Pole/zero cancellations in the general ∞ problem with reference to a two block design

Necessary and sufficient conditions are given for pole/zero cancellations in the close--loop transfer function from input disturbances to error signals in the general _∞ problem. These pole/zero cancellations can place restrictions on the suitability of particular _∞ design procedures. This will be shown by reference to a typical two-block design procedure.     

Systematic design of weighting matrices for the H∞ mixed sensitivity problem

A systematic design of weighting matrices for the H_∞ mixed sensitivity problem is presented in this paper. Once a nominal model has been chosen, an initial design of the weighting matrices based on the multiplicative output uncertainty is proposed. The final weighting matrices (which permit that an appropriate closed loop behavior is achieved)are obtained by tuning just one parameter for each output of the system. A multivariable control of two temperatures of a pilot plant (which constitutes a typical example of an industrial process) is included as an application where the validity of the proposed methodology has been tested.     

On the worst-case divergence of the least-squares algorithm

In this paper, we provide a _∞-norm lower bound on the worst-case identification error of least-squares estimation when using FIR model structures. This bound increases as a logarithmic function of model complexity and is valid for a wide class of inputs characterized as being quasi-stationary with covariance function falling off sufficiently quickly.     

H∞ control design in bilinear systems: a tensor formal series approach

In this paper, the H_∞ disturbance attenuation problem of bilinear system is discussed. Dynamic game theory is used to solve this bilinear minimax problem. The solvability of H_∞ disturbance attenuation in bilinear system is also discussed. The techniques of tensor products and formal power series are employed to solve the nonlinear Bellman-Isaac differential equation. Furthermore, the convergence for the tensor formal series approach of this H_∞ control problem is discussed, and the radius of convergence for this H_∞ control to be well defined is also obtained.     

An alternative solution to the H∞-optimal control problem

In this paper we present an alternative solution to the problem min _{X ε H^n×n∞} |A + BXC|_∞ where A, B, rmand C are rational matrices in H^n×n_∞. The solution circumvents the need to extract the matrix inner factors of B and C, providing a multivariable extension of Sarason's H^∞-interpolation theory [1] to the case of matrix-valued B(s) and C(s). The result has application to the diagonally-scaled optimization problem int |D(A + BXC)D⁻¹|_∞, where the infimum is over D, X εH^n×n_∞, D diagonal.     

Minimizing a convex combination of the overshoot and undershoot

We study the problem of minimizing a parameterized convex combination of the overshoot and undershoot of SISO continuous time system in response to a known input. From a dual formulation, we develop a condition for solution existence and specify the structure of optimal solution. In addition, an interrelation between the overshoot and undershoot in controller synthesis is analytically explained in our framework.     

H∞-control with time domain constraints: the infinite horizon case

It has been shown recently that time domain constraints can be incorporated into control problems when the constraints are enforced over a finite horizon, i.e. over a finite number of sample instants. This is done in (Sideris and Rotstein, 1993; Rotstein and Sideris, to appear) where the finite horizon problem is reduced into a finite dimensional, convex, but generically nondifferentiable optimization program.In this paper, the infinite horizon case is addressed. It is shown that when the horizon length goes to infinity, the solutions to the finite horizon problems converge to a solution of the infinite horizon problem. Moreover, a simple modification of the finite horizon problems guarantees convergence to a solution that is easily approximated by a finite dimensional transfer function.     

Exact computation of infimum for a class of continuous-time H∞ optimal control problem with a nonzero direct feedthrough term from the disturbance input to the controlled output

A noniterative method for the computation of infimum for a class of continuous-time H_∞ optimal control problem is considered in this paper. The problem formulation is fairly general and does not place any restrictions on any direct feedthrough terms of the given systems. The method is applicable to systems where (i) the transfer function from the disturbance input to the measurement output is free of imaginary axis invariant zeros and left invertible, and (ii) the transfer function from the control input to the controlled output of the given system is free of imaginary axis invariant zeros and right invertible. The result presented in this paper is a continuation of the previous work of the author and his co-workers (Chen et al., 1992), in which the direct feedthrough term from the disturbance input to the measurement output of the given system are required to be zero.     

An LMI solution to the robust synthesis problem for multi-rate sampled-data systems

In this paper we address the asynchronous multi-rate sampled-data H_∞ synthesis problem. Necessary and sufficient conditions are given for the existence of a controller achieving the desired performance, and the problem is shown to be equivalent to a convex optimization problem expressed in the form of linear operator inequalities. In the case where the sample and hold rates are synchronous, these operator inequalities reduce to linear matrix inequalities, for which standard numerical software is available.     

A model-based sliding mode control methodology applied to the HDA-plant

A sliding mode control methodology using output information is demonstrated in application to the HDA-plant, a plant for production of benzene. This process is a highly integrated, non-linear large scale process with non-minimum phase and relative degree zero characteristics. The non-linear control law is designed on the basis of a linear observer-based control system. The non-linear control law uses the states of the linear observer. The performance in the sliding mode is determined by a linear stable sub-manifold of the linear closed loop control system chosen via a robust pole selection scheme. The sliding mode control is optimized to operate in a wide operating region.     

H∞ model reduction of Markovian jump linear systems

In this paper, the H_∞ model reduction problem for linear systems that possess randomly jumping parameters is studied. The development includes both the continuous and discrete cases. It is shown that the reduced order models exist if a set of matrix inequalities is feasible. An effective iterative algorithm involving linear matrix inequalities is suggested to solve the matrix inequalities characterizing the model reduction solutions. Using the numerical solutions of the matrix inequalities, the reduced order models can be obtained. An example is given to illustrate the proposed model reduction method.     

Robust H∞ estimation of stationary discrete-time linear processes with stochastic uncertainties

The problem of H_∞ filtering of stationary discrete-time linear systems with stochastic uncertainties in the state space matrices is addressed, where the uncertainties are modeled as white noise. The relevant cost function is the expected value, with respect to the uncertain parameters, of the standard H_∞ performance. A previously developed stochastic bounded real lemma is applied that results in a modified Riccati inequality. This inequality is expressed in a linear matrix inequality form whose solution provides the filter parameters. The method proposed is applied also to the case where, in addition to the stochastic uncertainty, other deterministic parameters of the system are not perfectly known and are assumed to lie in a given polytope. The problem of mixed H₂/H_∞ filtering for the above system is also treated. The theory developed is demonstrated by a simple tracking example.     

On the stability of the information state system

The purpose of this paper is to present some preliminary results on the stability of the information state system. The information state system underlies the (infinite dimensional) dynamics of an H_∞ controller for a nonlinear system. Thus it is important to understand its stability and the structure of its equilibrium points. We analyse the important case corresponding to the mixed sensitivity problem. We prove the existence of an equilibrium information state, convergence under very general conditions to such an equilibrium state p_e and uniqueness of this state (up to an irrelevant constant). In this case the equilibrium p_e is usually singular in the sense that it takes on the value − ∞ except on a low dimensional subset of its domain.This meshes with the article [9] which analysed the effect of using p_e to initialize the information state controller and gave explicit formulas which in many cases produce a dramatic reduction in the amount of computation required to implement the controller. What this article suggests is that indeed p_e is the only equilibrium initialization possible.     

Reduced-order filtering for linear systems with Markovian jump parameters

This paper addresses the reduced-order H_∞ filtering problem for continuous-time Makovian jump linear systems, where the jump parameters are modelled by a discrete-time Markov process. Sufficient conditions for the existence of the reduced-order H_∞ filter are proposed in terms of linear matrix inequalities (LMIs) and a coupling non-convex matrix rank constraint. In particular, the sufficient conditions for the existence of the zero-order H_∞ filter can be expressed in terms of a set of strict LMIs. The explicit parameterization of the desired filter is also given. Finally, a numerical example is given to illustrate the proposed approach.     

On computing the worst-case peak gain of linear systems

Based on the bounds due to Doyle and Boyd, we present simple upper and lower bounds for the l¹-norm of the ‘tail’ of the impulse response of finite-dimensional discrete-time linear time-invariant systems. Using these bounds, we may in turn compute the l^∞-gain of these systems to any desired accuracy. By combining these bounds with results due to Khammash and Pearson, we derive upper and lower bounds for the worst-case l^∞-gain of discrete-time systems with diagonal perturbations.     

On the representation of shift-invariant operators by transfer functions

We give simple sufficient conditions on a function space on (0, ∞) to ensure that all shift-invariant operators defined on it are represented by transfer functions. This enables us to extend the result of Weiss on L_p(0, ∞) for 1 p < ∞ (but not p = ∞) to more general situations.     

Robust H∞ observer design of linear time-delay systems with parametric uncertainty

This paper deals with the problem of H_∞ observer design for a class of uncertain linear systems with delayed state and parameter uncertainties. This problem aims at designing the linear state observers such that, for all admissible parameter uncertainties, the observation process remains robustly stable and the transfer function from exogenous disturbances to error state outputs meets the prespecified H_∞ norm upper bound constraint, independently of the time delay. The time delay is assumed to be unknown, and the parameter uncertainties are allowed to be norm-bounded and appear in all the matrices of the state-space model. An effective matrix inequality methodology is developed to solve the proposed problem. We derive the conditions for the existence of the desired robust H_∞ observers, and then characterize the analytical expression of these observers in terms of some free parameters. A numerical example demonstrates the validity and applicability of the present approach.     

Computation of the ∞ norm for nonlinear systems A convergence result

In this paper we show that _∞ norm for a class of nonlinear continuous-time systems is the limit of the _∞ norm for approximating discrete-time systems. The convergence result is proved by means of a characterization of the _∞ norm in terms of the value for an ergodic control problem.     

An H-minimax approach to the design of robust control systems, part II: All solutions, all-pass form solutions and the ‘best’ solution

We characterize all solutions to a robustness optimization problem as the solutions of a two-parameter interpolation problem. From this characterization it is easy to show that an all-pass form solution always exists as long as a solution exists. We also study the possibility of using non-all-pass form solutions and by introducing other optimization objectives (motivated by improvements in disturbance rejection and robust stability) we search for the 'best' solution.     

Static output feedback stabilisation with H∞ performance for a class of plants

In this paper, the problem of static output feedback control of a linear system is considered. The existence of a static output feedback control law is given in terms of the solvability of two coupled Lyapunov inequalities which result in a non-linear optimisation problem. However, using state-coordinate and congruence transformations and by imposing a block-diagonal structure on the Lyapunov matrix, we will see that the determination of a static output feedback gain reduces, for a specific class of plants, to finding the solution of a system of linear matrix inequalities. The class of plants considered is those which are minimum phase with a full row rank Markov parameter. The method is extended to incorporate H_∞ performance objectives. This results in a sub-optimal static H_∞ control law found by non-iterative means. The simplicity of the method is demonstrated by a numerical example.