Switching Control for Parameter Identifiability of Uncertain Systems

Identifying the parameters of an uncertain system from input-output data is a problem of long-standing fundamental interest in control engineering. This problem is often referred to as the problem of parameter identifiability [1, 2]. This paper considers the identifiability problem with respect to uncertain linear systems where the uncertainty set consists of a known bounded set possibly containing a continuum of parameters. For this class of systems, we address the problem of ensuring the identifiability of the unknown parameters of the system by means of feedback controllers.

The motivations for studying this problem are immense. For instance, parameter identifiability of a feedback loop can be of interest in the context of fault detection/isolation for systems subjects to failures, in order to make it possible to promptly detect any departure from the nominal behavior and to precisely identify the parameter variation [3, 4]. Another application is control reconfiguration wherein the objective is that of replacing the active controller (typically designed in order to ensure robust stability in all the uncertainty region) with a different one providing enhanced (possibly optimized) performance [5, 6]. Finally, on-line estimation of the uncertain parameters under feedback can be exploited when dealing with systems which naturally exhibit multiple operating conditions for constructing adaptive control laws. In fact, many existing adaptive control techniques rely on the idea of certainty equivalence, which amounts to applying at each instant of time the controller designed for the model that best fits the available data [7, 8].

In this paper, the problem of parameter identifiability is approached by searching for feedback control laws under which closed-loop behaviors obtained with different system parameters can be distinguished one from another. To this end, we introduce a notion of discerning control. Parameter identifiability is then defined precisely in terms of discerning control.

In principle, parameter identifiability under feedback can always be ensured by means of a probing signal injected into the plant as an additive perturbation input, superimposed to the control variable [7, 8]. Nevertheless, in many contexts, such a solution should be avoided due to the inherent drawback of leading the feedback loop away from the desired behavior, thus destroying regulation properties. This is especially true when the behavior of the feedback loop has to be monitored continuously as in the contexts of fault/detection isolation and adaptive control. Then, a natural question arises on whether or not it is possible to guarantee parameter identifiability directly by means of a feedback control law, possibly designed also to satisfy other control objectives (e.g., stability in nominal operating conditions). An affirmative answer to this question was given in [9, 10] for the special case of switching linear systems, i.e., when the uncertain parameters can take on only a finite number of possible values. Specifically, in [9, 10], it is shown that, under quite mild assumptions, for switching linear systems almost all linear time-invariant (LTI) controllers are discerning.

In general, an analogous result cannot be established in case of continuously parameterized systems, i.e., when the uncertainty set is not finite. The reason is inherently tied to the fact that LTI controllers do not generally provide a sufficient level of excitation to the loop [7, Chapter 2]. The problem of loss of identifiability due to feedback can be overcome by means of time-varying controllers, and one possibility is given by switching control [11, 12]. In this paper, we exploit this property to show that parameter identifiability can in fact be generically ensured by switching among a finite number of LTI controllers (hereafter called modes), provided that the number of different controller modes is sufficiently large. Specifically, an upper bound on the number of controller modes needed for parameter identifiability is given in terms of the dimension of the uncertainty set. Moreover, we show that the result remains true even if we restrict the controller modes to be of a given fixed order and to satisfy certain stability requirements. The latter result is perhaps surprising as it indicates that the seemingly conflicting goals of ensuring parameter identifiability as well as a satisfactory behavior of the feedback system (at least under nominal conditions) can be simultaneously accomplished by means of switching control.

As a further contribution, we analyze the properties of least-squares parameter estimation in connection with the use of discerning controllers. Specifically, in order to ensure the practical applicability of the estimation technique, we focus on a multi-model approach wherein the estimate is selected among a finite number of possible values of the parameter vector (obtained by suitably sampling the uncertainty set). In this context, a bound on the worst-case parameter estimation error is derived, which accounts also for the presence of unknown but bounded disturbances and measurement noises. This latter result is of special interest in the context of multi-model adaptive switching control (MMASC) of uncertain systems [9], [13]-[16], of which multi-model least-square parameter estimation constitutes one of the key elements. In this respect, it has been shown in [9, 16] that, in the case of a finite uncertainty set, by employing discerning controllers it is possible to construct MMASC schemes which enjoy quite strong stability properties, namely exponential input-to-state stability. Hence, the results of the paper suggest that similar stability properties could be achieved also in the case of continuously parameterized uncertainty. This issue will be the subject of further research.

The remainder of the paper is organized as follows. In Section II, we describe the framework under consideration. In Section III, the main results of the paper are given concerning the existence and genericity of switching controllers ensuring parameter identifiability. Section IV analyzes the properties of multi-model least-squares parameter estimation. Examples are finally given in Section V.

For the sake of clarity, all the proofs are reported in the Appendix section.

Notation. Before concluding this section, let us introduce some notations and basic definitions. Given a vector $v\in\mathbb{R}^{n}$, $|v|$ denotes its Euclidean norm. Given a symmetric, positive semi-definite matrix $\,P$, we denote by $\,{\lambda}_{\rm min}(P)$ and $\,{\lambda}_{\rm max}(P)$ the minimum and maximum eigenvalues of $P$, respectively. Given a matrix $\,M$, $\,M^{\top}$ is its transpose and $\,|M|=\left[\lambda_{\rm max}(M^{\top}M)\right]^{1/2}$ its spectral norm. Given a measurable time function $v:\mathbb{R}^{+}\rightarrow\mathbb{R}^{n}$ and a time interval $\mathcal{I}\subseteq\mathbb{R}^{+}$, we denote the $\mathcal{L}_{2}$ and $\mathcal{L}_{\infty}$ norms of $v(\cdot)$ on $\mathcal{I}$ as $\|v\|_{2,\mathcal{I}}=\sqrt{\int_{\mathcal{I}}|v(t)|^{2}dt}$ and $\|v\|_{\infty,\mathcal{I}}=\mbox{ess sup}_{t\in\mathcal{I}}{|v(t)|}$ respectively. When $\mathcal{I}=R^{+}$, we simply write $\|v\|_{2}$ and $\|v\|_{\infty}$. Finally, we let $\mathcal{L}_{2}(\mathcal{I})$ and $\mathcal{L}_{\infty}(\mathcal{I})$ denote the sets of square integrable and, respectively, (essentially) bounded time functions on $\mathcal{I}$.

We consider a process described by an uncertain linear system $\mathcal{P}(\theta)$

where $x\in{\mathbb{R}}^{n_{x}}$ is the state, $u\in{\mathbb{R}}^{n_{u}}$ is the input, $y\in{\mathbb{R}}^{n_{y}}$ is the output, and $\theta\in{\mathbb{R}}^{n_{\theta}}$ is an unknown parameter vector belonging to the known bounded set $\Theta\subseteq{\mathbb{R}}^{n_{\theta}}$.

The problem of interest is that of designing a controller $\mathcal{C}$ ensuring global discernibility, i.e., identifiability of the unknown parameter vector $\theta$ from observations of the plant input/output data

where col stands for column vector. In [10], the problem was addressed in the special case when the set $\Theta$ is finite and it was shown that, under mild assumption, global discernibility can be ensured by means of a LTI controller. When the set $\Theta$ is not finite, a single LTI controller in general cannot ensure global discernibility by itself. Nevertheless, as will be shown in the following, it turns out that global discernibility can be achieved by switching among a finite number of LTI controllers. Accordingly, let the controller be described by a switching linear system

where $\xi\in{\mathbb{R}}^{n_{\xi}}$ is the controller state and $\sigma:{\mathbb{R}}_{+}\mapsto{\mathcal{N}}:=\left\{1,2,\ldots,N\right\}$ is the switching signal, i.e. the signal (right continuous) which identifies the index of the active system at each instant of time. Hereafter, it will be supposed that the switching signal $\sigma$ is generated so as to have a finite number of discontinuity points in every finite time interval. For any $i\in\mathcal{N}$, $F_{i}$, $G_{i}$, $H_{i}$, and $K_{i}$ are constant matrices of appropriate dimensions. In the sequel, we shall denote by $\mathcal{C}_{i}$ the LTI system with state-space representation $\{F_{i},G_{i},H_{i},K_{i}\}$ and by $\mathcal{C}_{\sigma}$ the control law associated with the switching signal $\sigma$.

Denote by $\chi=\textrm{col}(x,\xi)$ and $z=\textrm{col}(u,y)$ the state and the output of the closed-loop system $(\mathcal{P}(\theta)/\mathcal{C}_{\sigma})$ resulting from the interconnection of (3) and (6) when the unknown parameter takes value $\theta$ and the controller switching signal is $\sigma$. The corresponding dynamics can be therefore expressed as

where, for any $i\in\mathcal{N}$ and $\theta\in\Theta$,

Finally, let $z(t,t_{0},x_{0},\xi_{0},\theta,\sigma)$ denote the value at time $t$ of $z$ when the plant initial state at time $t_{0}$ is $x_{0}$, the controller initial state is $\xi_{0}$, the unknown parameter vector takes value $\theta$, and the controller switching signal is $\sigma$. The following notions can be introduced.

In words, when the switching controller (6) is ($\theta$,$\theta^{\prime}$)-discerning and a discerning switching signal $\sigma$ is adopted, then $\mathcal{P}(\theta)$ and $\mathcal{P}(\theta^{\prime})$ cannot give rise to the same observation data when $\mathcal{C}_{\sigma}$ is in the feedback loop (unless the initial conditions are null). As a consequence, under global discernibility, it is possible to uniquely identify the unknown parameter vector $\theta$ by observing $z$ on the interval $\mathcal{I}$.

In this section, we derive sufficient conditions for a switching control to be discerning and we show that, under mild assumptions, almost every switching controller is discerning provided that the number $N$ of controller modes is sufficiently large.

To this end, notice preliminarily that a necessary condition for the existence of a discerning controller is that all the pairs $(A(\theta),C(\theta))$ are observable. In fact, the presence of unobservable dynamics would entail the existence of non-zero trajectories of the closed-loop state $\chi$ corresponding to zero trajectories of the closed-loop output $z$ and, hence, for which it would be impossible to infer the plant mode. Accordingly, the following assumption is considered.

1. A1.

   The pair $(A(\theta),C(\theta))$ is observable for all $\theta\in\Theta$.

Let now $\varphi_{i}(s,\theta)$ denote the characteristic polynomial of the closed-loop system $(\mathcal{P}(\theta)/\mathcal{C}_{i})$ resulting from the feedback interconnection of the plant $\mathcal{P}(\theta)$ with the $i$-th controller mode $\mathcal{C}_{i}$. The following result holds.

Let now $\bar{n}_{\xi}$ denotes the total number of elements of the controller matrices $(F_{i},G_{i},H_{i},K_{i})$ when the controller order is $n_{\xi}$, and let $\mathcal{D}(\theta,\theta^{\prime})\subseteq\mathbb{R}^{\bar{n}_{\xi}}$ be the set of controller matrices¹¹1Here, with a little abuse of notation, we identify the quadruples $(F_{i},G_{i},H_{i},K_{i})$ with an $\bar{n}_{\xi}$-dimensional vector containing all the elements of the matrices $(F_{i},G_{i},H_{i},K_{i})$ according to a given order. $(F_{i},G_{i},H_{i},K_{i})$ for which the two closed-loop polynomials $\varphi_{i}(s,\theta)$ and $\varphi_{i}(s,\theta^{\prime})$ are coprime. Further, for a given number $N$ of controller modes, let $\mathcal{D}_{N}\subseteq\mathbb{R}^{N\bar{n}_{\xi}}$ be the set of switching controllers $(F_{i},G_{i},H_{i},K_{i}),i\in\mathcal{N}$ satisfying the hypothesis of Lemma 1 (i.e., such that for any pair of distinct parameter vectors $\theta,\theta^{\prime}\in\Theta$, there exist at least one index $i\in\mathcal{N}$ such that the two closed-loop characteristic polynomials $\varphi_{i}(s,\theta)$ and $\varphi_{i}(s,\theta^{\prime})$ are coprime). Since, in view of Lemma 1, all switching controllers belonging to $\mathcal{D}_{N}$ are globally discerning, we are now interested in studying the properties of the sets $\mathcal{D}(\theta,\theta^{\prime})$ and $\mathcal{D}_{N}$.

To this end, recall that coprimeness of the two polynomials $\varphi_{i}(s,\theta)$ and $\varphi_{i}(s,\theta^{\prime})$ is equivalent to the fact that their Sylvester resultant $R_{i}(\theta,\theta^{\prime})$ (i.e., the determinant of the Sylvester matrix associated with the two polynomials) is different from $0$. With this respect, we note that $R_{i}(\theta,\theta^{\prime})$ depends polynomially on the elements of the controller matrices $(F_{i},G_{i},H_{i},K_{i})$. Hence, when only a single pair of distinct parameter vectors $\theta,\theta^{\prime}\in\Theta$ is taken into account, the set of controller matrices $(F_{i},G_{i},H_{i},K_{i})$ for which $R_{i}(\theta,\theta^{\prime})=0$, which is the complement of $\mathcal{D}(\theta,\theta^{\prime})$, is an algebraic set. Then, as well known, only two situations may occur: either $R_{i}(\theta,\theta^{\prime})=0$ is always satisfied; or $R_{i}(\theta,\theta^{\prime})=0$ is satisfied on a set with zero Lebesgue measure. In this latter case, the set $\mathcal{D}(\theta,\theta^{\prime})$ is generic ²²2Recall that a subset $\mathcal{X}$ of a topological space is generic when it is open and dense: for any $x\in\mathcal{X}$, then there exists a neighborhood of $x$ contained in $\mathcal{X}$; for any $x\notin\mathcal{X}$, then every neighborhood of $x$ contains an element of $\mathcal{X}$. (since it is the complement of a proper algebraic set). The following lemma, proved in [9], provides sufficient conditions for such a favorable situation to occur.

Building on the above lemmas, under suitable regularity assumptions for the functions $A(\theta),B(\theta),C(\theta)$, it is possible to derive conditions for the existence of a global discerning switching controller on the whole uncertainty set $\Theta$. In particular, by exploiting the results of [17], the following theorem can be stated.

A few remarks are in order. First of all, notice that Theorem 1 provides a bound on the number $N$ of controller modes that may be needed in order to ensure identifiability of the unknown parameter $\theta$ in $\Theta$. Such a bound is consistent with the results of [9, 10] where it is shown that when the set $\Theta$ is finite, i.e., it is a $0$-dimensional manifold, one single LTI controller is generically discerning. As discussed in [17], the bound $N\geq 2\,M+1$ for parameter distinguishability is tight for general maps (in the sense that when $N<2\,M+1$ one can find counterexamples). However, for specific cases, fewer controller modes can be sufficient. For instance, when $\theta$ is a scalar parameter, one can consider a single LTI controller $(F_{i},G_{i},H_{i},K_{i})$ and plot the root locus of the closed loop polynomial $\varphi_{i}(s,\theta)$ as a function of $\theta$. Then, global discernibility is guaranteed provided that such a root locus never cross itself.

Notice finally that the set of analytic functions considered in the statement of Theorem 1 is quite general as it captures many function classes of interest (e.g., polynomials, trigonometric functions, exponentials, and also rational functions as long as away from singularities).

In this section, we discuss how the unknown parameter vector $\theta$ can be estimated from the closed-loop data $z$ on an interval $\mathcal{I}=[t_{0},t_{0}+T]$ and we show that, when the data result from application of a discerning switching controller, the resulting estimate enjoys some nice properties even in the presence of unknown disturbances and measurement noises.

To this end, recall that, when the unknown parameter vector takes value $\theta$, the evolution of $z$ on the interval $\mathcal{I}$ takes the form $z(t,t_{0},x_{0},\xi_{0},\theta,\sigma)$. Then the set $\mathcal{S}_{\sigma}(\theta)$ of all possible closed-loop data on the interval $\mathcal{I}$ associated with $\theta$ and with the switching signal $\sigma$ can be written as

Hence a natural approach for estimating the plant unknown parameters is the least-squares one, which amounts to selecting the parameter vector $\theta$ for which the distance between the observed close-loop data $z$ on the interval $\mathcal{I}$ and the set $\mathcal{S}_{\sigma}(\theta)$ is minimal. Accordingly, the optimal least-squares estimate $\hat{\theta}^{\circ}$ can be obtained as

where

Concerning the computation of the distance (20), for any possible feedback loop $(\mathcal{P}(\theta),\mathcal{C}_{\sigma})$, let $\Phi_{\sigma}(t,t_{0},\theta)$ denote its state transition matrix and let $W_{\sigma}(\theta)$ be its observability Gramian on the interval $\mathcal{I}$, i.e.,

Notice that, for any globally discerning switching control law, the observability Gramian $W_{\sigma}(\theta)$ turns out to be positive definite for any $\theta\in\Theta$ (otherwise there would be zero output trajectories corresponding to non-zero state trajectories and parameter identification would not be possible). Hence, in this case, the minimization in (20) yields

For further considerations on how this quantity can be computed in practice the interested reader is referred to Appendix A of [16], where a similar problem is addressed.

From Definition 1, it is immediately clear that when a globally discerning switching control law is adopted, for any non null $z$ the $\delta_{\sigma}(z,\hat{\theta})$ is zero if and only if $\hat{\theta}$ coincides with $\theta$, the true parameter vector.

While the above proposition illustrates the theoretical effectiveness of the leas-squares estimation criterion in ideal conditions, in practice computation of the minimum in (19) can be a quite challenging task when the set $\Theta$ is not finite. In this case, a standard approach is the multi-model one which amounts to considering only a finite number, say L, of possible parameter values by constructing the finite set $\Theta_{L}=\{\theta_{\ell},\,\ell=1,\ldots,L\}\subseteq\Theta$. Typically, $\Theta_{L}$ is obtained by sampling the set $\Theta$ with a given guaranteed density $\varepsilon$, so that for any $\theta\in\Theta$ there exists at least one $\theta_{\ell}\in\Theta_{L}$ such that $|\theta-\theta_{\ell}|\leq\epsilon$. When such a condition is satisfied, we say that $\Theta_{L}$ is $\epsilon$-dense in $\Theta$. Accordingly, the following multi-model least squares criterion can be used to estimate the unknown parameter vector $\theta$

as an alternative to (19).

In the following, a simple example is provided in order to illustrate how identifiability of an uncertain parameter vector can be achieved my means of switching control. To this end, consider an LTI plant with system matrices

with $\theta=(a,b)$, and let the switching controller be a purely proportional one

Since, the plant transfer matrix is $P(s,\theta)=b/[s\,(s+a)]$, each closed-loop characteristic polynomial takes the form $\varphi_{i}(s,\theta)=s^{2}+a\,s+b\,K_{i}\,.$

In this paper, we have addressed the problem of identifying the parameters of an uncertain linear system by means of switching control. It was shown that even when the uncertainty set is not finite, parameter identifiability can be generically ensured by switching among a finite number of linear time-invariant controllers. In particular, the results show that an upper bound on the number of controller modes needed for parameter identifiability can be given in terms of the dimension of the uncertainty set. The results also indicate that the seemingly conflicting goals of ensuring parameter identifiability as well as a satisfactory behavior of the feedback system can be simultaneously accomplished by means of switching control.

Several practical aspects have also been discussed. In particular, we have analyzed the properties of least-squares parameter estimation in connection with the use of discerning controllers, providing bounds on the worst-case parameter estimation error in the presence of: i) finite covering of the uncertainty set; and ii) bounded disturbances affecting the process dynamics as well as measurement noises.

The results lend themselves to be extended in various directions. Most notably, these results find a very natural application in the context of switching control for uncertain systems. In this respect, we envision that the analysis tools introduced in this paper should lead to the development of novel control reconfiguration algorithms capable of achieving input-to-state stability for uncertain systems even when the uncertainty set is described by a continuum.

Proof of Lemma 1: Consider two distinct parameter vectors $\theta,\theta^{\prime}\in\Theta$ and consider an index $i$ for which $\varphi_{i}(s,\theta)$ and $\varphi_{i}(s,\theta^{\prime})$ are coprime. Let the controller $\mathcal{C}_{i}$ be active on an interval $\mathcal{I}_{i}=[\underline{t},\bar{t}]\subset\mathcal{I}$. Consider now a nonzero quadruples of vectors $(x_{0},\xi_{0},x_{0}^{\prime},\xi^{\prime}_{0})$ representing possible initial states of the two feedback loops $(\mathcal{P}(\theta)/\mathcal{C}_{\sigma})$ and $(\mathcal{P}(\theta^{\prime})/\mathcal{C}_{\sigma})$ at time $t_{0}$. Let $(\underline{x},\underline{\xi},\underline{x}^{\prime},\underline{\xi}^{\prime})$ be the corresponding states that are reached at time $\underline{t}$, i.e., at the beginning of the time interval $\mathcal{I}_{i}$, under the switching law $\sigma$. Suppose now the switching signal $\sigma$ is chosen so as to satisfy a dwell-time condition, i.e., in such a way that there exists a lower bound $\tau_{\rm dwell}$ on the time interval between subsequent variations of the controller index. Then, it is immediate to see that, under such a switching law, when $(x_{0},\xi_{0},x_{0}^{\prime},\xi^{\prime}_{0})\neq 0$ then also $(\underline{x},\underline{\xi},\underline{x}^{\prime},\underline{\xi}^{\prime})\neq 0$. In fact, such a state is reached after switching a finite number of times between autonomous linear systems, i.e., the feedback loops, and it is known that an autonomous linear system cannot reach the zero state in finite time starting from a non-zero initial state. Notice now that, under assumption A1, coprimeness of the polynomials $\varphi_{i}(s,\theta)$ and $\varphi_{i}(s,\theta^{\prime})$ implies observability of the parallel system

(see for instance Proposition 1 of [9]). Then, if we initialize such a system as $(\chi(\underline{t}),\chi^{\prime}(\underline{t}))=(\underline{x},\underline{\xi},\underline{x}^{\prime},\underline{\xi}^{\prime})\neq 0$ at time $\underline{t}$, we have that $\tilde{z}$ is different from $0$ a.e. on $\mathcal{I}_{i}=[\underline{t},\bar{t}]$, where “a.e.” stands for “almost everywhere”, i.e. everywhere except on a set of zero Lebesgue measure. This, in turn, implies that $z(t,\underline{t},\underline{x},\underline{\xi},\theta,i)\neq z(t,\underline{t},\underline{x}^{\prime},\underline{\xi}^{\prime},\theta^{\prime},i)$, or equivalently, $z(t,t_{0},x_{0},\xi_{0},\theta,\sigma)\neq z(t,t_{0},x_{0}^{\prime},\xi_{0}^{\prime},\theta^{\prime},i)$, a.e. on $\mathcal{I}_{i}=[\underline{t},\bar{t}]$. Then, by choosing a switching signal $\sigma$ which satisfies a dwell-time condition and is such that each controller mode $i$ is active, at least, on an interval $\mathcal{I}_{i}\subset\mathcal{I}$ of positive measure, the same line reasoning can be repeated for any pair $\theta,\theta^{\prime}\in\Theta$ with $\theta\neq\theta^{\prime}$, thus concluding the proof.  $\blacksquare$

Proof of Theorem 1: Notice first that the resultant $R_{i}(\theta,\theta^{\prime})$ of the two polynomials $\varphi_{i}(\theta)$ and $\varphi_{i}(\theta^{\prime})$ is a polynomial (and hence analytic) function of the elements of the matrices $(A(\theta),B(\theta),C(\theta))$, $(A(\theta^{\prime}),B(\theta^{\prime}),C(\theta^{\prime}))$, and $(F_{i},G_{i},H_{i},K_{i})$. This, in turn, implies that $R_{i}(\theta,\theta^{\prime})$ is an analytic function of $\theta$ and $\theta^{\prime}$ (since the composition of analytic functions is analytic). Notice now that, under the stated hypotheses, Lemma 2 ensures that, for any pair $\theta,\theta^{\prime}\in\mathcal{M}$ with $\theta\neq\theta^{\prime}$, it is possible to find at least one set of matrices $(F_{i},G_{i},H_{i},K_{i})$ such that $\varphi_{i}(\theta)$ and $\varphi_{i}(\theta^{\prime})$ are coprime and, hence, $R_{i}(\theta,\theta^{\prime})\neq 0$. Recall, finally, that the set $\mathcal{D}_{N}$ corresponds to the set of switching controllers $(F_{i},G_{i},H_{i},K_{i})i\in\mathcal{N}$ for which the vector function ${\rm col}\left(R_{i}(\theta,\theta^{\prime}),i\in\mathcal{N}\right)$ is different from $0$ for any pair $\theta,\theta^{\prime}\in\mathcal{M}$ with $\theta\neq\theta^{\prime}$. Then, proceeding as in the proof of Theorem 2 of [17], we can conclude that $\mathcal{D}_{N}$ is generic and of full measure on $\mathbb{R}^{N\bar{n}_{\xi}}$ whenever $N\geq 2\,M+1$.  $\blacksquare$

Proof of Theorem 2: Let $(F_{i},G_{i},H_{i},K_{i})$ be a controller for which conditions (17) and (18) are satisfied with $\Pi(\theta)$ continuous in $\theta$. Further, consider a closed ball $\mathcal{B}(\varepsilon)$ in the controller parameter space $\mathbb{R}^{\bar{n}_{\xi}}$ centered in $(F_{i},G_{i},H_{i},K_{i})$ and with radius $\varepsilon$ and let

Note that, in view of the compactness of $\bar{\Theta}$ and of the continuity of $\Psi(\theta)$ and $\Pi(\theta)$, we have that $\max_{\theta\in\bar{\Theta}}\lambda_{\rm max}\left\{\Psi_{i}(\theta)^{\top}\Pi(\theta)+\Pi(\theta)\Psi_{i}(\theta)\right\}=\beta(0)<0$. Moreover, under the considered hypotheses, it is easy to show that $\beta(\varepsilon)$ depends continuously on $\varepsilon$ (in this respect, notice that $\Psi_{i}(\theta)$ is an affine function of the controller matrices $(F_{i},G_{i},H_{i},K_{i})$). Hence, this implies the existence of $\varepsilon>0$ such that $\beta(\varepsilon)<0$, i.e., such that all the controllers in $\mathcal{B}(\varepsilon)$ satisfies (18) with the same Lyapunov matrix $\Pi(\theta)$. As a consequence, the set $\mathcal{G}\subset\mathbb{R}^{\bar{n}_{\xi}}$ of all controllers satisfying (18) with the Lyapunov matrix $\Pi(\theta)$ turns out to be open, and $\mathcal{G}^{N}$ will be open as well. Finally, when $N\geq 2M+1$, the set $\mathcal{D}_{N}$ is generic and of full measure on $\mathbb{R}^{N\,\bar{n}_{\xi}}$ and, hence, $\mathcal{G}^{N}\cap\mathcal{D}_{N}$ is non-negligible.  $\blacksquare$