The H2 Control Problem A Detailed Comparison of State-Space and Transfer-Function Solutions Vladimír Kučera Czech Technical University in Prague SWAN 2006 The University of Texas at Arlington

Motivation For analytical design of control systems, it is often convenient to measure system performance in terms of norm of the closed-loop system transmittance from the exogenous signals to the regulated variables One common measure of performance for a linear system is the H2 norm of its transfer function The H2 norm is relevant when minimizing the variance of stochastic signals the peak amplitude of deterministic signals

Problem Formulation Given a linear, finite dimensional, time invariant plant P, find a controller C that stabilizes the control system and minimizes the norm of its transfer function T from v to z in RH2 (the space of strictly proper stable rational matrices).

Systems in state space form Methods of Solution Systems in state space form An optimal controller is obtained in observer form by solving two algebraic Riccati equations Systems described by transfer functions The optimal controller transfer function is obtained via two inner-outer factorizations and two proper-stable projections

Comparison It is well understood that the inner-outer factorization corresponds to solving an algebraic Riccati equation. However, why are the proper-stable projections not needed in the state-space approach? Efficient numerical algorithms are now available to handle operations on and among polynomial matrices. However, why is the state space algorithm still preferred?

Standard Assumptions Given a state-space realization of the plant It is assumed that ( F, G2) is stabilizable, ( F, H2) is detectable, the matrices have full column and row rank, respectively, for all finite ω and

Fractional Representations Firstly, P is represented in the form of doubly coprime factorizations over RH (the space of proper stable rational matrices) where and Note the block-triangular structure of D and .

Stabilizing Controllers Then all controllers that stabilize P are parametrized as where X, Y and are proper stable rational matrices that satisfy the Bézout identity and where W is a free parameter that ranges over RH .

Transfer Function Solution The closed loop transfer function is where CS is any stabilizing controller. The strategy to minimize is to express T as a function of the parameter W using any but fixed doubly coprime factorization of P, then manipulate the expression so that as a function of W has no linear term. Two inner-outer factorizations and two proper-stable projections are used in the process.

Norm Minimization (1) Using doubly coprime factors, where is outer, is inner ( ), and where S is the proper stable part of . Then

Norm Minimization (2) Now V1 is a function of W , where is outer, is inner ( ), and where is the proper stable part of . Then

Optimal Controller To summarize, where only the final term depends on W. Thus the optimal controller C0 corresponds to hence and

State Space Solution The strategy to minimize is to express T as a function of the parameter W using special doubly coprime factorizations of P (and the corresponding solutions to the Bézout identity), then manipulate the expression so that as a function of W has no linear term. Solution of two algebraic Riccati equations is used in the process.

Operations with Systems Denote a system transfer function Then

Plant Matrix Fractions (1) Construction of right coprime, proper stable factors where L is a matrix such that F + G2L is stable. w u y x G2 F L H2

Plant Matrix Fractions (2) Construction of left coprime, proper stable factors where K is a matrix such that F + KH2 is stable. u y G2 F K H2 – e

Bézout Equations Solution pairs to the Bézout equations can be explicitly constructed as Proof by direct verification, using the rules for operations with systems.

Inner Functions Select L so as to make inner with the notation similarity transformation on putting

Strictly Proper Stable Rational Functions For such a gain L, belongs to By duality, a gain K can be selected so as to make inner and belong to . with the notation similarity transformation

Optimal Controller Now and yield so that minimum is achieved for W = 0. The optimal controller results in the observer-based form applying the rules for operations with systems.

Comparison (1) The transfer-function approach takes any but fixed doubly coprime factors of P, while the state-space approach parametrizes all such factors using stabilizing gains K a L. extracts the inner factors from and , shapes these matrices to make them inner by selecting K and L. This selection makes and belong to , hence no need to extract their proper stable parts S and .

Comparison (2) Thus, the difference between the two approaches derives from a different construction and use of doubly coprime factors. No need to solve the Bézout equations in the state-space approach; a particular solution can be explicitly constructed. The observer-based form of the optimal controller is a result of taking that particular solution to the Bézout equations. In addition, the design parameters K and L directly define the optimal controller R0. Hence, the doubly coprime factors need not be calculated.

Computational Aspects A wind gust disturbance rejection controller for an F-8 aircraft is designed to compare the two approaches. The linearized, longitudinal state equations have 5 states, 2 control inputs, 3 external inputs, 2 measurement outputs, 4 performance outputs. The function h2syn of the MATLAB Robust Control Toolbox is compared with the upcoming Version 3 of the Polynomial Toolbox for MATLAB.

Comparison (3) Robust Control Toolbox contains a dedicated function h2syn while Polynomial Toolbox offers general purpose functions: ldf and rdf to create polynomial matrix fractions, spf and spcof to calculate spectral factors, axybc to extract proper stable parts, fact and xab and axb to calculate the optimal controller. The toolboxes return different optimal controllers, in the observer-based form and in the matrix-fraction form. A good match is observed: the minimum H2 norm achieved 0.2778 9787 for the state-space controller 0.2778 9788 for the transfer-function controller.

Comparison (4) Although yielding almost identical results, the two synthesis procedures are not equivalent. In fact, the state-space algorithm is more efficient than the transfer-function one. The critical part of the transfer-function algorithm is the final substitution of the optimal W into CS to obtain C0 . This substitution generates common factors that must be cancelled in order to obtain the optimal controller in reduced form. The state-space algorithm fixes the order of the controller to equal that of the plant.

Comparison (5) The computational complexity of the state-space design depends critically on the size of the state vector x, while the transfer-function algorithm depends largely on the sizes of the control inputs u and the measurement outputs y. That is why the transfer-function algorithm is most efficient in the single-input single-output case. Why in general is the state space design simpler? Because the state model carries more information, which makes it possible to parametrize the plant matrix fractions through the stabilizing gains K and L, and this information is fully exploited in the process.

References V. Kučera, „The H2 control problem: state-apace and transfer-function solutions,“ in Proc. IEEE Mediterranean Conference on Control and Automation, Ancona 2006. „The H2 control problem: a general transfer-function solution ,“ International Journal of Control, to be published.