Data-Driven Pole Placement in LMI Regions with Robustness Constraints

Contribution. The main contribution of the paper is to propose a data-driven robust control methodology that can achieve desired closed-loop pole placement in convex regions of the complex plane along with sufficient robust and optimal system performance constrained by imposing mixed $H_{2}/H_{\infty}$ performance metrics. We use the fundamental lemma based data-driven parametrized representation to formulate the convex formulations of the robust pole placement problem in LMI regions that can achieve very close to the model-based design characteristics. We provide theoretical guarantees on the performance of the proposed algorithm along with validation on a third order dynamic system example.

Notations. $S\succ(\succeq)0$ denotes positive definite (semidefinite) matrix; $H_{2}$ norm : The $H_{2}$ norm of system $G$ in time domain is given by, $||G||_{2}=(\int_{0}^{\infty}[\mbox{trace}(h(t)^{T}h(t))]dt)^{\frac{1}{2}}$ where $h(t)$ is the impulse response; $H_{\infty}$ norm : The system $H_{\infty}$ norm is given by $||G||_{\infty}=\mbox{sup}_{w}\sigma_{\mbox{max}}(G(jw))$, where $\sigma_{\mbox{max}}$ denotes maximum singular value, and $G(jw)$ is the system transfer matrix.

Definition 1: LMI pole placement regions - A subset $\mathcal{D}$ can be characterized as the desired pole placement regions if there exist a symmetric matrix $\alpha=[a_{ij}]\in\mathbb{R}^{n\times n}$, and a matrix $\beta=[b_{ij}]\in\mathbb{R}^{n\times n}$ such that,

$\displaystyle\mathcal{D}(z)=\{z\in\mathbb{C},\psi_{\mathcal{D}}<0\},$

(2)

where,

$\displaystyle\psi_{\mathcal{D}}(z)=\alpha+z\beta+z^{*}\beta^{T}=[a_{ij}+b_{ij}z+\beta_{ji}z^{*}]_{1\leq i,j\leq n}.$

(3)

Lemma 3[22]: For the system $\dot{x}=Ax+B_{2}u$, given an LMI region $\mathcal{D}$ defined by (4), the closed loop system $\tilde{A}=A+B_{2}K$ is said to be $\mathcal{D}$-stable if there exists a real symmetric matrix $X_{D}\succ 0$ satisfying the following condition.

$\displaystyle\begin{bmatrix}\tilde{A}X_{D}+X_{D}\tilde{A}^{T}&\alpha(\tilde{A}X_{D}-X_{D}\tilde{A}^{T})\\ \alpha(X_{D}\tilde{A}^{T}-\tilde{A}X_{D})&\tilde{A}X_{D}+X_{D}\tilde{A}^{T}\end{bmatrix}\prec 0,$

(25)

$\square$
Proof[22]: The above condition can easily be obtained by replacing $z$ with $\tilde{A}X_{D}$ and $z^{*}$ with $X_{D}\tilde{A}^{T}$ in (4). Also, the detailed discussion of this condition can be found in [22]. This guarantees that the eigen values of $\tilde{A}=A+B_{2}K$ belong to the conic region $\psi_{\mathcal{D}}(z;\theta)$ on the left half of complex plane. $\square$
Note that, the condition given in (25) is not an LMI, because if we replace $\tilde{A}$ with $A+B_{2}K$, we get a product term $KX_{D}$ of two unknown quantities $K$ and $X_{D}$. But, with a simple change of variable $Y=KX_{D}$, (25) can be converted in to an LMI condition.

Next, we utilize the relation given in (24), and derive the data based condition for placing the poles of closed loop system in the region defined by the LMI condition (4).

Theorem 1: For the system $\dot{x}=Ax+B_{2}u$ with unknown state dynamics, let the input sequence $U_{0,1,T}$ is persistently exciting, i.e. the rank condition (16) holds, then any matrix $Q$ satisfying (26), resulting in feedback gain $K=U_{0,1,T}Q(X_{0,T}Q)^{-1}$, will make the system $\mathcal{D}$-stable for the conic region defined by (4).

$\displaystyle\begin{bmatrix}X_{1,T}Q+Q^{T}X_{1,T}^{T}&\alpha(X_{1,T}Q-Q^{T}X_{1,T}^{T})\\ \alpha(Q^{T}X_{1,T}^{T}-X_{1,T}Q)&X_{1,T}Q+Q^{T}X_{1,T}^{T}\end{bmatrix}\prec 0.$

(26)

$\square$
Proof: We recall Lemma 3 giving us the condition (25), to place the closed loops of $A+B_{2}K$ in the desired conic region. However, we make the assumption that the system dynamics is unknown, therefore we rely upon the data driven persistency of excitation condition as defined in Section II. It can be seen from (24) that under a persistently exciting input, the closed loop dynamics can be represented in parameterized form derived from time evolution of system dynamics $X_{1,T}$, i.e., $A+B_{2}K=X_{1,T}G$. As such, we get the following inequality,

$\displaystyle\begin{bmatrix}X_{1,T}GX_{D}+X_{D}(X_{1,T}G)^{T}&\alpha(X_{1,T}GX_{D}-X_{D}(X_{1,T}G)^{T})\\ \alpha(X_{D}(X_{1,T}G)^{T}-(X_{1,T}G)X_{D})&X_{1,T}GX_{D}+X_{D}(X_{1,T}G)^{T}\end{bmatrix}\prec 0$

(27)

To make this inequality an LMI, we consider $GX_{D}=Q$, resulting in (26). We also have $G=Q(X_{D})^{-1}$. Now recalling Section II, $K$ can be computed as $K=U_{0,1,T}G=U_{0,1,T}Q(X_{D})^{-1}$. Using (21) we have $X_{0,T}G=I$, which implies, $X_{D}=X_{0,T}Q$. Therefore, the feedback gain turns out to be $K=U_{0,1,T}Q(X_{0,T}Q)^{-1}$. Please note that this condition completely relies upon collected data from system trajectories and the design parameter $\alpha$ which determines the LMI region (4). $\square$
Remark 1: It is essential to note that the above pole placement problem is a feasibility problem. Therefore, it is apparent that there can be many feasible $X_{D}$ which lies in the convex set defined by (25). This gives rise to different controller gains which all satisfy the $\mathcal{D}$-stability condition. Similar characteristics will be observed during data-driven design using Theorem 1, as different exploration trajectories can result in different possible control gains, however, they all satisfy the desired closed-loop pole placement constraint (4).

As described in Remark 1, we will now consider system (10) along with desired $H_{2}/H_{\infty}$ constraints. Please note, we are now considering an extraneous input $w$ and transfer functions associated with the $H_{2}$ and $H_{\infty}$ problem represent the gain from $w$ to regulated outputs $z_{2}$ and $z_{1}$, respectively. As we consider the state matrix $A$ and input matrix $B$ are unknown, the extraneous disturbance needs to pre-specified in a controlled environment during the design of the feedback gains. Recalling Section II, the trajectory based system dynamics turns out to be

	$\displaystyle X_{1,T}=AX_{0,T}+B_{1}W_{0,T}+B_{2}U_{0,1,T},$		(28)
	$\displaystyle X_{1,T}-B_{1}W_{0,T}=AX_{0,T}+B_{2}U_{0,1,T}.$		(29)

Therefore, the closed loop parameterized representation modifies to (30). Note, like $X_{0,T}$, $W_{0,T}$ can also be defined using (12). The closed loop parameterization with extraneous input becomes

$\displaystyle A+B_{2}K=[B_{2}\;\;A]\begin{bmatrix}K\\ I\end{bmatrix}=(X_{1,T}-B_{1}W_{0,T})G=\tilde{X}_{1,T}G.$

(30)

We now state the following theorem to solve problem P.

Theorem 2: For the system (10), to place the closed loop poles in the desired LMI region (4) along with sufficient mixed $H_{2}/H_{\infty}$ the following set of data driven LMIs needs to be solved, where $C_{2}=\begin{bmatrix}Q_{x}^{\frac{1}{2}}\\ 0\end{bmatrix}$, $D_{22}=\begin{bmatrix}0\\ R^{\frac{1}{2}}\end{bmatrix}$ and $Q_{x}\succeq 0$, $R\succ 0$ are the designable state and input penalty factors.

Proof: We start with considering the $H_{2}$ performance objective. Please note in (10), $H_{2}$ performance objective is to minimize $||T_{wz_{2}}||_{2}$. Following [23, 24], the model-based optimization problem for the $H_{2}$ performance is given as follows:

$$\min_{K,X_{2}}\operatorname{trace}\left(Q_{x}X_{2}\right)+\operatorname{trace}(R^{\frac{1}{2}}KX_{2}K^{T}R^{\frac{1}{2}})$$

(35)

subject to

$$\left\{\begin{array}[]{l}(A+B_{2}K)X_{2}+X_{2}(A+B_{2}K)^{T}+B_{1}B_{1}^{T}\prec 0,\\ X_{2}=X_{2}^{T}>0.\end{array}\right.$$

We now consider the $H_{\infty}$ performance objective which intends to minimize $||T_{wz_{1}}||_{\infty}$, and the use of KYP lemma (Bounded real lemma) results into the following optimization problem, [25, 26, 27]

$$\min_{\gamma,X_{\infty}}\gamma$$

(36)

subject to

	$\displaystyle\begin{bmatrix}\tilde{A}X_{\infty}+X_{\infty}\tilde{A}^{T}&B_{1}&X_{\infty}(C_{1}+D_{12}K)^{T}\\ B_{1}^{T}&-\gamma I&D_{11}^{T}\\ (C_{1}+D_{12}K)X_{\infty}&D_{11}&-\gamma I\end{bmatrix}\prec 0,$		(37)
	$\displaystyle\text{where,}\;\tilde{A}=A+B_{2}K,\normalsize$

$\displaystyle X_{\infty}\succ 0.$

(38)

Recalling Lemma 3, the pole placement condition needs to satisfy the following inequality,

	$\displaystyle\begin{bmatrix}\tilde{A}X_{D}+X_{D}\tilde{A}^{T}&\alpha(\tilde{A}X_{D}-X_{D}\tilde{A}^{T})\\ \alpha(X_{D}\tilde{A}^{T}-\tilde{A}X_{D})&\tilde{A}X_{D}+X_{D}\tilde{A}^{T}\end{bmatrix}\prec 0,$
	$\displaystyle\text{where,}\;\tilde{A}=A+B_{2}K.$

Next, we convert the above conditions in an optimization problem by seeking a common solution of $X$, where $X=X_{2}=X_{\infty}=X_{D}\succ 0$. Using a single Lyapunov matrix X that enforces multiple constraints has been studied in [22, 25]. As we are considering the mixed $H_{2}/H_{\infty}$ objective the stabilization constraint provided by the $H_{\infty}$ problem will serve as a conservative unifying condition for both $H_{2}$ and $H_{\infty}$ problem. Therefore, the model-based solution of problem P, can be written as,

$$\min_{K,X,S,\gamma}\operatorname{trace}\left(Q_{x}X\right)+\operatorname{trace}(S)+\gamma$$

(39)

subject to

	$\displaystyle\begin{bmatrix}\tilde{A}X+X\tilde{A}^{T}&B_{1}&X(C_{1}+D_{12}K)^{T}\\ B_{1}^{T}&-\gamma I&D_{11}^{T}\\ (C_{1}+D_{11}K)X&D_{11}&-\gamma I\end{bmatrix}\prec 0,$		(40)
	$\displaystyle\text{where,}\;\tilde{A}=A+B_{2}K,$

	$\displaystyle\begin{bmatrix}\tilde{A}X+X\tilde{A}^{T}&\alpha(\tilde{A}X-X\tilde{A}^{T})\\ \alpha(X\tilde{A}^{T}-\tilde{A}X)&\tilde{A}X+X\tilde{A}^{T}\end{bmatrix}\prec 0,$
	$\displaystyle\text{where,}\;\tilde{A}=A+B_{2}K,$		(41)

	$\displaystyle X=X^{T}>0,$		(42)
	$\displaystyle S-R^{1/2}KXK^{T}R^{1/2}\succeq 0.$		(43)

We represented the second term in the objective of (35) by the second term of (39) and the corresponding inequality (43). To this end, we now move into converting these model-based expressions to their data-driven counter part. Considering (40) and using (30), we can have,

$$\begin{bmatrix}(\tilde{X}_{1,T}G)X+X(\tilde{X}_{1,T}G^{T}&B_{1}&XC_{1}+XK^{T}D_{12}^{T}\\ B_{1}^{T}&-\gamma I&D_{11}^{T}\\ C_{1}X+D_{12}KX&D_{11}&-\gamma I\end{bmatrix}\prec 0.$$

(44)

We now substitute, $GX=Q$ and $K=U_{0,1,T}G$, this results in $XK^{T}D_{12}^{T}=Q^{T}U_{0,1,T}^{T}D_{12}^{T}$, giving us (32). The substitution of $K=U_{0,1,T}G$ also converts (43) into (45).

$\displaystyle S-R^{1/2}U_{0,1,T}GXG^{T}U_{0,1,T}^{T}R^{1/2}\succeq 0.$

(45)

Next, replace $GX=Q$ and $G^{T}=X^{-T}Q^{T}=(X_{0,T}Q)^{-T}Q^{T}$. Note, from (21) we can write $X_{0,T}G=I$, therefore pre-multiplying $GX=Q$ with $X_{0,T}$ results in $X=X_{0,T}Q$. Now, we have $S-R^{1/2}U_{0,1,T}Q(X_{0,T}Q)^{-T}Q^{T}U_{0,1,T}^{T}R^{1/2}\succeq 0$, and after applying Schur complement, we get (34). The first term of objective(39) uses the relation $X=X_{0,T}Q$ and converts into first term of (31). Finally, we supplement these LMI conditions with the data-driven $\mathcal{D}$-stability condition provided in Theorem 1, and this completes the proof of Theorem 2. $\square$
Remark 2 Please note that Theorem 2 considers the $H_{\infty}$ performance margin $\gamma$ as an optimization variable. However, in many scenarios the designer may be interested in a pre-specified robustness performance gain $(\bar{\gamma})\geq\gamma_{min}$, where $\gamma_{min}$ is the solution of the problem in Theorem 2. To use a pre-specified $(\bar{\gamma})$, the objective function in Theorem 2 modifies into $\operatorname{trace}\left(Q_{x}X_{0,T}Q\right)+\operatorname{trace}(S)$ along with the $\gamma$ in (32) will be replaced by $(\bar{\gamma})$.

In this section, we present an illustrative example of a pole placement problem with robust performance criteria using the data-driven method discussed in Section 4. We compare the obtained results using the data-driven approach with its model-based counterpart. We define the system given in (10) with the following state and input matrices, taken from [14]:

$$A=\begin{bmatrix}-0.5&1.4&0.4\\ -0.9&0.3&-1.5\\ 1.1&1&-0.4\end{bmatrix};B_{2}=\begin{bmatrix}0.1&-0.3\\ -0.1&-0.7\\ 0.7&-1\end{bmatrix}$$

. Two of the eigenvalues of $A$ are located on the right side of the $j\omega$ axis resulting in an unstable open-loop system. Note while designing the data-driven control, it is assumed that $A$ and $B_{2}$ are unknown. We choose the system matrix $B_{1}$ for extraneous inputs $w$ and other performance matrices $C_{1},D_{11},D_{12},C_{2}$ and $D_{22}$ as follows:

$$B_{1}=\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix};C_{1}=\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix};D_{11}=\begin{bmatrix}1&1&1\\ 1&1&1\\ 1&1&1\end{bmatrix};D_{12}=\begin{bmatrix}1&1&1\\ 1&1&1\\ \end{bmatrix}$$

,

$$C_{2}=\begin{bmatrix}Q_{x}^{\frac{1}{2}}\\ \mathbf{0}\end{bmatrix},\;\;\text{where}\;\;Q_{x}=\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix};D_{22}=\begin{bmatrix}\mathbf{0}\\ R^{\frac{1}{2}}\end{bmatrix},\;\;\text{where}\;\;R=\begin{bmatrix}1&0\\ 0&1\\ \end{bmatrix}$$

. We choose a random initial condition $x_{0}$ and a random input sequence $u\in\mathbb{R}^{2}$ with the magnitudes of both channels constrained at $[-0.5,0.5]$. Here, the interesting part is in deciding the length of the input sequence $u$. For this example $n=3$, $m=2$, therefore to satisfy the rank condition given in (16) as a requirement of the persistently excitation, the length of the input sequence $T$ is selected as $15$, as we require $T\geq(m+1)n+m$. Next, to generate the trajectory rollout as defined in (28), we choose $w$ randomly from a norm ball defined by $\lvert\lvert w\rvert\rvert_{2}\leq 0.05$. Trapezoidal approximations are used to generate the rollouts using continuous time dynamics. Now, for a given $\alpha=2$, the solution of problem P using Theorem 2 is as follows,

$$K_{\text{data}}=\begin{bmatrix}-3.627984&1.257298&-3.803731\\ 1.433545&0.837496&2.065323\end{bmatrix},\\ \gamma_{\text{min}}=4.832$$

The poles of the closed-loop system are -4.2545, -1.9539, and -0.6244. These poles (marked as red* in Fig. 1) are located on the negative real axis, which includes the conic region defined by $\alpha=2$ (shaded area in Fig. 1). To verify our designed data-driven controller gain and the location of closed-loop poles, we run the same experiments with the model-based equations given in (39) to (43). The computed model-based gain is shown below:

$$K_{\text{mod}}=\begin{bmatrix}-3.627994&1.257302&-3.803737\\ 1.433540&0.837497&2.065325\end{bmatrix},\\ \gamma_{\text{min}}=4.832$$

The computed gain $K_{\text{data}}$ and $K_{\text{mod}}$ are identical with a difference $\lvert\lvert K_{\text{data}}-K_{\text{mod}}\rvert\rvert\approx 10^{-4}$, which validates the accuracy of our proposed data-driven design.

We do an ablation study to further check the robustness of the data driven method. We removed the pole placement constraints from the conditions given in Theorem 2. This simply converts the problem into a mixed $H_{2}/H_{\infty}$ problem. Solving the LMIs (31) to (34), except (33), we obtained,

$$\bar{K}_{\text{data}}=\begin{bmatrix}-3.627984&1.257298&-3.803731\\ 1.433545&0.837496&2.065323\end{bmatrix},$$

$$\text{Poles:}-0.9062+j1.533,-0.9062-j1.533,-1.4182$$

Fig. 1 clearly indicates that these poles (marked as blue*) are located outside the LMI region (shaded area). Like previous experiments, the same results can be found in case of model-based optimization using (39) to (43) eliminating (4).