Discrete period Riccati equation solving method based on non-inverse iteration and latest estimation information

Abstract

The invention discloses a discrete period Riccati equation solving method based on non-inverse iteration and newest estimation information, which comprises the following steps of: firstly, converting an original equation into an equation with only one inversion operation through a Schur complementary theorem to reduce the inversion operation, and then carrying out non-inverse iteration on the inversion operation to further realize a non-inverse solving algorithm, so that the discrete period Riccati equation is obtained. The solving speed of the equation is improved, and the solving speed of the equation is further improved in combination with latest estimation information obtained through calculation in iterative calculation. According to the method, by optimizing the solving algorithm, the calculation complexity is reduced, the efficiency of solving the discrete period Riccati equation and the numerical stability are improved, and an effective solution is provided for solving the discrete period Riccati equation in the related fields.

Description

Technical Field

[0001] The present invention belongs to the field of linear periodic system control, and relates to a method for solving discrete periodic Riccati equations, specifically to a method for solving discrete periodic Riccati equations based on inverse-free iteration and the latest estimation information. Background Art

[0002] As the simplest time-varying linear system, linear discrete periodic systems play a bridging role in connecting linear time-invariant systems, linear time-varying systems, and even general non-linear systems, and are of great significance both in theory and practice. At present, various different control strategies for linear discrete periodic systems have been widely applied in many fields such as aerospace, signal processing, and industrial production, and control problems related to this system have also received extensive attention and research. Discrete periodic Riccati matrix equations play an important role in the analysis and design of linear discrete periodic systems. It is of great significance to study fast and accurate methods for solving discrete periodic Riccati matrix equations.

[0003] However, in the existing technologies, the inverse operation in discrete periodic Riccati equations has not been well solved. Especially when the dimension of the matrix is relatively high, the inverse operation will consume a large amount of computing resources. How to effectively handle the inverse operation in the solution is still a problem worthy of in-depth study. Summary of the Invention

[0004] In order to overcome the problems brought by matrix inverse calculation in traditional iterative algorithms and provide reliable support for the advancement of control problems of linear periodic systems, the present invention provides a method for solving discrete periodic Riccati equations based on inverse-free iteration and the latest estimation information. Through the Schur complement theorem, this method transforms the original equation into an equation with only one inverse operation, realizing the reduction of inverse operations. Then, inverse-free iteration is performed on this inverse operation, thereby realizing an inverse-free solution algorithm, improving the solution speed of the equation, and combining the latest estimation information obtained in the iterative calculation to further improve the solution speed of the equation, providing solid technical support for the optimal control problem of linear periodic systems.

[0005] The object of the present invention is achieved through the following technical solutions:

[0006] A method for solving discrete periodic Riccati equations based on inverse-free iteration and the latest estimation information, comprising the following steps:

[0007] Step 1: Rewrite the discrete periodic Riccati equation:

[0008] For a periodic discrete system, consider the following model:

[0009] x(k + 1) = A(k)x(k) + B(k)u(k)

[0010] where x(k) ∈ R n is the system state variable, u(k) ∈ R m is the control input of the system, n is the state dimension of the system, m is the input dimension of the system, and A and B are system matrices that vary periodically with the positive scalar ω;

[0011] In a linear discrete - time periodic system, the problem of the linear - quadratic optimal regulator is to find a state - feedback controller that minimizes the following performance function:

[0012]

[0013] where the matrix Q(k) ∈ R n×n , R(k) ∈ R m×m , Q T (k) = Q(k) > 0, R T (k) = R(k) > 0 and both vary periodically with the integer ω, and k represents the discrete - time index;

[0014] The Riccati equation obtained in solving the LQR problem is:

[0015]

[0016] where k = 1, 2,..., ω, A k := A(k), B k := B(k), P k := P(k), Q k := Q(k), R k := R(k), S k = B k R k -1 B k T ;

[0017] For positive - definite S and P:

[0018] A T (P -1 +S) -1 A = A T S -1 A - A T S -1 (P + S -1 ) -1 S -1 A

[0019] The discrete - time periodic Riccati equation is rewritten as:

[0020]

[0021] Step 2. Construct a non-inverse iterative solution algorithm:

[0022] Step 2-1. Initialization:

[0023] Set the initial value where λ is a positive number, Let i = 0;

[0024] Step 2-2. Iterative calculation:

[0025] In each iteration, first, calculate Y ω (i + 1) according to the current P1(i), and the specific calculation formula is:

[0026]

[0027] Then, calculate P ω (i + 1) according to the currently calculated Y ω (i + 1), and the specific calculation formula is:

[0028]

[0029] Next, use the latest estimation information to calculate Y k (i + 1) and P k (i + 1) in sequence, and the specific calculation formula is:

[0030]

[0031] Step 2-3: Update the iteration index: i = i + 1, return to Step 2-2 to continue the iteration until the convergence condition is met or the specified number of iteration steps is reached.

[0032] Compared with the prior art, the present invention has the following advantages:

[0033] 1. There is no inverse operation in the iterative process, which greatly saves computing resources. Especially when the matrix dimension is high, the time required for solving is greatly reduced compared with the conventional iterative algorithm.

[0034] 2. The latest estimation information is used in the iterative process, which makes the solution speed further faster.

[0035] 3. By optimizing the solution algorithm, the computational complexity is reduced, the efficiency and numerical stability of solving the discrete-time periodic Riccati equation are improved, and an effective solution is provided for the problems involving the solution of the discrete-time periodic Riccati equation in related fields. Description of the Drawings

[0036] Figure 1It is a flow chart of each iteration of a method for solving a discrete periodic Riccati equation based on inverse-free iteration and the latest estimation information;

[0037] Figure 2 It is the simulation error convergence curve;

[0038] Figure 3 It is the comparison of computer running time. Specific implementation manner

[0039] The technical solution of the present invention will be further described below in conjunction with the accompanying drawings, but it is not limited thereto. Any modification or equivalent replacement of the technical solution of the present invention without departing from the spirit and scope of the technical solution of the present invention shall be covered by the protection scope of the present invention.

[0040] For the solution of the discrete periodic Riccati equation in the optimal control problem of discrete periodic systems, the present invention provides a method for solving the discrete periodic Riccati equation based on inverse-free iteration and the latest estimation information. First, through the Schur complement theorem, the original equation is transformed into an equation with only one inverse operation, realizing the reduction of inverse operations. Then, inverse-free iteration is performed on this inverse operation, thereby realizing an inverse-free solution algorithm, improving the solution speed of the equation, and combining the latest estimation information obtained in the iterative calculation to further improve the solution speed of the equation. As Figure 1 shown, it specifically includes the following steps:

[0041] Step 1: Rewrite the discrete periodic Riccati equation, and the specific steps are as follows:

[0042] For a periodic discrete system, consider the following model:

[0043] x(k + 1) = A(k)x(k) + B(k)u(k)

[0044] In the formula, x(k) ∈ R n is the system state variable, u(k) ∈ R m is the control input of the system, where n is the state dimension of the system, m is the input dimension of the system, and A and B are system matrices that vary periodically with a positive scalar ω, that is, A(k + ω) = A(k), B(k + ω) = B(k).

[0045] Definition 1: In a linear discrete periodic system, the problem of a linear quadratic optimal regulator is to find a state feedback controller to minimize the following performance function:

[0046]

[0047] Among them, the matrix Q(k) ∈ R n×n , R(k) ∈ R m×m , QT (k) = Q(k) > 0, R T (k) = R(k) > 0 and both vary periodically with integer ω, where k represents the discrete-time identifier.

[0048] The Riccati equation obtained in solving the LQR problem is:

[0049]

[0050] where k = 1, 2,..., ω, A k := A(k), B k := B(k), P k := P(k), Q k := Q(k), R k := R(k), S k = B k R k -1 B k T 。

[0051] Lemma 1: For positive definite S and P:

[0052] A T (P -1 + S) -1 A = A T S -1 A - A T S -1 (P + S -1 ) -1 S -1 A

[0053] Using this lemma, the discrete periodic Riccati equation can be rewritten as:

[0054]

[0055] Note that in Lemma 1, it is required that S is positive definite, which requires that the input dimension of the system be greater than or equal to the output dimension of the system. In this way, the two inverse operations become one inverse operation Then, using a matrix Y k to approximate the inverse-free solution algorithm can be obtained.

[0056] Step 2: Construct the inverse-free iterative solution algorithm:

[0057] Let According to the matrix operation rules, we can obtain:

[0058]

[0059] Based on the above form, we can construct a non-inverse iterative solution algorithm:

[0060]

[0061] Among them, i represents the iteration step number.

[0062] We find that during the iterative solution process, the latest estimated information P k (i + 1) can be used in the iterative update calculation formula of Y k-1 (i + 1). Using this idea, we can obtain a discrete periodic Riccati equation solution algorithm based on non-inverse iteration and the latest estimated information:

[0063]

[0064] Step 2 - 1. Initialization: Set the initial value where λ is a relatively large positive number, Let i = 0.

[0065] Step 2 - 2. Iterative calculation: In each iteration, first, calculate Y ω (i + 1) according to the current P1(i), and the specific calculation formula is:

[0066]

[0067] Then, calculate P ω (i + 1) based on the currently calculated Y ω (i + 1), and the specific calculation formula is:

[0068]

[0069] Next, use the latest estimated information to calculate Y k (i + 1) and P k (i + 1) in sequence, and the specific calculation formula is:

[0070]

[0071]

[0072] Step 2 - 3: Update the iteration index: i = i + 1, return to Step 2 - 2 to continue the iteration until the convergence condition is met or the specified number of iteration steps is reached.

[0073] Step 3. Convergence analysis:

[0074] Step 3 - 1. Initial condition analysis:

[0075] Assume that the positive definite solution of the equation is First, when i = 0,

[0076]

[0077] Then, according to the iterative solution algorithm, we can get:

[0078] When k = ω:

[0079]

[0080] When k = ω - 1, ω - 2,..., 1:

[0081]

[0082] Therefore, we have:

[0083]

[0084] Step Three Two. Prove the monotonicity and boundedness by mathematical induction:

[0085] Assume that when i = n, we have:

[0086]

[0087] To prove that when i = n + 1, we still have:

[0088]

[0089] Lemma 2: CPC + P -1 ≥ 2C

[0090] According to Lemma 2 and the iterative formula, we can get When k = ω:

[0091]

[0092] When k = ω - 1, ω - 2,..., 1:

[0093]

[0094] According to mathematical induction, we have:

[0095]

[0096] The obtained sequences P and Y are both monotonic and bounded, so they can converge to a limit value, denoted as:

[0097]

[0098] Then, according to the iterative formula, taking the limit on both sides, we can get:

[0099]

[0100] Sequence P can converge to the unique positive definite solution of the equation.

[0101] Example:

[0102] For a periodic discrete system, consider the following model:

[0103] x(k + 1) = A(k)x(k) + B(k)u(k)

[0104] where x(k) ∈ R n is the system state variable, u(k) ∈ R m is the control input of the system, and A, B are system matrices that vary periodically with the positive scalar ω.

[0105] That is to say:

[0106]

[0107] The Riccati equation obtained in solving the LQR problem is:

[0108]

[0109] Using Lemma 1, the above equation can be transformed into:

[0110]

[0111] Select the initial iteration value:

[0112]

[0113] The iterative update formula is:

[0114]

[0115] Calculate the iterative update in sequence until the convergence condition is satisfied or the number of iterations is reached.

[0116] Simulate the discrete periodic system.

[0117] In this example, to solve the discrete periodic Riccati equation, the relevant parameters of the discrete periodic system are set. First, determine that the state dimension of the system is 500, which means that the system state at each moment is described by 500 state variables; the input dimension is also set to 500, that is, the system has 500 different input signals that can affect the system state; the system period is set to 3, indicating that the dynamic characteristics of the system cycle with a period of 3 time steps. For the periodic matrix, at each period k, the state transition matrix A kis a 500*500 stable matrix, which is randomly generated and adjusted to have a spectral radius less than 1 to ensure system stability; the input matrix B k is a 500*500 random matrix; the state weight matrix Q k and the control weight matrix R k are randomly generated diagonal positive definite matrices. The error is defined as The error convergence curve of the simulation results is as Figure 2 shown, further verifying the ability of the method of the present invention to solve the discrete periodic Riccati equation. Figure 3 shows the computer running time. Compared with the traditional iterative method, the method of the present invention requires less time and fewer computing resources, and the advantage is more obvious when the equation dimension is higher.

Claims

1. A method for solving discrete-time periodic Riccati equations based on non-inverse iteration and the latest estimation information, characterized in that The method includes the following steps: Step 1, rewrite the discrete periodic Riccati equation: For a periodic discrete system, consider the following model: x(k + 1) = A(k)x(k) + B(k)u(k) where \(x(k)\in\mathbb{R}\) n is the system state variable, \(u(k)\in\mathbb{R}\) m is the control input of the system, \(n\) is the state dimension of the system, \(m\) is the input dimension of the system, and \(A\) and \(B\) are system matrices that vary periodically with a positive scalar \(\omega\); In a linear discrete periodic system, the problem of the linear quadratic optimal regulator is to find a state feedback controller to minimize the following performance function: where the matrix Q(k) ∈ R n×n , R(k) ∈ R m×m , Q T (k) = Q(k) > 0, R T (k) = R(k) > 0 and both vary periodically with integer ω, where k represents the discrete-time index; The Riccati equation obtained in solving the LQR problem is: where k = 1, 2,..., ω, A k := A(k), B k := B(k), P k := P(k), Q k := Q(k), R k := R(k), S k = B k R k -1 B k T ; For positive definite S and P: A T (P -1 +S) -1 A = A T S -1 A - A T S -1 (P + S -1 ) -1 S -1 A Rewrite the discrete periodic Riccati equation as: Step 2, construct an inverse-free iterative solution algorithm:

2. The discrete periodic Riccati equation solving method based on non-inverse iteration and the latest estimation information according to claim 1, characterized in that In the first step described above, 3. The discrete periodic Riccati equation solving method based on non-inverse iteration and the latest estimation information according to claim 1, characterized in that The specific steps of Step 2 are as follows: Step 2-1, initialization: Set the initial value where λ is a positive number, Let i = 0; Step 2-2, iterative calculation: In each iteration, first, calculate Y ω (i + 1) according to the current P1(i). The specific calculation formula is as follows: Then, based on the currently calculated Y ω (i + 1) to calculate P ω (i + 1), and the specific calculation formula is: Next, calculate Y in sequence using the latest estimated information k (i + 1) and P k (i + 1), and the specific calculation formula is as follows: Step 2-3: Update the iteration index: i = i + 1, return to Step 2-2 to continue the iteration until the convergence condition is satisfied or the specified number of iteration steps is reached.

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