Strict positive and real output feedback controller design method based on linear matrix inequality

Abstract

The invention discloses a strict positive real output feedback controller design method based on a linear matrix inequality, and relates to the technical field of output feedback controller design. The method comprises the following steps: acquiring matrix input of a linear time-invariant system, and establishing a linear time-invariant system model based on a state-space equation; determining that the linear time-invariant system is a strictly positive satisfaction condition; judging whether the matrix B and the matrix C are full-rank and CB positive definite is true or not; if yes, a strict positive-real static output feedback controller is designed for the system; and if not, the system does not have a strict positive real output feedback controller. According to the controller design method provided by the invention, the strict positive real and static output feedback controller with H2 or H infinity performance can be designed based on the linear matrix inequality, and the limitation and solving complexity of the existing design method of the strict positive real and static output feedback controller are effectively solved.

Description

Technical Field

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

[0002] Passivity provides a powerful analysis and control tool for practical engineering systems. The theory of passivity and its application in control systems have attracted the research interests of many scholars. An important result of passivity is that negative feedback interconnected systems of passive dynamic systems and strictly passive dynamic systems are unconditionally internally stable. For finite-dimensional linear time-varying systems, passivity is equivalent to positive reality. Therefore, a famous stability theorem states that a strictly positive real system can robustly stabilize a positive real system interconnected with its negative feedback. This result has been applied to variable structure control, adaptive control, and robust switching control. 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 strictly positive real controller is to find a control law that makes the controlled system strictly positive real, and then use the stability results of 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 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 paid attention to. In order to achieve the good performance of strictly positive real control in terms of robustness and anti-disturbance, the closed-loop system H2 or H ∞Performance indicators are crucial for the extended application of positive real theory in engineering. For example, the literature Jos′e 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 realoptimal control of variable stiffness actuators. Control Engineering Practice, 82: 142-150, 2019 designed a positive real H ∞ Controller.

[0004] However, the existing research methods for positive real controller design 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, the systems in actual engineering are often more complex and difficult to meet such special requirements. Therefore, the scope of application of the existing methods is severely limited. ∞ There are few studies on strictly positive real controllers with high performance indicators, and existing design methods are only used to design state feedback controllers based on system states. Compared with measured outputs, system states are usually more difficult to obtain, and the design of output feedback controllers based on system outputs needs further study.

[0005] The present invention proposes a strictly positive real output feedback controller design method based on linear matrix inequality, aiming to provide a strictly positive real output feedback controller design scheme, so that the closed-loop system satisfies the strictly positive real property and H2 or H ∞ Performance indicators. Summary of the invention

[0006] The purpose of the present invention is to provide a strictly positive real output feedback controller design method based on linear matrix inequality to solve the problem that most of the existing design methods proposed in the above background technology are limited to being 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.

[0007] To achieve the above object, the present invention adopts the following technical solutions:

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

[0009] S1. Obtain the matrix input of the linear time-invariant system, the matrix input includes 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 whether the linear time-invariant system is strictly positive and real and satisfies the conditions;

[0010] S2, determine whether the matrix B and the matrix C are full rank and CB is positive definite;

[0011] S3, if it is established, then a strictly positive real static output feedback controller is designed for the system; based on the matrix input and linear matrix inequality of the linear time-invariant system, the closed-loop system composed of the static output feedback controller and the open-loop system satisfies the strictly positive real property; H2 and H can also be introduced ∞ Performance indicators, design strict positive real static output feedback H2 controller and H ∞ Controller;

[0012] S4. If it is not true, then there is no strictly positive real output feedback controller in the system.

[0013] Preferably, a linear time-invariant system model based on the state-space equation is established in S1, which is as follows:

[0014]

[0015] in: is the state vector, represents the differential 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 the coefficient matrices 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 expressed as:

[0018] G(s)=C(sI-A) -1 B

[0019] where I represents the identity matrix of appropriate dimensions, and A -1 Represents the inverse of the matrix A.

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

[0021] A linear time-invariant system is positive real if: G(s) has no poles in the right half-closed complex plane; for all s in the right half-open complex plane, G(s)+G * (s)≥0;

[0022] A linear time-invariant system is strictly positive real if the following conditions are met: G(s) is strictly positive real when there exists a scalar ∈>0 such that G(s-∈) is positive real;

[0023] Where: G * (s) represents the complex conjugate transpose of G(s), and A≥0 represents that A is a positive semidefinite symmetric matrix.

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

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

[0026] Determine whether the state space model of a linear time-invariant system satisfies the condition: For a given nonzero scalar ε, does the matrix P=P T >0, and The following conditions must be met:

[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, then there exists a static output feedback controller u(t)=Ky(t) satisfying K=HR -1 , so 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, so that the closed-loop system composed of the designed static output feedback controller and the open-loop system satisfies both the strictly positive real property and the H2 performance.

[0034] Determine whether the state space model of a linear time-invariant system satisfies the condition: For a given nonzero scalar ε, does the matrix P=P T >0, and The following conditions must be met:

[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 , so 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, μ is a A scalar of , where trace(A) represents the trace of the matrix A.

[0041] S303, design a strictly positive real static output feedback H for the system ∞ The controller makes the closed-loop system composed of the designed static output feedback controller and the open-loop system satisfy the strict positive real property and H ∞ performance.

[0042] Determine whether the state space model of the linear time-invariant system satisfies the condition: for a given non-zero scalar ε, does the matrix P=P T >0, and The following conditions must be met:

[0043]

[0044] CP=B T

[0045] Where: H = KR;

[0046] 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 , so 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, γ is a given scalar.

[0049] The second aspect of the present invention proposes the application of a strictly positive real output feedback controller design method based on linear matrix inequality in the control of a flexible robotic arm. The flexible robotic arm has a isotropic force actuator and a velocity sensor. A strictly positive real static output feedback controller is designed by the strictly positive real output feedback controller design method.

[0050] The third aspect of the present invention proposes the application of a strictly positive real output feedback controller design method based on linear matrix inequality in the control of a flexible robotic arm. The flexible robotic arm has a isotropic force actuator and a velocity sensor. A strictly positive real static output feedback H2 controller is designed by the strictly positive real output feedback controller design method.

[0051] The fourth aspect of the present invention proposes an application of a strictly positive real output feedback controller design method based on linear matrix inequality in a 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 H is designed by a strictly positive real output feedback controller design method. ∞ Controller.

[0052] Compared with the prior art, the present invention has the following beneficial effects:

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

[0054] (2) The present invention proposes a method based on static output feedback for the linear time-invariant system model so that the closed-loop system satisfies the strict positive real property and H2 or H ∞ Strictly positive real static output feedback H2 or H ∞Compared with the existing results, the static output feedback controller based on system output designed by the present invention is easier to obtain and has a wider range of applications in engineering practice. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] Figure 1 A flow chart of a design method for a strictly positive real output feedback controller based on linear matrix inequality in the present invention;

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

[0057] Figure 3 Schematic diagram of a flexible mechanical arm with an isostatic force actuator and a velocity sensor in Example 2 of the present invention;

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

[0059] Figure 5 This is a block diagram of the open-loop system of the flexible robotic arm in Example 2 of the present invention. DETAILED DESCRIPTION

[0060] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

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

[0062] Step 1: Model the system.

[0063] Obtain the matrix input of the linear time-invariant system, which includes 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 obtain the linear time-invariant system model based on the state space equation:

[0064]

[0065] in: is the state vector, represents the differential 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 the coefficient matrices

[0066] When u(t) = 0, through Laplace transformation, 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, A -1 Represents the inverse of the matrix A.

[0069] The following are 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 right half-closed complex plane; (2) for all s in the right half-open complex plane, G(s)+G * (s)≥0, where: G * represents the complex conjugate transpose of G, G * (s) represents the complex conjugate transpose of G(s).

[0071] Definition 2: A linear time-invariant system is strictly positive real if the following conditions are met: G(s) is strictly positive real when there exists a scalar ∈>0 such that G(s-∈) is positive real.

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

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

[0074] If not, proceed directly to step 4 to determine if there is no strictly positive real output feedback controller in the system.

[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 a linear time-invariant system satisfies the condition: For a given nonzero scalar ε, does the matrix P=P T >0, and The following conditions must be met:

[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, then there exists a static output feedback controller u(t)=Ky(t) satisfying K=HR -1 , so that the following closed-loop system satisfies the strictly positive real property:

[0082]

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

[0084] Determine whether the state space model of a linear time-invariant system satisfies the condition: For a given nonzero scalar ε, does the matrix P=P T >0, and The following conditions must be met:

[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, then there exists a static output feedback controller u(t) = Kt(t) satisfying K = HR -1 , so that the following closed-loop system:

[0089]

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

[0091] S303, design a strictly positive real static output feedback H for the system ∞ Controller:

[0092] Determine whether the state space model of the linear time-invariant system satisfies the condition: for a given non-zero scalar ε, does the matrix P=P T >0, and The following conditions must be met:

[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 , so that 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, γ is a given scalar.

[0099] Step 4: Determine that 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 * represent the transpose of matrix A and the complex conjugate transpose of matrix A respectively; 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] The following closed-loop system:

[0107]

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

[0109] Lemma 1: Consider an observable and controllable 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 makes the following conditions true:

[0110] AX+XA T <0

[0111] CX=B T

[0112] Lemma 2: (Projection Lemma) Consider the matrix Q∈H n , Existence Matrix Make

[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] Established.

[0121] The proof process is: Matrix inequality (3) is multiplied on the left and right by a full-rank matrix The following inequality is obtained:

[0122]

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

[0124]

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

[0126] So far, Lemma 3 has been proved.

[0127] The following are the H2 and H used to measure the linear time-invariant system. ∞ Two lemmas on performance indicators.

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

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

[0130] Lemma 5: For a given scalar γ>0, consider 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 makes

[0131]

[0132] The present invention assumes that the matrices B and C in step 2 are full rank and CB is positive definite. Theorem 1 and its proof process are given:

[0133] Theorem 1: Consider the open-loop system (1). If for a given nonzero scalar ε, there exists a matrix P=P T >0, Make

[0134]

[0135] CP=B T (8)

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

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

[0138]

[0139] By applying Lemma 3, we can obtain the following inequality from (7):

[0140]

[0141] (10) After sorting, it can be re-

[0142]

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

[0144] So far, Theorem 1 has been proved.

[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 free parameter ε should be chosen to make the feasibility problem as solvable as possible.

[0146] Theorem 2 and its proof are given:

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

[0148]

[0149] CP=B T (12)

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

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

[0152]

[0153] Established. According to (12) and the inequality relation μ 2 >trace(CPC T ) can derive the following equivalence relation:

[0154]

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

[0156]

[0157] So far, Theorem 2 has been proved.

[0158] Theorem 3 and its proof are given:

[0159] Theorem 3: Consider 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, Make

[0160]

[0161] CP=B T (14)

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

[0163] Proof: The proof process of Theorem 3 is similar to that of Theorem 2. According to Lemma 3, we can get (13)

[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] So far, Theorem 3 has been proved.

[0168] Theorem 3 is to design the suboptimal strictly positive real static output feedback H ∞ The controller provides sufficient conditions. By solving the convex optimization problem, the following inference is obtained: ∞ The optimal control problem provides a local optimal solution.

[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] Constraints: (13) and (14),

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

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

[0174] It should be noted that in Corollary 1, the value of Γ depends to some extent on the given scalar ε. For a fixed suitable scalar ε, the controller gain obtained by solving the convex optimization problem (15) is simply H ∞A local optimal solution to the optimal control problem. In addition, a search procedure can be used to determine the optimal value of the scalar ε and thus obtain the optimal solution to the problem. The effectiveness of this search procedure has been proven in other literature. First, the LMI toolbox is used to solve the feasibility problems (13) and (14) and obtain a set of initial scalar parameters; then, the numerical optimization algorithm fminsearch in the MATLAB optimization toolbox is used to obtain a local convergent solution to the scalar ε. On the other hand, the related state input 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 state feedback controller synthesis problems, the extended results of the present invention in state output feedback control provide a new method for this type of problem.

[0175] Next, two specific embodiments are given to illustrate the strictly positive real output feedback H based on the linear matrix inequality. ∞ Controller and design method of strictly positive real output feedback H2 controller.

[0176] Embodiment 1:

[0177] like Figure 2 As shown, Example 1 is a mathematical model of the leg under electrical stimulation of the quadriceps femoris. The model takes into account the gravity and inertia characteristics of the anatomical part and the damping and stiffness characteristics of the knee joint. The controller is designed to change the angle of the knee joint from 0° to 60° when electrical stimulation is applied to the quadriceps femoris. The equation describing the dynamics of the paraplegic patient is established in the state space as follows:

[0178]

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

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

[0181] is a nonlinear parameter of the system, defined as

[0182]

[0183] in: θ v is the knee joint angle, is the knee joint angular velocity, M a It is the torque generated by the electrical stimulation applied to the muscle. Input u = P N =PM a0 / G, P is the pulse width. Other related parameters are shown in the table below.

[0184] J Moment of inertia of the calf-foot complex <![CDATA[0.362[Kgm 2 ]]]> m Quality of the calf-foot complex 4.37[Kg] l Distance from knee joint to center of mass of calf-foot complex 23.8[cm] B Viscosity 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 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 located at 60°. The following system model can be established:

[0186]

[0187] The coefficient matrix of the 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 a given parameter γ=0.5, this embodiment simulates Theorem 3 using Matlab's YALMIP toolbox and MOSEK solver, and obtains a set of feasible solutions that satisfy Theorem 3:

[0192] R=2192.2,H=-38267.1

[0193] Therefore, this embodiment obtains a strictly positive real static output feedback H ∞ Controller gain K = HR -1 =-17.456, so that the closed-loop system is a strictly positive real system, and its transfer function matrix G cl1 (s) satisfies ‖G cl1 (s)‖ ∞ <0.5. Compared with the open-loop transfer function matrix G1(s) ∞ The norm is 42501.0, and the strictly positive real static output feedback H designed by S303 in step 3 ∞ The controller K effectively improves the robustness and performance of the closed-loop system in the face of uncertainty 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, the open-loop system is also strictly positive real. Therefore, the controller ensures that the closed-loop system satisfies H ∞ While improving the performance indicators, the strict positive real property of the system is also maintained. Then, 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 femoris.

[0196] Embodiment 2:

[0197] like Figure 3 As shown, Example 2 is a flexible mechanical arm with an isostatic force actuator and a speed sensor. Figure 4 Based on the block diagram of the flexible robotic arm feedback control system in the paper, the specific implementation steps of the design method 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 Figure 4 In the system block diagram shown, P(s) is the system model that is accurately modeled, and Δ(s) is the overflow dynamics that satisfies the positive real characteristic with uncertain parameters. Figure 5 As shown, for the passive object P(s), its input is the external disturbance w, and the output is the system controlled output z. The static output feedback controller K is designed, 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 interconnected system, and the internal stability of the entire closed-loop interconnected 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 the 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 the 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. Afterwards, this embodiment simulates Theorem 1 through Matlab's YALMIP toolbox and MOSEK solver, and obtains a set of feasible solutions that satisfy Theorem 1:

[0205]

[0206] Therefore, this embodiment obtains a strictly positive real static output feedback controller gain

[0207]

[0208] This makes the closed-loop system Q(s) a strictly positive real system, thereby ensuring the internal stability of the negative feedback interconnected system composed of Q(s) and Δ(s).

[0209] Next, select the parameters This example simulates Theorem 2 using Matlab's YALMIP toolbox and MOSEK solver to obtain a set of feasible solutions that satisfy Theorem 2:

[0210]

[0211] Therefore, this embodiment obtains a strictly positive real static output feedback H2 controller gain

[0212]

[0213] The closed-loop system is a strictly positive real system, and its transfer function matrix G cl2 (s) Satisfaction

[0214] ‖G cl2 (s)‖2=0.2908<1.6. The strictly positive real static output feedback H2 controller designed by S302 in step 3 effectively improves the performance of the closed-loop system in dealing with random aspects such as measurement noise and random noise. In addition, based on Lemma 1, considering the strictly positive real property of the closed-loop system, there exists a symmetric positive definite matrix

[0215]

[0216] Satisfies (A+B1KC1)P+P(A+B1KC1) T <0, CP=B T , therefore, the controller also ensures that the resulting closed-loop system Q(s) is strictly positive and real.

[0217] The above description 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 skilled in the art in the art, equivalent replacement or change according to the technical solution and inventive concept of the present invention within the technical scope disclosed by the present invention should be included in the protection scope of the present invention. In summary, the content of this specification should not be understood as limiting the present invention.

Claims

1. A strictly positive real output feedback controller design method based on linear matrix inequality, characterized in that: The steps include: S1. Obtain the matrix input of the linear time-invariant system, the matrix input includes 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 whether the linear time-invariant system is strictly positive and real and satisfies the conditions; S2, determine whether the matrix B and the matrix C are full rank and CB is positive definite; S3, if it is established, then a strictly positive real static output feedback controller is designed for the system; based on the matrix input and linear matrix inequality of the linear time-invariant system, the closed-loop system composed of the static output feedback controller and the open-loop system satisfies the strictly positive real property; H2 and H can also be introduced ∞ Performance indicators, design strict positive real static output feedback H2 controller and H ∞ Controller; S4. If it is not true, then there is no strictly positive real output feedback controller in the system.

2. The design method of a strictly positive real output feedback controller based on linear matrix inequality according to claim 1, characterized in that: In S1, a linear time-invariant system model based on the state space equation is established, which is as follows: in: is the state vector, represents the differential 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 the coefficient matrices represents an n×m dimensional real matrix, represents an n-dimensional real vector; When u(t) = 0, the transfer function matrix of the linear time-invariant open-loop system is expressed as: G(s)=C(sI-As) -1 B where I represents the identity matrix of appropriate dimensions, and A -1 Represents the inverse of the matrix A.

3. The design method of a strictly positive real output feedback controller based on linear matrix inequality according to claim 2, characterized in that: The conditions for determining that the linear time-invariant system is strictly positive and real in S1 are as follows: A linear time-invariant system is positive real if: G(s) has no poles in the right half-closed complex plane; for all s in the right half-open complex plane, G(s)+G * (s)≥0; A linear time-invariant system is strictly positive real if the following conditions are met: G(s) is strictly positive real when there exists a scalar ∈>0 such that G(s-∈) is positive real; Where: G * (s) represents the complex conjugate transpose of G(s), and A≥0 represents that A is a positive semidefinite symmetric matrix.

4. The design method of a strictly positive real output feedback controller based on linear matrix inequality according to claim 3, characterized in that: In S3, strictly positive real static output feedback controllers with different performances can be designed, as follows: S301, designing a strictly positive real static output feedback controller for the system, so that the closed-loop system composed of the designed static output feedback controller and the open-loop system satisfies the strictly positive real property; S302, designing a strictly positive real static output feedback H2 controller for the system, so that the closed-loop system composed of the designed static output feedback controller and the open-loop system satisfies both the strictly positive real property and the H2 performance; S303, design a strictly positive real static output feedback H for the system ∞ The controller makes the closed-loop system composed of the designed static output feedback controller and the open-loop system satisfy the strict positive real property and H ∞ performance.

5. The design method of a strictly positive real output feedback controller based on linear matrix inequality according to claim 4, characterized in that: The S301 is specifically as follows: Determine whether the state space model of a linear time-invariant system satisfies the condition: For a given nonzero scalar ε, does the matrix P=P T >0, and The following conditions must be met: CP=B T Where: H = KR; 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 , so that the following closed-loop system satisfies the strictly positive real property: Where: x(t) represents the state vector.

6. The method for designing a strictly positive real output feedback controller based on linear matrix inequality according to claim 4, characterized in that: The S302 is specifically as follows: Determine whether the state space model of a linear time-invariant system satisfies the condition: For a given nonzero scalar ε, does the matrix P=P T >0, and The following conditions must be met: Where: H = KR; 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 , so that the following closed-loop system: Satisfies the strictly positive real property and the closed-loop transfer function G cl (s) satisfies ||G cl (s)||2<μ, μ is a A scalar of , where trace(A) represents the trace of the matrix A.

7. The design method of a strictly positive real output feedback controller based on linear matrix inequality according to claim 4, characterized in that: The S303 is specifically as follows: Determine whether the state space model of the linear time-invariant system satisfies the condition: for a given non-zero scalar ε, does the matrix P=P T >0, and The following conditions must be met: Where: H = KR; 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 , so that the following closed-loop system: Satisfies the strictly positive real property and the closed-loop transfer function G cl (s) satisfies ||G cl (s)|| ∞ <γ, γ is a given scalar.

8. The design method of a strictly positive real output feedback controller based on linear matrix inequality as claimed in claim 5 is applied in the control of a flexible manipulator, characterized in that: The flexible robotic arm has a isostatic force actuator and a speed sensor, and a strictly positive real static output feedback controller is designed by the strictly positive real output feedback controller design method in claim 5.

9. The design method of a strictly positive real output feedback controller based on linear matrix inequality as claimed in claim 6 is applied to the control of a flexible manipulator, characterized in that: The flexible robotic arm has a isostatic force actuator and a speed sensor, and a strictly positive real static output feedback H2 controller is designed by the strictly positive real output feedback controller design method in claim 6.

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

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