Finite-time stabilization and H∞ control of Port-controlled Hamiltonian systems with disturbances and saturation

The finite-time stabilization and finite-time H∞ control problems of Port-controlled Hamiltonian (PCH) systems with disturbances and input saturation (IS) are studied in this paper. First, by designing an appropriate output feedback, a strictly dissipative PCH system is obtained and finite-time stabilization result for nominal system is given. Second, with the help of the Hamilton function method and truncation inequality technique, a novel output feedback controller is developed to make the PCH system finite-time stable when IS occurs. Further, a finite-time H∞ controller is designed to attenuate disturbances for PCH systems with IS, and sufficient conditions are presented. Finally, a numerical example and a circuit example are given to reveal the feasibility of the obtained theoretical results.


Introduction
The Port-controlled Hamiltonian (PCH) system has been studied well [1][2][3][4] since it was put forward [5,6]. In practical systems, the Hamilton function [7][8][9], as the total energy containing kinetic energy and potential energy, is a good candidate of Lyapunov function. Apart from the significant Hamilton function, the PCH system's other structures also have important physical meaning. Thus, many practical systems can be expressed as PCH systems and the PCH system has received wide attention in nonlinear analysis and synthesis [10][11][12][13]. Up to now, lots of results on stability analysis and control designs for PCH systems have been presented based on Hamilton function method [14][15][16][17][18].
Under asymptotically stabilized controller, system states can only converge to desired equilibrium points in infinite time, which is a common result. In order to optimize control time, the concept of finite-time stability naturally arises and further finite-time stability theory is developed to improve control performance. In fact, the finite-time control approach has the following significant superiorities: the closed-loop system possesses faster convergence speed and better robustness against uncertainties and disturbances [19][20][21][22]. Because of these advantages, finite-time control problems have received a great deal of attention and lots of results without and with IS, and the H 1 control problem of the case with IS and disturbances. Two examples with simulations are presented in Section 4. Section 5 gives the conclusion in final. Notation: The transposition of matrix g(z) is denoted by g T (z). The positive definite matrix R(z) is denoted by R(z)>0. The positive semi-definite matrix R(z) is denoted by R(z)�0. |a| stands for the absolute value of real number a. kRk represents the Euclidean norm of R. rE(z) denotes @EðzÞ @z .

Problem statement and preliminaries
A PCH system subject to disturbances and IS is considered _ z ¼ ½JðzÞ À RðzÞ�rEðzÞ þ g 1 ðzÞsatðUÞ þ g 2 ðzÞdðtÞ; where z 2 R n is the state, U 2 R m is the control input, y 2 R m is the output, dðtÞ 2 R s is the disturbance in L 2 , and z 2 R q is the penalty signal. The Hamilton function E(z) has a minimum point at z = 0, and rE(z) is the gradient of E(z). À J T ðzÞ ¼ JðzÞ 2 R n�n is the interconnection matrix, 0 � RðzÞ ¼ R T ðzÞ 2 R n�n is the damping matrix, M(z) is the weighting matrix, g 1 ðzÞ 2 R n�m is the full column rank gain matrix, (R, g 1 ) is a full row rank matrix, g 2 ðzÞ 2 R n�s , and satðUÞ ¼ ½satðU 1 Þ; satðU 2 Þ; . . . ; satðU m Þ� T is the IS function with p i is a positive real number which represents the upper bound of the saturated function sat (U i ).
For the subsequent analysis, the following four lemmas are adopted. Lemma 1 ([39]) Jensen's inequality: where r 1 , r 2 and a j are real numbers. Let r 1 ¼ 1 c and r 2 = 1 in Lemma 1, then the following inequality is derived The following system is considered If there is a C 1 radially unbounded Lyapunov function V(ψ) and a real number b > 1 making inequality (6) holds along system (5) with any c 0 2 R n ,  (2). Then, the inequality holds The following system is given where ψ is the state, z is the penalty signal and D is the disturbance. If there is a function V(ψ) satisfying the following Hamiltonian-Jacobian inequality, then the L 2 gain from D to z is no bigger than γ, i.e., where V(ψ) > 0 with ψ 6 ¼ 0, V(0) = 0, and γ > 0.

Remark 1 As the total energy function of PCH systems, the Hamilton function E(z) is usually selected as the form in Assumption 1, and it represents a very important class of Hamilton functions in mechanical systems.
In order to deal with the finite-time control problems of PCH system (1) with disturbances and IS, several novel control schemes are presented via output feedback strategies and truncation-inequality technique.

Main results
In this part, the finite-time control problems of PCH system (1) subject to disturbances and IS are considered. For nominal PCH system, the finite-time stabilization result is given first. Next, the finite-time stabilization result of PCH system is also proposed when IS occurs. Finally, the finite-time H 1 control problem for the case with disturbances and IS is studied.

Finite-time stabilization for nominal PCH systems
PCH system (1) with d(t) = 0 and sat(U) = U is considered in this subsection, i.e., the nominal system (11) is obtained ( _ z ¼ ½JðzÞ À RðzÞ�rEðzÞ þ g 1 ðzÞU; For matrix R(z) of system (11), two cases on positive semi-definite and positive definite are first discussed.
Case 1: If R(z) is a positive definite matrix, then system (11) is a strictly dissipative PCH system.
is a positive semi-definite matrix, an appropriate output feedback controller is needed designing to make new matrix � RðzÞ positive definite. Then, system (11) can be transformed into the following strictly dissipative PCH system, where K is a symmetric matrix with proper dimensions, According to the above discussion, a result is given. Theorem 1 Consider nominal PCH system (11) with Assumption 1. Suppose the condition (14) holds, then the output feedback controller (12) can finite-time stabilize PCH system (11).
Proof. Substituting output feedback controller (12) into system (11), the strictly dissipative PCH system (13) is derived. Taking E(z) as the Lyapunov function and calculating its derivative along system (11), one obtains i.e., From Lemma 2, one gets that the closed-loop PCH system (11) with output feedback controller (12) is globally finite-time stable.
Remark 2 If matrix R(z) is positive definite, i.e., case 1 is considered, PCH system (11) is a strictly dissipative PCH system with U = 0. In output feedback controller (12), we just choose K = 0, and the PCH system (11) is also globally finite-time stable.

Finite-time stabilization for PCH systems with IS
PCH system (1) with d(t) = 0 is considered in this subsection, i.e., ( _ z ¼ ½JðzÞ À RðzÞ�rEðzÞ þ g 1 ðzÞsatðUÞ; Via the Hamilton function method and truncation-inequality method, a novel output feedback strategy is developed to solve the finite-time stabilization problem of system (19). Now, the relevant theorem is given as follows.
Theorem 2 Consider PCH system (19) with Assumption 1. If there exists a matrix K with K T = K and proper dimensions satisfying the following condition, then the output feedback controller can finite-time stabilize PCH system (19) globally. Proof. Define δ = sat(U) − U, then system (19) can be rewritten as Choosing Hamilton function E(z) as the Lyapunov function and calculating its derivative, we have Utilizing truncation-inequality technique in Lemma 3, it yields where sR i ðzÞ are the eigenvalues of matrixRðzÞ, i = 1, 2, . . ., n.
According to Lemma 1, one obtains According to the above analysis and Lemma 2, it can be concluded output feedback controller (21) can finite-time stabilize PCH system (20).
Remark 3 Under IS constraints, the strictly dissipative PCH system is not obtained and hence the finite-time stabilization result cannot be yielded directly. However, the positive definite damping matrix is obtained utilizing the idea of constructing strictly dissipative PCH systems and truncation-inequality technique, which helps to prove the finite-time stability of the closedloop system in Theorem 2.

Finite-time H 1 control for PCH systems with IS
In this subsection, PCH system (1) with d(t)6 ¼0 is considered. To investigate the finite-time H 1 control problem, a novel output feedback controller is designed using Hamiltonian-Jacobian inequality technique and truncation-inequality method. And the result is derived as follows.
Theorem 3 Consider PCH system (1) with Assumption 1. Suppose there exists a symmetric matrix K with proper dimensions and a number γ > 0 satisfying the following conditions, then the output feedback controller can effectively solve the finite-time H 1 control problem of PCH system (1).
Substituting output feedback control law (29) into system (1), the following closed-loop system is obtained The proof is divided into two parts. Part 1, system (30) has a finite-time L 2 gain. Part 2, when d(t) = 0, system (30) is finite-time stable.
We will prove part 1 first. Choose the Lyapunov function E(z), and define According to truncation-inequality and Young's inequality, one has Based on (32), it follows from condition (28) that According to the Hamiltonian-Jacobian inequality in Lemma 4, one gets the L 2 gain of system (30) is not bigger than γ.

PLOS ONE
Finite-time stabilization and H1 control of Port-controlled Hamiltonian systems Next, part 2 will be proved. Calculating the derivative of E(z) when d(t) = 0, one obtains the following result, Condition (27) where s � R i ðzÞ are the eigenvalues of matrix � RðzÞ, i = 1, 2, . . ., n. Thus, inequality (34) can be written as From Lemma 1, it is further deduced that According to Lemma 2, one gets that when d(t) = 0, the closed-loop system (30) is globally finite-time stable.
Therefore, the finite-time H 1 control problem of PCH system (1) is solved. Remark 4 In this paper, the salient features are reflected in the following several aspects compared with the existing research results [30,31,[35][36][37][38]. (i) Compared with the inequality technique in [31,36], the truncation inequality technique, borrowed to address IS, is more feasible.
(ii) Different from the the result that the state is bounded in finite-time [35], our aim is that the state converges to the equilibrium point in finite-time. (iii) Via the Hamilton function method and finite-time control technique, two novel output feedback control laws are designed to solve the finite-time stabilization and finite-time H 1 control problems of PCH systems in the presence of disturbances and IS, which is different from the stabilization of Hamiltonian systems without disturbances and with IS [38] and the finite-time H 1 control for Hamiltonian descriptor systems without IS [30].

Simulations
This section presents two simulation examples to reveal the feasibility of the proposed control methods. Example 1. The following PCH system with IS is considered ( _ x ¼ ½JðxÞ À RðxÞ�rEðxÞ þ g 1 ðxÞsatðuÞ; where the state x 2 R 2 , the Hamilton function EðxÞ ¼ x u; 0:2 � u � À 0:2; À 0:2; u < À 0:2: By choosing K = 2 and 2 ¼ 1 5 , we know the condition (20) holds, wherê and all the conditions hold in Theorem 2. Based on Eq (21), the output feedback law is designed as According to Theorem 2, it is easy to get that control law (41) can finite-time stabilize system (38).
To demonstrate the feasibility of the control strategy, the initial condition x 1 (0) = 1 = x 2 (0) is first given and further the simulation is carried out. Figs 1 to 3 present the simulation results. Fig 1 illustrates that the states x 1 and x 2 converge to the equilibrium point fast via the output feedback control (41) when IS is considered, i.e., closed-loop system (38) is finite-time stable. The response curve of the output feedback signal is given in Fig 2. Fig 3 is the response curve of the saturated control input sat(u), and the amplitudes of the response curve are all within the saturation range. The simulations reveal that the output feedback control strategy with IS is effective.
Example 2. Fig 4 ([18]) describes the nonlinear circuit system, where the magnetic flux ψ controls the inductance, the electric charge q controls the capacitance, i 3 = f 1 (ψ) is current, U 1 = f 2 (q) is voltage, and the current source disturbance is denoted by i w .
The system is written as follows by Kirchhoff's Law,

PLOS ONE
Finite-time stabilization and H1 control of Port-controlled Hamiltonian systems where EðxÞ ¼ x

PLOS ONE
Finite-time stabilization and H1 control of Port-controlled Hamiltonian systems Considering IS, system (43) with disturbances is rewritten as _ x ¼ ½JðxÞ À RðxÞ�rEðxÞ þ g 1 ðxÞsatðuÞ þ g 2 ðxÞdðtÞ; where y, d(t) and z are the output, the disturbance and the penalty signal, respectively. satðuÞ 2 R is the saturated control input and given as u; 0:8 � u � À 0:8; À 0:8; u < À 0:8: For disturbance attenuation level γ = 1, we choose symmetric matrix K ¼ . Through verification, we find that these conditions in Theorem 3 hold, where According to Eq (29), the output feedback law is obtained From Theorem 3, we know that controller (49) can effectively solve the problem of finitetime H 1 control of PCH system (44).
The simulation is carried out with initial condition x 1 (0) = −0.5 and x 2 (0) = 1.2. To verify the robustness of the proposed controller against disturbances, the current source disturbance d(t) = i w = 1.1sin t is added into system (44) when 4s � t � 6s.
Figs 5 to 8 give the simulation results. The response curves of states x 1 and x 2 with and without the saturated control input are shown in Figs 5 and 6, respectively. It can be seen that compared with Fig 6, the effect of the disturbance on the system is well suppressed when the disturbance appears in Fig 5. After the disturbance vanishes, the states converge to the equilibrium point faster in Fig 5. The response curves of the output feedback control are given in Fig  7. Fig 8 shows the response curves of the saturated control input sat(u), and the amplitudes of the response curves are all within the saturation range. The simulations illustrate that the output feedback controller is very effective against disturbances subject to IS. Therefore, when IS is considered, the proposed finite-time H 1 control strategy is valid.

Conclusion
The problems of finite-time stabilization and finite-time H 1 control of PCH systems subject to disturbances and IS have been studied in this paper. By using system's structure characteristics and an appropriate output feedback, the strictly dissipative PCH system has been obtained first. Second, the finite-time stabilization results for the case without and with IS have been presented using the Hamilton function method and truncation inequality technique. Next, the

PLOS ONE
Finite-time stabilization and H1 control of Port-controlled Hamiltonian systems finite-time H 1 control problem for PCH system with disturbances and IS has also been solved. Finally, two examples have been proposed to illustrative the effectiveness of the theoretical results.
The control method proposed has been applied to two simulation examples in this paper. In fact, compared with the simulation results, the experimental results can better illustrate the effectiveness of the proposed method. It is of great theoretical and engineering significance to investigate engineering systems' experiments using the proposed control scheme, which will be studied in the future.