9 Performance Specifications

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SNU MAE Multivariable Control

9 Performance Specifications

9.1 Design Tradeoffs

Consider again the feedback system shown in Fig. 1¹.

r

−

- j - K u- j?di

up-

P - j?d y -

?j¾n 6

Figure 1: Standard feedback control configuration

For further discussion, it is convenient to define theinput loop transfer matrix Li andoutput loop transfer matrix,L0, as

Li=KP, L0=P K

respectively, whereLi is obtained from breaking the loop at the inputuof the plant whileLo is obtained from breaking the loop at the outputy of the plant.

Theinput sensitivity matrix is defined as the transfer matrix fromdi toup: Si= (I+Li)⁻¹, up=Sidi

Theoutput sensitivity matrix is defined as the transfer matrix fromd to y:

So= (I+LO)⁻¹, y=Sod.

Theinput andoutput complementary sensitivity matrices are defined as Ti=I−Si =Li(I+Li)⁻¹

To=I−So=Lo(I+Lo)⁻¹,

respectively. The word complementary is used to signify the fact thatT is the complement of S, T = I−S. The matrix I+Li is called the input return difference matrix and I+Lois called theoutput return difference matrix.

1Taken from Chap. 6, Essentials of Robust Control, by Zhou

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It is easy to see that the closed-loop system, if internally stable, satisfies the following equations:

y = To(r−n) +SoP di+Sod (1) r−y = So(r−d) +Ton−SoP di (2) u = KSo(r−n)−KSod−Tidi (3) up = KSo(r−n)−KSod+Sidi (4) These four equations show the fundamental benefits and design objectives inherent in feedback loops. For example, equation (1) shows that the effects of don the plant outputy can be made “small” by making the output sensitivity functionSo small. Similarly, equation (4) shows that the effects of di on the plant inputup can be made small by making the input sensitivity function Si

small.

The notion of smallness for a transfer matrix in a certain range of frequencies can be made explicit using frequency-dependent singular values. For example,

¯

σ(So)<1 over a frequency range would mean that the effects of disturbance d at the plant output are effectively desensitized over that frequency range.

Hence, good disturbance rejection at the plant output (y) would require that σ(So) = σ((I+P K))⁻¹= 1

σ(I+P K) (for disturbance at plant output, d) σ(SoP) = σ((I+P K)⁻¹P) =σ(P Si) (for disturbance at plant input,di) be made small. And good disturbance rejection at the plant input (up) would require that

σ(Si) = σ((I+KP)⁻¹) = 1

σ(I+KP) (for disturbance at plant input,di), σ(SiK) = σ(K(I+P K)⁻¹) =σ(KSo) (for disturbance at plant output,d) be made small, particularly in the low-frequency range whered anddi are usually significant.

Note that

σ(P K)−1 ≤ σ(I+P K)≤σ(P K) + 1 σ(KP)−1 ≤ σ(I+KP)≤σ(KP) + 1 then

1

σ(P K) + 1≤σ(So)≤ 1

σ(P K)−1, if σ(P K)>1 1

σ(KP) + 1 ≤σ(Si)≤ 1

σ(KP)−1, if σ(KP)>1

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These equations imply that

σ(So)¿1 ⇐⇒ σ(P K)À1 (5) σ(Si)¿1 ⇐⇒ σ(KP)À1 (6) Now supposeP andK are invertible:then

σ(P K)À1or σ(KP)À1 ⇐⇒ σ(SoP) =σ((I+P K)⁻¹P)≈σ(K⁻¹) = 1 σ(K) (7) σ(P K)À1or σ(KP)À1 ⇐⇒ σ(KSo) =σ(K(I+P K)⁻¹)≈σ(P⁻¹) = 1

σ(P) (8) Hence good performance at plant output (y) requires, in general,

- large output loop gainσ(Lo) =σ(P K)À1 in the frequency range whered is significant for desensitizingd (from (5)), and

- large enough controller gain σ(K) À 1 in the frequency range where di is significant for desensitizingdi (from (7)).

Similarly, good performance at plant input (u_p) requires, in general,

- large input loop gainσ(Li) =σ(KP)À1 in the frequency range wheredi is significant for desensitizingdi (from (6)), and

- large enough plant gainσ(P)À1 in the frequency range wheredis significant, which cannot be changed by controller design, for desensitizingd (from (8)).

Remark. In general, So 6=Si (whereas = holds for a scalar P). Hence, small σ(So) does not necessarily imply smallσ(Si); in other words, good disturbance rejection at the output does not necessarily mean good disturbance rejection at the plant input.

Hence,good multivariable feedback loop design boils down to achieving high loop (and possibly controller) gain in the necessary frequency range.

Despite the simplicity of this statement, feedback design is by no means trivial. This is true because loop gains cannot be made arbitrarily high over arbitrarily large frequency ranges. Rather, they must satisfy certain performance tradeoff and design limitations.

• commands and disturbance error reduction vs. stability under the model uncertainty

Assume that the plant model is perturbed to (I+4)P with4stable, and assume that the system is nominally stable(i.e., the closed-loop system with4= 0 is stable). Now the perturbed closed-loop system is stable if

det(I+ (I+4)P K) = det(I+P K)det(I+ (I+P K)⁻¹4P K)

= det(I+P K)det(I+4P K(I+P K)⁻¹)

= det(I+P K)det(I+4To)

has no right-half plane zero. Here we used the property det(I+M N) = det(I+N M). This would, in general, amount to requiring thatk4Tokbe

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small or that σ(To) be small at those frequencies where4 is significant, typically at high-frequency range, which, in turn, implies that the loop gain,σ(Lo), should be small at those frequencies.

• disturbance rejection vs. the sensor noise reduction

Largeσ(Lo(jw)) values over a large frequency range make errors due to d small. However, in equation (1), they also make errors due ton large because this noise “passed through” over the same frequency range, that is,

y=To(r−n) +SoP di+Sod≈(r−n) Note thatn is typically significant in the high-frequency range.

• large loop gains vs. the magnitude of control

Large loop gains outside of the bandwidth of P - that is, σ(L_o(jw))À1 or σ(Li(jw))À1 whileσ(P(jw))¿1 - can make the control activity (u) quite unacceptable, which may cause the saturation of actuators. This follows from

u=KSo(r−n−d)−Tidi=SiK(r−n−d)−Tidi≈P⁻¹(r−n−d)−di

Here, we have assumed P to be square and invertible for convenience.

The resulting equation shows that disturbances and sensor noise are ac- tually amplified at u whenever the frequency range significantly exceeds the bandwidth ofP, since forw such thatσ(P(jw))¿1 we have

σ[P⁻¹(jw)] = 1

σ[P(jw)]À1

• Similarly, the controller gain, σ(K), should also be kept not too large in the frequency range where the loop gain is small in order not to saturate the actuators. This is because for small loop gain σ(Lo(jw)) ¿ 1or σ(Li(jw))¿1

u=KSo(r−n−d)−Tidi≈K(r−n−d)

Therefore, it is desirable to keepσ(K) not too large when the loop gain is small.

To summarize the above discussion, we note that good performance requires in some frequency range, typically some low-frequency range (0, wl),

σ(P K)À1, σ(KP)À1 σ(K)À1

and good robustness and good sensor noise rejection require in some frequency range, typically some high-frequency range (wh,∞),

σ(P K)¿1, σ(KP)¿1, σ(K)≤M

whereM is not too large. These design requirements are shown graphically in Fig. 2. The specific frequencieswl andwh depend on the specific applications and the knowledge one has of the disturbance characteristics, the modeling uncertainties, and the sensor noise levels.

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ω_h

logω

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AQ QQ

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¯ σ(L)

σ(L) XXXXX

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Figure 2: Desired loop gain

9.2 Weighted Performance

We saw that the performance objectives of a feedback sys can be specified by conditions onS or T, etc. Suppose that we have the following condition on a scalar system:

|S(jω)| ≤

½ ² ∀ω≤ω0

M ∀ω > ω0

whereS(jω) =1+P(jω)K(jω)¹ . Equivalently,

|We(jω)S(jω)| ≤1 ∀ω

where the weighting function (usually a rational transfer ftn) satisfies

|We(jω)| =

½ 1/² ∀ω≤ω0

1/M ∀ω > ω0

This idea is generalized to weighted performance specifications in MIMO control design. Advantage of using weighted performance includes:

• Some components of a vector signal might be more important than others.

• Each component might be in different units or orders of magnitude.

• We might be interested in error rejection in low freq range only, and interested in other objectives in other freq range, etc.

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For example, look at the figure 3. Weighting ftns can be chosen to reflect the design objectives and knowledge of the disturbances and sensor noise, expected system inputs, and the relative importance of the outputs. Then, we choose a contoller K such that certain weighted signals are made small in some sense, such asH₂ orH_∞.

-Wr r

−

- j - K u- j?d_i up-

P 6

6 ? ?

Wu Wi Wd

˜

u d˜i d˜

- j?d y -

?j¾n 6

We

Wn

-

¾ e

˜ n Figure 3: Closed-loop system with weighting blocks

H2 Performance.

For simplicity, assume di = 0n = 0, and ˜d(t) = ηδ(t) with Eηη^∗ = I, i.e.

impulse noise with random input direction.

• Want to minimize the expected energy of the error due to ˜d:

E||e||²₂=E Z _∞

0

||e||²dt=||WeSoWd||²₂

• Don’t want to saturate actuators:

E||˜u||²₂=||WuKSoWd||²₂ Thus, our perf criterion would look something like

E[||e||²₂+ρ||˜u||²₂] =

¯¯

· √WeSoWd

ρWuKSoWd

¸¯¯¯

¯

¯¯

2

ρ: tradeoff btween disturbance rejection at the output and control effort.

But remember that LQG formulation allows only additive noise, and the plant uncertainty cannot be formally addressed.

H∞ Performance.

• Want the tolerance to uncertainties by limiting high loop gains. e.g. minimize

sup

||d||˜2≤1

||e||2 =||WeSoWd||∞

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subject to some restrictions on sup

||˜u||2≤1

||˜u||₂ =||W_uKS_oW_d||_∞.

Thus,

sup

||d||˜2≤1

{||e||²₂+ρ||˜u||²₂} =

¯¯

· √WeSoWd

ρW_uKS_oW_d

¸¯¯

¯¯

2

∞

• Or, want to limit the weighted complementary sensitivity function:

¯¯

· √WeSoWd

ρW1ToW2

¸¯¯

¯¯

∞

.

9.3 Selection of Weighting Functions

Selection of weighting functions is very important and it is not trivial. Let’s consider a 2nd order SISO example.

L = P K = ω_n² s(s+ 2ξωn)

S = 1

1 +L = s(s+ 2ξωn) s²+ 2ξωns+ω_n² and recall that

• speed of the response!ωn

• overshoot of the response!ξ How to choose our weighting functions?

• Recall the tracking error Eq. (2) :

r−y=S(r−d) +T n−SP di .

We want to keep |S| small over a frequency range where r and d are significant. How to chooseWe?

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• Recall Eq. (3) :

u=KS(r−n−d)−T di.

We want |KS| to roll off as fast as possible beyond the desired control bandwidth so that the high-freq noises are attenuated as much as possible.

How to chooseWu?