Chapter 5 Robust stability and the H1 norm An important application of the H1 control problem arises when studying robustness against model uncertainties. It turns out that the condition that a control system is robustly stable in spite of a certain kind of model uncertainties can be expressed quantitatively in terms of an H1 norm bound which the control system should satisfy. The connection of the H1 norm and robust stability will be described in this chapter. We will consider a plant y = Gu where it is assumed that G can be characterized as G = G 0 +æ A è5.1è è5.2è Here, G 0 is the nominal plant model, which is assumed to be linear and ænite dimensional. The additive uncertainty æ A is unknown, but it is assumed that an upper bound on its magnitude can be estimated as a function of frequency. More precisely, for a SISO system we can specify a ælter W èj!è such that jæ A èj!èjjwèj!èj; all! èsisoè è5.3è The uncertainty bound jw èj!èj characterizes the assumed maximum model uncertainty at various frequencies, and it can often be estimated from experimental data. In particular, if the plant model has been determined by frequency response experiments, the maximum error of the frequency response can often directly be estimated from the available experimental frequency response data. For convenience, the characterization è5.2è, è5.3è of an uncertain plant is usually stated in an equivalent form obtained by deæning æ such that æ A = W æ. Then è5.2è, è5.3è is equivalent to G = G 0 + W æ; jæèj!èj1; all! èsisoè è5.4è Here, the uncertainty block æ is assumed to be uniformly bounded at all frequencies, and the varying amount of plant uncertainty at various frequencies is completely incorporated into the ælter W. The ælter W is therefore commonly called uncertainty weighting ælter or simply the uncertainty weight. 52
W 1 æw 2 G + 0 e? u K y Figure 5.1: Control of uncertain system. For MIMO systems, the normbounded uncertainty description is generalized by assuming that the uncertainty blockæ A is bounded in its induced matrix norm kæèj!èk = èæèj!èè according to èæ A èj!èè jwèj!èj; all! èmimoè è5.5è where W is a scalar weighting ælter. The characterization è5.4è is then replaced by G = G 0 + W æ; èæèj!èè 1; all! èmimoè è5.6è or, equivalently èrecalling the deænition of the H1 normè G = G 0 + W æ; kæk1 1 èmimoè è5.7è Although the uncertain plant in è5.4è or è5.7è has been formulated for the additive uncertainty è5.2è, it is straightforward to extend the uncertainty description to other forms of normbounded uncertainties. An important uncertainty type is the output multiplicative uncertainty, G =èi+æ M èg 0 è5.8è Assuming that èæ M èj!èè jwèj!èj; all! the uncertain plant can be written in the equivalent form G =èi +æwèg 0 è5.9è =G 0 +æwg 0 ; kæk1 1 è5.10è which is of the same general form as è5.7è, with uncertainty weight WG 0 and with the uncertainty and weight in reversed order. Input multiplicative uncertainties can be characterized in a similar way. 53
æ v i? + + P u y z K Figure 5.2: Standard representation of uncertain control system. A general description of an uncertain system with a normbounded uncertainty is given by G = G 0 + W 1 æw 2 ; kæk1 1 è5.11è See Figure 5.1. For the control system in Figure 5.1, it is important to study whether the closed loop is stable for all possible G, i.e., for all normbounded æ such that kæk1 1. In this case the control system is said to be robustly stable and the controller K is said to be robustly stabilizing. The problem of ænding such a robustly stabilizing controller is stated as follows. Robust stabilization problem. Consider the uncertain plant described by è5.11è. Find a controller such that the control system in Figure 5.1 is stable for all uncertainties æ which satisfy the norm bound kæk1 1. The robust stabilization problem can be stated in terms of an equivalent H1 optimal control problem. The result will be based on the observation that the control system in Figure 5.1 can be written in the equivalent form in Figure 5.2, where P is an augmented plant deæned as P11 P P = 12 0 W2 = è5.12è P 22 G 0 P 21 To see the equivalence of the control systems in Figures 5.1 and 5.2, it is suæcient to show that the block diagrams give the same transfer function from u to y. Notice that by Figure 5.2 and equation è5.12è we have W 1 y =ëp 22 + P 21 æèi, P 11 æè,1 P 12 ëu =ëg 0 +W 1 æèi, 0 æ æè,1 W 2 ëu =ëg 0 +W 1 æw 2 ëu è5.13è 54
æ v + i? F z Figure 5.3: Representation of uncertain system. which is the same as in Figure 5.1. In order to obtain a qualitative appreciation of the robust stability problem, notice that the uncertainty block æ appears in a feedback loop around the closedloop plant, and classical stability analysis can be applied to provide stability conditions for this feedback loop. More precisely, the control system in Figure 5.2 can be written in the equivalent form shown in Figure 5.3, z =èi,fæè,1 Fv; F = FèP; Kè è5.14è where F = F èp; Kè denotes the closedloop transfer from v to z for the nominal system èwith æ = 0è in Figure 5.2, i.e., F èp; Kè =P 11 + P 12 èi, KP 22 è,1 KP 21 è5.15è According to classical control theory of SISO systems, the feedback loop in Figure 5.3 is stable if F èp; Kè is stable and the loop transfer function F æ has magnitude less than one; jf èj!èæèj!èj é 1, all! èimplying that jf èj! c èæèj! c èj é 1 at the crossover frequency! c, where F æ has phase,è. This result is known as the small gain theorem. By the assumption that æ is normbounded, such that jæèj!èj1, robust stability is thus guaranteed if the closedloop transfer function F satisæes jf èj!èj é 1, all!, or equivalently, the H1norm bound kf k1 é 1 holds. It turns out that the assumption on the uncertainty block æ, kæk11, leaves suæcient freedom so that a normbounded æ which actually destabilizes the system can always be found if jf èj!èj1, for some!, i.e., if kf k1 1. Hence, kf k1 é 1 is both a suæcient and necessary condition for robust stability. For MIMO systems, we can use the following èsimpliæedè qualitative arguments. Notice that the transfer function in è5.14è can be written èi, F æè,1 F = 1 adjèi, F æè æ F detèi, F æè è5.16è Assuming that F and æ are both stable, the stability of the operator in è5.16è is determined by the zeros of detèi, F æè. Since the system in Figure 5.3 is stable for 55
suæciently small æ, it follows by interpolation that if there exists a normbounded æ such that the operator in è5.16è is unstable, i.e., it has a righthalfplane pole, then there also exists a normbounded æ such that the operator in è5.16è has a pole on the stability boundary, i.e., on the imaginary axis, implying detèi, F èj!èæèj!èè = 0 for some!. Hence the robust stability condition is equivalent to the condition that detèi, F èj!èæèj!èè 6= 0; all! and æ such that kæk1 1 è5.17è Some analysis shows that the condition è5.17è holds if and only if èf èj!èè é 1, all!, or equivalently, kf k1 é 1. The above results are summarized in the following robust stability condition. Theorem 5.1 Condition for robust stability. Consider the system in Figure 5.2 or 5.3. The system is stable for all uncertainties which satisfy the norm bound kæk1 1 if and only if the nominal closedloop transfer function F = F èp; Kè is stable and kf k1 é 1 è5.18è Remark 5.1. As mentioned above, the 'if'part, kf k1 é 1 è robust stability follows from the classical small gain theorem, because the H1norm bound guarantees jf èj!èæèj!èj é 1 for SISO systems and èf èj!èæèj!èè é 1 for MIMO systems. By contrast, the 'only if'part, robust stability è kf k1 é 1 is newer, and was proved only in the 1980's. The result follows basically from the fact that the uncertainty can be chosen to have arbitrary phase. Hence, if kf k1 1 it follows that a normbounded uncertainty kæk1 1 can always be found which destabilizes the system. Remark 5.2. The robust stability condition in è5.18è is formulated for normbounded uncertainties with a norm bound which is normalized so that kæk1 1. Any other magnitude of the norm bound is assumed to be incorporated as a corresponding factor in the weight ælters W 1 and W 2. Sometimes it is, however, convenient to formulate the robustness criterion explicitly for normbounded uncertainties which satisfy the general norm bound kæk1 æ. It is easy to see that the corresponding robust stability condition is then kf k1 éæ,1 è5.19è It may for example be of interest to determine the largest normbounded uncertainty that can be allowed such that it is still possible to ænd a controller which gives robust stability. The maximally robustly stabilizing controller can be found by minimizing kf k1 èby æiterationè. The controller is then robustly stabilizing for all æ such that kæk1 é kf k,1 1. 56
Remark 5.3. Robust stability against nonlinear andèor timevarying uncertainties. Although the robust stability result has above been formulated for linear, timeinvariant èltiè uncertainties, it can easily be generalized to nonlinear, and possibly timevarying uncertainties. Suppose that the uncertainty is not restricted to be linear, but is only assumed to satisfy the norm bound kæk 1 è5.20è where kæk denotes the L 2 induced norm kæk = sup kævk2 : v 6= 0 kvk 2 è5.21è Notice that this deænition of the L 2 induced norm coincides with the deænition of the H1 norm form for linear timeinvariant systems èequations è2.51è, è2.61èè, but now æ may be nonlinear andèor timevarying. Then it can be shown that the system in Figure 5.2 or 5.3 is stable for all, possibly nonlinear andèor timevarying uncertainties æ which satisfy the L 2 induced norm bound è5.20è if and only if the nominal closedloop transfer function F = F èp; Kè is stable and satisæes the H1norm bound è5.18è, i.e., kf k1 é 1. Hence the condition for robust stability against nonlinear, timevarying uncertainties is the same as the condition for robust stability against linear timeinvariant uncertainties. This result may appear counterintuitive, as one would expect that the ærst, more general, uncertainty class would correspond to a more restrictive condition on the closed loop. The result is, however, a consequence of the fact that the nominal system is linear and timeinvariant, and then the worst, destabilizing uncertainty is also linear and timeinvariant. By Remark 5.3 the robust stability condition provides a quantitativeway of treating plant nonlinearities as uncertainties about a nominal linear plant model. Such an approach is of course only feasible for not too strongly nonlinear plants, where the norm bound of the uncertainty which captures the nonlinearities is not excessive. The signiæcance of the robust stability condition è5.18è is that it gives a quantitative characterization of robust stability in terms of the H1 norm. By the procedure in Chapter 4 we know how to ænd a controller which achieves the norm bound è5.18è, and which is robustly stabilizing. Before the introduction of H1 control theory the solution of the robust stabilization problem was not known. Although theoretically important, the robust stabilization problem alone is perhaps of limited practical interest. Firstly, many processes, for example in chemical process control systems, are known to be openloop stable. The maximally robustly stabilizing control strategy is then to operate the plant in open loop, because any feedback controller gives a reduction in robust stability for an openloop stable plant. Secondly, even for an unstable uncertain plant, a controller which is only designed for robust stability will in general be conservative. In practical controller design, the robust stability condition è5.18è is useful as a robustness bound used in combination with other performance related design criteria. This will be discussed in Chapter 7. Notes and references The formulation of robust stability in terms of the H1 norm is due to a number of 57
researchers, see for example Doyle and Stein è1981è, Chen and Desoer è1982è, Kimura è1984è, Vidyasagar è1985è, and Vidyasagar and Kimura è1986è. Highly readable accounts of the robust stability issue are found in the books by Vidyasagar è1985è and McFarlane and Glover è1990è. References Chen, M. J. and C. A. Desoer è1982è. "Necessary and suæcient conditions for robust stability of linear distributed feedback systems," Int. J. Control 35, 255í267. Doyle, J. C. and G. Stein è1981è. "Multivariable feedback design: concepts for a classicalèmodern synthesis," IEEE Transactions on Automatic Control 26, 4í16. Kimura, H. è1984è. "Robust stabilizability for a class of transfer functions," IEEE Transactions on Automatic Control 29, 788í793. McFarlane, D. C. and K. Glover è1990è. Robust Controller Design Using Normalized Coprime Factor Plant Descriptions. SpringerVerlag, Berlin. Vidyasagar, M. è1985è. Control Systems Synthesis: A Factorization Approach. MIT Press, Cambridge, MA. Vidyasagar, M. and H. Kimura è1986è. "Robust controllers for uncertain linear multivariable systems," Automatica 22, 85í94. 58