MSc in Computer Engineering, Cybersecurity and Artificial Intelligence Course FDE , a.a. 2019/2020, Lecture 11 Brief introduction to controllability and observability for discrete-time linear systems Prof. Mauro Franceschelli Dept. of Electrical and Electronic Engineering University of Cagliari, Italy Wednsday, 22nd April 2020 1 / 33
Outline Introduction Controllability Observability Duality 2 / 33
Introduction Introduction • So far we have focused on the analysis of the behavior of dynamical systems • We are now interested in the next two fundamental problems: 1 There always exist control inputs such that we can determine the future behavior of dynamical systems? 2 Can estimate the current and past state of a dynamical system by measuring its outputs? 3 / 33
Introduction Introduction • So far we have focused on the analysis of the behavior of dynamical systems • We are now interested in the next two fundamental problems: 1 There always exist control inputs such that we can determine the future behavior of dynamical systems? 2 Can estimate the current and past state of a dynamical system by measuring its outputs? 3 / 33
Introduction Introduction • So far we have focused on the analysis of the behavior of dynamical systems • We are now interested in the next two fundamental problems: 1 There always exist control inputs such that we can determine the future behavior of dynamical systems? 2 Can estimate the current and past state of a dynamical system by measuring its outputs? 3 / 33
Introduction Introduction • We will answer this questions focusing on discrete-time linear systems ① ( k + 1) = ❆① ( k ) + ❇✉ ( k ) ② ( k ) = ❈① ( k ) + ❉✉ ( k ) where ① ( k ) is the state vector; ✉ ( k ) is the input vector; ② ( k ) is the output vector. • Nowadays the almost totality of control systems are digital and therefore practical implementations involve discrete-time control laws and algorithms. • Fundamental properties of discrete-time and continuous-time linear systems are similar. 4 / 33
Introduction Introduction • We will answer this questions focusing on discrete-time linear systems ① ( k + 1) = ❆① ( k ) + ❇✉ ( k ) ② ( k ) = ❈① ( k ) + ❉✉ ( k ) where ① ( k ) is the state vector; ✉ ( k ) is the input vector; ② ( k ) is the output vector. • Nowadays the almost totality of control systems are digital and therefore practical implementations involve discrete-time control laws and algorithms. • Fundamental properties of discrete-time and continuous-time linear systems are similar. 4 / 33
Introduction Introduction • We will answer this questions focusing on discrete-time linear systems ① ( k + 1) = ❆① ( k ) + ❇✉ ( k ) ② ( k ) = ❈① ( k ) + ❉✉ ( k ) where ① ( k ) is the state vector; ✉ ( k ) is the input vector; ② ( k ) is the output vector. • Nowadays the almost totality of control systems are digital and therefore practical implementations involve discrete-time control laws and algorithms. • Fundamental properties of discrete-time and continuous-time linear systems are similar. 4 / 33
Outline Introduction Controllability Observability Duality 5 / 33
Controllability Complete state controllability • A dynamical system is said to be completely state controllable if it is possible to bring the system from an arbitrary initial state ① (0) to an arbitrary final state x f in a finite time period by providing a suitable input. • Consider the next system ① ( k + 1) = ❆① ( k ) + ❇ u ( k ) (1) where ❆ is an n × n matrix; ❇ is an n × 1 matrix; u ( k ) is a scalar input. 6 / 33
Controllability Complete state controllability • By recalling the formula for the state response starting at the discrete-time k = 0 where ① (0) is the initial state and x f ( k ) is the final desired state: n − 1 ① f ( k ) = ❆ k ① (0) + � ❆ k − j − 1 ❇ u ( j ) j =0 = ❆ k ① (0) + ❆ k − 1 ❇✉ (0) + ❆ k − 2 ❇✉ (1) + . . . + ❇ u ( k − 1) thus ① f ( k ) − ❆ k ① (0) = ❆ k ① (0) + ❆ k − 1 ❇ u (0) + ❆ k − 2 ❇ u (1) + . . . + ❇ u ( k ) u ( k − 1) u ( k − 2) . � � ❇ , ❆❇ , ❆ 2 ❇ , . . . , ❆ k − 1 ❇ . = (2) . u (0) 7 / 33
Controllability Complete state controllability • By recalling the formula for the state response starting at the discrete-time k = 0 where ① (0) is the initial state and x f ( k ) is the final desired state: n − 1 ① f ( k ) = ❆ k ① (0) + � ❆ k − j − 1 ❇ u ( j ) j =0 = ❆ k ① (0) + ❆ k − 1 ❇✉ (0) + ❆ k − 2 ❇✉ (1) + . . . + ❇ u ( k − 1) thus ① f ( k ) − ❆ k ① (0) = ❆ k ① (0) + ❆ k − 1 ❇ u (0) + ❆ k − 2 ❇ u (1) + . . . + ❇ u ( k ) u ( k − 1) u ( k − 2) . � � ❇ , ❆❇ , ❆ 2 ❇ , . . . , ❆ k − 1 ❇ . = (2) . u (0) 7 / 33
Controllability The controllability matrix • Notice that each matrix B , AB , ..., A k − 1 B is a column vector. u ( k − 1) u ( k − 2) � � ① f ( k ) − ❆ k ① (0) = B , AB , A 2 B , . . . , A k − 1 B . . . u (0) • Now, let k = n , i.e., the order of the system. If B , AB , A 2 B , . . . , A n − 1 B �� �� = n rank B , AB , A 2 , . . . , A n − 1 B then the columns of matrix � � span the whole n -dimensional space. � B , AB , A 2 B , . . . , A n − 1 B � • Matrix T = is commonly called the controllability matrix. • In literature for discrete-time systems it is also referred to as the reachability matrix. 8 / 33
Controllability The controllability matrix • Notice that each matrix B , AB , ..., A k − 1 B is a column vector. u ( k − 1) u ( k − 2) � � ① f ( k ) − ❆ k ① (0) = B , AB , A 2 B , . . . , A k − 1 B . . . u (0) • Now, let k = n , i.e., the order of the system. If B , AB , A 2 B , . . . , A n − 1 B �� �� = n rank B , AB , A 2 , . . . , A n − 1 B then the columns of matrix � � span the whole n -dimensional space. � B , AB , A 2 B , . . . , A n − 1 B � • Matrix T = is commonly called the controllability matrix. • In literature for discrete-time systems it is also referred to as the reachability matrix. 8 / 33
Controllability The controllability matrix • Notice that each matrix B , AB , ..., A k − 1 B is a column vector. u ( k − 1) u ( k − 2) � � ① f ( k ) − ❆ k ① (0) = B , AB , A 2 B , . . . , A k − 1 B . . . u (0) • Now, let k = n , i.e., the order of the system. If B , AB , A 2 B , . . . , A n − 1 B �� �� = n rank B , AB , A 2 , . . . , A n − 1 B then the columns of matrix � � span the whole n -dimensional space. � B , AB , A 2 B , . . . , A n − 1 B � • Matrix T = is commonly called the controllability matrix. • In literature for discrete-time systems it is also referred to as the reachability matrix. 8 / 33
Controllability The controllability matrix • Notice that each matrix B , AB , ..., A k − 1 B is a column vector. u ( k − 1) u ( k − 2) � � ① f ( k ) − ❆ k ① (0) = B , AB , A 2 B , . . . , A k − 1 B . . . u (0) • Now, let k = n , i.e., the order of the system. If B , AB , A 2 B , . . . , A n − 1 B �� �� = n rank B , AB , A 2 , . . . , A n − 1 B then the columns of matrix � � span the whole n -dimensional space. � B , AB , A 2 B , . . . , A n − 1 B � • Matrix T = is commonly called the controllability matrix. • In literature for discrete-time systems it is also referred to as the reachability matrix. 8 / 33
Controllability Complete state controllability: a criterion If the rank of the controllability matrix T is n then from any initial state ① 0 and arbitrary final state ① f there exists an unbounded control signal u (0) , u (1) , . . . , u ( n − 1) which brings the dynamical system from the state x 0 to the state x f in n steps. Thus the system is said to be completely state controllable in n steps. • Thus, the condition rank ( T ) = n is proven by construction to be at least a sufficient condition for complete state controllability. In fact, it is also necessary. 9 / 33
Controllability Complete state controllability: a criterion If the rank of the controllability matrix T is n then from any initial state ① 0 and arbitrary final state ① f there exists an unbounded control signal u (0) , u (1) , . . . , u ( n − 1) which brings the dynamical system from the state x 0 to the state x f in n steps. Thus the system is said to be completely state controllable in n steps. • Thus, the condition rank ( T ) = n is proven by construction to be at least a sufficient condition for complete state controllability. In fact, it is also necessary. 9 / 33
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