On Symmetric Gauss-Seidel ADMM Algorithm
for $\mathcal{H}_{\infty}$ Guaranteed Cost Control with
Convex Parameterization

Robust control theory typically investigates the effect of disturbances, noises, and uncertainties on system performance; and continued great efforts have been devoted to robust stabilization and robust performance in the literature [1, 2, 3, 4, 5, 6, 7]. Quite remarkably, several significant results [8, 9, 10] have been reported which relate the notion of quadratic stabilization to robust stabilization for a class of uncertain linear systems, and by this concept, the stability of an uncertain system is established with a quadratic Lyapunov function. On the other hand, it is also the case that $\mathcal{H}_{\infty}$ control is commonly and extensively used to attenuate the effect of disturbances on the system performance [11, 12]. Additionally, it is further known and shown in [13] that a certain type of quadratic stabilization problem can be essentially expressed as an $\mathcal{H}_{\infty}$ control problem, where a Riccati inequality condition relates the determination of a stabilizing feedback gain that imposes a suitable $\gamma$ disturbance attenuation level [14, 15]. Also notably, the problem of finding the minimal disturbance attenuation level is recognized as an important and commonly-encountered problem, and stated as the optimal $\mathcal{H}_{\infty}$ control problem. On this, it is additionally noteworthy that the work in [16] shows that the problem can be tackled by an iterative algorithm based on the Riccati inequality condition. However here, nonlinear characteristics of the Riccati inequality condition typically result in significant complexity and difficulty in obtaining the optimal gain and disturbance attenuation level. Moreover, $\mathcal{H}_{\infty}$ filtering has been widely studied for state estimation [17, 18]. It is remarkable that $\mathcal{H}_{\infty}$ filtering allows the system disturbances to be unknown, and uncertainties are tolerated in the system.

As considerable efforts have been made on the well-known Youla-Kucera parameterization (also known as $Q$-parameterization) for the determination of stabilizing controllers [19, 20, 21], one may thus think about borrowing this idea to solve the $\mathcal{H}_{\infty}$ optimal control problem in the presence of parametric uncertainties. However, the derivations of the classical Youla-Kucera parameterization results rely on the fact that the plant is linear with no parametric uncertainty, and the order of the controller depends on the order of the plant model and that of $Q$. Alternative parameterization techniques based on the positive real lemma and the bounded real lemma [22, 23] have also been proposed to deal with parametric uncertainties. However, as the required transfer function representation there results in reduced stability in numerical computations, and high computational cost also incurs; it is not considered as a preferable choice for many practical applications. Hence, several other parameterization methods are presented instead in a state-space framework, for example [24]. In essence, these techniques are considered as variants of the Youla-Kucera parameterization, but with more flexibility to deal with the structural constraints and parametric uncertainties. Regarding the nonlinear constraints existing in such a parameterization (also noted to be convex in [24]), outer linearization is necessary for polyhedral approximation during iterative refinement [25]. Similarly, the cutting-plane method is presented in [26, 27] to solve generalized $\mathcal{H}_{\infty}$ control problems. This technique could be effective in some scenarios, but there could be other certain scenarios such as high-dimension systems or uncertain systems with a large volume of parametric uncertainties. For example, in [28], a large-scale $\mathcal{H}_{\infty}$ optimal control problem is simplified in a way that enables sparse solutions and efficient computation. In [29], a distributed $\mathcal{H}_{\infty}$ controller is developed for large-scale platoons. In the scenarios that high-dimension optimization is involved (which resulted from rather numerous extreme systems involved computationally), the convergence rate of the cutting-plane method can be unacceptably slow; and in some cases, the optimization process could even terminate abruptly with unsuccessful outcomes. This is because a sizeable number of cutting planes needs to be added computationally at each iteration, and in difficult scenarios, the optimization process can thus become unwieldy. It is also worth mentioning that this method can only guarantee the so-called $\epsilon$-optimality because the constraints are typically not exactly satisfied but violated by certain small values. Therefore, such a situation causes deviations from the “true" optimal result, and consequently the desired robustness is not perfectly guaranteed, and particularly so if the parametric uncertainties are significant.

Because of the typical computational burden arising from the growth of system dimensions and parametric uncertainties, several advanced optimization techniques are presented in the more recent works. Hence for large-scale and nonlinear optimization problems, the alternating direction method of multipliers (ADMM) [30, 31, 32, 33] has attracted considerable attention from researchers, and is widely used in various areas such as statistical learning [30], distributed computation [34, 35, 36], and multi-agent systems [37, 38]. ADMM demonstrates high efficiency in the determination of the optimal solution to many challenging problems. Remarkably too, some of these challenging optimization problems cannot even be solved by the existing conventional gradient-based approaches, and in these, ADMM demonstrates its superiority. Nevertheless, the conventional ADMM methodology only ensures appropriate convergence with utilization of a two-block optimization structure, and this constraint renders a serious impediment to practical execution [39]. To cater to this deficiency, the symmetric Gauss-Seidel technique can be used to conduct the ADMM optimization serially [40, 41], which significantly improves the feasibility of the ADMM in many large-scale optimization problems. Although these methodologies are reasonably well-established, nevertheless only rather generic procedures are given at the present stage. Therefore, it leaves an open problem on how to apply these advanced optimization techniques in control problems such that these methodologies can be extended beyond the theoretical level.

It is also rather essential at this point to note that in the presence of significant parametric uncertainties, the $\mathcal{H}_{\infty}$ optimization problem is usually of the large-scale type (because of the exponential growth of the number of extreme systems involved computationally with respect to the number of parametric uncertainties; and each of these extreme systems has a one-to-one correspondence to an inequality constraint to ensure the closed-loop stability). Therefore, our contributions are summarized as follows. Here, we propose a novel optimization technique to solve the resulting large-scale $\mathcal{H}_{\infty}$ guaranteed cost control problem resulting from parametric uncertainties, where the stabilizing controllers are characterized by an appropriate convex parameterization (which will be described and established analytically). Firstly, we construct a convex set such that all the stabilizing controller gains are mapped onto the parameter space, and the desired robust stability is then attained with the optimal disturbance attenuation level in the presence of convex-bounded parametric uncertainties. With this parameterization technique, the parametric uncertainties can be suitably considered in the problem formulation. Secondly, a suitably interesting problem re-formulation based on the Schur complement facilitates the use of the symmetric Gauss-Seidel ADMM algorithm. Comparing with the methods in the existing literature (as described previously), this approach alleviates the ofter-times prohibitively heavy computational burden typically in many large-scale $\mathcal{H}_{\infty}$ control problems.

The remainder of this paper is organized as follows. In Section II, the optimal $\mathcal{H}_{\infty}$ controller synthesis with convex parameterization is provided. Section III presents the symmetric Gauss-Seidel ADMM algorithm to solve the $\mathcal{H}_{\infty}$ guaranteed cost control problem. Then, to validate the proposed algorithm, appropriate illustrative examples are given in Section IV with simulation results. Finally, pertinent conclusions are drawn in Section V.

Notations: $\mathbb{R}^{m\times n}$ ($\mathbb{R}^{n}$) denotes the real matrix with $m$ rows and $n$ columns ($n$ dimensional real column vector). $\mathbb{S}^{n}$ ($\mathbb{S}^{n}_{+}$) denotes the $n$ dimensional (positive semi-definite) real symmetric matrix, and $\mathbb{S}^{n}_{++}$ denotes the $n$ dimensional positive definite real symmetric matrix. The symbol $A\succ 0$ ($A\succeq 0$) means that the matrix $A$ is positive definite (positive semi-definite). $A^{T}$ ($x^{T}$) denotes the transpose of the matrix $A$ (vector $x$). $I_{n}$ ($I$) represents the identity matrix with a dimension of $n\times n$ (appropriate dimensions). The operator $\operatorname{Tr}(A)$ refers to the trace of the square matrix $A$. The operator $\langle A,B\rangle$ denotes the Frobenius inner product i.e. $\langle A,B\rangle=\operatorname{Tr}\left(A^{T}B\right)$ for all $A,B\in\mathbb{R}^{m\times n}$. The norm operator based on the inner product operator is defined by $\|x\|=\sqrt{\langle x,x\rangle}$ for all $x\in\mathbb{R}^{m\times n}$. $\|H(s)\|_{\infty}$ represents the $\mathcal{H}_{\infty}$-norm of $H(s)$. The operator $\operatorname{vec}(\cdot)$ denotes the vectorization operator that expands a matrix by columns into a column vector. The symbol $\otimes$ denotes the Kronecker product. $\sigma_{\textup{M}}(\cdot)$ returns the maximum singular value.

Consider a linear time-invariant (LTI) system

with $x(0)=x_{0}$, $x\in\mathbb{R}^{n}$ is the state vector, $u\in\mathbb{R}^{m}$ is the control input vector, $w\in\mathbb{R}^{l}$ is the exogenous disturbance input, $z\in\mathbb{R}^{q}$ is the controlled output vector, $K\in\mathbb{R}^{m\times n}$ is the feedback gain matrix. As a usual practice, Assumption 1 is made.

Denote $A_{c}=A-B_{2}K$ and $C_{c}=C-DK$, the transfer function from $w$ to $z$ is given by

and the $\mathcal{H}_{\infty}$-norm is defined as

It is worth stating that the objective of the optimal $\mathcal{H}_{\infty}$ control problem is to design a feedback controller that minimizes the $\mathcal{H}_{\infty}$-norm while maintaining the closed-loop stability. When the plant is affected by parametric uncertainties, the minimization of the upper bound to the $\mathcal{H}_{\infty}$-norm under all feasible models is known as the $\mathcal{H}_{\infty}$ guaranteed cost control problem. Note that in this work, $\gamma$-attenuation means that the $\mathcal{H}_{\infty}$-norm of $H(s)$ is bounded by $\gamma$, i.e., $\|H(s)\|_{\infty}\leq\gamma$.

In this work, for brevity, we define $p=m+n$ and $r=m+2n$. Then, the following extended matrices are introduced to represent the open-loop model (1a)-(II):

Also, define the matrix

where $W_{1}\in\mathbb{S}^{n}_{++}$, $W_{2}\in\mathbb{R}^{n\times m}$, $W_{3}\in\mathbb{S}^{m}$, and then define the matrical function

with $\mu=1/{\gamma}^{2}$ [42]. Similarly, $\Theta(W,\mu)$ is partitioned as

with $\Theta_{1}(W,\mu)\in\mathbb{S}^{n},\Theta_{2}(W)\in\mathbb{R}^{n\times m},\Theta_{3}(W)\in\mathbb{S}^{m}$.

The following theoretical developments generalize the results in [42, 43, 44].

For Statement (b), the following lemma is introduced first to relate a Riccati inequality condition to $\mathcal{H}_{\infty}$-norm attenuation.

Notice that Assumption 1 implies that the pair $[A_{c},C_{c}]$ is observable. Then, from Lemma 1, there exists a symmetric positive definite solution $P=P^{T}\succ 0$ such that

Since $P$ is nonsingular, by pre-multiplying and post-multiplying $P^{-1}$ in (12), we have

Denote $W_{p}=P^{-1}$, (13) is equivalent to

Meanwhile, from (9), we have

Then, by setting $W_{1}=W_{p}$ and $W_{2}^{T}=KW_{p}$, we have $K=W_{2}^{T}W_{1}^{-1}$ and $\Theta_{1}(W,\mu)\preceq 0.$ It gives that $K=W_{2}^{T}W_{1}^{-1}$ is a feasible solution to ensure the stability with $\gamma$-attenuation [42]. By substituting $K=W_{2}^{T}W_{1}^{-1}$ to (7), we can construct

By Schur complement, we can ensure $W\succeq 0$ by choosing $W_{3}\succeq KW_{1}K^{T}$. Based on the analysis above, $K=W_{2}^{T}W_{1}^{-1}$ is a stabilizing gain generated from $(W,\mu)\in\mathscr{C}$, and it follows from Lemma 1 that $\|H(s)\|_{\infty}\leq\gamma$ is guaranteed [42].

Statement (c) is direct consequence of Statement (b). ∎

Then, it suffices to extend the above results to uncertain systems, and then we make the following assumption.

Followed by Assumption 2, we have $F=\sum_{i=1}^{N}\xi_{i}F_{i}$, $\xi_{i}\geq 0$, $\forall i=1,2,\cdots,N$, and $\sum_{i=1}^{N}\xi_{i}=1$. Notice that $F$ belongs to a polyhedral domain, which can be expressed as a convex combination of the extreme matrices $F_{i}$, where

Then, define the matrical function in terms of each extreme vertice, where

which can also be partitioned as

with $\Theta_{1i}(W,\mu)\in\mathbb{S}^{n},\Theta_{2i}(W)\in\mathbb{R}^{n\times m},\Theta_{3i}(W)\in\mathbb{S}^{m}$. Consequently, a mapping between $W$ and $K$ can be constructed, and the results are shown in Theorem 2.