Feedback Passivation of Linear Systems with Fixed-Structured Controllers

TL;DR

This paper develops a method to design optimal fixed-structured output feedback controllers for linear systems, maximizing passivity levels in both continuous and discrete time, using convex optimization.

Contribution

It provides necessary and sufficient conditions for passivation with fixed-structured controllers and formulates the controller design as a convex optimization problem.

Findings

Conditions for passivation with fixed-structure controllers are established.

The method maximizes passivity indices via convex optimization.

Applicable to both continuous-time and discrete-time systems.

Abstract

This paper addresses the problem of designing an optimal output feedback controller with a specified controller structure for linear time-invariant (LTI) systems to maximize the passivity level for the closed-loop system, in both continuous-time (CT) and discrete-time (DT). Specifically, the set of controllers under consideration is linearly parameterized with constrained parameters. Both input feedforward passivity (IFP) and output feedback passivity (OFP) indices are used to capture the level of passivity. Given a set of stabilizing controllers, a necessary and sufficient condition is proposed for the existence of such fixed-structured output feedback controllers that can passivate the closed-loop system. Moreover, it is shown that the condition can be used to obtain the controller that maximizes the IFP or the OFP index by solving a convex optimization problem.

Equations

$$F(s)={\left(b(s)\otimes I\right)}^T\left(M+L(\alpha )\right)\left(b(s)\otimes I\right)$$

$$F(s)={\left(b(s)\otimes I\right)}^{T}M\left(b(s)\otimes I\right),$$

$$\mathcal{L}=\{\tilde{L}={\tilde{L}}^T:{\left(b(s)\otimes I\right)}^T\tilde{L}\left(b(s)\otimes I\right)=0\},$$

$$M+L(\alpha )\ge 0.$$

$${⟨Hu,u⟩}_T\ge \beta ,\forall u\in {\mathcal{L}}_{2e},\forall T\ge 0.$$

$${⟨Hu,u⟩}_T\ge \nu ||u_T|{|}_2^2+\beta ,\forall u\in {\mathcal{L}}_{2e},\forall T\ge 0,$$

$${⟨Hu,u⟩}_T\ge \xi ||H{u}_T|{|}_2^2+\beta ,\forall u\in {\mathcal{L}}_{2e},\forall T\ge 0,$$

$$\nu=\left\{\begin{array}[]{ll}\frac{1}{2}\underset{\omega\in\mathbb{R}}{\text{min}}\;\;\underline{\lambda}\left(G(j\omega)+G^{*}(j\omega)\right)&CT\ case\\ \frac{1}{2}\underset{\omega\in[0,2\pi]}{\text{min}}\underline{\lambda}\left(G(e^{j\omega})+G^{*}(e^{j\omega})\right)&DT\ case\end{array}\right.$$

$$\xi=\left\{\begin{array}[]{ll}\frac{1}{2}\underset{\omega\in\mathbb{R}}{\text{min}}\;\;\underline{\lambda}\left(G^{-1}(j\omega)+[G^{-1}(j\omega)]^{*}\right)&CT\ case\\ \frac{1}{2}\underset{\omega\in[0,2\pi]}{\text{min}}\underline{\lambda}\left(G^{-1}(e^{j\omega})+[G^{-1}(e^{j\omega})]^{*}\right)&DT\ case\end{array}\right.$$

$$C(s,\rho )={\rho }^T\bar{C}(s),C(z,\rho )={\rho }^T\bar{C}(z)$$

$$\mathcal{P}=\{\rho \in {\mathbb{R}}^p:{\underline{\rho }}_i\le {\rho }_i\le {\overline{\rho }}_i,i=1,\dots ,p\}.$$

$$\mathcal{P}=\{\rho \in {\mathbb{R}}^p:c_i\ge 0,i=1,\dots ,p\}$$

$$\begin{array}[]{cl}\vspace{1mm}G(s,\rho)&=\displaystyle\frac{G_{0}(s)}{1+G_{0}(s)C(s,\rho)}\\ \vspace{1mm}=&\frac{N_{0}(s)\prod_{i=1}^{p}D_{i}(s)}{D_{0}(s)\prod_{i=1}^{p}D_{i}(s)+N_{0}(s)\sum_{i=1}^{p}\rho_{i}N_{i}(s)\left(\prod_{j\neq i}D_{j}(s)\right)}\\ \triangleq&\displaystyle\frac{p_{N}(s)}{p_{D}(s,\rho)}\end{array}$$

$$p_D(s,\rho )=a_n(\rho )s^n+a_{n-1}(\rho )s^{n-1}+\dots +a_1(\rho )s+a_0(\rho )$$

$$\begin{array}[]{cccc}a_{n}(\rho)&a_{n-2}(\rho)&a_{n-4}(\rho)&\cdots\\ a_{n-1}(\rho)&a_{n-3}(\rho)&a_{n-5}(\rho)&\cdots\\ a_{31}(\rho)&a_{32}(\rho)&a_{33}(\rho)&\cdots\\ \vdots&\vdots&\vdots&\ddots\end{array}$$

$$\begin{array}[]{c}a_{ij}(\rho)=a_{i-1,1}(\rho)a_{i-2,j+1}(\rho)-a_{i-1,j+1}(\rho)a_{i-2,1}(\rho)\\ \;i=3,\ldots,n+1,j=1,2,\ldots\end{array}$$

$$G(z,\rho )=\frac{p_N(z)}{p_D(z,\rho )}$$

$$p_D(z,\rho )=a_n(\rho )z^n+a_{n-1}(\rho )z^{n-1}+\dots +a_1(\rho )z+a_0(\rho )$$

$$\begin{array}[]{rl}&a_{ij}(\rho)\\ =&a_{i-1,j}(\rho)a_{i-1,1}(\rho)-a_{i-1,n+4-i-j}(\rho)a_{i-1,n+3-i}(\rho)\\ &i=3,\ldots,n+1,\>j=1,2\ldots\end{array}$$

$$f_i(\rho )>0,i=1,\dots ,n+1\forall \rho \in \mathcal{P}.$$

$$g_i(\rho )=f_i(\rho )-\sum _{j=1}^p{s}_{ij}(\rho )c_j(\rho )i=1,\dots ,n+1$$

$${\theta }^{\ast }>0$$

$$\theta^{*}=\underset{\theta,s_{ij}}{\text{max}}\;\;\theta\ \text{s.t.}\ \left\{\begin{array}[]{rcl}g_{i}(\rho)-\theta&\text{is}&\text{SOS}\\ s_{ij}(\rho)&\text{is}&\text{SOS}\\ \forall i&=&1,\ldots,n+1\\ \forall j&=&1,\ldots,p\end{array}\right.$$

$$\begin{array}[]{rcl}p_{N}(j\omega)&=&p_{N}^{e}(w)+jp_{N}^{o}(w)\\ p_{D}(j\omega,\rho)&=&p_{D}^{e}(\omega,\rho)+jp_{D}^{o}(\omega,\rho)\end{array}$$

$$G(j\omega ,\rho )=\frac{p_N^e(w)+j{p}_N^o(w)}{p_D^e(\omega ,\rho )+j{p}_D^o(\omega ,\rho )}$$

$$\begin{array}[]{rl}&G(j\omega,\rho)+G^{*}(j\omega,\rho)\\ =&\displaystyle\frac{2p_{N}^{e}(w)p_{D}^{e}(\omega,\rho)+2p_{N}^{o}(w)p_{D}^{o}(\omega,\rho)}{p_{D}^{e}(\omega,\rho)^{2}+p_{D}^{o}(\omega,\rho)^{2}}.\end{array}$$

$$p_N^e(w)p_D^e(\omega ,\rho )+p_N^o(w)p_D^o(\omega ,\rho )-ϵ\text{ is SOS}.$$

$$G(j\omega ,\rho )+G^{\ast }(jw,\rho )\ge 0,\forall \omega \in \mathbb{R},$$

$$p_N^e(w)p_D^e(\omega ,\rho )+p_N^o(w)p_D^o(\omega ,\rho )\ge 0,\forall \omega \in \mathbb{R}.$$

$$p_N^e(w)p_D^e(\omega ,\rho )+p_N^o(w)p_D^o(\omega ,\rho )-ϵ\ge 0,\forall \omega \in \mathbb{R},$$

Full text

††thanks: This paper was not presented at any IFAC meeting.

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1 Introduction

Passivity provides a physically meaningful interpretation of the energy dissipation of a system from the input-output perspective (Willems (1972)). Notions of input and output passivity indices give a widely used measure of the level of passivity for a system (Kottenstette et al. (2014)). When exploited properly, passivity indices provide a means to design feedback controllers via the process of compensating for the lack of passivity in one subsystem of a feedback configuration with passivity surplus in the other (Van Der Schaft 2000, Bao & Lee 2007, Antsaklis et al. 2013).

Feedback passivation of plants that may not be passive is a widely studied problem (Larsen & Kokotovic (2001), Zhu et al. (2014), Zhao & Gupta (2016)). In the existing works, the controller can be chosen without any constraint on its structure. In this paper, we study the problem of designing a controller to maximize the closed-loop passivity level (as measured by a passivity index) when the controller has to satisfy a fixed structure. Reconstruction of such a controller can be performed by solving a semidefinite programming (SDP). The results in this paper can be utilized to improve the robust stability margins of interconnected systems as measured from the perspective of passivity.

Our problem setup involves linearly parameterized sets of controllers with constrained parameters. It is generally known that synthesis of such controllers is a notoriously difficult problem. A naive application of the Kalman-Yakubovich-Popov (KYP) lemma to tackling the problem would result in bilinear matrix inequalities, which are intractable in general. Our proposed approach establishes and exploits relationships between the passivity of SISO systems and sum-of-square (SOS) polynomials, which are amenable to convex optimization formulations. The introduction of linearly parameterized sets of controllers is motivated by its ubiquity in practical engineering applications, the most common of which consists of proportional-integral-derivative (PID) controllers, where the parameters appear linearly. By tuning the linear parameters of the controller in order to optimize the level of passivity in a feedback is of interest in view of the popularity of such controllers. Such parameterized controllers belong to a broader class of the so-called fixed-structured controllers. See, for instance, Saeki (2006) which considers fixed-structured PID controller design for $H_{\infty}$ control problems with linear constraints on the control structure; Malik et al. (2008) wherein a set of stabilizing fixed-structure and fixed-order controllers is constructed; and Bazanella et al. (2011) which studies model-free fixed-structure controller synthesis. To the best of the authors’ knowledge, the problem of optimizing the passivity level of a system with a fixed-structure controller considered in this paper has not been considered elsewhere.

The rest of this paper is organized as follows. Section 2 introduces some preliminaries and states the problem formulation. Section 3 presents the main results. The main results are illustrated by two examples in Section IV. Some final remarks and future work are described in Section V.

2 Preliminaries and Problem Formulation

2.1 Notation

The notation used in the paper is as follows. The sets of real and complex numbers are denoted by $\mathbb{R}$ and $\mathbb{C}$, and the imaginary unit is denoted as $j$. The notation $Re(\lambda)$ and $|\lambda|$ denote the real part and the magnitude of a complex number $\lambda$. $A^{-1}$, $A^{{}^{\prime}}$ and $A^{*}$ denote the inverse, the transpose and the conjugate transpose of matrix $A$, respectively. Given a Hermitian matrix $A=A^{*}$, the notation $\underline{\lambda}(A)$ denotes the minimum eigenvalue of $A$. For Hermitian matrices $A,B$, the notation $A-B\geq 0$ denotes $A-B$ is positive semidefinite. The degree of a polynomial $p(\cdot)$ is denoted by $\text{deg}(p(\cdot))$. The symbol $\otimes$ denotes the Kronecker product. The function $f_{T}$ is the truncation of $f$ to the interval $[0,T]$. The operator $<f,g>_{T}$ is defined as the inner products of signal $f$ and $g$ over $[0,T]$. $\mathcal{L}_{2e}$ denotes the extended $L_{2}$ signal space and $||\cdot||$ denotes the $L_{2}$ norm. For briefness, the notation $\star$ denotes the symmetric entries in a symmetric matrix.

2.2 Sum of square (SOS) matrix polynomial

Let us briefly introduce the class of SOS matrix polynomials, see e.g., Chesi (2010) for details.

A symmetric matrix polynomial $F:\mathbb{R}^{r}\rightarrow\mathbb{R}^{n\times n}$ is said to be SOS if and only if there exist matrix polynomials $F_{1},\ldots,F_{k}:\mathbb{R}^{r}\rightarrow\mathbb{R}^{n\times n}$ such that $F(s)=\sum_{i=1}^{k}F_{i}(s)^{T}F_{i}(s).$ SOS matrix polynomials are postive-semidefinite, and it turns out that one can establish whether a symmetric matrix polynomial is SOS via an LMI feasibility test.

Indeed, let $d$ be a nonnegative integer such that $2d\geq\deg(F)$. By extending the Gram matrix method or SMR for scalar polynomials to the representation of matrix polynomials, $F(s)$ can be written as

[TABLE]

where $b(s):\mathbb{R}^{r}\rightarrow\mathbb{R}^{\sigma(r,d)}$ is a vector containing all the monomials of degree less than or equal to $d$ in $s$ with $\sigma(r,d)=\frac{(r+d)!}{r!d!}$ , and $M\in\mathbb{R}^{n\sigma(r,d)\times n\sigma(r,d)}$ is a symmetric matrix satisfying

[TABLE]

$L(\alpha):\mathbb{R}^{\omega(r,2d,n)}\rightarrow\mathbb{R}^{n\sigma(r,d)\times n\sigma(r,d)}$ is a linear parametrization of the linear set

[TABLE]

and $\alpha\in\mathbb{R}^{\omega(r,2d,n)}$ is a free vector with $\omega(r,2d,n)=\frac{1}{2}n\left(\sigma(r,d)(n\sigma(r,d)+1)-(n+1)\sigma(r,2d)\right).$ It follows that $F(s)$ is a SOS matrix polynomial if and only if there exists $\alpha$ satisfying the LMI

[TABLE]

When $n=1$ is considered, the above results are reduced to SOS polynomials.

2.3 Strictly Input and Output Passive Systems

We start by introducing the definitions of passivity and positive realness for LTI systems, followed by a lemma revealing their relation.

Definition 1**.**

(Passivity (Van Der Schaft 2000, Kottenstette et al. 2014)) Consider a CT or DT LTI system $H:\ u\in\mathcal{L}_{2e}\rightarrow y\in\mathcal{L}_{2e}$. Then the system $H$ is

• •

passive if there exists a constant $\beta$ such that

[TABLE]

• •

strictly input passive (SIP) if there exist $\nu>0$ and $\beta$ such that

[TABLE]

and the largest $\nu>0$ satisfying (4) is called the Input Feedforward Passivity (IFP) index, denoted as IFP($\nu$).

• •

strictly output passive (SOP) if there exist $\xi>0$ and $\beta$ such that

[TABLE]

and the largest $\xi>0$ satisfying (5) is called the Output Feedback Passivity (OFP) index, denoted as OFP($\xi$).

The IFP and OFP indices, defined in terms of an excess of passivty, are introduced to quantify the degree of passivity.

Definition 2**.**

(Positive realness (Khalil & Grizzle 2002)) A square, proper and rational transfer function $G(s)$ (or $G(z)$ for DT case) is said to be positive real if

• •

$G(s)$* is analytic in $\text{Re}(s)>0$ in CT case; $G(z)$ is analytic in $|z|>1$ in DT case;*

• •

$G(jw)+G^{*}(jw)\geq 0$* , $\forall\omega\in\mathbb{R}$ for which $j\omega$ is not a pole of $G(s)$ in CT case; $G(e^{j\omega})+G^{*}(e^{j\omega})>0,\forall\omega\in[0,2\pi]$ for which $e^{j\omega}$ is not a pole of $G(z)$ in DT case;*

• •

Any pure imaginary pole $j\omega_{o}$ of $G(s)$ is a simple pole, and the associated residue $G_{o}\triangleq\lim_{s\rightarrow j\omega_{o}}(s-j\omega_{o})G(s)$ satisfies $G_{o}=G_{o}^{*}\geq 0$ in CT case; If $e^{j\omega_{o}}$ is a pole of $G(z)$ it is at most a simple pole, and the associated residue $G_{o}\triangleq\lim_{z\rightarrow j\omega_{o}}(z-e^{j\omega_{o}})G(s)$ satisfies $G_{o}=G_{o}^{*}\geq 0$ in DT case.

For a stable111In this work, a LTI system is said to be stable if the system is asymptotically stable. LTI system with transfer function $G$, the following lemma states the relation between the passivity and positive realness.

Lemma 3**.**

(Bao & Lee (2007))A stable LTI system $H:\ u\in\mathcal{L}_{2e}\rightarrow y\in\mathcal{L}_{2e}$ is passive if and only if its transfer function $G$ is positive real.

For a stable LTI system with the transfer function $G(s)$ (or $G(z)$ for DT case) that is strictly input passive, its IFP index, $\nu$, is given as

[TABLE]

For a minimum phase LTI system $G(s)$ (or $G(z)$ for DT case) that is strictly output passive, its OFP index, $\xi$, is given as

[TABLE]

2.4 Problem Formulation

In this work, we consider the problem of output feedback passivation of a single-input single-output (SISO) linear system through a fixed-structured controller (depicted in Figure 1). Particularly, the objective is to design an output feedback fixed-structure controller with parameter $\rho^{*}$, which maximizes the IFP or the OFP index for the closed-loop system.

The SISO plant with transfer function $G_{0}$ can be either CT or DT systems. The set of the controllers which can be implemented has a specified controller structure represented by $\mathcal{C}=\{C(s,\rho):\rho\in\mathcal{P}\}$ for the CT case and $\mathcal{C}=\{C(z,\rho):\rho\in\mathcal{P}\}$ for the DT case, where $\mathcal{P}\subseteq\mathbb{R}^{p}$ is a set of admissible values of the controller parameter vector $\rho$. We assume that the controllers are linearly parameterized, i.e.,

[TABLE]

where $\rho$ is the parameter vector and $\bar{C}$ is the predefined parameter independent vector of transfer functions. It is also assumed that all entries in $\bar{C}$ are selected to have stable poles. A typical class of controllers with linear parameterization is PID controllers. The linearity makes the resulting design problem more amendable to analysis. Moreover, it is shown in Bazanella et al. (2011) that any parameteter-dependent transfer function can be approximated to any degree of accuracy desired by a transfer function of the form (6) with sufficiently large $p$. As it is often required to restrict the admissible controller parameters to some desired bounded sets, we assume that the admissible set of $\rho$ is described by222As it will be explained in Remark 19, the proposed methodology can be used also to design feedback controllers with any convex set $\mathcal{P}$.

[TABLE]

The problems addressed in this work are as follows.

Problem 4**.**

For a given set of controllers, $\mathcal{C}=\{C(\rho):\rho\in\mathcal{P}\}$, establish whether the closed-loop system is stable for all $\rho\in\mathcal{P}$.

With the set of stabilizing controllers in hand, we further investigate the following problem.

Problem 5**.**

Establish whether there exists a controller $C$ in the set $\mathcal{C}$ that can passivate the system $G_{0}$. If the answer is positive, determine the controller $C^{*}$ that maximizes the IFP index and the OFP index respectively for the closed-loop system.

It is well-known that a necessary condition for a linear system to be feedback passivated is that the system should have a relative degree less than 2 and is weakly minimum phase (i.e., it should not have zeros on right side in s-plane or outside the unit circle in z-plane). Thus, we assume throughout this work the following assumption.

Assumption 6**.**

The plant $G_{0}$ has a relative degree less than $2$, and has all its zeros in the closed left half of the s-plane in CT case (in DT case, respectively, inside or on the unit circle of the z-plane).

A slightly more restrictive assumption is made when the optimal OFP controller design is considered.

Assumption 7**.**

The plant $G_{0}$ has a relative degree less than $2$, and has all its zeros in the open left half of the s-plane in CT case (in DT case, respectively, strictly inside the unit circle of the z-plane).

3 Main Results

3.1 Stability Analysis

Let us start by addressing Problem 4, which is to establish the robust stability of the closed-loop system for all parameter $\rho\in\mathcal{P}$.

First, let us observe that the controller set (7) can be equivalently described as

[TABLE]

with $c_{i}=(\overline{\rho}_{i}-\rho)(\rho_{i}-\underline{\rho}_{i})$.

For CT case, let us denote the transfer function of the plant as $G_{0}(s)=\frac{N_{0}(s)}{D_{0}(s)}$, and denote the $i$-th component in the vector $\bar{C}(s)$ as $\bar{C}_{i}(s)=\frac{N_{i}(s)}{D_{i}(s)}$. It follows that the closed-loop system as shown in Figure 1 is represented as

[TABLE]

where the polynomials $p_{N}(s)$ and $p_{D}(s,\rho)$ denote the numerator and denominator of the closed-loop transfer function respectively. Under Assumption 6 or 7 , it can be observed that there is no unstable zero-pole cancellation in the above closed-loop transfer function.

Rewrite the denominator polynomial $p_{D}(s,\rho)$ as

[TABLE]

wherein the coefficients $a_{0},\ldots,a_{n}$ are linear functions of the vector variable $\rho$. In order to analyze the stability of the closed-loop system, it is necessary and sufficient to check whether all the roots of the polynomial $p_{D}(s,\rho)$ have negative real parts for all $\rho\in\mathcal{P}$. To this end, let us exploit the modified Routh-Hurwitz table for the polynomial $p_{D}(s,\rho)$. By multiplying each component by their denominator in the classical Routh-Hurwitz table, we can obtain the modified Routh-Hurwitz table defined as

[TABLE]

where the number of rows is $n+1$ and the $ij$-th component is

[TABLE]

It can be verified that all the roots of $p_{D}(s,\rho)$ have negative real parts for all $\rho\in\mathcal{P}$ if and only if the polynomials in the first column of the modified Routh-Hurwitz table (10) are positive for all $\rho\in\mathcal{P}$.

Now let us further consider a discrete-time transfer function of the plant as $G_{0}(z)=\frac{N_{0}(z)}{D_{0}(z)}$, which is in closed-loop with the linearly parameterized controller $C(z,\rho)=\sum_{i=1}^{p}\rho_{i}\frac{N_{i}(z)}{D_{i}(z)}$. With similar argument of the CT case, the closed-loop system is represented as

[TABLE]

with

[TABLE]

wherein the coefficients $a_{0},\ldots,a_{n}$ depend linearly on the vector variable $\rho$. Similarly, in order to establish the stability of the closed-loop system for all $\rho\in\mathcal{P}$, it is necessary and sufficient to check whether all the roots of the polynomial $p_{D}(z,\rho)$ have magnitude less than 1. By multiplying the odd rows by their denominator and removing the even rows in the traditional Jury table, we define the modified Jury table where

[TABLE]

It can be verified that all the roots of $p_{D}(z,\rho)$ have magnitude less than 1 for all $\rho\in\mathcal{P}$ if and only if all the polynomials in the first column of the modified Jury table are positive for all $\rho\in\mathcal{P}$.

Let us denote the entries of the first column in the modified Routh-Hurwitz table for CT case or in the modified Jury table for DT case as $f_{i}(\rho),i=1,\ldots,n+1$ where $f_{i}(\rho)$ denotes the $i$-th entry in the column. Based on the previous analysis, we have the following lemma.

Lemma 8**.**

The closed-loop system is stable for all $\rho\in\mathcal{P}$ defined in (8) if and only if

[TABLE]

Now let us define the polynomials

[TABLE]

where $s_{ij}(\rho)$ are auxiliary polynomials.

Theorem 9**.**

The closed-loop system is stable for all $\rho\in\mathcal{P}$ if and only if

[TABLE]

where

[TABLE]

Proof. $"\Rightarrow"$ It can be observed that the set $\mathcal{P}$ defined in (8) is a compact set, and $c_{1},\ldots c_{p}$ are polynomials of even degree and their highest degree forms do not have common zeros except zero. Given an arbitrarily small scalar $\theta>0$, it follows from Theorem 7 in Chesi (2010) that $f_{i}(\rho)>\theta,\ \forall\rho\in\mathcal{P}$ holds if and only if there exist SOS polynomials $s_{ij}(\rho)$ such that $g_{i}(\rho)-\theta$ is SOS polynomial. Therefore, the condition (16) is satisfied with $\theta>0$, and hence the condition (15) holds.

$"\Leftarrow"$ Let us suppose that (15)-(16) hold. Then, one has that $g_{i}(\rho)-\theta$ and $s_{ij}(\rho)$ are nonnegative. Since $c_{j}(\rho)\geq 0$ whenever $\rho\in\mathcal{P}$, it follows from (14) that $f_{i}(\rho)>0,i=1,\dots n+1$ for all $\rho\in\mathcal{P}$. $\hfill\square$

Remark 10**.**

Theorem 9 shows that one can establish the positivity of the polynomials in the first column in the modified tables for all $\rho\in\mathcal{P}$ by solving the optimization problem (16). It is worth mentioning that the condition for polynomials which depend on some decision variables linearly to be SOS polynomials can be solved equivalently via LMIs based on the Gram matrix method as described in Section 2.2. Therefore, for any chosen degrees of polynomials $s_{ij}(\rho)$, this theorem provides a sufficient condition solvable through LMIs, which is also necessary when the degrees are large enough.

Since fixed-structured controllers, including PID control as a typical example, are so widely used in industrial applications, it is important to develop a methodology to characterize the set of stabilizing controllers before carrying out the optimal control design. Theorem 9 provides a method to establish whether a given set of controllers is stabilizing. In the next subsection, we will design the controller by choosing its parameter $\rho$ from the set $\mathcal{P}$ to reach the maximized passivity level for the closed-loop system. Indeed, most existing modern optimal control techniques are incapable of accommodating constraints on the controller order or structure into their design methods, and consequently cannot be used for designing optimal or robust controllers.

3.2 Feedback Passivation

Given a set of stabilizing controllers $\mathcal{C}$, we proceed to address Problem 5 in this subsection. We first consider the CT case, which is then extended to the DT case.

3.2.1 CT case

Recall that the numerator and denominator of the closed-loop transfer function (9) are denoted as polynomials $p_{N}(s)$ and $p_{D}(s)$, respectively, wherein the coefficients of $p_{D}(s)$ depends linearly on the vector variable $\rho$. By substituting $s=j\omega$, $p_{N}$ and $p_{D}$ can be rewritten via even-odd decomposition as

[TABLE]

where $p_{N}^{e},p_{N}^{o},p_{D}^{e},p_{D}^{o}$ are all real polynomials in $\omega$, and $p_{D}^{e}(\omega,\rho)$ and $p_{D}^{o}(\omega,\rho)$ depend linearly on $\rho$. The frequency response of the closed-loop system (9) can be expressed as

[TABLE]

which yields that

[TABLE]

Lemma 11**.**

There exists a controller $C(s,\rho)$ in the controller set $\mathcal{C}$ that can feedback passivate the plant $G_{o}(s)$ if and only if there exists a vector $\rho\in\mathcal{P}$ and a scalar $\epsilon\geq 0$ such that

[TABLE]

Proof. Since the controller set $\mathcal{C}$ is stabilizing, it follows from Lemma 3 that a controller $C(s,\rho)$ can feedback passivate the plant if and only if the closed-loop system is positive real. For a stable closed-loop system, the first and the third condition in Definition 2 are trivially satisfied. Therefore, the closed-loop system is positive real if and only if

[TABLE]

which, according to (19), is equivalent to

[TABLE]

Therefore, there exists a controller in the set $\mathcal{C}$ that can feedback passivate the plant if and only if there exists a vector $\rho\in\mathcal{P}$ and a scalar $\epsilon\geq 0$ such that

[TABLE]

Since there is no gap between nonnegative polynomials and SOS polynomials when the polynomial is univariate, the above condition is is equivalent to

[TABLE]

which completes the proof. $\hfill\square$

Note that checking the feasibility of (20) can be solved by a SDP. Specifically, the condition in (20) can be rewritten based on Section 2.2 as

[TABLE]

where $M(\rho,\epsilon)$ is a matrix depending linearly on $(\rho,\epsilon)$ while $L(\alpha)$ a linear parametrization of the set defined in (2). Moreover, to take the constraint $\rho\in\mathcal{P}$ into account, the feasibility problem in (20) can be equivalently solved by checking the positivity of $\epsilon^{*}$, which is the optimal solution of the following SDP:

[TABLE]

If the condition in (20) is feasible, i.e., $\epsilon^{*}>0$, let us further address the second part of Problem 5. Consider the problem of desgining an optimal IFP controller. According to Lemma 3 and (19), the problem can be equivalently rephrased in the following mathematical form

[TABLE]

Theorem 12**.**

If the condition in (20) is feasible, the maximum IFP index $\nu^{*}$ that can be achieved by the feedback controller set $\mathcal{C}$ is given by $\nu^{*}=(\gamma^{*})^{2}$ with $\gamma^{*}$ defined as

[TABLE]

and the corresponding controller is given by the optimal solution $\rho^{*}$.

Proof. Suppose the condition in (20) is feasible, it follows that there exists $\bar{\rho}\in\mathcal{P}$ such that

[TABLE]

and hence,

[TABLE]

Therefore, a lower bound of the optimal $\nu^{*}$ in (22) is zero. Next, let us observe that the constraint in (22) can be rewritten as

[TABLE]

Since $\nu\geq 0$ and by exploiting the Schur complement lemma, the above inequality can be further equivalently rewritten as

[TABLE]

According to Theorem 4 in Chesi (2010), we have that a univariate matrix polynomial is positive semidefinite if and only if it is SOS. Therefore, by replacing $\sqrt{\nu}$ with $\gamma$, the optimization problem in (22) can be equivalently solved by (23), which completes the proof. $\hfill\square$

Remark 13**.**

Theorem 12 provides a method via solving a convex optimization problem to design the controller in the set $\mathcal{C}$ that maximizes the IFP index for the closed-loop system. Particularly, the maximum $\gamma^{*}$ in the convex optimization problem (23) can be obtained by bisection algorithm (i.e., at each step of the bisection algorithm, fix the value of $\gamma$ and check the feasibility of (23) ). To check the feasibility of (23) with fixed value of $\gamma$, let us observe that the matrix in (23) depends linearly on the decision variables $\rho$, and the constraint $\rho\in\mathcal{P}$ can be imposed by adding extra LMI constraints as done in (21). Similar to the scalar polynomial case in (20), the condition for a matrix polynomial which depends on some decision variables linearly to be SOS polynomials can be solved equivalently via a SDP, as shown in Section 2.2.

Next, we consider the optimal OFP controller design. To this end, let us observe that the zeros of the closed-loop transfer function (9) have negative real part under Assumption 7 and the stable controller base $\bar{C}$. Therefore, the closed-loop system $G(s,\rho)$ is minimum phase system. Now, we are ready the present the following theorem.

Theorem 14**.**

The maximum OFP index $\xi^{*}$ that can be achieved by the feedback controller set $\mathcal{C}$ is given by $\xi^{*}$ defined as

[TABLE]

and the corresponding controller is given by the optimal solution $\rho^{*}$.

Proof. Since the closed-loop system $G(s,\rho)$ is minimum phase, its inverse exists. From (18), we have that

[TABLE]

and

[TABLE]

It follows from Lemma 3 that the maximum OFP index that can be reached is

[TABLE]

which can be rewritten into (24). $\hfill\square$

Similar to the optimization problem (20), since the polynomial in (24) depends linearly on decision variables $\rho$ and $\xi$, it can be solved by a SDP.

Remark 15**.**

An alternative approach to address directly Problem 5 without assuming that the set of $\mathcal{C}$ is stabilizing is to solve the SDP presented in Theorem 12, and then check the stability of the closed-loop system with the derived controller $C(\rho^{*})$. See Example 2 in Section 4 for more details.

3.2.2 DT case

In the end, we consider Problem 5 for the discrete-time systems (12). In order to establish whether a given stable closed-loop system is passive, we need to check the positivity of the real part of the transfer function $G(z,\rho)$ over the complex unit circle $\{z\in\mathbb{C}:|z|=1\}$.

Let $y\in\mathbb{R}$ be an auxiliary variable, and define the rational function as $\phi:\mathbb{R}\rightarrow\mathbb{C}$ as $\phi(y)=\frac{1-y^{2}+j2y}{1+y^{2}}$. Note that the complex unit circle $|z|=1$ is parameterized by the variable $y\in\mathbb{R}$ (Chesi (2019)). Consequently, one has that

[TABLE]

is equivalent to

[TABLE]

Let us denote the numerator and denominator of the transfer function in (12) as the polynomials $p_{N}(z)=\sum_{i=0}^{d_{N}}q_{i}z^{i}$ and $p_{D}(z,\rho)=\sum_{i=0}^{d_{D}}b_{i}(\rho)z^{i}$, respectively, wherein the coefficients $b_{i}(\rho),i=1,\ldots,d_{D}$ depend linearly on the vector variable $\rho$. By substituting $z=\phi(y)$, we have

[TABLE]

By even-odd decomposition, it follows that $p_{N}(\phi(y))$ and $p_{D}(\phi(y),\rho)$ can be expressed as

[TABLE]

[TABLE]

where $p_{1},p_{2},p_{3},p_{4}$ are all real polynomials in $y$ with coefficients of $p_{3},p_{4}$ depending linearly on $\rho$. Now it is ready to see

[TABLE]

which follows that

[TABLE]

Since the given set of controllers $\mathcal{C}$ is stabilizing, it follows from Lemma 3 and Definition 2 that the closed-loop system is passive if and only if the condition (25) holds. Based on similar reasoning of Lemma 11, we can obtain the following result.

Lemma 16**.**

There exists a controller $C(z)$ in the controller set $\mathcal{C}$ that can feedback passivate the plant $G_{o}(z)$ if and only if there exists a vector $\rho\in\mathcal{P}$ and a scalar $\epsilon\geq 0$ such that

[TABLE]

Lemma 16 provides, for the DT case, a necessary and sufficient condition for determining the existence of a controller $C(z)$ in the set $\mathcal{C}$ such that the closed-loop system (12) is passive. Similar to the CT case, this condition can be verified by solving a SDP with the same form in (21).

When the condition (29) is satisfied, the next step is to determine the controller $C^{*}$ in the set $\mathcal{C}$ that can achieve the maximum IFP index $\nu^{*}$ for the closed-loop system (28).

Corollary 17**.**

If the condition in (29) is feasible, the maximum IFP index $\nu^{*}$ that can be achieved by the feedback controller set $\mathcal{C}$ is given by $\nu^{*}=(\gamma^{*})^{2}$ with $\gamma^{*}$ defined as

[TABLE]

where

[TABLE]

By taking the inverse of $G(\phi,\rho)$, it is obtained that

[TABLE]

Corollary 18**.**

The maximum OFP index $\xi^{*}$ that can be achieved by the feedback controller set $\mathcal{C}$ is given by $\xi^{*}$ defined as

[TABLE]

and the corresponding controller is given by the optimal solution $\rho^{*}$.

Remark 19**.**

It can be easily seen that the proposed methodology in this subsection can be used not only for a hyper-rectangle set $\mathcal{P}$ as defined in (7), but also any convex set $\mathcal{P}$. Indeed, this can be achieved by replacing the LMI the second constraint in (21) with appropriate LMI corresponding to the set $\rho\in\mathcal{P}$.

4 Numerical Examples

In this section, we provide two examples to illustrate the proposed methodologies. The computations are done by Matlab with toolbox SOSTOOLS and SeDuMi.

4.1 Example 1

Let us begin with considering a DT plant with transfer function $G_{0}(z)=\frac{z}{z-2}$, and the controller set $\mathcal{C}$ described by $C(z,\rho)=\rho_{1}+\rho_{2}\frac{1}{z-0.5}$ with the parameter $\rho\in\mathcal{P}=[0.1,1]\times[1,2]$. Note that the plant is unstable since its pole has magnitude larger than

1. The closed-loop system (12) is derived as

[TABLE]

The first problem is to establish whether the closed-loop system is stable for all $\rho\in\mathcal{P}.$ To address this, we first compute the modified Jury table for the denominator $p_{N}(z,\rho)$, and the first column of the table is obtained as $f_{1}=2\rho_{1}+2,f_{2}=4\rho_{1}^{2}+8\rho_{1},f_{3}=12\rho_{1}^{4}+16\rho_{1}^{3}\rho_{2}+24\rho_{1}^{3}-16\rho_{1}^{2}\rho_{2}^{2}+80\rho_{1}^{2}\rho_{2}-36\rho_{1}^{2}.$ Next, we examine the positivity of these polynomials over the set $\rho\in\mathcal{P}$ based on Theorem 9. It is obvious that $f_{1}(\rho)>0$ and $f_{2}(\rho)>0$ for all $\rho\in\mathcal{P}$, so we just need to solve the SOS program in (16) for $i=3$. By choosing the the degrees of the auxiliary polynomials $s_{31}(\rho),s_{32}(\rho)$ as 2, we find the optimal solution as $\theta^{*}=0.32$, which guarantees the positivity of $f_{3}(\rho)$ over $\rho\in\mathcal{P}$. Therefore, it can be concluded that the closed-loop system $G(z,\rho)$ is stable for all $\rho\in\mathcal{P}$.

With this set of stabilizing controllers, we further consider optimal IFP controller design in Problem 5. The first step is to determine the existence of controllers in the set $\mathcal{C}$ that can feedback passivate the plant. This can be done by solving the SDP in (29). Specifically, by replacing $z$ with $\phi(y)=\frac{1-y^{2}+j2y}{1+y^{2}}$, we have

[TABLE]

Then, we solve the SOS program in (29), which is converted to solving a SDP in the form of (21), and it is obtained that the optimal solution of $\epsilon$ in (21) is positive. Therefore, it can be concluded that there exists a controller in the set $\mathcal{C}$ that can feedback passivate the plant $G_{0}$. The next step is to derive the controller $C^{*}$ in the set $\mathcal{C}$ that maximizes IFP $(\nu)$ for the closed-loop system. This is accomplished by solving the SDP (30) at each step of the bisection algorithm, which leads to the maximum $\nu$ as $\nu^{*}=0.48$ with the optimal solution $\rho_{1}^{*}=0.1,\rho_{2}^{*}=1.5$.

To verify the resulting IFP index $\nu^{*}$, one can transform the closed-loop transfer function $G(z,\rho^{*})$ to a state space system $(A,B,C,D)$, and then exploits the necessary and sufficient LMI condition for dissipativity to obtain the IFP index for the closed-loop system. (See Lemma 2 in Kottenstette et al. (2014) for details) To be specific, the closed-loop system $G(z,\rho^{*})$ can be rewritten as the state space system as follows

[TABLE]

It can be verified by Lemma 3 in Kottenstette et al. (2014) that the IFP index for this state space system is obtained as $0.48$ as expected.

4.2 Example 2

In this example, we consider a CT plant with the transfer function $G_{0}(s)=\frac{(s+2)(s+3)}{(s-1)(s-2)}$, and the controller set $\mathcal{C}$ is chosen to be the class of PI controllers, described as $C(s,\rho)=\rho_{1}+\rho_{2}\frac{1}{s+1}$ with the parameter $\rho\in\mathcal{P}=\left[0,1\right]\times\left[0,1\right]$.

The problem is to directly determine the controller $C^{*}$ in the set $\mathcal{C}$ that maximize the IFP index and the OFP index for the closed-loop system, respectively. Let us observe that the plant is unstable since it has poles $\{1,2\}$.

First, by substituting $s=j\omega$, one can express the closed-loop system (18) as

[TABLE]

Then, we solve the SOS progam in (20), which is converted to solving the SDP (21), and it is obtained that the optimal solution of $\epsilon$ in (21) is positive.

To design the optimal IFP controller, we consider the optimization problem in (22). By solving the SDP (23) at each step of bisection algorithm, we obtain that the maximum $\nu$ as $\nu^{*}=0.658$ with the solution $\rho_{1}^{*}=0.516,\rho_{2}^{*}=0.669$. To design the optimal OFP controller, we solve the optimization problem in (24), and obtain that the maximum $\xi$ as $\xi^{*}=0.542$ with the solution $\rho_{1}=\rho_{2}=1$.

In the end, we need to check the stability of the closed-loop system (9) with the derived $\rho^{*}$. For both the optimal IFP controller $\rho_{1}^{*}=0.516,\rho_{2}^{*}=0.669$ and the optimal OFP controller $\rho_{1}=\rho_{2}=1$, the closed-loop system $G(s,\rho^{*})=\frac{G_{0}(s)}{1+G_{0}(s)C(s,\rho^{*})}$ can be easily verified via Routh-Hurwitz stability criterion or calculating the poles that the closed-loop system $G(s,\rho^{*})$ is stable. Therefore, based on Lemma 3, one has that the maximum IFP index that the closed-loop system can achieve is $\nu^{*}=0.658$, and the corresponding controller is $C(s,\rho^{*})=0.516+\frac{0.669}{s+1},$ and the maximum OFP index that the closed-loop system can achieve is $\xi^{*}=0.542$ and the corresponding controller is $C(s,\rho^{*})=1+\frac{1}{s+1}$.

Similar to the previous example, the resulting IFP index $\nu^{*}$ and OFP index $\xi^{*}$ can be verified by transforming the closed-loop transfer function $G(s,\rho^{*})$ to state space system $(A,B,C,D)$, and then exploit the necessary and sufficient LMI condition for dissipativity (Lemma 2 in Kottenstette et al. (2014)) to obtain the IFP or OFP index for the closed-loop system.

5 Conclusion

This paper has considered feedback passivation of SISO LTI systems with linearly parameterized controller with the objective of maximizing the passivity level for the closed-loop systems. First, we have proposed a method to test whether a given set of controllers is stabilizing. Second, we have shown that given a set of stabilizing controllers, the optimal controller in the sense of maximum IFP or OFP index can be obtained by solving a SDP. The proposed results also provide an alternative method without assuming the set of controllers to be stabilizing. Future work will consider extensions to the multi-input multi-output (MIMO) case.