Bode's Sensitivity Integral Constraints: The Waterbed Effect in Discrete Time

TL;DR

This paper explores the fundamental limitations of feedback control in discrete-time systems, revealing how the sensitivity integral constraints are intrinsically linked to unstable poles and zeros outside the unit circle, highlighting the waterbed effect.

Contribution

It extends Bode's sensitivity integral constraints to discrete-time systems, emphasizing the role of unstable poles and transmission zeros outside the unit circle in performance limitations.

Findings

Sensitivity integral constraints relate to unstable open-loop poles.

Complementary sensitivity constraints relate to zeros outside the unit circle.

Illustrative examples demonstrate the theoretical results.

Abstract

Bode's sensitivity integral constraints define a fundamental rule about the limitations of feedback and is referred to as the waterbed effect. In a companion paper, we took a fresh look at this problem using a direct approach to derive our results. In this paper, we will address the same problem, but now in discrete time. Although similar to the continuous case, the discrete-time case poses its own peculiarities and subtleties. The main result is that the sensitivity integral constraint is crucially related to the locations of the unstable open-loop poles of the system. This makes much intuitive sense. Similar results are also derived for the complementary sensitivity function. In that case the integral constraint is related to the locations of the transmission zeros outside the unit circle. Hence all performance limitations are inherently related to the open-loop poles and the transmission zeros outside the unit circle. A number of illustrative examples are presented.

Equations

$$E(s)=\frac{p(s)}{q(s)}\frac{a(s)d(s)}{a(s)d(s)+b(s)c(s)}$$

$$\dot{\mathbf{x}}$$

$$e$$

$$=$$

$$d(p)\dot{\mathbf{x}}$$

$$d(p)e$$

$$\dot{\mathbf{z}}$$

$$e$$

$$\left[\begin{array}[]{c}\mathbf{\dot{z}}\\ \mathbf{\dot{\eta}}\end{array}\right]=\left[\begin{array}[]{cc}F&0\\ BH&A\end{array}\right]\left[\begin{array}[]{c}\mathbf{z}\\ \mathbf{\eta}\end{array}\right]+\left[\begin{array}[]{c}G\\ 0\end{array}\right]\mu$$

$$\dot{\mathbf{x}}$$

$$u$$

$$\frac{d\hat{\mathbf{x}}}{dt}$$

$$\hat{u}$$

$$\dot{\mathbf{z}}$$

$$\dot{\eta }$$

$$e$$

$$\frac{d}{dt}\hat{\mathbf{z}}$$

$$\frac{d}{dt}\hat{\eta }$$

$$u$$

$$\frac{d\tilde{\mathbf{z}}}{dt}$$

$$\frac{d\tilde{\eta }}{dt}$$

$$\begin{array}[]{c}\frac{d}{dt}\left[\begin{array}[]{c}\mathbf{\hat{z}}\\ \mathbf{\hat{\eta}}\end{array}\right]=\left[\begin{array}[]{cc}F-L_{zx}H-GK_{zx}&0\\ L_{nx}H&A\end{array}\right]\left[\begin{array}[]{c}\mathbf{\hat{z}}\\ \mathbf{\hat{\eta}}\end{array}\right]+\left[\begin{array}[]{c}L_{zx}\\ L_{nx}\end{array}\right]e\\ u=-K_{zx}\mathbf{\hat{z}-}C\mathbf{\hat{\eta}}\end{array}$$

$$\dot{\mathbf{z}}$$

$$\dot{\mathbf{x}}$$

$$y$$

$$\dot{\mathbf{z}}$$

$$\eta$$

$$u=N\mathbf{z}-K(\mathbf{x}-M\mathbf{z})$$

$$FM-MA+GN$$

$$HM$$

$$\dot{\mathbf{x}}$$

$$\mathbf{X}(s)=(sI-F+GK{)}^{-1}(GN+GKM)(sI-A{)}^{-1}B$$

$$\mathbf{X}(s)$$

$$\mathbf{X}(s)=M(sI-A{)}^{-1}B-(sI-F+GK{)}^{-1}MB$$

$$Y(s)=HM(sI-A{)}^{-1}B-H(sI-F+GK{)}^{-1}MB$$

Full text

I

Introduction

Two robust methods to track a persistent reference and eliminate a persistent disturbance are the Internal Model method and the Extended Estimator method. The use of integral action to track a constant reference or to eliminate a constant bias has been part of the tools of control from the earliest times. This concept was generalized as the Internal Model principle by Wonham and Francis in 1975.111The Internal Model Principle of Control Theory, B.A. Francis and W. M. Wonham  Automatica, Vol. 12, 1975. In another development, the concept of an estimator of the state of a dynamic plant having random inputs was introduced by Kalman and Bucy in 1961.222New Results in Linear Filtering and Prediction Theory, R, E. Kalman and R. S. Bucy, Journal of Basic Engineering, March 1961 The deterministic case was introduced by Luenberger in 1964.333Observing the State of a Linear System, D. G Luenberger, IEEE Transactions on Military Electronics, vol. 8, 1964 One of the basic assumptions of the Kalman Filter is that the disturbance signal is a white noise, having a constant spectrum. If the disturbance signal has a shaped spectrum, it is said to be ‘colored’  and is modeled as a white signal shaped by a dynamic system or filter. Construction of the Kalman filter then requires that the state of the model filter is estimated along with the state of the plant. The Extended Estimator is the deterministic version of this case. While these methods are widely used, a simple derivation of their structure and a comparison of their properties is not readily available. In the Section 2, the equations and block diagram of the internal model are given. In the Section 3 the same is done for the extended estimator. In the Section 4 the non-robust Model Following method is described. In the final section the methods are compared and contrasted. Programs in MATLAB and SIMULINK used to make the calculations are available on line at the site fpe6e.com.

II Design of the Internal Model.

The idea of an internal model is quite simple: if one wants a plant output to reproduce, for example, a sinusoid with no error in the steady state, then the plant must be driven by a signal which will cause the plant output to match produce the sine wave with zero input. Such a signal can be generated by an oscillator which is said to be an internal model of the reference generator. By equations one can see the same thing. Suppose a plant with transfer function $\frac{\displaystyle b(s)}{\displaystyle a(s)}$ has a serial controller $\frac{\displaystyle c(s)}{\displaystyle d(s)}$ and is asked to track exactly a reference input described by $\frac{\displaystyle p(s)}{\displaystyle q(s)}.$ With unity feedback, the transform of the error for this system will be

[TABLE]

If this transform is to have no residue at a root of $\ q(s)$ then that polynomial must be canceled from the transform.  If it is canceled by the controller denominator, $d(s),$ then the controller is said to include an internal model. If it is canceled by the plant transfer function denominator $a(s)$ then the plant is said to include an implicit internal model. This is most often the case when the plant transfer function includes integral action, an implicit model for steps. The tracking result holds even with changes in the plant parameters just so long as the system remains stable. Thus the solution is said to be robust. However, with an implicit internal model, the poles of the plant used to cancel the input poles cannot change if the tracking is to be accurate  We note here that if the system is not unity feedback or has a disturbance which we wish to eliminate exactly, then the error transform will differ from the above so that although the method is the same different conclusions will need to be made.

The Internal Model system can be designed in a set of simple steps. We begin with the nth order system equations for the plant:

[TABLE]

For this problem, using ${}^{\prime}p^{\prime}$ as the differential operator, we assume that there is an mth degree scalar polynomial, $d(p)$ such that $d(p)r(t)=0.$ and $d(p)w(t)=0$  For example, if \ w=1\and $r=\sin(t),$ then $d(p)=p^{3}+p.$ If we operate on Eq.(2) and Eq. (3)  with $d(p)$ the reference and the disturbance are eliminated and the result is

[TABLE]

If we now define new variables as $d(p)\mathbf{x}=\mathbf{z}$ and $d(p)u=\mu,$ then the equations become:

[TABLE]

Eq.(6) represents two systems in series, as shown in Fig. LABEL:figure1.  The error is the output of a system that can be described in control canonical form by the state parameters $A,B,$ and has the transfer function $\frac{\displaystyle-1}{\displaystyle d(s)}.$ The state of this system is comprised of the error and its derivatives as \mathbf{\eta}=[\begin{array}[]{ccccc}e^{(m)}&e^{(m-1)}&...&\dot{e}&e\end{array}]^{T}

\FRAME

fhFU3.7766in0.9072in0pt\QcbBlock diagram for Internal Model design\Qlbfigure1fig7_1.jpg

To stabilize this overall system with state feedback, we select  $n+m$ control poles in a vector $p_{ci},$ and compute control gains [\begin{array}[]{cc}K_{zi}&K_{\eta i}\end{array}] using the MATLAB function *place  *on the composite system which is described by the equations:

[TABLE]

with the result that \mu=-\left[\begin{array}[]{cc}K_{zi}&K_{\eta i}\end{array}\right]\left[\begin{array}[]{c}\mathbf{z}\\ \mathbf{\eta}\end{array}\right] as shown in Fig.LABEL:figure2.

\FRAME

fhFU3.3624in1.6034in0pt\QcbBlock diagram showing the state feedback for the composite system\Qlbfigure2fig7_2.jpg

In view of the nature of the state $\mathbf{\eta},$  as composed of the error and its derivatives, a polynomial in $p$ can be defined, based on $K_{\eta i}$, as $c(p)$ such that $\mu=-K_{zi}\mathbf{z}+c(p)e.$  We can now unscramble the equations to recover the original variables. Starting with Eq.(6), dividing by $d(p),$ and using the just derived control law, we get

[TABLE]

Notice that in these equations we have restored both reference and disturbance because dividing by $d(p)$ we get, for example, $\frac{d(p)}{d(p)}w$ which must be taken as $w.$ The result, with the Internal Model shown, is sketched in Fig.LABEL:figure3.

\FRAME

fhFU3.4566in1.8343in0pt\QcbBlock diagram of the Internal Model design\Qlbfigure3fig7_3.jpg

Finally, since the plant state is rarely available, we replace the plant state feedback with feedback of the estimated state, as shown in Fig. LABEL:figure4. The standard state estimator equations are

[TABLE]

Notice that we have included saturation of the control signal in this structure. Before we give an example of this design, it is important to notice that the control law shown in Fig.LABEL:figure2 guarantees that the modified state and controls as well as the error go to zero in the steady state but that these are *not *the physical state, $\mathbf{x}$ or the physical control, $u.$ It is $\mathbf{z}=d(p)\mathbf{x}$ and $\mu=d(p)u$ in addition the to physical error that are guaranteed to be sent to zero.

\FRAME

fhFU3.6711in1.6829in0pt\QcbBlock Diagram of the final Internal Model system\Qlbfigure4fig7_4.jpg

III The Extended Estimator

As a second approach to the design of robust control with external inputs, we develop a method for tracking a reference input and rejecting disturbances by an Extended Estimator rather than with an explicit Internal Model. The internal model will show up as part of the estimator in this case. The method is based on augmenting the usual state estimator to include estimates of the external signals in a way that permits us to cancel out their effects on the system error asymptotically. The physical situation is sketched in Figure LABEL:Xeststep1 showing the plant with disturbance, $w$, introduced into the plant and a reference at the output.

\FRAME

fhFU2.1793in0.8726in0pt\QcbOriginal system for design of Extended Estimator\QlbXeststep1fig7_5.jpg

Taking the difference between reference and plant output, the system output is the error, $e.$ In order to cancel the effects of the reference and the disturbance as well, an equivalent input is introduced as shown in Figure LABEL:Xeststep2. This equivalent external signal generator system is described by the matrices $A,B,C,D$. That is to say, the dynamics of the system matrix $A,$ with suitable initial conditions, can reproduce the effects of both the disturbance and the reference input so as to produce the actual error signal at the output of the equivalent plant. For this to work, the plant must not have a zero at any of the eigenvalues of $A$ as that would prevent that portion of the equivalent input to get to the output. For notation, we define the state of the equivalent input generator as $\mathbf{\eta}$ and its output as $\rho=C\mathbf{\eta}.$ Notice in particular that the state of the ‘Plant’ in this set-up is not the state of the physical plant even though it is described by the same dynamics. Furthermore, it has a non-physical input although its output is the true system error, which is not the usual plant output, which is $y$.

\FRAME

fhFU2.2131in0.9124in0pt\QcbEquivalent system for desgn of the Extended Estimator\QlbXeststep2fig7_6.jpg

This equivalent system is clearly not controllable, as the control signal has no influence on the equivalent external input signal. The design plan is to first design a simple feedback control for the ‘plant’ part of the system and then to build an extended estimator that will provide an estimate of the complete system state, including both the ‘plant’ state and the state of the external input generator system. This latter estimate will be used to cancel the equivalent input asymptotically. The first step is to design the plant control law based on the equivalent ‘plant’ state, $\mathbf{z}.$ For this we select a suitable set of control poles for the eXtended estimator as $p_{cx}$ and compute $K_{zx}=place(F,G,p_{cx})$  and apply it as shown in Figure LABEL:Xest-step3a.

\FRAME

fhFU1.8472in1.5022in0pt\QcbControl law block diagram for the Extended Estimator\QlbXest-step3afig7_7.jpg

Next we estimate the combined state of this composite equivalent system and use these extended estimates to compute the control. We must emphasize here that the estimate of the state in this equivalent world is not an estimate of the physical plant state but is a state that produces the same error as the physical system but with the effects of both reference and disturbance external inputs generated at the ‘plant’ input. These estimates are constructed from the system error, not from the usual physical plant output, $y$. The equations on which the design is based are thus

[TABLE]

An estimator for this system is described by the standard equations:

[TABLE]

In the control equation we have introduced the estimate of the equivalent input state with its output matrix $C$ is place of the unavailable control law. It will be used to cancel the effects of the external signals in the steady state, as we will see presently. The estimator law, Lt=\left[\begin{array}[]{cc}L_{zx}&L_{nx}\end{array}\right]^{\prime}, is designed by selecting $n+m$ estimator poles, $p_{ex}$.and using place in the usual way.  From these results we construct the error equations as

[TABLE]

Based on the design of the estimator gain $Lt,$  these equations are stable and the entire estimator error state will go to zero asymptotically. It is informative to rewrite the estimator equations as the controller, having input $e$ and output $u.$

[TABLE]

From these equations, it is clear that the eigenvalues of the controller contain those of the matrix $A,$ which is to say it contains an internal model of all the external inputs. The equations for the equivalent plant are given by, in turn,

[TABLE]

In view of the fact that $K_{zx}$ was designed to make $F-GK_{zx}$ stable and that we have shown that both $\mathbf{\tilde{z}}$ and $\mathbf{\tilde{\eta}}$ go to zero, we can conclude that $\mathbf{z}$ and thus $e=H\mathbf{z}$ will go to zero, which is the object of all this, after all! Notice that this result is robust with respect to the plant parameters, $F,G,H$, but not with respect to the characteristic equation of $A,$ which is the model polynomial, $d(p).$ Again, it is $\mathbf{z}=d(p)\mathbf{x}$  that goes to zero, not $\mathbf{x}.$The block diagram of the resulting physical system is shown in Fig  LABEL:Xeststep4 where we have included a saturation element at the (physical!) plant input. It is interesting to notice that by comparing Fig. LABEL:Xeststep4 with Fig. LABEL:figure4 it can be seen that the Internal Model design results in a combined feedback-feed forward structure while the Extended Estimator results in a strictly feed forward structure. Thus, while they will both be robust in tracking, the transient responses will be different, often spectacularly so.

\FRAME

fhFU3.7542in1.4148in0pt\QcbFinal Block Diagram of the Extended Estimator Design\QlbXeststep4fig7_8.jpg

IV Model Following

A third method to track a persistent reference is Model Following (see .Fig. LABEL:modelfollowing2) This is an open loop method that uses the state of the model to construct that particular control input which will force the plant output to asymptotically track the output of the model. The model output may or may not be persistent. The method is described more fully in Bryson444Control of Spacecraft and Aircraft, A E Bryson, Princeton University Press, 1994, including the case of disturbance rejection, and used to synthesize the landing flare logic for the 747 aircraft. The idea is that with a plant described by F, G,H, J having state $\mathbf{x}$ and output $y$ and with a given model described by A,B,C,D with state $\mathbf{z}$ and output $\eta,$to use the states $\mathbf{x}$ and $\mathbf{z}$ to construct a control signal so that   the error $\ y-\eta$  ‘quickly’ approaches zero. Rather than derive the design from first principles, we state the equations and demonstrate that they work. .Consider the plant described by

[TABLE]

and the model given by

[TABLE]

In this case, the model is driven by an impulse or essentially initial conditions only. Let the control be

[TABLE]

where $K$ is designed in the usual control law way so that $F-GK$ is a satisfactory stable control and the parameters $M$ and $N$ are selected so that

[TABLE]

\FRAME

ftbpFU4.0041in2.4898in0pt\QcbBlock Diagram for the Model Following Design\Qlbmodelfollowing2modelfollowing2.eps

With the given control law, Eq.(13)  the plant equations become

[TABLE]

In the transform domain, noting that $\mathbf{Z}(s)=(sI-A)^{-1}B,$ this can be written as

[TABLE]

Now substituting for $GN$ from Eq.14 and adding and substracting $sM$ this can be written as

[TABLE]

If we now multiply this out, the result is

[TABLE]

The output, $Y(s)=H\mathbf{X}(s)$ is thus

[TABLE]

Finally, as $HM=C,$ we have

[TABLE]

and therefore, in the time domain,

[TABLE]

which was what we set out to show.

V Examples

A set of MATLAB scripts (.m files) are listed in the Appendix to this note to aid in the study of these two alternative approaches to robust design. To illustrate the methods, two designs will be compared for a plant with the transfer function $\frac{\displaystyle 1}{\displaystyle s(s+1)}$, typical of a servomechanism. In one design, the system will be designed to track a reference sinusoidal input and in the other case to track a reference step and also reject a disturbance step. In both cases the control signal will be limited by a saturation value selected by the designer. The plant, a sterotypical servomechanism, is described by the equations:

[TABLE]

The Internal Model system is designed to have control poles at

-1.0000 + 2.0000i

-1.0000 - 2.0000i

-1.7321 + 1.0000i

-1.7321 - 1.0000i

and plant estimator poles at

-5.0000 + 8.6603i

-5.0000 - 8.6603i

Control poles for the Extended Estimator system are placed at

-1.0000 + 1.7321i

-1.0000 - 1.7321i

and the estimator poles are placed at

-1.7321 + 1.0000i

-1.7321 - 1.0000i

-3.0000 + 5.1962i

-3.0000 - 5.1962i

The choice of these poles is based on a combination of informed guess using the design overshoot and bandwidth. Later experiments would guide the final selection. In this case the results are given by the response curves plotted.

For the case of a system designed to track a sine wave having $\omega=1$, the outputs are plotted below.

\FRAME

fhFU3.1644in2.3549in0pt\QcbOutput responses of the two robust designs to a sine wave\Qlbsineoutputssineoutputs.eps

Notice in the next plot that the internal model uses less control and that the control for the Extended Estimator is strongly limited by the saturation.

\FRAME

fhFU2.9101in2.3575in0pt\QcbControl signals for the two designs in response to a sine wave\Qlbsinecontrolssinecontrols.epsThe errors are plotted below, showing how the Internal Model suffers from using less control

\FRAME

fhFU2.9308in2.3739in0pt\QcbError signals for the two designs in response to a sine wane\Qlbsineerrorssineerrors.eps

For the system designed to track a step, the outputs, are plotted below.

\FRAME

fbhFU2.7337in2.0747in0pt\QcbOutputs of the two designs to a step reference\Qlbstepoutputsstepoutputs.eps

In this case, the Extended Estimator controls really bounce around as a result of the saturation but the response seems quite reasonable.  \FRAMEfhFU2.8176in2.1197in0pt\QcbControls for the two designs to a reference step\Qlbstepvonttoldstepcontrols.epsFinally, the errors indicate comparable results. The results are a bit mixed as the IM design does a bit better tracking the reference while the Extended Estimator rejects the disturbance better.

\FRAME

fhFU3.1038in2.2995in0pt\QcbError signals for the two designs to a reference step\Qlbsteperrprssteperrors.eps  In the next example, the system that was designed to track a sine wave was given a step input. The resulting errors are plotted below. In this case, notice that while the extended estimator can use the implicit internal model of the plant and still gives zero final error, the IM design has a steady state error because the plant estimator has moved the plant pole away from the origin so it is no longer an internal model of the step..

\FRAME

fhFU3.5085in2.6498in0pt\QcbError responses of the systems designed to track a sine wave when the input is a step\Qlbsine2stepsine2step.eps

For comparison with the Model Following case, the systems designed to track a sinewave were run but with the plant purturbed to be

[TABLE]

The errors are plotted in Fig.LABEL:perror.

\FRAME

fhFU2.9819in2.4146in0pt\QcbErrors of the three designs with a purturbed plant.\Qlbperrorperrors.epsNotice that the Model Following design has the smallest maximum error but, being non-robust, has a persistent error while the other two designs continue the track the sine wave exactly.

VI Problems

Compute the zeros from the disturbance to the system error for the Internal Model case and explain why the plant pole at the origin does not act as an implicit internal model for step inputs in this case. 2. 2.

For the Extended Estimator case, is the plant integrator an implicit internal model or not? Why or why not?. 3. 3.

Repeat the IM design with the same plant but take $d(p)=p^{3}+p.$ You’ll need another pole to make the design and you are to place it at $s=-3.$  Show that this design for the IM case has an internal model for both $sin(t)$ and $1(t)$ inputs. 4. 4.

Repeat Problem 3 with the Extended Estimator design and compare the results with those of Problem 3 for both step and $\sin(t)$ reference and disturbance. 5. 5.

Replace the saturation with a gain, $K$, and plot the root locus with respect to $K$ for both design methods..  Do these loci suggest that the systems can be driven unstable for very large inputs?  Which will do so for the smaller value of $K$?