State Space Representation Prof. Seungchul Lee Industrial AI Lab. - - PowerPoint PPT Presentation

State Space Representation

• Prof. Seungchul Lee

Industrial AI Lab.

State of a Dynamic System

• A minimum set of variables, known as state variables, that fully describe the system and its response

to any given set of inputs.

• The number of state variables, 𝑜, is equal to the number of independent "energy storage elements" in

the system.

• The state equations

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State Representation of LTI System

• We restrict attention primarily to linear and time-invariant (LTI) system. Then it becomes a set of 𝑜

coupled first-order linear differential equations with constant coefficients.

• Written compactly in a matrix form

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Output of LTI System

• Output equations

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Block Diagram of LTI System

• The complete system model for LTI system in the standard state space form
• Block diagram

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Homogeneous State Response

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Homogeneous State Response

• With zero input, 𝑣 𝑢 = 0
• Let's figure out how such a system behaves

– Start by ignoring the input term:

• What is the solution to this system?

– If everything is scalar: – How do we know?

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Homogeneous State Response

• For higher-order systems, we just get a matrix version of this
• The definition is just like for scalar exponentials
• Derivative:
• The matrix exponential plays such an important role that it has its own name: the state transition

matrix, Φ(𝑢)

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Properties of State Transition Matrix

• Properties of state transition matrix

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Example: State Transition Matrix

• Example

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Example: State Transition Matrix

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Forced State Response of LTI System

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Forced State Response of LTI System

• But what if we have the controlled system:
• Consider the complete response of a linear system to an input 𝑣(𝑢)
• Derivation
• Complete solution

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For Higher Order Systems

• Complete solution
• The output
• The first is a term similar to the system homogeneous response 𝑦ℎ 𝑢 = 𝑓𝐵𝑢𝑦(0) that is dependent
• nly on the system initial conditions 𝑦(0)
• The second term is in the form of a convolution integral, and it is the particular solution for the input

𝑣(𝑢) with zero initial conditions

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• Note

For Higher Order Systems

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(Leibniz Integral Rule)

Note: Leibniz Integral Rule

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Example

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The Response of LTI to the Singularity Input Functions

• Impulse response
• The effect of impulse inputs on the state response is similar to changing a set of initial conditions

𝑦 0 → 𝑦 0 + 𝐶𝐿

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The Response of LTI to the Singularity Input Functions

• Step response

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The Response of LTI to the Singularity Input Functions

• Step response

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