Robust control for a class of secondary regulation rotate speed systems via Hamiltonian function method 1

Abstract

This paper investigates the robust control for a class of secondary regulation rotate speed systems by applying Hamiltonian function method, and proposes a number of new results on the system. Firstly, we present some Hamiltonian realization methods for a class of nonlinear systems. Then, by transforming the nonlinear system under study into a Hamiltonian form, the paper presents several stability and H_∞ control results, respectively. Finally, an illustrative example shows the effectiveness of the result obtained in this paper.

Keywords

Secondary regulation rotate speed system stability robust control hamiltonian function method

1 Introduction

A kind of new energy-saving system, Hydraulic secondary regulation system, is developed in recent years [1], mainly because it can recycle and reuse kinetic and gravitational potential of the system working body generated by braking. Secondary regulation technology plays a key role and has a wide range of applications in many fields [1 , 12–14]. Particularly, to make one or several of them to keep different speed, the speed control system has obvious advantages of energy-saving in several hydraulic actuator driven by an oil source system. Hence, the secondary regulation rotate speed system has attracted considerable attention in the past few decades, many nice results have been obtained for the systems [3, 7] and the references therein. Based on the state-space method, [7] studied the control problem for a class of double-input and single-output rotate speed systems, gave a genetic algorithm PID control strategy. In [3], the authors investigated H_∞ control of a class of secondary regulation speed control systems, presented a robust control strategy. The simulation shows that the control method has good robust stability and anti-disturbance capability. However, it should be pointed out that the method in [3] cannot be applied to multiple input and multiple output systems. Moreover, since the secondary regulation system is a multi-variable one, it is impossible to obtain better control behavior of such systems by applying traditional methods. Therefore, it needs to develop new research method.

Recently, port-controlled Hamiltonian (PCH) system, proposed in [5], has been well studied in [6 , 15–18], and applied to many practical control problems [2 , 11]. The main advantage of the method is that the Hamiltonian function in a PCH system is considered as the total energy and can be used as a candidate of Lyapunov function in many physical systems.

In this paper, we investigate the stability and H_∞ control problem of a class of secondary regulation rotate speed systems via Hamiltonian function method, and propose a number of new results on the issues for the systems. First, the concept of Hamiltonian realization of general nonlinear systems is given, and several Hamiltonian realization methods are proposed. Then, based on the Hamiltonian realization form, the stability and H_∞ control of the secondary regulation rotate speed systems are investigated, and several sufficient conditions are derived for the issues. Finally, an illustrative example is presented to support our new results. It should be pointed out that the key step in our method is to transform the system under study into a Hamiltonian one, which is different from the existing methods [3, 7]. Furthermore, the method presented of the paper can be used to study other control problem for secondary regulation rotate speed systems and other nonlinear systems (Please see Remark 3 below).

The remainder of the paper is organized as follows. Section 2 is the preliminaries. In Section 3, the stability and H_∞ control problems are discussed, and some new results are proposed for the systems. In Section 4, we give an illustrative example to support our new results, which is followed by the conclusion in Section 5.

Notation: Throughout this paper, Rⁿ denotes the n-dimensional Euclidean space, and R^n×m is the set of n × m real matrices. For symmetric matrices P and Q, the notation P > Q (P > Q) means that matrix P - Q is positive definite (positive semi-definite), and similarly, P < Q (P ≤ Q) means that matrix P - Q is negative definite (negative semi-definite). A matrix-valued function P : X → R^n×n is positive definite if P (x) is positive definite for each x ∈ X and there exists α > 0 such that P (x) > αI_n for all x ∈ X, where I_n is the n × n identity matrix. ∥x∥ denotes the Euclidean norm of x, and < ·, ·>stands for the inner product of Euclidean space.

2 Preliminaries

In this section, we present some preliminaries, which will be used in the sequel.

Consider the following nonlinear system: $\dot{x} (t) = f (x) + g (x) u,$ (1) where x (t) ∈ Rⁿ is the state, f (x) is smooth vector field satisfying f (0) =0.

The main objective of the paper is to study the stability with u = 0 and H_∞ control problem of the system (1) via the Hamiltonian function method.

The following concepts will be used in our analysis.

Definition 1. [9, 15] A continuous differentiable function H (x) is called a regular positive definite one with respect to x if there exists a class-K function α such that H (x) ≥ a (||x||), H (0) =0, ${\frac{\partial H}{\partial x} |}_{x = 0} = 0$ and ${\frac{\partial H}{\partial x} |}_{x \neq 0} \neq 0$ . For example, $H (x) = x_{1}^{2} + x_{2}^{2}$ is a regular positive definite function with respect to x on Rⁿ.

Definition 2. [9, 15] The system (1) is said to have a generalized Hamiltonian realization (GHR) if there exist a suitable coordinate chart and a function H (x) such that the system (1) can be expressed in the following Hamiltonian form $\dot{x} (t) = T (x) \nabla_{x} H (x) + g (x) u,$ (2) Where $\nabla_{x} H (x) : = \frac{\partial H (x)}{\partial x}$ .

Remark 1. It is well worth pointing out that the GHR of an arbitrary given system may have different form. For example, consider a system as following: $\dot{x} = f (x) = [\begin{matrix} 2 x_{1} + x_{1} x_{2} \\ - x_{2} \end{matrix}]$ (3)

By choosing a Hamiltonian function as $H (x) = 0.5 (x_{1}^{2} + x_{2}^{2})$ , we have following different realizations of the system (3): $\dot{x} = [\begin{matrix} 2 + x_{2} & 0 \\ 0 & - 1 \end{matrix}] \nabla_{x} H (x),$ and $\dot{x} = [\begin{matrix} 2 & x_{1} \\ 0 & - 1 \end{matrix}] \nabla_{x} H (x)$

Note that the key point in applying the Hamiltonian function method is to express the systems under study as the Hamiltonian form. Therefore, we present some GHR methods of the system (1).

Lemma 1. [9] The system (1) with u = 0 has the following orthogonal decomposition Hamiltonian realization $\dot{x} = L (x) \nabla_{x} H (x),$ where L (x) = J (x) + S (x) for x ≠ 0; and if x = 0, then L (x) =0, $\begin{matrix} \nabla_{x} H (x) & : = & \frac{\partial H (x)}{\partial x} and \\ J (x) & : = & \frac{1}{{∥ \nabla_{x} H ∥}^{2}} [f_{td} \nabla_{x} H^{T} - \nabla_{x} {Hf}_{td}^{T}], \\ S (x) & : = & \frac{〈 f, \nabla_{x} H 〉}{{∥ \nabla_{x} H ∥}^{2}} I_{n}, \\ f_{gd} (x) & : = & \frac{〈 f, \nabla_{x} H 〉}{{∥ \nabla_{x} H ∥}^{2}} \nabla H, \\ f_{td} (x) & : = & f (x) - f_{gd} (x) . \end{matrix}$

Next, we list another result on the GHR of the system (1).

Consider the system (1), and set $\begin{matrix} A_{i} & = & {(\frac{\partial f}{\partial x_{i}})}^{T} = (\frac{\partial f_{1}}{\partial x_{i}}, \dots, \frac{\partial f_{n}}{\partial x_{i}}), and \\ a_{i} & = & \frac{\partial}{\partial x_{i}} (i = 1, 2, \dots, n) \end{matrix}$

Construct two equations as follows $[\begin{matrix} A_{2} & - A_{1} \\ A_{3} & - A_{1} \\ ⋮ & ⋱ \\ A_{n} & - A_{1} \\ A_{3} & - A_{2} \\ ⋮ & ⋱ \\ A_{n} & - A_{2} \\ ⋮ & ⋮ \\ A_{n} & - A_{n - 1} \end{matrix}] [\begin{matrix} X_{1} (x) \\ X_{2} (x) \\ ⋮ \\ ⋮ \\ X_{n} (x) \end{matrix}] = 0$ (4) $\begin{matrix} ([\begin{matrix} a_{2} & - a_{1} \\ a_{3} & - a_{1} \\ ⋮ & ⋱ \\ a_{n} & - a_{1} \\ a_{3} & - a_{2} \\ ⋮ & ⋱ \\ a_{n} & - a_{2} \\ ⋮ & ⋮ \\ a_{n} & - a_{n - 1} \end{matrix}] \otimes I_{n}) [\begin{matrix} X_{1} (x) \\ X_{2} (x) \\ ⋮ \\ ⋮ \\ X_{n} (x) \end{matrix}] = 0 \end{matrix}$ (5)

where X_i (x) (i = 1 ;2 ; … ; n) are n-dimensional column vector fields, ⊗ is the Kronecker product, and for an arbitrary scalar function h (x), we define $a_{i} h (x) = \frac{\partial h (x)}{\partial x_{i}}, i = 1, 2, \dots, n .$

Lemma 2. [9] If Equations (4) and (5) have a solution X₁ (x) , X₂ (x) , ⋯ , X_n (x) such that the matrix ${[X_{1} (x), \dots, X_{n} (x)]}^{T}$ is nonsingular, then there exists a Hamiltonian function H (x) such that the system (1) has a GHR as follows $\dot{x} (t) = L (x) \nabla_{x} H (x) + g (x) u,$ (6) where $L (x) : = {[X_{1} (x), \dots, X_{n} (x)]}^{- T}$ .

Using Lemma 2, we obtain the following result on a kind of constant GHR, whose structural matrix is a constant one of x for the system (1).

Corollary 1. If Equation (4) has a constant solution X₁, ⋯ , X_n such that the matrix ${[X_{1}, \dots, X_{n}]}^{T}$ is nonsingular, then there exists a Hamiltonian function H (x) such that the system (1) has a constant Hamiltonian realization as follows $\dot{x} (t) = L \nabla_{x} H (x) + g (x) u,$ where $L : = {[X_{1}, \dots, X_{n}]}^{- T}$

Remark 2. The Hamiltonian function H (x) in Lemma 2 or Corollary 1 can be calculated by $\begin{matrix} H (x) & = & \int_{x_{1}^{(0)}}^{x_{1}} h_{1} (x_{1}, x_{2}, \dots, x_{n}) d x_{1} \\ + \int_{x_{2}^{(0)}}^{x_{2}} h_{2} (x_{1}^{(0)}, x_{2}, \dots, x_{n}) {dx}_{2} + \dots \\ + \int_{x_{n}^{(0)}}^{x_{n}} h_{n} (x_{1}^{(0)}, x_{2}^{(0)}, \dots, x_{n - 1}^{(0)}, x_{n}) {dx}_{n} \end{matrix}$ where $x^{(0)} : = {(x_{1}^{(0)}, x_{2}^{(0)}, \dots, x_{n}^{(0)})}^{T} \in R^{n}$ is an arbitrary initial point.

In the following, we present a result on the H_∞ control problem of the Hamiltonian system.

Consider the following ported-control Hamiltonian system with z = h (x) g^T (x) ∇ xH (x) being the penalty signal of the system as: $\dot{x} = [J (x) + R (x)] \nabla_{x} H (x) + g_{1} (x) u + g_{2} (x) w,$ (7) where J^T (x) = - J (x) and R (x) ≥0, H (x) is a Hamiltonian function and H (0) =0, z, g₁ (x), g₂ (x), h (x) are weighted matrices with appropriate dimensions, u is the control input, w is the external disturbance.

The H_∞ control problem of the system (7) is to, for a given γ > 0, find a control law u = α (x) such that the closed-loop system consisting of the system (7) and the control law is asymptotically stable when w vanishes, and meantime, for any non-zero w ∈ L₂ [0, T], the zero state response of the closed-loop system satisfies. $\int_{0}^{T} {∥ z (t) ∥}^{2} dt \leq γ^{2} \int_{0}^{T} {∥ w (t) ∥}^{2} dt, \infty > T \geq 0 .$ (8)

Lemma 3. [10] For the give γ > 0, consider the system (7). Assume that the system (7) is generalized zero energy detectable (GZED) with u = 0 (that is, if z = 0 and w = 0 ⇒ ∇ _xH (x) =0), and

H (x) ∈ C², and its Hessian matrix Hess(H (x₀)) ≥0;

the following inequality 0 ≤ W (x) : = $R (x) + \frac{1}{2 γ^{2}} [g_{1} (x) g_{1}^{T} (x) - g_{2} (x) g_{2}^{T} (x)]$ (9)

Then, an H_∞ controller of the system (7) can be designed as $u = - 0.5 [h^{T} (x) h (x) + \frac{1}{γ^{2}} I_{m}] g_{1}^{T} (x) \nabla_{x} H (x) .$

3 Main result

In this section, we apply the results given in Section 2 to investigate the stability and robust control problem of the following secondary regulation rotate speed system [7]. $\dot{x} (t) = f (x) + g (x) u,$ (10) where $\begin{matrix} f (x) & = & [\begin{matrix} - \frac{K_{v} K_{cd}}{A} x_{1} - \frac{K_{v} K_{p} K_{cs} P_{0}}{{Ay}_{max}} x_{3} \\ x_{3} \\ \frac{1}{J} x_{1} - \frac{R_{N}}{J} x_{3} \end{matrix}] \\ : = & [\begin{matrix} - {ax}_{1} - {bx}_{3} \\ x_{3} \\ {cx}_{1} - R_{N} {cx}_{3} \end{matrix}], \\ g (x) & = & [\begin{matrix} \frac{K_{v} K_{p} V_{2 max} P_{0}}{{Ay}_{max}} : = d & 0 \\ 0 & 0 \\ 0 & - \frac{1}{J} \end{matrix}] \end{matrix}$

$u = [\begin{matrix} U (s) \\ M_{L} (s) \end{matrix}]$ , and the physical meaning of prameters can be found in [7].

To apply the result in Section 2, we first design a controller $u = [\begin{matrix} - {ex}_{1} \\ J (x_{2} + {kx}_{3}) \end{matrix}] + l$ , where $l = [\begin{matrix} U (s) \\ M_{L} (s) \end{matrix}]$ is a new input. Substituting u into the system (10), one can obtain $\dot{x} = f (x) + g (x) l,$ (11)

where $\begin{matrix} f (x) & = & [\begin{matrix} - (a + de) x_{1} - {bx}_{3} \\ x_{3} \\ {cx}_{1} - x_{2} - (R_{N} c + k) x_{3} \end{matrix}], \\ g (x) & = & [\begin{matrix} d & 0 \\ 0 & 0 \\ 0 & - c \end{matrix}] . \end{matrix}$

Obviously, the system (11) can be expressed as the following Hamiltonian form: $\dot{x} = [J (x) - R (x)] \nabla_{x} H (x) + g_{1} (x) u + g (x) l,$ (12)

Where $H (x) = 0.5 (x_{1}^{2} + x_{2}^{2} + x_{3}^{2})$ . $\begin{matrix} J (x) & = & [\begin{matrix} 0 & 0 & - \frac{b + c}{2} \\ 0 & 0 & 1 \\ \frac{b + c}{2} & - 1 & 0 \end{matrix}], \\ R (x) & = & [\begin{matrix} a + de & 0 & \frac{b - c}{2} \\ 0 & 0 & 0 \\ \frac{b - c}{2} & 0 & R_{N} c + k \end{matrix}] \end{matrix}$

First, we present a stability result on the system (12).

Theorem 1. Consider the system (12) with l = 0. Assume that there exist constant real numbers e > 0 and k > 0 such that a + de + cR_N + k > 0 and (a + de) (cR_N + k) - (c - b) ²/4 >0 hold. Then, the system (12) is stable.

Proof. Choose $H (x) = 0.5 (x_{1}^{2} + x_{2}^{2} + x_{3}^{2})$ as a Lyapunov function candidate. Computing the derivative of H (x) along the trajectory of the system (12), and using $\nabla_{x}^{T} H (x) J (x) \nabla_{x} H (x) = 0$ , we have

$\begin{matrix} \dot{H} (x) & = & \nabla_{x}^{T} H (x) [J (x) - R (x)] \nabla_{x} H (x) \\ = & - \nabla_{x}^{T} H (x) R (x) \nabla_{x} H (x) . \end{matrix}$ (13)

To prove that the system (12) is stable under the conditions of the theorem, it needs to show R (x) ≥0, which implies that $\dot{H} (x) \leq 0$ . To do this, by solving the characteristic root of the determinant, |λI - R (x) |=0, one can obtain the following equation $\begin{matrix} λ [λ^{2} - λ (a + de + {cR}_{N} + k) \\ + (a + de) ({cR}_{N} + k) - \frac{{(c - b)}^{2}}{4}] = 0 \end{matrix}$

From the conditions of the theorem, it is easy to obtain that all characteristic roots λ ≥ 0, that is, R (x) ≥0. Therefore, the system is stable.

Next, we present a robust control result on the system (12).

Consider the system (12) and decompose the system as the following form $\dot{x} = [J (x) - R (x)] \nabla_{x} H (x) + g (x) v + g (x) w,$ (14) where J (x), R (x), H (x) and g (x) are the same as (12), $v = [\begin{matrix} U (s) \\ U_{1} (s) \end{matrix}]$ is the control input, $w = [\begin{matrix} 0 \\ M_{L} (s) - U_{1} (s) \end{matrix}]$ is the external disturbance (Note:M_L (s) is a load torque, hence it can be considered as a disturbance [3]). Without loss of generality, let the disturbance attenuation level γ > 0 be given, and choose $z = h (x) g^{T} (x) \nabla_{x} H (x)$ (15) as the penalty signal of the system (14), where h (x) is a weighted matrix with appropriatedimension.

In the following, we apply Lemma 3 to give the robust control result of the system (14).

Theorem 2. Consider the system (14) with (15). Assume that there exist constant numbers γ > 0, e > 0 and k > 0 such that the inequalities a + de + cR_N + k > 0 and (a + de) (cR_N + k) - (c - b) ²/4 >0 hold. Then, an H_∞ controller of the system (14) with (15) can be designed as $v = - 0.5 [h^{T} (x) h (x) + \frac{1}{γ^{2}} I_{m}] g^{T} (x) \nabla_{x} H (x)$ .

Proof. We use Lemma 3 to prove the theorem. Obviously, the system is GZED and Condition (1) holds of Lemma 3.

Next, we prove that (9) is true under the conditions of the theorem. That is, we need to show W (x) ≥0.

Note that g₁ (x) = g₂ (x) = g (x), which implies that $g_{1} (x) g_{1}^{T} (x) - g_{2} (x) g_{2}^{T} (x) = 0 .$

In addition, from a + de + cR_N + k > 0 and (a + de) (cR_N + k) - (c - b) ²/4 >0, and the proof of Theorem 1, one can obtain R (x) ≥0. Therefore, we have $\begin{matrix} W (x) \\ : = R (x) + \frac{1}{2 γ^{2}} [g_{1} (x) g_{1}^{T} (x) - g_{2} (x) g_{2}^{T} (x)] \geq 0, \end{matrix}$ which implies that Condition (9) of Lemma 3 holds.

Thus, based on Lemma 3, one can design an H_∞ controller $u = - 0.5 [h^{T} (x) h (x) + \frac{1}{γ^{2}} I_{m}] g_{1}^{T} (x) \nabla_{x} H (x)$ of the system (14).

From the above, one can design an H_∞ controller for the secondary regulation rotate speed system (10).

First, decompose the system (10) as the following form: $\dot{x} (t) = f (x) + g (x) v + g (x) w,$ (16) where f (x) and g (x) are the same as (10), $v = [\begin{matrix} U (s) \\ U_{1} (x) \end{matrix}], w = [\begin{matrix} 0 \\ M_{L} (s) - U_{1} (x) \end{matrix}] .$

Let z = h (x) g^T (x) x as a penal sign, then we have the following H_∞ control result for the system (10).

Theorem 3. Consider the system (10). Assume that there exist constant numbers e > 0 and k > 0 such that a + de + cR_N + k > 0 and (a + de) (cR_N + k) - (c - b) ²/4 >0 hold. Then, an H_∞ controller of the system (10) can be designed as $\begin{matrix} v & = & [\begin{matrix} - {ex}_{1} \\ J (x_{2} + {kx}_{3}) \end{matrix}] \\ - 0.5 [h^{T} (x) h (x) + \frac{1}{γ^{2}} I_{m}] g^{T} (x) x . \end{matrix}$

Remark 3. From Theorems 1, 2 and 3, it is easy to see that the results obtained by applying the Hamiltonian function method are concise and have a unified form, which is an advantage of the method proposed in the paper. In addition, since the Hamiltonian method is a effective tool for the study of nonlinear systems, it may be extended to investigate the control problems of the nonlinear systems as Markovian jumping parameters [19 –22], fuzzy system [23, 24], Fault-Tolerant, fault detection and reliable control problems [25 –27]. This is our future research directions.

4 An illustrative example

In this section, we present an illustrative example of the system (10). To do this, let K_v = 4.45 × 10^-3, k_cd = 0.75, K_p = 0.2, K_cs = 1, V_max = 6.37 × 10^-6, P₀ = 20, A = 1.85 × 10^-3, y_max = 0.015, J = 2.27 and R_N = 500, then we have a = 1.8, b = 641.5, c = 0.44, d = 0.0041. Choose e = 5000 and k = 5000, one can obtain that the conditions of Theorem 3 hold. Thus, set h (x) = [1, 1] and γ = 0.5, then an H_∞ controller of the system (10) can be designed as $u = [\begin{matrix} - 5000.0185 x_{1} + 0.02 x_{3} \\ - 0.00205 x_{1} + 2.27 x_{2} + 11351.98 x_{3} \end{matrix}]$ (17)

To show the effectiveness of the control law (17), we carry out some numerical simulations for the system. The simulation results are shown in Figs. 1 and 2, which shows the response curves based on the input signal of step and sinusoidal signal, respectively, with the interference signal.

Seen from the figures, the dynamical characteristic of the secondary hydraulic transmission system is improved obviously via Hamiltonian function method, but also the system has strong anti-interference ability and good robustness under the controller of the paper. The square signal can be regarded as a serials of step signal, so the response curve is the same, anyhow, the sinusoidal signal shows the best response.

In addition, we also give a comparison with traditional PID control. Let the pressure of secondary transmission be 5.5 MPa, the accumulator charging pressure be 4.2 MPa, a volume be 16L, the load torque be 12.5 N·m and the total moment of inertia be 1.93 kg.m². By applying our method and the traditional PID control strategy, we give the simulation curve of the speed control system in Fig. 3.

From the simulation curve, it is easy to see that the rise time of the system is longer, the overshoot is obvious and the oscillation is serious under PID control, while under the controller designed in the paper, the system is no overshoot, has fast response, and static error is small, which implies that the controller of the paper has good characteristics.

5 Conclusion

In this paper, we have investigated the stability and robust control of a class of secondary regulation rotate speed systems via Hamiltonian function method, and proposed a number of new results on the issues for the systems. Some Hamiltonian realization methods have been given for general nonlinear system, and based on the forms, the stability and H_∞ control results have been obtained. Study of an illustrative example has supported the new result obtained in this paper.

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