* 

MARCH 1988 c- LIDS-P-1756 

b 

//fli .e - Design of Feedback Control Systems for Stable Plants ep ,f,p~,v 

with Saturating Actuators’ 

by 

Petros Kapasouris 

Michael Athans 

Gunter Stein *+ 

//v -6 3 -c< 

/30 

Room 35-406 

Laboratory for Information and Decision Systems 

Massachusetts Institute of Technology 

Cambridge, MA 02 139 

A systematic control design methodology is introduced for multi-inpudmulti-output stable 

open loop plants with multiple saturations. This new methodology is a substantial improvement 

over previous heuristic single-inpudsingle-output approaches. 

The idea is to introduce a supervisor loop so that when the references and/or disturbances are 

sufficiently small, the control system operates linearly as designed. For signals large enough to 

cause saturations, the control law is modified in such a way to ensure stability and to preserve, to 

the extent possible, the behavior of the linear control design. 

Key benefits of this methodology are: the modified compensator never produces saturating 

coneol signals, integrators and/or slow dynamics in the compensator never windup, the directional 

properties of the controls are maintained, and the closed loop system has certain guaranteed 

stability properties. 

The advantages of the new design methodology are illustrated in the simulation of an 

academic example and the simulation of the multivariable longitudinal conml of a modified model 

of the F-8 aircraft. 

1 

This research was conducted at the M.I.T. Laboratory for Informuion and Decision Systems with support provided by 

the General Electric Corporate Research and Development Center, and by the NASA Ames and Langley Research Centers 

under grant NASAjNAG 2-297. 

* Now with ALPHATECH Inc. ++ Also with HONEYWELL Inc. 

This paper has been submitted to the 27* IEEE Conference on Decision and Control. 

(NASA-CR-182613) DESIGN OF FEEDBACK COITBOL N 88-200 50 

SYSTECiS FOR STABLE PLANTS WITH SATfJRATlNti 

ACTUATORS [flassachusetts fnst. of Tech. ) 

39 P CSCL U9B Uaclas 

63/63 0130184 
Page 1 

1. Introduction 

Almost every physical system has maximum and minimum limits or saturations on its control 

signals. For multivariable systems, a major problem that arises (because of saturations) is the fact 

that control saturations alter the direction of the control vector. For example, let us assume that 

there are m control signals with m saturation elements. Each saturation element operates on its 

input signal independently of the other saturation elements; as we shall show in the performance 

analysis section, this can disturb the direction of the applied control vector. Consequently, 

erroneous controls can occur, causing degradation with the perfoxmance of the closed loop system 

over and above the expected fact that output transients will be "slower". 

Another performance degradation occurs when a linear compensator with integrators is used 

in a closed loop system and the phenomenon of reset-windup appears. During the time of 

saturation of the actuators, the error is continuously integrated even though the controls are not 

what they should be. The integrator, and other slow compensator states, attain values that lead to 

larger controls than the saturation limits. This leads to the phenomenon known as reset-windup, 

resulting in serious deterioration of the performance (large overshoots and large settling times.) 

Many attempts have been made to address this problem for SISO systems, but a general design 

process has not been formalized. No research has been found in the literature that addresses and 

solves the reset-windup problem for MIMO systems. 

In practice, the saturations are ignored in the first stage of the control design process, and 

then the final controller is designed using ad-hoc modifications and extensive simulations. A 

common classical remedy was to reduce the bandwidth of the control system so that control 

saturation seldom occurred Thus, even for small commands and disturbances, one intentionally 

degraded the possible performance of the system (longer settling times etc.). Although reduction in 

closed-loop bandwidth by reduction in the loop gain is an "easy" design tool, it clearly is not 

necessarily the best that could be done. Hence, a new design methodology is desirable which will 

generate transients consistent with the actuation levels available, but which maintains the rapid 
Page 2 

speed of response for small exogenous signals (reference commands and disturbances). 

One way to design controllers for systems with bounded controls, would be to solve an 

optimal control problem; for example, the time optimal control problem or the minimum energy 

problem etc. The solution to such problems usually leads to a bang-bang feedback controller [l]. 

Even though the problem has been solved completely in principle, the solution to even the simplest 

systems requires good modelling, is diflicult to calculate open loop solutions, or the resulting 

switching surfaces are complicated to work with. For these reasons, in most applications the 

optimal control solution is not used. 

Because of the problems with optimal control results, other design techniques have been 

attempted. Most of them are based on solving the Lyapunov equation and getting a feedback which 

will guarantee global stability when possible or local stability otherwise [2]-131. The problem with 

these techniques is that the solutions tend to be unnecessarily consentative and consequently the 

performance of the closed loop system may suffer. For example, when global stability is 

guaranteed, it is often required that the final open loop system is strictly positive-real with all the 

limitations that such systems possess. 

Attempts to solve the reset windup problems when integrators are present in the forward 

loop, have been made for SISO systems [4]-[lo]. Most of these attempts lead to controllers with 

substantially improved performance but not well understood stability properties. As part of this 

research, an initial investigation was made on the effects on performance of the reset windups for 

MIMO systems [ 1 1 J showing potential for improving the performance of the system. A simple 

case study was also recently conducted on the effects of saturations to MlMO systems where 

potential for improvement in the perfomance was demonstrated [12]. 

This research brings new advances in the theory concerning tbe design of control systems 

with multiple saturations. A systematic methodology is introduced to design control systems with 

multiple saturations for stable open loop plants. The idea is to design a linear control system 

ignoring the saturations and when necessary to modify that linear control law. When the 

exogenous signals are small, and they do not cause saturations, the system operates linearly as 
I 

Page 3 

designed. When the signals are large enough to cause saturations, the control law is then modified 

in such a way to preserve ("mimic") to the extent possible the responses of the linear design. Our 

modification to the linear compensator is introduced at the error via an Error Governor (EG). The 

main benefits of the methodology are that it leads to controllers with the following properties: 

(a) The signals that the modified compensator produces never caw saturation. The nonlinear 

response mimics the shape of the linear one with the difference that its speed of response may be, 

as expected, slower. Thus the output of the compensator (the controls) are not altered by the 

saturations. 

(b) Possible integrators or slow dynamics in the compensator never windup. That is true 

because the signals produced by the modified compensator never exceed the limits of the 

saturations. 

(c) For closed loop systems with stable plants finite gain stability is guaranteed for any 

reference, disturbance and any modelling error as long as the "true" plant is open loop stable. 

(d) The on-line computation requid to implement the control system is minimal and 

realizable in most of today's microprocessors. 

. 2. Performance Analysis 

Without loss of generality one can assume that each element $(t) of the control vector u(t) = [ 

T u,(t) . . . %(t)] has saturation limits fl and the saturation operator can be defmed as follows: 

1 Ui(t) 2 1 

-1 2 Ui(t) 5 1 

-1 Ui(t) 5-1 

Figure 2.1 shows the closed loop system with the saturation element at the controls. The 

compensator K(s) is designed using linear control system techniques and it is assumed that the 
4 

Page 4 

closed loop system without the saturations (the linear system) is stable with "good" properties. 

d. (t) 

1 

~~ 

Figure 2.1 : The closed loop system 

There are well developed methods for defining pexfomce criteria and for designing linear 

closed loop systems which meet the performance requirements. It would then be desirable, 

whenever the closed loop system operates in the linear region, to meet the a priori performance 

constraints (because it easy to defrne them and easy to design control systems satisfying these 

constraints). When the system operates in the nonlinear region new perfomance criteria have to be 

defined and new ways of achieving the desired perfonnance must be developed. 

There are two major problems that multiple satllfations can introduce to the perfoxmance of 

the system: (a) the reset windup problem, and (b) the fact that multiple saturations change the 

direction of the controls. 

When the linear compensator contains integrators and/or slow dynamics reset windups can 

occur. Whenever the controls are saturated the error is continuously integrated and this can lead to 

large overshoots in the response of the system. It is obvious that if the states of the compensator 

were such that the controls would never saturate, then reset windups would never appear. See 

references [8] and [9] for additional discussion of the reset windup problem. 

Almost every current design methodology for linear systems inverts the plant and replaces the 

open loop system with a desired design loop. The inversion is done through the controls with 

I 
Page 5 

signals at specific frequencies and directions. The saturations alter the direction and frequency of 

the control signal and thus interfere with the inversion process. The main problem is that although 

both the compensator and the plant are multivariable highly coupled systems, the saturations 

operate as SISO systems. Each saturation operates on its input signal independently from the other 

saturation elements. 

To see exactly what happens assume as an example that in a two input system the control 

signal at some time to is uI1 = [ 3 

direction of the ut1 signal at time 

be transformed through the saturation to us = [ 1 

illustration of four different control directions u'~, ut2, u" 1, utt2 which are mapped at only two 

directions u' and u" . . 

1.1 IT the saturated signal will be u' = [ 1 

is altered. In fact, any input control signal u = [ u1 u2 IT will 

1 IT. Notice that the 

1]T if u1 2 1 and u2 2 1. Figure 2.2 shows an 

u2 

t u'2 4 

AU;' 

-1 

Figure 2.2: Examples of control directions at the input of the saturation 

u'~, u'z, u"~, u'I2 and at the output of the saturation u', u". 

Since the saturations can alter the direction of the control signals, and in effect disturb the 

compensator/plant inversion process, the logical question to ask is, under what conditions the 

linearly designed compensator that inverts (or partially inverts) the linear plant also inverts the plant 
Page 6 

when the saturations are present. 

To solve the performance problem let us assume that a nonzem operator is added to the 

system. The operator O1 is applied to the error signals and for convenience purposes it will be 

called Emr Governor (EG). 

u = KOle 

The nonzero operator will be chosen, when possible, so that the control u(t) never saturates, 

i.e. Ilu(t)ll, S 1, for any reference and/or disturbances. Figure 2.3 shows the closed loop system 

with the added operator. 

compensator saturation Plant 

Figure 2.3: General stxucture for the control system 

Effectively, with the introduction of the EG operator, the saturation is transferred from the 

controls to the errors and it makes the control analysis and design process easier. 

The selection of the EG operator will be such that the controls will never saturate; and if, for 

example, the compensator was designed to invert or partially invert the plant, then the inversion 

process will not be distorted by the saturation and GsatK will remain linear and equal to GK. In I 

I the closed loop system with the operator EG the compensator will never cause windups. The 

integrators and slow dynamics of the compensator will never cause the controls to exceed the limits 

of the saturation and thus windups never occur. 
3.Mathematical preliminaries 

Page 7 

This section is an introduction to the new design methodology. Some necessary mathematical 

preliminaries will be given and a basic problem will be introduced. The basic problem will be 

solved and it's solution will lead to the design of the EG operator that was introduced in section 2. 

For the proofs of the theorems given in this section see reference [13]. 

Consider the following linear time invariant system 

x(t) = Ax(t) A E Etm", x(t) E Wn (3.1) 

x(0) = xo (3.2) 

Y(t) = Cx(0 c E Id=", y(t) E Wrn (3.3) 

y(xo,t) = CeAtxo (3.4) 

where eAt is the state transition matrix (matrix exponential) for A.. 

Definition 3.1 : The scalar-valued function g(x) is defined as follows: 

&o): It" + w, g(x0) = IlY(xoJ)ll, (3.5) 

Theorem 3.1 : Let h,(A) be an observable mode of (A,C) and let the multiplicity of hi(A)) be ni. 

The function g(x) is fhte VXE IRn if and only if 

a) Re(hi(A)) IO, Vi, and 

b) The modes hi(A) with Re(ki(A)) = 0 and ni > 1 have independent 

eigenvectors ( Le. the order of the Jordan blocks associated with the 

eigenvalues of A with Re(Xi(A)) = 0 and ni > 1 is 1.). 

The systems that satisfy conditions (a) and (b) of theorem 3.1 are called neutrally stabIe. 

Definition 3.2: The set Pg is defrned as: 

P, = { [x,v] : XER", V€R, v 2 g(x) } 
Page 8 

From this definition we see that P, is the interior of the graph of the function g(x) in Rn+', as 

shown in figure 3.1. 

Def~tion 3.3: BA,C is the set of all XE lRn with 0 5 g(x) I; 1, Le. 

BA,C=(X: OSg(X)I;l} (3.7) 

Suppose that the system (3.1)-(3.4) has an initial condition %E Bkc. From this definition 

we see that for such an initial condition the output of the system, y(t), will satisfy Ily(t)lloo I 1. 

For neutrally stable systems the function g(x), the set P, and the set BA,C have the following 

properties. 

(a) The function g(x) is continuous and even. 

(b) The function g(x) is not necessarily differentiable at all points in Rn. 

(c) The set P, is a convex cone. 

(d) The BA,C set is symmetric with respect to the origin and convex. 

The proofs for these properties are given in reference [13]. 

One might expect that P, would be a convex cone from the linearity (g(0ur) = ag(x)) of the 

system (3.1)-(3.4). Figure 3.1 gives a visualization of the function g(q) and the sets BA,c and P, 

in Rn and IRn*l respectively. 

Definition 3.4 f141: The upper right Dini derivative is defined as 

(3.8) 
Page 9 

4 x2 

Figure 3.1: Visualization of the function g(x) and the sets P, and BA,c. 

Definitions of the lower right, upper left and lower left Dini derivatives are given in reference 

[ 141. In the sequel only the upper right Dini derivative will be used as in definition 3.4. The Dff(b) 

is finite at b if the function f satisfies the Lipschitz condition locally around to [ 141. Note that the 

function g(x) given in definition 3.1 satisfies the Lipschitz condition locally if the conditions of 

theorem 3.1 are met. This is obvious because g(x) is the boundary of the cone P,. 

Theorem 3.2 r141: Suppose that f(t) is continuous on (a,b), then f(t) is nonincreasing on (a,b) iff 

D'f(t) 5 0 for every tE (a,b). 

. 

3.1 Desim of a Time-Varyin9 Gain such that the Out~uts o fa Linear Svstem are Bounded 

Assume that a hear system is defined by the following equations 

;(t) = Ax(t)+Bu(t) AE Wnxn, BE WnXm 

YO) = Cx(t> CE Itrnxn 

(3.9) 

(3.10) 
Page 10 

and also assume that the linear system is neutrally stable. Then, if one were to construct the 

function g(x) (definition 3.1) for the system (3.9)-(3.10) with B = 0, the following is true; g(x) < 

00, Vx€Rn. This follows from theorem 3.1. 

The goal here, is to keep the outputs of the linear system (3.9)-(3.10) bounded (i.e. lyi(t)l 2 

1, V t, i) for any input u(t). To achieve our goal, consider the following system with a timevarying scalar gain h(t) 

x'(t) = Ax(t) + Bh(t)u(t) (3.11) 

Y(t) = Wt) (3.12) 

r - - -(Logicj4. - 7 

I 

I I 

I 

I 

Figure 3.2: The basic system for calculating h(t). 

Figure 3.2 shows the basic system and the location of the time-varying gain h(t). In this 

framework a basic problem can be defined. 

The Basic Problem: 

At time to, find the maximum gain h(to), 0 Ih(t,) S 1, such that Vu(t), t > to 3 h(t), t > 

tosuch that the output will satisfy [yi(t)l I1 V i, t > tw 

A solution to this problem can be obtained by using a function g(x) given in definition 3.1 

and by using a set BA,, given in definition 3.3. To be more specific, for the system (3.11)-(3.12), 

, with u(t) = 0, one can define g(x) and BA,c as in eqs. (3.13)-(3.15). The function g(x) is finite 

I because the system (3.9)-(3.10) is assumed to be neutrally stable (theorem 3.1). 
Page 11 

g(X0): Rn+R g(x0) = Ilr(xo,t>llpo (3.13) 

x(0) = xo (3.14) 

BA,C= {x: g(x) 1) (3.15) 

By defining g(x) and BA,C as in eqs. (3.13)-(3.15) one can construct h(t) as follows: 

Construction of xt): 

For every time t choose h(t) as follows 

a) if x(t)E IntBA,c then h(t) = 1 

b) if x(t)E BdBA,c then choose the largest h(t) such that 

(3.16) 

(3.17) 

0 s X(t) s 1 (3.18) 

or for the points where g(x) is differentiable choose the largest h(t) such that 

0 I; X(t) s 1 

Dg(x(t))[Ax(t) + Bh(t)Wl s 0 

(3.20) 

(3.21) 

where Dg(x(t)) is the Jacobian matrix of g(x(t)). 

c) if x(t)e BA,C then choose X(t), 0 I; X(t) 5 1 such that the expression in (3.19) is 

minimum. 

In the construction of h(t) if x(h)e BA,C then the basic problem cannot be solved because 

there exists a u(b) fort > b (Le. u(t) = 0) where it will lead to lly(x(t,)),t)llpo > 1. In such a case, the 

best that can be done is to find X(t) such that the states x(t) will be driven into BA,C as soon as 

possible. 

With the h(t) defined as above let us examine some properties of the system (3.11)-(3.12). 

To be more specific it will be shown that 

(a) There is always exists a h(t) that satisfies all the constraints in the construction of h(t). 

(b) If h(t) is constructed as specified above and x(~)E BA,C, then x(t)E BA,C Vt > to and for 
Page 12 

all U(t), t > t@ 

(c) The construction of h(t) solves the basic problem when that is possible (i.e. x(t)E BA,C 

for all t). 

Theorem 33: For the system given in eqs. (3.1 1)-(3.12) the following is always true VXE Itn. 

and at the points where g(x) is differentiable 

Dg(x) Ax S 0 VxeRn (3.23) 

where Dg(x(t)) is the Jacobian matrix of g(x(t)). 

Proof Assume that the inequality (3.22) is not true for some x(t) = xo. If the xo is used as an 

initial condition to the ;(t) = Ax(t) system then because of theorem 3.2 3t'M such that g(x(t')) > 

g(x(t)). But g(%) = IICx(t)lloo so this is a contradiction. Therefore, inequality (3.22) is me 

VX€ lRn. ///I 

The construction of x(t) is always possible because of theorem 3.3, namely one can choose 

h(t) = 0 Vt and the inequality (3.19) is always true. 

Lemma 3.1: In the system (3.11)-(3.12) if %E B,,c and l(t) is constructed as it was described 

above, then x(t)E BA,C for all t and for all u(t). 

Proof: The proof of this kmma follows from the construction of h(t). 

- 

ifif 
Page 13 

Theorem 3. 4: For the system (3.1 1)-(3.12) with h(t) constructed as above the following is always 

true 

if %E BA,c 

if xoe BA,c 

then Ily(t)ll, I1 Vinput u(t) 

then Ily(t)ll, <g(xo) Vinput u(t) 

- Proof If xOs BA,C, then 

The construction of h(t) guarantees that x(t)E BA,, Vt. (see Lemma 3.1). It is also true that 

for any state x(t)E BA,c IICx(t)ll, S 1. If IICx(t)ll, > 1 and x(t) is used as an initial condition in the 

system the following will be true, g(x(t)) > 1 and x(t)e BA,c which is a conmdiction. Since y(t) = 

Cx(t) and x(t)E BA,c Vt then Ily(t)ll, SI Vinput u(t). 

If XO~ BA,c, then g(x0) > 1 and from the construction of h(t) g(x(t)) < g(xo) (g(x) is 

decreasing by theorem 3.2). Thus Ily(t)ll, S g(x(t)) g(xo). ///I 

Theorem 39: At every time to, if x(~)E BA,C then the time-varying gain h(@ is the maximum 

possible such gain that 0 I h(h) S 1 and Vu(t), ob 3 h(t), t > to such that the output lyi(t)l I 1 V 

i, Ob. If x(b)e BA,C then such a gain h(b) does not exist 

proof: If X(@E BA,c, then from the construction of X(t), at any time to the maximum gain h(b) is 

chosen such that 0 S h(b) S 1 and x(t)E BA,CVt > b. If a greater gain A(@ is used then g(x(b) 

will be increasing (see theorem 3.2) and x(t)e B,,,V'ok; consequently there exists u(t) (i.e. u(t) = 

0 t 2 b) where Ily(t)ll, > 1. 

If x(b)e BA,,, then there exists u(t) (i.e. u(t)=O t 2 to) where Ily(t)ll, > 1 and thus for any 

h(b) the basic problem does not have a solution. ///I 

The solution to the basic problem which was given above assumed that h(t) is a scalar. A 

similar solution can be obtained if a time-varying diagonal matrix A(t) is employed. The 

construction of A(t) and all the properties that were described previously can easily be extended 

for the matrix case, Similar analysis can be done for systems with a feedforward term from the 

controls to the outputs [13]. 
Page 14 

4. Description of the Contrd Structure with the Operator EG 

In section 2 (performance analysis) the need for an operator EG to achieve better control 

system performance was shown. In section 3, it was shown how to choose a time varying gain 

h(t), at the inputs of a linear time invariant system, such that the outputs of that system will remain 

bounded. In this section, we combine the results of sections 2 and 3 to obtain, a control structure 

with an EG operator (i.e. a time gain-varying gain). This structure will be introduced and analyzed. 

With the EG operator at the emr signal, the system will remain unaltered (linear) when the . 

references and disturbances are such that they don't cause saturation. For "large" reference and 

disturbance signals the operator EG will ensure that the controls will never saturate. This control 

structure is useful for feedback systems with stable open loop plants and neutrally stable linear 

compensators. 

The new control structure has inherent good properties (stability, no reset windups etc.) 

which will be discussed and demonstrated in simulations of two examples. The examples chosen 

are an academic example (with pathological directional properties) and a model of the F8 aircraft 

longitudial dynamics. 

Consider a feedback control system with a linear plant G(s), a linear compensator K(s) and a 

magnitude saturation at the controls. The plant and the compensator are modelled by the following 

state space representations: 

Plant: ;(t) = Ax(t) + Bu,(t) (4.1) 

Y(t) = Cx(t) (4.2) 

U,(O = sat(u(t)> (4.3) 

Compensator: dc(t) = At+&) + B&) (4.4) 

u(t) = C,X,(t) (4.5) 

= r(t) - yo) (4.6) 

where r(t) is the reference, u(t) is the control and y(t) is the output signal. 

The compensator can be thought of as an independent linear system with input e(t) (error 
Page 15 

signal) and output u(t) (control signal). The objective is to introduce a time-varying gain h(t) (EG 

operator) at the error, e(t), such that the control, u(t), will never saturate. Following the discussion 

of section 3 the gain, h(t), is injected at the error signal and the resulting compensator is given by 

Figure 4.1: The basic system for calculating h(t). 

In analogy to figure 3.2, figure 4.1 shows the basic system for computing h(t). A function 

g(x) and a set BA,c are defrned and then the construction of X(t) follows in accordance with the 

results presented in section 3. 

g(X0): g(%) = Ilu(t)lloo (4.10) 

where ;c(t> = Acxx,(t); x,(O)=Xo (4.11) 

u(t) = CcxCO) (4.12) 

BA,C = (x: g(x> 5 1) (4.13) 

For g(x) to be finite, for all x, the compensator has to be neutrally stable (theorem 3.1). This 

is not an overly restrictive constraint because most compensators are usually neutrally stable. With 

finite g(x) the EG operator (h(t)) is given by 
Page 16 

Construction of 42) : 

For every time t choose X(t) as follows 

a) if x,(t)E IntBA,c then Ut) = 1 

b) if x,(t)E BdBA,c then choose the largest h(t) such that 

(4.14) 

(4.15) 

0 5 h(t) 5 1 

E+O E (4.16) 

or for the points where g(x) is differentiable choose the largest h(t) such that 

0 I; h(t) I; 1 

Dg(x,(t))[Acx,(t)+B,3i(t)e(t)] IO V t > 0 

(4.17) 

(4.18) 

where Dg(x,(t)) is the Jacobian matrix of g(xc(t)). 

c) if x,(t)e BA c then choose h(t), 0 5 h(t) 5 1 such that the expression (4.16) is 

minimum. 

From the results in section 3 it can be proven that if, at time t = 0, the compensator states, 

x,(t), belong in the BA,c set, then the EG operator exists and the signal u(t) remains bounded for 

any signal e(t). Hence, the controls will never saturate for any reference, any input disturbance, 

and any output disturbance. 
Page 17 

;--+ Logic w-7 

I ErrorGovernor I 

Figure 4.2 Control structure with the EG operator. 

Figure 4.2 shows the control structure obtained with the operator EG at the error signal. With 

this control smcture the feedback system will never suffer from the reset windup problems which 

occur when open loop integrators or "slow" poles are present. The reason for the absence of reset 

windups is that the Error Governor will prevent any states associated with integrators or the "slow" 

poles from reaching a value which will cause the controls to exceed the saturation limits. 

Another important property of the new control structure, is that the saturation does not alter 

either the direction of the control vector or the magnitude of the controls. Thus, if the compensator 

inverts part of the plant the saturation does not alter the inversion process. 

.. 4.1 mllitv for the Congo1 Svs~ with & EG 

When the plant is stable and the compensator includes the EG operator the following theorem 

can be proven. 

Theorem 4.1: The feedback system with a stable plant given by eqs. (4.1)-(4.3) and a compensator 

given by eqs.(4.7)-(4.9) is finite gain stable. 

Proof 3ro 3 Ilrll, 5 ro Ilull, I 1 
Page 18 

if Ilrll, I ro then h(t) = 1 and the linear system is stable, thus finite gain stable 

3yo 3 Ilyll, I yo Vr(t) because G(s) is stable with bounded inputs 

if llrllw > ro then Ilyll, 5 (llrll~ro)y~ and Ilyll, I (ydro>llrllw 

Thus, fork = (ydr0) then !lylloo 5 kllrll, /Ill 

Every stable system G(s) with bounded inputs is BIB0 stable because the outputs are always 

bounded. The system in figure 4.2 is finite gain stable because in addition to being BIB0 stable it 

is known that there exists a class of "small" inputs, Ilr(t)ll, S ro, for which the system remains 

linear. 

For unstable plants one cannot guarantee closed loop stability because when h(t) = 0 the 

system operates open loop. This is the reason why the control structure with the EG should be 

used for feedback systems with stable open loop plants. Another control structure can be used for 

systems with open loop unstable plants [ 131. This problem will be addressed separately in a future 

publication. 

For stable plants the closed loop system remains finite gain stable in the presence of any input 

and/or output disturbance. This is true because the controls never saturate for any input and/or 

output disturbance. In addition, it is easy to see that the closed loop system will remain finite gain 

stable for any stable unmodelled dynamics. In fact, the controls will never saturate if the model is 

replaced by the "true" stable plant; thus, integrator windups and/or control direction problems 

cannot occur. 

4.2 Simulation of the Academic Example #1 

The purpose of this example is to illustrate how the saturation can disturb the directionality of 

the controls and alter the compensator inversion of the plant. The "academic" plant G(s) has two 

zeros with low damping which the designed compensator K(s) cancels. Consider the following 

state space representation of the plant G(s) 
Page 19 

-1.5 1 0 

. 12 -3 2 

m= 1 .5 -2 

1 1 -1.5 0 

r o 2.4 -3.1 

1 6 -.5 Y (0 = 1 ] x(t> -2.8 

0 

0 

1 

1.8 

Singular values of the plant 

lo

10 

1 .o 

0.1 

0.01 I I I I I ,I, I I I ,111, I I I I, I,, I I Ill," 

0.01 0.1 1 .o 10 100 

logw (xndlsec) 

Figure 4.3: Singular values of the plant in the academic example #l. 

(4.19) 

(4.20) 

(4.21) 

Figure 4.3 shows the singular values of the open loop plant. Notice the effect of the two 

resonant zeros of the plant in the singular values at approximately 2.5 radsec. A compensator was 

designed to cancel the two resonant zeros of the plant. The compensator state space representation 

is given by the following model 
Page 20 

r -2.6093 , 1.4180 1 r -29.8308 2.989 1 

Xc(t) + h(t) e(t) (4.22) 

Xc(') = 1 -7.1476 1.5213 1 1 -68.7543 10.8387 1 

(4.23) 

The compensator has two states with poles at 0.544 f j2.422. The eigenvectors of the poles 

are collinear with the control direction of the transmission zero of the plant and thus, the 

compensator cancels the zeros of the plant. 

Loop singular values 

100 

10 

5 1.0 

F 0.1 

-2 

0.01 

3 

I I I I1 I,, I I I I I Ill I I 1111 

0.01 0.1 1 .o 10 100 

log o (radlsec) 

Figure 4.4: Singular values of the loop transfer function in the academic example #l. 

Figure 4.4 shows the singular values of the G(s)K(s) transfer function matrix. Since the 

compensator cancels the poorly damped zero the antiresonance present in figure 4.3 is not present 

in figure 4.4. 

In this example, the saturation can disturb the cancellation of the plant zeros by the 

compensator. Since both the plant and the compensator are stable the control structure with the 
Page 21 

operator EG can be used to correct the problem. Three simulations were performed for the closed 

loop system, these different simulations are as follows: 

1) In the first simulation X(t) = 1 and u(t) = u,(t). This is a simulation for a linear time 

invariant closed loop system and is refed to as the simulation for the linear system. 

2) In the second simulation X(t) = 1 and uS(t) = sat(u(t)). This is a simulation where the 

saturation element is added to the linear system without any other modification. This simulation is 

referred to as the simulation for the system with sawation. 

3) In the third simulation u,(t) = sat(u(t)), and h(t) served as the EG operator. This type of 

simulation is referred to as the simulation of the system with sawation and the EG. 

Figure 4.5 shows the state trajectory of the compensator states for the simulation of the linear 

system. Note that the states of the compensator do not remain within the B,,, set so there is a 

potential for the controls to saturate. 

Figures 4.6 and 4.7 show the linear response of the outputs y(t) and the controls u(t) 

respectively. The controls satisfy Ilu(t)ll, > 1 at certain times and saturation is expected. It is 

assumed that the output responses meet the specifications. Thus, we would like the outputs to 

retain the relative shapes of figure 4.6 when we introduce the nonlinear saturations. 

Figure 4.8 shows the state trajectory of the compensator states for the simulation of the 

system with saturation, it is clear that he states of the compensator do not remain within the B,,, 

set. When the controls are saturated the direction of the controls is disturbed and the state trajectory 

changes dramatically (compare figures 4.5 and 4.8). 

Figures 4.9 and 4.10 show the response of the outputs and the controls respectively. The 

controls have magnitude greater than one and consequently are saturating. In this example, when 

saturation occurs, the direction of the controls is altered in such a way that even though the original 

reference is [ .3 

resulting in oscillatory behavior. The compensator does not have any integrators to cause windups 

and the problems in the performance of the system are solely due to the effects of the saturation 

upon the direction of the control vector. 

.3IT, the control direction at saturation drives the system towards [.3 -.3IT 
Page 22 

Comparing the outputs, i.e. figures 4.6 and 4.9, we see that the shapes of the outputs in 

figure 4.9 do not match those desired and shown in figure 4.6. Thus, in this case the impact of 

saturation has produced an unacceptable output response. 

Figure 4.1 1 shows the compensator state trajectory for the simulation of the system with 

saturation and the EG operator. The states of the compensator do remain within the B,,c set so 

control saturation is not expected. In fact, the state trajectory remains on the boundary of the B,,, 

set for a long period of time which implies that the controls will stay at their maximum level for a 

long period of time. 

Figures 4.12 and 4.13 show the response of the outputs and the controls respectively. Note 

that the controls (the inputs to the saturation operator) do not cause saturation. Also note that when 

u2 reaches the value of -1, the control u1 is reduced to the appropriate level so that both controls 

will drive the output towards [.3 .3IT as desired. In effect, it is like having a "smart multivariable 

saturation" instead of the SISO saturations in each channel. The net effect can be seen easier in the 

output responses. Comparison of figure 4.12 with figure 4.6, shows that the outputs have similar 

shapes (as desired), except that the outputs in figure 4.12 are "slower" because the control 

magnitudes arc smaller than those in the linear case (compare figures 4.7 and 4.13). 

Figure 4.14 shows the real-time behavior of the gain h(t). At the beginning, h(t) is 1 and the 

system is linear. When the states of the compensator are such that they may lead the controls to 

saturate, h(t) becomes zero preventing the large errors to be driven by the compensator. The 

controls at the Same time remain at their maximum possible level ( Ilu(t)ll, = 1 ). Eventually, X(t) 

allows the compensator to accept more and more error, while at the same time the controls are kept 

at maximum level. At the end, h(t) becomes 1 and the system becomes linear time invariant again. 
Page 23 

state trajectory for the academic example with r=[ . 3 . 3 lT 

1.75 

1.05 

0.35 

x2 

-0.35 

- 1.05 

- 1.75 

-1.75 -1.05 -0.35 0.35 1.05 1.75 

x1 

Figure 4.5: State trajectory of the compensator states in the linear system, (r = [.3 .3IT). 

Academic example (linear) 

0.40 

0.30 

0.20 

5 

B 

g 0.10 

0.00 

-0.10 

0.00 2.00 4.00 6.00 8.00 10.00 

Time (sec.) 

Figure 4.6: Output response for the linear system, (r = [.3 .3]T). 
x2 

Page 24 

Academic example (linear) 

1 .so 

0.90 

9 0.30 

-0.90 

- 1 .SO 

0.00 2.00 4.00 6.00 8.00 10.00 

Time (sec.) 

Figure 4.7: Controls in the linear system, (r = [.3 .3IT). 

State trajectory for the academic example with r =[. 3 . 3IT 

3*00 

1.80 

* 

0.60 

-0.60 

- 1.80 

-3.00 

-3.00 -1.80 -0.60 0.60 1.80 3.00 

Figure 4.8: State trajectory of the compensator states in the system with saturation, (r = [.3 .3IT). 
Page 25 

Academic example with saturation f 1 

4.00 

3.00 

g 2.00 

9 

5 

0 1.00 

0.00 

-1.00 

0.00 2.00 4.00 6.00 8.00 10.00 

The (sec.) 

Figure 4.9: Output response for the system with saturation, (r = [.3 .3]T). 

4.00 

1.80 

.c. -0.40 3 

3 

-2.60 

u 

-4.80 

-7.00 

0.00 2.00 4.00 6.00 8.00 10.00 

?'ime (sec.) 

Figure 4.10: Conttols in the system with saturation, (r = [.3 .3]r>. 
Page 26 

state trajectory for the academic example with r =[ .3 .3 1' 

1.75 

1 .OS 

0.35 

-0.35 

- 1.05 

- 1.75 

-1.75 -1.05 -0.35 0.35 1.05 1.75 

Figure 4.11: State trajectory of the compensator states in the system 

with saturation and the EG, (r = [.3 .3IT). 

Academic example uith r =[ .3 .3] 

0.40 

0.30 

0.20 

0.10 

0.00 

-0.10 

0.00 2.00 4.00 6.00 8.00 10.00 

Time (sec.) 

Figure 4.12: Output response for the system with saturation and the EG, (r = [.3 .3IT>. 
Page 27 

Academic example with r=[. 3 .3 J 

1 .so 

0.90 

9 0.30 

3 

-0.30 V 

-0.90 

I .vw 

0.00 2.00 4.00 6.00 8.00 10.00 

Time (see.) 

Figure 4.13: Controls in the system with satuxation and the EG, (r = C.3 .3IT>. 

A (t) for the academic example with r=[ .3 .3 1 

0.88 

0.66 

0.44 

0.22 

0.00 

1.2 I 1 

0.9 

0.6 

0.3 

0 .o 0.0 0.3 0.6 0.9 1.2 1.5 

0.0 2.0 4.0 6.0 8.0 10.0 

Time (sec.) 

Figure 4.14: h(t) in the system with saturation and the EG; (r = [.3 

Insert: Blowup with OItS1.5 sec. 

.3IT>. 
Page 28 

- 

- -0.8 -.0006 -12 0 - -19 -3 - 

0 -.014 -16.64 -32.2 -.66 -.5 

kt) = 1 -.0001 -1.5 0 x(t)+ -.5 

1 0 0 0 0 0 

- - - .. 

4.3 Simulation of a Model of the F8 Aircrafr 

us (0 (4.24) 

The purpose of this example is to illusmte the effects of multiple saturations on the directions 

of the controls and consequently on the response of the control system and the integrator windup 

phenomenon. The simulation confirms our claim that the integrators in the control system with the 

EG never windup, and that the saturation does not effect the direction of the controls when the EG 

operator is used. 

Consider a model of the longitudinal dynamics of the F8 aircraft A flaperon has been added 

which does not exist in the F8 prototype. The state equations are given by 

000 

00-11 

(4.25) 

u,(t) = sat(u(t)) (4.26) 

and in compact form 

where 

I 6,(t) elevator angle (deg) limit at 25' 

6,(t) flaperon angle (deg) limit at 25' i Controls u (t) = 

(4.27) 

(4.28) 

(4.29) 
pitch angle (rad) 

flight path angle (rad) 1 

States x(t) = 

a(t) 

1000 

0.001 

0. 

pitch rate (radsec) 

forward velocity (Wsec) 

angle of attack (rad) 

pitch angle (rad) 

Singular values of the F8 model 

01 ‘ 0.1 1 .o 10 100 

log w (radlsec) 

Page 29 

(4.30) 

(4.3 1) 

Figure 4.15: Singular values of the F8 model. 

Figure 4.15 shows the singular values of the F8 linear model. Assume that a closed loop 

system has to be designed for the F8 model to follow pitch and flight path angle commands. Also 

assume that zero steady state error is required for step commands. The control system to be 

designed, should be thought as a semi-realistic MIMO conwoller so as to test the new design 

methodology introduced in this section. 

The design process is the following. First, linear control theory will be used to design the 

closed loop system. Then the linear compensator will be modified with the EG operator. Finally, 
Page 30 

' -.844 .819- 

-11.54 13.47 

-.86 .25 

-47.4 15 

4.68 -4.8 

4.82 .14 - 

simulations of the closed loop system will be performed to assess the benefits of the new design 

methodology. 

To obtain the kquired linear control system the saturation is ignored (u,(t) = u(t)) and, two 

integrators were added at the controls. The augmented system (sixth order) is given by the 

following 

1 -52.23 -3.36 73.1 -.0006 -94.3 1072 

-3.36 -29.7 -2.19 -.006 908.9 -921 G=[ 

where 

(4.32) 

(4.33) 

(4.34) 

Next, a linear compensator was designed for the augmented system to control the pitch angle 

and flight path angle. The LQGLTR methodology was used to design the compensator which is 

computed as follows:. 

K(s) = G[ sI-Aa-BaG-HC,]-' H (4.35) 

(4.36) I Ka(s) = - K(s) S 

where 

H= 
Page 31 

The LQG/LTR compensator K(s) cancels part of the F8 dynamics. From now on we assume 

that the G(s)KJs) is the desired forward loop transfer matrix, and that we would like to mimic (to 

the extent possible) the transient response of this linear feedback system even in the presence of 

saturations. Figure 4.16 shows the singular values of the resulting loop transfer function matrix 

G (SI K&). 

Loop singular values 

E+04 

100 

i! 

f! 2 1.0 

bo 

0.01 

I I I I 1111 I I I I I Ill I 1111 

0.01 0.1 1 .o 10 100 

log o (radlsec) 

Figure 4.16: Singular values of the loop transfer function in the F8 closed loop system 

To prevent control saturations, the Error Governor (the h(t) time-varying gain) is added to 

the feedback system at the emr signal e(t). The construction of h(t) is possible because the 

compensator K(s) is neutrally stable and finite gain stability is guaranteed because in addition the 

plant G(s) is stable. 

The result is a multivariable control system with integrators in the forward loop. In the 

presence of saturation, and without the EG operator, integrator windups would be expected and the 

direction of the control vector would be distorted. Three simulations were performed to show the 

integrator windup problem and how the problem is resolved by the operator EG. 

First, the closed loop system was simulated with reference vector r = [ 10 10IT. Figures 

4.17 and 4.18 show the linear output and control responses. As expected from the singular values 
Page 32 

of G(s)&(s), both outputs behave similarly and it is assumed that this type of an output response 

satisfies the posed constraints. Note that the controls have "impulsive" action at the beginning, and 

they violate the k25O limit; thus saturation is expected. 

Figures 4.19 and 4.20 show the outputs and controls of the system with saturation. From the 

oscillations in the output response it can be inferred that the integrators windup. In addition, the 

direction of the output is disturbed and the outputs are "not matched" any more (compare figures 

4.17 and 4.19). 

Figures 4.21 and 4.22 show the output and control responses of the system with saturation 

and the EG operator. Compare figures 4.17 and 4.21 and notice how the outputs are similar in 

shape (as it was desired), in addition to the fact that there are no integrator windups. The output 

response has of course slower rise time, since we must use smaller controls, but the nature of the 

response is similar to the hear one. The controls u(t) in figure 4.22 never exceed the limits of the 

saturation; and when the flaperon 6dt) reaches 25' the elevator 6Jt) remains almost constant until 

8dt) unsaturates. The direction of the controls during that period of time is such that drives the 

plant output towards the command [lo 

multivariable saturation". 

10IT. The system behaves like having "a smart 

Figure 4.23 shows the k(t). Note that the error is almost completely "turned-off at about .05 

seconds. The gain k(t) then increases slowly towards unity and the system operates linearly again. 
Page 33 

output 

15.00 

12.00 

9.00 

8 6.00 

n 

E 

3.00 

s 

0.00 

for the F8 closed loop system with r=[ 1 0 101' 

0.00 1.00 2.00 3.00 4.00 5.00 

TSme (sec) 

Figure 4.17: Output response for the FS linear system, (r = [ 10 10 IT>. 

Controls in the F8 closed loop system with r =[ 1 0 1 01 

-00 

Figure 4.18: Controls in the F8 linear system, (r = [ 10 10 19. 
Page 34 

-20.00 

n 

OI 

E 

M 

u 

3 

C 

5 

0 

' b S,(9 

Output of the closed loop system with r =[ 1 0 1 01 

12.00 

9.00 

6.00 

3.00 

0.00 

0.00 1.00 2.00 3.00 

Time (sec) 

4.00 5.00 

ControlsintheF8closedloopsystemuithr=[10 10f 

Figure 4.19: Output response for the F8 system with saturation, (r = [ 10 10 IT). 

Figure 4.20: Controls in the F8 system with saturation, (r = [ 10 10 IT). 

0.00 1.00 2.00 3.00 4.00 5.00 

Time (sec.) 
Page 35 

Output in the F8 closed loop system with r=[ 1 0 1 0 3 

15.00 

12.00 

n 

m 

E 9.00 

3 

s 

5 

6.00 

9 

3.00 

0.00 

0.00 1.00 2.00 3.00 4.00 5.00 

Time (scc.) 

Figure 4.21: Output response of the F8 system with saturation and the EG, (r = [ 10 lo]*). 

Controls in the F8 closed loop system with r=[ 10 101 

Figure 4.22: Controls in the F8 system with saturation and the EG, (r = [ 10 10IT). 
Page 36 

A(t) for the F8 closed loop withr=[ 10 10 3 

1.10 

0.88 

0.66 

A (9 

0.44 

0.22 

0.00 

I 

1.2 

0.9 I 0.6 

0.3 

0 .o 

0.0 0.15 0.30 0.45 0.60 0.75 

0.00 1.00 2.00 3.00 4.00 5.00 

Time (sec.) 

Figure 4.23: h(t) in the F8 system with saturation and the EG, (r = [ 10 101"). 

Insert: Blowup with 0 I t I .75 sec. 

5. Conclusion 

t 

Saturations exist in almost every physical system. In this research, the effects of multiple 

saturations present in a closed loop control system were studied extensively. In the presence of 

saturations the performance of a linear control system can suffer. For example, a linear control 

system that is closed loop stable can become unstable when saturations are present for certain 

references and disturbances. Saturations can also affect the performance of the control system by 

introducing reset windups and by changing the direction of the control signal. Large overshoots 

and oscillatory outputs are the consequence. 

A systematic methodology was introduced for the design of control systems with multiple 

saturations. The idea was to introduce a supervisor loop; and when the references and/or 

disturbances are "small" enough so as not to cause saturations, the system operates linearly as 
, 

Page 37 

designed. When the signals are large enough to cause saturations, then the control law is modified 

in such a way to preserve, to the extent possible, the behavior of the linear control design. 

The main benefits of the methodology are that it leads to controllers with the following 

properties: 

(a) The signals that the modified compensator produces never cause saturation. 

(b) Possible integrators or slow dynamics in the compensator never windup. 

(c) The closed loop system has inherent stability properties. 

(d) The on-line computation required to implement the control system is feasible. 

These properties were demonstrated in simulations of the F8 aircraft (stable) model and an 

academic example. 

Extensions of the methodology can be made to address the class of systems with open loop 

unstable plants [13]. Future publication will cover this problem in detail. 

6. References 

[ 13 M. Athans, P.L. Falb, Optimal Conrrol, New York, McGraw-Hill, 1966. 

[2] C.A. Harvey, On Feedback Systems Possessing Integrity With Respect to Actuators 

Outages", Proceed ings of the ONR./M IT WorkshoD on Resent DeveloDments in the Robustness 

Theory of Multivariable Systems, LIDS-R-954, M.LT., Cambridge, MA, April 25-27, 1979. 

[3] P. Molander and J.C. Willems, 'I Robustness Results For State Feedback Regulators", 

Proceed ings oft he ONFUMIT Workshop on Resent DeveloDments in the Robust ness Theory of 

Multivariable Svstems, LIDS-R-954, M.I.T., Cambridge, MA, April 25-27, 1979. 

[4] A. Weinreb and A.E. Bryson," Optimal Control of Systems with Hard Control Bounds" IEEE 

Transactions o n Automatic Control, Vol. AC-30, No. 11, November 1985, pp. 1135-1138. 

[SI I. Horowitz, "Feedback Systems with Rate and Amplitude Limiting", Int. J. Co ntrol, 
Page 38 

Vol. 40, NO. 6, 1984, pp. 1215-1229. 

[6] H.W. Thomas, D.J. Sandoz and M. Thomson, New desaturation strategy for digital PID 

controlers", IEE Proceed ina, Vol. 130, Pt. D, No. 4, July 1983, pp.113. 

[7] P. Gutman and P. Hagander, 'I A New Design of Constrained Controllers for Linear Systems", 

IEEE Transactions on Automatic Control, Vol. AC-30, No. 1, January 1985, pp. 22-33. 

[8] A. H. Glattfelder and W. Scaufelberger," Stability Analysis Of Single Loop Control Systems 

with Saturation and Antireset-Windup Circuits", IEEE Transactio ns on Automatic Co ntrol, Vol. 

AC-28, No. 12, December 1983, pp. 1074-1081. 

[9] R. Hanus, I' A New Technique for Preventing Windup Nuisances", Proc. IFIP Conf. on Auto. 

for Safetv in ShippinF and 0 ffshore Petrol. ODe rations, 1980, pp. 221-224. 

[lo] N.J. Krikelis, "State Feedback Integral Control with 'Intelligent.' Integrators", Int. J. Contrd, 

Vol. 32, NO. 3, 1980, pp. 465-473. 

[ 111 P. Kapasouris and M. Athans, Multivariable Control Systems with Saturating Actuators 

Antireset Windup Strategies", Proceed ings of the American Co ntrol Conference, Boston, MA, 

1985, pp. 1579-1584. 

[ 121 J.C. Doyle, R.S. Smith and D.F. Enns, 'I Control of Plants with Input Saturation Nonlinearities", 

Proceedings of the American Control Conference. Minneapolis, MN, 1987, pp. 1034- 1039. 

[ 131 P. Kapasouris, Design for Perforinance Enhancement in Feedback Control Systems with 

Multipfe Saturating Nonlinearities, Ph.D. Thesis, Department of Electrical Engineering, M.I.T., 

Boston, MA, February 1988. 

[ 141 N. Rouche, P. Habets and M Laloy, Stability Theory by Lyapunov's Direct Method, New York, 

Springer-Verlag, 1977.