Motivations and Perspectives 1. Classification of Control Theory General control system: input u Process output y Controller (1) Linear model Continuous-time (2) Nonlinear model Process (3) Linear model Discrete-time (4) Nonlinear model 1 (1.1) Model known (1.2) Model unknown (2.1) Model known (2.2) Model unknown (3.1) Model known (3.2) Model unknown (4.1) Model known (4.2) Model unknown Case (1) Process model: y(n) (t) = a 1y(n-1) (t) + … + a n y(t) + b1u(t) + … + bm u (m-1) (t) (1.1) The model parameters a1 ,..., an , b1 ,..., bm are known: Linear control theory (ELEC271 and ELEC560): transfer functions, stability test, pole-zero analysis, pole placement controller design, etc.. (1.2) The model parameters a1 ,..., an , b1 ,..., bm are unknown: Linear adaptive control theory (Lyapunov synthesis approach): propose a controller based on the unknown parameters a1 ,..., an , b1 ,..., bm , then develop an algorithm to estimate the parameters a1 ,..., an , b1 ,..., bm by minimizing a Lyapunov function candidate. input u Process (linear with unknown parameters) output y Controller (with free parameters) Parameter estimation Key issues: How to choose the controller structure How to choose a Lyapunov function candidate How to design the parameter estimation algorithm Convergence of the parameter estimation algorithm (whether the estimated parameters converge to the true values, how fast, …) Stability of the overall closed-loop system 2 Case (2) Process model: y (n) (t) = g[y (n -1) (t), …, y(t), u(t), …, u (m-1) (t)] (2.1) The nonlinear model g [*] is known: Nonlinear control theory (ELEC562): Lyapunov stability theory, linearization around equilibrium point, differential geometric methods, input-output linearization, etc.. (2.2) The nonlinear model g [*] is unknown: PID control u(t ) ae(t ) b e(t )dt c de(t ) , dt e( t ) y ( t ) y d ( t ) Fuzzy control Rule-based controller, using knowledge from human experts. Nonlinear adaptive control o Neural network based control o Adaptive fuzzy control (ELEC541) Indirect adaptive fuzzy control: Develop a fuzzy system model with free parameters for the unknown g [*] first, and then design a controller based on the fuzzy system model. The free parameters of the fuzzy system model are estimated on-line based on an adaptation law. Direct adaptive fuzzy control: Use a fuzzy system with free parameters directly as the controller, and then develop an algorithm to estimate the parameters of the controller. 3 Indirect adaptive fuzzy control input u Process (nonlinear and unknown) output y Controller (based on fuzzy model of process) fuzzy system model of the process (with free parameters estimated by an adaptation law) Key issues: What is the structure of the fuzzy system model What are the properties of the fuzzy system model Why use the fuzzy system models How to construct the fuzzy system model How to design the controller based on the fuzzy model What is the performance of the closed-loop control system (stability, convergence, robustness, optimality, etc..) 4 Direct adaptive fuzzy control input u Process (nonlinear and unknown) output y Fuzzy controller (fuzzy system with free parameters) Adaptation law (provide fuzzy controller parameters) Key issues: How to design the adaptation law How to guarantee the stability of the closed-loop control system 5 Case (3) Process model: y(t 1) = a 1y(t) + …+ a n y(t - n 1) + b1u(t) + …+ bm u(t - m 1) (3.1) The model parameters a1 ,..., an , b1 ,..., bm are known: Digital control theory (ELEC377): z-transform models, sampling continuous-time systems, stability test, pole-zero analysis, pole placement controller design, etc.. (3.2) The model parameters a1 ,..., an , b1 ,..., bm are unknown: Linear adaptive control theory (self-tuning controllers): design a controller as if the parameters a1 ,..., an , b1 ,..., bm are known, then develop an algorithm to estimate the parameters a1 ,..., an , b1 ,..., bm . input u Process (linear with unknown parameters) output y Controller Parameter estimation Key issues: Parameter estimation algorithms (least squares, gradient descent, projection based, …) Convergence of the parameter estimation algorithms (whether the estimated parameters converge to the true values, how fast, …) Controller design (pole placement, optimal control, …) Stability of the overall closed-loop system 6 Case (4) Process model: y(t + 1) = g[y(t), …, y(t - n + 1), u(t), …, u(t - m + 1)] (4.1) The nonlinear model g [*] is known: Nonlinear discrete-time control theory: Extensions of the concepts and theory for continuous-time systems (4.2) The nonlinear model g [*] is unknown: PID control u(t ) ae(t ) b e(t ) ce(t ), e(t ) y(t ) yd (t ) Fuzzy control Same as the continuous-time case Nonlinear adaptive control o Neural network based control o Adaptive fuzzy control (ELEC541) Indirect adaptive fuzzy control Direct adaptive fuzzy control Key issues: Similar to the continuous-time case 7 2. The gaps between control theory and control engineering practice The evolution of control theory 1940-1950s: the early days, classical control theory, more heuristic based approach. 1960-1970s: modern control theory, more mathematical approach. 1980-1990s: in-depth study versus playing mathematics. The control engineering practice PID is still the most commonly-used controller Why: o No need to know the model of the process under control o Good-enough performance o Easy to tune Systems are getting more and more complex with the extensive use of computers, networks, smart sensors and other new technologies, to: o Increase productivity o Reduce cost Control is just a small part of the whole system The gaps Unrealistic assumptions: linear, known models, etc., which make the theory unusable for practical systems. The real hard problems (nonlinear) are still unsolved. Nonlinear adaptive control: targetting the real hard problem with weakest assumptions. 8 3. Key issues in nonlinear adaptive control <1> How to find a good nonlinear structure f [*] to model the unknown nonlinear function g[*] in y (n) (t) = g[y (n -1) (t), …, y(t), u(t), …, u (m-1) (t)] or y(t + 1) = g[y(t), …, y(t - n + 1), u(t), …, u(t - m + 1)] The requirements: f [*] should be general enough to approximate a wide variety of nonlinear functions, that is, Can the f [*] achieves sup x | f [ x ] g[ x ] | for any small ? f [*] should be efficient to represent general nonlinear functions. The structure and parameters of f[*] should have clear meanings so that humans can easily understand it. The main choices: Polynomials f [x] = constant term + first-order terms + secondorder terms + … Advantages: o Direct extension of linear functions o Mathematically well-understood Disadvantages: o Efficiency is low, too many terms are needed to approximate nonlinear functions 9 Neural networks ( 2) (1) f [ x ] wi wij x j i j where w are weights and is a -function Network representation of a neural network f [x ] x1 x2 … xp Advantages: o Support from human brain research o Good mathematical properties Disadvantages: o Black-box, difficult to understand what is going on inside 10 Fuzzy system Advantages: o Rule-based systems, easy to understand for humans o Good mathematical properties o Capable of incorporating human knowledge o A variety of structures to choose Disadvantages: o Mathematically not well-studied <2> How to determine the structures of the choosing nonlinear models and develop adaptation laws to identify the parameters <3> How to develop controllers based on the identified nonlinear models 11 4. Different types of fuzzy systems <1> Standard fuzzy systems fuzzy rule base x f[x] fuzzifier defuzzifier fuzzy inference engine fuzzy set fuzzy set <2> Takagi-Sugeno fuzzy systems f[x] = (THEN-part of rule i) * (strength of rule i) i strength IF-part of rule 1 THEN-part of rule 1 strength IF-part of rule 2 THEN-part of rule 2 x f[x] + … strength IF-part of rule M THEN-part of rule M 12 <3> Hierarchical fuzzy systems f[x] fuzzy sub-system fuzzy sub-system fuzzy sub-systems fuzzy sub-system x1 xp x2 <4> Fuzzy adaptive rule systems <5> Projection pursuit fuzzy systems 13