Concept of Passive Systems

This chapter focuses on the stability of nonlinear systems, beginning with an overview of their dynamics We consider a nonlinear system defined by the equation \( \frac{dx}{dt} = f(x, u) \), where \( x \) represents the state vector and \( u \) denotes the input vector The stability analysis centers on the system's behavior when the input \( u \) is set to zero, leading to the function \( f^*(x) = f(x, 0) \) The components of \( f^*(x) \) are local Lipschitz functions, which satisfy the Lipschitz condition: \( |f^*(x_1) - f^*(x_2)| \leq L |x_1 - x_2| \) for all \( x_1, x_2 \) in the vicinity of \( x_0 \), where \( L \) is a positive constant This condition ensures that the dynamics of the system can be analyzed through the equation \( \frac{dx}{dt} = f^*(x) \).

Lyapunov stability ensures a unique solution with the initial condition x(0) = x0, where a point x* ∈ X is termed an equilibrium point if ∗(x*) = 0 The equilibrium point x = 0 is considered stable if, for every ε > 0, there exists a δ = δ(ε) > 0 such that x(0) < δ implies x(t) < ε for all t ≥ 0 It is asymptotically stable (AS) if stability holds and δ can be chosen so that x(0) < δ leads to x(t) approaching the origin as t approaches infinity The region of attraction is defined as the set of initial points x(0) for which the solution of (2.4) converges to the origin over time If this region encompasses the entire state-space X, the origin is classified as globally asymptotically stable (GAS) Unlike linear systems, nonlinear systems may exhibit multiple equilibrium points, some stable and others unstable The Lyapunov stability criterion provides a sufficient condition for determining the stability of an equilibrium point without needing to solve the state equation, using a continuously differentiable scalar function V(x) that is positive definite and defined in X, including the origin.

It is said to bepositive semidefinite if

Similarly, a function V (x) is said to be negative definite if V(0) = 0 and

V(x)0 [123].

Here,G ∗ (jω) is the complex conjugate transpose ofG(jω).

Theorem 2.25 ([139]) A linear system as given in (2.40) is passive (or strictly passive) if and only if its transfer functionG(s) :=C(sI−A) −1 B+D is positive real (or strictly positive real).

The theorem presented, along with Definition 2.24, establishes an input-output perspective of the positive-real lemma in the frequency domain, serving as a foundational definition for linear passive systems Notably, Theorem 2.25 identifies G1(s) = s + 1 as a strictly passive system, while G2(s) = 1/s is categorized as a passive system Additionally, it is important to highlight that any PID controller fits within this framework.

, k c >0, (2.56) is passive So is any multiloop PID controller.

Phase-related Properties

Passive systems exhibit a notable characteristic known as phase boundedness, particularly evident in Single Input Single Output (SISO) systems This is highlighted by the condition that the sum of the system's transfer function and its complex conjugate must be non-negative, expressed as G(jω) + G∗(jω) ≥ 0.

The condition Re(G(jω)) ≥ 0 indicates that the real part of the frequency response remains nonnegative, defining the concept of "positive real." In stable Single Input Single Output (SISO) passive systems, the phase shift in response to a sinusoidal input is confined within the range of [−90°, 90°], while strictly passive SISO systems exhibit a phase shift that lies within the interval (−90°, 90°).

The above statement is also true for multi-input multi-output (MIMO) lin- ear systems Here we adopt the following phase definition for MIMO systems given by Postlethwaiteet al.:

Definition 2.26 (Phase of MIMO LTI systems [96]) Consider an MIMO LTI system with a transfer functionG(s)∈C m × m Perform the polar decomposition on its frequency response:

= [X(jω)V ∗ (jω)] [V(jω)Λ(jω)V ∗ (jω)] =U(jω)H(jω), (2.57) whereΛ(jω)is an m×m diagonal, real and nonnegative matrix; X(jω)and

V(jω) are unitary matrices.U(jω) =X(jω)V ∗ (jω) is also a unitary matrix andH =V(jω)Λ(jω)V ∗ (jω)is a Hermitian matrix The phase of the system at frequency ω is defined as the principal arguments of the eigenvalues of

Theorem 2.27 (Phase condition for MIMO LTI strictly passive sys- tems [13]) Consider an MIMO LTI system with a transfer functionG(s)∈

C m × m If the system is strictly passive, then its phase shift lies in the open interval(−90 ◦ ,90 ◦ ) for any realω.

The proof of the theorem can be found in Section B.1 A stable linear system exhibits a frequency response with a phase shift confined to the range of [−90 ◦ ,90 ◦ ] across all frequencies, thereby fulfilling both specified conditions.

2 the difference between the degree of the denominator polynomial and the degree of the numerator polynomial (i.e., the relative degree) is less than 2.

In a simple SISO case, a stable and minimum phase transfer function G(s) = p q ( ( s s ) ) with G(0) > 0 demonstrates that with only left half plane (LHP) zeros and poles at frequency ω = ∞, the phase shift is determined to be 90 ◦ (n−m) For the system to maintain a phase bounded by [−90 ◦ ,0 ◦ ] across all frequencies, it is essential that n−m < 2 While phase is not defined for nonlinear systems, the phase-related conditions can be adapted to them The relative degree of a nonlinear system can be understood as the number of differentiations required for the output to explicitly include the input, leading to a definition of relative degree in this context.

Definition 2.28 (Relative degree [61]) A SISO control affine nonlinear system ˙ x=f(x) +g(x)u y =h(x), (2.58) is said to have relative degreer at point x 0 if

1.L g L k f h(x) = 0 for all x in a neighbourhood ofx 0 and allk < r−1;

2.L g L r f −1 h(x 0 ) = 0, whereL k f h(x)is the kth order Lie derivative of halong f.

A multivariable nonlinear control affine system as in the following equa- tion: ˙ x=f(x) + q j =1 g j (x)u j , y i =h i (x), i= 1, , p,

(2.59) has a vector relative degree given by{r 1 , r 2 ,ã ã ã, r p } at a pointx 0 if

1.L g i L k f h i (x) = 0, i= 1, , p,k= 0, , r i −2 for allxin a neighbour- hood ofx 0

2 The characteristic matrixC(x), given by

⎦ p × p is nonsingular at x 0 The total relative degree is defined asr= p i =1 r i

In a linear Single Input Single Output (SISO) system described by the equations ˙x = Ax + Bu and y = Cx, the relative degree is determined by calculating the difference between the degree of the denominator polynomial and the degree of the numerator polynomial in the transfer function.

H(s) =C(sI−A) −1 B of the system To extend the concept of minimum phase systems to nonlinear systems, we need to look at the zero dynamics:

Definition 2.29 (Zero dynamics) Consider the system in (2.32) with the constrainty= 0, i.e., ˙ x=f(x) +g(x)u,

The constrained system (2.60) is called the zero-output dynamics, or briefly,the zero dynamics.

If the matrix \( L g h(0) \frac{\partial h}{\partial x}(x) g(x) x = 0 \) is nonsingular and the vector fields \( g_1(x), \ldots, g_m(x) \) form an involutive distribution near \( x = 0 \), then it is possible to introduce new local coordinates \( (z, y) \) that allow the system to be expressed in a normal form In this representation, the dynamics are described by \( \dot{z} = q(z, y) \) and \( \dot{y} = b(z, y) + a(z, y)u \) The zero dynamics of the system are characterized by \( \dot{z} = q(z, 0) \).

Denote q(z,0) byf 0 (z) Then, the function q(z, y) can be expressed in the form q(z, y) =f 0 (z) +p(z, y)y, (2.63) wherep(z, y) is a smooth function (see [24]).

Definition 2.30 (Minimum phase nonlinear systems [24]) Consider the system in (2.32) Suppose thatL g h(0)is nonsingular Then the system is said to be:

1 minimum phase if its zero dynamics are asymptotically stable in a neigh- bourhood ofz= 0;

2 weakly minimum phase if there exists a positive differentiable function

We can define globally minimum phase and globally weakly minimum phase systems when both the normal form and minimum phase characteristics are considered on a global scale This understanding allows us to explore the phase-related properties of nonlinear passive systems.

Theorem 2.31 ([24]) Consider system H given in (2.32) Assume that rank{L g h(x)}is constant in a neighbourhood ofx= 0 If systemH is passive with aC 2 storage function S(x)which is positive definite, then

1.L g h(0) is nonsingular andH has relative degree{1,ã ã ã,1}.

2 The zero dynamics ofH exist locally atx= 0, andH is weakly minimum phase.

System H lacks a feedthrough term, ensuring its relative degree remains at least 1 In passive SISO nonlinear systems, the relative degree can be either 0 or 1, depending on the presence of a feedthrough term This theorem indicates that nonlinear passive systems exhibit phase-related input-output properties akin to those of linear systems These characteristics lead to specific output feedback stability conditions, which will be explored in the following section.

Interconnection of Passive Systems

Fig 2.3.Interconnections of passive systems

The phase-related characteristics of passive systems highlight crucial output feedback stability conditions, essential for assessing the stability of interconnected networks Passive systems are notably simple to control through output feedback, exemplified by linear passive systems, such as G(s) = 1/s, which can be stabilized using any proportional controller with a positive gain This principle extends to nonlinear systems, reinforcing the importance of stability conditions in various system types.

Theorem 2.32 For a nonlinear passive system H given in (2.32), a propor- tional only output feedback control lawu=−ky asymptotically stabilizes the equilibriumx= 0for anyk >0, provided thatH is ZSD.

Proof Assume thatH is passive with storage functionS(x) Foru=−y, the time derivative ofS satisfies

The bounded solution of ˙x=f(x,−y) is confined in{x|h(x) = 0} IfH is ZSD, thenx→0.

The output feedback stability condition is not limited to static feedback:

Theorem 2.33 states that when two passive systems, H1 and H2, are interconnected, both parallel and feedback configurations maintain passivity If these systems are zero-state detectable (ZSD) and possess continuously differentiable storage functions S1(x1) and S2(x2), they will retain their equilibrium properties.

(x 1 , x 2 ) = (0,0) of both interconnections is stable.

Proof Passivity: Because H 1 and H 2 are passive, there exist two positive semidefinite storage functionsS 1 (x 1 ) andS 2 (x 2 ) such that

S i (x i (t 1 ))−S i (x i (t 0 ))≤ t 1 t 0 u T i y i dt, i= 1, 2, (2.66) where x 1 , x 2 are the state variables of H 1 and H 2 , respectively Define x x T 1 , x T 2 T and S(x) = S 1 (x 1 ) +S 2 (x 2 ) Note that S(x) is positive semidefinite and

For the parallel interconnection,u=u 1 =u 2 andy=y 1 +y 2 Therefore,

For the feedback case,u 2 =y 1 andu 1 =r−y 2 :

Therefore, both interconnections are passive.

If systems H 1 and H 2 are ZSD, the equilibrium (x 1 , x 2 ) = (0,0) of both interconnections is Lyapunov stable, according to Theorem 2.11.

The above conditions can be extended to partial parallel and feedback connections:

Proposition 2.34 (Partial interconnection of passive systems) Con- sider systems H 1 : u 1 −→ y 1 and H 2 : u 2 −→ y 2 , where u 1 u T 11 , u T 12 T

If systems H1 and H2 are passive, both partial parallel interconnection and partial feedback interconnection will also result in passive systems Additionally, if systems H1 and H2 are zero-state detectable (ZSD) and their storage functions S1(x1) and S2(x2) are continuous, the equilibrium is maintained.

(x 1 , x 2 ) = (0,0)of both interconnections is stable.

Proof Similar to the proof of Theorem 2.33, becauseH 1 andH 2 are passive, there exist two positive semidefinite storage functionsS 1 (x 1 ) andS 2 (x 2 ) such that

S i (x i (t 1 ))−S i (x i (t 0 ))≤ t 1 t 0 u T i y i dt, i= 1,2, (2.70) where x 1 , x 2 are the state variables of H 1 and H 2 , respectively Define x x T 1 , x T 2 T and S(x) = S 1 (x 1 ) +S 2 (x 2 ) Note that S(x) is positive semidefinite and

For the partial parallel interconnection, define u 3 = u 12 = u 21 , y 3 = y 12 + y 21 The overall system inputs and outputs areu u T 11 , u T 3 , u T 22 T and y y 11 T , y T 3 , y T 22 T

For the feedback case,u 12 =−y 21 and u 21 =y 12 The overall system inputs and outputs areu u T 11 , u T 22 T andy y 11 T , y T 22 T

Therefore, both interconnections are passive If systemsH 1 andH 2 are ZSD, from Theorem 2.11, the equilibrium (x 1 , x 2 ) = (0,0) of both interconnections is Lyapunov stable.

A passive process can be stabilized at the equilibrium point (x=0) using any passive controller, regardless of its nonlinearity or coupling For instance, a gravity tank can achieve stability with a PID controller that has a positive gain, which can be increased to enhance response time without leading to instability This highlights the importance of stability analysis and control design rooted in passivity, with the potential to expand the stability conditions through the introduction of a passivity index.

Passivity Indices

Excess and Shortage of Passivity

To broaden the applicability of passivity-based stability conditions to both passive and nonpassive systems, it is essential to establish passivity indices that measure the level of passivity These indices can be characterized by assessing either an excess or a deficiency of passivity.

Let system H, as given in (2.32), be passive with a C 1 storage function

In a static feedforward system defined by f = -νu (with ν > 0), the overall system ˜H produces an output of ˜y = y - νu, as illustrated in Figure 2.5a Since the feedforward mechanism is static, it lacks a state-space, meaning the storage function of the overall system continues to be S(x) If the system ˜H is also passive, it retains these characteristics.

For a system H to be considered dissipative, it must adhere to the supply rate w(u, y) = u^T y − νu^T u, indicating that it possesses excessive input feedforward passivity, denoted as IFP(ν) However, the feedforward system −νI fails to be passive, as it violates the positive real condition with the integral t1 t0 u^T y f f dt = t1 t0 −νu^T u dt < 0 This highlights that the excess passivity in system H can offset the lack of passivity in the feedforward system Conversely, if H is nonpassive but remains dissipative with respect to the supply rate w(u, y) = u^T y + νu^T u (where ν > 0), then the combined system H + νI becomes passive, indicating that H lacks input feedforward passivity, represented as IFP(−ν).

In a negative feedback interconnection, as illustrated in Figure 2.5b, let ˜H represent the closed-loop system of H with a positive feedback of ρI, where ρ is greater than zero If ˜H is passive and possesses a C1 storage function S(x), then certain implications arise regarding its stability and performance characteristics.

Fig 2.5.Excess and shortage of passivity

The dissipativity of system H is characterized by the supply rate w(u, y) = u^T y - ρy^T y, indicating that H exhibits excessive output feedback passivity of ρ, referred to as OFP(ρ) Conversely, if H is not passive but is dissipative with respect to the supply rate w(u, y) = u^T y + ρy^T y (where ρ > 0), it can be made passive through the application of negative feedback, represented as ρI In this scenario, H is described as lacking output feedback passivity, denoted as OFP(−ρ).

Definition 2.35 (Excess/shortage of passivity [110]) LetH :u−→y. SystemH is said to be:

1 Input feedforward passive (IFP) if it is dissipative with respect to supply ratew(u, y) =u T y−νu T ufor someν∈R, denoted as IFP(ν).

2 Output feedback passive (OFP) if it is dissipative with respect to supply ratew(u, y) =u T y−ρy T y for someρ∈R, denoted as OFP(ρ).

In this book, a positive value of ν or ρ indicates that the system exhibits an excess of passivity, categorizing it as strictly input passive (IFP) or strictly output passive (OFP) It follows that if a system is classified as IFP(ν) or OFP(ρ), it will also remain IFP(ν−ε) or OFP(ρ−ε) for any ε greater than zero.

The IFP and OFP can also be defined on the input-output version of passivity:

Definition 2.36 ([130]) Let H : L m 2 e → L m 2 e System H is strictly input passive if there existβ andδ >0 such that

H is strictly output passive if there existβ and ε >0 such that

A strictly output passive system demonstrates a finite L2 gain, indicating its stability Additionally, a system exhibiting excessive output feedback passive (OFP) with a continuous storage function maintains a stable equilibrium at x = 0 when the input u = 0, given that the system is zero-state detectable (ZSD).

Following a proof similar to Theorem 2.32,x→0 when t→ ∞.

IFP and OFP systems have the following scaling property:

Proposition 2.37 (IFP/OFP Scaling [110]) For systems H and αH, whereαis a constant, the following statements are true:

1 If H is OFP(ρ), thenαH is OFP 1 α ρ

2 If H is IFP(ν), then αH is IFP(αν).

A linear system is considered strictly passive when it meets two key criteria: it must be stable and exhibit input-to-state stability (IFP) with a positive ν (ν > 0) This definition is distinct from the linear version of state strict passivity applicable to nonlinear systems, as outlined in Theorem 2.25.

Example 2.38.To illustrate the definition of IFP and OFP, let us consider a linear integrating system:

This system is lossless (passive but not strictly passive) By definition, system

H 1 : ˙ x =u y =x+νu (2.80) will have excessive IFP of ν This can be seen by using a storage function

From an input-output point of view,H(s) = 1/sis passive, and

H 1 (s) =H(s) +ν = (νs+ 1)/s (2.82) has excessive IFP of ν According to Theorem 2.25, H 1 (s) is not strictly passive because it is not stable.

Similarly,H(s) with a negative feedback ofρ(ρ >0),

Simultaneous Input-Output Feedback Passivity (IFP) and Output Feedback Passivity (OFP) will result in a strictly passive system, denoted as OFP(ρ) It's important to note that linear strictly output passive systems may not always exhibit strict passivity, as strict passivity for linear systems necessitates strict IFP.

H 3 (s) = s s+ 1 (2.84) is OFP(1), but it is not strictly passive becauseH 3 (0) +H 3 ∗ (0) = 0.

Simultaneous Input Feedforward (IFP) and Output Feedback (OFP) can be defined using general supply rates In a system denoted as H, which incorporates both input feedforward (νI) and output feedback (ρI), the overall system ˜H must be passive for these definitions to hold true.

H is dissipative with respect to the supply rate: w(u, y) = (1 +ρν)y T u−νu T u−ρy T y (2.85)

The previous discussion assumes that feedforward and feedback mechanisms are static and decentralized However, a more complex scenario arises when these mechanisms are represented as arbitrary nonlinear multivariable functions For example, the function can be expressed as w(u, y) = y^T u - ν^T(u)u - ρ^T(y)y, where v(u) = [v1(u), , vm(u)]^T and ρ(u) = [ρ1(u), , ρm(u)]^T.

Hill and Moylan proposed a generalized supply rate expressed as w(u(t), y(t)) = y^T(t)Qy(t) + 2u^T(t)Sy(t) + u^T(t)Ru(t), where Q, R, and S are constant symmetrical weighting matrices in R^m × m This formulation applies to multivariable, linear, and static feedforward and feedback systems necessary for ensuring the passivity of the process system.

Passivity Indices for Linear Systems

In a stable linear system characterized by the transfer function G(s), the IFP index, represented as ν(G(s)), can be determined using the KYP lemma When G(s) exhibits high IFP, there exists a maximum value of ν > 0, ensuring that the process with the feedforward term −νI is positive real.

G(jω)−νI+ [G(jω)−νI] ∗ >0, ∀ω (2.88) Therefore, we can have the following definition:

Definition 2.39 The input feedforward passivity index for a stable linear systemG(s)is defined as 2 ν(G(s))1

2min ω ∈R λ(G(jω) +G ∗ (jω)), (2.89) whereλdenotes the minimum eigenvalue.

When ν is negative, the minimum feedforward needed to make the process passive is νI This definition also provides a numerical method for calculating the IFP index In the case of linear systems, a more precise IFP index can be established by using a frequency-dependent passivity index.

Definition 2.40 ([11]) The input feedforward passivity index for a stable linear systemG(s)at frequencyω is given by ν F (G(s), ω) 1

By using the above definition, we can specify the condition that a dynamic feedforwardG f f (s) needs to satisfy so thatG(s) +G f f (s) is passive For a stable processG(s), a stableG f f (s) should be chosen such that ν F (G f f (s), ω) +ν F (G(s), ω)>0 ∀ω∈R (2.91)

Calculating the OFP index numerically is challenging due to the presence of feedback loops When dealing with a minimum phase process G(s), which ensures the existence and stability of G −1 (s), the stability of G(s) itself is not a requirement In this scenario, incorporating a positive feedback of ρI leads to a closed-loop system that can be analyzed effectively.

According to Proposition 2.23,G cl (s) is passive if and only if

G −1 cl (s) =G(s) −1 −ρI (2.93) is passive Therefore, the OFP index ofG(s) is the IFP index ofG −1 (s) We can have the following definition:

2 This definition is similar to the passivity index proposed in [135], except that in

[135] a positive value ofνimplies that the system lacks passivity.

Passivation

Input Feedforward Passivation

Static feedforward control can effectively passivate many stable processes For instance, the linear system G1(s) = s³ + 1 - s² + s can be passivated through a static unit feedforward This results in G(s) = G1(s) + 1 = s³/(s + 3) + s²/(s + 2) + 2/(s + 1), which is a minimum phase system with a relative degree of 0 and is positive real.

Consider a control affine processH as in (2.32) Assume that the process has a globally stable equilibrium at x= 0 with a Lyapunov function V(x). UseV (x) as a storage function, then dV(x) dt = ∂V(x)

As shown in Figure 2.5a with the feedforward νI, ˜y = h(x) +νu Then ˜ y T u=h T (x)u+νu T u As long as there exists aν such that νu T u >

The findings indicate that the results can be applied to dynamic feedforward systems Any stable control affine process, for which a Lyapunov function exists, can be effectively passivated using a feedforward dynamic system This concept is illustrated in Figure 2.3a.

H 1 : ˙ x =f 1 (x) +g 1 (x)u 1 y 1 =h 1 (x), (2.98) is nonpassive but has a globally stable equilibrium pointx = 0 with a Lya- punov functionV(x) A feedforward systemH 2 can be designed to passivate

To design a feedforward passivator, we can assume that the passivated system \( H \) shares the same state equation as \( H_1 \) The goal is to identify an appropriate output function \( y(t) = h(x) \) that ensures \( H \) remains passive According to the KYP lemma (Proposition 2.14), utilizing \( V(x) \) as a storage function guarantees that the condition \( L_f V(x) = \frac{\partial V}{\partial x}(x) f_1(x) \leq 0 \) is consistently met By selecting \( h(x) \), we can effectively maintain the passivity of the system.

, thenH is passive The feedforward systemH 2 can be obtained by subtractingy fromy 1 :

Such a feedforward will stabilize the zero dynamics ofH 1 (so thatH is made weakly minimum phase) and reduce its relative degree to no greater than

In linear systems, the feedforward approach can be effectively derived using the linear KYP lemma, which will be explored in subsequent chapters However, it is important to note that feedforward cannot stabilize an unstable process, as it does not influence the system's free dynamics when the input is zero Therefore, passivation of such systems can only be achieved through feedback mechanisms.

Output Feedback Passivation

Passivation of unstable processes has garnered significant interest as an effective method for stabilizing nonlinear systems Most studies focus on passivation through state feedback, with comprehensive discussions available in the literature A control affine system is considered feedback passive if there exists a state feedback transformation that allows for an invertible function, leading to a modified system representation that incorporates feedback mechanisms.

2.5 Passivation 31 is passive The condition for feedback passivity is given in the following the- orem:

Theorem 2.42 (State feedback passivity [24]) Consider the control affine system in (2.32) Assume rank

A system is considered feedback passive with a positive definite storage function S(x) if it has a relative degree of {1, 1, , 1} at x = 0 and is weakly minimum phase.

A nonminimum phase system or a system with a relative degree greater than 1 cannot be made passive through feedback, as a passive system must be weakly minimum phase and have a relative degree of 1 or less Since feedback cannot change the relative degree or the zero dynamics, passivation can only be achieved through feedforward methods.

For the output feedback case, an additional condition is required:

1 Necessary condition: If the system in (2.32) can be rendered passive with aC 2 storage functionS(x), then it has relative degree{1,ã ã ã,1}atx= 0 and is weakly minimum phase, and L g h(x)| x =0 is symmetrical and posi- tive definite.

2 Sufficient condition: The system in (2.32) can be rendered locally passive with aC 2 positive definite storage functionS(x)by an output feedback if its Jacobian linearization atx= 0 is minimum phase and ∂h ∂x ( x ) g(x) x =0 is symmetrical and positive definite.

To get some intuition from the above conditions, let us look at the case of linear systems For a linear system ˙ x=Ax+Bu, y=Cx, (2.102)

Theorem 2.42 says the linear system is feedback passive if it (1) has a relative degree of 1 (due to the assumption D = 0, the relative degree cannot be

A weakly minimum phase system may possess zeros in the left half-plane (LHP) and on the imaginary axis, while maintaining a rank of (CB) = m If the matrix CB is nonsingular, the linear system exhibits a relative degree of 1, indicating that Condition (3) satisfies Condition (1) for linear systems.

State feedback cannot alter the established conditions, as it is defined by the equation u = r - Kx, where r represents an exogenous input or reference Consequently, the closed-loop system is described by the equations ˙x = (A - BK)x + Br and y = Cx.

+1 cannot be passivated by any state feedback controllers.

For output feedback passivity, CB must also be (1) symmetrical and

To ensure a system is positive definite, it is essential that the input and output are correctly paired, while also adhering to the constraints on the steady-state gain's sign For instance, a linear system represented as G2(s) = -1/(s + 1) can achieve stabilization through a negative feedback controller K(s) that has a negative steady-state gain However, it cannot be classified as positive real due to the characteristics of the closed-loop system defined by 1 + G2(s).

2 ( s ) K ( s ) will have a negative steady-state gain.

Passivity Theorem

The stability condition for passive systems in feedback is established in Theorem 2.33 By applying the concepts of strict input passivity and strict output passivity, we can derive asymptotic stability conditions for interconnected passive systems, known as the Passivity Theorem The most straightforward formulation of the Passivity Theorem is presented here.

Theorem 2.44 (Passivity Theorem [110]) Assume that systemsH 1 and

H 2 are ZSD and dissipative with C 1 storage functions S 1 (x 1 ) and S 2 (x 2 ).

Then the equilibrium (x 1 , x 2 ) = (0,0) of their feedback connection (as shown in Figure 2.7a) with r≡0is asymptotically stable (AS) if

1.H 1 andH 2 are strictly output passive; or,

2.H 1 andH 2 are strictly input passive; or,

3.H 1 is GAS and strictly input passive andH 2 is passive.

If storage functions S 1 (x 1 )and S 2 (x 2 ) are radially unbounded, then the feedback connection is globally asymptotically stable (GAS).

The proof of the aforementioned theorem is detailed in reference [110] To enhance clarity, we present a simplified version of the proof In this context, the storage function for the closed-loop system is defined as S(x₁, x₂) = S₁(x₁) + S₂(x₂).

1 Since H 1 andH 2 are strictly output passive, there exist ρ 1 , ρ 2 >0 such that

The bounded solution of (x 1 , x 2 ) is confined in{(x 1 , x 2 )|(y 1 , y 2 ) = (0,0)}. BecauseH 1 andH 2 are ZSD, (x 1 , x 2 )→(0,0).

2 Since H 1 and H 2 are strictly input passive, there exist ν 1 , ν 2 > 0 such that

3 In this case, there exists aν 1 >0 such that

Because ˙S is bounded only by y 2 T y 2 , the bounded solution of (x 1 , x 2 ) is confined in{(x 1 , x 2 )|y 2 = 0} and u 1 = 0 BecauseH 1 is GAS andH 2 isZSD, (x 1 , x 2 )→(0,0).

If storage functions S 1 (x 1 ) and S 2 (x 2 ) are radially unbounded, then all the above results hold globally.

The input-output version of the Passivity Theorem can be presented as follows:

Theorem 2.45 ([130]) Consider the closed-loop system shown in Figure 2.7a with H 1 , H 2 : L m 2 e → L m 2 e Assume that for any r ∈ L m 2 there are solutions u 1 , u 2 ∈ L m 2 e If

1.H 1 is passive andH 2 is strictly input passive; or,

2.H 1 is strictly output passive andH 2 is passive, then, u 2 =y 1 =H 1 (u 1 )∈ L m 2 , i.e., the closed-loop system fromr toy 1 is

Furthermore, if the input-output stability of systems H 1 and/or H 2 is assumed, we have

Theorem 2.46 ([130]) Consider the closed-loop system shown in Figure 2.7a with H 1 , H 2 : L m 2 e → L m 2 e Assume for any r ∈ L m 2 that there are solutions u 1 , u 2 ∈ L m 2 e If

1.H 1 is passive andH 2 is strictly input passive andL 2 stable; or

2 Both H 1 andH 2 are strictly output passive, then, y 1 , y 2 ∈ L m 2 , i.e., both of the closed-loop systems from r to y 1 and fromr toy 2 areL 2 stable.

For linear systems, Condition 1 of the above theorem simply means:

Proposition 2.47 states that in a negative feedback configuration involving two linear time-invariant (LTI) systems, H1 and H2, the closed-loop system achieves asymptotic stability when system H1 is strictly passive and system H2 is passive.

This can be clearly seen from the example of two SISO systems H 1 and

In this analysis, the phase shifts of H1 and H2 are confined within the ranges (−90°, 90°) and [−90°, 90°], respectively Consequently, the total phase shift of the open-loop system does not reach −180°, which results in the absence of a critical frequency in the open-loop Bode diagram Based on the Nyquist-Bode stability criterion, the closed-loop system remains stable irrespective of the amplitude ratio of H1(jω)H2(jω), indicating that the system possesses an infinite gain margin.

By applying the concepts of excess and shortage of passivity, we can extend our findings to general systems that may not be passive For instance, if system H1 is globally asymptotically stable (GAS) but lacks input feedback passivity (IFP), specifically IFP(−ν1) with ν1 > 0, introducing a feedforward of νI—where ν equals ν1 plus an arbitrarily small positive number ε—will transform H1 into a strictly input passive system This adjustment ensures that the feedback system becomes equivalent to the original system depicted in the previous figure.

The Passivity Theorem 35 states that when a positive feedback of νI is introduced to H2, the equilibrium point (x1, x2) = (0,0) of the closed-loop system is globally asymptotically stable (GAS) if H2 exhibits passive behavior with excessive output feedback passivity of ν Conversely, if H2 lacks sufficient output feedback passivity, it can be balanced by an excess of input feedforward passivity from H1, ensuring that the closed-loop system remains GAS.

Theorem 2.48 states that in the feedback interconnection depicted in Figure 2.7a, if system H1 is globally asymptotically stable (GAS) and input-to-state stable (IFP(ν)), while system H2 is zero-state detectable (ZSD) and output-to-state stable (OFP(ρ)), then the equilibrium point (x1, x2) = (0,0) is asymptotically stable (AS) provided that ν + ρ > 0 Furthermore, if the storage functions of both H1 and H2 are radially unbounded, the equilibrium point (x1, x2) = (0,0) is guaranteed to be GAS.

If the systems are characterized by a more general supply rate as in (2.86), the above condition can be further extended:

Theorem 2.49 establishes that if systems H1 and H2 are dissipative with respect to the specified supply rates, defined as w_i(u_i, y_i) = u^T_i y_i - ρ^T_i(y_i)y_i - ν_i^T(u_i)(u_i) for i = 1, 2, and both systems are zero-state detectable (ZSD) with continuously differentiable storage functions S1(x1) and S2(x2), then the equilibrium point (x1, x2) = (0,0) of the feedback interconnection depicted in Figure 2.7a is valid.

One special case of the supply rates is ν i (u i ) = ¯ν i u i and ρ i (y i ) = ¯ρ i y i , where ¯ν i and ¯ρ i are scalar constants In this case, ν 1 T (v)v+ρ T 2 (v)v= ¯ν 1 v T v+ ¯ρ 2 v T v= (¯ν 1 + ¯ρ 2 )v T v, (2.116) ν 2 T (v)v+ρ T 1 (v)v= ¯ν 2 v T v+ ¯ρ 1 v T v= (¯ν 2 + ¯ρ 1 )v T v (2.117) Then, the equilibrium (x 1 , x 2 ) = (0,0) of the feedback interconnection is

Another special case is ν 1 (v) =ρ 1 (v) = 0, v 2 (v) =νv andρ 2 (v) =ρv (2.118) This leads to the following stability condition:

Proposition 2.50 states that if system H1 is passive and dissipative with respect to the supply rate \( w_1 = u^T_1 y_1 \), and system H2 is dissipative concerning the supply rate \( w_2 = u^T_2 y_2 - \rho y^T_2 y_2 - \nu u^T_2 u_2 \), then under the assumption that both systems H1 and H2 are zero-state detectable (ZSD) and their storage functions \( S_1(x_1) \) and \( S_2(x_2) \) are continuously differentiable (C1), the equilibrium point \( (x_1, x_2) = (0,0) \) of the feedback interconnection depicted in Figure 2.7a will be asymptotically stable provided that \( \rho > 0 \) and \( \nu > 0 \).

This condition does not require systemH 1 to be AS.

Heat Exchanger Example

Certain process systems, like heat exchangers, are naturally passive after appropriate rescaling of inputs and outputs Heat exchangers facilitate efficient heat transfer between two fluids, which are kept separate by a solid wall to prevent mixing These devices are essential in various applications, including air conditioning, refrigeration, space heating, power generation, and nearly every chemical manufacturing facility.

A single tube-in-shell heat exchanger effectively utilizes cooling water to extract heat from a process stream, as illustrated in Figure 2.8 The volumetric flow rates for the hot process stream and the cold service stream are denoted as v_h and v_c, respectively The temperatures at the inlet and outlet for the hot stream are represented as T_hi and T_ho.

A tube-in-shell heat exchanger operates as a distributed parameter system, characterized by the temperatures of hot and cold streams in the tube varying with location, which can be described using partial differential equations To facilitate our discussion, we will utilize an approximate lumped parameter model as proposed by Hangos et al [54].

The model was built under the following assumptions:

1 Constant volume of the hot and cold streams in the heat exchanger (V h andV c );

2 Constant physicochemical properties, including density of the hot and cold streams (ρ h andρ c ) and their specific heat (c P h andc P c );

3 Constant heat transfer coefficientU and areaA;

Both hot and cold streams are thoroughly mixed within the tube, with the outlet temperatures approximated as T ho for the hot stream and T co for the cold stream.

The state equations of the heat exchanger can be developed based on energy balance [54]:

The process relies on the inlet temperatures and flow rates of both hot and cold streams as inputs, while the outlet temperatures serve as the outputs and states Various models can be developed based on the selection of manipulated variables.

In a linear model, controlling outlet temperatures by manipulating inlet temperatures can be achieved under the assumption that the flow rates of both cold and hot streams remain constant.

0 V v h h u(t), (2.121) y(t) =x(t), (2.122) wherex= [x 1 , x 2 ] T = [T co , T ho ] T andu= [u 1 , u 2 ] T = [T ci , T hi ] T Define the following constantsk 1 = c U A

P h ρ h V h , a 1 = v V c c anda 2 = v V h h Clearly, these constants are positive for any design and operating conditions Then, the state equation becomes ˙ x −a 1 −k 1 k 1 k 2 −a 2 −k 2 x+ a 1 0

To study the passivity of the above system, we define the following storage function:

Note that the coefficients k 1 , k 2 , a 1 , a 2 > 0 If the outputs are rescaled as y ∗ = [y 1 ∗ , y 2 ∗ ] T a 1 k 1 y 1 , a k 2

The inequality S˙(x) < u T y ∗ for all x = 0 indicates that the heat exchanger is inherently passive, independent of various design parameters such as overall heat transfer coefficient (U), volume flow rates (V c and V h), and surface area (A) Additionally, it remains unaffected by fluid properties like specific heat capacity (c P c and density ρ c) and operating conditions, including flow velocities (v c and v h) By applying the parameters specified in reference [65], the flow rates are determined to be v c = 2.29 × 10³ ft³/h and v h = 6.24 × 10³ ft³/h.

V c = 5.57 ft 3 , V h = 20.40 ft 3 , A = 521.5 ft 2 , c P h = 0.58 Btu/(lbãF), c P c = 0.56 Btu/(lbãF), U = 75 Btu/(hã ft 2 ãF), ρ h = 47.74 lb/ft 3 and ρ c = 44.93 lb/ft 3 In this case, (2.121) and (2.122) become ˙ x −690.87 279.17 69.254 −375.29 x+

It is easy to verify that the above process is passive, because matrices

The conditions outlined in equation (2.41) are satisfied by the parameters Q=W=0 (2.129) The IFP index plot for this process is illustrated in Figure 2.9a, while Figure 2.9b presents the corresponding phase plot, indicating that the phase shift remains within the range of (−90 ◦, 90 ◦) across all frequencies.

In this nonlinear model, we consider the flow rates of hot and cold streams as manipulated variables, represented as u = [u₁, u₂]ᵀ = [vₕ, v𝚌]ᵀ, while assuming constant inlet temperatures Tᶜᵢ and Tʰᵢ This leads to a nonlinear representation of the system To analyze the process's passivity concerning the equilibrium point x₀ = [x₁₀, x₂₀]ᵀ = [Tᶜ₀, Tʰ₀]ᵀ, we define deviation variables as x = [x₁, x₂]ᵀ = x - x₀ and u = [u₁, u₂]ᵀ = u - u₀, where u₀ = [vₕ₀, v𝚌₀]ᵀ It is important to note that these deviation variables may assume negative values, leading to the equation ˙x₁ = -k₁(x₁ + x₁₀) + k₁(x₂ + x₂₀).

(2.130)Assume that (x 0 , u 0 ) is at steady state,i.e.,

Fig 2.9.Linear heat exchanger model

Also noteu 1 =u 1 +u 10 ≥0 andu 2 =u 2 +u 20 ≥0 becauseu 1 andu 2 are physical flow rates Then,

Therefore, the process is passive with respect to the equilibriumx = [0,0] T

It is interesting to point out that

The heat exchanger operates passively, similar to the linear case, as the passivity condition remains applicable across various design parameters, fluid types, and operating conditions, including different inlet and outlet temperatures.

2 The system is passive with respect to any physical equilibrium point [x 10 , x 20 ] T because (2.135) holds for anyx 0

3 The equilibrium pointx 0 is GS but not GAS Ifx 1 =x 2 = 0, the unforced system does not converge tox = 0.

Output rescaling serves as a form of sensor calibration, as the temperature of the inlet cold stream (T ci) can never exceed that of the outlet (T co) Consequently, the rescaling coefficient for y ∗ 1 is always non-positive Moreover, an increase in the flow rate of the inlet cold stream results in decreased outlet temperatures (T co).

T ho ) This implies that the direction of x movement has to be reversed to obtain a minimum phase condition.

Summary

The system described is a Zero State Descriptor (ZSD), allowing for straightforward control of the heat exchanger due to its simple rescaling of states According to Proposition 2.50, any output feedback controller that is dissipative in relation to a specific supply rate will asymptotically stabilize the equilibrium point at x = [0,0] T Notably, a proportional-only controller, defined as u = -ky ∗ for any k > 0, serves as a particular example of this stabilization method.

This chapter introduces the fundamental concepts of dissipative and passive systems, focusing on the input-output properties of passive systems that lead to essential stability conditions for interconnected systems While stability conditions derived from passivity may appear conservative compared to those from dissipativity, this conservativeness is mitigated by the introduction of Input Flow Properties (IFP) and Output Flow Properties (OFP) Dissipative systems, when analyzed with varying supply rates, can be represented as passive systems with specific IFP and OFP Additionally, the excess and shortage of IFP and OFP are utilized to characterize processes in terms of their passivity Subsequent chapters will explore passivity-based system analysis and control design for linear processes, offering numerical implementation strategies for routine process control practices.

Chapter 2 discusses input feedforward passivity (IFP) as a phase-related property, allowing for the characterization of uncertainties through their IFP This approach enables the utilization of both phase and gain information, potentially resulting in a less conservative control design The chapter presents robust control design methods grounded in the passivity uncertainty bound, supplemented by case studies and illustrative examples These advancements focus on linear systems and provide systematic approaches that can be effectively implemented in process control practices.

Introduction

Uncertainties

Uncertainties can be divided into two main types: parametric uncertainty and unstructured uncertainty Parametric uncertainty occurs when the model's structure is established, yet specific parameters remain unknown This situation can be represented by an actual but unidentified linear plant.

G t (s) := (A t , B t , C t , D t ), (3.1) where G t (s) is the transfer function and (A t , B t , C t , D t ) is its state-space representation, and the nominal plant (the model) as

The parametric uncertainty represents parametric variations in plant dy- namics, e.g., the uncertainties in certain entries of the state-space matrices

Unstructured uncertainty reflects a limited understanding of a process, often due to model errors that can arise from inaccurate or missing dynamics This type of uncertainty is typically represented through specific loop gains, poles, and zeros of the plant transfer function, highlighting the challenges in accurately modeling complex systems.

• Additive Uncertainty.The simplest way to express the difference between the model and the true system is additive representation (as shown in Figure 3.1):

G t (s) =G(s) +∆ A (s), (3.3) where the model uncertainty is given by

Fig 3.2.Model with multiplicative uncertainty

• Multiplicative Uncertainty.The model uncertainty may also be represented in the multiplicative form (as shown in Figure 3.2):

G t (s) = [I+∆ M (s)]G(s), (3.5) so that∆ M (s) is the modelling error relative to the nominal model, where,

Uncertainty systems, while often unknown, can frequently be bounded through estimation methods In H∞ control, the maximum potential magnitudes of these uncertainties help gauge their overall impact Represented as dynamic systems, uncertainties such as ∆A(s) and ∆M(s) can be analyzed across various frequencies using frequency responses, for instance, ∆A(jω) = Gt(jω) - G(jω) The magnitudes of these uncertainties are quantified using maximum singular values, denoted as ¯σ(∆A(jω)) This frequency-dependent characterization is particularly valuable, as model uncertainties typically remain small at low frequencies but tend to escalate at higher frequencies Consequently, the uncertainty regions are generally assumed to be disc-shaped, with the condition ¯σ[∆A(jω)] < γ(ω) applied to different frequencies.

A weighting function can be utilized to ensure that the uncertainty remains below 1 in magnitude across all frequencies As illustrated in Figures 3.1 and 3.2, stable and rational transfer functions, denoted as W A (s) and W M (s), can effectively represent the uncertainties ∆ A (s) and ∆ M (s).

46 3 Passivity-based Robust Control where ¯σ(∆(jω)) 0), a controller can be designed to ensure that the closed-loop system M(s) remains stable and strictly input passive, thereby achieving robust stability Conversely, if the uncertainty exhibits a negative IFP (ν < 0), the closed-loop system M(s) must possess excessive output feedforward passivity (OFP) (ρ > −ν) to maintain robustness, indicating that system M(s) has limited gain.

The frequency-dependent IFP index, as outlined in equation (2.90), effectively characterizes uncertainty across various frequencies in a dynamic input feedforward system This index integrates multiple factors, providing a comprehensive measure of uncertainty.

Fig 3.4.Uncertainty regions gain and phase information on the uncertainty This can be seen by a sin- gle input single output (SISO) example For a SISO uncertainty system

The IFP index of ∆(s) is defined as the real part of its frequency response, represented mathematically as ν F (∆(s), ω) = Re (∆(jω)) This expression can be further detailed by decomposing ∆(jω) into its gain and phase components, where ∆(jω) = r(ω) [cosθ(ω) + jsinθ(ω)], with r(ω) indicating the gain and θ(ω) denoting the phase of the uncertainty Consequently, the IFP index can be simplified to ν F (∆(s), ω) = r(ω) cosθ(ω).

Because we are concerned only with the shortage of IFP of the uncertainty, the following uncertainty bound can be defined.

Definition 3.1 (Passivity-based uncertainty measure) The passivity- based uncertainty measure is defined as ν − (∆(s), ω)−ν F (∆(s), ω)

(3.18) whereλdenotes the minimum eigenvalue.

The uncertainty measure discussed reflects the maximum potential shortage of input passivity To better understand this passivity-based uncertainty measure, we can examine a specific set of uncertain plants.

Example 3.2.Assume thatΠ is the set of uncertain plants generated by vary- ing the parameters of the following transfer function:

Fig 3.5.Passivity-based uncertainty bound

The nominal plant model is

The Nyquist regions for G t (s) ∈ Π are generated as shown in Figure 3.4.

In particular, the frequency response of the additive uncertainty ∆ A (jω) G t (jω)−G(jω) atω= 0.2 rad/s is shown in Figure 3.5.

The passivity-based uncertainty bound, ν − (∆(s), ω), is consistently less than or equal to the gain bound, indicating a potential for less conservative robust control In instances like Example 3.2, where the frequency response of the nominal model is not centered in the Nyquist region, the IFP index bound can be considerably smaller than the gain bound This discrepancy highlights the advantages of using passivity-based methods in robust control design.

For more general multivariable uncertainties, the passivity-based uncer- tainty bound has the following property:

Property 3.3 (IFP Index) For any stable multivariable linear system∆(s),

2 If ∆(s) = ∆ p (s) +∆ np (s), where ∆ p is passive and ∆ np is non-passive then, ν − (∆(s), ω)≤σ¯(∆ np (jω)).

Based on the above frequency-dependent passivity-based uncertainty mea- sure, the following robust stability condition can be obtained:

Proposition 3.4 Consider the feedback system shown in Figure 3.3, where

M(s) and ∆(s) are linear time-invariant (LTI) systems, with the assumption that the uncertainty ∆(s) remains stable, satisfying the condition ν − (∆(s), ω) ≤ ν F (W(s), ω) Here, W(s) represents a stable and minimum phase transfer function The stability of the closed-loop system is guaranteed under these conditions.

M (s) =M(s)[I−W(s)M(s)] −1 (3.25) is stable and strictly input feedforward passive.

In linear systems, the term "strictly passive systems" is synonymous with stable and strictly input passive systems The condition that ∆(s) + W(s) is positive real is evident, as it satisfies the inequality ν − (∆(s), ω) < ν F (W(s), ω) To analyze this, we perform loop shifting as illustrated in Figure 3.6 According to Proposition 2.47, for M(s) to maintain closed-loop stability with positive feedback from W(s), it must be strictly passive This establishes a generalized excessive output feedback (OFP) condition on M(s), where the feedback operates as a dynamic system.

Uncertainty Bounds Based on Simultaneous IFP and

Proposition 3.4 outlines the robust stability conditions applicable to any stable system, even when faced with uncertainties that are limited solely by the passivity-based IFP measure, which can include uncertainties with significantly high gain However, these conditions may impose strict requirements on the overall system.

• M(s) must have excessive OFP and must be minimum phase For example, if∆(s) is a multiplicative uncertainty, then,

If the processG(s) is not minimum phase, then it is not possible to find a controllerK(s) such thatM(s) is minimum phase.

In robust control design, it is essential to identify a controller K(s) that ensures the closed-loop system M(s) is strictly passive or strictly positive real Achieving an explicit solution can be challenging, particularly when M(s) has a relative degree of 1, which may cause numerical issues in applying the positive-real lemma A practical solution is to aim for M(s) to be extended strictly positive real (ESPR) However, if G(s) is strictly proper, it is not possible for a proper control K(s) to make M(s) ESPR.

To remove the above restrictions, we can characterize the uncertainty by using both IFP and OFP indices If the uncertainty ∆(s) is stable and gain

In the scenario where both the input feedforward (IFP) and output feedback (OFP) are simultaneously bounded, there exists a positive constant ρ that allows the function ∆(s) to remain passive with negative feedforward νI and positive feedback of ρI Due to the excessive OFP in ∆(s), as outlined in Theorem 2.49, the system M(s) is only required to meet the IFP condition with νM greater than -ρ, while neither M(s) nor M(s) needs to satisfy the exponential stability property (ESPR) This simultaneous bounding of IFP and OFP can be effectively represented through the concept of a system sector.

Definition 3.5 (Sector of a system [102]) A stable linear systemT(s)is said to be inside the sector[a, b], whereaand bare real numbers with a < b,

≥0, ∀s=jω (3.27) andT(s)is said to be outside the sector[a, b]if

The system sector is related to both the passivity bound and the gain bound [44] For any stable systemT(s),

2 T(s) is strictly passive ⇐⇒ T(s) is inside [δ, +∞ −ζ] (where δ andζ are arbitrarily small positive real numbers).

4 ||T(s)|| ∞ ≤γ ⇐⇒T(s) is confined to a symmetrical sector [−γ,+γ].

5 T(s) with simultaneous negative feedforwardνI and positive feedback of ρI is passive⇐⇒ T(s) is inside [ν, 1/ρ].

Property 3.6 (Properties of Sectors [147]).If a linear systemT is inside sector [a, b] and a linear systemT 1 is inside [a 1 , b 1 ], then,

1 SystemkT is inside [ka, kb] (wherekis a constant).

Theorem 3.7 (Sector stability theorem [147]) Consider linear systems

In a negative output feedback connection involving systems G1 and G2, where both systems are restricted to specific sectors, the stability of the closed-loop system is contingent upon certain conditions Let ζ and δ be constants, with one being strictly positive and the other being zero If G2 is constrained within the sector [a+ζ, b−ζ] (where b > 0), then G1 must meet one of the specified criteria for the closed-loop system to maintain stability.

The above condition is fundamentally equivalent to the stability con- dition based on simultaneous IFP and OFP (as given in Theorem 2.49).

Both the Small Gain Theorem and the Passivity Theorem are specific instances of sector stability, demonstrating that a stable system G(s) can be restricted within various sectors, such as [ν,+∞] or [−γ, γ] Here, ν and γ are defined as ν = 1/2 min ω ∈R λ(G(jω) + G ∗ (jω)) and γ = max ω ∈R σ¯(G(jω)), respectively The correlation between lower and upper sector bounds indicates that a decrease in the lower bound results in a corresponding decrease in the upper bound For a system with uncertainty ∆(s) confined within the sector [a, b] (where b |a| ≥ 0 ≥ a), closed-loop stability can be maintained if T(s) remains within the sector of [−1/b + σ, −1/a − ζ], where σ and ζ are any small positive constants This passivity-based uncertainty measure encompasses both the lower sector bound, associated with Input-to-State Stability (IFP), and the upper sector bound, related to Output-to-State Stability (OFP).

∆(s), preferably frequency-dependent so that system robustness and perfor- mance can be optimized at different frequencies However, implementation of both frequency-dependent lower and upper bounds is very complicated.

A simplified approach is to characterize the uncertainty using a frequency- dependent lower sector bound with a constant upper bound This is equivalent to a frequency-dependent IFP index with a constant OFP index:

Definition 3.8 (Sector-bounded passivity uncertainty measure[10]).

Given a stable uncertainty system ∆(s), for a constant b |ν(∆(s))|, the sector-bounded passivity uncertainty measure is defined as ν S − (∆(s), ω, b)−sup

Fig 3.8 System confined in different sectors

The sector-bounded passivity index represents the minimum feedforward needed to ensure that the uncertainty associated with positive feedback of 1/b remains passive This index can be estimated by analyzing specific conditions related to system behavior and stability.

Theorem 3.9 ([8]) For a given stable system T(s) and the upper sector bound b, the sector-bounded passivity index is bounded by the following in- equality: ν S − (T(s), ω, b)≤ 2¯σ 2 (T(jω))−bλ(T(jω) +T ∗ (jω))

Since N is a Hermitian matrix, the necessary and sufficient condition for N to be positive semidefinite is that its eigenvalues are all nonnegative, i.e., λ(N) ≥ 0 Denote λ m (ω) = λ(T(jω) +T ∗ (jω)) Because T(jω) +T ∗ (jω),

−2T ∗ (jω)T(jω)/band 2aIare all Hermitian matrices, from the Weyl inequal- ity [4], λ(N)≥λ m (ω) +a bλ m (ω)−2a−2¯σ 2 (T(jω)) b

Therefore, matrixN is positive semidefinite From Definition 3.5,T(jω) is in the sector [a, b] This proves (3.31).

The sector-bounded passivity measure has the following properties:

Property 3.10 (Properties of sector-bounded passivity index [10]).

1 This property can be concluded directly from Property 3.6.

2 Denote λ(ω) = λ(T(jω) +T ∗ (jω)) Since b ≥ ||T(s)|| ∞ and b≥ 1 2 λ(ω) for anyω∈R, from Theorem 3.9, ν S − (T(s), ω, b)≤ 2¯σ 2 (T(jω))−bλ(ω)

(3.36) For anyω, ¯ σ(T(jω))≥0, σ¯(T(jω))≥λ(ω)/2, b≥σ max (T(jω)) (3.37) This leads to

BecauseT(s) is diagonal, so isN(ω) = diag{n i (ω)} (i= 1, , m), where n i (ω) = Re

(3.41) The sector-bounded passivity indexν S − (T(s), ω, b) is the minimum value such that N(ω) is positive semidefinite for all ω, that is,n i (ω)≥0,∀ω, i= 1, , m Define n ∗ i (ω) = Re

, i= 1, , m (3.42) where a i = ν S − (t i (s), ω, b) is the minimum value such that n ∗ i (ω) ≥ 0; then max i a i = max i ν S − (t i (s), ω, b) (3.43) is the minimum value such that all the inequalities n i (ω)≥0 hold for all i= 1, , m Therefore, (3.35) holds.

Similar to the IFP index ν − , the sector-based passivity index also com- prises both the phase and gain information of the uncertainty, although it is less obvious.

Passivity-based Robust Control Framework

Robust Stability Condition

The robust stability condition on the basis of passivity can be described as follows:

Theorem 3.11 (Passivity-based robust stability condition) Consider an interconnected system (as shown in Figure 3.6) comprised of M(s)and a stable uncertainty system∆(s) Given a stable and minimum phase weighting function W(s)whose sector-bounded passivity index ν S − (W(s), ω, b w )≤ −ν S − (∆(s), ω, b) (3.44) the closed-loop system will be stable if

I (3.45) is strictly passive or ESPR, where M (s)is defined in (3.25).

Assuming ν S − (∆, ω, b) = α(ω), where α(ω) is a nonnegative real function of frequency, the systems ∆(s) and W(s) are restricted within the sectors [−α(ω), b] and [α(ω), b w], respectively According to Property 3.6, the combined system ∆(s) + W(s) lies within the sector [0, b + b w] Furthermore, Theorem 3.7 establishes that for the closed-loop system of M − ∆ to maintain stability, System M (s) must be positioned within the sector of − b + 1 b w + δ to +∞ − δ.

(whereδ is an arbitrarily small positive real number). Again from Property 3.6,M (s) + b 1

+ b w Ineeds to be confined to [δ,+∞ −δ].

The PBRC problem involves determining a controller for a plant model characterized by uncertainty, constrained by its sector-bounded passivity index, to ensure stability as outlined in Theorem 3.11 This theorem simplifies to Proposition 3.4 when the upper bound \( b \) approaches infinity Typically, \( b \) is selected significantly larger than the system's infinity norm, positioning Theorem 3.11 as a weak passivity stability condition While the sector-bounded passivity index closely aligns with the IFP passivity index when a large upper bound is utilized, the limitation on the relative degree of \( M(s) \) is eliminated as long as \( b \) remains finite.

Robust Stability and Nominal Performance

In H ∞ control, there is a competition between robustness and nominal performance, a principle that also applies to passivity-based robust control The robust stability condition outlined earlier indicates that gain constraints must be imposed on the closed-loop system as perceived by the uncertainty.

Proposition 3.12 The robust stability condition in Theorem 3.11 implies that for all frequencies, ¯ σ M(jω)−1

Proof Denote a(ω) =ν S − (∆(s), ω, b) Define a complex function

+ a ( ω ) >0, the above condition implies for allω∈Rthat λ [I−a(ω)M ∗ (jω)]

>0 (3.49) The above inequality can be rewritten as follows: λ¯ 2a(ω)M ∗ (jω)M(jω) +a(ω) b M(jω)

0, (4.13) but may encounter numerical problems when the minimum eigenvalue of

To address the numerical challenges in the SDP approach, we can transform the positive real condition into a gain condition utilizing the Cayley transformation This process involves analyzing the structured singular value of the modified steady-state gain matrix Additionally, a diagonal sign matrix is defined to facilitate this transformation.

The modified gain matrix G+(0) is derived from the original gain matrix G(0) by applying a sign function to its elements, resulting in a matrix V where each element is +1 for positive values and -1 for negative values This transformation ensures that all diagonal elements of G+(0) are positive Consequently, the decision variable is adjusted to D+ = DV > 0, aligning with the findings of Theorem 4.4, which pertains to the stability of the system represented by G(s).

DIC ifG + (0) is nonsingular and a real and positive definite diagonal matrix

Dcan be found such that,

Following a proof similar to [26], the following proposition can be proved:

Proposition 4.5 DefineH = [I−G + (0)] [I+G + (0)] −1 and F= (D + ) − 1 2 Inequality 4.14 holds (thusG(s)is DIC) if and only if ¯ σ&

≤1, (4.15) whereσ¯{ã} denotes the maximum singular value.

The expression F HF −1 ' serves as the upper limit for the diagonally scaled structured singular value of H Consequently, Proposition 4.5 offers a computational approach to verify the DIC condition Additionally, methods for calculating structured singular values can be employed to evaluate (4.15).

In MATLAB Robust Control Toolbox, this can be done by using the func- tionPSV, which calculates the maximum diagonally scaled structured singular value via the Perron eigenvector approach.

Example 4.6 ([26]).Consider a system with the following transfer function:

⎦, (4.16) with steady-state gain matrix given by

Here we test the DIC conditions mentioned in this section:

1 RGA-based necessary condition (Theorem 4.3): The RGA matrix ofG(0) is Λ(G(0)) ⎡

Because all diagonal elements are positive, this process may be DIC.

2 Small gain-based sufficient condition (Theorem 4.2):

Because ¯à(E d (0)) = 2>1, the small gain-based condition is not satisfied.

3 The strictly positive real condition in (4.13): G(0) has two imaginary eigenvalues and thus there does not exist a matrixDthat satisfies (4.13).

4 Passivity-based DIC condition (Theorem 4.4): Using the Cayley transfor- mation,

It can be found that ¯à(H) = 1 One of the possible values of the diagonal scaling matrix is

⎦, such that (4.14) is satisfied This leads to the conclusion that this process is DIC.

The above process was found to be DIC [26] This example shows that the passivity-based DIC condition (Theorem 4.4) is less conservative than the small gain-based condition.

DIC Analysis for Nonlinear Processes

DIC for Nonlinear Systems

The DIC property of nonlinear systems, akin to its linear counterpart, assesses the ability of multiloop linear or nonlinear controllers with integral action to stabilize a nonlinear multivariable process, ensuring zero steady-state error It also evaluates the stability of the closed-loop system when any subset of loops is detuned This analysis focuses exclusively on "square" processes, where the number of inputs equals the number of outputs The discussion involves a nonlinear model characterized by an input vector \( u \in \mathbb{R}^m \), an output vector \( y \in \mathbb{R}^m \), and a state vector \( x \in \mathbb{R}^n \).

The control system configuration mirrors that of Figure 4.1, but both the controller and process are nonlinear The decentralized nonlinear controller, which incorporates integral action, consists of three main components: a stable diagonal nonlinear controller, a gain matrix represented as K g = diag{k i } (where k i > 0 for i = 1, 2, , m), and an integrator Unlike linear Decentralized Integral Control (DIC) analysis, the stability of nonlinear systems must be evaluated at a specific equilibrium point It is also assumed that the state x(t) is solely determined by its initial value x(0) and the input function u(t).

Another assumption made for convenience is that the system (4.22) has equilibrium at the origin, that is f(0,0) = 0 and g(0,0) = 0 (4.23)

If the equilibriumx e is not at the origin, a translation is needed by redefining the statexasx−x e The nonlinear DIC is defined as follows:

Definition 4.7 (Decentralized integral controllability for nonlinear processes [119]).

Consider the closed-loop system shown in Figure 4.1 For the nonlinear process

1 If there exists a decentralized integral controller C, such that the unforced closed-loop system (r = 0) is globally asymptotically stable (GAS) for the equilibrium x = 0 and such that the globally asymptotic stability is maintained if each individual loop of controllerCis detuned independently by a factor ε i (0 ≤ε i ≤ 1, i= 1,ã ã ã, m), then the nonlinear process G is said to be decentralized integral controllable(DIC) for the equilibrium x= 0.

2 If the closed-loop system is asymptotically stable (AS) near the region of the equilibrium x= 0, then the nonlinear process G is said to be locally decentralized integral controllable (LDIC) around the equilibrium x= 0.

The process G must operate as GAS around the equilibrium point x=0 to qualify as DIC System N is typically nonlinear; however, if a linear diagonal system N can be identified that achieves closed-loop stability as defined in DIC, the process is considered DIC with a linear controller This is advantageous because linear control systems generally require less design, implementation, and maintenance effort compared to their nonlinear counterparts The concept of local DIC focuses on the area surrounding the equilibrium point, which can often be evaluated by applying the linear DIC condition on linearized models near that specific equilibrium point.

Sufficient DIC Condition for Nonlinear Processes

The sufficient DIC condition for nonlinear systems was established in prior research, indicating that, like linear processes, the DIC property of nonlinear processes can be evaluated through their steady-state input-output relationship To define the diagonal matrix K gε = diag{ε i k i }, where ε i represents the detuning factor for the ith loop, we also introduce K gε = ηK gε with η being a small positive scalar The diagonal matrix K g = diag{k i } is configured such that k i > 0 for all i from 1 to m This sign adjustment is incorporated into the diagonal controller N The state equation of the generalized process P, which represents the serial connection of process G and the diagonal controller N, is modeled accordingly.

The state equation for the linear integral controller is expressed as

The sufficient DIC condition for nonlinear processes can be presented as fol- lows:

Theorem 4.8 (DIC conditions for nonlinear processes [119]) Con- sider the closed-loop system in Figure 4.1 Assume that the generalized process

The controller's linear component C l and the process G are defined by equations (4.24) and (4.25) The nonlinear process G is considered to be Decentralized Input-Output Controllable (DIC) at the equilibrium point x = x e, provided that a decentralized controller N exists, ensuring that the generalized plant P meets specific criteria.

1 The equation 0 =f(x, u 1 ) obtained by setting η = 0 in (4.24) implicitly defines a uniqueC 2 functionx=h(u 1 ) for u 1 ∈U 1 ⊂R m

2 For any fixed u 1 ∈ U 1 ⊂R m , the equilibrium x e = h(u 1 ) of the system ˙ x=f(x, u 1 )is GAS and locally exponentially stable (LES) (as defined in Definition 2.2).

3 The steady-state input output functiong(h(u 1 ), u 1 )of the generalized pro- cess P satisfies the following conditions: u T 1 g(h(u 1 ), u 1 )>0 (4.26)

(for some scalarρ >0) for u 1 in a neighbourhood of u 1 = 0.

The theorem's proof, originally published in [119] and detailed in Section B.3, utilizes the Singular Perturbation Theorem [70] to ensure rigor Notably, the critical conditions outlined in (4.26) and (4.27) represent a strictly input passivity condition concerning the steady-state input-output function g(h(u₁), u₁).

The conditions associated with Theorem 4.8 are all for the general process

P (which is the serial connection of process G and the diagonal controller

N) rather than the nonlinear plant G To verify the above DIC conditions, a diagonal nonlinear controllerN needs to be constructed Searching for a linear diagonal systemN is often attempted first because it is much easier to develop than a nonlinear system In this case, the overall controller will be linear.

If we denoteG 0 as the steady-state gain matrix of thelinearised model of

Garound the equilibrium, and simply chooseN as a real diagonal matrixD, then (4.26) and (4.27) are reduced to the existence of a real diagonal matrix

G 0 D+DG T 0 >0, (4.28) which is a sufficient condition for local DIC This is the linear DIC condition given in Theorem 4.4 If there does not exist a D matrix such that (4.28)

If the 100 4 Passivity-based Decentralized Control is satisfied, the global conditions outlined in Theorem 4.8 may not be applicable Since the local Decentralized Input-to-State Stability (DIC) condition is generally simpler to verify, it can serve as a necessary condition for Theorem 4.8 However, it is important to note that this does not imply it is a necessary condition for DIC, as Theorem 4.8 is itself a sufficient condition for DIC.

The necessary conditions for linear processes, as outlined in Theorem 4.3, serve as essential criteria for nonlinear Deterministic Input-Output Control (DIC) These conditions are particularly beneficial in nonlinear scenarios, as testing Theorem 4.8 can be challenging A fundamental requirement for process P to be DIC, both locally and globally around the equilibrium, can be derived from Theorem 8 in reference [26] Furthermore, through singular perturbation analysis, it has been established that the linear DIC conditions are indeed necessary for local nonlinear DIC.

Computational Method for Nonlinear DIC Analysis

Testing the DIC conditions of Theorem 4.8 analytically can be challenging or even unfeasible, similar to other nonlinear analyses For physical processes, DIC analysis should focus on the specific operating region rather than a theoretical global space This section outlines a computational method to evaluate the DIC for a given nonlinear process G: u→y, within the operating region defined by u∈U ⊂R m and y∈Y ⊂R m, in relation to the steady-state equilibrium point (u e , y e ).

Procedure 4.9 (Numerical analysis of nonlinear DIC [119])

1 Check the stability (GAS and LES) of the nonlinear processG (Con- dition 2 of Theorem 4.8) If process P is unstable, it is not DIC.

2 Linearise the nonlinear process G around the equilibrium point, and check whether the steady-state gain matrixG(0)of the linearised model satisfies the necessary conditions for linear systems (e.g., the necessary conditions in [26, 76, 145]) If any of these necessary conditions are not satisfied, then processGis not DIC Otherwise, proceed to the next step.

3 Test the sufficient condition for linear DIC in (4.11) using the lin- earised model around the equilibrium point, as described in Section 4.2.2.

If this sufficient DIC condition is not satisfied, DIC of the nonlinear processG is not conclusive based on Theorem 4.8 Otherwise, proceed to the next step.

4 Check whether Condition 1 of Theorem 4.8 is satisfied by the nonlinear model (4.24) of the generalized process P : u 1 → y (u 1 ∈U 1 ⊂R m , y ∈ Y ⊂ R m ), where P is process G in series connection with the diagonal system N U 1 is the region of u 1 corresponding to u ∈ U andu 1 e is the steady-state equilibrium point corresponding tou=u e

An easy and often effective choice of system N is the real diagonal constant matrix D obtained in Step 3 Solve the steady-state equation and find a unique functionx=h(u 1 )from the equation 0 =f(x, u 1 ).

Check whether x =h(u 1 ) ∈C 2 for any u 1 ∈ U 1 If not, DIC is not conclusive Otherwise, proceed to the next step.

5 Redefine the input∆uasu 1 −u 1 e and output∆yasy−y e such that the steady-state input output function ∆y=g(h(∆u), ∆u) is unbiased in the sense that0 =g(h(0),0) Then, check Condition 3 of Theorem 4.8 for positiveness of the inner product∆y T ∆u=∆u T g(h(∆u), ∆u)nu- merically in the region of interest The condition (to ensure LES) of

The inequality ∆u T g(h(∆u), ∆u)≥ρ|∆u|² (for some scalar ρ > 0) ensures stability for ∆u in the vicinity of ∆u = 0 when D is utilized as system N Additionally, if the inner product ∆y T ∆u meets Condition 3 within the relevant area, then process G qualifies as DIC.

Nonlinear DIC Analysis for a Dual Tank System

In this article, we demonstrate the application of the DIC analysis procedure through a dual tank level control problem, as referenced in [119] The example involves water flowing into two tanks, regulated by Pump 1 and Pump 2, as depicted in Figure 4.2.

In a system with two tanks, the flow rates are designated as f1 and f2, while the outlet flow rates are represented as fo1 and fo2 for tank 1 and tank 2, respectively The liquid levels in the tanks are indicated by h1 and h2, with the assumption that h1 is greater than h2 Consequently, the flow rate from tank 1 to tank 2 is denoted as f12.

The liquid levels in tanks 1 and 2 can be described by the following equa- tions: dh 1 dt = 1

(4.29) whereA = 1 m 2 , k 1 = 0.26 m − 1 2/min and k 2 = 0.13 m − 1 2/min If we select the flow ratesf 1 and f 2 as control inputs and the heights of the liquid level

Fig 4.2.Schematic diagram of a dual tank level control process

102 4 Passivity-based Decentralized Control h 1 andh 2 as control outputs, then the process model in (4.29) can be written as ˙ x 1 = 1

The process model described in (4.30) is identified as Linear Exponential Stable (LES) based on its linearized representation around the equilibrium point, which is also Globally Asymptotically Stable (GAS) within the specified region (h1 > 0, h2 > 0, and h1 - h2 > 0) The Discrete Input-Output (DIC) property of this process at the equilibrium point xe = [h1e, h2e]T = [8.8, 5.8]T m is examined, confirming that the condition (4.11) holds true with D = I, indicating that the linearized system is DIC Furthermore, the conditions outlined in Theorem 4.8 for the nonlinear model in (4.30) are evaluated with D = I.

By redefining the state, input and output variables as ˜ x=x−x e = [x 1 −8.8, x 2 −5.8] T m,

(4.33) the steady-state input output mapping can be found as follows:

Condition 3 in Theorem 4.8 is reduced to ∆y T ∆u > 0, which can be verified numerically using discrete points in the region of interest A three- dimensional plot is given in Figure 4.3, from which it can be seen that the DIC conditions in Theorem 4.8 are satisfied in the following region:

∆u 2 ∈[−0.2, 0.2] m 3 /min (f 2 ∈[0.2, 0.6] m 3 /min) (4.35)Therefore, this nonlinear process is DIC in the above input space. Ŧ0.4 Ŧ0.2

Block Decentralized Integral Controllability

Conditions for BDIC

Sufficient condition based on the Passivity Theorem

Since the Passivity Theorem is applicable to both decentralized and multi- variable systems, the passivity-based DIC condition given in Theorem 4.4 can be extended to BDIC as follows:

Theorem 4.12 (Sufficient condition for BDIC[151]) A multivariable stable process with a transfer functionG(s)∈C m × m is block decentralized in- tegral controllable with respect to the prespecified controller structure in (4.38) if a nonsingular real block diagonal matrix,

W = diag{W 1 , , W i , , W k } ∈R m × m , (4.41) whereW i ∈R m i × m i ,can be found such that

Proof Any nonsingular square block diagonal matrixW given above can be factorized into two nonsingular, square matrices M and N with the same block diagonal structure ofW,i.e.,

N i ∈R m i × m i Thus (4.42) can be written as

As shown in Figure 4.5, define a virtual processG (s) =N T G(s)M It is easy to see thatG (0) is nonsingular Inequality 4.46 indicates thatG (0) satisfies the following inequality:

G (0)I m +I m G T (0)≥0, (4.47) whereI m is anm×midentity matrix From Theorem 4.4, the virtual process

G (s) is DIC As a result, there exists a multiloop controller,

C d (s) = 1 sdiag{K d 1 (s), , K dj (s), , K dm (s)}, (4.48) which stabilizes G (s) and maintains closed-loop stability when each diago- nal subcontroller is independently detuned by an arbitrary factor of ξ dj ∈

[0,1], ∀ j = 1, , m By choosing the detuning factors (ξ i for the ith con- troller block,i= 1, , k), it can be seen that processG (s) is also stabilized by

(4.50) Therefore, the real controller “seen” by the original processG(s) is

C(s) =M C d (s)N T = 1 sdiag{K 1 (s), , K i (s), , K k (s)}, (4.51) which can be detuned to

C(s) =ˆ ΞC(s) = 1 sdiag{ξ 1 K 1 (s), , ξ i K i (s), , ξ k K k (s)}, (4.52) without causing closed-loop instability Therefore, processG(s) is BDIC.

For the specified process G(s), the equation (4.42) transforms into a Linear Matrix Inequality (LMI) involving the decision variable matrix W, exhibiting a distinct block diagonal structure This LMI can be expressed in a manner akin to equation (4.12) The solution to this LMI problem can be efficiently obtained using MATLAB.

Obviously, the following necessary conditions for DCLI are also necessary conditions for BDIC:

Theorem 4.13 (Necessary conditions for DCLI [31]) Given a stable multivariable process with a transfer functionG(s), which can be partitioned intok×k block subsystems,G(s)possesses decentralized closed-loop integrity only if

1 The following block relative gain (BRG) condition is satisfied: det [Λ i (G(0))]>0,∀i= 1, , k, (4.53) whereΛ i (G(0))is the BRG ofG ii (s)[81].

2 The Niederlinski Index (NI) [88] is positive, i.e.,

Pairing Based on BDIC

BDIC represents a condition that is less stringent than DIC yet more stringent than DCLI The conditions outlined in Theorem 4.13 serve as a preliminary filter to eliminate unworkable pairings prior to assessing the sufficiency conditions presented in Theorem 4.12.

In a given process, multiple pairing schemes can meet the BDIC conditions, making it essential to select the optimal scheme based on additional criteria A key factor in this selection is resilience, which reflects the effectiveness of regulatory and servo behaviors achieved through feedback control This resilience can be quantified by assessing the minimum singular values of each subsystem of G b (s) at steady state or across a specific frequency band Higher minimum singular values indicate greater resilience in the subsystems, as they enable the controller to manage larger disturbances within the constraints of manipulated variables.

BDIC Analysis of the SFE Process

In this section, we apply the BDIC conditions presented in Section 4.4.1 to analyse the control schemes for a supercritical fluid extraction (SFE) process

The SFE process, illustrated in Figure 4.6, comprises three main physical units: the extractor, the stripper combined with the reboiler, and the trim-cooler The significant interconnection between the controlled variables and the differing response times across these units has made effective control of the SFE process challenging.

In this process, the manipulated variables are u 1 = solvent flow rate (extractor), u 2 = reflux flow rate (stripper), u 3 = boilup rate (stripper), and u 4 = shell-tube temperature (trim-cooler).

The controlled variables are y 1 = raffinate composition (extractor), y 2 = overhead composition (stripper), y 3 = bottoms composition (stripper), and y 4 = solvent temperature (trim-cooler).

Flashing to lower pressure Trim cooler C

Fig 4.6.Supercritical fluid extraction process

The linearized model of the SFE process, developed by Samyudia et al., is utilized for the BDIC analysis under steady-state operating conditions This model is expressed mathematically as ˙x = Ax + Bu and y = Cx, where u represents the input vector [u1, u2, u3, u4]ᵀ and y denotes the output vector [y1, y2, y3, y4]ᵀ For more details on the model and the nominal steady-state conditions, please refer to source [104].

The analysis of various control strategies for the SFE process, based on the proposed BDIC conditions, indicates that the control modes DAU1, DAU2, DAU3, DAU5, and DAU6 are BDIC and represent promising control schemes Conversely, modes SA, ESA, and DAU4 do not meet BDIC criteria and are therefore less favorable This finding aligns with conclusions reached by other researchers, who conducted complex performance analyses through dynamic simulations.

Modes SA and ESA, which reflect the control structures related to the physical decomposition of the plant, prove to be unsuitable due to the significant coupling between the extractor and stripper units, as confirmed by Samyudia et al.

Table 4.1 BDIC-based analysis for the SFE process

2 Stripper+reboiler No No No No

2 Trim-cooler Yes Yes Yes Yes

2 Stripper+trim-cooler Yes Yes Yes Yes

DAU3 1 Extractor+stripper+trim-cooler

2 Reboiler Yes Yes Yes Yes

2 Stripper+reboiler+trim-cooler No No No No

2 Reboiler+trim-cooler Yes Yes Yes Yes

Intuitive block diagonal control schemes, derived from the physical decomposition of a plant, are not always advisable, as they can result in control systems that exhibit suboptimal performance.

Dynamic Interaction Measure

Representing Dynamic Interactions

In earlier sections, we explored various steady-state interaction measures that, while easy to apply, fail to reflect the effects of interactions on dynamic control performance in decentralized systems To assess dynamic performance, one effective approach is to model these interactions as process uncertainty, allowing us to analyze their influence on the performance and stability of closed-loop systems through the robust control framework.

For a system described by a full model G(s) with a diagonal submodel

The output \( y_i(s) \) of the ith channel can be represented by the equation \( y_i(s) = g_{ii}(s)u_i(s) + \sum_{j=1, j \neq i}^m g_{ij}(s)u_j(s) \) In this equation, \( g_{ii}(s) \) represents the diagonal subsystem, while the summation term accounts for the interactions between the ith channel and all other channels, highlighting the interconnected nature of the system.

The ith decentralized controller \( c_i(s) \) is specifically designed based on the model \( g_{ii}(s) \), treating the second term as a perturbation In this context, the diagonal system model serves as the nominal model, with interactions represented as uncertainties This allows for a systematic analysis of interactions and closed-loop stability under decentralized control through robust control theory, a method initially developed by Grosdider and Morari Interactions can be modeled as either additive or multiplicative uncertainty, as illustrated in Figure 4.7, which characterizes the interaction as additive uncertainty.

Then the system “seen” by the off-diagonal system is

Note that both the controllerC(s) andG d (s) are diagonal; therefore, system

From the Small Gain Theorem, the closed-loop system is stable ifM A (s) and

This analysis may be conservative in its treatment of known off-diagonal systems as uncertainty, yet it offers valuable advantages that enhance its utility.

1 The interaction measure is much simpler and the stability condition is easier to apply than other approaches.

2 The “true” model uncertainties can be dealt with within the same frame- work Thus, decentralized controllers can be made robust.

To reduce the conservativeness of the approach, the diagonal structure of the closed-loop system \( M_A(s) \) can be considered By utilizing the diagonally structured singular value \( \bar{\sigma} \) as an interaction measure instead of relying solely on maximum singular values, we can replace the previous inequality with a less conservative alternative: \( \bar{\sigma}(M_A(j\omega)) \cdot \bar{a}(\Delta_A(j\omega)) < 1, \forall \omega \in \mathbb{R} \) This new inequality takes advantage of the diagonal structure of \( M_A(s) \), allowing for a more accurate representation of interactions, which can also be modeled as multiplicative uncertainty.

Fig 4.7.Representing interaction as additive uncertainty

In this case, the closed-loop system is stable ifM M (s) and∆ M (s) are stable and ¯ σ(M M (jω))à(∆ M (jω))0, (4.67) where D(ω) ∈ R m × m is a nonsingular diagonal matrix and t is a real scalar variable.

Problem 4.14 cannot be solved directly by an SDP solver because (4.67) is nonlinear and complex This problem can be converted into a real LMI problem as shown below [16] SinceD(ω) is nonsingular, (4.67) is equivalent to the following inequality:

∆(jω)H˜ +H∆˜ ∗ (jω) +tH >0 (4.70) Assume that ˜∆(jω) =X(ω) +jY(ω), where both X(ω) and Y(ω) are real matrices This leads to

The above inequality holds if and only if

Therefore, Problem 4.14 can be converted into the following generalized eigen- value problem with constraints described in real matrix inequalities.

For each frequency ω, a real matrix ofH(ω) can be obtained by solving the above optimization problem using an SDP solver The passivity-based interaction measure can then be defined as

Definition 4.16 (Passivity-based interaction measure (PB-IM)) Con- sider an LTI system with a transfer functionG(s) The diagonal elements of

In the context of a diagonal subsystem G d (s), the interaction representation ∆(s) can be expressed in two ways: ∆ A (s) = G(s) - G d (s) and ∆ M (s) = [G(s) - G d (s)]G −1 d (s) When ∆(s) is stable, the passivity-based interaction measure (PB-IM) at frequency ω is defined as ν I (G(s), ω)min.

(4.74) where∆˜(jω)is defined by (4.57), (4.62) and (4.65).

MatrixD represents a frequency-dependent decision variable derived from solving Problem 4.15 Throughout the book, we refer to ν IA (G(s), ω) and ν IM (G(s), ω) as interaction measures that utilize additive and multiplicative uncertainty models, denoted as ∆ A (s) and ∆ M (s), respectively.

The passivity-based interaction measure shares the same characteristics as the frequency-dependent IFP index outlined in Property 3.3 Consequently, a decentralized stability condition can be directly derived from Proposition 3.4.

Theorem 4.17 Consider an LTI process system with a transfer function G(s)and a stable and minimum phase transfer function W(s).

1 Assume that ν IA (G(s), ω) ≤ ν F (W(s), ω) The process can be stabilized by a decentralized controllerC(s) ifM A (s) =M A (s)[I−W(s)M A (s)] −1 is strictly positive real (i.e., stable and strictly input feedforward passive), where

2 Assume that ν IM (G(s), ω)≤ ν F (W(s), ω) The process can be stabilized by a decentralized controllerC(s)ifM M (s) =M M (s)[I−W(s)M M (s)] −1 is strictly positive real, where

Similarly, the sector-bounded passivity index (simultaneous IFP and OFP) given in Definition 3.8 can also be used as an interaction measure:

Definition 4.18 (Sector-based interaction measure (SB-IM)) Con- sider an LTI system with a transfer function G(s) The diagonal elements of G(s) form a diagonal subsystemG d (s) If ∆ A (s) and∆ M (s) are stable, the sector-based interaction measure (SB-IM) at frequencyω is defined as ν sIA (G(s), ω, b)max (ν S − (∆ A (s), ω, b),0), (4.77) ν sIM (G(s), ω, b)max (ν S − (∆ M (s), ω, b),0) (4.78)

The SB-IM allows for scalable interactions during computation, much like the PB-IM However, direct scaling poses challenges due to the proximity of PB-IM and SB-IM values when 'b' is large Therefore, it is often adequate to apply the same scaling matrices optimized for the PB-IM to the SB-IM.

IM can be calculated from the PB-IM using Theorem 3.9 The definition of SB-IM immediately leads to the decentralized stability condition based on Theorem 3.11:

Theorem 4.19 Consider an LTI process system with a transfer functionG(s)and a stable and minimum phase transfer function W(s).

1 Assume that ν sIA (G(s), ω, b) ≤ −ν S − (W(s), ω, b w ) The process can be stabilized by a decentralized controller C(s)if

I (4.79) is strictly positive real (or extended strictly positive real), whereM A (s)is given in (4.75).

2 Assume that ν sIM (G(s), ω, b) ≤ −ν S − (W(s), ω, b w ) The process can be stabilized by a decentralized controller C(s)if

M M (s) =M M (s) [I−W(s)M M (s)] −1 + 1 b+b w I (4.80) is strictly positive real (or extended strictly positive real), where M M (s) is given in (4.76).

The PB-IM serves as a more refined interaction measure derived from IFP, while the SB-IM offers a less conservative theoretical framework This makes the SB-IM particularly advantageous when implementing the robust control design approach outlined in Section 3.3.4 for decentralized control systems.

The passivity-based interaction measure indicates the potential performance of decentralized control systems According to Theorem 4.17, if the interaction measure ν IM (G(s), ω) is known, it can be concluded that ¯ σ(M M (jω)) is less than 1 divided by ν IM (G(s), ω) for all ω.

As a result, it is possible to make the sensitivity function of each loop

|S i (jω)| = |1−m M,i (jω)| (i = 1, , m) arbitrarily small at frequencies whenν IM (G(s), ω)0 (i 1, , m).

3 Screen out the non-DIC pairing schemes by using the necessary DIC condition given in Theorem 4.3.

4 Calculate the diagonally scaled passivity indices ν D (G + (s), ω) at a number of frequency points.

5 Compare the passivity index profiles of different pairings The best pair- ing should correspond to the one with the largest frequency bandwidth ω b such thatν D (G + (s), ω)≤0for anyω∈[0, ω b ] This pairing scheme would allow using controllers with integral action and the fastest dy- namic response.

Now we illustrate how to use the above pairing procedure for DUS control with two examples.

Example 5.3 (Distillation column [15]).Consider the distillation column de- scribed by the following 3×3 transfer matrix [79]:

The pairing schemes 1-2/2-3/3-1, 1-2/2-1/3-3 and 1-3/2-2/3-1 are not desir- able because they do not satisfy the DIC condition, asν D (G + (s),0)>0 As a result, only three pairing schemes remain,i.e., 1-1/2-2/3-3, 1-3/2-1/3-2 and

The diagonally scaled passivity indices for the transfer function were calculated and illustrated in Figure 5.6 for each pairing scheme The preferred pairing scheme is 1-1/2-3/3-2, as it exhibits the largest passivity bandwidth This preference aligns with the findings from the generalized dynamic relative gain (GDRG) method, which indicates that the 1-1/2-3/3-2 pairing also has the lowest loop interactions, as measured by the total interaction potential.

Note that suitable pairing schemes for DUS control may not always lead to small loop interactions This is clearly demonstrated in the following example:

Example 5.4 ([15]).Consider another 2×2 process with the following transfer function [60]:

The diagonally scaled passivity indices of the process with two different pairing schemes are shown in Figure 5.7 Off-diagonal pairing has smaller

134 5 Passivity-based Fault-tolerant Control

Fig 5.6.Example 5.3: Scaled passivity index

The analysis of scaled passivity index loop interactions reveals that off-diagonal pairing exhibits a smaller total interaction potential, making it preferable according to the GDRG pairing criterion In contrast, the DUS pairing rule favors diagonal pairing Although diagonal pairing results in larger dynamic interactions, the destabilizing effects are minimal, as confirmed by simulation studies These studies demonstrated that a DUS controller can effectively stabilize the process, while also highlighting that the diagonal pairing scheme enables the use of controllers that deliver high control performance.

Fault-tolerant Control Design for Stable Processes

Fault-tolerant PI Control

The fault-tolerant multiloop PI controller tuning problem is a significant topic for process control engineers, as PI controllers are prevalent in process industries These multiloop PI controllers, characterized by positive gains, are passive and can be optimized to meet passivity-based DUS conditions This section introduces an optimization-based tuning approach that effectively balances DUS requirements with specific performance specifications.

For multiloop PI controllers, Theorem 5.1 is reduced to the following condi- tion:

Proposition 5.5 ([14]) Given a stable LTI MIMO process with its transfer functionG(s)∈C m × m , any multiloop PI controller

, ∀i= 1, , m, (5.19) with the following parameter relation will unconditionally stabilize the closed- loop system: τ I,i 2 ≥ k c,i + ν

136 5 Passivity-based Fault-tolerant Control

Multiloop control performance can be represented in various ways, with one effective method being the use of the sensitivity function S_i(s) for each loop, combined with a weighting function w_i(s) that emphasizes control error reduction at low frequencies To optimize this approach, we define γ_i as a scalar decision variable for each loop, allowing for the design of controller loops that aim to maximize γ_i while adhering to specific constraints.

|γ i S i (jω)w i (jω)| 0, then

This design focuses on minimizing the H 2-norm of the closed-loop system, which is a key indicator of control performance The approach is primarily derived from the research conducted by the authors and their collaborators [16].

The decentralized fault-tolerant H 2 controller design problem can be stated as follows: Given anm×mLTI stable process with a transfer function

G(s), find a decentralized controllerK(s) = diag{k i (s)} (i= 1, , m) such that the DUS condition is satisfied and anH 2 nominal performance for each

The optimization of the 140 5 Passivity-based Fault-tolerant Control loop is achieved by selecting a stable and minimum phase weighting function w(s) This selection ensures that the condition ν − (w(s), ω) < −ν D (G + (s), ω) is satisfied, where G + (s) is defined in equation (5.6) This approach addresses the controller design challenge for the ith controller effectively.

(i= 1, , m) can be mathematically formulated as follows:

Problem 5.9. min k i ( s ) {γ i }, (5.29) subject to k i (s) =V ii k i (s) [1−w(s)V ii k i (s)] −1 is passive,∀ i= 1, , m, (5.30) and

The closed-loop transfer function T i (G ii (s), k i (s)), which represents control performance, must have its H 2-norm minimized, as indicated by the inequality T i (G ii , k i ) 2 < γ i Here, G ii (s) refers to the ith diagonal element of the matrix G(s), while V ii denotes the ith diagonal element of the sign matrix.

A typical choice of the closed-loop transfer function is as follows:

T i (G ii (s), k i (s)) w ki k i (s) [I+G ii (s)k i (s)] −1 w si (s) [I+G ii (s)k i (s)] −1

In the design of control systems, the weighting function for sensitivity, denoted as w si (s), and a constant weight, w ki, are utilized to penalize the controller gain This independent design approach ensures that each control loop is crafted to meet the specified condition outlined in equation (5.30).

Selecting the Weighting Function w(s)

For effective integration into final controllers, the weighting function w(s) should be both stable and minimum phase Additionally, it is advantageous for this transfer function to be simple and of low order When G+(s) represents a Dynamic Inversion Controller (DIC), an appropriate choice for w(s) could be expressed as w(s) = ks(s+a).

(s+b)(s+c), (5.33) wherea,b,candkare positive real parameters to be determined.

IfG + (s) is not DIC, the weighting functionw(s) does not possess a zero ats= 0 In this case, the following form can be used: w(s) =k(s+a)(s+b)

With such a weighting functionw(s), the final controllerK(s) = diag{k i (s)}

(i= 1, , m) does not have integral action.

If the passivity indices atn ω frequency points ν D (G + (s), ω 1 ), ν D (G + (s), ω 2 ),ã ã ã , ν D (G + (s), ω n ω )

(5.35) are obtained, the parameters ofw(s) can be found by solving the following optimization problem:

Like Problem 5.6, the above problem can be solved by using any nonlinear optimization solver, such as the MATLAB Optimization Toolbox.

Control Synthesis

We can simplify the control design task by constructing the DUS controller

K(s) = diag{k i (s)} (i = 1, , m) indirectly From Figure 5.3, we can see that

Therefore, we can first design the systemK (s), which is required to be pas- sive, and then obtain the final controller using (5.38) The passive controller

K (s) = diag{k i (s)} (i = 1, , m) can be found by solving the following problem:

Problem 5.11.For alli= 1, , m min k i ( s ) {γ i }, (5.39) subject to k i (s) is passive, (5.40) and

The closed-loop transfer function T i (G ii (s), k i (s)) gives a performance constraint for the ith loop as a function of k i (s) For the equivalent per- formance specification as given in (5.31),

T i (G ii (s), k i (s)) w ki V ii k i (s) [I+G ii (s)V ii k i (s) +w(s)k i (s)] −1 w si (s) [I+w(s)k i (s)] [I+G ii (s)V ii k i (s) +w(s)k i (s)] −1

142 5 Passivity-based Fault-tolerant Control

For a given LTI stable processG(s)∈C m × m , assume the following state-space representations (fori= 1, m):

G + ii (s) := (A gi , B gi , C gi , D gi ), (5.43) w(s) := (A p , B p , C p , D p ), and (5.44) w si (s) := (A wi , B wi , C wi , D wi ) (5.45) The following augmented plant can be constructed for theith loop:

The challenge lies in identifying a passive solution for the H2 control problem of the augmented plant P i (s) Many current H2/passive control design methods are unable to meet the performance specifications outlined in (5.32), as they rely on certain assumptions that limit their applicability.

This article introduces a method utilizing successive semidefinite programming (SSDP) techniques to address control synthesis challenges To streamline the process, an ad hoc Linear Quadratic Gaussian (LQG) control structure is employed, which consists of an observer and state feedback, akin to the approach outlined in reference [42] The system dynamics are described by the equation x˙ c = Ax c + B2 u i + L(y i − C2 x c − D22 u i), with the control input defined as u i = −K gi x c.

This approach does not require the assumptions given in (5.48) As a result,

H 2 problems with any performance specification, such as that in (5.42), can be solved From (5.49), controllerk i (s) should have the following state-space representation:

A ki =A+B 2 K gi −LC 2 −LD 22 K gi , B ki =L,

C ki =K gi , D ki = 0 (5.50) whereL ΠC 2 2 +B 1 D 21 T D 21 D 21 T −1 is the observer gain matrix andΠ is the solution to the Riccati equation below: Π

The aboveH 2 controller lacks integral action, which is essential for achieving offset-free control This limitation can be addressed by incorporating integral action into the controller design When adding an integrator to any passive controller \( k_i(s) \), the new structure \( k_i(s) + \frac{k_{si}}{s} \) remains passive as long as \( k_{si} \geq 0 \), ensuring that stability conditions are maintained Consequently, the updated structure for \( k_i(s) \) is established.

A Iki A+B 2 K gi −LC 2 −LD 22 K gi 0

The final controller for theith loop, k i (s) =U ii k i (s) [1 +w(s)k i (s)] −1 , (5.53) will retain the integral action as long as w(0) = 0, which implies that the process is decentralized integral controllable (DIC) Assume that

By using the positive-real lemma (Lemma 2.16) and property of system H 2- norm, the above control problem can be cast into a matrix inequality problem:

K gi ,k si ,P 1 ,P 2 ,Q {Tr(Q)}, (5.55) subject to

144 5 Passivity-based Fault-tolerant Control whereA ki is defined in (5.50) and

LC 2 A+B 2 K gi −LC 2 LD 22 k si

The problem involves four matrix decision variables: P1, P2, Q, and Kgi, along with a scalar variable ksi Inequalities 5.56 to 5.58 ensure the passivity of ki(s), while inequalities 5.59 and 5.60 establish the H2-norm condition The trace of matrix Q sets the upper limit for F1(Pi(s), ki(s))²² Since Acl, Bcl, Ccl, and Dcl depend on Kgi and ksi, both inequalities (5.56) and (5.59) form bilinear matrix inequalities (BMIs) A numerical solution to this problem can be achieved using the SSDP approach.

To address bilinear matrix inequalities, one effective method is to approximate the bilinear terms through Taylor expansion This approach assumes that X and Y are independent matrix decision variables, allowing their product to be approximated around the point (X₀, Y₀) using a specific equation.

The equation ≈X 0 δY + δXY 0 + X 0 Y 0 (5.63) defines a relationship where δX represents the difference between X and its reference value X 0, and δY signifies the difference between Y and its reference value Y 0 Both variables, X and Y, are constrained by their respective matrix norms, indicated by δX ≤ and δY ≤ (5.64), where is a small positive constant This ensures that the solution space of equation (5.63) remains closely aligned with that of equation (5.62).

Define δK gi = K gi −K gi 0, δk si = k si −k si 0, δP 1 = P 1 −P 10 and δP 2 =P 2 −P 20 By using (5.63), (5.56) to (5.60) can be approximated around

The approximated problem involves decision variables represented as deviation values, as outlined in Section A.2 under Problem A.1 To effectively solve this approximated problem, it is essential to impose restrictions on the solution radii of the deviation variables.

The iterative SSDP approach requires a feasible initial solution that satisfies all constraints outlined in Problem 5.12 A practical choice for this initial point is an arbitrary passive controller Under the assumption that K gi equals L T P 1, a specific inequality provides a sufficient condition for the subsequent analysis in equation (5.56).

With left and right multiplying ofP 1 −1 , and definingW =P 1 −1 , the following LMI can be obtained:

W(A−LC 2 ) T + (A−LC 2 )W+LB 2 T +B 2 L T −LD T 22 L T −LD 22 L T 0 is an m×m constant matrix and ¯C(s) is a stable transfer function.

The steady-state operating point (u ss, y ss) is considered feasible when y ss equals r ss The attainability of this steady state can be determined by applying the asymptotic stability condition associated with the Internal Model Control (IMC) controller.

Q, as shown in Figure 6.6, when an exogenous constant reference signalr(t) r ss is applied The steady-state attainability condition is given as follows:

174 6 Process Controllability Analysis Based on Passivity

Fig 6.6.Implementation of the IMC controllerQ

Theorem 6.6 (Steady-state attainability via linear feedback control [98]) Consider the closed-loop system shown in Figure 6.6 Assume that

1 The dynamics of the linear controller C¯(s)are described by the following state-space equations:

(where z∈R n z andξ∈R m ) and matrix A in (6.26) is Hurwitz.

2 The process G and the model G˜ are stable in the sense of Definition 6.5 andG˜=G.

3 The algebraic equationf(x ss , u ss ) = 0has a unique solution ¯ x ss =ψ(u ss ), (6.27) such that ψ(ã)isC 2

4 The steady-state relation between ξ ss andy ss is given by y ss =g(x ss ,Kξ¯ ss ) =g ψ( ¯Kξ ss ),Kξ¯ ss

=h( ¯Kξ ss )ϕ(ξ ss ), (6.28) where K¯ = −CA −1 B +D is the steady-state gain matrix of C¯(s) and ϕ(ã) is a mapping fromξ ss toy ss

Consider a constant reference r(t) =r ss such that the corresponding op- erating point (u ss , y ss )with y ss =r ss is feasible Let the integral action gain

K˜ =εK,ˆ (6.29) where Kˆ T = ˆK > 0 and ε > 0 is a detuning coefficient Assume that there exists a nonempty regionΛ ξ ⊂R m such that

In a steady state, the relationship is defined as y ss = ϕ(ξ ss) = h( ¯Kξ ss), where ξ is a variable belonging to the set Λ For a small positive value ε 0, the equilibrium point (u ss, y ss) demonstrates asymptotic stability when the closed-loop trajectory remains within the set Λ ξ or includes the steady state ξ ss for all time t ≥ 0.

The theorem outlines sufficient conditions for achieving asymptotic stability at a feasible equilibrium point (u ss, y ss) through linear output feedback control A crucial aspect is the strictly input passivity condition on the steady-state nonlinear mapping from ξ ss to y ss, which encompasses both the nonlinear mapping h(ã) and the steady-state gain matrix ¯K of the linear system.

C) This result can be derived from the Passivity Theorem (Theorem 2.44).¯

The authors and their co-worker developed an alternative proof utilizing the Singular Perturbation Theorem to determine the steady-state region of attraction This proof was initially published in [98] and can be found in Section B.4.2.

Theorem 6.6 establishes a steady-state condition where the integral action gain, ˜K=εKˆ (with 0< ε≤ε 0), can be minimized, rendering the dynamics of ¯C(s) and the nonlinear model ˜G irrelevant This theorem focuses on the controllability of nonlinear processes through linear feedback control, relying on a steady-state nonlinear model rather than a linearized version around a feasible operating point Consequently, we can mathematically define the steady-state region of attraction.

Definition 6.7 (Steady-state region of attraction under linear feed- back control [98]) Consider a stable nonlinear processGand a feasible equi- librium point(u ss , y ss ) Denote the following ellipsoidal region Π

K, γ, uˆ ss inR m centred atu ss as Π u

(6.31) whereγ >0 is a scalar parameter Then, the steady-state region of attraction under linear feedback control for the feasible operating point(u ss , y ss )is given by

The region Λ u is obtained by mapping the region Λ ξ, as specified in condition (6.30) of Theorem 6.6, into the input space through the linear steady-state relationship u ss = ¯Kξ ss Furthermore, the function à[ã] quantifies the hypervolume of the ellipsoid Π.

, similar to that used in (6.20) and(6.21).

176 6 Process Controllability Analysis Based on Passivity

The steady-state region of attraction under linear feedback control, denoted as Ω u (u ss ), represents the largest ellipsoid fully inscribed within the common area of the AIS and the region defined by the static input passivity condition (6.30) The size of Ω u (u ss ) is influenced by the selected operating point (u ss , y ss ), as the condition (6.30) varies with different operating points Unlike Λ u, the region of attraction Ω u (u ss ) includes steady-state initial operating conditions in the input space, ensuring that the closed loop with linear control converges asymptotically to the desired operating point (u ss , y ss ) Any closed-loop trajectory starting from an initial condition within Ω u (u ss ) remains within this region for all time t ≥ 0, demonstrating that Ω u (u ss ) is a positively invariant set that converges to the feasible operating point The optimization problem necessary to identify the steady-state region of attraction is articulated in Equation 6.32, aligning with standard practices in convex optimization theory.

The steady-state gain matrix ¯K can be selected to ensure that the set Ω u (u ss ) remains nonempty In this context, ¯K functions as a scaling matrix, ensuring that the relationship described by equation (6.30) holds true in the vicinity of the steady-state values (u ss , y ss ) Additionally, the Jacobian of the process's steady-state nonlinear mapping, evaluated at the equilibrium point (u ss , y ss ), is defined for further analysis.

One natural choice of ¯K is

It can be proved that with (6.34), the steady-state region of attraction under linear feedback controlΩ u (u ss ) is not empty [98].

The concept of the region of attraction can be illustrated by the following example from [98]:

In a stable nonlinear process with two inputs and two outputs, the steady-state nonlinear mapping can be illustrated through specific surfaces, as depicted in Figure 6.7 An example of a feasible equilibrium for this system is represented by the input value of u ss equal to 2 and 4.

Assume that the AIS is the following square region:

AIS ={(u 1 , u 2 )| −10≤u 1 ≤10 and −10≤u 2 ≤10} (6.36)From Figure 6.7, it can be seen that there are points in the input space that exhibit input multiplicity (i.e., there exist multiple different u ss values that

Fig 6.7.Steady-state nonlinear maph(ã) of the process in Example 6.8 [98]

178 6 Process Controllability Analysis Based on Passivity u 1 u 2 Ŧ 10 Ŧ 8 Ŧ 6 Ŧ 4 Ŧ 2 0 2 4 6 8 10 Ŧ 10 Ŧ8 Ŧ 6 Ŧ 4 Ŧ 2

Figure 6.8 illustrates the steady-state region of attraction and closed-loop trajectories, with the shaded area representing the input space region Λ u where condition (6.30) holds true The operating point (u ss, y ss) is marked by a filled circle within Λ u The steady-state region of attraction under linear feedback control, denoted as Ω u (u ss), is defined by the largest ellipse that fits entirely within Λ u and the AIS The dashed curves in the figure depict closed-loop trajectories initiated from steady-state conditions, represented by circles Notably, all closed-loop trajectories starting from points a, b, or c within Ω u (u ss) successfully converge to the equilibrium point (u ss, y ss) through the application of linear output feedback control.

Steady-state Output Space Achievable via Linear

Steady-state output space achievable via linear feedback control addresses a problem related to the region of attraction: given a feasible operating point

The servo controllability problem examines the set of operating points a closed-loop system can transition to from a specific point (u ss, y ss) using a linear controller Utilizing the IMC framework, we analyze the stability conditions of the IMC controller when the reference signal shifts from r(t) = y ss to a new value Assuming the system K is passive and G p = G = ˜G, we define the changes in input and output as ∆u = u - u ss and ∆y = G u - y ss, with ¯G representing the relationship between these changes According to the Passivity Theorem, the IMC controller achieves asymptotic stability if the system ¯G is strictly input passive for any (u ss, y ss) within the relevant region, establishing an incrementally strictly input passivity condition for the process system G.

Definition 6.9 (Incremental input passivity [32]) Let H : L m 2 e → L m 2 e SystemH is said to be incrementally input passive if

System H is said to be incrementally strictly input passive if there exists a constantν such that

HuưHu , uưu T ≥νuưu 2 T , ∀ u, u ∈ L m 2 e (6.38) Now we generalize the above result with the linear control shown in Fig- ure 6.6:

Theorem 6.10 (Steady-state attainability via linear feedback control for step changes in reference) Consider the closed-loop system shown in Figure 6.6 Assume that K >ˆ 0 and the feasible operating point (u ss , y ss ) is attainable using linear feedback control for all 0 < ε ≤ ε 0 , based on the assumptions and conditions outlined in Theorem 6.6 Define a static mapping ϕ(ξ ss ) =hKξ¯ ss

In this analysis, we consider a step change in the reference signal \( r(t) \), defined as \( r(t) = y_{ss} \) for \( t < 0 \) and \( r(t) = y_{ss} \) for \( t \geq 0 \) We assume that the new operating point \( (u_{ss}, y_{ss}) \) remains feasible Additionally, we impose a Lipschitz continuity condition on \( h(\hat{a}) \), expressed as \( h(u + \Delta u) - h(u) \leq \gamma \Delta u \) for all \( u \) within the AIS, where \( 0 \leq \gamma < \infty \) Furthermore, there exists a nonempty region \( \Theta_u \subset AIS \subset \mathbb{R}^m \) that plays a crucial role in our analysis.

1 The following condition is satisfied:

180 6 Process Controllability Analysis Based on Passivity

2 The steady-state mapping ϕ(ã)is incrementally strictly input passive for any ξ ss = ¯K −1 u ss , ∀ u ss ∈Θ u

Then the new operating point (u ss , y ss ) is asymptotically stable if the closed-loop trajectory is such thatu(t)∈Θ u for allt≥0.

The extended theorem builds upon Theorem 2.2 from [98], highlighting the intuitive nature of the incremental input passivity condition and its connection to controllability, despite its numerical verification challenges In contrast, Condition 1 offers a more straightforward approach for controllability analysis Section B.4.4 presents a proof for Condition 1, which differs slightly from the one in [98] Additionally, the equivalence of Conditions 1 and 2 was established in [97] The steady-state output space achievable through linear feedback control is defined accordingly.

Definition 6.11 (Steady-state output space achievable via linear feed- back control) Consider a stable nonlinear process Gand a feasible equilib- rium point (u ss , y ss ) Denote the following ellipsoidal region Π y

K, γ, yˆ ss inR m centred at y ss as Π y

A steady-state operating point y ss is achievable from (u ss , y ss ) via linear feedback control if there exists a Kˆ and γ such that the ellipsoidal region Π y

K, γ, yˆ ss is completely inscribed inΘ y and coversy ss , that is, y ss ∈Π y

⊂Θ y , (6.43) whereΘ y is the region in the output space that results from mapping the region Θ u where (6.41) holds, using the nonlinear steady-state relationy ss =h(u ss ).

The steady-state output space attainable through linear feedback control at the feasible operating point (u ss, y ss) encompasses all achievable y ss values derived from (u ss, y ss) using a single constant Kˆ, although different γ may be applied.

The definition expands on the achievable output space previously established As depicted in Figure 6.9 for an arbitrary region Θ y, the steady-state output space attainable through linear feedback control, denoted as Ω y (y ss), encompasses the set of steady-state operating points in the output space This set represents the points to which the closed-loop system with linear control is assured to converge, beginning from the specified operating point (u ss, y ss).

Now let us study the steady-state output space achievable of the process system in Example 6.8:

Example 6.12 ([98]).Consider stable nonlinear processGdescribed in Exam- ple 6.8 with the same available input space (AIS) and the feasible operating y ss y ss y 1 Θ u u 1 u 2 y 2 y=h(u) Θ y

In Figure 6.9, we explore the geometric interpretation of the output attainability point (u ss, y ss) as outlined in equation (6.35) By selecting the parameters ¯K and ˆK from Example 6.8, we identify the region Θ u where the condition ∂h/∂u K > ¯ 0 holds true This region Θ u is then translated into the output space through the nonlinear steady-state relationship y ss = h(u ss ) The outcome is depicted as the irregular shaded region Θ y in Figure 6.10 The points y ss that meet the criteria in equation (6.43) are derived numerically, illustrating the achievable output space through linear feedback control.

The region Ω y (y ss ) is depicted in Figure 6.10, centered around the steady-state output y ss = [17.63, 24.69] T, indicated by a filled circle A magnified view of Ω y (y ss ) is provided in Figure 6.11, where the dashed lines illustrate the simulated closed-loop trajectories of the process output as the reference signal r(t) transitions from y ss to the new value marked with squares Notably, since y ss is included in Ω y (y ss ), the closed-loop trajectories are assured to converge to the updated operating point However, unlike the steady-state region of attraction for linear feedback control Ω u (u ss ), the steady-state output space Ω y (y ss ) is not positively invariant for y(t), as clearly demonstrated in the figures.

The closed-loop output trajectory approaches the steady-state point located in the upper-left corner, exits the region, and subsequently reenters the area Ω y (y ss ) Throughout this process, the trajectory consistently stays within the defined region Θ y, in accordance with the conditions outlined in equation (6.41).

Steady-state Attainability by Nonlinear Control

Recent analyses have shed light on the relationship between nonlinearity and linear controllability in systems Various measures have been developed to assess the extent of nonlinearity in open-loop processes Eker and Nikolau highlight a prevailing belief that significant nonlinearity in an open-loop system typically leads to ineffective linear control in closed-loop scenarios However, recent findings suggest that the degree of nonlinearity may not necessarily correlate with suboptimal performance when using a linear controller.

182 6 Process Controllability Analysis Based on Passivity

Fig 6.11.Region Ω y (y ss ) and closed-loop trajectories [98]

Thermodynamic Variables and the Laws of Thermodynamics 194

The Structure of State Equations of Process Systems

Physically Motivated Supply Rates and Storage Functions

Hamiltonian Process Models

Case Study: Reaction Kinetic Systems

DUS H 2 Control Synthesis

Region of Steady-state Attainability