Hiroyuki Ichihara

Optimal Control for Polynomial Systems Using Matrix Sum of Squares Relaxations

This note deals with a computational approach to an optimal control problem for input-affine polynomial systems based on a state-dependent linear matrix inequality (SDLMI) from the Hamilton-Jacobi inequality. The design follows a two-step procedure to obtain an upper bound on the optimal value and a state feedback law. In the first step, a direct usage of the matrix sum of squares relaxations and semidefinite programming gives a feasible solution to the SDLMI. In the second step, two kinds of polynomial annihilators decrease the conservativeness of the first design. The note also deals with a control-oriented structural reduction method to reduce the computational effort. Numerical examples illustrate the resulting design method.

Necessary and sufficient conditions for finite-time boundedness of linear continuous-time systems

Finite-time stability (FTS) requires that the state of a system does not exceed a certain bound during a specified time interval for given bound on the initial state. The concept of FTS introduced exogenous inputs is called finite time boundedness (FTB). This paper gives necessary and sufficient conditions for FTB of linear time-varying continuous-time systems. The conditions are extensions of existing necessary and sufficient for FTS. The sufficient conditions for FTB are also given in terms of differential linear matrix inequalities, which can be relaxed by matrix sum of squares, univariate polynomials. Numerical examples show the solvability of the sufficient conditions.

Observer design for polynomial systems using convex optimization

This paper presents a computational technique of observer design for input-afflne polynomial systems based on Lyapunovs stability theorem and invariance principal by using convex optimization. Following some filter design results, an observer design method is discussed guaranteeing a regional stability of the closed-loop system for a given state estimate feedback law. Two performance improvements are also discussed with respect to the decay rate of the error dynamics, and the L2 gain between disturbances and the estimation errors. To compute these observer gains, scalar and matrix-valued sum of squares optimization are effectively used.

State feedback synthesis for polynomial systems with bounded disturbances

This paper deals with a state feedback synthesis for polynomial systems in the presence of disturbances with bounded peak or bounded energy. Positively invariant sets are composed of level sets of polynomial Lyapunov functions and are included in the region of the inputs and the state constraints under the disturbances. A two-step non-iterative design procedure is available by using matrix sum of squares (SOS) relaxations and semidefinite programming. At the first step, the matrix SOS technique is applied. Then to remove one of the causes of conservativeness of the first step, the polynomial annihilators are utilized in the second step. Numerical examples illustrate the presented design procedure.

Finite-time control for linear discrete-time systems with input constraints

Finite-time stabilization (FTS) and finite-time boundedness (FTB) control with input constraints are considered for linear discrete-time time-invariant systems. Design methods of state feedback and observer-based output feedback FTS/FTB controllers that satisfy input constraints are proposed based on reachable sets in finite-time period. Less conservative design method of controllers for the maximum magnitude of input signals is also given. Numerical examples are shown to illustrate the proposed design methods.

Minimax polynomial optimization by using sum of squares relaxation and its application to robust stability analysis of parameter-dependent systems

This paper gives a sufficient condition for a robust control problem in terms of minimax polynomial optimization. A minimax problem is investigated by using sum of squares relaxation of parametric polynomial optimization problems. The sufficient condition can be exact from the Positivstellensatx. The minimax approach is adopted for robust stability analysis of parameter-dependent systems and an equivalent stability condition is also given from quadratic version of distinguished representation of positive polynomials. Finally, a numerical example is shown.

State Feedback Synthesis for Polynomial Systems with Input Saturation using Convex Optimization

This paper proposes a convex optimization approach to state feedback synthesis for input-afflne polynomial systems with input saturation. A polytope model of the saturation is introduced to the system analysis, and leads a sufficient condition of the state feedback design as a convex problem from a viewpoint of robust control. The result is an extension of a linear matrix inequality approach for linear systems with input saturation. A redesign method is also proposed in which polynomial annihilators decrease the conservativeness of the first design. Both the design procedures can be computed by using matrix sum of squares relaxations and semidefinite programming. The proposed approach does not take any iterative strategies, but could take two steps of convex optimization to expand the invariant set.

L2-Gain Analysis and Control Design of Linear Systems with Input Saturation using Matrix Sum of Squares Relaxation

In this paper, matrix sum of squares (SOS) relaxation is used for analysis of linear systems with input saturation. On the basis of a polytopic type saturation model, we consider a polynomial type saturation model which leads less conservative analysis of Lscr2 -gain performance. The idea is also applicable to design problems. Both the analysis and the design problems are reduced to robust linear matrix inequality problems that can be solved by using a matrix SOS technique. A structural reduction for the relaxation is also discussed. Finally, numerical examples are shown

A descriptor system approach to estimating domain of attraction for non-polynomial systems via LMI optimizations

A computational methodology of estimating the domain of attraction (DA) is addressed for non-polynomial systems by a descriptor system approach. For an existing technique of approximating a non-polynomial function, a further investigation is conducted on the existence of upper and lower polynomial bounds of the non-polynomial function. In formulation of the DA analysis conditions, an implicit form and a generalized Lyapunov function are utilized for dealing with the non-polynomial systems in polynomial fashion and provide two stability conditions which can be reduced to linear matrix inequality (LMI) problems. A relation between these conditions is also discussed. Numerical examples illustrate our DA analysis method.

Observer design for polynomial systems with bounded disturbances

Computational methods of filter and observer design are presented for a class of polynomial systems with L2-bounded disturbance via convex optimization. A measurement and estimated state dependent polynomial filter gain stabilizes the origin of the error dynamics in an invariant set. In addition to the stability of the error dynamics, a polynomial observer gain guarantees a stability of the origin of the closedloop system in another invariant set for a given polynomial dependent estimated state feedback law. To compute the filter and observer gains and the invariant sets, matrix sum of squares relaxation and semidefinite programming are effectively applied. Numerical examples illustrate the design methods of the paper.