Control of single and multiple agent systems with input and communication delays.

Control of time-delay systems represents one important class of control problems. A lot of work has been done in recent years. Much of the research work has been focused on the stability analysis and stabilization of time-delay systems based on the Lyapunov functional and linear matrix inequality (LMI) approaches. While the LMI approach does provide an efficient tool, the results are mostly only sufficient and only numerical solutions are available. Our research aims to present analytical solutions by some algebraic approaches, which can provide an insightful understanding of control problem of time-delay systems. On the other hand, there has also been significant recent interest in networked multi-agent system (MAS) where unavoidably there exist delays in information acquisition as well as information exchange between agents in addition to possible input/state delays in each agent. This motivates the study of MASs with input and communication delays. The thesis is divided into two parts: Part I is focused on optimal control of single agent systems with multiple input delays; Part II is on distributed consensus control of MAS with communication and input delays. More specifically, in Part I, we shall study the linear quadratic regulation (LQR) and tracking for a single agent with linear dynamics. We first focus on LQR for linear discrete-time systems with multiple delays in a single input channel. By introducing a backward stochastic system, we establish a relationship between the LQR problem for the original system and optimal estimation for a backward stochastic system Nanyang Technological University Singapore iv which can be solved by using projection approach. The explicit optimal controllers are developed for both finite time and infinite time horizon cases. A parallel result for linear continuous-time systems is also established, where the optimal feedback gain is given in terms of the solution of Riccati-type partial differential equations (PDEs). Next, we study the optimal tracking problem for both discrete-time and continuoustime systems with multiple delays in a single input channel. Tracking problem requires the states and the input to track desired trajectories rather than converge to the origin. We convert the linear quadratic tracking problem into a white noise and state smoothing problem for an associated stochastic backward system. A deterministic backward delay system is introduced for establishing an explicit expression of the optimal controller. In Part II, we focus on distributed control of MASs with communication and input delays. We first comparatively study consensus protocols with and without control inputs information of neighbors. We will rigorously prove that knowing neighbors’ control inputs information can improve the consensus performance. However, the protocol with neighbors’ control inputs information cannot be implemented in practical systems. Therefore, a distributed protocol based on neighbors’ delayed control input information is proposed and investigated. We also investigate how the time delay affects the consensus performance. Next, we study distributed consensus for MAS with both delays and noises in transmission channels. Due to the random transmission noises, traditional protocol cannot be applied. By introducing a decaying consensus gain, we can provide necessary and sufficient conditions on the consensus gain and communication graph to guarantee that the whole system reaches the consensus in a stochastic sense. In a digital network, quantization is unavoidable. However, quantization error cannot be treated as a random transmission noise in most of cases. Therefore, we study distributed consensus for MAS with communication delays and limited data rate. A Nanyang Technological University Singapore v distributed consensus protocol is proposed based on dynamic encoding and decoding. It will be shown that for a connected network, as long as the time delays are bounded, the average consensus can be achieved with a finite communication data rate. Moreover, merely one bit information exchange between each pair of adjacent agents at each time step suffices to guarantee the average consensus. Nanyang Technological University Singapore Symbols and Acronyms Algebraic Operators [a, b] closed interval from a to b A′ transpose of A A−1 inverse of matrix A col{x1, . , xn} column vector formed by x1, x2, . , xn diag{x1, . , xn} block diagonal matrix formed by x1, x2, . , xn 〈x, y〉 mathematical expectation of xy′ 〈x, y〉H inner product over Hilbert space H δi,j Kronecker delta function δ(t) Dirac delta function χΩ(t) indicator function, i.e. χΩ(t) =    1, if t ∈ Ω 0, if t / ∈ Ω L{·} linear space spanned by the elements in the braces x̂(t1|t2) projection of x(t1) onto the linear space spanned by measurements up to time instant t2 x̃ projection error, i.e. x− x̂ σ(A) spectrum set of A ρ(A) resolvent set of A O(·) equal-order or higher order infinitesimal, i.e. f(τ) = O(τ) if and only if lim τ→0 ∣∣f(τ) τ ∣∣ < +∞ A the element in the i-th row and j-th column of matrix A In n× n identity matrix Nanyang Technological University Singapore viii 1n an n-dimensional column vector with all elements of 1 0n an n-dimensional column vector with all elements of 0 Re(·) the real component of a complex number ‖ · ‖2 Euclidean norm ‖ · ‖∞ infinity norm m1 ∧m2 the smaller number of m1 and m2, i.e. min{m1,m2} m1 ∨m2 the larger number of m1 and m2, i.e. max{m1,m2} em,i an m-dimensional column vector with the i-th entry being 1 and all the other entries 0 E(X) the mathematical expectation of X V ar(X) the variance of X P (A) the probability of event A Πi,j transition matrix generated by M , i.e. Πi,j =    M(i)M(i− 1) · · ·M(j), i ≥ j I, i < j d·e ceiling function b·c floor function Sets and Spaces C set of complex numbers R set of real numbers R set of positive real numbers R̄ set of nonnegative real numbers R Set of n dimensional real vectors Rm×n set of m× n real matrices Z set of integers Z set of positive integers Z̄ set of nonnegative integers Nanyang Technological University Singapore ix L2(a, b;R) space of all equivalent classes of square integrable n-dimensional real vector functions on [a, b] W (a, b;X ) Sobolev space of all functions x in L2(a, b;X ) with first order derivative