Yasuaki Oishi

Brief paper: Stability and stabilization of aperiodic sampled-data control systems using robust linear matrix inequalities

Stability analysis of an aperiodic sampled-data control system is considered for application to networked and embedded control. The stability condition is described in a linear matrix inequality to be satisfied for all possible sampling intervals. Although this condition is numerically intractable, a tractable sufficient condition can be constructed with the mean value theorem. Special attention is paid to tightness of the sufficient condition for less conservative stability analysis. A region-dividing technique for the reduction of conservatism and generalization to stabilization are also discussed. An example demonstrates the efficacy of the approach.

Brief paper: Polynomial-time algorithms for probabilistic solutions of parameter-dependent linear matrix inequalities

A randomized approach is considered for a feasibility problem on a parameter-dependent linear matrix inequality (LMI). In particular, a gradient-based and an ellipsoid-based randomized algorithms are improved by introduction of a stopping rule. The improved algorithms stop after a bounded number of iterations and this bound is of polynomial order in the problem size. When the algorithms stop, either of the following two events occurs: (i) they find with high confidence a probabilistic solution, which satisfies the given LMI for most of the parameter values; (ii) they detect in an approximate sense the non-existence of a deterministic solution, which satisfies the given LMI for all the parameter values. These results are important because the original randomized algorithms have issues to be settled on detection of convergence, on the speed of convergence, and on the assumption of feasibility. The improved algorithms can be adapted for an optimization problem constrained by a parameter-dependent LMI. A numerical example shows the efficacy of the proposed algorithms.

Mixed Deterministic/Randomized Methods for Fixed Order Controller Design

In this paper, we propose a general methodology for designing fixed order controllers for single-input single-output plants. The controller parameters are classified into two classes: randomized and deterministically designed. For the first class, we study randomized algorithms. In particular, we present two low-complexity algorithms based on the Chernoff bound and on a related bound (often called ldquolog-over-logrdquo bound) which is generally used for optimization problems. Secondly, for the deterministically designed parameters, we reformulate the original problem as a set of linear equations. Then, we develop a technique which efficiently solves it using a combination of matrix inversions and sensitivity methods. A detailed complexity analysis of this technique is carried on, showing its superiority (from the computational point of view) to existing algorithms based on linear programming. In the second part of the paper, these results are extended to H infin performance. One of the contributions is to prove that the deterministically designed parameters enjoy a special convex characterization. This characterization is then exploited in order to design fixed order controllers efficiently. We then show further extensions of these methods for stabilization of interval plants. In particular, we derive a simple one-parameter formula for computing the so-called critical frequencies which are required by the algorithms.

Stability and stabilization of aperiodic sampled-data control systems: An approach using robust linear matrix inequalities

Stability analysis of an aperiodic sampled-data control system is considered for application to network and embedded control. The stability condition is described in a linear matrix inequality to be satisfied for all possible sampling intervals. Although this condition is numerically intractable, a tractable sufficient condition can be constructed with the mean value theorem. Special attention is paid to tightness of the sufficient condition for less conservative stability analysis. A region-dividing technique for reduction of conservatism and generalization to stabilization are also discussed. Examples show the efficacy of the approach.

A region-dividing approach to robust semidefinite programming and its error bound

A new asymptotically exact approach is presented for robust semidefinite programming, where coefficient matrices polynomially depend on uncertain parameters. Since a robust semidefinite programming problem is difficult to solve directly, an approximate problem is constructed based on a division of the parameter region. The optimal value of the approximate problem converges to that of the original problem as the resolution of the division becomes finer. An advantage of this approach is that an upper bound on the approximation error is available before solving the approximate problem. This bound shows how the approximation error depends on the resolution of the division. Furthermore, it leads to construction of an efficient division that attains small approximation error with low computational complexity. Numerical examples show efficacy of the present approach. In particular, an exact optimal value is often found with a division of finite resolution

Randomized algorithms to solve parameter-dependent linear matrix inequalities and their computational complexity

The randomized algorithm of G. Calafiore and B. Polyak (2000), which consists of random sampling and sub-gradient descent, is analyzed in the case where it is used to solve parameter-dependent linear matrix inequalities. This paper shows that the expected time to achieve a solution is infinite if this algorithm is used in its original form. However, it is also shown that the algorithm can be improved so that its expected achievement time becomes finite. An explicit upper bound of the expected achievement time is given in a special case. A numerical example is provided.

Brief paper: Guaranteed cost regulator design: A probabilistic solution and a randomized algorithm

This paper presents a gradient-based randomized algorithm to design a guaranteed cost regulator for a plant with general parametric uncertainties. The algorithm either provides with high confidence a probabilistic solution that satisfies the design specification with high probability for a randomly sampled uncertainty or claims that the feasible set of the design parameters is too small to contain a ball with a given radius. In both cases, the number of iterations executed in the algorithm is of polynomial order of the problem size and is independent of the dimension of the uncertainty.

Probabilistic design of a robust state-feedback controller based on parameter-dependent Lyapunov functions

An ellipsoid-based randomized algorithm of Kanev et al. is extended for the use of parameter-dependent Lyapunov functions. The proposed algorithm is considered to be useful for a less conservative design of a robust state-feedback controller against nonlinear parametric uncertainty. Indeed, it enables us to avoid polytopic overbounding of uncertainty and employment of parameter-independent Lyapunov functions. After a bounded number of iterations, the proposed algorithm gives with high confidence a probabilistic solution that satisfies a provided inequality for a high-percentage of parameters. This algorithm can be used also for finding an optimal solution in an approximated sense. Convergence to a non-strict deterministic solution is considered and, especially, the expected number of iterations necessary to achieve a non-strict deterministic solution is provided infinite under some assumptions. A numerical example is provided.

Brief Computational complexity of randomized algorithms for solving parameter-dependent linear matrix inequalities

Randomized algorithms are proposed for solving parameter-dependent linear matrix inequalities and their computational complexity is analyzed. The first proposed algorithm is an adaptation of the algorithms of Polyak and Tempo [(Syst. Control Lett. 43(5) (2001) 343)] and Calafiore and Polyak [(IEEE Trans. Autom. Control 46 (11) (2001) 1755)] for the present problem. It is possible however to show that the expected number of iterations necessary to have a deterministic solution is infinite. In order to make this number finite, the improved algorithm is proposed. The number of iterations necessary to have a probabilistic solution is also considered and is shown to be independent of the parameter dimension. A numerical example is provided.

An Asymptotically Exact Approach to Robust Semidefinite Programming Problems with Function Variables

This technical note provides an approximate approach to a semidefinite programming problem with a parameter-dependent constraint and a function variable. This problem covers a variety of control problems including a robust stability/performance analysis with a parameter-dependent Lyapunov function. In the proposed approach, the original problem is approximated by a standard semidefinite programming problem through two steps: first, the function variable is approximated by a finite-dimensional variable; second, the parameter-dependent constraint is approximated by a finite number of parameter-independent constraints. Both steps produce approximation error. On the sum of these approximation errors, this technical note provides an upper bound. This bound enables quantitative analysis of the approach and gives an efficient way to reduction of the approximation error. Moreover, this technical note discusses how to verify that an optimal solution of the approximate problem is actually optimal also for the original problem.