On the norms used in computing the structured singular value

Different norms are considered to replace the Euclidean norm in an algorithm given by M. H. K. Fan and A. L. Tits (ibid., vol. 33, pp. 284-289, 1988), which is used for the computation of the structured singular value of any matrix. The algorithm is explained, and it is shown that the l1 norm is the best possible norm in a certain sense.published_or_final_versio

Computation of µ with real and complex uncertainties

The robustness analysis of system performance is one of the key issues in control theory, and one approach is to reduce this problem to that of computing the structured singular value, mu . When real parametric uncertainty is included, then mu must be computed with respect to a block structure containing both real and complex uncertainties. It is shown that mu is equivalent to a real eigenvalue maximization problem, and a power algorithm is developed to solve this problem. The algorithm has the property that mu is (almost) always an equilibrium point of the algorithm, and that whenever the algorithm converges a lower bound for mu results. This scheme has been found to have fairly good convergence properties. Each iteration of the scheme is very cheap, requiring only such operations as matrix-vector multiplications and vector inner products, and the method is sufficiently general to handle arbitrary numbers of repeated real scalars, repeated complex scalars, and full complex blocks

Computation of µ with real and complex uncertainties

The robustness analysis of system performance is one of the key issues in control theory, and one approach is to reduce this problem to that of computing the structured singular value, mu . When real parametric uncertainty is included, then mu must be computed with respect to a block structure containing both real and complex uncertainties. It is shown that mu is equivalent to a real eigenvalue maximization problem, and a power algorithm is developed to solve this problem. The algorithm has the property that mu is (almost) always an equilibrium point of the algorithm, and that whenever the algorithm converges a lower bound for mu results. This scheme has been found to have fairly good convergence properties. Each iteration of the scheme is very cheap, requiring only such operations as matrix-vector multiplications and vector inner products, and the method is sufficiently general to handle arbitrary numbers of repeated real scalars, repeated complex scalars, and full complex blocks

Complexity theoretic aspects of problems in control theory

Caption title.Includes bibliographical references (p. 5-6).Supported by the ARO. DAAL03-92-G-0115John N. Tsitsiklis

Quadratic stability with real and complex perturbations

It is shown that the equivalence between real and complex perturbations in the context of quadratic stability to linear, fractional, unstructured perturbations does not hold when the perturbations are block structured. For a limited class of problems, quadratic stability in the face of structured complex perturbations is equivalent to a particular class of scaled norms, and hence appropriate synthesis techniques, coupled with diagonal constant scalings, can be used to design quadratically stable systems

Quadratic stability with real and complex perturbations

It is shown that the equivalence between real and complex perturbations in the context of quadratic stability to linear, fractional, unstructured perturbations does not hold when the perturbations are block structured. For a limited class of problems, quadratic stability in the face of structured complex perturbations is equivalent to a particular class of scaled norms, and hence appropriate synthesis techniques, coupled with diagonal constant scalings, can be used to design quadratically stable systems

Robust control design with real parameter uncertainty using absolute stability theory

The purpose of this thesis is to investigate an extension of mu theory for robust control design by considering systems with linear and nonlinear real parameter uncertainties. In the process, explicit connections are made between mixed mu and absolute stability theory. In particular, it is shown that the upper bounds for mixed mu are a generalization of results from absolute stability theory. Both state space and frequency domain criteria are developed for several nonlinearities and stability multipliers using the wealth of literature on absolute stability theory and the concepts of supply rates and storage functions. The state space conditions are expressed in terms of Riccati equations and parameter-dependent Lyapunov functions. For controller synthesis, these stability conditions are used to form an overbound of the H2 performance objective. A geometric interpretation of the equivalent frequency domain criteria in terms of off-axis circles clarifies the important role of the multiplier and shows that both the magnitude and phase of the uncertainty are considered. A numerical algorithm is developed to design robust controllers that minimize the bound on an H2 cost functional and satisfy an analysis test based on the Popov stability multiplier. The controller and multiplier coefficients are optimized simultaneously, which avoids the iteration and curve-fitting procedures required by the D-K procedure of mu synthesis. Several benchmark problems and experiments on the Middeck Active Control Experiment at M.I.T. demonstrate that these controllers achieve good robust performance and guaranteed stability bounds

Verification of advanced controllers for safety-critical systems

In order to design and deploy a feedback controller in a real application, one must determine suitable specifications that the design must meet ("validate"), and then ensure that the chosen specifications have been met ("verify"). In this thesis, we investigate a verification paradigm based on formal methods, such as the Satisfiability Modulo Theories (SMT) and quantifier elimination (Weispfenning’s virtual term substitution and quantifier elimination by cylindrical algebraic decomposition) algorithms. Any control design requirement (such as satisfactory performance, robustness to uncertainties, stability, etc.) that can be expressed in a first order logic formula can be (in principle) verified by using one of these methods. Consequently, in principle, this allows us to consider problems like general non-convex optimisation, exact computation of structured singular value, and synthesis of non-convex feasible parameter sets. In practice, the generality of algorithms like quantifier elimination by cylindrical algebraic decomposition come with a downside of high running time when applied to more complex systems with more parameters. This, in some cases, limits the complexity of the system that we could consider. Therefore, we focused our attention on control problems such as obtaining an explicit MPC law for a linear time invariant system with a quadratic objective and polytopic constraints, or computation of the structured singular value for a system under parametric (and not norm-bounded) uncertainty. Such problems can be expressed as quantifier elimination problems with a particular quantification structure that allows us to take advantage of a specialised quantifier elimination algorithm - the quantifier elimination by Weispfenning’s virtual term substitution procedure that has much lower worst-case running time on these types of problems than quantifier elimination by cylindrical algebraic decomposition algorithm. Despite these constraints, we were able to apply a quantifier-elimination-based verification framework to clearance of a flight control law developed for a real world industrial system from the aerospace field not only at particular combination of parameters but throughout the whole flight envelope. In conclusion, while in principle formal methods are applicable to a large body of problems arising in control theory, more widespread practical application depends on further research in efficiency and running time improvement in the implementation of these algorithms.Full EC Project Title: Reconfiguration of control in flight for integral global upset recovery (RECONFIGURE) EC Project #: 314544 RG # & UFS Project Code: RG66745, NMZN/04

The complex structured singular value

A tutorial introduction to the complex structured singular value (μ) is presented, with an emphasis on the mathematical aspects of μ. The μ-based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems. Several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared, and a simple synthesis problem is studied. Uncertain systems are represented using Linear Fractional Transformations (LFTs) which naturally unify the frequency-domain and state space methods

Randomized algorithms for control of uncertain systems with application to hand disk drives

Ph.DDOCTOR OF PHILOSOPH