Control Systems/Standard Forms

{{#invoke:TScope|shiftLeft|BookCat/core|1 |namespace = |pagename =Control Systems/Standard Forms |fullpagename=Control Systems/Standard Forms |sortkey = }}

Companion Form

A companion form contains the coefficients of a corresponding characteristic polynomial along one of its far rows or columns. For example, one companion form matrix is:

解析失败 (带SVG或PNG备选的MathML（建议用于现代的浏览器和辅助工具）：从服务器“https://wikimedia.org/api/rest_v1/”返回无效的响应（“Math extension cannot connect to Restbase.”）：): {\displaystyle \begin{bmatrix} 0 & 0 & 0 & \cdots & 0 & -a_0 \\ 1 & 0 & 0 & \cdots & 0 & -a_1 \\ 0 & 1 & 0 & \cdots & 0 & -a_2 \\ 0 & 0 & 1 & \cdots & 0 & -a_3 \\ \vdots & \vdots & \vdots &\ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 1 & -a_{n-1} \end{bmatrix}}

and another is:

解析失败 (带SVG或PNG备选的MathML（建议用于现代的浏览器和辅助工具）：从服务器“https://wikimedia.org/api/rest_v1/”返回无效的响应（“Math extension cannot connect to Restbase.”）：): {\displaystyle \begin{bmatrix} -a_{n-1} & -a_{n-2} & -a_{n-3} & \cdots & -a_1 & -a_0 \\ 1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 1 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots &\ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 1 & 0 \end{bmatrix}}

There are two companion forms that are convenient to use in control theory, namely the observable canonical form and the controllable canonical form. These two forms are roughly transposes of each other (just as observability and controllability are dual ideas). When placed in one of these forms, the design of controllers or observers is simplified because the structure of the system is made apparent (and is easily modified with the desired control).

Observable Canonical Form

Observable-Canonical Form is useful in a number of cases, especially for designing observers.

The observable-canonical form is as follows:

解析失败 (带SVG或PNG备选的MathML（建议用于现代的浏览器和辅助工具）：从服务器“https://wikimedia.org/api/rest_v1/”返回无效的响应（“Math extension cannot connect to Restbase.”）：): {\displaystyle A = \begin{bmatrix} -a_1 & 1 & 0 & \cdots & 0 \\ -a_2 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ -a_{n-1} & 0 & 0 & \cdots & 1 \\ -a_n & 0 & 0 & \cdots & 0 \end{bmatrix}}

解析失败 (带SVG或PNG备选的MathML（建议用于现代的浏览器和辅助工具）：从服务器“https://wikimedia.org/api/rest_v1/”返回无效的响应（“Math extension cannot connect to Restbase.”）：): {\displaystyle B = \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{bmatrix}}

解析失败 (带SVG或PNG备选的MathML（建议用于现代的浏览器和辅助工具）：从服务器“https://wikimedia.org/api/rest_v1/”返回无效的响应（“Math extension cannot connect to Restbase.”）：): {\displaystyle C = \begin{bmatrix} 1 & 0 & \cdots & 0 \end{bmatrix}}

Controllable Canonical Form

Controllable-Canonical Form is useful in a number of cases, especially for designing controllers when the full state of the system is known.

The controllable-canonical form is as follows:

解析失败 (带SVG或PNG备选的MathML（建议用于现代的浏览器和辅助工具）：从服务器“https://wikimedia.org/api/rest_v1/”返回无效的响应（“Math extension cannot connect to Restbase.”）：): {\displaystyle A = \begin{bmatrix} -a_1 & -a_2 & -a_3 & \cdots & -a_{n-1} & -a_n \\ 1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 1 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots &\ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 1 & 0 \end{bmatrix}}

解析失败 (带SVG或PNG备选的MathML（建议用于现代的浏览器和辅助工具）：从服务器“https://wikimedia.org/api/rest_v1/”返回无效的响应（“Math extension cannot connect to Restbase.”）：): {\displaystyle B = \begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix}}

解析失败 (带SVG或PNG备选的MathML（建议用于现代的浏览器和辅助工具）：从服务器“https://wikimedia.org/api/rest_v1/”返回无效的响应（“Math extension cannot connect to Restbase.”）：): {\displaystyle C = \begin{bmatrix} b_1 & b_2 & b_3 & \cdots & b_n \end{bmatrix}}

解析失败 (带SVG或PNG备选的MathML（建议用于现代的浏览器和辅助工具）：从服务器“https://wikimedia.org/api/rest_v1/”返回无效的响应（“Math extension cannot connect to Restbase.”）：): {\displaystyle D = \begin{bmatrix} b_0 \end{bmatrix}}

If we have two spaces, space v which is the original space of the system (A, B, C, and D), then we can transform our system into the w space which is in controllable-canonical form (A_w, B_w, C_w, D_w) using a transformation matrix T_w. We define this transformation matrix as:

解析失败 (带SVG或PNG备选的MathML（建议用于现代的浏览器和辅助工具）：从服务器“https://wikimedia.org/api/rest_v1/”返回无效的响应（“Math extension cannot connect to Restbase.”）：): {\displaystyle T = \zeta_v \zeta_w^{-1}}

Where ζ is the controlability matrix.

Notice that we know beforehand A_w and B_w, since we know both the form of the matrices and the coefficients of the equation (e.g. a linear ODE with constant coefficients or a transfer function).

If we know these two matrices, then we can form ζ_w. We can then use this matrix to create our transformation matrix.

We will discuss the controllable canonical form later when we discuss state-feedback and closed-loop systems.

Phase Variable Form

The Phase Variable Form is obtained simply by renumbering the phase variables in the opposite order of the controllable canonical form. Thus:

解析失败 (带SVG或PNG备选的MathML（建议用于现代的浏览器和辅助工具）：从服务器“https://wikimedia.org/api/rest_v1/”返回无效的响应（“Math extension cannot connect to Restbase.”）：): {\displaystyle A = \begin{bmatrix} 0 & 1 & \cdots & 0 & 0 & 0\\ \vdots & \vdots &\ddots & \vdots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & 0 & 0\\ 0 & 0 & \cdots & 0 & 1 & 0\\ 0 & 0 & \cdots & 0 & 0 & 1\\ -a_n & -a_{n-1} & -a_{n-2} & \cdots & -a_2 & -a_1 \end{bmatrix}}

解析失败 (带SVG或PNG备选的MathML（建议用于现代的浏览器和辅助工具）：从服务器“https://wikimedia.org/api/rest_v1/”返回无效的响应（“Math extension cannot connect to Restbase.”）：): {\displaystyle B = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 1 \end{bmatrix}}

解析失败 (带SVG或PNG备选的MathML（建议用于现代的浏览器和辅助工具）：从服务器“https://wikimedia.org/api/rest_v1/”返回无效的响应（“Math extension cannot connect to Restbase.”）：): {\displaystyle C = \begin{bmatrix} b_n & b_{n-1} & \cdots & b_2 & b_1 \end{bmatrix}}

解析失败 (带SVG或PNG备选的MathML（建议用于现代的浏览器和辅助工具）：从服务器“https://wikimedia.org/api/rest_v1/”返回无效的响应（“Math extension cannot connect to Restbase.”）：): {\displaystyle D = \begin{bmatrix} b_0 \end{bmatrix}}

Modal Form

In this form, the state matrix is a diagonal matrix of its (non-repeated) eigenvalues. The control has a unitary influence on each eigenspace, and the output is a linear combination of the contributions from the eigenspaces (where the weights are the complex residuals at each pole).

解析失败 (带SVG或PNG备选的MathML（建议用于现代的浏览器和辅助工具）：从服务器“https://wikimedia.org/api/rest_v1/”返回无效的响应（“Math extension cannot connect to Restbase.”）：): {\displaystyle A = \begin{bmatrix} -p_1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & -p_2 & 0 & \cdots & 0 & 0 \\ 0 & 0 & -p_3 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots &\ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 0 & -p_n \end{bmatrix}}

解析失败 (带SVG或PNG备选的MathML（建议用于现代的浏览器和辅助工具）：从服务器“https://wikimedia.org/api/rest_v1/”返回无效的响应（“Math extension cannot connect to Restbase.”）：): {\displaystyle B = \begin{bmatrix} 1 \\ 1 \\ \vdots \\ 1 \end{bmatrix}}

解析失败 (带SVG或PNG备选的MathML（建议用于现代的浏览器和辅助工具）：从服务器“https://wikimedia.org/api/rest_v1/”返回无效的响应（“Math extension cannot connect to Restbase.”）：): {\displaystyle C = \begin{bmatrix} c_1 & c_2 & \cdots & c_n \end{bmatrix}}

Jordan Form

This "almost diagonal" form handles the case where eigenvalues are repeated. The repeated eigenvalues represent a multi-dimensional eigenspace, and so the control only enters the eigenspace once and its integrated through the other states of that small subsystem.

解析失败 (带SVG或PNG备选的MathML（建议用于现代的浏览器和辅助工具）：从服务器“https://wikimedia.org/api/rest_v1/”返回无效的响应（“Math extension cannot connect to Restbase.”）：): {\displaystyle A = \begin{bmatrix} -p_1 & 1 & 0 & 0 & 0 & \cdots & 0 & 0 \\ 0 & -p_1 & 1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 0 & -p_1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 0 & -p_4 & 0 &\cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots &\ddots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & 0 & \cdots & 0 & -p_n \end{bmatrix}}

解析失败 (带SVG或PNG备选的MathML（建议用于现代的浏览器和辅助工具）：从服务器“https://wikimedia.org/api/rest_v1/”返回无效的响应（“Math extension cannot connect to Restbase.”）：): {\displaystyle B = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 1 \\ \vdots \\ 1 \end{bmatrix}}

解析失败 (带SVG或PNG备选的MathML（建议用于现代的浏览器和辅助工具）：从服务器“https://wikimedia.org/api/rest_v1/”返回无效的响应（“Math extension cannot connect to Restbase.”）：): {\displaystyle C = \begin{bmatrix} c_1 & c_2 & \cdots & c_n \end{bmatrix}}

Computing Standard Forms in MATLAB

模板:Matlab CMD MATLAB contains a function for automatically transforming a state space equation into a companion (e.g., controllable or observable canonical form) form.

[Ap, Bp, Cp, Dp, P] = canon(A, B, C, D, 'companion');

模板:Matlab CMD Moving from one companion form to the other usually involves elementary operations on matrices and vectors (e.g., transposes or interchanging rows). Given a vector with the coefficients of a characteristic polynomial, MATLAB can compute a companion form with the coefficients in the top row (there are other 3 possible companion forms not generated by that function)

compan(P)

Given another vector with the coefficients of a transfer function's numerator polynomial, the canon command can do the same.

[Ap, Bp, Cp, Dp, P] = canon(tf(Pnum,Pden), 'companion');

To transform a state space equation into a modal (e.g., diagonal) form, the same command can be used.

[Ap, Bp, Cp, Dp, P] = canon(A, B, C, D, 'modal');

模板:Matlab CMD However, MATLAB also includes a command to compute the Jordan form of a matrix, which is a modified modal form suited for matrices that have repeated eigenvalues.

jordan(A)

← Eigenvalues and Eigenvectors

Control Systems

MIMO Systems →