One possible solution is
∆c(∆) =−W22+
W21 V21
U11 (W11+ ∆)T W11+ ∆ V11
U12 W12
Finally, controller parametersAc, Bc, Cc, Dc can be computed in the same way as in case of robust controllers in the previous section.
Chapter 4
Uncertainty modelling and robust control design for LTI systems
The analysis and synthesis procedures of robust controllers summarized in Sections 3.2 and 3.3 start from an a priori fixed augmented nominal model (denoted byGin Figure 3.3). So far we have not discussed on how to obtain this augmented model.
A typical procedure starts with the nominal plant model (denoted by U23 in the subsequent sections) that can be a result of some identification method or physical modelling. Based on physical considerations, the nominal model is augmented to de-scribe the relation to neglected dynamics and effects of disturbances (thus we get U of subsequent sections). The structure of neglected dynamics is also specified by defining S∆u. The most risky task is to specify – possibly frequency-dependent – bounds for the perturbation blocks (denoted by W∆) and disturbances (denoted by Wd). Without a careful choice of the uncertainty bounds, the set of assumed uncertainties can be too large causing conservatism of the designed controller, or too small bearing the risk of closed-loop instability. Determination of disturbance sets is a similarly effortful task and requires much physical insight into the system and the control problem. The next step is to further augment uncertain systemU to define the closed-loop interconnection.
New performance inputs (e.g. reference signals) can be added and performance outputs, which should be small, must be defined. All new inputs are normalized by weighting functions which are joined to augmented system G. The next effortful task is to chose penalizing weighting functions for the performance outputs. These weighting functions influence e.g. control effort and its frequency power distribution, tracking error and its frequency distribution, disturbance attenuation etc. In this way the control objectives and the model-reality relation can be specified and built into the generalized plant G.
There exist approaches supporting the selection of performance weighting functions when all other parameters are fixed, see e.g. [74], however, the design problem of struc-tured uncertainty weighting functions W∆, Wd is not automated yet. The modelling of uncertainty can be one of the most time-consuming task: it involves many experiments, trial and error settings of all weighting functions. In this chapter a design method is presented for uncertainty weighting functions with the assumption of fixed performance weighting functions. The presented algorithm can drastically decrease the necessary intuitive and heuristic design steps.
FOR LTI SYSTEMS
A natural idea for determining uncertainty sets is to use measurement data. One possibility to support weight selection is based on model validation methods [108, 123, 25, 28, 79]. One approach is that several weighting functions are fixed (e.g. by assuming known disturbance bounds) and the others of equal bounds are tuned to get a consistent model with the given data. Many approaches appeared in the literature to identify unstructured uncertainty sets applicable for robust control. Recently, the criterion of modelling is connected to the achievable control performance.
In the chapter, a concept is presented for modellingstructured uncertainties directly applicable for the synthesis techniques of the former chapter. No a priori information is assumed on the disturbances. BothW∆andWdare designed based on a criterion which is exactly the same as for the control design. The inputs of the presented algorithm are the nominal model of the system, the structure of the uncertainty model, and the specification of robust performance. The algorithm is to deliver the weighting functions of the uncertainty model and disturbances, and a controller satisfying robust performance specifications.
The overall algorithm reveals a double iterative solution. Since the design variables are involved in a non-convex optimization problem both in robust output-feedback con-trol and in joint modelling and concon-trol problems, the search for the variables is usually solved by iteration of convex steps. These steps compose the inner iteration loop. The outer loop involves measurement data of experiments. New controller may generate data which invalidate the actual model. A controller based on an invalid model may destabilize the plant. For that reason new experiments are required to test new con-trollers and to gain more information on the system.
The proposed method is built on the following three pillars:
• the uncertainty set is constrained by model validation conditions
• the uncertainty set is shaped based on robust performance specifications
• the uncertainty set and the robustness of the controller are iteratively improved based on new experiments
The first two items serve to ensure the minimality and optimality of uncertainty sets when a controller is given. The third item is responsible for handling unstable experi-ments and improving robustness.
A by-product of the algorithm that may have a self-contained value is the elabora-tion of a skewµsynthesis procedure that can easily be implemented based on existingµ synthesis methods available in Matlab [13]. Note that skewµanalysis tools are already available in Skew Mu Toolbox for Matlab, [41].
A simple numerical example presented in Appendix A is referred to in this chap-ter in order to demonstrate the key steps of the proposed algorithm through analytic calculations.
4.1. PROBLEM FORMULATION
4.1 Problem formulation
4.1.1 System configuration
The true plant to be controlled and which generates the data is described by operator T
y=T( ¯d, u), T ∈ ST, d¯∈ Sd¯⊂ L2, (4.1) where y and u denote the measured outputs and control inputs, respectively, d¯is a physical disturbance vector of an undetermined set and physical system T is a multi-variable stable system affected by variations of physical parameters and operating point changes which is expressed by the undetermined set of stable systems ST. No a priori information is assumed on the size of the disturbances and the variation of the system’s dynamics. Note that T is not required to be linear, it is only assumed that in finite
∆
Wd wu
-zu
wu0
d0
u ˆ
y
W∆
d
U =
U11 U12 U13
U21 U22 U23
Figure 4.1: Uncertain system model
time experiments the input-output behavior can be described with the help of a fictive disturbance d0∈ Ln2d and by an LFT of a given LTI model U ∈ RHn∞zu+ny×nwu+nd+nu and a structured dynamic uncertainty ∆0 in the form
zu ˆ y
=
U11 U12 U13 U21 U22 U23
| {z }
U
wu0
d0 u
, wu0 = ∆0zu, (4.2)
see also Figure 4.1. The unknown dynamics ∆0 ∈ S∆u belongs to a set of stable perturbations
S∆u ={∆∈ RHn∞wu×nzu|∆ = diag{∆1, . . . ,∆τ},∆i ∈ RHn∞wu,i×nzu,i, i= 1, ..., τ}(4.3) BlockU23∈ RHn∞y×nu is the nominal model, which can be the result of an identification method for restricted complexity models. Other blocksUij describe the interconnection structure of the nominal model and the structured uncertainty. The uncertainty can be normalized by weighting functions
W∆= diag{W∆,1Iwu,1, . . . , W∆,τIwu,τ} ∈ RHn∞wu×nwu
Wd= diag{w1, . . . , wnd} ∈ RHn∞d×nd
FOR LTI SYSTEMS
such that for all∆0 ∈ S∆uthere exists a perturbation ∆and for alld0 ∈ L2there exists a normalized disturbancedsuch that
∆0 =W∆∆, ∆∈BS∆u, d0 =Wdd, d∈BL2
The feedback signals wu := ∆zu = [wu,1, ..., wu,τ]T and zu = [zu,1, ..., zu,τ]T are parti-tioned according to block structure S∆u.
4.1.2 Experiments and model consistency
Information about the true system is gained by taking experiments: the system is ex-cited by input signalu(t)which can be either a predefined signal (open-loop experiment) or the output of a controller (closed-loop experiment). The inputs and outputs of the true system are measured and stored. We assume that the sampling is fast enough to have a good sampled data representation of the signals, i.e., the Shannon conditions hold. Then, Discrete Fourier Transform (DFT) can be applied, possibly with a window function, to compute the frequency-domain representation of the signals. Many text-books are dealing with the topic of sampling and Fourier transform, in the sequel we does not concern signal processing problems of this kind.
Suppose that the input-output data of system T in the lth experiment are given in the frequency domain by El = {(yl(jωk), ul(jωk))nk=1ω }, where nω is the number of frequency samples. The data containing N experiments are denoted byEN =SN
l=1El. Definition 4.1 (Consistency) The uncertain system (4.2) with weighting functions W∆ and Wd is consistent with respect to data set El if the following condition holds.
There exists a perturbation ∆l ∈ BS∆u and a disturbance dl ∈ BLn2d such that measured output yl is exactly reproduced by model output yˆl, i.e., for all frequencies k= 1, ..., nω
ˆ
yl(jωk) =yl(jωk) (4.4)
where ˆ
yl(jωk) :=FU(U(jωk), W∆(jωk)∆l(jωk))
Wd(jωk)dl(jωk) ul(jωk)
The model isunfalsified if it is consistent with all experiments EN.
It is required of the model structure to be able to represent the true system in all experiments. Certain rank conditions on systemU, detailed in Section 4.2, ensure the satisfaction of
Assumption 4.1 There exist a W∆ and a Wd such that the model is consistent with all experiments.
4.1. PROBLEM FORMULATION
K G
T
-¯
z
pd ¯
y
Ku
K¯ r
Figure 4.2: Closed-loop system configuration with true plant 4.1.3 Control performance
In modern control theory, the requirements on the control performance are defined by drawing the closed-loop control configuration. New inputs, outputs and weighting functions are introduced to build an augmented closed-loop system whose input-output mapping is to minimize in some chosen norm.
In order to specify the objectives of the design, true system T is augmented and placed in a closed-loop configuration as in Figure 4.2. New inputs, denoted by ¯r, con-taining known or measurable signals, e.g., reference inputs, can be added. Performance outputs, denoted by z¯p, which should be small during the control, must be defined.
SignalsyK and uK are defined as the inputs and outputs of controller K. They are not necessarily equal to yandu, respectively. Augmented plantGT contains systemT and weighting functions, which specify performance signal z¯p and normalize outer signals r¯ and d.¯
The control objective is to minimize
¯
γ(K) := sup
T ∈ST,d∈S¯ d¯,¯r∈BL2
kz¯pk2, z¯p =FL(GT, K)( ¯d,r).¯ (4.5) Since true plant T and sets ST and Sd¯are not known in advance, the controller will be designed based on model (4.2) which must be augmented in the same way, resulting in augmented plant G0 and the configuration in Figure 4.3. Without loss in generality, systemG0 and input ¯r can be scaled such that we can calculate with wpT := [dT rT]∈ BLn2d+nr instead of d∈BLn2d,r¯∈BLn2r.
In order to minimize the control objective (4.5) by calculations based on model (4.2), we will need the following assumption on performance specification.
Assumption 4.2 The consistency of the model (4.2) with respect to closed-loop data (yl(jωk), ul(jωk)) implies the consistency of the augmented plant model in Figure 4.3 with respect to data
zp,l(jωk) yK,l(jωk)
,
rl(jωk) uK,l(jωk)
and that zp,l(jωk) = ¯zp,l(jωk).
Assumption 4.2 expresses that plant augmentation preserves the consistency property.
We can summarize the problem statement as follows.
Problem 1. Given an unknown systemT, an LTI plant modelU of fixed structure for uncertainty and control specification defined by G0. Provide an iterative uncertainty model unfalsification and robust control design procedure that 1. constructs a consistent
FOR LTI SYSTEMS
∆
W∆
K G0
zu wu
Wd zp
-r
yK uK
d
d0 wu0
wp
Figure 4.3: Closed-loop system configuration with the model for robust control design model (U, W∆, Wd) for the true plant, based on closed-loop experiments, and 2. designs a H∞ controller K that minimizes objective (4.5).