Arash Hassibi

Quadratic stabilization and control of piecewise-linear systems

We consider analysis and controller synthesis of piecewise-linear systems. The method is based on constructing quadratic and piecewise-quadratic Lyapunov functions that prove stability and performance for the system. It is shown that proving stability and performance, or designing (state-feedback) controllers, can be cast, as convex optimization problems involving linear matrix inequalities that can be solved very efficiently. A couple of simple examples are included to demonstrate applications of the methods described.

Output feedback controller synthesis for piecewise-affine systems with multiple equilibria

This work builds on the stability analysis of piecewise-affine systems reported by Hassibi et al. (1998) and extends it to obtain a new synthesis tool for output feedback controllers. The proposed technique relies on formulating the search for a piecewise-quadratic Lyapunov function and a piecewise-affine controller as a bilinear matrix inequality. This can be solved iteratively as a set of two convex optimization problems involving linear matrix inequalities which can be solved numerically and very efficiently. A key point in this design technique is that it can be used to design controllers with different structures depending on the number of constraints that are added. In particular, it is shown that a controller with the structure of a regulator and estimator can be designed so that switching based on state estimates rather than on the output can be performed. It is also shown that many other desired features can be included in the design. Furthermore, the applicability of this design method to systems with multiple equilibrium points is shown in simulation examples.

Low-authority controller design via convex optimization

The premise in low-authority control (LAG) is that the actuators have limited authority, and hence cannot significantly shift the eigenvalues of the system. We introduce a near method for low authority controller design, based on convex programming. We formulate the LAC design problem as a nonlinear convex optimization problem, which can then be solved efficiently by interior-point methods. We show that by optimizing the l/sub 1/ norm of the gains, we can arrive at sparse designs, i.e., designs in which only a small number of the control gains are non-zero. Thus, we can also solve actuator/sensor placement or controller architecture design problems. Moreover, it is possible to address the robustness of the LAG, i.e., a closed-loop performance subject to uncertainties or variations in the plant model. Therefore, by combining all these, for example, we can solve the problem of robust actuator/sensor placement and LAC design in one step.

A KYP lemma and invariance principle for systems with multiple hysteresis non-linearities

Absolute stability criteria for systems with multiple hysteresis non-linearities are given in this paper. It is shown that the stability guarantee is achieved with a simple two part test on the linear subsystem. If the linear subsystem satisfies a particular linear matrix inequality and a simple residue condition, then, as is proven, the non-linear system will be asymptotically stable. The main stability theorem is developed using a combination of passivity, Lyapunov and Popov stability theories to show that the state describing the linear system dynamics must converge to an equilibrium position of the non-linear closed loop system. The invariant sets that contain all such possible equilibrium points are described in detail for several common types of hystereses. The class of non-linearities covered by the analysis is very general and includes multiple slope-restricted memoryless non-linearities as a special case. Simple numerical examples are used to demonstrate the effectiveness of the new analysis in comparison to other recent results, and graphically illustrate state asymptotic stability.

An implementation of discrete multi-tone over slowly time-varying multiple-input/multiple-output channels

We have investigated a method for data transmission over slowly time-varying MIMO channels. A low complexity method is introduced that effectively diagonalizes the MIMO channel. This enables the use of discrete multi-tone (DMT) modulation over the MIMO channel to achieve information transmission rates close to the Shannon capacity. DMT requires knowledge of the channel state information at the transmitter which is not always possible in practice. In this case the channel can be only made block diagonal and signal detection requires the solution to a least-squares problem with integer variables. This is a very challenging problem that is theoretically difficult (NP-hard). In this paper, a practically efficient method is proposed to solve this least squares problem.

Integer parameter estimation in linear models with applications to GPS

We consider parameter estimation in linear models when some of the parameters are known to be integers. Such problems arise, for example, in positioning using phase measurements in the global positioning system (GPS.) Given a linear model, we address two problems: (1) The problem of estimating the parameters. (2) The problem of verifying the parameter estimates. Under Gaussian measurement noise: Maximum likelihood estimates of the parameters are given by solving an integer least-squares problem (theoretically, this problem is very difficult to solve (NP-hard)); and Verifying the parameter estimates (computing the probability of correct integer parameter estimation) is related to computing the integral of a Gaussian PDF over the Voronoi cell of a lattice (this problem is also very difficult computationally). However, by using a polynomial-time algorithm due to Lenstra, Lenstra, and Lovasz (LLL algorithm), the integer least-squares problem associated with estimating the parameters can be solved efficiently in practice; and sharp upper and lower bounds can be found on the probability of correct integer parameter estimation. We conclude the paper with simulation results that are based on a GPS setup.

Low-Authority Controller Design by Means of Convex Optimization

The premise in low-authority control is that the actuators have limited authority and hence cannot signiecantly shift the eigenvalues of the system. As a result, the closed-loop eigenvalues can be well approximated analytically by perturbation theory. These analytical approximations may sufece to predict the behavior of the closed-loop systeminpractice.Weshowthat suchapproximationscanbeused tocastlow-authoritycontrollerdesignproblems for different objectives as convex optimization problems that can be solved efeciently in practice by using recently developed interior-point methods. Also, we show that, by optimizing the l1 norm of the feedback gains, we can arriveat sparsedesigns, i.e., designs inwhich only asmall number of the control gains are nonzero. Thus, in effect, we can also solve actuator/sensor placement or controller architecture design problems. Examples are also given that demonstrate the effectiveness of the design method.

Asymptotic stability for systems with multiple hysteresis nonlinearities

Absolute stability criteria for systems with multiple hysteresis nonlinearities are given in this paper. If the linear subsystem satisfies a simple two part test involving a linear matrix inequality and a simple residue condition, then the nonlinear system is proven to be asymptotically stable. The main stability theorem uses a combination of passivity, Lyapunov, and Popov stability theories to show that the state describing the linear system dynamics must converge to an equilibrium position of the nonlinear closed loop system. The stationary sets that contain all possible equilibrium points are detailed for common types of hystereses, and simple examples are used to illustrate the benefits of the new results.