infty }& = C{x}_{\mathrm{a}},&\end{array}$$

(7.70)

where w is a vector of disturbances. Matrix Γ has not been listed in Appendix B, but may be obtained through linearization using the CMAPSS-1 interface. The objective is to find a feedback gain K stabilizing the augmented closed-loop matrix ${A}_{\mathrm{c}} = {A}_{\mathrm{a}} - {B}_{\mathrm{a}}K$ while minimizing the infinity norm of the transfer function from w to ${z}_{\infty }$. The designer controls fan speed response by specifying a target region for the closed-loop eigenvalues of A _c. The ℋ _∞ minimization objective translates into transient suppression of the limited outputs. The presence of a disturbance

Fig. 7.11

Response of limited outputs for minimum interaction design: positive ?N _f demand

component introduces a design tradeoff. The larger the influence of disturbances (as measured through matrix Γ), the smaller the closed-loop allowable bandwidth. This can be observed when carrying the design using, for instance, the msfsyn command: Γ may need to be scaled to maintain feasibility.

In this example, Γ is scaled to 10% of their linearization value. A disk centered at − 10 with radius 7 is used as the target region, resulting in an ℋ _∞ norm of 169 and a feedback gain as follows:

$$K = \left [0.0001\;\; - 0.0008\;\;3.8233\right ],$$

which places the closed-loop eigenvalues of A _c at − 5. 444, − 3. 8419, and − 3. 0417. Using a fan speed demand of ΔN _f = 100 rpm, it can be verified that the corresponding steady values of the limited outputs are ΔT ₄₈ = 75. 835^∘ R, Δ Ps30=48.26 psi and ΔΦ = 1. 56 pps/psi. As Fig. 7.11 shows, the transient peak of ΔT ₄₈ remains under the limit of r ₂ = 200 that was used in the previous examples. A step health parameter disturbance was applied at t = 12 s. Figure 7.12 corresponds to $\Delta {N}_{\mathrm{f}} = -100$ rpm. In this case, all limited outputs remain within the bounds used in the previous examples.

Fig. 7.12

Response of limited outputs for minimum interaction design: negative ΔN _f demand

7.5.2 Example: Ellipsoidal Invariant Set: CMAPSS-1 Linearized Model

An ellipsoidal invariant set is now derived for the closed-loop system of the previous example, first using the feedback gain obtained through ℋ _∞ synthesis. Define a shifted state vector as done in Sect. 7.3.2:

$$\tilde{{x}}_{\mathrm{a}} = {x}_{\mathrm{a}} -\bar{ {x}}_{\mathrm{a}},$$

where $\bar{{x}}_{\mathrm{a}}$ represents a target state so that y ₁ = r ₁ when the system reaches steady-state. The state-feedback law for fan speed setpoint tracking is ${u}_{r} = -K{x}_{\mathrm{a}} + P{r}_{1}$, which can be written in terms of $\tilde{{x}}_{\mathrm{a}}$ as follows:

$${u}_{r} = -K{x}_{\mathrm{a}} + P{r}_{1} = -K(\tilde{{x}}_{\mathrm{a}} -\bar{ {x}}_{\mathrm{a}}) + P{r}_{1} = -K\tilde{{x}}_{\mathrm{a}},$$

where (7.33) has been used. Recalling that ${A}_{\mathrm{a}}\bar{{x}}_{\mathrm{a}} = 0$, substitution of the above control law into system (7.69) yields the following closed-loop dynamics

$$\dot{\tilde{{x}}}_{\mathrm{a}} = ({A}_{\mathrm{a}} - {B}_{\mathrm{a}}K)\tilde{{x}}_{\mathrm{a}}.$$

Now, the constraints are formulated in terms of x _a from the limited output specifications:

$$\begin{array}{rlrlrl} {y}_{2} = {C}_{a2}\tilde{{x}}_{\mathrm{a}} & \leq {r}_{2}, & & \\ {y}_{3} = {C}_{a3}\tilde{{x}}_{\mathrm{a}} & \geq {r}_{3}, & & \\ {y}_{4} = {C}_{a3}\tilde{{x}}_{\mathrm{a}} & \geq {r}_{4}. & & \end{array}$$

Note that the first output (incremental fan speed) has not been regarded as a limited output, but if necessary, it can be directly added to the list of constraints. The constraint vectors are now normalized so that they correspond to the assumed description G _i x ≤ 1. For this, inequalities reflecting positive upper bounds are simply divided by the value of the bound. For negative lower bounds, the left-hand and right-hand sides of the inequality are switched and the inequality divided by the negative of the bound. According to this, the three constraints adopt the desired form, with ${G}_{2} = {C}_{a2}/{r}_{2}$, ${G}_{3} = {C}_{a3}/{r}_{3}$, and ${G}_{4} = {C}_{a4}/{r}_{4}$.

An arbitrary symmetric, positive-definite matrix Q may be used to obtain a family of invariant ellipsoids from the solution of Lyapunov equation (7.24). For each constraint, the largest admissible ellipsoid from this family is found by calculating its V _i value. The smallest V _i , say V, is then selected, and the operating set for $\tilde{{x}}_{\mathrm{a}}$ is defined as $\tilde{{x}}_{\mathrm{a}}^{\mathrm{T}}P\tilde{{x}}_{\mathrm{a}} \leq V$. Although this guarantees invariance, the orientation of the ellipsoid may not be the best for the given constraints. An improvement is to find P to maximize the volume of the ellipsoid, subject to the invariance condition (7.23) and ${G}_{i}\tilde{{x}}_{\mathrm{a}} \leq 1$, see O’Dell [62].

As an example, take Q = I. The Lyapunov equation is solved using lyap((Aa-Ba*K)’,Q) to yield

$$ P = \left [\begin{array}{r@{\quad }r@{\quad }r} 0.1342\quad & 0.0303\quad & 7.3913\\ 0.0303\quad & 0.1184\quad &11.4089 \\ 7.3913\quad &11.4089\quad & 2397.3 \end{array} \right ]. $$

The values of V _i are calculated as V ₂ = 1438. 7, V ₃ = 8780. 9, and V ₄ = 9526. 1. Therefore, the operating set for $\tilde{{x}}_{\mathrm{a}}$ is described by the inequality $\tilde{{x}}_{\mathrm{a}}^{\mathrm{T}}P\tilde{{x}}_{\mathrm{a}} \leq 1438.7$.

This result can be given a practical interpretation by considering the meaning of $\tilde{{x}}_{\mathrm{a}}$. The operating set restricts the distance between x _a(0) and $\bar{{x}}_{\mathrm{a}}$, the target state. Recalling that x _a ^T = [ΔN _f  ΔN _c  ΔW _F] is the incremental state relative to a steady linearization point, we see that x _a(0) = 0 if the setpoint change maneuver starts at the linearization point. For such cases, we can restrict the target state to guarantee limit protection by enforcing $\bar{{x}}_{\mathrm{a}}^{\mathrm{T}}P\bar{{x}}_{\mathrm{a}} \leq 11.3$. Using (7.34), this inequality becomes

$${ \left ( \frac{{r}_{1}} {{G}_{1}(0)}\right )}^{2}{\left [\begin{array}{r} - {A}^{-1}B \\ 1 \end{array} \right ]}^{\mathrm{T}}P\left [\begin{array}{r} - {A}^{-1}B \\ 1 \end{array} \right ] \leq 1,$$

which in this example reduces to | r ₁ | ≤ 38. 3, a conservative figure. Now consider the problem of finding a stabilizing K that maximizes the volume of an invariant ellipsoid contained in the constrained set. This problem can be solved using O’Dell’s techniques [62], yielding the following combination of P and K:

$$P = \left [\begin{array}{l@{\quad }l@{\quad }l} 0.00000219\quad &0.00000537\quad &0.00064\\ 0.00000537\quad &0.00001354\quad &0.00121 \\ 0.00064366\quad &0.00121329\quad &1.12223 \end{array} \right ],$$

$$K = [0.000578\;\;0.001120\;\;0.877210].$$

This feedback gain places the closed-loop eigenvalues at − 1. 251, − 2. 965, and − 5. 165, producing a somewhat slower response in comparison with the value obtained through ℋ _∞ synthesis. The minimum value of V is now 1 and the bound for r ₁ becomes 78.8 rpm, a less conservative value. This value is still far from the maximum value of r ₁ for which a limited output reaches its limit. Indeed, since all closed-loop poles are real, no output will exhibit overshoot. Thus, an upper-limited variable y _j can only reach r _j in the steady state. For this, ${r}_{{i}^{{_\ast}}}$ must be chosen so that $\bar{{y}}_{j/{i}^{{_\ast}}} = {r}_{j}$. In this example, for y ₂ to reach r ₂ = 200, r ₁ must be chosen as ${r}_{2}{G}_{1}(0)/{G}_{2}(0) = 261.8$ rpm.

7.6 Acceleration and Deceleration Limiting

In addition to limits placed on the magnitudes of critical variables such as turbine temperature, shaft speeds, combustor pressure and engine pressure ratio, the core shaft acceleration must also be maintained between prescribed bounds. An upper bound is introduced to protect the engine against surge and stall, while the lower limit is introduced to provide safety against engine flame-out. Recalling (2.2), we see that core acceleration $\dot{{N}}_{\mathrm{c}}$ depends on fuel flow, W _F. A traditional way to maintain core acceleration below its prescribed upper bound is simply to override the value of u _r calculated by the min-selected regulators, replacing it with a constant rate of zero pps/sec whenever the acceleration reaches its upper limit. Minimum acceleration is likewise maintained by replacing the rate produced by the max stage with u _r  = 0 whenever the acceleration reaches its lower limit.

In CMAPSS-1, these operations are implemented by override switches triggered by $\dot{{N}}_{\mathrm{c}}$, as shown in Figs. 7.13 and 7.14. The threshold value for $\vert \dot{{N}}_{\mathrm{c}}\vert$ is 500 rpm/s. Note that a similar acceleration limiting scheme could, in principle, be applied to $\dot{{N}}_{\mathrm{f}}$. When thrust is being controlled indirectly by a feedback loop on N _f, however, such scheme is not necessary, since reference ramps are usually commanded for this variable. Hence, fan acceleration should follow response specifications. When an EPR loop is established as a means to achieve thrust control, N _f and N _c may be regarded as upper-limited variables, and corresponding acceleration limiters included.

Fig. 7.13

Schematic of CMAPSS-1 min–max arrangement with acceleration and deceleration limiters

Fig. 7.14

Schematic of CMAPSS-1 acceleration and deceleration override switches

7.6.1 “N-Dot” Control and Acceleration Scheduling

The so-called “N-dot” control concept exploits the algebraic relationship between shaft accelerations and fuel flow, as observed in (2.1) and (2.2). In principle, if the inverse of functions f ₁ and f ₂ were available, one could compute values of fuel flow resulting in a desired instantaneous value for $\dot{{N}}_{\mathrm{f}}$ or $\dot{{N}}_{\mathrm{c}}$. In practice, no such inverses are available for real-time computations, and uncertain and unmeasurable time-varying parameters participate in the definitions of f ₁ and f ₂.

Since fuel flow is related to acceleration by an uncertain algebraic relationship, the latter may be controlled by establishing a PI loop using the former as control input. Assuming that good tracking properties are attainable, thrust control is achieved by providing adequate acceleration references, or acceleration schedules. These references are shaped so that their integral corresponds to a desirable shaft speed response. In addition, acceleration schedules may be used to introduce limit protections in critical engine variables. For more details on Ndot control, readers are referred to Link and Jaw [14] and Spang and Brown’s survey [64].