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MSc in Computer Engineering, Cybersecurity and Artificial Intelligence Course FDE , a.a. 2019/2020, Lecture 20 Model Based Fault Diagnosis : Residual Generation via State Observers Prof. Mauro Franceschelli Dept. of Electrical and Electronic Engineering University of Cagliari, Italy Monday 25th May 2020 1 / 42

Outline Residual generation via state observers The Unknown Input Observer (UIO) UIO design procedure FDI scheme based on banks of observers 2 / 42

Residual generation via state observers Introduction Model-based FDI basically consists in the design of suitable residual generator based on the model of the plant/dynamical system and a residual evaluation method 3 / 42

Residual generation via state observers Introduction • Residual generation: How to generate residual signals using available inputs and outputs from the monitored system. This residual (or fault symptom) should indicate that a fault has occurred. It should normally be zero or close to zero under no fault condition, while significantly different from zero when a fault occurs. This means that the residual is characteristically independent of process inputs and outputs, in ideal conditions. • Residual evaluation: How to evaluate residuals for the likelihood of faults and apply a decision rule to determine if any faults have occurred. The residual evaluation block, may perform a simple threshold test (geometrical methods) on the instantaneous values or moving averages of the residuals. On the other hand, it may consist of more complex statistical methods, e.g., generalised likelihood ratio testing or sequential probability ratio testing. • Model-based FDI methods focus on the residual generation problem because the decision making can be considered as easy if the residual signals are well designed. 4 / 42

Residual generation via state observers Introduction • Residual generation: How to generate residual signals using available inputs and outputs from the monitored system. This residual (or fault symptom) should indicate that a fault has occurred. It should normally be zero or close to zero under no fault condition, while significantly different from zero when a fault occurs. This means that the residual is characteristically independent of process inputs and outputs, in ideal conditions. • Residual evaluation: How to evaluate residuals for the likelihood of faults and apply a decision rule to determine if any faults have occurred. The residual evaluation block, may perform a simple threshold test (geometrical methods) on the instantaneous values or moving averages of the residuals. On the other hand, it may consist of more complex statistical methods, e.g., generalised likelihood ratio testing or sequential probability ratio testing. • Model-based FDI methods focus on the residual generation problem because the decision making can be considered as easy if the residual signals are well designed. 4 / 42

Residual generation via state observers Introduction • Residual generation: How to generate residual signals using available inputs and outputs from the monitored system. This residual (or fault symptom) should indicate that a fault has occurred. It should normally be zero or close to zero under no fault condition, while significantly different from zero when a fault occurs. This means that the residual is characteristically independent of process inputs and outputs, in ideal conditions. • Residual evaluation: How to evaluate residuals for the likelihood of faults and apply a decision rule to determine if any faults have occurred. The residual evaluation block, may perform a simple threshold test (geometrical methods) on the instantaneous values or moving averages of the residuals. On the other hand, it may consist of more complex statistical methods, e.g., generalised likelihood ratio testing or sequential probability ratio testing. • Model-based FDI methods focus on the residual generation problem because the decision making can be considered as easy if the residual signals are well designed. 4 / 42

Residual generation via state observers Fault models • Signal f c ( k ) represents a component fault in the plant/system, and it can modeled both a multiplicative faults and additive faults • Signal f u ( k ) and f y ( k ) are input and output sensors faults and it can modeled both a multiplicative faults and additive faults 5 / 42

Residual generation via state observers Fault models The considered system and sensor fault model is then: ① ( k + 1) = ❆① ( k ) + ❇ ( ✉ ⋆ ( k ) + ❢ u ( k )) + ❢ c ( k ) ② ( k ) = ❈① ( k ) + ❢ y ( k ) where ✉ ( k ) = ✉ ⋆ ( k ) + ❢ u ( k ) and ② ( k ) = ② ⋆ ( k ) + ❢ y ( k ) . • If the process/component fault is multiplicative, then ❢ c ( k ) = δ A ① ( k ) where δ A is a n × n matrix. • If the sensor faults are multiplicative, then ❢ u ( k ) = δ u ✉ ⋆ ( k ) and ❢ y ( k ) = δ y ② ⋆ ( k ) where δ u and δ y are diagonal matrices of appropriate dimensions. 6 / 42

Residual generation via state observers Fault models The considered system and sensor fault model is then: ① ( k + 1) = ❆① ( k ) + ❇ ( ✉ ⋆ ( k ) + ❢ u ( k )) + ❢ c ( k ) ② ( k ) = ❈① ( k ) + ❢ y ( k ) where ✉ ( k ) = ✉ ⋆ ( k ) + ❢ u ( k ) and ② ( k ) = ② ⋆ ( k ) + ❢ y ( k ) . • If the process/component fault is multiplicative, then ❢ c ( k ) = δ A ① ( k ) where δ A is a n × n matrix. • If the sensor faults are multiplicative, then ❢ u ( k ) = δ u ✉ ⋆ ( k ) and ❢ y ( k ) = δ y ② ⋆ ( k ) where δ u and δ y are diagonal matrices of appropriate dimensions. 6 / 42

Residual generation via state observers Fault models The considered system and sensor fault model is then: ① ( k + 1) = ❆① ( k ) + ❇ ( ✉ ⋆ ( k ) + ❢ u ( k )) + ❢ c ( k ) ② ( k ) = ❈① ( k ) + ❢ y ( k ) where ✉ ( k ) = ✉ ⋆ ( k ) + ❢ u ( k ) and ② ( k ) = ② ⋆ ( k ) + ❢ y ( k ) . • If the process/component fault is multiplicative, then ❢ c ( k ) = δ A ① ( k ) where δ A is a n × n matrix. • If the sensor faults are multiplicative, then ❢ u ( k ) = δ u ✉ ⋆ ( k ) and ❢ y ( k ) = δ y ② ⋆ ( k ) where δ u and δ y are diagonal matrices of appropriate dimensions. 6 / 42

Residual generation via state observers Fault models • To simplify the notation, we can define a fault vector ❢ ( k ) = [ ❢ c ( k ) , ❢ u ( k ) , ❢ y ( k )] and define possibly rectangular matrices ▲ 1 , ▲ 2 , ▲ 3 such that ❢ c ( k ) = ▲ 1 ❢ ( k ), ❢ y ( k ) = ▲ 2 ❢ ( k ) and ❢ u ( k ) = ▲ 3 ❢ ( k ). • By this modeling choice, it holds ① ( k + 1) = ❆① ( k ) + ❇✉ ⋆ ( k ) + ▲ 1 ❢ ( k ) ② ( k ) = ❈① ( k ) + ▲ 2 ❢ ( k ) ✉ ( k ) = ✉ ⋆ ( k ) + ▲ 3 ❢ ( k ) 7 / 42

Residual generation via state observers Fault models • To simplify the notation, we can define a fault vector ❢ ( k ) = [ ❢ c ( k ) , ❢ u ( k ) , ❢ y ( k )] and define possibly rectangular matrices ▲ 1 , ▲ 2 , ▲ 3 such that ❢ c ( k ) = ▲ 1 ❢ ( k ), ❢ y ( k ) = ▲ 2 ❢ ( k ) and ❢ u ( k ) = ▲ 3 ❢ ( k ). • By this modeling choice, it holds ① ( k + 1) = ❆① ( k ) + ❇✉ ⋆ ( k ) + ▲ 1 ❢ ( k ) ② ( k ) = ❈① ( k ) + ▲ 2 ❢ ( k ) ✉ ( k ) = ✉ ⋆ ( k ) + ▲ 3 ❢ ( k ) 7 / 42

Residual generation via state observers State observer for residual generation • Now, consider the next state observer equations ˆ ① ( k + 1) = ❆ ˆ ① ( k ) + ❇✉ ( k ) + ❑ e ( ② ( k ) − ˆ ② ( k )) ˆ ② ( k ) = ❈ ˆ ① ( k ) Its state error dynamics with ❡ ( k ) = x ( k ) − ˆ x in presence of faults becomes ❡ ( k + 1) = ❆① ( k ) + ❇✉ ⋆ ( k ) + ▲ 1 ❢ ( k ) − ❆ ˆ ① ( k ) − ❇✉ ( k ) − ❑ e ( ② ( k ) − ˆ ② ( k )) = ❆❡ ( k ) + ❇✉ ⋆ ( k ) + ▲ 1 ❢ ( k ) − ❇✉ ⋆ ( k ) − ▲ 3 ❢ ( k ) − ❑ e ( ② ( k ) − ˆ ② ( k )) = ❆❡ ( k ) + ▲ 1 ❢ ( k ) − ▲ 3 ❢ ( k ) − ❑ e ( ❈① ( k ) + ▲ 2 ❢ ( k ) − ❈ ˆ ① ( k )) = ( ❆ − ❑ e ❈ ) ❡ ( k ) + ▲ 1 ❢ ( k ) − ▲ 3 ❢ ( k ) − ❑ e ▲ 2 ❢ ( k ) 8 / 42

Residual generation via state observers State observer for residual generation • Now, consider the next state observer equations ˆ ① ( k + 1) = ❆ ˆ ① ( k ) + ❇✉ ( k ) + ❑ e ( ② ( k ) − ˆ ② ( k )) ˆ ② ( k ) = ❈ ˆ ① ( k ) Its state error dynamics with ❡ ( k ) = x ( k ) − ˆ x in presence of faults becomes ❡ ( k + 1) = ❆① ( k ) + ❇✉ ⋆ ( k ) + ▲ 1 ❢ ( k ) − ❆ ˆ ① ( k ) − ❇✉ ( k ) − ❑ e ( ② ( k ) − ˆ ② ( k )) = ❆❡ ( k ) + ❇✉ ⋆ ( k ) + ▲ 1 ❢ ( k ) − ❇✉ ⋆ ( k ) − ▲ 3 ❢ ( k ) − ❑ e ( ② ( k ) − ˆ ② ( k )) = ❆❡ ( k ) + ▲ 1 ❢ ( k ) − ▲ 3 ❢ ( k ) − ❑ e ( ❈① ( k ) + ▲ 2 ❢ ( k ) − ❈ ˆ ① ( k )) = ( ❆ − ❑ e ❈ ) ❡ ( k ) + ▲ 1 ❢ ( k ) − ▲ 3 ❢ ( k ) − ❑ e ▲ 2 ❢ ( k ) 8 / 42