Nonlinear Systems Theory Hu Jiangping School of Automation Engineering University of Electronic Science and Technology of China E mail hjp lzu or f tGxs U s u t Y s y t Topic 4 Input Output stability 4 1 L stability 4 2 L stability of state model 4 3 L2gain 4 4 The small gain theorem Topic 4 Input Output stability Topic 4 Input Output stability Topic 4 Input Output stability Topic 4 Input Output stability Remark if the system is causal we can drop Topic 4 Input Output stability Topic 4 Input Output stability Topic 4 Input Output stability Topic 4 Input Output stability 4 1 L stability 4 2 L stability of state model 4 3 L2gain 4 4 The small gain theorem Topic 4 Input Output stability 4 2 L stability of state model Topic 4 Input Output stability Topic 4 Input Output stability Topic 4 Input Output stability Topic 4 Input Output stability Topic 4 Input Output stability Topic 4 Input Output stability Topic 4 Input Output stability 4 1 L stability 4 2 L stability of state model 4 3 L2gain 4 4 The small gain theorem Topic 4 Input Output stability 4 3 L2gain u t y t Measure energy of signal T ut u t dt Finite gain L2stability Topic 4 Input Output stability 4 3 L2gain Fourier transform of the output signal According to Parseval s Relation Topic 4 Input Output stability 4 3 L2gain Topic 4 Input Output stability 0 0 xf xG x u xx yh x Example check the finite gain L2stability of the following system 0 0 00 nm xfRhuR Topic 4 Input Output stability Remark in order to obtain V x we need solve the following Hamilton Jacobi inequality Topic 4 Input Output stability Topic 4 Input Output stability 2 3 12 0 1 x f x axkxu G Topic 4 Input Output stability 11 kk Remark Theorem 5 1 cannot be applied since the origin of the system is not exponentially stable Topic 4 Input Output stability Example If there exists a W x such that We can take Topic 4 Input Output stability Topic 4 Input Output stability Suppose the assumptions of Theorem 5 5 are satisfied on a domain that contains the origin f x is continuously differentiable and x 0 is an asymptotically stable equilibrium point of Then there is kl 0 such that for each with the system is small signal finite gain L2stable n DR xf x 0 x 01 xk Lemma 5 1 Topic 4 Input Output stability Proof x 0 is an asymptotically stable equilibrium Topic 4 Input Output stability Suppose the assumptions of Theorem 5 5 are satisfied on a domain that contains the origin f x is continuously differentiable no solution of can stay identically in other than the origin the system is small signal finite gain L2stable n DR xf x Lemma 5 2 Topic 4 Input Output stability LaSalle s invariance principle Asymptotically stable Example Topic 4 Input Output stability f x The condition is not hold globally we need other lemmas to show the small signal finite gain stability Method 1 By using Lemma 5 1 Topic 4 Input Output stability Linearization Method 2 By using Lemma 5 2 Topic 4 Input Output stability 4 1 L stability 4 2 L stability of state model 4 3 L2gain 4 4 The small gain theorem Topic 4 Input Output stability 4 4 Small gain theorem Topic 4 Input Output stability Topic 4 Input Output stability Topic 4 Input Output stability Example Let H1 be a linear time invariant system with a Hurwitz square transfer function matrix Topic 4 Input Output stability Example Assume we can design such that Topic 4 Input Output stability Will the control meet the design objective Topic 4 Input Output stability Topic 4 Input Output stability