7 Robustness Analysis with Real Parametric Uncertainty

Transcription

1 7 Robustness Analysis with Real Parametric Uncertainty Roberto Tempo IEIIT CNR Politecnico di Torino Franco Blanchini Università di Udine 7.1 Motivations and Preliminaries Motivating Example: DC Electric Motor with Uncertain Parameters 7.2 Description of the Uncertainty Structures Uncertainty Structure Preservation with Feedback Overbounding with Affine Uncertainty: The Issue of Conservatism Robustness Analysis for Affine Plants Value Set Construction for Affine Plants The DC-Electric Motor Example Revisited 7.6 Robustness Analysis for Affine Polynomials Value Set Construction for Affine Polynomials Example of Value Set Generation Interval Polynomials: Kharitonov s Theorem and Value Set Geometry From Robust Stability to Robust Performance Algebraic Criteria for Robust Stability Further Extensions: The Spectral Set 7.7 Multiaffine Uncertainty Structures General Uncertainty Structures and Controller Synthesis References Motivations and Preliminaries Over the last decades, stability and performance of control systems affected by bounded perturbations have been studied in depth. The attention of researchers and control engineers concentrated on robustness tools in the areas H, Kharitonov (or real parametric uncertainty), L 1, Lyapunov, μ, and Quantitative Feedback Control (QFT). For further discussions on these topics and on the exposition of the main technical results, the reader may consult different sections of this volume and the special issue on robust control of Automatica (1993). One of the key features of this chapter is the concept of robustness. To explain, instead of a single (nominal) system, we study a family of systems and we say that a certain property (e.g., stability or performance) is robustly satisfied if it is satisfied for all members of the family. In particular, we focus on linear, time-invariant, single-input, single-output systems affected by real parametric uncertainty. Stability of 7-1

2 7-2 Control System Advanced Methods interval polynomials (i.e., polynomials whose coefficients lie within given intervals) and the well-known Theorem of Kharitonov (Kharitonov, 1978) are at the core of this research area. This theorem states that an interval polynomial has all its roots in the open left half-plane if and only if four specially constructed polynomials have roots in the open left half-plane. Subsequently, the Edge Theorem (Bartlett et al., 1988) studied the problem of affine dependence between coefficients and uncertain parameters, and more general regions than the open left half-plane. This result provides an elegant solution proving that it suffices to check stability of the so-called one-dimensional exposed edges. We refer to the books (Ackermann, 1993; Barmish, 1994; Bhattacharyya et al., 1995; Djaferis, 1995; Kogan, 1995) for a discussion of the extensive literature on this subject. To explain robustness analysis more precisely with real parametric uncertainty, we consider a family of polynomials p(s, q) of degree n whose real coefficients a i (q) are continuous functions of an l-dimensional vector of real uncertain parameters q, each bounded in the interval [q i, q + i ]. More formally, we define p(s, q) =. a 0 (q) + a 1 (q)s + a 2 (q)s 2 + +a n (q)s n, q =[q. 1, q 2,..., q l ], and the set Q ={q. : qi q i q i +, i = 1, 2,..., l}. We assume that p(s, q) is of degree n for all q Q that is, we assume that a n (q) = 0 for all q Q. Whenever the relations between the polynomial coefficients a i (q) and the vector q are specified, we study the root location of p(s, q) for all q Q. Within this framework, the basic property we need to guarantee is robust stability. In particular, we say that p(s, q) is robustly stable if p(s, q) has roots in the open left half-plane for all q Q. The real parametric approach can be also formulated for control systems. In this case, we deal with a family of plants denoted by P(s, q). More precisely, we concentrate on robust stability or performance of a proper plant P(s, q) =. N P(s, q) D P (s, q), where N P (s, q) and D P (s, q) are the numerator and denominator polynomials whose real coefficients are continuous functions of q. We assume that D P (s, q) has invariant degree for all q Q. We also assume that there is no unstable pole zero cancellation for all q Q; the reader may refer to Chockalingam and Dasgupta (1993) for further discussions. Robustness analysis is clearly of interest when the plant requires compensation. In practice, if the compensator is designed on the basis of the nominal plant, then, robustness analysis can be performed by means of the closed-loop polynomial. That is, given a compensator transfer function C(s) =. N C(s) D C (s) connected in a feedback loop with P(s, q), we immediately write the closed-loop polynomial p(s, q) = N P (s, q)n C (s) + D P (s, q)d C (s) whose root location determines closed-loop stability. To conclude this preliminary discussion, we remark that one of the main technical tools described here is the so-called value set (or template in the QFT jargon, see Horowitz, 1991; see Barmish, 1994 for a detailed exposition of its properties). In particular, we show that if the polynomial or plant coefficients are affine functions of q, the value set can be easily constructed with 2D graphics. Consequently, robustness tests in the frequency domain can be readily performed. Finally, in this chapter, since the main goal is to introduce the basic concepts and tools available for robustness analysis with real parametric uncertainty, we do not provide formal proofs but we make reference to the specific literature on the subject.

3 Robustness Analysis with Real Parametric Uncertainty Motivating Example: DC Electric Motor with Uncertain Parameters For the sake of illustrative purposes, an example of a DC electric motor is formulated and carried out throughout this chapter in various forms. Consider the system represented in Figure 7.1 of an armature-controlled DC electric motor with independent excitation. The voltage to angle transfer function P(s) = Θ(s)/V(s) is given by K P(s) = LJs 3 + (RJ + BL)s 2 + (K 2 + RB)s, where L is the armature inductance, R the armature resistance, K the motor electromotive force-speed constant, J the moment of inertia, and B the mechanical friction. Clearly, the values of some of these parameters may be uncertain. For example, the moment of inertia and the mechanical friction are functions of the load. Therefore, depending on the specific application, if the load is not fixed, the values of J and B are not precisely known. Similarly, the armature resistence R is a parameter that can be measured very accurately but is subject to temperature variations, and the motor constant K is a function of the field magnetic flow which may vary. To summarize, it is reasonable to say that the motor parameters, or a subset of them, may be unknown but bounded within given intervals. More precisely, we can identify q 1 = L; q 2 = R; q 3 = K; q 4 = J; q 5 = B and specify a given interval [q i, q + i ] for each q i, i = 1, 2,..., 5. Then, instead of P(s), we write P(s, q) = q 3 q 1 q 4 s 3 + (q 2 q 4 + q 1 q 5 )s 2 + (q q 2q 5 )s. 7.2 Description of the Uncertainty Structures As discussed in the preliminaries in Section 7.1, we consider a proper plant P(s, q) whose coefficients are continuous functions of the uncertainty q which is confined to the set Q. Depending on the specific problem under consideration, the coefficients of N P (s, q) and D P (s, q) may be linear or nonlinear functions of q. To explain more precisely, we consider the example of Section Assume that the armature i = Armature current V = Armature voltage ω = Angular speed C m = Motor torque θ = Angle C r = Load torque FIGURE 7.1 DC electric motor.

4 7-4 Control System Advanced Methods inductance L, the armature resistance R, and the constant K are fixed while the moment of inertia J and the mechanical friction B are unknown. Then, we take q 1 = J and q 2 = B as uncertain parameters; the resulting set Q is a two-dimensional rectangle. In this case, the plant coefficients are affine functions of q 1 and q 2 N P (s, q) = K; D P (s, q) = Lq 1 s 3 + (Rq 1 + Lq 2 )s 2 + (K 2 + Rq 2 )s and we say that the plant has an affine uncertainty structure. This situation arises in practice whenever, for example, the load conditions are not known. Other cases, however, may be quite different from the point of view of the uncertainty description. For example, if L, K, and B are fixed and R and J are uncertain, we identify q 1 and q 2 with R and J, respectively. We observe that the denominator coefficient of s 2 contains the product of the two uncertain parameters q 1 and q 2 N P (s, q) = K; D P (s, q) = Lq 2 s 3 + (q 1 q 2 + BL)s 2 + (K 2 + Bq 1 )s. In this case, the plant coefficients are no longer affine functions of the uncertainties but they are multiaffine functions of q. This discussion can be further generalized. It is well known that the motor constant K is proportional to the magnetic flow. In an ideal machine with independent excitation, such a flow is constant, but in a real machine, the armature reaction phenomenon causes magnetic saturation with the consequence that the constant K drops when the armature current exceeds a certain value. Hence, we consider K as an uncertain parameter and we set q 1 = K. In turn, this implies that the plant coefficients are polynomial functions of the uncertainties. In addition, since q 1 enters in N P (s, q) and D P (s, q), we observe that there is coupling between numerator and denominator coefficients. In different situations when this coupling is not present, we say that the numerator and denominator uncertainties are independent. An important class of independent uncertainties, in which all the coefficients of the numerator and denominator change independently within given intervals, is the so-called interval plant; for example, see Barmish et al. (1992). In other words, an interval plant is the ratio of two independent interval polynomials; recall that an interval polynomial p(s, q) = q 0 + q 1 s + q 2 s 2 + +q n s n has independent coefficients bounded in given intervals q i q i q + i for i = 0, 1, 2,..., n. The choice of the uncertain parameters for a control system is a modeling problem, but robustness analysis is of increasing difficulty for more general uncertainty structures. In the following sections, we show that this analysis can be easily performed if the structure is affine and we demonstrate that a tight approximate solution can be readily computed in the multiaffine case. 7.3 Uncertainty Structure Preservation with Feedback In the previous sections, we described the classes of uncertainty structures entering into the open-loop plant. From the control system point of view, an important and closely related question arises: what are the conditions under which a certain uncertainty structure is preserved with feedback? To answer this question, we consider a plant P(s, q) and a compensator C(s) connected with the feedback structure shown in Figure 7.2. An affine function f : Q R is the sum of a linear function and a constant. For example, f (q) = 3q 1 + 2q 2 4 is affine. A function f : Q R is said to be multiaffine if the following condition holds: If all components q 1,..., q l except for one are fixed, then f is affine. For example, f (q) = 3q 1 q 2 q 3 6q 1 q 3 + 4q 2 q 3 + 2q 1 2q 2 + q 3 1 is multiaffine.

5 Robustness Analysis with Real Parametric Uncertainty 7-5 d r e C(s) P(s, q) y FIGURE 7.2 Closed-loop system. Depending on the specific problem under consideration (e.g., disturbance attenuation or tracking), we study sensitivity, complementary sensitivity, and output-disturbance transfer functions (e.g., see Doyle et al., 1992) S(s, q). = P(s, q)c(s) ; T(s, q). = P(s, q)c(s) 1 + P(s, q)c(s) ; R(s, q). = P(s, q) 1 + P(s, q)c(s). For example, it is immediate to show that the sensitivity function S(s, q) takes the form S(s, q) = D P (s, q)d C (s) N P (s, q)n C (s) + D P (s, q)d C (s). If the uncertainty q enters affinely into the plant numerator and denominator, q also enters affinely into S(s, q). We conclude that the affine structure is preserved with feedback. The same fact also holds for T(s, q) and R(s, q). Next, we consider the interval plant structure; recall that an interval plant has independent coefficients bounded in given intervals. It is easy to see that, in general, this structure is not preserved with compensation. Moreover, if the plant is affected by uncertainty entering independently into numerator and denominator coefficients, this decoupling is destroyed for all transfer functions S(s, q), T(s, q), and R(s, q). Finally, it is important to note that the multiaffine and polynomial uncertainty structures are preserved with feedback. Table 7.1 summarizes this discussion. In each entry of the first row of the table we specify the structure of the uncertain plant P(s, q) and in the entry below the corresponding structure of S(s, q), T(s, q), and R(s, q). 7.4 Overbounding with Affine Uncertainty: The Issue of Conservatism As briefly mentioned at the end of Section 7.3, the affine structure is very convenient for performing robustness analysis. However, in several real applications, the plant does not have this form; for example, see Abate et al. (1994). In such cases, the nonlinear uncertainty structure can always be embedded into an affine structure by replacing the original family by a larger one. Even though this process has the advantage that it handles much more general robustness problems, it has the obvious drawback that it gives only an approximate but guaranteed solution. Clearly, the goodness of the approximation depends on the TABLE 7.1 Uncertainty Structure with Feedback P(s, q) Independent Interval Affine Multiaffine Polynomial S(s, q), T(s, q), R(s, q) Dependent Affine Affine Multiaffine Polynomial

6 7-6 Control System Advanced Methods specific problem under consideration. To illustrate this simple overbounding methodology, we consider the DC-electric motor transfer function with two uncertain parameters and take q 1 = R and q 2 = J. As previously discussed, with this specific choice, the plant has a multiaffine uncertainty structure. That is, P(s, q) = K Lq 2 s 3 + (q 1 q 2 + BL)s 2 + (K 2 + Bq 1 )s. To overbound P(s, q) with an affine structure, we set q 3 = q 1 q 2. Given bounds [q 1, q+ 1 ] and [q 2, q+ 2 ] for q 1 and q 2, the range of variation [q 3, q+ 3 ] for q 3 can be easily computed: Clearly, the new uncertain plant q3 = min{q 1 q 2, q 1 q+ 2, q+ 1 q 2, q+ 1 q+ 2 }; q 3 + = max{q 1 q 2, q 1 q+ 2, q+ 1 q 2, q+ 1 q+ 2 }. P(s, q) = K Lq 2 s 3 + (q 3 + BL)s 2 + (K 2 + Bq 1 )s has three uncertain parameters q = (q 1, q 2, q 3 ) entering affinely into P(s, q). This new parameter is not independent, because q 3 = q 1 q 2 and not all values of [q3, q+ 3 ] are physically realizable. However, since we assume that the coefficients q i are independent, this technique leads to an affine overbounding of P(s, q) with P(s, q). We conclude that if a certain property is guaranteed for P(s, q), then, this same property is also guaranteed for P(s, q). Unfortunately, the converse is not true. The control systems interpretation of this fact is immediate: Suppose that a certain compensator C(s) does not stabilize P(s, q). It may turn out that this same compensator does stabilize the family P(s, q). Figure 7.3 illustrates the overbounding procedure for q1 = 1.2, q+ 1 = 1.7, q 2 = 1.7, q+ 2 = 2.2, q 3 = 2.04 and q+ 3 = To generalize this discussion, we restate the overbounding problem as follows: Given a plant P(s, q) having nonlinear uncertainty structure and a set Q, find a new uncertain plant P(s, q) with affine uncertainty structure and a new set Q. In general, there is no unique solution and there is no systematic procedure to construct an optimal overbounding. In practice, however, the control engineer may find via heuristic considerations a reasonably good method to perform it. The most natural way to obtain this bound may be to compute an interval overbounding for each coefficient of the numerator and denominator coefficients that is, an interval plant overbounding. To illustrate, letting a i (q) and b i (q) q q q FIGURE 7.3 Overbounding procedure.

7 Robustness Analysis with Real Parametric Uncertainty 7-7 denote the numerator and denominator coefficients of P(s, q), lower and upper bounds are given by ai = min a i(q); a + q Q i = max a i(q) q Q and bi = min b i(q); b + q Q i = max b i(q). q Q If a i (q) and b i (q) are affine or multiaffine functions and q lies in a rectangular set Q, these minimizations and maximizations can be easily performed. That is, if we denote by q 1, q 2,..., q L. = q 2l the vertices of Q, then and a i b i = min a i(q) = min a i(q k ); a q Q i + = max a i(q) = max a i(q k ) k=1,2,...,l q Q k=1,2,...,l = min b i(q) = min b i(q k ); b q Q i + = max b i(q) = max b i(q k ). k=1,2,...,l q Q k=1,2,...,l To conclude this section, we remark that for more general uncertainty structures than multiaffine, a tight interval plant overbounding may be difficult to construct. 7.5 Robustness Analysis for Affine Plants In this section, we study robustness analysis of a plant P(s, q) affected by affine uncertainty q Q. The approach taken here is an extension of the classical Nyquist criterion and requires the notion of value set. For fixed frequency s = jω, we define the value set P(jω, Q) C as P(jω, Q). ={P(jω, q) : D P (jω, q) = 0, q Q}. Roughly speaking, P( jω, Q) is a set in the complex plane which graphically represents the uncertain plant. Without uncertainty, P(jω) is a singleton and its plot for a range of frequencies is the Nyquist diagram. The nice feature is that this set is two-dimensional even if the number of uncertain parameters is large. Besides the issue of the value set construction (which is relegated to the next subsection for the specific case of affine plants), we now formally state a robustness criterion. This is an extension of the classical Nyquist criterion and holds for more general uncertainty structures than affine continuity of the plant coefficients with respect to the uncertain parameters suffices. However, for more general classes of plants than affine, the construction of the value set is a hard problem. Criterion 7.1: Robustness Criterion for Uncertain Plants The plant P(s, q) is robustly stable for all q Q if and only if the Nyquist stability criterion is satisfied for some q Q and 1 + j0 P(jω, Q) for all ω R. This criterion can be proved using continuity arguments; see Fu (1990). To detect robustness, one should check if the Nyquist stability criterion holds for some q Q; without loss of generality, this check can be performed for the nominal plant. Second, it should be verified that the value set does not go through the point 1 + j0 for all ω R. In practice, however, one can discretize a bounded interval Ω R with a sufficiently large number of samples continuity considerations guarantee that the intersampling is not a critical issue. Finally, by drawing the value set, the gain and phase margins can be graphically evaluated; similarly, the resonance peak of the closed-loop system can be computed using the well-known constant M-circles.

8 7-8 Control System Advanced Methods Value Set Construction for Affine Plants In this section, we discuss the generation of the value set P(jω, Q) in the case of affine plants. The reader familiar with Nyquist-type analysis and design is aware of the fact that a certain range of frequencies, generally close to the crossover frequencies, can be specified a priori. That is, a range Ω =[ω, ω + ] may be imposed by design specifications or estimated by performing a frequency analysis of the nominal system under consideration. In this section, we assume that D P (jω, q) = 0 for all q Q and ω Ω. We remark that if the frequency ω = 0 lies in the interval Ω, this assumption is not satisfied for type 1 or 2 systems however, these systems can be easily handled with contour indentation techniques as in the classical Nyquist analysis. We also observe that the assumption that P(s, q) does not have poles in the interval [ω, ω + ] simply implies that P(jω, Q) is bounded. To proceed with the value set construction, we first need a preliminary definition. The one-dimensional exposed edge e ik is a convex combination of the adjacent vertices q i and q k of Q e ik. = λq i + (1 λ)q k for λ [0, 1]. Denote by E the set of all q e ik for some i, k, and λ [0, 1]. This set is the collection of all one-dimensional exposed edges of Q. Under our assumption P(jω, Q) = 0, for q Q and all ω, the set P(jω, Q) is compact. Then, for fixed ω Ω, it can be shown (see Fu, 1990) that P(jω, Q) P(jω, E). ={P(jω, q) : q E} where P(jω, Q) denotes the boundary of the value set and P(jω, E) is the image in the complex plane of the exposed edges (remember that we assumed P(jω, Q) = 0, for q Q and all ω). This says that the construction of the value set only requires computations involving the one-dimensional exposed edges. The second important fact observed in Fu (1990) is that the image of the edge e ik in the complex plane is an arc of a circle or a line segment. To explain this claim, in view of the affine dependence of both N(s, q) and D(s, q) versus q, we write the uncertain plant corresponding to the edge e ik in the form P(s, e ik ) = N P(s, λq i + (1 λ)q k ) D P (s, λq i + (1 λ)q k ) = N P(s, q k ) + λn P (s,(q i q k )) D P (s, q k ) + λn P (s,(q i q k )) for λ [0, 1]. For fixed s = jω, it follows that the mapping from the edge e ik to the complex plane is bilinear. Then, it is immediate to conclude that the image of each edge is an arc of a circle or a line segment; the center and the radius of the circle and the extreme points of the segment can be also computed. Even though the number of one-dimensional exposed edges of the set Q is l2 l 1, the set E is one dimensional. Therefore, a fast computation of P( jω, E) can be easily performed and the boundary of the value set P(jω, Q) can be efficiently generated. Finally, an important extension of this approach is robustness analysis of systems with time delay. That is, instead of P(s, q), we consider P τ (s, q) =. N(s, q) D(s, q) e τs where τ 0 is a delay. It is immediate to see that the value set of P τ (s, q) at frequency s = jω is given by the value set of the plant N(s, q)/d(s, q) rotated with respect to the origin of the complex plane of an angle τω in clockwise direction. Therefore, Criterion 7.1 still applies; see Barmish and Shi (1990) for further details. Two vertices are adjacent if they differ for only one component. For example, in Figure 7.3 the vertices q 1 = (1.2, 2.2, 2.04) and q 2 = (1.2, 2.2, 3.74) are adjacent.

9 Robustness Analysis with Real Parametric Uncertainty The DC-Electric Motor Example Revisited To illustrate the concepts discussed in this section, we revisit the DC-electric motor example. We take two uncertain parameters q 1 = J; q 2 = B, where q 1 [0.03, 0.15] and q 2 [0.001, 0.03] with nominal values J = kg m 2 and B = N m/rps. The remaining parameters take values K = 0.9 V/rps, L = H, and R = 5Ω. The voltage to angle uncertain plant is 0.9 P(s, q) = 0.025q 1 s 3 + (5q q 2 )s 2 + ( q 2 )s. To proceed with robustness analysis, we first estimate the critical range of frequencies obtaining Ω = [10, 100]. We note that the denominator of P(s, q) is nonvanishing for all q Q in this range. Then, we study robust stability of the plant connected in a feedback loop with a PID compensator C(s) = K P + K I s + K Ds. For closed-loop stability, we recall that the Nyquist criterion requires that the Nyquist plot of the open-loop system does not go through the point 1 + j0 and that it does encircle this point (in counterclockwise direction) a number of times equal to the number of unstable poles; for example, see Horowitz (1963). In this specific case, setting K P = 200, K I = 5120, and K D = 20, we see that the closed-loop nominal system is stable with a phase margin φ 63.7 and a crossover frequency ω c 78.8 rad/s. As a result of the analysis carried out by sweeping the frequency, it turns out that the closed-loop system is not robustly stable, since at the frequency ω 16 rad/s the value set includes the point 1 + j0. Figure 7.4 shows the Nyquist plot of the nominal plant and the value set for 12 equispaced frequencies in the range (12,34) Im Re FIGURE 7.4 Nyquist plot and value set.

10 7-10 Control System Advanced Methods Im Re FIGURE 7.5 Nyquist plot and value set. To robustly stabilize P(s, q), we take K I = 2000 and the same values of K P and K D as before. The reasons for choosing this value of K I can be explained as follows: The compensator transfer function has phase zero at the frequency ω = K I /K D. Thus, reducing K I from 5120 to 2000 causes a drop of ω from 16 to 10 rad/s. This implies that the phase lead effect begins at lower frequencies pushing the value set out of the critical point 1 + j0. Since the nominal system is stable with a phase margin φ 63.7 and a crossover frequency ω c 80.8 rad/s, this new control system has nominal performance very close to the previous one. However, with this new PID compensator, the system becomes robustly stable. To see this, we generated the value sets for the same frequencies as before; see Figure 7.5. From this figure, we observe that P(jω, q) does not include the point 1 + j0. We conclude that the closed-loop system is now robustly stable; the worst-case phase margin is φ Robustness Analysis for Affine Polynomials In this section, we study robustness analysis of the closed-loop polynomial p(s, q) = N P (s, q)n C (s) + D P (s, q)d C (s) when the coefficients of N P (s, q) and D P (s, q) are affine functions of q. The main goal is to provide an alternative criterion for polynomials instead of plants. With this approach, we do not need the nonvanishing condition about D P (s, q); furthermore, unstable pole zero cancellations are not an issue. In this case, however, we lose the crucial insight given by the Nyquist plot. For fixed frequency s = jω, we define

11 Robustness Analysis with Real Parametric Uncertainty 7-11 the value set p(jω, Q) C as p(jω, Q). ={p(jω, q) : q Q}. As in the plant case, p(jω, Q) is a set in the complex plane which moves with frequency and which graphically represents the uncertain polynomial. Criterion 7.2: Zero-Exclusion Condition for Uncertain Polynomials The polynomial p(s, q) is robustly stable for all q Q if and only if p(s, q) is stable for some q Q and 0 p(jω, Q) for all ω R. The proof of this criterion requires elementary facts and, in particular, continuity of the roots of p(s, q) versus its coefficients; see Frazer and Duncan (1929). Similar to the discussion in Section 7.5, we note that Criterion 7.2 is easily implementable at least whenever the value set can be efficiently generated. That is, given an affine polynomial family, we take any element in this family and we check its stability. This step is straightforward using the Routh table or any root finding routine. Then, we sweep the frequency ω over a selected range of critical frequencies Ω =[ω, ω + ]. This interval can be estimated, for example, using some a priori information on the specific problem or by means of one of the bounds given in Marden (1966). If there is no intersection of p(jω, Q) with the origin of the complex plane for all ω Ω, then p(s, q) is robustly stable. Remark 7.1 A very similar zero exclusion condition can be stated for more general regions D than the open left half-plane. Meaningful examples of D regions are the open unit disk, a shifted left half-plane and a damping cone. In this case, instead of sweeping the imaginary axis, we need to discretize the boundary of D Value Set Construction for Affine Polynomials In this section, we discuss the generation of the value set p( jω, Q). Whenever the polynomial coefficients are affine functions of the uncertain parameters, the value set can be easily constructed. To this end, two key facts are very useful. First, for fixed frequency, we note that p(jω, Q) is a two-dimensional convex polygon. Second, letting q 1, q 2,..., q L denote the vertices of Q as in Section 7.4, we note that the vertices of the value set are a subset of the complex numbers p(jω, q 1 ), p(jω, q 2 ),..., p(jω, q L ). These two observations follow from the fact that, for fixed frequency, real and imaginary parts of p(s, q) are both affine functions of q. Then, the value set is a two-dimensional affine mapping of the set Q and its vertices are generated by vertices of Q. Thus, for fixed ω, it follows that p(jω, Q) = conv {p(jω, q 1 ), p(jω, q 2 ),..., p(jω, q L )} where conv denotes the convex hull. The conclusion is then immediate: For fixed frequency, one can generate the points p(jω, q 1 ), p(jω, q 2 ),..., p(jω, q L ) in the complex plane. The value set can be constructed by taking the convex hull of these points this can be readily done with 2D graphics. From the computational point of view, we observe that the number of edges of the polygon is at most 2l at each frequency. This follows from the observations that any edge of the value set is the image of an exposed edge of Q. In addition, parallel edges of Q are mapped into parallel edges in the complex plane and the edges of Q have only l distinct directions. These facts can be used to efficiently compute p(jω, Q). We now provide an example which illustrates the value set generation. A damping cone is a subset of the complex plane defined as {s : Re(s) α Im(s) } for α > 0. The convex hull conv S of a set S is the smallest convex set containing S.

12 7-12 Control System Advanced Methods Example of Value Set Generation Using the same data as in the example of Section and a PID controller with gains K P = 200, K I = 5120, and K D = 20, we study the closed-loop polynomial p(s, q) = 0.025q 1 s 4 + s 3 (5q q 2 ) + s 2 (5q ) + 180s Robustness analysis is performed for 29 equispaced frequencies in the range (2,30). Figure 7.6 shows the polygonality of the value set and zero inclusion for ω 16 rad/s which demonstrates instability. This conclusion is in agreement with that previously obtained in Section Interval Polynomials: Kharitonov s Theorem and Value Set Geometry In the special case of interval polynomials, robustness analysis can be greatly facilitated via Kharitonov s Theorem (Kharitonov, 1978). We now recall this result. Given an interval polynomial p(s, q) = q 0 + q 1 s + q 2 s 2 + +q n s n of order n (i.e., q n = 0) and bounds [q i, q + i ] for each coefficient q i, define the following four polynomials: p 1 (s) =. q q+ 1 s + q 2 s2 + q3 s3 + q 4 + s4 + q 5 + s5 + q6 s6 + q7 s7 + ; p 2 (s) =. q0 + q 1 s + q+ 2 s2 + q 3 + s3 + q4 s4 + q5 s5 + q 6 + s6 + q 7 + s7 + ; p 3 (s) =. q q 1 s + q 2 s2 + q 3 + s3 + q 4 + s4 + q5 s5 + q6 s6 + q 7 + s7 + ; p 4 (s) =. q0 + q+ 1 s + q+ 2 s2 + q3 s3 + q4 s4 + q 5 + s5 + q 6 + s6 + q7 s Im ,000 12,000 14,000 16,000 12,000 10, Re FIGURE 7.6 Value set plot.

13 Robustness Analysis with Real Parametric Uncertainty 7-13 Then, p(s, q) is stable for all q Q if and only if the four Kharitonov polynomials p 1 (s), p 2 (s), p 3 (s) and p 4 (s) are stable. To provide a geometrical interpretation of this result, we observe that the value set for fixed frequency s = jω is a rectangle with level edges parallel to real and imaginary axis. The four vertices of this set are the complex numbers p 1 (jω), p 2 (jω), p 3 (jω), and p 4 (jω); see Dasgupta (1988). If the four Kharitonov polynomials are stable, due to the classical Mikhailov s criterion (Mikhailov, 1938), their phase is strictly increasing for ω increasing. In turn, this implies that the level rectangular value set moves in a counterclockwise direction around the origin of the complex plane. Next, we argue that the strictly increasing phase of the vertices and the parallelism of the four edges of the value set with real or imaginary axis guarantee that the origin does not lie on the boundary of the value set. By continuity, we conclude that the origin is outside the value set and the zero exclusion condition is satisfied; see Minnichelli et al. (1989) From Robust Stability to Robust Performance In this section, we point out the important fact that the polynomial approach discussed in this chapter can be also used for robust performance. To explain, we take an uncertain plant with affine uncertainty and we show how to compute the largest peak of the Bode plot magnitude for all q Q that is, the worst-case H norm. Formally, for a stable strictly proper plant P(s, q), we define Given a performance level γ > 0, then max P(s, q) =. max q Q sup q Q ω max q Q P(s, q) < γ P(jω, q). if and only if N P (jω, q) D P (jω, q) < γ for all ω 0 and q Q. Since P(jω, q) 0 for ω, this is equivalent to check if the zero-exclusion condition N P (jω, q) γd P (jω, q)e jφ = 0 is satisfied for all ω R, q Q and φ [0, 2π]. In turn, this implies that the polynomial with complex coefficients p φ (s, q) = N P (s, q) γd P (s, q)e jφ has roots in the open left half-plane for all q Q and φ [0, 2π]. Clearly, for fixed φ [0, 2π], Criterion 7.2 can be readily used; however, since p φ (s, q) has complex coefficients, it should be necessarily checked for all ω R, including negative frequencies Algebraic Criteria for Robust Stability If the uncertain polynomial under consideration is affine, the well-known Edge Theorem applies (Bartlett et al., 1988). This algebraic criterion is alternative to the frequency domain approach studied in this section. Roughly speaking, this result says that an affine polynomial family is robustly stable if and only if all the polynomials associated with the one-dimensional exposed edges of the set Q are stable. Even though this result is of algebraic nature, it can be explained by means of value set arguments. For affine polynomials and for fixed ω, we have already observed in Section that the boundary of the value set p(jω, Q) is the image of the one-dimensional exposed edges of Q. Thus, to guarantee the zero-exclusion

14 7-14 Control System Advanced Methods condition, we need to guarantee that all edge polynomials are nonvanishing for all ω R otherwise an instability occurs. We conclude that stability detection for affine polynomials requires the solution of a number of one-dimensional stability problems. Each of these problems can be stated as follows: Given polynomials p 0 (s) and p 1 (s) of order n and m < n, respectively, we need to study the stability of p(s, λ) = p 0 (s) + λp 1 (s) for all λ [0, 1]. A problem of great interest is to ascertain when the robust stability of p(s, λ) can be deduced from the stability of the extreme polynomials p(s, 0) and p(s, 1). This problem can be formulated in more general terms: To construct classes of uncertain polynomials for which the stability of the vertex polynomials (or a subset of them) implies stability of the family. Clearly, the edges associated with an interval polynomial is one such class. Another important example is given by the edges of the closed-loop polynomial of a control system consisting of a first-order compensator and an interval plant; see Barmish et al. (1992). Finally, see Rantzer (1992) for generalizations and for the concept of convex directions polynomials Further Extensions: The Spectral Set In some cases, it is of interest to generate the entire root location of a family of polynomials. This leads to the concept of spectral set; see Barmish and Tempo (1991). Given a polynomial p(s, q), we define the spectral set as σ ={s. C : p(s, q) = 0 for some q Q}. The construction of this set is quite easy for affine polynomials. Basically, the key idea can be explained as follows: For fixed s C, checking if s is a member of the spectral set can be accomplished by means of the zero-exclusion condition; see also Remark Next, it is easy to compute a bounded root confinement region σ σ; for example, see Marden (1966). Then, the construction of the spectral set σ amounts to a two-dimensional gridding of σ and, for each grid point, checking zero exclusion. The spectral set concept can be further extended to control systems consisting of a plant P(s, q) with a feedback gain K P which needs tuning. In this case, we deal with the so-called robust root locus (Barmish and Tempo, 1990) that is, the generation of all the roots of the closed-loop polynomial when K P ranges in a given interval. To illustrate, we consider the same data as in Section and a feedback gain K P, thus obtaining the closed loop polynomial p(s, q) = 0.025q 1 s 3 + (5q q 2 )s 2 + ( q 2 )s + 0.9K P, where q 1 [0.03, 0.15] and q 2 [0.001, 0.03]. In Figure 7.7, we show the spectral set for K P = 200. Actually, only the portion associated with the dominant roots is visible since the spectral set, obviously, includes a real root in the interval [ , ] which is out of the plot. 7.7 Multiaffine Uncertainty Structures In this section, we discuss the generation of the value set for polynomials with more general uncertainty structures than affine. In particular, we study the case when the polynomial coefficients are multiaffine functions of the uncertain parameters. Besides the motivations provided in Section 7.3, we recall that this uncertainty structure is quite important for a number of reasons. For example, consider a linear state-space system of the form ẋ(t) = A(q)x(t) where each entry of the matrix A(q) R m m lies in a bounded interval, that is, it is an interval matrix. Then, the characteristic polynomial required for stability considerations has a multiaffine uncertainty structure. In the case of multiaffine uncertainty, the value set is generally not convex and its construction cannot be easily performed. However, we can easily generate a tight convex approximation of p( jω, Q) this approximation being its convex hull conv p( jω, Q). More precisely, the following fact, called the Mapping

15 Robustness Analysis with Real Parametric Uncertainty Im Re FIGURE 7.7 Spectral set. Theorem, holds: The convex hull of the value set conv p(jω, Q) is given by the convex hull of the vertex polynomials p(jω, q 1 ), p(jω, q 2 ),..., p(jω, q L ). In other words, the parts of the boundary of p(jω, Q) which are not line segments are always contained inside this convex hull. Clearly, if conv p(jω, Q) is used instead of p( jω, Q) for robustness analysis through zero exclusion, only a sufficient condition is obtained. That is, if the origin of the complex plane lies inside the convex hull, we do not know if it is also inside the value set. We now formally state the Mapping Theorem; see Zadeh and Desoer (1963). Theorem 7.1: Mapping Theorem For fixed frequency ω R, conv p(jω, Q) = conv {p(jω, q 1 ), p(jω, q 2 ),..., p(jω, q L )}. With regard to applicability and usefulness of this result, comments very similar to those made in Section 7.7 about the construction of the value set for affine polynomials can be stated. Figure 7.8 illustrates the Mapping Theorem for the polynomial p(s, q) = s 3 + (q 2 + 4q 3 + 2q 1 q 2 )s 2 + (4q 2 q 3 + q 1 q 2 q 4 )s + q 3 + 2q 1 q 3 q 1 q 2 (q 4 0.5) with four uncertain parameters q 1, q 2, q 3, and q 4 each bounded in the interval [0, 1] and frequency ω = 1 rad/s. The true value set shown in this figure is obtained via random generation of 10,000 samples uniformly distributed in the set Q.

16 7-16 Control System Advanced Methods Im Re FIGURE 7.8 Value set and its convex hull. 7.8 General Uncertainty Structures and Controller Synthesis For general uncertainty structures there is no analytical tool that enables us to construct the value set. For systems with a very limited number of uncertain parameters, a brute force approach, as in the example of Theorem 7.1, can be taken by simply gridding the set Q with a sufficient number of points. With this procedure, one can easily obtain a cloud in the complex plane which approximates the value set. Obviously, this method is not practical for a large number of uncertain parameters and provides no guarantee that robustness is satisfied outside the grid points. Several attempts in the literature aimed at solving this general problem. Among the pioneering contributions along this line we recall the parameter space approach (Ackermann, 1980), techniques for the multivariable gain margin computation (de Gaston and Safonov, 1988) and (Sideris and Sanchez Pena, 1989) and the geometric programming approach (Vicino et al., 1990). As far as the robust synthesis problem is concerned, it is worth mentioning that the methods described in this chapter are suitable for synthesis in a trial-and-error fashion. A practical use of the techniques can be summarized in the following two steps (as illustrated in Section 7.5.2): Synthesize a controller for the nominal plant with any proper technique. Perform robustness analysis taking into account the uncertainties and go back to the previous step if necessary. In the special case of low complexity compensators, the two steps can be combined. This is the case of PID synthesis for which efficient robust design tools have been proposed (Ho et al., 2000). To facilitate robustness analysis for general parametric uncertainty structures and for nonparametric uncertainty, different approaches have been proposed in the literature. Among the others, we recall the μ approach (Zhou et al., 1996; Sanchez Pena and Sznaier, 1998) and the probabilistic methods (Tempo et al., 2005). These approaches can be also used for designing a controller, but μ-synthesis is based on a

17 Robustness Analysis with Real Parametric Uncertainty 7-17 procedure which is not guaranteed to converge to a global solution and probabilistic methods provide a controller which satisfies the required specification only with a given probability. However, both methods are useful tools for many robust control problems. As a final remark, we emphasize that in this chapter we considered only uncertainties which are constant in time. It is known that parameter variations or, even worse, switching, may have a destabilizing effect. There are, indeed, examples of very simple systems that are (Hurwitz) stable for any fixed value of the parameters but can be destabilized by parameter variations. A class of systems which has been investigated is that of Linear Parameter-Varying (LPV) systems. For these systems, it has been established that the Lyapunov approach is crucial. For an overview of the problem of robustness of LPV systems and the link with switching systems, the reader is referred to Blanchini and Miani (2008). References Abate, M., Barmish, B.R., Murillo-Sanchez, C., and Tempo, R Application of some new tools to robust stability analysis of spark ignition engines: A case study. IEEE Transactions on Control Systems Technology, CST-2: Ackermann, J.E Parameter space design of robust control systems. IEEE Transactions on Automatic Control, AC-25: Ackermann, J. in cooperation with Bartlett, A., Kaesbauer, D., Sienel, W., and Steinhauser, R Robust Control, Systems with Uncertain Physical Parameters, Springer-Verlag, London. Ackermann, J., Curtain, R. F., Dorato, P., Francis, B. A., Kimura, H., and Kwakernaak, H. (Editors) Special Issue on robust control. Automatíca, 29: Barmish, B.R New Tools for Robustness of Linear Systems, Macmillan, New York. Barmish, B.R., Hollot, C.V., Kraus, F., and Tempo, R Extreme point results for robust stabilization of interval plants with first-order compensators. IEEE Transactions on Automatic Control, AC-37: Barmish, B.R. and Shi, Z Robust stability of perturbed systems with time delays. Automatica, 25: Barmish, B.R. and Tempo, R The robust root locus. Automatica, 26: Barmish, B.R. and Tempo, R On the spectral set for a family of polynomials. IEEE Transactions on Automatic Control, AC-36: Bartlett, A.C., Hollot, C.V., and Huang, L Root locations of an entire polytope of polynomials: It suffices to check the edges. Mathematics of Control, Signals and Systems, 1: Bhattacharyya, S.P., Chapellat, H., and Keel, L.H Robust Control: The Parametric Approach, Prentice-Hall, Englewood Cliffs, NJ. Blanchini, F. and Miani, S Set-Theoretic Methods in Control, Birkhäuser, Boston. Chockalingam, G. and Dasgupta, S Minimality, stabilizability and strong stabilizability of uncertain plants. IEEE Transactions on Automatic Control. AC-38: Dasgupta, S Kharitonov s theorem revisited. Systems and Control Letters. 11: de Gaston, R.R.E. and Safonov, M.G Exact calculation of the multiloop stability margin. IEEE Transactions on Automatic Control, AC-33: Djaferis, T.E Robust Control Design: A Polynomial Approach, Kluwer Academic Publishers, Boston. Doyle, J.C., Francis, B.A., and Tannenbaum, A.R Feedback Control Theory, Macmillan, New York. Frazer, R.A. and Duncan, W.J On the criteria for the stability of small motion. Proceedings of the Royal Society, A, 124: Fu, M Computing the frequency response of linear systems with parametric perturbations. Systems and Control Letters, 15: Ho, M.-T., Datta, A., and Bhattacharyya, S.P. 2000, Structure and Synthesis of PID Controllers, Springer-Verlag, London. Horowitz, I.M Synthesis of Feedback Systems, Academic Press, New York. Horowitz, I Survey of Quantitative Feedback Theory (QFT). International Journal of Control, 53: Kharitonov, V.L Asymptotic stability of an equilibrium position of a family of systems of linear differential equations. Differential nye Uraveniya, 14: Kogan, J Robust Stability and Convexity. Lecture Notes in Control and Information Sciences, Springer- Verlag, London. Marden, M Geometry of Polynomials, Mathematical Surveys, No. 3, American Mathematical Society, Providence, RI. Mikhailov, A.W Methods of harmonic analysis in control theory. Avtomatika i Telemekhanika, 3:27 81.

18 7-18 Control System Advanced Methods Minnichelli, R.J., Anagnost, J.J., and Desoer, C.A An elementary proof of Kharitonov s theorem with extensions. IEEE Transactions on Automatic Control. AC-34: Rantzer, A Stability conditions for polytopes of polynomials, IEEE Transactions on Automatic Control, AC-37: Sanchez Pena, R. and Sznaier, M Robust Systems, Theory and Applications, John Wiley and Sons, New York. Sideris, A. and Sanchez Pena, R.S Fast calculation of the multivariable stability margin for real interrelated uncertain parameters. IEEE Transactions on Automatic Control, AC-34: Tempo, R., Calafiore, G., and Dabbene, F Randomized Algorithms for Analysis and Control of Uncertain Systems, Springer-Verlag, London. Vicino, A., Tesi, A., and Milanese, M Computation of nonconservative stability perturbation bounds for systems with nonlinearly correlated uncertainties. IEEE Transactions on Automatic Control, AC- 35: Zadeh, L.A. and Desoer, C.A., Linear System Theory, McGraw-Hill, New York. Zhou, K., Doyle, J.C., and Glover, K Robust and Optimal Control, Prentice-Hall, Upper Saddle River, NJ.