2-Optimal Control of Discrete-Time and Sampled-Data Systems

Tongwen Chen    Dept. of Electrical and Computer Engineering, University of Calgary, Calgary, Alberta, Canada T2N 1N4

Bruce A. Francis    Dept. of Electrical Engineering, University of Toronto, Toronto, Ontario, Canada M5S 1A4

I Introduction

A recent trend in synthesizing sampled-data systems is to use the more natural continuous-time performance measures. This brought solutions to several new 2-optimal sampled-data control problems [1, 2, 3], each reducing to an 2 problem in discrete time.

Discrete-time 2 (LQG) theory was developed in the 1970’s, see, e.g., [4, 5, 6, 7, 8, 9]. As in the continuous-time case, the discrete optimal controller is closely related to the solutions of two Riccati equations. In [10], the solution to a continuous-time 2-Optimal control problem was rederived using the state-space approach. This gives a clean treatment of the problem and provides compact formulas for the optimal controller. Since complete, general formulas for the discrete optimal controller are not readily available in the literature, we ask the question here, can a state-space treatment be accomplished for discrete-time 2 problems?

The goal in this paper is twofold: to present a state-space solution to the discrete 2 control problem and to give direct formulas for an 2-Optimal sampled-data control problem with state feedback and disturbance feedforward. Though the results in the discrete-time case are known in various forms, we believe the derivation is new and quite self-contained, and therefore has some pedagogical value. Moreover, the formulas derived can be applied to the sampled-data problem via the powerful lifting technique [11, 12, 13, 14].

The organization of the paper is as follows. In the next section we collect and prove some preliminary results on Riccati equations; the presentation follows closely that in [10] in continuous time. Section III gives a complete state-space derivation of the discrete-time 2-Optimal control, first via state feedback and disturbance feedforward and then via dynamic output feedback. Section IV presents new direct formulas for a sampled-data 2 problem using state measurement. In Section V we apply the optimal sampled-data. control in Section IV to a two-motor system and compare with the optimal analog control. Finally, concluding remarks are contained in Section VI.

The notation in this paper is quite standard: is the complex plane, ⊂C is the open unit, disk, and D is the boundary of , namely, the unit circle. Also, is the set of all integers and + (−) is the nonnegative (negative) subset of , The space ℓ2(+), or simply ℓ2, consists of all square-summable sequences, perhaps vector-valued, defined on +. Similarly for ℓ2() and ℓ2(−). The discrete-time frequency-domain space 2D, or simply 2, is the Hardy space defined on . We use 2 for the real-rational subspace of 2. In discrete time, we use λ-transforms instead of z-transforms, where λ = z− 1. If a linear discrete system G has a state-space realization (A, B, C, D), then we denote the transfer matrix D + λC (I − λA)− 1B

by

^λ=ACBD.

Finally, ^~λ stands for the transposed matrix ĝ(1/λ)′.

II Riccati Equation

It is well-known that Riccati equations play an important role in the 2 optimization problem. The solution of a Riccati equation can be obtained via the stable eigenspace of the associated symplectic matrix if the state transition matrix of the plant is nonsingular. If this matrix is singular, as is the case when the plant has a time delay, then the symplectic matrix is not defined; but we can use the stable generalized eigenspace of a certain matrix pair [9].

Let A, Q, R be real n × n matrices with Q and R symmetric. Define the ordered pair of 2n × 2n matrices

=H1H2:=A0−QIIR0A′.

A pair of matrices of this form is called a symplectic pair. (This definition is not the most general one.) Note that if A is nonsingular, then 2−1H1 is a symplectic matrix.

Introduce the 2n × 2n matrix

:=0−II0

It is easily verified that 1JH1′=H2JH2′. Thus the generalized eigenvalues (including those at infinity) for the matrix pair H (i.e., those numbers λ satisfying H1x = λH2x for some nonzero x) are symmetric about the unit circle, i.e., λ is a generalized eigenvalue iff 1 / λ is [9].

Now we assume H has no generalized eigenvalues on D. Then it must have n inside and n outside. Thus the two generalized eigenspaces i(H) and o(H), corresponding to generalized eigenvalues inside and outside the unit circle respectively, both have dimension n. Let us focus on the stable subspace i(H). There exist n × n matrices X1 and X2 such that

iH=ImX1X2.

Then for some stable n × n matrix Hi,

1X1X2=H2X1X2Hi.

Some properties of the matrix X′1X2 are useful.

Now assume further that X1 is nonsingular, i.e., the two sub-spaces

iH,Im0I

are complementary. Set :=X2X1−1. Then

iH=ImIX.

Note that the n × n matrix X is uniquely determined by the pair H (though X1 and X2 are not), that is, ↦X is a function. We shall denote this function by Ric and write X = Ric(H).

To recap, Ric is a function 2n × 2n → n × n that maps H to X, where X is defined by equation (6). The domain of Ric, denoted dom Ric, consists of all symplectic pairs H with two properties, namely, H has no generalized eigenvalues on D and the two subspaces

iH,Im0I

are complementary.

Some properties of X are given next.