# ParameterizedH∞ Controller Design for Switched Polynomial Hamiltonian Systems Subject to Actuator Saturation

Authors: Huawei Zhu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This paper is concerned with the global asymptotical stabilization andL2 disturbance attenuation problem for the switched polynomial Hamiltonian systems subject to actuator saturation. To achieve the above objective, a parameterized controller is designed and a parameter solution algorithm is presented. Compared with the controller designed with the existing method, the parameterized controller has greater functionality. By choosing an appropriate value for the parameter according to its obtained feasible interval value, a better performance can be achieved. At last, two numerical examples are provided to verify the effectiveness of the proposed approach.


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## Body

## 1. Introduction

Switched systems are a class of hybrid systems, which can be used to describe a large number of practical systems, such as boost converter circuit[1], cyber-physical systems [2], aircraft systems [3, 4], and robotics [5]. Because of the importance of this kind of systems, the research on this kind of system has always been one of the most important fields of scholars. Large quantities of worthwhile results for such systems have been given, including stability analysis and stabilization [6], H∞ control [7], sliding mode control [8], and adaptive output-feedback neural tracking control [9].In general, external disturbance and nonlinearity are common in the practical control systems. Many scholars have paid much attention to theH∞ control problem for the nonlinear systems including switched nonlinear systems and obtained many achievements [10], [11–13], [7], [14–17], [4, 5, 18–20]. Still and all, there are still many problems needed to be solved. For example, in [4, 18, 19], some sum of squares (SOS)-based approaches are proposed to solve the H∞ control problem of nonlinear polynomial systems [4, 18, 19]. However, in these methods of [4, 18, 19], it needs that the polynomial inequality should be convex and the parameter cannot be restricted. In [20], the robust H∞ control problem of the switched system with stabilizable and unstabilizable subsystems is solvable if the stabilizable and unstabilizable subsystems satisfy certain conditions and admissible switching strategy among them. ADT and piecewise Lyapunov function approaches are applied to achieve the control design. However, in the method of [20], it needs to solve some classes of partial differential inequalities, which is difficult in most of the times [5]. A nonlinear H∞ control strategy-based dynamic distributed control law is developed to achieve synchronous tracking of networked robots in the joint space in [5]. In this method, it is avoided to obtain controller parameters by solving differential inequalities, but how to check the feasibility of the matrix inequalities is not indicated.As a factor that may cause system instability and affect system performance, actuator saturation (AS) widely exists in actual dynamic systems [21–28], [29]. Neglecting the adverse effect of AS will not only result in performance degradation of the closed-loop system but also give rise to instability. For this reason, many scholars have paid their attention to the stability analysis and controller design for switched systems subject to AS [21–27]. However, most of these results are aimed at switched linear systems subject to AS, and the relevant results are more limited for the nonlinear systems subject to AS. For nonlinear systems, because of the nonlinearity of these systems, the stabilization problem usually is difficult to solve [22, 30]. The results in regard to stability analysis and controller design of switched nonlinear systems subject to AS and external disturbance are much less. Most of the recent results are primarily focused at some kinds of system with special structures. In [11–13], some achievements for a kind of nonlinear system subject to AS with Hamiltonian structure were studied. As an important class of nonlinear systems with unified mathematical structure and clear physical meanings, port-controlled Hamiltonian systems have attracted the attention of scholars and have been studied in many different scenarios [10–13, 28, 31–36]. With the method obtained in [10, 28, 31, 32], lots of practical systems can be modeled as Hamiltonian systems. Many scholars have proposed some dissipative Hamiltonian realization methods, with which lots of practical systems can be modeled as Hamiltonian systems. In the method of [11–13], the Hamilton function is used as Lyapunov function and a kind of controller is designed with the Hamilton function. Although the solution of differential inequality is avoided, the controller is fixed at the same time, which limits its control performance, which is the motivation of the research of this paper.In the control system design, to obtain internal stability for the system, controller parameterization is usually a simple and effective solution. The parameters in the controller can not only provide more freedom to achieve a variety of control objectives, but also improve the performance of the system by changing the parameter values. The reliableH∞ state feedback control and controller parameterization problem for nonlinear systems with strictly redundant actuators were studied in [37]. However, in the method of [37], it needs to solve Hamilton-Jacobi-Issacs (HJI) inequalities (or equations), which is difficult in most times. In reference [38, 39], a kind of new controller parameterization method for Hamilton systems was studied. In the method of [38, 39], it does not need to solve HJI inequalities. In [5], controllers with dynamic structure were employed to attain high-quality transient behavior for a group of robotic manipulators with nonidentical dynamics and model uncertainties subject to time-varying communication delays. As far as the author knows, there is no research result of parameterized controller design method for the Hamiltonian systems subject to AS and external disturbance.Motivated by the facts above, considering the external disturbances and the actuator saturation, a controller parameterization-basedH∞ control strategy for switched polynomial Hamiltonian systems was presented in this paper. The contributions of the proposed method are as follows:(1)
A parameterized controller design method is presented to solve the global asymptotical stabilization andL2 disturbance attenuation problem for the switched polynomial Hamiltonian systems subject to actuator saturation. Comparing with the fixed-structure controller investigated by [11], on account of containing parameters whose values can be adjusted, the designed parameterized controller with the method of this paper possesses greater control ability and better control performance.(2)
A symbolic computation-based parameter solution algorithm is described to solve the interval value of the parameter. Comparing with the method of [11], the method of this paper is more effective to verify the validity of the polynomial matrix inequalities conditions. In addition, the obtained value of the parameter is an interval. The performance of the system can be improved by choosing an appropriate value for the parameter according to its obtained feasible interval value. Comparing with the method of SOS [4, 18, 19, 40], it does not need the polynomial inequality to be convex; in addition, the parameter value in the polynomial inequality can be restricted to an arbitrary range.The remainder of this paper is organized as follows. Section2 is the problem formulation. The main contribution of this paper is then given in Section 3, in which a parameterized controller is designed and a parameter solution algorithm is provided. In Section 4, two numerical examples to support the effectiveness of these results are given, which is followed by the conclusion in Section 5.Notations: the notations throughout this paper are quite standard. We useP>0 (P≥0) to denote a positive definite (semi-positive definite) matrix P. Rn denotes the n-dimensional Euclidean space. 0≠x∈Rn denotes x∈Rn&x≠0. L20,+∞ is the space of square integrable vector over 0,+∞. I denotes identity matrix with appropriate dimension. The superscript “T” stands for matrix transposition. The superscript “−T” of matrix P stands for the inverse of PT. diaga1,…,an means a diagonal matrix with diagonal terms a1,…,an. sign. denotes the signum function. In the symmetrical matrix, “∗” is used as an ellipsis for the term that is induced by symmetry. Matrices, if not explicitly stated, are assumed to have compatible dimensions.


## 2. Problem Formulation

Consider following switched polynomial Hamiltonian system (SPHS) subject to AS.(1)x˙=Jλtx−Rλtx∇Hλtx+gλtxsatu+g¯λtxw,where x∈Rn represent the state variables. The map λt:t0,+∞⟶Λ=1,2,…,m˜ is a piecewise right-continuous constant function called the switching law or switching path. λt=ii∈Λ means the i−th subsystem is active.(2)∑=tk¯,ik¯|ik¯∈Λ;k¯=0,1,…;0=t0≤t1≤…,which is the switching sequence under arbitrary switching law with dwell time τ>0. For an arbitrary switching path λt=ik¯∈Λ (t∈tk¯,tk¯+1, k¯=0,1,…), let tk¯k¯=0+∞ be the switching time sequence satisfying(3)t0<t1<⋯<tk¯<⋯<+∞,and xk¯k¯=0+∞ be the switching state sequence corresponding to the switching time sequence tk¯k¯=0+∞. For λt=i, i∈Λ, Hix represent the Hamiltonian function for the i−th subsystem with Hix>0 for ∀0≠x∈Rn and Hi0=0, ∇Hix=∂Hix/∂x, Jix∈Rn×n is skew-symmetric structure matrix, Rix∈Rn×n is positive semi-definite symmetric polynomial matrix, gix∈Rn×m˜, g¯ix∈Rn×s are polynomial matrices, w∈Rs is the external disturbance, and u∈Rm˜ is the control input. The saturation function sat.:Rm˜⟶Rm˜ is defined as(4)satu≔satu1,satu2,…,satum˜T,where(5)satuj=signujminρj,uj,with the saturation level ρj>0, j∈1,2,…,m˜.Let the penalty signal be defined as follows:(6)z=BλtgλtTx∇Hλtx,where Bi,i∈Λ are weighting matrix with full column rank.Lemma 1 (see [11]).
Letη=satu−u where satu is defined as (2). Then, there exists a real number ε∈0,1, such that ηTη≤εuTu where η=η1,η2,…,ηm˜T∈Rm˜ and ηi,i∈1,2,…,m˜ is the dead-zone nonlinearity function.Assumption 1 (see [34]).
For∀i∈Λ and ∀x,y∈Rn, the Hamiltonian function Hix satisfies(7)Hix>Hiy⇔xP>yP,where xP≜supx=1xTPx, and P>0 is a positive definite matrix.Proposition 1 (see [34]).
Assume that Assumption1 holds for system (1); if there exist at least one subsystem satisfying Hlx<0,l∈Λ and the other subsystem satisfying H˙ix≤0,∀i∈Λ,i≠l, then system (1) is globally asymptotically stable under an arbitrary switching path λt with dwell time τ>0.On the base of the aforementioned discussion, the solvability of theH∞ control problem of SPHSs subject to AS in this paper is as follows: for any given disturbance attenuation level γ>0 and arbitrary switching path λt with dwell time τ>0, design a parameterized controller u=uλtx,km, in which km is the parameter with serial number m, and there exists feasible interval value for each km, such thatA1. The closed-loop system (1) is globally asymptotically stable when w≡0.A2. ∀w∈L20,+∞, for each i∈Λ, γ−dissipation inequality,(8)H˙ix+Qix≤12γ2w2−z2.Holds with zero initial condition, whereQix≥0 is a scalar function.


## 3. Main Results

In this section, a parameterized controller design method is expounded and a parameter solution algorithm is described to solve theH∞ control problem of system (1). In what follows, they will be accomplished in two subsections.


### 3.1. Parameterized Controller Design

Definition 1.
Design symmetrical polynomial parameter matrixKix,km,i∈Λ with following terms:(9)Kinrnc=∑s=0r^μ˜mkm∏j˜=1nxj˜sj˜,where μ˜m is the coefficient of the item, km is the parameter, m∈1,2…, is the serial number of the parameter km, s is the serial number of the monomial with maximum number r^, nr is the row number, nc is the column number for the term in matrix Kix,km, j˜ is the serial number of the state value with the maximum number n, xj˜sj˜ is a power with degree Sj˜, and xj˜ is a state variable with serial number j˜. It should be noted that the value of μ˜m can be set 0 when the associated monomial is not existent.
With above designed parameter matrices, in view of (6), for i∈Λ, define the following parameterized controller for system (1):(10)u=u|λt=i=−Kix,km+12BiTBigiTx∇Hix.Remark 1.
In general, it is enough to contain only one parameterkm for each parameter matrix Kix,km, which do not affect the adjustability of the parameterized controller (6) too much.
In following subsection, the designed parameterized controller is utilized to solve theH∞ control problem of system (1).Theorem 1.
Assume that Assumption1 holds for the Hamiltonian functions of system (1). There exist real number εi∈0,1, parameter matrix Kix,km,∀i∈Λ, in which km is the parameter with serial number m, and there exists feasible interval value for each parameter km, such that(i)
For∀i∈Λ, ∀x∈Rn, Gi≥0 holds where(11)Gi=Rix+gixKix,kmgiTx−12γ−2g¯ixg¯iTx−12gixgiTx−12εigixKix,km+12BiTBi2giTx.(ii)
There exist at least one subsystem satisfyingUi>0,l∈Λ, ∀0≠x∈Rn, and the other subsystems satisfy Uj≥0,j∈Λ,j≠l, ∀x∈Rn, where(12)Ui=Rix+gixKix,kmgiTx+12gixBiTBigiTx−12gixgiTx−12εigixKix,km+12BiTBi2giTx.
Then, under parameterized controller (6) and arbitrary switching law with dwell time τ>0, the H∞ control problem of system (1) is solvable.Proof.
Assume thatλt is an arbitrary switching law with dwell time τ>0. For i∈Λ, substituting (6) into (1) leads to(13)x˙=Jix−Rix−gixKix,km+12BiTBigiTx∇Hix+gixηi+g¯ixw.
Considering thei−th subsystem of system (1), i∈Λ, noticing the fact that Jix is a skew-symmetric matrix, it gives ∂Hix/∂xJix∇Hix=0. Then, in view of Lemma 1, the differential of the Hamiltonian Hix along the trajectory of system (1) is obtained, i.e.,(14)H˙ix=∇HiTx−Rix−gixKix,km+12BiTBigiTx∇Hix+∇HiTxg¯ixw+∇HiTxgixηi.
Then, whenw≡0, by (14), in light of (12), it gives(15)H˙ix≤−∇HiTxgixKix,kmgiTx+Rix−12gixgiTx+12gixBiTBigiTx−12εigixKix,km+12BiTBi2giTx∇Hix=−∇HiTxUi∇Hix.
Considering the condition (ii), in view of Proposition1, when w≡0, the closed-loop system (1) is globally asymptotically stable, which means A1 holds.
Next, theL2 disturbance attenuation problem for system (1) is considered.
For thei−th subsystem, i∈Λ, it follows from (6) and (14) that(16)H˙i≤−∇HiTxRix+gixKix,kmgiTx−12γ−2g¯ixg¯iTx−12gixgiTx−12εigixKix,km+12BiTBi2giTx∇Hix+12γ2wTw−12∇HiTxgixBiTBigiTx∇Hix=−∇HiTxGi∇Hix−12zTz+12γ2wTw,which means(17)H˙ix+∇HiTxGi∇Hix≤12γ2w2−z2,holds. Then, in light of (i), letting Qix=∇HiTxGi∇Hix, γ−dissipation inequality H˙ix+Qix≤1/2γ2w2−z2,∀w∈L20,+∞ holds for each subsystem of system (1), which means A2 holds. Therefore, the H∞ control problem of system (1) under arbitrary switching law with dwell time τ>0 and parameterized controller (6) is solved.Remark 2.
Based on Assumption1, a global asymptotical stabilization method (see Proposition 1) for switched Hamiltonian system was studied by [34]. Then, this method was generalized to solve the globally asymptotically stabilization problem of switched Hamiltonian system subject to AS by [11]. However, the structure of the designed controller in [11] is fixed, which restricts the application of the method of [11]. In this paper, the fix structure controller is promoted into parameterized controller (6). Comparing with the controller designed with the method of [11], the advantage of the parameterized controller (6) includes the following several items: (a) it is more probably to satisfy the conditions (i), (ii) of Theorem 1 in this paper with the parameterized controller (6) than to satisfy the conditions (1) and (2) of the Theorem 2in [11] with the fixed-structure controller (6) in [11]. The controller (6) in [11] is a special form of the parameterized controller (6) in this paper when Ki=1/2γ2I, which means that when the conditions (1) and (2) of the Theorem 2 in [11] cannot be satisfied with the fixed-structure controller (6) in [11], the conditions (i), (ii) of Theorem 1 in this paper sometimes can be satisfied by adjusting the value of km in the parameterized controller (6) (see Example 1 in Section 1). (b) By adjusting the parameter value of the parameterized controller (6), some performances of the system (1) can be improved. It should be pointed out that the performance regulation function of the parameterized controller is restricted by the structure of gix and value of saturation level ρj, j∈1,2,…,m˜.Remark 3.
According to Theorem1, for system (1) under arbitrary switching law with dwell time τ>0, the H∞ control problem can be established by designing the parameterized controller (6) and obtaining the feasible interval value of the parameter km. However, the major difficulty is to design the parameter matrix Kix,km and find out the interval value of km such that (i), (ii) hold over the entire system state space. In general, parameter matrix Kix,km can be set as diagonal matrix, in which the parameter km is located in the leading diagonal. In order to solve the interval value of the parameter km, a symbolic computation-based parameter solution algorithm is produced in Section 3.2.Remark 4.
Considering that the single-structure polynomial Hamiltonian system is a special form of switched polynomial Hamiltonian system, the results of this paper can also be used to solve theH∞ control problem of nonswitched polynomial Hamiltonian system.Remark 5.
According to the proof of Theorem 3 in [34], the sequences xk¯Pk¯=0+∞ and Hixk¯k¯=0+∞ (∀i∈Λ) are all monotonically decreasing with respect to k¯. In addition, in view of condition (i) and (13), the following expression holds:(18)H˙ix≤12γ2wt2−zt2.
Integrating both sides of (17) over tk¯,tk¯+1, ∀k¯=0,1,…, leads to(19)Hixtk¯+1−Hixtk¯≤∫tk¯tk¯+112γ2wt2−zt2dt.
Fort∈tk¯,tk¯+1, let Mi=…,tm¯i,…,tk¯ be the switching time sequence over 0,t when the i−th subsystem is activated. m¯i∈0,1,…,k¯ is the serial number of switching with maximum number k¯ when the i−th subsystem is activated. Obviously, Mi is a time subsequence of t0,t1,…,tk¯. Noting that the sequences Hixk¯k¯=0+∞ (∀i∈Λ) are all monotonically decreasing with respect to k¯, for t∈tk¯,tk¯+1, according to the zero initial condition and Hixt>0, it is obtained that(20)0≤Hixt≤∑Mi∫tm¯itm¯i+112γ2wt2−zt2dt+∫tk¯t12γ2wt2−zt2dt.
Then,∀j∈Λ,j≠i, and let Mj=…,tm¯j,…,ts be the switching time sequence over 0,t when the j−th subsystem is activated. m¯j∈0,1,…,k¯ is the serial number of switching with maximum number s when the j−th subsystem is activated. Obviously, Mj is a time subsequence of t0,…t1,…,tk¯. With the same method as obtaining (16), the following inequality is obtained:(21)0≤Hjxts+1≤∑Mj∫tm¯jtm¯j+112γ2wt2−zt2dt.
Combining with (19) and (20), one gets(22)0≤∫0tγ2wt2−zt2dt,which means that the result A2 in Theorem 1 is consistent with the result (27) of [41].Remark 6.
Consider following switched polynomial nonlinear systems with AS:(23)x˙=fλtx+gλtxsatu+g¯λtxw,where x∈Rn is the state, for i∈Λ, fix:Rn⟶Rn is a polynomial continuous vector field with fi0=0, and the others are the same as those in system (1). In view of the dissipative Hamiltonian realization methods presented in [10, 28, 31, 32], many switched polynomial nonlinear systems with AS, such as (19), can be transformed into the form of system (1). Therefore, the result studied in this paper can be used to solve the H∞ control problem of general switched polynomial nonlinear systems with AS.


### 3.2. Parameter Solution Algorithm

In this subsection, a symbolic computation-based parameter solution (PS) algorithm to solve the feasible interval value of the parameter is investigated, which is proceeded as follows. Furthermore, Table1 shows the process of the PS algorithm.Table 1
Process of the PS algorithm.
PS algorithmInput: System (1), flag=0, RCI=zerosm˜Output: Controlleru#Step 1: InitializationGix,Ki.subsKi,Kix,kiUix,Ki.subsKi,Kix,km#Step 2: Obtain the set of constraintsGπi with Gi≥0fi0=εifi1=1−εiGSi=fi0,fi1Gπi=fi0>0,fi1>0GPi=PrincipalMinorGiforl⌢ in range0,lenGP˜i:ifdiffGPil⌢,x1,…,diffGPil⌢,xn≠0:GMil⌢ki,εi=QuadraticMatrixGPil⌢GPPil⌢=PrincipalMinorGMil⌢Gπil⌢=GPPil⌢1≥0,…,GPPil⌢j^≥0,…else:GPPil⌢=GPil⌢Gπil⌢=GPil⌢≥0GSi.unionGPPil⌢Gπi.unionGπil⌢#Step 3: Obtain the set of constraintsUςi with Ui>0USi=setUςi=setUO˜i=OrderPrincipalMinorUiforl⌢ in range0,lenUO˜i:ifdiffUO˜il⌢,x1,…,diffUO˜il⌢,xn≠0:UM˜il⌢ki,εi=QuadraticMatrixUO˜il⌢UOOil⌢=OrderPrincipalMinorUM˜il⌢Uς˜il⌢=UOOil⌢1>0,…,UOOil⌢j^>0,…else:UOOil⌢=UO˜il⌢Uς˜il⌢=UO˜il⌢>0USi.unionUOOil⌢Uςi.unionUς˜il⌢#Step 4: Obtain the set of constraintsUς^i with Ui≥0US^i=setUπ^i=setUP^i=PrincipalMinorUiforl⌢ in range0,lenUP^i:ifdiffUP^il⌢,x1,…,diffUP^il⌢,xn≠0:UM^il⌢ki,εi=QuadraticMatrixUP^il⌢UPPil⌢=PrincipalMinorUM^il⌢Uς^il⌢=UPPil⌢1≥0,…,UPPil⌢h⌢≥0,…else:UPPil⌢=UP^il⌢Uς^il⌢=UP^il⌢≥0US^i.unionUPPil⌢Uς^i.unionUς^il⌢#Step 5: Obtain the interval value of the parameterRCi=set;Si=GSi,USiπi=Gπi,UςiCi=PlotImplicitSiCPi=CheckPointCiforj⌢ in range0,lenCi:m^i=0forl⌢ in range0,lenπi:ifπil⌢.subsx,CPij⌢ is True:m^i=m^i+1ifm^i==lenπi:RCi. unionCij⌢RCIi=1flag=1ifRCIi=0:S^i=GSi,US^iπ^i=Gπi,Uς^iCi=PlotImplicitS^iCPi=CheckPointCiforj⌢ in range0,lenCi:m^i=0forl⌢ in range0,lenπ^i:ifπ^il⌢.subsx,CPij⌢ is True:m^i=m^i+1ifm^i==lenπ^i:RCi.unionCij⌢RCIi=1//Step 6: Obtain controller for thei-th subsystemifRCIi==1uiki.subski,k¯i#Step 7: Obtain controller for system (1)#Repeat steps 1–6 for each subsystem of system (1)ifRCI==1.all is true and flag==1:returnuPS algorithm:Step 1. Obtain matrix inequality conditions. Design a parameter matrix Kix,km as Definition 1, and obtain the polynomial matrix inequality conditions by substituting Kix,km into (7), (8) for the i−th subsystem.Step 2. Obtain constraint condition set Gπi.Step 2.1. Achieve the constraint condition set Gπi by making all the principal minor of Gi described by (7) being non-negative and the condition 0<εi<1.Step 2.2. Transform all the polynomial constraint conditions in Gπi into the form of quadratic form. Then, the condition that the quadratic form is non-negative is equivalent to the condition that the quadratic form matrix is positive semi-definite. Get new constraint conditions from the positive semi-definiteness of the quadratic form matrix with the way of Step 2.1, and substitute them with the primary constraint.Step 3. Obtain constraint condition set Uςi with Ui>0. Get the constraint condition set Uςi by making all the order principal minor of Ui described by (8) being positive. For the constraint condition containing system state in Uςi, transform it into the form of quadratic form. Then, obtain new constraint conditions by making all the order principal minors of the quadratic form matrix being positive and substitute it with the original condition.Step 4. Obtain constraint condition set Uς^i with Ui≥0. Reset the constraint condition set Uς^i by making all the principal minor of Ui described by (8) being non-negative. Then, obtain Uς^i with the same way of Step 3.Step 5. Decompose the parameter space and obtain the interval value of the parameter. Let all the inequalities in Gπi and Uςi be implicit function, and draw them in the parameter space. Then, the parameter space is partitioned into many cells. According to Remark 1, there always exists only one parameter km in each parameter matrix in general. Thus, the decomposition is always produced in two-dimensional space (made up by εi and ki) for the i−th subsystem. In view of the cylindrical algebraic decomposition algorithm [42], all the points containing in each cell have the same characteristics. Then, search out the cell Ci as the required region Ci from all the cells by verifying any one point in each cell with all the constraint conditions in Gπi and Uςi. If there exists no Ci satisfying Uςi found, partition the parameter space again according to Gπi and Uς^i. Then, search out the cell Ci as the required region Ci from all the cells by verifying any one point in each cell with all the constraint conditions in Gπi and Uς^i.Step 6. Obtain the controller for the i-th subsystem. According to the required regions Ci obtained in Step 5, obtain the feasible interval value of each parameter. Then, choose a value for each ki according to its feasible interval value. Obtain the controller by substituting the values of ki into the parameterized controller (6).Step 7. Obtain Ci and controller for each subsystem by running steps 1–6. If there exists at least one subsystem (assumed the j−th subsystem, j∈Λ), for which the Cj satisfies Uj>0, the obtained controllers will be the required controllers.Remark 7.
In Table1, RCIi=1 identifies that there exist RCi regions for the i-th subsystem. flag=1identifies that there exists subsystem containing constraints Uςi satisfying Ui>0. RCi represents the set of cells satisfying the desired conditions for the i-th subsystem. Gix,Ki.subsKi,Kix,ki means substituting Ki in Gi with Kix,ki. PrincipalMinorGi means getting the set of all principal minor of Gi. GMil⌢ki,εi=QuadraticMatrixGPil⌢ means obtaining a quadratic form matrix GMil⌢ki,εi by transforming GPil⌢ into quadratic form GPil⌢=z^il⌢TxGMil⌢ki,εiz^il⌢x, in which z^il⌢x is a column vector whose term is monomial expression in regard to x. OrderPrincipalMinorUi denotes all the order principal minor of Ui. Ci=PlotImplicitSi means decomposing the parameter space into many cells Ci by using the trajectories of the implicit functions shown by all the polynomial expressions in Si. CPi=CheckPointCi represents a set of test point coordinates CPi obtained by taking a test point from each cell Ci.Remark 8.
The method of solving the interval value ofki and εi described in this paper can be extended to solve the interval value of parameters in other control system design involving polynomial matrix inequality condition containing parameters. In Example 2 of this paper, the PS algorithm will be used to solve the interval value of ki and γ when arraigning a constant to each εi. Then, except that the condition 0<εi<1 is replaced by the condition 0<γ in obtaining constraint condition set Gπi, the other steps are the same as solving the interval value of ki and εi.Remark 9.
The PS algorithm is suitable to solve the interval value of the parameter contained in the matrix inequality with polynomial structure. Comparing with the method of SOS in solving interval values of parameters in polynomial matrix inequalities, it does not need the polynomial inequality being convex, which implies that the polynomial inequality can contain nonlinear combination of unknown values. In addition, the value of the parameter in the inequality can be restricted to an arbitrary range in the method of this paper.


## 3.1. Parameterized Controller Design

Definition 1.
Design symmetrical polynomial parameter matrixKix,km,i∈Λ with following terms:(9)Kinrnc=∑s=0r^μ˜mkm∏j˜=1nxj˜sj˜,where μ˜m is the coefficient of the item, km is the parameter, m∈1,2…, is the serial number of the parameter km, s is the serial number of the monomial with maximum number r^, nr is the row number, nc is the column number for the term in matrix Kix,km, j˜ is the serial number of the state value with the maximum number n, xj˜sj˜ is a power with degree Sj˜, and xj˜ is a state variable with serial number j˜. It should be noted that the value of μ˜m can be set 0 when the associated monomial is not existent.
With above designed parameter matrices, in view of (6), for i∈Λ, define the following parameterized controller for system (1):(10)u=u|λt=i=−Kix,km+12BiTBigiTx∇Hix.Remark 1.
In general, it is enough to contain only one parameterkm for each parameter matrix Kix,km, which do not affect the adjustability of the parameterized controller (6) too much.
In following subsection, the designed parameterized controller is utilized to solve theH∞ control problem of system (1).Theorem 1.
Assume that Assumption1 holds for the Hamiltonian functions of system (1). There exist real number εi∈0,1, parameter matrix Kix,km,∀i∈Λ, in which km is the parameter with serial number m, and there exists feasible interval value for each parameter km, such that(i)
For∀i∈Λ, ∀x∈Rn, Gi≥0 holds where(11)Gi=Rix+gixKix,kmgiTx−12γ−2g¯ixg¯iTx−12gixgiTx−12εigixKix,km+12BiTBi2giTx.(ii)
There exist at least one subsystem satisfyingUi>0,l∈Λ, ∀0≠x∈Rn, and the other subsystems satisfy Uj≥0,j∈Λ,j≠l, ∀x∈Rn, where(12)Ui=Rix+gixKix,kmgiTx+12gixBiTBigiTx−12gixgiTx−12εigixKix,km+12BiTBi2giTx.
Then, under parameterized controller (6) and arbitrary switching law with dwell time τ>0, the H∞ control problem of system (1) is solvable.Proof.
Assume thatλt is an arbitrary switching law with dwell time τ>0. For i∈Λ, substituting (6) into (1) leads to(13)x˙=Jix−Rix−gixKix,km+12BiTBigiTx∇Hix+gixηi+g¯ixw.
Considering thei−th subsystem of system (1), i∈Λ, noticing the fact that Jix is a skew-symmetric matrix, it gives ∂Hix/∂xJix∇Hix=0. Then, in view of Lemma 1, the differential of the Hamiltonian Hix along the trajectory of system (1) is obtained, i.e.,(14)H˙ix=∇HiTx−Rix−gixKix,km+12BiTBigiTx∇Hix+∇HiTxg¯ixw+∇HiTxgixηi.
Then, whenw≡0, by (14), in light of (12), it gives(15)H˙ix≤−∇HiTxgixKix,kmgiTx+Rix−12gixgiTx+12gixBiTBigiTx−12εigixKix,km+12BiTBi2giTx∇Hix=−∇HiTxUi∇Hix.
Considering the condition (ii), in view of Proposition1, when w≡0, the closed-loop system (1) is globally asymptotically stable, which means A1 holds.
Next, theL2 disturbance attenuation problem for system (1) is considered.
For thei−th subsystem, i∈Λ, it follows from (6) and (14) that(16)H˙i≤−∇HiTxRix+gixKix,kmgiTx−12γ−2g¯ixg¯iTx−12gixgiTx−12εigixKix,km+12BiTBi2giTx∇Hix+12γ2wTw−12∇HiTxgixBiTBigiTx∇Hix=−∇HiTxGi∇Hix−12zTz+12γ2wTw,which means(17)H˙ix+∇HiTxGi∇Hix≤12γ2w2−z2,holds. Then, in light of (i), letting Qix=∇HiTxGi∇Hix, γ−dissipation inequality H˙ix+Qix≤1/2γ2w2−z2,∀w∈L20,+∞ holds for each subsystem of system (1), which means A2 holds. Therefore, the H∞ control problem of system (1) under arbitrary switching law with dwell time τ>0 and parameterized controller (6) is solved.Remark 2.
Based on Assumption1, a global asymptotical stabilization method (see Proposition 1) for switched Hamiltonian system was studied by [34]. Then, this method was generalized to solve the globally asymptotically stabilization problem of switched Hamiltonian system subject to AS by [11]. However, the structure of the designed controller in [11] is fixed, which restricts the application of the method of [11]. In this paper, the fix structure controller is promoted into parameterized controller (6). Comparing with the controller designed with the method of [11], the advantage of the parameterized controller (6) includes the following several items: (a) it is more probably to satisfy the conditions (i), (ii) of Theorem 1 in this paper with the parameterized controller (6) than to satisfy the conditions (1) and (2) of the Theorem 2in [11] with the fixed-structure controller (6) in [11]. The controller (6) in [11] is a special form of the parameterized controller (6) in this paper when Ki=1/2γ2I, which means that when the conditions (1) and (2) of the Theorem 2 in [11] cannot be satisfied with the fixed-structure controller (6) in [11], the conditions (i), (ii) of Theorem 1 in this paper sometimes can be satisfied by adjusting the value of km in the parameterized controller (6) (see Example 1 in Section 1). (b) By adjusting the parameter value of the parameterized controller (6), some performances of the system (1) can be improved. It should be pointed out that the performance regulation function of the parameterized controller is restricted by the structure of gix and value of saturation level ρj, j∈1,2,…,m˜.Remark 3.
According to Theorem1, for system (1) under arbitrary switching law with dwell time τ>0, the H∞ control problem can be established by designing the parameterized controller (6) and obtaining the feasible interval value of the parameter km. However, the major difficulty is to design the parameter matrix Kix,km and find out the interval value of km such that (i), (ii) hold over the entire system state space. In general, parameter matrix Kix,km can be set as diagonal matrix, in which the parameter km is located in the leading diagonal. In order to solve the interval value of the parameter km, a symbolic computation-based parameter solution algorithm is produced in Section 3.2.Remark 4.
Considering that the single-structure polynomial Hamiltonian system is a special form of switched polynomial Hamiltonian system, the results of this paper can also be used to solve theH∞ control problem of nonswitched polynomial Hamiltonian system.Remark 5.
According to the proof of Theorem 3 in [34], the sequences xk¯Pk¯=0+∞ and Hixk¯k¯=0+∞ (∀i∈Λ) are all monotonically decreasing with respect to k¯. In addition, in view of condition (i) and (13), the following expression holds:(18)H˙ix≤12γ2wt2−zt2.
Integrating both sides of (17) over tk¯,tk¯+1, ∀k¯=0,1,…, leads to(19)Hixtk¯+1−Hixtk¯≤∫tk¯tk¯+112γ2wt2−zt2dt.
Fort∈tk¯,tk¯+1, let Mi=…,tm¯i,…,tk¯ be the switching time sequence over 0,t when the i−th subsystem is activated. m¯i∈0,1,…,k¯ is the serial number of switching with maximum number k¯ when the i−th subsystem is activated. Obviously, Mi is a time subsequence of t0,t1,…,tk¯. Noting that the sequences Hixk¯k¯=0+∞ (∀i∈Λ) are all monotonically decreasing with respect to k¯, for t∈tk¯,tk¯+1, according to the zero initial condition and Hixt>0, it is obtained that(20)0≤Hixt≤∑Mi∫tm¯itm¯i+112γ2wt2−zt2dt+∫tk¯t12γ2wt2−zt2dt.
Then,∀j∈Λ,j≠i, and let Mj=…,tm¯j,…,ts be the switching time sequence over 0,t when the j−th subsystem is activated. m¯j∈0,1,…,k¯ is the serial number of switching with maximum number s when the j−th subsystem is activated. Obviously, Mj is a time subsequence of t0,…t1,…,tk¯. With the same method as obtaining (16), the following inequality is obtained:(21)0≤Hjxts+1≤∑Mj∫tm¯jtm¯j+112γ2wt2−zt2dt.
Combining with (19) and (20), one gets(22)0≤∫0tγ2wt2−zt2dt,which means that the result A2 in Theorem 1 is consistent with the result (27) of [41].Remark 6.
Consider following switched polynomial nonlinear systems with AS:(23)x˙=fλtx+gλtxsatu+g¯λtxw,where x∈Rn is the state, for i∈Λ, fix:Rn⟶Rn is a polynomial continuous vector field with fi0=0, and the others are the same as those in system (1). In view of the dissipative Hamiltonian realization methods presented in [10, 28, 31, 32], many switched polynomial nonlinear systems with AS, such as (19), can be transformed into the form of system (1). Therefore, the result studied in this paper can be used to solve the H∞ control problem of general switched polynomial nonlinear systems with AS.


## 3.2. Parameter Solution Algorithm

In this subsection, a symbolic computation-based parameter solution (PS) algorithm to solve the feasible interval value of the parameter is investigated, which is proceeded as follows. Furthermore, Table1 shows the process of the PS algorithm.Table 1
Process of the PS algorithm.
PS algorithmInput: System (1), flag=0, RCI=zerosm˜Output: Controlleru#Step 1: InitializationGix,Ki.subsKi,Kix,kiUix,Ki.subsKi,Kix,km#Step 2: Obtain the set of constraintsGπi with Gi≥0fi0=εifi1=1−εiGSi=fi0,fi1Gπi=fi0>0,fi1>0GPi=PrincipalMinorGiforl⌢ in range0,lenGP˜i:ifdiffGPil⌢,x1,…,diffGPil⌢,xn≠0:GMil⌢ki,εi=QuadraticMatrixGPil⌢GPPil⌢=PrincipalMinorGMil⌢Gπil⌢=GPPil⌢1≥0,…,GPPil⌢j^≥0,…else:GPPil⌢=GPil⌢Gπil⌢=GPil⌢≥0GSi.unionGPPil⌢Gπi.unionGπil⌢#Step 3: Obtain the set of constraintsUςi with Ui>0USi=setUςi=setUO˜i=OrderPrincipalMinorUiforl⌢ in range0,lenUO˜i:ifdiffUO˜il⌢,x1,…,diffUO˜il⌢,xn≠0:UM˜il⌢ki,εi=QuadraticMatrixUO˜il⌢UOOil⌢=OrderPrincipalMinorUM˜il⌢Uς˜il⌢=UOOil⌢1>0,…,UOOil⌢j^>0,…else:UOOil⌢=UO˜il⌢Uς˜il⌢=UO˜il⌢>0USi.unionUOOil⌢Uςi.unionUς˜il⌢#Step 4: Obtain the set of constraintsUς^i with Ui≥0US^i=setUπ^i=setUP^i=PrincipalMinorUiforl⌢ in range0,lenUP^i:ifdiffUP^il⌢,x1,…,diffUP^il⌢,xn≠0:UM^il⌢ki,εi=QuadraticMatrixUP^il⌢UPPil⌢=PrincipalMinorUM^il⌢Uς^il⌢=UPPil⌢1≥0,…,UPPil⌢h⌢≥0,…else:UPPil⌢=UP^il⌢Uς^il⌢=UP^il⌢≥0US^i.unionUPPil⌢Uς^i.unionUς^il⌢#Step 5: Obtain the interval value of the parameterRCi=set;Si=GSi,USiπi=Gπi,UςiCi=PlotImplicitSiCPi=CheckPointCiforj⌢ in range0,lenCi:m^i=0forl⌢ in range0,lenπi:ifπil⌢.subsx,CPij⌢ is True:m^i=m^i+1ifm^i==lenπi:RCi. unionCij⌢RCIi=1flag=1ifRCIi=0:S^i=GSi,US^iπ^i=Gπi,Uς^iCi=PlotImplicitS^iCPi=CheckPointCiforj⌢ in range0,lenCi:m^i=0forl⌢ in range0,lenπ^i:ifπ^il⌢.subsx,CPij⌢ is True:m^i=m^i+1ifm^i==lenπ^i:RCi.unionCij⌢RCIi=1//Step 6: Obtain controller for thei-th subsystemifRCIi==1uiki.subski,k¯i#Step 7: Obtain controller for system (1)#Repeat steps 1–6 for each subsystem of system (1)ifRCI==1.all is true and flag==1:returnuPS algorithm:Step 1. Obtain matrix inequality conditions. Design a parameter matrix Kix,km as Definition 1, and obtain the polynomial matrix inequality conditions by substituting Kix,km into (7), (8) for the i−th subsystem.Step 2. Obtain constraint condition set Gπi.Step 2.1. Achieve the constraint condition set Gπi by making all the principal minor of Gi described by (7) being non-negative and the condition 0<εi<1.Step 2.2. Transform all the polynomial constraint conditions in Gπi into the form of quadratic form. Then, the condition that the quadratic form is non-negative is equivalent to the condition that the quadratic form matrix is positive semi-definite. Get new constraint conditions from the positive semi-definiteness of the quadratic form matrix with the way of Step 2.1, and substitute them with the primary constraint.Step 3. Obtain constraint condition set Uςi with Ui>0. Get the constraint condition set Uςi by making all the order principal minor of Ui described by (8) being positive. For the constraint condition containing system state in Uςi, transform it into the form of quadratic form. Then, obtain new constraint conditions by making all the order principal minors of the quadratic form matrix being positive and substitute it with the original condition.Step 4. Obtain constraint condition set Uς^i with Ui≥0. Reset the constraint condition set Uς^i by making all the principal minor of Ui described by (8) being non-negative. Then, obtain Uς^i with the same way of Step 3.Step 5. Decompose the parameter space and obtain the interval value of the parameter. Let all the inequalities in Gπi and Uςi be implicit function, and draw them in the parameter space. Then, the parameter space is partitioned into many cells. According to Remark 1, there always exists only one parameter km in each parameter matrix in general. Thus, the decomposition is always produced in two-dimensional space (made up by εi and ki) for the i−th subsystem. In view of the cylindrical algebraic decomposition algorithm [42], all the points containing in each cell have the same characteristics. Then, search out the cell Ci as the required region Ci from all the cells by verifying any one point in each cell with all the constraint conditions in Gπi and Uςi. If there exists no Ci satisfying Uςi found, partition the parameter space again according to Gπi and Uς^i. Then, search out the cell Ci as the required region Ci from all the cells by verifying any one point in each cell with all the constraint conditions in Gπi and Uς^i.Step 6. Obtain the controller for the i-th subsystem. According to the required regions Ci obtained in Step 5, obtain the feasible interval value of each parameter. Then, choose a value for each ki according to its feasible interval value. Obtain the controller by substituting the values of ki into the parameterized controller (6).Step 7. Obtain Ci and controller for each subsystem by running steps 1–6. If there exists at least one subsystem (assumed the j−th subsystem, j∈Λ), for which the Cj satisfies Uj>0, the obtained controllers will be the required controllers.Remark 7.
In Table1, RCIi=1 identifies that there exist RCi regions for the i-th subsystem. flag=1identifies that there exists subsystem containing constraints Uςi satisfying Ui>0. RCi represents the set of cells satisfying the desired conditions for the i-th subsystem. Gix,Ki.subsKi,Kix,ki means substituting Ki in Gi with Kix,ki. PrincipalMinorGi means getting the set of all principal minor of Gi. GMil⌢ki,εi=QuadraticMatrixGPil⌢ means obtaining a quadratic form matrix GMil⌢ki,εi by transforming GPil⌢ into quadratic form GPil⌢=z^il⌢TxGMil⌢ki,εiz^il⌢x, in which z^il⌢x is a column vector whose term is monomial expression in regard to x. OrderPrincipalMinorUi denotes all the order principal minor of Ui. Ci=PlotImplicitSi means decomposing the parameter space into many cells Ci by using the trajectories of the implicit functions shown by all the polynomial expressions in Si. CPi=CheckPointCi represents a set of test point coordinates CPi obtained by taking a test point from each cell Ci.Remark 8.
The method of solving the interval value ofki and εi described in this paper can be extended to solve the interval value of parameters in other control system design involving polynomial matrix inequality condition containing parameters. In Example 2 of this paper, the PS algorithm will be used to solve the interval value of ki and γ when arraigning a constant to each εi. Then, except that the condition 0<εi<1 is replaced by the condition 0<γ in obtaining constraint condition set Gπi, the other steps are the same as solving the interval value of ki and εi.Remark 9.
The PS algorithm is suitable to solve the interval value of the parameter contained in the matrix inequality with polynomial structure. Comparing with the method of SOS in solving interval values of parameters in polynomial matrix inequalities, it does not need the polynomial inequality being convex, which implies that the polynomial inequality can contain nonlinear combination of unknown values. In addition, the value of the parameter in the inequality can be restricted to an arbitrary range in the method of this paper.


## 4. Example and Simulations

In order to verify the effectiveness of the method in this paper, two examples are supplied.Example 1.
Consider system (1) consisting of two subsystems with following parameters:(24)f1x=−4x1+12x22x23x3−6x1−12x2−4x24,g1=131022132f2x=−x1x22+2−12x2−4x32x12x3+3x1−8x2−4x12x2+4x1x3,g2=020501021satuj=signujmin1,uj,j∈1,2,3,g¯1=10611T,g¯2=3175T,γ=1.
Transform Example1 into Hamiltonian form with R1x=diag2,3,0, R2x=diag2+x22,4,0, H1x=x12+2x22+x32, H2x=1/2x12+x22+1/2x32, J1x=030−30x230−x230, J2x=0−6−4x330x124x3−2x120.
LetP=diag1,2,1, and define(25)xP=supx=1xTPx=supx=1H1x=supx=112H2x.
Obviously, Assumption1 holds for Example 1.
LetB1=diag1,1,1, and B2=diag1,1,1. In the following subsection, the parameterized controllers and the values of the parameters will be obtained with the PS algorithm developed in Section 3.2.
LetΛ=1,2. By Definition 1, design the following parameter matrices:(26)Kix,km=diagki,ki,ki,i∈Λ.
Then, the parameterized controllers can be obtained by substituting (20) into (6):(27)u1x,k1=−2k1−1x1+x3−2k1−13x1+4x2+3x3−2k1−1x1+4x2+2x3,u2x,k2=−10k2−5x2−2k2−1x1+x3−k2−122x2+x3.
On the other hand, letΨ1=ε1k1+1/22, and Ψ2=ε2k2+1/22. Then, by substituting (20) into (7), (8), we can obtain Gi and Ui, which are shown by (23)–(26).(28)G1=−1072+11k1−112Ψ1−34+8k1−4Ψ1−6Ψ1+12k1−61∗−4Ψ1+8k1−19−5Ψ1−38+10k1∗∗14k1−7Ψ1−1352≥0,(29)G2=4k2−2Ψ2+x22−92−512−2Ψ2+4k2−192∗−13Ψ2+26k2−3072k2−12Ψ2−43∗∗5k2−52Ψ2−15≥0,(30)U1=2+11k1−112Ψ18k1−4Ψ1−6Ψ1+12k1∗−4Ψ1+8k1+3−5Ψ1+10k1∗∗14k1−7Ψ1>0,(31)U2=4k2−2Ψ2+x22+20−2Ψ2+4k2∗−13Ψ2+26k2+4k2−12Ψ2∗∗5k2−52Ψ2>0.
Then, with the way of steps 2 and 3 of the PS algorithm, the constraint condition setsGπi and Uςi can be directly obtained with Gi and Ui. The expressions of Gπi and Uςi are omitted here. Letting all the inequalities in Gπi and Uςi be implicit function, and drawing them in the parameter space, partition the εi−ki plane into many cells. The curves of the implicit functions for the constraint conditions of the 1−th subsystem are shown in Figure 1, and curves of the implicit function for the constraint conditions of the 2−th subsystem are shown in Figure 2. Search out the required cell Ci by verifying any one point in each cell with all the constraint conditions in Gπi and Uςi. The required cell Ci for each subsystem is marked by gray background in Figures 1 and 2 separately.
According toC1 and C2, choose(32)k1=15,k2=18.
Design the following switching law:(33)λt=1t∈t2k,t2k+1t2k+1−t2k=0.1s,2t∈t2k+1,t2k+2t2k+2−t2k+1=0.2s,where k∈0,1,….
Suppose initial statesxt0=−10.60.8T and external disturbance w=0. Simulate system (1) of Example 1 under parameterized controller (21) and (22) with parameter (27) and switching law (28). The simulation results are shown in Figure 3. The curves of the related saturated controllers are shown in Figure 4.
Suppose initial statesxt0=000T, and suppose external disturbance w=0.8 is inserted into system (1) of Example 1 at the time interval 2.1∼3.3s. Simulate system (1) of Example 1 under parameterized controller (21) and (22) with parameter (27) and switching law (28). The simulation results are shown in Figure 5. The curves of the related saturated controllers are shown in Figure 6.Figure 1
Curves of constraint conditions inε1−k1 plane.Figure 2
Curves of constraint conditions inε2−k2 plane.Figure 3
State response of the system without disturbance.Figure 4
Curves ofsatu for the system without disturbance.Figure 5
State response of the system with disturbance.Figure 6
Curves ofsatu for the system with disturbance.Summary 1.
Note that the conditions (i) and (ii) of Theorem1 in this paper are the same as the conditions (1) and (2) in Theorem 2 of [11] when ki=1/2γ2. According to Figures 1 and 2 and Gπ1, Gπ2, when k1=k2=1/2γ2=1/2, there exist no ε1∈0,1, ε2∈0,1 such that conditions (1) and (2) in Theorem 2 of [11] hold, which means that the method of [11] cannot be used to solve the H∞ control problem of system (1) in Example 1, which implies that the parameterized controller (6) has greater functionality than the fixed-structure controller designed with the method of [11] for the SPHSs. In addition, when verifying whether the symmetric matrix inequalities are satisfied for some systems with complex structure, this method is easier to operate comparing with the method of [11].Example 2.
In this subsection, the switched Hamiltonian system subject to AS described in Example1 of [11] is considered. The structure parameter values, which can be found in Example 1 of [11], are omitted in this paper.
Note that(34)H1x=12x12+x22,H2x=14x14+x12x22+x24.
LetP=diag1/2,1, and define(35)xP=supx=1xTPx=supx=1H1x=supx=1H2x.
Obviously, Assumption1 holds for the 1−th subsystem and 2−th subsystem of system (1) in Example 2.
LetΛ=1,2. By Definition 1, define the following parameter matrices(36)Kix,km=ki,i∈Λ.
As with [11], choose B1=1/2, B2=1/4. Then, the parameterized controllers can be obtained by substituting (29) into (6):(37)u1x,k1=−4k1−1/2x2,u2x,k2=−3k2−332x14+2x22x12.
Choosingε1=1/3, ε2=1/2 as same as [11], in view of (11), it leads to(38)G1=γ−2G^1=G111G112∗G122>0,where G111=x12γ2+4γ2−1/2, G112=0G122=−1/96γ264k12−368k1+193.(39)G2=γ−2G^2=G211G212∗G222>0,where G211=−3/4096γ2x123072k22−12096k2−2045, G212=0, G222=2x222γ2−1.
Noting the fact thatγ>0, the conditions Gi>0 must be equivalent with the conditions G^i>0. Then, the constraint condition set Gπi can be obtained with G^i>0 instead of Gi>0.
On the other hand, in view of (12), the matrix inequality (34) and (35) can be obtained.(40)U1=U111U112∗U122>0,where U111=x12+4, U112=0, U122=−145/96+23/6k1−2/3k12.(41)U2=U211U212∗U222>0,where U211=7287/4096x12+567/64x12k2−9/4x12k22, U212=0, U222=4x22.
In addition, with the way of steps 2 and 3 of the PS algorithm, the constraint condition setsGπ1 and Uς1 can be obtained. Besides, the constraint condition sets Gπ2 and Uς2 can be obtained with the way of steps 2 and 3 of the PS algorithm. The expressions of Gπi and Uςi are omitted here. As with Example 1, partition the γ−k1 plane into many cells (see Figure 7) and partition the γ−k2 plane into many cells (see Figure 8), respectively. Then, by the Gπi and Uςi, the required cell C1, C2 can be obtained with the way of step 5 of the PS algorithm, respectively. The required cell Ci for each subsystem is marked by gray background in Figures 7 and 8 separately. According to Ci, the interval value of γ is obtained, i.e.,(42)γ∈0.70711,γ¯,where γ¯≥5. Obviously, according to Figures 7 and 8, we know that the right boundary value of the interval value of γ can be adjusted by setting appropriate upper boundary of the vertical axis in the simulation.
It should be pointed out that the conditions (i) and (ii) of Theorem1 in this paper are the same as the conditions (1) and (2) in Theorem 2 of [11] when ki=1/2γ2, whose curve is drawn in Figures 7 and 8 with long dash line for each subsystem, respectively. The interval solution of conditions (1) and (2) in Theorem 2 of [11] is the part of the long dash line, which is located in Ci.
According toC1 and C2, choose(43)k1=5,k2=4.
Then, the controllers can be obtained by substituting (37) into (30), (31). In addition, design the same switching law as Example 1 of [11].
Suppose initial statesxt0=2−2T, γ=0.8, and external disturbance w=0. Simulate Example 2 under parameterized controller (30), (31) with parameter (37) and switching law as Example 1 of [11]. The response of the system states is shown in Figure 9 (solid lines). The curve of the related saturated controller is shown in Figure 10 (solid line). With the same initial conditions, Example 2 is simulated with the method of [11]. The simulations are shown in Figures 9 and 10 with dashed lines, respectively.
Suppose initial statesxt0=00T, γ=0.8, and suppose external disturbance w=8cost is inserted into system (1) at 4.5∼9s. Simulate system (1) of Example 2 under parameterized controller (30), (31) with parameter (37) and switching law as Example 1 of [11]. The response of the system states is shown in Figure 11 (solid lines). The curve of the related saturated controller is shown in Figure 12 (solid line). With the same initial conditions, Example 2 is simulated with the method of [11]. The simulations are shown in Figures 11 and 12 with dashed lines, respectively.Figure 7
Curves of constraint conditions inγ−k1 plane.Figure 8
Curves of constraint conditions inγ−k2 plane.Figure 9
State response for the system without disturbance.Figure 10
Curves ofsatu for the system without disturbance.Figure 11
State response for the system with disturbance.Figure 12
Curves ofsatu for the system with disturbance.Summary 2.
According to Figures9 and 10, under the controllers designed with the method of this paper or the method of [11], the system states converge to the origin when there exists no external disturbance. It is shown in Figures 11 and 12 that the system’s states converge to the origin quickly after the disturbance vanishes under the controller designed with the method of this paper or designed with the method of [11]. In addition, in view of Figures 11 and 12, we know that the disturbance attenuation performance of the parameterized controller (30), (31) with parameter (37) is better than the controller designed with the method of [11], especially for state x2. Considering the fact that the controller designed with the method of [11] is a special case of the parameterized controller (30), (31) when ki=1/2γ2=50/64, the disturbance attenuation performance of the system of Example 2 can be improved by choosing appropriate values for the parameters ki, which implies that the performance regulation function of the parameterized controller is effective.


## 5. Conclusions

In this paper, a parameterized controller design method and parameter solution algorithm are presented to solve theH∞ control problem of the SPHSs subject to AS. As for the H∞ controller design problem of some SPHSs with complex structure, the method produced in this paper is more effective. Future research directions will be focused on applying the conclusions of this paper to solve the H∞ problem of actual polynomial systems subject to AS.



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