Chapter 6 Deadbeat controller design with � ∞ norm
6.4 Approaches to solve the ∞ optimization problem
Having reviewed the description of ∞ minimization problem in previous section, we aim now to concentrate on the solution of the problem. Various approaches for tackling the ∞ problem have been introduced in the literature, some of which are developed in the frequency domain and some in the state space framework. In this section, the focus will be on the three major procedures, which will be discussed briefly as we proceed. The discussion pertains to the figure 6.2.1, and its associated mathematical characterization in (6.2.1).
One of the earliest approaches to treat the ∞ problem, is the so-called model-matching problem. As the name suggests, in this scheme the ∞ problem is considered to be equivalent to that of matching two models. With regard to the closed-loop description of the figure 6.1.1 as = + , it is readily observed that the ∞ minimization problem may be interpreted as matching the given transfer matrix − ∈ ∞, with the
cascade of the three transfer matrices , in which and are known transfer matrices in ∞, whereas the ∈ ∞ is the design parameter. This interpretation has been depicted in figure 6.4.1 [127].
Figure 6.4.1 Model-matching problem illustration
Based on the above interpretation, the main problem is now expressible as finding ∈
∞ such that the model-matching error ‖ + ‖∞ is minimized (or kept below
a specified level) [127]. According to [128], a sufficient condition to achieve the minimum is that the two matrices and have constant ranks for all the frequencies
< ∞. In practice, this condition for well-defined problems is fulfilled.
In order to compute a solution to the model-matching problem, it has been shown in [128] that the problem is equivalent to yet another problem, the so-called Hankel-norm approximation problem, also known as the Nehari extension problem. To see this, let us assume that and are square and inner (or all-pass), i.e. [1, 133]:
~ = and ~ = (6.4.1)
in which the tilde designates the parahermitian transpose of the transfer matrix. Now, the model-matching error, owing to the norm-preserving property of the inner matrices [1], may be reformulated as:
‖ + ‖∞ = ‖ ~ ~ + ‖∞= ‖ ~ ~ + ‖∞
= ‖ ~ + ~‖∞ (6.4.2)
− +
In (6.4.2), while ~ is unstable, ~ has only stable eigenvalues [129]. By denoting
~ as , the model-matching problem transforms into that of approximating a
stable transfer matrix by the unstable one − ~: [122]
min∈
∞‖ + ‖∞ = min∈ ∞‖ +
~‖
∞ (6.4.3)
The above results rely on the assumptions that and are inner. As it is stated in [127, 130], this may be accomplished through appropriate selection of the state feedback gain and the observer gain , in the parameterization of the system. However, the first assumption made on and , i.e. being square, is violated by some important classes of problems, e.g. the mixed performance and robustness problem [83]. In the general case that and are neither square nor inner, it is always possible to find orthogonal complements of the transfer matrices and , respectively denoted by
⊥ and ⊥, such that [ ⊥] and [ ⊥] are square and inner [1].
Following the same procedure in (6.4.2), we will have [133, 127]:
‖ + ‖∞ = ‖[ + ]‖ ∞ (6.4.4) in which: = ~ ~ , = ~ ⊥ ~ (6.4.5) = ~⊥ ~ , = ~⊥ ~⊥
Therefore, (6.4.4) converts the problem of minimizing the model-matching error to that of the ∞ norm minimization of the quantity on the right hand side of the equality as:
min∈
∞‖ + ‖∞ = min∈ ∞‖[
+ ]‖
∞
(6.4.6)
This is known as the four-block problem, compared to the special case of (6.4.3), which consists of just one block, hence the name one-block problem.
The right hand side identity in the equality (6.4.6) may also be inferred as the distance
between a transfer matrices = [ ] and − ∈ ∞, hence the name distance problem as an alternative [133]: min∈ ∞‖[ + ]‖ ∞ = dist , [ ∞ ] (6.4.7)
The solution to such problems is extensively elaborated in [130] for the case of continuous time systems. It is developed based on the notion of the norm of a certain operator, the so-called Hankel operator [132], usually designated by . It is shown that the norm of equals the spectral norm of the square root of the product of the controllability and observability Gramians of the transfer matrix , and that the minimal model-matching error equals the norm of the Hankel operator [133, 131, 130]. For a discrete time treatment of the distance problem the reader is referred to [134].
According to the fact that in the Hankel approximation problem, the procedure to attain the solution is both theoretically and computationally very involved, in [136] Glover et al. propose a new approach which relies on the solution to two algebraic Riccati equations with the same order as the system. Here, we will briefly describe their approach. The results are given in terms of the description of systems in the continuous time framework, which is regarded as the more standard framework.
It is well-known that associated with the continuous time algebraic Riccati equation:
∗ + + + = (6.4.8)
in which , and are real × matrices with and symmetric, there exists a × Hamiltonian matrix:
= [− − ∗] (6.4.9)
Assuming that has no eigenvalues on the imaginary axis, the spectrum of , i.e. � , will be symmetric about the imaginary axis. It is then possible to construct two invariant subspaces of dimension , corresponding to the stable and unstable modes of ,
respectively denoted by − and + . If we could find a basis for − and partition it as:
− = Im [ ] , , ∈ ℂ × (6.4.10)
such that is nonsingular, or equivalently, the two following subspaces are complement:
− , Im [ ] (6.4.11)
we can then define as = − . is uniquely determined by . In other words, → serves as a function, known as Ric, with the domain designated by dom Ric . Therefore, dom Ric encompasses Hamiltonian matrices with no purely imaginary eigenvalues, and those for which the two subspaces in (3.6.11) are complementary. These two features of the elements of dom Ric are usually recognized as the stability property and the complementarity, respectively. [1]
Theorem 6.4.1 [1] Suppose ∈ dom Ric and = Ric . Then: (i) is real symmetric.
(ii) satisfies the algebraic Riccati equation of (6.4.8). (iii) + is stable.
In the above theorem, is called a stabilizing solution to the Riccati equation of (6.4.8), i.e. the set of spectrum of + is in the open LHP.
The proposed approach in [136] to tackle the sub-optimal ∞ problem and the conditions for solvability of the problem, is based on the above way of constructing stabilizing solutions to the Riccati equation in terms of invariant subspaces of . In [135], Doyle et al. consider a simplified version of the problem stated in [136], by equating the and matrices in the plant state space description (expression (2.3.2)) to zero. For the sake of brevity, in here we just represent the results stated in [135] in the form of the theorem 6.4.2. It should be pointed out that the results are based on further assumptions on the
plant, which will not be restated here. These assumptions are quite standard and may be found in many references, e.g. [135, 136, 1, 85].
The solution and solvability conditions for the sub-optimal ∞ problem, i.e. ‖ ‖∞ < for some > , involves two Hamiltonian matrices:
∞ = [ − − − − ] (6.4.12) ∞ = [ − − − − ]
Theorem 6.4.2 [135] There exists an admissible controller such that ‖ ‖∞ < if and only if the following three conditions are satisfied:
i) ∞ ∈ dom Ric and ∞ = Ric ∞ ii) ∞ ∈ dom Ric and ∞ = Ric ∞ iii) � ∞ ∞ <
As can be seen, the feasibility condition is expressed in terms of the existence of unique positive definite stabilizing solutions to two algebraic Riccati equations, such that the spectral radius �, of their product is less than .
Conditions for the general case i.e. when and matrices are nonzero, are studied in [136]. The sub-optimal controller is parameterized in both [135] and [136], when the problem is feasible.
Having surveyed two of the major methods for tackling the ∞ problem, we conclude this section by introducing yet another scheme for treating the problem. In this approach, the ∞ norm minimization problem is transferred into a standard linear matrix inequality (LMI) feasibility problem. [142] The LMI characterization of the ∞ problem, is the so- called bounded real lemma, which is stated in the form of the following theorem:
Theorem 6.4.3 [106, 6] For an asymptotically stable discrete time system with the state space realization = − − + , ‖ ‖∞ < if and only if there exists a symmetric positive definite matrix such that:
[ − −
− ]
(6.4.13)
It is well-known that LMIs arise in many control analysis and synthesis problems and are reformulable as convex optimization problems, which correspondingly makes them readily amenable to computer solutions. [141] Moreover, LMI problems can be solved via efficient tractable numerical algorithms, e.g. interior-point method. [9, 103, 104, 105, 106] LMIs are especially beneficial for solving problems lacking analytical solution. Due to these peculiar attributes of LMIs, they have always been of special interest to many researchers and engineers.
In the next section, this approach will be exploited to synthesize a deadbeat compensator with ∞ norm constraint.