M. G. Safonov, #Stability margins of diagonally perturbed multivariable feedback systems", IEE Proc., 129-D, pp. 251#256, #1982#.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....function of a stable linear system, and A is a stable operator that represents the uncertainties that arise from various sources such as modeling errors, neglected or unmodeled dynamics or parameters, etc. Such control system models have found wide acceptance in robust control; see for example [1, 2, 3, 4]. From the physical laws governing the system and from the modeling procedures used to arrive at the paradigm in Fig. 1, the uncertainty A is usually known or assumed to possess various additional properties. Common examples are that A is structured (i.e. diagonal or block diagonal) that it is ....

....robust stability of the closed loop system shown in Fig. 9(a) P is the parameter dependent plant, with transfer function given by ai( q bi) P(s) diag(px(s) p2(s) pi(s) s2 2cis 1 and C is the controller with the transfer function s2 s 1 C(s) 0. 3 (s 1) s 2) ai 6 [0,1] bi 6 [1,2], ci 6 [1,2] The problem now is to ascertain the stability of this system for all allowable values of the parameters. a) Block diagram of the parameter dependent system. b) Block diagram redrawn in our framework. Figure 9: Example 2: Stability analysis of a parameter dependent system. ....

[Article contains additional citation context not shown here]

M. G. Safonov. Stability margins of diagonally perturbed multivariable feedback systems. lEE Proc, 129-D(6):251-256, November 1982.

....structure, while in (43) they do not. Avoiding this growth in the set of possible per turbations, and the corresponding unnecessary strength ening of the robust performance specification (43) is the motivation behind Doyle s introduction of the concept of the structured singular valve [72] See [69] for the case whe re the perturbation transfer matrix is restricted to be diagonal. This modern framework allows many different types of perturbations to be considered, and not just perturbations on the loop gain. To give one example, consider a system with a so called additive plant perturbation, ....

M.G. Safonov, "Stability margins of diagonally perturbed multivariable feed-back systems," lEE Proc., vol. 129-D, pp. 25'1 256, 1982.

.... These two functions must satisfy the two following conditions: R1) Phi lb (Q) Phi max (Q) Phi ub (Q) R2) As the maximum half length of the sides of Q, denoted by size(Q) goes to zero, the difference between upper and lower bounds uniformly converges to zero, i.e. 8 ffl 0 9 ffi 0 8 Q Q init size(Q) ffi = Phi ub (Q) Gamma Phi lb (Q) ffl: Roughly speaking, then, the bounds Phi lb and Phi ub become sharper as the rectangle shrinks to a point. We describe the algorithm briefly (for a detailed description as well as for a discussion of convergence issues, see [ 1 ] ....

....can be used to compute Phi ub . Of course, more sophisticated bounds can be used. A local optimization procedure can be used to search for a (locally) worst parameter value, which would give a good lower bound. The upper bound can be vastly improved by scaling (see Doyle [ 5 ] Safonov [ 8 ] or other techniques for approximating the structured singular value (see Fan and Tits [ 6 ] 4. An Example We consider a mechanical plant consisting of two masses connected by a spring with the lefthand mass driven by a force, as shown in figure 2 below. The parameters are the masses and ....

M. G. Safonov. Stability margins of diagonally perturbed multivariable feedback systems. IEE Proc., 129-D:251--256, 1982.

....in guaranteeing the robust stability of the exponentially weighted closed loop system. Eliminating or reducing this conservatism of the SGT that arises due to structured perturbations is a major area of research in itself (structured singular value [40] scaling or the scaled singular value [41]) The scaled singular value (which we will abbreviate as SSV) is directly relevant to our problem, and we will give a brief and informal discussion here. The motivation for the SSV arises from the following simple observation: The system shown in Figure 3 is equivalent to the system in Figure 1 ....

M. G. Safonov, "Stability margins of diagonally perturbed multivariable feedback systems", IEE Proc., 129-D, pp. 251--256, (1982).

.... drawback of closed loop convex loop shaping is that margin specifications for MAMS systems that are expressed in terms of singular values or more general sector conditions are often overly conservative when one considers more detail about the types of plant variations that can occur (see [9, 25, 26, 28, 12]) Scaling can greatly reduce the conservatism of these specifications. While a fixed scaling preserves the closed loop convexity of loop shaping specifications, specifications involving optimal (variable) scaling are not closed loop convex. This drawback is not present for SASS control systems. ....

M. G. Safonov. Stability margins of diagonally perturbed multivariable feedback systems. IEE Proc., 129-D:251--256, 1982.

....the thousands of papers and much real progress, robust control theory has not yet provided a straightforward or direct solution to simple and realistic problems of robust control. A notable development that addresses this issue is analysis and synthesis, pioneered by Doyle, Safonov, and others [16, 6]. They recognized that describing plant variations in terms of singular values is rarely realistic, and just a case of forcing a practical problem to fit into the hypothesis of a theorem (the Small Gain theorem in this case) Instead they describe uncertainties in much more realistic ways: they ....

M. G. Safonov. Stability margins of diagonally perturbed multivariable feedback systems. IEE Proc., 129-D:251--256, 1982.

.... Willems and many others [41, 42, 43, 29, 28, 31, 4, 26] the absolute stability theory with extraordinary contributions from Yakubovich and Popov [32, 33, 34, 35, 36, 37, 38, 24] and nally the robust control eld with contributions from, for example, Doyle, Safonov, Zames, and many others [5, 22, 1, 6, 44, 27]. The relationship is indicated in Figure 2 It was A. Megretski, originally from Yakubovich group at S.t Petersburg state university, who rst started to merge the western input output tradition with the absolute stability theory of Soviet Union into uni ed framework. Some of the early work was in ....

M.G. Safonov. Stability margins of diagonally perturbed multivariable feedback systems. IEE Proceedings, 129(6):251-256, November 1982.

....= j I A 0 e j A 1 ) 1 . The computation of kT (j )k 1 is, to our knowledge, still an open problem. Similar formulas are derived in [89] for the case of coupled delay di erential and di erence equations. 3. 2 Structured singular values At the beginning of the 1980 s, Doyle [24] and Safonov [91] pointed out the importance of robust stability of systems with structured uncertainties. For instance, block diagonal perturbation matrices appear in the case of simultaneous variations in the gains and phases of several system components. Doyle argued that any norm bounded perturbation problem, ....

M.G. Safonov. Stability margins of diagonally perturbed multivariable feedback systems. In IEE Proc. Pt. D, volume 129, pages 251-256. 1982.

....theorem. The method is illustated on two similar examples showing the virtues of the mixed approach. Keywords: Structured uncertainty, mixed mu synthesis, passivity. 1 Introduction Structured singular values ( is used for analysis and synthesis of systems with uncertainties or perturbations [1, 6]. Complex uncertainties representing dynamic uncertainties can be treated in the synthesis procedure using the D K iteration scheme. Even if this algorithm is not guaranteed to converge to the global minimum, it provides a good design in most applications. Recently also real (parametric) ....

M. Safonov. Stability margins of diagonally perturbed multivariables feedback systems. In IEEE Proceedings of the Conference on Decision and Control, 1981. 12

....which stability of the perturbed system is guaranteed. In analysis, the critical constraint of the resulting minimization problem has the form det[I M Delta] 0, where M is a given square matrix, and Delta is a diagonal matrix, containing the unknown parameters on its diagonal; see e.g. [6, 12, 16]. The constraint above is thus a multivariable polynomial in the parameters. The norm on the parameters is the max norm, kak 1 = max i ja i j. In [19] or [17] and [18] more general stability domains are considered, which is useful when a pole placement strategy is in mind. Performance ....

....bounds for 1 ; this procedure either gives the exact value of 1 , or enables to reduce the size of the problem. The method may fail, but has little computational cost. A secondary contribution concerns the quantity OE(M; r) Delta = inf D2D(r) k DMD Gamma1 k F : 1:5) It has been noted in [16], without theoretical justification, that the optimal scaling for problem (1.5) provides an excellent initial guess for problem (1.4) We shall justify theoretically this result. Some other approaches A different approach was initiated by Kharitonov theorem, where stability of a certain class of ....

[Article contains additional citation context not shown here]

M.G. Safonov. Stability margins of diagonally perturbed multivariable feedback systems. IEE Proceedings, 129-D(6), 1982.

....Many nonlinear quasiconvex functions can be represented in the form of a GEVP with appropriate A, B, and C (see [1] 3. LMI problems in systems and control 3.1. Matrix scaling problem The problem of similarity scaling a matrix to minimize its norm appears in several control applications [2, 3, 4] (see also [5] and [6] for a related problem) Given M 2 C m Thetam , the optimal diagonally scaled norm of M is defined as (M ) Delta = inf D 2 P fl fl DMD Gamma1 fl fl ; where P is the set of diagonal non singular matrices of size m. Note that (M ) fl if and only if there exists ....

M. G. Safonov. Stability margins of diagonally perturbed multivariable feedback systems. IEE Proc., 129-D:251--256, 1982.

....largest singular value of #. When D = #I : # # C , M) is equal to the spectral radius of M , and when D = C nn , M) is equal to the maximum singular value of M . For an introduction to structured singular values and their importance in system analysis and design, see [Doyle, 1982] [Safonov, 1982] and [Zhou et al. 1995] Many researchers had worked on algorithms for computing exactly, under various uncertainty structures, until it was established that the problem of deciding whether (M) # 1 is NP hard for the following cases: i) If M is real and each block # i is a real multiple of ....

Safonov, M. G. (1982). Stability margins of diagonally perturbed multivariable feedback systems. in Proc. IEE-D, 129, 251-256.

....bound, the small gain theorem provides a necessary and sufficient condition for robust stability. When Delta is structured say diagonal the small gain condition is no longer necessary for stability; diagonal scalings can then 2 be used to derive less conservative robust stability conditions [4, 5]. In addition, if Delta is a memoryless time invariant sector bounded nonlinearity, the celebrated Popov criterion yields a sufficient condition for robust stability (see for example, 2] When Delta is LTI or parametric, the well known analysis and Km analysis methods provide sufficient ....

M. G. Safonov, "Stability margins of diagonally perturbed multivariable feedback systems," IEE Proc., vol. 129-D, pp. 251--256, Nov. 1982.

....of a stable linear system, and Delta is a stable operator that represents the uncertainties that arise from various sources such as modeling errors, neglected or unmodeled dynamics or parameters, etc. Such control system models have found wide acceptance in robust control; see for example [1, 2, 3, 4]. From the physical laws governing the system and from the modeling procedures used to arrive at the paradigm in Fig. 1, the uncertainty Delta is usually known or assumed to possess various additional properties. Common examples are that Delta is structured (i.e. diagonal or block diagonal) ....

M. G. Safonov. Stability margins of diagonally perturbed multivariable feedback systems. IEE Proc., 129-D(6):251--256, November 1982.

....flDMD Gamma1 fl fl fl fi fi fi D 2 C n Thetan ; D is diagonal and nonsingular o : 1) We will refer to f min (M) as the optimally scaled singular value of M . Problem (1) arises in the robustness analysis of control systems with structured uncertainties. For further details, see references [6, 7, 2]. Reformulation as a convex optimization problem. We note that f(M; jDj) f(M;D) we also observe that f(M;D) is homogeneous of degree zero in D, that is, f(M; ffD) f(M;D) for all nonzero ff 2 C. Therefore, we may rewrite (1) as f min (M) inf nfl fl fle D Me GammaD fl fl fl fi fi fi ....

M. G. Safonov, Stability margins of diagonally perturbed multivariable feedback systems, IEE Proc., 129-D (1982), pp. 251--256.

.... multipliers) An introduction to these methods can be found in the book by Desoer and Vidyasagar [2] A second approach to the problem of robust stability of control systems with structured uncertainties is the modern (or structured singular value) approach, pioneered by Doyle [3, 4] and Safonov [5, 6]. This approach relies on deriving sufficient conditions for the robust stability of the system in Figure 1 through simple linear algebraic techniques. Our main objective in this paper is to show explicitly and rigorously the connections between these two approaches in the case when Delta is ....

M. G. Safonov. Stability margins of diagonally perturbed multivariable feedback systems. IEE Proc., 129-D:251--256, 1982.

....uncertainty Delta is assumed to possess various additional properties. Common examples are that Delta is structured (i.e. diagonal or block diagonal) that it is linear time invariant or real constant etc. Such control system models have found wide acceptance in robust control; see for example [D, S, PD, L, ZDG]. We will henceforth refer to the system in Figure 1 as the P Delta interconnection, and denote it by I(P; Delta) The Small Theorem is the foundation of the structured singular value approach towards assessing the stability of systems affected by structured uncertainties. Specifically, the ....

M. G. Safonov, Stability margins of diagonally perturbed multivariable feedback systems, IEE Proc., 129-D (1982), 251--256.

....We conclude again that the volume of the set of bad plants is at most ffl bad. For discrete time systems, the same argument holds taking u(q) kS(z; q)k 1 : sup 2[0;2 ] jS(e j ; q)j. Application 3: Let u(q) be equal to the inverse of the structured singular value ; see e.g. 6] and [7]. If the samples q i , i = 1; 2; N , are randomly generated in Q according to a uniform distribution and if u(q i ) 1= for all i = 1; 2; N and N N 0 , then, with a probability at least 1 Gamma ffi, the volume of the set of plants with robustness margin no greater than 1= ....

M. G. Safonov, "Stability margins of diagonally perturbed multivariable systems," IEE Proceedings, vol. 129, part D, pp. 251-256, 1982.

....are achieved for rank one problems with all the methods. Comparison of the three approaches for general random matrices will be given through numerical examples. 1 Motivation and Background The computation of , the structured singular value, which was introduced by Doyle in [3] also Safonov in [8]) has been of many researchers interests for years (see [9] and the references therein) By focusing on the worst case performance of the control systems with structured uncertainty, it gives guaranteed but sometimes conservative stability margins. The computation problem itself is in general NP ....

M.G. Safonov. Stability margins of diagonally perturbed multivariable feedback systems. In IEE Proceedings, volume Part D, 129(6), pages 251--256, 1982.

....parametric uncertainties, robustness 1 Introduction The structured singular value ( were introduced by Doyle [3] as a tool for analyzing linear systems with parametric (real) and dynamic (complex) uncertainties. Safonov introduced a similar stability margin (k m ) at about the same time [10]. The magnitude of the smallest uncertainty within a given class that could cause the system to become unstable is given by km = 1= The structured singular value is thus important when analyzing and designing robust controllers. Normally, when analyzing continuous time systems the analysis is ....

M. Safonov. Stability margins of diagonally perturbed multivariables feedback systems. In IEEE Proceedings of the Conference on Decision and Control, 1981.

....at least 1 Gamma ffi, the volume of the set of bad plants is at most ffl bad. For discrete time systems, the same argument holds taking u(q) kS(z; q)k 1 : sup 2[0;2 ] jS(e j ; q)j. Application 3: Let u(q) be equal to the inverse of the structured singular value ; see e.g. 4] and [11]. If the samples q i , i = 1; 2; N , are randomly generated in Q according to a uniform distribution and if u(q i ) 1= for all i = 1; 2; N and N N 0 , then, with a probability at least 1 Gamma ffi, the volume of the set of plants with robustness margin no greater than ....

M. G. Safonov, "Stability Margins of Diagonally Perturbed Multivariable Systems, " IEE Proceedings, vol. 129, part D, pp. 251-256, 1982.

....developed at USC by the authors. The early version of the Robust Control Toolbox called LINF was distributed widely [2] The Robust Control Toolbox includes tools which facilitate the following: Robust Analysis Singular Values [12, 29] Characteristic Gain Loci [25] Structured Singular Values [31, 32, 13]. Robust Synthesis synthesis [33, 15] LQG LTR, Frequency Weighted LQG [12, 29] H 2 , H [16, 34, 18, 36, 37, 28, 24, 19] Robust Model Reduction Optimal Descriptor Hankel (with Additive Error Bound) 37] Schur Balanced Truncation (with Additive Error Bound) 39] Schur Balanced ....

....is robustness criterion. The quantity is size of the smallest uncertainty , as measured by the singular value at each frequency, that can destabilize the closed loop system. The function is the so called diagonally perturbed multivariable stability margin (MSM) introduced by Safonov and Athans [30, 32], the reciprocal of which is known as , the structured singular value (SSV) 13] i.e. More precisely, when is not present, this problem is called the robust stability problem. Doyle, Wall and Stein [14] introduced the extra uncertainty to represent the performance specification which, ....

[Article contains additional citation context not shown here]

M. G. Safonov, "Stability Margins of Diagonally Perturbed Multivariable Feedback Systems," IEE Proc., 129 (Pt. D), 2, pp. 252-255, November 1982

....developed at USC by the authors. The early version of the Robust Control Toolbox called LINF was distributed widely [2] The Robust Control Toolbox includes tools which facilitate the following: Robust Analysis Singular Values [12, 29] Characteristic Gain Loci [25] Structured Singular Values [31, 32, 13]. Robust Synthesis synthesis [33, 15] LQG LTR, Frequency Weighted LQG [12, 29] H 2 , H [16, 34, 18, 36, 37, 28, 24, 19] Robust Model Reduction Optimal Descriptor Hankel (with Additive Error Bound) 37] Schur Balanced Truncation (with Additive Error Bound) 39] Schur Balanced ....

....structure as D. This is a nonconvex optimization problem, which is impractical to solve exactly as mentioned earlier. 9 Generalized Small Gain Theorem: If nominal M(s) is stable, then the perturbed system is stable for all stable if and only if for all . Diagonal Scaling In 1981, Safonov [31] introduced the concept of diagonal scaling to compute the upper bounds of MSM. See Figure 1 12. sD( IMD ( 1 det IM D ( 0= K m = aM( a=M( rM( M ( sM( Dd=Ifor some d C M( r=M( DC nn M( s=M( max U U r MU ( M( IM D ( 1 D i for which D i 1 K m Mjw( 1 wR ....

[Article contains additional citation context not shown here]

M. G. Safonov, "Stability Margins of Diagonally Perturbed Multivariable Feedback Systems," Proc. IEEE Conf. on Decision and Control, San Diego, CA December 16-18, 1981.

No context found.

M. G. Safonov, #Stability margins of diagonally perturbed multivariable feedback systems", IEE Proc., 129-D, pp. 251#256, #1982#.

No context found.

M.G. Safonov. Stability margins of diagonally perturbed multivariable feedback systems. In IEE Proceedings, volume Part D, 129(6), pages 251--256, 1982.

No context found.

M.G. SAFONOV. Stability margins of diagonally perturbed multivariable feedback systems. IEE Proc., 129-D(6):251--256, November 1982.

No context found.

M.G. Safonov, "Stability margins of diagonally perturbed multivariable feedback systems," IEE Proceedings, Part D, vol. 129, no. 6, pp:251--256, 1982.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC