Chapter 8 Robust performance problems In the previous chapters we have presented synthesis methods for optimal H 2 and H1 control problems, and studied the robust stabilization problem with respect to both unstructured and structured normbounded uncertainties. However, in a realistic practical controller design problem the controller is in most cases required to satisfy both performance and robustness criteria. This leads in general to more complex design procedures than the synthesis methods described in the previous chapters. In this chapter some approaches which are used will be described. We consider the control system in Fig. 8.1. Here æ denotes a normbounded uncertainty, which may be structured or unstructured. Denote the closedloop transfer function from to by F P èp; K; æè, such that = F P èp; K; æè è8.1è æ v æ u P z æ y K Figure 8.1. holds for the control system in Fig. 8.1. The control objective is to minimize a performance measure JèF P èp; K; æèè related to the closedloop transfer function. Typically, the cost JèF P è denotes the H1 norm or èthe square ofè the H 2 norm. In contrast to the optimal control problems studied in chapters 4 and 5, the actual value of the cost depends on the uncertainty æ, which is unknown. Therefore, some assumptions on the 61
uncertainty should be made in the formulation of an optimal control problem for the uncertain plant. A minimum requirement of any controller for the uncertain plant in Fig. 8.1 is, of course, that it be robustly stabilizing. There are two main formulations of an optimal control problem for uncertain plants: robust performance optimization and nominal performance optimization subject to robust stability. In the robust performance problem the worst cost obtained in the assumed uncertainty set is minimized. The general robust performance problem is deæned as follows. Robust performance problem. Find a robustly stabilizing controller for the uncertain plant depicted in Fig. 8.1, which minimizes the worstcase cost J worst èp; Kè deæned as èunstructured uncertaintiesè, or J worst èp; Kè = sup n o JèF P èp; K; æèè : kæk1 æ J worst èp; Kè = sup n JèF P èp; K; æèè : æ 2 æ s èæè o è8.2è è8.3è èstructured uncertaintiesè, where æ s èæè denotes the structured normbounded uncertainty set è7.8è. A controller designed for robust performance which achieves a given performance bound such that J worst èp; Kè éæ, guarantees a cost less æ for all normbounded uncertainties. There are systematic controller synthesis procedures for both the robust H1 and the robust H 2 performance problems. The robust H1 performance problem can be shown to be equivalent toasynthesis problem. The robust H 2 performance problem is more complex, as it mixes two diæerent system norms; the H 2 norm associated with performance, and the H1 norm associated with robustness. This leads to a mixed H 2 =H1 problem, for which special solution methods have been developed. The worstcase nature of the costs in è8.2è and è8.3è may, on the other hand lead to a conservative design for 'average uncertainties', which are not worstcase with respect to the cost JèF è. For this reason, an alternatve formulation of the optimal control problem for uncertain plants may be stated as follows. Nominal performance problem subject to robust stability. Find a robustly stabilizing controller for the uncertain plant depicted in Fig. 8.1, which minimizes the nominal cost JèF P èp; K; 0èè deæned for the nominal plant with æ=0. Here, the nominal cost is minimized, and the uncertainty is taken into account only by requiring robust stability. A potential problem with this formulation is that even though nominal performance may be good, there is no guarantee of acceptable performance when there are uncertainties æ 6= 0. The nominal performance problems subject to robust stability are in general more complex to solve than the robust performance problems. In particular, they lack closed solutions, and must be solved by numerical optimization techniques. See Pensar è1995è for an extensive treatment of the problems involved. A possible method for both avoiding too conservative design and ensuring acceptable overall performance for all uncertainties could be to combine the robust performance 62
and the nominal performance problems by minimizing a combination of nominal and worstcase costs. Not much has been done on this type of problems, however. The next sections give brief discussions of the robust H1 and H 2 problems. 8.1 The robust H 1 performance problem In this section, we study the robust H1 performance problem deæned for the uncertain plant in Fig. 8.1. For convenience, it is assumed that the uncertainty belongs to the set æ s èæè in equation è7.8è. This implies no restriction, as an unstructured uncertainty can always be characterized by the set è7.8è with one uncertainty block ès= 1è. The robust H1 performance problem is deæned as follows. Robust H1 performance problem. Find a robustly stabilizing controller for the uncertain plant in Fig. 8.1, which achieves the robust H1 performance bound sup n kf P èp; K; æèk1 :æ2æ s èæè o éæ,1 è8.4è For convenience, the H1norm bound has been taken equal to the inverse of the uncertainty magnitude æ. This is not restrictive, as it can always be achieved by suitable scaling of the variables. F Figure 8.2. In order to solve the robust H1 performance problem, consider the plant in Fig. 8.2. It follows from the arguments in Chapter 6 that the plant in Fig. 8.2 satisæes the H1 norm bound kf k1 éæ,1 è8.5è æ P F Figure 8.3. 63
if and only if the uncertain plant depicted in Fig. 8.3 is robustly stable with respect to normbounded èunstructuredè uncertainties æ P that satisfy the normbound kæ P k1 æ. Thus, an H1 performance problem is equivalent to a robust stability problem. Here, the uncertainty æ P is can be considered as a æctive uncertainty associated with the H1norm bound. Applying the equivalence of H1 performance and robust stability to the closedloop system in Fig. 8.1 shows that the control system in Fig. 8.1 satisæes the H1norm bound kf P èp; K; æèk1 éæ,1 è8.6è æ P æ v æ u P z æ y K Figure 8.4. if and only if the system in Fig. 8.4 is robustly stable with respect to normbounded èunstructuredè uncertainties æ P that satisfy the normbound kæ P k1 æ. Hence, the plant in Fig. 8.1 achieves the robust H1 performance bound è8.4è, or equivalently, satisæes the H1norm bound è8.6è for all æ 2 æ s èæè, if and only if the system in Fig. 8.4 is robustly stable with respect to all normbounded æ P which satisfy kæ P k1 æ and all æ 2 æ s èæè. But this is equivalent to robust stability with respect to the set æ P;s èæè of uncertainties deæned according to æ P;s èæè = n æ ~ = block diagèæ; æp è; æ 2 æ s ; æ P 2 H1 rpærp ; kæ P k1 æ o è8.7è where it is assumed that æ P is r p æ r p, cf. Fig. 8.5. Thus, the robust H1 performance problem is equivalent to a robust stability problem with respect to structured uncertainty belonging to the set æ P;s èæè, obtained by extending the original uncertainty set æ s èæè with a performancerelated uncertainty block æ P. To summarize, wehave the following result. Deæne the closedloop transfer function F = F èp; Kè in Fig. 8.5 from ëv T æ vt P ë T to ëz T æ zt P ë T, zæ væ = F èp; Kè è8.8è 64
væ u æ 0 0 æ P P zæ y K Figure 8.5. Condition for robust H1 performance. Consider the system in Fig. 8.1, where the uncertainty is assumed be in the set æ s èæè. The system is robustly stable and achieves the robust H1 performance bound è8.4è if and only if the closedloop tranfer function F = F èp; Kè from ëv T æ vt P ë T to ëz T æ zt P ë T in Fig. 8.5 is stable, and sup P èf èj!èè éæ,1! è8.9è where the structured singular value P èf è is taken to correspond to the structure of the extended uncertainty set æ P;s èæè in è8.7è. The problem of ænding a controller whichachieves robust H1 performance can thus be solved by the procedures for the robust stabilization problem described in Chapter 7. 8.2 The robust H 2 performance problem In the robust H 2 performance problem, the objective is to ænd a controller which achieves robust H 2 performance for the uncertain plant in Fig. 8.1. The robust H 2 performance problem. Find a robustly stabilizing controller for the uncertain plant in Fig. 8.1, which minimizes the worstcase cost H 2 cost as J 2;worst èp; Kè = sup n kf P èp; K; æèk 2 : o 2 kæk 1 æ è8.10è èunstructured uncertaintiesè, or J 2;worst èp; Kè = sup n kf P èp; K; æèk 2 2 :æ2æ s èæè o è8.11è èstructured uncertaintiesè, where æ s èæè denotes the structured normbounded uncertainty set è7.8è. The problem contains a mixture of a performancerelated H 2 cost and a robustnessrelated H1 cost, and in contrast to the robust H1 performance problem, it cannot be 65
reduced to any of the standard problems studied so far. Instead, the problem gives rise to a sort of mixed H 2 =H1 problem. In fact, the problem is too hard to solve exactly, but there is, however, a useful upper bound on the worstcase H 2 cost J 2;worst èp; Kè, which is minimized instead. The mixed H 2 =H1 optimal controller cannot be expressed in closed form. The structure of the optimal controller is, however, known, and its parameters can therefore be optimized with respect to the cost. The mixed H 2 =H1 optimal control problem and its relation to the robust H 2 performance problem has been discussed for example by Stoorvogel è1993è, Zhou et al. è1994è and Pensar è1995è. Khargonekar and Rotea è1991è have presented an eæcient optimizationbased solution procedure for the mixed H 2 =H1 optimal control problem. References Khargonekar, P. P. and M. A. Rotea è1991è. Mixed H 2 =H1 control: A convex optimization approach. IEEE Transactions on Automatic Control 36, 824í837. Pensar, J è1995è. Parametric Methods for Optimal and Robust Control. PhD Thesis. Process Control Laboratory, Abo Akademi University. Stoorvogel, A. A. è1993è. The robust H 2 control problem: A worstcase design. IEEE Transactions on Automatic Control 38, 1358í1370. Zhou, K., K. Glover, B. Bodenheimer and J. Doyle è1994è. Mixed H 2 and H1 performance objectives I: Robust performance analysis. IEEE Transactions on Automatic Control 39, 1564í1574. 66