21.7
伝達関数から状態空間への変換
状態空間表現は、デジタルコンピューター上で物理システムをシミュレートするための強力なツールであり、伝達関数を状態空間形式に変換する必要があります。RLC 回路で見られる定数係数を持つ n 次線形微分方程式を考えてみましょう。状態変数は出力とその n−1 導関数として選択されます。これらの変数を微分し…
状態空間表現は、デジタル コンピューター上の物理システムをシミュレートするために使用されます。このためには、伝達関数を最初に状態空間に変換する必要があります。
係数が一定の n 次線形微分方程式を考えます。
出力とその n-1 導関数が状態変数として選択されます。これらの一連の方程式を微分し、それらを元の方程式に代入すると、状態方程式が得られます。
後続の各状態変数は、前の変数の導関数として定義されます。
次に、結果の方程式はベクトル行列形式で表され、微分方程式の負の係数とともに 1 と 0 の明確なパターンが作成されます。このユニークな構造は、状態方程式の位相変数形式です。
伝達関数を考えてみましょう。方程式は交差乗算され、対応する微分方程式は、初期条件がゼロであると仮定して逆ラプラス変換を行うことによって求められます。
状態変数は、連続する導関数として選択されます。
次に、方程式の両側に微分が適用され、状態方程式と出力方程式が得られます。
これらの方程式は、ベクトル行列形式で表されます。
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Q1: Why is state-space representation necessary for simulating physical systems on digital computers?
State-space representation enables digital simulation of physical systems by converting transfer functions into a matrix form suitable for computational processing. This vector-matrix format allows computers to efficiently model and analyze dynamic system behavior. The structured representation with state variables and their derivatives provides a clear mathematical framework for numerical computation and system analysis.
Q2: How do you select state variables when converting a transfer function to state space?
State variables are chosen as the output and its n-1 derivatives for an nth-order system. In practical applications like RLC circuits, the voltage across the capacitor and current through the inductor serve as state variables. Each subsequent state variable is defined as the derivative of the previous one, creating a systematic hierarchy that captures the system's dynamic behavior.
Q3: What is the phase-variable form in state-space representation?
The phase-variable form is the unique matrix structure created when state equations are represented in vector-matrix form. It displays a distinct pattern of 1's and 0's along with the negative coefficients from the original differential equation. This standardized structure provides a clear and concise method for simulating and analyzing the dynamic behavior of linear systems.
Q4: What steps are involved in converting a transfer function to state-space form?
First, cross-multiply the transfer function and apply the inverse Laplace transform assuming zero initial conditions to obtain the differential equation. Next, select state variables as successive derivatives of the output. Differentiate both sides of the equation to generate state equations, then represent these equations in vector-matrix form to complete the conversion.
Q5: How does the state-space representation relate to the inverse conversion process?
State-space representation and its inverse conversion form a complementary pair in control systems analysis. While state-space converts transfer functions into matrix form for time-domain simulation, the reverse process converts state-space back to transfer functions. Understanding both transformations is essential for comprehensive system modeling and analysis in control engineering applications.
Q6: Why are zero initial conditions assumed when converting transfer functions to state space?
Zero initial conditions simplify the mathematical conversion by allowing direct application of the inverse Laplace transform without accounting for initial energy stored in the system. This assumption creates a standard baseline for the conversion process and ensures the resulting state equations accurately represent the system's response to external inputs from a relaxed initial state.
Q7: How does an RLC circuit example illustrate state-space conversion?
In a series RLC circuit, the voltage across the capacitor and current through the inductor are selected as state variables. The transfer function is cross-multiplied to obtain the differential equation, then converted to state-space form using successive derivatives. This practical example demonstrates how physical circuit components map to state variables and how the resulting matrix equations model the circuit's dynamic response.