Controller Design via Experimental Exploration with Robustness Guarantees

We assume that we are given an unknown real system $P_{0}$ described as

Here $e$ is the controlled output and $d$ is a generalized disturbance (both used to formulate performance specifications), while $y$ is the measurement output and $u$ the control input. The underlying control problem is to find a controller

such that the corresponding closed-loop system, which is referred to as $P_{0}\star F$, is stable and such that a closed-loop cost function $J$, which encodes the performance specifications, is minimized. Since $P_{0}$ is unknown, we aim to find such a controller based on evaluations of the cost function $J$. This amounts to the selection of suitable test controllers, their implementation on the real system $P_{0}$ and the evaluation of their achieved closed-loop performance.

This is the setting in [BerSch16, MarHen17, DriEng17, CalSey15, RohTri18], where the individual approaches differ, e.g., in the choice of cost, the available measurements from the plant, the assumed prior knowledge about the plant and the employed controller parametrization.

We confine the discussion to continuous-time linear time-invariant (LTI) systems $P_{0}$ and the design of state-feedback controller gains $F$ which motivates to choose $y$ as the state $x$ of $P_{0}$. Moreover, we choose the $H_{\infty}$-norm cost function

for which ample motivations are found in the robust control literature. Data-based techniques for estimating $H_{\infty}$-norms have been proposed, e.g., in [RalFor17]. To simplify the exposition, we assume that the measurements of the cost are exact, although it is possible to extend our framework to noisy ones.

We consider the case that $P_{0}$ is partially unknown. To this end, we adopt the framework of LFRs [ZhoDoy96] and describe $P_{0}$ as the interconnection of some known system $P$ given by

in feedback with some unknown part or uncertainty

Here $\Delta_{0}$ is a real matrix of suitable dimension and $w$, $z$ are the interconnection variables. Then (1) admits a state-space representation with the matrix

By slightly abusing the notation, the latter matrix and the system (1) are denoted as $\Delta_{0}\star P$. Note that the controlled unknown system, the interconnection of (4), (5) and (2), is then given by $P_{0}\star F=(\Delta_{0}\star P)\star F=\Delta_{0}\star P\star F$. Such representations are known to be highly flexible since they permit to effectively capture structural dependencies of models on uncertain scalar parameters or matrix sub-blocks, which are typically collected on the diagonal of the (structured) uncertainty $\Delta_{0}$. As an extra advantage, LFRs allow a seamless generalization to multiple heterogeneous (i.e., a mixture of time-varying, non-linear or infinite dimensional) uncertainties collected in a nonlinear feedback operator $w(t)=\Delta_{0}(t,z)(t)$, but this is not pursued here.

Instead, we adopt the point-of-view that $\Delta_{0}$ describes unknown parts of the system $P_{0}$, while the known parts (as, e.g., resulting from first-principle modeling) are captured by $P$. Moreover, we assume that $\Delta_{0}$ is contained in some known set $\mathbf{\Delta}$ of matrices that is compact and typically given by

with (repeated) diagonal and full unstructured blocks on the diagonal, all bounded in norm by one. As an extreme case, this description does capture the models in [BerSch15, MarHen17, BocMat18] and the ones in [FerUme20, MatPro19, FazGe19, ZhaHu20], in which it is assumed that nothing aside from linearity is known about $P_{0}$ and where $\Delta_{0}$ is just one large unstructured uncertain matrix.

Note that the development of modern robust control has been substantially motivated by the fact that facing completely unknown systems is often not realistic. By now LFRs are used in tandem with dedicated analysis and design tools from robust control, such as structural singular values or integral quadratic constraints (IQCs) [MegRan97], which permit to accurately exploit the fine structure of the unknown $\Delta_{0}$.

Therefore, in view of their modeling power, LFRs provide an ideal setting to incorporate prior structural knowledge about a system (through $P$) with unknown to-be-learnt components (through the elements of $\Delta_{0}$).

Clearly, guaranteeing stability is a critical issue in learning based approaches since probing the system with gains that are not stabilizing can lead to catastrophes. In contrast to many other approaches as, e.g., in [DriEng17, CalSey15, RohTri18] and aligned with [MarHen17], we propose to only select controllers that are guaranteed to be robustly stabilizing, i.e., that are taken from the set

This set is typically much smaller than the set of controllers that are merely required to stabilize $\Delta_{0}\star P$. However, since $\Delta_{0}\in\mathbf{\Delta}$ is unknown and since we can only rely on the prior knowledge about $\mathbf{\Delta}$, there is no other choice than to pick gains from $\mathbb{F}(\mathbf{\Delta})$ in order to ensure a safe operation of the system in closed-loop. The minimal value of $J(F)$ over the set $\mathbb{F}(\mathbf{\Delta})$ is related to the cost of interest as

in which the gap reflects the price to-be-paid for safety.

For optimizing the cost $J$ it is highly beneficial and an often seen strategy to parameterize a family of test controllers by a few parameters before applying an optimization algorithm, especially if the ambient space of controller gains has a large dimension [MarHen16, RobMan11, BanCal17]. Formally, such a parametrization is a mapping $\mathcal{F}$ with a domain $\mathrm{dom}(\mathcal{F})$ that is contained in a low dimensional ambient space and chosen in order to render the gap in the inequality

as small as possible. Then, the idea is to minimize the surrogate cost $J\circ\mathcal{F}$ over $\mathrm{dom}(\mathcal{F})$ instead of determining the minimum of the original cost $J$. Since the former minimization problem is formulated in a low dimensional space, it is expected to require substantially fewer evaluations of the cost function for its (approximate) solution. The gap in ($\mathcal{P^{\prime}}$) constitutes the price to-be-paid for this reduction of complexity and is rarely analyzed in the literature. We stress that it is instrumental to choose a parametrization $\mathcal{F}$ such that its values are contained in $\mathbb{F}(\mathbf{\Delta})$ for reasons of safety. Then the gap in ($\mathcal{P^{\prime}}$) can even be more precisely identified as the sum of that in ($\mathcal{S}$) and the one in

For some index set $\mathbb{I}$, we propose a novel parametrization $\mathcal{F}:\mathbb{I}\to\mathbb{F}(\mathbf{\Delta})$ of auspicious robustly stabilizing controllers based on a partition of $\mathbf{\Delta}$. It features an a priori safety guarantee without the need to ensure this property through the employed optimization algorithm as proposed, e.g., in [ZhaHu20]. For its construction, we use advanced robust control techniques that explicitly take the available prior knowledge into account. Based on this parametrization, we show how experimental controller probing allows for controlling the size of the gap in ($\mathcal{P}$) by varying the coarseness of the partition of $\mathbf{\Delta}$, and how to even reduce the gap in ($\mathcal{S}$) by systematically decreasing the size of $\mathbf{\Delta}$ without endangering safety.

In comparison to a standard robust design, which does not utilize data from closed-loop experiments, our approach naturally generates safe controllers with improved performance on the real plant $P_{0}$.

Let us choose the index set $\mathbb{I}:=\{1,\dots,N\}$ and subsets $\mathbf{\Delta}_{1},\dots,\mathbf{\Delta}_{N}$ of the uncertainty set $\mathbf{\Delta}$ that form the partition

With this partition, we construct $\mathcal{F}:\mathbb{I}\to\mathbb{F}(\mathbf{\Delta})$ based on the rationale to render $J(\mathcal{F}(k))$ for at least one index $k\in\mathbb{I}$ as small as possible, since this leads to the best possible reduction of the gap in ($\mathcal{P}$). The proposed parametrization assigns to $k\in\mathbb{I}$ a controller $F\in\mathbb{F}(\mathbf{\Delta})$ which reduces ${\sup_{\Delta\in\mathbf{\Delta}_{k}}\|\Delta\star P\star F\|_{\infty}}$ as much as possible. This means that we are facing a robust multi-objective synthesis problem involving robust stability w.r.t. $\Delta\in\mathbf{\Delta}$ and worst-case $H_{\infty}$ performance w.r.t. $\Delta\in\mathbf{\Delta}_{k}$. Such problems are usually nonconvex as well as nonsmooth and thus hard to solve systematically. Still, it is possible to compute good upper bounds on the corresponding optimal value by solving a linear SDP if relying on so-called multiplier relaxations in robust control. One such relaxation is given in Theorem 1 and requires to specify a set $\mathbb{P}(\mathbf{\Delta})$ of real symmetric matrices with an LMI description such that

We also assume that such multiplier classes $\mathbb{P}(\mathbf{\Delta}_{k})$ are available for the partition members $\mathbf{\Delta}_{k}$ and for $k=1,\ldots,N$. A more detailed discussion with concrete choices for such multiplier sets can be found in [Sch05, SchWei00].

The proof of this result is found in [SchWei00]. It shows that

is satisfied for $\gamma_{*}(k)\!:=\!\inf\{\gamma\in\mathbb{R}:\text{ LMIs \eqref{DC::lem::eq::lmi_part} are feasible}\}$.

All this leads us to the construction of the parametrization $\mathcal{F}$ as follows: For some fixed small $\varepsilon>0$ and $\gamma_{*}^{\varepsilon}(k):=(1+\varepsilon)\gamma_{*}(k)$, we assign to $k\in\mathbb{I}$ some gain $\mathcal{F}(k)$ with

We emphasize that both $\gamma_{*}(k)$ and $\mathcal{F}(k)$ can be computed by solving a standard semi-definite program. Still note that, in general, $\gamma_{*}(k)$ is not attained (no optimal controller exists), which motivates the introduction of $\varepsilon$.

In the sequel, we abbreviate the surrogate cost function resulting from the parametrization $\mathcal{F}$ as

Further, let us note at this point that $\Delta_{0}\in\mathbf{\Delta}_{k}$ for some index $k\in\mathbb{I}$ clearly implies

After having introduced the controller parametrization, the conceptual algorithm of this paper reads as follows. For each $k\in\mathbb{I}$, we can implement the controller $\mathcal{F}(k)$ on the system, since it is assured to be stabilizing for $P_{0}$, and measure the cost $L(k)$. A mere minimization over $k\in\mathbb{I}$ then leads to an optimal controller $\mathcal{F}(k_{\ast})$, and inequality ($\mathcal{P^{\prime}}$) now reads as

Fine partitions of $\mathbf{\Delta}$ lead to large index sets $\mathbb{I}$. Instead of considering all $k\in\mathbb{I}$, we can take fewer (random) samples $\{k_{1},\ldots,k_{M}\}$ of $\mathbb{I}$ and obtain a (rough) approximation $\min_{j=1,\dots,M}L(k_{j})$ of $L(k_{\ast})$. In particular for the partition with $\mathbb{I}=\mathbf{\Delta}$ as described in Remark 2, one can directly employ a whole variety of smarter (derivative free) sampling and optimization strategies, such as Bayesian optimization involving Gaussian processes discussed in [BerSch16, MarHen17].

Instead of considering a single (fine) partition, one can as well start from a coarse partition of $\mathbf{\Delta}$ and propose adaptive refinement strategies which generate a sequence of controller parametrizations as follows. Given $\mathbf{\Delta}=\bigcup_{k=1}^{N}\mathbf{\Delta}_{k}$, determine an index $k^{0}\in\operatorname*{arg\,min}_{j=1,\dots,N}L(j)$. Then generate a partition of $\mathbf{\Delta}_{k^{0}}$ denoted as $\bigcup_{j=1}^{N}\mathbf{\Delta}_{k^{0}_{j}}$ in order to obtain a refined partition of the original set as

This refined partition yields a new parametrization $\mathcal{F}^{1}$ with corresponding new cost $L^{1}$ and some next optimal index $k^{1}\in\operatorname*{arg\,min}_{j=1,\dots,N}L^{1}(k^{0}_{j})$. By construction it is guaranteed that $L^{1}(k^{1})\leq L(k^{0})$ holds. This step can be iterated in order to further decrease the value of the surrogate cost. In Section IV we propose a specific algorithm based on this approach which involves, in particular, a concrete strategy for refining given partitions.

Our setup allows for identifying the sources of the gap in ($\mathcal{P}$) and permits to generate systematic refinements towards its reduction. To illustrate this issue, let us suppose that the relaxation gap in (8) is small. Then we infer (by the definition of $\mathcal{F}$ and for small $\varepsilon>0$) that

On the other hand, for $k\in\mathbb{I}$ with $\Delta_{0}\in\mathbf{\Delta}_{k}$ and if this member $\mathbf{\Delta}_{k}$ of the partition is sufficiently small, we have

Hence $\inf_{k\in\mathbb{I}}L(k)$ is close to $\inf_{F\in\mathbb{F}(\mathbf{\Delta})}\|\Delta_{0}\star P\star F\|_{\infty}$ which shows that the gap in ($\mathcal{P}$) is small. In conclusion, it is essential that the size of the partition member containing $\Delta_{0}$ and the relaxation gap in (8) are both small. Without going into details, we emphasize that the latter can be controlled with the choices of the multiplier sets $\mathbb{P}(\mathbf{\Delta})$ and $\mathbb{P}(\mathbf{\Delta}_{k})$, through the use of more advanced multi-object control techniques and by applying further refinements in robust control [ArzPea00], such as incorporating S-variables [EbiPea15] or dynamic instead of static IQCs [MegRan97].

Our approach offers the opportunity to even reduce the gap in ($\mathcal{S}$) by identifying a smaller index set $\tilde{\mathbb{I}}\subset\mathbb{I}$ with

since $k\in\mathbb{I}/\tilde{\mathbb{I}}$ implies $\Delta_{0}\notin\mathbf{\Delta}_{k}$ by (10). Note that $\tilde{\mathbf{\Delta}}$ can be considerably smaller than the original $\mathbf{\Delta}$, which implies that the related set $\mathbb{F}(\tilde{\mathbf{\Delta}})$ of robustly stabilizing controllers is (much) larger than $\mathbb{F}(\mathbf{\Delta})$. Thus, replacing $\mathbf{\Delta}$ with $\tilde{\mathbf{\Delta}}$ reduces the cost of safety as expressed by the gap in ($\mathcal{S}$).

This suggests to repeat our design procedure for $\tilde{\mathbf{\Delta}}$, which amounts to constructing a new parametrization $\tilde{\mathcal{F}}$ giving controllers $\tilde{\mathcal{F}}(k)$ (via Theorem 1) with which we can perform new closed-loop experiments to evaluate $\tilde{L}=J\circ\tilde{\mathcal{F}}$. The controllers $\tilde{\mathcal{F}}(k)$ are expected to achieve (considerably) improved closed-loop performance with a smaller gap in ($\mathcal{P^{\prime}}$), just due to the reduction of the gap in ($\mathcal{S}$). The algorithm proposed in the next section is based on this strategy.

Note that the set $\mathbb{I}$ is constructed based on (10) which provides an upper bound on the cost $L(k)$ at the index $k$ with $\Delta_{0}\in\mathbf{\Delta}_{k}$. We can also devise a lower bound which can be exploited similarly in order to further reduce $\tilde{\mathbb{I}}$ and shrink the gap in ($\mathcal{S}$). To this end, observe that standard $H_{\infty}$ design permits to numerically determine

Then $\Delta_{0}\in\mathbf{\Delta}_{k}$ for $k\in\mathbb{I}$ indeed yields the lower bound

Note that this lower bound is not cheap to compute as it involves a numeric minimization of $\gamma_{\mathrm{nom}}$ on $\mathbf{\Delta}_{k}$ for each considered $k\in\mathbb{I}$. In contrast, the upper bound $\gamma_{\ast}^{\varepsilon}(k)$ is essentially obtained for free while constructing the map $\mathcal{F}$.