Automatic Control (TSRT15): Lecture 9

Tianshi ChenDivision of Automatic ControlDept. of Electrical EngineeringEmail: tschen@isy.liu.sePhone: 13-282226Office: B-house extrance 25-27

2Summary of the last lecture

Control Synthesis: Given the Bode plot of G(s), design F(s) to ensure that F(s)G(s) satisfies the design specifications on the open-loop system.

More specifically, design F(s) to place the gain crossover frequency (i.e. |F(iω)G(iω)|=1 at desired bandwidth), obtain a suitable phase margin(typically between 45º and 60º) at this frequency, and obtain a suitably largestatic gain at the frequency 0.

F(s)

-1

Σ

G(s)

3Summary of the last lecture

Desired phasemargin 50º

Desired crossoverfrequency:10 rad/s

Desired e0=0

4Summary of the last lecturePhase-lead compensator

β computed from phase required

τD computed to achieve maximum phase lead at the desired crossover frequency

K from |FleadG|=1 at the desired crossover frequency

5Summary of the last lecture

Phase-lag compensator

Example: To get e0=0 the loop-gain must have an integrator, hence γ=0.

Note that the phase-lag compensator changes the gain and phase slightly at the desired crossover frequency.

6Outline

Limitations in feedback control design

Robustness

A more general control structure

7Limitations in feedback control design

Σ

G(s)F(s)

-1

R(s) Σ Z(s)

U(s): Limited input. The input signals cannot be arbitrarily large. (e.g., limited engine torque, pump capacity etc).

V(s): Model errors and disturbances. Ideally, the output is G(s)U(s), butthe actual output is Z(s) = G(s)U(s) + V(s).

N(s): Measurement error. In parctice, the measurement Y(s) is not exact and is actually Y(s)+N(s).

ΣU(s)

N(s)

V(s)

Y(s)

8Input limitations

Block scheme calculations give

If we have made F(s)G(s) large, we have

I.e., if we have made the gain of the open-loop system large (in a frequency region) then the input signals will be large if G(s) is small.

9Disturbances

The disturbance is an additive signal V(s) affecting the output.

V(s) is thus the difference between the actual output and the ideal output given the input U(s)

Block scheme calculations give

To reduce the influence from disturbances we must make the sensitivity function S(s) small, i.e. the loop-gain GO(s)=F(s)G(s) must be large.

10Measurement errors

The transfer function T(s) that satisfies T(s) + S(s) = 1

T(s) is called complementary sensitivity function

Block scheme computations give

11Complementary sensitivity function

One of the central issues in control theory arises:

We cannot design a control system such that it is arbitrarily robust against disturbances and measurement errors in the same frequency region!

If |S(iω)| is small at a frequency (disturbances with that frequency are damped well), then |T(iω)| will be close to 1 (measurement errors with that frequency will not be damped).

Intuitively reasonable: If we have a good model, we must trust our modeland punish measurement errors. If we have a bad model, we may trust our measurements. If we have a bad model and bad measurements, we have no idea what is happening.

12Bode's sensitivity integral

S(s)+T(s)=1 says that we must have a compromise between the influence from disturbances and measurement errors.

Unfortunately, it is even worse. The sensitivity function cannot be arbitrarily small everywhere, it must satisfy Bode’s sensitivity integral.

Assume the loop-gain has m unstable poles p1,p2,…,pm and at least two more poles than zeros, and the closed-loop system is stable. It then holds that

This fact is known as the “waterbed effect”: This conservation law shows that to get lower sensitivity in one frequency range, we must get higher sensitivity in some other region.

13Waterbed effect

Pushing down the sensitivity function (disturbances better damped) gives a negative addition to the integral.

The sensitivity function must then increase in some other region (disturbances are amplified).

14Robustness

The models we use are only approximations of reality.

Model errors often exist (uncertain mass, unknown phenomena, nonlinear effects, time-delays in computations etc).

Our stability and performance calculations are however based on precise properties of the model (Bode plot of the open-loop system etc)

What happens if the model is subject to model error? Can we still guarantee the stability? Can we still guarantee the performance?

Robustness: The capability to keep a system’s stability and performance under perturbations or uncertainties when designing a controller.

15Robustness

We must make some kind of assumptions about the model errors, in order to be able to say something!

We assume there is a relative model error Δ(s) which we do not know

The actual system is thus G(s)(1+ Δ(s))

G(s)F(s)

-1

R(s) Σ ΣY(s)

Δ(s)

16Levitating ball

We approximated the model for the levitating ball with a double-integrator (i.e. a simple mass-force system)

y(t)

u(t)We do not know the mass exactly but have

The actual transfer function can after some manipulations be written as

17Stability with model errors

Assume we have designed a controller F(s) based on a model G(s)

We now use this controller on the true system G0(s)=G(s)(1+ Δ(s))

Phase and gain margin requirements tells us that the frequencies at which the gain of the open-loop system is 1 and phase is -180° is critical for stability.

In the stability limit we thus have that the open-loop system is equal to -1

This can be written as

18

Theorem: Assume G(s) and G0(s) have the same number of poles in the closed-right half plane, and F(s)G(s) and F(s)G0(s) tends to 0 when |s| goes towards infinity. A sufficient requirement for stability is

Robust stability criteria

A sufficient robust stability criteria is given by

In other words, the model error must be small in the frequency region where the complementary sensitivity function T(s) is large.

Given a feedback controller F(s) stabilizing a model G(s). Let T(s) denote the corresponding complementary sensitivity function and G0(s) = G(s)(1+ Δ(s)) denote the actual system.

19Levitating ball

y(t)

u(t)

Nominal model with

Controller based on nominal model(PD with approximate differentiation)

Complementary sensitivity function

20Levitating ball

|T(iω)|max = 1.172

20log|T(iω)|

Robst stability can be guaranteed!

21Robust performance

Let Y(s) be the ideal output with Δ(s)=0 and Y0(s) be the actual output obtained from G(s)(1+ Δ(s)). Then we have

G(s)F(s)

-1

R(s) Σ Σ

Δ(s)

where S0(s) is the actual sensitivity function from G(s)(1+ Δ(s))

22Robust performance

If we assume the ideal sensitivity function S(s) and the actual sensitivity function S0(s) to be close, the robustness of the closed-loop behavior can be seen to be dependent on the ideal sensitivity function S(s).

In the regard, the smaller S(s) the smaller due to

It is thus impossible to design the closed-loop system arbitrary:

For robust stability, a small complementary sensitivity function is required;

For robust performance, a small sensitivity function is required.

23More general control structure

Feedforward and feedback controllers (compensators)

Σ

G(s)R(s) Σ Z(s)ΣU(s)

N(s)

V(s)

Y(s)

Fr(s)

-Fy(s)

The closed-loop system and the complementary sensitivity function can be designed (almost) separately

24Container crane

Phase-lead and phase-lag compensators have been designed suchthat the loop-gain satisfies the desired phase margin and gaincrossover frequency.

25Container crane

T(iω) S(iω)

The disturbances are amplified in the region 1-3 rad/s, but less than 5dB.

26Container crane

The step-response was not perfect, too much oscillations

27Container crane

We solve this by using a separate feedforward compensator

K such that Gc(0)=1

All robustness properties of the original phase-leadand phase-lag compensator are kept, since T(s) and S(s) stay the same.

28Another example

It turns out that our model is probably wrong. We have missed some resonant frequencies

29Another example

Stability is still guaranteed under the robustness criterion!

The model is poor at high frequencies, giving a bandwidth limit on T(s).

30Summary of today’s lecture

Disturbances, model errors, input limits and measurement errors are the factors that limit the performance of a control system.

It is fundamentally impossible to design a controller which is disturbance and measurement error insensitive in the same frequency region.

Disturbance rejection can not be done arbitrarily good, even with perfect measurements, due to Bode’s sensitivity integral.

For robust stability, a small complementary sensitivity function is required

For robust performance, a small sensitivity function is required.

Controllers with 2 degrees of freedom make it possible to design the complementary sensitivity function and the closed-loop system separately.

31Summary of today’s lecture

Complementary sensitivity function T(s): T(s) + S(s) =1

Robustness: The capability to keep a system’s stability and performance under perturbations or uncertainties when designing a controller.

Important concepts