H ∞ control of periodic piecewise vibration systems with actuator saturation

Abstract

In this paper, an H_∞ controller with actuator saturation consideration is proposed to attenuate the vibration of periodic piecewise vibration systems. Based on a continuous Lyapunov function with a time-varying Lyapunov matrix, the H_∞ performance index of periodic piecewise vibration systems is studied first. On the basis of the obtained H_∞ criterion, the conditions of designing a state-feedback active vibration controller are proposed in matrix inequality form with actuator saturation taken into account. Because of the nonconvexity of the conditions, a corresponding algorithm to compute the controller gain is developed as well. A representative numerical example is used to verify the effectiveness of the proposed method.

Keywords

Actuator saturation periodic piecewise systems vibration control

1. Introduction

In engineering applications, some vibration systems exhibit periodic dynamic characteristics such as rotor-blade systems, such that under a constant rotational speed, the variation of the blade position will lead to periodical variations of inertial distribution. A periodic time-varying system is always used to model such vibration systems. A periodic piecewise system is a special kind of periodic time-varying system, with the property that during a period, the system retains time-invariant in each subinterval. This kind of periodic system may also be obtained as the approximated models of periodic time-varying systems by dividing the system into several intervals during a period, and in each interval, the system is approximated with a constant system with techniques such as the average method (Farhang and Midha, 1995; Selstad and Farhang, 1996). On the other hand, we can find prototypes of periodic piecewise systems in practice, such as conveying machines with periodic-varying loads or manufacturing machines which manipulate repetitive tasks.

Vibration control for periodic systems has attracted extensive attention in recent years. As a special kind of time-varying system, the techniques for general time-varying systems can be adopted to deal with the control problems of periodic time-varying systems. An active vibration control strategy combining online modeling or identification and controller updating as an effective and easy application method has been commonly employed in engineering applications. This technique always calls for a fast online modeling process to obtain the system dynamics and update the controller. The filtered-x least mean squares method has been broadly adopted to control the time-varying systems (Huang et al., 2012; Zhang et al., 2012) because of its merits of low computation burden and easy implementation. However, since the two processes, that is, system modeling and controller updating, are coupled, it may result in accumulated errors, amplified vibration and destabilized systems (Kim and Swanson, 2005). Moreover, the convergence of the two processes may not be synchronous; it is hard to tell whether the controller is optimal after both of them converge (Wang and Mak, 2014). A control strategy including adaptive identification combining with nonadaptive control was proposed in Wang and Mak (2014), which decouples the two processes, and the optimal controller can be guaranteed when the identification process converges. However, this kind of active vibration control calls for fast online processing, which leads to a high cost. An alternative approach for controlling periodic systems is based on the well-known Floquet theory. By implementing the Lyapunov–Floquet transformation, Floquet theory was used in the control of periodic systems in Sinha and Joseph (1994), and this method has been broadly adopted in engineering (Deshmukh et al., 1999; Sinha et al., 2000). However, the closed-loop system asymptotic stability cannot be guaranteed because of a generalized inverse of a rectangular matrix present in the control law. To solve this problem, a procedure consisting of the Lyapunov–Floquet transformation, the backstepping technique and Floquet theory was proposed in Deshmuhk and Sinha (2004). The optimal control with solution of a periodic Riccati equation is used in helicopter vibration control (Arcara et al., 2000) and for periodic delayed systems, optimal controls separately based on Floquet theory, Chebyshev spectral collocation and the Lyapunov–Floquet transformation were studied in Nazari et al. (2014).

Another focus in vibration control in practice is controller saturation, which has been broadly investigated over the past decades. Because of the unmeasured disturbance and practical actuator power limitation, it can be expected that the desired force may exceed the capacity of the practical actuator, which leads to actuator saturation. Saturation often has negative effects on the system performance and may result in system instability. Many works have been devoted to solving this problem. A method which guarantees performance with large control gains was developed by Nguyen and Jabbari (1999, 2000). This method has been successfully applied for linear parameter varying (LPV) systems and civil engineering structures with controller saturation (Du and Lam, 2006; Jabbri, 2001).

In this paper, we study the vibration control of periodic piecewise vibration systems with techniques borrowed from switched system theory. The L₂-gain performance of switched systems based on average dwell time was studied in Zhai et al. (2001), and with the average dwell time approach, asynchronous H_∞ control of discrete-time switched systems was studied in Zhang and Shi (2009). By relaxing the average dwell time constraint, mode-dependent average dwell time is adopted to formulate a weighted H_∞ performance for switched systems. Periodic piecewise systems can be treated as a special case of switched systems, with fixed periodic switching law and subsystem dwell time. For periodic piecewise systems with possibly non-Hurwitz susbsystems, the L₂-gain performance based on the Lyapunov function with a continuous and discontinuous time-varying Lyapunov matrix were studied in Li et al. (2015a,b), In this paper, a continuous Lyapunov function with a time-varying Lyapunov matrix is adopted here to study the H_∞ performance of periodic piecewise systems, and an H_∞ performance controller considering actuator saturation is proposed to attenuate the system vibration. Due to the nonconvexity of the conditions, an iterative algorithm is also developed to find possible solutions of controller gains. It is worth mentioning that the performance index developed in this paper is different from those in Li et al. (2015a,b), since all subsystems are stable in this study and this leads to a new characterization. Moreover, the H_∞ controller design problem under input saturation is also tackled here. These are the main contributions of the present work. The paper is organized as follows. The problems are formulated in Section 2. The H_∞ performance for a periodic piecewise vibration system is studied in Section 3, and an H_∞ controller under actuator saturation and the corresponding algorithm are developed in Section 4. Section 5 provides an illustrative example to validate the merit of the approach proposed in this paper. The paper is concluded in Section 6.

2. Vibration model description

A simplified lumped parameters vibration model is shown in Figure 1. It is assumed that this vibration system composes masses m₁, m₂, … , m_k, dampers c₁, c₂, … , c_k and springs k₁, k₂, … , k_k. The depicted system is excited by a disturbance vector $w = [f 1 (t) f 2 (t) \dots f ℓ (t)] T$ and ℓ is the number of disturbance input. Then the dynamic equation of this vibration system can be written in matrix form as

M \overset{··}{x} (t) + C \overset{\cdot}{x} (t) + K x (t) = G w (t)

(1)

where

x (t) = [x 1 (t) x 2 (t) \dots x k (t)] T

is the vector containing the displacements of each mass. M, C, and K are the mass, stiffness and damping matrices, respectively, which are given as

M = [m 1 00 \dots 00 m 2 0 \dots 0 : : : : 00 \dots 0 m k], C = [c 1 + c 2 - c 2 0 \dots 0 - c 2 c 2 + c 3 - c 3 00 : : : : 0 \dots 0 - c k c k] K = [k 1 + k 2 - k 2 0 \dots 0 - k 2 k 2 + k 3 - k 3 00 : : : : 0 \dots 0 - k k k k]

(2)

and G ∈ ℝ^k × ℓ gives the location of the disturbance. Suppose that there are some actuators installed between adjacent masses, then the dynamic equation can be given as

M \overset{··}{x} (t) + C \overset{\cdot}{x} (t) + K x (t) = G w (t) + H u (t)

(3)

where

u = [F c 1 F c 2 \dots F cn] T

is the vector of actuator forces. H ∈ ℝ^k×n gives the location of the actuators and n is the controller number.

Figure 1.

A lumped parameter vibration system model.

If we choose a state vector $q (t) = [x (t) \overset{\cdot}{x} (t)] T$ , then the vibration system can be written in the following state-space form

\overset{\cdot}{q} (t) = Aq (t) + Bu (t) + B w w (t) z (t) = Cq (t) + Du (t) + D w w (t)

(4)

where z(t) is the system output vector, u(t) and w(t) are the control input and disturbance input vectors, respectively, and we have that

A = [0 I - M - 1 K - M - 1 C], B = [0 M - 1 H], B w = [0 M - 1 G]

(5)

C, D, and D_w are matrices with appropriate dimensions. Assume that the vibration system (3) has periodic piecewise elements, for example, the system may have periodic piecewise mass M(t) = M(t + T_pm), periodic piecewise spring K(t) = K(t + T_pk) or periodic piecewise damper C(t) = C(t + T_pc), with a fundamental period T_p ∈ ℝ such that T_p = ℓ_mT_pm = ℓ_kT_pk = ℓ_cT_pc and {ℓ_m, ℓ_k, ℓ_c} is setwise coprime. Then the state-space model will display periodic piecewise characteristics, that is, for all t ≥ 0, A(t) = A(t + T_p), B(t) = B(t + T_p), B_w(t) = B_w(t + T_p), C(t) = C(t + T_p), D(t) = D(t + T_p), D_w(t) = D_w(t + T_p).

In the following paragraph, the description and analysis are based on the vibration system model in state-space form (4). Suppose that the system parameter has S subsets over a fundamental period. In this way, the interval [0,T_p) can be partitioned into S subintervals [t_i−1, t_i), i ∈ N, N = {1, 2, … , S}, where t₀ = 0, t_S = T_p. In the ith subinterval, the system elements are constant, system matrices can be represented by (A_i, B_i, B_wi, C_i, D_i, D_wi) and the dwell time is T_i = t_i − t_i−1 with $\sum_{i = 1}^{S} T i = T p$ . In this case, system (4) is equivalently represented by

\overset{\cdot}{q} (t) = A i q (t) + B i u (t) + B wi w (t), z (t) = C i q (t) + D i u (t) + D wi w (t), t \in [t i - 1, t i)

(6)

It is worth mentioning that we only consider the periodic piecewise vibration system which can be described by (6). Other periodic systems, such as having periodic piecewise masses, springs or dampers with different periodicities and no fundamental period for the whole system, are not considered in this paper.

To facilitate subsequent development, we give a definition concerning the exponential stability of system (4).

Definition 1

The periodic piecewise vibration system (4) with u(t) = 0, w(t) = 0 is said to be exponentially stable such that the state of the system satisfies $‖ x (t) ‖ \leq κ e - λ * t ‖ x (0) ‖, \forall t > 0$ , for some constants κ ≥ 1, λ* > 0.

3. H_∞ performance of periodic piecewise vibration system

In this section, based on a continuous time-varying Lyapunov function, the H_∞ performance of the periodic piecewise vibration system is studied.

Consider a Lyapunov function given by

V (t) = x T (t) P (t) x (t)

(7)

Here, P(t) is chosen to be a continuous time-varying function with a fundamental period T_p. For convenience, we consider the description over the first period, that is, for i ∈ N

P (t) = P i + (t - t i - 1) P i + 1 - P i T i, t \in [t i - 1, t i)

(8)

where P_i > 0, i = 1, 2, … , S + 1, are constant matrices. Notice that, for the continuity of P(t), we have

P S + 1 = P 1

(9)

and

0 < min_{i \in N} \underline{λ} (P i) \leq P (t) \leq max_{i \in N} \bar{λ} (P i)

(10)

Generally speaking, practical vibration systems with only passive elements are inherently stable. In this way, we assume that for a periodic piecewise vibration system (4), there exist λ_i > 0, i = 1, 2, … , S, such that $\overset{\cdot}{V} (t) < - λ i V (t), t \in [t i - 1, t i)$ . Then, the H_∞ performance condition for a piecewise periodic vibration system (4) based on the proposed Lyapunov function (7) can be given as follows. The result is developed along a similar line as in Li et al. (2015a), but with a tighter performance characterization.

Theorem 1

Consider periodic piecewise vibration system (4) with u = 0, given λ_i > 0, i = 1, 2, … , S and let λ_max = max λ_i, λ_min = min λ_i. If there exist P_i > 0 for i = 0, 1, … , S such that

[A_{i}^{T} P i + P i A i + P i + 1 - P i T i + λ i P i P i B wi C_{i}^{T} * - γ 2 I D_{wi}^{T} * * - I] < 0

(11)

[A_{i}^{T} P i + 1 + P i + 1 A i + P i + 1 - P i T i + λ i P i + 1 P i + 1 B wi C_{i}^{T} * - γ 2 I D_{wi}^{T} * * - I] < 0,

(12)

P S + 1 = P 1

(13)

then system (4) is exponentially stable and, under zero initial conditions, satisfies

\int_{0}^{\infty} z T (τ) z (τ) d τ \leq γ 2 λ max λ min \int_{0}^{\infty} w T (τ) w (τ) d τ

(14)

Proof

The exponential stability of system (4) can be guaranteed from Li et al. (2015a). Assume t ∈ [αT + t_m−1, αT + t_m), α = 1, 2, … , m = 1, … , S + 1. From (11) and (12), we have

\overset{\cdot}{V} (t) = (x T (t) A_{m}^{T} + w T (t) B_{wm}^{T}) P (t) x (t) + x T (t) P (t) (A m x (t) + B wm w (t)) + x T (t) \overset{\cdot}{P} (t) x (t) = x T (t) (A_{m}^{T} P m + P m A m + P m + 1 - P m T m + λ m P m + t - t m - 1 T m (A_{m}^{T} (P m + 1 - P m) + (P m + 1 - P m) A m + λ m (P m + 1 - P m)) + C_{m}^{T} C m) x (t) + w T (t) (B_{m}^{T} P m + B_{m}^{T} (t - t m - 1) P m + 1 - P m T m + D_{m}^{T} C m) x (t) + x T (t) (P m B m + (t - t m - 1) \times P m + 1 - P m T m B m + C_{m}^{T} D m) w (t) + w T (t) (- γ 2 + D_{m}^{T} D m) w (t) - λ m V (t) - Π (t) \leq - λ m V (t) - Π (t),

(15)

where Π(t) = z^T(t)z(t) − γ²w^T(t)w(t). Integrating (15), it holds that

V (t) \leq e - λ m (t - (α T p + t m - 1)) V (α T p + t m - 1) - \int_{α T p + t m - 1}^{t} e - λ m (t - τ) Π (τ) d τ \leq e - λ m (t - (α T p + t m - 1)) {V (α T p + t m - 2) e - λ m - 1 T m - 1 - \int_{α T p + t m - 2}^{α T p + t m - 1} e - λ m (α T p + t m - 1 - τ) Π (τ) d τ} - \int_{α T p + t m - 1}^{t} e - λ m (t - τ) Π (τ) d τ : \leq V (α T p) e - \sum_{i = 1}^{m - 1} λ i T i - λ m (t - (α T p + t m - 1)) - \int_{α T p}^{α T p + t 1} e - λ 1 (α T p + t 1 - τ) - \sum_{i = 1}^{m - 1} λ i T i - λ m (t - (α T p + t m - 1)) Π (τ) d τ \dots - \int_{α T p + t m - 1}^{t} e - λ m (t - τ) Π (τ) d τ : \leq V (0) e - α \sum_{i = 1}^{k} λ i T i - \sum_{i = 1}^{m - 1} λ i T i - λ m (t - (α T p + t m - 1)) - \int_{0}^{t 1} e - λ 1 (t 1 - τ) - \sum_{i = 2}^{S} λ j T i - (α - 1) \sum_{i = 1}^{S} λ i T i - \sum_{i = 1}^{m - 1} λ i T i - λ m (t - (α T p + t m - 1)) Π (τ) d τ - \int_{t 1}^{t 2} e - λ 2 (t 2 - τ) - \sum_{i = 3}^{S} λ i T i - (α - 1) \sum_{i = 1}^{S} λ i T i - \sum_{i = 1}^{m - 1} λ i T i - λ m (t - (α T p + t m - 1)) Π (τ) d τ \dots - \int_{t S - 1}^{T S} e - λ S (T s - τ) - (α - 1) \sum_{i = 1}^{S} λ i T i - \sum_{i = 1}^{m - 1} λ i T i - λ m (t - (α T p + t m - 1)) Π (τ) d τ \dots - \int_{α T p}^{α T p + t 1} e - λ 1 (α T p + t 1 - τ) - \sum_{i = 1}^{m - 1} λ i T i - λ m (t - (α T p + t m - 1)) Π (τ) d τ \dots - \int_{α T p + t m}^{t} e - λ m (t - τ) Π (τ) d τ = V (0) e - α \sum_{i = 1}^{S} λ i T i - \sum_{i = 1}^{m - 1} λ i T i - λ m (t - (α T p + t m - 1)) - \sum_{k = 1}^{α} \sum_{j = 1}^{S} \int_{(k - 1) T p + t j - 1}^{(k - 1) T p + t j} e - λ j ((k - 1) T p + t j - τ) - \sum_{i = j + 1}^{S} λ i T i - (α - k) \sum_{i = 1}^{S} λ i T i - \sum_{i = 1}^{m - 1} λ i T i - λ m (t - (α T p + t m - 1)) Π (τ) d τ - \sum_{j = 1}^{m - 1} \int_{α T p + t j - 1}^{α T p + t j} e - λ j (α T p + t j - τ) - \sum_{i = j + 1}^{m - 1} λ i T i - λ m (t - (α T p + t m - 1)) Π (τ) d τ - \int_{α T p + t m - 1}^{t} e - λ m (t - τ) Π (τ) d τ

(16)

Inequality (16) is equivalent to

V (t) + \sum_{k = 1}^{α} \sum_{j = 1}^{S} \int_{(k - 1) T p + t j - 1}^{(k - 1) T p + t j} e - λ j ((k - 1) T p + t j - τ) - \sum_{i = j + 1}^{S} λ i T i - (α - k) \sum_{i = 1}^{S} λ i T i e - \sum_{i = 1}^{m - 1} λ i T i - λ m (t - (α T p + t m - 1)) z T (τ) z (τ) d τ + \sum_{j = 1}^{m - 1} \int_{α T p + t j - 1}^{α T p + t j} e - λ j (α T p + t j - τ) - \sum_{i = j + 1}^{m - 1} λ i T i - λ m (t - (α T p + t m - 1)) z T (τ) z (τ) d τ + \int_{α T p + t m - 1}^{t} e - λ m (t - τ) z T (τ) z (τ) d τ \leq V (0) e - α \sum_{i = 1}^{S} λ i T i - \sum_{i = 1}^{m - 1} λ i T i - λ m (t - (α T + t m - 1)) + γ 2 (\sum_{k = 1}^{α} \sum_{j = 1}^{S} \int_{(k - 1) T p + t j - 1}^{(k - 1) T p + t j} e - λ j ((k - 1) T p + t j - τ) - \sum_{i = j + 1}^{S} λ i T i \times e - (α - k) \sum_{i = 1}^{S} λ i T i - \sum_{i = 1}^{m - 1} λ i T i - λ m (t - (α T p + t m - 1)) w T (τ) w (τ) d τ + \sum_{j = 1}^{m - 1} \int_{α T p + t j - 1}^{α T p + t j} e - λ j (α T p + t j - τ) - \sum_{i = j + 1}^{m - 1} λ i T i - λ m (t - (α T p + t m - 1)) w T (τ) w (τ) d τ + \int_{α T p + t m - 1}^{t} e - λ m (t - τ) w T (τ) w (τ) d τ)

(17)

with V(t) > 0, under zero initial conditions, and then we obtain

\int_{0}^{t} e - λ max (t - τ) z T (τ) z (τ) d τ \leq \sum_{k = 1}^{α} \sum_{j = 1}^{S} \int_{(k - 1) T p + t j - 1}^{(k - 1) T p + t j} e - λ j ((k - 1) T p + t j - τ) - \sum_{i = j + 1}^{S} λ i T i - (α - k) \sum_{i = 1}^{S} λ i T i \times e - \sum_{i = 1}^{m - 1} λ i T i - λ m (t - (α T p + t m - 1)) z T (τ) z (τ) d τ + \sum_{j = 1}^{m - 1} \int_{α T p + t j - 1}^{α T p + t j} e - λ j (α T p + t j - τ) - \sum_{i = j + 1}^{m - 1} λ i T i - λ m (t - (α T p + t m - 1)) z T (τ) z (τ) d τ + \int_{α T p + t m - 1}^{t} e - λ m (t - τ) z T (τ) z (τ) d τ \leq γ 2 (\sum_{k = 1}^{α} \sum_{j = 1}^{S} \int_{(k - 1) T p + t j - 1}^{(k - 1) T p + t j} e - λ j ((k - 1) T p + t j - τ) - \sum_{i = j + 1}^{S} λ i T i e - (α - k) \sum_{i = 1}^{S} λ i T i - \sum_{i = 1}^{m - 1} λ i T i - λ m (t - (α T p + t m - 1)) w T (τ) w (τ) d τ + \sum_{j = 1}^{m - 1} \int_{α T p + t j - 1}^{α T p + t j} e - λ j (α T p + t j - τ) - \sum_{i = j + 1}^{m - 1} λ i T i - λ m (t - (α T p + t m - 1)) w T (τ) w (τ) d τ + \int_{α T p + t m - 1}^{t} e - λ m (t - τ) w T (τ) w (τ) d τ) \leq γ 2 \int_{0}^{t} e - λ min (t - τ) w T (τ) w (τ) d τ

(18)

Integrating (18) from t = 0 to ∞, we have

\int_{0}^{\infty} z T (τ) z (τ) \leq λ max λ min γ 2 \int_{0}^{\infty} w T (τ) w (τ) d τ

4. H_∞ Controller synthesis

In this section, our objective is to design a state-feedback controller such that the closed-loop vibration system is exponentially stable and has an H_∞ disturbance attenuation performance. First, controller saturation will be studied independently. Then, an H_∞ controller with saturation consideration will be proposed for periodic piecewise vibration systems.

4.1. Controller saturation

Controller saturation is always encountered in practice. When considering saturation, periodic piecewise vibration system (4) can be rewritten as

\overset{\cdot}{q} (t) = A (t) q (t) + B (t) SAT (u) + B w (t) w (t), z (t) = C (t) q (t) + D (t) SAT (u) + D w (t) w (t)

(19)

where the actuator saturation expression SAT(u) is given by SAT(u) = [sat(u₁), … , sat(u_j), … , sat(u_n)]^T, and sat(u_j) is the saturation function with the limit of

u_{j}^{lim}

for the jth actuator, where j = 1,2, … , n, that is

sat (u j) = {u j, | u j | \leq u_{j}^{lim} sgn (u j) u_{j}^{lim}, | u j | > u_{j}^{lim}

(20)

Introducing a transforming matrix Φ(p) = diag[p₁, … , p_n] with

p j = {sat (u j) u j, u j \neq 0 1, u j = 0

(21)

such that SAT(u) = Φ(p)u. Allowing the command force of the jth actuator to be

δ j u_{j}^{lim}

for an arbitrary scalar δ_j > 1, controllers with large gains can be obtained (Nguyen and Jabbari, 1999). Therefore, p_j is bounded within

1 δ j

and 1, that is

p \in Θ {p : 1 δ j \leq p j \leq 1, j = 1, 2, \dots, n}

(22)

and then the corresponding vertex set can be established as

Θ vex {p : p j = 1 δ j or p j = 1, j = 1, 2, \dots, n}

(23)

Consider a state-feedback periodic controller u(t) = u(t + T_p) and u(t) = K_iq(t), i ∈ [t_i−1, t_i), and K_i is the controller gain for the ith subsystem to be determined. For convenience, we use the double subscripts ij to denote the order of the subsystem and the controller, that is, we have

u ij = K_{i}^{j} q (t)

, where u_ij is the jth controller in the ith subsystem and

K_{i}^{j}

is the jth row of the gain matrix K_i. Then the closed-loop system can be given as

\overset{\cdot}{q} (t) = (A i + B i Φ (p) K i) q (t) + B wi w z (t) = (C i + D i Φ (p) K i) q (t) + D wi w, t \in [t i - 1, t i)

(24)

Assume that the closed-loop system is stable and the disturbance satisfies ‖w‖_∞ ≤ w_max, and then the reachable set of closed-loop system (24) can be characterized by the following theorem.

Theorem 2

Consider periodic piecewise vibration system (19). If there exist β > 0, matrix Q > 0 and state-feedback controller u(t) = K_iq(t), i = 1, 2, … , S such that the following matrix inequalities are feasible

[A_{i}^{T} Q + Q A i + K_{i}^{T} Φ (p) B_{i}^{T} Q + QB i Φ (p) K i + β Q QB i B_{i}^{T} Q - β I] \leq 0

(25)

then an estimate of the reachable set of the closed-loop system is given via an ellipsoid

{q : q T Qq \leq w_{max}^{2}}

. Moreover, for given

δ j > 1, u_{j}^{lim} > 0, i = 1, 2, \dots, S, j = 1, 2, \dots, n

, if

[Q (K_{i}^{j}) T K_{i}^{j} (δ j u_{j}^{lim} w max) 2] > 0

(26)

then

‖ u ij ‖ \infty \leq δ j u_{j}^{lim}

within the ellipsoid

{q : q T Qq \leq w_{max}^{2}}

Proof

The proof is similar to the one in Nguyen and Jabbari (1999). Notice that, for the periodic piecewise vibration system (19) with zero initial conditions, according to (25), we have $q T (t_{1}^{-}) Qq (t_{1}^{-}) \leq w_{max}^{2}$ . Since the state is continuous, we have $q T (t_{1}^{+}) Qq (t_{1}^{+}) \leq w_{max}^{2}$ which indicates that the initial state of the second subsystem is within the ellipsoid, and then we obtain $q T (t_{2}^{-}) Qq (t_{2}^{-}) \leq w_{max}^{2}$ . This implies that the system state will not escape from the invariant ellipsoid $q T (t) Qq (t) \leq w_{max}^{2}$ .

Remark 1

It should be mentioned that a common Lyapunov function V(t) = q^T(t)Qq(t) is employed to develop the conditions, which easily guarantees an invariant ellipsoid for the state in all subsystems. On the other hand, only the vertices of Θ_vex need to be checked due to the affine dependence of the matrix inequalities.

Remark 2

To deal with controller saturation, the above technique requires each actuator limit being available and the estimation of the disturbance peak value being known in advance, which can be obtained easily in practice.

4.2. Controller synthesis with actuator saturation

Our purpose here is to find a controller with saturation considered such that for any peak-bounded disturbance w, the energy of the output z satisfies (14). The system block with controller saturation is depicted in Figure 2. The existence conditions for a set of H_∞ controllers with saturation can be given as follows.

Figure 2.

Schematic of a closed-loop system with controller saturation.

Theorem 3

Consider periodic piecewise vibration system (19), if there exist a set of λ_i > 0, i = 1, 2, … , S, γ, β > 0, matrices Q > 0, P_i > 0, i = 1, 2, … , S, and state feedback controller u(t) = K_ix(t), i = 1, 2, … , S such that

[{A_{i}^{T} P i + P i A i + K_{i}^{T} Φ (p) B_{i}^{T} P i + P i B i Φ (p) K i + P i + 1 - P i T i + λ i P i} P i B wi C_{i}^{T} + K_{i}^{T} Φ (p) D_{i}^{T} * - γ 2 I D_{wi}^{T} * * - I] < 0

(27)

[{A_{i}^{T} P i + 1 + P i + 1 A i + K_{i}^{T} Φ (p) B_{i}^{T} P i + 1 + P i + 1 B i Φ (p) K i + P i + 1 - P i T i + λ i P i + 1} P i + 1 B wi C_{i}^{T} + K_{i}^{T} Φ (p) D_{i}^{T} * - γ 2 I D_{wi}^{T} * * - I] < 0

(28)

P S + 1 = P 1

(29)

[A_{i}^{T} Q + Q A i + K_{i}^{T} Φ (p) B_{i}^{T} Q + QB i Φ (p) K i + β Q QB i B_{i}^{T} Q - β I] \leq 0

(30)

[Q (K_{i}^{j}) T K_{i}^{j} (δ j u_{j}^{lim} w max) 2] > 0

(31)

and p ∈ Θ_vex, then the closed-loop system is exponentially stable and satisfies

\int_{0}^{\infty} z T (τ) z (τ) d τ \leq λ max λ min γ 2 \int_{0}^{\infty} w T (τ) w (τ) d τ

(32)

where λ_max and λ_min are defined in Theorem 1.

The proof can be obtained easily by combining Theorem 1 and Theorem 2.

Remark 3

It should be noticed that different Lyapunov functions are used in the H_∞ performance requirement and controller saturation demand. A Lyapunov function with a continuous time-varying Lyapunov matrix is adopted to derive the H_∞ performance in order to achieve a less conservative result, while a quadratic Lyapunov function is employed to deal with the controller saturation, which can easily ensure an invariant ellipsoid for the system state. A Lyapunov function in the form of (8) can also be employed for controller saturation, but the derivation will become much more complicated, a quadratic Lyapunov function is more suitable to ensure the controller saturation performance. The coefficient $γ 2 λ max λ min$ is thus the performance index of the disturbance attenuation.

Notice that the condition in Theorem 3 is nonconvex with respect to the decision variables which cannot be solved directly. The following iteration algorithm is developed to obtain the controller gain.

Algorithm HC

Set ℓ = 1, select an initial set of $λ_{i}^{(1)} = 0, : K_{i}^{(1)} = 0, i = 1, 2, \dots, S$ , $λ_{min}^{0} = 0, λ_{max}^{0} > 0, γ_{2}^{0} = 0,$ and set β⁽¹⁾ > 0 to be sufficiently small.

For fixed $K_{i}^{(ℓ)}, λ_{i}^{(ℓ)}, β (ℓ)$ , solve the following convex optimization problem for $P_{i}^{(ℓ)}, P_{i + 1}^{(ℓ)}, Q (ℓ), γ$ . OP1: minimize γ subject to the following convex constraints

[Ψ (ℓ) (i) P_{i}^{(ℓ)} B wi C_{i}^{T} + K_{i}^{(ℓ) T} Φ (p) D_{i}^{T} * - γ 2 I D_{wi}^{T} * * - I] < 0

(33)

[Ψ (ℓ) (i + 1) P_{i + 1}^{(ℓ)} B wi C_{i}^{T} + K_{i}^{(ℓ) T} Φ (p) D_{i}^{T} * - γ 2 I D_{wi}^{T} * * - I] < 0

(34)

[{A_{i}^{T} Q (ℓ) + Q (ℓ) A i + K_{i}^{(ℓ) T} Φ (p) B_{i}^{T} Q + QB i Φ (p) K_{i}^{(ℓ)} + β (ℓ) Q (ℓ)} Q (ℓ) B i B_{i}^{T} Q (ℓ) - β (ℓ) I] \leq 0

(35)

[Q (ℓ) (K_{i}^{j (ℓ)}) T K_{i}^{j (ℓ)} (δ j u_{lim}^{j} w max) 2] > 0

(36)

P_{i}^{(ℓ)} > 0, P_{i + 1}^{(ℓ)} > 0, P_{S + 1}^{(ℓ)} = P_{1}^{(ℓ)}, Q (ℓ) > 0

(37)

where

Ψ (ℓ) (i) = A_{i}^{T} P_{i}^{(ℓ)} + P_{i}^{(ℓ)} A i + K_{i}^{(ℓ) T} Φ (p) B_{i}^{T} P_{i}^{(ℓ)} + P_{i}^{(ℓ)} B i Φ (p) K_{i}^{(ℓ)} + P_{i + 1}^{(ℓ)} - P_{i}^{(ℓ)} T i + λ_{i}^{(ℓ)} P_{i}^{(ℓ)}

Ψ (ℓ) (i + 1) = A_{i}^{T} P_{i + 1}^{(ℓ)} + P_{i + 1}^{(ℓ)} A i + K_{i}^{(ℓ) T} Φ (p) B_{i}^{T} P_{i + 1}^{(ℓ)} + P_{i + 1}^{(ℓ)} B i Φ (p) K_{i}^{(ℓ)} + P_{i + 1}^{(ℓ)} - P_{i - 1}^{(ℓ)} T i + λ_{i}^{(ℓ)} P_{i + 1}^{(ℓ)}

Denote

γ_{1}^{(ℓ)} = γ, P_{i}^{(ℓ)}, P_{i + 1}^{(ℓ)}, Q (ℓ)

as the solution to the optimization problem. If

| γ_{1}^{(ℓ)} - γ_{2}^{(ℓ - 1)} | < ɛ

, where ɛ > 0 is a prescribed bound, then set ℓ = ℓ + 1,

P_{i}^{(ℓ)} = P_{i}^{(ℓ - 1)}, P_{i + 1}^{(ℓ)} = P_{i + 1}^{(ℓ - 1)}, Q (ℓ) = Q (ℓ - 1)

. STOP.

For fixed $P_{i}^{(ℓ)}, P_{i + 1}^{(ℓ)}, Q (ℓ)$ , solve the following convex optimization problem for $K_{i}^{(ℓ)}, λ_{i}^{(ℓ)}, β (ℓ)$ . OP2: minimize γ subject to (33)–(37) and

σ 1 λ_{min}^{(ℓ - 1)} < λ_{i}^{ℓ} < σ 2 λ_{max}^{(ℓ - 1)}

(38)

where σ₁ ≤ 1, σ₂ ≥ 1 are prescribed constants. Denote

γ_{2}^{(ℓ)} = γ, K_{i}^{(ℓ)}, λ_{i}^{(ℓ)}, β (ℓ)

as the solution to the optimization problem.

If $| γ_{2}^{(ℓ)} - γ_{1}^{(ℓ)} | < ɛ$ , STOP, else set ℓ = ℓ + 1 and $λ_{i}^{(ℓ)} = λ_{i}^{(ℓ - 1)},$ β^(ℓ) = β^{(ℓ − 1)}, $K_{i}^{(ℓ)} = K_{i}^{(ℓ - 1)},$ then go to Step 2.

Remark 4

K_i, λ_i, λ_min, γ₂ are set to be zero and λ_max > 0 as the initial conditions for the H_∞ disturbance attenuation performance, and a sufficiently small β > 0 is chosen in the beginning for controller saturation performance. This initial set of K_i, λ_i, λ_max, λ_min and β guarantees the algorithm can proceed. During the iteration, the parameter γ can be reduced since the K, λ, β obtained in Step 3 will be utilized as the initial value in Step 2 to derive a smaller γ, the convergence of γ in this algorithm is naturally guaranteed. Notice that σ₁ can be chosen slightly less than one and σ₂ is slightly greater than one, thus giving more freedom in computation of λ_i. At the same time, (38) bounds the ratio of λ_max/λ_min. Combining the convergence of γ and bounded ratio of λ_max/λ_min, the performance index $γ 2 λ max λ min$ will converge during iteration.

Remark 5

δ^j > 1 is fixed in advance which is used to increase the controller gain. A greater δ^j can maximize the utilization of the available control capability Nguyen and Jabbari (1999). However, a greater δ^j will also increase the size of the polytope Θ_vex, which will deteriorate the H_∞ performance. A suitable δ^j needs to chosen through tuning to balance the trade-off between the H_∞ performance and the controller gain.

5. Numerical example

In this section, simulations are given to illustrate the merits of the proposed technique. A 2-degree-of-freedom (DOF) vibration system is shown in Figure 3, which includes masses m₁, m₂, springs k₁, k₂, dampers c₁, c₂ and one actuator which provides a force F_c. A disturbance is applied on m₂. Assume that m₂ is periodic piecewise linear. This kind of periodic vibration system is an epitome of some manufacturing machines or conveying structures which have periodic loads. The model is not complicated, but it is representative since other complicated models can be developed from it. The system parameters are given as m₁ = 4 kg, k₁ = 100 N/m, c₁ = 5 N s/m, k₂ = 50 N/m and c₂ = 5 N s/m (Filipovle and Schroder, 2001). Moreover, the system period is 15 s with t₁ = 5 s, t₂ = 10 s and t₃ = 15 s, respectively, and assume the period of m₂ variation is 15 s, and the value of m₂ over a period is given as

m 2 = {1.8 kg, t \in [0, 5) s 2 kg, t \in [5, 10) s 2.2 kg, t \in [10, 15) s

Figure 3.

A 2-DOF vibration system.

The system state-space form can be established according to (5) and the subsystem matrices can be given as

A 1 = [00100001 - 37.50 12.50 - 2.50 1.25 27.78 - 27.78 2.78 - 2.78], A 2 = [00100001 - 37.50 12.50 - 2.50 1.25 25.00 - 25.00 2.50 - 2.50] A 3 = [00100001 - 37.50 12.50 - 2.50 1.25 22.73 - 22.73 2.27 - 2.27], B 1 = B 2 = B 3 = [00 - 0.25 0.56] T B w 1 = B w 2 = B w 3 = [000 0.56] T

(39)

Our purpose here is to attenuate the displacement of m₂, that is, x₂. Hence, we have C₁ = C₂ = C₃ = [0, 1, 0, 0], D₁ = D₂ = D₃ = 0 and D_w1 = D_w2 = D_w3 = 0.

Given a sinusoidal disturbance as w = 10sin(1.2πt), t = [0,45), then w_max = 10. The disturbance frequency is chosen to be 0.6 Hz because the system vibrates heavily at this frequency. The control objective is attenuating the vibration at this frequency. First, we study the system response under control without considering actuator saturation, which is defined as Case I. The controller gain can be calculated according to Algorithm HC, only considering inequalities (33), (34) and (37). The controller gain is obtained as

K 1 = [194.429 - 527.668 - 50.317 - 143.802] K 2 = [337.016 - 795.486 - 64.245 - 242.142] K 3 = [184.994 - 473.979 - 43.200 - 137.999]

and ‖K₁‖ = 582.621, ‖K₂‖ = 899.521, ‖K₃‖ = 528.9493. The system response comparison is shown in Figure 4, and the control output is shown in Figure 5. It can be seen that the obtained controller can greatly attenuate the vibration of the periodic piecewise vibration system and the maximum actuator force is 20 N.

Figure 4.

Response comparison of open- and closed-loop systems in Case I.

Figure 5.

Actuator force in Case I.

Then we discuss the condition that the actuator has saturation. Two cases are studied here, that is, Case II: controller designing without considering saturation, and Case III: controller designing with considering saturation. The actuator limit is 10 N, which is half of the control demand in Case I. The controller gain in Case II is the same as in Case I and the controller gain in Case III is computed according to Algorithm HC, which is obtained as

K 1 = [26.9471 - 9.1840 15.2567 - 16.9415] K 2 = [26.9471 - 17.1655 12.1869 - 15.0335] K 3 = [41.9097 - 34.1406 13.5497 - 16.3764]

and ‖K₁‖ = 31.9639, ‖K₂‖ = 37.3541, ‖K₃‖ = 58.0842. It can be seen that the controller gains are much smaller than those in Case I and Case II. The evolutions of γ and the performance index

γ 2 λ max λ min

are shown in Figures 6 and 7, from which, the convergence of γ and

γ 2 λ max λ min

can be observed. The response comparison is shown in Figure 8 and the control demand and actuator force in Case II and Case III are given in Figure 9 and Figure 10, respectively. It can be observed that the system responses of the closed-loop system are significantly smaller than the open-loop system, and the improvements of the closed-loop system in Case II and Case III are almost at the same level. However, the controller demand in Case II is much greater than the demand in Case III and the actuator limitation. Huge control demand will result in a large voltage signal input which may damage the actuator. It illustrates the importance of considering saturation when designing the controller.

Figure 6.

Evolution of γ.

Figure 7.

Evolution of $γ 2 λ max λ min$ .

Figure 8.

Response of open- and closed-loop systems in Case II and Case III.

Figure 9.

Actuator force in Case II. (a) Control demand. (b) Controller output.

Figure 10.

Actuator force in Case III. (a) Control demand. (b) Control output.

6. Conclusion

This paper has provided a controller design approach with H_∞ disturbance attenuation performance and actuator saturation consideration for periodic piecewise vibration systems. The H_∞ performance criterion has been studied for periodic piecewise vibration system first, and then a state-feedback controller with actuator saturation taken into account has been developed based on matrix inequalities. Since the conditions cannot be solved directly, an algorithm has been proposed to obtain the controller gain. Simulation results have demonstrated the controller based on the proposed method being effective in reducing the response of the representative vibration system.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work was partially supported by GRF HKU 17205815 and HKU CRCG 201409176081.

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H ∞ control of periodic piecewise vibration systems with actuator saturation

Abstract

Keywords

1. Introduction

2. Vibration model description

Definition 1

3. H∞ performance of periodic piecewise vibration system

Theorem 1

Proof

4. H∞ Controller synthesis

4.1. Controller saturation

Theorem 2

Proof

Remark 1

Remark 2

4.2. Controller synthesis with actuator saturation

Theorem 3

Remark 3

Remark 4

Remark 5

5. Numerical example

6. Conclusion

Footnotes

Declaration of Conflicting Interests

Funding

References

3. H_∞ performance of periodic piecewise vibration system

4. H_∞ Controller synthesis