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Robust Control: The Parametric Approach
Copyright ©J IFAC Advances in Control Education. Tokyo. Japan. 1994
ROBUST CONTROL: THE PARAMETRIC APPROACH
s.
P. Bhattacharyya" and L. H. Keel""
" Department of Electriaal Engineering, Tezcu ABM Univerriq" College StGtion, Tezcu, USA ". Center of E:r:eellence in InformAtion S,Iltenu, Tenne.. ee StGte Univerriq" Ntuhville, Tenne.. _, USA
.hatr.et. This paper presents the development of a new course in the control curriculum dealinS with the control of systems subject to parametric uncertainty. This branch of Robust Control has been the subject ofintenaive research and development over the last 15 years, followinS a breakthroush in 1978, by the Ruuian control theorist V. L. Kharitonov. The result of this development is that there now exists a aipilicant number of elesant and practically results in the field of robust control under real parameter uncertainty. This subject is rich in theoretical content, easy to motivate from a practical standpoint and requires just the risht level of mathematics to be tausht as a fundamental discipline to ensineera and scientists.
Key Worda. Robustness, robust control, stability, parametric approach
1. INTRODUCTION
performance under various perturbations. In Classical Control Theory as developed by Nyquist and Bode in the 1930-50's, robustness specifications for control systems acquired the form of gain and phase margins and time delay tolerance of the closed loop system. These are measures of the ability of the proposed design to tolerate errors and uncertainty in the nominal model of the plant. These notions dominate engineering design in the control field till today.
Feedback is employed to obtain predictable outputs from a system despite the presence of uncertainty. Indeed it has been argued that the very objective of control system design is to combat the effects of uncertainty and therefore that the term Robust Control is tautological and should be abolished. However it is important to realize that the term Robust Control exists, because even in the recent past, seemingly attractive control design methodologies such as the Linear Quadratic Optimal Regulator (LQR) and the Geometric Approach to Linear Multivariable Control have been proposed that were later found to be fundamentally deficient because of the lack of robustness.
In the 1960's a new approach to control systems was developed. In this theory Robustness was initially ignored in preference to Optimality. An important outcome was Kalman's theory (1960) of the LQR wherein the dynamics of a linear time invariant state space system was optimised with respect to a quadratic cost index. Later it was shown, by Kalman (1964) that the optimal state feedback gains produced by the LQR theory also possessed excellent guaranteed gain and phase margins. It was only in the late 1970's that it was eventually established by Doyle and Stein (1979), using simple examples, that the guaranteed margins of the state feedback LQ regulators in fact vanished under an output feedback implementation. Although this came as a revelation to many in the control community, some control theorists had always felt suspicious of the promised benefits of optimality and the circuitous route towards obtaining output feedback controllers via state feedback and observers. In particular Rosenbrock (1966) showed the nonro-
Robust Control is a basic and old subject in Control Theory. It rests on the fundamental results in stability theory developed by Maxwell, Routh, Hurwitz , Hermite and Lyapunov in the 19th century and Nyquist, Bode and others in the 20th century. It is motivated by the practical need, in engineering systems, for the pre6ervation of 6tabilitll and performance under perturbation6 of the nominal 61J6tem. In particular it is necessary because almost all the existing techniques for the analysis and design of control systems are based on a linear time-invariant mathematical model of the object to be controlled, namely the "plant". Even the "simplest" plants do not adequately fit this description and therefore it is necessary to evaluate any proposed design by evaluating its
49
and nonlinear optimisation techniques to deal with this problem.
bustness of optimal controllers with respect to feedback loop failures, in his 1966 paper "Good, Bad or Optimal?" and Pearson (1968) advocated as far back as 1968, the direct design of output feedback compensators.
These isolated attempts were not sufficient to spark a large interest in this important but difficult problem area. If one were to attempt to use the mainstream results of robust control theory up until the late 1980's the only option available to systematically address real parametric uncertainty in control systems was to use the HOC or J.' framework. These approaches would over bound real parameter uncertainty by norm bounded perturbations, which essentially complexify real parameters and so are inherently conservative.
The Doyle and Stein's (1979) paper prepared the ground for the next phase of development of control theory, namely HOC optimal control (Zames, 1981). wherein optimisation of the sensitivity function was proposed as a design strategy for robustification. The goal of controller design, in this approach, is to minimise the norm of the "disturbance transfer function" over the set of output feedback stabilising controllers. Two important tools used in the HOC theory are the Small Gain Theorem (Zames, 1963) and the YJBK parametruation (Youla et 41. 1976). The Small Gain Theorem gives conditions for stability robustness of systems under transfer function norm bounded perturbations. The YJBK parametruation characterises the family of all stabilising feedback controllers for a fixed plant in terms of a free "parameter". This free parameter is any element of the set of stable proper matrices. The YJBK parametrisation thus reduces the solution of optimal control problems to the determination of the minimising value of the cost function over this set. An elegant theory of controller synthesis, utilising this phil08ophy, developed within the HOC framework is now in place (Doyle et 0.1. 1989).
In 1978 Kharitonov published a surprising result on the stability of interval polynomials (Kharitonov, 1978). Kharitonov's Theorem established that the Hurwitz stability of a real interval polynomial family of arbitrary degree could be ascertained from that of four fixed corner polynomials. The startling elegance and immediate usefulness of this result set off a flurry of interest worldwide in the real parametric robust control problem. For the first time control specialists felt that control problems with real parametric uncertainty could be dealt with in an elegant, insightful and computationally effective manner. It showed that a rich underlying structure is present in the stability region geometry, and proper exploitation of this structure yields far more meaningful results than blind formulation of design objective as optimisation problems. The number 4 of polynomials also showed that much of the discussion about computational complexity of Robust Control problems could be irrelevant. Rapid advances have taken place in the field of robust parametric stability and control since Kharitonov's breakthrough. These results provide the most effective tools to date to deal with the above class of problems. The objective of this course is to provide a reasonably complete account of these developments that can be used by researchers, students and engineers.
In 1982 Doyle (1982) pointed out that simply considering norms of transfer functions is conservative and it is necessary to study robustness under the constraint that the admissible perturbations respect the system structure, given by the definition of subsystems and interconnections. This led to the development of the socalled J.' theory where stability margins are proposed to be measured by using "norms" of transfer function matrices possessing, for instance, a fixed-zero structure. Relative to these hectic developments, the area of robustness under real parameter uncertainty remained, with two notable exceptions, largely undeveloped. In the early 1960's, Horowitz (1963) emphasised that the central objective offeedback control system design was combating real parameter uncertainty. Horowitz developed the socalled Quantitative Feedback Theory (QFT) approach to control system design where frequency response techniques were utilised to study the effect of real parametric uncertainty, to compensate for its adverse effects and to expose design trade-offs inherent to the system under consideration. In the late 1960's, Siljak (1969) also focussed on the problem of real parametric uncertainty and developed parameter plane methods
2. COURSE CONTENT The entire course consists of fourteen units. These units roughly correspond to the Chapters of the book "Robust Control: The Parametric Approach" coauthored by Bhattacharyya, Chapellat and Keel which will be published this year. Accompanying the book is a software package based on Matlab. This package has implemented m08t of the computations associated with the results and in particular can be used to solve the exercises given in the book. In addition to the 14 units described below the course also includes a summary of the main results in HOC and J.' theory. 50
We begin Unit 1 with a new look at classical stability criteria for a single polynomial. We consider a family of polynomials where the coefficients depend continuously on a set of parameters, and introduce the Boundary Crossing Theorem which establishes, roughly, that along any continuous path in parameter space connecting a stable polynomial to an unstable one the first encounter with an unstable polynomial must be with one which has unstable roots only on the stability boundary. This is a straightforward consequence of the continuity of the roots of a polynomial with respect to its coefficients. This simple theorem in fact serves as the unifying idea for the entire subject of robust parametric stability as presented in this course. In Unit 1 we give simple derivations of the Routh and Jury stability tests as well the Hermite-Bieler Theorem based on this result.
norms. The main conceptual tool is once again the Boundary Crossing Theorem, and its computational version the Zero Exclusion Principle. We consider the special case in which p varies in a box. For linearly parametrized systems this case gives rise to a polytope of polynoll1iala in coefficient space. For such families we establish the important fact that stability is determined by the exposed edges and in special cases by the vertices. This result carries over to complex polynomials as well as to quasipolynomials which arise in control systems containing time-delay. The Tsypkin-Polyak locWl for stability margin determination is also described for such systems. In Unit 5 we turn our attention to the robust stability of interval polynomial families. We state and prove Kharitonov's theorem which deals with the Hurwitz stability of such families, treating both the real and the complex cases. This theorem is interpreted as a generalization of the Hermite-Bieler Interlacing Theorem and a simple derivation is also given using the Vertex Lemma of Unit 2. An important extremal property of the Kharitonov polynomials is established, namely that the worst case real stability margin in the coefficient space over an interval family occurs on the Kharitonov vertices. This fact is used to give an application of Kharitonov polynomials to robust state feedback stabilisation. Finally the problem of Schur stability of interval polynomials is studied. Here it is established that a subset of the exposed edges of the underlying interval box suffices to determine the stability of the entire family.
In Unit 2 we study the problem of determining the stability of a line segment joining a pair of polynomials. The pair is said to be strongly stable if the entire segment is stable. This is the simplest case of robust stability of a parametrised family of polynomials. We develop necessary and sufficient conditions for strong stability in the form of the Segment Lemma treating both the Hurwitz and Schur cases. We also establish the Vertex Lemma which gives some useful sufficient conditions for strong stability of a pair based on certain standard forms for the difference polynomial. These forms are examples of the notion of Convex Directions which we also discuss. The Segment and Vertex Lemmas are used in proving the Generalized Kharitonov Theorem in Unit 7.
In Unit 6 we state and prove the Edge Theorem. This important result due to Bartlett, Hollot and Lin, shows that the root space boundary of a polytope of polynomials is exactly determined by the root loci of the exposed edges. Since each exposed edge is a one parameter family of polynomials this result allows us to constructively determine the root space of a family of linearly parametrised systems.
In Unit 3 we consider the problem of determining the robust stability of a parametrised family of polynomials where the parameter is just the set of polynomial coefficients. Using orthogonal projection we derive quasi-closed form expressions for the real stability radius in coefficient space in the Euclidean norm. We then describe the Tsypkin-Polyak locus which determines the stability radius in the [P norm for arbitrary p. Then we deal with a family of complex polynomials where each coefficient is allowed to vary in a disc in the complex plane and give a constructive solution to the problem of robust stability determination in terms of the HOC norm of two transfer functions.
In Unit 7 we generalize Kharitonov's problem by considering the robust Hurwitz stability of a linear combination of interval polynomials. This formulation is motivated by the problem of robust stability of a feedback control system containing a fixed compensator and an interval plant in its forward path. The solution is provided by the Generalized Kharitonov Theorem which shows that for a compensator to robustly stabilise the system it is sufficient that it stabilises a prescribed set of line 6egmen't8 in the plant parameter space. Under special conditions on the compensator it suffices to stabilise the Kharitonov vertices. These line segments, called
In Unit 4 we extend these results to the parameter space concentrating on the case of linear parametrisation where the polynomial coefficients are affine linear functions of the real parameter vector p. We develop the procedure for calculating the real parametric stability margin measured in the ll, l2 and too 51
In Unit 14 some examples of interval identification and design are desc:ibed as a demonstration of practical use of the theory described in the course.
generalised Kharitonov segDlents, play a fundamental characterizing role in later units. In Unit 8 we develop extreDlal frequency domain properties of linear interval control systems. The generalized Kharitonov segments are shown to possess boundary properties that are useful for generating the frequency domain templates and the Nyquiat, Bode and Nichols envelopes of linear interval systems. The extremal gain and phase margins of these systems occur on these segments. We show how these concepts are useful in extending classical design techniques to linear intervalaystems.
As stated earlier, the main results of H oo and ~ theory are also summarised in the course. In the accompanying text these topics are given in two appendices. We believe that this information along with the results in the area of real parametric uncertainty, that are covered in the course can be combined in a skillful and imaginative way to produce effective solutions to control system analysis, synthesis and design problems.
In Unit 9 we consider the robust stability and performance of control systems subjected to parametric uncertainty as well as unstructured perturbations. The parameter uncertainty is modeled through a linear interval system whereas two types of unstructured uncertainty are considered, namely H oo norm bounded uncertainty and nonlinear sector bounded perturbations. The latter class of problems is known as the absolute stability or Lur'e problem. We present robust versioDJI of the Small Gain TheoreDl and the Absolute Stability or Lur'e problem which allow us to quantify the worst case parametric or unstructured stability margins that the closed loop system can tolerate.
3. REFERENCES Kharitonov, V. L. {1978} Asymptotic stability of an equilibrium position of a family of systems of linear differential equations. Differential U7'IItmen, 14, 2086 - 2088. Kalman, R. E. {1960} Contribution to the theory of optimal control. Bol. Soc. Mtdem. Mezico, 102 - 119. Kalman, R. E. {1964} When is a linear control system optimal. ASME 7Nn.t. Ser. D (J. of Btuic Eng.), 51 - 60. Doyle, J. C. and G. Stein {1979} Robustness with observers. IEEE 7Nn.t. on Auto. Contr., AC - 24, 607 - 611. R08enbrock, H. H. {1966} Good, bad, or optimal. IEEE 7Nn.t. on Auto. Contr.. Pearson, J. B. (1968) Compensator for dynamic optimisation. Int. J. of Contr., 9, 473. Zames, G. (1981) Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms, and approximate inverses. IEEE 7Nn.t. on Auto. Contr., AC-26, 301 - 320. Zames, G. (1963) Functional analysis applied to nonlinear feedback systems. IEEE 7Nr08. on Gir. Theo., 392 - 404. Youla, D. C., H. A. Jabr, and J. J. Bongiorno {1976} Modem Wiener - Hopf design of optimal controllers - Part 11: the multivariable case. IEEE 7Nn.t. on Auto. Contr., AC-21, 319 - 338. Doyle, J. C., K. Glover, P. P. Khargonekar, and B. A. Francis (1989) State space solution to standard H2 and Hoo control problems. IEEE 7Nn.t. on Auto. Contr., AC - 34, 831 - 847. Doyle, J. C. (1982) Analysis of feedback systems with structured uncertainties. Proc. of lEE - D, 129, 242 - 250. Horowitz, 1. (1963) Synthe6u of Feedback Control Sy6tem.. New York, NY: Academic Press. Siljak, D. D. (1969) Nonlinear Sy6tem.: Pa7'llmetric Analy6u and De6ign. New York: Wiley, 1969.
Units 10 deals with the robust stability of polynomials containing uncertain interval parameters which appear affine multilinearly in the coefficients. The main tool to solve this problem is the Mapping Theorem described in the 1963 book of Zadeh and Desoer. We state and prove this theorem and apply it to the robust stability problem. In Unit 11 we continue to develop results on multilinear interval systems extending the Generalized Kharitonov Theorem and the frequency domain properties of Units 7, 8 and 9 to the multilinear case. In Unit 12 we deal with parameter perturbations in state space models. The same mapping theorem is used to give an effective solution to the robust stability of state space systems. We also include an important result on the calculation of the real stability radius defined in terms of the operator {induced} norm of a feedback matrix. In Unit 13 we describe some synthesis techniques. To begin with we show how standard results from H OO theory can be exploited to deal with parametric perturbations using the extremal properties developed earlier. Nat we demonstrate a direct procedure whereby any minimum phase interval plant of order n, with m zeros can be robustly stabilized by a fixed stable minimum phase compensator of order n - m - 1.
52
ROBUST CONTROL: THE PARAMETRIC APPROACH
s.
P. Bhattacharyya" and L. H. Keel""
" Department of Electriaal Engineering, Tezcu ABM Univerriq" College StGtion, Tezcu, USA ". Center of E:r:eellence in InformAtion S,Iltenu, Tenne.. ee StGte Univerriq" Ntuhville, Tenne.. _, USA
.hatr.et. This paper presents the development of a new course in the control curriculum dealinS with the control of systems subject to parametric uncertainty. This branch of Robust Control has been the subject ofintenaive research and development over the last 15 years, followinS a breakthroush in 1978, by the Ruuian control theorist V. L. Kharitonov. The result of this development is that there now exists a aipilicant number of elesant and practically results in the field of robust control under real parameter uncertainty. This subject is rich in theoretical content, easy to motivate from a practical standpoint and requires just the risht level of mathematics to be tausht as a fundamental discipline to ensineera and scientists.
Key Worda. Robustness, robust control, stability, parametric approach
1. INTRODUCTION
performance under various perturbations. In Classical Control Theory as developed by Nyquist and Bode in the 1930-50's, robustness specifications for control systems acquired the form of gain and phase margins and time delay tolerance of the closed loop system. These are measures of the ability of the proposed design to tolerate errors and uncertainty in the nominal model of the plant. These notions dominate engineering design in the control field till today.
Feedback is employed to obtain predictable outputs from a system despite the presence of uncertainty. Indeed it has been argued that the very objective of control system design is to combat the effects of uncertainty and therefore that the term Robust Control is tautological and should be abolished. However it is important to realize that the term Robust Control exists, because even in the recent past, seemingly attractive control design methodologies such as the Linear Quadratic Optimal Regulator (LQR) and the Geometric Approach to Linear Multivariable Control have been proposed that were later found to be fundamentally deficient because of the lack of robustness.
In the 1960's a new approach to control systems was developed. In this theory Robustness was initially ignored in preference to Optimality. An important outcome was Kalman's theory (1960) of the LQR wherein the dynamics of a linear time invariant state space system was optimised with respect to a quadratic cost index. Later it was shown, by Kalman (1964) that the optimal state feedback gains produced by the LQR theory also possessed excellent guaranteed gain and phase margins. It was only in the late 1970's that it was eventually established by Doyle and Stein (1979), using simple examples, that the guaranteed margins of the state feedback LQ regulators in fact vanished under an output feedback implementation. Although this came as a revelation to many in the control community, some control theorists had always felt suspicious of the promised benefits of optimality and the circuitous route towards obtaining output feedback controllers via state feedback and observers. In particular Rosenbrock (1966) showed the nonro-
Robust Control is a basic and old subject in Control Theory. It rests on the fundamental results in stability theory developed by Maxwell, Routh, Hurwitz , Hermite and Lyapunov in the 19th century and Nyquist, Bode and others in the 20th century. It is motivated by the practical need, in engineering systems, for the pre6ervation of 6tabilitll and performance under perturbation6 of the nominal 61J6tem. In particular it is necessary because almost all the existing techniques for the analysis and design of control systems are based on a linear time-invariant mathematical model of the object to be controlled, namely the "plant". Even the "simplest" plants do not adequately fit this description and therefore it is necessary to evaluate any proposed design by evaluating its
49
and nonlinear optimisation techniques to deal with this problem.
bustness of optimal controllers with respect to feedback loop failures, in his 1966 paper "Good, Bad or Optimal?" and Pearson (1968) advocated as far back as 1968, the direct design of output feedback compensators.
These isolated attempts were not sufficient to spark a large interest in this important but difficult problem area. If one were to attempt to use the mainstream results of robust control theory up until the late 1980's the only option available to systematically address real parametric uncertainty in control systems was to use the HOC or J.' framework. These approaches would over bound real parameter uncertainty by norm bounded perturbations, which essentially complexify real parameters and so are inherently conservative.
The Doyle and Stein's (1979) paper prepared the ground for the next phase of development of control theory, namely HOC optimal control (Zames, 1981). wherein optimisation of the sensitivity function was proposed as a design strategy for robustification. The goal of controller design, in this approach, is to minimise the norm of the "disturbance transfer function" over the set of output feedback stabilising controllers. Two important tools used in the HOC theory are the Small Gain Theorem (Zames, 1963) and the YJBK parametruation (Youla et 41. 1976). The Small Gain Theorem gives conditions for stability robustness of systems under transfer function norm bounded perturbations. The YJBK parametruation characterises the family of all stabilising feedback controllers for a fixed plant in terms of a free "parameter". This free parameter is any element of the set of stable proper matrices. The YJBK parametrisation thus reduces the solution of optimal control problems to the determination of the minimising value of the cost function over this set. An elegant theory of controller synthesis, utilising this phil08ophy, developed within the HOC framework is now in place (Doyle et 0.1. 1989).
In 1978 Kharitonov published a surprising result on the stability of interval polynomials (Kharitonov, 1978). Kharitonov's Theorem established that the Hurwitz stability of a real interval polynomial family of arbitrary degree could be ascertained from that of four fixed corner polynomials. The startling elegance and immediate usefulness of this result set off a flurry of interest worldwide in the real parametric robust control problem. For the first time control specialists felt that control problems with real parametric uncertainty could be dealt with in an elegant, insightful and computationally effective manner. It showed that a rich underlying structure is present in the stability region geometry, and proper exploitation of this structure yields far more meaningful results than blind formulation of design objective as optimisation problems. The number 4 of polynomials also showed that much of the discussion about computational complexity of Robust Control problems could be irrelevant. Rapid advances have taken place in the field of robust parametric stability and control since Kharitonov's breakthrough. These results provide the most effective tools to date to deal with the above class of problems. The objective of this course is to provide a reasonably complete account of these developments that can be used by researchers, students and engineers.
In 1982 Doyle (1982) pointed out that simply considering norms of transfer functions is conservative and it is necessary to study robustness under the constraint that the admissible perturbations respect the system structure, given by the definition of subsystems and interconnections. This led to the development of the socalled J.' theory where stability margins are proposed to be measured by using "norms" of transfer function matrices possessing, for instance, a fixed-zero structure. Relative to these hectic developments, the area of robustness under real parameter uncertainty remained, with two notable exceptions, largely undeveloped. In the early 1960's, Horowitz (1963) emphasised that the central objective offeedback control system design was combating real parameter uncertainty. Horowitz developed the socalled Quantitative Feedback Theory (QFT) approach to control system design where frequency response techniques were utilised to study the effect of real parametric uncertainty, to compensate for its adverse effects and to expose design trade-offs inherent to the system under consideration. In the late 1960's, Siljak (1969) also focussed on the problem of real parametric uncertainty and developed parameter plane methods
2. COURSE CONTENT The entire course consists of fourteen units. These units roughly correspond to the Chapters of the book "Robust Control: The Parametric Approach" coauthored by Bhattacharyya, Chapellat and Keel which will be published this year. Accompanying the book is a software package based on Matlab. This package has implemented m08t of the computations associated with the results and in particular can be used to solve the exercises given in the book. In addition to the 14 units described below the course also includes a summary of the main results in HOC and J.' theory. 50
We begin Unit 1 with a new look at classical stability criteria for a single polynomial. We consider a family of polynomials where the coefficients depend continuously on a set of parameters, and introduce the Boundary Crossing Theorem which establishes, roughly, that along any continuous path in parameter space connecting a stable polynomial to an unstable one the first encounter with an unstable polynomial must be with one which has unstable roots only on the stability boundary. This is a straightforward consequence of the continuity of the roots of a polynomial with respect to its coefficients. This simple theorem in fact serves as the unifying idea for the entire subject of robust parametric stability as presented in this course. In Unit 1 we give simple derivations of the Routh and Jury stability tests as well the Hermite-Bieler Theorem based on this result.
norms. The main conceptual tool is once again the Boundary Crossing Theorem, and its computational version the Zero Exclusion Principle. We consider the special case in which p varies in a box. For linearly parametrized systems this case gives rise to a polytope of polynoll1iala in coefficient space. For such families we establish the important fact that stability is determined by the exposed edges and in special cases by the vertices. This result carries over to complex polynomials as well as to quasipolynomials which arise in control systems containing time-delay. The Tsypkin-Polyak locWl for stability margin determination is also described for such systems. In Unit 5 we turn our attention to the robust stability of interval polynomial families. We state and prove Kharitonov's theorem which deals with the Hurwitz stability of such families, treating both the real and the complex cases. This theorem is interpreted as a generalization of the Hermite-Bieler Interlacing Theorem and a simple derivation is also given using the Vertex Lemma of Unit 2. An important extremal property of the Kharitonov polynomials is established, namely that the worst case real stability margin in the coefficient space over an interval family occurs on the Kharitonov vertices. This fact is used to give an application of Kharitonov polynomials to robust state feedback stabilisation. Finally the problem of Schur stability of interval polynomials is studied. Here it is established that a subset of the exposed edges of the underlying interval box suffices to determine the stability of the entire family.
In Unit 2 we study the problem of determining the stability of a line segment joining a pair of polynomials. The pair is said to be strongly stable if the entire segment is stable. This is the simplest case of robust stability of a parametrised family of polynomials. We develop necessary and sufficient conditions for strong stability in the form of the Segment Lemma treating both the Hurwitz and Schur cases. We also establish the Vertex Lemma which gives some useful sufficient conditions for strong stability of a pair based on certain standard forms for the difference polynomial. These forms are examples of the notion of Convex Directions which we also discuss. The Segment and Vertex Lemmas are used in proving the Generalized Kharitonov Theorem in Unit 7.
In Unit 6 we state and prove the Edge Theorem. This important result due to Bartlett, Hollot and Lin, shows that the root space boundary of a polytope of polynomials is exactly determined by the root loci of the exposed edges. Since each exposed edge is a one parameter family of polynomials this result allows us to constructively determine the root space of a family of linearly parametrised systems.
In Unit 3 we consider the problem of determining the robust stability of a parametrised family of polynomials where the parameter is just the set of polynomial coefficients. Using orthogonal projection we derive quasi-closed form expressions for the real stability radius in coefficient space in the Euclidean norm. We then describe the Tsypkin-Polyak locus which determines the stability radius in the [P norm for arbitrary p. Then we deal with a family of complex polynomials where each coefficient is allowed to vary in a disc in the complex plane and give a constructive solution to the problem of robust stability determination in terms of the HOC norm of two transfer functions.
In Unit 7 we generalize Kharitonov's problem by considering the robust Hurwitz stability of a linear combination of interval polynomials. This formulation is motivated by the problem of robust stability of a feedback control system containing a fixed compensator and an interval plant in its forward path. The solution is provided by the Generalized Kharitonov Theorem which shows that for a compensator to robustly stabilise the system it is sufficient that it stabilises a prescribed set of line 6egmen't8 in the plant parameter space. Under special conditions on the compensator it suffices to stabilise the Kharitonov vertices. These line segments, called
In Unit 4 we extend these results to the parameter space concentrating on the case of linear parametrisation where the polynomial coefficients are affine linear functions of the real parameter vector p. We develop the procedure for calculating the real parametric stability margin measured in the ll, l2 and too 51
In Unit 14 some examples of interval identification and design are desc:ibed as a demonstration of practical use of the theory described in the course.
generalised Kharitonov segDlents, play a fundamental characterizing role in later units. In Unit 8 we develop extreDlal frequency domain properties of linear interval control systems. The generalized Kharitonov segments are shown to possess boundary properties that are useful for generating the frequency domain templates and the Nyquiat, Bode and Nichols envelopes of linear interval systems. The extremal gain and phase margins of these systems occur on these segments. We show how these concepts are useful in extending classical design techniques to linear intervalaystems.
As stated earlier, the main results of H oo and ~ theory are also summarised in the course. In the accompanying text these topics are given in two appendices. We believe that this information along with the results in the area of real parametric uncertainty, that are covered in the course can be combined in a skillful and imaginative way to produce effective solutions to control system analysis, synthesis and design problems.
In Unit 9 we consider the robust stability and performance of control systems subjected to parametric uncertainty as well as unstructured perturbations. The parameter uncertainty is modeled through a linear interval system whereas two types of unstructured uncertainty are considered, namely H oo norm bounded uncertainty and nonlinear sector bounded perturbations. The latter class of problems is known as the absolute stability or Lur'e problem. We present robust versioDJI of the Small Gain TheoreDl and the Absolute Stability or Lur'e problem which allow us to quantify the worst case parametric or unstructured stability margins that the closed loop system can tolerate.
3. REFERENCES Kharitonov, V. L. {1978} Asymptotic stability of an equilibrium position of a family of systems of linear differential equations. Differential U7'IItmen, 14, 2086 - 2088. Kalman, R. E. {1960} Contribution to the theory of optimal control. Bol. Soc. Mtdem. Mezico, 102 - 119. Kalman, R. E. {1964} When is a linear control system optimal. ASME 7Nn.t. Ser. D (J. of Btuic Eng.), 51 - 60. Doyle, J. C. and G. Stein {1979} Robustness with observers. IEEE 7Nn.t. on Auto. Contr., AC - 24, 607 - 611. R08enbrock, H. H. {1966} Good, bad, or optimal. IEEE 7Nn.t. on Auto. Contr.. Pearson, J. B. (1968) Compensator for dynamic optimisation. Int. J. of Contr., 9, 473. Zames, G. (1981) Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms, and approximate inverses. IEEE 7Nn.t. on Auto. Contr., AC-26, 301 - 320. Zames, G. (1963) Functional analysis applied to nonlinear feedback systems. IEEE 7Nr08. on Gir. Theo., 392 - 404. Youla, D. C., H. A. Jabr, and J. J. Bongiorno {1976} Modem Wiener - Hopf design of optimal controllers - Part 11: the multivariable case. IEEE 7Nn.t. on Auto. Contr., AC-21, 319 - 338. Doyle, J. C., K. Glover, P. P. Khargonekar, and B. A. Francis (1989) State space solution to standard H2 and Hoo control problems. IEEE 7Nn.t. on Auto. Contr., AC - 34, 831 - 847. Doyle, J. C. (1982) Analysis of feedback systems with structured uncertainties. Proc. of lEE - D, 129, 242 - 250. Horowitz, 1. (1963) Synthe6u of Feedback Control Sy6tem.. New York, NY: Academic Press. Siljak, D. D. (1969) Nonlinear Sy6tem.: Pa7'llmetric Analy6u and De6ign. New York: Wiley, 1969.
Units 10 deals with the robust stability of polynomials containing uncertain interval parameters which appear affine multilinearly in the coefficients. The main tool to solve this problem is the Mapping Theorem described in the 1963 book of Zadeh and Desoer. We state and prove this theorem and apply it to the robust stability problem. In Unit 11 we continue to develop results on multilinear interval systems extending the Generalized Kharitonov Theorem and the frequency domain properties of Units 7, 8 and 9 to the multilinear case. In Unit 12 we deal with parameter perturbations in state space models. The same mapping theorem is used to give an effective solution to the robust stability of state space systems. We also include an important result on the calculation of the real stability radius defined in terms of the operator {induced} norm of a feedback matrix. In Unit 13 we describe some synthesis techniques. To begin with we show how standard results from H OO theory can be exploited to deal with parametric perturbations using the extremal properties developed earlier. Nat we demonstrate a direct procedure whereby any minimum phase interval plant of order n, with m zeros can be robustly stabilized by a fixed stable minimum phase compensator of order n - m - 1.
52