Part III Performance Limitations in Feedback Control with
8.4 Stochastic SISO Performance Limitations of RRS
analysed in this book, and, therefore, in the next section we consider the problem from a stochastic point of view.
8.4 Stochastic SISO Performance Limitations of RRS
In the previous section, we discussed the implications of NMP dynamics from a deterministic point of view. In this section we take the stochastic approach.
8.4.1 Limiting Optimal Control Performance Limitations
An optimal control problem (OCP) is that of obtaining a control law u(x) belonging to a class of admissible controlsU that minimizes a cost2associated with a system ˙x=f(x,u). The cost is usually a function of x, u, andλ (a vector of weighting parameters),i.e.V(x,u,λ). The optimal control problem can then be posed as
OCP : min
u V(x,u,λ)
subject to u∈ U and x˙ =f(x,u).
Then, we say that uopt is the optimal control if
V(x,u,λ)≥V(x,uopt,λ) ∀ u∈ U. We also denote the optimal control as
uopt= arg min
u∈U V(x,u,λ), and the optimal cost
Vopt(x,λ) = min
u∈UV(x,u,λ).
We say that an optimal control is a limiting optimal control if, in the limit as some of the components ofλtend to zero, the cost to be minimisedV(x,u,λ) becomes a function of either the state or the control action but not of both.
The use of optimal control to analyse performance limitations falls in the second type of results that this analysis yields. The results in the previous section showed that there are constraints and trade-offs associated with the dynamics of the plant that hold for every controller under the assumption that the closed loop is stable, and the architecture is that shown in Figure 8.1. In this section, we will analyse what is the best achievable performance. We do this usingcheap limiting optimal control, or simplycheap control.
2The cost is a mathematical expression that characterises deviations from the desired performance or a desired outcome; hence, should be minimised.
162 8 Linear Performance Limitations
Cheap Control is a case of limiting optimal control in which the weight of the control effort in the cost is (or tends to) zero. Since the control is cheap, there is no cost associated with using large control action—a high gain control. As previously mentioned, this setting is obviously unrealistic, but the fact that under these conditions the cost cannot be reduced to zero evidences fundamental limitations. Moreover, the cost obtained can be used as a benchmark to compare the value of the costs resulting from the application of other control strategies. Therefore, the result are a valuable tool for analysis.
In this section, we are interested in the following control problem.
Problem 1 (Minimum Variance Cheap Control Problem SISO case) For the SISO plantG(s)and the control scheme shown in Figure 8.1, we seek the proper stabilising controllerC(s)that minimises
V = lim
→0var[y] +var[u]. (8.21) We assume that the only disturbance acting on the system is the output dis- turbance d0, which is a Wide Sense Stationary (WSS) stochastic process. We further assume that d0 has a power spectral density Sdd(ω)that admits a ra- tional approximation so it can be represented as filtered white noise n with constant PSD Snn∀ω∈(−∞,∞), i.e.,
Sdd(ω)≈|H(jω)|2Snn, (8.22) where the filter H(s) is a rational stable transfer function. ◦ ◦ ◦ The problem defined above is one particular version of the so-called cheap optimal control problem. Note that as the weight in the control effort—
hereby represented as the variance of the control signal—tends to zero, there is no penalty on how much control is used.
For stable plants with one NMP zero, we have the following result that quantifies the best achievable performance in terms of the cost (8.21).
Theorem 8.2.Let G(s)be a stable SISO plant with one NMP zero, say q∈ R+. Then, the minimum value of the Cost (8.21)is
minV = 2q|H(q)|2Snn (8.23) or equivalently
var[y]≥2q|H(q)|2Snn (8.24)
◦ ◦ ◦ Proof. Since G(s) is stable, we can use the following form of the Youla pa- rameterization of all stabilising controllers [89]
C(s) = Q(s)
1−G(s)Q(s), (8.25)
8.4 Stochastic SISO Performance Limitations of RRS 163 where Q(s) is any stable, real, and rational transfer function. From this pa- rameterisation, it follows that
S(s) = 1−G(s)Q(s). (8.26)
Using the Fourier transform pair between autocorrelation and the power spec- tral density, the variance of the output can be expressed as
var[y] =Ryy(0) = 1 2π
∞
−∞
Syy(ω)dω
= 1 2π
∞
−∞|H(jω)|2|1−Q(ω)G(jω)|2Snndω
= Snn
2π ∞
−∞|H(jω)−Q(ω)H(jω)G(jω)|2dω
=SnnH(jω)−Q(ω)V(jω)22,
(8.27)
where in the last step we have made use of theH2 norm with
V(jω) =H(jω)G(jω). (8.28)
Let us factorise V(jω) as follows:
V(jω) = (H(jω)Gm(jω)) s−q
s+q
=Vm(jω)Va(jω), (8.29) where Gm(s) and consequently Vm(s) have no poles or zeros in the closed right-hand side of the complex plane (C+), and Va(s) is an all-pass term.
Using this factorisation, expression (8.27) can be manipulated as follows:
var[y] =SnnVa(jω)[Va(jω)−1H(jω)−Q(jω)Vm(jω)]22
=SnnVa(jω)−1H(jω)−Q(jω)Vm(jω)22
=Snn[Va(jω)−1H(jω)]u+ [Va(jω)−1H(jω)]s−Q(jω)Vm(jω)22, (8.30) where we have used the fact that|Va(jω)|= 1, and separatedVa(jω)−1W(jω) into its unstable and stable parts indicated by the subscriptsuandsrespec- tively.
Since [Va(jω)−1H(jω)]u ∈ H⊥2 and [Va(jω)−1H(jω)]s−Q(jω)Vm(jω)∈ H2, they are orthogonal. Then, we can express (8.30) as
var[y] =Snn[Va(jω)−1H(jω)]u22+Sww[Va(jω)−1H(jω)]s−Q(jω)Vm(jω)22, (8.31) from which it follows that theQ(s) that minimises the output variance is
Q(s)opt=Vm(s)−1[Va(jω)−1H(jω)]s, (8.32)
164 8 Linear Performance Limitations
and the minimum value of the output variance is
minvar[y] =Snn[Va(jω)−1H(jω)]u22. (8.33) The final result follows from (8.33) noting that
Va(s)−1H(s)
u=
s+σq s−σq
H(s)
u
= Resq
(s+σq s−σqH(s)
) s−σq
=2σqH(σq) s−σq
. Note thatQ(s)optin (8.32) would be improper becauseH(s)G(s) is usually strictly proper, and thusVm−1(s) is improper. This is not a problem since we can always make a proper approximation by adding stable fast poles. This, however, implies that the second term in the right-hand side of (8.30) can be made arbitrarily small as the fast poles go to infinity, but it will not vanish.
Therefore, in this case the “min” should be replaced by “ inf” in Expression (8.23), and the inequality (8.24) holds strictly. Since we will use the result of this theorem as a performance benchmark rather than a tool for synthesizing the controller, this issue shall not be of concern.
The result given in Theorem 8.2 is a novel characterisation, in stochastic terms, of the trade-offs given in the previous sections using integrals on the sensitivity function. Indeed, in the deterministic framework, we have seen that the Poisson integral formula evidences a trade-off between sensitivity reduc- tion at some frequencies and increase at others. We have also seen that this trade-off worsens when the sensitivity reduction is to be achieved at frequen- cies close to the NMP zero. The same effect is now quantified in the result of Theorem 8.2, in which the optimal controller tries to eliminate the effect of the disturbance over the output. However, the ability to do this is limited when the bulk of energy of the disturbance is concentrated close to the NMP zero (i.e.when the PSD or|H(s)|has large values ats=q).
Finally, it is well known that a NMP zero close to the imaginary axis limits the performance of a regulator with step output disturbances [184] [48]. The result can be inferred from (8.24), since the disturbance has power at low frequency. However, it follows from (8.24) that a NMP zero at low frequency may not represent a difficulty in general.
8.4.2 Stochastic SISO Results and RRS
If (8.24) is used in the context of RRS, we could say that
E[φ2]≥2q1Sφφ(−jq1), (8.34) if the power spectrum of the roll wave-induced motion can be obtained by filtering white noise:
Sφφ(ω)≈ |H(jω)|2Snn, (8.35)
8.5 Optimal Roll Reductionvs.Yaw Interference Trade-off 165