Design Method of Strictly Positive Real Output Feedback Controller Based on Linear Matrix Inequality

Abstract

The present invention discloses a design method of a strictly positive real output feedback controller based on linear matrix inequalities, which relates to the technical field of output feedback controller design; the method comprises the following steps: obtaining the matrix input of a linear time-invariant system, and establishing a linear time-invariant system model based on a state space equation; determining the conditions for the linear time-invariant system to be strictly positive real; judging whether the matrix B and the matrix C are full rank and CB is positive definite; if so, designing a strictly positive real static output feedback controller for the system; if not, there is no strictly positive real output feedback controller for the system. The controller design method provided by the present invention can design a strictly positive real static output feedback controller with H<subgt;2< / subgt; or H<subgt;∞< / subgt> performance based on linear matrix inequalities, effectively solving the limitations and solving complexity of the existing design methods of strictly positive real static output feedback controllers.

Description

Technical Field

[0001] The present invention belongs to the technical field of output feedback controller design, and particularly relates to a design method of a strictly positive real output feedback controller based on linear matrix inequalities. Background Art

[0002] Passivity provides a powerful analysis and control tool for engineering practical systems, and the passivity theory and its applications in control systems have attracted the research interests of many scholars. An important result of passivity is that the negative feedback interconnection system of a passive dynamic system and a strictly passive dynamic system is unconditionally internally stable. For finite-dimensional linear time-varying systems, passivity is equivalent to positive realness. Therefore, a well-known stability theorem states that a strictly positive real system can robustly stabilize a positive real system interconnected with it in negative feedback. This result has been applied to variable structure control, adaptive control, and robust switching control, etc. The key step in these applications is to establish a controller to transform a non-strictly positive real system into a strictly positive real system.

[0003] The design problem of a strictly positive real controller is to find a control law such that the controlled system is strictly positive real, and then use the stability result of the positive real system to solve the stabilization problem. In addition, compared with the system state, the measured output is often easier to obtain in engineering applications. Therefore, the design of a strictly positive real static output feedback controller has been widely studied. On the other hand, in many practical applications, the stability of the system is not the only issue that needs to be concerned about. In order to achieve good performance in terms of robustness and anti-disturbance of strictly positive real control, the H2 or H of the closed-loop system must be considered when designing the controller. ∞Performance metrics are crucial for the extended application of positive real theory in engineering. For example, the literature "José Claudio Geromeland PB Gapski. Synthesis of positive real H2 controllers. IEEE Transactions on Automatic Control, 42(7): 988 - 992, 1997" and "James Richard Forbes. Synthesis of strictly positive real H2 controllers using dilated LMIs. International Journal of Control, 92(11): 2584 - 2590, 2019" studied strictly positive real H2 controllers; the literature "Berno JEMisgeld, Lukas Hewing, Lin Liu, and Steffen Leonhardt. Closed - loop positive real optimal control of variable stiffness actuators. Control Engineering Practice, 82: 142 - 150, 2019" designed positive real H ∞ controllers.

[0004] However, the existing design methods for positive real controllers have the following limitations: For strictly positive real static output feedback controllers, most of the existing design methods are only applicable to the special case where the system input matrix B and the control input matrix B1 satisfy B = B1 and the system output matrix C and the measurement output matrix C1 satisfy C = C1. However, systems in engineering practice are often more complex and difficult to meet such special requirements. Therefore, the applicable scope of the existing methods is severely limited. For strictly positive real controllers that meet the H2 or H ∞ performance metrics, there is relatively little relevant research, and the existing design methods are only used to design state - feedback controllers based on the system state. Usually, the system state is more difficult to obtain than the measurement output. The design problem of output - feedback controllers based on the system output needs to be further studied.

[0005] The present invention proposes a design method for a strictly positive real output - feedback controller based on linear matrix inequalities, aiming to provide a design scheme for a strictly positive real output - feedback controller such that the closed - loop system simultaneously satisfies the strictly positive real property and the H2 or H ∞ performance metrics. Summary of the Invention

[0006] The object of the present invention is to provide a design method of a strictly positive real output feedback controller based on linear matrix inequalities, so as to solve the problems in the above-mentioned background technology that most of the existing design methods are only applicable to special cases where the system input matrix B and the control input matrix B1 satisfy B = B1 and the system output matrix C and the measurement output matrix C1 satisfy C = C1, etc.

[0007] To achieve the above object, the present invention is implemented by the following technical solutions:

[0008] The first aspect of the present invention proposes a design method of a strictly positive real output feedback controller based on linear matrix inequalities, including the following steps:

[0009] S1. Obtain the matrix inputs of a linear time-invariant system, where the matrix inputs include the system matrix A, the control input matrix B1, the measurement output matrix C1, the system input matrix B, and the system output matrix C, and establish a linear time-invariant system model based on the state space equation; determine the conditions for the linear time-invariant system to be strictly positive real;

[0010] S2. Judge whether the matrices B and C are full rank and CB is positive definite;

[0011] S3. If it holds, design a strictly positive real static output feedback controller for the system; based on the matrix inputs of the linear time-invariant system and linear matrix inequalities, make the closed-loop system composed of the static output feedback controller and the open-loop system satisfy the strictly positive real property; and it is also possible to introduce H2 and H ∞ performance indicators to design a strictly positive real static output feedback H2 controller and H ∞ controller;

[0012] S4. If it does not hold, there is no strictly positive real output feedback controller for the system.

[0013] Preferably, in the above S1, the establishment of the linear time-invariant system model based on the state space equation is specifically as follows:

[0014]

[0015] Where: is the state vector, represents the differentiation of the state vector, is the control input vector, is the system input vector, is the measurement output vector,

[0016] is the system output vector, and each coefficient matrix represents an n×m-dimensional real matrix, represents an n-dimensional real vector;

[0017] When \(u(t) = 0\), the transfer function matrix of the linear time-invariant open-loop system is represented as:

[0018] \(G(s)=C(sI - A)^{ -1}B

[0019] where \(I\) represents the identity matrix with appropriate dimensions, and \(A^{ -1}\) represents the inverse of matrix \(A\).

[0020] Preferably, the condition for determining that the linear time-invariant system in S1 is strictly positive real is as follows:

[0021] The condition for a linear time-invariant system to be positive real is as follows: \(G(s)\) has no poles in the closed right-half complex plane; for all \(s\) in the open right-half complex plane, \(G(s)+G^{ *}(s)\geq0\);

[0022] The condition for a linear time-invariant system to be strictly positive real is as follows: if there exists a scalar \(\epsilon>0\) such that \(G(s - \epsilon)\) is positive real, then \(G(s)\) is strictly positive real;

[0023] where: \(G^{ *}(s)\) represents the complex conjugate transpose of \(G(s)\), and \(A\geq0\) represents that \(A\) is a positive semi-definite symmetric matrix.

[0024] Preferably, different performance strictly positive real static output feedback controllers can be designed in S3, as follows:

[0025] S301. Design a strictly positive real static output feedback controller for the system.

[0026] Judge whether the state-space model of the linear time-invariant system satisfies the condition: for a given non-zero scalar \(\epsilon\), whether there exists a matrix \(P = P^{ T}>0\), and satisfy the following conditions:

[0027]

[0028] \(CP = B^{ T}

[0029] where: \(H = KR\);

[0030] If the state-space model of the linear time-invariant system satisfies the above linear matrix inequality conditions, there exists a static output feedback controller \(u(t)=Ky(t)\) satisfying \(K = HR^{ -1}\), such that the following closed-loop system satisfies the strictly positive real property:

[0031]

[0032] where: x(t) represents the state vector.

[0033] S302. Design a strictly positive real static output feedback H2 controller for the system, such that the closed-loop system composed of the designed static output feedback controller and the open-loop system simultaneously satisfies the strictly positive real property and H2 performance.

[0034] Judge whether the state-space model of the linear time-invariant system satisfies the condition: for a given non-zero scalar ε, whether there exists a matrix P = P T > 0, and satisfy the following conditions:

[0035]

[0036] CP = B T

[0037] where: H = KR;

[0038] If the state-space model of the linear time-invariant system satisfies the above linear matrix inequality conditions, then there exists a static output feedback controller u(t) = Ky(t) satisfying K = HR -1 , such that the following closed-loop system:

[0039]

[0040] satisfies the strictly positive real property and the closed-loop transfer function G cl (s) satisfies ||G cl (s)||2 < μ, where μ is a scalar satisfying and trace(A) represents the trace of matrix A.

[0041] S303. Design a strictly positive real static output feedback H ∞ controller for the system, such that the closed-loop system composed of the designed static output feedback controller and the open-loop system simultaneously satisfies the strictly positive real property and H ∞ performance.

[0042] Judge whether the state-space model of the described linear time-invariant system satisfies the condition: for a given non-zero scalar ε, whether there exists a matrix P = P T > 0, and satisfy the following conditions:

[0043]

[0044] CP = B T

[0045] where: H = KR;

[0046] For the state - space model of a linear time - invariant system, if it satisfies the above linear matrix inequality conditions, there exists a static output feedback controller u(t)=Ky(t) such that K = HR -1 , such that the following closed - loop system:

[0047]

[0048] satisfies the strictly positive real property and the closed - loop transfer function G cl (s) satisfies ||G cl (s)|| ∞ <γ, where γ is a given scalar.

[0049] In the second aspect of the present invention, an application of the strictly positive real output feedback controller design method based on linear matrix inequalities in the control of a flexible robotic arm is proposed. The flexible robotic arm has a co - located force actuator and a velocity sensor, and a strictly positive real static output feedback controller is designed by the strictly positive real output feedback controller design method.

[0050] In the third aspect of the present invention, an application of the strictly positive real output feedback controller design method based on linear matrix inequalities in the control of a flexible robotic arm is proposed. The flexible robotic arm has a co - located force actuator and a velocity sensor, and a strictly positive real static output feedback H2 controller is designed by the strictly positive real output feedback controller design method.

[0051] In the fourth aspect of the present invention, an application of the strictly positive real output feedback controller design method based on linear matrix inequalities in the leg mathematical model under electrical stimulation of the quadriceps femoris is proposed. It is characterized in that a leg mathematical model under electrical stimulation of the quadriceps femoris is established, and a strictly positive real static output feedback H ∞ controller is designed by the strictly positive real output feedback controller design method.

[0052] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0053] (1) The present invention provides a strictly positive real output feedback controller design method based on linear matrix inequalities for a linear time - invariant system model. The controller gain can be obtained by solving linear matrix inequalities. The design method is simple, the calculation cost is low, it is easy to implement in actual programming, and it has obvious advantages for multi - input multi - output systems.

[0054] (2) The present invention gives a strictly positive real static output feedback H2 or H ∞ for a linear time - invariant system model based on static output feedback, such that the closed - loop system simultaneously satisfies the strictly positive real property and the H2 or H ∞Controller. Compared with the existing results, for the static output feedback controller designed based on the system output in the present invention, the system output is easier to obtain and has a wider application in engineering practice. Description of the Drawings

[0055] Figure 1 is the flowchart of the design method of the strictly positive real output feedback controller based on linear matrix inequality in the present invention;

[0056] Figure 2 is the schematic diagram of the leg mathematical model under the electrical stimulation of the quadriceps femoris in Embodiment 1 of the present invention;

[0057] Figure 3 is the schematic diagram of the flexible robotic arm with a collocated force actuator and a velocity sensor in Embodiment 2 of the present invention;

[0058] Figure 4 is the block diagram of the feedback control system of the flexible robotic arm in Embodiment 2 of the present invention;

[0059] Figure 5 is the block diagram of the open-loop system of the flexible robotic arm in Embodiment 2 of the present invention. Detailed Embodiments

[0060] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0061] Refer to Figure 1 , the design method of the strictly positive real output feedback controller based on linear matrix inequality includes the following steps:

[0062] Step 1: Model the system.

[0063] Obtain the matrix inputs of the linear time-invariant system, including the system matrix A, the control input matrix B1, the measurement output matrix C1, the system input matrix B, and the system output matrix C, to obtain the linear time-invariant system model based on the state-space equation:

[0064]

[0065] Where: is the state vector, represents the differentiation of the state vector, is the control input vector, is the system input vector, is the measurement output vector, is the system output vector, and each coefficient matrix

[0066] When u(t)=0, through Laplace transform, the transfer function matrix of the linear time-invariant open-loop system is obtained as follows:

[0067] G(s)=C(sI - A) -1 B

[0068] where: I represents the identity matrix with appropriate dimensions, and A -1 represents the inverse of matrix A.

[0069] The following gives the definitions of linear time-invariant positive real systems and strictly positive real systems, specifically:

[0070] Definition 1: A linear time-invariant system is positive real if it satisfies the following conditions: (1) G(s) has no poles in the closed right-half complex plane; (2) for all s in the open right-half complex plane, G(s)+G * (s)≥0, where: G * represents the complex conjugate transpose of G, and G * (s) represents the complex conjugate transpose of G(s).

[0071] Definition 2: A linear time-invariant system is strictly positive real if it satisfies the following condition: When there exists a scalar ∈>0 such that G(s - ∈) is positive real, then G(s) is strictly positive real.

[0072] Step 2: Determine whether matrices B and C are full rank and CB is positive definite.

[0073] If so, proceed to Step 3 to design a strictly positive real output feedback controller for the linear time-invariant system;

[0074] If not, directly go to Step 4 to determine that the system does not have a strictly positive real output feedback controller.

[0075] Step 3: Design a strictly positive real output feedback controller for the linear time-invariant system.

[0076] S301. Design a strictly positive real static output feedback controller for the system:

[0077] Determine whether the state-space model of the linear time-invariant system satisfies the condition: For a given non-zero scalar ε, whether there exists a matrix P = P T >0, and satisfy the following conditions:

[0078]

[0079] CP = BT

[0080] Where: H = KR.

[0081] If the state - space model of the linear time - invariant system satisfies the above linear matrix inequality conditions, there exists a static output feedback controller u(t) = Ky(t) such that K = HR -1 , making the following closed - loop system satisfy the strictly positive real property:

[0082]

[0083] S302. Design a strictly positive real static output feedback H2 controller for the system:

[0084] Judge whether the state - space model of the linear time - invariant system satisfies the condition: for a given non - zero scalar ε, whether there exist matrices P = P T > 0, and satisfying the following conditions:

[0085]

[0086] CP = B T

[0087] Where: H = KR.

[0088] If the state - space model of the linear time - invariant system satisfies the above linear matrix inequality conditions, there exists a static output feedback controller u(t) = Kt(t) such that K = HR -1 , making the following closed - loop system:

[0089]

[0090] satisfy the strictly positive real property and the closed - loop transfer function G cl (s) satisfies ||G cl (s)||2 < μ, where μ is a scalar satisfying , and trace(A) represents the trace of matrix A.

[0091] S303. Design a strictly positive real static output feedback H ∞ controller for the system:

[0092] Judge whether the state - space model of the linear time - invariant system satisfies the condition: for a given non - zero scalar ε, whether there exist matrices P = P T > 0, and satisfying the following conditions:

[0093]

[0094] CP=B T

[0095] Where: H = KR.

[0096] If the state space model of the linear time-invariant system satisfies the above linear matrix inequality conditions, then there exists a static output feedback controller u(t) = Ky(t) satisfying K = HR -1 , making the following closed-loop system:

[0097]

[0098] Satisfies the strictly positive real property and the closed-loop transfer function G cl (s) satisfies ||G cl (s)|| ∞ <γ, where γ is a given scalar.

[0099] Step 4: Determine whether there is no strictly positive real output feedback controller for the linear time-invariant system.

[0100] In the present invention, the following standard symbols are used: A>(≥)0 indicates that A is a positive definite (semi-positive definite) symmetric matrix, A<(≤)0 indicates that A is a negative definite (semi-negative definite) symmetric matrix; A T and A * Respectively represent the transpose of matrix A and the complex conjugate transpose of matrix A; A -1 represents the inverse of the A matrix; and Represents n-dimensional real vector and m×n-dimensional real matrix respectively; H n represents an n-dimensional Hermitian matrix; trace represents the trace of the matrix; I and 0 represent the identity matrix and zero matrix with appropriate dimensions, respectively.

[0101] The following is a detailed introduction to the design method of the strictly positive real static output feedback controller in step 3. First, the strictly positive real property determination lemma of the linear time-invariant system based on state space realization is given.

[0102] For the linear time-invariant system model in step 1:

[0103]

[0104] The purpose of this invention is to design a static output feedback controller:

[0105] u(t)=Ky(t)

[0106] Make the following closed-loop system:

[0107]

[0108] is a strictly positive real system, and at the same time, the transfer function matrix G cl (s) = C(sI - A c ) -1 B has an H2 norm less than a given scalar μ, μ > 0 or an H ∞ norm less than a given scalar γ, γ > 0, where A c = A + B1KC1.

[0109] Lemma 1: Consider a controllable and observable system (A, B, C, 0), whose transfer function matrix G(s) is strictly positive real if and only if there exists a symmetric matrix X = X T > 0 such that the following conditions hold:

[0110] AX + XA T <0

[0111] CX = B T

[0112] Lemma 2: (Projection Lemma) Consider a matrix Q ∈ H n , there exists a matrix such that

[0113] Q + UKV * + VK * U * <0

[0114] if and only if:

[0115] U ⊥ QU ⊥* <0, V ⊥ QV ⊥* <0

[0116] Lemma 3: Consider matrices W, M, N of appropriate dimensions and a given non - zero scalar ε. If there exists a matrix R such that

[0117] <>

[0118] Then, the following inequality:

[0119] W < 0, W + MN + N T M T <0 (4)

[0120] holds.

[0121] The proof process is as follows: Multiply the matrix inequality (3) on the left and right by a full - rank matrix to obtain the following inequality:

[0122]

[0123] Applying Lemma 2, from (5) we can obtain:

[0124]

[0125] where: the matrix U in Lemma 2 is regarded as U ⊥ is regarded as [IM], and the matrix V is regarded as V ⊥ is regarded as [I0]. Therefore, (4) can be obtained from obtaining (6), that is, (4) can be deduced from (3).

[0126] Thus, Lemma 3 is proven.

[0127] Next, two lemmas are given to measure the H2 and H ∞ performance metrics of a linear time-invariant system.

[0128] Lemma 4: For a given scalar μ > 0, consider a continuous-time system with a transfer function matrix G(s) = C(sI - A) -1 B, then ‖G(s)‖2 < μ if and only if there exists a matrix X = X T > 0 such that

[0129] AX + XA T + BB T < 0, trace(CXC T ) < μ 2

[0130] Lemma 5: For a given scalar γ > 0, consider a continuous-time system with a transfer function matrix G(s) = C(sI - A) -1 B, then the system is asymptotically stable and ‖G(s)‖ ∞ < γ if and only if there exists a matrix X = X T > 0 such that

[0131]

[0132] This invention assumes that the matrices B and C in step two are full rank and CB is positive definite. Theorem 1 and its proof process are given as follows:

[0133] Theorem 1: Considering the open-loop system (1), if for a given non-zero scalar ε, there exists a matrix P = P T > 0, such that

[0134]

[0135] CP = B T (8)

[0136] where: H = KR, then there exists a static output feedback controller K = HR -1 such that the closed-loop system (2) is a strictly positive real system.

[0137] Proof: For a given non-zero scalar ε, from (7) we have -εR - εR T < 0, that is, R is non-singular. According to Lemma 1, first prove that there exists a matrix P = P T > 0 satisfying

[0138]

[0139] By applying Lemma 3, the following inequality can be obtained from (7)

[0140]

[0141] After rearrangement, (10) can be rewritten as

[0142]

[0143] This inequality implies that (9) holds. Therefore, from conditions (7) and (8), it can be obtained that the closed-loop system (2) is strictly positive real.

[0144] Thus, Theorem 1 is proven.

[0145] It should be noted that a non-zero scalar ε is introduced in the design of the static output feedback controller to reduce conservatism. For a fixed parameter ε, (7) is a linear matrix inequality, and its feasible solution can be obtained using the LMI toolbox of Matlab. The selection of the free parameter ε should make the feasibility problem solvable as much as possible.

[0146] Give Theorem 2 and its proof process:

[0147] Theorem 2: Consider the open-loop system (1) and the static output feedback controller K. If for a given non-zero scalar ε, there exists a matrix P = P T > 0, such that

[0148]

[0149] CP = B T (12)

[0150] where: H = KR, then the closed-loop system (2) is strictly positive real, and the closed-loop transfer function matrix G cl (s) satisfies ‖G cl (s)‖2 < μ, where is a given scalar. Thus, the static output feedback controller gain can be calculated by K = HR -1 obtained.

[0151] Proof: The proof process of Theorem 2 is similar to that of Theorem 1. By applying Lemma 3, (9) and

[0152]

[0153] hold. From (12) and the inequality relation μ 2 > trace(CPC T ), the following equivalent relation can be obtained:

[0154]

[0155] Therefore, the given scalar μ needs to satisfy According to Lemma 1, (9) and (12) show that the closed-loop system (2) is strictly positive real, and the closed-loop transfer function matrix satisfies ‖G(s)‖2 < μ,

[0156]

[0157] Thus far, Theorem 2 is proven.

[0158] Theorem 3 and its proof are given as follows:

[0159] Theorem 3: Considering the open-loop system (1) and the static output feedback controller K, if for a given non-zero scalar ε and a given scalar γ > 0, there exists a matrix P = P T > 0, such that

[0160]

[0161] CP = B T (14)

[0162] where: H = KR, then the closed-loop system (2) is strictly positive real, and the closed-loop transfer function matrix satisfies ‖G cl (s)‖ ∞ < γ. The static output feedback controller gain can be calculated by K = HR -1 obtained.

[0163] Proof: The proof process of Theorem 3 is similar to that of Theorem 2. By Lemma 3, (13) can be used to obtain

[0164]

[0165] Therefore, the closed-loop system (2) is strictly positive real, and the closed-loop transfer function matrix satisfies

[0166] ‖G cl (s)‖ ∞ <γ.

[0167] Thus, Theorem 3 is proven.

[0168] Theorem 3 provides a sufficient condition for designing a sub-optimal strictly positive real static output feedback H ∞ controller. Through solving a convex optimization problem, the following corollary provides a local optimal solution for the H ∞ optimal control problem.

[0169] Corollary 1: Consider the open-loop system (1) and the static output feedback controller K. For a given non-zero scalar ε, if the following convex optimization problem:

[0170] Γ = minγ (15)

[0171] Subject to: (13) and (14),

[0172] where: H = KR, has a set of feasible solutions: P = P T > 0, then the closed-loop system (2) is strictly positive real and satisfies ‖G cl (s)‖ ∞ < Γ, where: Γ > 0 is a local minimum of γ. Therefore, the controller gain can be obtained by K = HR -1 .

[0173] Proof: The proof process of Corollary 1 is similar to the proof process of Theorem 3.

[0174] It should be noted that in Corollary 1, the value of Γ depends to a certain extent on the given scalar ε. For a fixed and appropriate scalar ε, the controller gain obtained by solving the convex optimization problem (15) is only the H ∞A local optimal solution to the optimal control problem. In addition, the optimal value of the scalar ε can be determined through a search program to obtain the optimal solution to this problem. The effectiveness of this search program has been verified in other literature. First, use the LMI toolbox to solve the feasibility problems (13) and (14) and obtain a set of initial scalar parameters; then, use the numerical optimization algorithm fminsearch in the MATLAB optimization toolbox to obtain a locally convergent solution of the scalar ε. On the other hand, the related state feedback controller synthesis problem can be regarded as a special case of the static output feedback controller synthesis problem when C1 = I. Although there are existing methods for solving the state feedback controller synthesis problem, the extended results of the present invention in state output feedback control provide a new method for such problems.

[0175] Next, two specific embodiments are given to illustrate the design methods of the strictly positive real output feedback H ∞ controller and the strictly positive real output feedback H2 controller.

[0176] Embodiment 1:

[0177] As Figure 2 shown, Embodiment 1 is a mathematical model of the leg under electrical stimulation of the quadriceps muscle. The model considers the gravitational and inertial characteristics of the anatomical part and the damping and stiffness characteristics of the knee joint. The design purpose of the controller is to change the knee joint angle from 0° to 60° when applying electrical stimulation to the quadriceps muscle. The equations describing the dynamics of the paraplegic patient are established in the state space as:

[0178]

[0179] where: the state variable x1 = Δθ v = θ v - θ v0 represents the change in the knee joint angle,

[0180] represents the change in the knee joint angular velocity, x3 = ΔM a = M a - M a0 represents the change in the torque generated by the electrical stimulation applied to the muscle, P N = P - M a0 / G;

[0181] is a non - linear parameter of the system, defined as

[0182]

[0183] where: θ v is the knee joint angle, is the angular velocity of the knee joint, M a is the torque generated by the electrical stimulation applied to the muscle. The input u = P N = P - M a0 / G, where P is the pulse width. Other relevant parameters are shown in the following table.

[0184] J Inertia moment of the shank-foot complex <![CDATA[0.362[Kgm 2 > m Mass of the shank-foot complex 4.37 [Kg] l Distance from the knee joint to the center of mass of the shank-foot complex 23.8 [cm] B Viscous coefficient 0.27 [Nms / rad] λ Exponential term coefficient 41.208 [Nm / rad] E Exponential term coefficient 2.024 [1 / rad] ω Static elastic knee joint angle 2.918 [rad] τ Time constant of the pole 0.951[s] G Static gain 42500 [Nm / s]

[0185] According to the above parameter definitions, the state variable x1 (i.e., the change in the knee joint angle) is selected for measurement, and the operating point is at 60°. The following system model can be established:

[0186]

[0187] The coefficient matrix of this system model is as follows:

[0188]

[0189] C = C1 = [0 0 1]

[0190] It can be verified that the matrices B and C are full rank, and CB is positive definite.

[0191] For the given parameter γ = 0.5, in this embodiment, the theorem 3 is simulated through the YALMIP toolbox of Matlab and the MOSEK solver, and a set of feasible solutions that satisfy theorem 3 are obtained:

[0192] R = 2192.2, H = -38267.1

[0193] Therefore, this embodiment obtains a strictly positive real static output feedback H ∞ The controller gain K = HR -1 = -17.456, making the closed-loop system a strictly positive real system, and the G of its transfer function matrix cl1 (s) satisfies ‖G cl1 (s)‖ ∞ < 0.5. Compared with the H of the open-loop transfer function matrix G1(s) ∞ The norm is 42501.0. Through the strictly positive real static output feedback H designed in step three ∞ The controller K effectively improves the robustness and performance of the closed-loop system in the face of uncertainties and external disturbances. In addition, based on Lemma 1, considering the strictly positive real property of the closed-loop system, there exists a symmetric positive definite matrix

[0194]

[0195] Satisfies (A + B1KC1)P + P(A + B1KC1) T< 0, CP = B T . Similarly, it can be known that the open-loop system is also strictly positive real. Therefore, while ensuring that the closed-loop system meets the H ∞ performance index, the controller also maintains the strictly positive real property of the system. Furthermore, the passivity theorem can be used to design a positive real controller for the closed-loop system to ensure the control stability of the leg mathematical model under electrical stimulation of the quadriceps muscle.

[0196] Example 2:

[0197] As Figure 3 shown, Example 2 is a flexible robotic arm with a co-located force actuator and a velocity sensor. Through the Figure 4 block diagram of the flexible robotic arm feedback control system in, the specific implementation steps of the design methods of the strictly positive real static output feedback controller and the strictly positive real static output feedback H2 controller are introduced respectively.

[0198] First, in the Figure 4 shown system block diagram, P(s) is the accurately modeled system model, and Δ(s) is the spillover dynamics with uncertain parameters that satisfy the positive real property. As Figure 5 shown, for the passive object P(s), its input is the external disturbance w, and its output is the system controlled output z. Design a static output feedback controller K, and there is a control relationship u(t) = Ky(t), where y is the measured output of P(s) and u is the control input of P(s). The passive object P(s) and the static output feedback controller K form a closed-loop system Q(s), and Δ(s) and Q(s) form a closed-loop negative feedback interconnection system. Furthermore, the internal stability of the entire closed-loop interconnection system is guaranteed by the passivity theorem.

[0199] Assume that the controlled object P(s) has the following form:

[0200]

[0201] The coefficient matrix of this system model is as follows:

[0202]

[0203] It can be verified that the matrices B and C are full rank, and CB is positive definite.

[0204] The transfer function matrix G2(s) of this system is unstable and has eigenvalues λ1 = 4.0329, λ2 = -1.2556, λ3 = -1.7773. Therefore, the open-loop system is not strictly positive real. Then, in this example, the theorem 1 is simulated through the YALMIP toolbox and the MOSEK solver of Matlab, and a set of feasible solutions that satisfy theorem 1 are obtained:

[0205]

[0206] Therefore, a strictly positive real static output feedback controller gain is obtained in this embodiment.

[0207]

[0208] So that the closed-loop system Q(s) is a strictly positive real system, which further guarantees the internal stability of the negative feedback interconnection system composed of Q(s) and Δ(s).

[0209] Secondly, select the parameters In this embodiment, the theorem 2 is simulated by using the YALMIP toolbox and MOSEK solver of Matlab, and a set of feasible solutions satisfying theorem 2 are obtained:

[0210]

[0211] Therefore, a strictly positive real static output feedback H2 controller gain is obtained in this embodiment.

[0212]

[0213] So that the closed-loop system is a strictly positive real system, and the G cl2 (s) of its transfer function matrix satisfies

[0214] ‖G cl2 (s)‖2 = 0.2908 < 1.6. The strictly positive real static output feedback H2 controller designed in step S302 effectively improves the performance of the closed-loop system in dealing with random aspects such as measurement noise and random noise. In addition, considering the strictly positive real property of the closed-loop system based on Lemma 1, there exists a symmetric positive definite matrix

[0215]

[0216] Satisfying (A + B1KC1)P + P(A + B1KC1) T < 0, CP = B T . Therefore, this controller also guarantees that the obtained closed-loop system Q(s) is strictly positive real.

[0217] The above is only used to help understand the method of the present invention and its core essence, but the protection scope of the present invention is not limited thereto. For those of ordinary skill in the art in the technical field of the present invention, any equivalent replacement or change made according to the technical solution and inventive concept of the present invention should be covered within the protection scope of the present invention. In summary, the content of this specification should not be construed as a limitation to the present invention.

Claims

1. A design method for a strictly positive real output feedback controller based on linear matrix inequalities, characterized in that, Including the following steps: S1. Obtain the matrix inputs of the linear time-invariant system, where the matrix inputs include the system matrix , the control input matrix , the measurement output matrix , the system input matrix and the system output matrix , and establish a linear time-invariant system model based on the state space equation; determine the conditions that the linear time-invariant system satisfies to be strictly positive real; Establish a linear time-invariant system model based on the state-space equation, specifically as follows: Wherein: is the state vector, represents the differentiation of the state vector, is the system control input vector, is the system input vector, is the measurement output vector, is the system output vector, and each matrix , , , , ; represents a real matrix of dimension represents a real vector of dimension; When the transfer function matrix of the linear time-invariant system is expressed as: Among them, represents an identity matrix with appropriate dimensions; Determine the conditions for the linear time-invariant system to be strictly positive real, specifically as follows: A linear time-invariant system is positive real if it satisfies the following conditions: it has no poles in the closed right-half complex plane; for all in the open right-half complex plane, we have A linear time-invariant system is strictly positive real if it satisfies the following condition: when there exists a scalar such that is positive real, then is strictly positive real; Wherein: represents the complex conjugate transpose of; S2, Matrix and Matrix Judge when full rank holds Whether positive definite holds; S3. If it holds, design a strictly positive real static output feedback controller for the system; based on the matrix input of the linear time-invariant system and linear matrix inequalities, make the closed-loop system composed of the static output feedback controller and the linear time-invariant system model satisfy the strictly positive real property; Determine whether the linear time-invariant system model satisfies the condition: for a given non-zero scalar , whether there exist matrices , , and such that the following conditions are satisfied: Wherein: ; If the linear time-invariant system model satisfies the above linear matrix inequality conditions, there exists an output vector of the static output feedback controller satisfying , such that the following closed-loop system satisfies the strictly positive real property: Wherein: represents a state vector; S4. If it does not hold, there is no strictly positive real output feedback controller for the system.

2. The design method of a strictly positive real output feedback controller based on linear matrix inequalities according to claim 1, characterized in that Introduced in S3 and performance indicators, design a strictly positive real static output feedback controller and controller; it is possible to design strictly positive real static output feedback controllers with different performances, specifically as follows: S301. Design a strictly positive real static output feedback controller for the system, such that the closed-loop system composed of the designed static output feedback controller and the linear time-invariant system model satisfies the strictly positive real property; S302. Design a strictly positive real static output feedback controller for the system, such that the closed-loop system composed of the designed static output feedback controller and the linear time-invariant system model simultaneously satisfies the strictly positive real property and performance; S303. Design a strictly positive real static output feedback controller for the system such that the closed-loop system composed of the designed static output feedback controller and the linear time-invariant system model simultaneously satisfies the strictly positive real property and performance.

3. The method for designing a strictly positive real output feedback controller based on linear matrix inequalities according to claim 2, characterized in that, The specific content of S302 is as follows: Determine whether the linear time-invariant system model satisfies the condition: for a given non-zero scalar , whether there exist matrices , , and that satisfy the following conditions: Wherein: ; For the linear time-invariant system model, if the above linear matrix inequality conditions are satisfied, there exists an output vector of the static output feedback controller satisfying , such that the following closed-loop system: Satisfy the strictly positive real property and the closed-loop transfer function Satisfy , is a scalar that satisfies of 4. The design method of a strictly positive real output feedback controller based on linear matrix inequalities according to claim 2, characterized in that, The specific content of S303 is as follows: Determine whether the linear time-invariant system model satisfies the condition: for a given non-zero scalar , whether there exist matrices , , and that satisfy the following conditions: Wherein: ; If the linear time-invariant system model satisfies the above linear matrix inequality conditions, there exists an output vector of the static output feedback controller that satisfies , such that the following closed-loop system: Satisfy the strictly positive real property and the closed-loop transfer function Satisfy , is a given scalar.

5. The application of the design method of the strictly positive real output feedback controller based on linear matrix inequality in the control of a flexible manipulator is characterized in that The flexible robotic arm has a co-located force actuator and a velocity sensor, and designs a strictly positive real static output feedback controller through the strictly positive real output feedback controller design method in claim 1.

6. Application of the design method of a strictly positive real output feedback controller based on linear matrix inequalities in the control of a flexible robotic arm, characterized in that, The flexible robotic arm has a co-located force actuator and a velocity sensor, and a strictly positive real static output feedback controller is designed by the strictly positive real output feedback controller design method in claim 3. controller.

7. The method for designing a strictly positive real output feedback controller based on linear matrix inequalities is applied to the leg mathematical model under electrical stimulation of the quadriceps femoris, characterized in that A leg mathematical model under electrical stimulation of the quadriceps femoris is established, and a strictly positive real static output feedback controller is designed by the strictly positive real output feedback controller design method in claim 4. controller.

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